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absolute de Rham cohomology | https://ncatlab.org/nlab/source/absolute+de+Rham+cohomology |
#Contents#
* table of contents
{:toc}
## Idea
Absolute de Rham cohomology is a [[de Rham cohomology]] theory for [[arithmetic motives]] which provides a spectral interpretation of [[L-functions]]. It is viewed as a phenomenon in hypothetical "absolute" geometry "over a field of one element". It is hoped by some that it may help to approach [[Riemann hypothesis]].
## Related concepts
* [[absolute cohomology]]
## References
* Nagoya conference [Witt vectors, foliations, and absolute de Rham cohomology](http://kskedlaya.org/conference2010/), 2010, [participants](http://kskedlaya.org/conference2010/participants.html), [schedule](http://kskedlaya.org/conference2010/schedule.html)
* Kedlaya overview slides [pdf](http://kskedlaya.org/slides/nagoya2010.pdf) from Nagoya talk
* MathOverflow question [Riemann hypothesis via absolute geometry](http://mathoverflow.net/questions/69389/riemann-hypothesis-via-absolute-geometry) |
absolute differential form | https://ncatlab.org/nlab/source/absolute+differential+form |
# Absolute differential forms
* table of contents
{: toc}
## Idea
It\'s well known that one can integrate a [[differential form]] on an [[orientation|oriented]] [[submanifold]]. Less well known (but also true), one can integrate a [[differential pseudoform]] on an [[pseudoriented]] (transversely oriented) submanifold. But in classical differential geometry, one also sees forms that can be integrated on *unoriented* submanifolds.
I call these _absolute_ forms. The term 'absolute' suggests a lack of additional required structure, in this case some sort of orientation on the domain of integration. It also suggests [[absolute value]], since many of the examples from classical differential geometry involve absolute values. Indeed, we can define the absolute value of a form or a pseudoform to be an absolute form, although not every absolute form arises in this way.
The main theorem of absolute forms is that, if $\omega$ is a (pseudo)-$p$-form and $R$ is a (pseudo)-oriented $p$-dimensional submanifold, then
$$ {\left| \int_R \omega \right|} \leq \int_{|R|} {|\omega|} ,$$
where ${|\omega|}$ is an absolute $p$-form (the absolute value of $\omega$), $|{R}|$ is simply $R$ with its (pseudo)-orientation ignored, and the absolute value on the left is the ordinary absolute value of scalars. This theorem also applies if we start with an absolute $p$-form $\omega$, (although in that case $R$ starts out unoriented and so is the same as ${|R|}$). If $R$ is a [[de Rham cohomology|de Rham]] chain (a formal [[linear combination]] of appropriately oriented submanifolds), we also take absolute values of the formal coefficients in ${|R|}$. (This operation does not respect the usual notion of equality of chains, but the theorem is true all the same.)
## Definitions
Let $X$ be a [[differentiable manifold]] (or similar sort of space), and let $p$ be a [[natural number]] (typically $0 \leq p \leq n$, where $n$ is the [[dimension]] of $X$). Recall that an (exterior differential) __$p$-[[exterior differential form|form]]__ $\omega$ on $X$ is a [[function]] that assigns a [[real number]] (or whatever is the relevant sort of scalar) $\omega_c(v_1,\ldots,v_p)$ to a point $c$ in $X$ and a $p$-[[tuple]] $(v_1,\ldots,v_p)$ of [[tangent vectors]] at $c$, [[multilinear map|multilinearly]] and [[alternating form|alternating]] in the $v_i$. Similarly, a __$p$-[[pseudoform]]__ $\omega$ on $X$ is a function that assigns a scalar $\omega_c^o(v_1,\ldots,v_p)$ to a point $c$ in $X$, a local [[orientation]] $o$ at $c$, and a $p$-tuple $(v_1,\ldots,v_p)$ of tangent vectors at $c$, multilinearly and alternating in the $v_i$ and reversing sign under a reversal of $o$.
+-- {: .num_defn}
###### Definition
An __absolute $p$-form__ $\omega$ on $X$ is a function that assigns a scalar $\omega_c(v_1,\ldots,v_p)$ to a point $c$ in $X$ and a $p$-[[tuple]] $(v_1,\ldots,v_p)$ of [[tangent vectors]] at $c$ and that satisfies the following conditions:
1. Fixing $c$, $\omega_c({-})$ shall be [[uniformly continuous map|uniformly continuous]].
2. The $p$-tuple $(v_1,\ldots,v_p)$ shall be [[linearly independent subset|linearly independent]] if $\omega_c(v_1,\ldots,v_p) \ne 0$. Thus, although $\omega_c$ is not linear, we may still call it [[alternating form|alternating]]; however (as a consequence of 3), it is actually [[symmetric function|symmetric]].
3. Fix a $p$-dimensional subspace $S$ of the tangent space at $c$ and an orientation $o$ of $S$. Now given a linearly independent $p$-tuple $(v_1,\ldots,v_p)$ from $S$ (that is a [[basis]] of $S$), let $\omega_c(v_1,\ldots,v_p)_S^o$ be $\pm\omega_c(v_1,\ldots,v_p)$ according to whether the orientation of $S$ induced by the $v_i$ matches $o$, and extend this by continuity to all $p$-tuples from $S$ (which extension must be unique and exists by 1&2). The resulting function $\omega_c({-})_S^o$ shall be [[multilinear map|multilinear]] (and so also alternating, by 2).
=--
The multilinearity condition here is rather weaker than for a (pseudo)-form, since it applies only within a $p$-dimensional subspace. Shifting one vector even slightly outside of $S$ loses all connection provided by multilinearity, which is why we need a continuity condition; continuity holds for (pseudo)-forms automatically.
An absolute $p$-form $\omega$ is __[[continuous map|continuous]]__ if it is jointly continuous in all of its data ($c$ as well as the $v_i$). Since the domain of the function $\omega$ is a manifold (a [[vector bundle]] over $X$, although $\omega$ is not a map of vector bundles), we can even discuss [[differentiable map|differentiability]], [[smooth map|smoothness]], and even [[analytic map|analyticity]] of $\omega$ when $X$ has the relevant structure.
An absolute $0$-form is the same thing as a $0$-form. An absolute $n$-form on an $n$-dimensional manifold $X$ is essentially the same thing as an $n$-pseudoform; with the notation from condition 3, the only possibility for $S$ is the entire tangent space $T_c{X}$, and we have
$$ \tilde\omega_c^o(v_1,\ldots,v_n) = \omega_c(v_1,\ldots,v_n)_{T_c{X}}^o $$
to relate the $n$-pseudoform $\tilde{\omega}$ to the absolute $n$-form $\omega$. Finally, the only absolute $p$-form for $p \gt n$ is $0$.
At a point $c$, an absolute $p$-form $\omega$ is:
* __indefinite__ if $\omega_c(v_1,\ldots,v_p) \gt 0$ for some (necessarily [[linearly independent subset|linearly independent]]) $p$-tuple of vectors and $\omega_c(v_1,\ldots,v_p) \lt 0$ for some $p$-tuple,
* __semidefinite__ if not indefinite,
* __definite__ (and hence semidefinite) if $\omega_c(v_1,\ldots,v_p) \ne 0$ for every independent $p$-tuple of vectors at $c$,
* __positive__ (and hence semidefinite) if $\omega_c(v_1,\ldots,v_p) \geq 0$ for every $p$-tuple of vectors (it is enough when they are independent),
* __negative__ (and hence semidefinite) if $\omega_c(v_1,\ldots,v_p) \leq 0$ for every (independent) $p$-tuple of vectors.
All these are at a point $c$; $\omega$ satisfies the condition tout court if it holds for all $c$.
Given an absolute $p$-form $\omega$, its __[[absolute value]]__ ${|\omega|}$ is a positive semidefinite absolute $p$-form:
$$ {|\omega|}_c(v_1,\ldots,v_p) \coloneqq {|\omega_c(v_1,\ldots,v_p)|} .$$
If we start with a $p$-form $\omega$, then the same definition defines a positive absolute $p$-form ${|\omega|}$. If we start with a $p$-pseudoform $\omega$, then essentially the same definition still works; we use either orientation to evaluate $\omega$ with the same result. Note that ${|\omega|}$ is continuous if $\omega$ is. However, we may *not* conclude that ${|\omega|}$ is differentiable just because $\omega$ is differentiable (or even analytic). On the other hand, ${|\omega|}$ inherits differentiability properties from $\omega$ wherever $\omega \ne 0$. (Even then, however, we cannot inherit analyticity, except in $1$ dimension.)
Given two absolute $p$-forms $\omega$ and $\eta$, their __sum__ $\omega + \eta$ is an absolute $p$-form:
$$ (\omega + \eta)_c(v_1,\ldots,v_p) \coloneqq \omega_c(v_1,\ldots,v_p) + \eta_c(v_1,\ldots,v_p) .$$
Given an absolute $p$-form $\omega$ and a scalar field $f$, their __product__ $f \omega$ is an absolute $p$-form:
$$ (f \omega)_c(v_1,\ldots,v_p) \coloneqq f(c) \omega_c(v_1,\ldots,v_p) .$$
In this way, the space of absolute $p$-forms is a [[module]] over the [[associative algebra|algebra]] of scalar fields and the space of [[sections]] of a [[vector bundle]]. For now, we decline to define products of absolute forms of aribtrary rank.
Given an absolute $p$-form $\omega$ on $X$, a manifold $U$, and a [[continuously differentiable map]] $R\colon U \to X$, the __pullback__ $R^*\omega$ is an absolute $p$-form on $U$:
$$ (R^*\omega)_c(v_1,\ldots,v_p) \coloneqq \omega_{R(c)}(R_*v_1,\ldots,R_*v_p) .$$
Here, $R_*v_i$ is the [[pushforward]] of $v_i$ under $R$. Note that $R^*\omega$ is continuous if $\omega$ is; we can also pull back differentiability and analyticity properties that $\omega$ and $R$ both have.
Given a continuous absolute $p$-form $\omega$ on $X$, a $p$-dimensional manifold $U$, and a continuously differentiable map $R\colon U \to X$, the __integral__ $\int_R \omega$ is a scalar:
$$ \int_R \omega \coloneqq \int_U R^*\omega .$$
On the right-hand side, $R^*\omega$ is a continuous absolute $p$-form on $U$, but since $U$ is $p$-dimensional, this is essentially the same as a continuous $p$-pseudoform on $U$, and we already know how to integrate this (see [[integration of differential forms]]).
## Examples
Examples of absolute forms from classical differential geometry include:
* Absolute $0$-forms are the same as ordinary $0$-forms.
* Absolute $n$-forms on an $n$-[[dimensional]] manifold are the same as $n$-pseudoforms (and hence the same as [[absolutely continuous measure|absolutely continuous]] [[Radon measures]]).
* In [[complex analysis]], ${|\mathrm{d}z|}$ is an absolute $1$-form sometimes used in [[contour integration]]. This literally is the absolute value of the differential of the identity map $z$.
* More generally, the [[arclength]] element $\mathrm{d}s = {\|\mathrm{d}\mathbf{x}\|}$ on a [[Riemannian manifold]] is an absolute $1$-form. Neither $\mathrm{d}s$ nor (in general) $\mathrm{d}\mathbf{x}$ is actually the differential of anything, but $\mathrm{d}\mathbf{x}$ is the canonical [[tangent vector|vector]]-valued $1$-form (which, on an [[affine space]], really is the differential of the [[identity map]] $\mathbf{x}$), and we really can use the metric to take the norm of such a form to get an absolute $1$-form.
* Similarly, the [[surface area]] element $\mathrm{d}S$ on a Riemannian manifold is an absolute $2$-form, and we can continue into higher dimensions (although the classical [[volume element]] $\mathrm{d}V$ in $\mathbb{R}^3$ is already covered as a $3$-pseudoform). In principle, we ought to be able to write down expression for $\mathrm{d}S$ etc in terms of $\mathrm{d}s$, although so far the only thing that I know how to do is $\mathrm{d}S = {\|{\mathrm{d}\mathbf{x} \hat\times \mathrm{d}\mathbf{x}}\|} / 2$, where $\hat\times$ indicates a wedge product of vector-valued forms whose vectors are multiplied by the [[cross product]]. (This can be generalized to any finite-dimensional area in any finite-dimensional Riemannian manifold; in particular, $\mathrm{d}V = {|{\mathrm{d}\mathbf{x} \hat\cdot \mathrm{d}\mathbf{x} \hat\times \mathrm{d}\mathbf{x}}|} / 6$.)
+-- {: .num_remark}
The reason that $\mathrm{d}s$ determines $\mathrm{d}S$ is that lengths determine areas, as through Heron\'s Formula. However, our usual ways of deriving one form from another (thinking of them, at a given point, as maps from lists of vectors to numbers) involve only rerranging (permuting, duplicating, and contracting) the inputs and applying operations to the outputs. But this cannot be sufficient; knowing only the lengths of two vectors $v$ and $w$ (which vectors are the inputs to $\mathrm{d}S$, and which lengths are the outputs of $\mathrm{d}s$ at those inputs) does not tell you the area of the parallelogram that they span (which is the output of $\mathrm{d}S$). You also need the length of $v - w$ (or of $v + w$). So while we can express $\mathrm{d}S_c$ ($\mathrm{d}S$ at a point $c$) as $(v, w \mapsto 2 H(\mathrm{d}s(v), \mathrm{d}s(w), \mathrm{d}s(v - w))$ (where $H$ is the Heron function, the real-valued function of three real variables that takes the lengths of the three sides of a triangle and returns its area), or equivalently as $(v, w \mapsto 2 H({\|v\|}, {\|w\|}, {\|{v - w}\|}))$, we cannot express $\mathrm{d}S$ with a formula anything along the lines of ${\|{\mathrm{d}\mathbf{x} \hat\times \mathrm{d}\mathbf{x}}\|} / 2$ using $\mathrm{d}s$ directly instead of $\mathrm{d}\mathbf{x}$. (And so you may as well describe $\mathrm{d}S_c$ as $(v, w \mapsto {\|{v \times w}\|})$ instead.)
=--
## Related concepts
* exterior [[differential forms]] and [[pseudoforms]] (the more well-known variations);
* [[cojet differential forms]] (and [[cogerm differential forms]] more generally) generalize both exterior and absolute $1$-forms and pseudo-$1$-forms;
* [[coflare differential form]]s are a common generalization of exterior forms, absolute forms, and cojet forms (although not all cogerm forms).
## References
Near the end of a Usenet post from 2002, we see a definition of $\int_R {|\omega|}$ for $\omega$ a (pseudo)-$p$-form and $R$ a $p$-dimensional submanifold, but without a broader context for ${|\omega|}$ itself:
* [[Toby Bartels]] and Ralph Hartley; [Densitized Pseudo Twisted Forms](https://groups.google.com/group/sci.physics.research/msg/424da828e75b6b90?dmode=source)
{#Usenet}
Apparently absolute $p$-forms (at least if continuous) are the same as even $p$-[[densities]] as defined by Gelfand; see this MathOverflow answer:
* Juan Carlos Álvarez Paiva; answer to [Why do I need densities in order to integrate on a non-orientable manifold?](http://mathoverflow.net/questions/90455/why-do-i-need-densities-in-order-to-integrate-on-a-non-orientable-manifold/90714#90714).
{#MathOverflow}
[[!redirects absolute differential form]]
[[!redirects absolute differential forms]]
[[!redirects absolute form]]
[[!redirects absolute forms]]
[[!redirects differential absolute form]]
[[!redirects differential absolute forms]]
[[!redirects absolute value of a form]]
[[!redirects absolute value of a differential form]]
[[!redirects absolute value of a pseudoform]]
[[!redirects absolute value of a pseudo-form]]
|
absolute extensor | https://ncatlab.org/nlab/source/absolute+extensor |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
A [[topological space]] $K$ is called an _absolute extensor_ if for
1. $X$ any [[nice topological space|nice]] [[topological space]]
1. $A \overset{i}{\hookrightarrow} X$ any [[closed subspace]]
1. $f\colon A \longrightarrow K$ any [[continuous function]]
there is an [[extension]] to a [[continuous function]] $\tilde{f}:X\to K$, i.e., such that $\tilde{f}=f\circ i$:
$$
\array{
A &\overset{f}{\longrightarrow}& K
\\
{}^{\mathllap{i}}\downarrow & \nearrow_{\mathrlap{\exists \tilde f}}
\\
X
}
$$
Here "[[nice topological space]]" is variously taken to mean [[metrizable topological space]] or at least [[normal topological space]].
A variation of this concept, **absolute neighborhood extension**, only requires the extension to exist over a neighborhood of $A$ in $X$.
## Examples
The [[Tietze extension theorem]] implies that the [[real line]] $\mathbb{R}$ equipped with its [[Euclidean space]] [[metric topology]] is an absolute extensor. It follows that so are the [[closed interval]] [[subspace]] $[0,1] \subset \mathbb{R}$ and the [[circle]] $S^1$.
Products of absolute extensors are absolute extensors, including the [[Hilbert cube]].
The two point discrete space $S^0$ as well as any sphere $S^n$ is an absolute neighborhood extensor, but not an absolute extensor.
## Related concepts
* [[Tietze extension theorem]]
* [[quasi-finite CW-complex]]
* [[absolute retract]]
## References
* [[eom]], _[Absolute neighbourhood extensor](https://www.encyclopediaofmath.org/index.php/Absolute_neighbourhood_extensor)_
[[!redirects absolute extensors]]
[[!redirects absolute neighbourhood extensor]]
[[!redirects absolute neighbourhood extensors]]
[[!redirects absolute neighborhood extensor]]
[[!redirects absolute neighborhood extensors]]
|
absolute Galois group | https://ncatlab.org/nlab/source/absolute+Galois+group |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _absolute Galois group_ of a [[field]] $k$ is that of the [[field extension]] $k \hookrightarrow k_s$ which is the [[separable closure]] of $k$. When $k$ is a [[perfect field]] this is equivalently the Galois group of the [[algebraic closure]] $k \hookrightarrow \overline{k}$.
## Definition
+-- {: .num_defn}
###### Definition
Let $k$ be a [[field]]. Let $k_s$ denote the [[separable closure]] of $k$. Then the [[Galois group]] $Gal(k\hookrightarrow k_s)$ of the [[field extension]] $k\hookrightarrow k_s$ is called *absolute Galois group of $k$*.
=--
## Properties
+-- {: .num_remark}
###### Remark
By general [[Galois theory]]
we have $Gal(K\hookrightarrow K_s)\simeq \pi_1(Spec\; K)$ is equivalent to the [[fundamental group]] of the [[spectrum of a commutative ring|spectrum]] [[scheme]] $Spec K$
=--
An instance of [[Grothendieck's Galois theory]] is the following:
+-- {: .num_prop}
###### Proposition
The functor
$$\begin{cases}
Sch_{et}\to Gal(k\hookrightarrow k_s)-Set
\\
X\mapsto X(k_s)
\end{cases}$$
from the category of [[étale scheme|étale schemes]] to the category of sets equipped with an [[action]] of the absolute Galois group is an equivalence of categories.
=--
+-- {: .num_prop}
###### Proposition
Recall the every [[profinite group]] appears as the [[Galois group]] of some [[Galois extension]]. Moreover we have:
Every [[projective object|projective]] [[profinite group]] appears as an absolute [[Galois group]] of a [[pseudo algebraically closed field]].
=--
## Examples
### Of the rational numbers
{#OfTheRationalNumbers}
+-- {: .num_remark}
###### Remark
There is no direct description (for example in terms of [[generators and relations]]) known for the absolute Galois group $G_\mathbb{Q} \coloneqq Gal(\mathbb{Q}\hookrightarrow \overline{\mathbb{Q}})$ of the [[rational numbers]] (with $\overline{\mathbb{Q}}$ being the [[algebraic numbers]]).
However [[Belyi's theorem]] implies that there is a faithful [[action]] of $G_\mathbb{Q}$ on the [[children's drawing|children's drawings]].
=--
+-- {: .num_theorem}
###### Theorem
**(Drinfeld, Ihara, Deligne)**
There is an inclusion of the absolute Galois group of the rational numbers into the [[Grothendieck-Teichmüller group]] (recalled e.g. as [Stix 04, theorem 6](#Stix04)).
=--
## References
### General
* {#Stix04} [[Jakob Stix]], _The Grothendieck-Teichmüller group and Galois theory of the rational numbers_, 2004 ([[StiXGaloisAndGT.pdf:file]])
Discussion of the [[p-adic numbers|p-adic]] absolute Galois group as the [[etale fundamental group]] of a [[quotient]] of some [[perfectoid space]] is in
* {#Weinstein04} [[Jared Weinstein]], _$Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ as a geometric fundamental group_ ([arXiv:1404.7192](https://arxiv.org/abs/1404.7192), [workshop](https://www.mathi.uni-heidelberg.de/~G.QpAsPi1geom/))
See also
* Wikipedia, _[Absolute Galois group](https://en.wikipedia.org/wiki/Absolute_Galois_group)_
### In string theory
Discussion in the context of [[string theory]] includes
* [[Gregory Moore]], _Arithmetic and Attractors_ ([arXiv:hep-th/9807087](https://arxiv.org/abs/hep-th/9807087), abridged version in [arXiv:hep-th/9807056](https://arxiv.org/abs/hep-th/9807056))
category: Galois theory
[[!redirects absolute Galois groups]]
[[!redirects absolute Galois theory]]
|
absolute idealism | https://ncatlab.org/nlab/source/absolute+idealism |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--
## Related concepts
* [[doctrine of ideas]]
* [[idealism]]
[[objective idealism]], [[subjective idealism]]
* [[exceptional naturalism]]
* [[empiricism]]
## References
* Wikipedia, _[Absolute idealism](http://en.wikipedia.org/wiki/Absolute_idealism)_
* [[Hegel]], _[[Science of Logic]]_
* [[Roger Penrose]]'s _[Mind-Maths-Matter](Roger+Penrose#MindMathsMatter)_
|
absolute pushout | https://ncatlab.org/nlab/source/absolute+pushout | [[!redirects absolute pushouts]]
#Contents#
* table of contents
{:toc}
## Idea
An **absolute pushout** is a [[pushout]] which is [[preserved limit|preserved]] by any [[functor]] whatsoever. In general this happens because the pushout is a pushout for purely "diagrammatic" reasons. See [[absolute colimit]] for more.
## Definitions
+-- {: .num_defn #DefBasic}
###### Definition
A particular [[pushout]] [[diagram]] in a particular [[category]] $C$ is an **absolute pushout** if it is [[preserved limit|preserved]] by every [[functor]] with [[domain]] $C$.
=--
Equivalently, since the [[Yoneda embedding]] is the [[free cocompletion]] of $C$:
+-- {: .num_defn #DefYoneda}
###### Definition
A particular [[pushout]] diagram in a particular [[category]] $C$ is an **absolute pushout** if it is [[preserved limit|preserved]] by the [[Yoneda embedding]] $C \hookrightarrow [C^{op},Set]$.
=--
## Split pushouts
We propose the following notion of **split pushout**. See [Isaacson, Def. 3.22](#Isaacson).
+-- {: .num_defn #DefSplitPushout}
###### Definition
A [[commutative diagram|commutative]] square
\begin{center}\begin{tikzcd}
A \ar[r, "p"] \ar[d, "q"'] & B \ar[d, "m"] \\ C \ar[r, "n"'] & P
\end{tikzcd}\end{center}
defines a **split pushout** if there exist [[sections]] $p s = 1$, $q t = 1$, $m u = 1$
\begin{center}\begin{tikzcd}
A \ar[r, "p"'] \ar[d, "q"] & B \ar[d, "m"'] \ar[l, bend right, "s"'] \\ C \ar[r, "n"] \ar[u, bend left, "t"] & P \ar[u, bend right, "u"']
\end{tikzcd}\end{center}
so that $p t = u n$.
=--
## Split pushouts are absolute pushouts
+-- {: .num_prop }
###### Proposition
Split pushouts are absolute pushouts.
=--
+-- {: .proof}
###### Proof
Note that split pushouts are preserved by arbitrary functors, so it suffices to show that a split pushout is a pushout in the category in which it lives. To that end consider, a [[cone]] under the [[span]] $(p,q)$:
\begin{center}\begin{tikzcd}
A \ar[r, "p"] \ar[d, "q"'] & B \ar[d, "b"] \\ C \ar[r, "c"'] & X
\end{tikzcd}\end{center}
Upon composing with the sections to $q$ and $m$
\begin{center}\begin{tikzcd}
C \ar[d, "t"'] \ar[r, "n"]\ar[dd, bend right, equals]& P \ar[d, "u"] \\
A \ar[r, "p"] \ar[d, "q"'] & B \ar[d, "b"] \\ C \ar[r, "c"'] & X
\end{tikzcd}\end{center}
we see that $c$ factors through the claimed pushout $P$ as $c = (b u) n$. We must verify that $b$ also factors as $b = (b u) m$. Since $p$ is an epimorphism, it suffices to prove that $b p = b u m p$, which follows easily:
$$ b p = c q = c q t q = b p t q = b u n q = b u m p.$$
This produces the desired factorization. Finally, since $m$ is an epimorphism, such factorizations are unique.
=--
Note the proof that a split pushout defines a pushout square in the category in which it lives did not require $p$ to be a _split_ epimorphism. However, arbitrary functors do not preserve epimorphisms. They do however preserve split epimorphisms, and thus the section guarantees that the image of $p$ will define an epimorphism in any category.
## General characterization
\begin{theorem}
A commutative square
\begin{center}\begin{tikzcd}
A \ar[r, "p"] \ar[d, "q"'] & B \ar[d, "m"] \\ C \ar[r, "n"'] & P
\end{tikzcd}\end{center}
is an absolute pushout if and only if either there exist
1. A section $u:P\to B$, such that $m u = 1_P$.
2. Morphisms $r_1,\dots,r_k : B \to A$ and $s_1,\dots,s_k : B\to A$, for some $k\ge 1$, such that $p s_1 = 1_B$, $q s_i = q r_i$ for all $i$, $p r_i = p s_{i+1}$ for all $i\lt k$, and $p r_k = u m$.
3. Morphisms $t_1,\dots,t_{\ell+1} : C \to A$ and $v_1,\dots,v_{\ell} : C\to A$, for some $\ell \ge 0$, such that $q t_1 = 1_C$, $p t_i = p v_i$ for all $i\lt \ell$, $q v_i = q t_{i+1}$ for all $i\le \ell$, and $p t_{\ell+1} = u n$.
or the transpose thereof (i.e. interchanging $B$ with $C$ and so on).
\end{theorem}
\begin{proof}
For "if", suppose given a commutative square
\begin{center}\begin{tikzcd}
A \ar[r, "p"] \ar[d, "q"'] & B \ar[d, "b"] \\ C \ar[r, "c"'] & X.
\end{tikzcd}\end{center}
Since $m$ is a (split) epimorphism (by $u$), any factorization of this square through the given one will be unique, so it suffices to show that such a factorization exists. Define $x = b u:P\to X$. Then we have
$$ x m = b u m = b p r_k = c q r_k = c q s_k = b p s_k = b t r_{k-1} = \dots = b p s_1 = b $$
and
$$ x n = b u n = b p t_\ell = c q t_\ell = c q v_{\ell-1} = b p v_{\ell-1} = b p t_{\ell-1} = \dots = c q t_1 = c. $$
The transposed case is of course dual.
Conversely, suppose the given square is an absolute pushout. Thus, in particular the induced square
\begin{center}\begin{tikzcd}
\hom(P,A) \ar[r, "p"] \ar[d, "q"'] & \hom(P,B) \ar[d, "m"] \\ \hom(P,C) \ar[r, "n"'] & \hom(P,P)
\end{tikzcd}\end{center}
is a pushout in $Set$. Thus, in particular, the function $\hom(P,B)+\hom(P,C) \to \hom(P,P)$ is surjective, and thus for $1_P\in \hom(P,P)$ there must be either a $u:P\to B$ such that $m u = 1_P$ or a $u':P\to C$ such that $n u' = 1_P$. WLOG assume the former.
Now the induced square
\begin{center}\begin{tikzcd}
\hom(B,A) \ar[r, "p"] \ar[d, "q"'] & \hom(B,B) \ar[d, "m"] \\ \hom(B,C) \ar[r, "n"'] & \hom(B,P)
\end{tikzcd}\end{center}
is also a pushout in $Set$. We have two elements $1_B, u m \in \hom(B,B)$ that become equal in $\hom(B,P)$ (since $m 1_B = m = (1_P) m = m u m$), and in a pushout in $Set$ this means they must be related by a zigzag of elements of the vertex $\hom(B,A)$. Unraveling this explicitly produces the morphisms $r_i,s_i$.
Similarly, from the induced square
\begin{center}\begin{tikzcd}
\hom(C,A) \ar[r, "p"] \ar[d, "q"'] & \hom(C,B) \ar[d, "m"] \\ \hom(C,C) \ar[r, "n"'] & \hom(C,P)
\end{tikzcd}\end{center}
and the elements $1_C\in\hom(C,C)$ and $s n \in \hom(C,B)$, we obtain the morphisms $t_i,v_i$.
\end{proof}
In particular, when $k=1$ and $\ell=0$, the above data reduces to
1. A section $u:P\to B$, such that $m u = 1_P$.
2. Morphisms $r,s : B \to A$ such that $p s = 1_B$, $q s = q r$, and $p r = u m$.
3. A morphism $t : C \to A$ such that $q t = 1_C$ and $p t = u n$.
This is precisely the data of the above-defined notion of split pushout, together with the additional morphism $r:B\to A$ such that $q r = q s$ and $p r = u m$. However, given a split pushout as above we can define $r = t q s$ and check $q r = q t q s = q s$ and $p r = p t q s = u n q s = u m p s = u m$. This gives another proof that any split pushout is an absolute pushout.
## Examples
In their study of [[generalized Reedy categories]], Berger and Moerdijk introduce the notion of an [[Eilenberg-Zilber category]], one of the axioms of which demands that spans of split epimorphisms admit absolute pushouts. In practice, this seems to be the case because the pushout of these split epimorphisms is a split epimorphism as above, often with an additional section $v$ of $n$ satisfying the additional equation that $v m = q s$.
## References
The general characterization of absolute pushouts appears as Proposition 5.5 in:
* Robert Paré, Robert On absolute colimits. J. Algebra 19 (1971), 80–95.
The above notion of split pushout appears in Definition 3.22 of:
* {#Isaacson} Isaacson, Samuel B. "Symmetric cubical sets." Journal of Pure and Applied Algebra 215.6 (2011): 1146-1173. ([doi](https://doi.org/10.1016/j.jpaa.2010.08.001))
The Berger-Moerdijk definition of an Eilenberg-Zilber category appears in:
* {#BergerMoerdijk} [[Clemens Berger]] and [[Ieke Moerdijk]], _On an extension of the notion of Reedy category_ (2008) ([arXiv:0809.3341](http://arxiv.org/abs/0809.3341))
|
absolute retract | https://ncatlab.org/nlab/source/absolute+retract |
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###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
A [[metrisable space|metrisable]] [[topological space]] $Y$ is an _absolute neighborhood retract_ (ANR) if ([Borsuk 32, p. 222](#Borsuk32)) for _any_ [[embedding of topological spaces|embedding]] $Y \subset Z$ as a [[closed subspace]] in a [[metrisable topological space]] $Z$, $Y$ is a [[neighborhood retract]] of $Z$.
A [[metrisable space|metrisable]] topological space $Y$ is an _absolute retract_ if for _any_ embedding $Y\subset Z$ as a [[closed subspace]] in a [[metrisable topological space]] $Z$, $Y$ is a [[retract]] of $Z$.
## Properties
{#Properties}
\begin{prop}\label{ANRIsLocalProperty}
**(ANR is a local property for metrizable spaces)** \linebreak
A [[metrizable topological space]] which admits an [[open cover]] by [[absolute neighbourhood retracts]] is itself an [[absolute neighbourhood retract]].
\end{prop}
(review in [Hu 65, III Thm. 8.1](#Hu65))
\begin{prop}
A [[metrisable topological space]] is an [[absolute retract]] precisely if it is
1. a [[contractible topological space]],
1. an [[absolute neighborhood retract]].
\end{prop}
([Hu 65, Prop. II.7.2](#Hu65))
## Examples
{#Examples}
\begin{example}
Every ([[finite number|finite]]-[[dimension of a manifold|dimensional]])
[[metrizable topological space|metrizable]] [[locally Euclidean topological space]] -- in particular every [[topological manifold]] -- is an [[absolute neighbourhood retract]].
\end{example}
By Prop. \ref{ANRIsLocalProperty} (see also [Hu 65, III Cor. 8.3](#Hu65)).
In fact:
\begin{example}\label{ParacompactBanachManifoldsAreANRs}
Every [[paracompact topological space|paracompact]] [[Banach manifold]] is an [[absolute neighbourhood retract]].
\end{example}
By [Palais 1966, Cor. to Thm. 5 on p. 3](#Palais66).
\begin{proposition}\label{ClosedInclusionIntoANRIsANRIffhCofibration}
Let $X$ be an [[absolute neighbourhood retract]] (ANR) and $A \xhookrightarrow{i} X$ a [[closed subspace]]-inclusion. Then $A$ is an ANR precisely iff the inclusion $i$ is a [[Hurewicz cofibration]].
\end{proposition}
([Aguilar, Gitler & Prieto 2002, Thm. 4.2.15](#AGP02))
\begin{example}
Every [[finite number|finite]]-[[dimension of a CW complex|dimensional]] [[locally finite CW complex|locally finite]] [[CW-complex]] is an [[absolute neighbourhood retract]].
\end{example}
([Dugundji 52](#Dugundji52), [Kodama 56](#Kodama56), review in [Hu 65, III Cor. 8.4](#Hu65))
## Related concepts
* [[retract]],
* [[neighborhood retract]]
* [[deformation retract]]
## References
The notion of _absolute neighbourhood retract_ is due to
* {#Borsuk32} [[Karol Borsuk]], _Über eine Klasse von lokal zusammenhängenden Räumen_, Fund. Math **19** (1932) 220-242 ([dml:212574](https://eudml.org/doc/212574), [pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm19/fm19119.pdf))
Further development:
* [[Olof Hanner]], _Some theorems on absolute neighbourhood retracts_, Arkiv För Matematik Band 1 nr 30 (1950) ([doi:10.1007/BF02591376](https://doi.org/10.1007/BF02591376))
* {#Dugundji52} [[James Dugundji]], _Note on CW polytopes_, Portugaliae mathematica (1952) **11** 1 (1952) 7-10-b ([dml:114693](https://eudml.org/doc/114693))
* {#Kodama56} Yukihiro Kodama, _Note on an absolute neighborhood extensor for metric spaces_, Journal of the Mathematical Society of Japan **8** 3 (1956) 206-215 ([doi:10.2969/jmsj/00830206](https://doi.org/10.2969/jmsj/00830206))
* [[Karol Borsuk]], _Concerning the classification of topological spaces from the stand point of the theory of retracts_, Fundamenta Mathematicae **46** (3) (1959) 321-330 ([dml:213516](https://eudml.org/doc/213516))
Discussion for [[infinite-dimensional manifolds]]:
* {#Palais66} [[Richard S. Palais]], *Homotopy theory of infinite dimensional manifolds*, Topology **5** 1 (1966) 1-16 (<a href="https://doi.org/10.1016/0040-9383(66)90002-4">doi:10.1016/0040-9383(66)90002-4</a>)
Textbook accounts and review:
* [[Karol Borsuk]], _Theory of retracts_, Vol. 44. Państwowe Wydawn. Naukowe, 1967
* {#Hu65} [[Sze-Tsen Hu]], _Theory of Retracts_, Wayne State University Press (1965) ([google-books](https://books.google.ae/books/about/Theory_of_Retracts.html?id=GVTvAAAAMAAJ&redir_esc=y))
* [[Sibe Mardešić]], _Absolute Neighborhood Retracts and Shape Theory_ ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/mardesic.pdf))
* {#AGP02} Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, Def. 4.2.10 in: _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([doi:10.1007/b97586](https://link.springer.com/book/10.1007/b97586), [toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf))
See also:
* [[eom]], _[Absolute retract for normal spaces](https://www.encyclopediaofmath.org/index.php/Absolute_retract_for_normal_spaces)_
* [[eom]], _[Retract of a topological space](https://encyclopediaofmath.org/wiki/Retract_of_a_topological_space)_
* Wikipedia, _<a href="https://en.wikipedia.org/wiki/Retract#Absolute_neighborhood_retract_(ANR)">Absolute neighbourhood retract</a>_
[[!redirects absolute retracts]]
[[!redirects absolute neighborhood retract]]
[[!redirects absolute neighborhood retracts]]
[[!redirects absolute neighbourhood retract]]
[[!redirects absolute neighbourhood retracts]]
[[!redirects ANR]]
[[!redirects ANRs]]
|
absolute value | https://ncatlab.org/nlab/source/absolute+value |
+-- {: .rightHandSide}
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### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Terminology
In [[field]] theory, what we call an 'absolute value' here is often called a 'valuation'. However, there is also a more general notion of [[valuation]] used in field theory, which is what we call 'valuation'. The notion of absolute value is also used in [[functional analysis]], where it may be called a 'multiplicative norm' (rather than merely submultiplicative, as norms on [[Banach algebras]] are required to be).
## Definition
For $k$ a [[rig]] (typically either a [[field]] or at least an [[integral domain]], or else an [[associative algebra]] over such), an **absolute value** on $k$ is a (non-trivial) multiplicative [[seminorm]], or equivalently a finite real-valued [[valuation]].
This means it is a [[function]]
$$
{\vert {-} \vert}\colon k \to \mathbb{R}
$$
to the [[real numbers]] such that for all $x, y \in k$
1. ${\vert x \vert} \geq 0$;
2. ${\vert x \vert} = 0$ precisely if $x = 0$;
3. ${\vert x \cdot y \vert} = {\vert x \vert} {\vert y \vert}$;
4. ${\vert x + y \vert} \leq {\vert x \vert} + {\vert y \vert}$ (the [[triangle inequality]]).
If the last triangle inequality is strengthened to
* ${\vert x + y \vert} \leq max({\vert x \vert}, {\vert y \vert})$
then ${\vert {-} \vert}$ is called an [[ultrametric]] or **non-archimedean** absolute value, since then for any $x, y \in k$ with $\vert x \vert \lt \vert y \vert$ then for all natural numbers $n$, $\vert n x \vert \leq \vert x \vert \lt \vert y \vert$. If the opposite holds, that whenever $\vert x \vert \lt \vert y \vert$ (and $x\neq 0$) there exists a natural number $n$ with $\vert n x \vert \gt \vert y \vert$, then it is called **archimedean**.
Two absolute values ${\vert {-} \vert}_1$ and ${\vert {-} \vert}_2$ are called _equivalent_ if for all $x \in k$
$$
({\vert x \vert}_1 \lt 1)
\Leftrightarrow
({\vert x \vert}_2 \lt 1)
\,.
$$
An [[equivalence class]] of absolute values is also called a **[[place]]**.
A [[field]] equipped with an absolute value which is a [[complete metric space]] with respect to the corresponding [[metric]] is called a [[complete field]].
## Examples
### Trivial absolute value
Every field admits the trivial absolute value ${\vert {-} \vert}_0$ defined by
$$
{\vert x \vert}_0 =
\left\{
\array{
0 & if\; x = 0
\\
1 & otherwise
}
\right.
\,.
$$
This is non-archimedean.
### On the real and complex numbers
{#OnTheRealAndComplexNumbers}
Since the [[real numbers]] are a [[sequentially Cauchy complete]] [[Archimedean field]], the standard absolute value ${\vert {-} \vert_\infty}$ on the [[real numbers]] is
$$
{\vert x \vert_\infty} = \lim_{n \to \infty} x \tanh(n x)
$$
With the standard absolute value function defined, the maximum and principal square root functions could be defined on the real numbers as well.
The standard absolute value on the [[complex numbers]] is
$$
{\vert x + i y \vert_\infty}
=
\sqrt{x^2 + y^2}
\,.
$$
These standard absolute values are archimedean, and with respect to these standard absolute values, both $\mathbb{R}$ and $\mathbb{C}$ are [[complete field|complete]] and hence are complete [[archimedean valued fields]]. Notice that $\mathbb{R}$ is in addition an [[ordered field]] and as such also an [[archimedean field]].
Similar norms exist on the [[quaternions]] and [[octonions]], showing that absolute values can be of interest on noncommutative and even nonassociative [[division rings]].
### On the rational numbers
The standard absolute value [above](#OnTheRealAndComplexNumbers) restricts to the standard absolute value on the [[rational numbers]]
$$
{\vert {-} \vert_\infty}\colon \mathbb{Q} \to \mathbb{R}
\,.
$$
Moreover, for any [[prime number]] $p$ and [[positive number]] $\epsilon \lt 1$, there is an absolute value ${\vert {-} \vert_{p,\epsilon}}$ on $\mathbb{Q}$ defined by
$$
\left\vert \frac{k}{l} p^n\right\vert_{p,\epsilon}
=
\epsilon^n
$$
whenever $n$ is an [[integer]] and $k$ and $l$ are nonzero [[integers]] not divisible by $p$ (and ${\vert 0 \vert_{p,\epsilon}} = 0$).
These are called the **$p$-adic absolute values**. Given $p$, they are all equivalent (the open unit ball consists of all rational numbers whose denominator in lowest terms is not divisible by $p$), so there is a unique **$p$-adic [[place]]**. For most purposes, only the place matters, and one may write simply $|q|_p$; however, if one wants a specific absolute value, then the usual choice is to use $\epsilon = 1/p$ (so that ${|p^n|_p} = p^{-n}$ whenever $n$ is an integer).
The $p$-adic absolute value is non-archimedean. The [[complete field|completion]] $\mathbb{Q}_p$ of $\mathbb{Q}$ under this absolute value is called the field of [[p-adic numbers]], which is therefore a [[non-archimedean field]].
[[Ostrowski's theorem]] says that these examples exhaust the non-trivial absolute values on the [[rational numbers]]. Therefore the [[real numbers]] and the [[p-adic numbers]] are the only possible field completions of $\mathbb{Q}$.
### On Laurent power series
The field of [[Laurent series]] $k[ [ T] ]$ over a [[field]] $k$ is a [[complete field]] with respect to the absolute value that sends a series to $\epsilon^n$ for a fixed $0 \lt \epsilon \lt 1$ and with $n$ the lowest integer such that the $n$th coefficient of the series is not $0$.
## See also
* [[real square root]]
* [[sign function]]
## References
* {#BoschGuntzerRemmert84} [[Siegfried Bosch]], [[Ulrich Güntzer]], [[Reinhold Remmert]], §1.5, §1.6 _[[Non-Archimedean Analysis]] -- A systematic approach to rigid analytic geometry_, 1984 ([pdf](http://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf))
Discussion in [[point-free topology]]:
* [[Ming Ng]], [[Steven Vickers]], *A Point-Free Look at Ostrowski's Theorem and Absolute Values* [[arXiv:2308.14758](https://arxiv.org/abs/2308.14758)]
[[!redirects absolute value]]
[[!redirects absolute values]]
[[!redirects archimedean absolute value]]
[[!redirects Archimedean absolute value]]
[[!redirects archimedean absolute values]]
[[!redirects Archimedean absolute values]]
[[!redirects archimedean]]
[[!redirects non-archimedean]]
[[!redirects non-archimedean valuation]]
[[!redirects non-archimedean valuations]]
[[!redirects real absolute value]]
[[!redirects real absolute values]] |
absolute zero | https://ncatlab.org/nlab/source/absolute+zero |
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### Context
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[thermodynamics]] the term "absolute zero", short for "absolute zero temperature", refers to absolute vanishing of the [[temperature]].
## Related concepts
* [[quantum phase transition]]
* [[temperature]], [[inverse temperature]]
* [[topological order]], [[topological entropy]]
## References
See also:
* Wikipedia, *[Absolute zero](https://en.wikipedia.org/wiki/Absolute_zero)*
[[!redirects absolute zero temperature]]
|
absolutely continuous function | https://ncatlab.org/nlab/source/absolutely+continuous+function |
# Absolutely continuous functions
* table of contents
{: toc}
## Idea
The basic idea behind a [[continuous function]] is that the output of the function can be made to change by only a small amount so long as the input is allowed to change by only a small amount. There are, of course, different ways to make this precise, including [[uniformly continuous functions]] and [[Lipschitz continuous functions]]. With an absolutely continuous function, you allow multiple changes to multiple inputs to be combined into a single total change (and you consider the [[absolute values]] of the changes, so that they won\'t cancel).
The result is a notion of function that gets along well with the [[fundamental theorem of calculus]] in the context of the [[Lebesgue integral]] on the [[real line]]. Absolute continuity is weaker than Lipschitz continuity but stronger than mere (pointwise) continuity.
## Definitions
The first definition below is the most elementary; that the others are equivalent are important theorems.
Let $a$ and $b$ be [[real numbers]], and let $f$ be a real-valued [[function]] on the [[interval]] $[a,b]$. Then $f$ is __absolutely continuous__ on $[a,b]$ iff:
+-- {: .num_defn #elementary}
###### Definition
Given any [[positive number]] $\epsilon$, for some positive number $\delta$, given any [[natural number]] $n$ and any $2n$-tuple of elements of $[a,b]$, interpreted as an increasing $n$-tuple of nonoverlapping subintervals of $[a,b]$, if the total length of the intervals is less than $\delta$, then the total variation of $f$ on the intervals is less than $\epsilon$. That is (after $\epsilon$ and $\delta$), given $a \leq a_1 \leq b_1 \leq a_2 \leq b_2 \leq \cdots \leq a_n \leq b_n \leq b$, if
$$ \sum_{i = 1}^n (b_i - a_i) \lt \delta ,$$
then
$$ \sum_{i = 1}^n {|{f(b_i) - f(a_i)}|} \lt \epsilon .$$
=--
Various trivial variations of this may be met with: the comparison with $\delta$ and/or $\epsilon$ may be weak instead of strict; the number of subintervals may be infinite (so long as they are still nonoverlapping), since an infinite sum (of nonnegative numbers, as we have here) is simply a [[supremum]] of finite sums; and of course we may start by specifying that the $2n$ numbers come in order as the endpoints of the $n$ subintervals, rather than starting with any $2n$ numbers and then putting them in order and forming the subintervals from those. (Note that putting them in order is fine even in [[constructive analysis]], since choosing the $i$th element in order from a list of [[rational numbers]] is continuous, so may be extended constructively to real numbers, although we can\'t assume that the final list is a permutation of the original list.)
For the next definition, fix a model of [[nonstandard analysis]].
+-- {: .num_defn #nonstandard}
###### Definition
Given any [[hypernatural number]] $n$ in the model and any $2n$-tuple of elements of the nonstandard extension of $[a,b]$, interpreted as an increasing $n$-tuple of nonoverlapping subintervals of $[a,b]$, if the total length of the intervals is [[infinitesimal number|infinitesimal]], then the total variation of $f$ on the intervals is infinitesimal. That is, given [[hyperreal numbers]] $a \leq a_1 \leq b_1 \leq a_2 \leq b_2 \leq \cdots \leq a_n \leq b_n \leq b$, if
$$ \sum_{i = 1}^n (b_i - a_i) \approx 0 ,$$
then
$$ \sum_{i = 1}^n {|{f^*(b_i) - f^*(a_i)}|} \approx 0 .$$
=--
See [Tuckey 1993](#Tuckey1993), pages 34--36.
That the next definition is equivalent is the [[fundamental theorem of calculus]] for the [[Lebesgue integral]] on the [[real line]].
+-- {: .num_defn #Lebesgue}
###### Definition
There exists a Lebesgue-integrable function $g$ on $[a,b]$ such that $f(x) = f(a) + \int_{x=a} g(x) \,\mathrm{d}x$ for $x \in [a,b]$. (This is a [[semidefinite integral]].)
=--
In this case, $g$ must equal the [[derivative]] $f'$ [[almost everywhere]] on $[a,b]$. (So in particular, $f$ is differentiable almost everywhere with a Lebesgue-integrable derivative, although this is not enough without requiring that $f$ be an indefinite integral of its derivative.)
+-- {: .num_defn #Luzin}
###### Definition
The function $f$ is [[uniformly continuous]] on $[a,b]$, $f$ is of [[bounded variation]] on $[a,b]$, and the [[direct image]] under $f$ of any [[null subset]] of $[a,b]$ is null.
=--
+-- {: .num_defn #Stieltjes}
###### Definition
The [[Stieltjes measure]] $\mathrm{d}f$ is [[absolutely continuous measure|absolutely continuous]] with respect to [[Lebesgue measure]] on $[a,b]$.
=--
The last of these is the source of the term 'absolutely continuous' as applied to measures.
## Generalizations
One may easily generalize the [[codomain]] of the elementary definition of absolutely continuous functions to any [[metric space]].
## Examples
The cube-root function is absolutely continuous (on any bounded interval) but not Lipschitz continuous (on any interval containing $0$).
The [[Cantor function]] is *not* absolutely continuous, even though it is continuous, and differentiable almost everywhere, with a Lebesgue-integrable derivative.
## References
* Curtis Tuckey. 1993. Nonstandard Methods in the Calculus of Variations. CRC Press. [Google books snippet](https://books.google.com/books?id=RvrC1mw9tIEC&q=%22absolutely+continuous%22#v=snippet&q=%22absolutely%20continuous%22&f=false).
{#Tuckey1993}
[[!redirects absolutely continuous function]]
[[!redirects absolutely continuous functions]]
[[!redirects absolutely continuous mapping]]
[[!redirects absolutely continuous mappings]]
[[!redirects absolutely continuous map]]
[[!redirects absolutely continuous maps]]
|
absolutely continuous measure | https://ncatlab.org/nlab/source/absolutely+continuous+measure |
# Absolutely continuous measures
* table of contents
{: toc}
## Idea
A [[measure]] $\mu$ is absolutely continuous with respect to $\nu$ if we can think of $\mu$ as a weighted variation on $\nu$.
## Definition
Fix a [[measurable space]] $X$ and let $\mu$ and $\nu$ be two [[measures]] on $X$.
+-- {: .num_defn}
###### Definition
The measure $\mu$ is __absolutely continuous__ with respect to $\nu$ if every $\nu$-[[full set]] is also $\mu$-full.
=--
Because [[null sets]] are more familiar than [[full sets]], we may equivalently express things as follows (but this is not correct in [[constructive mathematics]]):
+-- {: .num_defn}
###### Definition (classical)
The measure $\mu$ is __absolutely continuous__ with respect to $\nu$ if every $\nu$-[[null set]] is also $\mu$-null.
=--
If $\mu$ and $\nu$ are [[positive measures]], then we may also express this as follows:
+-- {: .num_defn}
###### Definition (positive, classical)
The positive measure $\mu$ is __absolutely continuous__ with respect to $\nu$ if, for every [[measurable set]] $A$, $\mu(A) = 0$ if $\nu(A) = 0$.
=--
Since the [[absolute value]] of a measure is a positive measure, we can also express the general definition as follows:
+-- {: .num_defn}
###### Definition (classical)
The measure $\mu$ is __absolutely continuous__ with respect to $\nu$ if, for every [[measurable set]] $A$, ${|\mu|}(A) = 0$ if ${|\nu|}(A) = 0$.
=--
## Generalisation
Since only the [[full sets]] (or, classically, the [[null sets]]) matter, we do not need to have the full structure of a [[measure]]. Sometimes we equip a [[measurable space]] with a $\delta$-[[delta-filter|filter]] $\mathcal{F}$ of full sets (or a $\sigma$-[[sigma-ideal|ideal]] $\mathcal{N}$ of null sets) without specifying a measure that produces these. (For example, a [[smooth manifold]] is so equipped, effectively the full/null sets under [[Lebesgue measure]], even though there is no canonical such measure, since these sets are the same regardless of [[coordinate chart]].) Then we say:
+-- {: .num_defn}
###### Definition
The measure $\mu$ is __absolutely continuous__ with respect to $\mathcal{F}$ (or $\mathcal{N}$) if every element of $\mathcal{F}$ is $\mu$-[[full set|full]] (or every element of $\mathcal{N}$ is $\mu$-[[null set|null]]).
=--
In fact, only the full/null sets of $\mu$ matter either, but until somebody has use for the notion of one $\delta$-filter (or $\sigma$-ideal) being absolutely continuous with respect to another, I will refrain from writing it down. (See [[centipede mathematics]].)
## Defaults
If one calls a measure on the [[real line]] 'absolutely continuous', this means with respect to [[Lebesgue measure]].
This generalises to any [[cartesian space]] (with Lebesgue measure) or indeed to any [[smooth manifold]] of [[finite dimension]] (where there is no canonical Lebesgue measure but a family of local ones and so still a notion of Lebesgue-full and Lebesgue-null sets).
Or, this generalises to any [[compact group]] (with [[Haar measure]]) or indeed to any [[locally compact group]] (where there is no canonical Haar measure but a proportional family of them and so still a canonical notion of Haar-full and Haar-null sets).
## Properties
Let $\nu$ be a measure, and let $f \in L^1(\nu)$ be an [[absolutely integrable function]] with respect to $\nu$; then [[integration]] defines a measure $f \nu$:
$$ (f \nu)(E) = \int_E f \nu = \int_E f(x) \nu(\mathrm{d}x) .$$
This measure $f \nu$ is absolutely continuous with respect to $\nu$. Conversely, given any absolutely continuous measure $\mu$, there is (at most) a unique (up to [[almost equality]]) absolutely integrable function $f$ such that $\mu = f \nu$; and this function must exist if $\mu$ and $\nu$ are [[localizable measure|localizable]]. This converse is the subject of the [[Radon–Nikodym theorem]].
A function $f\colon \mathbb{R} \to \mathbb{R}$ is [[absolutely continuous function|absolutely continuous]] iff the [[Stieltjes integral|Lebesgue–Stieltjes measure]] $\mathrm{d}f$ is absolutely continuous with respect to [[Lebesgue measure]]. (All such functions are [[continuous function|continuous]], and this example is actually the origin of the term.)
[[!redirects absolutely continuous measure]]
[[!redirects absolutely continuous measures]]
|
absolutely convex subset | https://ncatlab.org/nlab/source/absolutely+convex+subset |
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### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}
## Definition
A subset $B$ of a [[vector space]] $V$ over $\mathbb{R}$ or $\mathbb{C}$ is **absolutely convex** if $\lambda x + \mu y \in B$ whenever $x, y \in B$ and ${|\lambda|} + {|\mu|} \le 1$.
## Properties
Absolutely convex subsets are closely related to [[semi-norms]].
Given a vector space $V$ and an absolutely convex subset $B \subseteq V$, we define $V_B$ to be the linear span of $B$ in $V$. Then let $\mu_B \colon V_B \to \mathbb{R}$ be the [[Minkowski functional]] of $B$. That is, $\mu_B$ is given by:
$$
\mu_B(v) = \inf\{ t \gt 0 : t v \in B\}
$$
Then $\mu_B$ is a semi-norm on $V_B$ and $E_B \coloneqq V_B/\ker \mu_B$ is a [[normed space]].
# Related concepts
* [[convex set]]
* [[balanced set]]
[[!redirects absolutely convex]]
[[!redirects absolutely convex set]]
[[!redirects absolutely convex sets]]
[[!redirects absolutely convex subset]]
[[!redirects absolutely convex subsets]]
|
absolutely dense functor | https://ncatlab.org/nlab/source/absolutely+dense+functor | [[!redirects absolute dense functor]]
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### Context
#### Category theory
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# Contents
* table of contents
{:toc}
## Definition
An **absolutely dense functor** $F \colon A \to B$ is a [[functor]] which is equivalently characterised by the following conditions:
1. $F$ is [[dense]] and $Lan_F F$ is an [[absolute Kan extension]].
1. $F^* = [F^{op}, Set] \colon [B^{op}, Set] \to [A^{op}, Set]$ is [[fully faithful]].
1. The [[counit]] of the adjunction $F_* \dashv F^* : [B^{op}, Set] \to [A^{op}, Set]$ is invertible.
1. $F$ is [[corepresentably fully faithful]] (also called a [[lax epimorphism]]) in [[Cat]].
1. For every functor $G : B^{op} \times B \to C$, there is a canonical isomorphism ([MathOverflow answer](https://mathoverflow.net/a/354097))
$$ \int_{b \in B} G(b, b) \cong \int_{a \in A} G(F a, F a) $$
(This is a notion of [[final functor|initiality]] for [[ends]].)
More generally, an **absolutely dense morphism** in a [[proarrow equipment]] $K \to M$ is a 1-cell $f : a \to b$ in $K$ for which the counit $f_* \circ f^* \cong 1_b$.
## Properties
Every simultaneously reflective and coreflective subcategory of a presheaf category is itself a presheaf category and is induced by precomposition along an absolutely dense functor: this is the main result of the paper of El Bashir--Velebil below.
## References
(Absolutely dense functors are called **Cauchy dense** in the following paper of [[Brian Day]].)
- [[Brian Day]]. _Density presentations of functors_. Bulletin of the Australian Mathematical Society 16.3 (1977): 427-448.
- [[Jiri Adamek]], [[Robert El Bashir]], [[Manuela Sobral]], [[Jiri Velebil]]. _On functors which are lax epimorphisms_. TAC. (2001)
- Robert El Bashir and [[Jiri Velebil]]. _Simultaneously reflective and coreflective subcategories of presheaves_. Theory and Applications of Categories 10.16 (2002): 410-423.
- [[Fernando Lucatelli Nunes]] and [[Sousa Lurdes]]. _On lax epimorphisms and the associated factorization_. Journal of Pure and Applied Algebra 226.12 (2022)
[[!redirects absolutely dense]]
[[!redirects absolutely dense functors]]
[[!redirects absolutely dense morphism]]
[[!redirects absolutely dense morphisms]]
|
absorbing subset | https://ncatlab.org/nlab/source/absorbing+subset |
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### Context
#### Functional analysis
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#Contents#
* table of contents
{:toc}
## Definition
Let $V$ be a [[vector space]] over $\mathbb{R}$ or $\mathbb{C}$. A subset $B$ of $V$ is said to be **absorbing** if $\bigcup_{r \gt 0} r B = V$.
## Examples
* The unit ball in a [[seminormed vector space]] is absorbing.
* The linear span of absolutely convex set $C$ in a vector space has $C$ as an absorbing set.
## Related concepts
* [[convex set]]
* [[balanced set]]
* [[absolutely convex set]]
category: functional analysis
[[!redirects absorbing]]
[[!redirects absorbing set]]
[[!redirects absorbing sets]]
[[!redirects absorbing subset]]
[[!redirects absorbing subsets]]
|
absorption category | https://ncatlab.org/nlab/source/absorption+category | +-- {: .rightHandSide}
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#### Algebra
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#### Category theory
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#### Categorification
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[[!include categorification - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
The concept of an absorption category should be the [[oidification]] of an [[absorption monoid]].
## Definition
An __absorption category__ or __annihilation category__ $C$ is a [[category]] where for every two objects $a, b \in Ob(C)$ there is a morphism $0_{a\to b}: a \to b$, such that for any objects $a, b, c, d \in Ob(C)$, for any morphism $f:b \to c$, $f \circ 0_{a\to b} = 0_{a\to c}$, and for any morphism $g:d \to a$, $0_{a\to b} \circ g = 0_{d\to b}$.
Such a structure is the same thing as a [[enriched category theory|category enriched]] in the category of [[pointed sets]], taking the [[monoidal category|monoidal product]] to be the [[smash product]].
## Examples
* Every [[ringoid]] and [[algebroid]] is an absorption category.
* An [[absorption monoid]] is an absorption category with only one object.
## Related concepts
[[!include oidification - table]]
[[!redirects absorption categories]]
[[!redirects annihilation category]]
[[!redirects annihilation category]] |
absorption magma | https://ncatlab.org/nlab/source/absorption+magma | # Contents
* table of contents
{: toc}
## Definition
An __absorption magma__ or __annihilation magma__ is a [[magma]] $(M,\cdot)$ with an element $0\in M$ satisfying the absorption/annihilation axioms: for all $a \in M$, $0 \cdot a = 0$ and $a \cdot 0 = 0$.
Equivalently, this is a [[magma object]] in the [[category]] of [[pointed sets]].
## Properties
### Zero divisors
A non-zero element $a \in M$ is a __[[zero divisor]]__ if thete exists a non-zero element $b \in M$ such that $a \cdot b = 0$ or $b \cdot a = 0$.
## Examples
* [[absorption monoids]]
* Every [[magma object]] in the category of [[commutative magma|commutative]] [[unital magmas]] is an absorption magma. Particular examples of this include the multiplicative monoid in a rig (monoid objects in [[CMon]]) or a ring (monoid objects in [[Ab]].
* The multiplicative magmas of the [[octonions]] and the [[sedenions]] are absorption magmas.
## See also
* [[magma]]
* [[zero divisor]]
* [[rig]], [[ring]], [[lattice]], [[integral domain]] [[field]]
[[!redirects absorption magmas]]
[[!redirects annihilation magma]]
[[!redirects annihilation magmas]]
[[!redirects integral magma]]
[[!redirects integral magmas]]
[[!redirects division magma]]
[[!redirects division magmas]] |
absorption monoid | https://ncatlab.org/nlab/source/absorption+monoid | +-- {: .rightHandSide}
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###Context###
#### Algebra
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#### Monoid theory
+-- {: .hide}
[[!include monoid theory - contents]]
=--
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# Contents
* table of contents
{: toc}
## Idea
Absorption monoids are the [[monoid objects]] in [[pointed sets]], in the same way that [[rings]] are the [[monoid objects]] in [[abelian groups]]. Thus, the theory of absorption monoids and the theory of rings are very similar to each other, except that rings have additive structure whereas absorption monoids do not have additive structure.
## Definition
An __absorption monoid__ or __annihilation monoid__ is a [[monoid]] $(M,1,\cdot)$ that is also an [[absorption magma]] $(M,0)$.
Equivalently, it is a [[monoid object]] in the [[category]] of [[pointed sets]], since left and right multiplication $\cdot$ by any element $x$ preserves the point $0$.
## Properties
### Initial and terminal absorption monoids
The [[initial object|initial]] absorption monoid is the [[boolean domain]] $\mathbb{2}$ with elements $0 \in \mathbb{2}$ representing [[false]], $1 \in \mathbb{2}$ representing [[true]], and $(-)\cdot(-):\mathbb{2} \times \mathbb{2} \to \mathbb{2}$ representing [[conjunction]].
The [[terminal object|terminal]] absorption monoid is the [[trivial monoid]] $\mathbb{1}$, the monoid whose underlying [[set]] is a [[singleton]]. The trivial monoid is also [[strictly terminal]].
### Absorption monoid homomorphisms
Given absorption monoids $M$ and $N$, an absorption monoid homomorphism is a function $h:M \to N$ such that
* $h(0) = 0$
* $h(1) = 1$
* for all $a \in M$ and $b \in M$, $h(a \cdot b) = h(a) \cdot h(b)$.
### Ideals and anti-ideals
A two-sided [[ideal]] of an absorption monoid $M$ is a subset $I$ of $M$ such that
* $0 \in I$
* for all elements $a \in M$ and $b \in M$, if $a \cdot b \in I$, then either $a \in I$ or $b \in I$.
A two-sided anti-ideal of an absorption monoid $M$ is a subset $A$ of $M$ such that
* $0 \notin I$
* for all elements $a \in M$ and $b \in M$, if $a \in I$ and $b \in I$, then $a \cdot b \in I$.
### Quotient absorption monoids
Given an absorption monoid $M$ and a two-sided ideal $I$, the quotient of $M$ by $I$ is the [[initial object|initial]] absorption monoid $M/I$ with absorption monoid homomorphism $i:M \to M/I$ such that for all elements $a \in I$, $i(a) = 0$.
### Invertible elements
An element $a \in M$ is an **invertible element** or a **unit** if there exists an element $b \in M$ such that $a \cdot b = 1$ and $b \cdot a = 1$.
The set of invertible elements $M^\times$ in an absorption monoid $M$ is always closed under multiplication; i.e. $M^\times$ is a [[submonoid]] of $M$. In fact, since every element is invertible, $M^\times$ forms a [[subgroup]] of $M$, called the [[group of units]].
### Division monoids
An absorption monoid $M$ is a [[division monoid]] if every non-invertible element in $M$ is equal to zero. $M$ is Heyting if there is a [[tight apartness relation]] on $M$ such that every invertible element is apart from zero, and $M$ is discrete if every element in $M$ is either zero or invertible.
### Regular elements
An element $a \in M$ is a **regular element**, **cancellative element**, or **cancellable element** if for all elements $b \in M$ and $c \in M$, $b = c$ if and only if $a \cdot b = a \cdot c$ and $c \cdot a = c \cdot b$.
The set of regular elements $\mathrm{Reg}(M)$ in an absorption monoid $M$ is always closed under multiplication; i.e. $\mathrm{Reg}(M)$ is a [[submonoid]] of $M$.
### Integral monoids
An absorption monoid $M$ is an [[integral monoid]] if every non-regular element in $M$ is equal to zero. $M$ is Heyting if there is a [[tight apartness relation]] on $M$ such that every regular element is apart from zero, and $M$ is discrete if every element in $M$ is either zero or regular.
### Ore sets and Ore absorption monoids
Given an absorption monoid $M$, an [[Ore set]] is a [[submonoid]] $S$ of $\mathrm{Reg}(M)$ such that every element of $S$ satisfies the left and right Ore conditions:
* for all $a \in S$ and $b \in M$, there exists $c \in S$ and $d \in M$ such that $a \cdot d = b \cdot c$
* for all $a \in S$ and $b \in M$, there exists $c \in S$ and $d \in M$ such that $d \cdot a = c \cdot b$
A absorption monoid is an Ore absorption monoid if $\mathrm{Reg}(M)$ is an Ore set.
### Localization and group completion
The localization of an Ore integral monoid $M$ at $\mathrm{Reg}(M)$ is a [[division monoid]]. The [[localization of a monoid|localization]] of an absorption monoid at $0$ is the [[trivial group]]; thus, the [[group completion]] of any absorption monoid is the [[trivial group]].
### Actions and modules
Given an absorption monoid $M$, an left $M$-action on a pointed set $(P, 0)$ is an ternary function $\alpha_L:M \times P \to P$ such that:
* for all elements $p \in P$, $\alpha_L(1, p) = p$
* for all elements $a \in M$, $b \in M$, and $c \in P$, $\alpha_L(a, \alpha_L(b, c)) = \alpha_L(a \cdot b, c)$
* for all elements $p \in P$, $\alpha_L(0, p) = 0$
* for all elements $a \in M$, $\alpha_L(a, 0) = 0$
A right $M$-action on a pointed set $(P, 0)$ is a binary function $\alpha_R:P \times M \to P$ such that:
* for all elements $p \in P$, $\alpha_R(p, 1) = p$
* for all elements $a \in M$, $b \in M$, and $c \in P$, $\alpha_R(\alpha_R(c, a), b) = \alpha_R(c, a \cdot b)$
* for all elements $p \in P$, $\alpha_R(p, 0) = 0$
* for all elements $a \in M$, $\alpha_R(0, a) = 0$
Given absorption monoids $M$ and $N$, an $M$-$N$-[[biaction]] on a pointed set $(P, 0)$ is a ternary function $\alpha:M \times P \times N \to P$ such that:
* for all elements $p \in P$, $\alpha(1, p, 1) = p$
* for all elements $a \in M$, $b \in M$, $c \in P$, $d \in N$, $e \in N$, $\alpha(a, \alpha(b, c, d), e) = \alpha_L(a \cdot b, c, d \cdot e)$
* for all elements $p \in P$ and $d \in N$, $\alpha_L(0, p, d) = 0$
* for all elements $a \in M$ and $d \in N$, $\alpha_L(a, 0, d) = 0$
* for all elements $a \in M$ and $p \in P$, $\alpha_L(a, p, 0) = 0$
Pointed sets equipped with left or right $M$-actions are called left or right $M$-[[module object|modules]], and pointed sets equipped with $M$-$N$-biactions are called $M$-$N$-[[bimodule object|bimodules]].
## Examples
### The multiplicative monoid of the natural numbers
The multiplicative monoid of the natural numbers $\mathbb{N}^\times$ is the [[free object|free]] commutative absorption monoid on the natural numbers, the [[initial object|initial]] commutative absorption monoid $\mathbb{N}^\times$ with a function $\mathrm{prime}:\mathbb{N} \to \mathbb{N}^\times$. $\mathbb{N}^\times$ has [[decidable equality]]. The localization of $\mathbb{N}^\times$ at the image of $\mathrm{prime}$, or equivalently at the non-zero elements of $\mathbb{N}^\times$, is the multiplicative monoid of the non-negative rational numbers, $\mathbb{Q}_{\geq 0}^\times$.
### Other examples
* The [[extended natural numbers]] $(\bar{\mathbb{N}}, 0, +, \infty)$ are an absorption monoid.
* Every [[integral monoid]] is an absorption monoid.
* The multiplicative monoid of every [[rig]] is an absorption monoid.
* Every join-[[semilattice]] with a [[top element]] and evety meet-semilattice with a [[bottom element]] is an absorption monoid.
## Related concepts
* [[monoid]]
* [[integral monoid]]
* [[zero divisor]]
* [[rig]], [[ring]], [[lattice]], [[field]]
[[!include oidification - table]]
[[!redirects absorption monoids]]
[[!redirects annihilation monoid]]
[[!redirects annihilation monoids]]
[[!redirects absorbing element]]
[[!redirects absorbing elements]]
[[!redirects annihilating element]]
[[!redirects annihilating elements]] |
Abstract and Concrete Categories | https://ncatlab.org/nlab/source/Abstract+and+Concrete+Categories |
This page collects links related to the book
* [[Jiri Adamek|Jiří Adámek]], [[Horst Herrlich]], [[George Strecker]]:
**Abstract and Concrete Categories -- The Joy of Cats**
John Wiley and Sons, New York (1990)
reprinted as:
Reprints in Theory and Applications of Categories **17** (2006) 1-507
[tac:tr17](http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html)
[book webpage](http://katmat.math.uni-bremen.de/acc/), [pdf](http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf)
on [[category theory]]
Originally published in the series
Pure and Applied Mathematics. A Wiley-Interscience Publication.
John Wiley & Sons, Inc., New York, 1990. xiv+482 pp. ISBN: 0-471-60922-6
Online edition appeared in 2004:
* Reprints in Theory and Applications of Categories, No. 17 (2006), 1–507.
[abstract](http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html),
[pdf](http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf).
Beware that the book uses the term "quasicategory" for what _[[Categories for the Working Mathematician]]_ calls a [[metacategory]]: a model (in the informal sense) of the first-order theory of a category, with no regard for size issues. These days a [[quasicategory]] is mostly understood to be a particular type of [[simplicial set]] that models an [[(∞,1)-category]].
category: reference
[[!redirects Abstract and Concrete Categories: The Joy of Cats]]
[[!redirects Abstract and Concrete Categories -- The Joy of Cats]]
[[!redirects Joy of Cats]]
[[!redirects The Joy of Cats]]
|
abstract circle | https://ncatlab.org/nlab/source/abstract+circle |
#Contents#
* table of contents
{:toc}
## Idea
An _abstract circle_ in the sense of ([Moerdijk 96](#Moerdijk96)) is essentially a cellular structure of the form of a [[circle]] $S^1$, in direct analogy to how a [[linear interval]] in the abstract sense is a cellular model of the actual topological interval $[0,1]$.
Indeed, just as linear intervals have as [[classifying topos]] the category of [[simplicial sets]], so abstract circles have as classifying topos the category of [[cyclic sets]] ([Moerdijk 96](#Moerdijk96)).
## References
* {#Moerdijk96} [[Ieke Moerdijk]], _Cyclic sets as a classifying topos_, 1996 ([[MoerdijkCyclic.pdf:file]])
[[!redirects abstract circles]]
|
abstract elementary class | https://ncatlab.org/nlab/source/abstract+elementary+class |
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#### Model theory
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#Contents#
* table of contents
{:toc}
## Idea
In [[model theory]] the notion of *abstract elementary classes* is a vast generalizations of that of [[elementary classes]] of [[structure in model theory|structures]] beyond [[first-order theory|first-order theories]] (e.g. for the infinitary logic $L_{\omega_1,\omega}$) as introduced by [[Saharon Shelah]]. Their theory is also more general than the homogeneous model theory.
An __abstract elementary class__ is a __nonempty__ class $K$ of structures for a given signature with language $L(K)$, that is closed under [[isomorphisms]] and equipped with a strong substructure relation $\prec_K$ (strong substructure relation means that if $M\prec_K N$ and $M_0\subset M$ is a substructure, then $M_0 \prec_K N$) that is a [[partial order]] satisfying the axioms on union of chains (Tarski-Vaught), coherence and downward Loewenheim-Skolem properties. More precisely, $\prec_K$ is a partial order such that
(A0) if $M,N\in K$, $M\prec_K N$ then $M\subset N$
(A1) (closure under isomorphisms)
* (a) $M\in K$ and $N$ an $L(K)$ structure with $N\cong M$, then $N\in K$
* (b) if $N_1,N_2,M_1,M_2\in K$, $f_i : N_i\cong M_i$, $i = 1,2$, $f_1\subset f_2$, with $M_1\prec_K M_2$ then $N_1\prec_K N_2$
(A2) for $M,N,P\in K$, if $M\prec_K P$, $N\prec_K P$, and $M\subset N$, then $M\prec_K N$
(A3) downward Loewenheim-Skolem. There exist a cardinal $LS(K) = LS(K,\prec_K)\geq |L(K)|+\aleph_0$ such that $\forall M\in K$, $\forall A\subset |M|$, $\exists N\in K$ with $A\subset |N|$, $N\prec_K M$, $\|N\|\leq |A|+LS(K)$.
(A4) (Tarski-Vaught chain condition) for every regular cardinal $\mu$
....
The usual elementary classes, i.e. the classes of the form $K = Mod(T)$ for a first-order theory $T$, are abstract elementary with respect to the relation $\prec_K$ of being an elementary submodel, with ${|LS(K)|} = {|L(T)|}+\aleph_0$ ($L(T)$ is the underlying language of the theory $T$).
## Related entries
* [[amalgamation]]
## References
* wikipedia [abstract elementary class](http://en.wikipedia.org/wiki/Abstract_elementary_class)
* [[Saharon Shelah]], _Classification theory for elementary abstract classes I, II, Studies in Logic (London), __18__, __20__, College Publications, London 2009
* John Baldwin, _Categoricity_, Amer. Math. Soc. 2011, [pdf](http://www.math.uic.edu/~jbaldwin/pub/AEClec.pdf)
* D. W. Kueker, _Abstract elementary classes and infinitary logic_, Ann. Pure Appl. Logic __156__ (2008), 274-286.
AECs can also be essentially identified with [[accessible categories]] in which all morphisms are [[monomorphisms]]. Some recent papers which study them from this viewpoint include:
* [[Tibor Beke]], [[Jiří Rosický]] _Abstract elementary classes and accessible categories_, Annals of Pure and Applied Logic __163__ (2012) 2008-2017, [arxiv/1005.2910](http://arxiv.org/abs/1005.2910)
* Michael Lieberman, [[Jiří Rosický]], Sebastien Vasey, *Internal sizes in μ-abstract elementary classes*, [arxiv/1708.06782](https://arxiv.org/abs/1708.06782); *Set-theoretic aspects of accessible categories*, [arxiv/1902.06777](https://arxiv.org/abs/1902.06777)
* M. J. Lieberman, _Topological and category-theoretic aspects of abstract elementary classes_, Thesis, The University of Michigan 2009, [pdf](http://deepblue.lib.umich.edu/bitstream/2027.42/63854/1/liebermm_1.pdf); defense slides [pdf](http://www.math.upenn.edu/~mlieb/defense.pdf); _Category theoretic aspects of abstract elementary classes_, Annals Pure Appl. Logic __162__ (2011), 903-915; _A topology for Galois types in AECs_, [arxiv/0906.3573](http://arxiv.org/abs/0906.3573)
On [[topos theory|topos theoretic]] methods in model theory of AECs:
* Christian Espíndola, _A topos-theoretic proof of Shelah's eventual categoricity conjecture for abstract elementary classes_, [arxiv/1906.09169](https://arxiv.org/abs/1906.09169); _A short proof of Shelah's eventual categoricity conjecture for AEC's with amalgamation, under GCH_, [arxiv/1909.13713](https://arxiv.org/abs/1909.13713)
On the [[homotopy types]] of [[classifying spaces]] of abstract elementary classes:
* [[Tim Campion]], Jinhe Ye, *Homotopy Types of Abstract Elementary Classes*, Journal of Pure and Applied Algebra
**225** 5 (2021) 106461 [[arXiv:1909.07965](https://arxiv.org/abs/1909.07965), [doi:10.1016/j.jpaa.2020.106461](https://doi.org/10.1016/j.jpaa.2020.106461)]
[[!redirects abstract elementary classes]] |
abstract general, concrete general and concrete particular | https://ncatlab.org/nlab/source/abstract+general%2C+concrete+general+and+concrete+particular |
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#### Category theory
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=--
=--
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#Contents#
* table of contents
{:toc}
## Idea
The [[category theory|category theoretic]] notions of
* _[[category]]_ and _[[object]]_
on the one hand and of
* _[[theory]]_ and _[[model]]_
on the other have been suggested ([Lawvere](#Lawvere)) to usefully formalize, respectively, the heuristic notions
* "general" and "particular"
as well as
* "abstract" and "concrete", respectively.
We have:
* a ([[syntactic category]] of a) [[Lawvere theory]] $T$ (or the equivalent in any [[doctrine]]) $T$ is an _abstract general_ or _abstract universal_ (*abstraktes Allgemeines*)
* the [[category]] $T Mod(E)$ of $T$-[[models]]/[[algebra over an algebraic theory|algebras]] in any context $E$ is a _concrete general_ or _concrete universal_
* an [[object]] of any $T Mod(E)$ is a _particular_.
That seems to be roughly what is suggested in [Lawvere](#Lawvere). Of course one could play with this further and consider further refinement such as
* a (generating) [[object]] in $T$ is an _abstract particular_ ;
* an [[object]] of any $T Mod(E)$ is a _concrete particular_.
## Examples
### Groups
The [[syntactic category]] $T_{Grp}$ of the [[theory]] of [[group]]s is the "general abstract" of groups. Its essentially unique generating object is _the abstract particular_ group.
The category $T_{Grp} Mod(Set) = $ [[Grp]] of all groups is the _concrete general_ of groups.
An object in there is some [[group]]: a concrete particular.
## References
The [[category theory|category-theoretic]] formalization of these notions as proposed by Bill Lawvere is disussed in print for instance in
* {#Lawvere} [[Bill Lawvere]], _Categorical refinement of a Hegelian principle_, section 1 of [[Bill Lawvere]], _[[Tools for the advancement of objective logic]] -- Closed categories and toposes_, in John Macnamara, [[Gonzalo Reyes]], _the logical foundations of cognition_, Oxford University Press (1994)
See also an email comment recorded [here](http://conceptualmathematics.wordpress.com/2012/06/09/general-concepts-and-reality-prof-f-william-lawvere/).
For discussion of "particular" and related in [[philosophy]] see also
The terminology is inspired by
* [[Georg Hegel]], _[[Science of Logic]]_,
{#HegelQuotes} for instance
> [EL§61](Science+of+Logic#EL61) If we are to believe the Critical philosophy, thought is subjective, and its ultimate and invincible mode is abstract universality or formal identity. Thought is thus set in opposition to Truth, which is no abstraction, but concrete universality. In this highest mode of thought, which is entitled Reason, the Categories are left out of account. The extreme theory on the opposite side holds thought to be an act of the particular only, and on that ground declares it incapable of apprehending the Truth. This is the Intuitional theory.
> [§71](Science+of+Logic#71) It is only after profounder acquaintance with the other sciences that logic ceases to be for subjective spirit a merely abstract universal and reveals itself as the universal which embraces within itself the wealth of the particular
> [§1323](Science+of+Logic#1323) This universal Notion, which we have now to consider here, contains the three moments: universality, particularity and individuality.
> [§1337b](Science+of+Logic#1337b) When people talk of the determinate Notion, what is usually meant is merely such an abstract universal.
> [§1599](Science+of+Logic#1599) Such a universal which merely subsumes, is an abstraction which only becomes concrete in something else, in the particular. End, on the contrary, is the concrete universal, which possesses in its own self the moment of particularity
> [PS§456b](Science+of+Logic#PS456b) This common element is either any one particular side of the object raised to the form of universality, such as, for example, in the rose, the red colour; or the concrete universal, the genus, for example, in the rose, the plant;
See also in and around the section [The genus](Science+of+Logic#TheGenus)
Survey of these Hegelian ideas includes
* {#HibbenLuft} John Grier Hibben, Eric v.d. Luft, _Hegel's Shorter Logic: An Introduction and Commentary_
where on p. 143 it says about the _Shorter Logic_:
> Particularity and individuality are related as "abstract" and "concrete", respectively. The particular is the "abstract individual". The individual is the "concrete particular". The universal is their union, and may be either "abstract" or "concrete". The so-called "concrete universal" is Hegel's gold standard for conceptual thought $[$...$]$.
For a discussion of proponents of the concrete universal in British philosophy, and of their divergences from, and continuities with, Hegelian thought, see
* Robert Stern, _Hegel, British idealism, and the curious case of the concrete universal_, British Journal for the History of Philosophy 15(1) 2007: 115 – 153 ([pdf](http://www.sheffield.ac.uk/polopoly_fs/1.101730!/file/concrete-universal-published.pdf))
where he cites the philosopher Bernard Bosanquet
> A world or cosmos is a system of members, such that every member, being _ex hypothesi_ distinct, nevertheless contributes to the unity of the whole in virtue of the peculiarities which constitute its distinctness. And the important point for us at present is the difference of principle between a world and a class. It takes all sorts to make a world; a class is essentially of one sort only. In a word, the difference is that the ultimate principle of unity and community is fully exemplified in the former, but only superficially in the latter. The ultimate principle, we may say, is sameness in the other; generality is sameness in spite of the other; universality is sameness by means of the other. (Bosanquet, The Principle of Individuality and Value, 3)
and T. M. Knox
> An abstract universal has no organic connexion with its particulars. Mind, or reason, as a concrete universal, particularizes itself into differences which are interconnected by its universality in the same way in which parts of the organism are held together by the single life which all things share. The parts depend on the whole for their life, but on the other hand the persistence of life necessitates the differentiation of the part. (Translator’s notes to Hegel’s Philosophy of Right (Oxford: Oxford University Press, 1952))
For general related discussion see also
* Wikipedia, _[Particular](http://en.wikipedia.org/wiki/Particular)_, _[Abstract particular](http://en.wikipedia.org/wiki/Abstract_particulars)_
[[!redirects abstract general]]
[[!redirects abstract generals]]
[[!redirects abstract universal]]
[[!redirects general abstract]]
[[!redirects concrete general]]
[[!redirects general concrete]]
[[!redirects concrete general]]
[[!redirects concrete generals]]
[[!redirects concrete universal]]
[[!redirects concrete particular]]
[[!redirects concrete particulars]]
[[!redirects abstract particular]]
[[!redirects abstract particulars]]
[[!redirects abstract general, concrete general and concrete particular]]
[[!redirects abstract general, concrete general, and concrete particular]]
[[!redirects abstract general, concrete general, concrete particular]] |
Abstract Homotopy and Simple Homotopy Theory | https://ncatlab.org/nlab/source/Abstract+Homotopy+and+Simple+Homotopy+Theory | This is a book by [[K. H. Kamps]] and [[T. Porter]] that introduces and discusses several different approaches to abstract homotopy theory.
##Reference
* K. H. Kamps and T. Porter, _Abstract Homotopy and Simple Homotopy Theory_ ([GoogleBooks](http://books.google.de/books?id=7JYKxInRMdAC&dq=Porter+Kamps&printsec=frontcover&source=bl&ots=uuyl_tIjs4&sig=Lt8I92xQBZ4DNKVXD0x76WkcxCE&hl=de&sa=X&oi=book_result&resnum=3&ct=result#PPP1,M1)) |
abstract model theory | https://ncatlab.org/nlab/source/abstract+model+theory | +-- {: .rightHandSide}
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#### Model theory
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# Contents
* table of contents
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## Idea
_Abstract model theory_ is the study of the general properties of the [[model theory]] of extensions of (classical untyped) [[first-order logic]].
Originally motivated by _[[Lindström's theorem]]_ that characterizes first-order logic, the field has subsequently been extended to provide alternative characterizations and include different logics within its range.
The basic concept of abstract model theory is that of an _abstract logic_
which is a triple $\mathcal{L}=(S,\Phi ,\models)$ where $\models$ is a binary relation between the class of $\mathcal{L}$-'structures' $S$ and the class of $\mathcal{L}$-'sentences' $\Phi$ to be thought of as minimalistic version of the _satisfaction_ relation.
##Remark
As the logical relations studied by abstract model theory are of a functorial nature, some category theory entered the picture already in Barwise (1974). The theory of [[institution|institutions]], aka _institution-independent model theory_ (Diaconescu 2008), constitutes abstract categorical model theory proper. In a similar abstract categorical vein is the [[geometric theory#FunctorialDefinition|functorial approach to geometric theories]] described in Johnstone (2002, sec. B4.2).
## Related entries
* [[model theory]]
* [[Lindström's theorem]]
* [[compactness theorem]]
* [[Löwenheim-Skolem theorem]]
* [[institution]]
* [[Per Lindström]]
## Link
* Wikipedia, _[Abstract model theory](http://en.wikipedia.org/wiki/Abstract_model_theory)_
## References
* Jon Barwise, _Axioms for abstract model theory_ , Annals of Mathematical Logic 7 pp.221-265, 1974.
* Barwise, Feferman (eds.), _Model-theoretic Logics_ , Springer Heidelberg 1985 (freely available online: [toc](http://projecteuclid.org/euclid.pl/1235417263#toc)) .
* Răzvan Diaconescu, _Institution-independent Model Theory_ , Birkhäuser Basel 2008.
* Marta García-Matos, [[Jouko Väänänen]], _Abstract Model Theory as a Framework for Universal Logic_ , Logica Universalis 2005 pp.1-33 ([draft](http://www.math.helsinki.fi/logic/opetus/lt/unilog2005_proofs.pdf)) .
* [[Peter Johnstone]], _[[Sketches of an Elephant]] vol. I_ , Oxford UP 2002.
* [[Jouko Väänänen]], _The Craig Interpolation Theorem and abstract model theory_ , Synthese 164:401 (2008) (freely available online) |
abstract rewriting system | https://ncatlab.org/nlab/source/abstract+rewriting+system | # Contents
* table of contents
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## Idea
The notion of abstract rewriting system is the simplest mathematical formalization of what is a [[rewriting system]]. It doesn't presuppose anything on the nature of the [[syntax|syntactical]] objects which are rewritten, thus the word "abstract".
## Definition
An **abstract rewriting system** $(X, \rightarrow)$ is given by a [[set]] $X$ and a [[binary relation]] $\rightarrow$ on $X$, ie. a [[subset]] of the [[cartesian product]] $X \times X$.
So this is a [[concept with an attitude]]: While an abstract re-writing system is just a [[relation]], calling this relation an abstract rewriting system indicates that one is interested in studying the behaviour of chains of related elements $x \to x_1 \to x_2 \to \cdots$ (thought of as successive stages of [[rewriting]] $x$), for instance to see if they are [[confluent category|confluent]].
## Related entries
* [[rewriting system]]
## References
* Wikipedia, [Abstract rewriting system](https://en.wikipedia.org/wiki/Abstract_rewriting_system)
* Wikipedia, <a href="https://en.wikipedia.org/wiki/Confluence_(abstract_rewriting)">Confluence_(abstract_rewriting)</a>
[[!redirects abstract rewriting systems]]
[[!redirects abstract re-writing system]]
[[!redirects abstract re-writing systems]]
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abstract scattering theory | https://ncatlab.org/nlab/source/abstract+scattering+theory |
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#### Physics
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## Idea
In [[quantum mechanics]] based on a [[Hilbert space]] $H$ wth quantum evolution semigroup of [[unitary operators]] $U(t)$, $t\in \mathbf{R}$, Lax and Phillips formulated an abstract formulation of [[scattering]] theory, based on two distinguished subspaces of
states $\mathcal{D}_-$ of ingoing states and $\mathcal{D}_+$ of outgoing states satisfying certain system of axioms.
Apart from [[quantum mechanics]], abstract scattering theory has other interesting special cases, applying to the theory of [[automorphic form|automorphic functions]] (due to an idea of [[Israel Gelfand]], later realized by B. S. Pavlov and L. D. Faddeev), zeta function analysis and so on.
## Related concepts
* [[scattering amplitude]]
* [[scattering matrix]]
* [[scattering cross section]]
## References
* Peter D. Lax, Ralph S. Phillips, _Scattering theory_, Bull. Amer. Math. Soc. Volume 70, Number 1 (1964), 130-142 [euclid](http://projecteuclid.org/euclid.bams/1183525789) [MR0167868](http://www.ams.org/mathscinet-getitem?mr=0167868); _Scattering theory_ (book), Academic Press 1990, 2nd ed. [gBooks](http://books.google.hr/books?id=vgqMqqCnf5AC&lpg=PP1&pg=PP1#v=onepage&q&f=false); _Scattering theory for automorphic functions, Annals of Mathematics Studies __87__, Princeton Univ. Press 1976
* B. S. Pavlov, L. D. Faddeev, _Scattering theory and automorphic functions_, Zapiski Sem. LOMI __27__ (1972) 161-193
Extension of the method which Lax and Phillips used for automorphic forms ($SL(2,\mathbf{R})$-case) to the case of more general harmonic analysis on more general Riemannian [[symmetric spaces]] is studied in
* M. A. [[Semenov-Tian-Shansky]], _Harmonic analysis on Riemannian symmetric spaces of negative curvature and the scattering theory_, Izvestiya Akademii Nauk USSR __40__:3 (1976) 562-592
category: physics |
abstract Stone duality | https://ncatlab.org/nlab/source/abstract+Stone+duality |
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## Idea
Due to [[Paul Taylor]], Abstract Stone Duality (ASD) is a re-[[axiom|axiomatisation]] of the notions of *[[topological space]]* and *[[continuous function]]* in [[general topology]] in terms of a [[lambda calculus|lambda-calculus]] of [[computable function|computable]] [[continuous functions]] and [[predicates]].
Abstract Stone duality is both [[constructive mathematics|constructive]] and [[computable mathematics|computable]], thus being one approach to [[synthetic topology]].
The topology on a space is treated not as a discrete [[lattice]], but as an [[exponential object]] of the same [[category]] as the original space, with an associated λ-calculus (which includes an [[internalization|internal]] lattice structure). Every expression in the λ-calculus denotes both a continuous function and a program. ASD does not use the [[Set|category of sets]] (or any [[topos]]), but the [[full subcategory]] of [[overt space|overt]] [[discrete space|discrete]] objects plays this role (an overt object is the dual to a [[compact space|compact]] object), forming an [[arithmetic universe]] (a [[pretopos]] with [[lists]]) with general [[recursion]]; an optional 'underlying set' axiom (which is not [[predicative mathematics|predicative]]) will make this a topos.
The classical (but not constructive) theory of [[locally compact space|locally compact]] [[sober space|sober]] [[topological space]]s is a model of ASD, as is the theory of locally compact [[locale]]s over any topos (even constructively). In "Beyond Local Compactness" on the ASD website, Taylor removes the restriction of local compactness.
## Related concepts
* [[Stone duality]]
## References
* [[Paul Taylor]], *Abstract Stone Duality* ([www.paultaylor.eu/ASD](http://www.paultaylor.eu/ASD))
* [[Paul Taylor]], *Review of Abstract Stone Duality* (2004) [[pdf](https://cdc.ioc.ee/appsem04/webproc/short/taylor.pdf)]
On [[Dedekind real numbers]] via abstract Stone duality:
* [[Paul Taylor]], *[Dedekind cuts](http://www.paultaylor.eu/ASD/dedras/classical)* (2007-2009?)
* [[Andrej Bauer]], [[Paul Taylor]], *The Dedekind reals in abstract Stone duality*, Mathematical Structures in Computer Science **19** 4 (2009) 757-838 [[doi:10.1017/S0960129509007695](https://doi.org/10.1017/S0960129509007695)]
review in:
* [[Andrej Bauer]], *Efficient Computation with Dedekind Reals*, talk at *[Computability and complexity in analysis 2008](https://math.andrej.com/2008/08/24/efficient-computation-with-dedekind-reals/)* and at *[Mathematics, Algorithms and Proofs 2008](http://cdsagenda5.ictp.trieste.it/full_display.php?smr=0&ida=a07167)* (2008) [[web](https://math.andrej.com/2008/08/24/efficient-computation-with-dedekind-reals/), slides: [pdf](https://math.andrej.com/wp-content/uploads/2008/08/slides-map2008.pdf), extended abstract: [pdf](https://math.andrej.com/wp-content/uploads/2008/08/abstract-cca2008.pdf)]
See also:
* [Café discussion](http://golem.ph.utexas.edu/category/2009/01/abstract_stone_duality.html)
[[!redirects Abstract Stone Duality]]
[[!redirects abstract stone duality]]
[[!redirects ASD]] |
abstract syntax tree | https://ncatlab.org/nlab/source/abstract+syntax+tree |
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## Idea
In [[formal logic]] and [[computer science]], by an *abstract syntax tree* one means a labelled [[tree]] whose nodes represent *expressions* of a [[formal language]], such as as [[programming language]], and whose edges indicate sub-expressions, meant to reflect the "abstract" [[syntax]] of the language, in contrast to the "concrete syntax" given by linear [[string (computer science)|strings]] of characters.
## Related concepts
* [[variable]], [[substitution]]
* [[word]], [[string (computer science)]]
* [[higher-order abstract syntax]]
## References
Textbook account:
* [[Robert Harper]], §1.1 in: _[[Practical Foundations for Programming Languages]]_, Cambridge University Press (2016) [[ISBN:9781107150300](http://www.cambridge.org/us/academic/subjects/computer-science/programming-languages-and-applied-logic/practical-foundations-programming-languages-2nd-edition?format=HB), [pdf](https://www.cs.cmu.edu/~rwh/pfpl/2nded.pdf)]
See also:
* Wikipedia, *[Abstract syntax tree](https://en.wikipedia.org/wiki/Abstract_syntax_tree)*
[[!redirects abstract syntax trees]]
[[!redirects abstract syntax]]
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abstraction | https://ncatlab.org/nlab/source/abstraction | [[!redirects centipede mathematics]]
# Contents
* table of contents
{:toc}
## Introduction
Abstraction or generalization is a basic tactic in mathematics research. By formulating an argument or placing a concept in 'proper generality', one not only strengthens results and widens applicability, but one also perceives much more clearly what is truly at stake within the sphere of concepts where the mathematics takes place.
Without too much of an exaggeration, one could say that the development of modern algebra generally, and categorical algebra in particular, has been driven by the need to extract the essence of mathematical structures and situate them within axiomatic frameworks of general type. This is certain true of such concepts as [[topos]], [[abelian category]], [[scheme]], [[cohomology theory]], and countless others.
Finding the 'right' generality (as opposed to merely maximal generality) of concepts is a kind of dialectical process, in which generality in the strictly logical sense must be balanced against other considerations such as simplicity, ease of application, suggestiveness, aesthetics, etc. The result of such persistent conceptual polishing is often a theory that becomes concentrated in its well-chosen definitions, from which key summary results flow naturally.
Occasionally throughout the history of mathematics (and physics, etc.), the drive to abstraction has met with suspicion. To some extent this may be a generational phenomenon, where shifting conceptual landscapes and pressure to relearn the foundations of a subject may meet with resistance from mathematicians long accustomed to older insights. There can also be justice in the charge that generality for the sake of generality is a sterile pursuit, and a typical counter-reaction is that further layers of abstraction should "prove themselves", e.g., by leading to solutions of venerable problems.
## Styles of abstraction
### Centipede mathematics
<div style="float:left;margin:0 10px 10px 0;"><img src="http://math.ucr.edu/home/baez/centipede.jpg" alt="" width="208" height="150" /></div>
**Centipede mathematics** is where you take an extant mathematical concept and see how many parts you can strip away without completely destroying its ability to function properly. Steven Krantz has attributed this term to Antoni Zygmund.
From _[Mathematical Apocrypha Redux](http://books.google.com/books?id=LMC5UeaStKwC&pg=PA6&lpg=PA6&dq=%22centipede+mathematics%22&source=bl&ots=aivICIZzMU&sig=2rMkg3u6tMhjQyfI-C46BZThiyk&hl=en&ei=kF46Ss_vC4yMsgPb0JD-Bg&sa=X&oi=book_result&ct=result&resnum=9)_: 'You take a centipede and pull off ninety-nine of its legs and see what it can do.' (We do not recommend doing this to an actual centipede, unless it is a [land-mine destroying robot](http://www.washingtonpost.com/wp-dyn/content/article/2007/05/05/AR2007050501009.html?hpid=topnews).)
For example, you can start with the concept of [[abelian group]] and try removing some things:
* remove commutativity to get [[groups]];
* then remove inverses to get [[monoids]];
* or remove associativity instead of inverses to get [[loop(algebra)|loops]];
* etc (see [[group]] for more).
What still works? Abelian groups form an [[abelian category]], while groups only form a [[semi-abelian category]], so [[cohomology]] gets more complicated (see [[nonabelian cohomology]]) but still makes sense. Monoid [[representation]]s (say on [[vector space]]s) make as much sense as group representations, but loop representations are not apparently meaningful. (But like nonabelian cohomology, maybe somebody will make sense of them some day.) Describing homomorphisms and presentations of monoids takes a little more care than for groups, while the naïve definitions continue to work for loops. Sometimes things actually work better; [[group object]]s make sense only in a [[cartesian monoidal category]], while [[monoid object]]s make sense in any [[monoidal category]]. By seeing what are the salient features for making sense of things like cohomology and representation theory, we also see that we can generalise these, not by simply removing clauses from the definition of group, but by reworking the concept into such things as [[category|categories]] or [[quantum group]]s.
Centipede mathematics in the context of [[foundations]] is often called __[[reverse mathematics]]__. Here we remove axioms from [[set theory]] (or some other form of foundations) and consider what theorems can still be proved, which can be proved if reformulated into (obviously) classically equivalent forms, and which are lost entirely (ideally provably so). On the Lab, some of us like to do this for [[constructive mathematics]], but it is also important for [[internalization]]. (Mainstream reverse mathematics has more to do with [[predicative mathematics]] than with either of these and works with subsystems of [[higher-order arithmetic]].)
Like '[[general abstract nonsense]]', the term 'centipede mathematics' can vary from tongue-in-cheek to derogatory. Certainly there is little value in proving isolated theorems like 'Hemidemisemicontinuous functions are Leibniz-integrable.' when one never really needs to integrate such general functions in such a restrictive way (both terms here are made up, to avoid insulting the creators of any real-life useless theorems). But sometimes thinking about what is really necessary points the way to really fruitful generalisations; like every judgement in mathematics, good taste is required.
## Illustrative quotes
From [[Lawvere]], in [An interview with F. William Lawvere](#picado):
> What is the primary tool for such summing up of the essence of ongoing
mathematics? Algebra! Nodal points in the progress of this kind of research occur when, as in the case with the finite number of axioms for the metacategory of categories, all that we know so far can be expressed in a single sort of algebra. I am proud to have participated with Eilenberg, Mac Lane, Freyd, and many others, in bringing about the contemporary awareness of Algebra as Category Theory.
From [Miles Reid](#reid), p. 123:
> Many students of the time could apparently not think of any higher ambition than Étudier les EGAs. The study of category theory for its own sake (surely one of the most sterile of all intellectual pursuits) also dates from this time; Grothendieck himself can't necessarily be blamed for this, since his own use of categories was very successful in solving problems.
From [[Tom Leinster]], referring to the quotation from Reid, [Leinster2010](#leinster):
> I've become rather fond of that quotation, though not for the reason that Reid intended. 'Sterile' doesn't only mean infertile or unproductive. It's also what you want surgical instruments to be: clean, uncontaminated, disease-free. No one wants to be operated on with a dirty scalpel.
> I'm going to clarify that point, because I can hear a sarcastic voice saying 'oh, so you want to do category theory uncontaminated by real mathematics'. Category theory is at its best, I think, when it reflects what mathematicians actually do and shows how to do it better, when it functions as the [mathematics of mathematics](http://cheng.staff.shef.ac.uk/misc/4000.pdf).
From [[Categories for the Working Mathematician]], p. 108:
> One may speculate as to why the discovery of adjoint functors was so delayed. (...) Bourbaki just missed... Bourbaki's idea of universal construction was devised to be so general as to include more [than the problem of finding a left adjoint] ... [his] formulation lacks the symmetry of the adjunction problem ... and so missed a basic discovery; this discovery was left to a younger man, perhaps one less beholden to tradition or to fashion. Put differently, good general theory does not search for the maximum generality, but for the right generality.
## Related entries
* [[abstract general, concrete general and concrete particular]]
* [[concrete structure]]
## References
* Jorge Picado, _An interview with F. William Lawvere_, ([pdf](http://www.mat.uc.pt/~picado/lawvere/interview.pdf))
{#picado}
* Miles Reid, _Undergraduate Algebraic Geometry_, available from [Reid's web page](https://homepages.warwick.ac.uk/staff/Miles.Reid/) as ([pdf](https://homepages.warwick.ac.uk/staff/Miles.Reid/MA4A5/UAG.pdf))
{#reid}
* Tom Leinster, _A Perspective on Higher Category Theory_, $n$-Category Café post (March 8, 2010), ([link](https://golem.ph.utexas.edu/category/2010/03/a_perspective_on_higher_catego.html))
{#leinster}
* [[Saunders Mac Lane]], _Categories for the Working Mathematician_ ($2^{nd}$ edition), Springer Verlag New York, 1998. ([pdf](http://www.maths.ed.ac.uk/~aar/papers/maclanecat.pdf))
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AccCat | https://ncatlab.org/nlab/source/AccCat |
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## Idea
By $AccCat$ (often just $Acc$ or $\mathbf{Acc}$ etc.) one typically means the ([[very large category|very large]]) [[2-category]] whose
* [[objects]] are [[accessible categories]],
* [[1-morphisms]] are [[accessible functors]]
* [[2-morphisms]] are [[natural transformations]].
Hence [[forgetful functor|forgetting]] the accessiblity conditions yields a (non-full) [[2-functor]] from $AccCat$ to the 2-category [[Cat]] of all categories, [[functors]] and [[natural transformations]].
## Properties
### 2-Limits
\begin{proposition}
\label{AccCatHasAllPIELimits}
The [[2-category]] [[AccCat]] has all (lax) [[2-limits]] and these are [[preserved limit|preserved]] by the inclusion $AccCat \to$ [[Cat]].
\end{proposition}
This appears as [Makkai & Paré (1989), Thm. 5.1.6, Cor. 5.1.8](#MakkaiParé1989), [Adámek & Rosický (1994) around Thm. 2.77](#AdámekRosický1994).
\begin{proposition}
Given a [[cosmos for enrichment]] $\mathcal{V}$ which is ([[symmetric monoidal category|symmetric monoidal]] [[closed monoidal category|closed]] and) [[locally presentable category|locally presentable]], then the [[2-category]] $\mathcal{V}$-[[AccCat]] of $\mathcal{V}$-[[enriched category|enriched]] [[accessible categories]] has all [[PIE 2-limits]] and [[split idempotent|splittings]] of [[idempotent]] [[equivalence in a 2-category|equivalences]], equivalently it has all [[flexible 2-limits]] as well as [[2-pullbacks]] along [[isofibrations]].
The analogous statements holds for $\mathcal{V}$-enriched and *[[conical limit|conically]]* accessible categories, in which case the [[forgetful functor]] $\mathcal{V}\text{-}ConAccCat \to \mathcal{V}\text{-}Cat$ [[preserved limit|preserves]] these [[2-limits]].
\end{proposition}
This is [Lack & Tendas (2023), Thm. 5.5, Thm. 5.9](#LackTendas23).
\begin{proposition}
**(directed unions)**
\linebreak
The [[2-category]] [[AccCat]] has [[directed colimits|directed]] [[2-colimits]] of systems of [[fully faithful functors]]. If there is a proper class of [[strongly compact cardinals]], then it has directed colimits of systems of [[faithful functors]].
\end{proposition}
[Paré & Rosický (2013)](#ParéRosický13)
## Related entries
* [[accessible category]] (see the respective section [there](accessible+category#TheTwoCategoryOfAccessibleCategories))
* [[Cat]]
## References
* {#MakkaiParé1989} [[Michael Makkai]], [[Robert Paré]], _Accessible categories: The foundations of categorical model theory_, Contemporary Mathematics 104. American Mathematical Society (1989) [[ISBN:978-0-8218-7692-3](https://bookstore.ams.org/conm-104)]
* {#AdámekRosický1994} [[Jiří Adámek]], [[Jiří Rosický]], *[[Locally presentable and accessible categories]]*, London Mathematical Society Lecture Note Series **189**, Cambridge University Press (1994) [[doi:10.1017/CBO9780511600579](https://doi.org/10.1017/CBO9780511600579)]
* {#ParéRosický13} [[Robert Paré]], [[Jiří Rosický]], *Colimits of accessible categories*, Math. Proc. Cambr. Phil. Soc. **155** (2013) 47-50 [[doi:10.1017/S0305004113000030](https://doi.org/10.1017/S0305004113000030), [arXiv:1110.0767](http://arxiv.org/abs/1110.0767)]
* {#LackTendas23} [[Stephen Lack]], [[Giacomo Tendas]], *Virtual concepts in the theory of accessible categories*, Journal of Pure and Applied Algebra **227** 2 (2023) 107196 [[arXiv:10.1016/j.jpaa.2022.107196](https://doi.org/10.1016/j.jpaa.2022.107196), [arXiv:2205.11056](https://arxiv.org/abs/2205.11056)]
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{: toc}
## Definition
Given a [[Cartesian space]] $V$ and a [[smooth curve]] $r:\mathbb{R} \to V$, the **acceleration** of $r$ is given by the [[second derivative]] of $r$: $a \coloneqq \partial_t^2 r$.
## See also
* [[curve]]
* [[velocity]]
* [[jerk]]
## References
See the references at *[[differential geometry of curves and surfaces]]*.
* Wikipedia, _[Acceleration](https://en.wikipedia.org/wiki/Acceleration)_
[[!redirects accelerations]] |
accessible (infinity,1)-category | https://ncatlab.org/nlab/source/accessible+%28infinity%2C1%29-category |
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# Contents#
* table of contents
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## Idea
The notion of **accessible $(\infty,1)$-category** is the generalization of the notion of [[accessible category]] from [[category theory]] to [[(∞,1)-category]] theory.
It is a means to handle $(\infty,1)$-categories that are not [[essentially small (∞,1)-category|essentially small]] in terms of small data.
An _accessible_ $(\infty,1)$-category is one which may be [[large category|large]], but can entirely be _accessed_ as an $(\infty,1)$-category of "conglomerates of objects" in a small $(\infty,1)$-category -- precisely: that it is a category of $\kappa$-small [[ind-object]]s in some small $(\infty,1)$-category $C$.
A $\kappa$-accessible $(\infty,1)$-category which in addition has all [[(∞,1)-colimits]] is called a _[[locally presentable (∞,1)-category|locally ∞-presentable]]_ or a _$\kappa$-[[compactly generated (∞,1)-category]]_.
## Definition
Let $\kappa$ be a [[regular cardinal]].
+-- {: .num_defn}
###### Definition
A [[(∞,1)-category]] $\mathcal{C}$ is **$\kappa$-accessible** if it satisfies the following equivalent conditions:
1. There is a [[small (∞,1)-category]] $\mathcal{C}^0$ and an [[equivalence of (∞,1)-categories]]
$$
\mathcal{C} \simeq Ind_\kappa(C^0)
$$
of $\mathcal{C}$ with the [[(∞,1)-category of ind-objects]], relative $\kappa$, in $\mathcal{C}^0$.
1. The $(\infty,1)$-category $\mathcal{C}$
1. is [[locally small (∞,1)-category|locally small]]
1. has all $\kappa$-[[filtered colimits]]
1. the full [[sub-(∞,1)-category]] $\mathcal{C}^\kappa \hookrightarrow \mathcal{C}$ of $\kappa$-[[compact objects]] is an [[essentially small (∞,1)-category]];
1. $\mathcal{C}^\kappa \hookrightarrow \mathcal{C}$ generates $\mathcal{C}$ under $\kappa$-[[filtered (∞,1)-colimits]].
1. The $(\infty,1)$-category $\mathcal{C}$
1. is [[locally small (∞,1)-category|locally small]]
1. has all $\kappa$-[[filtered colimits]]
1. there is _some_ [[essentially small (∞,1)-category|essentially small]]$\,$ [[sub-(∞,1)-category]] $\mathcal{C}' \hookrightarrow \mathcal{C}$ of $\kappa$-[[compact objects]] which generates $\mathcal{C}$ under $\kappa$-[[filtered (∞,1)-colimits]].
The notion of accessibility is mostly interesting for _large_ (∞,1)-categories. For
* If $\mathcal{C}$ is small, then there exists a $\kappa$ such that $\mathcal{C}$ is $\kappa$-accessible if and only if $\mathcal{C}$ is an [[idempotent-complete (∞,1)-category]].
Generally, $\mathcal{C}$ is called an **accessible $(\infty,1)$-category** if it is $\kappa$-accessible for some regular cardinal $\kappa$.
=--
+-- {: .num_prop}
###### Proposition
These conditions are indeed equivalent.
=--
For the first few this is [[Higher Topos Theory|HTT, prop. 5.4.2.2]]. The last one is in [[HTT|HTT, section 5.4.3]].
+-- {: .num_defn}
###### Definition
An [[(∞,1)-functor]] between accessible $(\infty,1)$-categories that preserves $\kappa$-filtered colimits is called an **[[accessible (∞,1)-functor]]** .
=--
+-- {: .num_defn}
###### Definition
Write $(\infty,1)AccCat \subset (\infty,1)Cat$ for the 2-[[sub-(∞,1)-category]] of [[(∞,1)Cat]] on
* those objects that are accessible $(\infty,1)$-categories;
* those morphisms for which there is a $\kappa$ such that the [[(∞,1)-functor]] is $\kappa$-continuous and preserves $\kappa$-[[compact object]]s.
=--
So morphisms are the [[accessible (∞,1)-functor]]s that also preserves [[compact object]]s. (?)
This is [[Higher Topos Theory|HTT, def. 5.4.2.16]].
## Properties
### Stability under various operations
{#StabilityUnderOperations}
+-- {: .num_theorem}
###### Theorem
If $C$ is an accessible $(\infty,1)$-category then so are
* for $K$ a small simplicial set the [[(∞,1)-category of (∞,1)-functors]] $Func(K,C)$;
* for $p : K \to C$ a small [[diagram]], the [[over quasi-category]] $C_{/p}$ and under-quasi-category $C_{p/}$.
=--
This is [[Higher Topos Theory|HTT]] section 5.4.4, 5.4.5 and 5.4.6.
+-- {: .num_theorem}
###### Theorem
The [[(∞,1)-pullback]] of accessible $(\infty,1)$-categories in [[(∞,1)Cat]] is again accessible.
=--
This is [[Higher Topos Theory|HTT, section 5.4.6]].
Generally:
+-- {: .num_theorem}
###### Theorem
The $(\infty,1)$-category $(\infty,1)AccCat$ has all small [[(∞,1)-limit]]s and the inclusion
$$
(\infty,1)AccCAT \hookrightarrow (\infty,1)CAT
$$
preserves these.
=--
This is [[Higher Topos Theory|HTT, proposition 5.4.7.3]].
## Related concepts
* [[compactly generated (∞,1)-category]]
[[!include locally presentable categories - table]]
## References
Theory of [[accessible categories|accessible 1-categories]]:
* {#AdamekRosicky} [[Jiří Adámek]], [[Jiří Rosický]], _[[Locally presentable and accessible categories]]_, Cambridge University Press, (1994)
Theory of accessible $(\infty,1)$-categories:
* [[Jacob Lurie]], Section 5.4 of _[[Higher Topos Theory]]_
See also:
* {#Rezk2021} [[Charles Rezk]], *Generalizing accessible ∞-categories*, 2021 ([pdf](https://faculty.math.illinois.edu/~rezk/accessible-cat-thoughts.pdf)).
[[!redirects accessible (infinity,1)-categories]]
[[!redirects accessible (∞,1)-category]]
[[!redirects accessible (∞,1)-categories]] |
accessible (infinity,1)-functor | https://ncatlab.org/nlab/source/accessible+%28infinity%2C1%29-functor |
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#Contents#
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## Idea
The generalization of the notion of [[accessible functor]] from [[category theory]] to [[(∞,1)-category]] theory.
## Definition
\begin{definition}
An [[(∞,1)-functor]]
$F \;\colon\; C \to D$ is **accessible** if $C$ is an [[accessible (∞,1)-category]] and there is a [[regular cardinal]] $\kappa$ such that $F$ preserves $\kappa$-small [[filtered (infinity,1)-category|filtered]]$\,$[[(infinity,1)-colimits|$(\infty,1)$-colimits]].
\end{definition}
This appears as [[Higher Topos Theory|HTT, def. 5.4.2.5]].
## Properties
\begin{proposition}\label{AdjointFunctorsAreAccessible}
**([[adjoint (infinity,1)-functors|adjoint $(\infty,1)$-functors]] are accessible)**
\linebreak
If an [[(infinity,1)-functor|$(\infty,1)$]]-functor between [[accessible (∞,1)-categories]] has a left or right [[adjoint (∞,1)-functor]], then it is itself accessible.
\end{proposition}
([[Higher Topos Theory|HTT, prop. 5.4.7.7]])
## References
* [[Jacob Lurie]], Section 5.4.2 of: _[[Higher Topos Theory]]_ (2009)
[[!redirects accessible (infinity,1)-functor]]
[[!redirects accessible (∞,1)-functor]]
[[!redirects accessible (infinity,1)-functors]]
[[!redirects accessible (∞,1)-functors]]
[[!redirects accessible (∞,1)-functor]]
[[!redirects accessible infinity1-functor]]
|
accessible category | https://ncatlab.org/nlab/source/accessible+category |
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#### Compact objects
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[[!include compact object - contents]]
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# Contents
* table of contents
{:toc}
## Idea
An accessible [[category]] is a possibly [[large category]] which is however essentially determined by a [[small category]], in a certain way.
## Definition
Let $\kappa$ be a [[regular cardinal]]. Recall an object $X: \mathcal{C}$ is $\kappa$-compact iff $\mathcal{C}(X,-)$ commutes with $\kappa$-filtered colimits.
+-- {: .num_defn }
###### Definition
A [[locally small category]] $\mathcal{C}$ is **$\kappa$-accessible** if:
1. the category has $\kappa$-[[directed colimits]] (or, equivalently, $\kappa$-filtered colimits), and
1. there is a [[set]] of $\kappa$-[[compact objects]] that generate the category under $\kappa$-directed colimits.
Then $\mathcal{C}$ is an **accessible category** if there exists a $\kappa$ such that it is $\kappa$-accessible.
=--
+-- {: .num_remark }
###### Remark
Unlike for [[locally presentable categories]], it does not follow that if $\mathcal{C}$ is $\kappa$-accessible and $\kappa\lt \lambda$ then $\mathcal{C}$ is also $\lambda$-accessible. It is true, however, that for any accessible category, there are arbitrarily large cardinals $\lambda$ such that $C$ is $\lambda$-accessible.
=--
+-- {: .num_prop }
###### Proposition
Equivalent characterizations include that $C$ is accessible iff:
* it is the category of [[models]] (in [[Set]]) of some small [[sketch]].
* it is of the form $\mathrm{Ind}_\kappa(S)$ for $S$ small, i.e. the $\kappa$-[[ind-object|ind-completion]] of a small category, for some $\kappa$.
* it is of the form $\kappa\,\mathrm{Flat}(S)$ for $S$ small and some $\kappa$, i.e. the category of $\kappa$-[[flat functor|flat]] functors from some small category to $Set$.
* it is the category of models (in $\mathbf{Set}$) of a suitable type of logical theory.
=--
The relevant notion of [[functor]] between accessible categories is
+-- {: .num_defn }
###### Definition
A [[functor]] $F\colon C\to D$ between accessible categories is
an **[[accessible functor]]** if there exists a $\kappa$ such that $C$ and $D$ are both $\kappa$-accessible and $F$ preserves $\kappa$-[[filtered colimits]].
=--
## Properties
{#Properties}
### Raising the index of accessibility
If $\mathcal{C}$ is $\lambda$-accessible and $\lambda\unlhd\mu$ (see [[sharply smaller cardinal]]), then $\mathcal{C}$ is $\mu$-accessible. Thus, any accessible category is $\mu$-accessible for arbitrarily large cardinals $\mu$.
### Stability under various constructions
+-- {: .num_prop}
###### Proposition
If $\mathcal{C}$ is an accessible category and $K$ is a [[small category]], then the [[functor category]] $Func(K^{op}, \mathcal{C})$ is again accessible.
=--
([Lurie, prop. 5.4.4.3](#Lurie))
+-- {: .num_prop #StabilityUnderInverseImage}
###### Proposition
**(preservation of accessibility under inverse images)**
Let $F : \mathcal{C} \to \mathcal{D}$ be a [[functor]] between [[locally presentable categories]] which preserves $\kappa$-[[filtered category|filtered]] [[colimits]], and let $\mathcal{D}_0 \subset \mathcal{D}$ be an accessible subcategory. Then the inverse image $f^{-1}(\mathcal{D}_0) \subset C$ is a $\kappa$-accessible subcategory.
=--
This appears as [[Higher Topos Theory|HTT, corollary A.2.6.5]].
+-- {: .num_prop}
###### Proposition
**(accessibility of fibrations and weak equivalences in a combinatorial model category)**
Let $\mathcal{C}$ be a [[combinatorial model category]], $Arr(\mathcal{C})$ its [[arrow category]], $W \subset Arr(\mathcal{C})$ the [[full subcategory]] on the weak equivalences and $F \subset Arr(\mathcal{C})$ the full subcategory on the fibrations. Then $F$, $W$ and $F \cap W$ are accessible subcategories of $Arr(\mathcal{C})$.
=--
This appears as [[Higher Topos Theory|HTT, corollary A.2.6.6]].
### Adjoint functor theorem
+-- {: .num_prop}
###### Proposition
**(adjoint functors)**
Every accessible functor satisfies the [[solution set condition]], and every left or right adjoint between accessible categories is accessible. Therefore, the [[adjoint functor theorem]] takes an especially pleasing form for accessible categories that are complete and cocomplete (i.e. are [[locally presentable categories|locally presentable]]): a functor between such categories is a left (resp. right) adjoint iff it is accessible and preserves all small colimits (resp. limits).
=--
### Idempotence completeness
+-- {: .num_prop}
###### Proposition
A [[small category]] is accessible precisely when it is [[idempotent complete category|idempotent complete]].
=--
[Makkai & Paré (1989)](#MakkaiParé1989) say that this means accessibility is an "almost pure smallness condition."
### Categories of models over a theory
+-- {: .num_prop }
###### Proposition
For a [[category]] $\mathcal{M}$ the following are equivalent:
* $\mathcal{M}$ is finitely [[accessible category|accessible]].
* $\mathcal{M}\simeq \mathbb{T}\text{-}Mod(Set)$ for some [[theory of presheaf type|theory $\mathbb{T}$ of presheaf type]].
=--
Moreover, one has the following result due to [[Christian Lair]]:
+-- {: .num_prop }
###### Proposition
For a [[category]] $\mathcal{M}$ the following are equivalent:
* $\mathcal{M}$ is [[accessible category|accessible]].
* $\mathcal{M}$ is [[sketch|sketchable]].
=--
See also at _[categorical model theory](model+theory#CategoricalModelTheory)_.
### Well-poweredness and well-copoweredness
* Every accessible category $\mathcal{C}$ is [[well-powered category|well-powered]], since it has a small [[dense subcategory]] $\mathcal{A}$, for which the [[restricted Yoneda embedding]] $\mathcal{C} \to [\mathcal{A}^{op},Set]$ is fully faithful and preserves monomorphisms, hence embeds the subobject posets of $\mathcal{C}$ as sub-posets of those of $[\mathcal{A}^{op},Set]$.
* Every accessible category with [[pushouts]] is well-*copowered*. This is shown in [Adamek-Rosicky, Proposition 1.57 and Theorem 2.49](#AdámekRosický1994). Whether this is true for all accessible categories depends on what [[large cardinal]] properties hold: by Corollary 6.8 of Adamek-Rosicky, if [[Vopenka's principle]] holds then all accessible categories are well-copowered, while by Example A.19 of Adamek-Rosicky, if all accessible categories are well-copowered then there exist arbitrarily large [[measurable cardinals]].
### The 2-category of accessible categories
{#TheTwoCategoryOfAccessibleCategories}
Write [[AccCat]] for the [[2-category]] whose
* [[objects]] are [[accessible categories]],
* [[1-morphisms]] are [[accessible functors]]
* [[2-morphisms]] are [[natural transformations]].
\begin{proposition}
\label{AccCatHasAllPIELimits}
The [[2-category]] [[AccCat]] has all (lax) [[2-limits]] and these are [[preserved limit|preserved]] by the inclusion $AccCat \to$ [[Cat]].
\end{proposition}
This appears as [Makkai & Paré (1989), Thm. 5.1.6, Cor. 5.1.8](#MakkaiParé1989), [Adámek & Rosický (1994) around Thm. 2.77](#AdámekRosický1994).
\begin{proposition}
Given a [[cosmos for enrichment]] $\mathcal{V}$ which is ([[symmetric monoidal category|symmetric monoidal]] [[closed monoidal category|closed]] and) [[locally presentable category|locally presentable]], then the [[2-category]] $\mathcal{V}$-[[AccCat]] of $\mathcal{V}$-[[enriched category|enriched]] [[accessible categories]] has all [[PIE 2-limits]] and [[split idempotent|splittings]] of [[idempotent]] [[equivalence in a 2-category|equivalences]] (equivalently it has all [[flexible 2-limits]]), as well as [[2-pullbacks]] along [[isofibrations]].
The analogous statements holds for $\mathcal{V}$-enriched and *[[conical limit|conically]]* accessible categories, in which case the [[forgetful functor]] $\mathcal{V}\text{-}ConAccCat \to \mathcal{V}\text{-}Cat$ [[preserved limit|preserves]] these [[2-limits]].
\end{proposition}
This is [Lack & Tendas (2023), Thm. 5.5, Thm. 5.9](#LackTendas23).
\begin{proposition}
**(directed unions)**
\linebreak
The [[2-category]] [[AccCat]] has [[directed colimits|directed]] [[2-colimits]] of systems of [[fully faithful functors]]. If there is a proper class of [[strongly compact cardinals]], then it has directed colimits of systems of [[faithful functors]].
\end{proposition}
[Paré & Rosický (2013)](#ParéRosický13)
## Examples
- Every small [[discrete category]] is $\kappa$-accessible for every [[regular cardinal]] $\kappa$, since every discrete filtered diagram is trivial.
### Functor categories
See at _[Functor category -- Accessibility](functor+category#LocalPresentability)_.
## Related concepts
* [[class-accessible category]]
[[!include locally presentable categories - table]]
## References
### General
The term _accessible category_ is due to
* {#MakkaiParé1989} [[Michael Makkai]], [[Robert Paré]], _Accessible categories: The foundations of categorical model theory_, Contemporary Mathematics **104**, American Mathematical Society, (1989) [[ISBN:978-0-8218-7692-3](https://bookstore.ams.org/conm-104)]
Further monographs (with focus on [[locally presentable categories]]):
* {#AdámekRosický1994} [[Jiří Adámek]], [[Jiří Rosický]], *[[Locally presentable and accessible categories]]*, London Mathematical Society Lecture Note Series **189**, Cambridge University Press (1994) [[doi:10.1017/CBO9780511600579](https://doi.org/10.1017/CBO9780511600579)]
See also
* {#ParéRosický13} [[Robert Paré]], [[Jiří Rosický]], *Colimits of accessible categories*, Math. Proc. Cambr. Phil. Soc. **155** (2013) 47-50 [[doi:10.1017/S0305004113000030](https://doi.org/10.1017/S0305004113000030), [arXiv:1110.0767](http://arxiv.org/abs/1110.0767)]
* [[Jiří Adámek]], [[Francis Borceux]], [[Stephen Lack]], [[Jiří Rosický]], _A classification of accessible categories,_ Journal of Pure and Applied Algebra 175:7-30, 2002. [abstract](http://maths.mq.edu.au/~slack/papers/acc.html)
which further stratifies the accessible categories in terms of [[sound doctrines]].
The concept is studied in a 2-categorical setting in
* [[John Bourke]], _Accessible aspects of 2-category theory_ , arXiv:2003.06375 (2020). ([abstract](https://arxiv.org/abs/2003.06375))
A discussion of [[accessible (∞,1)-categories]] is in [section 5.4, p. 341](http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=341)
of
* {#Lurie} [[Jacob Lurie]], _[[Higher Topos Theory]]_
Some recent developments in the theory of accessible categories can be found in a series of papers on [categorical model theory](model+theory#CategoricalModelTheory) and [[abstract elementary classes]] (many of which also contain some results about arbitrary accessible categories), such as:
* [[Tibor Beke]], [[Jiří Rosický]], *Abstract elementary classes and accessible categories*, 2011, [arxiv](https://arxiv.org/abs/1005.2910)
* Michael Lieberman, [[Jiří Rosický]], Sebastien Vasey, *Internal sizes in μ-abstract elementary classes*, [arxiv](https://arxiv.org/abs/1708.06782)
* Michael Lieberman, [[Jiří Rosický]], Sebastien Vasey , *Set-theoretic aspects of accessible categories*, [arxiv](https://arxiv.org/abs/1902.06777)
See also:
* {#Rezk2021} [[Charles Rezk]], *Generalizing accessible ∞-categories*, 2021 ([pdf](https://faculty.math.illinois.edu/~rezk/accessible-cat-thoughts.pdf))
### In enriched category theory
{#ReferencesInEnrichedCategoryTheory}
Discussion of accessible [[enriched categories]] (see also [references on enriched locally presentable categories](locally+presentable+category#ReferencesInEnrichedCategoryTheory)):
* [[Francis Borceux]], [[Carmen Quinteriro]], *Enriched accessible categories*, Bull. Austral. Math. Soc. __54__ (1996) 489-501 [[doi:10.1017/S0004972700021900](https://doi.org/10.1017/S0004972700021900)]
* [[Francis Borceux]], [[Carmen Quinteriro]], [[Jiří Rosický]], *A theory of enriched sketches*, Theory and Applications of Categories, **4** 3 (1998) 47-72 [[tac:4-03](http://www.tac.mta.ca/tac/volumes/1998/n3/4-03abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/1998/n3/n3.pdf)]
* {#LackTendas23} [[Stephen Lack]], [[Giacomo Tendas]], *Virtual concepts in the theory of accessible categories*, Journal of Pure and Applied Algebra **227** 2 (2023) 107196 [[arXiv:10.1016/j.jpaa.2022.107196](https://doi.org/10.1016/j.jpaa.2022.107196), [arXiv:2205.11056](https://arxiv.org/abs/2205.11056)]
> New characterizations of (enriched) accessible categories and introducing the notions of virtual orthogonality and virtual reflectivity.
[[!redirects accessible categories]]
[[!redirects enriched accessible category]]
[[!redirects enriched accessible categories]]
|
accessible functor | https://ncatlab.org/nlab/source/accessible+functor |
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#Contents#
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## Definition
A [[functor]]
$$
F\colon C\to D
$$
is a **$\kappa$-accessible functor** (for $\kappa$ a [[regular cardinal]]) if $C$ and $D$ are both $\kappa$-[[accessible categories]] and $F$ preserves $\kappa$-[[filtered colimit]]s. $F$ is an **accessible functor** if it is $\kappa$-accessible for some regular cardinal $\kappa$.
## Properties
It is immediate from the definition that accessible functors are closed under composition.
### Raising the index of accessibility
If $\lambda\le\kappa$, then every $\kappa$-filtered colimit is also $\lambda$-filtered, and thus if $F$ preserves $\lambda$-filtered colimits then it also preserves $\kappa$-filtered ones. Therefore, if $F$ is $\lambda$-accessible and $C$ and $D$ are $\kappa$-accessible, then $F$ is $\kappa$-accessible. Two conditions under which this happens are:
1. $C$ and $D$ are [[locally presentable categories]].
2. $\lambda$ is [[sharply smaller cardinal|sharply smaller]] than $\kappa$, i.e. $\lambda\lhd\kappa$.
In particular, for any accessible functor $F$ there are arbitrarily large cardinals $\kappa$ such that $F$ is $\kappa$-accessible, and if the domain and codomain of $F$ are locally presentable then $F$ is $\kappa$-accessible for all sufficiently large $\kappa$.
### Preserving presentable objects
For any accessible functor $F$, there are arbitrarily large cardinals $\kappa$ such that $F$ is $\kappa$-accessible and preserves $\kappa$-[[presentable objects]]. Indeed, this can be achieved simultaneously for any set of accessible functors. See [Adamek-Rosicky, Theorem 2.19](#AdamekRosicky).
### Essential images of accessible functors
\begin{theorem}
Assuming the existence of a [[proper class]] of [[strongly compact cardinals]],
the following are equivalent for the [[essential image]] $K$ of an accessible functor:
* $K$ is [[complete]];
* $K$ is [[cocomplete]];
* $K$ is [[locally presentable]].
\end{theorem}
\begin{theorem}
Assuming the existence of a [[proper class]] of [[strongly compact cardinals]],
the closure of the image of an accessible functor under passage to subobjects is an accessible subcategory.
\end{theorem}
The existence of a [[proper class]] of [[strongly compact cardinals]]
can be weakened, see the paper of [Brooke-Taylor and Rosický](#BrookeTaylorRosicky).
## Examples
\begin{example}\label{AdjointsAreAccecible}
Given [[locally presentable categories]] $C$ and $D$ and a [[functor]] $F\colon C\to D$, if $F$ has a [[left adjoint|left]] or [[right adjoint]], then it is an accessible functor.
\end{example}
\begin{example}
By Example \ref{AdjointsAreAccecible} it follows that [[polynomial endofunctors]] of [[Set|$Set$]] are accessible, as they are [[composition|composites]] of [[adjoint functors]].
\end{example}
## Related concepts
* [[accessible (∞,1)-functor]].
* [[accessible monad]]
## References
The theory of accessible 1-categories is described in
* [[Michael Makkai]], [[Robert Paré]], _Accessible categories: The foundations of categorical model theory_ Contemporary Mathematics 104. American Mathematical Society, Rhode Island, 1989.
{#MakkaiPare}
* [[Jiří Adámek]], [[Jiří Rosický]], _[[Locally presentable and accessible categories]]_, Cambridge University Press, (1994)
{#AdamekRosicky}
Essential images of accessible functors are considered in
* [[Jiří Rosický]], _More on directed colimits of models_, Applied Categorical Structures volume 2 (1994) pages 71–76, doi:[10.1007/BF00878503](https://doi.org/10.1007/BF00878503)
An improvement of Rosický's result is in
* {#BrookeTaylorRosicky} Andrew Brooke-Taylor, [[Jiří Rosický]], _Accessible images revisited_ ([arXiv:1506.01986](https://arxiv.org/abs/1506.01986)).
The theory of accessible $(\infty,1)$-categories is the topic of section 5.4 of
* [[Jacob Lurie]], _[[Higher Topos Theory]]_
[[!redirects accessible functors]]
|
accessible model category | https://ncatlab.org/nlab/source/accessible+model+category | # Accessible model categories
* table of contents
{: toc}
## Definition
An **accessible model category** is a [[model structure]] on a [[locally presentable category]] whose two [[weak factorization systems]] can be realized by [[functorial factorizations]] that are [[accessible functors]], that is that they are [[accessible weak factorization systems]].
This implies that in fact its weak factorization systems can be enhanced to [[algebraic weak factorization systems]] that are also accessible (see [[accessible wfs]]). However, such algebraic structure is not given as data in the notion of accessible model category: for that, see [[algebraic model category]].
## Properties
### Left and right lifting
Accessible model structures can be both left- and right-lifted along adjunctions as long as the relevant "acyclicity condition" holds. That is, let $U:A\to B$ be a functor between locally presentable categories and suppose $B$ is an accessible model category. Then:
1. If $U$ is a right adjoint, then there is a model structure on $A$ in which the weak equivalences and fibrations are created by $U$ (i.e. $W_A = U^{-1}(W_B)$ and $F_A = U^{-1}(F_B)$) if and only if every map having the left lifting property with respect to $F_A$ lies in $W_A$.
1. If $U$ is a left adjoint, then there is a model structure on $A$ in which the weak equivalences and cofibrations are created by $U$ (i.e. $W_A = U^{-1}(W_B)$ and $C_A = U^{-1}(C_B)$) if and only if every map having the right lifting property with respect to $C_A$ lies in $W_A$.
Both of the lifted model structures are then again accessible. See [[transferred model structure]]. For proofs, see [HKRS](#HKRS15) and its correction in [GKR](#GKR18).
## Related concepts
* [[accessible weak factorization system]]
* [[algebraic model category]]
[[!include algebraic model structures - table]]
## References
* {#Rosicky17} [[Jiri Rosicky]], *Accessible model categories*, Appl Categor Struct (2017) 25: 187. [doi](https://doi.org/10.1007/s10485-015-9419-6), [arxiv](https://arxiv.org/abs/1503.05010)
* {#HKRS15} [[Kathryn Hess]], Magdalena Kędziorek, [[Emily Riehl]], [[Brooke Shipley]], _A necessary and sufficient condition for induced model structures_ ([arXiv:1509.08154](http://arxiv.org/abs/1509.08154)). This paper contains an error, corrected by:
* {#GKR18} [[Richard Garner]], Magdalena Kedziorek, [[Emily Riehl]], _Lifting accessible model structures_, [arXiv:1802.09889](https://arxiv.org/abs/1802.09889)
* [[John Bourke]], *Equipping weak equivalences with algebraic structure*, [arxiv:1712.02523](https://arxiv.org/abs/1712.02523)
[[!redirects accessible model categories]]
[[!redirects accessible model structure]]
[[!redirects accessible model structures]]
|
accessible monad | https://ncatlab.org/nlab/source/accessible+monad |
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###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
+-- {: .num_defn #AccessibleMonad}
###### Definition
An _[[accessible monad]]_ is a [[monad]] on an [[accessible category]] whose underlying [[functor]] is an [[accessible functor]].
=--
## Properties
### Category of algebras over an accessible monad
{#CategoryOfAlgebras}
+-- {: .num_prop}
###### Proposition
The Eilenberg-Moore category of a $\kappa$-accessible monad, def. \ref{AccessibleMonad}, is a $\kappa$-[[accessible category]]. If in addition the category on which the monad acts is a $\kappa$-[[locally presentable category]] then so is the EM-category.
=--
([Adamek-Rosicky, 2.78](#AdamekRosicky))
Moreover, let $C$ be a [[topos]]. Then
* if a [[monad]] $T : C \to C$ has a [[right adjoint]] then $T Alg(C)= C^T$ is itself a topos;
* if a [[comonad]] $T : C \to C$ is [[exact functor|left exact]], then $T CoAlg(C) = C_T$ is itself a topos.
See at _[[topos of algebras over a monad]]_ for details.
## References
* [[Jiří Adámek]], [[Jiří Rosický]], _[[Locally presentable and accessible categories]]_, Cambridge University Press, (1994)
{#AdamekRosicky}
|
accessible topological space | https://ncatlab.org/nlab/source/accessible+topological+space |
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### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
A [[topological space]] $(X,\tau)$ is called an **accessible topological space** or a **$T_1$-topological space** if it is both a [[Kolmogorov topological space]] and a [[symmetric topological space]]. Or equivalently, if its [[specialization preorder]] is [[equality]]. Or equivalently still, if all [[singletons]] are [[closed subsets]] ("[[closed point|points are closed]]").
[[!include main separation axioms -- table]]
## Properties
{#Properties}
[[!include main separation axioms -- as lifting properties]]
## Examples
[[Leibniz's identity of indiscernibles]] implies that every [[set]] $X$ with its [[power set]] $\mathcal{P}(X)$ is an accessible topological space:
$$\left(\forall P \in \mathcal{P}(X).(x \in P) \iff (y \in P)\right) \implies x = y$$
for all $x \in X$ and $y \in X$, since the relation
$$\forall P \in \mathcal{P}(X).(x \in P) \iff (y \in P)$$
is an [[equivalence relation]].
## Related concepts
* [[specialization order]]
* [[separation axiom]]
* [[identity of indiscernibles]]
| [[preorder]] | [[partial order]] | [[equivalence relation]] | [[equality]] |
| - | - | - | - |
| [[topological space]] | [[ Kolmogorov topological space]] | [[symmetric topological space]] | [[accessible topological space]] |
## References
* Wikipedia, [T1 space](https://en.wikipedia.org/wiki/T1_space)
[[!redirects T1]]
[[!redirects T_1]]
[[!redirects T1 space]]
[[!redirects T1 spaces]]
[[!redirects T1-space]]
[[!redirects T1-spaces]]
[[!redirects T_1 space]]
[[!redirects T_1 spaces]]
[[!redirects T_1-space]]
[[!redirects T_1-spaces]]
[[!redirects symmetric topological space]]
[[!redirects symmetric topological spaces]]
[[!redirects T1 topological space]]
[[!redirects T1 topological spaces]]
[[!redirects T_1 topological space]]
[[!redirects T_1 topological spaces]]
[[!redirects T1-topological space]]
[[!redirects T1-topological spaces]]
[[!redirects T_1-topological space]]
[[!redirects T_1-topological spaces]]
|
accessible weak factorization system | https://ncatlab.org/nlab/source/accessible+weak+factorization+system |
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###Context###
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
=--
=--
# Accessible weak factorization systems
* table of contents
{: toc}
## Definition
A [[weak factorization system]] on a [[locally presentable category]] is **accessible** if it admits a [[functorial factorization]] that is an [[accessible functor]].
## Equivalent forms
\begin{theorem}
For a weak factorization system $(L,R)$ on a locally presentable category $M$, the following are equivalent.
1. $(L,R)$ is accessible.
2. There is a small category $J \to M^\to$ over the [[arrow category]] such that $R$ consists of the morphisms with coherent right lifting functions relative to $J$.
3. $(L,R)$ can be generated by the [[algebraic small object argument]].
4. $(L,R)$ can be equipped with the structure of an accessible [[algebraic weak factorization system]].
\end{theorem}
\begin{proof}
Properties of the algebraic small object argument yield (2)$\Rightarrow$(3)$\Rightarrow$(4), and (4)$\Rightarrow$(1) since an algebraic wfs is in particular a functorial factorization. The remaining implication is the following lemma, which is Remark 3.1.8 of [HKRS15](#HKRS15), which relies on Theorem 4.3 of [Rosicky17](#Rosicky17).
\end{proof}
\begin{lemma}
\label{EnhancementToAlgebaicWeakFactorization}
Suppose $E$ is an accessible functorial factorization on a [[locally presentable category]] $M$, realizing a [[weak factorization system]]. Then there is an accessible [[algebraic weak factorization system]] realizing the same weak factorization system.
\end{lemma}
\begin{proof}
\label{ProofOfEnhancementToAlgebaicWeakFactorization}
Let $Coalg(L)$ be the category of [[algebra over an endofunctor|coalgebras]] for the [[pointed endofunctor|copointed endofunctor]] of the [[arrow category]] $M^\to$ induced by $E$, i.e. morphisms equipped with a section of their $E$-factorization exhibiting them as a retract of the first factor. Then $Coalg(L)$ is locally presentable (being complete and a [[PIE limit]] construction from $M$). Thus, it has a small dense subcategory $X$. We can then apply [[Garner's small object argument]] to generate an algebraically-free algebraic weak factorization system from $X$. The algebraic right-maps in this awfs are the morphisms with coherent lifting functions against $X$, which by density is the same as having a coherent lifting function against all of $Coalg(L)$, which is the same as being an algebra for the pointed endofunctor of $M^\to$ corresponding to $E$. Thus this is an awfs with the same right-maps, hence the same underlying weak factorization system.
\end{proof}
Regarding point (4), note that being accessible is a *property* of a wfs, while being algebraic is a *structure* on it. A given accessible wfs can admit many different algebraic realizations, and not all of them (even the accessible ones) may be produced by the ordinary algebraic small object argument (although they can all be produced by the fancier version involving a double category of maps as input).
## Properties
### Left and right lifting
Any accessible weak factorization system can be right-lifted along a right adjoint, or also left-lifted along a left adjoint, between locally presentable categories. That is, if $U:A\to B$ is a functor between locally presentable categories and $(L,R)$ is an accessible weak factorization system of $B$, then:
1. If $U$ is a right adjoint, then there is an accessible wfs $(L',R')$ on $A$ such that $R' = U^{-1}(R)$.
1. If $U$ is a left adjoint, then there is an accessible wfs $(L',R')$ on $A$ such that $L' = U^{-1}(L)$.
See [HKRS](#HKRS15) and its correction in [GKR](#GKR18) for details. In particular, this is useful for the construction of [[transferred model structures]].
## Related concepts
* [[algebraic weak factorization system]]
* [[accessible model structure]]
[[!include algebraic model structures - table]]
## References
* {#Rosicky17} [[Jiri Rosicky]], *Accessible model categories*, Appl Categor Struct (2017) 25: 187. [doi](https://doi.org/10.1007/s10485-015-9419-6), [arxiv](https://arxiv.org/abs/1503.05010)
* {#HKRS15} [[Kathryn Hess]], Magdalena Kędziorek, [[Emily Riehl]], [[Brooke Shipley]], _A necessary and sufficient condition for induced model structures_ ([arXiv:1509.08154](http://arxiv.org/abs/1509.08154)). This paper contains an error, corrected by:
* {#GKR18} [[Richard Garner]], Magdalena Kedziorek, [[Emily Riehl]], _Lifting accessible model structures_, [arXiv:1802.09889](https://arxiv.org/abs/1802.09889)
[[!redirects accessible weak factorization systems]]
[[!redirects accessible weak factorisation system]]
[[!redirects accessible weak factorisation systems]]
[[!redirects accessible wfs]]
[[!redirects accessible WFS]]
|
Achim Jung | https://ncatlab.org/nlab/source/Achim+Jung |
* [personal page](https://www.cs.bham.ac.uk/~axj/)
## Selected writings
On [[domain theory]]:
* [[Samson Abramsky]], [[Achim Jung]], *Domain Theory*, in: *Handbook of Logic in Computer Science* **3**, Oxford University Press (1995) [[ISBN:9780198537625](https://global.oup.com/academic/product/handbook-of-logic-in-computer-science-9780198537625?cc=de&lang=en&), [pdf](https://www.cs.bham.ac.uk/~axj/pub/papers/handy1.pdf)]
category: people |
Achim Krause | https://ncatlab.org/nlab/source/Achim+Krause |
* [personal page](https://www.uni-muenster.de/IVV5WS/WebHop/user/krauseac/)
## Selected writings
On [[prismatic cohomology]] applied to [[algebraic K-theory]]:
* {#AKN22} [[Benjamin Antieau]], [[Achim Krause]], [[Thomas Nikolaus]], _On the K-theory of $\mathbb{Z}/p^{n}$ -- Announcement_ [[arXiv:2204.03420](https://arxiv.org/abs/2204.03420)[
category: people
|
Ackermann groupoid | https://ncatlab.org/nlab/source/Ackermann+groupoid |
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### Context
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An *Ackermann groupoid* is a particular type of [[algebra|algebraic]] [[mathematical structure]] that provides [[semantics]] for a flavour of [[relevance logic]], a weak form of [[substructural logic]].
Note that "groupoid" here does not mean *[[groupoid]]*, but *[[magma]]*. The terminology comes from logic, rather than category theory.
## Definition
An *Ackermann groupoid* is a [[partially ordered set|partially ordered]] [[magma]] $(M,\circ, 1,\leq)$ that is left unital ($1\circ a = a$ for all $a\in M$), and has a binary operation, "implication", written $b\to c$ satisfying $a \leq b\to c$ if and only if $a\circ b \leq c$.
This might be called an _implicational Ackermann groupoid_, since it provides semantic models for an implicational fragment of logic, together with [[intensional conjunction]] (here $\to$ models implication, analogous to [[linear implication]] in [[linear logic]]). A _positive_ Ackermann groupoid upgrades the underlying [[poset]] to a [[distributive lattice]], permitting the interpretation of additional logical connectives, namely (classical) [[logical conjunction]] and [[logical disjunction]].
## Example
Every [[Church monoid]] is an Ackermann groupoid.
## Related concepts
* [[relevance monoidal category]]
* [[relevance logic]]
* [[Church monoid]]
* [[Heyting algebra]]
## References
Ackermann groupoids were introduced in
* Robert K. Meyer and Richard Routley, _Algebraic analysis of entailment I_, Logique et Analyse NOUVELLE SÉRIE, Vol. 15, No. 59/60 (1972) pp407-428, [JSTOR](https://www.jstor.org/stable/44083856)
and named for [[Wilhelm Ackermann]].
[[!redirects Ackermann groupoids]]
[[!redirects Ackermann magma]]
[[!redirects Ackermann magmas]] |
acoustics | https://ncatlab.org/nlab/source/acoustics |
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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
\tableofcontents
\section{Idea}
A branch of [[physics]] that studies [[mechanical waves]].
\section{See also}
* [[sound]]
* [[vibration]]
* [[noise]]
* [[music]]
* [[wave]]
* [[wave equation]]
\section{References}
See also:
* Wikipedia, _[Acoustics](https://en.wikipedia.org/wiki/Acoustics)_ |
action | https://ncatlab.org/nlab/source/action | > this entry is about the notion of _action_ in algebra (of one algebraic object on another object). For the notion of _[[action functional]]_ in [[physics]] see there.
***
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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
#### Monoid theory
+-- {: .hide}
[[!include monoid theory - contents]]
=--
=--
=--
# Actions
* table of contents
{: toc}
## Idea
{#Idea}
There are various variants of the notion of something _acting_ on something else. They are all closely related.
The simplest concept of an action involves one [[set]], $X$, acting on another set $Y$ and such an action is given by a [[function]] from the [[product]] of $X$ with $Y$ to $Y$
$$
act\colon X \times Y \to Y
\,.
$$
For fixed $x \in X$ this produces an [[endofunction]] $act(x,-) \colon Y \to Y$ and hence some "transformation" or "action" on $Y$. In this way the whole of $X$ acts on $Y$.
Here $(x\mapsto act(x,-))$ is the _[[currying|curried]]_ function $\widehat{act}\colon X \to Y^Y$ of the original $act$, which maps $X$ to the [[function set|set]] of [[endofunctions]] on $Y$. Quite generally one has these two perspectives on actions.
Usually the key aspect of an action of some $X$ is that $X$ itself carries an algebraic structure, such as being a [[group]] (or just a [[monoid]]) or being a [[ring]] or an [[associative algebra]], which is also possessed by $Y^Y$ and preserved by the curried action $\widehat{act}$. Note that if $Y$ is any set then $Y^Y$ is a monoid, and when $X$ acts on it one calls it an [[MSet|X-set]]. For $Y^Y$ to have a ring/algebra structure, $Y$ must be some sort of [[abelian group]] or [[vector space]] with the action by [[linear functions]]; then one calls the action also a _[[module]]_ or _[[representation]]_.
In terms of the uncurried action $X\times Y\to Y$, the "preservation" condition says roughly speaking that acting consecutively with two elements in $X$ is the same as first multiplying them and then acting with the result:
\[
\label{ActionProperty}
act(x_2,act(x_1,y)) = act(x_2\cdot x_1, y)
\,.
\]
To be precise, this is the condition for a *left action*; a *right action* is defined dually in terms of a map $Y\times X\to Y$. If $X$ has no algebraic structure, or if its relevant structure is commutative, then there is no essential difference between the two; but in general they can be quite different.
This _action property_ can also often be identified with a [[functor]] property: it characterizes a [[functor]] from the [[delooping]] $\mathbf{B}X$ of the [[monoid]] $X$ to the [[category]] (such as [[Set]]) of which $Y$ is an object. The distinction between left and right actions is mirrored in the variance; acting on the left yields a covariant functor, whereas acting on the right is expressed via contravariance.
In this way essentially every kind of [[functor]], [[n-functor]] and [[enriched functor]] may be thought of as defining a generalized kind of action. This perspective on actions is particularly prevalent in [[enriched category theory]], where for instance [[coends]] may be thought of as producing [[tensor products]] of actions in this general functorial sense.
Under the [[Grothendieck construction]] (or one of its variants), this perspective turns into the perspective where an action of $X$ is some [[bundle]] $Y/X$ over $\mathbf{B}X$, whose [[fiber]] is $Y$:
$$
\array{
Y &\longrightarrow& Y/X
\\
&& \downarrow
\\
&& \mathbf{B}X
}
\,.
$$
Here the total space $Y/X$ of this bundle is typically the "weak" [[quotient]] (for instance: [[homotopy quotient]]) of the action, whence the notation. If one thinks of $\mathbf{B}X$ as the [[classifying space]] for the $X$-[[universal principal bundle]], then this bundle $Y/X \to \mathbf{B}X$ is the $Y$-[[fiber bundle]] which is [[associated bundle|associated]] via the action to this universal bundle. For more on this perspective on actions see at _[[∞-action]]_.
## Definitions
### Actions of a group
{#ActionsOfAGroup}
An **action** of a [[group]] $G$ on an [[object]] $S$ in a [[category]] $\mathcal{C}$ is a [[representation]] of $G$ on $S$, that is a [[group homomorphism]] $\rho \colon G \to Aut(S)$, where $Aut(S)$ is the [[automorphism group]] of $S$ in $\mathcal{C}$.
Group actions, especially [[continuous function|continuous]] actions on [[topological spaces]], are also known as _transformation groups_ (starting around [Klein 1872, Sec. 1](#Klein1872), see also [Koszul 65](#Koszul65) [Bredon 72](#Bredon72), [tom Dieck 79](#tomDieck79), [tom Dieck 87](#tomDieck87)). Alternatively, if the group $G$ that acts is understood, one calls ([Bredon 72, Ch. II](#Bredon72)) the space $X$ equipped with an action by $G$ a _[[topological G-space]]_ (or *[[G-set]]*, *[[G-manifold]]*, etc., as the case may be).
As indicated above, a more abstract but equivalent definition regards the group $G$ as a category (a [[groupoid]]), denoted $\mathbf{B} G$, with a single object $\ast$. Then an _action_ of $G$ in the category $C$ is equivalently a [[functor]] of the form
$$
\rho \colon \mathbf{B} G \to \mathcal{C}
$$
Here the object $S$ of the previous definition is the $\rho(\ast)$ of that single object.
Concretely, if $\mathcal{C}$ is a category like [[Set]], then an action is equivalently a [[function]]
$$
\array{
G \times S &\longrightarrow& S
\\
(g,s) &\mapsto& \rho(g)(s)
}
$$
which satisfies the _action property_
\[
\label{ActionPropertyOfGroupActions}
\underset{
g_1, g_2, s
}{\forall}
\;\;\;
\rho(g_1 \cdot g_2)(s)
\;=\;
\rho(g_1)
\big(
\rho(g_2)(s)
\big)
\]
and
$$
\underset{s}{\forall}
\;\;\;
\rho(e)(s) \;=\; s
$$
### Actions of a monoid
More generally we can define an _action_ of a [[monoid]] $M$ in the category $C$ to be a functor
$$\rho: \mathbf{B} M \to C $$
where $\mathbf{B} M$ is (again) $M$ regarded as a one-object category.
The _category of actions_ of $M$ in $C$ is then defined to be the [[functor category]] $C^{\mathbf{B} M}$. When $C$ is $Set$ this is called [[MSet]].
Considering this in [[enriched category theory]] yields the [[internalization|internal]] notion of _[[action objects]]_.
### Actions of a category
One can[^2] also define an [[action of a category on a set|action of a category]] $D$ on the category $C$ as a functor from $D$ to $C$, but usually one just calls this a [[functor]].
[^2]: One example of this relatively rare usage is [[William Lawvere]]: _Qualitative Distinctions Between Some Toposes of Generalized Graphs_, Contemporary Mathematics 92 (1989) in which this sense of action is routinely used.
Another perspective on the same situation is: a (small) category is a [[monad]] in the category of [[span]]s in [[Set]]. An action of the category is an algebra for this monad. See [[action of a category on a set]].
On the other hand, an action of a [[monoidal category]] (not _in_ a monoidal category, as above) is called an [[module category]] (also "[[actegory]]"). This notion can be expanded of course to actions _in_ a [[monoidal bicategory]], where in the case of $Cat$ as monoidal bicategory it specializes to the notion of module category.
### Actions of a group object
Suppose we have a category, $C$, with binary [[product]]s and a [[terminal object]] $*$. There is an alternative way of viewing group actions in [[Set]], so that we can talk about an action of a [[group object]], $G$, in $C$ on an object, $X$, of $C$.
By the adjointness relation between cartesian product, $A\times ?$, and function set, $?^A$, in [[Set]], a group homomorphism
$$\alpha: G\to Aut(X)$$
corresponds to a function
$$act: G\times X\to X$$
which will have various properties encoding that $\alpha$ was a homomorphism of groups:
$$act(g_1g_2,x) = act(g_1,act(g_2,x))$$
$$act(1,x) = x$$
and these can be encoded diagrammatically.
Because of this, we can **define** an action of a group object, $G$, in $C$ on an object, $X$, of $C$ to be a morphism
$$act: G\times X\to X$$
satisfying conditions that certain diagrams (left to the reader) encoding these two rules, commute.
The advantage of this is that it does not require the category $C$ to have internal automorphism group objects for all objects being considered.
As an example, only [[locally compact topological space|locally compact topological spaces]] have well-behaved topological automorphism groups, and thus actions of topological spaces on topological spaces must either be restricted to actions on locally compact spaces, or else be defined as above.
As another example, within the category of [[profinite group|profinite groups]] viewed as topological groups, not all objects have automorphism groups for which the natural topology is profinite. Thus profinite group actions on (the underlying topological space of) a profinite group must either be given in this form, or else be restricted to actions on profinite groups for which the automorphism group is naturally profinite.
### Actions of a monoid object
Suppose we have a category, $C$, with binary [[product]]s and a [[terminal object]] $*$. There is an alternative way of viewing monoid actions in [[Set]], so that we can talk about an action of a [[monoid object]], $M$, in $C$ on an object, $X$, of $C$.
By the adjointness relation between cartesian product, $A\times ?$, and function set, $?^A$, in [[Set]], a monoid homomorphism
$$\alpha: M\to End(X)$$
corresponds to a function
$$act: M\times X\to X$$
which will have various properties encoding that $\alpha$ was a homomorphism of monoids:
$$act(m_1m_2,x) = act(m_1,act(m_2,x))$$
$$act(1,x) = x$$
and these can be encoded diagrammatically.
Because of this, we can **define** an action of a monoid object, $M$, in $C$ on an object, $X$, of $C$ to be a morphism
$$act:M\times X\to X$$
satisfying conditions that certain diagrams (left to the reader) encoding these two rules, commute.
The advantage of this is that it does not require the category $C$ to have internal endomorphism monoid objects for all objects being considered.
### Actions of a set
The action of a set on a set was defined above; it consists of a function $act: X\times Y\to Y$. This can equivalently be represented by a [[quiver]] with $Y$ as its vertices, with its edges labeled by elements of $X$, and such that each vertex has exactly one arrow leaving it with each label. (This is a sort of "Grothendieck construction".) It is also the same as a simple (non halting) [[deterministic automaton]], with $Y$ the set of states and $X$ the set of inputs.
That an action is a type of edge labeled quiver can be seen by explicitly giving the product [[projection]] functions, $p_1$ and $p_2$, of $X\times Y$.
$$X\overset{\quad p_1 \quad}{ \leftarrow}X\times Y\underoverset{\quad act \quad}{p_2}{\rightrightarrows}Y$$
The shape of this diagram corresponds to that of an edge labeled quiver:
$$Labels\overset{\quad label \quad}{ \leftarrow}Edges\underoverset{\quad target \quad}{source}{\rightrightarrows}Vertices$$
While the set $X$ has no algebraic structure to be preserved, the action $act$ generates a unique **[[free category]] action** $act^{*}:X^{*}\times Y\to Y$ where $X^{*}$ is the [[free monoid]] on $X$ containing [[path|paths]] of $X$ elements. The monoidal structure of $X^*$ is preserved: two actions in succession is equal to the action of the concatenation of their paths.
$$act^{*}(x^{*}_2,act(x^{*}_1,y)) = act^{*}(x^{*}_2\cdot x^{*}_1, y) $$
An action of a set $X$ in itself is also called a __[[binary operation]]__, and the set $X$ is called a __[[magma]]__.
## Examples
* A [[representation]] is a "linear action".
* In [[symplectic geometry]] one considers [[Hamiltonian actions]].
* In [[topology]]: [[topological G-space]]
* [[circle action]]
* [[group action on an n-sphere]]
(...)
## Related concepts
{#RelatedConcepts}
* [[action object]]
* **action**, [[∞-action]],
* [[conjugation action]], [[adjoint action]]
* [[diagonal action]]
* [[transitive action]], [[free action]], [[regular action]]
* [[proper action]], [[properly discontinuous action]]
* [[coaction]]
* [[representation]], [[∞-representation]]
* [[module]], [[∞-module]]
* [[action monad]]
* [[associated bundle]], [[associated ∞-bundle]]
* [[quotient]], [[quotient stack]], [[quotient type]]
* [[representation theory]], [[invariant theory]]
* [[equivariant homotopy theory]]
* [[Borel model structure]]
* [[biaction]]
## References
### Group actions
On [[group actions]], mostly in [[TopologicalSpaces]], hence in the form of [[topological G-spaces]]:
Historical origins:
* {#Klein1872} [[Felix Klein]], _[[Vergleichende Betrachtungen über neuere geometrische Forschungen]]_ (1872) Mathematische Annalen volume 43, pages 63–100 1893 ([doi:10.1007/BF01446615](https://doi.org/10.1007/BF01446615))
English translation by M. W. Haskell:
_A comparative review of recent researches in geometry_, Bull. New York Math. Soc. 2, (1892-1893), 215-249. ([euclid:1183407629](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-2/issue-10/A-comparative-review-of-recent-researches-in-geometry/bams/1183407629.full), LaTeX version retyped by Nitin C. Rughoonauth: [arXiv:0807.3161](https://arxiv.org/abs/0807.3161))
Introduction of group actions into ([[quantum physics|quantum]]) [[physics]] (cf. *[[Gruppenpest]]*):
* [[Hermann Weyl]], §III.1 in: *Gruppentheorie und Quantenmechanik*, S. Hirzel, Leipzig (1931), translated by H. P. Robertson: *The Theory of Groups and Quantum Mechanics*, Dover (1950) [[ISBN:0486602699](https://store.doverpublications.com/0486602699.html), [ark:/13960/t1kh1w36w](https://archive.org/details/ost-chemistry-quantumtheoryofa029235mbp/page/n15/mode/2up)]
Textbook accounts:
* {#Bredon72} [[Glen Bredon]], *[[Introduction to compact transformation groups]]*, Academic Press 1972 ([ISBN 9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1)
, [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))
* {#tomDieck79} [[Tammo tom Dieck]], *[[Transformation Groups and Representation Theory]]*, Lecture Notes in Mathematics 766, Springer 1979 ([doi:10.1007/BFb0085965](https://link.springer.com/book/10.1007/BFb0085965))
* {#tomDieck87} [[Tammo tom Dieck]], *[[Transformation Groups]]*, de Gruyter 1987 ([doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372))
Lecture notes:
* {#Koszul65} [[Jean-Louis Koszul]], _Lectures on Groups of Transformations_, Tata Institute 1965 ([pdf](http://www.math.tifr.res.in/~publ/ln/tifr32.pdf), [[KoszulGroupsOfTransformations.pdf:file]])
* [[Patrick Morandi]], _Group actions_ ([[MorandiGroupActions.pdf:file]])
[[!redirects action]]
[[!redirects actions]]
[[!redirects group action]]
[[!redirects group actions]]
[[!redirects monoid action]]
[[!redirects monoid actions]]
[[!redirects left action]]
[[!redirects left actions]]
[[!redirects right action]]
[[!redirects right actions]]
[[!redirects transformation group]]
[[!redirects transformation groups]] |
action (physics) - table | https://ncatlab.org/nlab/source/action+%28physics%29+-+table |
| [[action functional]] | [[kinetic action]] | [[interaction]] | [[path integral]] [[measure]] |
|--|--|--|--|
| $\exp(-S(\phi)) \cdot \mu = $ | $\exp(-(\phi, Q \phi)) \cdot$ | $\exp(I(\phi)) \cdot$ | $\mu$ |
| [[BV-complex|BV]] [[differential]] | [[elliptic complex]] + | [[antibracket]] with interaction + | [[BV-Laplacian]] |
| $d_q =$ | $Q$ + | $\{I,-\}$ + | $\hbar \Delta$ |
|
action functional | https://ncatlab.org/nlab/source/action+functional |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Variational calculus
+-- {: .hide}
[[!include variational calculus - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
In [[physics]] the [[dynamics]] of a [[physical system]] may be encoded by a [[nonlinear functional]] -- called the **action functional** -- on its [[configuration space]]:
* in [[classical mechanics]] and [[classical field theory]] -- by the **action principle** or **[[principle of least action]]** -- the extrema of the action functional -- obtained by [[variational calculus]] and given by [[Euler-Lagrange equation]]s -- encode the physically observable configurations ;
* in [[quantum mechanics]] and [[quantum field theory]] the evolution of the [[quantum state]]s is encoded by the integral -- the [[path integral]] -- of the exponentiated action functional over the space of field configurations.
For emphasis the description of dynamics by action functionals is called the **Lagrangean** approach. Another formulation of dynamics in physics that does not involve an action functional explicitly is [[Hamiltonian mechanics]] on [[phase space]]. At least in certain classes of cases the relation and equivalence of both approaches is understood. Generally the formulation of [[quantum field theory]] in terms of action functionals suffers from a lack of precise understanding of what the [[path integral]] over the action functional really means.
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"float": "left",
"margin": {
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"right": 10,
"bottom": 10,
"left": 0,
"unit": "px"
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"alt": "Action functional on a napkin",
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## Definition
Let $\mathbf{H}$ be the ambient [[(∞,1)-topos]] with a [[natural numbers object]] and equipped with an additive [[continuum]] [[line object]] $\mathbb{A}^1$ (see there). Let $C \in \mathbf{H}$ be the [[configuration space]] of a physical system. Then an **action functional** is a morphism
$$
\exp(\tfrac{i}{\hbar} S(-)) : C \to \mathbb{A}^1 / \mathbb{Z}
\,
$$
(here $\hbar$ refers to [[Planck's constant]]).
If $\mathbf{H}$ is a [[cohesive (∞,1)-topos]] then there is an <a href="https://ncatlab.org/nlab/show/cohesive+(infinity,1)-topos+--+structures#CurvatureCharacteristics">intrinsic differential</a> of the action functional to a morphism
$$
\mathbf{d} \exp(\tfrac{i}{\hbar} S(-)) : C \to \mathbf{\flat}_{dR}\mathbb{A}^1/\mathbb{Z}
\,.
$$
The [[equation]]
$$
\mathbf{d} \exp(\tfrac{i}{\hbar} S(-)) = 0
$$
is the [[Euler-Lagrange equation]] of the system. It characterizes the [[critical locus]] of $S$ is the [[covariant phase space]] inside the configuration space: the space of classically realized trajectories/histories of the system. If $\mathbf{H}$ models [[derived geometry]] then this critical locus is presented by a [[BRST-BV complex]].
### Local action functionals (traditional theory)
{#LocalActionFunctional}
An action functional is called **local** if it arises from integration of a [[Lagrangian]].
In traditional theory this is interpreted as follows: an action functional $S : C \to \mathbb{A}^1$ is called local if
* the [[configuration space]] $C$ is the space $C = \Gamma_X(E)$ of [[section]]s of a [[fiber bundle]] $E \to X$ over some parameter space ([[spacetime]] $X$);
* there is a [[Lagrangian]] density $J_\infty(E) \to \Omega^{\dim X}(X)$ on the [[jet bundle]] of $E$;
* on a section/field configuration $\phi : X \to E$ the action $S$ takes the value
$$
S(\phi) = \int_X L(j_\infty(\phi))
\,,
$$
where $j_\infty(\phi) = (\phi, \partial_i \phi, \cdots)$ is the jet-prolongation of $\phi$ (the collection of all its higher partial derivatives).
Consider action functional for on a configuration space of
[[smooth function]]s from the line to a [[smooth manifold]] $X$.
We can consider
1. $ S(q) = \int_a^b L(q,\dot{q}) \,\mathrm{d}t $, where $q$ is a path through configuration space, on the time interval $[a,b]$, with derivative $\dot{q} = \mathrm{d}q/\mathrm{d}t$. When minimising the action, we fix the values of $q(a)$ and $q(b)$.
2. $ L(q,\dot{q}) = \int_{S} \mathcal{L}(q,\dot{q}) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z $, where now $q$ is a configuration of fields on $S$, which is a region of space. We fix boundary conditions on the boundary of $S$ (typically that $q$ and $\dot{q}$ go to zero if $S$ is all of space).
3. $ S(q) = \int_{R} \mathcal{L}(q,\dot{q}) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}t $, where now $q$ is a configuration of fields on $R$, which a region of spacetime, with time derivative $\dot{q} = \partial{q}/\partial{t}$. We fix boundary conditions on the boundary of $R$.
The formulation of (3) above is still not manifestly coordinate independent. However, $\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}t$ is simply the [[volume form]] on spacetime and $\dot{q}$ is merely one choice of coordinate on [[configuration space|state space]] and could just as easily be replaced by a derivative with respect to any timelike coordinate on spacetime (or drop coordinates altogether).
### Extended local action functionals in (higher) gauge theory
{#ExtendedLocalInGaugeTheory}
For [[gauge theories]] and [[higher gauge theories]] the configuration spaces of the physical system are in general not plain [[manifolds]] or similar, but are [[orbifolds]] or more generally [[smooth groupoids]], [[smooth ∞-groupoids]]. (An exposition of and introduction to much of the following is at _[[geometry of physics]]_.)
For instance for $G$ a [[Lie group]] and $\mathbf{B}G_{conn}$ the smooth [[moduli stack]] of $G$-[[principal connections]] (see at [[connection on a bundle]]), then the [[smooth groupoid]] of $G$-[[gauge field]] configurations is the [[internal hom]]/[[mapping stack]] $[\Sigma, \mathbf{B}G_{conn}] \in $[[Smooth∞Grpd]] (or some concretification thereof, see at _[geometric of physics -- differential moduli](geometry%20of%20physics#DifferentialModuli)_: this is the [[smooth groupoid]] whose [[objects]] are $G$-[[gauge field]]-configurations on $\Sigma$ ([[connection on a bundle|connections]] on $G$-[[principal bundles]] over $\Sigma$), and whose [[morphisms]] are [[gauge transformations]] between these. The [[infinitesimal space|infinitesimal]] approximation to this [[smooth ∞-groupoid]], its _[[∞-Lie algebroid]]_ is the (off-shell) [[BRST complex]] of the theory. The tangent to the $n$-fold [[higher gauge transformations]] becomes the $n$-fold _ghosts_ in the BRST complex.
More generally $G$ here can by any [[smooth ∞-group]], such as the [[circle n-group]] $\mathbf{B}^{n-1}U(1)$ or the [[String 2-group]] or the [[Fivebrane 6-group]], and so on, in which case $[\Sigma, \mathbf{B}G_{conn}]$ is the [[smooth ∞-groupoid]] of [[higher gauge theory|higher gauge field]], [[gauge transformations]] between these, [[higher gauge transformations]] between those, and so on.
Notice that this means in particular that in [[higher geometry]] a [[gauge theory]] is a [[sigma-model]] [[quantum field theory]]: one whose [[target space]] is not just a plain [[manifold]] but is a [[moduli stack]] of gauge field configurations.
A [[gauge invariance|gauge invariant]] action functional is then a morphism of [[smooth ∞-groupoids]]
$$
\exp( i S(-)) \colon [\Sigma, \mathbf{B}G_{conn}] \to U(1)
\,.
$$
This is of particular interest, again, if it is _local_. In fact, in this context now we can also ask that it is "extended" in the sense of [[extended topological quantum field theory]]: that we have an action functional not only in top dimension, being a function, but also in codimension 1, being a [[prequantum bundle]], and in higher codimension, being a [[prequantum n-bundle]].
This is notably the case for all (higher) gauge theories of [[schreiber:infinity-Chern-Simons theory]] type, such as ordinary [[Chern-Simons theory]] and such as ordinary [[Dijkgraaf-Witten theory]], as well as its higher generalizations. In these cases the action functional $\exp(i S(-)) \colon [\Sigma, \mathbf{B}G_{conn}]$ arises itself from [[transgression]] of an [[extended Lagrangian]] that is defined on the universal [[moduli stack]] of gauge field configurations $\mathbf{B}G_{conn}$ itself, namely from a [[universal characteristic class]] in higher nonabelian [[differential cohomology]] of the form
$$
\mathbf{L} \colon \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn}
\,.
$$
Here $\mathbf{B}^n U(1)_{conn}$ is the universal [[smooth infinity-groupoid|smooth]] [[moduli infinity-stack]] for [[circle n-bundles with connection]]. Such a morphism of moduli stacks locally takes a [[connection on a bundle|connection]] [[differential form]] $A$ to a [[Chern-Simons form]] $CS(A)$, but globally it sends the underlying [[principal bundle]] to a [[circle n-group|circle (n-1)-group]] [[principal ∞-bundle]] and accordingly acts globally on the connection.
This is hence a fully local [[Lagrangian]]: an _[[extended Lagrangian]]_. Alternatively, one may think of this whole morphism as modulating a [[prequantum circle n-bundle]] on the universal moduli stack $\mathbf{B}G_{conn}$ of [[gauge fields]] itself.
For instance for ordinary [[Chern-Simons theory]] here $n = 3$ $G$ is a [[semisimple Lie group]] and $\mathbf{L}$ is a smooth and differential refinement of the [[first Pontryagin class]]/[[second Chern class]], or of an integral multiple of that (the "level" of the theory). In this case $\mathbf{L}$ may also be thought of as modulating the universal [[Chern-Simons circle 3-bundle]]. If instead $G$ is a [[discrete group]] then $\mathbf{L}$ is a [[cocycle]] in the $U(1)$-[[group cohomology]] and this is the [[extended Lagrangian]] of [[Dijkgraaf-Witten theory]].
This [[extended Lagrangian]] becomes an extended action functional after [[transgression]]: the operation of [[fiber integration in ordinary differential cohomology]] refines to a morphism of moduli stacks of the form
$$
\exp(2 \pi i \int_{\Sigma_k} (-))
\colon
[\Sigma_k, \mathbf{B}^n U(1)_{\mathrm{conn}} ]
\to
\mathbf{B}^{n-k}U(1)_{conn}
\,,
$$
where $\Sigma$ is an [[orientation|oriented]] [[closed manifold]] of [[dimension]] $k$. This morphism locally simply takes a [[differential n-form]] to its ordinary [[integration of differential forms]] over $\Sigma_k$, but globally it takes the correct [[higher holonomy]] of [[circle n-bundles with connection]].
Combining this with an extended [[schreiber:∞-Chern-Simons theory]] [[Lagrangian]] $\mathbf{L} \colon \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn}$ as above yields for each dimension $k$ a [[prequantum circle n-bundle]] on the space of gauge field configurations over $\Sigma_k$, by forming the [[transgression]] composite
$$
\exp(i S(-))
\coloneqq
\exp(2 \pi i \int_{\Sigma_k} [\Sigma_k, \mathbf{L}])
\;\; \colon \;\;
[\Sigma_k, \mathbf{B}G_{conn}]
\stackrel{[\Sigma_k, \mathbf{L}]}{\to}
[\Sigma_k, \mathbf{B}^n U(1)_{conn}]
\stackrel{\exp(2 \pi i \int_{\Sigma_k}(-))}{\to}
\mathbf{B}^{n-k}U(1)_{conn}
\,.
$$
This morphism locally takes the local [[differential form]] incarnation $A$ of a [[connection on an ∞-bundle]] to the exponentiation of the [[integration of differential forms]] $\int_\Sigma CS(A)$ of some higher [[Chern-Simons form]], but globally it computes the correct [[higher holonomy]] of the higher [[circle n-bundle with connection]] over the universal moduli stack of fields, as modulated by the extended [[Lagrangian]] $\mathbf{L}$.
## Examples
For [[spacetime]] [[field theory]]:
* [[Einstein-Hilbert action]]
* [[Einstein-Maxwell theory]]
* [[Einstein-Yang-Mills theory]]
* [[Einstein-Yang-Mills-Dirac theory]]
* [[Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory]]
For [[branes]]:
* [[Nambu-Goto action]]
* [[Dirac-Born-Infeld action]]
* [[Perry-Schwarz action]]
* A large class of examples of action functionals arises in [[schreiber:∞-Chern-Simons theory]]. See there for details.
## Related concepts
[[!include action (physics) - table]]
* [[principle of extremal action]], [[Euler-Lagrange equations]]
* [[path integral]] [[quantization]]
* [[effective action functional]], [[background field formalism]]
* [[parent action functional]]
[[!include extended prequantum field theory - table]]
## References
Historical and expository accounts of the "[[principle of extremal action]]" or "action principle", for short:
* Agamenon R. E. Oliveira, *History of Two Fundamental Principles of Physics: Least Action and Conservation of Energy*, Advances in Historical Studies **3** 2 (2014) [[doi:10.4236/ahs.2014.32008](http://dx.doi.org/10.4236/ahs.2014.32008)]
* Walter Dittrich, *The Development of the Action Principle -- A Didactic History from Euler-Lagrange to Schwinger*, SpringerBriefs in Physics, Springer (2021) [[doi:10.1007/978-3-030-69105-9](https://doi.org/10.1007/978-3-030-69105-9)]
* [[Douglas Cline]], *Variational Principles in Classical Mechanics*, University of Rochester (2021) [[pdf](http://classicalmechanics.lib.rochester.edu/pdf/Variational_Principles_in_Classical_Mechanics_3e_final_fbcover.pdf), <a href="https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)">online version</a>]
Textbook account in the context of [[gauge theories]]:
* [[Marc Henneaux]], [[Claudio Teitelboim]], §1.1.1 in: _[[Quantization of Gauge Systems]]_, Princeton University Press (1992) [[ISBN:9780691037691](https://press.princeton.edu/books/paperback/9780691037691/quantization-of-gauge-systems), [jstor:j.ctv10crg0r](https://www.jstor.org/stable/j.ctv10crg0r)]
Lecture notes with more details are in the section _[Lagrangians and Action functionals](geometry+of+physics#LagrangiansAndActionFunctionals)_ of
* _[[geometry of physics]]_ .
Discussion of extended higher local action functional for (higher) gauge theories of generalized [[schreiber:∞-Chern-Simons theory]] type are discussed in
* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Hisham Sati]], _[[schreiber:Extended higher cup-product Chern-Simons theories]]_
The extended local action functionals for ordinary 3d [[Chern-Simons theory]]/[[Dijkgraaf-Witten theory]] and for 7d [[String 2-group]] Chern-Simons theory are constructed in
* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Cech cocycles for differential characteristic classes]]_
and discussed further in
* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Hisham Sati]], _[[schreiber:7d Chern-Simons theory and the 5-brane]]_
A comprehensive discussion in a general context of higher differential geometry is in
* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_
[[!redirects action functional]]
[[!redirects action functionals]]
[[!redirects action functinal]]
[[!redirects local action functional]]
[[!redirects local action functionals]] |
action group > history | https://ncatlab.org/nlab/source/action+group+%3E+history | < [[action group]]
[[!redirects action group -- history]] |
action groupoid | https://ncatlab.org/nlab/source/action+groupoid |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Given an [[action]] $\rho$ of a [[group]] $G$ on a [[set]] $S$, the action groupoid $S//G$ is a bit like the quotient set $S/G$ (the set of $G$-orbits). But, instead of taking elements of $S$ in the same $G$-orbit as being [[equality|equal]] in $S/G$, in the action groupoid they are just [[isomorphism|isomorphic]]. We may think of the action groupoid as a [[homological resolution|resolution]] of the usual quotient. When the action of $G$ on $S$ fails to be free, the action groupoid is generally better-behaved than the quotient set.
The action groupoid also goes by other names, including '[[weak quotient]]'. It is a special case of a '[[pseudo colimit]]', as explained below. It is also called a "[[semidirect product]]" and then written $S \rtimes G$. The advantage of this is that it accords with the generalisation to the action of a group $G$ on a groupoid $S$, which is relevant to orbit space considerations, since if $G$ acts on a space $X$ it also acts on the fundamental groupoid of $X$; this is fully developed in "Topology and Groupoids", Chapter 11.
## Definition
### In category theory
Given an [[action]]
$\rho : S \times G \to S$ of a group $G$ on the set $S$, the _action groupoid_ $S//G$ (or, more precisely, $S//_\rho G$) is the [[groupoid]] for which:
* an object is an element of $S$
* a morphism from $s \in S$ to $s' \in S$ is a group element $g \in G$ with $g s = s'$. So, a general morphism is a pair $(g,s) : s \to g s$.
* The composite of $(g,s) : s \to g s = s'$ and $(g',s'):
s' \to g's'$ is $(g' g, s) : s \to g' g s$.
Equivalently, we may define the _action groupoid_ $S//G$ to be the groupoid
$$
\array{
&& S \times G
\\
& {}^{s := p_1}\swarrow
&&
\searrow^{t = \rho}
\\
S
&&&&
S
}
$$
with composition
$$(S \times G) \times_{t,s} (S \times G) \simeq S \times G \times G \to S \times G$$
given by the product in $G$.
We can denote the morphisms in $S//G$ by
$$S//G:=\{s\stackrel{g}{\to} \rho(s,g) | s\in S, g\in G\}.$$
### In (∞,1)-category theory
+-- {: .num_defn #ActionInfinityGroupoid}
###### Definition
Let $C$ be an ($\infty$,1)-category, let $G\in Grpd(C)$ be a groupoid object in $C$, let $X\in C$ be an object. Then the [[simplicial object]]
$$
\array{
\cdots
&
\underoverset{\to}{\to}{\to}
&
X\times_{G_0}G\times_p G
&
\rightrightarrows
&
X\times_{G_0}G
&
\to
&
X
}
$$
such that the degree-wise projections give a simplicial map
$$
\array{
\cdots
&
\underoverset{\to}{\to}{\to}
&
X\times_{G_0}G\times_p G
&
\rightrightarrows
&
X\times_{G_0}G
&
\to
&
X
\\
&&
\downarrow
&&
\downarrow
&&
\downarrow^a
\\
\cdots
&
\underoverset{\to}{\to}{\to}
&
G\times_p G
&
\rightrightarrows
&
G
&
\xrightarrow{p}
&
G_0
}
$$
is called an _action of_ $G$ _on_ $X$. The colimit $colim\; X\times_{G_0}^{\times_\bullet}$ is called _action $\infty$-groupoid of_ $G$ _on_ $X$.
=--
## Interpretations
On top of the above explicit definitions, there are several useful ways to think of action groupoids.
Recall that the action $\rho$ is equivalently thought of as a functor
$$
\rho : \mathbf{B}G \to Sets
$$
from the [[group]] $G$ regarded as a one-object groupoid, denoted $\mathbf{B}G$.
This functor sends the single object of $\mathbf{B}G$ to the set $S$.
### As a pseudo colimit
$S//G$ is the [[2-limit|2-colimit]] of $\rho$, i.e., the [[category of elements]] of $\rho$.
$$
S//G \simeq colim_{\mathbf{B}G} \rho
\,.
$$
The universal cocone consists of cells of the form
$$
\array{
S &&\stackrel{\rho(g)}{\to}&& S
\\
& \searrow &\stackrel{\simeq}{\Leftarrow}& \swarrow
\\
&& S//G
}
\,,
$$
where the 2-morphism is uniquely specified and in components given by $s \mapsto (s \stackrel{g}{\to} \rho(s,g))$.
### As associated universal bundle
Let $Set_*$ be the category of pointed sets and $Sets_* \to Sets$ be the canonical forgetful functor.
We can think of this as the "universal $Set$-bundle".
Then $S//G$ is the pullback
$$
\array{
S//G
&\to&
Sets_*
\\
\downarrow
&&
\downarrow
\\
\mathbf{B}G
&\stackrel{\rho}{\to}&
Sets
}
\,.
$$
One place where we discussed this is the comment [It was David Roberts who apparently first noticed...](http://golem.ph.utexas.edu/category/2007/08/on_hess_and_lack_on_bundles_of_1.html#c019094).
Notice also that an action of $G$ on the set $S$ gives rise to a morphism $p: S \rtimes G \to G$ which has the property of unique path lifting, or in other words is a discrete opfibration. It is also called a covering morphism of groupoids, and models nicely covering maps of spaces.
Higgins used this idea to lift presentations of a group $G$ to presentations of the covering morphism of $G$ derived from the action of $G$ on cosets, and so to apply graph theory to obtain old and new subgroup theorems in group theory.
### As a stack
In the case where the action is [[internalization|internal]] to sets with structure, such as internal to [[Diff]] one wants to realize the action groupoid as a [[Lie groupoid]]. That Lie groupoid in turn may be taken to present a [[differentiable stack]] which then usually goes by the same name $S//G$.
## Properties
### Relation to representation theory
The action groupoids $X//G$ of a group $G$ come equipped with a canonical map to $\mathbf{B}G \simeq \ast //G$. Regarded via this map as objects in the [[slice (infinity,1)-category|slice]] of groupoids over $\mathbf{B}G$, action groupoids are in fact [[equivalence|equivalent]] to the actions that they arise from.
For more on this see at _[[infinity-action]]_ and also at _[[geometry of physics -- representations and associated bundles]]_.
## Action $\infty$-groupoid {#ActionooGrpd}
All of this goes through almost verbatim for actions in the context of [[(∞,1)-category theory]].
Let $G$ be an [[∞-group]] in that $\mathbf{B}G$ is an [[∞-groupoid]] with a single object. An action of $G$ on an [[(∞,1)-category]] is an [[(∞,1)-functor]]
$$
\rho : \mathbf{B}G \to (\infty,1)Cat
$$
to [[(∞,1)Cat]]. This takes the single object of $\mathbf{B}G$ to some $(\infty,1)$-category $V$.
Again we want to **define** the _action groupoid_ $V//G$ as the [[limit in a quasi-category|(∞,1)-categorical colimit]] over the action:
$$
V//G := \lim_\to \rho
\,.
$$
By the result [described here](http://ncatlab.org/nlab/show/limit+in+a+quasi-category#WithValInooGrpd) this is, as before, equivalent to the pullback of the "universal $(\infty,1)Cat$-bundle" $Z \to (\infty,1)Cat$, namely to the [[coCartesian fibration]]
$$
\array{
V//G &\to& Z
\\
\downarrow && \downarrow
\\
\mathbf{B}G &\stackrel{\rho}{\to}& (\infty,1)Cat
}
$$
classified by $\rho$ under the [[(∞,1)-Grothendieck construction]]. As before, we can continue a [[fiber sequence]] to the left by adjoining the $(\infty,1)$-categorical pullback along the point inclusion $* \to \mathbf{B}G$
$$
\array{
V&\to& V//G &\to& Z
\\
\downarrow && \downarrow && \downarrow
\\
{*} &\to& \mathbf{B}G &\stackrel{\rho}{\to}& (\infty,1)Cat
}
\,.
$$
The resulting total $(\infty,1)$-pullback rectangle is the fiber of $Z \to (\infty,1)Cat$ over the $(\infty,1)$-category $V$, which is $V$ itself, as indicated.
## Some comments
* If the action of a Lie group $G$ on the manifold $X$ is free and proper, what you get is a manifold $X/G$.
* If the action of a Lie group $G$ on the manifold $X$ is not necesssarily free and proper, what you get is a Lie groupoid, denoted (among other symbols) by $X//G$.
## Related concepts
* [[global quotient orbifold]], [[quotient stack]], [[homotopy quotient]]
* the [[groupoid cardinality]] of action groupoids is given by the [[class formula]]
## References
* P.J. Higgins, 1971, "Categories and Groupoids", van
Nostrand, {New York}. Reprints in Theory and Applications of Categories, 7 (2005) pp 1--195.
* [[Ronnie Brown]], *Topology and groupoids*, Booksurge (2006) [[webpage](http://groupoids.org.uk/topgpds.html), [pdf](https://groupoids.org.uk/pdffiles/topgrpds-e.pdf)]
* John Armstrong's article, [Groupoids (and more group actions)](http://unapologetic.wordpress.com/2007/06/09/groupoids-and-more-group-actions/)
* John Baez, [TWF 249](http://math.ucr.edu/home/baez/week249.html)
[[!redirects action groupoids]]
[[!redirects action group]]
[[!redirects action groups]]
[[!redirects groupoid quotient]] |
action Lie algebroid | https://ncatlab.org/nlab/source/action+Lie+algebroid |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
For $V$ a [[space]], $G$ a [[group]] and $\rho : G\times V \to V$ a [[action]] of $G$ on $V$, we have the corresponding [[action groupoid]]. If everything is sufficiently smooth, this is a [[Lie groupoid]] denoted $V//_\rho G$.
The _action Lie algebroid_ of $\rho$ is the [[Lie algebroid]] that corresponds to this Lie groupoid (under [[Lie integration]]).
The [[Chevalley-Eilenberg algebra]] of an action Lie algebroid is in [[physics]] known as a [[BRST complex]].
## Details
Let $G$ be a [[Lie group]], $V$ a [[smooth manifold]] and $\rho : G \times V \to V$ a smooth [[action]]. Write $V//G$ for the corresponding [[action groupoid]], itself a Lie groupoid. The [[Lie algebroid]] $Lie(V//G)$ corresponding to this is the [[action Lie algebroid]].
The [[Chevalley-Eilenberg algebra]] of the action Lie algebroid is
$$
CE(Lie(V//G)) = (\wedge^\bullet_{C^\infty(V)} (C^\infty(V) \otimes\mathfrak{g}^*), d_{\rho})
\,,
$$
where the differential acts on functions $f \in C^\infty(V)$ by
$$
d_\rho : f \mapsto \rho(-)(-)^* f \in C^\infty(V)\otimes \mathfrak{g}^*
\,.
$$
Explicitly, for $t \in \mathfrak{g}$ this sends $f$ to the function
$(d_\rho f)(t)$ which is the derivative along $t \in T_e G$ of the function $G \times V \stackrel{\rho}{\to}V \stackrel{f}{\to} \mathbb{R}$.
Even more explicitly, if we choose local coordinates $\{v^k\} : \mathbb{R}^{dim V} \to V$ on a patch, and choose a basis $\{t^a\}$ of $\mathfrak{g}^*$ then we have that restricted to this patch the differential is on generators given by
$$
d_\rho : f \mapsto \rho^k{}_a t^a \wedge \partial_k f
$$
$$
d_\rho : t^a \mapsto - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c
\,.
$$
Specifically for $V$ a finite dimensional [[vector space]], $\rho : G$ a _linear_ action, $\{v^k\}$ a choice of basis of that vector space and $f$ a _[[linear function]]_ $f= f_k v^k$ , we have that $(f_k := \partial_k f) \in \mathbb{R}^{dim V}$ are the components vector of the dual vector given by $V$ in this basis, and the above gives the [[matrix multiplication]] form of the action
$$
d_\rho : v^k \mapsto t^a \rho_a{}^k{}_l v^l
\,.
$$
Notice for completeness that the equation $(d_\rho)^2 = 0$ is equivalent to the [[Jacobi identity]] of the Lie bracket and the action property of $\rho$:
$$
d_\rho d_\rho v^k =
(t^a \wedge t^b \rho_a{}^k{}_r \rho_b{}^r{}_l
-
\frac{1}{2}C^a{}_{b c}t^b \wedge t^c \rho_a{}^k{}_l )
v^l
\,.
$$
These local formulas shall be useful below for recognizing from our general abstract definition of covariant derivative the formulas traditionally given in the literature. For that notice that in the above local coordinates further restricting attention to linear actions, the [[Weil algebra]] of the action Lie algebroid is given by
$$
W(Lie(V//G)) = (\wedge^\bullet_{C^\infty(\mathbb{R}^{dim V})} ( \Gamma(T^* \mathbb{R}^{dim V}) \oplus \mathfrak{g}^* \oplus \mathfrak{g}^*[1]), d_{W_\rho})
$$
where the differential is given on generators by
$$
d_{W_\rho} : v^k \mapsto \rho_a{}^k{}_l t^a \wedge v^l + d_{dR} v^k
$$
$$
d_{W_\rho} : t^a \mapsto - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a
$$
and where the uniquely induced differential on the shifted generators -- the one encoding [[Bianchi identities]] -- is
$$
d_{W_\rho} : d_{dR} v^k
\mapsto
\rho_a{}^k{}_k r^a \wedge v^l
- \rho_a{}^k{}_l t^a \wedge d_{dR} v^l
$$
and
$$
d_{W} : r^a \mapsto C^a{}_{b c} t^b \wedge r^c
\,.
$$
Notice that we may identify the [[delooping]] Lie groupoid $\mathbf{B}G$ of $G$ with the action groupoid of the trivial action on the point, $\mathbf{B}G \simeq *//G$. On Lie algebroids this morphism is dually the inclusion
$$
CE(Lie(V//G)) \leftarrow CE(\mathfrak{g})
$$
that is the identity on $\mathfrak{g}^*$.
## Applications
* A [[covariant derivative]] is the 1-form [[curvature]] of [[∞-Lie algebra valued forms|Lie algebroid valued differential forms]] with values in an action Lie algebroid.
* [[equivariant de Rham cohomology]]
## Related concepts
* [[action groupoid]], [[homotopy quotient]], [[Borel construction]]
* [[BRST complex]]
[[!redirects action Lie algebroids]] |
action monad | https://ncatlab.org/nlab/source/action+monad | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _action monad_ or _writer monad_ is a construction generalizing many seemingly different concepts across mathematics and [[computer science]].
It may intuitively be understood in the following ways, where throughout we fix a [[group]] or [[monoid]] $M$.
* It is the [[monad]] associated to the [[free-forgetful adjunction]] between [[G-set|M-sets]] (sets equipped with an $M$-[[action]]) and [[sets]].
* It is the monad whose [[algebra over a monad|algebras]] are $M$-sets and whose morphisms are [[equivariant maps]].
* It is the [[monad in computer science|monad modeling computational effects]] that can be added or concatenated (using the multiplication map of $M$), such as writing into a log file or standard output to screen. In this case, $M$ is usually the [[free monoid]] on a fixed [[alphabet]] (i.e. the [[data type]] "[[string (in computer science)|String]]").
If one, more generally, replaces [[sets]] and a [[monoid]] with a [[monoidal category]] and an [[internal monoid]], a similar construction can be given. This generalizes, for example, the [[action]] that [[rings]] have on their [[modules]], and that [[Lie groups]] have on [[manifolds]].
## Definition
### In Sets
{#DefinitionInSets}
Let $M$ be a [[monoid]] with [[unit]] $e:1\to M$ and [[multiplication]] $\cdot:M\times M \to M$. The (left) **$M$-action monad** is a [[monad]] on [[Set]] where
* The [[endofunctor]] maps a set $X$ to the set $M\times X$;
* The unit is given by
\begin{tikzcd}
X \ar{r}{\cong} & 1\times X \ar{r}{e\times\mathrm{id}} & M\times X
\end{tikzcd}
for each object $X$;
* The multiplication is given by
\begin{tikzcd}
M\times (M\times X) \ar{r}{\cong} & (M\times M)\times X \ar{r}{\cdot\times\mathrm{id}} & M\times X
\end{tikzcd}
for each object $X$.
The monad axioms follow from the monoid axioms for $M$.
There exists an analogous monad for _right_ actions, whose endofunctor maps $X$ to $X\times M$.
### In any monoidal category
{#DefinitionInMonoidalCategory}
More generally, let $(C,\otimes,1)$ be a [[monoidal category]].
Let $M$ be a [[monoid object]] in $C$ with [[unit]] $e:1\to M$ and [[multiplication]] $\cdot:M\otimes M \to M$. The (left) **$M$-action monad** is a [[monad]] on $C$ where
* The [[endofunctor]] maps an object $X$ of $C$ to the object $M\otimes X$;
* The unit is given by
\begin{tikzcd}
X \ar{r}{\cong} & 1\otimes X \ar{r}{e\times\mathrm{id}} & M\otimes X
\end{tikzcd}
for each object $X$;
* The multiplication is given by
\begin{tikzcd}
M\otimes (M\otimes X) \ar{r}{\cong} & (M\otimes M)\otimes X \ar{r}{\cdot\otimes\mathrm{id}} & M\otimes X
\end{tikzcd}
for each object $X$.
Again, the monad axioms follow from the monoid axioms for $M$. (And again there is an analogous notion for right actions).
## Algebras
The [[algebra over a monad|algebras]] over the action monad for a monoid (or group) $M$ can be seen as the [[G-space|$M$-spaces]], i.e. sets or spaces equipped with an [[action]] of $M$ (hence the name).
Plugging in the definition, an algebra over this monad is then a set $A$ together with a map $e:M\times A\to A$, such that the following diagrams commute.
The algebra diagrams
\begin{tikzcd}
A \ar{dr}[swap]{\mathrm{id}} \ar{r}{\eta} & M\times A \ar{d}{e} && M \times M \times A \ar{d}{\mu} \ar{r}{\mathrm{id}_M\times e} & M\times A \ar{d}{e} \\
& A && M \times A\ar{r}{e} & A .
\end{tikzcd}
say equivalently that for all $a\in A$ and $m,n\in M$,
$$
1\cdot a= a, \qquad (m n)\cdot a = m\cdot (n\cdot a) .
$$
In other words, a $T_M$-algebra is exactly a set equipped with an $M$-[[action]] in the traditional group-theoretical sense.
This gives a way to talk about monoid and group [[actions]] [[internally]] to any [[monoidal category]], giving the notion of a [[module object]].
### Examples
{#Examples}
* In the category of [[manifolds]] with their [[cartesian product]], an [[internal group]] is a [[Lie group]]. The algebras of the corresponding action monad are the manifolds equipped with the action of the Lie group, such as [[homogeneous spaces]].
* The same can be said about [[topological spaces]] and continuous actions of [[topological groups]].
* In the category [[Ab]] of [[abelian groups]] with their [[tensor product]], an [[internal monoid]] is a [[ring]]. The algebras of the resulting action monad are the [[modules]] over that ring.
* In the category of [[endofunctors]] of a category $C$, an [[internal monoid]] is the same as a [[monad]] $T$. An algebra of the resulting action monad is an instance of a [[module over a monad|module over a monad]] $T$.
## Properties
* For the case of [[group actions]] on [[sets]], a [[G-set|$G$-set]] is [[free action|free as an action]] if and only if it is [[algebra over a monad#free_algebras|free as an algebra]] over the action monad.
* The action monad $X\mapsto M\times X$ on [[Set]] is canonically [[strong monad|strong]], with the strength given by the [[associator]] (see [[strong monad#examples]]). It is [[commutative monad|commutative]] if and only if $M$ is commutative as a monoid.
* More generally, in an arbitrary [[symmetric monoidal category]], the action monad induced by tensoring with an [[internal monoid]] is commutative if and only the monoid object is commutative. (See for example [Brandenburg 2014, Exp. 6.3.12](#Brandenburg2014).)
* Any action monad canonically [[distributive law|distributes]] over a [[strong monad]] $T$, with the distributive law $M \otimes T X \to T(M\otimes X)$ induced by the strength.
## Examples
### As logging-effect in computer science
{#AsLoggingEffectInComputerScience}
As a [[monad in computer science]], the action monad is known under the name of **writer monad**, since it encodes the computational effect of programs "writing logging messages" like into a [[stream]].
The following explanation is taken from [Perrone, Example 5.1.14](#notesperrone).
Let $M$ be a [[monoid]], and let's write it additively. Denote by $T_M$ its _right_ writer monad.
A [[Kleisli morphism]] of $T_M$ is a [[morphism]] $k \colon X \to Y \times M$. We can interpret it as a process which, when given an input $x\in X$, does not just produce an output $y\in Y$, but also an element of $M$.
For example, it could be energy released by a chemical reaction, or waste, or a cost of the transaction. In [[computer science]], this is the behaviour of a function that computes a certain value, but that also writes into a log file (or to the standard output) that something has happened (the monoid operation being the concatenation of strings). For example, when you compile a TeX document, a log file is produced alongside your output file. Hence the name "writer monad".
Let's now look at the [[Kleisli composition]]. If we have processes $k \colon X \to Y\times M$ and $h \colon Y\to Z\times M$, then $k\circ_{kl} Y \colon X\to Z\times M$ is given by
\begin{tikzcd}[row sep=0, column sep=large, nodes={scale=1.25}]
X
\ar{r}{k}
&
Y \times M
\ar{r}{h\times \mathrm{id}_M}
&
Z \times M \times M
\ar{r}{\mathrm{id}_X\times +}
&
Z \times M
\\
x \ar[mapsto]{r}
&
(y,m)
\ar[mapsto]{r}
&
(z,n,m)
\ar[mapsto]{r}
&
(z, n+m)
\mathrlap{\,.}
\end{tikzcd}
What it does is the following:
1. It executes the process $k$ with an input $x\in X$, giving as output an element of $y\in Y$ as well as a cost (or extra output) $m\in M$.
1. It executes the process $k$ taking as input the $y\in Y$ produced by $k$, giving an element $z\in Z$ as well as an extra cost $n\in M$. (All of this while keeping track of the first cost $m$.)
1. The two costs $m$ and $n$ are summed (or the extra outputs are concatenated).
So, for example, the cost of executing two processes one after another is the sum of the costs. The same is true about the release of energy in a chemical reaction, and about waste.
Just as well, executing two programs one after another will produce a concatenation of text in a log file (or two log files).
### Quantum measurement
{#QuantumMeasurement}
For $B$ a [[finite set]], and fixing any [[ground field]] $k$ (such as the [[complex numbers]]), consider the [[commutative monoid]] in $k$-[[vector spaces]] (i.e. the ordinary [[commutative algebra]])
$$
\mathbb{1}^B
\;=\;
\oplus_B k
\;\in\;
CMon\big(Vect_k\big)
\,=\,
CAlg_k
$$
which may be thought of as [[generators and relations|generated]] from a $B$-[[indexed set]] of mutually orthogonal [[projection operators]].
Then the induxed $\mathbb{1}^B$-writer monad on $k$-[[VectorSpaces]] (in the sense [above](#DefinitionInMonoidalCategory)) is [[isomorphism|isomorphic]] to the linear version of the $B$-[[reader monad]].
This is related to the notion of [[quantum measurement]], and as such is discussed at: *[[quantum reader monad]]*.
## Related concepts
* [[stream]]
* [[list monad]]
* [[action]], [[G-set]], [[equivariant map]]
* [[internalization]]
* [[monoidal category]], [[monoid object]], [[group object]], [[module object]]
* [[monad]], [[algebra over a monad]]
* [[monad in computer science]],
* [[function monad]]
[[!include reader-writer (co)monads -- table]]
## References
### General
Elementary exposition:
* {#notesperrone} [[Paolo Perrone]], Section 5 of: _Notes on Category Theory with examples from basic mathematics_, . [[arXiv:1912.10642](http://arxiv.org/abs/1912.10642)]
Proof that the construction of actions monads from monoids is part of an [[adjunction]] between monoids and [[strong monads]]:
* {#Wolff73} [[Harvey Wolff]], *Monads and Monoids on Symmetric Monoidal Closed Categories*, Archiv der Mathematik **24** (1973) 113–120 [[doi:10.1007/BF01228184](https://doi.org/10.1007/BF01228184)]
Generalization of this discussion to [[Frobenius algebras]]/[[Frobenius monads]] (cf. *[[cowriter comonads]]*):
* [[Martti Karvonen]], *Frobenius algebras in functor categories*, Oxford (2014) [[pdf](https://www.cs.ox.ac.uk/people/aleks.kissinger/theses/bob/Karvonen.pdf), [[Karvonen-FrobeniusMonads.pdf:file]]]
and analogous discussion in [[dagger-categories]]:
* [[Chris Heunen]], [[Martti Karvonen]], *Monads on dagger categories*, Theory and Applications of Categories **31** 35 (2016) 1016-1043 [[arXiv:1602.04324](https://arxiv.org/abs/1602.04324), [tac:31-35](http://www.tac.mta.ca/tac/volumes/31/35/31-35abs.html)]
Discussion of the [[Eilenberg-Moore category]] of action monads:
* {#Seal12} [[Gavin J. Seal]], Section 3 of: *Tensors, monads and actions*, Theory and Applications of Categories **28** 15 (2013) 403-434. [[arXiv:1205.0101](http://arxiv.org/abs/1205.0101), [tac:28-15](http://www.tac.mta.ca/tac/volumes/28/15/28-15abs.html)]
Brief discussion in the context of [[algebraic geometry]]:
* {#Brandenburg2014} [[Martin Brandenburg]], Exp. 6.3.4 in: _Tensor categorical foundations of algebraic geometry_ [[arXiv:1410.1716](https://arxiv.org/abs/1410.1716), [urn:nbn:de:hbz:6-22359532742](https://nbn-resolving.de/urn:nbn:de:hbz:6-22359532742)]
### In computer science
Discussion of the [[writer monad]] as an effect-[[monad in computer science]]:
* [[Tarmo Uustalu]], p.23 of: *Monads and Interaction Lecture 1* [[pdf](https://cs.ioc.ee/~tarmo/mgs21/mgs1.pdf), [[Uustalu-Monads1.pdf:file]]], lecture notes for [MGS 2021](https://staffwww.dcs.shef.ac.uk/people/G.Struth/mgs21.html) (2021):
* {#Milewski19} [[Bartosz Milewski]] (compiled by Igal Tabachnik), §4.1 and §21.2.4 in: *Category Theory for Programmers*, Blurb (2019) [[pdf](https://github.com/hmemcpy/milewski-ctfp-pdf/releases/download/v1.3.0/category-theory-for-programmers.pdf), [github](https://github.com/hmemcpy/milewski-ctfp-pdf), [webpage](https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/), [ISBN:9780464243878](https://www.blurb.com/b/9621951-category-theory-for-programmers-new-edition-hardco)]
[[!redirects action monads]]
[[!redirects monoid action monad]]
[[!redirects monoid action monads]]
[[!redirects group action monad]]
[[!redirects group action monads]]
[[!redirects writer monad]]
[[!redirects writer monads]]
|
action object > history | https://ncatlab.org/nlab/source/action+object+%3E+history |
see _[[module object]]_ |
action of a category on a set | https://ncatlab.org/nlab/source/action+of+a+category+on+a+set | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
Given a (small) [[category]] $C$ and given a [[set]] $S$ there are (at least) the following two equivalent ways to define an [[action]] of $C$ on $S$.
### Action as a functor
An **action of a category** $C$ on a [[set]] $S$ is nothing but a [[functor]] $\rho : C \to$ [[Set]].
The particular set $S$ that this functor defines an [[action]] on is the disjoint union of sets that the functor assigns to the objects of $C$:
$$
S = \bigsqcup_{c \in Obj(C)} \rho(c)
\,.
$$
Given an element $s \in S$ which sits in the subset $\rho(c) \subset S$ associated with the object $c$ of $C$, it is acted on by all [[morphism]]s $c \stackrel{f}{\to} d$ in $C$ whose [[source]] is $c$. By the definition of [[functor]] every such morphism defines a map of sets
$$
\rho(f) : (\rho(c) \subset S) \to (\rho(d) \subset S)
$$
and the the action of $f$ on $s \in \rho(c)$ under $\rho$ is
$$
\rho(f) : (s \in \rho(c)) \mapsto (\rho(f)(s) \in \rho(d))
\,.
$$
In the case that $C$ has just a single object $\bullet$ the category $C$ is just a [[monoid]] (might for instance be a [[group]]), there is just a single set $S = \rho(\bullet)$ and we recover the ordinary notion of a [[monoid]] or [[group]] acting on a set.
Indeed this generalizes the instance (the motivating example for the notion of action) where $\rho:G\rightarrow \mathbf{Aut}(S)$ is a group action on a set $S$, since the notion of coproduct is a generalization of the notion of automorphism group since naively a [[cardinal]] is an isomorphism class of sets and the notion of coproduct in turn generalizes that of cardinal (see [[cardinal|there]]).
### Action as an algebra for a monad
An equivalent perspective on the above situation is often useful. To motivate this, notice that the decomposition $S = \sqcup_{c \in Obj(c)} \rho(c)$ of the set $S$ into subsets corresponding to objects of the category $C$ can equivalently be encoded in a map of sets
$$
\lambda : S \to Obj(C)
$$
which sends each element of $S$ to the object of $c$ it corresponds to under the action.
(In the case that our category $C$ is a [[groupoid]] or even a [[Lie groupoid]] this map may be familiar as the _anchor map_ or _moment map_ of the action.)
But also the category $C$ itself comes with maps to $Obj(C)$: the [[source]] map $s$ and [[target]] map $t$, which are suggestively drawn as a [[span]] in [[Set]] by writing:
$$
\array{
&& Mor(C)
\\
& {}^{s}\swarrow
&& \searrow^{t}
\\
Obj(C)
&&&&
Obj(C)
}
\,.
$$
Recall from the above discussion that a morphism $f : c \to d$ in $C$ could act on an element $s \in S$ if the image of $s$ under the anchor map $\lambda$ coincides with the source of $f$, i.e. with the image of $f$ under the source map $s$. Formally this means that the pairs of elements of $S$ and morphisms of $C$ which can be paired by the action live in the [[pullback]] set $S {}_\lambda \times_s Mor(C)$ (the fiber product):
$$
\array{
&&
S {}_\lambda \times_s Mor(C)
\\
& {}^{pr_1}\swarrow && \searrow^{pr_2}
\\
S
&&
&& Mor(C)
\\
& \searrow^{\lambda}&
& {}^{s}\swarrow
&& \searrow^{t}
\\
&&
Obj(C)
&&&&
Obj(C)
}
\,.
$$
Above we have seen that the action of $C$ on $S$ sends every element in this fiber product, which is a pair
$$
(s \in \rho(c) \subset S, (c \stackrel{f}{\to} d) \in Mor(C))
$$
to an element $\rho(f)(s) \in \rho(d)$. So this is a map of sets $\rho : S {}_\lambda \times_s C \to S$. But a special such map, in that it satisfies a couple of conditions. One condition is that $s \in \rho(c)$ is taken to $\rho(d)$ by $f : c \to d$. This can be encoded by saying that $\rho$ extends to a morphism of [[span]]s from the pullback span above back to $S$:
$$
\array{
&& S {}_\lambda \times_s Mor(C)
\\
& \swarrow && \searrow^{t \circ pr_2}
\\
pt &&\downarrow^{\rho}&& Obj(C)
\\
& \nwarrow && \nearrow_{\lambda}
\\
&&
S
}
$$
But $\rho$ satisfies yet another compatibility condition: so far we have only used the source-target mathcing condition of the functor $\rho : C \to Set$. There is also its _functoriality_, i.e. its respect for composition.
But composition in the category $C$ is itself naturally expressed in terms of morphisms of spans:
the set of composable morphisms $Mor(C) {}_t \times_s Mor(C)$ is itself the tip of a [[span]] arising from composing the [[span]] of $C$ with itself by [[pullback]]:
$$
\array{
&&&& Mor(C) {}_t\times_s Mor(C)
\\
&&& \swarrow
&& \searrow
\\
&& Mor(C)
&&&&
Mor(C)
\\
&
{}^s\swarrow
&& \searrow^t
&&
{}^s\swarrow
&& \searrow^t
\\
Obj(C)
&&&&
Obj(C)
&&&&
Obj(C)
}
$$
and the composition operation $\circ$ in $C$ is a morphism from this composed span to the original span
$$
\array{
&& Mor(C) {}_t \times_s Mor(C)
\\
& {}^{s \circ pr_1}\swarrow && \searrow^{t \circ pr_2}
\\
Obj(C) &&\downarrow^{\circ}&& Obj(C)
\\
& {}^{s}\nwarrow && \nearrow_{t}
\\
&&
Mor(C)
}
\,.
$$
In total this gives us two different ways to map the total span with tip $S {}_\lambda \times_s Mor(C) {}_t \times_s Mor(C)$ obtained by composing the anchor map span with two copies of the span of $C$ back to the anchor map span
$$
\array{
&& S {}_\lambda \times_s Mor(C) {}_t \times_s Mor(C)
\\
& {}^{}\swarrow && \searrow^{t \circ pr_3}
\\
pr &&\downarrow && Obj(C)
\\
& {}^{s}\nwarrow && \nearrow_{\lambda}
\\
&&
S
}
\,.
$$
The **action property** of $\rho$, which is nothing but the functoriality of $\rho$ in the above description, says precisely that these two morphisms coincide.
Abstractly this says that
* a (small) [[category]] is a [[monad]] in [[span|spans]] in [[Set]];
* the [[action]] of a [[category]] on a [[set]] is an [[algebra]] for this [[monad]].
+--{.query}
Generalizing this slightly, it should be possible to associate an action of a category $C$ on a category $\coprod_{c\in C_0}\rho(c)$ to a functor $\rho:C\rightarrow \Cat$ with the expectation, that this then is just a module for $C$ as a monad.
=--
|
action of a monoidal category | https://ncatlab.org/nlab/source/action+of+a+monoidal+category | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
## Idea
{#Idea}
In [[vertical categorification]] of how [[monoids]]/[[monoid objects]] $A$ may [[action|act]] on other objects $N$ ([[action objects]], [[module objects]]) inside an ambient [[monoidal category]] by [[maps]]
$$
A \otimes N \longrightarrow N
\,.
$$
satisfying the [action property](action#eq:ActionProperty), so a [[monoidal category]] $\mathcal{A}$ may act on other *[[categories]]* $\mathcal{N}$ by [[functors]]
$$
\oslash
\;\colon\;
\mathcal{A} \times \mathcal{N}
\longrightarrow
\mathcal{N}
$$
subject to [[associators]], [[unitors]] and [[coherence conditions]] for [[action objects]] [[coherence|coherently]] [[internalization|internalized]] into the [[2-category]] [[Cat]], analogous to the laws in a [[monoidal category]].
At least if some linear structure is present and respected (such as when $\mathcal{A}$ qualifies as a [[2-ring]]) it is natural to speak of *[[module categories]]* over $\mathcal{A}$ (see also at *[[n-module|$n$-module]]*).
Similarly, compatible actions from both sides, such as for a [[bimodule]], give a notion of *[[bimodule category]].
Note that the term **actegory**, introduced by [McCrudden (2000)](#McCrudden00)[^1], is often used in the literature for this concept, and consequently **[[biactegory]]** for the two-sided case. However, since "actegory" is a single transposition away from "category", we prefer to use the explicit terminology on this page and elsewhere.
[^1]: From [Cockett & Pastro (2007)](#CockettPastro07): "The term *actegory* is used to describe the situation of a monoidal category “acting” on a category. They first appeared (under a different name) in the work of Bénabou as a simple example of a bicategory. B. Pareigis developed the theory of actegories (again under a different name) and showed there usefulness in the representation theory of monoids and comonoids. The word “actegory” was first suggested at the Australian Category Seminar and first appeared in print in the thesis of P. McCrudden where they were used to study categories of representations of coalgebroids."
## Definition
For any [[category]] $A$, the category of [[endofunctors]] $End(A)$ is [[monoidal category|monoidal]] with respect to the (horizontal) composition (the composition of functors and the [[Godement product]] for natural transformations).
Given a [[monoidal category]] $(C,\otimes,I,l,r,a)$ a (left or right) __$C$-module category__ is a category $A$ together with a (left or right) coherent action of $C$ on $A$. Depending on author and context, the left coherent action of $C$ on $A$ is a morphism of monoidal categories $C\to End(A)$ in the lax, colax, pseudo or strict sense (most often in pseudo-sense) or, in another terminology, a monoidal, comonoidal, strong monoidal or strict [[monoidal functor]]. Right coherent actions correspond to the monoidal functors into the category $End(A)$ with the opposite tensor product.
$C$-module categories, colax $C$-equivariant functors and natural transformations of colax $C$-equivariant functors form a [[strict 2-category]] $_C Act^c$. A [[monad]] in $_C Act^c$ amounts to a pair of a monad in $Cat$ and a [[distributive law]] between the monad and an action of $C$.
The notion of $C$-action (hence a $C$-module category) is easily extendable to [[bicategories]] (see Baković's thesis).
+-- {: .num_defn #ModuleCategory}
###### Definition
A **(left) $\mathcal{C}$-module category** is
1. a category $\mathcal{A}$;
1. a functor $\oslash : \mathcal{C} \times \mathcal{A} \to \mathcal{A}$ called the *action*;
1. a natural isomorphism $\lambda_a : a \to I \oslash a$ called the *unitor*;
1. a natural isomorphism $\alpha_{c,d,a} : c \oslash (d \oslash a) \to (c \otimes d) \oslash a$ called the *actor*;
satisfying a pentagonal and two triangular laws (see [KJ01](#KJ01), diagg. (1.1)-(1.3)) that witness the coherence of $\lambda$ and $\alpha$ with the unitors and associators of $\mathcal{C}$.
=--
## Connection with enrichment
If a category $D$ is [[enriched category|enriched]] in $C$ with [[copowers]], then the copower structure forms a module category on the ordinary category underlying $D$.
Conversely, if module category is such that the functor $(-)\oslash d:C\to D$ has a [[right adjoint]] for all objects $d$ of $D$, then the right adjoints $D(d,-):D\to C$ provide an enrichment of $D$ in $C$ for which the action is a copower. See [KJ01](#KJ01).
More generally, supposing $C$ is small, the following data are equivalent: (1) a $C$-module category $D$; (2) an enrichment of $D$ in the category $\hat C$ of [[presheaves]] on $C$ (i.e. a [[locally graded category]]), with copowers by [[representables]]. Here we regard $\hat C$ as a monoidal category with the [[Day convolution]].
Starting from a $C$-module category $D$, consider the enrichment given by $D(d,d')=hom(-\oslash d,d'):C^{op}\to Set$. If these presheaves are representable, this is what it means to be enriched in $C$ for which the action is a copower. If they are not representable, it is still an enrichment, and the copowers by representables are the action.
For this reason, many concepts from enriched category theory make sense for module categories too.
## Related Concepts
* [[action object]], [[module object]]
* [[2-module]], [[n-module]]
* [[monoidal actegory]]
## References
* [[Bodo Pareigis]], _Non-additive ring and module theory I. General theory of monoids_, Publ. Math. Debrecen 24 (1977), 189--204. MR 56:8656; _Non-additive ring and module theory II. C-categories, C-functors, and C-morphisms_, Publ. Math. Debrecen 24 (351--361) 1977.
* {#KJ01} [[Max Kelly]], [[George Janelidze]], _A note on actions of a monoidal category_, Theory and Applications of Categories, Vol. 9, 2001, No. 4, pp 61--91 [link](http://www.tac.mta.ca/tac/volumes/9/n4/9-04abs.html)
* {#McCrudden00} P. McCrudden, *Categories of representations of coalgebroids*, Advances in Mathematics __154__ 2 (2000) 299--332 [[doi:10.1006/aima.2000.1926](https://doi.org/10.1006/aima.2000.1926)]
* P. Schauenburg, _Actions of monoidal categories and generalized Hopf smash products_, J. Algebra __270__ (2003) 521--563 (remark: actegories with action in the strong monoidal sense)
* [[Zoran Škoda]], _Distributive laws for actions of monoidal categories_, [arXiv:0406310](https://arxiv.org/abs/math/0406310), _Equivariant monads and equivariant lifts versus a 2-category of distributive laws_, [arXiv:0707.1609](https://arxiv.org/abs/0707.1609)
* [[J. R. B. Cockett]], Craig Pastro, _The logic of message-passing_ [arXiv:math/0703713](https://arxiv.org/abs/math/0703713).
A recent survey of many basic definitions and operations on actegories is
* [[Matteo Capucci]], Bruno Gavranović, _Actegories for the working amthematician_, [arXiv:2203.16351](https://arxiv.org/abs/2203.16351)
A generalisation from [[monoidal categories]] to [[bicategories]] is studied in, where actions are called **representations**:
* [[Robert Gordon]], and [[John Power]]. _Enrichment through variation_, Journal of Pure and Applied Algebra 120.2 (1997): 167-185.
[[!redirects actegory]]
[[!redirects actegories]]
|
acyclic assembly lemma | https://ncatlab.org/nlab/source/acyclic+assembly+lemma | ##Idea
The **Acyclic Assembly Lemma** gives conditions for the [[total complex]] of a [[double complex]] of [[modules]] over a ring to be acyclic.
A full statement and proof can be found in [[An Introduction to Homological Algebra]] by [[Charles Weibel|Chuck Weibel]] on pages 59 and 60 (Lemma 2.7.3).
##Reference:
* [[Charles Weibel]], _[[An introduction to homological algebra]]_, Cambridge Studies in Adv. Math. 38, CUP 1994 |
acyclic fibration | https://ncatlab.org/nlab/source/acyclic+fibration |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
=--
=--
\tableofcontents
## Definition
In a [[model category]], an **acyclic fibration** or **trivial fibration** is a [[morphism]] which is both a [[fibration]] and a [[weak equivalence]].
Dually, an **acyclic cofibration** or **trivial cofibration** is a morphism which is both a [[cofibration]] and a weak equivalence.
## Examples
* [[acyclic Kan fibration]]
[[!redirects acyclic fibration]]
[[!redirects acyclic fibrations]]
[[!redirects acyclic cofibration]]
[[!redirects acyclic cofibrations]]
[[!redirects trivial fibration]]
[[!redirects trivial fibrations]]
[[!redirects trivial cofibration]]
[[!redirects trivial cofibrations]]
|
acyclic graph | https://ncatlab.org/nlab/source/acyclic+graph |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Graph theory
+-- {: .hide}
[[!include graph theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Recall that an [[undirected graph]] $G$ consists of a set $V$ of _vertices_ and a set $E$ of unordered pairs of vertices. A **path** in $G$ is a finite sequence of vertices $x_0, \ldots, x_n$ such that each pair $x_i x_{i+1}$ belongs to $E$. If $x_n = x_0$, then the path is called a **cycle**. A graph is **connected** if it is nonempty and if for every distinct pair of vertices $x, y$, there is a path for which $x_0 = x$ and $x_n = y$. A graph is **acyclic** if the only cycles are paths
$$x_0, \ldots, x_{n-1}, x_n, x_{n-1}, \ldots, x_0$$
where steps are retraced; an [[acyclic graph]] is also called a _[[forest]]_. A *[[tree]]* is a connected forest.
Each forest is a [[disjoint union|disjoint sum]] (a [[coproduct]] in the category of graphs) of trees. Removal of an edge of a tree results in a forest.
## References
See also:
* Wikipedia, *[Polytree](https://en.wikipedia.org/wiki/Polytree)*
[[!redirects acyclic graphs]]
[[!redirects forest]]
[[!redirects forests]]
[[!redirects rooted forest]]
[[!redirects rooted forests]]
|
acyclic group | https://ncatlab.org/nlab/source/acyclic+group |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A [[discrete group]] $\Gamma$ is called an _acyclic group_ if its [[classifying space]] $B \Gamma$ is an "acyclic space" in that its [[ordinary cohomology]] $H^\bullet(B \Gamma, \mathbb{Z})$ vanishes in all positive degree, so that
$$
H^\bullet(B \Gamma, \mathbb{Z}) \simeq \mathbb{Z}
\,.
$$
Hence to cohomology these spaces look like [[contractible topological spaces]].
The existence of acyclic groups is one reason why [[rational homotopy theory]] restricts attention to [[simply-connected topological spaces]].
## References
The first two non-trivial examples of an acyclic group were apparently given in
* G. Baumslag, K.W. Gruenberg, _Some reflections on cohomological dimension and freeness, J. Algebra 6 (1967), 394–409.
* D. B. A. Epstein, _A group with zero homology, Proc. Camb. Phil. Soc. 64
(1968), 599–601
See also
* A. J. Berrick, _The acyclic group dichotomy_ ([arXiv:1006.4009](https://arxiv.org/abs/1006.4009))
* Wikipedia, _[Acyclic group](https://en.wikipedia.org/wiki/Acyclic_space#Acyclic_groups)_
[[!redirects acyclic groups]] |
acyclic Kan fibration | https://ncatlab.org/nlab/source/acyclic+Kan+fibration |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
A [[Kan fibration]] that at the same time is a [[weak homotopy equivalence]] is called an **acyclic Kan fibration**. (Also: a _trivial Kan fibration_.)
## Properties
Acyclic Kan fibrations are the [[acyclic fibrations]] in the [[classical model structure on simplicial sets]] $sSet_{Qu}$, hence those morphisms which have the [[right lifting property]] against [[monomorphisms]] (degreewise [[injections]]) of [[simplicial sets]].
In particular, this implies that:
\begin{proposition}\label{AcyclicKanFibrationsAreSurjective}
Acyclic Kan fibrations are (degreewise) [[surjective]], in that they have the [[right lifting property]] against any [[empty bundle]] $\varnothing \xhookrightarrow{\;} S$.
\end{proposition}
\begin{remark}
Prop. \ref{AcyclicKanFibrationsAreSurjective} is in contrast to plain [[Kan fibrations]], see [this remark](horn#ZeroSimplexHasNoHorns) about the empty horn, and see the discussion on surjective Kan fibrations [there](Kan+fibration#SurjectiveKanFibrations).
\end{remark}
In fact:
\begin{prop}
Acyclic Kan fibrations are precisely the morphisms of [[simplicial sets]] that have the [[right lifting property]] against all [[boundary of a simplex|simplex boundary inclusions]].
\end{prop}
See [this Prop.](classical+model+structure+on+simplicial+sets#AcyclicKanFibrationsAsRLPAgainstBoundaryInclusions) at *[[classical model structure on simplicial sets]]*.
This is part of the statement that $sSet_{Qu}$ is a [[cofibrantly generated model category|cofibrantly generated]] (see [this Prop.](classical+model+structure+on+simplicial+sets#AsACofibrantlyGeneratedModelCategory)).
## Related entries
* [[simplicial homotopy theory]]
[[!redirects acyclic Kan fibrations]]
[[!redirects trivial Kan fibration]]
[[!redirects trivial Kan fibrations]]
|
acyclic object | https://ncatlab.org/nlab/source/acyclic+object |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
In [[homological algebra]]:
For $F : \mathcal{A} \to \mathcal{B}$ a [[left exact functor|left exact]] [[additive functor]] between [[abelian categories]], an [[object]] $A \in \mathcal{A}$ is $F$-**acyclic** if the right [[derived functor]] of $F$ has no cohomology on $A$ in positive degree
$$
(p \gt 0) \Rightarrow R^p F A = 0
\,.
$$
## Properties
A [[resolution]] by $F$-acyclic objects serves to compute the [[derived functor]] of $F$. See at _[derived functor in homological algebra -- Via acyclic resolutions](derived+functor+in+homological+algebra#ViaAcyclicResolutions)_
## References
See most references on [[homological algebra]].
Also:
* [[Michael Barr]], *Acyclic models*, Canadian J. Math. **48** 2 (1996) 258–273 [[doi:10.4153/CJM-1996-013-x](https://doi.org/10.4153/CJM-1996-013-x), [pdf](https://www.math.mcgill.ca/barr/papers/acmod.pdf), [[Barr-AcyclicModels1996.pdf:file]]]
* [[Michael Barr]], *Acyclic models*, CRM Monograph Series **17** (2002) [[ams:crmm-17](https://bookstore.ams.org/crmm-17), [pdf](https://www.math.mcgill.ca/barr/papers/acycmod.pdf), [[Barr-AcyclicModels2002.pdf:file]]]
[[!redirects acyclic objects]]
[[!redirects acyclic model]]
[[!redirects acyclic models]]
|
acyclic type | https://ncatlab.org/nlab/source/acyclic+type |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--
\tableofcontents
## Definition
In [[univalent type theory]] with [[pushout types]], a type $A$ is an **acyclic type** if its [[suspension type]] $\Sigma A$ is a [[contractible type]]. Equivalently, a type $A$ is acyclic if its reduced integral homology vanishes.
## Related concepts
* [[suspension type]]
* [[epimorphism]]
* [[acyclic object]]
* [[acyclic group]]
## References
* [[Tom de Jong]], *Acyclic types and epimorphisms in HoTT*, [[Homotopy Type Theory Electronic Seminar Talks]], 17 November 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Jong-2022-11-17-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=vohAYPAAWxs))
* [[Ulrik Buchholtz]], [[Tom de Jong]], [[Egbert Rijke]], *Epimorphisms and Acyclic Types in Univalent Mathematics*, ([arXiv:2401.14106](https://arxiv.org/abs/2401.14106))
[[!redirects acyclic type]]
[[!redirects acyclic types]] |
Adam | https://ncatlab.org/nlab/source/Adam |
Contributor _Adam_ is somebody who once was a grad student at UC Berkeley. Not a member of the Logic group. That's all we know.
Possibly [Adam Megacz](http://www.megacz.com).
category: people
|
Adam Brandenburger | https://ncatlab.org/nlab/source/Adam+Brandenburger |
* [personal page](https://www.adambrandenburger.com/)
* [institute page](https://www.stern.nyu.edu/faculty/bio/adam-brandenburger)
## Selected writings
[[sheaf theory|Sheaf theoretic]] discussion of [[quantum contextuality]] ([[hidden variable theories]], [[Kochen-Specker theorem]]):
* {#AbramskyBrandenburger11} [[Samson Abramsky]], [[Adam Brandenburger]], _The Sheaf-Theoretic Structure Of Non-Locality and Contextuality_, New Journal of Physics **13** (2011) 113036 [[arXiv:1102.0264](https://arxiv.org/abs/1102.0264), [doi:10.1088/1367-2630/13/11/113036](https://doi.org/10.1088/1367-2630/13/11/113036)]
category: people
|
Adam Chlipala | https://ncatlab.org/nlab/source/Adam+Chlipala |
* [webpage](http://adam.chlipala.net/)
## Selected writings
On [[software verification]] via [[dependent type theory]] with [[Coq]]:
* [[Adam Chlipala]], *Implementing Certified Programming Language Tools in Dependent Type Theory*, Berkeley (2007) [[UCB/EECS-2007-113](https://www2.eecs.berkeley.edu/Pubs/TechRpts/2007/EECS-2007-113.html), [pdf](https://www2.eecs.berkeley.edu/Pubs/TechRpts/2007/EECS-2007-113.pdf), [web](http://adam.chlipala.net/papers/ChlipalaPhD/)]
* [[Adam Chlipala]], _Certified programming with dependent types_, MIT Press 2013 ([ISBN:9780262026659 ](https://mitpress.mit.edu/books/certified-programming-dependent-types), [pdf](http://adam.chlipala.net/cpdt/cpdt.pdf), [book webpage](http://adam.chlipala.net/cpdt/))
On [[internal categories in homotopy type theory]]:
* [[Jason Gross]], [[Adam Chlipala]], [[David Spivak]], *Experience Implementing a Performant Category-Theory Library in Coq*, In: *Interactive Theorem Proving. ITP 2014* Lecture Notes in Computer Science, **8558** Springer (2014) $[$[arXiv:1401.7694](http://arxiv.org/abs/1401.7694), [doi:10.1007/978-3-319-08970-6_18](https://doi.org/10.1007/978-3-319-08970-6_18)$]$
On [[cryptography]]:
* Andres Erbsen, Jade Philipoom, [[Jason Gross]], Robert Sloan, [[Adam Chlipala]], *Simple High-Level Code for Cryptographic Arithmetic - With Proofs, Without Compromises*, [2019 IEEE Symposium on Security and Privacy (SP)](https://ieeexplore.ieee.org/xpl/conhome/8826229/proceeding) $[$[doi:10.1109/SP.2019.00005](https://doi.org/10.1109/SP.2019.00005)$]$
category: people |
Adam Falkowski | https://ncatlab.org/nlab/source/Adam+Falkowski |
* [webpage](http://www.fuw.edu.pl/~afalkows/Work/work.html)
* particle physics blog: _[Resonaances](http://resonaances.blogspot.fr/)_
## Related $n$Lab pages
* [[Horava-Witten theory]]
* [[flavour anomaly]]
* [[hydrogen line]]
category: people |
Adam G. Riess | https://ncatlab.org/nlab/source/Adam+G.+Riess |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Adam_Riess)
* [Institute page](https://physics-astronomy.jhu.edu/directory/adam-riess/)
## Selected writings
On observation of [[supernovae]] interpreted as indicating a [[positive number|positive]] [[cosmological constant]] ([[dark energy]], the origin of the modern [[standard model of cosmology]]):
* {#Riess1998} [[Adam G. Riess]] et al., *Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant*, The Astronomical Journal, **116** 3 (1998) [[doi:10.1086/300499](https://iopscience.iop.org/article/10.1086/300499)]
* {#RiessEtAl16} [[Adam G. Riess]] et al., _A 2.4% Determination of the Local Value of the Hubble Constant_, The Astrophysical Journal **826** 1 (2016) [[arXiv:1604.01424](https://arxiv.org/abs/1604.01424), [doi:10.3847/0004-637X/826/1/56](https://iopscience.iop.org/article/10.3847/0004-637X/826/1/56)]
category: people
[[!redirects Adam Riess]] |
Adam Keenan | https://ncatlab.org/nlab/source/Adam+Keenan |
* [webpage](http://www.math.utah.edu/~keenan/)
## Selected writings
On [[differential topology]]:
* [[Mladen Bestvina]] (notes by [[Adam Keenan]]), _Differentiable Topology and Geometry_, 2002 ([pdf](http://www.math.utah.edu/~keenan/manifoldsnotes.pdf), [[BestvinaKeenanDifferentialTopology.pdf:file]])
On the [[ADE classification]] of [[finite subgroups of SU(2)]] and [[free action|free]] [[finite group]] [[actions]] on [[n-spheres]]:
* {#Keenan03} [[Adam Keenan]], _Which finite groups act freely on spheres?_, 2003 ([pdf](http://www.math.utah.edu/~keenan/actions.pdf))
category: people
|
Adam Krupicka | https://ncatlab.org/nlab/source/Adam+Krupicka | |
Adam Lewandowski | https://ncatlab.org/nlab/source/Adam+Lewandowski |
## Selected writings
On [[glueballs]] via [[AdS-QCD]]:
* [[Emanuel Katz]], [[Adam Lewandowski]], [[Matthew Schwartz]], _Tensors Mesons in AdS/QCD_, Phys. Rev. D74:086004, 2006 ([arXiv:hep-ph/0510388](https://arxiv.org/abs/hep-ph/0510388))
category: people |
Adam Marsh | https://ncatlab.org/nlab/source/Adam+Marsh |
* [webpage](https://cal.berkeley.edu/adamjmarsh)
## Selected writings
On [[differential geometry]] and [[topology]]/[[homotopy theory]] in [[mathematical physics]] (such as [[fiber bundles in physics]]):
* [[Adam Marsh]], _Mathematics for Physics: An Illustrated Handbook_, World Scientific 2018 ([doi:10.1142/10816](https://doi.org/10.1142/10816), [book webpage](https://www.mathphysicsbook.com/))
chapter 10: _Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results_ ([arXiv:1607.03089](https://arxiv.org/abs/1607.03089))
category: people
|
Adam Petcher | https://ncatlab.org/nlab/source/Adam+Petcher |
* [personal page](https://adam.petcher.net/)
## Selected writings
On [[verified software]] in [[cryptography]]:
* [[Adam Petcher]], *A Foundational Proof Framework for Cryptography*, 2014 ([pdf](https://dash.harvard.edu/bitstream/handle/1/17463136/PETCHER-DISSERTATION-2015.pdf), [harvard:17463136](https://dash.harvard.edu/handle/1/17463136))
* [[Adam Petcher]], Greg Morrisett, *The Foundational Cryptography Framework*, In: R. Focardi, A. Myers (eds.) *Principles of Security and Trust* POST 2015. Lecture Notes in Computer Science, vol 9036. Springer, Berlin, Heidelberg ([doi:10.1007/978-3-662-46666-7_4](https://doi.org/10.1007/978-3-662-46666-7_4))
category: people
|
Adams category | https://ncatlab.org/nlab/source/Adams+category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Adams category_, named after ([Adams 74](#Adams74)), was historically one of the first constructions of the [[stable homotopy category]], following ideas in ([Boardman 65](#Boardman65)), see ([Lewis-May-Steinberger 86, pages 1-2](#LewisMaySteinberger86)) for recollection of the historical development and critical comments on the definition.
The Adams category is defined to be the category of [[CW-spectra]] together with [[left homotopy|left]] [[homotopy classes]] (via [[cylinder spectra]]) of "eventually defined" functions between them.
This was originally advertised as being a definition not involving tools from [[category theory]]. Arguably this is also its main deficiency when it comes to working with it (this is the "polemic" of [Lewis-May-Steinberger 86, preamble](#LewisMaySteinberger86)). For modern alternatives see at _[[stable homotopy category]]_.
## References
The definition is due to
* {#Adams74} [[Frank Adams]], Part III, section 2 of _[[Stable homotopy and generalised homology]]_, 1974
following
* {#Boardman65} [[Michael Boardman]], _Stable homotopy theory_, mimeographed notes, University of Warwick, 1965 onward
The term "Adams category" for this starts to be used for instance in
* [[Harold Hastings]], _On function spectra_, Proceedings of the AMS, volume 44, Number 1, May 1974 ([pdf](http://www.ams.org/journals/proc/1974-044-01/S0002-9939-1974-0343258-4/S0002-9939-1974-0343258-4.pdf))
An account following ([Adams 74](#Adams74)) is also in
* {#Switzer75} [[Robert Switzer]], section 8 of _Algebraic Topology - Homotopy and Homology_, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.
Comments on the historical development are in
* {#LewisMaySteinberger86} [[L. Gaunce Lewis]], [[Peter May]], M. Steinberger, preamble of _Equivariant stable homotopy theory_, Springer Lecture Notes in Mathematics, 1986 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/equi.pdf))
in the spirit of
> There is much to love in [[Stable homotopy and generalised homology|Adams' book]], but not in the foundational part on CW spectra. ([[Peter May]], [MO comment](http://math.stackexchange.com/a/53783/58526))
More recent textbook accounts include
* {#Kochman96} [[Stanley Kochman]], section 3.3 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996
Lecture notes include
* [[Cary Malkiewich]], section 2.1 of _The stable homotopy category_, 2014 ([pdf](http://math.stanford.edu/~carym/stable.pdf))
[[!redirects Adams' category]]
|
Adams conjecture | https://ncatlab.org/nlab/source/Adams+conjecture |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Adams conjecture_ is a statement about triviality of [[spherical fibrations]] associated to certain formal differences of [[vector bundles]] ([[K-theory]] classes) via the [[J-homomorphism]]. The conjecture was stated in ([Adams 63, conjecture 1.2](#Adams63)), for vector bundles of rank up to two over a [[finite CW-complex]], which was proven in ([Adams 63, theorem 1.4](#Adams63)). A general proof was then given in ([Quillen 71](Quillen71)).
The Adams conjecture/Adams-Quillen theorem serves a central role in the identification of the [[image of the J-homomorphism]].
## Statements
Let $X$ be (the [[homotopy type]] of) a [[topological space]]. For $V \;\colon\; X \longrightarrow B O$ classifying a real [[vector bundle]] on $X$, the corresponding [[spherical fibration]] is classified by the composite
$$
J(V)
\;\colon\;
X \stackrel{V}{\longrightarrow} B O \stackrel{J}{\longrightarrow} B GL_1(\mathbb{S})
$$
with the delooped [[J-homomorphism]].
This descends to a map from [[topological K-theory]] to [[spherical fibrations]].
Now for $L$ a [[line bundle]] on some $X$ and for non-vanishing $k \in \mathbb{Z}$, [[John Adams]] observed that the [[spherical fibration]] associated with the difference $L^{\otimes k} - L \in K O(X)$ has the property that some $k$-fold multiple of it has trivial spherical fibration, hence that there is $N \in \mathbb{N}$ for which
$$
J\left(
\oplus^{k^N} (L^{\otimes k} - L)
\right)
= 0
\,.
$$
Noticing that $L \mapsto L^{\otimes^k} = \Psi^k(L)$ is the $k$th [[Adams operation]] on [[K-theory]] applied to the [[line bundle]] $L$, [[John Adams]] then conjectured that the above is true for all [[vector bundles]] $V$ in the form
$$
J\left(
\oplus^{k^N} (\Psi^k(V) - V)
\right)
= 0
\,.
$$
## Related entries
* [[J-homomorphism]]
* [[Schur index]]
## References
### General
The conjecture originates in:
* {#Adams63} [[John Adams]], _On the groups $J(X)$ I: Topology_, 2 (1963) pp. 181–195
Textbook accounts:
* [[Dai Tamaki]], [[Akira Kono]], Section 4.7 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))
Review:
* [[Akhil Mathew]], _The Adams conjecture I_ ([web](http://amathew.wordpress.com/2013/01/23/the-adams-conjecture-i/))
* [[Michael Hopkins]] (notes by [[Akhil Mathew]]), Lecture 30 in: _Spectra and stable homotopy theory_, 2012 ([pdf](http://math.uchicago.edu/~amathew/256y.pdf), [[HopkinsMathewStableHomotopyTheory.pdf:file]])
The [[proof]] of the Adams conjecture is originally due to
* {#Quillen71} [[Daniel Quillen]], _The Adams conjecture_, Topology, vol 10, 1971 ([pdf](http://math1.unice.fr/~cazanave/Gdt/ImJ/Quillen.pdf))
The proof using [[algebraic geometry]] is due to
* [[Dennis Sullivan]], _Genetics of homotopy theory and the Adams conjecture_, The Annals of Mathematics, Second Series, Vol. 100, No. 1 (Jul., 1974), pp. 1-79 ([JSTOR](http://www.jstor.org/stable/1970841), [pdf](http://math1.unice.fr/~cazanave/Gdt/ImJ/Sullivan.pdf))
Yet another proof via [[Becker-Gottlieb transfer]] is due to
* J. Becker, D. Gottlieb, _The transfer map and fiber bundles_ Topology , 14 (1975) {#BeckerGottlieb75}
### In equivariant cohomology
{#ReferencesInEquivariantCohomology}
The generalization to [[equivariant cohomology]] ([[equivariant K-theory]]) is discussed in
* [[Tammo tom Dieck]], theorem 11.3.8 in _[[Transformation Groups and Representation Theory]]_ Lecture Notes in Mathematics 766 Springer 1979
* Z. Fiedorowicz, H. Hauschild, [[Peter May]], theorem 0.4 of _Equivariant algebraic K-theory_, _Equivariant algebraic K-theory_, Algebraic K-Theory. Springer, Berlin, Heidelberg, 1982. 23-80 ([pdf](http://math.uchicago.edu/~may/PAPERS/40.pdf))
* Henning Hauschild, [[Stefan Waner]], theorem 0.1 of _The equivariant Dold theorem mod $k$ and the Adams conjecture_, Illinois J. Math. Volume 27, Issue 1 (1983), 52-66. ([euclid:1256065410](https://projecteuclid.org/euclid.ijm/1256065410))
* Kuzuhisa Shimakawa, _Note on the equivariant $K$-theory spectrum_, Publ. RIMS, Kyoto Univ. **29** (1993), 449-453 ([pdf](http://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=29&iss=3&rank=5), [doi](https://doi.org/10.2977/prims/1195167052))
* Christopher French, theorem 2.4 in _The equivariant $J$–homomorphism for finite groups at certain primes_, Algebr. Geom. Topol. Volume 9, Number 4 (2009), 1885-1949 ([euclid:1513797069](https://projecteuclid.org/euclid.agt/1513797069))
[[!redirects Adams conjectures]]
[[!redirects Adams' conjecture]]
[[!redirects Adams' conjectures]]
[[!redirects equivariant Adams conjecture]]
[[!redirects equivariant Adams conjectures]]
|
Adams e-invariant | https://ncatlab.org/nlab/source/Adams+e-invariant |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _$e$-invariant_ ([Adams 66, Sections 3,7](#Adams66), short for "extension invariant", see Def. \ref{EInvariantAsExtensionClassInECohomology} below) is the second in a sequence of homotopy invariants of "stable maps", i.e. of [[morphisms]] in the [[stable homotopy category]] (in particular: of [[stable homotopy groups of spheres]]), being elements of [[Ext-groups]] between the [[homology groups]]/[[cohomology groups]] of the [[domain]] and [[codomain]] of the map, with respect to some suitable choice of [[Whitehead-generalized cohomology theory]] $E$.
The previous invariant in the sequence is the _[[d-invariant]]_, the next is the _[[f-invariant]]_. These are the elements that appear in the first lines on the second page of the $E$-[[Adams spectral sequence]] for $[X,Y]_\bullet$.
## Definition
Let $X \overset{f}{\longrightarrow} Y$ be morphism in the [[stable homotopy category]] out of a [[finite spectrum]] $X$ (for instance the image under [[suspension]] $\Sigma^\infty$ of a morphism in the [[classical homotopy category]] of [[pointed homotopy types]] out of a [[finite CW-complex]]).
Let $E$ be a [[multiplicative cohomology theory]], such that the [[d-invariant]] of $f$ in $E$ vanishes, hence such that pullback $f^\ast \;\colon\; E^\bullet(Y) \to E^\bullet(X)$ in $E$-cohomology is the [[zero morphism]].
The archetypical example is $f \;\colon\; S^{2n-1} \to S^{2n}$ a map out of an [[odd integer|odd]]-[[dimension|dimensional]] [[sphere]] and $E = KU$ [[complex topological K-theory]].
\linebreak
### As an extension of generalized Adams-operation modules
{#AsAnExtensionOfAdamsOperationModules}
+-- {: .num_defn #EInvariantAsExtensionClassInECohomology}
###### Definition
**(e-invariant as extension class in E-cohomology)**
Writing $C_f$ for the [[homotopy cofiber]] of $f$
$$
\cdots
\to
X
\overset{f}{\longrightarrow}
Y
\overset{}{\longrightarrow}
C_f
\overset{}{\longrightarrow}
\Sigma X
\to
\cdots
\,,
$$
this implies that the [[long exact sequence in cohomology]] corresponding to the pair $(Y, C_f)$ truncates to a [[short exact sequence]] of the form
\[
\label{TheShortExactSequenceInETheory}
0
\to
E^\bullet(\Sigma X)
\overset{}{\longrightarrow}
E^\bullet(C_f)
\overset{}{\longrightarrow}
E^\bullet(Y)
\to
0
\,.
\]
This is hence an [[algebra extension|extension]] of $E^\bullet(Y)$ by $E^\bullet(\Sigma X)$ in any [[category]] in which $f^\ast$ is a [[homomorphism]], for instance that of modules over the [[E-Steenrod algebra]]. For the case of $E = KU$ take the category of [[graded abelian groups]] equipped with [[Adams operations]].
Thus this short exact sequence defines an element in the [[Ext group]] formed in this category
\[
\label{GeneraleInvariantAsElementOfExtGroup}
e(f)
\;\in\;
Ext^1\big(
E^\bullet(Y),
\,
E^\bullet(\Sigma X)
\big)
\]
and this is the _e-invariant_ of $f$ seen in $E$-theory.
=--
([Adams 66, Section 3, p. 27](#Adams66), review includes [BL 09, Sec. 2](#BL09)).
\linebreak
## For $E = KU$
### As an extension of K-groups with Adams operations
{#AsAnExtensionOfComplexTopologicalKTheoryWithAdamsOperations}
We discuss this in more detail for the case of [[complex topological K-theory]] $E = KU$, and for $f \;\colon\; S^{2(n + \bullet)-1} \to S^{2n}$ a map between [[spheres]]; and we show how the resulting extension is characterized by a single [[rational number]] [[modulo]] the [[integers]], this being the e-invariant in the form of a [[Q/Z]]-valued character
$$
e_{\mathbb{C}}
\;\colon\;
\pi^s_{\bullet}
\overset{\;\;\;\;\;}{\longrightarrow}
\mathbb{Q}/\mathbb{Z}
$$
on [[stable homotopy groups of spheres]] (Def. \ref{eInvariantInComplexTopologicalKtheoryAsRationalNumberModuloIntegers} below).
We follow [Hopkins-Mathew 12, Lecture 11](#HopkinsMathew12).
\linebreak
#### Abelian groups with Adams operations
+-- {: .num_defn #AbelianGroupWithAdamsOperations}
###### Definition
**(abelian group with Adams operations)**
We say that an _abelian group with Adams operation_ $\big(A,\{\psi_A^k\}_{k \in \mathbb{N}}\big)$ is an [[abelian group]] $A$ equipped with an [[action]] of the [[multiplicative group|multiplicative]] [[monoid]] $(\mathbb{N}, \cdot)$ of [[natural numbers]], hence equipped with [[group homomorphisms]]
$$
\psi_A^k \;\colon\; A \to A
\,,
\;\;\;\;\;\;\;
\text{for all}
\;
k \in \mathbb{N}
$$
such that
\[
\label{ActionPropertyOfAbstractAdamsOperations}
\psi_A^{k_1}
\circ
\psi_A^{k_2}
\;=\;
\psi_A^{ k_1 \cdot k_2 }
\,,
\;\;\;\;\;\;\;\;\;
\text{for all}
\;
k_1, k_2 \,\in\, \mathbb{N}
\,.
\]
Moreover, for $\big(A,\{\psi_A^k\}_{k \in \mathbb{N}}\big), \, \big(A',\{\psi_{A'}^k\}_{k \in \mathbb{N}}\big)$ two abelian groups with Adams operations, a [[homomorphism]] between them is a [[group homomorphism]] (a [[linear map]]) $\phi \;\colon\; A \to A'$ that respects these operations, hence such that the following [[commuting square|squares commute]]:
\[
\label{HomomorphismOfAbelianGroupsWithAdamsOperations}
\array{
A &\overset{\;\;\;\phi\;\;\;}{\longrightarrow}& A'
\\
{}^{\mathllap{ \psi^k_A }}
\big\downarrow
&&
\big\downarrow
{}^{\mathrlap{\psi_{A'}^k}}
\\
A &\overset{\;\;\;\phi\;\;\;}{\longrightarrow}& A'
\,,
}
\;\;\;\;\;\;\;\;
\text{for all}
\;
k \in \mathbb{N}
\,.
\]
This makes an [[abelian category]] which we denoted $Ab_{Adams}$, canonically equipped with a [[forgetful functor]] to [[Ab]]:
\[
\label{ForgetfulFunctorFromAbelianGroupsWithAdamsOperations}
\array{
Ab_{Adams}
&\overset{\;\;\;\;\;\;}{\longrightarrow}&
Ab
\\
\big(
A,
\,
\{\psi^k_A\}_{k \in \mathbb{N}}
\big)
&\mapsto&
A
\,.
}
\]
=--
+-- {: .num_example #IntegersEquippedWithHomogeneousAdamsOperations}
###### Example
For $n \in \mathbb{N}$, the additive abelian group of [[integers]] $\mathbb{Z}$ becomes an _abelian group with Adams operations_
$$
\mathbb{Z}(n)
\;\coloneqq\;
\big(
\mathbb{Z},\,
\{\psi^k_{\mathbb{Z}(n)}\}
\big)
\,,
$$
in the sense of Def. \ref{AbelianGroupWithAdamsOperations}, by setting
$$
\array{
\mathbb{Z}
&
\overset{\;\;\; \psi^k_{\mathbb{Z}(n)}\;\;\;}{\longrightarrow}
&
\mathbb{Z}
\\
r &\mapsto& k^n \cdot r
\,.
}
$$
=--
+-- {: .num_example #AdamsOperationsOnComplexTopologicalKTheoryGroups}
###### Example
**([[Adams operations]] on [[complex topological K-theory]] [[cohomology groups|groups]])**
For $X$ a [[compact topological space|compact]] [[pointed topological space]], the [[complex topological K-theory]] [[cohomology group|group]] $K(X)$ becomes an abelian group with Adams operations in the sense of Def. \ref{AbelianGroupWithAdamsOperations},
via the actual [[Adams operations]],
$$
K(X),
\widetilde K(X)
\;\in\;
Ab_{Adams}
$$
and hence so does the [[reduced K-theory]] $\widetilde K(X)$:
$$
\array{
\widetilde K(X)
&\overset{\;\;ker(i^\ast)\;\;}{\longrightarrow}&
K(X)
&\overset{\;i^\ast\;}{\longrightarrow}&
K(\ast)
\\
\big\downarrow
{}^{\mathrlap{\psi^k_{\widetilde K(X)}}}
&&
\big\downarrow
{}^{\mathrlap{\psi^k_{K(X)}}}
&&
\big\downarrow
{}^{\mathrlap{\psi^k_{K(\ast)}}}
\\
\widetilde K(X)
&\underset{\;\;ker(i^\ast)\;\;}{\longrightarrow}&
K(X)
&\underset{\;i^\ast\;}{\longrightarrow}&
K(\ast)
\,,
}
$$
Moreover, for each (pointed) [[continuous function]] $X \overset{f}{\longrightarrow} Y$ the corresponding [[pullback in cohomology]] respects the Adams operations and hence yields a [[homomorphism]] (eq:HomomorphismOfAbelianGroupsWithAdamsOperations):
\[
\label{PullbackInKCohomologyRespectsAdamsOperation}
\widetilde K(X)
\overset{\;\;f^\ast\;\;}{\longrightarrow}
\widetilde K(Y)
\,,
\;\;\;\;
\in
\;
Ab_{Adams}
\,.
\]
=--
+-- {: .num_prop #AdamsOperationOnComplexTopologicalKtheoryOfnSpheres}
###### Proposition
**([[Adams operations]] on [[complex topological K-theory]] of [[n-spheres]])**
For $n \in \mathbb{N}$, the [[Adams operations]] on the [[reduced K-theory]] (Example \ref{AdamsOperationsOnComplexTopologicalKTheoryGroups}) of the [[n-sphere|2n-sphere]] are given by:
$$
\array{
\widetilde K
\big(
S^{2n}
\big)
&
\overset{ \;\;\; \psi^k\;\;\; }{\longrightarrow}
&
\widetilde K
\big(
S^{2n}
\big)
\\
V &\mapsto& k^n \cdot V
}
$$
and hence are [[isomorphism|isomorphic]] in $Ab_{Adams}$ (Def. \ref{AbelianGroupWithAdamsOperations}) to the [[objects]] from Example \ref{IntegersEquippedWithHomogeneousAdamsOperations}:
$$
\widetilde K
\big(
S^{2n}
\big)
\;\simeq\;
\mathbb{Z}(n)
\;\;\;\;
\in
\;
Ab_{Adams}
\,.
$$
=--
#### The Adams $e_{\mathbb{C}}$-invariant in $\mathbb{Q}/\mathbb{Z}$
{#TheEInvariantAsAnElementOfQModZ}
+-- {: .num_example #TheDefinignSESInComplexTopologicalKTheory}
###### Example
**(the defining short exact sequence in complex topological K-theory)**
For $n, n' \,\in\, \mathbb{N}$ let
$$
f \;\colon\; S^{2(n + n') - 1} \longrightarrow S^{2n}
$$
be a [[continuous function]] between [[spheres]] (representing its image in the [[classical homotopy category]] and in fact in the [[stable homotopy category]], which is all that its image in a [[Whitehead-generalized cohomology theory]] such as [[complex topological K-theory]] depends on), such that the [[d-invariant]] of $f$ vanishes under [[complex topological K-theory]] $E = KU$, hence such that the [[pullback in cohomology|pullback]] $f^\ast$ is the [[zero map]] in K-theory:
$$
d(f) \;\coloneqq\; f^\ast \;=\; 0 \;\;\;\colon\;
\widetilde K\big(S^{2n}\big)
\longrightarrow
\widetilde K\big( S^{2(n + n') - 1} \big)
\,.
$$
Writing $C_f$ for the [[homotopy type]] of the [[homotopy cofiber]]/[[attaching space]] of $f$:
\[
\label{ThePushoutDiagramsForTheCofiberSpaceAndSuspension}
\array{
S^{2(n + n') - 1}
&\longrightarrow&
\ast
\\
{}^{\mathllap{f}}
\big\downarrow
&
{}^{_{(hpo)}}
&
\big\downarrow
\\
S^{2n}
&\underset{i_{2n}}{\longrightarrow}&
C_f
\\
{}^{}
\big\downarrow
&
{}^{_{(hpo)}}
&
\big\downarrow
{}^{\mathrlap{ p_{2(n+n')} }}
\\
\ast
&\longrightarrow&
S^{2(n + n')}
}
\;\;\;\;\;\;\;\;
\simeq
\;\;\;\;\;\;\;\;
\array{
S^{2(n + n') - 1}
&\longrightarrow&
D^{2(n + n')}
\\
{}^{\mathllap{f}}
\big\downarrow
&
{}^{_{(po)}}
&
\big\downarrow
\\
S^{2n}
&\underset{i_{2n}}{\longrightarrow}&
C_f
\\
{}^{}
\big\downarrow
&
{}^{_{(po)}}
&
\big\downarrow
{}^{\mathrlap{ p_{2(n+n')} }}
\\
\ast
&\longrightarrow&
S^{2(n + n')}
}
\]
this implies that the [[long exact sequence in cohomology]] induced by the [[CW-pair]] $S^{2n} \hookrightarrow C_f$ truncates to the [[short exact sequence]] (eq:TheShortExactSequenceInETheory), here regarded, by Example \ref{AdamsOperationsOnComplexTopologicalKTheoryGroups}, in the [[abelian category]] $Ab_{Adams}$ of abelian groups with Adams operations (Def. \ref{AbelianGroupWithAdamsOperations}):
\[
\label{ShortExactSequenceForComplexTopologicalKTheory}
\array{
0
&\to&
\widetilde K\big( S^{2(n + n')} \big)
&\overset{\;\;\; p_{2(n+n')}^\ast \;\;\;}{\longrightarrow}&
\widetilde K\big( C_f \big)
&\overset{\;\;\; i_{2n}^\ast \;\;\;}{\longrightarrow}&
\widetilde K\big( S^{2n} \big)
&\to&
0
\\
&&
\big\downarrow
{}^{\mathrlap{\simeq}}
&&
\big\downarrow
{}^{\mathrlap{=}}
&&
\big\downarrow
{}^{\mathrlap{\simeq}}
\\
0
&\to&
\mathbb{Z}(n + n')
&\underset{\;\;\; \;\;\;\; \;\;\;}{\longrightarrow}&
\widetilde K\big( C_f \big)
&\underset{\;\;\; \;\;\;\; \;\;\;}{\longrightarrow}&
\mathbb{Z}(n)
&\to&
0
}
\;\;\;\;\;\;\;\;
\in
\;
Ab_{Adams}
\,,
\]
where in the second line we have identified the outer groups via Prop. \ref{AdamsOperationOnComplexTopologicalKtheoryOfnSpheres}.
By definition of [[Ext-groups]], the [[isomorphism class]] of this short exact sequence with the outer groups fixed is an element
\[
\label{AdamsInvariantInComplexKTheoryAsElementOfExt1}
e_{\mathbb{C}}(f)
\;\;\in\;\;
Ext^1_{Ab_{Adams}}
\big(
\mathbb{Z}(n),
\,
\mathbb{Z}(n + n')
\big)
\,.
\]
This is the [[Adams e-invariant]] of $f$ as seen in [[complex topological K-theory]]; in specialization of (eq:GeneraleInvariantAsElementOfExtGroup).
=--
More concretely, it turns out that the extension (eq:AdamsInvariantInComplexKTheoryAsElementOfExt1) is completely characterized by a single [[rational number]] modulo this [[integers]]. This we discuss next (Def. \ref{eInvariantInComplexTopologicalKtheoryAsRationalNumberModuloIntegers} below).
+-- {: .num_remark #ExtensionAfterForgettingAdamsModuleStructureIsTrivial}
###### Remark
**(extension after forgetting the Adams module structure is trivial)**
After forgetting the action of the Adams operations via (eq:ForgetfulFunctorFromAbelianGroupsWithAdamsOperations), the sequence (eq:ShortExactSequenceForComplexTopologicalKTheory) is still a [[short exact sequence]], now of plain [[abelian groups]]. However, since $Ext^1_{Ab}(\mathbb{Z},-) = 0$ ([this Prop.](Ext#ExtensionsOfTheIntegersAreTrivial)), it is necessarily trivial as an extension, showing that the underlying abelian cohomlogy group of the cofiber space is just
\[
\label{UnderlyingAbelianGroupOfKTheoryOfCofiberSpace}
\array{
\mathbb{Z} \oplus \mathbb{Z}
&
\simeq
&
\widetilde K \big( S^{2n} \big)
\,\oplus\,
\widetilde K \big( S^{2(n + n')} \big)
&
\simeq
&
\widetilde K
\big(
C_f
\big)
\\
\big(
1,
\,
1
\big)
&\mapsto&
\big(
\Sigma^{2n} 1,
\,
\Sigma^{2(n + n')} 1
\big)
&\mapsto&
\big(
V_{2n},
\,
V_{2(n+n')}
\big)
}
\;\;
\;\in\;
Ab
\,,
\]
where $V_{2(n + n')} \;\coloneqq\; p_{2(n+n')}^\ast \Sigma^{2(n+n')} 1$ and where $V_{2n}$ is a choice of lift of $\Sigma^{2n} 1$ through $i_{2n}^\ast$ in the short exact sequence (eq:ShortExactSequenceForComplexTopologicalKTheory):
\[
\label{TheSplittingOfExactSequenceOfAbelianGroups}
\array{
0
\to
&
\widetilde K\big( S^{2(n+n')} \big)
&
\overset{p^\ast_{2(n+n')}}{\longrightarrow}
&
\widetilde K\big( C_f \big)
&
\overset{i^\ast_{2n}}{\longrightarrow}
&
\widetilde K\big( S^{2n} \big)
&
\to
0
\\
&
\Sigma^{2(n+n')} 1
&\mapsto&
V_{2(n+n')}
\\
& &&
V_{2n}
&\mapsto&
\Sigma^{2n} 1
\,.
}
\]
Notice that the [[isomorphism]] (eq:UnderlyingAbelianGroupOfKTheoryOfCofiberSpace) depends on a choice of [[splitting]] (eq:TheSplittingOfExactSequenceOfAbelianGroups) of the short exact sequence (eq:ShortExactSequenceForComplexTopologicalKTheory) in [[Ab]]: any two choices $V_{2n}$, $V'_{2n}$ differ by a multiple $s \in \mathbb{Z}$ of the generator $V_{2(n + n')}$:
\[
\label{DependenceOfDegree2nGeneratorOnChoiceOfSplitting}
V'_{2n}
\;=\;
V_{2n} + s \cdot V_{2(n + n')}
\,.
\]
=--
Conversely, this means that all the information in the extension (eq:ShortExactSequenceForComplexTopologicalKTheory) is in how the Adams operations act on the K-theory of the cofiber space:
+-- {: .num_defn #eInvariantInComplexTopologicalKtheoryAsRationalNumberModuloIntegers}
###### Definition
**([[e-invariant]] in [[complex topological K-theory]] as [[rational number]] [[modulo]] [[integers]])**
Given a map $f \;\colon\; S^{2(n + n') - 1} \longrightarrow S^{2n}$
with vanishing $KU$-[[d-invariant]],
as in Example \ref{TheDefinignSESInComplexTopologicalKTheory}, we have by
Remark \ref{ExtensionAfterForgettingAdamsModuleStructureIsTrivial} that the [[Adams operations]] (Example \ref{AdamsOperationsOnComplexTopologicalKTheoryGroups}) on the cofiber space $C_f$ (eq:ThePushoutDiagramsForTheCofiberSpaceAndSuspension) must be of the following form ([Adams 66, Prop. 7.5, (7.10) (9.2)](#Adams66)):
\[
\label{ActionOfAdamsOperationsOnCofiberSpace}
\array{
\widetilde K\big( C_f\big)
&\overset{ \;\;\; \psi^k\;\;\; }{\longrightarrow}&
\widetilde K\big( C_f\big)
\\
V_{2n}
&\mapsto&
k^n \cdot V_{2n}
\;+\;
{\color{blue}
\mu_k(f)
}
\cdot V_{2(n+ n')}
\\
V_{2(n + n')}
&\mapsto&
k^{n + n'} \cdot V_{2(n + n')}
\;\;\;\;\;\;\;\;
\,.
}
\]
Namely, the first summands on the right are constrained to be as shown, by Prop. \ref{AdamsOperationOnComplexTopologicalKtheoryOfnSpheres} and using
that [[pullback in cohomology]] $i^\ast_{2n}$, $p^\ast_{2(n + n')}$(eq:PullbackInKCohomologyRespectsAdamsOperation) respects the [[Adams operations]] (Example \ref{AdamsOperationsOnComplexTopologicalKTheoryGroups}); while the second summand, which vanishes under $i^\ast_{2n}$, must be some multiple
$$
\mu_k(f) \;\in\; \mathbb{Z}
$$
of the only other generator $V_{2(n + n')}$ (eq:UnderlyingAbelianGroupOfKTheoryOfCofiberSpace). This $\mu$ is the only part of the data that is not completely fixed by the Adams module structure, and which may depend on the map $f$.
We say that the _[[Adams e-invariant]] of $f$_ is this multiple $\mu$, normalized as a [[rational number]] as follows, and then regarded modulo addition of [[integers]] as an element in [[Q/Z]]:
\[
\label{eInvariantAsRationalNumberModuloIntegers}
e_{\mathbb{C}}(f)
\;\coloneqq\;
\left[
\frac{
\mu_k(f)
}{
k^{n}
\big(
k^{n'} - 1
\big)
}
\right]
\;\;
\in
\;\;
\mathbb{Q}/\mathbb{Z}
\,.
\]
=--
+-- {: .num_prop #eInvariantInKTheoryAsRationalNumberModuloIntegersIsWellDefined}
###### Proposition
**(e-invariant as rational number modulo integers is well defined)**
The e-invariant $e_{\mathbb{C}}(f)$ (eq:eInvariantAsRationalNumberModuloIntegers) from Def. \ref{eInvariantInComplexTopologicalKtheoryAsRationalNumberModuloIntegers}
is well-defined, in that it is
1. independent of the choice of splitting $\Sigma^{2n} 1 \,\mapsto\, V_{2n}$ in (eq:UnderlyingAbelianGroupOfKTheoryOfCofiberSpace);
1. independent of the choice of $k$ on the right of (eq:eInvariantAsRationalNumberModuloIntegers).
=--
+-- {: .proof}
###### Proof
**On 1.** Under a different choice of splitting, $V_{2n}$ changes to (eq:DependenceOfDegree2nGeneratorOnChoiceOfSplitting)
$$
V'_{2n} \;=\; V_{2n} + s \cdot V_{2(n + n')}
$$
for some $s \in \mathbb{Z}$. By inspection of (eq:ActionOfAdamsOperationsOnCofiberSpace) this implies that $\mu_k(f)$ changes to
$$
\mu'_k(f)
\;=\;
\mu_k(f) + s \cdot \big( k^{n + n'} - k^n \big)
\,;
$$
and so in (eq:eInvariantAsRationalNumberModuloIntegers) we have
$$
e'_{\mathbb{C}}(f)
\;=\;
\left[
\frac{
\mu'_k(f)
}{
k^n(k^{n'} -1 )
}
\right]
\;=\;
\left[
\frac{
\mu_k(f)
}{
k^n(k^{n'} -1 )
}
+ s
\right]
\;=\;
\left[
\frac{
\mu_k(f)
}{
k^n(k^{n'} -1 )
}
\right]
\;=\;
e_{\mathbb{C}}(f)
\,.
$$
**On 2.** Use the commutativity
(eq:ActionPropertyOfAbstractAdamsOperations) of the Adams operation
together with the formula (eq:ActionOfAdamsOperationsOnCofiberSpace) to find for any $k_1, k_2 \,\in\, \mathbb{N}$:
$$
\begin{aligned}
&
\phantom{\;=\;\;}
\psi^{k_1} \circ \psi^{k_2}
\big(
V_{2n}
\big)
\\
&
\;=\;
\psi^{k_2} \circ \psi^{k_1}
\big(
V_{2n}
\big)
\\
\\
\Leftrightarrow
\;\;\;
&
\phantom{\;=\;\;}
\psi^{k_1}
\big(
k_2^n \cdot V_{2n}
+
\mu_{k_2}(f) \cdot V_{2(n + n')}
\big)
\\
&
\;=\;
\psi^{k_2}
\big(
k_1^n \cdot V_{2n}
+
\mu_{k_1}(f) \cdot V_{2(n + n')}
\big)
\\
\\
\Leftrightarrow
\;\;\;
&
\phantom{\;=\;\;}
(k_1 k_2)^n \cdot V_{2n}
+
\Big(
k_2^n \mu_{k_1}(f)
+
k_1^{n + n'} \mu_{k_2}(f)
\Big)
\cdot
V_{2(n + n')}
\\
&
\;=\;
(k_1 k_2)^n \cdot V_{2n}
+
\Big(
k_1^n \mu_{k_2}(f)
+
k_2^{n + n'} \mu_{k_1}(f)
\Big)
\cdot
V_{2(n + n')}
\\
\\
\Leftrightarrow
\;\;\;
&
\phantom{\;=\;\;}
k_1^n
\big(
k_1^{n'}
-
1
\big)
\mu_{k_2}(f)
\\
&
\;=\;
k_2^n
\big(
k_2^{n'}
-
1
\big)
\mu_{k_1}(f)
\\
\\
\Leftrightarrow
\;\;\;
&
\phantom{\;=\;\;}
\frac{
\mu_{k_1}(f)
}{
k_1^n
\big(
k_1^{n'}
-
1
\big)
}
\\
&
\;=\;
\frac{
\mu_{k_2}(f)
}{
k_2^n
\big(
k_2^{n'}
-
1
\big)
}
\end{aligned}
$$
=--
\linebreak
### As the top degree Chern character on cofiber space
{#AsThe2nFormComponentOfTheChernCharacterOnCofiberSpaces}
+-- {: .num_prop #QModZValuedEInvariantIsTopDegreeCoefficientOfChernCharacterOnCofiberSpace}
###### Proposition
**([[Q/Z]]-valued [[e-invariant]] is top-degree coefficient of [[Chern character]] on cofiber space)**
In the situation of Example \ref{TheDefinignSESInComplexTopologicalKTheory}, with
$$
f \;\colon\; S^{2(n+n')-1 } \longrightarrow S^{2n}
$$
a map between spheres, and with $V_{2n} \,\in\, \widetilde K\big( C_f \big)$ any lift (eq:TheSplittingOfExactSequenceOfAbelianGroups)
of $\Sigma^{2 n} 1 \,\in\, \widetilde K \big( S^{2n} \big) $ to its homotopy cofiber space, we have that the [[e-invariant]] $e_{\mathbb{C}}(f)$ (Def. \ref{eInvariantInComplexTopologicalKtheoryAsRationalNumberModuloIntegers}) is equivalently the [[Kronecker pairing|evaluation]] [[modulo]] [[integers]] of the [[Chern character]] $ch(V_{2n}) \,\in\, H^{ev}\big( C_f;\, \mathbb{Q} \big)$ on the [[fundamental class]] of the cofiber space:
$$
\exp
\left(
2 \pi \mathrm{i}
\int_{C_f}
ch\big( V_{2n} \big)
\right)
\;=\;
\exp
\left(
2 \pi \mathrm{i}
\,
{
\color{blue}
e_{\mathbb{C}}(f)
}
\right)
\;\;\;
\in
\mathrm{U}(1)
\,.
$$
=--
[Adams 66, Prop. 7.5](#Adams66)
+-- {: .proof}
###### Proof
By (eq:ActionOfAdamsOperationsOnCofiberSpace) we have,
now in [[matrix calculus]]-notation:
$$
\psi^k
\;\colon\;
\left[
\array{
V_{2 n }
\\
V_{2(n + n')}
}
\right]
\;\;\;
\mapsto
\;\;\;
\left[
\array{
k^n
&
e(f) \, k^n (k^{n'} - 1)
\\
0
&
k^{n + n'}
}
\right]
\cdot
\left[
\array{
V_{2 n }
\\
V_{2(n + n')}
}
\right]
\;\;\;\;\;
\in
\;
\widetilde K\big( C_f \big)
\,.
$$
This [[matrix]] has two [[eigenvectors]] over the [[rational numbers]] (in general). Therefore we now consider the image of these K-theory classes under the [[Chern character]] map
$$
ch
\;\colon\;
\widetilde K\big( C_f \big)
\longrightarrow
H^{ev}\big( C_f;\, \mathbb{Q}\big)
\,.
$$
Since the [[Adams operations are compatible with the Chern character]], we then have the following [[eigenvectors]] of the [[Adams operations]] under $ch$:
\[
\label{EigenvectorsOfAdamsOperationsAfterPassageThroughChernCharacter}
\psi^k_H
\;\;\colon\;\;
\left\{
\array{
ch
\big(
V_{2 n}
\big)
-
e(f)
\cdot
ch
\big(
V_{2(n + n')}
\big)
&
\mapsto
&
k^{n }
&
\cdot
&
\big(
ch
(
V_{2 n}
)
-
e(f)
\cdot
ch
(
V_{2(n + n')}
)
\big)
\\
ch
(
V_{2(n+n')}
)
&
\mapsto
&
k^{n + n'}
&
\cdot
&
ch
(
V_{2 (n + n')}
)
}
\right.
\;\;\;\;
\in
\;
H^{ev}\big( C_f; \, \mathbb{Q} \big)/H^{2(n+n')}\big( C_f; \, \mathbb{Z} \big)
\,.
\]
Here, since $V_{2n}$ is well defined modulo addition (eq:DependenceOfDegree2nGeneratorOnChoiceOfSplitting) of integral multiples of $V_{2(n+n')}$, and since
\[
\label{ChernCharacterOnCofiverSpaceOfGeneratorofTopCell}
ch\big( V_{2(n+n')} \big)
\;=\;
1
\,\in\,
\mathbb{Z}
\,\simeq\,
H^{2(n+n')}\big( C_f;\, \mathbb{Z} \big)
\,,
\]
this expression (eq:EigenvectorsOfAdamsOperationsAfterPassageThroughChernCharacter) is well-defined in [[ordinary cohomology|ordinary]] [[rational cohomology]] in even degrees [[modulo]] [[integral cohomology]] in top degree.
But since the [[eigenvectors]] of $\psi^k_H $ to [[eigenvalue]] $k^r$ are precisely the ordinary cohomology classes in homogeneous degree $H^{2r}\big( C_f;\, \mathbb{Q} \big) \,\subset\, H^{ev}\big( C_f;\, \mathbb{Q} \big)$ (see [there](Adams+operations+compatible+with+the+Chern+character#eq:AdamsOperationOnOrdinaryCohomologyInDegree2r)), this means that
$$
ch
\big(
V_{2n}
\big)
\;\;=\;\;
\underset{
\in \; H^{2n}\big( C_f; \, \mathbb{Q} \big)
}{
\underbrace{
ch
\big(
V_{2n}
\big)
-
e(f)
\cdot
ch
\big(
V_{2(n+n')}
\big)
}
}
\;+\;
\underset{
\in \; H^{2(n + n')}\big( C_f; \, \mathbb{Q}/\mathbb{Z} \big)
}{
\underbrace{
e(f)
\cdot
ch
\big(
V_{2(n+n')}
\big)
\,.
}
}
$$
(See also [Conner-Floyd 66, p. 100](#ConnerFloyd66).)
The evaluation of this cohomology class on the [[fundamental class]] of $C_f$ picks out the [[coefficient]] of $ch\big( V_{2(n+n')} \big)$, by (eq:ChernCharacterOnCofiverSpaceOfGeneratorofTopCell):
$$
\int_{C_f}
ch
\big(
V_{2n}
\big)
\;=\;
e(f)
\;\;\;
\in
\;
\mathbb{Q}/\mathbb{Z}
\,,
$$
and hence the claim follows.
=--
\begin{remark}
\label{eRInvariantIsNotInGeneralTopDegreeChernCharacter}
The analogue statement of Prop. \ref{QModZValuedEInvariantIsTopDegreeCoefficientOfChernCharacterOnCofiberSpace}
for the $e_{\mathbb{R}}$-invariant (Def. \ref{eRInvariantInTermsOfAdamsOperations}) may _fail_:
The $e_{\mathbb{R}}$-invariant defined in terms of Adams operations (Def. \ref{eRInvariantInTermsOfAdamsOperations}) is $c$ times the top degree coefficient of the [[Chern character]] on $K \mathrm{O} \to K \mathrm{U}$ (the [[Pontrjagin character]]) with ([Adams 66 (7.3) ](#Adams66)):
1. $c = 1$ for $(n + n') = \,0\, mod \, 8$ (here they coincide)
1. $c = \tfrac{1}{2}$ for $(n + n') = \,4\, mod \, 8$ (here the $e_{\mathbb{R}}$-invariant is finer).
Similarly ([Adams 66 Prop. 7.14](#Adams66)):
* the $e_{\mathbb{R}}$-invariant equals the $e_{\mathbb{C}}$-invariant for $n' = \,0\, mod \, 8$,
* but equals $\tfrac{1}{2}e_{\mathbb{C}}$ for $n + n' = \,4\, mod \, 8$ .
This means that the $e_{\mathbb{R}}$-invariant is _finer_ than the $e_{\mathbb{C}}$-invariant.
\end{remark}
\linebreak
### As a cobordism invariant of U-manifolds with framed boundary
{#AsACobordismInvariantOfManifoldsWithBoundary}
We discuss how the [[e-invariant]] in its [[Q/Z]]-incarnation (Def. \ref{eInvariantInComplexTopologicalKtheoryAsRationalNumberModuloIntegers}) has a natural formulation in [[cobordism theory]] ([Conner-Floyd 66](#ConnerFloyd66)).
This is Prop. \ref{EInvariantIsToddClassOnCoboundingUFrManifold} below; but first to recall some background:
+-- {: .num_remark}
###### Remark
In generalization to how the [[U-bordism ring]] $\Omega^U_{2k}$ is [[Brown representability theorem|represented]] by [[homotopy classes]] of [[maps]] into the [[Thom spectrum]] [[MU]], so the [[(U,fr)-bordism ring]] $\Omega^{U,fr}_{2k}$ is represented by maps into the [[quotient spaces]] $MU_{2k}/S^{2k}$ (for $S^{2k} = Th(\mathbb{C}^{k}) \to Th( \mathbb{C}^k \times_{U(k)} E U(k) ) = MU_{2k}$ the canonical inclusion):
\[
\label{InTermsOfHomotopyGroupsOfQuotientedThomSpace}
\Omega^{(U,fr)}_\bullet
\;=\;
\pi_{\bullet + 2k}
\big(
MU_{2k}/S^{2k}
\big)
\,,
\;\;\;\;\;
\text{for any}
\;
2k \geq \bullet + 2
\,.
\]
=--
([Conner-Floyd 66, p. 97](#ConnerFloyd66))
+-- {: .num_remark}
###### Remark
The [[bordism rings]] for [[MU]], [[MUFr]] and [[MFr]] sit in a [[short exact sequence]] of the form
\[
\label{ShortExactSequenceOfUFrBordismRings}
0
\to
\Omega^U_{\bullet+1}
\overset{i}{\longrightarrow}
\Omega^{U,fr}_{\bullet+1}
\overset{\partial}{
\longrightarrow
}
\Omega^{fr}_\bullet
\to
0
\,,
\]
where $i$ is the evident inclusion, while $\partial$ is restriction to the [[boundary]].
(By [this Prop.](MUFr#AShortExactSequenceOfUFrBordismRings) at _[[MUFr]]_.)
In particular, this means that $\partial$ is [[surjective function|surjective]], hence that every $Fr$-manifold is the boundary of a [[(U,fr)-manifold]].
=--
+-- {: .num_prop #EInvariantIsToddClassOnCoboundingUFrManifold}
###### Proposition
**([[e-invariant is Todd class of cobounding (U,fr)-manifold]])**
Evaluation of the [[Todd class]] on [[(U,fr)-manifolds]] yields [[rational numbers]] which are [[integers]] on actual $U$-manifolds. It follows with the [[short exact sequence]] (eq:ShortExactSequenceOfUFrBordismRings) that assigning to $Fr$-manifolds the Todd class of any of their cobounding $(U,fr)$-manifolds yields a well-defined element in [[Q/Z]].
Under the [[Pontrjagin-Thom isomorphism]] between the [[framed bordism ring]] and the [[stable homotopy group of spheres]] $\pi^s_\bullet$, this assignment coincides with the [[Adams e-invariant]] in [its Q/Z-incarnation](#TheEInvariantAsAnElementOfQModZ):
\[
\label{ToddClassesOnShortExactSequenceOfUFrBordismRings}
\array{
0
\to
&
\Omega^U_{\bullet+1}
&
\overset{i}{\longrightarrow}
&
\Omega^{U,fr}_{\bullet+1}
&
\overset{\partial}{
\longrightarrow
}
&
\Omega^{fr}_\bullet
&
\simeq
&
\pi^s_\bullet
\\
&
\big\downarrow{}^{\mathrlap{Td}}
&&
\big\downarrow{}^{\mathrlap{Td}}
&&
\big\downarrow{}^{}
&&
\big\downarrow{}^{e}
\\
0
\to
&
\mathbb{Z}
&\overset{\;\;\;\;\;}{\hookrightarrow}&
\mathbb{Q}
&\overset{\;\;\;\;}{\longrightarrow}&
\mathbb{Q}/\mathbb{Z}
&=&
\mathbb{Q}/\mathbb{Z}
}
\,,
\]
=--
([Conner-Floyd 66, Theorem 16.2](#ConnerFloyd66))
The first step in the proof of (eq:ToddClassesOnShortExactSequenceOfUFrBordismRings) is the observation ([Conner-Floyd 66, p. 100-101](#ConnerFloyd66)) that the representing map (eq:InTermsOfHomotopyGroupsOfQuotientedThomSpace) for a $(U,fr)$-manifold $M^{2k}$ cobounding a $Fr$-manifold represented by a map $f$ is given by the following [[homotopy coherent diagram|homotopy]] [[pasting diagram]] (see also at _[[Hopf invariant]] -- [In generalized cohomology](Hopf+invariant#InGeneralizedCohomology)_):
\begin{imagefromfile}
"file_name": "CoboundingUFrManifoldDiagrammatically.jpg",
"web": "nlab",
"width": 470,
"unit": "px",
"margin": {
"top": -20,
"right": 0,
"bottom": 20,
"left": 20,
"unit": "px"
},
"alt": "homotopy pasting diagram exhibiting cobounding UFr-manifolds",
"caption": "from [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)"
\end{imagefromfile}
\linebreak
## For $E = KO$
{#KOVersion}
### The Adams $e_{\mathbb{R}}$-invariant in $\mathbb{Q}/\mathbb{Z}$
\begin{defn}
\label{eRInvariantInTermsOfAdamsOperations}
Definition \ref{eInvariantInComplexTopologicalKtheoryAsRationalNumberModuloIntegers} (via [[Adams operations]]) applies verbatim also with real (i.e. orthogonal) [[topological K-theory]] [[KO]] in place of complex topological K-theory [[KU]]. The resulting invariant is denoted $e_{\mathbb{R}}$ ([Adams 66, p. 39](#Adams66)).
\end{defn}
But beware that $e_{\mathbb{R}}$ may fail to be equal to the coefficient of the top degree Chern character on [[KO]] (the [[Pontrjagin character]]), see Remark \ref{eRInvariantIsNotInGeneralTopDegreeChernCharacter}. That is, in fact, what makes $e_{\mathbb{R}}$ a finer invariant: It is either equal to $e_{\mathbb{C}}$ or to $\tfrac{1}{2}e_{\mathbb{C}}$. ([Adams 66 (7.3)](#Adams66)).
### As a cobordism invariant of $SU$-manifolds with framed boundary
An analogous but finer version of the [[cobordism theory|cobordism]]-theoretic construction ([above](#AsACobordismInvariantOfManifoldsWithBoundary)) works for [[special unitary group]]-structure instead of [[unitary group]]-structure and in dimensions $8\bullet + 4$:
Since on $(8 \bullet + 4)$-dimensional $SU$-manifolds the [[Todd class]] is divisible by 2 [Conner-Floyd 66, Prop. 16.4](#ConnerFloyd66), we have ([Conner-Floyd 66, p. 104](#ConnerFloyd66)) the following variant of (eq:ToddClassesOnShortExactSequenceOfUFrBordismRings):
\[
\label{HalfToddClassesOnShortExactSequenceOfSUFrBordismRings}
\array{
0
\to
&
\Omega^{SU}_{8\bullet+4}
&
\overset{i}{\longrightarrow}
&
\Omega^{SU,fr}_{8\bullet+4}
&
\overset{\partial}{
\longrightarrow
}
&
\Omega^{fr}_{8 \bullet + 3}
&
\simeq
&
\pi^s_{8 \bullet + 3}
\\
&
\big\downarrow{}^{\tfrac{1}{2}\mathrlap{Td}}
&&
\big\downarrow{}^{\tfrac{1}{2}\mathrlap{Td}}
&&
\big\downarrow{}^{}
&&
\big\downarrow{}^{e_{\mathbb{R}}}
\\
0
\to
&
\mathbb{Z}
&\overset{\;\;\;\;\;}{\hookrightarrow}&
\mathbb{Q}
&\overset{\;\;\;\;}{\longrightarrow}&
\mathbb{Q}/\mathbb{Z}
&=&
\mathbb{Q}/\mathbb{Z}
}
\,.
\]
This produces $e_{\mathbb{R}}$, the [[Adams e-invariant]] with respect to [[KO]]-theory instead of [[KU]] ([Adams 66, p. 39](#Adams66)), which, in degrees $8k + 3$, is indeed half of the e-invariant $e_{\mathbb{C}}$ for $KU$ (by [Adams 66, Prop. 7.14](#Adams66)).
\linebreak
## Diagrammatic construction
{#ConstructionViaUnitCofiberTheories}
> under construction -- not complete yet
While the Adams e-invariant of a map exists when the its [[d-invariant]] vanishes, the classical constructions above implicitly proceed via a finer invariant of maps _equipped with a trivialization_ of their d-invariant. In order to bring this out more explicitly, the following is meant to be another, more abstractly homotopy theoretic, way to approach the construction of the e-invariant (following [[schreiber:M-Theory as Mf-Theory|SS21]]). This perspective turns out to make various properties immediately manifest, notably the equality between Adams's construction via the Chern character on [[KU]] and Conner-Floyd's construction via the Todd character on [[MUFr]].
1. [Preliminaries](#DiagrammaticConstructionPreliminaries)
1. [Unit cofiber cohomology theory](#UnitCofiberCohomologyTheory)
1. [The refined e-invariant](#TheDiagrammaticEInvariant)
1. [Recovering Adams's construction](#DiagrammaticEInvariantRecoveringAdamsConstruction)
1. [Recovering Conner-Floyd's construction](#ConnerFloydTheoremViaPastingDiagram)
### Preliminaries
{#DiagrammaticConstructionPreliminaries}
First some **Notation:**
* For $E$ a [[spectrum]] and $n \in \mathbb{Z}$ we write
* $E^n \coloneqq \Omega^\infty \Sigma^n E$ for the $n$th component space
* $E_n \coloneqq \pi_n(E) \simeq \widetilde E{}^0(S^n) \simeq \pi_0(E^{-n})$ for its [[homotopy groups of a spectrum|homotopy groups]].
* For $R$ a [[ring]] we write $H R$ for its [[Eilenberg-MacLane spectrum]]
and $ H^{\mathrm{ev}}R \;\coloneqq\; \underset{k \in \mathbb{N}}{\oplus} \Sigma^{2k} H R$ for its [[connective spectrum|connective]] even 2-periodic version.
The **Setup** is as above, which we recall for completeness:
* Let $n, d \in \mathbb{N}$, with $d \geq 1$,
* consider a map $ S^{2(n+d)-1} \xrightarrow{\;c\;} S^{2n}$ representing a class in $\mathbb{S}_{2d-1}$, which we will denote by the same name.
Without restriction of generality we may assume that this class is non-trivial $c \neq 0 \,\in\, \mathbb{S}_{2d-1}\,.$
* We write $C_c$ for the cofiber space of $c$:
\begin{xymatrix}
S^{2(n+d)-1}
\ar[r]^-c
&
S^{2n}
\ar[r]
&
C_c
\ar[r]
&
S^{2(n+d)}
\ar[r]^{ \Sigma c }
&
S^{2 n + 1}
\ar[r]
&
\cdots
\end{xymatrix}
### Unit cofiber cohomology theory
{#UnitCofiberCohomologyTheory}
\begin{definition}[Unit cofiber cohomology theory]
\label{UnitCofiberCohomologyTheory}
For $E$ a [[multiplicative cohomology theory]] ([[Brown representability theorem|represented]] by a [[homotopy-commutative ring spectrum]]) with unit operation $\mathbb{S} \overset{ e^E }{\longrightarrow} E$ we denote the corresponding [[homotopy cofiber]]-theory $E/\mathbb{S}$.
(In notation common around the [[Adams spectral sequence]] this would be
"$\Sigma \overline {E}$" -- as in [Adams 74, theorem 15.1 page 319](Adams+spectral+sequence#Adams74) -- or just "$\overline{ E }$" -- as in [Hopkins 99, Cor. 5.3](Adams+spectral+sequence#Hopkins99)).
Via the induced [[homotopy cofiber sequence]]
\begin{xymatrix}
\mathbb{S}
\ar[r]^{1^E}
&
E
\ar[r]
&
E/\mathbb{S}
\ar[r]
&
\Sigma \mathbb{S}
\ar[r]^{ \Sigma (1^E) }
&
\Sigma E
\ar[r]
&
\cdots
\end{xymatrix}
this comes with a canonical [[cohomology operation]] $\partial \;\colon\; E/\mathbb{S} \longrightarrow \Sigma \mathbb{S}$ to shifted [[stable Cohomotopy]]:
\begin{xymatrix@C=50pt@R=12pt}
\mathbb{S}
\ar[dd]
^>>{\ }="t"
\ar[rr]
^-{
\;1^E\;
}
_>>{\ }="s"
&&
\overset{
\mathclap{
\raisebox{4pt}{
\tiny
\color{blue}
\bf
\begin{tabular}{c}
$E$-cohomology
\end{tabular}
}
}
}{
E
}
\ar[dd]
_-{
\mathclap{\phantom{\vert^{\vert^{\vert}}}}
i^E
\mathclap{\phantom{\vert_{\vert}}}
}
^>>{\ }="t2"
\ar[rr]
^-{
}
_>>{\ }="s2"
&&
\ast
\ar[dd]
\\
{\phantom{A}}
\\
\ast
\ar[rr]
&&
\underset{
\mathclap{
\raisebox{-4pt}{
\tiny
\color{blue}
\bf
\begin{tabular}{c}
$E$-cofiber
\\
cohomology
\\
theory
\end{tabular}
}
}
}{
E/\mathbb{S}
}
\ar[rr]
|-{ \;\partial^E\; }
_-{
\mbox{
\tiny
\color{green}
\bf
\begin{tabular}{c}
\phantom{-}
\\
boundary
\\
operation
\end{tabular}
}
}
&&
\underset{
\mathclap{
\raisebox{-6pt}{
\tiny
\color{blue}
\bf
\begin{tabular}{c}
shifted stable
\\
Cohomotopy
\end{tabular}
}
}
}{
\Sigma \mathbb{S}
}
\,,
&
{\phantom{AAA}}
%
\ar@{=>}^-{ \mbox{\tiny\rm(po)} }
"s"; "t"
\ar@{=>}^-{ \mbox{\tiny\rm(po)} }
"s2"; "t2"
\end{xymatrix}
\end{definition}
\begin{definition}\label{InducedCohomologyOperationsOnCofiberCohomology}
[Induced cohomology operations on cofiber cohomology]
Let $E \overset{\phi}{\longrightarrow} F$ be a
[[multiplicative cohomology theory|multiplicative]]
[[cohomology operation]], so that in particular
it preserves the units, witnessed
by a homotopy-commutative square on the
left here:
\begin{xymatrix@=24pt}
\mathbb{S}
\ar[rr]|-{ \;1^{E}\; }
\ar@{=}[d]
&&
E
\ar[d]
|-{
\mathclap{\phantom{\vert^{\vert}}}
\phi
\mathclap{\phantom{\vert_{\vert}}}
}
\ar[rr]
&&
E/\mathbb{S}
\ar[rr]|-{ \;\partial^E\; }
\ar[d]
|-{
\mathclap{\phantom{\vert^{\vert}}}
\phi/\mathbb{S}
\mathclap{\phantom{\vert_{\vert}}}
}
&&
\Sigma \mathbb{S}
\ar@{=}[d]
\\
\mathbb{S}
\ar[rr]|-{ \;1^F\; }
&&
F
\ar[rr]
&&
F/\mathbb{S}
\ar[rr]|-{ \;\partial^F\; }
&&
\Sigma\mathbb{S}
\,.
\end{xymatrix}
Then passing to [[homotopy cofibers]] yields the induced cohomology operation $\phi/\mathbb{S}$ on cofiber theories (Def. \ref{UnitCofiberCohomologyTheory}).
\end{definition}
\begin{example}\label{ChernCharacterOnCofiberOfKTheory}
[Chern character on cofiber of K-theory]
The [[Chern character]] operation
$\mathrm{KU} \overset{\phi}{\longrightarrow} H^{\mathrm{ev}}\mathbb{Q}$
is multiplicative, hence passes to an operation
$\mathrm{KU}/\mathbb{S}
\overset{ \;\mathrm{ch}/\mathbb{S}\; }{\longrightarrow}
H^{\mathrm{ev}}\mathbb{Q}$
via Def. \ref{InducedCohomologyOperationsOnCofiberCohomology}.
\end{example}
In a similar fashion we get:
\begin{remark}
\label{CanonicalSplittingOfCofiberOfPeriodicRationalCohomology}
There is a canonical [[split exact sequence|splitting]] $\mathrm{spl}_0$ of the [[short exact sequence]]
\begin{xymatrix@=18pt}
0
\ar[r]
&
\big(
\widetilde {H^{\mathrm{ev}}\mathbb{Q}}
\big)
{}^{2n}
\big(
S^{2(n+d)}
\big)
\ar[rr]
\ar@{=}[d]
&&
\big(
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S}}
\big)
{}^{2n}
\big(
S^{2(n+d)}
\big)
\ar[rr]
\ar[d]
|-{
\hspace{2.7pt}
\scalebox{.7}{
\rotatebox{-90}{$
\,
\simeq
\,
$}
}
}
_{\mathrm{spl}_0}
&&
\widetilde {\mathbb{S}}
{}^{2n+1}
\big(
S^{2(n+d)}
\big)
\ar[r]
\ar@{=}[d]
&
0
\\
0
\ar[r]
&
\mathbb{Q}
\;
\ar@{^{(}->}[rr]
&&
\mathbb{Q}
\oplus
\mathbb{S}_{2d-1}
\ar@{->>}[rr]
&&
\mathbb{S}_{2d-1}
\ar[r]
&
0
\,,
\end{xymatrix}
namely that coming from the inclusion
$H\mathbb{Q} \hookrightarrow H^{\mathrm{ev}}\mathbb{Q}$ (which is multiplicative, in particular preserves the unit):
\begin{xymatrix@=18pt}
0
\ar[r]
&
\big(
\widetilde {H\mathbb{Q}}
\big)
{}^{2n}
\big(
S^{2(n+d)}
\big)
\ar[rr]
\ar@{=}[d]
&&
\big(
\widetilde {(H\mathbb{Q})/\mathbb{S}}
\big)
{}^{2n}
\big(
S^{2(n+d)}
\big)
\ar[rr]
\ar[d]|-{
\hspace{2.7pt}
\scalebox{.7}{
\rotatebox{-90}{$
\,
\simeq
\,
$}
}
}
&&
\widetilde {\mathbb{S}}
{}^{2n+1}
\big(
S^{2(n+d)}
\big)
\ar[r]
\ar@{=}[d]
&
0
\\
0
\ar[r]
&
0
\;
\ar@{^{(}->}[rr]
&&
\mathbb{S}_{2d-1}
\ar@{=}[rr]
&&
\mathbb{S}_{2d-1}
\ar[r]
&
0
\,,
\end{xymatrix}
where on the left we used our assumption that $d \neq 0$.
We will use this canonical splitting together with the canonical splitting of the even rational cohomology of a cofiber space
$S^{2(n+d)-1} \xrightarrow{ \;c\; } S^{2n} \xrightarrow{ \;\; } C_c$
given by the splitting into degrees $2(n+d)$ and $2n$
\begin{xymatrix@C=8pt}
0
\ar[r]
&
\big(
\widetilde {H^{\mathrm{ev}}\mathbb{Q}}
\big)
{}^{2n}
\big(
S^{2(n+d)}
\big)
\ar[rr]
\ar@{=}[d]
&&
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}
{}^{2n}
\big(
C_c
\big)
\ar[rr]
\ar[d]
|-{
\hspace{2.7pt}
\scalebox{.7}{
\rotatebox{-90}{$
\,
\simeq
\,
$}
}
}
_{\mathrm{spl}_0}
&&
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}
{}^{2n}
\big(
S^{2n}
\big)
\ar[r]
\ar@{=}[d]
&
0
\\
0
\ar[r]
&
\mathbb{Q}
\;
\ar@{^{(}->}[rr]
&&
\mathbb{Q}
\oplus
\mathbb{Q}
\ar@{->>}[rr]
&&
\mathbb{Q}
\ar[r]
&
0
\,,
\end{xymatrix}
which is induced in the same fashion from the inclusion $H\mathbb{Q} \hookrightarrow H^{\mathrm{ev}}\mathbb{Q}$.
Both of these splittings are hence characterized by the fact that
their corresponding retraction (see [there](split+exact+sequence#InAbelianCategory)) is
*projection onto rational cohomology in degree $2(n+d)$*.
\end{remark}
\begin{lemma}
\label{CofiberECohomologyAsExtensionOfStableCohomotopyByECohomology}
(Cofiber $E$-cohomology as extension of stable Cohomotopy by $E$-cohomology)
For $E$ a [[multiplicative cohomology theory]] and
$X$ a space, assume that the
$E$-[[Boardman homomorphism]]
$\widetilde {\mathbb{S}}{}^\bullet(X) \xrightarrow{ \;\beta^\bullet_X\; }
\widetilde E^\bullet(X)$
is [[zero morphism|zero]] in degrees $n$ and $n + 1$
-- for instance in that $X \simeq S^{k \geq 2}$,
$n = 0$ and the
groups $\widetilde E^0(S^k) = \pi_k(E)$ have no [[torsion subgroup|torsion]] --
then the [[cohomology operations]] $i^E$, $\partial^E$
form a [[short exact sequence]] of [[cohomology groups]]:
\begin{xymatrix@C=12pt}
0
\ar[r]
&
\widetilde E
\big(
X
\big)
\;
\ar@{^{(}->}[rr]^-{ i }
&&
(
\widetilde
{
E/\mathbb{S}
}
){}^n(X)
\ar@{->>}[rr]^-{ \partial }
&&
\widetilde {\mathbb{S}}{}^{n+1}(X)
\ar[r]
&
0
\end{xymatrix}
\end{lemma}
(e.g. [Stong 68, p. 102](MOFr#Stong68))
\begin{proof}
Generally, the long cofiber sequence of cohomology theories
induces a long exact sequence of cohomology groups (...)
\begin{xymatrix@C=12pt}
\cdots
\ar[r]
&
\widetilde {\mathbb{S}}{}^{n}(X)
\ar[rr]^-{ 1^n_X }
&&
\widetilde E^n(X)
\ar[rr]^-{ i^n_X }
&&
(\widetilde{E/\mathbb{S}})^n(X)
\ar[rr]^-{ \partial^n_X }
&&
\widetilde S^{n+1}(X)
\ar[rr]^-{ 1^{n+1}_X }
&&
\widetilde {E}^{n+1}(X)
\ar[r]
&
\cdots
\end{xymatrix}
Under the given assumption the two outermost morphisms
shown are zero, and hence the sequence truncates as claimed.
\end{proof}
### The refined e-invariant
{#TheDiagrammaticEInvariant}
\begin{proposition}\label{EFluxesAreCocycleInCofiberTheory}
Let $E$ be a [[multiplicative cohomology theory]]
and $n,d \in \mathbb{N}$.
Consider the [[function]] that sends [[pairs]] consisting of
1. the stable class $\big[ G^{\mathbb{S}}_n (c) \big]$ of a map $S^{n+d-1} \overset{\;c\;}{\longrightarrow} S^n$
1. the class of a trivialization $H^E_{n-1}\!(c)$ of its [[d-invariant]] in $E$-cohomology
\begin{xymatrix@C=18pt}
S^{n + d - 1}
\ar[dd]
|-{
\mathclap{\phantom{\vert^{\vert}}}
\color{green}
c
\mathclap{\phantom{\vert_{\vert}}}
}
^>>>{\ }="t"
\ar[rr]_>>>{\ }="s"
&&
\ast
\ar[dd]
|-{
\mathclap{\phantom{\vert^{\vert}}}
0
\mathclap{\phantom{\vert_{\vert}}}
}
\\
\\
S^n
\ar[rr]
|-{
\;
\Sigma^n (1^E)
\;
}
&&
E^{n}
%
\ar@{=>}
|-{
\mathclap{\phantom{\vert^{\vert^{\vert}}}}
\color{orange}
H^E_{n-1}\!(c)
\mathclap{\phantom{\vert_{\vert_{\vert}}}}
}
"s"; "t"
\end{xymatrix}
to the class in $(E/\mathbb{S})_d$ of this diagram:
\begin{xymatrix@=30pt}
S^{n + d - 1}
\ar[dd]
|-{
\mathclap{\phantom{\vert^{\vert}}}
\color{green}
c
\mathclap{\phantom{\vert_{\vert}}}
}
\ar[rr]
\ar@{}[ddrr]|-{
\rotatebox[origin=c]{-45}{\color{orange}$\big\Downarrow$}
\mbox{\tiny (po)}
}
&&
\ast
\ar[dr]
\ar[dd]
\\
&& &
\ast
\ar[dd]|-{
\mathclap{\phantom{\vert^{\vert}}}
0
\mathclap{\phantom{\vert_{\vert}}}
}
\\
S^{n}
\ar[dd]
\ar[rr]
\ar@{}[ddrr]|-{
\rotatebox[origin=c]{-45}{$\big\Downarrow$}
\mbox{\tiny (po)}
}
\ar@/_.7pc/[drrr]
|<<<<<{ \;\Sigma^{n} (1^{E})\;\; }
|>>>>>>>>>>{ {\phantom{AA}} \atop {\phantom{AA}} }
&&
C_c
\ar[dd]
\ar@{-->}[dr]|-{
\vdash
{
\color{orange}
H^E_{n-1}\!(c)
}
}
&&
\\
&& &
E^{n}
\ar[dd]
\\
\ast
\ar@/_.7pc/[drrr]|-{ \;0\; }
\ar[rr]
&&
S^{n + d}
\ar@{-->}[dr]|-{
\mathllap{\vdash}
(
{\color{green}
G^{\mathbb{S}}_n\!(c)
}
\,,\,
{\color{orange}
H^E_{n-1}\!(c)
}
)
}
&&
\\
&& &
(E/\mathbb{S})^{n}
\end{xymatrix}
Then: This function respects the canonical fibrations of both sides over $\mathbb{S}_{d-1}$, i.e. it is a [[lift]] through the boundary map $\partial$.
\end{proposition}
See also [this Prop.](d-invariant#EFluxesAreCocycleInCofiberTheory) at _[[d-invariant]]_.
\begin{proof}
**First**, here is a quick formal argument to see that _some_ such map does exist:
By Definition \ref{SetOfTrivializationsOfTheDInvariant}, an element in $H_{n-1}Fluxes^E\big( S^{n+d-1} \big)$ is equivalently the class of a homotopy [[cone]] with tip $\Sigma^{n+d-1} \mathbb{S}$ over the [[cospan]] formed by the ring spectrum unit $e^E$ and the [[zero morphism]]:
\begin{xymatrix@=26pt}
\Sigma^{n+d-1}\mathbb{S}
\ar[rr]_>>>{\ }="s"
\ar[dd]^>>>{\ }="t"
|-{
\mathclap{\phantom{\vert^{\vert}}}
c
\mathclap{\phantom{\vert_{\vert}}}
}
&&
\ast
\ar[dd]
|-{
\mathclap{\phantom{\vert^{\vert}}}
0
\mathclap{\phantom{\vert_{\vert}}}
}
\\
\\
\Sigma^n \mathbb{S}
\ar[rr]
|-{
\; \Sigma^n(e^E) \;
}
&&
\Sigma^n E
%
\ar@{=>}
|-{
\mathclap{\phantom{\vert^{\vert^{\vert}}}}
{
\color{orange}
H^E_{n-1}\!(c)
}
\mathclap{\phantom{\vert_{\vert_{\vert}}}}
}
"s"; "t"
\end{xymatrix}
But in [[Spectra]] [[homotopy cofiber sequences]] are [[homotopy fiber sequences]] (by [this Prop.](Spectrum#InSpectraHomotopyFiberSequencesAreHomotopyCofiberSequences)), so that by the [[universal property]] of [[homotopy fibers]] the class of the above diagram induces the class of a map
$\vdash {\color{orange} H^E_{n-1}(c) } \;\colon\; \Sigma^{n+d-1}\mathbb{S} \longrightarrow fib\big( \Sigma^{n} e^E \big) \,\simeq\, \Sigma^{n-1} (E/\mathbb{S}$):
\begin{xymatrix@=26pt}
\Sigma^{n+d-1}\mathbb{S}
\ar@/_1pc/[dddr]
|-{
\mathclap{\phantom{\vert^{\vert}}}
c
\mathclap{\phantom{\vert_{\vert_{\vert}}}}
}
^-{\ }="t2"
\ar@/^1pc/[rrrd]
\ar@{-->}[dr]
|-{
\mathclap{\phantom{\vert^{\vert}}}
\vdash
{
\color{orange}
H^E_{n-1}\!(c)
}
\mathclap{\phantom{\vert_{\vert}}}
}
_>>>>{\ }="s2"
\\
&
\Sigma^{n-1}
(\mathbb{S}/E)
\ar[rr]_>>>{\ }="s"
\ar[dd]^>>>{\ }="t"
|-{
\mathclap{\phantom{\vert^{\vert}}}
\partial
\mathclap{\phantom{\vert_{\vert}}}
}
&&
\ast
\ar[dd]
|-{
\mathclap{\phantom{\vert^{\vert}}}
0
\mathclap{\phantom{\vert_{\vert}}}
}
\\
\\
&
\Sigma^n \mathbb{S}
\ar[rr]
|-{
\;\Sigma^n(e^E)\;
}
&&
\Sigma^n E
%
\ar@{=>}
^-{ \mbox{\tiny\color{orange}(pb)} }
"s"; "t"
\ar@{=>}
"s2"; "t2"
\end{xymatrix}
equipped with a homotopy from its image under $\partial$ to $c$.
This implies the claim, by
$$
\pi_0
Maps
\Big(
\Sigma^{n+d - 1}\mathbb{S}
\,,\,
\Sigma^{n-1} (E/\mathbb{S})
\Big)
\;\simeq\;
(E/\mathbb{S})_{d}
\,.
$$
\linebreak
**Second,** to see that this map is realized as claimed (the following construction is close to the proof of [Conner-Floyd 66, Theorem 16.2](MUFr#ConnerFloyd66)):
Let
$\big[ S^{n + d - 1} \overset{c}{\longrightarrow} S^{n} \big] \;\in\; \pi^n\big(S^{n+d-1}\big)$ be a given class in Cohomotopy.
We need to produce a map of the form
\begin{xymatrix@R=10pt}
{
\color{orange}
H^E_{n-1}\!(c)
}
\ar@{}[rr]|-{\longmapsto}
&&
M^d
\\
H_{n-1}\mbox{Fluxes}(X)_c
\ar[rr]
\ar[dd]
&&
\big(
E\!/\mathbb{S}
\big)_d
\ar[dd]^-{ \partial }
\\
{\phantom{A}}
\\
\big\{
[
\Sigma^\infty c
]
\big\}
\;
\ar@{^{(}->}[rr]
&&
\;
\mathbb{S}_{d-1}
\end{xymatrix}
and show that it is a bijection onto this fiber, hence that the square is [[cartesian square|cartesian]]. To this end, we discuss the following [[homotopy coherent diagram|homotopy]] [[pasting diagram]], all of whose cells are homotopy cartesian:
\begin{xymatrix@R=15pt@C=60pt}
\Sigma^\infty S^{n + d - 1}
\ar[dd]_-{
\Sigma^\infty
{\color{green}
c
}
}
\ar[rr]
&&
\ast
\ar[dd]
\\
\\
\Sigma^\infty S^{n}
\ar[dd]
\ar[rr]|-{
%\;\Sigma^\infty q_c\;
}
\ar@/_1.4pc/[rrr]
|<<<<<<<<<<<<<<<<<<<{
\; \Sigma^{n}(e^{E}) \;
}
&&
\Sigma^\infty C_c
\ar[dd]|<<<<{ \phantom{A} }
\ar@{-->}[r]
\ar@{}[r]
|-{
\scalebox{.7}{$
\begin{array}{c}
\vdash
{\color{orange}
H^E_{n-1}\!(c)
}
\\
{\phantom{a}}
\end{array}
$}
}
\ar@/^2pc/[rr]
&
\Sigma^n E
\ar[r]
\ar[dd]
&
\ast
\ar[dd]
\\
\\
\ast
\ar[rr]
&&
\Sigma^{\infty\scalebox{.6}{$+1$}} S^{n + d - 1}
\ar[r]^-{
\color{magenta}
M^{d}
}
\ar@/_1.2pc/[rr]|-{
\;
\Sigma^{\infty\scalebox{.5}{$+1$}}
{\color{green}
c
}
\;
}
&
\Sigma^{n}(E\!/\mathbb{S})
\ar[r]^{ \partial }
&
\Sigma^{\infty\scalebox{.6}{$+1$}} S^{n}
\end{xymatrix}
For given $H^E_{n-1}\!(c)$, this diagram is constructed as follows
(where we say "square" for any _single_ cell and "rectangle" for the pasting composite of any adjacent _pair_ of them):
* The two squares on the left are the stabilization of the homotopy pushout squares defining the cofiber space $C_c$ and the suspension of $S^{n + d - 1}$
* The bottom left rectangle (with $\Sigma^n(e^E)$ at its top)
is the homotopy pushout defining $\Sigma^n(E\!/\mathbb{S})$.
* The classifying map for the given $(n-1)$-flux, shown as a dashed arrow,
completes a co-cone under the bottom left square. Thus the map ${\color{magenta}M^d}$ forming the bottom middle square is uniquely implied by the homotopy pushout property of the bottom left square.
Moreover, the [[pasting law]] implies that this bottom middle square is itself homotopy cartesian.
* The bottom right square is the homotopy pushout defining $\partial$.
* By the [[pasting law]] it follows that also the bottom right rectangle is homotopy cocartesian, hence that, after the two squares on the left, it exhibits the third step in the long homotopy cofiber sequence of $\Sigma^\infty c$. This means that its total bottom morphism is $\Sigma^{\infty + 1} c$, and hence that $\partial \big[ M^d \big] = [c]$.
In conclusion, these construction steps yield a map $H^E_{n-1}\!(c) \mapsto M^d$ over $[\Sigma^\infty c]$, as required.
\end{proof}
\begin{definition}
\label{LiftedEInvariantDiagrammatically}
The _refined e-invariant_ $\widehat e_{KU}$ is the composite of
1. the equivalence from Prop. \ref{EFluxesAreCocycleInCofiberTheory} for $E = $ [[KU]],
1. the cofiber Chern character from Example \ref{ChernCharacterOnCofiberOfKTheory},
1. the canonical splitting from Remark \ref{CanonicalSplittingOfCofiberOfPeriodicRationalCohomology}:
\begin{xymatrix@C=40pt@R=6pt}
H^{\mathrm{KU}}_{n-1}\mbox{Fluxes}
\big(
S^{2(n+d)-1}
\big)
\ar[r]|-{
\raisebox{2.4pt}{
\scalebox{.7}{$\simeq$}
}
}
&
(\mathrm{KU}/\mathbb{S})_{2d}
\ar[r]_-{\;\mathrm{ch}/\mathbb{S}\;}
&
\big((H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S}\big)_{2d}
\ar[r]
|-{
\raisebox{2.2pt}{
\scalebox{.7}{$
\simeq
$}
}
}
_-{ \mathrm{spl}_0 }
&
\mathbb{S}_{2(n+d)-1}
\oplus
\mathbb{Q}
\\
\Big(
[{\color{green}c}]
,
\big[
{\color{orange}
H^{\mathrm{KU}}_{2n-1}\!(c)
}
\big]
\Big)
\ar@{|->}[rrr]
&
&&
\Big(
[{\color{green}c}]
,
{\color{orange}
\widehat e_{\mathrm{KU}}(c)
}
\Big)
\end{xymatrix}
sending a trivialization of the d-invariant
\begin{xymatrix@=30pt}
S^{2(n + d) - 1}
\ar[dd]
|-{
\mathclap{\phantom{\vert^{\vert}}}
\color{green}
c
\mathclap{\phantom{\vert_{\vert}}}
}
^>>>{\ }="t"
\ar[rr]_>>>{\ }="s"
&&
\ast
\ar[dd]
|-{
\mathclap{\phantom{\vert^{\vert}}}
0
\mathclap{\phantom{\vert_{\vert}}}
}
\\
\\
S^{2n}
\ar[rr]
|-{
\;
\Sigma^{2n} (1^{\mathrm{KU}})
\;
}
&&
\mathrm{KU}^{2n}
%
\ar@{=>}
|-{
\mathclap{\phantom{\vert^{\vert^{\vert}}}}
\color{orange}
H^{\mathrm{KU}}_{2n-1}\!(c)
\mathclap{\phantom{\vert_{\vert_{\vert}}}}
}
"s"; "t"
\end{xymatrix}
to the class of this pasting composite:
\begin{xymatrix@R=1.8em@C=24pt}
S^{2(n + d) - 1}
\ar[dd]
|-{
\mathclap{\phantom{\vert^{\vert}}}
\color{green}
c
\mathclap{\phantom{\vert_{\vert}}}
}
\ar[rr]
\ar@{}[ddrr]|-{
\rotatebox[origin=c]{-45}{\color{orange}$\big\Downarrow$}
\mbox{\tiny (po)}
}
&&
\ast
\ar[dr]
\ar[dd]
\\
&& &
\ast
\ar[dd]|-{
\mathclap{\phantom{\vert^{\vert}}}
0
\mathclap{\phantom{\vert_{\vert}}}
}
\\
S^{n}
\ar[dd]
\ar[rr]|-{ \;q_c\; }
\ar@{}[ddrr]|-{
\rotatebox[origin=c]{-45}{$\big\Downarrow$}
\mbox{\tiny (po)}
}
\ar@/_.7pc/[drrr]
|<<<<<{ \;\Sigma^{n} (1^{\mathrm{MU}})\;\; }
|>>>>>>>>>>{ {\phantom{AA}} \atop {\phantom{AA}} }
&&
C_c
\ar[dd]
|>>>>>>{
\mathclap{\phantom{\vert^{\vert}}}
p_c
\mathclap{\phantom{\vert_{\vert}}}
}
\ar@{-->}[dr]|-{
\vdash
{
\color{orange}
H^{\mathrm{KU}}_{2n-1}\!(c)
}
}
&&
\\
&& &
\mathrm{KU}^{n}
\ar[dd]
|-{
\mathclap{\phantom{\vert^{\vert^{\vert}}}}
i^{\mathrm{KU}}
\mathclap{\phantom{\vert^{\vert}}}
}
\ar[dr]
|-{
\;\mathrm{ch}\;
}
\\
\ast
\ar@/_.7pc/[drrr]|-{ \;0\; }
\ar[rr]
&&
S^{2(n + d)}
\ar@{-->}[dr]|-{
\mathllap{\vdash}
(
{\color{green}
G^{\mathbb{S}}_n\!(c)
}
\,,\,
{\color{orange}
H^{\mathrm{KU}}_{n-1}\!(c)
}
)
}
&&
(H^{\mathrm{ev}}\mathbb{Q})^{2n}
\ar[dd]
|-{
\mathclap{\phantom{\vert^{\vert^{\vert}}}}
i^{H^{\mathrm{ev}}\mathbb{Q}}
\mathclap{\phantom{\vert^{\vert}}}
}
\\
&& &
(\mathrm{KU}/\mathbb{S})^{n}
\ar[dr]
|-{
\;\mathrm{ch}/\mathbb{S}\;
}
\\
&& &&
\big(
(H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S}
\big)^{2n}
\end{xymatrix}
\end{definition}
\begin{lemma}\label{DiagrammaticecInvariantTakesIntegerLattiveValues}
The values of the refined e-invariant from Def. \ref{LiftedEInvariantDiagrammatically} for fixed [[Cohomotopy]] class $[c]$ form an integer lattice inside the rational numbers:
$$
\widehat e_{\mathrm{KU}}
\Big(
[c],
\big[
H^{\mathrm{KU}}_{2d-1}\!(c)
\big]
\Big)
-
\widehat e_{\mathrm{KU}}
\Big(
[c],
\big[
{H}^\prime{}^{\mathrm{KU}}_{2d-1}\!(c)
\big]
\Big)
\;\in\;
\mathbb{Z}
$$
\end{lemma}
\begin{proof}
Using the same three ingredients that enter Def. \ref{LiftedEInvariantDiagrammatically}, we paste together the following [[commuting diagram]], whose vertical middle composite is the refined $\widehat e_{KU}$-invariant:
\begin{xymatrix@R=16pt}
\mathbb{Z}
\;
\ar[rr]
\ar@{=}[d]
&&
H^{\mathrm{KU}}_{2n-1}\mbox{Fluxes}
\big(
S^{2(n+d)}
\big)
\ar[rr]
\ar@{=}[d]
&&
\widetilde {\mathbb{S}}{}^{2n+1}
\big(
S^{2(n+d)}
\big)
\ar@{=}[d]
\\
\widetilde { \mathrm{KU} }{}^{2n}
\big(
S^{2(n+d)}
\big)
\ar[rr]
\ar[dd]|-{
\mathclap{\phantom{\vert^{\vert}}}
\mathrm{ch}
\mathclap{\phantom{\vert_{\vert}}}
}
&&
\big(\widetilde { (\mathrm{KU}/\mathbb{S}) }\big){}^{2n}
\big(
S^{2(n+d)}
\big)
\ar[dd]
|-{
\mathclap{\phantom{\vert^{\vert}}}
\mathrm{ch}/\mathbb{S}
\mathclap{\phantom{\vert_{\vert}}}
}
\ar[rr]
&&
\widetilde {\mathbb{S}}{}^{2n+1}
\big(
S^{2(n+d)}
\big)
\ar@{=}[dd]
\\
&&
&
\\
\big(
\widetilde {H^{\mathrm{ev}}\mathbb{Q}}
\big){}^{2n}
\big(
S^{2(n+d)}
\big)
\ar@{=}[d]
\ar[rr]
&&
\big(
\widetilde { (H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S} }
\big){}^{2n}
\big(
S^{2(n+d)}
\big)
\ar[rr]
\ar[d]
|-{
\hspace{2.7pt}
\scalebox{.7}{
\rotatebox{-90}{$
\,
\simeq
\,
$}
}
}
_{\mathrm{spl}_0}
&&
\widetilde {\mathbb{S}}{}^{2n+1}
\big(
S^{2(n+d)}
\big)
\ar@{=}[d]
\\
\mathbb{Q}
\;
\ar@{^{(}->}[rr]
&&
\mathbb{Q}
\oplus
\mathbb{S}_{2d-1}
\ar@{->>}[rr]
&&
\mathbb{S}_{2d-1}
\end{xymatrix}
Now observe that:
1. by Lemma \ref{CofiberECohomologyAsExtensionOfStableCohomotopyByECohomology} the middle rows of this diagram are [[short exact sequence|exact]];
1. the left vertical map is the canonical [[injection]], since the [[Chern character]] is multiplicative, and in particular respects the unit.
By (1) the difference between any two choices of trivializations of the d-invariant (of $H^E_{n-1}$-fluxes) is an integer in the top left, and by (2) this translates to an integer difference between their $\widehat e$-invariants as we pass to the bottom of the diagram.
\end{proof}
Hence we have a diagrammatic construction of an invariant of $[c]$ in
$\mathbb{Q}/\mathbb{Z}$. It just remains to see that this actually coincides with the classical Adams invariant:
\linebreak
### Recovering Adams's construction
{#DiagrammaticEInvariantRecoveringAdamsConstruction}
\begin{remark}
\label{eCInvaiantViaCanonicalSplitting}
Observe that both the classical Adams invariant
and diagrammatic construction from Def. \ref{LiftedEInvariantDiagrammatically} above
extract a rational number via a canonical splitting
in both cases retracting onto rational cohomology
$\simeq \mathbb{Q}$ in degree $2(n+d)$ (Remark \ref{CanonicalSplittingOfCofiberOfPeriodicRationalCohomology}).
For the classical Adams invariant this is the content of Prop. \ref{QModZValuedEInvariantIsTopDegreeCoefficientOfChernCharacterOnCofiberSpace}, which we may summarize more succinctly:
\begin{xymatrix@R=8pt@C=14pt}
\widetilde {\mathrm{KU}}{}^{2n}
\big(
S^{2n}
\big)
\ar[rr]
|-{ \;\mathrm{spl}^{\mathrm{KU}} \circ i_r \; }
^-{
\mbox{
\tiny
\color{green}
\bf
\begin{tabular}{c}
choose lift
\\
\phantom{a}
\end{tabular}
}
}
&&
\widetilde {\mathrm{KU}}{}^{2n}
\big(
C_c
\big)
\ar[rr]
|-{ \;\mathrm{ch}\; }
^-{
\mbox{
\tiny
\color{green}
\bf
\begin{tabular}{c}
Chern character
\\
\phantom{a}
\end{tabular}
}
}
&&
\widetilde {H^{\mathrm{ev}}\mathbb{Q}}{}^{2n}
\big(
C_c
\big)
\ar[rr]
|-{ \;p_l \circ \mathrm{spl}^{H^{\mathrm{ev}}\mathbb{Q}}_0\; }
^-{
\mathclap{
\mbox{
\tiny
\color{green}
\bf
\begin{tabular}{c}
projection onto degree $2(n+d)$
\\
\phantom{A}
\end{tabular}
}
}
}
&&
\widetilde {
H\mathbb{Q}
}{}^{2(n+d)}(C_c)
\ar@{=}[r]
&
\mathbb{Q}
\\
\big[
G^{\mathrm{KU}}_{2n,\mathrm{unit}}
\big]
\ar@{|->}[rr]
&&
\big[
\vdash C^{\mathrm{KU}}_{2n-1}\!(c)
\big]
\ar@{|->}[rr]
&&
\mathrm{ch}
\big[
\vdash C^{\mathrm{KU}}_{2n-1}\!(c)
\big]
\ar@{|->}[rr]
&&
{\color{orange}
\widehat e_{\mathbb{C}}(c)
}
\cdot
\mathrm{ch}
\big[
\vdash H^{\mathrm{KU}}_{2n-1,\mathrm{unit}}
\big]
\ar@{|->}[r]
&
{\color{orange}
\widehat e_{\mathbb{C}}(c)
}
\end{xymatrix}
\end{remark}
Therefore we check that these two canonical splittings are compatible:
\begin{lemma}
\label{CompatibilityOfTheTwoCanonicalSplittings}
Given non-trivial
$\big[ S^{2(n+d)-1} \xrightarrow{\;c\;} S^{2n}\big]
\,\in \, \mathbb{S}_{2d-1}$
for any $n,d \in \mathbb{N}$ with $d \geq 1$,
we have a [[commuting diagram]] of the following form:
\begin{xymatrix@=7pt}
&&
\big(
\widetilde { (H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S} }
\big)^{2n}
\big(
S^{2(n+d)}
\big)
\ar[dd]
|-{
\mathclap{\phantom{\vert^{\vert}}}
p^\ast
\mathclap{\phantom{\vert_{\vert}}}
}
\ar@{=}[dr]|-{ \;\mathrm{spl}_0\; }
\\
&&
&
\mathbb{Q} \oplus \mathbb{S}_{2d-1}
\ar[dd]|-{
\scalebox{.68}{$
\left(
\!\!
\begin{array}{cc}
\mathrm{id} & \cdot
\\
0 & \cdot
\end{array}
\!\!
\right)
$}
}
\\
\big(
\widetilde { H^{\mathrm{ev}}\mathbb{Q} }
\big)^{2n}
\big(
C_c
\big)
\ar@{=}[dr]|-{ \;\mathrm{spl}_0\; }
\ar[rr]|-{ \;i_\ast\; }
&&
\big(
\widetilde { (H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S} }
\big)^{2n}
\big(
C_c
\big)
\ar@{=}[dr]
\\
&
\mathbb{Q} \oplus \mathbb{Q}
\ar[rr]
|-{
\;
\scalebox{.68}{$
\left(
\!\!
\begin{array}{cc}
\mathrm{id} & \cdot
\\
0 & \cdot
\end{array}
\!\!
\right)
$}
\;
}
&&
\mathbb{Q} \oplus
\big(
\mathbb{Q}/\mathbb{Z}
\oplus'
\mathbb{S}_{2d-1}/\mathbb{Z}
\big)
\,,
\end{xymatrix}
\end{lemma}
Here:
* The rear morphisms are pushforward along
the projection $E \xrightarrow{i} E/\mathbb{S}$
and pullback along $C_c \xrightarrow{p} S^{2(n+d)}$,
respectively;
* $\mathrm{spl}_0$ denotes the canonical isomorphisms
from Remark \ref{CanonicalSplittingOfCofiberOfPeriodicRationalCohomology};
* $\mathbb{S}_{2d-1}/\mathbb{Z}$ denotes the [[cofiber]] of $\mathbb{Z} \xrightarrow{\;c\;} \mathbb{S}_{2d-c}$;
* the symbol $\oplus'$ denotes some possibly non-trivial [[group extension|extension]], left undetermined, of [[Q/Z]] by $\mathbb{S}_{d-1}/\mathbb{Z}$, and vice versa;
* the [[matrices]] act as usual, on row vectors by [[matrix multiplication]] from the right.
\begin{proof}
Consider the diagram which is the image under $\pi_0 \mathrm{Maps}^{\ast/}(-,-)$ of the sequences
\begin{xymatrix@R=12pt@C=17pt}
\vdots
\ar@{<-}[d]
\\
S^{2(n+d)}
\ar@{<-}[dd]
\\
\\
C_c
\ar@{<-}[dd]
\ar@{}[r]|-{ , }
&
(H^{\mathrm{ev}}\mathbb{Q})^{2n}
\ar[r]
&
\big(
(H^{\mathrm{ev}}\mathbb{Q})
/
\mathbb{S}
\big)^{2n}
\ar[r]
&
\mathbb{S}^{2n-1}
\\
\\
S^{2n}
\ar@{<-}[d]
\\
\vdots
\end{xymatrix}
In this diagram all rows and columns are [[long exact sequences of generalized cohomology groups]] (since $\mathrm{Maps}^{\ast/}$ sends both homotopy cofiber sequences in the first argument as well as homotopy fiber sequences in the second argument to homotopy fiber sequences, and using that these induce [[long exact sequences of homotopy groups]]).
By the definition (or [[Brown representability theorem|characterization]]) of [[reduced cohomology|reduced]] [[Whitehead-generalized cohomology theory|generalized cohomology]] [[cohomology group|groups]], this diagram is equal (in the sector shown) to the following [[commuting diagram]] of [[abelian groups]]:
\begin{xymatrix@C=18pt}
\widetilde {\mathbb{S}}{}^{2n}
\big(
S^{2n+1}
\big)
\ar[d]
\ar[r]
&
\widetilde { (H^{\mathrm{ev}}\mathbb{Q}) }{}^{2n}
\big(
S^{2n+1}
\big)
\ar[r]
\ar[d]
&
(\widetilde {(H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S}}){}^{2n}
\big(
S^{2n+1}
\big)
\ar[d]
\ar[r]
&
\widetilde {\mathbb{S}}{}^{2n+1}
\big(
S^{2n+1}
\big)
\ar[d]
\\
\widetilde {\mathbb{S}}{}^{2n}
\big(
S^{2(n+d)}
\big)
\ar[r]
\ar[d]
&
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n}
\big(
S^{2(n+d)}
\big)
\;
\ar[r]
\ar[d]
&
(\widetilde {(H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S}}){}^{2n}
\big(
S^{2(n+d)}
\big)
\ar[r]
\ar[d]
&
\widetilde {\mathbb{S}}{}^{2n+1}
\big(
S^{2(n+d)}
\big)
\ar[d]
\ar[r]
&
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n+1}
\big(
S^{2(n+d)}
\big)
\ar[d]
\\
\widetilde {\mathbb{S}}{}^{2n}
\big(
C_c
\big)
\ar[d]
\ar[r]
&
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n}
\big(
C_c
\big)
\;
\ar[r]
\ar[d]
&
(\widetilde {(H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S}}){}^{2n}
\big(
C_c
\big)
\ar[r]
\ar[d]
&
\widetilde {\mathbb{S}}{}^{2n+1}
\big(
C_c
\big)
\ar[d]
\ar[r]
&
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n+1}
\big(
C_c
\big)
\\
\widetilde {\mathbb{S}}{}^{2n}
\big(
S^{2n}
\big)
\ar[d]
\ar[r]
&
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n}
\big(
S^{2n}
\big)
\ar[r]
\ar[d]
&
(\widetilde {(H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S}}){}^{2n}
\big(
S^{2n}
\big)
\ar[r]
&
\widetilde {\mathbb{S}}{}^{2n+1}
\big(
S^{2n}
\big)
\\
\widetilde {\mathbb{S}}{}^{2n}
\big(
S^{2(n+d)-1}
\big)
\ar[r]
&
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n}
\big(
S^{2(n+d)-1}
\big)
\end{xymatrix}
Evaluating here all the cohomology groups on spheres yields:
\begin{xymatrix@=18pt}
{\phantom{
\widetilde {\mathbb{S}}{}^{2n}
\big(
S^{2n+1}
\big)
}}
\hspace{-28pt}
\mathclap{
\color{blue}
\mathbb{S}_{1} \oplus 0
}
\hspace{+28pt}
\ar[d]
\ar[r]
&
{\phantom{
\widetilde { (H^{\mathrm{ev}}\mathbb{Q}) }{}^{2n}
\big(
S^{2n+1}
\big)
}}
\hspace{-43pt}
\mathclap{
\color{blue}
0
}
\hspace{+43pt}
\ar[r]
\ar[d]
&
(\widetilde {(H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S}}){}^{2n}
\big(
S^{2n+1}
\big)
\ar[d]
\ar[r]
&
{\phantom{
\widetilde {\mathbb{S}}{}^{2n+1}
\big(
S^{2n+1}
\big)
}}
\hspace{-35pt}
\mathclap{
\color{blue}
0 \oplus \mathbb{Z}
}
\hspace{+35pt}
\ar[d]
|-{
\mathclap{\phantom{\vert^{\vert}}}
\color{blue}
c
\mathclap{\phantom{\vert_{\vert}}}
}
\\
{\phantom{
\widetilde {\mathbb{S}}{}^{2n}
\big(
S^{2(n+d)}
\big)
}}
\hspace{-37pt}
\mathclap{
\color{blue}
\mathbb{S}_{2d} \oplus 0
}
\hspace{+37pt}
\ar[r]|<<<{ \color{blue}\;0\; }
\ar[d]
&
{\phantom{
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n}
\big(
S^{2(n+d)}
\big)
}}
\hspace{-46pt}
\mathclap{
\color{blue}
\mathbb{Q} \oplus 0
}
\hspace{+46pt}
\;
\ar[r]
\ar[d]
&
(\widetilde {(H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S}}){}^{2n}
\big(
S^{2(n+d)}
\big)
\ar[r]
\ar[d]
&
{\phantom{
\widetilde {\mathbb{S}}{}^{2n+1}
\big(
S^{2(n+d)}
\big)}}
\hspace{-35pt}
\mathclap{
\color{blue}
0 \oplus \mathbb{S}_{2d-1}
}
\hspace{+35pt}
\ar[d]
\ar[r]
&
{\phantom{
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n+1}
\big(
S^{2(n+d)}
\big)
}}
\hspace{-52pt}
\mathclap{
\color{blue}
0
}
\hspace{+52pt}
\ar[d]
\\
\widetilde {\mathbb{S}}{}^{2n}
\big(
C_c
\big)
\ar[d]
\ar[r]
&
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n}
\big(
C_c
\big)
\;
\ar[r]
\ar[d]
&
(\widetilde {(H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S}}){}^{2n}
\big(
C_c
\big)
\ar[r]
\ar[d]
&
\widetilde {\mathbb{S}}{}^{2n+1}
\big(
C_c
\big)
\ar[d]
\ar[r]
&
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n+1}
\big(
C_c
\big)
\\
{\phantom{
\widetilde {\mathbb{S}}{}^{2n}
\big(
S^{2n}
\big)
}}
\hspace{-24pt}
\mathclap{
\color{blue}
0 \oplus \mathbb{Z}
}
\hspace{+24pt}
\ar[d]
|-{
\mathclap{\phantom{\vert^\vert}}
\color{blue}
c
\mathclap{\phantom{\vert_\vert}}
}
\ar[r]
|-{
\color{blue}
\; 1 \mapsto 1\;
}
&
{\phantom{
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n}
\big(
S^{2n}
\big)
}}
\hspace{-37pt}
\mathclap{
\color{blue}
0 \oplus \mathbb{Q}
}
\hspace{+37pt}
\ar[r]
\ar[d]
&
(\widetilde {(H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S}}){}^{2n}
\big(
S^{2n}
\big)
\ar[r]
&
{\phantom{
\widetilde {\mathbb{S}}{}^{2n+1}
\big(
S^{2n}
\big)
}}
\hspace{-30pt}
\mathclap{
\color{blue}
0
}
\hspace{+30pt}
\\
{\phantom{
\widetilde {\mathbb{S}}{}^{2n}
\big(
S^{2(n+d)-1}
\big)
}}
\hspace{-29pt}
\mathclap{
\color{blue}
0 \oplus \mathbb{S}_{2d-1}
}
\hspace{+29pt}
\ar[r]
&
{\phantom{
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n}
\big(
S^{2(n+d)-1}
\big)
}}
\hspace{-52pt}
\mathclap{
\color{blue}
0
}
\hspace{+52pt}
\end{xymatrix}
Now recognizing [[split exact sequences]] using the vanishing [[Ext]]-groups $Ext^1(-,\mathbb{Q}) = 0$ and $Ext^1(\mathbb{Z},-) = 0$ (see [here](Ext#Examples)) yields:
\begin{xymatrix@=18pt}
{\phantom{
\widetilde {\mathbb{S}}{}^{2n}
\big(
S^{2n+1}
\big)
}}
\hspace{-28pt}
\mathclap{
\mathbb{S}_{1} \oplus 0
}
\hspace{+28pt}
\ar[d]
\ar[r]
&
{\phantom{
\widetilde { (H^{\mathrm{ev}}\mathbb{Q}) }{}^{2n}
\big(
S^{2n+1}
\big)
}}
\hspace{-43pt}
\mathclap{
0
}
\hspace{+43pt}
\ar[r]
\ar[d]
&
{\phantom{
(\widetilde {(H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S}}){}^{2n}
\big(
S^{2n+1}
\big)
}}
\hspace{-53pt}
\mathclap{
\color{blue}
0 \oplus \mathbb{Z}
}
\hspace{+53pt}
\ar[d]
\ar[r]
&
{\phantom{
\widetilde {\mathbb{S}}{}^{2n+1}
\big(
S^{2n+1}
\big)
}}
\hspace{-35pt}
\mathclap{
0 \oplus \mathbb{Z}
}
\hspace{+35pt}
\ar[d]
|-{
\mathclap{\phantom{\vert^{\vert}}}
c
\mathclap{\phantom{\vert_{\vert}}}
}
\\
{\phantom{
\widetilde {\mathbb{S}}{}^{2n}
\big(
S^{2(n+d)}
\big)
}}
\hspace{-37pt}
\mathclap{
\mathbb{S}_{2d} \oplus 0
}
\hspace{+37pt}
\ar[r]|<<<{ \;0\; }
\ar[d]
&
{\phantom{
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n}
\big(
S^{2(n+d)}
\big)
}}
\hspace{-46pt}
\mathclap{
\mathbb{Q} \oplus 0
}
\hspace{+46pt}
\;
\ar[r]
\ar[d]
&
{\phantom{
(\widetilde {(H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S}}){}^{2n}
\big(
S^{2(n+d)}
\big)
}}
\hspace{-50pt}
\mathclap{
\color{magenta}
\mathbb{Q} \oplus \mathbb{S}_{2d-1}
}
\hspace{+50pt}
\ar[r]
\ar[d]
&
{\phantom{
\widetilde {\mathbb{S}}{}^{2n+1}
\big(
S^{2(n+d)}
\big)}}
\hspace{-35pt}
\mathclap{
0 \oplus \mathbb{S}_{2d-1}
}
\hspace{+35pt}
\ar[d]
\ar[r]
&
{\phantom{
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n+1}
\big(
S^{2(n+d)}
\big)
}}
\hspace{-52pt}
\mathclap{
0
}
\hspace{+52pt}
\ar[d]
\\
\color{blue}
(\mathbb{S}_{2d}/\mathbb{S}_1)
\oplus
\mathbb{Z}
\ar[d]|-{
\color{blue}
\mathclap{\phantom{\vert^{\vert}}}
0 \,\oplus\, \mathrm{ord}(c)
\mathclap{\phantom{\vert_{\vert}}}
}
\ar[r]
&
{\phantom{
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n}
\big(
C_c
\big)
}}
\hspace{-35pt}
\mathclap{
\color{magenta}
\mathbb{Q} \oplus \mathbb{Q}
}
\hspace{+35pt}
\;
\ar[r]
\ar[d]
&
(\widetilde {(H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S}}){}^{2n}
\big(
C_c
\big)
\ar[r]
\ar[d]
&
\color{blue}
0 \oplus
\mathbb{S}_{2d-1}/\mathbb{Z}
\ar[d]
\ar[r]
&
{\phantom{
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n+1}
\big(
C_c
\big)
}}
\hspace{-41pt}
\mathclap{
\color{blue}
0
}
\hspace{+41pt}
\\
{\phantom{
\widetilde {\mathbb{S}}{}^{2n}
\big(
S^{2n}
\big)
}}
\hspace{-24pt}
\mathclap{
0 \oplus \mathbb{Z}
}
\hspace{+24pt}
\ar[d]
|-{
\mathclap{\phantom{\vert^\vert}}
c
\mathclap{\phantom{\vert_\vert}}
}
\ar[r]
|-{
\; 1 \mapsto 1\;
}
&
{\phantom{
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n}
\big(
S^{2n}
\big)
}}
\hspace{-37pt}
\mathclap{
0 \oplus \mathbb{Q}
}
\hspace{+37pt}
\ar[r]
\ar[d]
&
{\phantom{
(\widetilde {(H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S}}){}^{2n}
\big(
S^{2n}
\big)
}}
\hspace{-48pt}
\mathclap{
\color{blue}
0 \oplus \mathbb{Q}/\mathbb{Z}
}
\hspace{+48pt}
\ar[r]
&
{\phantom{
\widetilde {\mathbb{S}}{}^{2n+1}
\big(
S^{2n}
\big)
}}
\hspace{-30pt}
\mathclap{
0
}
\hspace{+30pt}
\\
{\phantom{
\widetilde {\mathbb{S}}{}^{2n}
\big(
S^{2(n+d)-1}
\big)
}}
\hspace{-29pt}
\mathclap{
0 \oplus \mathbb{S}_{2d-1}
}
\hspace{+29pt}
\ar[r]
&
{\phantom{
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n}
\big(
S^{2(n+d)-1}
\big)
}}
\hspace{-52pt}
\mathclap{
0
}
\hspace{+52pt}
\end{xymatrix}
Here the two identifications shown in magenta
we may *choose to be the canonical ones*
from Remark \ref{CanonicalSplittingOfCofiberOfPeriodicRationalCohomology}.
From this, and using again $\mathrm{Ext}^1(-,\mathbb{Q}) = 0$
together with
$\mathrm{Ext}^1(-, A \oplus B )
\simeq
\mathrm{Ext}^1(-,A) \oplus \mathrm{Ext}^1(-,B)$
(by [this Prop.](Ext#ExtCompatibleWithDirectSumsAndProducts)),
the remaining entry and the maps into it must be as claimed:
\begin{xymatrix@=18pt}
{\phantom{
\widetilde {\mathbb{S}}{}^{2n}
\big(
S^{2n+1}
\big)
}}
\hspace{-23pt}
\mathclap{
\mathbb{S}_{1} \oplus 0
}
\hspace{+23pt}
\ar[d]
\ar[r]
&
{\phantom{
\widetilde { (H^{\mathrm{ev}}\mathbb{Q}) }{}^{2n}
\big(
S^{2n+1}
\big)
}}
\hspace{-38pt}
\mathclap{
0
}
\hspace{+38pt}
\ar[r]
\ar[d]
&
{\phantom{
(\widetilde {(H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S}}){}^{2n}
\big(
S^{2n+1}
\big)
}}
\hspace{-48pt}
\mathclap{
0 \oplus \mathbb{Z}
}
\hspace{+48pt}
\ar[d]
\ar[r]
&
{\phantom{
\widetilde {\mathbb{S}}{}^{2n+1}
\big(
S^{2n+1}
\big)
}}
\hspace{-30pt}
\mathclap{
0 \oplus \mathbb{Z}
}
\hspace{+30pt}
\ar[d]
_-{
\mathclap{\phantom{\vert^{\vert}}}
c
\mathclap{\phantom{\vert_{\vert}}}
}
\\
{\phantom{
\widetilde {\mathbb{S}}{}^{2n}
\big(
S^{2(n+d)}
\big)
}}
\hspace{-32pt}
\mathclap{
\mathbb{S}_{2d} \oplus 0
}
\hspace{+32pt}
\ar[r]^-{ \;0\; }
\ar[d]
&
{\phantom{
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n}
\big(
S^{2(n+d)}
\big)
}}
\hspace{-41pt}
\mathclap{
\mathbb{Q} \oplus 0
}
\hspace{+41pt}
\;
\ar[r]
\ar[d]
&
{\phantom{
(\widetilde {(H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S}}){}^{2n}
\big(
S^{2(n+d)}
\big)
}}
\hspace{-45pt}
\mathclap{
\mathbb{Q} \oplus \mathbb{S}_{2d-1}
}
\hspace{+45pt}
\ar[r]
\ar[d]
_-{
\scalebox{.65}{$
\left(
\!\!
\begin{array}{cc}
\mathrm{id} & \cdot
\\
0 & \cdot
\end{array}
\!\!
\right)
$}
}
&
{\phantom{
\widetilde {\mathbb{S}}{}^{2n+1}
\big(
S^{2(n+d)}
\big)}}
\hspace{-30pt}
\mathclap{
0 \oplus \mathbb{S}_{2d-1}
}
\hspace{+30pt}
\ar[d]
\ar[r]
&
{\phantom{
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n+1}
\big(
S^{2(n+d)}
\big)
}}
\hspace{-47pt}
\mathclap{
0
}
\hspace{+47pt}
\ar[d]
\\
(\mathbb{S}_{2d}/\mathbb{S}_1)
\oplus
\mathbb{Z}
\ar[d]_-{
\mathclap{\phantom{\vert^{\vert}}}
0 \,\oplus\, \mathrm{ord}(c)
\mathclap{\phantom{\vert_{\vert}}}
}
\ar[r]
&
{\phantom{
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n}
\big(
C_c
\big)
}}
\hspace{-30pt}
\mathclap{
\mathbb{Q} \oplus \mathbb{Q}
}
\hspace{+30pt}
\;
\ar[r]
^-{
\scalebox{.65}{$
\left(
\!\!
\begin{array}{cc}
\mathrm{id} & \cdot
\\
0 & \cdot
\end{array}
\!\!
\right)
$}
}
\ar[d]
&
\color{magenta}
\mathbb{Q}
\oplus
\!
\big(
\mathbb{Q}/\mathbb{Z}
\oplus'
\mathbb{S}_{2d-1}/\mathbb{Z}
\big)
\ar[r]
\ar[d]
&
0 \oplus
\mathbb{S}_{2d-1}\!/\mathbb{Z}
\ar[d]
\ar[r]
&
{\phantom{
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n+1}
\big(
C_c
\big)
}}
\hspace{-36pt}
\mathclap{
0
}
\hspace{+36pt}
\\
{\phantom{
\widetilde {\mathbb{S}}{}^{2n}
\big(
S^{2n}
\big)
}}
\hspace{-19pt}
\mathclap{
0 \oplus \mathbb{Z}
}
\hspace{+19pt}
\ar[d]
_-{
\mathclap{\phantom{\vert^\vert}}
c
\mathclap{\phantom{\vert_\vert}}
}
\ar[r]
|-{
\; 1 \mapsto 1\;
}
&
{\phantom{
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n}
\big(
S^{2n}
\big)
}}
\hspace{-32pt}
\mathclap{
0 \oplus \mathbb{Q}
}
\hspace{+32pt}
\ar[r]
\ar[d]
&
{\phantom{
(\widetilde {(H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S}}){}^{2n}
\big(
S^{2n}
\big)
}}
\hspace{-43pt}
\mathclap{
0 \oplus \mathbb{Q}/\mathbb{Z}
}
\hspace{+43pt}
\ar[r]
&
{\phantom{
\widetilde {\mathbb{S}}{}^{2n+1}
\big(
S^{2n}
\big)
}}
\hspace{-25pt}
\mathclap{
0
}
\hspace{+25pt}
\\
{\phantom{
\widetilde {\mathbb{S}}{}^{2n}
\big(
S^{2(n+d)-1}
\big)
}}
\hspace{-24pt}
\mathclap{
0 \oplus \mathbb{S}_{2d-1}
}
\hspace{+24pt}
\ar[r]
&
{\phantom{
\widetilde {(H^{\mathrm{ev}}\mathbb{Q})}{}^{2n}
\big(
S^{2(n+d)-1}
\big)
}}
\hspace{-47pt}
\mathclap{
0
}
\hspace{+47pt}
\end{xymatrix}
\end{proof}
\begin{theorem}
\label{DiagrammaticeCInavriantReproducedClassicaleCInvariant}
The diagrammatic $\widehat e_{\mathrm{KU}}$-invariant
from Def. \ref{LiftedEInvariantDiagrammatically}
reproduces the classical Adams $e_{\mathbb{C}}$-invariant.
\end{theorem}
\begin{proof}
The homotopy-commuting rectangle in the
bottom right part of the defining pasting diagram
Def. \ref{LiftedEInvariantDiagrammatically}
says that
$$
p^\ast
\Big(
\widehat e_{\mathrm{KU}}(c)
,
[c]
\Big)
\;=\;
i_\ast
\Big(
\mathrm{ch}
\big[
\vdash H^{\mathrm{KU}}_{2n-1}(c)
\big]
\Big)
\,.
$$
By Lemma \ref{CompatibilityOfTheTwoCanonicalSplittings}
this means that the image of both sides
along their canonical retractions
(Remark \ref{CanonicalSplittingOfCofiberOfPeriodicRationalCohomology})
onto degree=$2(n+d)$ rational cohomology $\simeq \mathbb{Q}$
coincide. But by Definition
\ref{LiftedEInvariantDiagrammatically} and
Remark \ref{eCInvaiantViaCanonicalSplitting},
this is the claimed equality
(even at the level of refined invariants, i.e. before forgetting choices of trivializations of the d-invaariant, hence before quotienting by $\mathbb{Q}$).
\end{proof}
\linebreak
### Recovering Conner-Floyd's e-invariant
{#ConnerFloydTheoremViaPastingDiagram}
With this diagrammatic formulation of the e-invariant, the Conner-Floyd theorem -- that the [[e-invariant is the Todd class of cobounding (U,fr)-manifolds]] -- follows as an immediate corollary: It just amounts to factoring the above pasting composite further through [[MU]], as follows:
\begin{xymatrix@C=15pt}
S^{2(n + d)-1}
\ar[dd]_-{ c }
\ar[rr]
\ar@{}[ddrr]|-{
\rotatebox[origin=c]{-45}{\color{orange}$\big\Downarrow$}
\mbox{\tiny (po)}
}
&&
\ast
\ar[dr]
\ar[dd]
\\
&& &
\ast
\ar[dd]|-{
\mathclap{\phantom{\vert^{\vert}}}
0
\mathclap{\phantom{\vert_{\vert}}}
}
\\
S^{2n}
\ar[dd]
\ar[rr]
\ar@{}[ddrr]|-{
\rotatebox[origin=c]{-45}{\color{orange}$\big\Downarrow$}
\mbox{\tiny (po)}
}
\ar@/_.7pc/[drrr]
|<<<<<{ \;\Sigma^{2n} (1^{M\mathrm{U}})\;\; }
|>>>>>>>>>{ {\phantom{AA}} \atop {\phantom{AA}} }
&&
C_c
\ar[dd]
\ar@{-->}[dr]^-{
}
%\ar@/_2pc/[dddrrr]
% |>>>>>>>>>>>>>>>{
% \mathclap{\phantom{\vert}}
% \;{\color{magenta} e(f)}\;
% }
&&
\\
&& &
M\mathrm{U}^{2n}
\ar[dd]
\ar[dr]|-{
\;\sigma\;
}
\ar@/^1.7pc/[ddrr]|>>>>>>>>>>>{
\;
\mathclap{\phantom{\vert^{\vert}}}
\mathrm{Td}
\mathclap{\phantom{\vert_{\vert}}}
\;
}
\\
\ast
\ar@/_.7pc/[drrr]|-{ \;0\; }
\ar[rr]
&&
S^{2(n+d)}
\ar@{-->}[dr]|-{
\;
\mathclap{\phantom{\vert^{\vert}}}
\color{magenta}M^{2d}
\mathclap{\phantom{\vert_{\vert}}}
}
&&
\mathrm{KU}^{2n}
\ar[dr]|-{
\;
\mathrm{ch}
\;
\mathclap{\phantom{\vert_{\vert}}}
}
\ar[dd]|>>>>>>{ \phantom{\vert_{\vert}} }
\\
&& &
(M\mathrm{U}/\mathbb{S})^{2n}
\ar@/^1.7pc/[ddrr]
|>>>>>>>>>{
\mathclap{\phantom{\vert^{\vert}}}
\;\mathrm{Td}/\mathbb{S}\;
\mathclap{\phantom{\vert_{\vert}}}
}
\ar[dr]|-{ \;\sigma/\mathbb{S}\; }
&&
(H^{\mathrm{ev}}\mathbb{Q})^{2n}
\ar[dd]
\\
&& &&
(\mathrm{KU}/\mathbb{S})^{2n}
\ar[dr]|-{ \;\mathrm{ch}/\mathbb{S}\; }
\\
&& && &
\big(
(H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S}
\big)^{2n}
\end{xymatrix}
using that [[the rational Todd class is the Chern character of the Thom class]] and that this identification is represented by a corresponding factorization of maps of multiplicative cohomology theories, as shown (e.g. [Smith 73, Section 1](rational+Todd+class+is+Chern+character+of+Thom+class#Smith73))
\linebreak
## Examples
### Third stable homotopy group of spheres
{#ThirdStableStem}
+-- {: .num_prop #AdamseRInvariantDetectsThirdStableHomotopyGroupOfSpheres}
###### Proposition
**([Adams 66, Example 7.17 and p. 46](#Adams66))**
In degree 3, the [[KO]]-theoretic [[e-invariant]] $e_{\mathbb{R}}$ takes the value $\left[\tfrac{1}{24}\right] \in \mathbb{Q}/\mathbb{Z}$ on the [[quaternionic Hopf fibration]] $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$ and hence reflects the full [[third stable homotopy group of spheres]]:
$$
\array{
\pi^s_3
&
\underoverset{
\simeq
}{
e_{\mathbb{R}}
}{
\;\;\longrightarrow\;\;
}
&
\mathbb{Z}/24
&
\subset
&
\mathbb{Q}/\mathbb{Z}
\\
[h_{\mathbb{H}}]
&&\mapsto&&
\left[\tfrac{1}{24}\right]
}
$$
while $e_{\mathbb{C}}$ sees only "half" of it (by [Adams 66, Prop. 7.14](#Adams66)).
=--
## Related concepts
* [[eta-invariant]]
* [[d-invariant]], [[f-invariant]]
* [[Hopf invariant]], [[Hopf invariant one problem]]
* [[J-homomorphism]]
* [[stable homotopy groups of spheres]]
## References
The definition is due to:
* {#Adams66} [[John Adams]], Sections 3,7 of: _On the groups $J(X)$ IV_, Topology 5: 21 (1966) ([pdf](http://math.unice.fr/~cazanave/Gdt/ImJ/J-IV.pdf), <a href="https://doi.org/10.1016/0040-9383(66)90004-8">doi:10.1016/0040-9383(66)90004-8</a>)
_Correction_, Topology 7 (3): 331 (1968)
Discussion in more general [[Whitehead generalized cohomology theories]]:
* {#Krueger73} Warren M. Krueger, _Generalized Steenrod-Hopf Invariants for Stable Homotopy Theory_, Proceedings of the American Mathematical Society, Vol. 39, No. 3 (Aug., 1973), pp. 609-615 ([jstor:2039603](https://www.jstor.org/stable/2039603))
* Warren M. Krueger, _Relation with the Hopf invariant revisited_, Illinois J. Math.
Volume 24, Issue 2 (1980), 188-191 ([euclid:ijm/1256047713](https://projecteuclid.org/euclid.ijm/1256047713))
and in relation to the [[Adams spectral sequence]]:
* {#Switzer75} [[Robert Switzer]], Section 19.19 in: _Algebraic Topology - Homotopy and Homology_, Grundlehren der Mathematischen Wissenschaften, Vol. 212, Springer, 1975 ([doi:10.1007/978-3-642-61923-6](https://link.springer.com/book/10.1007/978-3-642-61923-6))
and to the [[f-invariant]]:
* {#BL09} Mark Behrens, Gerd Laures, _$\beta$-Family congruences and the $f$-invariant_, Geometry & Topology Monographs 16 (2009) 9–29 ([arXiv:0809.1125](https://arxiv.org/abs/0809.1125), [doi: 10.2140/gtm.2009.16.9](https://msp.org/gtm/2009/16/p002.xhtml))
Interpretation in [[MUFr]] [[bordism theory]]:
* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], Section 16 of: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__, Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))
Interpretation via [[index theory]]:
* [[Michael Atiyah]], [[Vijay Patodi]], [[Isadore Singer]], p. 18 onwards in: _Spectral asymmetry and Riemannian geometry. II_, Volume 78, Issue 3 November 1975 , pp. 405-432 ([doi:10.1017/S0305004100051872](https://doi.org/10.1017/S0305004100051872))
Review and exposition:
* {#Weibel} [[Charles Weibel]], chapter VI, section 2 of _[The K-book](http://www.math.rutgers.edu/~weibel/Kbook.html)_ ([pdf](http://www.math.rutgers.edu/~weibel/Kbook/Kbook.VI.pdf))
* [[Ulrich Bunke]], Section 2 of: _Differential cohomology in geometry and analysis_, talk 2008 ([pdf](https://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/Bunke/Vortrag-Erlangen.pdf), [[BunkeEInvariantErlangen2008.pdf:file]])
* {#Wirthmuller12} [[Klaus Wirthmüller]], p. 52 (54 of 67) in: _Vector bundles and K-theory_, 2012 ([[wirthmueller-vector-bundles-and-k-theory.pdf:file]])
* {#HopkinsMathew12} [[Michael Hopkins]] (notes by [[Akhil Mathew]]), Lecture 11 in: _Spectra and stable homotopy theory_, 2012 ([pdf](http://math.uchicago.edu/~amathew/256y.pdf), [[HopkinsMathewStableHomotopyTheory.pdf:file]])
* {#Quick14EInvariant} [[Gereon Quick]], _The $e$-invariant_, lecture notes in: _[Advanced algebraic topology](https://folk.ntnu.no/gereonq/Math231br.html)_, 2014 ([[QuickEInvariant.pdf:file]])
* {#Quick14JHomomorphism} [[Gereon Quick]], _The $e$-invariant and the $J$-homomorphism_, lecture notes in: _[Advanced algebraic topology](https://folk.ntnu.no/gereonq/Math231br.html)_, 2014 ([[QuickEInvariantAndJHomomorphism.pdf:file]])
Discussion via [[Toda brackets]]:
* Hiroaki Hamanaka, _Adams $e$-invariant, Toda bracket and $[X, U(n)]$_, J. Math. Kyoto Univ. Volume 43, Number 4 (2003), 815-827. ([euclid:kjm/1250281737]( http://projecteuclid.org/euclid.kjm/1250281737))
Discussion in [[MU-theory]]:
* N. V. Panov, _Characteristic numbers in $U$-theory_, Akad. Nauk SSSR Ser. Mat., 1971 Volume 35, Issue 6 ([mathnet:2174](http://m.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=2174&option_lang=eng))
Discussion in [[BP-theory]]:
* Yasumasa Hirashima, _On the $BP_\ast$-Hopf invariant_, Osaka J. Math., Volume 12, Number 1 (1975), 187-196 ([euclid:ojm/1200757733](https://projecteuclid.org/euclid.ojm/1200757733))
* [[Martin Bendersky]], _The BP Hopf Invariant_, American Journal of Mathematics, Vol. 108, No. 5 (Oct., 1986) ([jstor:2374595](https://www.jstor.org/stable/2374595))
[[!redirects Adams e-invariants]]
[[!redirects e-invariant]]
[[!redirects e-invariants]]
[[!redirects Adams E-invariant]]
[[!redirects E-invariant]]
[[!redirects Adams E-invariants]]
[[!redirects E-invariants]]
|
Adams operation | https://ncatlab.org/nlab/source/Adams+operation |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Given a [[complex line bundle]] $L$ over a [[space]] $X$ its $k$th [[tensor product of vector bundles|tensor power]] $L^{\otimes k}$ is another line bundle for any $k \in \mathbb{N}$. The line bundles define certain elements of [[complex topological K-theory]] group $K(X)$, and there is a unique [[cohomology operation]]
$$
\psi^k \colon K(X) \to K(X)
\,,
$$
the _$k$th Adams operation_, such that:
* $\psi^k([L]) = [L^{\otimes k}]$ if $[L]$ is the K-theory class of any line bundle,
* $\psi^k \colon K(X) \to K(X)$ is a [[group homomorphism]],
* $\psi^k$ is a [[natural transformation]]: any map $f: X \to Y$ induces a map $f^* : K(Y) \to K(X)$ on $K$-theory, and $\psi^k \circ f^* = f^* \circ \psi^k$.
More abstractly, Adams operations can be defined on any [[Lambda-ring]]. They are an example of [[power operations]].
## Definition
{#Definition}
The Adams operations have an explicit definition in terms of the [[Lambda-ring]] structure on [[topological K-theory]], this we state as def. \ref{DefinitionInTermsOfLambdaRing} below. While explicit, this definition may look contrived on first sight. But it turns out that it satisfies a list of properties, of which two simple ones already uniquely characterize the Adams operations. This is proposition \ref{TheBasicProperties} below.
+-- {: .num_prop #LambdaRingStructureOnKTheory}
###### Definition
**(Lambda-ring structure on topological K-theory)**
Let $X$ be a [[compact topological space]] and write $K(X)$ for its [[topological K-theory]] [[ring]]. For $E$ a [[vector bundle]] over $X$, write $[E] \in K(X)$ for its class in K-theory. Given $E$, write
$$
\lambda_t[E]
\;\coloneqq\;
\underoverset{k = 0}{\infty}{\sum}
[\wedge^k_X E] t^k
\;\;\in\;\;
K(X)[ [t] ]
$$
for the [[formal power series]] with [[coefficients]] in the ring $K(X)$ being the K-theory classes of the skew-symmetrized [[tensor product of vector bundles]] of $E$ with itself.
Since the constant term of this power series is always the unit $[\wedge^0 E] = 1$, hence
$$
\lambda_t[E] \in 1 + (t) \cdot K(X)[ [t] ]
$$
there exists a multiplicative inverse formal power series $\lambda_t[E]^{-1}$.
Then given the class of a [[virtual vector bundle]] $[E] - [F] \in K(X)$, define more generally
$$
\lambda_t[[E- F]]
\;\;\coloneqq\;\,
\lambda_t[E] \cdot \lambda_t[F]^{-1}
\;\;\in\;\;
K(X)[ [t] ]
\,.
$$
=--
+-- {: .num_prop #DefinitionInTermsOfLambdaRing}
###### Definition
**(explicit definition of Adams operation)**
For $E$ a [[vector bundle]] over some [[topological space]] $X$, write
$$
\psi^0(E) \coloneqq rank(E)
$$
for the [[bundle]] which over each [[connected component]] of $X$ is the [[trivial bundle|trivial]] vector bundle of the same [[rank of a vector bundle|rank]] as $E$ over that component.
Define a [[formal power series]] with [[coefficients]] in the K-theory ring $K(X)$ by
$$
\begin{aligned}
\psi_t(E)
& \coloneqq
\underoverset{\infty}{k = 0}{\sum}
\psi^k(E) t^k
\\
& \coloneqq
\psi^0(E)
-
t \frac{d}{d t} log \lambda_{-t}(E)
\;\;\in\;\;
K(X)[ [t] ]
\end{aligned}
\,,
$$
where $\lambda_t$ is the [[Lambda-ring]] operation from def. \ref{LambdaRingStructureOnKTheory}.
Here the [[derivative]] of the [[logarithm]] of formal power series stands for the usual expression in terms of the [[geometric series]]:
$$
\begin{aligned}
\frac{d}{d t} log \lambda_{-t}(E)
& =
\frac{1}{\lambda_{-t}(E)}
\frac{d}{d t}
\lambda_{-t}(E)
\\
& =
\underoverset{\infty}{k = 0}{\sum}
\left(
1 - \lambda_{-t}(E)
\right)^k
\cdot
\frac{d}{d t} \lambda_{-t}(E)
\end{aligned}
\,.
$$
The **$k$th Adams operation** is the [[cohomology operation]] on [[topological K-theory]]
$$
\psi^k \;\colon\; K(-) \longrightarrow K(-)
$$
which is the coefficient of $t^k$ in $\psi_t$.
=--
+-- {: .num_prop #TheBasicProperties}
###### Proposition
**(basic properties and characterization of Adams operations)**
The Adams operations
$$
\psi^k \;\colon\; K(X) \longrightarrow K(X)
$$
have the following properties, for all elements $x,y \in K(X)$ and $k, l \in \mathbb{N}$ and $p \; \text{prime}$:
1. $\psi^k(x + y) = \psi^k(x) + \psi^k(y)$
($\psi^k$ is a [[natural transformation|natural]] [[abelian group]] [[homomorphism]])
1. $x \,\text{a line} \;\;\Rightarrow\;\; \psi^k(x) = x^k$
(applied to a class $x \coloneqq [L]$ represented by a [[line bundle]] $L$, $\psi^k$ is the $k$th [[tensor power]])
1. $\psi^k(x \cdot y) = \psi^k(x) \cdot \psi^k(y)$
($\psi^k$ is in fact a [[natural transformation|natural]] [[ring]] [[homomorphism]])
1. $\psi^k(\psi^l(x)) = \psi^{k l}(x)$
1. $\psi^p(x) = x^p \, \text{mod}\, p$
1. if $x \in \tilde K(S^{2n})$ ([[reduced cohomology]]) then
$x \in \tilde K(S^{2n}) \hookrightarrow K(S^{2n}) \;\;\Rightarrow\;\;\psi^k(x) = k^n \cdot x$.
Moreover, the first two of these already uniquely characterize the Adams operations.
=--
e.g. [Wirthmuller 12, section 11](#Wirthmuller12)
## Properties
### Basic properties
+-- {: .num_prop #AdamsOperationOnComplexTopologicalKtheoryOfnSpheres}
###### Proposition
**(Adams operations on [[complex topological K-theory]] of [[n-spheres]])**
For $n \in \mathbb{N}$, the [[Adams operations]] on the [[reduced K-theory]] of the [[n-sphere|2n-sphere]] are given by:
$$
\array{
\widetilde K
\big(
S^{2n}
\big)
&
\overset{ \;\;\; \psi^k\;\;\; }{\longrightarrow}
&
\widetilde K
\big(
S^{2n}
\big)
\\
V &\mapsto& k^n \cdot V
}
$$
=--
(e.g. [Wirthmüller, Prop. on p. 45 (47 of 67)](#Wirthmuller12))
### Compatibility with complexification
{#CompatibilityWithComplexification}
The Adams operations are compatible with the [[complexification]] map $(-) \otimes_{\mathbb{R}} \mathbb{C}$ from [[real vector bundles]] to [[complex vector bundles]], hence from [[KO-cohomology]] to [[KU-cohomology]], in that the following [[commutative diagram|diagram commutes]], for all $k$:
$$
\array{
KO(X)
&\overset{ \psi^k }{\longrightarrow}&
KO(X)
\\
{}^{ \mathllap{ (-) \otimes_{\mathbb{R}} \mathbb{C} } }
\big\downarrow
&&
\big\downarrow
{}^{ \mathrlap{ (-) \otimes_{\mathbb{R}} \mathbb{C} } }
\\
KO(X)
&\overset{ \psi^k }{\longrightarrow}&
KO(X)
}
$$
([Adams 62, Thm. 5.1. (iv)](#Adams62), [Karoubi 78, Prop. IV.7.25](#Karoubi78))
### Compatibility with the Chern character
{#CompatibilityWithTheChernCharacter}
The Adams operation are compatible with the [[Chern character map]] in the following way:
+-- {: .num_defn #AdamsLikeOperationsOnRationalCohomology}
###### Definition
**(Adams-like operations on [[rational cohomology]])**
For $X$ a [[topological space]], with [[rational cohomology]] in even degrees denoted
$$
H^{ev}(X;\, \mathbb{Q})
\;\coloneqq\;
\underset{r \in \mathbb{N}}{\prod}
H^{2 r}(X;\, \mathbb{Q})
$$
define graded [[linear maps]]
$$
\psi^k_{H} \;\colon\; H^{ev}(X) \longrightarrow H^{ev}(X)
$$
for $k \in \mathbb{N}$ by taking their restriction to degree $2r$ to act by multiplication with $k^r$
$$
\array{
H^{2r}(X;\mathbb{Q})
&\overset{\;\;\;\psi^k_H\;\;\;}{\longrightarrow}&
H^{2r}(X;\mathbb{Q})
\\
\alpha_{2k} &\mapsto& k^{r} \cdot \alpha_{2r}
\,.
}
$$
=--
+-- {: .num_prop #AdamsOperationsAreCompatibleWithTheChernCharacter }
###### Proposition
**([[Adams operations compatible with the Chern character]])**
For $X$ a [[topological space]] with a finite [[CW-complex]]-[[mathematical structure]], the [[Chern character]] $ch$ on the [[complex topological K-theory]] of $X$ intertwines the [[Adams operations]] $\psi^n$ on K-theory with the Adams-like operations $\psi^n_H$ on [[rational cohomology]] from Def. \ref{AdamsLikeOperationsOnRationalCohomology}, for $k \geq 1$, in that the following [[commuting diagram|diagram commutes]]:
$$
\array{
K(X)
&\overset{\;\;\;ch\;\;\;}{\longrightarrow}&
H^{ev}(X;\,\mathbb{Q})
\\
{}^{ \mathllap{ \psi^k } }
\big\downarrow
&&
\big\downarrow
{}^{ \mathrlap{ \psi^k_H } }
\\
K(X)
&\overset{\;\;\;ch\;\;\;}{\longrightarrow}&
H^{ev}(X;\,\mathbb{Q})
\,,
}
$$
=--
([Adams 62, Thm. 5.1. (vi)](#Adams62), review in [Karoubi 78, Chapter V, Theorem 3.27](#Karoubi78), [Maakestad 06, Thm. 4.9](#Maakestad06))
+-- {: .proof}
###### Proof idea
Use the exponentional-formula for the [[Chern character]] with the [[splitting principle]].
=--
\linebreak
### Adams conjecture
The [[Adams conjecture]] (a [[theorem]]) says that for all $k \in \mathbb{N}$ and $V \in K(X)$ there is $n \in \mathbb{N}$ such that the [[spherical fibration]] assigned to the [[K-theory]] class $k^n (\psi^k(V)-V)$ under the [[J-homomorphism]] is trivial, hence that
$$
J
\left(
k^n \left(
\psi^k(V) - V
\right)
\right)
= 0
\,.
$$
## Related concepts
* [[Lambda-ring]]
* [[Steenrod square]]
* [[Steenrod operation]]
* [[power operation]]
* [[Adams operation on Jacobi diagrams]]
## References
### General
The original article:
* {#Adams62} [[John Adams]], Section 5 of: _Vector fields on spheres_, Bull. Amer. Math. Soc. Volume 68, Number 1 (1962), 39-41 ([euclid:bams/1183524456](https://projecteuclid.org/euclid.bams/1183524456), [pdf](https://www.math.ens.fr/~benoist/refs/Adams.pdf))
See also:
* [[Victor M. Buchstaber]], [[Sergei P. Novikov]], _Formal groups, power systems and Adams operators_, Math. USSR-Sb.13 (1971) 80-116 [[doi:10.1070/SM1971v013n01ABEH001030](https://doi.org/10.1070/SM1971v013n01ABEH001030)]
Review:
* {#Karoubi78} [[Max Karoubi]], Section IV.7 in in: _K-Theory -- An introduction_, Grundlehren der mathematischen Wissenschaften 226, Springer 1978 ([pdf](https://webusers.imj-prg.fr/~max.karoubi/K.book/MK.book.pdf), [doi:10.1007/978-3-540-79890-3](https://link.springer.com/book/10.1007%2F978-3-540-79890-3))
* {#AguilarGitlerPrieto} Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, Section 10 of: _Algebraic topology from a homotopical viewpoint_, Springer (2002)
* {#Maakestad06} Helge Maakestad, _Notes on the Chern-character_, Journal of Generalized Lie Theory and Applications, 2017, 11:1 ([arXiv:math/0612060](https://arxiv.org/abs/math/0612060), [doi:10.4172/1736-4337.100025](https://www.hilarispublisher.com/open-access/notes-on-the-cherncharacter-.pdf))
* [[Michael Hopkins]] (notes by [[Akhil Mathew]]), Lecture 10 in: _Spectra and stable homotopy theory_, 2012 ([pdf](http://math.uchicago.edu/~amathew/256y.pdf), [[HopkinsMathewStableHomotopyTheory.pdf:file]])
* {#Wirthmuller12} [[Klaus Wirthmüller]], Section 11 of: _Vector Bundles and K-theory_, 2012 ([[wirthmueller-vector-bundles-and-k-theory.pdf:file]])
* {#Hatcher} [[Allen Hatcher]], section 2.3 of _Vector bundles and K-theory_ ([web](https://www.math.cornell.edu/~hatcher/VBKT/VBpage.html))
* Wikipedia, _[Adams operation](http://en.wikipedia.org/wiki/Adams_operation)_
* [[Jacob Lurie]], remark 2 in: _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 35 _The image of $J$_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture35.pdf))
### In representation theory
{#ReferencesInRepresentationTheory}
Adams operations on the [[representation ring]] (the [[equivariant K-theory]] of the point) are discussed in
* {#tomDieck79} [[Tammo tom Dieck]], section 3.5 of _[[Transformation Groups and Representation Theory]]_, Lecture Notes in Mathematics 766 Springer 1979
* Robert Boltje, _A characterization of Adams operations on representation rings_, 2001 ([pdf](https://boltje.math.ucsc.edu/publications/p01v.pdf))
* [[Dai Tamaki]], [[Akira Kono]], Section 4.3 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))
* {#tomDieck09} [[Tammo tom Dieck]], section 6.2 of _Representation theory_, 2009 ([pdf](http://www.uni-math.gwdg.de/tammo/rep.pdf))
* [[Michael Boardman]], _Adams operations on Group representations_, 2007 ([pdf](http://www.math.jhu.edu/~jmb/note/adamrept.pdf))
* Ehud Meir, Markus Szymik, _Adams operations and symmetries of representation categories_ ([arXiv:1704.03389](https://arxiv.org/abs/1704.03389))
### In knot theory
Adams operations on [[Jacobi diagrams]] [[quotient vector space|modulo]] [[STU relations]] ([[Adams operation on Jacobi diagrams]]) are discussed in
* {#BarNatan95} [[Dror Bar-Natan]], Theorem 7, Def. 3.11 and Section 6.3.4 of: _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (<a href="https://doi.org/10.1016/0040-9383(95)93237-2">doi:10.1016/0040-9383(95)93237-2</a>, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))
[[!redirects Adams operations]] |
Adams operation on Jacobi diagrams | https://ncatlab.org/nlab/source/Adams+operation+on+Jacobi+diagrams |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
On the [[vector space]] $\mathcal{A}$ of [[Jacobi diagrams]] [[quotient vector space|modulo]] [[STU-relations]] ([[chord diagrams modulo 4T are Jacobi diagrams modulo STU|equivalently]] [[chord diagrams]] [[quotient vector space|modulo]] [[4T relations]]) there is a system of [[linear maps]] (for $q \in \mathbb{Z}$, $q \neq 0$)
$$
\psi^q
\;\colon\;
\mathcal{A}
\longrightarrow
\mathcal{A}
$$
which respect the [[coalgebra]] [[mathematical structure|structure]] and satisfy
$$
\psi^{q_2} \circ \psi^{q_1}
\;=\;
\psi^{q_1 \cdot q_2}
$$
and as such are (dually) analogous to the [[Adams operations]] on [[topological K-theory]].
([Bar-Natan 95, Def. 3.11 & Theorem 7](#BarNatan95))
In fact, when evaluated in [[Lie algebra weight systems]] $w_{\mathbf{N}}$ and under the identification (see [here](equivariant+K-theory#RelationToRepresentationTheory)) of the [[representation ring]] of a [[compact Lie group]] $G$ with the $G$-[[equivariant K-theory]] of the point, these Adams operations on [[Jacobi diagrams]] correspond to the [[Adams operations]] on [[equivariant K-theory]]:
$$
w_{\mathbf{N}}(\psi^q D)
\;=\;
w_{\psi^q \mathbf{N}}(D)
\,.
$$
([Bar-Natan 95, Exc. 6.24](#BarNatan95)
)
## References
* {#BarNatan95} [[Dror Bar-Natan]], _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (<a href="https://doi.org/10.1016/0040-9383(95)93237-2">doi:10.1016/0040-9383(95)93237-2</a>, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))
[[!redirects Adams operations on Jacobi diagrams]]
|
Adams operations compatible with the Chern character | https://ncatlab.org/nlab/source/Adams+operations+compatible+with+the+Chern+character |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[Adams operations]] $\psi^k$ on [[complex topological K-theory]] are compatible with the [[Chern character map]] to [[rational cohomology]] in that the effect of $\psi^k$ on the Chern character image in degree $2r$ is multiplication by $k^r$.
## Statement
+-- {: .num_defn #AdamsLikeOperationsOnRationalCohomology}
###### Definition
**(Adams-like operations on [[rational cohomology]])**
For $X$ a [[topological space]], with [[rational cohomology]] in even degrees denoted
$$
H^{ev}(X;\, \mathbb{Q})
\;\colon\;
\underset{r \in \mathbb{N}}{\prod}
H^{2 r}(X;\, \mathbb{Q})
$$
define graded [[linear maps]]
$$
\psi^k_{H} \;\colon\; H^{ev}(X) \longrightarrow H^{ev}(X)
$$
for $k \in \mathbb{N}$ by taking their restriction to degree $2r$ to act by multiplication with $k^r$:
\[
\label{AdamsOperationOnOrdinaryCohomologyInDegree2r}
\array{
H^{2r}(X;\mathbb{Q})
&\overset{\;\;\;\psi^k_H\;\;\;}{\longrightarrow}&
H^{2r}(X;\mathbb{Q})
\\
\alpha_{2k} &\mapsto& k^{r} \cdot \alpha_{2k}
\,.
}
\]
=--
+-- {: .num_prop #AdamsOperationsAreCompatibleWithTheChernCharacter }
###### Proposition
**([[Adams operations compatible with the Chern character]])**
For $X$ a [[topological space]] with a finite [[CW-complex]]-[[mathematical structure]], the [[Chern character]] $ch$ on the [[complex topological K-theory]] of $X$ intertwines the [[Adams operations]] $\psi^n$ on K-theory with the Adams-like operations $\psi^n_H$ on [[rational cohomology]] from Def. \ref{AdamsLikeOperationsOnRationalCohomology}, for $k \geq 1$, in that the following [[commuting diagram|diagram commutes]]:
$$
\array{
K(X)
&\overset{\;\;\;ch\;\;\;}{\longrightarrow}&
H^{ev}(X;\,\mathbb{Q})
\\
{}^{ \mathllap{ \psi^k } }
\big\downarrow
&&
\big\downarrow
{}^{ \mathrlap{ \psi^k_H } }
\\
K(X)
&\underset{\;\;\;ch\;\;\;}{\longrightarrow}&
H^{ev}(X;\,\mathbb{Q})
\,,
}
$$
=--
([Adams 62, Thm. 5.1. (vi)](#Adams62), review in [Karoubi 78, Chapter V, Theorem 3.27](#Karoubi78), [Maakestad 06, Thm. 4.9](#Maakestad06))
+-- {: .proof}
###### Proof idea
Use the exponentional-formula for the [[Chern character]] with the [[splitting principle]].
=--
## Related entries
* [[e-invariant]]
## References
The original statement:
* {#Adams62} [[John Adams]], Theorem 5.1 (vi) _Vector fields on spheres_, Bull. Amer. Math. Soc. Volume 68, Number 1 (1962), 39-41 ([euclid:bams/1183524456](https://projecteuclid.org/euclid.bams/1183524456), [pdf](https://www.math.ens.fr/~benoist/refs/Adams.pdf))
Textbook accounts:
* {#Karoubi78} [[Max Karoubi]], Theorem V3.27 in: _K-Theory -- An introduction_, Grundlehren der mathematischen Wissenschaften 226, Springer 1978 ([pdf](https://webusers.imj-prg.fr/~max.karoubi/K.book/MK.book.pdf), [doi:10.1007/978-3-540-79890-3](https://link.springer.com/book/10.1007%2F978-3-540-79890-3))
Review and exposition:
* {#Maakestad06} Helge Maakestad, _Notes on the Chern-character_, Journal of Generalized Lie Theory and Applications, 2017, 11:1 ([arXiv:math/0612060](https://arxiv.org/abs/math/0612060), [doi:10.4172/1736-4337.100025](https://www.hilarispublisher.com/open-access/notes-on-the-cherncharacter-.pdf))
[[!redirects Adams operations are compatible with the Chern character]]
[[!redirects the Adams operations are compatible with the Chern character map]]
|
Adams resolution | https://ncatlab.org/nlab/source/Adams+resolution |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
* [Classical definition](#ClassicalDefinition)
* [Via injective resolutions](#ViaInjectiveResolutions)
### Classical definition
{#ClassicalDefinition}
For $X$ a [[spectrum]] and $E^\bullet$ a [[generalized cohomology theory]] [[Brown representability theorem|represented]] by a [[spectrum]] $E$, then an _$E$-Adams resolution_ of $X$ is a [[diagram]] of the form
$$
\array{
\vdots
\\
\downarrow
\\
F_2 &\stackrel{f_2}{\longrightarrow}& K_2
\\
\downarrow
\\
F_1 &\stackrel{f_1}{\longrightarrow}& K_1
\\
\downarrow
\\
X &\stackrel{f_0}{\longrightarrow}& K_0
}
$$
where
* each $K_i$ is a wedge of [[suspensions]] of $E$;
* each $F_{n+1} \to F_n \to K_n$ is a [[homotopy fiber sequence]];
* each $f_n$ is a surjection on [[cohomology]].
The original and default case is that where $E = H \mathbb{F}_p$ is an [[Eilenberg-MacLane spectrum]] with mod $p$ [[coefficients]], in which case $E^\bullet$ is [[ordinary cohomology]] with these coefficients.
In this case the $K_i$ are [[generalized Eilenberg-MacLane spectra]].
The [[long exact sequences of homotopy groups]] for all the [[homotopy fibers]] in this diagram arrange into a diagram of the form
$$
\array{
\vdots
\\
\downarrow & \nwarrow
\\
\pi_\bullet(F_2) &\stackrel{\pi_\bullet(f_2)}{\longrightarrow}& \pi_\bullet(K_2)
\\
\downarrow & \nwarrow^{\mathrlap{\pi_\bullet(\partial_2)}}
\\
\pi_\bullet(F_1) &\stackrel{\pi_\bullet(f_1)}{\longrightarrow}& \pi_\bullet(K_1)
\\
\downarrow & \nwarrow^{\mathrlap{\pi_\bullet(\partial_1)}}
\\
\pi_\bullet(X) &\stackrel{\pi_\bullet(f_0)}{\longrightarrow}& \pi_\bullet(K_0)
}
\,,
$$
where the diagonal maps are the images of the [[connecting homomorphisms]] and hence decrease degree in $\pi_\bullet$ by one. This is an (unrolled) [[exact couple]]. The corresponding [[spectral sequence]] is the [[Adams spectral sequence]] induced by the given Adams resolution.
In the case of $E = H \mathbb{F}_p$, applying [[cohomology]] $H^\bullet(-, \mathbb{F}_p)$ to the original diagram
yields a [[free resolution]] of the [[cohomology ring]] $H^\bullet(X,\mathbb{Z}_p)$ by a [[chain complex]] of [[free modules]] over the [[Steenrod algebra]] $A_p$.
$$
\array{
H^\bullet(K_0) &\leftarrow& H^\bullet(\Sigma K_1) &\leftarrow& H^\bullet(\Sigma^2 K_2) &\leftarrow& \cdots
\\
\downarrow && \downarrow && \downarrow
\\
H^\bullet(X)
&\leftarrow& 0
&\leftarrow& 0
&\leftarrow& \cdots
}
$$
The computation of the [[cohomology]] of $X$ by means of this resolution is given by the [[Adams spectral sequence]].
### Via injective resolutions
{#ViaInjectiveResolutions}
A streamlined discussion of $E$-Adams resolutions in close analogy to [[injective resolutions]] in [[homological algebra]] was given in ([Miller 81](#Miller81)), advertized in ([Hopkins 99](#Hopkins99)) and worked out in more detail in ([Aramian](#Aramian)).
Write $HoSpectra$ for the [[stable homotopy category]] and write
$$
[-,-] \;\colon\; HoSpectra^{op} \times HoSpectra \longrightarrow Ab
$$
for the [[hom-functor]] with values in [[abelian groups]].
+-- {: .num_defn #HomotopyFunctor}
###### Definition
For $S \in HoSpectra$, the _homotopy functor it represents_ is the [[representable functor]]
$$
[S,-] \;\colon\; HoSpectra \longrightarrow Ab
$$
(as opposed to the other, contravariant, functor).
=--
+-- {: .num_example}
###### Example
For $S = \Sigma^\infty S^n \simeq \Sigma^n \mathbb{S}$ we have
$$
[\Sigma^\infty S^n ,- ]\simeq \pi_n
$$
is the $n$th [[stable homotopy group]]-[[functor]].
=--
Throughout, let $E$ be a [[ring spectrum]].
#### $E$-Injective spectra
First we consider a concept of $E$-[[injective objects]] in [[Spectra]].
+-- {: .num_defn #ExactSequences}
###### Definition
Say that
1. a sequence of spectra
$$
A_1 \longrightarrow A_2 \longrightarrow \cdots \longrightarrow A_n
$$
is
1. a (long) _exact sequence_ if the induced sequence of homotopy functors, def. \ref{HomotopyFunctor}, is a [[long exact sequence]] in $[HoSpectra,Ab]$;
2. (for $n = 2$) a _short exact sequence_ if
$$
0 \longrightarrow A_1 \longrightarrow A_2 \longrightarrow A_3 \longrightarrow 0
$$
is (long) exact;
1. a morphism $A \longrightarrow B$ is
1. a _monomorphism_ if $0 \longrightarrow A \longrightarrow B$ is an exact sequence;
1. an _epimorphism_ if $A \longrightarrow B \longrightarrow 0$ is an exact sequence.
For $E$ a [[ring spectrum]], then a sequence of spectra is (long/short) _$E$-exact_ and a morphism is epi/mono, respectively, if becomes long/short exact or epi/mono, respectively, after taking [[smash product of spectra|smash product]] with $E$.
=--
+-- {: .num_example}
###### Example
Every [[homotopy cofiber sequence]] of spectra is exact in the sense of def. \ref{ExactSequences}.
=--
+-- {: .num_remark}
###### Remark/Warning
Consecutive morphisms in an $E$-exact sequence according to def. \ref{ExactSequences} in general need not compose up to homotopy, to the [[zero morphism]]. But this does become true for sequences of $E$-injective objects, defined below in def. \ref{EInjective}.
=--
+-- {: .num_lemma}
###### Lemma
1. If $f \colon B\longrightarrow A$ is a monomorphism in the sense of def. \ref{ExactSequences}, then there exists a morphism $g \colon C \longrightarrow A$ such that the [[wedge sum]] morphism is a [[weak homotopy equivalence]]
$$
f \vee g \;\colon\; B \wedge C \stackrel{\simeq}{\longrightarrow} A
\,.
$$
1. If $f \colon A \longrightarrow B$ is an epimorpimsm in the sense of def. \ref{ExactSequences}, then there exists a homotopy [[section]] $s \colon B\to A$, i.e. $f\circ s\simeq Id$, together with a morphism $g \colon C \to A$ such that the [[wedge sum]] morphism is a [[weak homotopy equivalence]]
$$
s \vee f \colon B\vee C \stackrel{\simeq}{\longrightarrow} A
\,.
$$
=--
+-- {: .num_defn #EInjective}
###### Definition
For $E$ a [[ring spectrum]], say that a spectrum $S$ is _$E$-injective_ if for each morphism $A \longrightarrow S$ and each $E$-monomorphism $f \colon A \longrightarrow B$ in the sense of def. \ref{ExactSequences}, there is a [[diagram]] in [[HoSpectra]] of the form
$$
\array{
A &\longrightarrow & S
\\
\downarrow & \nearrow_{\mathrlap{\exists}}
\\
B
}
\,.
$$
=--
+-- {: .num_lemma}
###### Lemma
If $S$ is $E$-injective in the sense of def. \ref{EInjective}, then there exists a spectrum $X$ such that $S$ is a [[retract]] in [[HoSpectra]] of $E \wedge X$.
=--
#### $E$-Adams resolutions
+-- {: .num_defn #EAdamsResolution}
###### Definition
For $E$ a [[ring spectrum]], then an _$E$-Adams resolution_ of an spectrum $S$ is a long exact sequence, in the sense of def. \ref{ExactSequences}, of the form
$$
0 \longrightarrow S \longrightarrow I_0 \longrightarrow I_1 \longrightarrow I_2 \longrightarrow \cdots
$$
such that each $I_j$ is $E$-injective, def. \ref{EInjective}.
=--
+-- {: .num_lemma}
###### Lemma
Any two consecutive maps in an $E$-Adams resolution compose to the [[zero morphism]].
=--
+-- {: .num_lemma}
###### Lemma
For $X \to X_\bullet$ an $E$-Adams resolution, def. \ref{EAdamsResolution}, and for $X \longrightarrow Y$ any morphism, then there exists an $E$-Adams resolution $Y \to J_\bullet$ and a [[commuting diagram]]
$$
\array{
X &\longrightarrow& I_\bullet
\\
\downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g_\bullet}}
\\
Y &\longrightarrow& J_\bullet
}
\,.
$$
=--
+-- {: .num_example}
###### Example
**(standard resolution)**
Consider the augmented [[cosimplicial object|cosimplicial]] which is the $\mathbb{S} \to E$-[[Amitsur complex]] [[smash product of spectra|smashed]] with $X$:
$$
X \longrightarrow E \wedge X \stackrel{\longrightarrow}{\longrightarrow} E \wedge E \wedge X \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
E \wedge E \wedge E \wedge X
\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}}
\cdots
\,.
$$
Its corresponding [[Moore complex]] (the sequence whose maps are the alternating sum of the above coface maps) is an $E$-Adams resolution, def. \ref{EAdamsResolution}.
=--
#### $E$-Adams towers
+-- {: .num_defn #EAdamsTower}
###### Definition
An _$E$-Adams tower_ of a spectrum $X$ is a [[commuting diagram]] in [[HoSpectra]] of the form
$$
\array{
&& \vdots
\\
&& \downarrow^{\mathrlap{p_2}}
\\
&& X_2 &\stackrel{\kappa_2}{\longrightarrow}& \Omega^2 I_3
\\
&\nearrow& \downarrow^{\mathrlap{p_1}}
\\
&& X_1 &\stackrel{\kappa_1}{\longrightarrow}& \Omega I_2
\\
&\nearrow& \downarrow^{\mathrlap{p_0}}
\\
X
&\underset{}{\longrightarrow}&
X_0 = I_0
&\stackrel{\kappa_0}{\longrightarrow}&
I_1
}
$$
such that
1. each hook is a [[homotopy fiber sequence]] (hence it is a [[tower of homotopy fibers]]);
1. the [[composition]] of the $(\Sigma \dashv \Omega)$-[[adjuncts]] of $\Sigma_{p_{n-1}}$ with $\Sigma^n \kappa_n$
$$
i_{n+1} \;\colon\; I_n \stackrel{\widetilde {\Sigma p_{n-1}}}{\longrightarrow}
\Sigma^n X_n \stackrel{\Sigma^{n}\kappa_n}{\longrightarrow} I_{n+1}
$$
constitute an $E$-Adams resolution of $X$, def. \ref{EAdamsResolution}:
$$
0 \to X \stackrel{i_0}{\to} I_0 \stackrel{i_2}{\to} I_2 \stackrel{}{\to} \cdots
\,.
$$
Call this the _associated $E$-Adams resolution_ of the $E$-Adams tower.
The _associated inverse sequence_ is
$$
X = X_0 \stackrel{\gamma_0}{\longleftarrow} \Omega C_1 \stackrel{\gamma_1}{\longleftarrow} C_2 \longleftarrow \cdots
$$
where $C_{k+1} \coloneqq hocofib(i_k)$.
=--
(In ([Ravenel](#Ravenel)) it is is the associated inverse sequence that is called the associated resolution.)
=--
+-- {: .num_example}
###### Example
Every $E$-Adams resolution of $X$, def. \ref{EAdamsResolution}, induces an $E$-Adams tower, def. \ref{EAdamsTower} of which it is the associated $E$-Adams resolution.
=--
## Related concepts
* [[Whitehead tower]], [[Postnikov tower]]
* [[E-nilpotent completion]]
* [[Adams spectral sequence]]
## References
Reviews include
* {#Ravenel} [[Doug Ravenel]], around Chapter 2, def. 2.1.3 of _[[Complex cobordism and stable homotopy groups of spheres]]_
* {#Kochmann96} [[Stanley Kochmann]], section 3.6 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996
A streamlined presentation close in spirit to constructions in [[homological algebra]] was given in
* {#Miller81} [[Haynes Miller]], _On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space_, J. Pure Appl. Algebra 20 (1981) ([pdf](http://math.mit.edu/~hrm/papers/miller-relations-between-adams-spectral-sequences.pdf))
and is reproduced and expanded on in
* {#Hopkins99} [[Mike Hopkins]], section 5 of _Complex oriented cohomology theories and the language of stacks_, course notes 1999 ([pdf](http://www.math.rochester.edu/u/faculty/doug/otherpapers/coctalos.pdf))
* {#Aramian} [[Nersés Aramian]], _The Adams spectral sequence_ ([[AramianANSS.pdf:file]])
[[!redirects Adams resolutions]]
[[!redirects Adams tower]]
[[!redirects Adams towers]]
|
Adams spectral sequence | https://ncatlab.org/nlab/source/Adams+spectral+sequence |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Adams spectral sequence_ ([Adams 58](#Adams58)) is a type of [[spectral sequences]] used for computations of [[stable homotopy groups]] of [[spectra]] in terms of their [[generalized homology]]/[[generalized cohomology]]. Given a spectrum $X$ and a [[ring spectrum]] $E$, then under mild assumptions the Adams spectral sequence converges to the [[homotopy groups]] of the $E$-[[nilpotent completion]] of $X$, while under stronger assumptions the latter is the $E$-[[Bousfield localization of spectra|Bousfield localization of spectra]]. The second page of the spectral sequence is given by the $E$-[[homology]] of $X$ as [[modules]] over the dual $E$-[[Steenrod operations]]. The Adams spectral sequence may be seen as a variant of the [[Serre spectral sequence]] obtained by replacing a single fibration by an "[[Adams resolution]]".
The original _[[classical Adams spectral sequence]]_ is the case where $E = H\mathbb{F}_p$ is [[ordinary homology]] mod $p$, while the _[[Adams-Novikov spectral sequence]]_ ([Novikov 67](#Novikov67)) is the case where $E = $ [[MU]] is [[complex cobordism cohomology theory]] or $E = $ [[BP]], [[Brown-Peterson theory]].
Generally, for $E$ a suitable [[E-infinity algebra]] there is a corresponding _$E$-Adams(-Novikov) spectral sequence_ whose second page is given by $E$-[[generalized cohomology]] and which arises as the [[spectral sequence of a simplicial stable homotopy type]] of the [[cosimplicial object|cosimplicial]] object which is the [[Cech nerve]]/[[Sweedler coring]]/[[Amitsur complex]] of $E$. As such the Adams spectral sequence is an analog in [[stable homotopy theory]] of the [[Bousfield-Kan spectral sequence|Bousfield-Kan]] [[homotopy spectral sequence]] in unstable [[homotopy theory]].
For a detailed introduction see at _[[Introduction to the Adams Spectral Sequence]]_.
Working with the Adams spectral sequence tends to be fairly involved, as is clear from the subtlety of the results it computes (notably [[stable homotopy groups of spheres]]) and as witnessed by the fact that one uses further [[spectral sequences]] just to compute the low pages of the Adams spectral sequence, e.g. the [[May spectral sequence]] and the [[chromatic spectral sequence]].
A clear conceptual picture in [[higher algebra]] of what happens in the Adams spectral sequence ([Lurie 10](#Lurie10)) has emerged via the re-formulation in ([Miller 81](#Miller81), [Hopkins 99](#Hopkins99)). Survey of this perspective includes ([Wilson 13](#Wilson13)).
Here one observes that for $E$ a [[ring spectrum]], hence an [[E-∞ ring]], the [[totalization]] of its [[Amitsur complex]] [[cosimplicial object|cosimplicial]] spectrum is really the algebraic dual incarnation of the [[1-image]] factorization of the terminal morphism
$$
\array{
Spec(E) &\longrightarrow& Spec(Tot(E^{\wedge^\bullet}))
\\
\downarrow & \swarrow_{\mathrlap{p}}
\\
Spec(\mathbb{S})
}
$$
in [[E-∞ geometry]]/[[spectral geometry]].
Moreover, a [[spectrum]] $X$ is equivalently a [[quasicoherent sheaf]] on $Spec(S)$ and $E^{\wedge^\bullet} \wedge X$ is accordingly the [[Sweedler coring]] that expresses the [[descent]] property of $X$ pullled back along the cover $p$, dually the $E$-[[localization]] of $X$. The Adams spectral sequence may then be seen to be the computation of the [[homotopy groups]] of the $E$-localization of $X$ in terms of its restriction to that cover.
In general, notably for $E = H \mathbb{F}_p$, the [[1-image]] of $Spec(E) \to Spec(\mathbb{S})$ is smaller than $Spec(\mathbb{S})$ and therefore this process computes not all of $X$, but just the restriction to that one image (for instance just the $p$-local component). Examples of ring spectra which are "complete" with respect to the sphere spectrum in that the above 1-image coincides with $Spec(S)$ notably includes the [[complex cobordism cohomology]] spectrum $E = $[[MU]] ([Hopkins 99, p. 70](#Hopkins99)).
That explains the relevance of the [[Adams-Novikov spectral sequence]] (noticing that the wedge summands of $MU_{(p)}$ are the [[Brown-Peterson spectrum|BP-spectra]]) and the close interplay between the ANSS and [[chromatic homotopy theory]].
## Details
1. [Via the canonical resolution](#CanonicalResolution)
1. [Via injective resolutions](#ViaInjectiveResolutions)
1. [As derived descent in higher algebra](#DefinitionInHigherAlgebra)
### The $E$-Adams spectral sequence
{#CanonicalResolution}
We here discuss Adams spectral sequences for computation of $E$-[[Bousfield localization of spectra|localization]] of [[mapping spectra]] $[Y,X]$ for $E$ a [[commutative ring spectrum]] which is flat in a certain sense (def. \ref{FlatE} below), via the "canonical" $E$-Adams resolution introduced in ([Adams 74, theorem 15.1](#Adams74)). There are other resolutions which lead to the same spectral sequence, this we discuss below in the section on [E-Injective resolutions](#ViaInjectiveResolutions).
The _[[classical Adams spectral sequence]]_ is the special case of this general concept of $E$-Adams spectral sequences given by setting $Y = X = \mathbb{S}$ the [[sphere spectrum]] and $E = $ [[HA|H]]$\mathbb{F}_p$ the [[Eilenberg-MacLane spectrum]] of a [[prime field]]. This is discussed [below](#ClassicalCase).
The _[[Adams-Novikov spectral sequence]]_ is the special case given by setting $Y = X = \mathbb{S}$ and $E = $ [[MU]], discussed [below](#TheAdamsNovikovSpectralSequence).
#### The spectral sequence
##### Filtered spectra
{#SpectralSequenceOfAFilteredSpectrum}
We introduce the types of [[spectral sequences]] of which the $E$-Adams spectral sequences (def. \ref{AdamsEAdamsSpectralSequence} below) are an example.
+-- {: .num_defn #FilteredSpectrum}
###### Definition
A **[[filtered spectrum]]** is a [[spectrum]] $Y \in Ho(Spectra)$ equipped with a sequence $Y_\bullet \colon (\mathbb{N}, \gt) \longrightarrow Ho(Spectra)$ in the [[stable homotopy category]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#GradedAbelianGroupStructureOnHomsInTheHomotopyCategory)) of the form
$$
\cdots
\longrightarrow
Y_3
\overset{f_2}{\longrightarrow}
Y_2
\overset{f_1}{\longrightarrow}
Y_1
\overset{f_0}{\longrightarrow}
Y_0
\coloneqq
Y
\,.
$$
=--
+-- {: .num_remark}
###### Remark
More generally a [[filtered object in an (infinity,1)-category|filtering]] on an object $X$ in (stable or not) [[homotopy theory]] is a $\mathbb{Z}$-graded sequence $X_\bullet $ such that $X$ is the [[homotopy colimit]] $X\simeq \underset{\longrightarrow}{\lim} X_\bullet$. But for the present purpose we stick with the simpler special case of def. \ref{FilteredSpectrum}.
=--
+-- {: .num_remark}
###### Remark
There is _no_ condition on the [[morphisms]] in def. \ref{FilteredSpectrum}. In particular, they are _not_ required to be [[n-monomorphisms]] or [[n-epimorphisms]] for any $n$.
On the other hand, while they are also not explicitly required to have a presentation by [[cofibrations]] or [[fibrations]], this follows automatically: by the existence of the [[model structure on topological sequential spectra]] ([thm.](Introduction+to+Stable+homotopy+theory+--+1-1#StableModelStructureOnSequentialSpectraIsModelCategory)) or equivalently ([thm.](Introduction+to+Stable+homotopy+theory+--+1-2#SequentialSpectraQuillenEquivalence)) the [[model structure on orthogonal spectra]] ([thm.](Introduction+to+Stable+homotopy+theory+--+1-2#OrthogonalSpectraStableModelStructure)), every filtering on a spectrum is equivalent to one in which all morphisms are represented by [[cofibrations]] or by [[fibrations]].
This means that we may think of a filtration on a spectrum in the sense of def. \ref{FilteredSpectrum} as equivalently being a [[tower of fibrations]] over that spectrum.
=--
The following definition \ref{UnrolledExactCoupleOfAFiltrationOnASpectrum} unravels the structure encoded in a filtration on a spectrum, and motivates the concepts of [[exact couples]] and their [[spectral sequences]] from these.
+-- {: .num_defn #UnrolledExactCoupleOfAFiltrationOnASpectrum}
###### Definition
**(exact couple of a filtered spectrum)**
Consider a spectrum $X \in Ho(Spectra)$ and a [[filtered spectrum]] $Y_\bullet$ as in def. \ref{FilteredSpectrum}.
Write $A_k$ for the [[homotopy cofiber]] of its $k$th stage, such as to obtain the diagram
$$
\array{
\cdots
&\stackrel{}{\longrightarrow}&
Y_3
&\stackrel{f_2}{\longrightarrow}&
Y_2
&\stackrel{f_1}{\longrightarrow} &
Y_1
&\stackrel{f_0}{\longrightarrow}&
Y
\\
&& \downarrow^{\mathrlap{g_3}}
&& \downarrow^{\mathrlap{g_2}}
&& \downarrow^{\mathrlap{g_1}}
&& \downarrow^{\mathrlap{g_0}}
\\
&& A_3 && A_2 && A_1 && A_0
}
$$
where each stage
$$
\array{
Y_{k+1} &\stackrel{f_k}{\longrightarrow}& Y_k
\\
&& \downarrow^{\mathrlap{g_k}}
\\
&& A_k
}
$$
is a [[homotopy cofiber sequence]] ([def.](Introduction+to+Stable+homotopy+theory+--+P#HomotopyFiber)), hence equivalently ([prop.](Introduction+to+Stable+homotopy+theory+--+1-1#HomotopyCofiberSequencesAreHomotopyFiberSequencesInSpectra)) a [[homotopy fiber sequence]], hence where
$$
Y_{k+1}
\overset{f_k}{\longrightarrow}
Y_k
\overset{g_k}{\longrightarrow}
A_k
\overset{h_k}{\longrightarrow}
\Sigma Y_{k+1}
$$
is an exact triangle ([prop.](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryIsTriangulated)).
Apply the graded hom-group functor $[X,-]_\bullet$ ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#GradedAbelianGroupStructureOnHomsInTheHomotopyCategory)) to the above tower. This yields a diagram of $\mathbb{Z}$-[[graded abelian groups]] of the form
$$
\array{
\cdots
&\stackrel{}{\longrightarrow}&
[X,Y_3]_\bullet
&\stackrel{[X,f_2]_\bullet)}{\longrightarrow}&
[X,Y_2]_\bullet
&\stackrel{[X,f_1]_\bullet}{\longrightarrow} &
[X,Y_1]_\bullet
&\stackrel{[X,f_0]_\bullet}{\longrightarrow}&
[X,Y_0]_\bullet
\\
&& \downarrow^{\mathrlap{[X,g_3]_\bullet}}
&& \downarrow^{\mathrlap{[X,g_2]_\bullet}}
&& \downarrow^{\mathrlap{[X,g_1]_\bullet}}
&& \downarrow^{\mathrlap{[X,g_0]_\bullet}}
\\
&& [X,A_3]_\bullet
&& [X,A_2]_\bullet
&& [X,A_1]_\bullet
&& [X,A_0]_\bullet
}
\,,
$$
where each hook at stage $k$ extends to a [[long exact sequence of homotopy groups]] ([prop.](Introduction+to+Stable+homotopy+theory+--+1-1#LongFiberSequencesOfMapsOfSpectra)) via [[connecting homomorphisms]] $[X,h_k]_\bullet$
$$
\cdots
\to
[X,A_k]_{\bullet+1}
\stackrel{ [X,h_k]_{\bullet+1} }{\longrightarrow}
[X,Y_{k+1}]_{\bullet}
\stackrel{[X,f_k]_\bullet}{\longrightarrow}
[X,Y_k]_\bullet
\stackrel{[X,g_k]_\bullet}{\longrightarrow}
[X,A_k]_\bullet
\stackrel{ [X,h_k]_\bullet }{\longrightarrow}
[X,Y_{k+1}]_{\bullet-1}
\to
\cdots
\,.
$$
If we regard the [[connecting homomorphism]] $[X,h_k]$ as a morphism of degree -1, then all this information fits into one diagram of the form
$$
\array{
\cdots
&\stackrel{}{\longrightarrow}&
[X,Y_3]_\bullet
&\stackrel{[X,f_2]_\bullet}{\longrightarrow}&
[X,Y_2]_\bullet
&\stackrel{[X,f_1]_\bullet}{\longrightarrow} &
[X,Y_1]_\bullet
&\stackrel{[X,f_0]_\bullet}{\longrightarrow}&
[X,Y_0]_\bullet
\\
&&
\downarrow &{}_{\mathllap{ }} \underset{[X,h_2]_\bullet}{\nwarrow} &
\downarrow &{}_{\mathllap{ }} \underset{[X,h_1]_\bullet}{\nwarrow} &
\downarrow &{}_{\mathllap{ }} \underset{[X,h_0]_\bullet}{\nwarrow}
& \downarrow^{\mathrlap{[X,g_0]_\bullet}}
\\
&& [X,A_3]_\bullet
&& [X,A_2]_\bullet
&& [X,A_1]_\bullet
&& [X,A_0]_\bullet
}
\,,
$$
where each triangle is a rolled-up incarnation of a [[long exact sequence of homotopy groups]] (and in particular is _not_ a commuting diagram!).
If we furthermore consider the [[bigraded object|bigraded]] [[abelian groups]] $[X,Y_\bullet]_{\bullet}$ and $[X,A_\bullet]_{\bullet}$, then this information may further be rolled-up to a single diagram of the form
$$
\array{
[X,Y_\bullet]_\bullet
& \stackrel{[X,f_\bullet]_\bullet}{\longrightarrow} &
[X,Y_\bullet]_{\bullet}
\\
& {}_{\mathllap{ [X, h_\bullet]_\bullet }}\nwarrow
& \downarrow^{\mathrlap{[X, g_\bullet]_\bullet }}
\\
&&
[X,A_\bullet]_\bullet
}
\,.
$$
Specifically, regard the terms here as bigraded in the following way
$$
\begin{aligned}
D^{s,t}(X,Y) & \coloneqq [X,Y_s]_{t-s}
\\
E^{s,t}(X,Y) & \coloneqq [X,A_s]_{t-s}
\end{aligned}
\,.
$$
Then the bidegree of the morphisms is
| morphism | bidegree |
|----------|----------|
| $[X,f]$ | $(-1,-1)$ |
| $[X,g]$ | $(0,0)$ |
| $[X,h]$ | $(1,0)$ |
This way $t$ counts the cycles of going around the triangles:
$$
\cdots
\to
D^{s+1,t+1}(X,Y)
\stackrel{[X,f]}{\longrightarrow}
D^{s,t}(X,Y)
\stackrel{[X,g]}{\longrightarrow}
E^{s,t}(X,Y)
\stackrel{[X,h]}{\longrightarrow}
D^{s+1,t}(X,Y)
\to
\cdots
$$
Data of this form is called an _[[exact couple]]_, def. \ref{ExactCouple} below.
=--
+-- {: .num_defn #UnrolledExactCouple}
###### Definition
An _unrolled [[exact couple]]_ (of _Adams-type_) is a diagram of [[abelian groups]] of the form
$$
\array{
\cdots
&\stackrel{}{\longrightarrow}&
\mathcal{D}^{3,\bullet}
&\stackrel{i_2}{\longrightarrow}&
\mathcal{D}^{2,\bullet}
&\stackrel{i_1}{\longrightarrow} &
\mathcal{D}^{1,\bullet}
&\stackrel{i_0}{\longrightarrow}&
\mathcal{D}^{0,\bullet}
\\
&&
\downarrow^{\mathrlap{}}
&{}_{ } \underset{\mathllap{k_2}}{\nwarrow} &
{}^{\mathllap{j_2}}\downarrow &
\underset{k_1}{\nwarrow}
&
{}^{\mathllap{j_1}}\downarrow &{}_{} \underset{\mathllap{k_0}}{\nwarrow}
& \downarrow_{\mathrlap{j_0}}
\\
&& \mathcal{E}^{3,\bullet} && \mathcal{E}^{2,\bullet}
&& \mathcal{E}^{1,\bullet} && \mathcal{E}^{0,\bullet}
}
$$
such that each triangle is a rolled-up [[long exact sequence]] of [[abelian groups]] of the form
$$
\cdots
\to
\mathcal{D}^{s+1,t+1}
\stackrel{i_s}{\longrightarrow}
\mathcal{D}^{s,t}
\stackrel{j_s}{\longrightarrow}
\mathcal{E}^{s,t}
\stackrel{k_s}{\longrightarrow}
\mathcal{D}^{s+1,t}
\to
\cdots
\,.
$$
=--
The collection of this "un-rolled" data into a single diagram of [[abelian groups]] is called the corresponding _[[exact couple]]_.
+-- {: .num_defn #ExactCouple}
###### Definition
An _[[exact couple]]_ is a [[diagram]] (non-commuting) of [[abelian groups]] of the form
$$
\array{
\mathcal{D}
&\stackrel{i}{\longrightarrow}&
\mathcal{D}
fto \\
& {}_{\mathllap{k}}\nwarrow & \downarrow^{\mathrlap{j}}
\\
&& \mathcal{E}
}
\,,
$$
such that this is [[exact sequence|exact]] in each position, hence such that the [[kernel]] of every [[morphism]] is the [[image]] of the preceding one.
=--
The concept of exact couple so far just collects the sequences of long exact sequences given by a filtration. Next we turn to extracting information from this sequence of sequences.
+-- {: .num_remark #Observingd1}
###### Remark
The sequence of long exact sequences in def. \ref{UnrolledExactCoupleOfAFiltrationOnASpectrum} is inter-locking, in that every $[X,Y_s]_{t-s}$ appears _twice_:
$$
\array{
&& & \searrow && \nearrow
\\
&& && [X,Y_{s+1}]_{t-s-1}
\\
&& & {}^{\mathllap{ [X,h] }}\nearrow
&& \searrow^{\mathrlap{ [X,g] }}
&& && && \nearrow
\\
&& [X,A_s]_{t-s}
&& \underset{d_1 }{\longrightarrow}
&& [X, A_{s+1}]_{t-s-1}
&& \stackrel{d_1 }{\longrightarrow}
&& [X,A_{s+2}]_{t-s-2}
\\
& \nearrow && && && {}_{\mathllap{ [X,h] }}\searrow
&& \nearrow_{\mathrlap{ [X,g] }}
\\
&& && && && [X,Y_{s+2}]_{t-s-2}
\\
&& && && & \nearrow && \searrow
}
$$
This gives rise to the horizontal ("[[splicing of short exact sequences|splicing]]") composites $d_1$, as shown, and by the fact that the diagonal sequences are long exact, these are [[differentials]] in that they square to zero: $(d_1)^2 = 0$. Hence there is a [[cochain complex]]:
$$
\array{
\cdots
&
\overset{}{\longrightarrow}
&&
[X,A_s]_{t-s}
&&
\overset{d_1}{\longrightarrow}
&&
[X,A_{S+1}]_{t-s-1}
&&
\stackrel{d_1 }{\longrightarrow}
&&
[X,A_{s+2}]_{t-s-2}
&&
\longrightarrow & \cdots
}
\,.
$$
We may read off from these interlocking long exact sequences what these differentials _mean_, as follows. An element $c \in [X,A_s]_{t-s}$ lifts to an element
$\hat c \in[X,Y_{s+2}]_{t-s-1} $ precisely if $d_1 c = 0$:
$$
\array{
&\hat c \in & [X, Y_{s+2}]_{t-s-1}
\\
&& & \searrow^{\mathrlap{ [X,f] }}
\\
&& && [X,Y_{s+1}]_{t-s-1}
\\
&& & {}^{\mathllap{ [X,h] }}\nearrow
&& \searrow^{\mathrlap{ [X,g] }}
\\
& c \in & [X,A_s]_{t-s}
&&
\underset{ d_1 }{\longrightarrow}
&& [X,A_{s+1}]_{t-s-1}
}
$$
In order to organize this observation, notice that in terms of the exact couple of remark \ref{UnrolledExactCoupleOfAFiltrationOnASpectrum}, the differential
$$
d_1 \;\coloneqq \; [X,g] \circ [X,h]
$$
is the composite
$$
d \coloneqq j \circ k
\,.
$$
=--
Some terminology:
+-- {: .num_defn #PageOfAnExactCouple}
###### Definition
Given an exact couple, def. \ref{ExactCouple},
$$
\array{
\mathcal{D}^{\bullet,\bullet}
&\stackrel{i}{\longrightarrow}&
\mathcal{D}^{\bullet,\bullet}
\\
& {}_{\mathllap{k}}\nwarrow & \downarrow^{\mathrlap{j}}
\\
&& \mathcal{E}^{\bullet,\bullet}
}
$$
observe that the composite
$$
d \coloneqq j \circ k
$$
is a [[differential]] in that it squares to 0, due to the exactness of the exact couple:
$$
\begin{aligned}
d \circ d & = j \circ \underset{= 0}{\underbrace{k \circ j}} \circ k
\\
& = 0
\end{aligned}
\,.
$$
One says that the **page** of the exact couple is the [[graded object|graded]] [[chain complex]]
$$
(\mathcal{E}^{\bullet,\bullet}, d \coloneqq j \circ k)
\,.
$$
Given a cochain complex like this, we are to pass to its [[cochain cohomology]]. Since the cochain complex here has the extra structure that it arises from an exact couple, its cohomology groups should still remember some of that extra structure. Indeed, the following says that the remaining extract structure on the cohomology of the page of an exact couple is itself again an exact couple, called the "derived exact couple".
=--
+-- {: .num_defn #DerivedExactCouple}
###### Definition
Given an exact couple, def. \ref{ExactCouple}, then its _derived exact couple_ is the diagram
$$
\array{
\widetilde {\mathcal{D}}
&\stackrel{\tilde i}{\longrightarrow}&
\widetilde {\mathcal{D}}
\\
& {}_{\mathllap{\tilde k}}\nwarrow & \downarrow^{\mathrlap{\tilde j}}
\\
&& \widetilde{\mathcal{E}}
}
\;\;\;\;
\coloneqq
\;\;\;\;
\array{
im(i)
&\stackrel{i}{\longrightarrow}&
im(i)
\\
& {}_{\mathllap{k}}\nwarrow & \downarrow {\mathrlap{j \circ i^{-1}}}
\\
&& H(\mathcal{E}, j \circ k)
}
$$
with
1. $\tilde{\mathcal{E}} \coloneqq ker(d)/im(d)$ (with $d \coloneqq j \circ k$ from def. \ref{PageOfAnExactCouple});
1. $\tilde {\mathcal{D}} \coloneqq im(i)$;
1. $\tilde i \coloneqq i|_{im(i)}$;
1. $\tilde j \coloneqq j \circ i^{-1}$ (where $i^{-1}$ is the operation of choosing any preimage under $i$);
1. $\tilde k \coloneqq k|_{ker(d)}$.
=--
+-- {: .num_lemma #DerivedExactCoupleIsExactCouple}
###### Lemma
The derived exact couple in def. \ref{DerivedExactCouple}
is well defined and is itself an exact couple, def. \ref{ExactCouple}.
=--
+-- {: .proof}
###### Proof
This is straightforward to check. For completeness we spell it out:
First consider that the morphisms are well defined in the first place.
It is clear that $\tilde i$ is well-defined.
That $\tilde j$ lands in $ker(d)$: it lands in the image of $j$ which is in the kernel of $k$, by exactness, hence in the kernel of $d$ by definition.
That $\tilde j$ is independent of the choice of preimage: For any $x \in \tilde {\mathcal{D}} = im(i)$, let $y, y' \in \mathcal{D}$ be two preimages under $i$, hence $i(y) = i(y') = x$. This means that $i(y'-y) = 0$, hence that $y'-y \in ker(i)$, hence that $y'-y \in im(k)$, hence there exists $z \in \mathcal{E}$ such that $y' = y + k(z)$, hence $j(y') = j(y) + j(k(z)) = j(y) + d(z)$, but $d(z) = 0$ in $\tilde{\mathcal{E}}$.
That $\tilde k$ vanishes on $im(d)$: because $im(d) \subset im(j)$ and hence by exactness.
That $\tilde k$ lands in $im(i)$: since it is defined on $ker(d) = ker(j \circ k)$ it lands in $ker(j)$. By exactness this is $im(i)$.
That the sequence of maps is again exact:
The kernel of $\tilde j$ is those $x \in \im(i)$ such that their preimage $i^{-1}(x)$ is still in $im(x)$ (by exactness of the original exact couple) hence such that $x \in im(i|_{im(i)})$, hence such that $x \in im(\tilde i)$.
The kernel of $\tilde k$ is the intersection of the kernel of $k$ with the kernel of $d = j \circ k$, wich is still the kernel of $k$, hence the image of $j$, by exactness. Indeed this is also still the image of $\tilde j$, since for every $x \in \mathcal{D}$ then $\tilde j(i(x)) = j(x)$.
The kernel of $\tilde i$ is $ker(i) \cap im(i) \simeq im(k) \cap im(i)$, by exactness. Let $x \in \mathcal{E}$ such that $k(x) \in im(i)$, then by exactness $k(x) \in ker(j)$ hence $j(k(x)) = d(x) = 0$, hence $x \in ker(d)$ and so $k(x) = \tilde k(x)$.
=--
+-- {: .num_defn #ExactCoupleSpectralSequence}
###### Definition
Given an exact couple, def. \ref{ExactCouple},
then the induced
**[[spectral sequence]] of the exact couple**
is the sequence of pages, def. \ref{PageOfAnExactCouple}, of the induced sequence of derived exact couples, def. \ref{DerivedExactCouple}, lemma \ref{DerivedExactCoupleIsExactCouple}.
The **$r$th page** of the spectral sequence is the page (def. \ref{PageOfAnExactCouple}) of the $r$th exact couple, denoted
$$
\{\mathcal{E}_r, d_r\}
\,.
$$
=--
+-- {: .num_remark #TermsOfNextPageOnSpectralSequence}
###### Remark
So the spectral sequence of an exact couple (def. \ref{ExactCoupleSpectralSequence}) is a sequence of cochain complexes $(\mathcal{E}_r, d_r)$, where the cohomology of one is the terms of the next one:
$$
\mathcal{E}_{r+1} \simeq H(\mathcal{E}_r,d_r)
\,.
$$
In practice this is used as a successive stagewise approximation to the computation of a limiting term $\mathcal{E}_\infty$. What that limiting term is, if it exists at all, is the subject of _convergence_ of the spectral sequence, we come to this [below](#Convergence).
=--
Def. \ref{ExactCoupleSpectralSequence} makes sense without a (bi-)grading on the terms of the exact couple, but much of the power of spectral sequences comes from the cases where such a bigrading is given, since together with the sequence of pages of the spectral sequence, this tends to organize computation of the successive cohomology groups in an efficient way. Therefore consider:
+-- {: .num_defn #AdamsTypeSpectralSequenceOfATower}
###### Definition
Given a [[filtered spectrum]] as in def. \ref{FilteredSpectrum},
$$
\array{
\cdots
&\stackrel{}{\longrightarrow}&
X_3
&\stackrel{f_2}{\longrightarrow}&
X_2
&\stackrel{f_1}{\longrightarrow} &
X_1
&\stackrel{f_0}{\longrightarrow}&
X
\\
&& \downarrow^{\mathrlap{g_3}}
&& \downarrow^{\mathrlap{g_2}}
&& \downarrow^{\mathrlap{g_1}}
&& \downarrow^{\mathrlap{g_0}}
\\
&& A_3 && A_2 && A_1 && A_0
}
$$
and given another spectrum $X \in Ho(Spectra)$, the induced **[[spectral sequence of a filtered spectrum]]** is the [[spectral sequence]] that is induced, by def. \ref{ExactCoupleSpectralSequence} from the [[exact couple]] (def. \ref{ExactCouple}) given by def. \ref{UnrolledExactCoupleOfAFiltrationOnASpectrum}:
$$
\array{
\mathcal{D} &\stackrel{i}{\longrightarrow}& \mathcal{D}
\\
&{}_{\mathllap{k}}\nwarrow& \downarrow^{\mathrlap{j}}
\\
&& \mathcal{E}
}
\;\;\;\;\;\;
\coloneqq
\;\;\;\;\;\;
\array{
\underset{s,t}{\oplus} D^{s,t}(X,Y)
&\overset{[X,f]}{\longrightarrow}&
\underset{s,t}{\oplus} D^{s,t}(X,Y)
\\
& {}_{\mathllap{ [X, h] }}\nwarrow & \downarrow^{[X,g]}
\\
&& \underset{s,t}{\oplus} E^{s,t}(X,Y)
}
\;\;\;\;
\coloneqq
\;\;\;\;
\array{
\underset{s,t}{\oplus} [X,Y_s]_{t-s}
&\overset{[X,f]}{\longrightarrow}&
\underset{s,t}{\oplus} [X,Y_s]_{t-s}
\\
& {}_{\mathllap{ [X,h] }}\nwarrow & \downarrow^{\mathrlap{[X,g]}}
\\
&& [X,A_s]_{t-s}
}
$$
with the following bidegree of the differentials:
$$
deg = \;\;\;\;
\array{
\mathcal{D} &\stackrel{(-1,-1)}{\longrightarrow}& \mathcal{D}
\\
&{}_{\mathllap{(1,0)}}\nwarrow& \downarrow^{\mathrlap{(0,0)}}
\\
&& \mathcal{E}
}
\,.
$$
In particular the first page is
$$
\mathcal{E}^{s,t}_1
=
[X,A_s]_{t-s}
$$
$$
d_1
=
[X, g \circ h]
\,.
$$
As we pass to derived exact couples, by def. \ref{DerivedExactCouple},
the bidegree of $i$ and $k$ is preserved, but that of $j$ increases by
$(1,1)$ with each page, since (by def. \ref{PageOfAnExactCouple})
$$
\begin{aligned}
deg(\tilde j)
&
= deg( j \circ i^{-1})
\\
&= deg(j) - deg(i)
\\
& = deg(j) + (1,1)
\end{aligned}
\,.
$$
Similarly the first differential has degree
$$
\begin{aligned}
deg(j \circ k )
& =
deg(j) + deg(k)
\\
&= (1,0) + (0,0)
\\
& = (1,0)
\end{aligned}
$$
<div style="float:right;margin:0 10px 10px 0;">
<img src="http://ncatlab.org/nlab/files/adamstypedifferentials.jpg" width="360" >
</div>
and so the differentials on the $r$th page are of the form
$$
d_r
\;\colon\;
\mathcal{E}_r^{s,t}
\longrightarrow
\mathcal{E}_r^{s+r, t+r-1}
\,.
$$
It is conventional to depict this in tables where $s$ increases vertically and upwards and $t-s$ increases horizontally and to the right, so that $d_r$ goes up $r$ steps and always one step to the left. This is the "Adams type" grading convention for spectral sequences (different from the [[Serre spectral sequence|Serre-]][[Atiyah-Hirzebruch spectral sequence]] convention ([prop.](Introduction+to+Stable+homotopy+theory+--+S#AHSSExistence))). One also says that
* $s$ is the _filtration degree_;
* $t-s$ is the _total degree_;
* $t$ is the _internal degree_.
A priori all this is $\mathbb{N}\times \mathbb{Z}$-graded, but we regard it as being $\mathbb{Z} \times \mathbb{Z}$-graded by setting
$$
\mathcal{D}^{s \lt 0,\bullet} \coloneqq 0
\;\;\;\;\,,
\;\;\;\;
\mathcal{E}^{s \lt 0, \bullet} \coloneqq 0
$$
and trivially extending the definition of the differentials to these zero-groups.
=--
##### $E$-Adams filtrations
{#AdamsFiltration}
Given a [[homotopy commutative ring spectrum]] $(E,\mu,e)$, then an _$E$-Adams spectral sequence_ is a [[spectral sequence]] as in def. \ref{AdamsTypeSpectralSequenceOfATower}, where each cofiber is induced from the unit morphism $e \;\colon\; \mathbb{S} \longrightarrow E$:
+-- {: .num_defn #AdamsEAdamsSpectralSequence}
###### Definition
Let $X,Y \in Ho(Spectra)$ be two [[spectra]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#TheStableHomotopyCategory)), and let $(E,\mu,e) \in CMon(Ho(Spectra),\wedge, \mathbb{S})$ be a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)) in the [[tensor triangulated category|tensor triangulated]] [[stable homotopy category]] $(Ho(Spectra), \wedge, \mathbb{S})$ ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#TensorTriangulatedStructureOnStableHomotopyCategory)).
Then the **$E$-[[Adams spectral sequence]]** for the computation of the [[graded abelian group]]
$$
[X,Y]_\bullet
\coloneqq
[X, \Sigma^{-\bullet} Y]
$$
of morphisms in the [[stable homotopy category]]
([def.](Introduction+to+Stable+homotopy+theory+--+1-1#GradedAbelianGroupStructureOnHomsInTheHomotopyCategory)) is the [[spectral sequence of a filtered spectrum]] (def. \ref{AdamsTypeSpectralSequenceOfATower}) of the image under $[X,-]_\bullet$ of the tower
$$
\array{
& \vdots
\\
& {}^{\mathllap{f_0}}\downarrow
\\
& Y_3 &\overset{g_3}{\longrightarrow}& E \wedge Y_3 = A_3
\\
& {}^{\mathllap{f_0}}\downarrow
\\
& Y_2 &\overset{g_2}{\longrightarrow}& E \wedge Y_2 = A_2
\\
& {}^{\mathllap{f_0}}\downarrow
\\
& Y_1 &\overset{g_1}{\longrightarrow}& E \wedge Y_1 = A_1
\\
& {}^{\mathllap{f_0}}\downarrow
\\
Y = & Y_0 &\overset{g_0}{\longrightarrow}& E \wedge Y_0 = A_0
}
\,,
$$
where each hook is a [[homotopy fiber sequence]] (equivalently a [[homotopy cofiber sequence]], [prop.](Introduction+to+Stable+homotopy+theory+--+1-1#HomotopyCofiberSequencesAreHomotopyFiberSequencesInSpectra)), hence where each
$$
Y_{n+1}
\overset{f_n}{\longrightarrow}
Y_n
\overset{g_n}{\longrightarrow}
A_n
\overset{h_n}{\longrightarrow}
\Sigma Y_{n+1}
$$
is an exact triangle ([prop.](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryIsTriangulated)), where inductively
$$
A_n \coloneqq E \wedge Y_n
$$
is the derived [[smash product of spectra]] ([corollary](Introduction+to+Stable+homotopy+theory+--+1-2#MonoidalStableHomotopyCategory)) of $E$ with the stage $Y_n$ ([cor.](Introduction+to+Stable+homotopy+theory+--+1-2#MonoidalStableHomotopyCategory)), and where
$$
g_n
\;\colon\;
Y_n
\underoverset{\simeq}{\ell^{-1}_{Y_n}}{\longrightarrow}
\mathbb{S} \wedge Y_n
\overset{e \wedge id}{\longrightarrow}
E \wedge Y_{n}
$$
is the composition of the inverse derived [[unitor]] on $Y_n$ ([cor.](Introduction+to+Stable+homotopy+theory+--+1-2#MonoidalStableHomotopyCategory)) with the derived [[smash product of spectra]] of the unit $e$ of $E$ and the identity on $Y_n$.
Hence, by def \ref{AdamsTypeSpectralSequenceOfATower}, the first page is
$$
E_1^{s,t}(X,Y)
\;\coloneqq\;
[X, A_s ]_{t-s}
$$
$$
d_1 = [X, g \circ h]
$$
and the differentials are of the form
$$
d_r
\;\colon\;
E_r^{s,t}(X,Y)
\longrightarrow
E_r^{s+r, t+r-1}(X,Y)
\,.
$$
A priori $E_r^{\bullet,\bullet}(X,Y)$ is $\mathbb{N}\times \mathbb{Z}$-graded, but we regard it as being $\mathbb{Z} \times \mathbb{Z}$-graded by setting
$$
E_r^{s \lt 0,\bullet}(X,Y) \coloneqq 0
$$
and trivially extending the definition of the differentials to these zero-groups.
=--
([Adams 74, theorem 15.1 page 318](#Adams74))
+-- {: .num_remark}
###### Remark
The morphism
$$
[X,g_k]
\;\colon\;
[X,Y_k]_\bullet
\overset{[X,e \wedge id_{Y_k}] }{\longrightarrow}
[X, E \wedge Y_k]_\bullet
$$
in def. \ref{AdamsEAdamsSpectralSequence} is sometimes called the **[[Boardman homomorphism]]** ([Adams 74, p. 58](#Adams74)).
For $X = \mathbb{S}$ the [[sphere spectrum]] it reduces to a canonical morphism from stable homotopy to [[generalized homology]] ([rmk.](Introduction+to+Stable+homotopy+theory+--+1-2#EMHomology))
$$
\pi_\bullet(g_k)
\;\colon\;
\pi_\bullet(Y_k)
\longrightarrow
E_\bullet(Y_k)
\,.
$$
For $E = $ [[HA]] an [[Eilenberg-MacLane spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#ReducedALinearizationOfnSphere)) this in turn reduces to the **[[Hurewicz homomorphism]]** for spectra.
This way one may think of the $E$-Adams filtration on $Y$ in def. \ref{AdamsEAdamsSpectralSequence} as the result of filtering any spectrum $Y$ by iteratively projecting out all its $E$-homology.
This idea was historically the original motivation for the construction of the [[classical Adams spectral sequence]] by [[John Frank Adams]], see the first pages of ([Bruner 09](#Adams+spectral+sequence#Bruner09)) for a historical approach.
=--
It is convenient to adopt the following notation for $E$-Adams spectral sequences (def. \ref{AdamsEAdamsSpectralSequence}):
+-- {: .num_defn #HomotopyFiberOfUnitOfCommutativeRingSpectrum}
###### Definition
For $(E,\mu,e) \in CMon(Ho(Spectra),\wedge, \mathbb{S})$ a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)), write $\overline{E}$ for the [[homotopy fiber]] of its unit $e \colon \mathbb{S}\to E$, i.e. such that there is a [[homotopy fiber sequence]] (equivalently a [[homotopy cofiber sequence]], [prop.](Introduction+to+Stable+homotopy+theory+--+1-1#HomotopyCofiberSequencesAreHomotopyFiberSequencesInSpectra)) in the [[stable homotopy category]] $Ho(Spectra)$ of the form
$$
\overline{E}
\longrightarrow
\mathbb{S}
\overset{e}{\longrightarrow}
E
\,,
$$
equivalently an [[triangulated category|exact triangle]] ([prop.](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryIsTriangulated)) of the form
$$
\overline{E}
\longrightarrow
\mathbb{S}
\overset{e}{\longrightarrow}
E
\longrightarrow
\Sigma \overline{E}
\,.
$$
=--
([Adams 74, theorem 15.1 page 319](#Adams74)) Beware that for instance ([Hopkins 99, proof of corollary 5.3](#Hopkins99)) uses "$\overline{E}$" not for the [[homotopy fiber]] of $\mathbb{S} \overset{e}{\to} E$ but for its [[homotopy cofiber]], hence for what is $\Sigma \overline{E}$ according to ([Adams 74](#Adams74)).
+-- {: .num_lemma #Wp}
###### Lemma
In terms of def. \ref{HomotopyFiberOfUnitOfCommutativeRingSpectrum}, the spectra entering the definition of the $E$-[[Adams spectral sequence]] in def. \ref{AdamsEAdamsSpectralSequence} are equivalently
$$
Y_p
\;\simeq\;
\overline{E}^p \wedge Y
$$
and
$$
A_p
\;\simeq\;
E \wedge \overline{E}^p \wedge Y
\,.
$$
where we write
$$
\overline{E}^p
\coloneqq
\underset{
p\; factors
}{
\underbrace{
\overline{E} \wedge \cdots \wedge \overline{E}
}} \wedge Y
\,.
$$
Hence the first page of the $E$-Adams spectral sequence reads equivalently
$$
E^{s,t}_1(X,Y)
\simeq
[X, E \wedge {\overline{E}^s} \wedge Y]_{t-s}
\,.
$$
=--
([Adams 74, theorem 15.1 page 319](#Adams74))
+-- {: .proof}
###### Proof
By definition the statement holds for $p = 0$. Assume then by [[induction]] that it holds for some $p \geq 0$. Since the [[smash product of spectra]]-functor $(-) \wedge \overline{E}^p \wedge Y$ preserves [[homotopy cofiber sequences]] ([lemma](Introduction+to+Stable+homotopy+theory+--+1-2#SmashTensoringWithSpectrumDerivedPreserveshomotopycofibers), this is part of the [[tensor triangulated category|tensor triangulated]] structure of the [[stable homotopy category]]), its application to the [[homotopy cofiber sequence]]
$$
\overline{E}
\longrightarrow
\mathbb{S}
\overset{e}{\longrightarrow}
E
$$
from def. \ref{HomotopyFiberOfUnitOfCommutativeRingSpectrum} yields another homotopy cofiber sequence, now of the form
$$
\overline{E}^{p+1} \wedge Y
\longrightarrow
\overline{E}^p \wedge Y
\overset{g_p}{\longrightarrow}
E \wedge \overline{E}^p \wedge Y
$$
where the morphism on the right is identified as $g_p$ by the induction assumption, hence $A_{p+1}\simeq E \wedge \overline{E}^p \wedge Y$. Since $Y_{p+1}$ is defined to be the homotopy fiber of $g_p$, it follows that $Y_{p+1} \simeq \overline{E}^{p+1} \wedge Y$.
=--
+-- {: .num_remark}
###### Remark
Terminology differs across authors. The filtration in def. \ref{AdamsEAdamsSpectralSequence} in the rewriting by lemma \ref{Wp} is due to ([Adams 74, theorem 15.1](#Adams74)), where it is not give any name. In ([Ravenel 84, p. 356](Adams+spectral+sequence#Ravenel84)) it is called the (canonical) **Adams tower** while in ([Ravenel 86, def. 2.21](Adams+spectral+sequence#Ravenel86)) it is called the canonical **Adams resolution**. Several authors follow the latter usage, for instance ([Rognes 12, def. 4.1](#Rognes12)). But ([Hopkins 99](Adams+spectral+sequence#Hopkins99)) uses "Adams resolution" for the "$E$-injective resolutions" that we discuss [below](#ViaInjectiveResolutions) and uses "Adams tower" for yet another concept, def. \ref{EAdamsTower} below. See also remark \ref{TerminologyAssociatedInverseSequence}.
=--
We proceed now to analyze the first two pages and then the convergence properties of $E$-Adams spectral sequences of def. \ref{AdamsEAdamsSpectralSequence}.
#### The first page
{#FirstPageAndHopfAlgebroid}
By lemma \ref{Wp} the first page of an $E$-Adams spectral sequence (def. \ref{AdamsEAdamsSpectralSequence}) looks like
$$
\begin{aligned}
E^{s,t}_1(X,Y)
&
\simeq
[X, E \wedge \overline{E}^s \wedge Y]_{s-t}
\end{aligned}
\,.
$$
We discuss now how, under favorable conditions, these hom-groups may alternatively be computed as morphisms of $E$-[[generalized homology|homology]] equipped with suitable [[comodule]] structure over a [[Hopf algebroid]] structure on the dual $E$-[[Steenrod operations]] $E_\bullet(E)$ (The $E$-[[generalized homology]] of $E$ ([rmk.](Introduction+to+Stable+homotopy+theory+--+1-2#EMHomology))). Then [below](#TheE2TermOfTheEAdamsSpectralSequence) we discuss that, as a result, the $d_1$-cohomology of the first page computes the [[Ext]]-groups from the $E$-homology of $Y$ to the $E$-homology of $X$, regarded as $E_\bullet(E)$-comodules.
The condition needed for this to work is the following.
##### Flat homotopy ring spectra
+-- {: .num_defn #FlatE}
###### Definition
Call a [[homotopy commutative ring spectrum]] $(E,\mu,e)$ ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)) **flat** if the canonical right $\pi_\bullet(E)$-[[module]] structure on $E_\bullet(E)$ ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum)) (equivalently the canonical left module struture, see prop. \ref{EETwoLeftModuleStructures} below) is a [[flat module]].
=--
The key consequence of the assumption that $E$ is flat in the sense of def. \ref{FlatE} is the following.
+-- {: .num_prop #EnHomology}
###### Proposition
Let $(E,\mu,e)$ be a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)) and let $X \in Ho(Spectra)$ be any [[spectrum]]. Then there is a [[homomorphism]] of [[graded abelian groups]] of the form
$$
E_\bullet(E)
\otimes_{\pi_\bullet(E)}
E_\bullet(X)
\longrightarrow
[\mathbb{S}, E \wedge E \wedge X]_\bullet
=
\pi_\bullet(E \wedge E \wedge X)
$$
(for $E_\bullet(-)$ the canonical $\pi_\bullet(E)$-modules from [this prop.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum)) given on elements
$$
\Sigma^{n_1}\mathbb{S}
\overset{\alpha_1}{\longrightarrow}
E \wedge E
\;\;\,,
\;\;
\Sigma^{n_2} \mathbb{S}
\overset{\alpha_2}{\longrightarrow}
E \wedge X
$$
by
$$
\alpha_1 \cdot \alpha_2
\;\colon\;
\Sigma^{n_1 + n_2}\mathbb{S}
\overset{\simeq}{\longrightarrow}
\Sigma^{n_1} \mathbb{S} \wedge \Sigma^{n_2}\mathbb{S}
\overset{\alpha_1 \wedge \alpha_2}{\longrightarrow}
E \wedge E \wedge E \wedge X
\overset{id_E \wedge \mu \wedge id_X}{\longrightarrow}
E \wedge E \wedge X
\,.
$$
If $E_\bullet(E)$ is a [[flat module]] over $\pi_\bullet(E)$ then this is an [[isomorphism]].
=--
([Adams 69, lecture 3, lemma 1 (p. 68)](#Adams69), [Adams 74, part III, lemma 12.5](#Adams74))
+-- {: .proof}
###### Proof
First of all, that the given pairing is a well defined homomorphism (descends from $E_\bullet(E) \times E_\bullet(X)$ to $E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(X)$) follows from the associativity of $\mu$.
We discuss that it is an isomorphism when $E_\bullet(E)$ is flat over $\pi_\bullet(E)$:
First consider the case that $X \simeq \Sigma^{n} \mathbb{S}$ is a suspension of the [[sphere spectrum]]. Then (by [this example](Introduction+to+Stable+homotopy+theory+--+1-2#SuspendedSphereSpectrumHomology), using the [[tensor triangulated category|tensor triangulated]] stucture on the [[stable homotopy category]] ([prop.] (Introduction+to+Stable+homotopy+theory+--+1-2#TensorTriangulatedStructureOnStableHomotopyCategory)))
$$
E_\bullet(X) = E_\bullet(\Sigma^n X) \simeq \pi_{\bullet-n}(E)
$$
and
$$
\pi_\bullet(E \wedge E \wedge X)
=
\pi_\bullet(E \wedge E \wedge \Sigma^n \mathbb{S})
\simeq
E_{\bullet-n}(E)
$$
and
$$
E_\bullet(E) \otimes_{\pi_\bullet(E)} \pi_{\bullet-n}(E)
\simeq
E_{\bullet-n}(E)
$$
Therefore in this case we have an isomorphism for all $E$.
For general $X$, we may without restriction assume that $X$ is represented by a sequential [[CW-spectrum]] ([prop.](Introduction+to+Stable+homotopy+theory+--+1-1#CWApproximationForSequentialSpectra)). Then the [[homotopy cofibers]] of its cell attachment maps are suspensions of the [[sphere spectrum]] ([rmk.](Introduction+to+Stable+homotopy+theory+--+1-1#StrictModelStructureCellAttachmentToSpectra)).
First consider the case that $X$ is a CW-spectrum with finitely many cells. Consider the [[homotopy cofiber sequence]] of the $(k+1)$st cell attachment (by that [remark](Introduction+to+Stable+homotopy+theory+--+1-1#StrictModelStructureCellAttachmentToSpectra)):
$$
\array{
\Sigma^{n_k-1}\mathbb{S}
&\longrightarrow&
X_k
&\longrightarrow&
X_{k+1}
&\longrightarrow&
\Sigma^{n_k}\mathbb{S}
&\longrightarrow&
\Sigma X_k
}
$$
and its image under the natural morphism $E_\bullet(E) \otimes_{\pi_\bullet(E)}E_\bullet(-) \to \pi_\bullet([\mathbb{S}, E \wedge E \wedge (-)])$, which is a [[commuting diagram]] of the form
$$
\array{
E_\bullet(E) \otimes_{\pi_\bullet(E)}E_\bullet(\Sigma^{n_k-1}\mathbb{S})
&\longrightarrow&
E_\bullet(E) \otimes_{\pi_\bullet(E)}E_\bullet(X_k)
&\longrightarrow&
E_\bullet(E) \otimes_{\pi_\bullet(E)}E_\bullet(X_{k+1})
&\longrightarrow&
E_\bullet(E) \otimes_{\pi_\bullet(E)}E_\bullet(\Sigma^{n_k}\mathbb{S})
&\longrightarrow&
E_\bullet(E) \otimes_{\pi_\bullet(E)}E_\bullet(\Sigma X_k)
\\
\downarrow
&&
\downarrow
&&
\downarrow
&&
\downarrow
&&
\downarrow
\\
[\mathbb{S}, E \wedge E \wedge \Sigma^{n_k-1}\mathbb{S}]_\bullet
&\longrightarrow&
[\mathbb{S}, E \wedge E \wedge X_k]_{\bullet}
&\longrightarrow&
[\mathbb{S}, E \wedge E \wedge X_{k+1}]_{\bullet}
&\longrightarrow&
[\mathbb{S}, E \wedge E \wedge \Sigma^{n_k}\mathbb{S}]_{\bullet}
&\longrightarrow&
[\mathbb{S}, E \wedge E \wedge \Sigma X_k]_{\bullet}
}
\,.
$$
Here the bottom row is a [[long exact sequence]] since $E \wedge E \wedge (-)$ preserves homotopy cofiber sequences (by [this lemma](Introduction+to+Stable+homotopy+theory+--+1-2#SmashTensoringWithSpectrumDerivedPreserveshomotopycofibers), part of the [[tensor triangulated category|tensor triangulated]] structure on $Ho(Spectra)$ [prop.](Introduction+to+Stable+homotopy+theory+--+1-2#TensorTriangulatedStructureOnStableHomotopyCategory)), and since $[\mathbb{S},-]_\bullet \simeq \pi_\bullet(-)$ sends [[homotopy cofiber sequences]] to [[long exact sequences]] ([prop.](Introduction+to+Stable+homotopy+theory+--+1-1#LongExactSequenceOfStableHomotopyGroups)). By the same reasoning, $E_\bullet(-)$ of the homotopy cofiber sequence is long exact; and by the assumption that $E_\bullet(E)$ is [[flat module|flat]], the functor $E_\bullet(E)\otimes_{\pi_\bullet(E)}(-)$ preserves this exactness, so that also the top row is a [[long exact sequence]].
Now by [[induction]] over the cells of $X$, the outer four vertical morphisms are [[isomorphisms]]. Hence the [[5-lemma]] implies that also the middle morphism is an isomorphism.
This shows the claim inductively for all finite CW-spectra. For the general statement, now use that
1. every CW-spectrum is the [[filtered colimit]] over its finite CW-subspectra;
1. the [[symmetric monoidal smash product of spectra]] $\wedge$ ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#SsymModuleSymmetricSpectra)) preserves colimits in its arguments separately (since it has a [[right adjoint]] ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#MonoidalCategoryOfModules)));
1. $[\mathbb{S},-]_\bullet \simeq \pi_\bullet(-)$ commutes over filtered colimits of CW-spectrum inclusions (by [this lemma](Introduction+to+Stable+homotopy+theory+--+P#CompactSubsetsAreSmallInCellComplexes), since spheres are [[compact topological space|compact]]);
1. $E_\bullet(E) \otimes_{\pi_\bullet(E)}(-)$ distributes over colimits (it being a [[left adjoint]]).
=--
Using prop. \ref{EnHomology}, we find below (theorem \ref{ComoduleHomsInE1PageOfEAdamsSpectralSequence}) that the first page of the $E$-Adams spectral sequence may be equivalently rewritten as hom-groups of [[comodules]] over $E_\bullet(E)$ regarded as a [[graded commutative Hopf algebroid]]. We now first discuss what this means.
##### The $E$-Steenrod algebra
{#DualESteenrodAlgebra}
We discuss here all the extra structure that exists on the $E$-self homology $E_\bullet(E)$ of a flat homotopy commutative ring spectrum. For $E = H \mathbb{F}_p$ the [[Eilenberg-MacLane spectrum]] on a [[prime field]] this reduces to the classical structure in [[algebraic topology]] called the _dual [[Steenrod algebra]]_ $\mathcal{A}^\ast_p$. Therefore one may generally speak of $E_\bullet(E)$ as being the _dual $E$-Steenrod algebra_.
Without the qualifier "dual" then "$E$-Steenrod algebra" refers to the $E$-self-cohomology $E^\bullet(E)$. For $E = H \mathbb{F}_p$ this _Steenrod algebra_ $\mathcal{A}_p$ (without "dual") is traditionally considered first, and the [[classical Adams spectral sequence]] was originally formulated in terms of $\mathcal{A}_p$ instead of $\mathcal{A}_p^\ast$. But one observes ([Adams 74, p. 280](#Adams74)) that the "dual" Steenrod algebra $E_\bullet(E)$ is much better behaved, at least as long as $E$ is flat in the sense of def. \ref{FlatE}.
Moreover, the dual $E$-Steenrod algebra $E_\bullet(E)$ is more fundamental in that it reflects a [[higher geometry|stacky geometry]] secretly underlying the $E$-Adams spectral sequence ([Hopkins 99](#Hopkins99)). This is the content of the concept of "[[commutative Hopf algebroid]]" (def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents} below) which is equivalently the [[formal dual]] of a [[groupoid]] internal to [[affine schemes]], def. \ref{CommutativeHopfAlgebroid}.
A simple illustrative archetype of the following construction of commutative Hopf algebroids from homotopy commutative ring spectra is the following situation:
For $X$ a [[finite set]] consider
$$
\array{
X \times X \times X
\\
\downarrow^{\mathrlap{\circ = (pr_1, pr_3)}}
\\
X \times X
\\
{}^{\mathllap{s = pr_1}}\downarrow
\uparrow
\downarrow^{\mathrlap{t = pr_2}}
\\
X
}
$$
as the ("[[codiscrete groupoid|codiscrete]]") [[groupoid]] with $X$ as [[objects]] and precisely one morphism from every object to every other. Hence the [[composition]] operation $\circ$, and the [[source]] and [[target]] maps are simply projections as shown. The identity morphism (going upwards in the above diagram) is the [[diagonal]].
Then consider the image of this structure under forming the [[free abelian groups]] $\mathbb{Z}[X]$, regarded as [[commutative rings]] under pointwise multiplication.
Since
$$
\mathbb{Z}[X \times X]
\simeq
\mathbb{Z}[X] \otimes \mathbb{Z}[X]
$$
this yields a diagram of homomorphisms of commutative rings of the form
$$
\array{
(\mathbb{Z}[X] \otimes \mathbb{Z}[X] )
\otimes_{\mathbb{Z}[X]}
(\mathbb{Z}[X] \otimes \mathbb{Z}[X])
\\
\uparrow^{\mathrlap{} }
\\
\mathbb{Z}[X] \otimes \mathbb{Z}[X]
\\
\uparrow
\downarrow
\uparrow
\\
\mathbb{Z}[X]
}
$$
satisfying some obvious conditions. Observe that here
1. the two morphisms $\mathbb{Z}[X] \rightrightarrows \mathbb{Z}[X] \otimes \mathbb{Z}[X]$ are $f \mapsto f \otimes e$ and $f \mapsto e \otimes f$, respectively, where $e$ denotes the unit element in $\mathbb{Z}[X]$;
1. the morphism $\mathbb{Z}[X] \otimes \mathbb{Z}[X] \to \mathbb{Z}[X]$ is the multiplication in the ring $\mathbb{Z}[X]$;
1. the morphism
$$
\mathbb{Z}[X] \otimes \mathbb{Z}[X]
\longrightarrow
\mathbb{Z}[X] \otimes \mathbb{Z}[C] \otimes \mathbb{Z}[C]
\overset{\simeq}{\longrightarrow}
(\mathbb{Z}[X] \otimes \mathbb{Z}[X] ) \otimes_{\mathbb{Z}[X]} (\mathbb{Z}[X] \otimes \mathbb{Z}[X])
$$
is given by $f \otimes g \mapsto f \otimes e \otimes g$.
All of the following rich structure is directly modeled on this simplistic example. We simply
1. replace the commutative ring $\mathbb{Z}[X]$ with any flat [[homotopy commutative ring spectrum]] $E$,
1. replace [[tensor product of abelian groups]] $\otimes$ with derived [[smash product of spectra]];
1. and form the [[stable homotopy groups]] $\pi_\bullet(-)$ of all resulting expressions.
+-- {: .num_defn #HopfAlgebroidStructureOnDualEOperations}
###### Definition
Let $(E, \mu, e)$ be a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)) which is flat according to def. \ref{FlatE}.
Then the **dual $E$-[[Steenrod algebra]]** is the pair of [[graded abelian groups]]
$$
(E_\bullet(E), \pi_\bullet(E))
$$
([rmk.](Introduction+to+Stable+homotopy+theory+--+1-2#EMHomology)) equipped with the following structure:
1. the [[graded commutative ring]] structure
$$
\pi_\bullet(E) \otimes \pi_\bullet(E)
\longrightarrow
\pi_\bullet(E)
$$
induced from $E$ being a [[homotopy commutative ring spectrum]] ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum));
1. the [[graded commutative ring]] structure
$$
E_\bullet(E) \otimes E_\bullet(E) \longrightarrow E_\bullet(E)
$$
induced from the fact that with $E$ also $E \wedge E$ is canonically a [[homotopy commutative ring spectrum]] ([exmpl.](Introduction+to+Stable+homotopy+theory+--+1-2#TensorProductOfTwoCommutativeMonoids)), so that also $E_\bullet(E) = \pi_\bullet(E \wedge E)$ is a graded commutative ring ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum));
1. the [[homomorphism]] of [[graded commutative rings]]
$$
\Psi
\;\colon\;
E_\bullet(E)
\longrightarrow
E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(E)
$$
induced under $\pi_\bullet(-)$ from
$$
E \wedge E
\overset{id \wedge e \wedge id}{\longrightarrow}
E \wedge E \wedge E
$$
via prop. \ref{EnHomology};
1. the [[homomorphisms]] of graded commutative rings
$$
\eta_L
\;\colon\;
\pi_\bullet(E)
\longrightarrow
E_\bullet(E)
$$
and
$$
\eta_R
\;\colon\;
\pi_\bullet(E)
\longrightarrow
E_\bullet(E)
$$
induced under $\pi_\bullet(-)$ from the homomorphisms of commutative ring spectra
$$
E \underoverset{\simeq}{r_{E}^{-1}}{\to} E \wedge \mathbb{S}
\overset{id \wedge e}{\longrightarrow}
E \wedge E
$$
and
$$
E \underoverset{\simeq}{\ell_E^{-1}}{\to} \mathbb{S} \wedge E
\overset{id \wedge e}{\longrightarrow}
E \wedge E
\,,
$$
respectively ([exmpl.](Introduction+to+Stable+homotopy+theory+--+1-2#TensorProductOfTwoCommutativeMonoids));
1. the homomorphism of graded commutative rings
$$
\epsilon
\;\colon\;
E_\bullet(E)
\longrightarrow
\pi_\bullet(E)
$$
induced under $\pi_\bullet(-)$ from
$$
\mu \;\colon\; E \wedge E \longrightarrow E
$$
regarded as a homomorphism of homotopy commutative ring spectra ([exmpl.](Introduction+to+Stable+homotopy+theory+--+1-2#TensorProductOfTwoCommutativeMonoids));
1. the homomorphisms graded commutative rings
$$
c
\;\colon\;
E_\bullet(E)
\longrightarrow
E_\bullet(E)
$$
induced under $\pi_\bullet(-)$ from the [[braiding]]
$$
\tau_{E,E}
\;\colon\;
E \wedge E
\longrightarrow
E \wedge E
$$
regarded as a homomorphism of homotopy commutative ring spectra ([exmpl.](Introduction+to+Stable+homotopy+theory+--+1-2#TensorProductOfTwoCommutativeMonoids)).
=--
([Adams 69, lecture 3, pages 66-68](#Adams69))
Notice that (as verified by direct unwinding of the definitions):
+-- {: .num_lemma}
###### Lemma
For $(E, \mu, e)$ a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)), consider $E_\bullet(E)$ with its canonical left and right $\pi_\bullet(E)$-module structure as in [this prop.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum). These module structures coincide with those induced by the ring homomorphisms $\eta_L$ and $\eta_R$ from def. \ref{HopfAlgebroidStructureOnDualEOperations}.
=--
These two actions need not strictly coincide, but they are isomorphic:
+-- {: .num_prop #EETwoLeftModuleStructures}
###### Proposition
For $(E, \mu, e)$ a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)), consider $E_\bullet(E)$ with its canonical left and right $\pi_\bullet(E)$-module structure ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum)). Since $E$ is a [[commutative monoid]], this right module structure may equivalently be regarded as a left-module, too. Then the [[braiding]]
$$
E_\bullet(E)
\simeq
\pi_\bullet(E \wedge E)
\overset{\pi_\bullet(\tau_{E,E})}{\longrightarrow}
\pi_\bullet(E \wedge E)
\simeq
E_\bullet(E)
$$
constitutes a module isomorphism ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#ModulesInMonoidalCategory)) between these two left module structures.
=--
+-- {: .proof}
###### Proof
On representatives as in the proof of ([this propo.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum)), the original left action is given by (we are notationally suppressing [[associators]] throughout)
$$
E \wedge E \wedge E
\overset{\mu \wedge id}{\longrightarrow}
E \wedge E
\,,
$$
while the other left action, induced from the canonical right action, is given by
$$
E \wedge E \wedge E
\underoverset{\simeq}{\tau_{E, E \wedge E}}{\longrightarrow}
E \wedge E \wedge E
\overset{id \wedge \mu}{\longrightarrow}
E \wedge
\,.
$$
So in order that $\tau_{E,E}$ represents a module homomorphism under $\pi_\bullet(-)$, it is sufficient that the following diagram commutes (we write $E_i \coloneqq E$ for $i \in \{1,2,3\}$ to make the action of the [[braiding]] more manifest)
$$
\array{
E_1 \wedge E_2 \wedge E_3
&\overset{id \wedge \tau_{E_2,E_3}}{\longrightarrow}&
E_1 \wedge E_3 \wedge E_2
\\
{}^{\mathllap{id}}\downarrow
&&
\downarrow^{\mathrlap{\tau_{E_1, E_3 \wedge E_2}}}
\\
&&
E_3 \wedge E_2 \wedge E_1
\\
{}^{\mathllap{\mu \wedge id}}\downarrow
&&
\downarrow^{\mathrlap{id \wedge \mu}}
\\
E \wedge E_3
&\underset{\tau_{E,E_3}}{\longrightarrow}&
E_3 \wedge E
}
\,.
$$
But since $(E,\mu,e)$ is a [[commutative monoid]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#MonoidsInMonoidalCategory)), it satisfies $\mu = \mu \circ \tau$ so that we may factor this diagram as follows:
$$
\array{
E_1 \wedge E_2 \wedge E_3
&\overset{id \wedge \tau_{E_2,E_3}}{\longrightarrow}&
E_1 \wedge E_3 \wedge E_2
\\
{}^{\mathllap{\tau_{E_1, E_2} \wedge id}}\downarrow
&&
\downarrow^{\mathrlap{\tau_{E_1, E_3 \wedge E_2}}}
\\
E_2 \wedge E_1 \wedge E_3
&\overset{\tau_{E_2 \wedge E_1, E_3}}{\longrightarrow}&
E_3 \wedge E_2 \wedge E_1
\\
{}^{\mathllap{\mu \wedge id}}\downarrow
&&
\downarrow^{\mathrlap{id \wedge \mu}}
\\
E \wedge E_3
&\underset{\tau_{E,E_3}}{\longrightarrow}&
E_3 \wedge E
}
\,.
$$
Here the top square commutes by [[coherence theorem for symmetric monoidal categories|coherence]] of the braiding ([rmk](Introduction+to+Stable+homotopy+theory+--+1-2#SymmetricMonoidalCategoriesCoherenceTheorem)) since both composite morphisms correspond to the same [[permutation]], while the bottom square commutesm due to the [[natural transformation|naturality]] of the braiding. Hence the total rectangle commutes.
=--
The dual $E$-[[Steenrod algebras]] of def. \ref{HopfAlgebroidStructureOnDualEOperations} evidently carry a lot of structure. The concept organizing this is that of_[[commutative Hopf algebroids]]_.
+-- {: .num_defn #CommutativeHopfAlgebroid}
###### Definition
A **[[graded commutative Hopf algebroid]]** is an [[internal groupoid]] in the [[opposite category]] $gCRing^{op}$ of $\mathbb{Z}$-[[graded commutative rings]], regarded with its [[cartesian monoidal category]] structure.
=--
(e.g. [Ravenel 86, def. A1.1.1](#Ravenel86))
+-- {: .num_remark #CommutativeHopfAlgebroidSpelledOut}
###### Remark
We unwind def. \ref{CommutativeHopfAlgebroid}. For $R \in gCRing$, write $Spec(R)$ for the same object, but regarded as an object in $gCRing^{op}$.
An [[internal category]] in $gCRing^{op}$ is a [[diagram]] in $gCRing^{op}$ of the form
$$
\array{
Spec(\Gamma) \underset{Spec(A)}{\times} Spec(\Gamma)
\\
\downarrow^{\mathrlap{\circ}}
\\
Spec(\Gamma)
\\
{}^{\mathllap{s}}\downarrow \; \uparrow_{\mathrlap{i}} \downarrow^{\mathrlap{t}}
\\
Spec(A)
}
\,,
$$
(where the [[fiber product]] at the top is over $s$ on the left and $t$ on the right) such that the pairing $\circ$ defines an [[associativity law|associative]] [[composition]] over $Spec(A)$, [[unitality|unital]] with respect to $i$. This is an [[internal groupoid]] if it is furthemore equipped with a morphism
$$
inv \;\colon\; Spec(\Gamma) \longrightarrow Spec(\Gamma)
$$
acting as assigning [[inverses]] with respect to $\circ$.
The key basic fact to use in order to express this equivalently in terms of algebra is that [[tensor product]] of commutative rings exhibits the [[cartesian monoidal category]] structure on $CRing^{op}$, see at _[CRing -- Properties -- Cocartesian comonoidal structure](CRing#CocartesianComnonoidalStructure)_:
$$
Spec(R_1) \underset{Spec(R_3)}{\times} Spec(R_2)
\simeq
Spec(R_1 \otimes_{R_3} R_2)
\,.
$$
This means that the above is equivalently a diagram in $gCRing$ of the form
$$
\array{
\Gamma \underset{A}{\otimes} \Gamma
\\
\uparrow^{\mathrlap{\Psi}}
\\
\Gamma
\\
{}^{\mathllap{\eta_L}}\uparrow
\downarrow^{\mathrlap{\epsilon}} \;
\uparrow^{\mathrlap{\eta_R}}
\\
A
}
$$
as well as
$$
c \; \colon \; \Gamma \longrightarrow \Gamma
$$
and satisfying [[formal duality|formally dual]] conditions, spelled out as def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents} below. Here
* $\eta_L, \eta_R$ are called the left and right _[[unit]] maps_;
* $\epsilon$ is called the _co-unit_;
* $\Psi$ is called the _[[comultiplication]]_;
* $c$ is called the _[[antipode]]_ or _conjugation_
=--
+-- {: .num_remark #HopfAlgebrasAsHopfAlgebroids}
###### Remark
Generally, in a commutative Hopf algebroid, def. \ref{CommutativeHopfAlgebroid}, the two morphisms $\eta_L, \eta_R\colon A \to \Gamma$ from remark \ref{CommutativeHopfAlgebroidSpelledOut} need not coincide, they make $\Gamma$ genuinely into a [[bimodule]] over $A$, and it is the [[tensor product]] of [[bimodules]] that appears in remark \ref{CommutativeHopfAlgebroidSpelledOut}. But it may happen that they coincide:
An [[internal groupoid]] $\mathcal{G}_1 \stackrel{\overset{s}{\longrightarrow}}{\underset{t}{\longrightarrow}} \mathcal{G}_0$ for which the [[domain]] and [[codomain]] morphisms coincide, $s = t$, is euqivalently a [[group object]] in the [[slice category]] over $\mathcal{G}_0$.
Dually, a [[commutative Hopf algebroid]] $\Gamma \stackrel{\overset{\eta_L}{\longleftarrow}}{\underset{\eta_R}{\longleftarrow}} A$ for which $\eta_L$ and $\eta_R$ happen to coincide is equivalently a commutative **[[Hopf algebra]]** $\Gamma$ over $A$.
=--
Writing out the formally dual axioms of an [[internal groupoid]] as in remark \ref{CommutativeHopfAlgebroidSpelledOut} yields the following equivalent but maybe more explicit definition of commutative Hopf algebroids, def. \ref{CommutativeHopfAlgebroid}
+-- {: .num_defn #CommutativeHopfAlgebroidDefinitionInExplicitComponents}
###### Definition
A **[[commutative Hopf algebroid]]** is
1. two [[commutative rings]], $A$ and $\Gamma$;
1. ring [[homomorphisms]]
1. (left/right unit)
$\eta_L,\eta_R \colon A \longrightarrow \Gamma$;
1. (comultiplication)
$\Psi \colon \Gamma \longrightarrow \Gamma \underset{A}{\otimes} \Gamma$;
1. (counit)
$\epsilon \colon \Gamma \longrightarrow A$;
1. (conjugation)
$c \colon \Gamma \longrightarrow \Gamma$
such that
1. (co-[[unitality]])
1. (identity morphisms respect source and target)
$\epsilon \circ \eta_L = \epsilon \circ \eta_R = id_A$;
1. (identity morphisms are units for composition)
$(id_\Gamma \otimes_A \epsilon) \circ \Psi = (\epsilon \otimes_A id_\Gamma) \circ \Psi = id_\Gamma$;
1. (composition respects source and target)
1. $\Psi \circ \eta_R = (id \otimes_A \eta_R) \circ \eta_R$;
1. $\Psi \circ \eta_L = (\eta_L \otimes_A id) \circ \eta_L$
1. (co-[[associativity]]) $(id_\Gamma \otimes_A \Psi) \circ \Psi = (\Psi \otimes_A id_\Gamma) \circ \Psi$;
1. ([[inverses]])
1. (inverting twice is the identity)
$c \circ c = id_\Gamma$;
1. (inversion swaps source and target)
$c \circ \eta_L = \eta_R$; $c \circ \eta_R = \eta_L$;
1. (inverse morphisms are indeed left and right inverses for composition)
the morphisms $\alpha$ and $\beta$ induced via the [[coequalizer]] property of the [[tensor product]] from $(-) \cdot c(-)$ and $c(-)\cdot (-)$, respectively
$$
\array{
\Gamma \otimes A \otimes \Gamma
&
\underoverset
{\longrightarrow}
{\longrightarrow}
{}
&
\Gamma \otimes \Gamma
&
\overset{coeq}{\longrightarrow}
&
\Gamma \otimes_A \Gamma
\\
&&
{}_{\mathllap{(-)\cdot c(-)}}\downarrow
&
\swarrow_{\mathrlap{\alpha}}
\\
&& \Gamma
}
$$
and
$$
\array{
\Gamma \otimes A \otimes \Gamma
&
\underoverset
{\longrightarrow}
{\longrightarrow}
{}
&
\Gamma \otimes \Gamma
&
\overset{coeq}{\longrightarrow}
&
\Gamma \otimes_A \Gamma
\\
&&
{}_{\mathllap{c(-)\cdot (-)}}\downarrow
&
\swarrow_{\mathrlap{\beta}}
\\
&& \Gamma
}
$$
satisfy
$\alpha \circ \Psi = \eta_L \circ \epsilon $
and
$\beta \circ \Psi = \eta_R \circ \epsilon $.
=--
([Adams 69, lecture 3, pages 62-66](#Adams69), [Ravenel 86, def. A1.1.1](#Ravenel86))
+-- {: .num_remark}
###### Remark
In ([Adams 69, lecture 3, page 60](#Adams69)) the terminology used is "Hopf algebra in a fully satisfactory sense" with emphasis that the left and right module structure may differ. According to ([Ravenel 86, first page of appendix A1](#Ravenel86)) the terminology "Hopf algebroid" for this situation is due to [[Haynes Miller]].
=--
+-- {: .num_example #CommutativeHopfAlgebroidCodiscreteOverCommutativeRing}
###### Example
For $R$ a [[commutative ring]], then $R \otimes R$ becomes a [[commutative Hopf algebroid]] over $R$, formally dual (via def. \ref{CommutativeHopfAlgebroid}) to the [[pair groupoid]] on $Spec(R) \in CRing^{op}$.
For $X$ a [[finite set]] and $R = \mathbb{Z}[X]$, then this reduces to the motivating example from [above](#DualESteenrodAlgebra).
=--
It is now straightforward, if somewhat tedious, to check that:
+-- {: .num_prop #DualESteenrodAlgebrasIsCommutativeHopfAlgebroid}
###### Proposition
Let $(E, \mu, e)$ be a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)) which is flat according to def. \ref{FlatE}, then the dual $E$-[[Steenrod algebra]] $ (E_\bullet(E), \pi_\bullet(E))$ with the
structure maps $(\eta_L, \eta_R, \epsilon, c, \Psi)$ from prop. \ref{HopfAlgebroidStructureOnDualEOperations} is a graded commutative Hopf algebroid according to def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents}:
$$
(E_\bullet(E), \pi_\bullet(E))
\;\in\;
CommHopfAlgd
$$
=--
([Adams 69, lecture 3, pages 67-71](#Adams69), [Ravenel 86, chapter II, prop. 2.2.8](#Ravenel86))
+-- {: .proof}
###### Proof
One observes that $E \wedge E$ satisfies the axioms of a commutative Hopf algebroid object in homotopy commutative ring spectra, over $E$, by direct analogy to example \ref{CommutativeHopfAlgebroidCodiscreteOverCommutativeRing} (one just has to verify that the symmetric [[braidings]] go along coherently, which works by use of the [[coherence theorem for symmetric monoidal categories]] ([rmk.](Introduction+to+Stable+homotopy+theory+--+1-2#SymmetricMonoidalCategoriesCoherenceTheorem))). Applying the functor $\pi_\bullet(-)$ that forms [[stable homotopy groups]] to all structure morphisms of $E \wedge E$ yields the claimed structure morphisms of $E_\bullet(E)$.
=--
We close this subsection on [[commutative Hopf algebroids]] by discussion of their [[isomorphism classes]], when regarded dually as affine [[groupoids]]:
+-- {: .num_defn #IsomorphismClassesOfCOmmutativeHopfAlgebroid}
###### Definition
Given an [[internal groupoid]] in $gCRing^{op}$ (def. \ref{CommutativeHopfAlgebroid}, remark \ref{CommutativeHopfAlgebroidSpelledOut})
$$
\array{
Spec(\Gamma) \underset{Spec(A)}{\times} Spec(\Gamma)
\\
\downarrow^{\mathrlap{\circ}}
\\
Spec(\Gamma)
\\
{}^{\mathllap{s}}\downarrow \; \uparrow_{\mathrlap{i}} \downarrow^{\mathrlap{t}}
\\
Spec(A)
}
\,,
$$
then its affife scheme $Spec(A)_{/\sim}$ of **[[isomorphism classes]] of objects** is the [[coequlizer]] of the source and target morphisms
$$
Spec(\Gamma)
\underoverset
{\underset{t}{\longrightarrow}}
{\overset{s}{\longleftarrow}}
{\phantom{AA}}
Spec(A)
\overset{equ}{\longrightarrow}
Spec(A)_{/\sim}
\,.
$$
Hence this is the [[formal dual]] of the [[equalizer]] of the left and right unit (def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents})
$$
A
\underoverset
{\underset{\eta_R}{\longrightarrow}}
{\overset{\eta_L}{\longrightarrow}}
{\phantom{AA}}
\Gamma
\,.
$$
=--
By example \ref{CommutativeHopfAlgebroidCodiscreteOverCommutativeRing} every [[commutative ring]] gives rise to a commutative Hopf algebroid $R \otimes R$ over $R$. The _[[core of a ring|core]]_ of $R$ is the formal dual of the corresponding affine scheme of isomorphism classes according to def. \ref{IsomorphismClassesOfCOmmutativeHopfAlgebroid}:
+-- {: .num_defn #CoreOfARing}
###### Definition
For $R$ a [[commutative ring]], its **[[core of a ring|core]]** $c R$ is the [[equalizer]] in
$$
c R
\longrightarrow
R
\stackrel{\longrightarrow}{\longrightarrow}
R \otimes R
\,.
$$
A ring which is isomorphic to its core is called a **[[solid ring]]**.
=--
([Bousfield-Kan 72, §1, def. 2.1](#BousfieldKan72), [Bousfield 79, 6.4](#Bousfield79))
+-- {: .num_prop #FormingCoreOfARingIsIdempotent}
###### Proposition
The [[core of a ring|core]] of any ring $R$ is solid (def. \ref{CoreOfARing}):
$$
c c R \simeq c R
\,.
$$
=--
([Bousfield-Kan 72, prop. 2.2](#BousfieldKan72))
+-- {: .num_prop #ClassificationOfSolidRings}
###### Proposition
The following is the complete list of solid rings (def. \ref{CoreOfARing}) up to [[isomorphism]]:
1. The [[localization of a ring|localization]] of the ring of [[integers]] at a set $J$ of [[prime numbers]] (def. \ref{InvertingPrimes})
$$
\mathbb{Z}[J^{-1}]
\,;
$$
1. the [[integers modulo n]]
$$
\mathbb{Z}/n\mathbb{Z}
$$
for $n \geq 2$;
1. the [[Cartesian product|product]] rings
$$
\mathbb{Z}[J^{-1}]
\times
\mathbb{Z}/n\mathbb{Z}
\,,
$$
for $n \geq 2$ such that each [[prime factor]] of $n$ is contained in the set of primes $J$;
1. the ring cores of product rings
$$
c(\mathbb{Z}[J^{-1}] \times \underset{p \in K}{\prod} \mathbb{Z}/p^{e(p)})
\,,
$$
where $K \subset J$ are infinite sets of primes and $e(p)$ are positive natural numbers.
=--
([Bousfield-Kan 72, prop. 3.5](#BousfieldKan72), [Bousfield 79, p. 276](#Bousfield79))
##### Comodules over the $E$-Steenrod algebra
+-- {: .num_defn #SteenrodComoduleStructureOnSpectrum}
###### Definition
Let $(E, \mu, e)$ be a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)) which is flat according to def. \ref{FlatE}.
For $X \in Ho(Spectra)$ any spectrum, say that the **comodule structure** on $E_\bullet(X)$ ([rmk.](Introduction+to+Stable+homotopy+theory+--+1-2#EMHomology))) **over the dual $E$-Steenrod algebra** (def. \ref{HopfAlgebroidStructureOnDualEOperations}) is
1. the canonical structure of a $\pi_\bullet(E)$-[[module]] (according to [this prop.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum));
1. the homomorphism of $\pi_\bullet(E)$-modules
$$
\Psi_{E_\bullet(X)}
\;\colon\;
E_\bullet(X)
\longrightarrow
E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(X)
$$
induced under $\pi_\bullet(-)$ and via prop. \ref{EnHomology} from the morphism of spectra
$$
E \wedge X
\simeq
E \wedge \mathbb{S} \wedge X
\overset{id \wedge e \wedge id}{\longrightarrow}
E \wedge E \wedge X
\,.
$$
=--
+-- {: .num_defn #HopfComoduleRing}
###### Definition
Given a [[graded commutative Hopf algebroid]] $\Gamma$ over $A$ as in def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents}, hence an [[internal groupoid]] in $gCRing^{op}$, then a **comodule ring** over it is an [[action]] in $CRing^{op}$ of that internal groupoid.
=--
In the same spirit, a **[[comodule]]** over a commutative Hopf algebroid (not necessarily a comodule ring) is a [[quasicoherent sheaf]] on the corresponding [[internal groupoid]] (regarded as a ([[algebraic stack|algebraic]]) [[stack]]) (e.g. [Hopkins 99, prop. 11.6](#Hopkins99)). Explicitly in components:
+-- {: .num_defn #CommutativeHopfAlgebroidComodule}
###### Definition
Given a $\mathbb{Z}$-[[graded commutative Hopf algebroid]] $\Gamma$ over $A$ (def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents})
then a **left [[comodule]]** over $\Gamma$ is
1. a $\mathbb{Z}$-[[graded module|graded]] $A$-[[module]] $N$;
1. ([[co-action]]) a [[homomorphism]] of graded $A$-modules
$\Psi_N \;\colon\; N \longrightarrow \Gamma \otimes_A N$;
such that
1. (co-[[unitality]])
$(\epsilon \otimes_A id_N) \circ \Psi_N = id_N$;
1. (co-action property)
$(\Psi \otimes_A id_N) \circ \Psi_N = (id_\Gamma \otimes_A \Psi_N)\circ \Psi_N$.
A [[homomorphism]] between graded comodules $f \colon N_1 \to N_2$ is a homomorphism of underlying graded $A$-modules such that the following [[commuting diagram|diagram commutes]]
$$
\array{
N_1 &\overset{f}{\longrightarrow}& N_1
\\
{}^{\mathllap{\Psi_{N_1}}}\downarrow
&&
\downarrow^{\mathrlap{\Psi_{N_2}}}
\\
\Gamma \otimes_A N_1
&\underset{id \otimes_A f}{\longrightarrow}&
\Gamma \otimes_A N_2
}
\,.
$$
We write
$$
\Gamma CoMod
$$
for the resulting [[category]] of left comodules over $\Gamma$. Analogously for right comodules. The notation for the hom-sets in this category is abbreviated to
$$
Hom_\Gamma(-,-)
\coloneqq
Hom_{\Gamma CoMod}(-,-)
\,.
$$
A priori this is an [[Ab-enriched category]], but it is naturally further [[enriched category|enriched]] in [[graded abelian groups]]:
we may drop in the above definition of comodule homomorphisms $f\colon N_1 \to N_2$ the condition that the underlying morphism be grading-preserving. Say that $f$ has degree $n$ if it increases degree by $n$. This gives a $\mathbb{Z}$-graded hom-group
$$
Hom^\bullet_\Gamma(-,-)
\,.
$$
=--
+-- {: .num_example #ComoduleStructureOnGroundRing}
###### Example
For $(\Gamma,A)$ a [[commutative Hopf algebroid]], then $A$ becomes a left $\Gamma$-comodule (def. \ref{CommutativeHopfAlgebroidComodule}) with coaction given by the right unit
$$
A \overset{\eta_R}{\longrightarrow} \Gamma \simeq \Gamma \otimes_A A
\,.
$$
=--
+-- {: .proof}
###### Proof
The required co-unitality property is the dual condition in def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents}
$$
\epsilon \circ \eta_R = id_A
$$
of the fact in def. \ref{CommutativeHopfAlgebroid} that identity morphisms respect sources:
$$
id
\;\colon\;
A
\overset{\eta_R}{\longrightarrow}
\Gamma
\simeq
\Gamma \otimes_A A
\overset{\epsilon \otimes_A id}{\longrightarrow}
A \otimes_A A
\simeq
A
$$
The required co-action property is the dual condition
$$
\Psi \circ \eta_R = (id \otimes_A \eta_R) \circ \eta_R
$$
of the fact in def. \ref{CommutativeHopfAlgebroid} that composition of morphisms in a groupoid respects sources
$$
\array{
A
&\overset{\eta_R}{\longrightarrow}&
\Gamma
\\
{}^{\mathllap{\eta_R}}\downarrow
&&
\downarrow^{\mathrlap{\Psi}}
\\
\Gamma \simeq \Gamma \otimes_A A
&\underset{id \otimes_A \eta_R}{\longrightarrow}&
\Gamma \otimes_A \Gamma
}
\,.
$$
=--
+-- {: .num_prop #IndeedComoduleStructureOnEX}
###### Proposition
Let $(E, \mu, e)$ be a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)) which is flat according to def. \ref{FlatE}, and for $X \in Ho(Spectra)$ any spectrum, then the morphism $\Psi_{E_\bullet(X)}$ from def. \ref{SteenrodComoduleStructureOnSpectrum} makes $E_\bullet(X)$ into a [[comodule]] (def. \ref{CommutativeHopfAlgebroidComodule}) over the dual $E$-[[Steenrod algebra]] (def. \ref{HopfAlgebroidStructureOnDualEOperations})
$$
E_\bullet(X)
\;\in\;
E_\bullet(E) CoMod
\,.
$$
=--
([Adams 69, lecture 3, pages 67-71](#Adams69), [Ravenel 86, chapter II, prop. 2.2.8](#Ravenel86))
+-- {: .num_example #GroundRingIsCanonicalComoduleOverHopfAlgebroid}
###### Example
Given a [[commutative Hopf algebroid]] $\Gamma$ over $A$, def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents}, then $A$ itself becomes a left $\Gamma$-[[comodule]] (def. \ref{CommutativeHopfAlgebroidComodule}) with [[coaction]] given by
$$
\Psi_A
\;\colon\;
A
\overset{\eta_L}{\longrightarrow}
\Gamma
\simeq
\Gamma \otimes_A A
$$
and a right $\Gamma$-comodule with coaction given by
$$
\Psi_A
\;\colon\;
A
\overset{\eta_R}{\longrightarrow}
\Gamma
\simeq
\Gamma \otimes_A A
\,.
$$
=--
More generally:
+-- {: .num_prop #CoFreeComodules}
###### Proposition
Given a [[commutative Hopf algebroid]] $\Gamma$ over $A$, there is a [[free-forgetful adjunction]]
$$
A Mod
\underoverset
{\underset{co-free}{\longrightarrow}}
{\overset{forget}{\longleftarrow}}
{\bot}
\Gamma CoMod
$$
between the [[category]] of $\Gamma$-[[comodules]], def. \ref{CommutativeHopfAlgebroidComodule} and the [[category of modules]] over $A$, where the [[cofree functor]] is [[right adjoint]].
Moreover:
1. The co-free $\Gamma$-[[comodule]] on an $A$-module $C$ is $\Gamma \otimes_A C$ equipped with the [[coaction]] induced by the [[comultiplication]] $\Psi$ in $\Gamma$.
1. The [[adjunct]] $\tilde f$ of a comodule homomorphism
$$
N \overset{f}{\longrightarrow} \Gamma \otimes_A C
$$
is its composite with the counit $\epsilon$ of $\Gamma$
$$
\tilde f
\;\colon\;
N
\overset{f}{\longrightarrow}
\Gamma \otimes_A C
\overset{\epsilon \otimes_A id}{\longrightarrow}
A \otimes_A C
\simeq
C
\,.
$$
=--
The **proof** is [[formal dual|formally dual]] to the proof that shows that constructing [[free modules]] is [[left adjoint]] to the [[forgetful functor]] from a [[category of modules]] to the underlying [[monoidal category]] ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#MonoidModuleOverItself)). But since the details of the adjunction isomorphism are important for the following discussion, we spell it out:
+-- {: .proof}
###### Proof
A homomorphism into a co-free $\Gamma$-comodule is a morphism of $A$-modules of the form
$$
f \;\colon\; N \longrightarrow \Gamma \otimes_A C
$$
making the following [[commuting diagram|diagram commute]]
$$
\array{
N &\overset{f}{\longrightarrow}& \Gamma \otimes_A C
\\
{}^{\mathllap{\Psi_N}}\downarrow
&&
\downarrow^{\mathrlap{\Psi \otimes_A id}}
\\
\Gamma \otimes_A N
&\underset{id \otimes_A f}{\longrightarrow}&
\Gamma \otimes_A \Gamma \otimes_A C
}
\,.
$$
Consider the composite
$$
\tilde f
\;\colon\;
N
\overset{f}{\longrightarrow}
\Gamma \otimes_A C
\overset{\epsilon \otimes_A id}{\longrightarrow}
A \otimes_A C
\simeq
C
\,,
$$
i.e. the "corestriction" of $f$ along the counit of $\Gamma$. By definition this makes the following square commute
$$
\array{
\Gamma \otimes_A N
&\overset{id \otimes_A f}{\longrightarrow}&
\Gamma \otimes_A \Gamma \otimes_A C
\\
{}^{\mathllap{=}}\downarrow
&&
\downarrow^{\mathrlap{id \otimes_A \epsilon \otimes_A id}}
\\
\Gamma \otimes_A N
&\underset{id \otimes_A \tilde f}{\longrightarrow}&
\Gamma \otimes_A C
}
\,.
$$
Pasting this square onto the bottom of the previous one yields
$$
\array{
N &\overset{f}{\longrightarrow}& \Gamma \otimes_A C
\\
{}^{\mathllap{\Psi_N}}\downarrow
&&
\downarrow^{\mathrlap{\Psi \otimes_A id}}
\\
\Gamma \otimes_A N
&\underset{id \otimes_A f}{\longrightarrow}&
\Gamma \otimes_A \Gamma \otimes_A C
\\
{}^{\mathllap{=}}\downarrow
&&
\downarrow^{\mathrlap{id \otimes_A \epsilon \otimes_A id}}
\\
\Gamma \otimes_A N
&\underset{id \otimes_A \tilde f}{\longrightarrow}&
\Gamma \otimes_A C
}
\,.
$$
Now due to co-unitality, the right right vertical composite is the identity on $\Gamma \otimes_A C$. But this means by the commutativity of the outer rectangle that $f$ is uniquely fixed in terms of $\tilde f$ by the relation
$$
f = (id \otimes_A f) \circ \Psi
\,.
$$
This establishes a [[natural bijection]]
$$
\frac{
N \overset{f}{\longrightarrow} \Gamma \otimes_A C
}{
N \overset{\tilde f}{\longrightarrow} C
}
$$
and hence the adjunction in question.
=--
+-- {: .num_prop #LeftComodulesToRightComodules}
###### Proposition
Consider a [[commutative Hopf algebroid]] $\Gamma$ over $A$, def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents}.
Any left comodule $N$ over $\Gamma$ (def. \ref{CommutativeHopfAlgebroidComodule}) becomes a right comodule via the coaction
$$
N
\overset{\Psi}{\longrightarrow}
\Gamma \otimes_A N
\overset{\simeq}{\longrightarrow}
N \otimes_A \Gamma
\overset{id \otimes_A c}{\longrightarrow}
N \otimes_A \Gamma
\,,
$$
where the isomorphism in the middle the is [[braiding]] in $A Mod$ and where $c$ is the conjugation map of $\Gamma$.
Dually, a right comodule $N$ becoomes a left comodule with the coaction
$$
N
\overset{\Psi}{\longrightarrow}
N \otimes_A \Gamma
\overset{\simeq}{\longrightarrow}
\Gamma \otimes_A N
\overset{c \otimes_A id}{\longrightarrow}
\Gamma \otimes_A N
\,.
$$
=--
+-- {: .num_defn #CotensorProductOfComodules}
###### Definition
Given a [[commutative Hopf algebroid]] $\Gamma$ over $A$, def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents},
and given $N_1$ a right $\Gamma$-comodule and $N_2$ a left comodule (def. \ref{CommutativeHopfAlgebroidComodule}), then their **[[cotensor product]]** $N_1 \Box_\Gamma N_2$ is the [[kernel]] of the difference of the two coaction morphisms:
$$
N_1 \Box_\Gamma N_2
\;\coloneqq\;
ker
\left(
N_1 \otimes_A N_2
\overset{\Psi_{N_1}\otimes_{A} id - id \otimes_A \Psi_{N_2} }{\longrightarrow}
N_1 \otimes_A \Gamma \otimes_A N_2
\right)
\,.
$$
If both $N_1$ and $N_2$ are left comodules, then their cotensor product is the cotensor product of $N_2$ with $N_1$ regarded as a right comodule via prop. \ref{LeftComodulesToRightComodules}.
=--
e.g. ([Ravenel 86, def. A1.1.4](#Ravenel86)).
+-- {: .num_example #PrimitiveElementInAComodule}
###### Example
Given a [[commutative Hopf algebroid]] $\Gamma$ over $A$, ([def.](commutative+Hopf+algebroid#CommutativeHopfAlgebroidDefinitionInExplicitComponents)), and given $N$ a left $\Gamma$-comodule ([def.](commutative+Hopf+algebroid#CommutativeHopfAlgebroidComodule)). Regard $A$ itself canonically as a right $\Gamma$-comodule via example \ref{GroundRingIsCanonicalComoduleOverHopfAlgebroid}. Then the cotensor product
$$
Prim(N) \coloneqq A \Box_\Gamma N
$$
is called the **[[primitive elements]]** of $N$:
$$
Prim(N) = \{ n \in N \;\vert\; \Psi_N(n) = 1 \otimes n \}
\,.
$$
=--
+-- {: .num_prop}
###### Proposition
Given a [[commutative Hopf algebroid]] $\Gamma$ over $A$, def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents},
and given $N_1, N_2$ two left $\Gamma$-comodules (def. \ref{CommutativeHopfAlgebroidComodule}), then their [[cotensor product]] (def. \ref{CotensorProductOfComodules}) is commutative, in that there is an [[isomorphism]]
$$
N_1 \Box N_2 \;\simeq\; N_2 \Box N_1
\,.
$$
=--
(e.g. [Ravenel 86, prop. A1.1.5](#Ravenel86))
+-- {: .num_lemma #ComoduleHomInTermsOfCotensorProduct}
###### Lemma
Given a [[commutative Hopf algebroid]] $\Gamma$ over $A$, def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents},
and given $N_1, N_2$ two left $\Gamma$-comodules (def. \ref{CommutativeHopfAlgebroidComodule}), such that $N_1$ is [[projective module|projective]] as an $A$-[[module]], then
1. The morphism
$$
Hom_A(N_1, A)
\overset{f \mapsto (id \otimes_A f) \circ \Psi_{N_1}}{\longrightarrow}
Hom_A(N_1, \Gamma \otimes_A A)
\simeq
Hom_A(N_1, \Gamma)
\simeq
Hom_A(N_1, A) \otimes_A \Gamma
$$
gives $Hom_A(N_1,A)$ the structure of a right $\Gamma$-comodule;
1. The [[cotensor product]] (def. \ref{CotensorProductOfComodules}) with respect to this right comodule structure is isomorphic to the hom of $\Gamma$-comodules:
$$
Hom_A(N_1, A) \Box_\Gamma N_2
\simeq
Hom_\Gamma(N_1, N_2)
\,.
$$
Hence in particular
$$
A \Box_\Gamma N_2
\;\simeq\;
Hom_\Gamma(A,N_2)
$$
=--
(e.g. [Ravenel 86, lemma A1.1.6](#Ravenel86))
+-- {: .num_remark}
###### Remark
In computing the second page of $E$-[[Adams spectral sequences]], the second statement in lemma \ref{ComoduleHomInTermsOfCotensorProduct} is the key translation that makes the comodule [[Ext]]-groups on the page be equivalent to a [[Cotor]]-groups. The latter lend themselves to computation, for instance via [[Lambda-algebra]] or via the [[May spectral sequence]].
=--
##### Universal coefficient theorem
The key use of the Hopf coalgebroid structure of prop. \ref{HopfAlgebroidStructureOnDualEOperations} for the present purpose is that it is extra structure inherited by morphisms in $E$-homology from morphisms of spectra. Namely forming $E$-homology $f_\ast \colon E_\bullet(X) \to E_\bullet(Y)$ of a morphism of a spectra $f \colon X \to Y$ does not just produce a morphism of $E$-homology groups
$$
[X,Y]_\bullet
\longrightarrow
Hom_{Ab^{\mathbb{Z}}}(E_\bullet(X), E_\bullet(Y))
$$
but in fact produces homomorphisms of comodules over $E_\bullet(E)$
$$
\alpha
\;\colon\;
[X,Y]_\bullet
\longrightarrow
Hom_{E_\bullet(E)}(E_\bullet(X), E_\bullet(Y))
\,.
$$
This is the statement of lemma \ref{SmashingMapsWithEFactorsThroughSteenrodComoduleHomomorphisms} below. The point is that $E_\bullet(E)$-comodule homomorphism are much more rigid than general abelian group homomorphisms and hence closer to reflecting the underlying morphism of spectra $f \colon X \to Y$.
In good cases such an approximation of _homotopy_ by _homology_ is in fact accurate, in that $\alpha$ is an [[isomorphism]]. In such a case ([Adams 74, part III, section 13](#Adams74)) speaks of a "[[universal coefficient theorem]]" (the [[coefficients]] here being $E$.)
One such case is exhibited by prop. \ref{ComoduleHomForENCohomology} below. This allows to equivalently re-write the first page of the $E$-Adams spectral sequence in terms of $E$-homology homomorphisms in theorem \ref{ComoduleHomsInE1PageOfEAdamsSpectralSequence} below.
+-- {: .num_lemma #SmashingMapsWithEFactorsThroughSteenrodComoduleHomomorphisms}
###### Lemma
For $X,Y \in Ho(Spectra)$ any two [[spectra]], the morphism (of $\mathbb{Z}$-[[graded abelian groups|graded abelian]]) [[generalized homology|generalized]] [[homology groups]] given by [[smash product of spectra|smash product]] with $E$ ([rmk.](Introduction+to+Stable+homotopy+theory+--+1-2#EMHomology))
$$
\array{
\pi_\bullet(E \wedge -)
&\colon&
[X,Y]_\bullet
&\longrightarrow&
Hom^\bullet_{Ab^{\mathbb{Z}}}(E_\bullet(X), E_\bullet(Y))
\\
&&
(X \overset{f}{\longrightarrow} Y)
&\mapsto&
\left(
E_\bullet(X)
\overset{f_\ast}{\longrightarrow}
E_\bullet(Y)
\right)
}
$$
factors through the [[forgetful functor]] from $E_\bullet(E)$-[[comodule]] [[homomorphisms]] (def. \ref{CommutativeHopfAlgebroidComodule}) over the dual $E$-[[Steenrod algebra]] (def. \ref{HopfAlgebroidStructureOnDualEOperations}):
$$
\array{
&& Hom^\bullet_{E_\bullet(E)}(E_\bullet(X), E_\bullet(Y))
\\
& {}^{\mathllap{\exists}}\nearrow
& \downarrow^{\mathrlap{forget}}
\\
[X,N]_\bullet
&\underset{\pi_\bullet(E \wedge -)}{\longrightarrow}&
Hom^\bullet_{Ab^{\mathbb{Z}}}(E_\bullet(X), E_\bullet(Y))
}
\,,
$$
where $E_\bullet(X)$ and $E_\bullet(Y)$ are regarded as $E$-Steenrod comodules according to def. \ref{CommutativeHopfAlgebroidComodule}, prop. \ref{IndeedComoduleStructureOnEX}.
=--
+-- {: .proof}
###### Proof
By def. \ref{CommutativeHopfAlgebroidComodule} we need to show that for $X \overset{f}{\longrightarrow} Y$ a morphism in $Ho(Spectra)$ then the following [[commuting diagram|diagram commutes]]
$$
\array{
E_\bullet(X)
&\overset{f_\ast}{\longrightarrow}&
E_\bullet(Y)
\\
{}^{\mathllap{\Psi_{E_\bullet(X)}}}\downarrow
&&
\downarrow^{\mathrlap{\Psi_{E_\bullet(Y)}}}
\\
E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(X)
&\underset{id \otimes_{\pi_\bullet(E)} f_\ast }{\longrightarrow}&
E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(Y)
}
\,.
$$
By def. \ref{CommutativeHopfAlgebroidComodule} and prop. \ref{IndeedComoduleStructureOnEX} this is the image under foming [[stable homotopy groups]] $\pi_\bullet(-)$ of the following diagram in $Ho(Spectra)$:
$$
\array{
E \wedge X
&\overset{id \wedge f}{\longrightarrow}&
E \wedge Y
\\
{}^{\mathllap{\simeq}}\downarrow
&&
\downarrow^{\mathrlap{\simeq}}
\\
E \wedge \mathbb{S} \wedge X
&&
E \wedge \mathbb{S} \wedge Y
\\
{}^{\mathllap{id \wedge e \wedge id}}\downarrow
&&
\downarrow^{\mathrlap{id \wedge e \wedge id}}
\\
E \wedge E \wedge X
&\underset{id \wedge id \wedge f}{\longrightarrow}&
E \wedge E \wedge Y
}
\,.
$$
But that this diagram commutes is simply the [[functor|functoriality]] of the derived [[smash product of spectra]] as a functor on the [[product category]] $Ho(Spectra) \times Ho(Spectra)$.
=--
+-- {: .num_prop #AdamsUCT}
###### Proposition
Let $(E, \mu, e)$ be a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)), and let $X, Y \in Ho(Spectra)$ be two [[spectra]] such that $E_\bullet(X)$ is a [[projective module]] over $\pi_\bullet(E)$ (via [this prop.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum)).
Then the homomorphism of [[graded abelian groups]]
$$
\phi_{UC}
\;\colon\;
[X, E \wedge Y]_\bullet
\stackrel{}{\longrightarrow}
Hom_{\pi_\bullet(E)}^\bullet(E_\bullet(X), E_\bullet(Y))_\bullet
$$
given by
$$
(X \overset{f}{\longrightarrow} E \wedge Y)
\;\mapsto\;
\pi_\bullet
( E \wedge X
\overset{id \wedge f}{\longrightarrow}
E \wedge E \wedge Y
\overset{\mu \wedge id}{\longrightarrow}
E \wedge Y
)
$$
is an [[isomorphism]].
=--
([Schwede 12, chapter II, prop. 6.20](#Schwede12))
+-- {: .proof}
###### Proof
First of all we claim that the morphism in question factors as
$$
\beta
\;\colon\;
[X, E \wedge Y]_\bullet
\overset{\simeq}{\longrightarrow}
Hom^\bullet_{E Mod}( E \wedge X , E \wedge Y)
\overset{\pi_\bullet}{\longrightarrow}
Hom^\bullet_{\pi_\bullet(E)}( E_\bullet(X), E_\bullet(Y) )
\,,
$$
where
1. $E Mod = E Mod(Ho(Spectra), \wedge, \mathbb{S})$ denotes the category of [[homotopy module spectra]] over $E$ ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum))
1. the first morphisms is the [[free-forgetful adjunction]] isomorphism for forming [[free modules|free]] ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#MonoidModuleOverItself)) $E$-[[homotopy module spectra]]
1. the second morphism is the respective component of the composite of the [[forgetful functor]] from $E$-[[homotopy module spectra]] back to $Ho(Spectra)$ with the functor $\pi_\bullet$ that forms [[stable homotopy groups]].
This is because (by [this prop.](Introduction+to+Stable+homotopy+theory+--+1-2#MonoidModuleOverItself)) the first map is given by first smashing with $E$ and then postcomposing with the $E$-action on the free module $E \wedge X$, which is the pairing $E \wedge E \overset{\mu}{\to} E$ ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#MonoidModuleOverItself)).
Hence it is sufficient to show that the morphism on the right is an isomorphism.
We show more generally that for $N_1, N_2$ any two $E$-[[homotopy module spectra]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)) such that $\pi_\bullet(N_1)$ is a [[projective module]] over $\pi_\bullet(E)$, then
$$
Hom^\bullet_{E Mod}( N_1 , N_2)
\overset{\pi_\bullet}{\longrightarrow}
Hom^\bullet_{\pi_\bullet(E)}( \pi_\bullet(N_1), \pi_\bullet(N_2) )
$$
is an isomorphism.
To see this, first consider the case that $\pi_\bullet(N_1)$ is in fact a $\pi_\bullet(E)$-[[free module]].
This implies that there is a basis $\mathcal{B} = \{x_i\}_{i \in I}$ and a homomorphism
$$
\underset{i \in I}{\vee}
\Sigma^{\vert x_i\vert}
E \longrightarrow
N_1
$$
of $E$-[[homotopy module spectra]], such that this is a [[stable weak homotopy equivalence]].
Observe that this sits in a [[commuting diagram]] of the form
$$
\array{
Hom^\bullet_{E Mod}( \underset{i \in I}{\vee} \Sigma^{\vert x_i\vert} E, N_2 )
&\overset{\pi_\bullet}{\longrightarrow}&
Hom^\bullet_{\pi_\bullet(E)}(\pi_\bullet(\underset{i \in I}{\vee}\Sigma^{\vert x_i\vert} E) , \pi_\bullet(N_2) )
\\
{}^{\mathllap{\simeq}}\downarrow
&&
\downarrow^{\mathrlap{\simeq}}
\\
\underset{i \in I}{\prod} [ \Sigma^{\vert x_i\vert}\mathbb{S}, N_2 ]_\bullet
&\underset{\simeq}{\longrightarrow}&
\underset{i \in I}{\prod}
\pi_{\bullet + \vert x_i \vert}(N_2)
}
$$
where
1. the left vertical isomorphism exhibits [[wedge sum]] of spectra as the [[coproduct]] in the [[stable homotopy category]] ([lemma](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryHasCoproducts));
1. the bottom isomorphism is from [this prop.](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyGrouspAsHomsOutOfSphereSpectrum);
1. the right vertical isomorphism is that of the [[free-forgetful adjunction]] for modules over $\pi_\bullet(E)$.
Hence the top horizontal morphism is an isomorphism, which was to be shown.
Now consider the general case that $\pi_\bullet(N_1)$ is a [[projective module]] over $\pi_\bullet(E)$. Since (graded) projective modules are precisely the [[retracts]] of (graded) [[free modules]] ([prop.](projective+module#NProjectiveIFFHomNExact)), there exists a diagram of $\pi_\bullet(E)$-modules of the form
$$
id
\;\colon\;
\pi_\bullet(N_1)
\longrightarrow
\pi_\bullet( \underset{i \in I}{\vee} \Sigma^{\vert x_i \vert} E )
\longrightarrow
\pi_\bullet(N_1)
$$
which induces the corresponding [[split idempotent]] of $\pi_\bullet(E)$-modules
$$
\pi_\bullet( \underset{i \in I}{\vee} \Sigma^{\vert x_i \vert} E )
\longrightarrow
\pi_\bullet(N_1)
\longrightarrow
\pi_\bullet( \underset{i \in I}{\vee} \Sigma^{\vert x_i \vert} E )
\,.
$$
As before, by freeness this is actually the image under $\pi_\bullet$ of an idempotent of [[homotopy ring spectra]]
$$
e_\bullet
\;\colon\;
\underset{i \in I}{\vee} \Sigma^{\vert x_i \vert} E
\longrightarrow
\underset{i \in I}{\vee} \Sigma^{\vert x_i \vert} E
$$
and so in particular of [[spectra]].
Now in the [[stable homotopy category]] $Ho(Spectra)$ all [[split idempotent|idempotents split]] ([prop.](split+idempotent#InTrinagulatedCategoryWithDirectSumsOfTrianglesAllIdempotentsSplit)), hence there exists a diagram of spectra of the form
$$
e
\;\colon\;
\underset{i \in I}{\vee} \Sigma^{\vert x_i \vert} E
\longrightarrow
X
\longrightarrow
\underset{i \in I}{\vee} \Sigma^{\vert x_i \vert} E
$$
with $\pi_\bullet(e) = e_\bullet$.
Consider the composite
$$
X
\longrightarrow
\underset{i \in I}{\vee} \Sigma^{\vert x_i \vert} E
\longrightarrow
N_1
\,.
$$
Since $\pi_\bullet(e) = e_\bullet$ it follows that under $\pi_\bullet$ this is an isomorphism, then that $X \simeq N_1$ in the [[stable homotopy category]].
In conclusion this exhibits $N_1$ as a [[retract]] of an free $E$-homotopy module spectrum
$$
id
\;\colon\;
N_1
\longrightarrow
\underset{i \in I}{\vee} \Sigma^{\vert x_i \vert} E
\longrightarrow
N_1
\,,
$$
hence of a spectrum for which the morphism in question is an isomorphism.
Since the morphism in question is [[natural transformation|natural]], its value on $N_1$ is a retract in the [[arrow category]] of an isomorphism, hence itself an isomorphism ([lemma](Introduction+to+Stable+homotopy+theory+--+P#RetractPreservesIsomorphism)).
=--
+-- {: .num_remark}
###### Remark
A stronger version of the statement of prop. \ref{AdamsUCT}, with the free homotopy $E$-module spectrum $E \wedge Y$ replaced by any homotopy $E$-module spectrum $F$, is considered in ([Adams 74, chapter III, prop. 13.5](#Adams74)) ("[[universal coefficient theorem]]"). Strong conditions are considered that ensure that
$$
F^\bullet(X) = [X,F]_\bullet
\longrightarrow
Hom^\bullet_{\pi_\bullet(E)}(E_\bullet(X), \pi_\bullet(F))
$$
is an isomormphism (expressing the $F$-cohomology of $X$ as the $\pi_\bullet(E)$-linear dual of the $E$-homology of $X$).
For the following we need only the weaker but much more general statement of prop. \ref{AdamsUCT}, and in fact this is all that ([Adams 74, p. 323](#Adams74)) ends up using, too.
=--
With this we finally get the following statement, which serves to identify maps of certain spectra with their induced maps on $E$-homology:
+-- {: .num_prop #ComoduleHomForENCohomology}
###### Proposition
Let $(E, \mu, e)$ be a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)), and let $X, Y \in Ho(Spectra)$ be two [[spectra]] such that
1. $E$ is flat according to def. \ref{FlatE};
1. $E_\bullet(X)$ is a [[projective module]] over $\pi_\bullet(E)$ (via [this prop.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum)).
Then the morphism from lemma \ref{SmashingMapsWithEFactorsThroughSteenrodComoduleHomomorphisms}
$$
[X, E \wedge Y]_\bullet
\stackrel{\pi_\bullet(E \wedge -)}{\longrightarrow}
Hom_{E_\bullet(E)}^\bullet(E_\bullet(X), E_\bullet( E \wedge Y)))
\simeq
Hom_{E_\bullet(E)}^\bullet(E_\bullet(X), E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(Y) ))
$$
is an [[isomorphism]] (where the isomophism on the right is that of prop. \ref{EnHomology}).
=--
([Adams 74, part III, page 323](#Adams74))
+-- {: .proof}
###### Proof
Observe that the following [[commuting diagram|diagram commutes]]:
$$
\array{
[X, E \wedge Y]_\bullet
&& \overset{\pi_\bullet(E \wedge -)}{\longrightarrow} &&
Hom_{E_\bullet(E)}^\bullet(E_\bullet(X), E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(Y) ))
\\
& {}_{\mathllap{\phi_{UC}}}\searrow
&&
\swarrow_{\mathrlap{\epsilon \otimes id \circ (-) }}
\\
&&
Hom^\bullet_{\pi_\bullet(E)}(E_\bullet(X), E_\bullet(Y))
}
\,,
$$
where
1. the top morphism is the one from lemma \ref{SmashingMapsWithEFactorsThroughSteenrodComoduleHomomorphisms};
1. the right vertical morphism is the adjunction isomorphism from prop. \ref{CoFreeComodules};
1. the left diagonal morphism is the one from prop. \ref{AdamsUCT}.
To see that this indeed commutes, notice that
1. the top morphism sends $(X \overset{f}{\to} E \wedge Y)$ to $E_\bullet(X) \overset{E_\bullet(f)}{\to} E_\bullet(E \wedge Y) \simeq \pi_\bullet(E \wedge E \wedge Y) $ by definition;
1. the right vertical morphism sends this further to $E_\bullet(X) \overset{E_\bullet(f)}{\to} \pi_\bullet(E \wedge E \wedge Y) \overset{\pi_\bullet(\mu \wedge id)}{\to} \pi_\bullet(E \wedge Y)$, by the proof of prop. \ref{CoFreeComodules} (which says that the map is given by postcomposition with the counit of $E_\bullet(E)$) and def. \ref{HopfAlgebroidStructureOnDualEOperations} (which says that this counit is represented by $\mu$);
1. by prop. \ref{AdamsUCT} this is the same as the action of the left diagonal morphism.
But now
1. the right vertical morphism is an isomorphism by prop. \ref{EnHomology};
1. the left diagonal morphism is an isomorphism by prop. \ref{AdamsUCT}
and so it follows that the top horizontal morphism is an isomorphism, too.
=--
In conclusion:
+-- {: .num_theorem #ComoduleHomsInE1PageOfEAdamsSpectralSequence}
###### Theorem
Let $(E, \mu, e)$ be a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)), and let $X, Y \in Ho(Spectra)$ be two [[spectra]] such that
1. $E$ is flat according to def. \ref{FlatE};
1. $E_\bullet(X)$ is a [[projective module]] over $\pi_\bullet(E)$ (via [this prop.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum)).
Then the first page of the $E$-Adams spectral sequence, def. \ref{AdamsEAdamsSpectralSequence}, for $[Y,X]_\bullet$ is isomorphic to the following chain complex of graded homs of [[comodules]] (def. \ref{CommutativeHopfAlgebroidComodule}) over the dual $E$-[[Steenrod algebra]] $(E_\bullet(E), \pi_\bullet(E))$ (prop. \ref{HopfAlgebroidStructureOnDualEOperations}):
$$
E_1^{s,t}(X,Y)
\;\simeq\;
Hom^t_{E_\bullet(E)}(E_\bullet(X), E_{\bullet-s}(A_s))
\;\;\,,
\;\;\;\;\;
d_1
\;=\;
Hom_{E_\bullet(E)}(E_\bullet(X), E_\bullet( g \circ h ))
$$
$\,$
$$
0
\to
Hom_{E_\bullet(E)}^t(E_\bullet(X),E_\bullet(A_0))
\stackrel{d_1}{\longrightarrow}
Hom_{E_\bullet(E)}^t(
E_\bullet(X),
E_{\bullet-1}(A_1)
)
\stackrel{d_1}{\longrightarrow}
Hom_{E_\bullet(E)}^t(
E_\bullet(X),
E_{\bullet-2}(A_2)
)
\stackrel{d_1}{\longrightarrow}
\cdots
\,.
$$
=--
([Adams 74, theorem 15.1 page 323](#Adams74))
+-- {: .proof}
###### Proof
This is prop. \ref{ComoduleHomForENCohomology} applied to def. \ref{AdamsEAdamsSpectralSequence}:
$$
\begin{aligned}
E_1^{s,t}(X,Y)
& =
[X, \underset{A_s}{\underbrace{E \wedge Y_s}}]_{t-s}
\\
& \simeq
Hom^{t-s}_{E_\bullet(E)}( E_\bullet(X), E_\bullet(\underset{A_s}{\underbrace{E \wedge Y_s}}) )
\\
&\simeq
Hom^{t}_{E_\bullet(E)}( E_\bullet(X), E_{\bullet-s}(A_s) )
\end{aligned}
$$
=--
#### The second page
{#TheE2TermOfTheEAdamsSpectralSequence}
+-- {: .num_theorem #SecondPageOfEAdamsSpectralSequence}
###### Theorem
Let $(E, \mu, e)$ be a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)), and let $X, Y \in Ho(Spectra)$ be two [[spectra]] such that
1. $E$ is flat according to def. \ref{FlatE};
1. $E_\bullet(X)$ is a [[projective module]] over $\pi_\bullet(E)$ (via [this prop.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum)).
Then the entries of the second page of the $E$-Adams spectral sequence for $[X,Y]_\bullet$ (def. \ref{AdamsEAdamsSpectralSequence}) are the [[Ext]]-groups of [[commutative Hopf algebroid]]-[[comodules]] (def. \ref{CommutativeHopfAlgebroidComodule}) over the [[commutative Hopf algebroid]] structure on the dual $E$-[[Steenrod algebra]] $E_\bullet(E)$ from prop. \ref{HopfAlgebroidStructureOnDualEOperations}:
$$
E_2^{s,t}(X,Y)
\simeq
Ext^{s,t}_{E_\bullet(E)}(E_\bullet(X), E_\bullet(Y))
\,.
$$
(On the right $s$ is the degree that goes with any [[Ext]]-functor, and the "internal degree" $t$ is the additional degree of morphisms between graded modules from def. \ref{CommutativeHopfAlgebroidComodule}.)
In the special case that $X = \mathbb{S}$ is the [[sphere spectrum]], then (by prop. \ref{ComoduleHomInTermsOfCotensorProduct}) these are equivalently [[Cotor]]-groups
$$
E^{s,t}_2(X,Y)
\simeq
Cotor^{s,t}_{E_\bullet(E)}(\pi_\bullet(E), E_\bullet(Y))
\,.
$$
=--
([Adams 74, theorem 15.1, page 323](#Adams74))
+-- {: .proof}
###### Proof
By theorem \ref{ComoduleHomsInE1PageOfEAdamsSpectralSequence}, under the given assumptions the first page reads
$$
E_1^{s,t}(X,Y)
\;\simeq\;
Hom^t_{E_\bullet(E)}(E_\bullet(X), E_{\bullet-s}(A_s))
\;\;\,,
\;\;\;\;\;
d_1
\;=\;
Hom_{E_\bullet(E)}(E_\bullet(X), E_\bullet( g \circ h ))
$$
$\,$
$$
0
\to
Hom_{E_\bullet(E)}^t(E_\bullet(X),E_\bullet(A_0))
\stackrel{d_1}{\longrightarrow}
Hom_{E_\bullet(E)}^t(
E_\bullet(X),
E_{\bullet-1}(A_1)
)
\stackrel{d_1}{\longrightarrow}
Hom_{E_\bullet(E)}^t(
E_\bullet(X),
E_{\bullet-2}(A_2)
)
\stackrel{d_1}{\longrightarrow}
\cdots
\,.
$$
By remark \ref{TermsOfNextPageOnSpectralSequence} the second page is the [[cochain cohomology]] of this complex. Hence by the standard theory of [[derived functors in homological algebra]] (see the section _[Via acyclic resolutions](derived+functor+in+homological+algebra#ViaAcyclicResolutions)_), it is now sufficient to see that:
1. the category $E_\bullet(E) CoMod$ (def. \ref{CommutativeHopfAlgebroidComodule}, prop. \ref{DualESteenrodAlgebrasIsCommutativeHopfAlgebroid}) is an [[abelian category]] with [[enough injectives]] (so that all [[right derived functors|right]] [[derived functor in homological algebra|derived functors]] on $E_\bullet(E) CoMod$ exist);
1. the first page graded chain complex $(E^{\bullet,t}_1(X,Y), d_1)$ is the image under the [[hom-functor]] $F \coloneqq Hom_{E_\bullet(E)}(E_\bullet(Y),-)$ of an $F$-[[acyclic resolution]] of $E_\bullet(X)$ (so that its cohomology indeed computes the [[Ext]]-derived functor ([theorem](derived+functor+in+homological+algebra#DerivedFFromFInjectiveResolution))).
That $E_\bullet(E) CoMod$ is an [[abelian category]] is lemma \ref{CategoryOfHopfComodulesIsAbelianIfHopfAlgebroidIsFlat} below, and that it has enough injectives is lemma \ref{IfCounitOfHopfAlgebroidIsFlatThenCofreeModulesAreInjective}.
Lemma \ref{ResolutionEWp} below shows that $E_\bullet(A_\bullet)$ is a resolution of $E_\bullet(Y)$ in $E_\bullet(E) CoMod$. By prop. \ref{EnHomology} it is a resolution by cofree comodules (def. \ref{CoFreeComodules}). That these are $F$-acyclic is lemma \ref{CoFreeHopfComodulesAreHomNAcyclicForProjectiveN} below.
=--
##### $E$-Adams resolutions
We discuss that the first page of the $E$-Adams spectral sequence indeed exhibits a [[resolution]] as required by the proof of theorem \ref{SecondPageOfEAdamsSpectralSequence}.
+-- {: .num_lemma #ResolutionEWp}
###### Lemma
Given an $E$-Adams spectral sequence $(E^{s,t}_r(X,Y),d_r)$ as in def. \ref{AdamsEAdamsSpectralSequence}, then the sequences of morphisms
$$
0
\to
E_\bullet(Y_p)
\overset{E_\bullet(g_p)}{\longrightarrow}
E_\bullet(A_p)
\overset{E_\bullet(h_p)}{\longrightarrow}
E_{\bullet-1}(Y_{p+1})
\to
0
$$
are [[short exact sequences|short exact]], hence their [[splicing of short exact sequences]]
$$
\array{
0
&\to&
E_\bullet(Y)
&& \overset{E_\bullet(g_0)}{\longrightarrow} &&
E_\bullet(A_0)
&& \overset{\partial}{\longrightarrow} &&
E_{\bullet-1}(A_1)
&& \overset{\partial}{\longrightarrow} &&
E_{\bullet-2}(A_2)
&& \longrightarrow &&
\cdots
\\
&&
&&
&&
& {}_{\mathllap{E_\bullet(h_0)}}\searrow && \nearrow_{\mathrlap{E_\bullet(g_1)}}
&&
{}_{\mathllap{E_\bullet(h_1)}}\searrow
&& \nearrow_{\mathrlap{E_\bullet(g_2)}}
\\
&&
&&
&&
&&
E_{\bullet-1}(Y_1)
&&
&&
E_{\bullet-2}(Y_2)
}
$$
is a [[long exact sequence]], exhibiting the graded [[chain complex]] $(E_\bullet(A_\bullet), \partial)$ as a [[resolution]] of $E_\bullet(Y)$.
=--
([Adams 74, theorem 15.1, page 322](#Adams74))
+-- {: .proof}
###### Proof
Consider the image of the defining [[homotopy cofiber sequence]]
$$
Y_p
\overset{g_p}{\longrightarrow}
E \wedge Y_p
\overset{h_p}{\longrightarrow}
\Sigma Y_{p+1}
$$
under the functor $E \wedge (-)$. This is itself a homotopy cofiber sequence of the form
$$
E \wedge Y_p
\overset{E \wedge g_p}{\longrightarrow}
E \wedge E \wedge Y_p
\overset{E \wedge h_p}{\longrightarrow}
\Sigma E \wedge Y_{p + 1}
$$
(due to the [[tensor triangulated category|tensor triangulated]] structure of the [[stable homotopy category]], [prop.](Introduction+to+Stable+homotopy+theory+--+1-2#TensorTriangulatedStructureOnStableHomotopyCategory)).
Applying the [[stable homotopy groups]] functor $\pi_\bullet(-) \simeq [\mathbb{S},-]_\bullet$ ([lemma](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyGrouspAsHomsOutOfSphereSpectrum)) to this yields a [[long exact sequence]] ([prop.](Introduction+to+Stable+homotopy+theory+--+1-1#LongExactSequenceOfStableHomotopyGroups))
$$
\cdots
\longrightarrow
E_{\bullet}(Y_{p+1})
\overset{ E_\bullet(f_p) }{\longrightarrow}
E_\bullet(Y_p)
\overset{ E_\bullet(g_p) }{\longrightarrow}
E_\bullet(A_p)
\overset{ E_\bullet(h_p) }{\longrightarrow}
E_{\bullet-1}(Y_{p+1})
\overset{ E_{\bullet-1}(f_p) }{\longrightarrow}
E_{\bullet-1}(Y_{p})
\overset{ E_{\bullet-1}(g_p) }{\longrightarrow}
E_{\bullet-1}(A_p)
\longrightarrow
\cdots
\,.
$$
But in fact this [[split exact sequence|splits]]: by [[unitality]] of $(E,\mu,e)$, the product operation $\mu$ on the [[homotopy commutative ring spectrum]] $E$ is a [[left inverse]] to $g_p$ in that
$$
id
\;\colon\;
E \wedge Y_p
\overset{E \wedge g_p}{\longrightarrow}
E \wedge E \wedge Y_p
\overset{\mu \wedge id}{\longrightarrow}
E \wedge Y_p
\,.
$$
Therefore $E_\bullet(g_p)$ is a [[monomorphism]], hence its [[kernel]] is trivial, and so by exactness $E_\bullet(f_p) = 0$. This means that the above long exact sequence collapses to [[short exact sequences]].
=--
##### Homological co-algebra
We discuss basic aspects of [[homological algebra]] in [[categories]] of [[comodules]] (def. \ref{CommutativeHopfAlgebroidComodule}) over [[commutative Hopf algebroids]] (def. \ref{CommutativeHopfAlgebroid}), needed in the proof of theorem \ref{SecondPageOfEAdamsSpectralSequence}.
+-- {: .num_lemma #CategoryOfHopfComodulesIsAbelianIfHopfAlgebroidIsFlat}
###### Lemma
Let $(\Gamma, A)$ be a [[commutative Hopf algebroid]] $\Gamma$ over $A$ (def. \ref{CommutativeHopfAlgebroid}, \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents}), such that the right $A$-module structure on $\Gamma$ induced by $\eta_R$ is a [[flat module]].
Then the [[category]] $\Gamma CoMod$ of [[comodules]] over $\Gamma$ (def. \ref{CommutativeHopfAlgebroidComodule}) is an [[abelian category]].
=--
(e.g. [Ravenel 86, theorem A1.1.3](#Ravenel86))
+-- {: .proof}
###### Proof
It is clear that, without any condition on the Hopf algebroid, $\Gamma CoMod$ is an [[additive category]].
Next we need to show if $\Gamma$ is flat over $A$, that then this is also a [[pre-abelian category]], in that [[kernels]] and [[cokernels]] exist.
To that end, let $f \colon (N_1,\Psi_{N_1}) \longrightarrow (N_2,\Psi_{N_2})$ be a morphism of comodules, hence a [[commuting diagram]] in $A$[[Mod]] of the form
$$
\array{
N_1 &\stackrel{f}{\longrightarrow}& N_2
\\
\downarrow^{\mathrlap{\Psi_{N_1}}} && \downarrow^{\mathrlap{\Psi_{N_2}}}
\\
\Gamma \otimes_A N_1
&\stackrel{id_\Gamma \otimes_A f}{\longrightarrow}&
\Gamma \otimes_A N_2
}
\,.
$$
Consider the kernel $ker(f)$ of $f$ in $A$[[Mod]] and its image under $\Gamma \otimes_A (-)$
$$
\array{
ker(f) &\longrightarrow& N_1 &\stackrel{f}{\longrightarrow}& N_2
\\
\mathllap{\exists}\downarrow
&&
\downarrow^{\mathrlap{\Psi_{N_1}}}
&&
\downarrow^{\mathrlap{\Psi_{N_2}}}
\\
\Gamma \otimes_A ker(f)
&\longrightarrow&
\Gamma \otimes_A N_1
&\stackrel{id_\Gamma \otimes_A f}{\longrightarrow}&
\Gamma \otimes_A N_2
}
\,.
$$
By the assumption that $\Gamma$ is a [[flat module]] over $A$, also $\Gamma \otimes_A ker(f) \simeq ker(\Gamma \otimes_A f)$ is a [[kernel]]. Hence by the [[universal property]] of kernels and the commutativity of the square o the right, there exists a unique vertical morphism as shown on the left, making the left [[commuting square|square commute]]. This means that the $A$-module $ker(f)$ uniquely inherits the structure of a $\Gamma$-comodule such as to make $ker(f) \to N_1$ a comodule homomorphism. By the same universal property it follows that $ker(f)$ with this comodule structure is in fact the kernel of $f$ in $\Gamma CoMod$.
The argument for the existence of [[cokernels]] proceeds [[formal dual|formally dually]]. Hence $\Gamma CoMod$ is a [[pre-abelian category]].
But it also follows from this construction that the comparison morphism
$$
coker(ker(f)) \longrightarrow ker(coker(f))
$$
formed in $\Gamma CoMod$ has underlying it the corresponding comparison morphism in $A Mod$. There this is an [[isomorphism]] by the fact that the [[category of modules]] $A Mod$ is an [[abelian category]], hence it is an isomorphism also in $\Gamma CoMod$. So the latter is in fact an [[abelian category]] itself.
=--
+-- {: .num_lemma #IfCounitOfHopfAlgebroidIsFlatThenCofreeModulesAreInjective}
###### Lemma
Let $(\Gamma, A)$ be a [[commutative Hopf algebroid]] $\Gamma$ over $A$ (def. \ref{CommutativeHopfAlgebroid}, \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents}), such that the right $A$-module structure on $\Gamma$ induced by $\eta_R$ is a [[flat module]].
Then
1. every co-free $\Gamma$-[[comodule]] (def. \ref{CoFreeComodules}) on an [[injective module]] over $A$ is an [[injective object]] in $\Gamma CoMod$;
1. $\Gamma CoMod$ has [[enough injectives]] ([def.](injective+object#EnoughInjectives)) if the [[axiom of choice]] holds in the ambient [[set theory]].
=--
(e.g. [Ravenel 86, lemma A1.2.2](#Ravenel86))
+-- {: .proof}
###### Proof
First of all, assuming the [[axiom of choice]], then the [[category of modules]] $A Mod$ has [[enough injectives]] (by [this proposition](injective+object#AbHasEnoughInjectives)).
Now by prop. \ref{CoFreeComodules} we have the [[adjunction]]
$$
A Mod
\underoverset
{\underset{co-free}{\longrightarrow}}
{\overset{forget}{\longleftarrow}}
{\bot}
\Gamma CoMod
\,.
$$
Observe that the [[left adjoint]] is a [[faithful functor]] (being a [[forgetful functor]]) and that, by the proof of lemma \ref{CategoryOfHopfComodulesIsAbelianIfHopfAlgebroidIsFlat}, it is an [[exact functor]]. This implies that
1. for $I \in A Mod$ an [[injective module]], then the co-free comodule $\Gamma \otimes_A I$ is an [[injective object]] in $\Gamma CoMod$ (by [this lemma](injective+module#RightAdjointsOfExactFunctorsPreserveInjectives));
1. for $N \in \Gamma CoMod$ any object, and for $i \colon forget(N) \hookrightarrow I$ a monomorphism of $A$-modules into an injective $A$-module, then the [[adjunct]] $\tilde i \colon N \hookrightarrow \Gamma\otimes_A I$ is a monomorphism (by [this lemma](injective+object#TransferOfEnoughInjectivesAlongAdjunctions))) hence a monomorpism into an injective comodule, by the previous item.
Hence $\Gamma CoMod$ has enough injective objects ([def.](injective+object#EnoughInjectives)).
=--
+-- {: .num_lemma #CoFreeHopfComodulesAreHomNAcyclicForProjectiveN}
###### Lemma
Let $(\Gamma, A)$ be a [[commutative Hopf algebroid]] $\Gamma$ over $A$ (def. \ref{CommutativeHopfAlgebroid}, \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents}), such that the right $A$-module structure on $\Gamma$ induced by $\eta_R$ is a [[flat module]]. Let $N \in \Gamma CoMod$ be a $\Gamma$-[[comodule]] (def. \ref{CommutativeHopfAlgebroidComodule}) such that the underlying $A$-module is a [[projective module]] (a [[projective object]] in $A$[[Mod]]).
Then (assuming the [[axiom of choice]] in the ambient set theory) every co-free comodule (prop. \ref{CoFreeComodules}) is an $F$-[[acyclic object]] for $F$ the [[hom functor]] $Hom_{\Gamma CoMod}(N,-)$.
=--
+-- {: .proof}
###### Proof
We need to show that the [[derived functors in homological algebra|derived functors]] $\mathbb{R}^{\bullet} Hom_{\Gamma}(N,-)$ vanish in positive degree on all co-free comodules, hence on $\Gamma \otimes_A K$, for all $K \in A Mod$.
To that end, let $I^\bullet$ be an [[injective resolution]] of $K$ in $A Mod$. By lemma \ref{IfCounitOfHopfAlgebroidIsFlatThenCofreeModulesAreInjective} then $\Gamma \otimes_A I^\bullet$ is a sequence of [[injective objects]] in $\Gamma CoMod$ and by the assumption that $\Gamma$ is flat over $A$ it is an [[injective resolution]] of $\Gamma \otimes_A K$ in $\Gamma CoMod$. Therefore the derived functor in question is given by
$$
\begin{aligned}
\mathbb{R}^{\bullet \geq 1} Hom_\Gamma(N, \Gamma \otimes_A K)
& \simeq
H_{\bullet \geq 1}( Hom_\Gamma( N, \Gamma \otimes_A I^\bullet ) )
\\
& \simeq H_{\bullet \geq 1}( Hom_A(N, I^\bullet) )
\\
& \simeq 0
\end{aligned}
\,.
$$
Here the second equivalence is the cofree/forgetful adjunction isomorphism of prop. \ref{CoFreeComodules}, while the last equality then follows from the assumption that the $A$-module underlying $N$ is a [[projective module]] (since [[hom functors]] out of [[projective objects]] are [[exact functors]] ([here](projective+object#EquivalenceOfDefinitionsInAbelian)) and since derived functors of exact functors vanish in positive degree ([here](derived+functor+in+homological+algebra#DerivedFunctorOfExactFunctor))).
=--
With lemma \ref{CoFreeHopfComodulesAreHomNAcyclicForProjectiveN} the proof of theorem \ref{SecondPageOfEAdamsSpectralSequence} is completed.
#### Convergence
{#Convergence}
We discuss the convergence of $E$-Adams spectral sequences (def. \ref{AdamsEAdamsSpectralSequence}), i.e. the identification of the "limit", in an appropriate sense, of the terms $E_r^{s,t}(X,Y)$ on the $r$th page of the spectral sequence as $r$ goes to infinity.
If an $E$-Adams spectral sequence converges, then it converges not necessarily to the full stable homotopy groups $[X,Y]_\bullet$, but to some [[localization of abelian groups|localization]] of them. This typically means, roughly, that only certain $p$-[[torsion subgroups]] in $[X,Y]_\bullet$ for some [[prime numbers]] $p$ are retained. We give a precise discussion below in _[Localization and adic completion of abelian groups](#LocalizationOfAbelianGroups)_.
If one knows that $[X,Y]_q$ is a [[finitely generated object|finitely generated]] [[abelian group]] (as is the case notably for $\pi_q^s = [\mathbb{S},\mathbb{S}]_q$ by the _[[Serre finiteness theorem]]_) then this allows to recover the full information from its pieces: by the _[[fundamental theorem of finitely generated abelian groups]]_ (prop. \ref{FundamentalTheoremOfFinitelyGeneratedAbelianGroups} below) these groups are [[direct sums]] of powers $\mathbb{Z}^n$ of the [[infinite cyclic group]] $\mathbb{Z}$ with finite [[cyclic groups]] of the form $\mathbb{Z}/p^k \mathbb{Z}$, and so all one needs to compute is the powers $k$ "one prime $p$ at a time". This we review below in _[Primary decomposition of abelian groups](#PrimaryDecompositionOfAbelianGroups)_.
The deeper reason that $E$-Adams spectral sequences tend to converge to [[localization of an abelian group|localizations]] of the abelian groups $[X,Y]_\bullet$ of morphisms of spectra, is that they really converge to (the actual homotopy groups of) [[localizations of spectra]]. This is more than just a reformulation, because the localization at the level of spectra determies the [[filtration]] which controls the nature of the convergence. We discuss this localization of spectra below in _[Localization and nilpotent completion of spectra](#LocalizationOfSpectra)_.
Then we state convergence properties of $E$-Adams spectral sequences below in _[Convergence statements](#ConvergenceStatements)_.
##### Primary decomposition of abelian groups
{#PrimaryDecompositionOfAbelianGroups}
An $E$-[[Adams spectral sequence]] _typically_ converges (discussed [below](#ConvergenceStatements)) not to the full [[stable homotopy groups]] $[X,Y]_\bullet$, but just to some piece which on the [[finite group|finite]] [[direct summands]] consists only of [[p-primary groups]] for some [[prime numbers]] $p$ that depend on the nature of the [[homotopy ring spectrum]] $E$ . Here we review basic facts about $p$-primary decomposition of finite abelian groups and introduce their graphical calculus (remark \ref{pprimarygraphical} below) which will allow to read off these $p$-primary pieces from the stable page of the $E$-Adams spectral sequnce.
+-- {: .num_theorem #FundamentalTheoremOfFinitelyGeneratedAbelianGroups}
###### Theorem
**(fundamental theorem of finitely generated abelian groups)**
Every [[finitely generated object|finitely generated]] [[abelian group]] $A$ is [[isomorphism|isomorphic]] to a [[direct sum]] of [[p-primary groups|p-primary]] [[cyclic groups]] $\mathbb{Z}/p^k \mathbb{Z}$ (for $p$ a [[prime number]] and $k$ a [[natural number]] ) and copies of the [[infinite cyclic group]] $\mathbb{Z}$:
$$
A
\;\simeq\;
\mathbb{Z}^n
\oplus
\underset{i}{\bigoplus}
\mathbb{Z}/p_i^{k_i} \mathbb{Z}
\,.
$$
The summands of the form $\mathbb{Z}/p^k \mathbb{Z}$ are also called the _[[p-primary group|p-primary]] components_ of $A$. Notice that the $p_i$ need not all be distinct.
**fundamental theorem of finite abelian groups**:
In particular every [[finite group|finite]] [[abelian group]] is of this form for $n = 0$, hence is a [[direct sum]] of [[cyclic groups]].
**fundamental theorem of cyclic groups**:
In particular every [[cyclic group]] $\mathbb{Z}/n\mathbb{Z}$ is a [[direct sum]] of cyclic groups of the form
$$
\mathbb{Z}/n\mathbb{Z}
\simeq
\underset{i}{\bigoplus} \mathbb{Z}/ p_i^{k_i} \mathbb{Z}
$$
where all the $p_i$ are distinct and $k_i$ is the maximal power of the [[prime factor]] $p_i$ in the prime decomposition of $n$.
Specifically, for each natural number $d$ dividing $n$ it contains $\mathbb{Z}/d\mathbb{Z}$ as the [[subgroup]] generated by $n/d \in \mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}$. In fact the [[lattice of subgroups]] of $\mathbb{Z}/n\mathbb{Z}$ is the [[formal dual]] of the lattice of natural numbers $\leq n$ ordered by inclusion.
=--
(e.g. [Roman 12, theorem 13.4](#Roman12), [Navarro 03](#Navarro03)) for cyclic groups e.g. ([Aluffi 09, pages 83-84](#Aluffi09))
This is a special case of the _[[ structure theorem for finitely generated modules over a principal ideal domain]]_.
+-- {: .num_example #TwoFiniteGroupsOfOrderp2}
###### Example
For $p$ a [[prime number]], there are, up to [[isomorphism]], two [[abelian groups]] of [[order of a group|order]] $p^2$, namely
$$
(\mathbb{Z}/p\mathbb{Z}) \oplus (\mathbb{Z}/p\mathbb{Z})
$$
and
$$
\mathbb{Z}/p^2 \mathbb{Z}
\,.
$$
=--
+-- {: .num_example}
###### Example
For $p_1$ and $p_2$ two distinct [[prime numbers]], $p_1 \neq p_2$, then there is, up to isomorphism, precisely one [[abelian group]] of order $p_1 p_2$, namely
$$
\mathbb{Z}/p_1 \mathbb{Z}
\oplus
\mathbb{Z}/p_2 \mathbb{Z}
\,.
$$
This is equivalently the [[cyclic group]]
$$
\mathbb{Z}/p_1 p_2 \mathbb{Z}
\simeq
\mathbb{Z}/p_1 \mathbb{Z}
\oplus
\mathbb{Z}/p_2 \mathbb{Z}
\,.
$$
The isomorphism is given by sending $1$ to $(p_2,p_1)$.
=--
+-- {: .num_example }
###### Example
Moving up, for two distinct prime numbers $p_1$ and $p_2$, there are exactly two abelian groups of order $p_1^2 p_2$, namely $(\mathbb{Z}/p_1 \mathbb{Z})\oplus (\mathbb{Z}/p_1 \mathbb{Z}) \oplus (\mathbb{Z}/p_2 \mathbb{Z})$ and $(\mathbb{Z}/p_1^2 \mathbb{Z})\oplus (\mathbb{Z}/p_2 \mathbb{Z})$. The latter is the [[cyclic group]] of order $p_1^2 p_2$. For instance, $\mathbb{Z}/12\mathbb{Z} \cong (\mathbb{Z}/4 \mathbb{Z})\oplus (\mathbb{Z}/3 \mathbb{Z})$.
=--
+-- {: .num_example}
###### Example
Similarly, there are four abelian groups of order $p_1^2 p_2^2$, three abelian groups of order $p_1^3 p_2$, and so on.
More generally, theorem \ref{FundamentalTheoremOfFinitelyGeneratedAbelianGroups} may be used to compute exactly how many abelian groups there are of any finite [[order of a group|order]] $n$ (up to [[isomorphism]]): write down its [[prime factorization]], and then for each prime power $p^k$ appearing therein, consider how many ways it can be written as a product of positive powers of $p$. That is, each [[partition]] of $k$ yields an abelian group of order $p^k$. Since the choices can be made independently for each $p$, the numbers of such partitions for each $p$ are then multiplied.
Of all these abelian groups of order $n$, of course, one of them is the [[cyclic group]] of order $n$. The fundamental theorem of cyclic groups says it is the one that involves the one-element partitions $k= [k]$, i.e. the cyclic groups of order $p^k$ for each $p$.
=--
+-- {: .num_remark #PrimaryDecompositionGraphicalRepresentation}
###### Remark
**(graphical representation of $p$-primary decomposition)**
Theorem \ref{FundamentalTheoremOfFinitelyGeneratedAbelianGroups} says that for any [[prime number]] $p$, the [[p-primary group|p-primary part]] of any finitely generated abelian group is determined uniquely up to [[isomorphism]] by
* a total number $k \in \mathbb{N}$ of powers of $p$;
* a [[partition]] $k = k_1 + k_2 + \cdots + k_q$.
The corresponding [[p-primary group]] is
$$
\underoverset{i = 1}{q}{\bigoplus} \mathbb{Z}/p^{k_i} \mathbb{Z}
\,.
$$
In the context of [[Adams spectral sequences]] it is conventional to depict this information graphically by
* $k$ dots;
* of which sequences of length $k_i$ are connected by vertical lines, for $i \in \{1, \cdots, q\}$.
For example the graphical representation of the $p$-primary group
$$
\mathbb{Z}/p\mathbb{Z}
\oplus
\mathbb{Z}/p\mathbb{Z}
\oplus
\mathbb{Z}/p^3 \mathbb{Z}
\oplus
\mathbb{Z}/p^4\mathbb{Z}
$$
is
$$
\array{
&&& \bullet
\\
&& & \vert
\\
&& \bullet & \bullet
\\
&& \vert & \vert
\\
&& \bullet & \bullet
\\
&& \vert & \vert
\\
\bullet & \bullet & \bullet & \bullet
}
\,.
$$
This notation comes from the convention of drawing stable pages of [[multiplicative spectral sequence|multiplicative]] [[Adams spectral sequences]] and reading them as encoding the [extension problem](Introduction+to+Stable+homotopy+theory+--+S#ExtensionProblemForSpectralSequences) for computing the homotopy groups that the spectral sequence converges to:
* a dot at the top of a vertical sequence of dots denotes the group $\mathbb{Z}/p\mathbb{Z}$;
* inductively, a dot vetically below a sequence of dots denotes a [[group extension]] of $\mathbb{Z}/p\mathbb{Z}$ by the group represented by the sequence of dots above;
* a vertical line between two dots means that the generator of the group corresponding to the upper dot is, regarded after inclusion into the group extension, the product by $p$ of the generator of the group corresponding to the lower dot, regarded as represented by the generator of the group extension.
So for instance
$$
\array{
\bullet
\\
\vert
\\
\bullet
}
$$
stands for an [[abelian group]] $A$ which forms a [[group extension]] of the form
$$
\array{
\mathbb{Z}/p\mathbb{Z}
\\
\downarrow
\\
A
\\
\downarrow
\\
\mathbb{Z}/p\mathbb{Z}
}
$$
such that multiplication by $p$ takes the generator of the bottom copy of $\mathbb{Z}/p\mathbb{Z}$, regarded as represented by the generator of $A$, to the generator of the image of the top copy of $\mathbb{Z}/p\mathbb{Z}$.
This means that of the two possible choices of extensions (by example \ref{TwoFiniteGroupsOfOrderp2}) $A$ corresponds to the non-trivial extension $A = \mathbb{Z}/p^2\mathbb{Z}$. Because then in
$$
\array{
\mathbb{Z}/p\mathbb{Z} &
\\
\downarrow
\\
\mathbb{Z}/p^2 \mathbb{Z} &
\\
\downarrow
\\
\mathbb{Z}/p\mathbb{Z} &
}
$$
the image of the generator 1 of the first group in the middle group is $p = p \cdot 1$.
Conversely, the notation
$$
\array{
\bullet
\\
\\
\bullet
}
$$
stands for an [[abelian group]] $A$ which forms a [[group extension]] of the form
$$
\array{
\mathbb{Z}/p\mathbb{Z}
\\
\downarrow
\\
A
\\
\downarrow
\\
\mathbb{Z}/p\mathbb{Z}
}
$$
such that multiplication by $p$ of the generator of the top group in the middle group does _not_ yield the generator of the bottom group.
This means that of the two possible choices (by example \ref{TwoFiniteGroupsOfOrderp2}) $A$ corresponds to the _trivial_ extension $A = \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z}$. Because then in
$$
\array{
\mathbb{Z}/p\mathbb{Z}
\\
\downarrow
\\
\mathbb{Z}/p\mathbb{Z}
\oplus
\mathbb{Z}/p\mathbb{Z}
\\
\downarrow
\\
\mathbb{Z}/p\mathbb{Z}
}
$$
the generator 1 of the top group maps to the element $(1,0)$ in the middle group, and multiplication of that by $p$ is $(0,0)$ instead of $(0,1)$, where the latter is the generator of the bottom group.
Similarly
$$
\array{
\bullet
\\
\vert
\\
\bullet
\\
\vert
\\
\bullet
}
$$
is to be read as the result of appending to the previous case a dot _below_, so that this now indicates a group extension of the form
$$
\array{
\mathbb{Z}/p^2 \mathbb{Z}
\\
\downarrow
\\
A
\\
\downarrow
\\
\mathbb{Z}/p \mathbb{Z}
}
$$
such that $p$-times the generator of the bottom group, regarded as represented by the generator of the middle group, is the image of the generator of the top group. This is again the case for the unique non-trivial extension, and hence in this case the diagram stands for $A = \mathbb{Z}/p^3 \mathbb{Z}$.
And so on.
For example the stable page of the $\mathbb{F}_2$-[[classical Adams spectral sequence]] for computation of the [[p-primary group|2-primary part]] of the [[stable homotopy groups of spheres]] $\pi_{t-s}(\mathbb{S})$ has in ("internal") degree $t-s \leq 13$ the following non-trivial entries:
<img src="http://ncatlab.org/nlab/files/ClassicalAdamsSpectralSequence.jpg" width="600" >
(graphics taken from ([[Symmetric spectra|Schwede 12]]))
Ignoring here the diagonal lines (which denote multiplication by the element $h_1$ that encodes the additional [[ring]] structure on $\pi_\bullet(\mathbb{S})$ which here we are not concerned with) and applying the above prescription, we read off for instance that $\pi_3(\mathbb{S}) \simeq \mathbb{Z}/8\mathbb{Z}$ (because all three dots are connected) while $\pi_8(\mathbb{S}) \simeq \mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z} $ (because here the two dots are not connected). In total
| $k =$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
| $\pi_k(\mathbb{S})_{(2)} = $ | $\mathbb{Z}_{(2)}$ | $\mathbb{Z}/2$ | $\mathbb{Z}/2$ | $\mathbb{Z}/8$ | $0$ | $0$ | $\mathbb{Z}/2$ | $\mathbb{Z}/16$ | $(\mathbb{Z}/2)^2$ | $(\mathbb{Z}/2)^3$ | $\mathbb{Z}/2$ | $\mathbb{Z}/8$ | $0$ | $0$ |
Here the only entry that needs further explanation is the one for $k = 0$. We discuss the relevant concepts for this below in the section _[Localization and adic completion of abelian groups](#LocalizationOfAbelianGroups)_, but for completeness, here is the quick idea:
The symbol $\mathbb{Z}_{(2)}$ refers to the [[p-adic integers|2-adic integers]] (def. \ref{pAdicIntegers}), i.e. for the [[limit]] of abelian groups
$$
\mathbb{Z}_{(2)}
=
\underset{\longleftarrow}{\lim}_{n \geq 1} \mathbb{Z}/2^n \mathbb{Z}
$$
This is not [[p-primary group|2-primary]], but it does arise when applying [[p-adic completion|2-adic completion]] of abelian groups (def. \ref{AdicCompletionOfAbelingGroups}) to finitely generated abelian groups as in theorem \ref{FundamentalTheoremOfFinitelyGeneratedAbelianGroups}. The 2-adic integers is the abelian group associated to the diagram
$$
\array{
\vdots
\\
\vert
\\
\bullet
\\
\vert
\\
\bullet
\\
\vert
\\
\bullet
\\
\vert
\\
\bullet
}
$$
as in the above figure. Namely by the above prescrption, this infinite sequence should encode an abelian group $A$ such that it is an [[group extension|extension]] of $\mathbb{Z}/p\mathbb{Z}$ by itself of the form
$$
0 \to A \overset{p \cdot(-)}{\longrightarrow} A \longrightarrow \mathbb{Z}/p\mathbb{Z}
$$
(Because it is supposed to encode an extension of $\mathbb{Z}/p\mathbb{Z}$ by the group corresponding to the result of chopping off the lowest dot, which however in this case does not change the figure.)
Indeed, by lemma \ref{pAdicIntegersAspExtensionofFpByThemselves} below we have a short exact sequence
$$
0
\to
\mathbb{Z}_{(p)}
\overset{p \cdot (-)}{\longrightarrow}
\mathbb{Z}_{(p)}
\longrightarrow
\mathbb{Z}/p\mathbb{Z}
\to
0
\,.
$$
=--
##### Localization and adic completion of abelian groups
{#LocalizationOfAbelianGroups}
+-- {: .num_remark #ExtAb}
###### Remark
Recall that [[Ext]]-groups $\Ext^\bullet(A,B)$ between [[abelian groups]] $A, B \in $ [[Ab]] are concentrated in degrees 0 and 1 ([prop.](free+resolution#AbelianGroupHasFreeResolutionOfLength2)). Since
$$
Ext^0(A,B) \simeq Hom(A,B)
$$
is the plain [[hom-functor]], this means that there is only one possibly non-vanishing Ext-group $Ext^1$, therefore often abbreviated to just "$Ext$":
$$
Ext(A,B)
\coloneqq
Ext^1(A,B)
\,.
$$
=--
+-- {: .num_defn #AbelianGroupLocal}
###### Definition
Let $K$ be an [[abelian group]].
Then an [[abelian group]] $A$ is called **$K$-local** if all the [[Ext]]-groups from $K$ to $A$ vanish:
$$
Ext^\bullet(K,A) \simeq 0
$$
hence equivalently (remark \ref{ExtAb}) if
$$
Hom(K,A) \simeq 0
\;\;\;\;\;
and
\;\;\;\;\;
Ext(K,A) \simeq 0
\,.
$$
A [[homomorphism]] of abelian groups $f \colon B \longrightarrow C$ is called **$K$-local** if for all $K$-local groups $A$ the function
$$
Hom(f,A) \;\colon\; Hom(B,A) \longrightarrow Hom(A,A)
$$
is a [[bijection]].
> (**Beware** that here it would seem more natural to use $Ext^\bullet$ instead of $Hom$, but we do use $Hom$. See ([Neisendorfer 08, remark 3.2](localization+of+abelian+groups#Neisendorfer08)).
A homomorphism of abelian groups
$$
\eta \;\colon\; A \longrightarrow L_K A
$$
is called a **$K$-localization** if
1. $L_K A$ is $K$-local;
1. $\eta$ is a $K$-local morphism.
=--
We now discuss two classes of examples of localization of abelian groups
1. _[Classical localization at/away from primes](#ClassicalLocalizationOfAbelianGroups)_;
1. _[Formal completion at primes](#FormalCompletionOfAbelianGroups)_.
$\,$
**Classical localization at/away from primes**
{#ClassicalLocalizationOfAbelianGroups}
For $n \in \mathbb{N}$, write $\mathbb{Z}/n\mathbb{Z}$ for the [[cyclic group]] of [[order of a group|order]] $n$.
+-- {: .num_lemma #OutOfCyclicGroupExt1}
###### Lemma
For $n \in \mathbb{N}$ and $A \in Ab$ any [[abelian group]], then
1. the [[hom-object|hom-group]] out of $\mathbb{Z}/n\mathbb{Z}$ into $A$ is the $n$-[[torsion subgroup]] of $A$
$$
Hom(\mathbb{Z}/n\mathbb{Z}, A)
\simeq
\{
a \in A \;\vert\; p \cdot a = 0
\}
$$
1. the first [[Ext]]-group out of $\mathbb{Z}/n\mathbb{Z}$ into $A$ is
$$
Ext^1(\mathbb{Z}/n\mathbb{Z},A)
\simeq
A/n A
\,.
$$
=--
+-- {: .proof}
###### Proof
Regarding the first item: Since $\mathbb{Z}/p\mathbb{Z}$ is generated by its element 1 a morphism $\mathbb{Z}/p\mathbb{Z} \to A$ is fixed by the image $a$ of this element, and the only relation on 1 in $\mathbb{Z}/p\mathbb{Z}$ is that $p \cdot 1 = 0$.
Regarding the second item:
Consider the canonical [[free resolution]]
$$
0 \to \mathbb{Z} \overset{p \cdot (-)}{\longrightarrow} \mathbb{Z} \longrightarrow \mathbb{Z}/p\mathbb{Z} \to 0
\,.
$$
By the general discusson of [[derived functors in homological algebra]] this exhibits the [[Ext]]-group in degree 1 as part of the following [[short exact sequence]]
$$
0
\to
Hom(\mathbb{Z},A)
\overset{Hom(n\cdot(-),A)}{\longrightarrow}
Hom(\mathbb{Z}, A)
\longrightarrow
Ext^1(\mathbb{Z}/n\mathbb{Z},A)
\to 0
\,,
$$
where the morphism on the left is equivalently $A \overset{n \cdot (-)}{\to} A$.
=--
+-- {: .num_example}
###### Example
An [[abelian group]] $A$ is $\mathbb{Z}/p\mathbb{Z}$-local precisely if the operation
$$
p \cdot (-)
\;\colon\;
A \longrightarrow A
$$
of multiplication by $p$ is an [[isomorphism]], hence if "$p$ is invertible in $A$".
=--
+-- {: .proof}
###### Proof
By the first item of lemma \ref{OutOfCyclicGroupExt1} we have
$$
Hom(\mathbb{Z}/p\mathbb{Z}, A)
\simeq
\{
a \in A \;\vert\; p \cdot a = 0
\}
$$
By the second item of lemma \ref{OutOfCyclicGroupExt1} we have
$$
Ext^1(\mathbb{Z}/p\mathbb{Z},A)
\simeq
A/p A
\,.
$$
Hence by def. \ref{AbelianGroupLocal} $A$ is $\mathbb{Z}/p\mathbb{Z}$-local precisely if
$$
\{
a \in A \;\vert\; p \cdot a = 0
\}
\simeq
0
$$
and if
$$
A / p A \simeq 0
\,.
$$
Both these conditions are equivalent to multiplication by $p$ being invertible.
=--
+-- {: .num_defn #InvertingPrimes}
###### Definition
For $J \subset \mathbb{N}$ a set of [[prime numbers]], consider the [[direct sum]] $\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}$ of [[cyclic groups]] of [[order of a group|order]] $p$.
The operation of $\underset{p \in J}{\otimes} \mathbb{Z}/p\mathbb{Z}$-localization of abelian groups according to def. \ref{AbelianGroupLocal} is called **inverting the primes** in $J$.
Specifically
1. for $J = \{p\}$ a single prime then $\mathbb{Z}/p\mathbb{Z}$-localization is called **localization away from $p$**;
1. for $J$ the set of all primes except $p$ then $\underset{p \in J}{\otimes} \mathbb{Z}/p\mathbb{Z}$-localization is called **localization at $p$;
1. for $J$ the set of all primes, then $\underset{p \in J}{\otimes} \mathbb{Z}/p\mathbb{Z}$-localizaton is called **[[rationalization]]**..
=--
+-- {: .num_defn #LocalizationOfIntegersAtSetOfPrimes}
###### Definition
For $J \subset Primes \subset \mathbb{N}$ a [[set]] of [[prime numbers]], then
$$
\mathbb{Z}[J^{-1}]
\hookrightarrow
\mathbb{Q}
$$
denotes the [[subgroup]] of the [[rational numbers]] on those elements which have an expression as a fraction of natural numbers with denominator a product of elements in $J$.
This is the abelian group underlying the [[localization of a commutative ring]] of the ring of integers at the set $J$ of primes.
If $J = Primes - \{p\}$ is the set of all primes _except_ $p$ one also writes
$$
\mathbb{Z}_{(p)}
\coloneqq
\mathbb{Z}[Primes - \{p\}]
\,.
$$
Notice the parenthesis, to distinguish from the notation $\mathbb{Z}_{p}$ for the [[p-adic integers]] (def. \ref{pAdicIntegers} below).
=--
+-- {: .num_remark #ClassicalLocalizationSeenFromSpecZ}
###### Remark
The terminology in def. \ref{InvertingPrimes} is motivated by the following perspective of [[arithmetic geometry]]:
Generally for $R$ a [[commutative ring]], then an $R$-[[module]] is equivalently a [[quasicoherent sheaf]] on the [[spectrum of a commutative ring|spectrum of the ring]] $Spec(R)$.
In the present case $R = \mathbb{Z}$ is the [[integers]] and [[abelian groups]] are identified with $\mathbb{Z}$-modules. Hence we may think of an abelian group $A$ equivalently as a [[quasicoherent sheaf]] on [[Spec(Z)]].
The "[[ring of functions]]" on [[Spec(Z)]] is the integers, and a point in $Spec(\mathbb{Z})$ is labeled by a [[prime number]] $p$, thought of as generating the ideal of functions on [[Spec(Z)]] which vanish at that point. The [[residue field]] at that point is $\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$.
Inverting a prime means [[forcing]] $p$ to become invertible, which, by this characterization, it is precisely _away_ from that point which it labels. The localization of an abelian group at $\mathbb{Z}/p\mathbb{Z}$ hence corresponds to the restriction of the corresponding quasicoherent sheaf over $Spec(\mathbb{Z})$ to the complement of the point labeled by $p$.
Similarly localization _at_ $p$ is localization away from all points except $p$.
See also at _[[function field analogy]]_ for more on this.
=--
+-- {: .num_prop}
###### Proposition
For $J \subset \mathbb{N}$ a set of [[prime numbers]], a homomorphism of abelian groups $f \;\colon\; A \lookrightarrow B $ is local (def. \ref{AbelianGroupLocal}) with respect to $\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}$ (def. \ref{InvertingPrimes}) if under [[tensor product of abelian groups]] with $\mathbb{Z}[J^{-1}]$ (def. \ref{LocalizationOfIntegersAtSetOfPrimes}) it becomes an [[isomorphism]]
$$
f \otimes \mathbb{Z}[J^{-1}]
\;\colon\;
X \otimes \mathbb{Z}[J^{-1}]
\overset{\simeq}{\longrightarrow}
Y \otimes \mathbb{Z}[J^{-1}]
\,.
$$
Moreover, for $A$ any abelian group then its $\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}$-localization exists and is given by the canonical projection morphism
$$
A
\longrightarrow
A \otimes \mathbb{Z}[J^{-1}]
\,.
$$
=--
(e.g. [Neisendorfer 08, theorem 4.2](localization+of+abelian+groups#Neisendorfer08))
$\,$
**Formal completion at primes**
{#FormalCompletionOfAbelianGroups}
We have seen above in remark \ref{ClassicalLocalizationSeenFromSpecZ} that classical localization of abelian groups at a prime number is geometrically interpreted as restricting a [[quasicoherent sheaf]] over [[Spec(Z)]] to a single point, the point that is labeled by that prime number.
Alternatively one may restrict to the "infinitesimal neighbourhood" of such a point. Technically this is called the _[[formal neighbourhood]]_, because its ring of functions is, by definition, the ring of [[formal power series]] (regarded as an [[adic ring]] or [[pro-ring]]). The corresponding operation on abelian groups is accordingly called _[[formal completion]]_ or _[[adic completion]]_ or just _completion_, for short, of abelian groups.
It turns out that if the abelian group is [[finitely generated object|finitely generated]] then the operation of [[p-completion]] coincides with an operation of _localization_ in the sense of def. \ref{AbelianGroupLocal}, namely localization at the [[p-primary group|p-primary component]] $\mathbb{Z}(p^\infty)$ of the group $\mathbb{Q}/\mathbb{Z}$ (def. \ref{ZpInfinity} below). On the one hand this equivalence is useful for deducing some key properties of [[p-completion]], this we discuss below. On the other hand this situation is a shadow of the relation between [[localization of spectra]] and [[nilpotent completion of spectra]], which is important for characterizing the convergence properties of [[Adams spectral sequences]].
+-- {: .num_defn #AdicCompletionOfAbelingGroups}
###### Definition
For $p$ a [[prime number]], then the **[[p-adic completion]]** of an [[abelian group]] $A$ is the abelian group given by the [[limit]]
$$
A^\wedge_p
\coloneqq
\underset{\longleftarrow}{\lim}
\left(
\cdots
\longrightarrow
A / p^3 A
\longrightarrow
A / p^2 A
\longrightarrow
A/p A
\right)
\,,
$$
where the morphisms are the evident [[quotient]] morphisms. With these understood we often write
$$
A^\wedge_p \coloneqq \underset{\longleftarrow}{\lim}_n A/p^n A
$$
for short. Notice that here the indexing starts at $n = 1$.
=--
+-- {: .num_example #pAdicIntegers}
###### Example
The [[p-adic completion]] (def. \ref{AdicCompletionOfAbelingGroups}) of the [[integers]] $\mathbb{Z}$ is called the **[[p-adic integers]]**, often written
$$
\mathbb{Z}_p
\coloneqq
\mathbb{Z}^\wedge_p
\coloneqq
\underset{\longleftarrow}{\lim}_n \mathbb{Z}/p^n \mathbb{Z}
\,,
$$
which is the original example that gives the general concept its name.
With respect to the canonical [[ring]]-structure on the integers, of course $p \mathbb{Z}$ is a prime ideal.
Compare this to the ring $\mathcal{O}_{\mathbb{C}}$ of [[holomorphic functions]] on the [[complex plane]]. For $x \in \mathbb{C}$ any point, it contains the prime ideal generated by $(z-x)$ (for $z$ the canonical [[coordinate]] function on $\mathbb{z}$). The [[formal power series ring]] $\mathbb{C}[ [(z.x)] ]$ is the [[adic completion]] of $\mathcal{O}_{\mathbb{C}}$ at this ideal. It has the interpretation of functions defined on a [[formal neighbourhood]] of $X$ in $\mathbb{C}$.
Analogously, the [[p-adic integers]] $\mathbb{Z}_p$ may be thought of as the functions defined on a [[formal neighbourhood]] of the point labeled by $p$ in [[Spec(Z)]].
=--
+-- {: .num_lemma #pAdicIntegersAspExtensionofFpByThemselves}
###### Lemma
There is a [[short exact sequence]]
$$
0
\to
\mathbb{Z}_p
\overset{p \cdot (-)}{\longrightarrow}
\mathbb{Z}_p
\longrightarrow
\mathbb{Z}/p\mathbb{Z}
\to
0
\,.
$$
=--
+-- {: .proof}
###### Proof
Consider the following [[commuting diagram]]
$$
\array{
\vdots && \vdots && \vdots
\\
\downarrow
&&
\downarrow
&&
\downarrow
\\
\mathbb{Z}/p^3\mathbb{Z}
&\overset{p\cdot (-)}{\longrightarrow}&
\mathbb{Z}/p^4 \mathbb{Z}
&\longrightarrow&
\mathbb{Z}/p\mathbb{Z}
\\
\downarrow
&&
\downarrow
&&
\downarrow
\\
\mathbb{Z}/p^2\mathbb{Z}
&\overset{p\cdot (-)}{\longrightarrow}&
\mathbb{Z}/p^3 \mathbb{Z}
&\longrightarrow&
\mathbb{Z}/p\mathbb{Z}
\\
\downarrow
&&
\downarrow
&&
\downarrow
\\
\mathbb{Z}/p\mathbb{Z}
&\overset{p\cdot (-)}{\longrightarrow}&
\mathbb{Z}/p^2 \mathbb{Z}
&\longrightarrow&
\mathbb{Z}/p\mathbb{Z}
\\
\downarrow
&&
\downarrow
&&
\downarrow
\\
0
&\longrightarrow&
\mathbb{Z}/p\mathbb{Z}
&\longrightarrow&
\mathbb{Z}/p\mathbb{Z}
}
\,.
$$
Each horizontal sequence is exact. Taking the [[limit]] over the vertical sequences yields the sequence in question. Since limits commute over limits, the result follows.
=--
We now consider a concept of $p$-completion that is in general different from def. \ref{AdicCompletionOfAbelingGroups}, but turns out to coincide with it in [[finitely generated object|finitely generated]] abelian groups.
+-- {: .num_defn #pInvertedInZ}
###### Definition
For $p$ a [[prime number]], write
$$
\mathbb{Z}[1/p]
\coloneqq
\underset{\longrightarrow}{\lim}
\left(
\mathbb{Z}
\overset{p \cdot (-)}{\longrightarrow}
\mathbb{Z}
\overset{p \cdot (-)}{\longrightarrow}
\mathbb{Z}
\overset{}{\longrightarrow}
\cdots
\right)
$$
for the [[colimit]] (in [[Ab]]) over iterative applications of multiplication by $p$ on the [[integers]].
This is the [[abelian group]] generated by formal expressions $\frac{1}{p^k}$ for $k \in \mathbb{N}$, subject to the relations
$$
(p \cdot n) \frac{1}{p^{k+1}}
=
n \frac{1}{p^k}
\,.
$$
Equivalently it is the abelian group underlying the [[localization of a ring|ring localization]] $\mathbb{Z}[1/p]$.
=--
+-- {: .num_defn #AbelianGrouppComplete}
###### Definition
For $p$ a [[prime number]], then localization of abelian groups (def. \ref{AbelianGroupLocal}) at $\mathbb{Z}[1/p]$ (def. \ref{pInvertedInZ}) is called **$p$-completion of abelian groups**.
=--
+-- {: .num_lemma #CharacterizingpCompleteAbelianGroupsBypSequence}
###### Lemma
An [[abelian group]] $A$ is $p$-complete according to def. \ref{AbelianGrouppComplete} precisely if both the [[limit]] as well as the [[lim^1]] of the sequence
$$
\cdots \overset{}{\longrightarrow}
A \overset{p}{\longrightarrow}
A \overset{p}{\longrightarrow}
A \overset{p}{\longrightarrow} A
$$
vanishes:
$$
\underset{\longleftarrow}{\lim}
\left(
\cdots \overset{}{\longrightarrow}
A \overset{p}{\longrightarrow}
A \overset{p}{\longrightarrow}
A \overset{p}{\longrightarrow} A
\right)
\simeq
0
$$
and
$$
\underset{\longleftarrow}{\lim}^1
\left(
\cdots \overset{}{\longrightarrow}
A \overset{p}{\longrightarrow}
A \overset{p}{\longrightarrow}
A \overset{p}{\longrightarrow} A
\right)
\simeq
0
\,.
$$
=--
+-- {: .proof}
###### Proof
By def. \ref{AbelianGroupLocal} the group $A$ is $\mathbb{Z}[1/p]$-local precisely if
$$
Hom(\mathbb{Z}[1/p], A) \simeq 0
\;\;\;\;\;\;\;
and
\;\;\;\;\;\;\;
Ext^1(\mathbb{Z}[1/p], A) \simeq 0
\,.
$$
Now use the colimit definition $\mathbb{Z}[1/p] = \underset{\longrightarrow}{\lim}_n \mathbb{Z}$ (def. \ref{pInvertedInZ}) and the fact that the [[hom-functor]] sends colimits in the first argument to limits to deduce that
$$
\begin{aligned}
Hom(\mathbb{Z}[1/p], A)
& =
Hom( \underset{\longrightarrow}{\lim}_n \mathbb{Z}, A )
\\
& \simeq
\underset{\longleftarrow}{\lim}_n Hom(\mathbb{Z},A)
\\
& \simeq
\underset{\longleftarrow}{\lim}_n A
\end{aligned}
\,.
$$
This yields the first statement. For the second, use that for every [[cotower]] over abelian groups $B_\bullet$ there is a [[short exact sequence]] of the form
$$
0
\to
\underset{\longleftarrow}{\lim}^1_n Hom(B_n, A)
\longrightarrow
Ext^1( \underset{\longrightarrow}{\lim}_n B_n, A )
\longrightarrow
\underset{\longleftarrow}{\lim}_n Ext^1( B_n, A)
\to
0
$$
(by [this lemma](Introduction+to+Stable+homotopy+theory+--+S#lim1AndExt1)).
In the present case all $B_n \simeq \mathbb{Z}$, which is a [[free abelian group]], hence a [[projective object]], so that all the [[Ext]]-groups out of it vannish. Hence by exactness there is an isomorphism
$$
Ext^1( \underset{\longrightarrow}{\lim}_n \mathbb{Z}, A )
\simeq
\underset{\longleftarrow}{\lim}^1_n Hom(\mathbb{Z}, A)
\simeq
\underset{\longleftarrow}{\lim}^1_n A
\,.
$$
This gives the second statement.
=--
+-- {: .num_example #pPrimaryGroupsArePComplete}
###### Example
For $p$ a [[prime number]], the [[p-primary group|p-primary]] [[cyclic groups]] of the form $\mathbb{Z}/p^n \mathbb{Z}$ are $p$-complete (def. \ref{AbelianGrouppComplete}).
=--
+-- {: .proof}
###### Proof
By lemma \ref{CharacterizingpCompleteAbelianGroupsBypSequence} we need to check that
$$
\underset{\longleftarrow}{\lim}
\left(
\cdots
\overset{p}{\longrightarrow}
\mathbb{Z}/p^n \mathbb{Z}
\overset{p}{\longrightarrow}
\mathbb{Z}/p^n \mathbb{Z}
\overset{p}{\longrightarrow}
\mathbb{Z}/p^n \mathbb{Z}
\right)
\simeq
0
$$
and
$$
\underset{\longleftarrow}{\lim}^1
\left(
\cdots
\overset{p}{\longrightarrow}
\mathbb{Z}/p^n \mathbb{Z}
\overset{p}{\longrightarrow}
\mathbb{Z}/p^n \mathbb{Z}
\overset{p}{\longrightarrow}
\mathbb{Z}/p^n \mathbb{Z}
\right)
\simeq
0
\,.
$$
For the first statement observe that $n$ consecutive stages of the tower compose to the [[zero morphism]]. First of all this directly implies that the limit vanishes, secondly it means that the [[tower]] satisfies the [[Mittag-Leffler condition]] ([def.](Introduction+to+Stable+homotopy+theory+--+S#MittagLefflerCondition)) and this implies that the $\lim^1$ also vanishes ([prop.](Introduction+to+Stable+homotopy+theory+--+S#Lim1VanihesUnderMittagLeffler)).
=--
+-- {: .num_defn #ZpInfinity}
###### Definition
For $p$ a [[prime number]], write
$$
\mathbb{Z}(p^\infty)
\coloneqq
\mathbb{Z}[1/p]/\mathbb{Z}
$$
(the [[p-primary group|p-primary]] part of $\mathbb{Q}/\mathbb{Z}$), where $\mathbb{Z}[1/p] = \underset{\longrightarrow}{\lim}(\mathbb{Z}\overset{p}{\to} \mathbb{Z} \overset{p}{\to} \mathbb{Z} \to \cdots )$ from def. \ref{pInvertedInZ}.
Since [[colimits]] commute over each other, this is equivalently
$$
\mathbb{Z}(p^\infty)
\simeq
\underset{\longrightarrow}{\lim}
( 0 \hookrightarrow \mathbb{Z}/p\mathbb{Z} \hookrightarrow \mathbb{Z}/p^2 \mathbb{Z} \hookrightarrow \cdots )
\,.
$$
=--
+-- {: .num_theorem #pCompletionOfAbelianGroupsByHomsOutOfZpinfinity}
###### Theorem
For $p$ a [[prime number]], the $\mathbb{Z}[1/p]$-localization
$$
A \longrightarrow L_{\mathbb{Z}[1/p]} A
$$
of an abelian group $A$ (def. \ref{pInvertedInZ}, def. \ref{AbelianGroupLocal}), hence the $p$-completion of $A$ according to def. \ref{AbelianGrouppComplete}, is given by the morphism
$$
A \longrightarrow Ext^1( \mathbb{Z}(p^\infty), A )
$$
into the first [[Ext]]-group into $A$ out of $\mathbb{Z}(p^\infty)$ (def. \ref{ZpInfinity}), which appears as the first [[connecting homomorphism]] $\delta$ in the [[long exact sequence]] of [[Ext]]-groups
$$
0
\to
Hom(\mathbb{Z}(p^\infty),A)
\longrightarrow
Hom(\mathbb{Z}[1/p],A)
\longrightarrow
Hom(\mathbb{Z},A)
\overset{\delta)}{\longrightarrow}
Ext^1(\mathbb{Z}(p^\infty), A)
\to
\cdots
\,.
$$
that is induced (via [this prop.](derived+functor+in+homological+algebra#LongExactSequenceOfRightDerivedFunctorsFromShortExactSequence)) from the defining [[short exact sequence]]
$$
0
\to
\mathbb{Z}
\longrightarrow
\mathbb{Z}[1/p]
\longrightarrow
\mathbb{Z}(p^\infty)
\to
0
$$
(def. \ref{ZpInfinity}).
=--
e.g. ([Neisendorfer 08, p. 16](localizatioon+of+abelian+groups#Neisendorfer08))
+-- {: .num_prop}
###### Proposition
If $A$ is a [[finitely generated module|finitely generated]] [[abelian group]], then its $p$-completion (def. \ref{AbelianGrouppComplete}, for any [[prime number]] $p$) is equivalently its [[p-adic completion]] (def. \ref{AdicCompletionOfAbelingGroups})
$$
\mathbb{Z}[1/p] A
\simeq
A^\wedge_p
\,.
$$
=--
+-- {: .proof}
###### Proof
By theorem \ref{pCompletionOfAbelianGroupsByHomsOutOfZpinfinity} the $p$-completion is $Ext^1(\mathbb{Z}(p^\infty),A)$. By def. \ref{ZpInfinity} there is a [[colimit]]
$$
\mathbb{Z}(p^\infty)
=
\underset{\longrightarrow}{\lim}
\left(
\mathbb{Z}/p\mathbb{Z}
\to
\mathbb{Z}/p^2 \mathbb{Z}
\to
\mathbb{Z}/p^3 \mathbb{Z}
\to
\cdots
\right)
\,.
$$
Together this implies, by [this lemma](Introduction+to+Stable+homotopy+theory+--+S#lim1AndExt1), that there is a [[short exact sequence]] of the form
$$
0
\to
\underset{\longleftarrow}{\lim}^1 Hom(\mathbb{Z}/p^n \mathbb{Z},A)
\longrightarrow
X^\wedge_p
\longrightarrow
\underset{\longleftarrow}{\lim}_n Ext^1(\mathbb{Z}/p^n \mathbb{Z}, A)
\to
0
\,.
$$
By lemma \ref{OutOfCyclicGroupExt1} the [[lim^1]] on the left is over the $p^n$-[[torsion subgroups]] of $A$, as $n$ ranges. By the assumption that $A$ is finitely generated, there is a [[maximum]] $n$ such that all torsion elements in $A$ are annihilated by $p^n$. This means that the [[Mittag-Leffler condition]] ([def.](Introduction+to+Stable+homotopy+theory+--+S#MittagLefflerCondition)) is satisfied by the [[tower]] of $p$-torsion subgroups, and hence the [[lim^1]]-term vanishes ([prop.](Introduction+to+Stable+homotopy+theory+--+S#Lim1VanihesUnderMittagLeffler)).
Therefore by exactness of the above sequence there is an [[isomorphism]]
$$
\begin{aligned}
L_{\mathbb{Z}[1/p]}X
& \simeq
\underset{\longleftarrow}{\lim}_n Ext^1(\mathbb{Z}/p^n \mathbb{Z}, A)
\\
& \simeq
\underset{\longleftarrow}{\lim}_n A/p^n A
\end{aligned}
\,,
$$
where the second isomorphism is by lemma \ref{OutOfCyclicGroupExt1}.
=--
+-- {: .num_prop #pDivisibleGroupsHaveVanishingpCompletion}
###### Proposition
If $A$ is a $p$-[[divisible group]] in that $A \overset{p \cdot (-)}{\longrightarrow} A$ is an [[isomorphism]], then its $p$-completion (def. \ref{AbelianGrouppComplete}) vanishes.
=--
+-- {: .proof}
###### Proof
By theorem \ref{pCompletionOfAbelianGroupsByHomsOutOfZpinfinity} the localization morphism $\delta$ sits in a [[long exact sequence]] of the form
$$
0
\to
Hom(\mathbb{Z}(p^\infty),A)
\longrightarrow
Hom(\mathbb{Z}[1/p],A)
\overset{\phi}{\longrightarrow}
Hom(\mathbb{Z},A)
\overset{\delta}{\longrightarrow}
Ext^1(\mathbb{Z}(p^\infty), A)
\to
\cdots
\,.
$$
Here by def. \ref{pInvertedInZ} and since the [[hom-functor]] takes [[colimits]] in the first argument to [[limits]], the second term is equivalently the [[limit]]
$$
Hom(\mathbb{Z}[1/p],A)
\simeq
\underset{\longleftarrow}{\lim}
\left(
\cdots
\to
A \overset{p \cdot (-)}{\longrightarrow} A \overset{p \cdot (-)}{\longrightarrow} A
\right)
\,.
$$
But by assumption all these morphisms $p \cdot (-)$ that the limit is over are [[isomorphisms]], so that the limit collapses to its first term, which means that the morphism $\phi$ in the above sequence is an [[isomorphism]]. But by exactness of the sequence this means that $\delta = 0$.
=--
+-- {: .num_cor}
###### Corollary
Let $p$ be a [[prime number]]. If $A$ is a [[finite abelian group]], then its $p$-completion (def. \ref{AbelianGrouppComplete}) is equivalently its [[p-primary group|p-primary part]].
=--
+-- {: .proof}
###### Proof
By the [[fundamental theorem of finite abelian groups]], $A$ is a finite [[direct sum]]
$$
A
\simeq
\underset{i}{\oplus}
\mathbb{Z}/p_i^{k_i}\mathbb{Z}
$$
of [[cyclic groups]] of [[order of a group|ordr]] $p_i^{k_1}$ for $p_i$ [[prime numbers]] and $k_i \in \mathbb{N}$ ([thm.](finite+abelian+group#FiniteAbelianGroupIsDirectSumOfCyclics)).
Since finite direct sums are equivalently [[products]] ([[biproducts]]: [[Ab]] is an [[additive category]]) this means that
$$
Ext^1( \mathbb{Z}(p^\infty), A )
\simeq
\underset{i}{\prod}
Ext^1( \mathbb{Z}(p^\infty), \mathbb{Z}/p_i^{k_1}\mathbb{Z} )
\,.
$$
By theorem \ref{pCompletionOfAbelianGroupsByHomsOutOfZpinfinity} the $i$th factor here is the $p$-completion of $\mathbb{Z}/p_i^{k_i}\mathbb{Z}$, and since $p \cdot(-)$ is an isomorphism on $\mathbb{Z}/p_i^{k_i}\mathbb{Z}$ if $p_i \neq p$ (because its [[kernel]] evidently vanishes), prop. \ref{pDivisibleGroupsHaveVanishingpCompletion} says that in this case the factor vanishes, so that only the factors with $p_i = p$ remain. On these however $Ext^1(\mathbb{Z}(p^\infty),-)$ is the identity by example \ref{pPrimaryGroupsArePComplete}.
=--
##### Localization and nilpotent completion of spectra
{#LocalizationOfSpectra}
We discuss
1. _[Bousfield localization of spectra](#BousfieldLocalizationOfSpectra)_
1. _[Nilpotent completion of spectra](#NilpotentCompletionOfSpectra)_
which are the analogs in [[stable homotopy theory]] of the construction of [[localization of abelian groups]] discussed [above](#LocalizationOfAbelianGroups).
**Literature**: ([Bousfield 79](#Bousfield79))
$\,,$
**Localization of spectra**
{#BousfieldLocalizationOfSpectra}
+-- {: .num_defn #EAcyclicAndLocalSpectra}
###### Definition
Let $E \in Ho(Spectra)$ be be a [[spectrum]]. Say that
1. a spectrum $X$ is **$E$-acyclic** if the [[smash product of spectra|smash product]] with $E$ is [[zero object|zero]], $E \wedge X \simeq 0$;
1. a morphism $f \colon X \to Y$ of spectra is an **$E$-equivalence** if $E \wedge f \;\colon\; E \wedge X \to E \wedge Y$ is an [[isomorphism]] in $Ho(Spectra)$, hence if $E_\bullet(f)$ is an isomorphism in $E$-[[generalized homology]];
1. a spectrum $X$ is **$E$-local** if the following equivalent conditions hold
1. for every $E$-equivalence $f$ then $[f,X]_\bullet$ is an isomorphism;
1. every [[morphism]] $Y \longrightarrow X$ out of an $E$-acyclic spectrum $Y$ is [[zero morphism|zero]] in $Ho(Spectra)$;
=--
([Bousfield 79, §1](#Bousfield79)) see also for instance ([Lurie, Lecture 20, example 4](#Lurie10))
+-- {: .num_lemma #TwoConditionsOnELocalSpectrumAreEquivalent}
###### Lemma
The two conditions in the last item of def. \ref{EAcyclicAndLocalSpectra} are indeed equivalent.
=--
+-- {: .proof}
###### Proof
Notice that $A \in Ho(Spectra)$ being $E$-acyclic means equivalently that the unique morphism $0 \longrightarrow A$ is an $E$-equivalence.
Hence one direction of the claim is trivial. For the other direction we need to show that for $[-,X]_\bullet$ to give an isomorphism on all $E$-equivalences $f$, it is sufficient that it gives an isomorphism on all $E$-equivalences of the form $0 \to A$.
Given a morphism $f \colon A \to B$, write $B \longrightarrow B/A$ for its [[homotopy cofiber]]. Then since $Ho(Spectra)$ is a [[triangulated category]] ([prop.](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryIsTriangulated)) the defining axioms of triangulated categories ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#CategoryWithCofiberSequences), [lemma](Introduction+to+Stable+homotopy+theory+--+1-1#TrianglesMayBeShiftedToTheLeft)) give that there is a [[commuting diagram]] of the form
$$
\array{
0
&\longrightarrow&
A
&\overset{id}{\longrightarrow}&
A
&\overset{}{\longrightarrow}&
0
&\overset{}{\longrightarrow}&
\Sigma A
\\
\downarrow
&&
\downarrow^{\mathrlap{id}}
&&
\downarrow^{\mathrlap{f}}
&&
\downarrow
&&
\downarrow^{\mathrlap{id}}
\\
\Sigma^{-1} B/A
&\longrightarrow&
A
&\underset{f}{\longrightarrow}&
B
&\longrightarrow&
B/A
&\longrightarrow&
\Sigma A
}
\,,
$$
where both the top as well as the bottom are [[homotopy cofiber sequences]]. Hence applying $[-,X]_\bullet$ to this diagram in $Ho(Spectra)$ yields a diagram of [[graded abelian groups]] of the form
$$
\array{
0
&\longleftarrow&
[A,X]_\bullet
&\longleftarrow&
[A,X]_\bullet
&\longleftarrow&
0
&\longleftarrow&
[A,X]_{\bullet+1}
\\
\uparrow
&&
\uparrow^{\mathrlap{id}}
&&
\uparrow^{\mathrlap{[f,X]_\bullet}}
&&
\uparrow
&&
\uparrow^{\mathrlap{id}}
\\
[B/A,X]_{\bullet+1}
&\longleftarrow&
[A,X]_\bullet
&\longleftarrow&
[B,X]_\bullet
&\longleftarrow&
[B/A,X]_\bullet
&\longleftarrow&
[A,X]_{\bullet+1}
}
\,,
$$
where now both horizontal sequences are [[long exact sequences]] ([prop.](Introduction+to+Stable+homotopy+theory+--+1-1#LongFiberSequencesOfMapsOfSpectra)).
Hence if $[B/A,X]_\bullet \longrightarrow 0$ is an isomorphism, then all four outer vertical morphisms in this diagram are isomorphisms, and then the [[five-lemma]] implies that also $[f,X]_\bullet$ is an isomorphism.
Hence it is now sufficient to observe that with $f \colon A \to B$ an $E$-equivalence, then its homotopy cofiber $B/A$ is $E$-acyclic.
To see this, notice that by the [[tensor triangulated category|tensor triangulated]] structure on $Ho(Spectra)$ ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#TensorTriangulatedStructureOnStableHomotopyCategory)) the [[smash product of spectra|smash product]] with $E$ preserves homotopy cofiber sequences, so that there is a homotopy cofiber sequence
$$
E \wedge A
\overset{E \wedge f}{\longrightarrow}
E \wedge B
\longrightarrow
E \wedge (B/A)
\longrightarrow
E \wedge \Sigma A
\,.
$$
But if the first morphism here is an isomorphism, then the axioms of a [[triangulated category]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#CategoryWithCofiberSequences)) imply that $E \wedge B / A \simeq 0$. In detail: by the axioms we may form the morphism of homotopy cofiber sequences
$$
\array{
E \wedge A
&\overset{E \wedge f}{\longrightarrow}&
E \wedge B
&\longrightarrow&
E \wedge B/A
&\longrightarrow&
E \wedge \Sigma A
\\
\downarrow^{\mathrlap{id}}
&&
\downarrow^{\mathrlap{(E\wedge f)^{-1}}}
&&
\downarrow
&&
\downarrow^{\mathrlap{id}}
\\
E \wedge A
&\underset{id}{\longrightarrow}&
E \wedge A
&\longrightarrow&
0
&\longrightarrow&
E \wedge \Sigma A
}
\,.
$$
Then since two of the three vertical morphisms on the left are isomorphisms, so is the third ([lemma](Introduction+to+Stable+homotopy+theory+--+1-1#TwoOutOfThreeForMorphismsOfDistinguishedTriangles)).
=--
+-- {: .num_defn #ELocalizationOfSpectra}
###### Definition
Given $E,X \in Ho(Spectra)$, then an **$E$-[[Bousfield localization of spectra]]** of $X$ is
1. an $E$-local spectrum $L_E X$
1. an $E$-equivalence $X \longrightarrow L_E X$.
according to def. \ref{EAcyclicAndLocalSpectra}.
=--
We discuss now that $E$-Localizations always exist. The key to this is the following lemma \ref{KappaCellSpectrumWitnessingELocalization}, which asserts that a spectrum being $E$-local is equivalent to it being $A$-null, for some "small" spectrum $A$:
+-- {: .num_lemma #KappaCellSpectrumWitnessingELocalization}
###### Lemma
For every [[spectrum]] $E$ there exists a spectrum $A$ such that any spectrum $X$ is $E$-local (def. \ref{EAcyclicAndLocalSpectra}) precisely if it is $A$-null, i.e.
$$
X \;is\; E\text{-local}
\;\;\;\;
\Leftrightarrow
\;\;\;\;
[A,X]_\ast = 0
$$
and such that
1. $A$ is $E$-acyclic (def. \ref{EAcyclicAndLocalSpectra});
1. there exists an infinite [[cardinal number]] $\kappa$ such that $A$ is a $\kappa$-[[CW spectrum]] (hence a [[CW spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#CWSpectrum)) with at most $\kappa$ many cells);
1. the class of $E$-acyclic spectra (def. \ref{EAcyclicAndLocalSpectra}) is the class generated by $A$ under
1. [[wedge sum]]
1. the relation that if in a [[homotopy cofiber sequence]] $X_1 \to X_2 \to X_3$ two of the spectra are in the class, then so is the third.
=--
([Bousfield 79, lemma 1.13 with lemma 1.14](#Bousfield79)) review includes ([Bauer 11, p.2,3](Bauer11), [VanKoughnett 13, p. 8](#VanKoughnett13))
+-- {: .num_prop #ELocalizationCofiber}
###### Proposition
For $E \in Ho(Spectra)$ any [[spectrum]], every spectrum $X$ sits in a [[homotopy cofiber sequence]] of the form
$$
G_E(X)
\longrightarrow
X
\overset{\eta_X}{\longrightarrow}
L_E(X)
\,,
$$
and [[natural transformation|natural]] in $X$, such that
1. $G_E(X)$ is $E$-acyclic,
1. $L_E(X)$ is $E$-local,
according to def. \ref{EAcyclicAndLocalSpectra}.
=--
([Bousfield 79, theorem 1.1](#Bousfield79)) see also for instance ([Lurie, Lecture 20, example 4](#Lurie10))
+-- {: .proof}
###### Proof
Consider the $\kappa$-[[CW-spectrum]] spectrum $A$ whose existence is asserted by lemma \ref{KappaCellSpectrumWitnessingELocalization}. Let
$$
I_A
\coloneqq
\{A \to Cone(A)\}
$$
denote the set containing as its single element the canonical morphism (of [[sequential spectra]]) from $A$ into the standard [[cone]] of $A$, i.e. the cofiber
$$
Cone(A)
\coloneqq
cofib( A \to A \wedge I_+ )
\simeq
A \wedge I
$$
of the inclusion of $A$ into its standard [[cylinder spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#StandardCylinderSpectrumSequential)).
Since the standard cylinder spectrum on a CW-spectrum is a [[good cylinder object]] ([prop.](Introduction+to+Stable+homotopy+theory+--+1-1#CylinderSpectrumOverCWSpectrumIsGood)) this means ([lemma](Introduction+to+Stable+homotopy+theory+--+P#HomsOutOfCofibrantIntoFibrantComputeHomotopyCategory)) that for $X$ any fibrant sequential spectrum, and for $A \longrightarrow X$ any morphism, then an extension along the cone inclusion
$$
\array{
A &\longrightarrow& X
\\
\downarrow & \nearrow
\\
Cone(A)
}
$$
equivalently exhibits a null-homotopy of the top morphism. Hence the $(A \to Cone(A))$-[[injective objects]] in $Ho(Spectra)$ are precisely those spectra $X$ for which $[A,X]_\bullet \simeq 0$.
Moreover, due to the degreewise nature of the smash tensoring $Cone(A) = A \wedge I$ ([def](Introduction+to+Stable+homotopy+theory+--+1-1#TensoringAndPoweringOfSequentialSpectra)), the inclusion morphism $A \to Cone(A)$ is degreewise the inclusion of a [[CW-complex]] into its standard cone, which is a [[relative cell complex]] inclusion ([prop.](Introduction+to+Stable+homotopy+theory+--+P#TopologicalCylinderOnCWComplexIsGoodCylinderObject)).
By [this lemma](Introduction+to+Stable+homotopy+theory+--+1-1#kappaCellSpectrumIsKappaSmall) the $\kappa$-[[cell spectrum]] $A$ is _$\kappa$-small object_ ([def.](Introduction+to+Stable+homotopy+theory+--+P#ClassOfMorphismsWithSmallDomains)) with respect to morphisms of spectra which are degreewise [[relative cell complex]] inclusion [[small object argument]] .
Hence the [[small object argument]] applies ([prop.](Introduction+to+Stable+homotopy+theory+--+P#SmallObjectArgument)) and gives for every $X$ a factorization of the terminal morphism $X \to \ast$ as an $I_A$-[[relative cell complex]] ([def.](Introduction+to+Stable+homotopy+theory+--+P#TopologicalCCellComplex)) followed by an $I_A$-[[injective morphism]] ([def.](Introduction+to+Stable+homotopy+theory+--+P#RightLiftingProperty))
$$
X
\overset{I_A Cell}{\longrightarrow}
L_E X
\overset{I_A Inj}{\longrightarrow}
\ast
\,.
$$
By the above, this means that $[A, L_E X] = 0$, hence by lemma \ref{KappaCellSpectrumWitnessingELocalization} that $L_E X$ is $E$-local.
It remains to see that the [[homotopy fiber]] of $X \to L_E X$ is $E$-acyclic: By the [[tensor triangulated category|tensor triangulated]] structure on $Ho(Spectra)$ ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#TensorTriangulatedStructureOnStableHomotopyCategory)) it is sufficient to show that the [[homotopy cofiber]] is $E$-acyclic (since it differs from the homotopy fiber only by suspension). By the [[pasting law]], the homotopy cofiber of a [[transfinite composition]] is the transfinite composition of a sequence of homotopy pushouts. By lemma \ref{KappaCellSpectrumWitnessingELocalization} and applying the pasting law again, all these homotopy pushouts produce $E$-acyclic objects. Hence we conclude by observing that the transfinite composition of the morphisms between these $E$-acyclic objects is $E$-acyclic. Since by construction all these morphisms are relative cell complex inclusions, this follows again with the compactness of the $n$-spheres ([lemma](Introduction+to+Stable+homotopy+theory+--+P#CompactSubsetsAreSmallInCellComplexes)).
=--
+-- {: .num_lemma}
###### Lemma
The morphism $X \to L_E (X)$ in prop. \ref{ELocalizationCofiber} exhibits an $E$-localization of $X$ according to def. \ref{ELocalizationOfSpectra}
=--
+-- {: .proof}
###### Proof
It only remains to show that $X \to L_E X$ is an $E$-equivalence. By the [[tensor triangulated category|tensor triangulated]] structure on $Ho(Spectra)$ ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#TensorTriangulatedStructureOnStableHomotopyCategory)) the [[smash product of spectra|smash product]] with $E$ preserves homotopy cofiber sequences, so that
$$
E \wedge G_E X
\longrightarrow
E \wedge X
\overset{E \wedge \eta_X}{\longrightarrow}
E \wedge L_E X
\longrightarrow
E \wedge \Sigma G_E X
$$
is also a homotopy cofiber sequence. But now $E \wedge G_E X \simeq 0$ by prop. \ref{ELocalizationCofiber}, and so the axioms ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#CategoryWithCofiberSequences)) of the [[triangulated category|triangulated structure]] on $Ho(Spectra)$ ([prop.](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryIsTriangulated)) imply that $E \wedge \eta$ is an isomorphism.
=--
$\,$
**Nilpotent completion of spectra**
{#NilpotentCompletionOfSpectra}
+-- {: .num_defn #ENilpotentCompletion}
###### Definition
Let $(E, \mu, e)$ be a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)) and $Y \in Ho(Spectra)$ any spectrum. Write $\overline{E}$ for the [[homotopy fiber]] of the unit $\mathbb{S}\overset{e}{\to} E$ as in def. \ref{HomotopyFiberOfUnitOfCommutativeRingSpectrum} such that the $E$-Adams filtration of $Y$ (def. \ref{AdamsEAdamsSpectralSequence}) reads (according to lemma \ref{Wp})
$$
\array{
\vdots
\\
\downarrow
\\
\overline{E}^3 \wedge Y
\\
\downarrow
\\
\overline{E}^2 \wedge Y
\\
\downarrow
\\
\overline{E} \wedge Y
\\
\downarrow
\\
Y
}
\,.
$$
For $s \in \mathbb{N}$, write
$$
\overline{E}_{s-1}
\coloneqq
hocof( \overline{E}^s \overset{i^s}{\longrightarrow} \mathbb{S})
$$
for the homotopy cofiber. Here $\overline{E}_{-1} \simeq 0$.
By the [[tensor triangulated category|tensor triangulated]] structure of $Ho(Spectra)$ ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#TensorTriangulatedStructureOnStableHomotopyCategory)), this homotopy cofiber is preserved by forming [[smash product of spectra|smash product]] with $Y$, and so also
$$
\overline{E}_n \wedge Y
\simeq
hocof( \overline{E}^n \wedge Y \overset{}{\longrightarrow} Y)
\,.
$$
Now let
$$
\overline{E}_s
\overset{p_{s-1}}{\longrightarrow}
\overline{E}_{s-1}
$$
be the morphism implied by the [[octahedral axiom]] of the [[triangulated category]] $Ho(Spectra)$ ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#CategoryWithCofiberSequences), [prop.](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryIsTriangulated)):
$$
\array{
\overline{E}^{s+1}
&\overset{i}{\longrightarrow}&
\overline{E}^s
&\longrightarrow&
E \wedge \overline{E}^s
&\longrightarrow&
\Sigma \overline{E}^{s+1}
\\
{}^{\mathllap{=}}\downarrow
&&
\downarrow^{\mathrlap{i^s}}
&&
\downarrow^{}
&&
\downarrow
\\
\overline{E}^{s+1}
&\longrightarrow&
\mathbb{S}
&\longrightarrow&
\overline{E}_s
&\longrightarrow&
\Sigma \overline{E}^{s+1}
\\
&&
\downarrow
&&
\downarrow^{\mathrlap{p_{s-1}}}
\\
&&
\overline{E}_{s-1}
&\overset{=}{\longrightarrow}&
\overline{E}_{s-1}
\\
&&
\downarrow
&&
\downarrow
\\
&&
\Sigma \overline{E}^s
&\longrightarrow&
\Sigma E \wedge \overline{E}^s
}
\,.
$$
By the [[commuting square]] in the middle and using again the [[tensor triangulated category|tensor triangulated]] structure, this yields an inverse sequence under $Y$:
$$
Y
\simeq
\mathbb{S} \wedge Y
\longrightarrow
\cdots
\overset{p_3 \wedge id}{\longrightarrow}
\overline{E}_3 \wedge Y
\overset{p_2 \wedge id}{\longrightarrow}
\overline{E}_2 \wedge Y
\overset{p_1 \wedge id}{\longrightarrow}
\overline{E}_1 \wedge Y
$$
The **[[E-nilpotent completion]]** $Y^\wedge_E$ of $Y$ is the [[homotopy limit]] over the resulting inverse sequence
$$
Y^\wedge_E
\coloneqq
\mathbb{R}\underset{\longleftarrow}{\lim}_n \overline{E}_n \wedge Y
$$
or rather the canonical morphism into it
$$
Y \longrightarrow Y^\wedge_E
\,.
$$
Concretely, if
$$
Y
\simeq
\mathbb{S} \wedge Y
\longrightarrow
\cdots
\overset{p_3 \wedge id}{\longrightarrow}
\overline{E}_3 \wedge Y
\overset{p_2 \wedge id}{\longrightarrow}
\overline{E}_2 \wedge Y
\overset{p_1 \wedge id}{\longrightarrow}
\overline{E}_1 \wedge Y
$$
is presented by a tower of fibrations between fibrant spectra in the [[model structure on topological sequential spectra]], then $Y^\wedge_E$ is represented by the ordinary [[sequential limit]] over this tower.
=--
([Bousfield 79, top, middle and bottom of page 272](#Bousfield79))
+-- {: .num_remark }
###### Remark
In ([Bousfield 79](#Bousfield79)) the $E$-nilpotent completion of $X$ (def. \ref{ENilpotentCompletion}) is denoted "$E^\wedge X$". The notation "$X^\wedge_E$" which we use here is more common among modern authors. It emphasizes the conceptual relation to [[p-adic completion]] $A^\wedge_p$ of abelian groups (def. \ref{AdicCompletionOfAbelingGroups}) and is less likely to lead to confusion with the smash product of $E$ with $X$.
=--
+-- {: .num_remark #CanonicalMapFromELocalizationToTotalization}
###### Remark
The nilpotent completion $X^\wedge_E$ is $E$-local.
This induces a universal morphism
$$
L_E X
\overset{}{\longrightarrow}
X^\wedge_E
$$
from the $E$-[[Bousfield localization of spectra]] of $X$ into the $E$-nilmpotent completion
=--
([Bousfield 79, top of page 273](#Bousfield79))
We consider now conditions for this morphism to be an [[equivalence]].
+-- {: .num_prop #SufficientConditionsForTotalizationToBeELocalization}
###### Proposition
Let $E$ be a [[connective spectrum|connective]] [[ring spectrum]] such that the core of $\pi_0(E)$, def. \ref{CoreOfARing}, is either of
* the [[localization of a ring|localization]] of the [[integers]] at a set $J$ of [[primes]], $c \pi_0(E) \simeq \mathbb{Z}[J^{-1}]$;
* a ring of [[integers modulo n]] $c \pi_0(E) \simeq \mathbb{Z}/n\mathbb{Z}$, for $n \geq 2$.
Then the map in remark \ref{CanonicalMapFromELocalizationToTotalization} is an equivalence
$$
L_E X
\stackrel{\simeq}{\longrightarrow}
X^\wedge_E
\,.
$$
=--
([Bousfield 79, theorem 6.5, theorem 6.6](#Bousfield79)).
##### Convergence theorems
{#ConvergenceStatements}
We state the two main versions of [[Aldridge Bousfield|Bousfield]]'s convergence theorems for the $E$-[[Adams spectral sequence]], below as theorem \ref{EAdamsConvergenceForCorepi0EBeingZLocalizedAtPrimes} and theorem \ref{EAdamsConvergenceForCorepi0EBeingCyclicGroup}.
First we need to define the concepts that enter the convergence statement:
1. the infinity-page $E_\infty^{s,t}(X,Y)$ (def. \ref{InfinityPageEAdams}),
1. a filtration on $[X,Y^\wedge_E]_\bullet$ (def. \ref{EAdamsFiltration})
1. what it means for the former to converge to the latter (def. \ref{EAdamsConvergingCompletely}).
Broadly the statement will be that typically
1. the $E$-Adams spectral sequence $E_r^{s,t}(X,Y)$ computes the [[stable homotopy groups]] $[X,Y^\wedge_E]$ of maps from $X$ into the [[E-nilpotent completion]] of $Y$;
1. these groups are [[localization of an abelian group|localizations]] of the full groups $[X,Y]_\bullet$ depending on the [[core of a ring|core]] of $\pi_0(E)$.
**Literature**: ([Bousfield 79](#Bousfield79))
$\,$
+-- {: .num_defn #InfinityPageEAdams}
###### Definition
Let $(E, \mu, e)$ be a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)) and $X,Y \in Ho(Spectra)$ two [[spectra]] with associated $E$-[[Adams spectral sequence]] $\{E_r^{s,t}, d_r\}$ (def. \ref{AdamsEAdamsSpectralSequence}).
Observe that
$$
if \; r \gt s \; then \; \;
E^{s,\bullet}_{r+1}(X,Y)
\simeq
ker(d_r|_{E_r^{s,\bullet}(X,Y)})
\subset
E_r^{s,\bullet}(X,Y)
$$
since the differential $d_r$ on the $r$th page has bidegree $(r,r-1)$, and since $E_r^{s \lt 0,\bullet(X,Y)} \simeq 0$, so that for $r \gt s$ the image of $d_r$ in $E_r^{s,t}(X,Y)$ vanishes.
Thus define the bigraded abelian group
$$
E_\infty^{s,t}(X,Y)
\coloneqq
\underset{r \gt s}{\lim} E_r^{s,t}(X,Y)
=
\underset{r \gt s}{\cap} E_r^{s,t}(X,Y)
$$
called the "**infinity page**" of the $E$-Adams spectral sequence.
=--
+-- {: .num_defn #EAdamsFiltration}
###### Definition
Let $(E, \mu, e)$ be a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)) and $X,Y \in Ho(Spectra)$ two [[spectra]] with associated $E$-[[Adams spectral sequence]] $\{E_r^{s,t}, d_r\}$ (def. \ref{AdamsEAdamsSpectralSequence}) and [[E-nilpotent completion]] $Y^\wedge_E$ (def. \ref{ENilpotentCompletion}).
Define a [[filtered object|filtration]]
$$
\cdots
\hookrightarrow
F^3 [X, Y^\wedge_E]_\bullet
\hookrightarrow
F^2 [X, Y^\wedge_E]_\bullet
\hookrightarrow
F^1 [X,Y^\wedge_E]_\bullet
=
[X, Y^\wedge_E ]_\bullet
$$
on the [[graded abelian group]] $[X,Y^\wedge_E]_\bullet$ by
$$
F^s [X, Y^\wedge_E]_\bullet
\;\coloneqq\;
ker(
\;
[X, Y^\wedge_E]_\bullet
\overset{[X, Y^\wedge_E \to \overline{E}_{s-1} \wedge Y ]}{\longrightarrow}
[X, \overline{E}_{s-1} \wedge Y]_\bullet
\;
)
\,,
$$
where the morphisms $Y^\wedge_E \to \overline{E}_{s-1} \wedge Y$ is the canonical one from def. \ref{ENilpotentCompletion}.
=--
+-- {: .num_defn #EAdamsConvergingCompletely}
###### Definition
Let $(E, \mu, e)$ be a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)) and $X,Y \in Ho(Spectra)$ two [[spectra]] with associated $E$-[[Adams spectral sequence]] $\{E_r^{s,t}, d_r\}$ (def. \ref{AdamsEAdamsSpectralSequence}) and [[E-nilpotent completion]] $Y^\wedge_E$ (def. \ref{ENilpotentCompletion}).
Say that the $E$-[[Adams spectral sequence]] $\{E_r^{s,t}, d_r\}$ **converges completely** to the [[E-nilpotent completion]] $[X,Y^\wedge_E]_\bullet$ if the following two canonical morphisms are [[isomorphisms]]
1. $
[X, Y^\wedge_E]_\bullet
\longrightarrow
\underset{\longleftarrow}{\lim}_s
[X, Y^\wedge_E]_\bullet / F^s [X, Y^\wedge_E]_\bullet
$
(where on the right we have the limit over the tower of [[quotients]] by the stages of the [[filtration]] from def. \ref{EAdamsFiltration})
1. $
F^s [X, Y^\wedge_E]_{t-s}
/
F^{s+1}[X, Y^\wedge]_{t-s}
\longrightarrow
E^{s,t}_\infty(X,Y)
\;\;\;\;\;\;\;\;
\forall s,t
$
(where $F^s [X, Y^\wedge_E]_\bullet$ is the filtration stage from def. \ref{EAdamsFiltration} and $E^{s,t}_\infty(X,Y)$ is the infinity-page from def. \ref{InfinityPageEAdams}).
Notice that the first morphism is always surjective, while the second is necessarily injective, hence the condition is equivalently that the first is also injective, and the second also surjective.
=--
([Bousfield 79, §6](#Bousfield79))
$\,$
Now we state sufficient conditions for complete convergence of the $E$-Adams spectral sequence. It turns out that convergence is controled by the [[core of a ring|core]] (def. \ref{CoreOfARing}) of the ring $\pi_0(E)$. By prop. \ref{ClassificationOfSolidRings} these cores are either localizations of the integers $\mathbb{Z}[J^{-1}]$ at a set $J$ of primes (def. \ref{InvertingPrimes}) or are rings of [[integers modulo n]], or cores of products of these. We discuss the first two cases.
+-- {: .num_theorem #EAdamsConvergenceForCorepi0EBeingZLocalizedAtPrimes}
###### Theorem
Let $(E,\mu,e)$ be a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)) and let $X,Y \in Ho(Spectra)$ be two [[spectra]] such that
1. the [[core of a ring|core]] (def. \ref{CoreOfARing}) of the 0-th [[stable homotopy group]] ring of $E$ ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommuta
tiveRingSpectrum)) is the [[localization of a ring|localization]] of the [[integers]] at a set $J$ of primes (def. \ref{InvertingPrimes})
$$
c \pi_0(E) \simeq \mathbb{Z}[J^{-1}] \subset \mathbb{Q}
$$
1. $X$ is a [[CW-spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#CWSpectrum)) with a [[finite number]] of cells ([rmk.](Introduction+to+Stable+homotopy+theory+--+1-1#StrictModelStructureCellAttachmentToSpectra));
then the $E$-[[Adams spectral sequence]] for $[X,Y]_\bullet$ (def. \ref{AdamsEAdamsSpectralSequence}) converges completely (def. \ref{EAdamsConvergingCompletely}) to the localization
$$
[X, Y^\wedge_E]_\bullet
=
\mathbb{Z}[J^{-1}] \otimes [X,Y]_\bullet
$$
of $[X,Y]_\bullet$.
=--
([Bousfield 79, theorem 6.5](#Bousfield79))
+-- {: .num_theorem #EAdamsConvergenceForCorepi0EBeingCyclicGroup}
###### Theorem
Let $(E,\mu,e)$ be a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)) and let $X,Y \in Ho(Spectra)$ be two [[spectra]] such that
1. the [[core of a ring|core]] (def. \ref{CoreOfARing}) of the 0-th [[stable homotopy group]] ring of $E$ ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum)) is a [[prime field]]
$c \pi_0(E) \simeq \mathbb{F}_p$
for some [[prime number]] $p$;
1. $Y$ is a [[connective spectrum]] in that its [[stable homotopy groups]] $\pi_\bullet(Y)$ vanish in negative degree;
1. $X$ is a [[CW-spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#CWSpectrum)) with a [[finite number]] of cells ([rmk.](Introduction+to+Stable+homotopy+theory+--+1-1#StrictModelStructureCellAttachmentToSpectra));
1. $[X,Y]_\bullet$ is degreewise a [[finitely generated object|finitely generated]] group
then the $E$-[[Adams spectral sequence]] for $[X,Y]_\bullet$ (def. \ref{AdamsEAdamsSpectralSequence}) converges completely (def. \ref{EAdamsConvergingCompletely}) to the $p$-[[adic completion]] (def. \ref{AdicCompletionOfAbelingGroups})
$$
[X, Y^\wedge_E]_\bullet
\simeq
\underset{\longleftarrow}{\lim}_n [X,Y]_\bullet/p^n[X,Y]_\bullet
$$
of $[X,Y]_\bullet$.
=--
([Bousfield 79, theorem 6.6](#Bousfield79))
### Examples
+-- {: .num_example}
###### Example
Examples of [[commutative ring spectra]] $E$ for which the dual $E$-[[Steenrod algebra]] $E_\bullet(E)$ over $\pi_\bullet(E)$ of def. \ref{HopfAlgebroidStructureOnDualEOperations} where the left and right action of $\pi_\bullet(E)$ are not just isomorphic (via prop. \ref{EETwoLeftModuleStructures}) but actually equal according to remark \ref{HopfAlgebrasAsHopfAlgebroids}, includes the case $E = $ [[HA|H]]$\mathbb{F}_p$.
=--
+-- {: .num_example #ExamplesOfFlatRingSpectra}
###### Example
Examples of [[commutative ring spectra]] that are flat according to def. \ref{FlatE} include
$E = $
* $\mathbb{S}$ -- the [[sphere spectrum]];
* $H \mathbb{F}_p$ -- [[Eilenberg-MacLane spectra]] for [[prime fields]];
* [[MO]], [[MU]], [[MSp]] -- [[Thom spectra]];
* [[KO]], [[KU]] -- [[topological K-theory]] spectra.
=--
([Adams 69, lecture 1, lemma 28 (p. 45)](#Adams69))
+-- {: .proof}
###### Proof of the first two items
For $E = \mathbb{S}$ we have $\mathbb{S}_\bullet(\mathbb{S}) \coloneqq \pi_\bullet(\mathbb{S} \wedge \mathbb{S}) \simeq \pi_\bullet(\mathbb{S})$, since the [[sphere spectrum]] $\mathbb{S}$ is the [[tensor unit]] for the derived [[smash product of spectra]] ([cor.](Introduction+to+Stable+homotopy+theory+--+1-2#MonoidalStableHomotopyCategory)). Hence the statement follows since every ring is, clearly, flat over itself.
For $E = H \mathbb{F}_p$ we have that $\pi_\bullet(H \mathbb{F}_p) \simeq \mathbb{F}_p$ ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#StableHomotopyGroupsOfEMSpectrum)), hence a [[field]] (a [[prime field]]). Every module over a field is a [[projective module]] ([prop.](projective+module#ModuleOverAFieldIsProjective)) and every projective module is flat ([prop.](flat+module#ProjectiveModulesAreFlat)).
=--
+-- {: .num_example}
###### Example
Examples of ring spectra that are _not_ flat in the sense of def. \ref{FlatE} include [[HA|H]][[integers|Z]], and $M S U$.
=--
+-- {: .num_examples #ExamplesOfEnilpotentLocalizations}
###### Example
* $E = H \mathbb{F}_p$ the [[Eilenberg-MacLane spectrum]] of a [[prime field]]. For $X$ a [[connective spectrum]], its $H \mathbb{F}_p$-nilpotent completion is its [[p-completion]]
$$
X^\wedge_{(H\mathbb{F}_p)}
\simeq
X^{\hat{}}_p
\coloneqq
\underset{\longleftarrow}{\lim}_{n \in \mathbb{N}}
X \wedge M(\mathbb{Z}/p^n)
$$
(where $M A$ denotes the [[Moore spectrum]] of the [[abelian group]] $A$).
* $E = $ [[MU]]. Every spectrum is alreay $MU$-nilpotently complete
$$
X^\wedge_{MU} \simeq X
\,.
$$
* $E = $ [[BP]] at prime $p$. For every spectrum $X$ its $BP$-nilpotent completion is its [[p-localization]]
$$
X^\wedge_{BP} \simeq X_{(p)} \coloneqq X \wedge M \mathbb{Z}_{(p)}
$$
(where $\mathbb{Z}_{(p)}\subset \mathbb{Q}$ is the result of inverting all primes different from $p$).
=--
For more discussion of [[E-infinity geometry|E-infinity]] (derived) [[formal completions]] via totalizations of [[Amitsur complexes]], see ([Carlsson 07](completion+of+a+module#Carlsson07)).
#### Classical Adams spectral sequence ($E = H \mathbb{F}_2$, $X = \mathbb{S}$)
{#ClassicalAdamsSpectralSequence}
We consider now the example of the $E$-[[Adams spectral sequence]] $\{E_r^{s,t}(X,Y), d_r\}$ (def. \ref{AdamsEAdamsSpectralSequence}) for the case that
1. $E = H \mathbb{F}_p$ is the [[Eilenberg-MacLane spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#OrthogonalEilenbergMacLaneRingSpectrum)) with [[coefficients]] in a [[prime field]], regarded in $Ho(Spectra)$ with its canonical struture of a [[homotopy commutative ring spectrum]] induced (via [this corollary](Introduction+to+Stable+homotopy+theory+--+1-2#MonoidalStableHomotopyCategory)) from its canonical structure of an [[orthogonal ring spectrum]] (from [this def.](Introduction+to+Stable+homotopy+theory+--+1-2#OrthogonalEilenbergMacLaneRingSpectrum));
1. $X = Y = \mathbb{S}$ are both the [[sphere spectrum]].
This example is called the _[[classical Adams spectral sequence]]_.
The $H\mathbb{F}_p$-dual Steenrod algebra according to the general definition \ref{HopfAlgebroidStructureOnDualEOperations} turns out to be the classical dual [[Steenrod algebra]] $\mathcal{A}_p^\ast$ recalled [below](#TheClassicalDualSteenrodAlgebra) .
Notice that $H \mathbb{F}_2$ satisfies the two assumptions needed to identify the second page of the $H\mathbb{F}_p$-Adams spectral sequence according to theorem \ref{SecondPageOfEAdamsSpectralSequence}:
+-- {: .num_lemma #HFpSatisfiesConditionsForSecondPage}
###### Lemma
The [[Eilenberg-MacLane spectrum]] $H\mathbb{F}_p$ is flat according to \ref{FlatE}, and $H \mathbb{F}_p(\mathbb{S})$ is a [[projective module]] over $\pi_\bullet(H \mathbb{F}_p)$.
=--
+-- {: .proof}
###### Proof
The [[stable homotopy groups]] of $H \mathbb{F}_p$ is the [[prime field]] $\mathbb{F}_p$ itself, regarded as a graded commutative ring concentrated in degree 0 ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#StableHomotopyGroupsOfEMSpectrum))
$$
\pi_\bullet(H\mathbb{F}_p)
\simeq
\mathbb{F}_p
\,.
$$
Since this is a [[field]], all [[modules]] over it are [[projective modules]] ([prop.](projective+module#ModuleOverAFieldIsProjective)), hence in particular [[flat modules]] ([prop.](flat+module#ProjectiveModulesAreFlat)).
=--
+-- {: .num_cor #ClassicalAdamsSpectralSequenceEstablished}
###### Corollary
The [[classical Adams spectral sequence]], i.e. the $E$-Adams spectral sequence (def. \ref{AdamsEAdamsSpectralSequence}) for $E = H \mathbb{F}_p$ ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#OrthogonalEilenbergMacLaneRingSpectrum)) and $X = Y = \mathbb{S}$, has on its second page the [[Ext]]-groups of classical dual [[Steenrod algebra]] [[comodules]] from $\mathbb{F}_p \simeq H\mathbb{F}_p(\mathbb{S})$ to itself, and converges completely (def. \ref{EAdamsConvergingCompletely}) to the [[p-adic completion]] (def. \ref{AdicCompletionOfAbelingGroups}) of the [[stable homotopy groups of spheres]], hence in degree 0 to the [[p-adic integers]] and in all other degrees to the $p$-primary part (theorem \ref{FundamentalTheoremOfFinitelyGeneratedAbelianGroups})
$$
E_2^{s,t}(\mathbb{S},\mathbb{S})
\;=\;
Ext^{s,t}_{\mathcal{A}_p^\ast}(\mathbb{F}_p, \mathbb{F}_p)
\;\Rightarrow\;
(\pi_{\bullet}(\mathbb{S}))_p
\,.
$$
=--
+-- {: .proof}
###### Proof
By lemma \ref{HFpSatisfiesConditionsForSecondPage} the conditions of theorem \ref{SecondPageOfEAdamsSpectralSequence} are satisfied, which implies the form of the second page.
For the convergence statement, we check the assumptions in theorem \ref{EAdamsConvergenceForCorepi0EBeingCyclicGroup}:
1. By prop. \ref{FormingCoreOfARingIsIdempotent} and prop. \ref{ClassificationOfSolidRings} the ring $\mathbb{F}_p = \pi_0(H \mathbb{F}_p)$ coincides with its [[core of a ring|core]]: $c \mathbb{F}_p \simeq \mathbb{F}_p$;
1. $\mathbb{S}$ is clearly a [[connective spectrum]];
1. $\mathbb{S}$ is clearly a finite [[CW-spectrum]];
1. the groups $\pi_\bullet(\mathbb{S}) \simeq [\mathbb{S},\mathbb{S}]_\bullet$ are degreewise finitely generated, by [[Serre's finiteness theorem]].
Hence theorem \ref{EAdamsConvergenceForCorepi0EBeingCyclicGroup} applies and gives the convergence as stated.
Finally, by prop. \ref{DualSteenrodAlgebrasIsHFpHopfAlgebra} the dual $E$-Steenrod algebra in the present case is the classical dual [[Steenrod algebra]].
=--
We now use the [[classical Adams spectral sequence]] from corollary \ref{ClassicalAdamsSpectralSequenceEstablished} to compute the first dozen [[stable homotopy groups of spheres]].
##### The dual Steenrod algebra
{#TheClassicalDualSteenrodAlgebra}
+-- {: .num_defn #SteenrodAlgebraForHFp}
###### Definition
Let $p$ be a [[prime number]]. Write $\mathbb{F}_p$ for the corresponding [[prime field]].
The **mod $p$-[[Steenrod algebra]]** $\mathcal{A}_{p}$ is the graded co-commutative [[Hopf algebra]] over $\mathbb{F}_p$ which is
* for $p = 2$ generated by elements denoted $Sq^n$ for $n \in \mathbb{N}$, $n \geq 1$;
* for $p \gt 2$ generated by elements denoted $\beta$ and $P^n$ for $\in \mathbb{N}$, $n \geq 1$
(called the **Serre-Cartan basis elements**)
whose product is subject to the following relations (called the **Ádem relations**):
**for $p = 2$**:
for $0 \lt h \lt 2k$ the
$$
Sq^h Sq^k
\;=\;
\underoverset{i = 0}{[h/2]}{\sum}
\left(
\array{
k -i - 1
\\
h - 2i
}
\right)
Sq^{h + k -i} Sq^i
\,,
$$
**for $p \gt 2$**:
for $0 \lt h \lt p k$ then
$$
P^h P^k
\;=\;
\underoverset{i = 0}{[h/p]}{\sum}
(-1)^{h+i}
\left(
\array{
(p-1)(k-i) - 1
\\
h - p i
}
\right)
P^{h +k - i}P^i
$$
and if $0 \lt h \lt p k$ then
$$
\begin{aligned}
P^h \beta P^k
& =\;
\underoverset{[h/p]}{i = 0}{\sum}
(-1)^{h+i}
\left(
\array{
(p-1)(k-i)
\\
h - p i
}
\right)
\beta P^{h+k-i}P^i
\\
& +
\underoverset{[(h-1)/p]}{i = 0}{\sum}
(-1)^{h+i-1}
\left(
\array{
(p-1)(k-i) - 1
\\
h - p i - 1
}
\right)
P^{h+k-i} \beta P^i
\end{aligned}
$$
and whose coproduct $\Psi$ is subject to the following relations:
**for $p = 2$**:
$$
\Psi(Sq^n)
\;=\;
\underoverset{k = 0}{n}{\sum} Sq^k \otimes Sq^{n-k}
$$
**for $p \gt 2$**:
$$
\Psi(P^n)
\;=\;
\underoverset{n}{k = 0}{\sum}
P^k \otimes P^{n-k}
$$
and
$$
\Psi(\beta) = \beta \otimes 1 + 1 \otimes \beta
\,.
$$
=--
e.g. ([Kochmann 96, p. 52](#Kochmann96))
+-- {: .num_defn #DualSteenrodAlgebraForHPf}
###### Definition
The $\mathbb{F}_p$-[[linear dual]] of the mod $p$-Steenrod algebra (def. \ref{SteenrodAlgebraForHFp}) is itself naturally a graded [[commutative Hopf algebroid|commutative Hopf algebra]] (with coproduct the linear dual of the original product, and vice versa), called the **dual Steenrod algebra** $\mathbb{A}_{\mathbb{F}_p}^\ast$.
=--
+-- {: .num_prop #DualSteenrodAlgebrasIsHFpHopfAlgebra}
###### Proposition
There is an isomorphism
$$
\mathcal{A}^\ast_{p}
\simeq
H_\bullet( H \mathbb{F}_p, \mathbb{F}_p )
=
\pi_\bullet( H \mathbb{F}_p \wedge H \mathbb{F}_p )
\,.
$$
=--
(e.g. [Ravenel 86, p. 49](#Ravenel86), [Rognes 12, remark 7.24](#Rognes12))
We now give the generators-and-relations description of the dual Steenrod algebra $\mathcal{A}^\ast_{p}$ from def. \ref{DualSteenrodAlgebraForHPf}, in terms of linear duals of the generators for $\mathcal{A}_{p}$ itself, according to def. \ref{SteenrodAlgebraForHFp}.
+-- {: .num_theorem #MilnorTheoremOnDualSteenrodAlgebra}
###### Theorem
**(Milnor's theorem)**
The dual mod $2$-Steenrod algebra $\mathcal{A}^\ast_{2}$ (def. \ref{DualSteenrodAlgebraForHPf}) is, as an [[associative algebra]], the free [[graded commutative algebra]]
$$
\mathcal{A}^\ast_{p}
\simeq
Sym_{\mathbb{F}_p}(\xi_1, \xi_2, \cdots, )
$$
on generators:
* $\xi_n$, $n \geq 1$ being the linear dual to $Sq^{p^{n-1}} Sq^{p^{n-2}} \cdots Sq^p Sq^1$,
of degree $2^n -1$.
The dual mod $p$-Steenrod algebra $\mathcal{A}^\ast_{p}$ (def. \ref{DualSteenrodAlgebraForHPf}) is, as an [[associative algebra]], the free [[graded commutative algebra]]
$$
\mathcal{A}^\ast_{p}
\simeq
Sym_{\mathbb{F}_p}(\xi_1, \xi_2, \cdots, \;\tau_0, \tau_1, \cdots)
$$
on generators:
* $\xi_n$, $n \geq 1$ being the linear dual to $P^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1$,
of degree $2(p^n-1)$.
* $\tau_n$ being linear dual to $P^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1\beta$.
Moreover, the coproduct on $\mathcal{A}^\ast_{p}$ is given on generators by
$$
\Psi(\xi_n)
=
\underoverset{k = 0}{n}{\sum} \xi_{n-k}^{p^k} \otimes \xi_k
$$
and
$$
\Psi(\tau_n)
=
\tau_n \otimes 1
+
\underoverset{k=0}{n}{\sum}
\xi_{n-k}^{p^k} \xi_{n-k}^{p^k}\otimes \tau_k
\,,
$$
where we set $\xi_0 \coloneqq 1$.
(This defines the coproduct on the full algbra by it being an algebra homomorphism.)
=--
This is due to ([Milnor 58](Steenrod+algebra#Milnor58)). See for instance ([Kochmann 96, theorem 2.5.1](#Kochmann96), [Ravenel 86, chapter III, theorem 3.1.1](#Ravenel86))
##### The cobar complex
In order to compute the second page of the classical $H \mathbb{F}_p$-Adams spectral sequence (cor. \ref{ClassicalAdamsSpectralSequenceEstablished}) we consider a suitable [[cochain complex]] whose [[cochain cohomology]] gives the relevant [[Ext]]-groups.
+-- {: .num_defn #UnitCoidealInCommutativeHopfAlgebra}
###### Definition
Let $(\Gamma,A)$ be a [[graded commutative Hopf algebra]], hence a [[commutative Hopf algebroid]] for which the left and right units coincide $\eta \;\colon\; A \longrightarrow \Gamma$ (remark \ref{HopfAlgebrasAsHopfAlgebroids}).
Then the **unit coideal** of $\Gamma$ is the [[cokernel]]
$$
\overline{\Gamma}
\coloneqq
coker( A \overset{\eta}{\longrightarrow} \Gamma)
\,.
$$
=--
+-- {: .num_remak #DirectSumDecompositionUnitCoideal}
###### Remark
By co-unitality of graded commutative Hopf algebras (def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents}) $\epsilon \circ \eta = id_A$ the defining projection of the unit coideal (def. \ref{UnitCoidealInCommutativeHopfAlgebra})
$$
A
\overset{\eta}{\longrightarrow}
\Gamma
\overset{}{\longrightarrow}
\overline{\Gamma}
$$
forms a [[split exact sequence]] which exhibits a [[direct sum]] decomposition
$$
\Gamma \simeq A \oplus \overline{\Gamma}
\,.
$$
=--
+-- {: .num_lemma #PropertiesOfUnitCoideal}
###### Lemma
Let $(\Gamma,A)$ be a [[commutative Hopf algebra]], hence a [[commutative Hopf algebroid]] for which the left and right units coincide $\eta \;\colon\;
A \longrightarrow \Gamma$.
Then the unit coideal $\overline{\Gamma}$ (def. \ref{UnitCoidealInCommutativeHopfAlgebra}) carries the structure of an $A$-[[bimodule]] such that the [[projection]] morphism
$$
\Gamma \longrightarrow \overline{\Gamma}
$$
is an $A$-bimodule homomorphism. Moreover, the coproduct $\Psi \;\colon\; \Gamma \longrightarrow \Gamma \otimes_A \Gamma$ descends to a morphism $\overline{\Gamma} \;\colon\; \overline{\Gamma} \longrightarrow \overline{\Gamma} \otimes_A \overline{\Gamma}$ such that the projection intertwines the two coproducts.
=--
+-- {: .proof}
###### Proof
For the first statement, consider the [[commuting diagram]]
$$
\array{
A \otimes A
&\overset{A \otimes \eta}{\longrightarrow}&
A \otimes \Gamma
&\longrightarrow&
A \otimes \overline{\Gamma}
\\
\downarrow && \downarrow && \downarrow^{\mathrlap{\exists}}
\\
A
&\underset{\eta}{\longrightarrow}&
\Gamma
&\longrightarrow&
\overline{\Gamma}
}
\,,
$$
where the left [[commuting square]] exhibits the fact that $\eta$ is a homomorphism of left $A$-modules.
Since the [[tensor product of abelian groups]] $\otimes$ is a [[right exact functor]] it preserves cokernels, hence $A \otimes \overline{\Gamma}$ is the cokernel of $A \otimes A \to A\otimes \Gamma$ and hence the right vertical morphisms exists by the [[universal property]] of cokernels. This is the compatible left module structure on $\overline{\Gamma}$. Similarly the right $A$-module structure is obtained.
For the second statement, consider the [[commuting diagram]]
$$
\array{
A
&\overset{\eta}{\longrightarrow}&
\Gamma
&\longrightarrow&
\overline{\Gamma}
\\
{}^{\mathllap{\eta}}\downarrow
&&
\downarrow^{\mathrlap{\Psi}}
&&
\downarrow^{\mathrlap{\exists}}
\\
\Gamma \simeq \Gamma \otimes_A A
&\underset{id \otimes_A \eta}{\longrightarrow}&
\Gamma \otimes_A \Gamma
&\longrightarrow&
\overline{\Gamma} \otimes_A \overline{\Gamma}
}
\,.
$$
Here the left square commutes by one of the co-unitality conditions on $(\Gamma,A)$, equivalently this is the co-action property of $A$ regarded canonically as a $\Gamma$-comodule.
Since also the bottom morphism factors through zero, the [[universal property]] of the cokernel $\overline{\Gamma}$ implies the existence of the right vertical morphism as shown.
=--
+-- {: .num_defn #CobarComplex}
###### Definition
**(cobar complex)**
Let $(\Gamma,A)$ be a [[commutative Hopf algebra]], hence a [[commutative Hopf algebroid]] for which the left and right units coincide $A \overset{\eta}{\longrightarrow} \Gamma$. Let $N$ be a left $\Gamma$-comodule.
The **cobar complex** $C^\bullet_\Gamma(N)$ is the [[cochain complex]] of abelian groups with terms
$$
C^s_\Gamma(N)
\coloneqq
\underset{s\; factors}{
\underbrace{
\overline{\Gamma}
\otimes_A
\cdots
\otimes_A
\overline{\Gamma}
}
}
\otimes_A N
$$
(for $\overline{\Gamma}$ the unit coideal of def. \ref{UnitCoidealInCommutativeHopfAlgebra}, with its $A$-bimodule structure via lemma \ref{PropertiesOfUnitCoideal})
and with [[differentials]] $d_s \colon C^s_\Gamma(N) \longrightarrow C^{s+1}_\Gamma(N)$ given by the alternating sum of the coproducts via lemma \ref{PropertiesOfUnitCoideal}.
=--
([Ravenel 86, def. A1.2.11](#Ravenel86))
+-- {: .num_prop #CobarComplexGivesExtGroupsOutOfA}
###### Proposition
Let $(\Gamma,A)$ be a [[commutative Hopf algebra]], hence a [[commutative Hopf algebroid]] for which the left and right units coincide $A \overset{\eta}{\longrightarrow} \Gamma$. Let $N$ be a left $\Gamma$-comodule.
Then the [[cochain cohomology]] of the cobar complex $C^\bullet_\Gamma(N)$ (def. \ref{CobarComplex}) is the [[Ext]]-groups of comodules from $A$ (regarded as a left comodule via def. \ref{ComoduleStructureOnGroundRing}) into $N$
$$
H^\bullet(C^\bullet_\Gamma(N))
\;\simeq\;
Ext^\bullet_\Gamma(A,N)
\,.
$$
=--
([Ravenel 86, cor. A1.2.12](#Ravenel86), [Kochman 96, prop. 5.2.1](#Kochman96))
+-- {: .proof}
###### Proof idea
One first shows that there is a resolution of $N$ by co-free comodules given by the complex
$$
D^\bullet_\Gamma(N)
\coloneqq
\Gamma \otimes_A \overline{\Gamma}^{\otimes_A^{\bullet}} \otimes_A N
$$
with differentials given by the alternating sum of the coproducts. This is called the _cobar resolution_ of $N$.
To see that this is indeed a resolution, one observes that a contracting homotopy is given by
$$
s (\gamma \gamma_1\vert \cdots \vert \gamma_s n)
\coloneqq
\epsilon(\gamma) \gamma_1\vert \cdots \vert \gamma_s n
$$
for $s \gt 0$ and
$$
s(\gamma n) \coloneqq 0
\,.
$$
Now from lemma \ref{CoFreeHopfComodulesAreHomNAcyclicForProjectiveN}, in view of remark \ref{DirectSumDecompositionUnitCoideal}, and since $A$ is trivially projective over itself, it follows that this is an $F$-acyclic resolution for $F \coloneqq Hom_\Gamma(A,-)$.
This means that the resolution serves to compute the [[Ext]]-functor in question and we get
$$
\begin{aligned}
Ext^\bullet_\Gamma(A,N)
& \simeq
H^\bullet(Hom_\Gamma(A, D^\bullet_\Gamma(N)))
\\
& =
H^\bullet( Hom_\Gamma(A, \Gamma \otimes_A \overline{\Gamma}^{\otimes_A^{\bullet}} \otimes_A N ) )
\\
&\simeq
H^\bullet( Hom_A(A, \overline{\Gamma}^{\otimes_A^{\bullet}} \otimes_A N ) )
\\
& \simeq
H^\bullet( \overline{\Gamma}^{\otimes_A^{\bullet}} \otimes_A N )
\,,
\end{aligned}
\,,
$$
where the second-but-last equivalence is the isomorphism of the co-free/forgetful adjunction
$$
A Mod
\underoverset
{\underset{co-free}{\longrightarrow}}
{\overset{forget}{\longleftarrow}}
{\bot}
\Gamma CoMod
$$
from prop. \ref{CoFreeComodules}, while the last equivalence is the isomorphism of the free/forgetful adjunction
$$
A Mod
\underoverset
{\underset{forget}{\longrightarrow}}
{\overset{free}{\longleftarrow}}
{\bot}
Ab
$$
=--
##### The May spectral sequence
The cobar complex (def. \ref{CobarComplex}) realizes the second page of the [[classical Adams spectral sequence]] (cor. \ref{ClassicalAdamsSpectralSequenceEstablished}) as the [[cochain cohomology]] of a [[cochain complex]]. This is still hard to compute directly, but we now discuss that this cochain complex admits a [[filtered complex|filtration]] so that the induced [[spectral sequence of a filtered complex]] is computable and has trivial extension problem ([rmk.](Introduction+to+Stable+homotopy+theory+--+S#ExtensionProblemForSpectralSequences)). This is called the _[[May spectral sequence]]_.
We obtain this spectral sequence in prop. \ref{MaySpectralSequenceForClassicalAdamsSpectralSequence} below. First we need to consider some prerequisites.
+-- {: .num_lemma #CoModuleExtFromAToAForPrimitivelyGeneratedExteriorAlgebra}
###### Lemma
Let $(\Gamma,A)$ be a graded commutative Hopf algebra, i.e. a [[graded commutative Hopf algebroid]] with left and right unit coinciding for which the underlying $A$-algebra of $\Gamma$ is a free graded commutative $A$-algebra on a set of generators $\{x_i\}_{i \in I}$
such that
1. all generators $x_i$ are [[primitive elements]];
1. $A$ is in degree 0;
1. $(i \lt j) \Rightarrow (deg(x_i) \leq deg(x_j))$;
1. there are only finitely many $x_i$ in a given degree,
then the [[Ext]] of $\Gamma$-comodules from $A$ to itself is the free graded commutative algebra on these generators
$$
Ext_\Gamma(A,A) \simeq A[\{x_i\}_{i \in I}]
\,.
$$
=--
([Ravenel 86, lemma 3.1.9](#Ravenel86), [Kochman 96, prop. 3.7.5](#Kochman96))
+-- {: .proof}
###### Proof
Consider the co-free left $\Gamma$-comodule ([prop.](commutative+Hopf+algebroid#CoFreeComodules))
$$
\Gamma \otimes_A A[\{y_i\}_{i \in I}]
$$
and regard it as a [[chain complex]] of left comodules by defining a [[differential]] via
$$
d \colon x_i \mapsto y_i
$$
$$
d \colon y_i \mapsto 0
$$
and extending as a graded [[derivation]].
We claim that $d$ is a homomorphism of left comodules: Due to the assumption that all the $x_i$ are primitive we have on generators that
$$
\begin{aligned}
(id,d) ( \Psi(x_i) )
& =
(id,d) ( x_i \otimes 1 + 1 \otimes x_i )
\\
& =
\underset{= 0}{x_i \otimes \underbrace{(d 1)} }
+
\underset{= y_i}{ 1 \otimes \underbrace{(d x_i)} }
\\
& =
\Psi( d x_i )
\end{aligned}
$$
and
$$
\begin{aligned}
(id,d)( \Psi(y_i) )
& =
(id,d) ( 1, y_i )
\\
& =
(1, d y_i)
\\
& =
0
\\
& =
\Psi( 0 )
\\
& =
\Psi(d y_i)
\end{aligned}
\,.
$$
Since $d$ is a graded derivation on a free graded commutative algbra, and $\Psi$ is an algebra homomorphism, this implies the statement for all other elements.
Now observe that the canonical [[chain map]]
$$
(\Gamma \otimes_A A[\{y_i\}_{i \in I}] ,\; d)
\overset{\simeq_{qi}}{\longrightarrow}
A
$$
(which projects out the generators $x_i$ and $y_i$ and is the identity on $A$), is a [[quasi-isomorphism]], by construction. Therefore it constitutes a co-free resolution of $A$ in left $\Gamma$-comodules.
Since the counit $\eta$ is assumed to be flat, and since $A[\{y_i\}_{i \in I}]$ is degreewise a [[free module]] over $A$, hence in particular a [[projective module]], prop. \ref{CoFreeHopfComodulesAreHomNAcyclicForProjectiveN} says that the above is an [[acyclic resolution]] with respect to the functor $Hom_{\Gamma}(A,-) \colon \Gamma CoMod \longrightarrow A Mod$. Therefore it computes the [[Ext]]-functor. Using that forming co-free comodules is [[right adjoint]] to forgetting $\Gamma$-comodule structure over $A$ (prop. \ref{CoFreeComodules}), this yields:
$$
\begin{aligned}
Ext^\bullet_\Gamma(A,A)
& \simeq
H_\bullet(Hom_\Gamma(A, \Gamma \otimes_A A[\{y_i\}_{i \in I}] ), d)
\\
& \simeq
H_\bullet(Hom_A(A, A[\{y_i\}_{i \in I}] ), d= 0 )
\\
& \simeq
Hom_A(A, A[\{y_i\}_{i \in I}] )
\\
& \simeq
A[\{x_i\}_{i \in I}]
\end{aligned}
\,.
$$
=--
+-- {: .num_lemma #SpectralSequenceConvergingToExtForFilteredHopfAlgebra}
###### Lemma
If $(\Gamma,A)$ as above is equipped with a [[filtering]], then there is a [[spectral sequence]]
$$
\mathcal{E}_1
\;=\;
Ext_{gr_\bullet \Gamma}(gr_\bullet A, gr_\bullet A)
\;\Rightarrow\;
Ext_{\Gamma}(A, A)
$$
converging to the [[Ext]] over $\Gamma$ from $A$ to itself, whose first page is the $Ext$ over the [[associated graded]] Hopf algebra $gr_\bullet \Gamma$.
=--
([Ravenel 86, lemma 3.1.9](#Ravenel86), [Kochman 96, prop. 3.7.5](#Kochman96))
+-- {: .proof}
###### Proof
The filtering induces a filtering on the cobar complex (def. \ref{CobarComplex}) which computes $Ext_\Gamma$ (prop. \ref{CobarComplexGivesExtGroupsOutOfA}). The spectral sequence in question is the corresponding [[spectral sequence of a filtered complex]]. Its first page is the homology of the associated graded complex (by this [prop.](spectral+sequence+of+a+filtered+complex#FirstPages)), which hence is the homology of the cobar complex (def. \ref{CobarComplex}) of the [[associated graded]] Hopf algebra $gr_\bullet \Gamma$. By prop. \ref{CobarComplexGivesExtGroupsOutOfA} this is the [[Ext]]-groups as shown.
=--
Let now $A \coloneqq \mathbb{F}_2$, $\Gamma \coloneqq \mathcal{A}^\bullet_{2}$ be the mod 2 [[dual Steenrod algebra]]. By [[Milnor's theorem]] (prop. \ref{MilnorTheoremOnDualSteenrodAlgebra}), as an $\mathbb{F}_2$-algebra this is
$$
\mathcal{A}^\bullet_{2}
=
Sym_{\mathbb{F}_2}(\xi_1, \xi_2, \cdots)
\,.
$$
and the coproduct is given by
$$
\Psi(\xi_n)
=
\underoverset{k = 0}{i}{\sum} \xi_{i - k}^{2^k} \otimes \xi_k
\,,
$$
where we set $\xi_0 \coloneqq 1$.
+-- {: .num_defn #hGeneratorsinClassicalAdamsSpectralSequence}
###### Definition
Introduce new generators
$$
h_{i,n}
\coloneqq
\left\{
\array{
\xi_i^{2^n} & for \; i \geq 1, \, k \geq 0
\\
1 & for \; i = 0
}
\right.
$$
=--
+-- {: .num_remark #hGeneratorsForSteenrodAlgebraGiveUniqueDecompositionOfMonomials}
###### Remark
By binary expansion of powers, there is a unique way to express every monomial in $\mathbb{F}_2[\xi_1, \xi_2, \cdots]$ as a product of the new generators in def. \ref{hGeneratorsinClassicalAdamsSpectralSequence}
such that each such element appears at most once in the product. E.g.
$$
\begin{aligned}
\xi_i^5 \xi_j^7
& =
\xi_i^{2^0 + 2^2} \xi_j^{2^0 + 2^1 + 2^2}
\\
& =
h_{i,0} h_{i,1} h_{j,0} h_{j,1} h_{j,2}
\end{aligned}
\,.
$$
=--
+-- {: .num_prop #CoproductOnDualSteenrodInTermsOfAdaptedGenerators}
###### Proposition
In terms of the generators $\{h_{i,n}\}$ from def. \ref{hGeneratorsinClassicalAdamsSpectralSequence}, the coproduct on the dual [[Steenrod algebra]] $\mathcal{A}^\ast_{2}$ takes the following simple form
$$
\Psi(h_{i,n})
\;=\;
\underoverset{k = 0}{i}{\sum} h_{i-k,n+k}\otimes h_{k,n}
\,.
$$
=--
+-- {: .proof}
###### Proof
Using that the coproduct of a [[bialgebra]] is a [[homomorphism]] for the algebra structure and using [[freshman's dream]] arithmetic over $\mathbb{F}_2$, one computes:
$$
\begin{aligned}
\Psi(h_{i,n})
& =
\Psi\left(\xi_i^{2^n}\right)
\\
& = (\Psi(\xi_i))^{2^n}
\\
& =
\left(\underoverset{k = 0}{i}{\sum} \xi_{i-k}^{2^k} \otimes \xi_k\right)^{2^n}
\\
& = \underoverset{k = 0}{i}{\sum} \left(\xi_{i-k}^{2^k}\right)^{2^n} \otimes \xi^{2^n}_k
\\
& = \underoverset{k = 0}{i}{\sum} \xi_{i-k}^{2^k \cdot 2^n} \otimes \xi^{2^n}_k
\\
& = \underoverset{k = 0}{i}{\sum} \xi_{i-k}^{2^{(k+n)}} \otimes \xi^{2^n}_k
\\
& = \underoverset{k = 0}{i}{\sum} h_{i-k,n+k}\otimes h_{k,n}
\end{aligned}
\,.
$$
=--
+-- {: .num_prop #MaySpectralSequenceForClassicalAdamsSpectralSequence}
###### Proposition
There exists a converging [[spectral sequence]] of graded $\mathbb{F}_2$-vector spaces of the form
$$
E_1^{s,t,p}
=
\mathbb{F}_2[ \{ h_{i,n} \}_{{i \geq 1,} \atop {n \geq 0}}]
\;\Rightarrow\;
Ext^{s,t}_{\mathcal{A}_2^\ast}(\mathbb{F}_2, \mathbb{F}_2)
\,,
$$
called the **[[May spectral sequence]]** (where $s$ and $t$ are from the bigrading of the spectral sequence itself, while the index $p$ is that of the graded $\mathbb{F}_2$-vector spaces), with
1. $h_{i,n} \in E_1^{1, 2^{2^{i+n} - 2^n - 1, 2i - 1 }}$
1. first differential given by
$$
d_1 (h_{i,n})
=
\underoverset{k = 0}{i}{\sum} h_{i-k,n+k}\otimes h_{k,n}
\,;
$$
1. higher differentials of the form
$$
d_r
\;\colon\;
E_r^{s,t, p}
\longrightarrow
E_r^{s+1, t-1, p-2r+1}
\,,
$$
where the filtration is by maximal degree.
Notice that since everything is $\mathbb{F}_2$-linear, the [extension problem](Introduction+to+Stable+homotopy+theory+--+S#ExtensionProblemForSpectralSequences) of this spectral sequence is trivial.
=--
([Kochman 96, prop. 5.3.1](#Kochman96))
+-- {: .proof}
###### Proof
Define a [[graded algebra|grading]] on the dual [[Steenrod algebra]] $\mathcal{A}^\bullet_{2}$ (theorem \ref{MilnorTheoremOnDualSteenrodAlgebra}) by taking the degree of the generators from def.\ref{hGeneratorsinClassicalAdamsSpectralSequence} to be (this idea is due to ([Ravenel 86, p.69](#Ravenel86)))
$$
{\vert h_{i,n} \vert}
\coloneqq
2i-1
$$
and extending this additively to monomials, via the unique decomposition of remark \ref{hGeneratorsForSteenrodAlgebraGiveUniqueDecompositionOfMonomials}.
For example
$$
\begin{aligned}
\vert \xi_i^5 \xi_j^7\vert
& =
{\vert h_{i,0} h_{i,1} h_{j,0} h_{j,1} h_{j,2} \vert}
\\
& =
2(2i-1) + 3(2j-1)
\end{aligned}
\,.
$$
Consider the corresponding increasing [[filtration]]
$$
\cdots
\subset
F_p \mathcal{A}^\ast_{2}
\subset
F_{p+1} \mathcal{A}^\ast_{2}
\subset
\cdots
\subset
\mathcal{A}^\ast_{2}
$$
with filtering stage $p$ containing all elements of total degree $\leq p$.
Observe via prop. \ref{CoproductOnDualSteenrodInTermsOfAdaptedGenerators} that
$$
\begin{aligned}
\Psi(h_{i,n})
&
=
\underset{deg = 2i-1}{\underbrace{h_{i,n} \otimes 1}}
+
\underoverset{0 \lt k \lt i}{}{\sum}
\underset{deg = 2i-2}{\underbrace{h_{i-k,n+k} \otimes h_{k,n}}}
+
\underset{deg = 2i-1}{\underbrace{1 \otimes h_{i,n} }}
\end{aligned}
\,.
$$
This means that after projection to the [[associated graded]] Hopf algebra
$$
F_\bullet \mathcal{A}^\ast_{2}
\longrightarrow
gr_\bullet \mathcal{A}^\ast_{2}
\coloneqq
F_\bullet( \mathcal{A}^\ast_2)/F_{\bullet-1}( \mathcal{A}^\ast_2 )
$$
all the generators $h_{i,n}$ become [[primitive elements]]:
$$
\begin{aligned}
\Psi(h_{i,n})
&
=
h_{i,n}\otimes 1
+
1 \otimes h_{i,n}
\;\;\;\;\;
\in
gr_\bullet \mathcal{A}^\ast_{2}
\otimes
gr_\bullet \mathcal{A}^\ast_{2}
\end{aligned}
\,.
$$
Hence lemma \ref{CoModuleExtFromAToAForPrimitivelyGeneratedExteriorAlgebra} applies and says that the $Ext$ from $\mathbb{F}_2$ to itself over the [[associated graded]] Hopf algebra is the polynomial algebra in these generators:
$$
Ext_{gr_\bullet \mathcal{A}^\ast_{2}}(\mathbb{F}_2,\mathbb{F}_2)
\simeq
\mathbb{F}_2[\{h_{i,n}\}_{{i \geq 1,} \atop {n \geq 0}}]
\,.
$$
Moreover, lemma \ref{SpectralSequenceConvergingToExtForFilteredHopfAlgebra} says that this is the first page of a spectral sequence that converges to the $Ext$ over the original Hopf algebra:
$$
\mathcal{E}_1
=
\mathbb{F}_2[\{h_{i,n}\}_{{i \geq 1} \atop {n \geq 0}}]
\;\Rightarrow\;
Ext_{\mathcal{A}^\ast_{2}}(\mathbb{F}_2,\mathbb{F}_2)
\,.
$$
Moreover, again by lemma \ref{SpectralSequenceConvergingToExtForFilteredHopfAlgebra}, the differentials on any $r$-page are the restriction of the differentials of the bar complex to the $r$-almost cycles ([prop.](Introduction+to+Stable+homotopy+theory+--+I#DifferentialsOnAlmostChains)). Now the differential of the cobar complex is the alternating sum of the coproduct on $\mathcal{A}^\ast_{2}$, hence by prop. \ref{CoproductOnDualSteenrodInTermsOfAdaptedGenerators} this is:
$$
d_1 (h_{i,n})
=
\underoverset{k = 0}{i}{\sum} h_{i-k,n+k}\otimes h_{k,n} \,.
$$
=--
##### The second page
Now we use the [[May spectral sequence]] (prop. \ref{MaySpectralSequenceForClassicalAdamsSpectralSequence}) to compute the second page and in fact the stable page of the [[classical Adams spectral sequence]] (cor. \ref{ClassicalAdamsSpectralSequenceEstablished}) in low internal degrees $t-s$.
+-- {: .num_lemma #TermsOnSecondPageOfMaySpectralSequenceInLowDegrees}
###### Lemma
**(terms on the second page of May spectral sequence)**
In the range $t - s \leq 13$, the second page of the May spectral sequence for $Ext_{\mathbb{A}^\ast_{\mathbb{F}_2}}(\mathbb{F}_2,\mathbb{F}_2)$ has as generators all the
* $h_n$
* $b_{i,n} \coloneqq (h_{i,n})^2$
as well as the element
* $x_7 \coloneqq h_{2,0} h_{2,1} + h_{1,1} h_{3,0}$
subject to the relations
* $h_n h_{n+1} = 0$
* $h_2 b_{2,0} = h_0 x_7$
* $h_2 x_7 = h_0 b_{2,1}$.
=--
e.g. ([Ravenel 86, lemma 3.2.8 and lemma 3.2.10](#Ravenel86), [Kochman 96, lemma 5.3.2](#Kochman96))
+-- {: .proof}
###### Proof
Remember that the differential in the cobar complex (def. \ref{CobarComplex}) lands not in $\Gamma = \mathcal{A}^\ast_2$ itself, but in the unit coideal $\overline{\Gamma} \coloneqq coker(\eta)$ where the generator $h_{0,n} = \xi_0 = 1$ disappears.
Using this we find for the differential $d_1$ of the generators in low degree on the first page of the [[May spectral sequence]] (prop. \ref{MaySpectralSequenceForClassicalAdamsSpectralSequence}) via the formula for the differential from prop. \ref{CoproductOnDualSteenrodInTermsOfAdaptedGenerators}, the following expressions:
$$
\begin{aligned}
d_1(h_n)
& \coloneqq
d_1(h_{1,n})
\\
& =
\overline{\Psi}(h_{1,n})
\\
& =
h_{1,n} \otimes \underset{= 0}{\underbrace{ h_{0,n} }}
+
\underset{= 0}{\underbrace{ h_{0,n+1} }} \otimes h_{1,n}
\\
& =
0
\end{aligned}
$$
and hence all the elements $h_n$ are cocycles on the first page of the May spectral sequence.
Also, since $d_1$ is a [[derivation]] (by definition of the cobar complex, def. \ref{CobarComplex}) and since the product of the image of the cobar complex in the first page of the May spectral sequence is graded commutative, we have for all $n,k$ that
$$
\begin{aligned}
d_1 (h_{n,k})^2
& =
2 h_{n,k} (d_1 (h_{n,k}))
\\
& = 0
\end{aligned}
$$
(since $2 = 0 \; mod \; 2$).
Similarly we compute $d_1$ on the other generators. These terms do not vanish, but so they impose relations on products in the cobar complex:
$$
\begin{aligned}
d_1(h_{2,0})
& =
h_{1,1} \otimes h_{1,0}
\\
d_1( h_{2,1} )
& =
h_{1,2} \otimes h_{1,1}
\\
d_1( h_{2,2} )
& =
h_{1,3} \otimes h_{1,2}
\\
d_1( h_{2,3} )
& =
h_{1,4} \otimes h_{1,3}
\\
d_1( h_{3,0} )
& =
h_{2,1} \otimes h_{1,0}
+
h_{1,2} \otimes h_{2,0}
\end{aligned}
$$
This shows that $h_n h_{n+1} = 0$ in the given range.
The remaining statements follow similarly.
=--
+-- {: .num_remark #PictureFromTermsOnSecondPageOfMaySpectralSequence}
###### Remark
With lemma \ref{TermsOnSecondPageOfMaySpectralSequenceInLowDegrees}, so far we see the following picture in low degrees.
$$
\array{
\vdots & \vdots
\\
3 & h_0^4 & & & {h_1^3},\; {h_0^2 h_2}
\\
2 & h_0^2 & & h_1^2 & h_0 h_2
\\
1& h_0 & h_1 & & h_2
\\
& 0 & 1 & 2& 3 & 4
}
$$
Here the relation
$$
h_0 \otimes h_1 = 0
$$
removes a vertical tower of elements above $h_1$.
So far there are two different terms in degree $(s,t-s) = (3,3)$. The next lemma shows that these become identified on the next page.
=--
+-- {: .num_lemma #DifferentialsOnSecondPageOfMaySpectralSequence}
###### Lemma
**(differentials on the second page of the May spectral sequence)**
The differentials on the second page of the [[May spectral sequence]] (prop. \ref{MaySpectralSequenceForClassicalAdamsSpectralSequence}) relevant for internal degrees $t-s \leq 12$ are
1. $d_2(h_{n}) = 0$
1. $d_2(b_{2,n}) = h_n^2 h_{n+2} + h_{n+1}^3$
1. $d_2(x_7) = h_0 h_2^2$
1. $d_2(b_{3,0}) = h_1 b_{2,1} + h_3 b_{2,0}$
=--
([Kochman 96, lemma 5.3.3](#Kochman96))
+-- {: .proof}
###### Proof
The first point follows as before in lemma \ref{TermsOnSecondPageOfMaySpectralSequenceInLowDegrees}, in fact the $h_n$ are infinite cycles in the May spectral sequence.
We spell out the computation for the second item:
We may represent $b_{2,k}$ by $\xi_2^{2^k} \times \xi_2^{2^k}$
plus terms of lower degree. Choose the representative
$$
B_{2,k}
=
\xi_2^{2^k} \otimes \xi_2^{2^k}
\;+\;
\xi_1^{2^{k + 1}} \otimes \xi_1^{2^k} \xi_2^{2^k}
\;+\;
\xi_1^{2^{k+1}} \xi^{2^k} \otimes \xi_1^{2^k}
\,.
$$
Then we compute $d B_{2,k}$, using the definition of the cobar complex (def. \ref{CobarComplex}), the value of the coproduct on dual generators (theorem \ref{MilnorTheoremOnDualSteenrodAlgebra}), remembering that the coproduct $\Psi$ on a Hopf algebra is a homomorphism for the underlying commutative ring, and using [[freshman's dream]] arithmetic to evaluate prime-2 powers of sums. For the three summands we obtain
$$
\begin{aligned}
d ( \xi_2^{2^k} \otimes \xi_2^{2^k})
& =
\overline{\Psi} (\xi_2^{2^k}) \otimes \xi_2^{2^k}
+
\xi_2^{2^k} \otimes \overline{\Psi}(\xi_2^{2 k})
\\
& =
\underset{c_1}{\underbrace{\xi_1^{2^{k+1}} \otimes \xi_1^{2^k} \otimes \xi_2^{2^k}}}
\;+\;
\underset{c_2}{\underbrace{\xi_2^{2^k} \otimes \xi_1^{2^{k+1}} \otimes \xi_1^{2^k}}}
\end{aligned}
$$
and
$$
\begin{aligned}
d ( \xi_1^{2^{k + 1}} \otimes \xi_1^{2^k} \xi_2^{2^k} )
& =
\xi_1^{2^k} \otimes \overline{\Psi}( \xi_1^{2^k} \xi_2^{2^k})
\\
& =
\xi_1^{2^{k + 1}}
\otimes
\left(
\xi_1^{2^k} \otimes 1
+
1 \otimes \xi_1^{2^k}
\right)
\left(
\xi_2^{2^k} \otimes 1
+
\xi_1^{2^{k+1}} \otimes \xi_1^{2^k}
+
1 \otimes \xi_2^{2^k}
\right)
\\
& =
\underset{b}{
\underbrace{
\xi_1^{2^{k + 1}} \otimes \xi_1^{2^{k+1}+ 2^k} \otimes \xi_1^{2^k}
}}
\;+\;
\underset{c_1}{\underbrace{
\xi_1^{2^{k+1}} \otimes \xi_1^{2^k} \otimes \xi_2^{2^k}
}}
\;+\;
\underset{a}{\underbrace{
\xi_1^{2^{k+1}} \otimes \xi_2^{2^k} \otimes \xi_1^{2^k}
}}
\;+\;
\xi_1^{2^{k+1}} \otimes \xi_1^{2^{k+1}} \otimes \xi_1^{2^{k+1}}
\end{aligned}
$$
and
$$
\begin{aligned}
d( \xi_1^{2^{k+1}} \xi^{2^k} \otimes \xi_1^{2^k} )
&
=
\overline{\Psi}( \xi_1^{2^{k+1}} \xi^{2^k} ) \otimes \xi_1^{2^k}
\\
& =
\left(
\xi_1^{2^{k+1}} \otimes 1
+
1 \otimes \xi_1^{2^{k+1}}
\right)
\left(
\xi_2^{2^k} \otimes 1
+
\xi_1^{2^{k+1}} \otimes \xi_1^{2^k}
+
1 \otimes \xi_2^{2^k}
\right)
\otimes
\xi_1^{2^k}
\\
& =
\xi_1^{2^{k+2}} \otimes \xi_1^{2^k} \otimes \xi_1^{2^k}
\;+\;
\underset{a}{\underbrace{
\xi_1^{2^{k+1}} \otimes \xi_2^{2^k} \otimes \xi_1^{2^k}
}}
\;+\;
\underset{c_2}{\underbrace{
\xi_2^{2^k} \otimes \xi_1^{2^{k+1}} \otimes \xi_1^{2^k}
}}
\;+\;
\underset{b}{\underbrace{
\xi_1^{2^{k+1}} \otimes \xi_1^{2^{k+1} + 2^k} \otimes \xi_1^{2^k}
}}
\end{aligned}
\,.
$$
The labeled summands appear twice in $d B_{2,k}$ hence vanish (mod 2). The remaining terms are
$$
d B_{2,k}
=
\xi_1^{2^{k+1}} \otimes \xi_1^{2^{k+1}} \otimes \xi_1^{2^{k+1}}
\;+\;
\xi_1^{2^{k+2}} \otimes \xi_1^{2^k} \otimes \xi_1^{2^k}
$$
and these indeed represent the claimed elements.
=--
+-- {: .num_remark }
###### Remark
With lemma \ref{DifferentialsOnSecondPageOfMaySpectralSequence} the picture from remark \ref{PictureFromTermsOnSecondPageOfMaySpectralSequence} is further refined:
For $k = 0$ the differentia $d_2(b_{2,n}) = h_n^2 h_{n+2} + h_{n+1}^3$ means that on the third page of the May spectral sequence there is an identification
$$
h_1^3 = h_0^2 h_2
\,.
$$
Hence where on page two we saw two distinct elements in bidegree $(s,t-s) = (3,3)$, on the next page these merge:
$$
\array{
\vdots & \vdots
\\
3 & h_0^4 & & & {h_1^3} = {h_0^2 h_2}
\\
2 & h_0^2 & & h_1^2 & h_0 h_2
\\
1& h_0 & h_1 & & h_2
\\
& 0 & 1 & 2& 3 & 4
}
$$
=--
Proceeding in this fashion, one keeps going until the 4-page of the May spectral sequence ([Kochman 96, lemma 5.3.5](#Kochman96)). Inspection of degrees shows that this is sufficient, and one obtains:
+-- {: .num_theorem #TheStablePageOfTheClassicalAdamsSpectralSequenceInLowDegree}
###### Theorem
**(stable page of classical Adams spectral sequence)**
In internal degree $t-s \leq 12$ the infinity page (def. \ref{InfinityPageEAdams}) of the [[classical Adams spectral sequence]] (cor. \ref{ClassicalAdamsSpectralSequenceEstablished}) is spanned by the items in the following table
<img src="http://ncatlab.org/nlab/files/ClassicalAdamsSpectralSequence.jpg" width="600" >
Here every dot is a generator for a copy of $\mathbb{Z}/2\mathbb{Z}$. Vertical edges denote multiplication with $h_0$ and diagonal edges denotes multiplication with $h_1$.
=--
e.g. ([Ravenel 86, theorem 3.2.11](#Ravenel86), [Kochman 96, prop. 5.3.6](#Kochman96)), graphics taken from ([[Symmetric spectra|Schwede 12]]))
##### The first dozen stable stems
{#StableStems}
Theorem \ref{TheStablePageOfTheClassicalAdamsSpectralSequenceInLowDegree} gives the stable page of the [[classical Adams spectral sequence]] in low degree. By corollary \ref{ClassicalAdamsSpectralSequenceEstablished} and def. \ref{EAdamsConvergingCompletely} we have that a vertical sequence of dots encodes an 2-primary part of the stable homotopy groups of spheres according to the graphical calculus of remark \ref{PrimaryDecompositionGraphicalRepresentation} (the rules for determining group extensions there is just the solution to the extension problem ([rmk.](Introduction+to+Stable+homotopy+theory+--+S#ExtensionProblemForSpectralSequences)) in view of def. \ref{EAdamsConvergingCompletely}):
| $k =$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
| $\pi_k(\mathbb{S}\otimes \mathbb{Z}_{(2)}) = $ | $\mathbb{Z}_{(2)}$ | $\mathbb{Z}/2$ | $\mathbb{Z}/2$ | $\mathbb{Z}/8$ | $0$ | $0$ | $\mathbb{Z}/2$ | $\mathbb{Z}/16$ | $(\mathbb{Z}/2)^2$ | $(\mathbb{Z}/2)^3$ | $\mathbb{Z}/2$ | $\mathbb{Z}/8$ | $0$ | $0$ |
The full answer in this range turns out to be this:
| $k =$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | $\cdots$ |
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
| $\pi_k(\mathbb{S}) = $ | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | $\mathbb{Z}_{24}$ | $0$ | $0$ | $\mathbb{Z}_2$ | $\mathbb{Z}_{240}$ | $(\mathbb{Z}_2)^2$ | $(\mathbb{Z}_2)^3$ | $\mathbb{Z}_6$ | $\mathbb{Z}_{504}$ | $0$ | $\mathbb{Z}_3$ | $(\mathbb{Z}_2)^2$ | $\mathbb{Z}_{480} \oplus \mathbb{Z}_2$ | $\cdots$ |
And expanding the range yields this :
<img src="http://www.math.cornell.edu/~hatcher/stemfigs/p%3D2pic.gif" alt="stable homotopy groups of spheres at 2" width="800" />
(graphics taken from [Hatcher's website](http://www.math.cornell.edu/~hatcher/))
### $E$-injective resolutions
{#ViaInjectiveResolutions}
A streamlined discussion of $E$-[[Adams resolutions]] in close analogy to [[injective resolutions]] in [[homological algebra]] was given in ([Miller 81](#Miller81)), advertized in ([Hopkins 99](#Hopkins99)) and worked out in more detail in ([Aramian](#Aramian)).
Notice that the standard concept of [[exact sequences]] and [[injective objects]] makes sense in [[abelian categories]], but not in the [[stable homotopy category]] of [[spectra]], as the latter is only an [[additive category]]. Of course this is because the [[stable homotopy theory|stable homotopy theoretic]] analog of what are [[exact sequences]] in abelian categories are [[homotopy fiber sequences]] of spectra. But for computational purposes it turns out useful to consider a blend between these two concepts (due to [Miller 81](#Miller81)), where a sequence of spectra $X_\bullet$ is regarded as exact if the [[homotopical functor]] to the abelian category of [[abelian groups]] that it [[representable functor|represents]] takes values in [[exact sequences]]. With respect to this hybrid concept, $E$-Adams resolutions in the [[stable homotopy category]] are the direct analog of [[injective resolutions]] in an [[abelian category]].
+-- {: .num_defn #HomotopicalFunctorCoRepresentedByASpectrum}
###### Definition
For $X$ a [[spectrum]], we say that the _[[homotopical functor]]_
that it [[representable functor|co-represents]] is the functor of [[stable homotopy groups]] of the [[mapping spectrum]]-construction out of $X$, regarded as a functor on the [[stable homotopy category]]:
$$
\pi_\bullet[X, -] \colon Ho(Spectra) \longrightarrow Ab^{\mathbb{Z}}
\,.
$$
(Regarded as an [[(∞,1)-functor]] on the [[stable (∞,1)-category of spectra]] this is also called the _[[homological functor]]_ co-represented by $X$.)
=--
+-- {: .num_example}
###### Example
For $X = \mathbb{S}$ the [[sphere spectrum]] then the homotopical functor that it co-represents according to def. \ref{HomotopicalFunctorCoRepresentedByASpectrum}
$$
\pi_\bullet[\mathbb{S},- ]\simeq \pi_\bullet(-)
$$
is the [[stable homotopy group]]-functor.
=--
Throughout,$E$ is a [[ring spectrum]].
+-- {: .num_defn #ExactSequences}
###### Definition
Say that
1. a sequence of [[spectra]]
$$
A_1 \longrightarrow A_2 \longrightarrow \cdots \longrightarrow A_n
$$
is
1. a (long) _exact sequence_ if the induced sequence of [[homotopy functors]] according to def. \ref{HomotopicalFunctorCoRepresentedByASpectrum}, is a [[long exact sequence]] in $[HoSpectra,Ab^{\mathbb{Z}}]$;
2. (for $n = 2$) a _short exact sequence_ if
$$
0 \longrightarrow A_1 \longrightarrow A_2 \longrightarrow A_3 \longrightarrow 0
$$
is (long) exact in the above sense;
1. a morphism $A \longrightarrow B$ is
1. a _monomorphism_ if $0 \longrightarrow A \longrightarrow B$ is an exact sequence in the above sense;
1. an _epimorphism_ if $A \longrightarrow B \longrightarrow 0$ is an exact sequence in the above sense.
For $E$ a [[ring spectrum]], then a sequence of spectra is called (long/short) _$E$-exact_ and a morphism is epi/mono, respectively, if becomes long/short exact or epi/mono, respectively, after taking [[smash product of spectra|smash product]] with $E$.
=--
+-- {: .num_example #HomotopyCofiberSequencesAreExactSequencesOfSpectra}
###### Example
Every [[homotopy cofiber sequence]] of spectra is exact in the sense of def. \ref{ExactSequences}.
=--
+-- {: .num_example #SuspensionPreservesExactSequencesOfSpectra}
###### Example
The [[suspension]] functor $\Sigma \colon Ho(Spectra) \to Ho(Spectra)$ preserves exact sequences in the sense of def. \ref{ExactSequences}.
=--
+-- {: .proof}
###### Proof
By the suspension/looping [[adjunction]]-[[isomorphism]] $[\Sigma A_\bullet, -]\simeq [A_\bullet, \Omega(-)]$ ([prop.](Introduction+to+Stable+homotopy+theory+--+P#AlternativeLoopingAndSuspensionIsQuillenEquivalenceOnStableModelStructure)) and so the statement follows from the assumption that $A_\bullet$ is long exact.
=--
+-- {: .num_example #HomotopyRetractionsAreMonomorphismsOfSpectra}
###### Example
If a morphism, $s \colon A \to B$ has a [[retraction]] $r \colon B \to A$ in [[Ho(Spectra)]] then it is a monomorphism in the sense of def. \ref{ExactSequences}.
=--
+-- {: .proof}
###### Proof
We need to check that for every $X$ the morphism $i^\ast \colon [B, X]\to [A,X]$ is surjective. By retraction, given $f \colon A \to X$, then $r \circ f \colon B \stackrel{r}{\to} A \stackrel{f}{\to} X$ is a preimage.
=--
+-- {: .num_example #TensoringWithUnitOfRingSpectrumIsMonomorphismOfSpectra}
###### Example
For any spectrum $X$ the morphism
$$
X \simeq \mathbb{S} \wedge X \stackrel{e \wedge id}{\longrightarrow}
E \wedge X
$$
is an $E$-monomorphism in the sense of def. \ref{ExactSequences}.
=--
+-- {: .proof}
###### Proof
We need to check that $E \wedge X \stackrel{id \wedge e \wedge id}{\longrightarrow} E \wedge E \wedge X$ is a monomorphism in the sense of def. \ref{ExactSequences}. Observe that this morphism has a [[retraction]] given by $\mu \wedge id$. Therefore it is a monomorphism by
example \ref{HomotopyRetractionsAreMonomorphismsOfSpectra}.
=--
+-- {: .num_remark}
###### Remark/Warning
Consecutive morphisms in an $E$-exact sequence according to def. \ref{ExactSequences} in general need not compose up to homotopy, to the [[zero morphism]]. But this does become true (lemma \ref{ConsecutiveMapsInEAdamsResolutionComposeToZero} below) for sequences of $E$-injective objects, defined below in def. \ref{EInjective}.
=--
+-- {: .num_lemma #CharacterizationOfEpiMonomorphismsOfSpectra}
###### Lemma
1. If $f \colon B\longrightarrow A$ is a monomorphism in the sense of def. \ref{ExactSequences}, then there exists a morphism $g \colon C \longrightarrow A$ such that the [[wedge sum]] morphism is a [[weak homotopy equivalence]]
$$
f \vee g \;\colon\; B \wedge C \stackrel{\simeq}{\longrightarrow} A
\,.
$$
In particular, every morphism in [[Ho(Spectra)]] has an [[extension]] along a monomorphism in this sense.
1. If $f \colon A \longrightarrow B$ is an epimorpimsm in the sense of def. \ref{ExactSequences}, then there exists a homotopy [[section]] $s \colon B\to A$, i.e. $f\circ s\simeq Id$, together with a morphism $g \colon C \to A$ such that the [[wedge sum]] morphism is a [[weak homotopy equivalence]]
$$
s \vee f \colon B\vee C \stackrel{\simeq}{\longrightarrow} A
\,.
$$
=--
+-- {: .proof}
###### Proof
Given a monomorphism $f \colon A \longrightarrow B$, consider the correspondiing [[homotopy cofiber sequence]]
$$
A\stackrel{f}{\longrightarrow} B \stackrel{r}{\longrightarrow} C \stackrel{\delta}{\longrightarrow} \Sigma A \stackrel{-\Sigma f}{\longrightarrow} \Sigma B
\,.
$$
We first observe that the [[connecting homomorphism]] is equivalent to the [[zero morphism]] $\delta \simeq 0$. This follows because by example \ref{HomotopyCofiberSequencesAreExactSequencesOfSpectra} the sequence
$$
[C,X]
\stackrel{\delta^\ast_\bullet}{\longleftarrow}
[\Sigma A, X]
\stackrel{(-\Sigma f)^\ast_\bullet}{\longleftarrow}
[\Sigma B, X]
$$
is an [[exact sequence]] (of homotopy groups) for every $X$, while by example \ref{SuspensionPreservesExactSequencesOfSpectra} the morphism on the right is epi, so that $\delta^\ast_\bullet = 0$.
Now since $B \stackrel{r}{\longrightarrow} C \stackrel{\delta \simeq 0}{\longrightarrow}$ is also a [[homotopy fiber sequence]], the [[pasting law]] identifies $B \simeq C \times A \simeq C \vee A$:
$$
\array{
B \simeq C \times A &\longrightarrow& A &\longrightarrow& 0
\\
\downarrow && \downarrow && \downarrow
\\
C &\longrightarrow& 0 &\longrightarrow& \Sigma A
}
\,.
$$
=--
+-- {: .num_defn #EInjective}
###### Definition
For $E$ a [[ring spectrum]], say that a spectrum $S$ is _$E$-injective_ if for each morphism $A \longrightarrow S$ and each $E$-monomorphism $f \colon A \longrightarrow S$ in the sense of def. \ref{ExactSequences}, there is a [[diagram]] in [[HoSpectra]] of the form
$$
\array{
A &\longrightarrow & S
\\
\downarrow & \nearrow_{\mathrlap{\exists}}
\\
B
}
\,.
$$
=--
+-- {: .num_lemma #EInjectiveSpectraAreRetractsOfFreeEModules}
###### Lemma
A spectrum is $E$-injective in the sense of def. \ref{EInjective}, precisely if it is a [[retract]] in [[HoSpectra]] of a free $E$-modules, hence of $E \wedge X$ for some spectrum $X$.
=--
+-- {: .proof}
###### Proof
In one direction, assume that $S$ is $E$-injective and consider the diagram
$$
\array{
S &\stackrel{id}{\longrightarrow}& S
\\
{}^{\mathllap{e \wedge id}}\downarrow
\\
E \wedge S
}
\,.
$$
By example \ref{TensoringWithUnitOfRingSpectrumIsMonomorphismOfSpectra} here the vertical morphism is an $E$-monomorphism, and so by assumption there is a lift
$$
\array{
S &\stackrel{id}{\longrightarrow}& S
\\
\downarrow & \nearrow
\\
E \wedge S
}
$$
which exhibits $S$ as a retract of $E \wedge S$.
In the other direction, given a retraction $S \stackrel{\overset{r}{\longleftarrow}}{\underset{s}{\longrightarrow}} E \wedge X$ we show that there exist extensions in
$$
\array{
A &\stackrel{g}{\longrightarrow} & S
\\
{}^{\mathllap{f}}\downarrow
\\
B
}
$$
whenever the vertical morphism is an $E$-monomorphism. To see this, complete the extension problem to the following [[commuting diagram]]
$$
\array{
A &&& \stackrel{e \wedge id}{\longrightarrow} &&& E \wedge A
\\
{}^{\mathllap{f}}\downarrow &\searrow^{\mathrlap{g}} & && & {}^{\mathllap{{\mu \wedge id} \atop {\circ id \wedge s g }}}\swarrow & \downarrow^{\mathrlap{id \wedge f}}
\\
&& S &\stackrel{r}{\longleftarrow}& E \wedge X
\\
\downarrow && && && \downarrow
\\
B &&& \stackrel{e \wedge id}{\longrightarrow} &&& E \wedge B
}
\,.
$$
Now, since $f$ is assumed to be an $E$-monomorphism, the morphism $Eid\wedge f$ on the right is a monomorphism in the sense of def. \ref{ExactSequences}, and so by lemma \ref{CharacterizationOfEpiMonomorphismsOfSpectra} there exists an extension $h$ in
$$
\array{
A &&& \stackrel{e \wedge id}{\longrightarrow} &&& E \wedge A
\\
{}^{\mathllap{f}}\downarrow &\searrow^{\mathrlap{g}} & && & {}^{\mathllap{{\mu \wedge id} \atop {\circ id \wedge s g }}}\swarrow & \downarrow^{\mathrlap{id \wedge f}}
\\
&& S &\stackrel{r}{\longleftarrow}& E \wedge X
\\
\downarrow && && &\nwarrow^{\mathrlap{h}}& \downarrow
\\
B &&& \stackrel{e \wedge id}{\longrightarrow} &&& E \wedge B
}
\,.
$$
By composition and commutativity, this gives the required extension of $g$ along $f$.
=--
+-- {: .num_defn #EAdamsResolution}
###### Definition
For $E$ a [[ring spectrum]], then an _$E$-Adams resolution_ of an spectrum $S$ is a long exact sequence, in the sense of def. \ref{ExactSequences}, of the form
$$
0 \longrightarrow S \longrightarrow I_0 \longrightarrow I_1 \longrightarrow I_2 \longrightarrow \cdots
$$
such that each $I_j$ is $E$-injective, def. \ref{EInjective}.
=--
+-- {: .num_lemma #ConsecutiveMapsInEAdamsResolutionComposeToZero}
###### Lemma
Any two consecutive maps in an $E$-Adams resolution, def. \ref{EAdamsResolution}, compose to the [[zero morphism]].
=--
The following lemma says that $E$-Adams resolutions may be [[extension|extended]] along morphisms.
+-- {: .num_lemma}
###### Lemma
For $X \to X_\bullet$ an $E$-Adams resolution, def. \ref{EAdamsResolution}, and for $X \longrightarrow Y$ any morphism, then there exists an $E$-Adams resolution $Y \to J_\bullet$ and a [[commuting diagram]]
$$
\array{
X &\longrightarrow& I_\bullet
\\
\downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g_\bullet}}
\\
Y &\longrightarrow& J_\bullet
}
\,.
$$
=--
There are two $E$-Adams resolutions that we will consider. Following ([Hopkins 99](#Hopkins99)) we call them the "normalized resolution" and the "standard resolution". But beware that what all the traditional literature ([Adams 74](#Adams74), [Bousfield 79](#Bousfield79), [Ravenel 86](#Ravenel86), ...) considers (and somewhat implicitly) is the "normalized" resolution, not the standard resolution. The standard resolution is standard only from the more recent perspective of [[E-∞ geometry]]: it is the [[Amitsur complex]] of the $\mathbb{S}$-algebra $E$, exhibiting the formal dual of the [[Cech nerve]] of $Spec(E) \to Spec(\mathbb{S})$.
+-- {: .num_example #NormalizedEResolution}
###### Example
**(normalized $E$-Adams resolution)**
Let $\overline{E}$ denote the [[homotopy fiber]] of the unit of the ring spectrum $E$, fitting into a [[homotopy fiber sequence]]
$$
\overline{E}
\overset{i}{\longrightarrow}
\mathbb{S}
\overset{e}{\longrightarrow}
E
\longrightarrow
\Sigma \overline{E}
\,.
$$
For $X$ a spectrum, its **normalized $E$-Adams resolution** is the top row of
$$
\array{
X
&\overset{(e,id)}{\longrightarrow}&
E \wedge X
&& \longrightarrow &&
E \wedge (\Sigma \overline{E}) \wedge X
&& \longrightarrow &&
E \wedge (\Sigma \overline{E}) \wedge (\Sigma \overline{E}) \wedge X
&& \longrightarrow &&
\cdots
\\
&& & \searrow && \nearrow_{\mathrlap{(e,id)}}
&& \searrow && \nearrow_{\mathrlap{(e,id)}}
\\
&& && (\Sigma \overline{E})\wedge X
&&&& (\Sigma \overline{E}) \wedge \overline{E} \wedge X
}
$$
=--
(e.g. [Hopkins 99, corollary 5.3](#Hopkins99)).
+-- {: .num_remark}
###### Remark
The notation for $\overline{E}$ in def. \ref{NormalizedEResolution} follows ([Bousfield 79, section 5](#Bousfield79)). In ([Hopkins 99, proof of corollary 5.3](#Hopkins99)) the same notation is used not for the homotopy fiber of $\mathbb{S} \overset{e}{\to} E$ but for its homotopy cofiber. While our notation makes plenty of "$\Sigma$"s appear in the above resolution, the advantage is that in the induced inverse sequence of a normalized resolution below in example \ref{NormalizedEResolutionAssociatedSequence} these all drop out and we are left with the original form of the expressions as considered by ([Adams 74](#Adams74)) and followed in most of the literature.
=--
+-- {: .num_example #StandardEResolution}
###### Example
**(standard $E$-Adams resolution)**
Any [[ring spectrum]] $E$ gives rise to an [[augmentation|augmented]] [[cosimplicial object|cosimplicial]] [[spectrum]] (its _[[bar construction]]_)
$$
\mathbb{S} \longrightarrow E
\stackrel{\longrightarrow}{\longrightarrow}
E \wedge E
\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
E \wedge E \wedge E
\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}}
\cdots
$$
whose
* coface maps are given by inserting the unit $\mathbb{S} \stackrel{e}{\to} E$
$$
d^i
\colon
E^{\wedge^n}
\simeq
E^{\wedge^i} \wedge \mathbb{S} \wedge E^{\wedge^{n-i}}
\stackrel{id \wedge e \wedge id}{\longrightarrow}
E^{\wedge^{n+1}}
\,;
$$
* codegeneracy maps are given by the product operation $\mu \colon E \wedge E \to E$
$$
s^j
\colon
E^{\wedge^{n+1}}
\simeq
E^{\wedge^{i}} \wedge E \wedge E \wedge E^{\wedge n-1-i}
\stackrel{id \wedge \mu \wedge id}{\longrightarrow}
E^n
\,.
$$
The corresponding [[Amitsur complex]] is given by forming alternating sums of the coface maps
$$
\mathbb{S}
\stackrel{e}{\longrightarrow}
E
\stackrel{e\wedge id - id \wedge e}{\longrightarrow}
E \wedge E
\stackrel{e \wedge id - id\wedge e \wedge id + id \wedge e}{\longrightarrow}
E \wedge E \wedge E
\stackrel{}{\longrightarrow}
\cdots
\,.
$$
Given any [[spectrum]] $X$, then forming the [[smash product]] $(-)\wedge X$ with this sequence yields a sequence of the form
$$
X
\stackrel{}{\longrightarrow}
E \wedge X
\stackrel{}{\longrightarrow}
E \wedge E \wedge X
\stackrel{}{\longrightarrow}
E \wedge E \wedge E \wedge X
\stackrel{}{\longrightarrow}
\cdots
\,.
$$
This is called the **standard $E$-Adams resolution** of $X$.
=--
(e.g. [Hopkins 99, def. 5.4](#Hopkins99)).
+-- {: .num_prop #StandardAdamsResolutionIsIndeedAdamsResolution}
###### Proposition
The standard resolution of example \ref{StandardEResolution}
is indeed an $E$-Adams resolution of $X$ in the sense of def. \ref{EAdamsResolution}.
=--
+-- {: .proof}
###### Proof
As generally for [[bar resolutions]], one checks that the alternating sum of the codegeneracy maps constitute homotopy operators that give contracting homotopies when commuted with the alternating sum of the coface maps. This gives that the sequence is $E$-exact.
Moreover, the terms in the sequence are all $E$-injective by lemma \ref{EInjectiveSpectraAreRetractsOfFreeEModules}.
=--
#### $E$-Adams towers
+-- {: .num_defn #EAdamsTower}
###### Definition
An _$E$-Adams tower_ of a spectrum $X$ is a [[commuting diagram]] in the [[stable homotopy category]] of the form
$$
\array{
&& \vdots
\\
&& \downarrow^{\mathrlap{p_2}}
\\
&& X_2 &\stackrel{\kappa_2}{\longrightarrow}& \Omega^2 I_3
\\
&\nearrow& \downarrow^{\mathrlap{p_1}}
\\
&& X_1 &\stackrel{\kappa_1}{\longrightarrow}& \Omega I_2
\\
&\nearrow& \downarrow^{\mathrlap{p_0}}
\\
X
&\underset{}{\longrightarrow}&
X_0 = I_0
&\stackrel{\kappa_0}{\longrightarrow}&
I_1
}
$$
such that
1. each hook is a [[homotopy fiber sequence]];
1. the [[composition]] of the $(\Sigma \dashv \Omega)$-[[adjuncts]] of $\Sigma_{p_{n-1}}$ with $\Sigma^n \kappa_n$
$$
i_{n+1} \;\colon\; I_n \stackrel{\widetilde {\Sigma p_{n-1}}}{\longrightarrow}
\Sigma^n X_n \stackrel{\Sigma^{n}\kappa_n}{\longrightarrow} I_{n+1}
$$
constitute an $E$-Adams resolution of $X$, def. \ref{EAdamsResolution}:
$$
0 \to X \stackrel{i_0}{\to} I_0 \stackrel{i_1}{\to} I_1 \stackrel{}{\to} \cdots
\,.
$$
Call this the **associated $E$-Adams resolution** of the $E$-Adams tower.
=--
([Hopkins 99, def. 4.10](#Hopkins99))
The following is the main statement of the above little theory of $E$-injective spectra.
+-- {: .num_prop #RelationBetweenEAdamsTowersAndEAdamsResolutions}
###### Proposition
Every $E$-Adams resolution $X \to I_\bullet$ (def. \ref{EAdamsResolution}) induces an $E$-Adams tower, def. \ref{EAdamsTower} of which it is the associated $E$-Adams resolution.
=--
+-- {: .proof}
###### Proof idea
Given an $E$-Adams resolution
$$
X \overset{i_0}{\longrightarrow} I_0 \overset{i_1}{\longrightarrow} I_1 \longrightarrow \cdots
$$
consider the induced diagram
$$
\array{
&& && C_1 && && && && C_3
\\
&& & {}^{\mathllap{\rho_1}}\nearrow && \searrow^{\mathrlap{\sigma_1}}
&& && && {}^{\mathllap{\rho_3}}\nearrow
\\
&&
I_0
&&
\underset{i_1}{\longrightarrow}
&&
I_1
&&
\overset{i_2}{\longrightarrow}
&&
I_2
&&
\underset{i_3}{\longrightarrow}
&
\cdots
\\
& {}^{\mathllap{\sigma_0 \coloneqq i_0}}\nearrow
&& && && {}_{\mathllap{\rho_2}}\searrow
&& \nearrow_{\mathrlap{\sigma_2}}
\\
C_0 \coloneqq X && && && && C_2
}
$$
constructed [[induction|inductively]] as follows:
To start with, $\rho_1$ is the [[homotopy cofiber]] of $i_0$, and $\sigma_1$ is the morphism universally induced from this by the fact that $i_1 \circ i_0 \simeq 0$, by lemma \ref{ConsecutiveMapsInEAdamsResolutionComposeToZero}. Observe that $\sigma_1$ is an $E$-monomorphism and $\rho_1$ is an $E$-epimorphism in the sense of def. \ref{ExactSequences}.
Then assume that an $E$-epi/mono factorization
$$
i_n \colon I_{n_1} \overset{\rho_n}{\longrightarrow} C_n \overset{\sigma_n}{\to} I_n
$$
has been constructed. Let now $\rho_{n+1}$ be its homotopy cofiber. Since $\rho_{n}$ is $E$-epi, the equivalence $0 \simeq i_{n+1} \circ i_n = i_{n+1}\circ \sigma_n \circ \rho_n$ from lemma \ref{ConsecutiveMapsInEAdamsResolutionComposeToZero} implies that already $i_{n+1} \circ \sigma_n \simeq 0$. With this, the universal property of the homotopy cofiber induces a morphism $\sigma_{n+1}\colon C_{n+1}\to I_{n+1}$. As before, $\rho_{n+1}$ is $E$-epi and $\sigma_{n+1}$ is $E$-mono, and so the induction proceeds.
Using this, we now construct an $E$-Adams tower as follows (...).
=--
There is another tower associated with an $E$-Adams resolutions:
+-- {: .num_defn #AssociatedInverseSequence}
###### Definition
Given an $E$-Adams resolutions $X \to I_\bullet$ (def. \ref{EAdamsResolution}), its **associated inverse sequence** is
$$
\array{
X
=
C_0
&\stackrel{\gamma_0}{\longleftarrow}&
\Sigma^{-1} C_1
&\stackrel{\gamma_1}{\longleftarrow}&
\Sigma^{-2} C_2
&\longleftarrow&
\cdots
\\
\downarrow && \downarrow && \downarrow
\\
I_0 && \Sigma^{-1} I_1 && \Sigma^{-2} I_2
}
$$
with the $C_i$ as in the proof of prop. \ref{RelationBetweenEAdamsTowersAndEAdamsResolutions} and
$\gamma_n \coloneqq \Sigma^{-} hofib(\sigma_n)$.
=--
+-- {: .num_example #NormalizedEResolutionAssociatedSequence}
###### Example
Let $X \to I_\bullet = (E \wedge (\Sigma \overline{E})^{\wedge^{\bullet-1}}\wedge E)$ be a _normalized $E$-Adams resolution_ according to example \ref{NormalizedEResolution}. Then its associated inverse sequence according to def. \ref{AssociatedInverseSequence} is the sequence from def. \ref{EAdamsSpectralSequence}
$$
\array{
X
&\stackrel{\gamma_0}{\longleftarrow}&
\overline{E} \wedge X
&\stackrel{\gamma_1}{\longleftarrow}&
\overline{E} \wedge \overline{E} \wedge X
&\longleftarrow&
\cdots
\\
\downarrow && \downarrow && \downarrow
\\
E \wedge X
&&
\Sigma^{-1}(E \wedge (\Sigma \overline{E}) \wedge X)
&&
\Sigma^{-2}(E \wedge (\Sigma\overline{E}) \wedge (\Sigma \overline{E}) \wedge X
\\
&& \simeq && \simeq
\\
&&
E \wedge \overline{E} \wedge X
&&
E \wedge \overline{E} \wedge \overline{E} \wedge X
}
$$
This is the tower of spectra considered in the original texts ([Adams 74, p. 318](#Adams74)) and ([Bousfield 79, p. 271](#Bousfield79)).
=--
+-- {: .num_remark #TerminologyAssociatedInverseSequence}
###### Remark
In ([Ravenel 84, p. 356](#Ravenel84)) it is the associated inverse sequence as in example \ref{NormalizedEResolutionAssociatedSequence} that is called the "Adams tower", while in ([Ravenel 86, def. 2.21](#Ravenel86)) this is called an "$E$-Adams resolution". We instead follow ([Hopkins 99](#Hopkins99)) in using "$E$-Adams resolution" for "$E$-injective resolution" as in def. \ref{EAdamsResolution}, "$E$-Adams tower" for def. \ref{EAdamsTower} and follow ([Aramian](#Aramian)) in saying "associated inverse sequence" for the above.
=--
#### $E$-Adams spectral sequences
+-- {: .num_defn #EAdamsSpectralSequence}
###### Definition
Given [[spectra]] $X$ and $Y$, and given an $E$-Adams resolution of $X$, def. \ref{EAdamsResolution}, or equivalently (by prop. \ref{RelationBetweenEAdamsTowersAndEAdamsResolutions}) an $E$-Adams tower over $X$, def. \ref{EAdamsTower},
$$
\array{
&& \vdots
\\
&& \downarrow^{\mathrlap{p_2}}
\\
&& X_2 &\stackrel{\kappa_2}{\longrightarrow}& \Omega^2 I_3
\\
&\nearrow& \downarrow^{\mathrlap{p_1}}
\\
&& X_1 &\stackrel{\kappa_1}{\longrightarrow}& \Omega I_2
\\
&\nearrow& \downarrow^{\mathrlap{p_0}}
\\
X
&\underset{}{\longrightarrow}&
X_0 = I_0
&\stackrel{\kappa_0}{\longrightarrow}&
I_1
}
$$
then the corresponding **$E$-Adams spectral sequence** for the [[mapping spectrum]] $[Y,X]$ is the associated [[spectral sequence of a tower of fibrations]] of the image of that [[tower of fibrations]] under the [[mapping spectrum]] operation $[Y,-]$:
$$
\array{
&& \vdots
\\
&& \downarrow^{\mathrlap{[Y,p_2]}}
\\
&& [Y,X_2] &\stackrel{[Y,\kappa_2]}{\longrightarrow}& [Y,\Omega^2 I_3]
\\
&\nearrow& \downarrow^{\mathrlap{[Y,p_1]}}
\\
&& [Y,X_1] &\stackrel{[Y,\kappa_1]}{\longrightarrow}& [Y,\Omega I_2]
\\
&\nearrow& \downarrow^{\mathrlap{[Y,p_0]}}
\\
[Y,X]
&\underset{}{\longrightarrow}&
[Y,X_0] = [Y,I_0]
&\stackrel{[Y,\kappa_0]}{\longrightarrow}&
[Y,I_1]
}
\,.
$$
More in detail, the associated [[exact couple]] of the tower is
$$
\array{
\mathcal{D} && \stackrel{p}{\longrightarrow} && \mathcal{D}
\\
& {}_{\mathllap{\partial}}\nwarrow && \swarrow_{\mathrlap{\kappa}}
\\
&& \mathcal{E}
}
$$
with
$$
\mathcal{D} \coloneqq
\oplus_{s,t} \mathcal{D}^{s,t}
\coloneqq
\oplus_{s,t} \pi_{t-s}([Y,X_s])
$$
$$
\mathcal{E} \coloneqq
\oplus_{s,t} \mathcal{E}^{s+1,t}
\coloneqq
\oplus_{s,t} \pi_{t-s}([Y,\Omega^s I_{s+1}])
$$
and
$$
p
\colon
\pi_{t-s}([Y,X_{s+1}])
\stackrel{\pi_{t-s}([Y,p_s])}{\longrightarrow}
X_{t-s}([Y,X_s])
$$
$$
\kappa
\colon
\pi_{t-s}([Y,X_s])
\stackrel{\pi_{t-s}([Y,\kappa_s])}{\longrightarrow}
\pi_{t-s}([Y,\Omega^s I_{s+1}])
$$
$$
\partial
\colon
\pi_{t-s}([Y,\Omega^s I_{s+1}])
\stackrel{\pi_{t-s}([Y,\partial_s])}{\longrightarrow}
\pi_{t-s}([Y,\Sigma X_{s+1}])
\,.
$$
The _$E$-Adams spectral sequence_ of the $E$-Adams tower is the [spectral sequence induced](exact+couple#SpectralSequencesFromExactCouples) by this [[exact couple]].
=--
+-- {: .num_prop #UniquenessOfEAdamsSpectralSequence}
###### Proposition
Given two $E$-Adams towers, def. \ref{EAdamsTower}, for some $X$, then the corresponding two $E$-Adams spectral sequences, def. \ref{EAdamsSpectralSequence}, are [[isomorphism|isomorphic]] from the $\mathcal{E}_2$-page on.
=--
+-- {: .num_prop #TowerSpectralSequencesOfAdamsTowerAndInverseSequenceCoincide}
###### Proposition
Given an $E$-Adams resolution (def. \ref{EAdamsResolution}), there is an [[isomorphism]] of spectral sequences between
1. the [[tower spectral sequence]] of its associated $E$-Adams tower (def. \ref{EAdamsTower}), i.e. the spectral sequence of def. \ref{EAdamsSpectralSequence};
1. the [[tower spectral sequence]] of its associated inverse sequence (def. \ref{AssociatedInverseSequence}).
=--
+-- {: .num_remark}
###### Remark
Hence both of these construction are to be called the $E$-Adams spectral sequence. It is in fact the second construction -- for the case of the normalized resolution as in example \ref{NormalizedEResolutionAssociatedSequence} -- that is considered in the original sources ([Adams 74, p. 318](#Adams74), [Bousfield 79, p. 271](#Bousfield79)).
=--
### As derived descent in $E_\infty$-geometry
{#DefinitionInHigherAlgebra}
It turns out that the [[E-nilpotent completion]] $X^\wedge_E$ according to [[Aldridge Bousfield|Bousfield]]'s original definition \ref{ENilpotentCompletion} -- to which the $E$-Adams spectrum seequence converges under good conditions (theorem \ref{EAdamsConvergenceForCorepi0EBeingZLocalizedAtPrimes}, theorem \ref{EAdamsConvergenceForCorepi0EBeingCyclicGroup}) -- is equivalently the homotopy limit over the tower of [[totalizations]] of the canonical [[cosimplicial object|cosimplicial]] spectrum $E^\bullet \wedge X$ ([prop.](nilpotent+completion+of+spectra#ENilpotentCompletionIsHolimOverTotTower)).
This means that the $E$-[[Adams spectral sequence]] may equivalently be regarded as computing [[descent]] of [[quasicoherent infinity-stacks]] in [[E-infinity geometry]] along the canonical morphisms $Spec(E)\longrightarrow $ [[Spec(S)]].
This perspective seems to originate in ([Hopkins 99, remark 5.5 (ii)](#Hopkins99)) and is greatly expanded on in
([Lurie 10, lectures 8, 9](#Lurie10)). Exposition of the basic idea is in [Mathew 12](#Mathew12) and with more details in ([Wilson 13](#Wilson13)). The first published proof of the equivalence of this new perspective to the Adams-Bousfield theory is ([Mathew-Naumann-Noel 15, prop. 2.14](#MathewNaumannNoel15)).
We give a survey here.
> under construction
First we recall
* [Spectral sequences computing homotopy groups of filtered objects](#SpectralSequencesForHomotopyGroupsOfFilteredObjects)
for the general case of filtered objects in suitable [[stable (∞,1)-categories]]. Then we consider the specialization of that to the
* [Homotopy groups of cosimplicial totalizations filtered by coskeleta](#HomotopyGroupsOfOfTotalizationsFilteredByCoskeleta).
Finally we consider specifically the examples of such given by
* [Canonical cosimplicial resolutions of E-∞ algebras](#CanonicalCosimplicialResolutionOfEInfinityAlgebras).
In conclusion this yields for each suitable [[E-∞ algebra]] $E$ over $S$ and $S$-[[∞-module]] $X$ a [[spectral sequence]] converging to the [[homotopy groups]] of the $E$-[[Bousfield localization of spectra|localization]] of $X$, and this is
* [The E-Adams-Novikov spectral sequence](#TheEAdamsSpectralSequence).
The quick idea is this: Given an $S$ -module $X$, regard it as a quasicoherent sheaf on $Spec(S)$. Choose a map $Spec(E) \to Spec(S)$. This is a cover of its [[1-image]]
$Spec(E) \to Spec(S)^\wedge_{Spec(E)}$, which is the derived [[formal completion]] of $Spec(S)$ at the image of $Spec(E)$. Restrict attention then to the restriction of $X$ to that formal completion $X^\wedge_{Spec(E)}$. (So if $Spec(E) \to Spec(S)$ was already an atlas, hence was already complete, we stick with the original $X$). Then pull back $S$ to the [[Cech nerve]] of the cover $Spec(E) \to Spec(S)^\wedge_{Spec(E)}$. The realization of this Cech nerve reproduces the completed image, and hence the canonical filtration on the Cech nerve gives a filtration spectral sequence for $X^\wedge_{Spec(E)}$.
#### Spectral sequences computing homotopy groups of filtered objects
{#SpectralSequencesForHomotopyGroupsOfFilteredObjects}
Let thoughout $\mathcal{C}$ be a [[stable (∞,1)-category]] equipped with a [[t-structure]] such that its [[heart of a stable (∞,1)-category|heart]] is an [[abelian category]].
+-- {: .num_example}
###### Example
For instance
* $\mathcal{C} = Spec$ the [[stable (∞,1)-category of spectra]].
=--
+-- {: .num_defn #GeneralizedFilteredObject}
###### Definition
A _generalized [[filtered object]]_ in $\mathcal{C}$ is
simply a sequential diagram $X \colon (\mathbb{Z}, \lt) \to \mathcal{C}$
$$
\cdots
X_{n+1} \to X_n \to X_{n-1} \to \cdots
\,.
$$
Or rather, the object being filtered is the [[homotopy limit]]
$$
X \coloneqq \underset{\leftarrow}{\lim}_n X_n
$$
and the sequential diagram exhibits the filtering.
=--
This appears as ([[Higher Algebra|Higher Algebra, def. 1.2.2.9]]).
+-- {: .num_defn #ExactCoupleForFilteredObject}
###### Definition
For a generalized filtered object $X_\bullet$, def. \ref{GeneralizedFilteredObject}, write
$$
F_n \coloneqq fib(X_n \to X_{n+1})
$$
for the [[homotopy fiber]] of the $n$th structure map, for all $n \in \mathbb{Z}$, and define an [[exact couple]]
$$
\array{
&& \pi_\bullet(F_\bullet)
\\
& \swarrow && \nwarrow
\\
\pi_\bullet(X_\bullet)
&&
\stackrel{}{\longrightarrow}
&&
\pi_\bullet(X_\bullet)
}
$$
where the maps are given by the [[long exact sequences of homotopy groups]]
$$
\cdots
\to
\pi_\bullet(X_{n+1})
\to
\pi_\bullet(F_n) \to \pi_\bullet(X_n) \to \pi_\bullet(X_{n+1}) \to \pi_{\bullet+1}(F_n) \to \cdots
$$
=--
We now have the [[spectral sequence of a filtered stable homotopy type]].
+-- {: .num_prop #FiltrationSpectralSequence}
###### Proposition
Let $\mathcal{C}$ be a [[stable (∞,1)-category]] equipped with a [[t-structure]] such that its [[heart of a stable (∞,1)-category|heart]] is an [[abelian category]].
If $\mathcal{C}$ has [[sequential limits]] and if $X_n \simeq 0$ for all $n \gt n_0$ then the [[spectral sequence]] induced by the [[exact couple]]
of def. \ref{ExactCoupleForFilteredObject} converges to the [[homotopy groups]] of the [[homotopy limit]] $\underset{\leftarrow}{\lim}_n X_n$ of the generalized filtered object:
$$
E^{p,q}_1
=
\pi_{p+q} F_{p-1}
\Rightarrow
\pi_{p+q} (\underset{\leftarrow}{\lim} X_\bullet)
$$
=--
This is due to ([[Higher Algebra|Higher Algebra, prop. 1.2.2.14]]). Review is in ([Wilson 13, theorem 1.2.1](#Wilson13)).
For the traditional statement in the [[category of chain complexes]] see at _[[spectral sequence of a filtered complex]]_.
#### Homotopy groups of cosimplicial totalizations filtered by coskeleta
{#HomotopyGroupsOfOfTotalizationsFilteredByCoskeleta}
+-- {: .num_defn #FiltrationOfTotalizationByTotalizationOfCoskeleta}
###### Definition
Given an [[cosimplicial object]]
$$
Y \;\colon\; \Delta \longrightarrow \mathcal{C}
$$
its [[totalization]] $Tot Y \simeq \underset{\leftarrow}{\lim}_n Y_n$
is filtered, def. \ref{GeneralizedFilteredObject}, by the
totalizations of its [[coskeleton|coskeleta]]
$$
Tot Y
\to
\cdots
\to
Tot (cosk_2 Y)
\to
Tot (cosk_1 Y)
\to
Tot (cosk_0 Y)
\to 0
\,.
$$
=--
+-- {: .num_prop #E2PageByMooreComplex}
###### Proposition
The filtration spectral sequence, prop. \ref{FiltrationSpectralSequence},
applied to the filtration of a [[totalization]] by [[coskeleton|coskeleta]] as in def. \ref{FiltrationOfTotalizationByTotalizationOfCoskeleta}, has as $E_2$-term the [[cohomology groups]] of the [[Moore complex]] associated with the cosimplicial object
$$
E_2^{p,q}
=
H^p(\pi_q(Tot (cosk_\bullet(Y))))
\Rightarrow
\pi_{p-q} Tot(Y)
\,.
$$
=--
This is ([[Higher Algebra|Higher Algebra, remark 1.2.4.4]]). Review is around ([Wilson 13, theorem 1.2.4](#Wilson13)).
#### Canonical cosimplicial resolution of $E_\infty$-algebras
{#CanonicalCosimplicialResolutionOfEInfinityAlgebras}
We discuss now the special case of coskeletally filtered
totalizations coming from the canonical cosimplicial objects
induced from [[E-∞ algebras]] ("[[Amitsur complexes]]", "[[Sweedler corings]]").
In this form this appears as ([Lurie 10, theorem 2](#Lurie10)). A review is in ([Wilson 13, 1.3](#Wilson13)). For the analog of this in the traditional formulation see ([Ravenel 86, ch. 3, prop. 3.1.2](#Ravenel86)).
+-- {: .num_defn}
###### Definition
Let $S$ be an [[E-∞ ring]] and let $E$ be an [[E-∞ algebra]] over $S$, hence an [[E-∞ ring]] equipped with a [[homomorphism]]
$$
S \longrightarrow E
\,.
$$
The _canonical [[cosimplicial object]]_
associated to this (the "$\infty$-[[Sweedler coring]]" or "[[Amitsur complex]]") is that given by the iterated [[smash product]]/[[tensor product]] over $S$:
$$
E^{\wedge^{\bullet+1}_S} \;\colon\; \Delta \to \mathcal{C}
\,.
$$
More generally, for $X$ an $S$-[[∞-module]], the canonical [[cosimplicial object]] is
$$
E^{\wedge^{\bullet+1}_S}\wedge_S X \;\colon\; \Delta \to \mathcal{C}
\,.
$$
=--
+-- {: .num_prop #FlatnessCondition}
###### Proposition
If $E$ is such that the self-[[generalized homology]]
$E_\bullet(E) \coloneqq \pi_\bullet(E \wedge_S E)$ (the dual $E$-[[Steenrod operations]]) is such that as a [[module]] over $E_\bullet \coloneqq \pi_\bullet(E)$ it is a [[flat module]], then there is a [[natural equivalence]]
$$
\pi_\bullet
\left(
E^{\wedge^{n+1}_S}
\wedge_S
X
\right)
\simeq
E_\bullet(E^{\wedge^n_S})
\otimes_{E_\bullet}
E_\bullet(X)
\,.
$$
=--
+-- {: .num_remark}
###### Remark
This makes $(E_\bullet, E_\bullet(E))$ be the [[Hopf algebroid]]
formed by the $E$-[[Steenrod algebra]]. See there for more on this.
=--
+-- {: .num_example}
###### Example
The condition in prop. \ref{FlatnessCondition} is
satisfied for
* $E = H \mathbb{F}_p$ an [[Eilenberg-MacLane spectrum]] with $mod\;p$ [[coefficients]];
* $E = B P$ the [[Brown-Peterson spectrum]];
* $E = MU$ the [[complex cobordism cohomology theory|complex cobordism spectrum]].
It is NOT satisfied for
* $E = H \mathbb{Z}$ the [[Eilenberg-MacLane spectrum]] for [[integers|integer]] [[coefficients]];
* $E = M S U$.
=--
+-- {: .num_remark #ExtGroupsByMooreComplex}
###### Remark
Under good conditions (...), $\pi_\bullet$ of the canonical [[cosimplicial object]] provides a [[resolution]] of [[comodule]] [[tensor product]] and hence computes the [[Ext]]-groups over the [[Hopf algebroid]]:
$$
H^p(\pi_q(Tot(cosk_\bullet(E^{\wedge^{\bullet+1}_S } \wedge_S X))))
\simeq
Ext^p_{E_\bullet(E)}(\Sigma^q E_\bullet, E_\bullet(X))
\,.
$$
(...)
=--
Here the homotopy groups are expressed by [[Ext]]-groups using a [[universal coefficient theorem]]
[for generalized cohomology](universal+coefficient+theorem#ForGeneralizedCohomology) ([Adams 74, III.13](#Adams74)).
(e.g. [Wilson 13, theorem 1.3.5](#Wilson13), based on [Bousfield 79](Bousfield+localization+of+spectra#Bousfield79))
+-- {: .num_remark #CanonicalMapFromELocalizationToTotalization}
###### Remark
There is a canonical map
$$
L_E X \stackrel{}{\longrightarrow} \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X)
$$
from the $E$-[[Bousfield localization of spectra]] of $X$ into the [[totalization]].
=--
#### The $E$-Adams-Novikov spectral sequence
{#TheEAdamsSpectralSequence}
Summing this up yields the general $E$-Adams(-Novikov) spectral sequence
+-- {: .num_cor}
###### Corollary
Let $E$ a [[connective spectrum|connective]] [[E-∞ ring]]
that satisfies the conditions of prop. \ref{SufficientConditionsForTotalizationToBeELocalization}.
Then by prop. \ref{FiltrationSpectralSequence} and prop. \ref{SufficientConditionsForTotalizationToBeELocalization} there is a strongly convergent multiplicative
[[spectral sequence]]
$$
E^{p,q}_\bullet \Rightarrow \pi_{q-p} L_{c \pi_0 E} X
$$
converging to the [[homotopy groups]] of the $ c \pi_0(E)$-[[Bousfield localization of spectra|localization]] of $X$. If moreover the dual $E$-[[Steenrod algebra]] $E_\bullet(E)$ is [[flat module|flat]] as a [[module]] over $E_\bullet$, then, by prop. \ref{E2PageByMooreComplex} and remark \ref{ExtGroupsByMooreComplex}, the $E_2$-term of this spectral sequence is given by the [[Ext]]-groups over the $E$-[[Steenrod algebra|Steenrod]] [[Hopf algebroid]].
$$
E^{p,q}_\bullet
=
Ext^p_{E_\bullet(E)}(\Sigma^q E_\bullet, E_\bullet X)
\,.
$$
=--
## Related concepts
* [[d-invariant]], [[e-invariant]], [[f-invariant]]
* [[change of rings theorem]]
* [[Adams-Novikov spectral sequence]]
* [[Bousfield-Kan spectral sequence]]
* [[May spectral sequence]], [[chromatic spectral sequence]]
* [[Steenrod algebra]]
* [[descent spectral sequence]]
* [[equivariant Adams spectral sequence]]
[[!include Lurie spectral sequences -- table]]
## References
### General
The original sources are
* {#Adams58} [[Frank Adams]], _On the structure and applications of the Steenrod algebra_, Comm. Math. Helv. 32 (1958), 180–214.
* {#Novikov67} [[Sergei Novikov]], _The methods of algebraic topology from the viewpoint of cobordism theories_, Izv. Akad. Nauk. SSSR. Ser. Mat. 31 (1967), 855–951 (Russian). ([[Novikov67.pdf:file]])
* {#Adams74} [[John Frank Adams]], _[[Stable homotopy and generalized homology]]_, Chicago Lectures in mathematics, 1974
based in parts on
* {#Adams69} [[John Frank Adams]], _Lectures on generalised cohomology_, in [[Peter Hilton]] (ed.) _Category Theory, Homology Theory and Their Applications III_, volume 99 of Lecture Notes in Mathematics (1969), Springer-Verlag Berlin-Heidelberg-New York.
The [[spectrum]]-level discussion of the ASS goes back to around
* R. M. F. Moss, _On the composition pairing of Adams spectral sequences_, Proceedings of the London Mathematical Society 3.1 (1968): 179-192.
The general convergence theorems are due to
* {#Bousfield79} [[Aldridge Bousfield]], _The localization of spectra with respect to homology_, Topology 18 (1979), no. 4, 257--281. ([pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/bousfield-topology-1979.pdf))
* {#Ravenel84} [[Douglas Ravenel]], _Localization with respect to certain periodic homology theories_, American Journal of Mathematics, Vol. 106, No. 2, (Apr., 1984), pp. 351-414 ([pdf](https://www.math.rochester.edu/people/faculty/doug/mypapers/loc.pdf))
based on
* {#BousfieldKan72} [[Aldridge Bousfield]], [[Daniel Kan]], _The core of a ring_, Journal of Pure and Applied Algebra, Volume 2, Issue 1, April 1972, Pages 73-81
A streamlined presentation of Adams resolutions and their invariance properties, close in spirit to the theory of injective resolutions in [[homological algebra]] was given in
* {#Miller81} [[Haynes Miller]], _On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space_, J. Pure Appl. Algebra 20 (1981) ([pdf](http://math.mit.edu/~hrm/papers/miller-relations-between-adams-spectral-sequences.pdf))
further highlighted in
* {#Hopkins99} [[Mike Hopkins]], section 5 of _[[Complex oriented cohomology theories and the language of stacks]]_, course notes 1999 ([[HopkinsComplexOrientedCohomology.pdf:file]])
and worked out in some more detail in
* {#Aramian} [[Nersés Aramian]], _The Adams spectral sequence_ ([[AramianANSS.pdf:file]])
For full details of some of the steps involved see also ([Schwede 12](#Schwede12)).
Reviews of the traditional theory:
* {#Ravenel78} [[Douglas Ravenel]], _A Novice's guide to the Adams-Novikov spectral sequence_, in: [[Michael Barratt]], [[Mark Mahowald]] (eds.) _Geometric Applications of Homotopy Theory II_. Lecture Notes in Mathematics, vol 658, Springer 1978 ([doi:10.1007/BFb0068728](https://doi.org/10.1007/BFb0068728), [pdf](https://people.math.rochester.edu/faculty/doug/mypapers/Novice.pdf))
* {#Ravenel86} [[Doug Ravenel]], _[[Complex cobordism and stable homotopy groups of spheres]]_, 1986 onwards
* {#Kochmann96} [[Stanley Kochman]], section 3.6 of: _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996
* {#McCleary01} [[John McCleary]], chapter 9 of: _[A user's guide to spectral sequences](https://pages.vassar.edu/mccleary/books/users-guide-to-spectral-sequences/)_, Cambridge University Press, 2001
* {#Goerss2007} [[Paul Goerss]], _The Adams-Novikov spectral sequence and the Homotopy Groups of Spheres_, lecture notes 2007 ([pdf](http://www.math.northwestern.edu/~pgoerss/papers/stras1.pdf))
* [[Alan Hatcher]], _[Spectral sequences in algebraic topology](http://www.math.cornell.edu/~hatcher/SSAT/SSATpage.html)_ _II: The Adams spectral sequence_ ([pdf](http://www.math.cornell.edu/~hatcher/SSAT/SSch2.pdf))
* {#Bruner09} [[Robert Bruner]], _An Adams spectral sequence primer_, 2009 ([pdf](http://www.math.wayne.edu/~rrb/papers/adams.pdf))
* {#Schwede12} [[Stefan Schwede]], chapter II, section 10.3 of _[[Symmetric spectra]]_, 2012
* {#Rognes12} [[John Rognes]], _The Adams spectral sequence_ (following [Bruner 09](#Bruner09)), 2012 ([pdf](http://folk.uio.no/rognes/papers/notes.050612.pdf))
* _[[Introduction to the Adams Spectral Sequence]]_
* [[Agnès Beaudry]], [[Jonathan A. Campbell]], §4 in: *A Guide for Computing Stable Homotopy Groups*, in *Topology and Quantum Theory in Interaction*, Contemporary Mathematics **718**, Amer. Math. Soc. (2018) [[arXiv:1801.07530](https://arxiv.org/abs/1801.07530), [ams:conm/718](https://bookstore.ams.org/view?ProductCode=CONM/718)]
> (with an eye towards [[functorial quantum field theory]])
The modern point of view in terms of derived $E_\infty$-descent is hinted at in ([Hopkins 99, remark 5.5 (ii)](#Hopkins99)) and is consistently adopted in
* {#Lurie10} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_ (2010)
* lecture 8, _The Adams Spectral Sequence_ ([pdf] (http://www.math.harvard.edu/~lurie/252xnotes/Lecture8.pdf))
* lecture 9, _The Adams Spectral Sequence for $MU$_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture9.pdf))
* lecture 10, _The proof of Quillen's theorem_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture10.pdf))
* lecture 30, _Localizations and the Adams-Novikov Spectral Sequence_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture30.pdf))
based on
* {#Lurie} [[Jacob Lurie]], _[[Higher Algebra]]_
This is nicely surveyed in
* {#Mathew12} [[Akhil Mathew]], _[The Adams spectral sequence as derived descent, and chromotopy](https://amathew.wordpress.com/2012/09/21/3844/)_, 2012
* {#Wilson13} [[Dylan Wilson]], _Spectral Sequences from Sequences of Spectra: Towards the Spectrum of the Category of Spectra_, lecture at [2013 Pre-Talbot Seminar](https://www.hiroleetanaka.com/pretalbot2013/), March 2013 ([[DylanWilsonOnANSS.pdf:file]])
The actual proof that this new perspecive is equivalent the original Adams-Bousfield theory appears as prop. 2.14 in
* {#MathewNaumannNoel15} [[Akhil Mathew]], [[Niko Naumann]], [[Justin Noel]], _Nilpotence and descent in equivariant stable homotopy theory_, Advances in Mathematics Volume 305, 10 January 2017, Pages 994-1084 ([arXiv:1507.06869](http://arxiv.org/abs/1507.06869), [doi:10.1016/j.aim.2016.09.027](https://doi.org/10.1016/j.aim.2016.09.027))
### Hopf algebroid $Ext$-structure on $E^2$
{#ReferencesHopfAlgebroidStructure}
* appendix A.1 of [Ravenel 86](#Ravenel86)
* {#Baker} [[Andrew Baker]], _Brave new Hopf algebroids and the Adams spectral sequence for $R$-modules_ ([pdf](http://www.maths.gla.ac.uk/~ajb/dvi-ps/brave-ha.pdf))
* {#BakerLazarev01} [[Andrew Baker]], [[Andrey Lazarev]], _On the Adams Spectral Sequence for $R$-modules_, Algebr. Geom. Topol. 1 (2001) 173-199 ([arXiv:math/0105079](http://arxiv.org/abs/math/0105079))
* {#BakerJeanneret02} [[Andrew Baker]] and Alain Jeanneret, _Brave new Hopf algebroids and extensions of $MU$-algebras_, Homology Homotopy Appl. Volume 4, Number 1 (2002), 163-173. ([Euclid](http://projecteuclid.org/euclid.hha/1139840059))
* {#Hovey03} [[Mark Hovey]], _Homotopy theory of comodules over a Hopf algebroid_ ([arXiv:math/0301229](http://arxiv.org/abs/math/0301229))
### Further examples with more general coefficients
For [[tmf]]:
* [[Mark Behrens]], _The Adams spectral sequence for $tmf$_ ([[BehrensANSSfortmf.pdf:file]])
* [[Michael Hill]], _The 3-local $tmf$-homology of $B \Sigma_3$_, Proceedings of the AMS, 2007 ([pdf](http://www.ams.org/journals/proc/2007-135-12/S0002-9939-07-08937-X/S0002-9939-07-08937-X.pdf))
[[!redirects Adams spectral sequences]]
[[!redirects Adams spectral sequences]]
|
Adams-Quillen theorem | https://ncatlab.org/nlab/source/Adams-Quillen+theorem |
#Contents#
* table of contents
{:toc}
## Idea
The Adams-Quillen theorem describes the structure of a [[Hopf algebroid over a commutative base]] on the dual [[BP]]-[[Steenrod algebra]] $BP_\bullet(BP)$.
This is the analog for [[BP]] of what the [[Landweber-Novikov theorem]] is for [[MU]].
## Related concepts
* [[Landweber exact functor theorem]]
## References
* [[Doug Ravenel]], chapter 4, section 1 of _[[Complex cobordism and stable homotopy groups of spheres]]_, [pdf](http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenelA1.pdf)
|
Adams–Novikov spectral sequence | https://ncatlab.org/nlab/source/Adams%E2%80%93Novikov+spectral+sequence |
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###Context###
#### Homological algebra
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[[!include homological algebra - contents]]
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#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Adams-Novikov spectral sequence_ is the $E$-[[Adams spectral sequence]], for $E = $ [[MU]].
More in detail, the _Adams-Novikov spectral sequence_ is a class of [[spectral sequences]], which refine the _[[classical Adams spectral sequence]]_ by replacing [[ordinary cohomology]] with [[coefficients]] in [[Fp]] by [[complex cobordism cohomology theory]] [[MU]]. This makes the Adams-Novikov spectral sequence converge to the full [[homotopy groups]] of [[connective spectra]], hence in particular to the full [[stable homotopy groups of spheres]] ([Novikov 67, Theorems 3.1, 3.3](#Novikov67), reviewed as [Ravenel 78, Thm. 3.1](#Ravenel78)).
More generally for $E$ any suitable [[E-infinity ring]] there is an Adams-Novikov-type spectral sequence involving $E$-[[generalized cohomology]]/[[generalized homology]]. This fully general notion is often again just referred to as the _$E$-[[Adams spectral sequence]]_. Accordingly, see there for more. For detailed introduction, see at _[[Introduction to the Adams Spectral Sequence]]_.
## Properties
### Adams Chart
{#AdamsChart}
[Isaksen 14](#Isaksen14):
\begin{imagefromfile}
"file_name": "IsaksenANSSE2p.jpg"
\end{imagefromfile}
\begin{imagefromfile}
"file_name": "IsaksenANSSEinfinityp.jpg"
\end{imagefromfile}
### Relation to Brown-Peterson spectrum
The $p$-component of the $E^2$-term of the Adams-Novikov spectral sequence for the [[sphere spectrum]], hence the one converging to the
[[stable homotopy groups of spheres]] $\pi_\ast(\mathbb{S})$ is
$$
Ext_{BP_\ast(BP)}(BP_\ast, BP_\ast)
\,,
$$
where $BP$ denotes the [[Brown-Peterson spectrum]] at prime $p$.
recalled e.g. as [Ravenel, theorem 1.4.2](#Ravenel)
## Related concepts
* [[stable homotopy groups of spheres]]
* [[chromatic spectral sequence]]
* [[EHP spectral sequence]]
## References
The generalization of the [[Adams spectral sequence]] from $E = $ [[HA]] to $E = $ [[MU]] is due to
* {#Novikov67} [[Sergei Novikov]], _The methods of algebraic topology from the viewpoint of cobordism theories_, Izv. Akad. Nauk. SSSR. Ser. Mat. 31 (1967), 855–951 (Russian) ([[Novikov67.pdf:file]])
While that article was still being translated to English, [[Frank Adams]] learned of and then lectured about this work in 1967 in Chicago. This lecture together with two later lectures in 1970 and in 1971 constitute the book
* {#Adams74} [[Frank Adams]], _[[Stable homotopy and generalized homology]]_, Chicago Lectures in mathematics, 1974
Further early development of this Adams-Novikov spectral sequence:
* {#Zahler72} [[Raphael Zahler]], _The Adams-Novikov Spectral Sequence for the Spheres_, Annals of Mathematics Second Series, Vol. 96, No. 3 (Nov., 1972), pp. 480-504 ([doi:10.2307/1970821](https://doi.org/10.2307/1970821))
Review:
* {#Ravenel78} [[Douglas Ravenel]], _A Novice's guide to the Adams-Novikov spectral sequence_, in: [[Michael Barratt]], [[Mark Mahowald]] (eds.) _Geometric Applications of Homotopy Theory II_. Lecture Notes in Mathematics, vol 658, Springer 1978 ([doi:10.1007/BFb0068728](https://doi.org/10.1007/BFb0068728), [pdf](https://people.math.rochester.edu/faculty/doug/mypapers/Novice.pdf))
* {#Ravenel} [[Douglas Ravenel]], _[[Complex cobordism and stable homotopy groups of spheres]]_, page 11 of chapter I _An introduction to the homotopy groups of spheres_ ([pdf](http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenel1.pdf))
* [[Doug Ravenel]], _[Complex cobordism and stable homotopy groups of spheres](http://www.math.rochester.edu/people/faculty/doug/mu.html)_, chapter IV _$B P$-Theory and the Adams-Novikov Spectral Sequence_ ([pdf](http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenel4.pdf))
* {#Kochman96} [[Stanley Kochman]], sections 4.7 and 5.6 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996
* [[Jacob Lurie]], _Localizations and the Adams-Novikov Spectral Sequence_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture30.pdf))
The Adams chart of the ANSS for large order has been presented in
* {#Isaksen14} [[Daniel Isaksen]], _Classical and motivic Adams-Novikov charts_ ([arXiv:1408.0248](http://arxiv.org/abs/1408.0248))
[[!redirects Adams-Novikov spectral sequence]]
[[!redirects Adams-Novikov spectral sequences]] |
additive (infinity,1)-category | https://ncatlab.org/nlab/source/additive+%28infinity%2C1%29-category |
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###Context###
#### Additive and abelian categories
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[[!include additive and abelian categories - contents]]
=--
#### $(\infty,1)$-Category theory
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[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The analog in [[(∞,1)-category]] theory of the notion of [[additive category]] in [[category theory]].
## Related concepts
* [[semiadditive (∞,1)-category]]
* [[stable (∞,1)-category]]
## References
* {#GepnerGrothNikolaus13} [[David Gepner]], [[Moritz Groth]], [[Thomas Nikolaus]], _Universality of multiplicative infinite loop space machines_, [arXiv:1305.4550](http://arxiv.org/abs/1305.4550).
[[!redirects additive (infinity,1)-categories]]
[[!redirects additive (∞,1)-category]]
[[!redirects additive (∞,1)-categories]] |
additive analytic geometry | https://ncatlab.org/nlab/source/additive+analytic+geometry | +-- {: .rightHandSide}
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#### Geometry
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[[!include higher geometry - contents]]
=--
#### Additive analytic geometry
+--{: .hide}
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Additive analytic geometry over the [[complex number]] is an analog of classical (called here multiplicative) [[complex analytic geometry]] obtained by replacing the convolution algebra $\ell^1(\mathbb{N},\mathbb{C})$ of converging power series of radius $1$ (discrete [[Mellin transforms]]) by the algebra $L^1(\mathbb{R}_+,\mathbb{C})$ of integrable functions (continuous Mellin transforms) or other variants.
The analog of the complex unit disc $D_m(0,1)=D(0,1)$ is the hyperbolic/additive unit disc
$$D_a(-\infty,0)=\{s,Re(s)\leq 0\}\cup \{-\infty\},$$
(if one choses the class of integrable function, we actually have $-\infty=\infty$, since the Fourier transform of an $L^1$ function vanishes at infinity; this may not be the case if we work with another class of functions/distributions).
The main interest of this new geometry is that it has a built in flow (action of the additive monoid $\mathbb{R}_+$ on itself) and its functions are locally given by continuous Mellin transforms.
One may work, if needed with other kinds of functions/distributions on $\mathbb{R}_+$, for example if one wants to study distributional trace formulas.
In any case, one needs to use [[Dirac distribution|Dirac measures/distributions]] to relate directly the above two types of analytic geometries, by the relation on coordinates given by $z=e^s$.
Higher dimensional spaces may also be considered, but the lack of a unit element in $L^1$ makes the extension of the semi and sub-analytic topologies to this geometry a bit tricky. Unitarization may be necessary.
A full complex geometry with various Grothendieck topologies seems to emerge from the consideration of the simple analogy/link with classical complex analysis given by the complex exponential map, but it is not so easy to find the right class of functions/distributions to be used here. One may simply try out with $L^1$ functions, and see where one arrives, but a regularity condition on an additive disk may also be imposed by asking that $(\log x)^n f(x)x^\delta$ is $L^1$ for every $n$, in order to obtain smoothness of functions on the boundary; using schwartz functions or overconvergent ones may also be a solution to the problem of having a nice notion of differential calculus on rational domains, that one must also solve in classical (multiplicative) analytic geometry.
Additive analytic geometry also has a non-commutative and automorphic extension
(formulated in spectral terms), that may be useful to study and formulate some problems of automorphic representation theory in a geometric context. For example, the results of Godement and Jacquet may look like a kind of pasting of two "Mellin automorphic spectra" along the usual automorphic spectrum, in this context.
## A simple example
One may paste two additive unit discs along their boundary $\mathbb{R}$,
that is the spectrum of $L^1(\mathbb{R})^+$ by $s\mapsto -s$ on $\mathbb{C}$,
to get the additive version of the projective space
$\mathbb{P}^1_a(\mathbb{C})=\{\infty\}\cup \mathbb{C}$. One may also paste two different discs along an annulus.
Remark that there is also a way to use other pastings, such as
$s\mapsto 1-s$.
## Remarks
To be able to work with $L$-functions, seen as additive functions, it seems one needs to pass to the setting of distributions of a certain kind, that may be analytic on a half space and distributional on the boundary. The study of the spectrum of distribution convolution algebras probably requires further work.
If such a function may be extended to a type of additive projective space using a functional equation, then the complement of its set $S$ of zeroes gives an open subspace $U$ of an additive projective space. In good situations, this gives an additive subspace whose sheaf of functions is equipped with the given flow, and its embedding into projective space induces an identification between the associated additive de Rham cohomology spaces in degree $0$ and $2$. The Poincaré group of this space is generated by looping around each of the given points in the set $S$, that may be infinite.
It seems that the L-function has to be automorphic so that one may study it using additive geometry, because the functional equation of theta functions is at the heart of the above construction. Indeed, even if higher dimensional complex varieties allow one to define various types of generalized multiplicative spaces (using Mellin transforms of $\delta$ distributions, for example), there is yet no clear additive versions of global arithmetic schemes, because there is no convolution operation on $L^1(\mathbb{R}_+,\mathbb{Z})$.
A possible approach to this problem of finding a higher dimensional version of the above construction is to first try to give a spectral interpretation for theta functions, so that one may study more general $L$ functions. This corresponds to decomposing the $L$-function of an arithmetic scheme with respect to the motivic weight (cohomology relative to the spectrum of $\mathbb{Z}$), and not its absolute motivic weight (which would allow to interpret, for example, the factor $s$ in the $\xi$ function).
Proving that the given spectral $\theta$ functions fulfils a functional equation is not an easy task and can't be solved directly without showing that they are decomposed in $L$-functions of cuspidal automorphic type for some $\GL_m$, and using Godement and Jacquet's generalization of Tate's thesis.
## Additive Hodge theory
One may develop an analog of [[Hodge theory]] in this context (working with real analytic additive functions in the $s$ and $\bar{s}$ coordinates, and the corresponding analog of the Laplacian; the formal theory with [[Sobolev spaces]] seems to carry through to this context as one may see on wikipedia). This will not give finite dimensionality of [[de Rham cohomology]], because of the above remark on the Poincaré group. However, such a Hodge theory may still have interesting applications related to its interaction with Poincaré duality in additive de Rham cohomology, since, in good cases, all the given construction are compatible with the given flow. |
additive and abelian categories | https://ncatlab.org/nlab/source/additive+and+abelian+categories |
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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Homological algebra
+-- {: .hide}
[[!include homological algebra - contents]]
=--
#### Additive and abelian categories
+-- {: .hide}
[[!include additive and abelian categories - contents]]
=--
=--
=--
* table of contents
{: toc}
## Definitions
There is a sequence of extra [[stuff, structure, property|structure and property]] on a [[category]] that makes this category behave like a general context for [[homological algebra]]. In order of increasing structure and property this is:
1. **[[Ab-enriched category]]**:
a [[category]] that is [[Ab]]-[[enriched category|enriched]];
2. **[[pre-additive category]]**:
an $Ab$-enriched category that has a [[terminal object]] or [[initial object]] and therefore a [[zero object]]; Notice however that many authors (e.g. Weibel, Popescu) by preadditive (or pre-additive) simply mean $Ab$-enriched.
* [[pseudo-abelian category]]: a pre-additive category such that every [[idempotent]] has a [[kernel]] and [[cokernel]]
3. **[[additive category]]**: a pre-additive category that admits binary [[product]]s or binary [[coproduct]]s and therefore binary [[biproduct]]s (equivalently, an $Ab$-enriched category with all finite products or coproducts);
4. **[[pre-abelian category]]**:
an additive category that admits [[kernel]]s and [[cokernel]]s (equivalently, an $Ab$-enriched category with all finite [[limit]]s and [[colimit]]s);
5. **[[abelian category]]**:
a pre-abelian category such that every [[monomorphism]] is a kernel and every [[epimorphism]] is a cokernel (and many other equivalent definitions).
Pre-abelian and abelian categories are sometimes called (AB1) and (AB2) categories, after the sequence of additional axioms on top of additive categories introduced by Grothendieck in [[Tohoku]]. AB1 and AB2 are self-dual axioms (AB1 is existence of kernels and cokernels, and AB2 requires that, for any $f$, the canonical morphism $\mathrm{Coim}\,f\to \mathrm{Im}\,f$ is an isomorphism). These continue in non-selfdual manner:
* AB3: an abelian category with all coproducts (hence with all [[colimit]]s);
* AB4: an (AB3) category in which coproducts of monomorphisms are monic;
* AB5: an (AB3) category in which [[filtered colimits]] of [[exact sequence]]s exist and are exact;
* AB6: an (AB3) category such that
* for every object $A$ in $C$ and any family $B^j$ with $j \in J$ of directed families $B^j = B^j_i$ with $i in I_j$ the intersections of [[subobject]]s over $j$ commute with direct sums over $j$.
* Notice that this implies that inf for any family of subobjects exists.
The concepts (AB3--AB6) also have dual forms (AB3\*--AB6\*).
There are further refinements along these lines. In particular
* [[Grothendieck category]]: an (AB5) category with a [[generator]].
## Further refinements
Various further axiom structures are considered for additive (sometimes abelian) categories.
* [[suspended category]]
* [[triangulated category]]
* [[Quillen exact category]]
* [[Grothendieck category]]: an AB5-category with a [[generator]]
* Gabriel's [[property sup]]
## Generalizations
* Abelian groups are to general [[group]]s as [[abelian category|abelian categories]] are to [[semi-abelian category|semi-abelian categories]].
## Examples
Various generic classes of examples of additive and abelian categories are of relevance:
## Related concepts
* [[delta-functor]]
* [[category of chain complexes]]
* [[derived category]]
* [[stable (infinity,1)-category]]
[[!redirects additive and abelian categories]]
[[!redirects additive or abelian categories]]
[[!redirects additive or abelian category]]
[[!redirects AB5-category]]
[[!redirects AB5-categories]]
|
additive and abelian categories - contents | https://ncatlab.org/nlab/source/additive+and+abelian+categories+-+contents |
**[[additive and abelian categories]]**
## Context and background
* [[enriched category theory]]
* [[homological algebra]]
* [[stable homotopy theory]]
## Categories
* [[Ab-enriched category]]
* [[pre-additive category]]
* [[pseudo-abelian category]]
* [[additive category]],
* (AB1) [[pre-abelian category]]
* (AB2) [[abelian category]]
* (AB5) [[Grothendieck category]]
## Functors
* [[additive functor]]
* left/right [[exact functor]]
## Derived categories
* [[derived category]]
* [[derived functor]]
* [[homotopy category]]
* [[triangulated category]]
<div markdown="1">[Edit this sidebar](/nlab/edit/additive+and+abelian+categories+-+contents)</div> |
additive category | https://ncatlab.org/nlab/source/additive+category |
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###Context###
#### Enriched category theory
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#### Additive and abelian categories
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#### Homological algebra
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[[!include homological algebra - contents]]
=--
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#Contents#
* table of contents
{:toc}
## Definition
+-- {: .num_defn #AdditiveCategory}
###### Definition
An **additive category** is a [[category]] which is
1. an [[Ab-enriched category]];
(sometimes called a [[pre-additive category]]--this means that each [[hom-set]] carries the structure of an [[abelian group]] and composition is [[bilinear map|bilinear]])
1. which admits [[finite limit|finite]] [[coproducts]]
(and hence, by prop. \ref{ProductsAreBiproducts} below, finite [[products]] which coincide with the coproducts, hence finite [[biproducts]]).
The natural [[morphisms]] *between* additive categories are [[additive functors]].
=--
+-- {: .num_remark }
###### Remark
A [[pre-abelian category]] is an additive category which also has [[kernels]] and [[cokernels]]. Equivalently, it is an Ab-enriched category with all [[finite limits]] and finite [[colimits]]. An especially important sort of additive category is an [[abelian category]], which is a pre-abelian one satisfying the extra exactness property that all [[monomorphisms]] are [[kernels]] and all [[epimorphisms]] are [[cokernels]]. See at _[[additive and abelian categories]]_ for more.
=--
+-- {: .num_remark }
###### Remark
The [[Ab]]-enrichment of an additive category does not have to be given a priori. Every [[semiadditive category]] (a category with finite [[biproducts]]) is automatically [[enriched category|enriched]] over [[commutative monoids]] (as described at _[[biproduct]]_), so an additive category may be defined as a category with finite biproducts whose [[hom-object|hom-monoids]] happen to be [[groups]]. (The requirement that the hom-monoids be groups can even be stated in elementary terms without discussing enrichment at all, but to do so is not very enlightening.) Note that the entire $Ab$-enriched structure follows automatically for [[abelian category|abelian categories]].
=--
+-- {: .num_remark }
###### Remark
Some authors use **additive category** to simply mean an Ab-enriched category, with no further assumptions. It can also be used to mean a $CMon$-enriched (commutative monoid enriched) category, with or without assumptions of products.
=--
## Properties
+-- {: .num_prop #ProductsAreBiproducts}
###### Proposition
In an [[Ab-enriched category]] (or even just a $CMon$-enriched category), a [[finite product|finite]] [[product]] is also a [[coproduct]], and dually (hence a [[biproduct]]).
This statement includes the zero-ary case: any [[terminal object]] is also an [[initial object]], hence a [[zero object]] (and dually), hence every additive category has a [[zero object]].
More precisely, for $\{X_i\}_{i \in I}$ a [[finite set]] of objects in an Ab-enriched category, the unique morphism
$$
\underset{i \in I}{\coprod} X_i \longrightarrow \underset{j \in I}{\prod} X_j
$$
whose components are identities for $i = j$ and are [[zero morphism|zero]] otherwise is an [[isomorphism]].
=--
+-- {: .proof}
###### Proof
Consider first the nullary (i.e., zero-ary) case. Given a [[terminal object]] $\ast$, the unique morphism $id_\ast: \ast \to \ast$ is the zero morphism $0$ in its hom-object. For any object $A$, the zero morphism $0_A: \ast \to A$ must equal any morphism $f: \ast \to A$ on account of $f = f id_\ast = f 0 = 0_A$ where the last equation is by $CMon$-enrichment. Hence $\ast$ is initial. (N.B.: this argument applies more generally to categories [[enriched category|enriched]] in [[pointed sets]], and is self-[[formal duality|dual]].)
Consider now the case of binary (co-)products. Using [[zero morphisms]], in addition to its canonical [[projection]] maps $p_i \colon X_1 \times X_2 \to X_i$, any binary [[product]] also admits "injection" maps $X_i \to X_1 \times X_2$, and dually for the [[coproduct]]:
$$
\array{
X_1 && && X_2
\\
& \searrow^{\mathrlap{(id,0)}} && {}^{\mathllap{(0,id)}}\swarrow
\\
{}^{\mathllap{id_{X_1}}}\downarrow && X_1 \times X_2 && \downarrow^{\mathrlap{id_{X_2}}}
\\
& \swarrow_{\mathrlap{p_{X_1}}} && {}_{\mathllap{p_{X_2}}}\searrow
\\
X_1 && && X_2
}
\;\;\;\;\;\;\;\;\;\;\;\;\,,\;\;\;\;\;\;\;\;\;\;\;\;
\array{
X_1 && && X_2
\\
& \searrow^{\mathrlap{i_{X_1}}} && {}^{\mathllap{i_{X_2}}}\swarrow
\\
{}^{\mathllap{id_{X_1}}}\downarrow
&& X_1 \sqcup X_2 && \downarrow^{\mathrlap{id_{X_2}}}
\\
& \swarrow_{\mathrlap{(id,0)}} && {}_{\mathllap{(0,id)}}\searrow
\\
X_1 && && X_2
}
\,.
$$
Observe some basic compatibility of the $Ab$-enrichment with the product:
First, for $(\alpha_1,\beta_1), (\alpha_2, \beta_2)\colon R \to X_1 \times X_2$ then
$$
(\star)
\;\;\;\;\;\;
(\alpha_1,\beta_1) + (\alpha_2, \beta_2)
=
(\alpha_1+ \alpha_2 , \; \beta_1 + \beta_2)
$$
(using that the projections $p_1$ and $p_2$ are linear and by the universal property of the product).
Second, $(id,0) \circ p_1$ and $(0,id) \circ p_2$ are two projections on $X_1\times X_2$ whose sum is the identity:
$$
(\star\star)
\;\;\;\;\;\;
(id, 0) \circ p_1
+
(0, id) \circ p_2
=
id_{X_1 \times X_2}
\,.
$$
(We may check this, via the [[Yoneda lemma]] on [[generalized elements]]: for $(\alpha, \beta) \colon R \to X_1\times X_2$ any morphism, then $(id,0)\circ p_1 \circ (\alpha,\beta) = (\alpha,0)$ and $(0,id)\circ p_2\circ (\alpha,\beta) = (0,\beta)$, so the statement follows with equation $(\star)$.)
Now observe that for $f_i \;\colon\; X_i \to Q$ any two morphisms, the sum
$$
\phi
\;\coloneqq\;
f_1 \circ p_1 + f_2 \circ p_2
\;\colon\;
X_1 \times X_2
\longrightarrow
Q
$$
gives a morphism of [[cocones]]
$$
\array{
X_1 && && X_2
\\
& \searrow^{\mathrlap{(id,0)}} && {}^{\mathllap{(0,id)}}\swarrow
\\
{}^{\mathllap{id_{X_1}}}\downarrow && X_1 \times X_2 && \downarrow^{\mathrlap{id_{X_2}}}
\\
& &&
\\
X_1 && \downarrow^{\mathrlap{\phi}} && X_2
\\
& {}_{\mathllap{f_1}}\searrow && \swarrow_{\mathrlap{f_2}}
\\
&& Q
}
\,.
$$
Moreover, this is unique: suppose $\phi'$ is another morphism filling this diagram, then, by using equation $(\star \star)$, we get
$$
\begin{aligned}
\phi
& =
\phi \circ id_{X_1 \times X_2}
\\
&=
\phi \circ ( (id_{X_1},0) \circ p_1 + (0,id_{X_2})\circ p_2 )
\\
& =
\phi \circ (id_{X_1}, 0) \circ p_1 + \phi \circ (0, id_{X_2}) \circ p_2
\\
& =
f_1 \circ p_1 + f_2 \circ p_2
\\
& =
\phi' \circ (id_{X_1}, 0) \circ p_1 + \phi' \circ (0, id_{X_2}) \circ p_2
\\
& =
\phi' \circ ( (id_{X_1},0) \circ p_1 + (0,id_{X_2})\circ p_2 )
\\
&=
\phi' \circ id_{X_1 \times X_2}
\\
&=
\phi'
\end{aligned}
\,.
$$
This means that $X_1\times X_2$ satisfies the [[universal property]] of a [[coproduct]].
By a [[formal dual|dual]] argument, the binary coproduct $X_1 \sqcup X_2$ is seen to also satisfy the universal property of the binary product. By [[induction]], this implies the statement for all finite (co-)products. (If a particular finite (co-)product exists but binary ones do not, one can adapt the above argument directly to that case.)
=--
+-- {: .num_remark}
###### Remark
Such products which are also coproducts as in prop. \ref{ProductsAreBiproducts} are sometimes called _[[biproducts]]_ or _[[direct sums]]_; they are [[absolute limit|absolute limits]] for [[Ab]]-[[enriched category|enrichment]].
=--
+-- {: .num_remark}
###### Remark
The coincidence of products with biproducts in prop. \ref{ProductsAreBiproducts} does _not_ extend to infinite products and coproducts.) In fact, an [[Ab-enriched category]] is [[Cauchy complete category|Cauchy complete]] just when it is additive and moreover its [[idempotents]] split.
=--
Conversely:
+-- {: .num_defn #SemiadditiveCategory}
###### Definition
A **[[semiadditive category]]** is a [[category]] that has all [[finite products]] which, moreover, are [[biproducts]] in that they coincide with finite [[coproducts]] as in def. \ref{ProductsAreBiproducts}.
=--
+-- {: .num_prop #SemiAdditivityInducesAbelianMonoidEnrichment}
###### Proposition
In a [[semiadditive category]], def. \ref{SemiadditiveCategory}, the [[hom-sets]] acquire the structure of [[commutative monoids]] by defining the sum of two morphisms $f,g \;\colon\; X \longrightarrow Y$ to be
$$
f + g
\;\coloneqq\;
X \overset{\Delta_X}{\to} X \times X
\simeq
X \oplus X
\overset{f \oplus g}{\longrightarrow}
Y \oplus Y
\simeq
Y \sqcup Y
\overset{\nabla_Y}{\to}
Y
\,.
$$
With respect to this operation, [[composition]] is [[bilinear map|bilinear]].
=--
+-- {: .proof}
###### Proof
The [[associativity]] and commutativity of $+$ follows directly from the corresponding properties of $\oplus$. Bilinearity of composition follows from [[natural transformation|naturality]] of the [[diagonal]] $\Delta_X$ and [[codiagonal]] $\nabla_X$:
$$
\array{
W
&\overset{\Delta_W}{\longrightarrow}&
W \times W
&\overset{\simeq}{\longrightarrow}&
W \oplus W
\\
\downarrow^{\mathrlap{e}}
&&
\downarrow^{\mathrlap{e \times e}}
&&
\downarrow^{\mathrlap{e \oplus e}}
\\
X
&\overset{\Delta_X}{\to}&
X \times X
&\simeq&
X \oplus X
&\overset{f \oplus g}{\longrightarrow}&
Y \oplus Y
&\simeq&
Y \sqcup Y
&\overset{\nabla_X}{\to}&
Y
\\
&&
&&
&&
\downarrow^{\mathrlap{h \oplus h}}
&&
\downarrow^{\mathrlap{h \sqcup h}}
&&
\downarrow^{\mathrlap{h}}
\\
&&
&&
&&
Z \oplus Z
&\simeq&
Z \sqcup Z
&\overset{\nabla_Z}{\to}&
Z
}
$$
=--
+-- {: .num_prop #SemiaddtiveStructureUnderlyingAdditiveInducesOriginalEnrichment}
###### Proposition
Given an additive category according to def. \ref{AdditiveCategory}, then the enrichment in [[commutative monoids]] which is induced on it via prop. \ref{ProductsAreBiproducts} and prop. \ref{SemiAdditivityInducesAbelianMonoidEnrichment} from its underlying [[semiadditive category]] structure coincides with the original enrichment.
=--
+-- {: .proof}
###### Proof
By the proof of prop. \ref{ProductsAreBiproducts}, the [[codiagonal]] on any object in an additive category is the sum of the two projections:
$$
\nabla_X \;\colon\; X \oplus X \overset{p_1 + p_2}{\longrightarrow} X
\,.
$$
Therefore (checking on [[generalized elements]], as in the proof of prop. \ref{ProductsAreBiproducts}) for all morphisms $f,g \colon X \to Y$ we have [[commuting squares]] of the form
$$
\array{
X &\overset{f+g}{\longrightarrow}& Y
\\
{}^{\mathllap{\Delta_X}}\downarrow && \uparrow^{\mathrlap{\nabla_Y =}}_{\mathrlap{p_1 + p_2}}
\\
X \oplus X
&\underset{f \oplus g}{\longrightarrow}&
Y\oplus Y
}
\,.
$$
=--
+-- {: .num_remark}
###### Remark
Prop. \ref{SemiaddtiveStructureUnderlyingAdditiveInducesOriginalEnrichment} says that being an [[additive category]] is an extra [[property]] on a category, not extra [[structure]]. We may ask whether a given category is additive or not, without specifying with respect to which abelian group structure on the hom-sets.
=--
## Related concepts
* [[semiadditive category]]
* [[abelian category]]
* [[additive (∞,1)-category]]
* [[triangulated category]]
## References
Textbook accounts:
* [[Masaki Kashiwara]], [[Pierre Schapira]], Section 8 of: *[[Categories and Sheaves]]*, Grundlehren der Mathematischen Wissenschaften **332**, Springer (2006) [[doi:10.1007/3-540-27950-4](https://link.springer.com/book/10.1007/3-540-27950-4), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/kashiwara2.pdf)]
Discussion in [[homological algebra]]:
* [[Charles Weibel]], *[[An Introduction to Homological Algebra]]*, Cambridge University Press (1994) [[doi:10.1017/CBO9781139644136](https://doi.org/10.1017/CBO9781139644136), [pdf](https://web.math.rochester.edu/people/faculty/doug//otherpapers/weibel-hom.pdf)]
See also:
* {#IntroductionToLinearCategories} [[William Lawvere]], *Introduction to Linear Categories and Applications*, course lecture notes (1992) [[pdf](https://github.com/mattearnshaw/lawvere/blob/192dac273e8bf352f307f87b9ec4fe8ef7dc85b9/pdfs/1992-introduction-to-linear-categories-and-applications.pdf), [[Lawvere-LinearCategories.pdf:file]]]
Discussion of [[model category]] structures on additive categories is around def. 4.3 of
* {#Beligiannis} Apostolos Beligiannis, _Homotopy theory of modules and Gorenstein rings_, Math. Scand. 89 (2001) ([pdf](http://users.uoi.gr/abeligia/mathscand.pdf))
Formalization of additive categories as [[univalent categories]] in [[univalent foundations of mathematics]] ([[homotopy type theory]]):
* [[unimath]], *Additive categories* [[UniMath.CategoryTheory.Additive](https://unimath.github.io/doc/UniMath/4dd5c17/UniMath.CategoryTheory.Additive.html)]
A characterisation of preadditive categories in terms of commutative monoids in cartesian multicategories is given in:
* [[Claudio Pisani]], _Sequential multicategories_, Theory and Applications of Categories 29.19 (2014), [arXiv:1402.0253](https://arxiv.org/abs/1402.0253)
[[!redirects additive categories]] |
additive disjunction | https://ncatlab.org/nlab/source/additive+disjunction |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In the context of [[linear logic]] or [[linear type theory]], the *additive disjunction* "$\oplus$" is a form of [[disjunction]] whose [[categorical semantics]] is typically that of the [[direct sum]] (e.g. [Mihályi & Novitzká (2013), p. 9](#MihályiNovitzká13)).
## Related concepts
* [[direct sum]], [[quantum parallelism]]
[[!include logic symbols -- table]]
## References
* {#MihályiNovitzká13} Daniel Mihályi, Valerie Novitzká, *What about Linear Logic in Computer Science?*, Acta Polytechnica Hungarica **10** 4 (2013) 147-160 [[pdf](http://acta.uni-obuda.hu/Mihalyi_Novitzka_42.pdf), [[MihalyiNovitzka-LinearLogic.pdf:file]]]
[[!redirects additive disjunctions]]
[[!redirects linear disjunction]]
[[!redirects linear disjunctions]]
|
additive envelope | https://ncatlab.org/nlab/source/additive+envelope | The additive envelope of an [[Ab-enriched category]] $A$ is defined by taking the objects as formal direct sums of objects in $A$, and morphisms as matrices of coefficients, giving an [[additive category]]. This is a universal construction.
For instance, a [[semisimple category]] is the additive envelope of its full subcategory consisting of simple objects. Also, any additive category is equivalent to its own additive envelope.
By further taking the [[Karoubi envelope]] (i.e. formally adding images of idempotent elements), one constructs a [[Karoubian category]] called the pseudo-abelian envelope of $A$. In general, the pseudo-abelian envelope of $A$ is not abelian.
The combination of additive envelope and Karoubi envelope gives the [[Cauchy completion]] in the case of [[Ab-enriched categories]].
|
additive functor | https://ncatlab.org/nlab/source/additive+functor |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory - contents]]
=--
#### Additive and abelian categories
+--{: .hide}
[[!include additive and abelian categories - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
+-- {: .num_defn}
###### Definition
A [[functor]] $F: \mathcal{A} \to \mathcal{B}$ between [[additive category|additive categories]] is itself called __additive__ if it preserves finite [[biproducts]].
That is,
1. $F$ maps a [[zero object]] to a zero object, $F(0) \simeq 0 \in \mathcal{B}$;
1. given any two [[objects]] $x, y \in \mathcal{A}$, there is an [[isomorphism]] $F(x \oplus y) \cong F(x) \oplus F(y)$, and this respects the inclusion and projection maps of the [[direct sum]]:
$$
\array { x & & & & y \\
& {}_{\mathllap{i_x}}\searrow & & \swarrow_{\mathrlap{i_y}} \\
& & x \oplus y \\
& {}^{\mathllap{p_x}}\swarrow & & \searrow^{\mathrlap{p_y}} \\
x & & & & y }
\quad\quad\stackrel{F}{\mapsto}\quad\quad
\array { F(x) & & & & F(y) \\
& {}_{\mathllap{i_{F(x)}}}\searrow & & \swarrow_{\mathrlap{i_{F(y)}}} \\
& & F(x \oplus y) \cong F(x) \oplus F(y) \\
& {}^{\mathllap{p_{F(X)}}}\swarrow & & \searrow^{\mathrlap{p_{F(y)}}} \\
F(x) & & & & F(y) }
$$
=--
+-- {: .num_remark}
###### Remark
In practice, functors between additive categories are generally assumed to be additive.
=--
+-- {: .num_remark #SufficientConditions}
###### Remark
Each of the following conditions is sufficient for guaranteeing that a functor $\mathcal{A} \to \mathcal{B}$ preserves biproducts (where $\mathcal{A}$ and $\mathcal{B}$ are categories with a zero object):
* The functor preserves finite products (for instance, because it's a right adjoint) and any product in $\mathcal{B}$ is a biproduct.
* The functor preserves finite coproducts (for instance, because it's a left adjoint) and any coproduct in $\mathcal{B}$ is a biproduct.
* The functor preserves finite products and coproducts.
=--
## Examples
\begin{example}
\label{HomFunctor}
The [[hom-functor]] $Hom(-,-) \colon \mathcal{A}^{op}\times \mathcal{A} \to Ab$ is additive in both arguments separately (using the nature of [[biproducts]] and that [[hom-functors preserve limits]] in each variable separately).
\end{example}
+-- {: .num_example}
###### Example
For $\mathcal{A} = R$[[Mod]] and $N \in \mathcal{A}$, the functor that forms [[tensor product of modules]] $(-)\otimes N \colon \mathcal{A} \to \mathcal{A}$.
=--
In fact these examples are generic, see prop. \ref{RightExactAdditiveIsTensor} below.
+-- {: .num_example}
###### Example
Every [[solid abelian group]] is by definition an [[additive functor]].
=--
## Properties
### Relation to $Ab$-enriched functors
An [[additive category]] canonically carries the structure of an [[Ab-enriched category]] where the $Ab$-enrichment structure is induced from the biproducts as described at _[[biproduct]]_.
+-- {: .num_prop #AdditiveIsAbEnriched}
###### Proposition
With respect to the canonical [[Ab-enriched category]]-structure on additive categories $\mathcal{A}$, $\mathcal{B}$, additive functors $F : \mathcal{A} \to \mathcal{B}$ are equivalently [[Ab]]-[[enriched functors]].
=--
+-- {: .proof}
###### Proof
An $Ab$-enriched functor preserves all finite biproducts that exist, since finite biproducts in [[Ab-enriched category|Ab-enriched categories]] are [[Cauchy complete category|Cauchy colimits]].
=--
### Characterization of right exact additive functors
{#CharacterizationByExactness}
Let $R, R'$ be [[rings]].
The following is the _[[Eilenberg-Watts theorem]]_. See there for more.
+-- {: .num_prop #RightExactAdditiveIsTensor}
###### Proposition
If an additive functor $F : R$[[Mod]] $\to R'$[[Mod]] is a [[right exact functor]], then there exists an $R'$-$R$-[[bimodule]] $B$ and a [[natural isomorphism]]
$$
F \simeq B \otimes_R (-)
$$
with the functor that forms the [[tensor product]] with $B$.
=--
This is ([Watts, theorem 1](#Watts)),
## Related concepts
* [[additive monad]]
* [[enriched functor]]
* In the context of [[derived functors in homological algebra]] one considers functors that are additive and in addition left/right [[exact functors]], as discussed above in _[Characterization by exactness](#CharacterizationByExactness)_.
* [[satellite]]
## References
* {#Watts} Charles Watts, _Intrinsic characterizations of some additive functors_, Proceedings of the American Mathematical Society, **11** 1 (1960) 5-8 (1959) [[jstor:2032707](http://www.jstor.org/stable/2032707)]
[[!redirects additive functors]] |
additive K-theory | https://ncatlab.org/nlab/source/additive+K-theory | Term __additive K-theory__ is a synonym for [[cyclic homology]], used in early articles mainly of Russian and a bit by French school. Sometimes, additive K-theory denotes more specifially a different packing of cyclic homology, namely with appropriately shifted degrees (see the reference by Loday-Quillen at [[Loday-Quillen-Tsygan theorem]] and the article by Kuribayashi below).
__Additive K-theory__ is also a title of a historical article
* [[Boris Tsygan]], [[Boris Feigin]], _Additive K-theory_, in K-theory, arithmetic and geometry, LNM 1289 (1987), edited by Yu. I. Manin, pp. 67--209, seminar 1984-1986 in Moscow), [MR89a:18017](http://www.ams.org/mathscinet-getitem?mr=923136)
The additive K-theory is here studied in relation to the algebraic K-theory and Hochschild homology. Like there is a K-theory spectrum, one also constructs an additive K-theory/cyclic homology spectrum.
Contents of Tsygan-Feigin
* Introduction.
* Ch. 1. Additive K-functors.
* Ch. 2. Derived functors and relative additive K-functors.
* Ch. 3. Generalized free products.
* Ch. A. Lie algebra homology.
* Ch. 5. Operations in additive K-theory.
* Ch. 6. Additive K-functors of the commutative noetherian algebras.
* Ch. 7. Characteristic classes.
* Appendix. Cyclic objects.
Related articles include
* Б. Л. Фейгин, Б. Л. Цыган, "Аддитивная K-теория и кристальные когомологии", Функц. анализ и его прил., 19:2 (1985), 52–-62, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=1358&what=fullt&option_lang=rus), [MR88e:18008](http://www.ams.org/mathscinet-getitem?mr=800920); Engl. transl. in B. L. Feĭgin, B. L. Tsygan, _Additive $K$-theory and crystalline cohomology_, Functional Analysis and Its Applications, 1985, 19:2, 124–132.
* Katsuhiko Kuribayashi, Toshihiro Yamaguchi, _On additive K-theory with the Loday-Quillen $\ast$-product_, Math. Scand. __87__, No 1 (2000) [article](http://www.mscand.dk/article/view/14296) [doi](http://dx.doi.org/10.7146/math.scand.a-14296)
[[!redirects Additive K-theory]] |
additive monad | https://ncatlab.org/nlab/source/additive+monad |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
#### Additive and abelian categories
+--{: .hide}
[[!include additive and abelian categories - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A [[monad]] $\mathbf{G}=(G,\mu,\eta)$ on an [[additive category]] $A$ is **additive** if its underlying [[endofunctor]] $G:A\to A$ is an [[additive functor]]. One defines an **additive [[comonad]]** in the same vein.
Note that every additive category is [[Ab-enriched category|Ab-enriched]], and an additive monad is then the same as an [[Ab]]-[[enriched monad]].
## Related concepts
* [[strong monad]]
[[!redirects additive monads]]
[[!redirects additive comonad]]
[[!redirects additive comonads]]
|
ADE -- table | https://ncatlab.org/nlab/source/ADE+--+table | **[[ADE classification]]** and **[[McKay correspondence]]**
| [[Dynkin diagram]]/ <br/> [[Dynkin quiver]] | [[dihedron]],<br/> [[Platonic solid]] | [[classification of finite rotation groups|finite subgroups of SO(3)]] | [[classification of finite rotation groups|finite subgroups of SU(2)]] | [[simple Lie group]] |
|---------------------|---|-----|---------------|----------------------|
| $A_{n \geq 1}$ | | [[cyclic group]] <br/> $\mathbb{Z}_{n+1}$ | [[cyclic group]] <br/> $\mathbb{Z}_{n+1}$ | [[special unitary group]] <br/> $SU(n+1)$ |
| [[A1]] | | [[cyclic group of order 2]] <br/> $\mathbb{Z}_2$ | [[cyclic group of order 2]] <br/> $\mathbb{Z}_2$ | [[SU(2)]] |
| [[A2]] | | [[cyclic group of order 3]] <br/> $\mathbb{Z}_3$ | [[cyclic group of order 3]] <br/> $\mathbb{Z}_3$ | [[SU(3)]] |
| [[A3]] <br/> = <br/> [[D3]] | | [[cyclic group of order 4]] <br/> $\mathbb{Z}_4$ | [[cyclic group of order 4]] <br/> $2 D_2 \simeq \mathbb{Z}_4$ | [[SU(4)]] <br/> $\simeq$ <br/> [[Spin(6)]] |
| [[D4]] | [[dihedron]] on <br/> [[bigon]] | [[Klein four-group]] <br/> $D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$ | [[quaternion group]] <br/> $2 D_4 \simeq$ [[Q8]] | [[SO(8)]], [[Spin(8)]] |
| [[D5]] | [[dihedron]] on <br/> [[triangle]] | [[dihedral group of order 6]] <br/> $D_6$ | [[binary dihedral group of order 12]] <br/> $2 D_6$ | [[SO(10)]], [[Spin(10)]] |
| [[D6]] | [[dihedron]] on <br/> [[square]] | [[dihedral group of order 8]] <br/> $D_8$ | [[binary dihedral group of order 16]] <br/> $2 D_{8}$ | [[SO(12)]], [[Spin(12)]] |
| $D_{n \geq 4}$ | [[dihedron]], <br/> [[hosohedron]] | [[dihedral group]] <br/> $D_{2(n-2)}$ | [[binary dihedral group]] <br/> $2 D_{2(n-2)}$ | [[special orthogonal group]], [[spin group]] <br/> $SO(2n)$, $Spin(2n)$ |
| $E_6$ | [[tetrahedron]] | [[tetrahedral group]] <br/> $T$ | [[binary tetrahedral group]] <br/> $2T$ | [[E6]] |
| $E_7$ | [[cube]], <br/> [[octahedron]] | [[octahedral group]] <br/> $O$ | [[binary octahedral group]] <br/> $2O$ | [[E7]] |
| $E_8$ | [[dodecahedron]], <br/> [[icosahedron]] | [[icosahedral group]] <br/> $I$ | [[binary icosahedral group]] <br/> $2I$ | [[E8]] | |
ADE classification | https://ncatlab.org/nlab/source/ADE+classification |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A long list of mathematical structures happens to have a classification that is in [[bijection]] with the [[simply laced Dynkin diagrams]] of types A, D and E (but excluding type B and C), for instance
* [[Platonic solids]]
* [[finite group|finite]] [[subgroups]] of the [[special orthogonal group]] $SO(3)$ and of the [[special unitary group]] $SU(2)$ (see at [[classification of finite rotation groups]]) ([Milnor 57](#Milnor57), see e.g. [Keenan 03, theorem 4](#Keenan03))
* [[simply laced Dynkin diagrams]] (their [[simple Lie groups]])
\begin{imagefromfile}
"file_name": "ADEDynkin.jpg",
"width": 490
\end{imagefromfile}
* connected [[quivers]] with a [[finite number]] of [[indecomposable object|indecomposable]] [[quiver representations]] over an [[algebraically closed field]]
([[Gabriel's theorem]])
* 7d [[spherical space forms]] with [[spin structure]] carrying $N \geq 4$ [[Killing vectors]] (see at [spherical space form -- 7d with spin structure](spherical+space+form#7DSphericalSpaceFormsWithSpinStructure))
equivalently: [[near horizon geometries]] of smooth (i.e. non-[[orbifold]]) $\geq \tfrac{1}{2}$ [[BPS state|BPS]] [[black brane|black]] [[M2-brane]]-solutions of the [[equations of motion]] of [[11-dimensional supergravity]]
This is due to [MFFGME 09](#MFFGME09).
[[!include 7d spherical space forms -- table]]
* [[du Val singularities]] (see [[ADE singularity]])
* [[singularities]] of [[elliptic fibrations]] (see there are _[classification of singular fibers](elliptic+fibration#ClassificationOfSingularFibers)_)
* certain 4d [[ALE spaces]] ([Kronheimer 89](#Kronheimer89))
* certain [[2d CFTs]]
* certain [[6d (2,0)-supersymmetric QFT|6d CFTs]]
* intersection diagrams of vanishing 2-cycles in [[K3]]s (e.g. [BBS 07, p.423](#BBS07))
and many more.
[[!include ADE -- table]]
The obvious question for what might be the conceptual origin of this joint classification is attributed to ([Arnold 76](#Arnold76)).
Starting with ([Douglas-Moore 96](#DouglasMoore96)) is the observation that many of these structures are naturally aspects of the description of [[string theory]] [[KK-compactification|KK-compactified]] on _[[orbifolds]] with [[ADE singularities]]_ of the form $\mathbb{C}^n \sslash \Gamma$ for $\Gamma$ a [[finite group|finite]] [[subgroup]] of $SL_2(\mathbb{C})$.
## Via $N=2 $ super Yang-Mills theory
{#ViaSuperYangMillsTheory}
Various seemingly unrelated structures in mathematics fall into an "ADE classification". Notably [[finite group|finite]] [[subgroups]] of [[special unitary group|SU(2)]] and [[compact Lie group|compact]] [[simple Lie groups]] do. The way this works usually is that one tries to classify these structures somehow, and ends up finding that the classification is governed by the combinatorics of [[Dynkin diagrams]] (see also _[[McKay correspondence]]_).
While that does explain a bit, it seems the statement that both the [[icosahedral group]] and the Lie group [[E8]] are related to the same [[Dynkin diagram]] somehow is still more a question than an answer. Why is that so?
The first key insight is due to [Kronheimer 89](ADE+singularity#Kronheimer89a). He showed that the (resolutions of) the [[orbifold]] quotients $\mathbb{C}^2/\Gamma$ for finite subgroups $\Gamma$ of $SU(2)$ are precisely the generic form of the [[gauge group|gauge]] [[orbits]] of the [[direct product group]] of $U(n_i)$s acting in the evident way on the [[direct sum]] of $Hom(\mathbb{C}^{n_i}, \mathbb{C}^{n_j})$-s, where $i$ and $j$ range over the vertices of the [[Dynkin diagram]], and $(i,j)$ over its edges.
This becomes more illuminating when interpreted in terms of [[gauge theory]]: in a [[quiver gauge theory]] the [[gauge group]] is a [[direct product group]] of $U(n_i)$ factors associated with vertices of a [[quiver]], and the [[particles]] which are [[charged particle|charged]] under this gauge group arrange, as a [[linear representation]], into a [[direct sum]] of $Hom(\mathbb{C}^{n_i}, \mathbb{C}^{n_j})$-s, for each edge of the quiver.
Pick one such particle, and follow it around as the gauge group transforms it. The space swept out is its gauge [[orbit]], and [Kronheimer 89](#Kronheimer89) says that if the quiver is a Dynkin diagram, then this gauge orbit looks like $\mathbb{C}^2/\Gamma$.
On the other extreme, gauge theories are of interest whose gauge group is not a big direct product, but is a [[simple Lie group]], such as [[special unitary group|SU(N)]] or [[E8]]. The mechanism that relates the two classes of examples is [[spontaneous symmetry breaking]] ("[[Higgs field|Higgsing]]"): the ground state energy of the field theory may happen to be achieved by putting the fields at any one point in a higher dimensional space of field configurations, acted on by the gauge group, and fixing any one such point "spontaneously" singles out the corresponding [[stabilizer subgroup]].
Now here is the final ingredient: it is [[N=2 D=4 super Yang-Mills theory]] ("[[Seiberg-Witten theory]]") which have a potential that is such that its [[vacua]] break a simple gauge group such as $SU(N)$ down to a Dynkin diagram [[quiver gauge theory]]. One place where this is reviewed, physics style, is in [Albertsson 03, section 2.3.4](N=2+D=4+super+Yang-Mills+theory#Albertsson03).
More precisely, these theories have two different kinds of vacua, those on the "[[Coulomb branch]]" and those on the "[[Higgs branch]]" depending on whether the scalars of the "[[vector multiplets]]" (the gauge field sector) or of the "[[hypermultiplet]]" (the matter field sector) vanish. The statement above is for the Higgs branch, but the Coulomb branch is supposed to behave "dually".
So that then finally is the relation, in the ADE classification, between the simple Lie groups and the finite subgroups of SU(2): start with an N=2 super Yang Mills theory with gauge group a simple Lie group. Let it spontaneously find its vacuum and consider the orbit space of the remaining spontaneously broken symmetry group. That is (a resolution of) the orbifold quotient of $\mathbb{C}^2$ by a discrete subgroup of $SU(2)$.
## Related concepts
* [[24]]
* [[McKay correspondence]]
## References
### General Surveys
* {#Arnold76} [[Vladimir Arnold]], _Problems in present day mathematics_, (1976) in Felix E. Browder, _Mathematical developments arising from Hilbert problems_, Proceedings of symposia in pure mathematics 28, American Mathematical Society, p. 46, _Problem VIII. The A-D-E classifications_ (V. Arnold).
A survey is in
* Michael Hazewinkel, W Hesseling, Dirk Siersma, and Ferdinand Veldkamp, _The ubiquity of Coxeter Dynkin diagrams (an introduction to the ADE problem)_, Nieuw Archief voor Wiskunde 25 (1977), 257-307. ([pdf](http://oai.cwi.nl/oai/asset/10039/10039A.pdf))
which in turn is summarized in
* {#Siegel14} Kyler Siegel, _The Ubiquity of the ADE classification in Nature_ , 2014 ([pdf](http://math.stanford.edu/~ksiegel/ADEClassifications.pdf))
See also
* Wikipedia, _[ADE classification](http://en.wikipedia.org/wiki/ADE_classification)_
Discussion of the free finite group actions on spheres goes back to
* {#Milnor57} [[John Milnor]], _Groups which act on $S^n$ without fixed points_, American Journal of Mathematics Vol. 79, No. 3 (Jul., 1957), pp. 623-630 ([JSTOR](http://www.jstor.org/stable/2372566))
Review inclues
* {#Keenan03} [[Adam Keenan]], _Which finite groups act freely on spheres?_, 2003 ([pdf](http://www.math.utah.edu/~keenan/actions.pdf))
Discussion of [[ALE spaces]] via ADE include
* {#Kronheimer89} [[Peter Kronheimer]], _The construction of ALE spaces as hyper-Kähler quotients_, J. Differential Geom. Volume 29, Number 3 (1989), 665-683. ([Euclid](https://projecteuclid.org/euclid.jdg/1214443066))
Related stuff includes...
on [[immersions]] of [[3-spheres]] into $\mathbb{R}^4$:
* Shumi Kinjo, _Immersions of 3-sphere into 4-space associated with Dynkin diagrams of types A and D_ ([arXiv:1309.6526](http://arxiv.org/abs/1309.6526))
### In string theory
The original articles explaining the appearance of ADE classification from within [[string theory]] include
* {#DouglasMoore96} [[Michael Douglas]], [[Gregory Moore]], _D-branes, Quivers, and ALE Instantons_ ([arXiv:hep-th/9603167](http://arxiv.org/abs/hep-th/9603167))
* [[Clifford Johnson]], [[Robert Myers]], _Aspects of Type IIB Theory on ALE Spaces_, Phys.Rev. D55 (1997) 6382-6393 ([arXiv:hep-th/9610140](http://arxiv.org/abs/hep-th/9610140))
* [[Michael Douglas]], [[Brian Greene]], [[David Morrison]], _Orbifold Resolution by D-Branes_, Nucl.Phys. B506:84-106,1997 ([arXiv:hep-th/9704151](http://arxiv.org/abs/hep-th/9704151))
* [[Brian Greene]], [[Calin Lazaroiu]], Mark Raugas, _D-branes on Nonabelian Threefold Quotient Singularities_, Nucl.Phys. B553 (1999) 711-749 ([arXiv:hep-th/9811201](http://arxiv.org/abs/hep-th/9811201))
* Andrea Cappelli, Jean-Bernard Zuber, _A-D-E Classification of Conformal Field Theories_ ([arXiv:0911.3242](http://arxiv.org/abs/0911.3242))
Surveys include
* MO discussion, _[ADE classification from string theory](http://mathoverflow.net/a/34680/381)_
Discussion of an ADE-classification of BPS [[Freund-Rubin compactifications]] is in
* {#MFFGME09} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Sunil Gadhia]], [[Elena Méndez-Escobar]], _Half-BPS quotients in M-theory: ADE with a twist_, JHEP 0910:038,2009 ([arXiv:0909.0163](http://arxiv.org/abs/0909.0163), [pdf slides](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf))
Specifically the ADE classfication involved in the [[6d (2,0)-supersymmetric QFT]] on the [[M5-brane]] is discussed in
* {#Witten95} [[Edward Witten]], _Some Comments On String Dynamics_ ([arXiv:hep-th/9507121](http://arxiv.org/abs/hep-th/9507121))
* {#HeckmannMorrisonVafa13} [[Jonathan Heckman]], [[David Morrison]], [[Cumrun Vafa]], _On the Classification of 6D SCFTs and Generalized ADE Orbifolds_ ([arXiv:1312.5746](http://arxiv.org/abs/1312.5746))
Discussion in the context of [[M-theory on G2-manifolds]] includes
* {#Acharya02} [[Bobby Acharya]], section 3.1.1 of _M Theory, $G_2$-manifolds and Four Dimensional Physics_, Classical and Quantum Gravity Volume 19 Number 22, 2002 ([pdf](http://users.ictp.it/~pub_off/lectures/lns013/Acharya/Acharya_Final.pdf))
* {#BBS07} [[Katrin Becker]], [[Melanie Becker]], [[John Schwarz]], p. 423 of _String Theory and M-Theory: A Modern Introduction_, 2007
### In solid state physics
{#ReferencesInSolidStatePhysics}
On ADE-classifications in/of [[topological phases of matter]]/[[topological order]]:
* Mayukh Nilay Khan, Jeffrey C. Y. Teo, Taylor L. Hughes, *Anyonic Symmetries and Topological Defects in Abelian Topological Phases: an application to the ADE Classification*, Phys. Rev. B **90** 235149 (2014) $[$[arXiv:1403.6478](https://arxiv.org/abs/1403.6478)$]$
[[!redirects ADE classifications]]
[[!redirects ADE-classification]]
[[!redirects ADE-classifications]]
[[!redirects A-D-E classification]]
[[!redirects A-D-E classifications]]
|
ADE singularity | https://ncatlab.org/nlab/source/ADE+singularity |
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#### Geometry
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[[!include higher geometry - contents]]
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#Contents#
* table of contents
{:toc}
## Idea
An _ADE singularity_ is an [[orbifold]] [[fixed point]] locally of the form $\mathbb{C}^2\sslash\Gamma$ with $\Gamma \hookrightarrow SU(2)$ a [[finite subgroup of SU(2)]] given by the [[ADE classification]] (and $SU(2)$ is understood with its defining linear [[action]] on the [[complex numbers|complex]] [[vector space]] $\mathbb{C}^2$).
## Properties
### Resolution by spheres touching along a Dynkin diagram
{#ResolutionBySpheresTouchingAlongADynkinDiagram}
The [[blow-up]] of an ADE-singularity is given by a [[union]] of [[Riemann spheres]] that touch each other such as to form the shape of the [[Dynkin diagram]] whose A-D-E label corresponds to that of the given [[finite subgroup of SU(2)]].
This statement is originally due to ([duVal 1934 I, p. 1-3 (453-455)](#duVal1934I)). A description in terms of [[hyper-Kähler geometry]] is due to [Kronheimer 89a](#Kronheimer89a).
Quick survey of this fact is in [Reid 87](#Reid87), a textbook account is [Slodowy 80](#Slodowy80).
{#InStringTheory} In [[string theory]] this fact is interpreted in terms of [[gauge enhancement]] of the [[M-theory]]-lift of coincident [[black brane|black]] [[D6-branes]] to an [[MK6]] at an ADE-singularity ([Sen 97](enhanced+gauge+symmetry#Sen97)):
\begin{center}
\begin{imagefromfile}
"file_name": "ADESingularity.jpg",
"width": 760
\end{imagefromfile}
\end{center}
> graphics grabbed from [HSS18](#HSS18)
See at _[[M-theory lift of gauge enhancement on D6-branes]]_ for more.
$\,$
[[!include ADE -- table]]
### From coincident KK-monopoles
[[!include KK-monopole geometries -- table]]
### Bridgeland stability conditions
For $G_{ADE} \subset SU(2)$ a [[finite subgroup of SU(2)]],
let $\tilde X$ be the [[resolution of singularities|resolution]] of the corresponding ADE-singularity as [above](#ResolutionBySpheresTouchingAlongADynkinDiagram).
Then the [[connected component]] of the space of [[Bridgeland stability conditions]] on the bounded [[derived category]] of [[coherent sheaves]] over $\tilde X$ can be described explicitly ([Bridgeland 05](#Bridgeland05)).
Specifically for type-A singularities the space of stability conditions is in fact [[connected topological space|connected]] and [[simply-connected topological space]] ([Ishii-Ueda-Uehara 10](#IshiiUedaUehara10)).
Brief review is in [Bridgeland 09, section 6.3](#Bridgeland09).
## Related concepts
* [[singularity]]
* [[conical singularity]]
* [[quiver gauge theory]]
* [[M-theory on G2-manifolds]]
* [[D7-brane]]
## References
### General
Original articles include
* {#duVal1934I} [[Patrick du Val]], _On isolated singularities of surfaces which do not affect the conditions of adjunction. I_, Proceedings of the Cambridge Philosophical Society, 30 (4): 453–459 (1934a) ([doi:10.1017/S030500410001269X](https://doi.org/10.1017/S030500410001269X))
* {#duVal1934II} [[Patrick du Val]], _On isolated singularities of surfaces which do not affect the conditions of adjunction. II_, Proceedings of the Cambridge Philosophical Society, 30 (4): 460–465 (1934) ([doi:10.1017/S0305004100012706](https://doi.org/10.1017/S0305004100012706))
* {#duVal1934III} [[Patrick du Val]], _On isolated singularities of surfaces which do not affect the conditions of adjunction. III_, Proceedings of the Cambridge Philosophical Society, 30 (4): 483–491 (1934) ([doi:10.1017/S030500410001272X](https://doi.org/10.1017/S030500410001272X))
Textbook accounts include
* {#Durfee79} Alan H. Durfee, _Fifteen characterizations of rational double points and simple critical points_, L'Enseignement Mathématique Volume: 25 (1979) ([doi:10.5169/seals-50375](http://dx.doi.org/10.5169/seals-50375), [pdf](http://www.maths.ed.ac.uk/~v1ranick/papers/durfee15.pdf))
* {#Slodowy80} [[Peter Slodowy]], _Simple singularities and simple algebraic groups_, in Lecture Notes in Mathematics 815, Springer, Berlin, 1980.
* {#Lamotke86} [[Klaus Lamotke]], chapter IV of _Regular Solids and Isolated Singularities_, Vieweg, Braunschweig, Wiesbaden 1986.
* {#Reid87} [[Miles Reid]], _Young persons guide to canonical singularities, in [[Spencer Bloch]] (ed.), _[Algebraic geometry -- Bowdoin 1985, Part 1](https://www.ams.org/books/pspum/046.1/)_, Proc. Sympos. Pure Math. 46 Part 1, Amer. Math. Soc., Providence, RI, 1987, pp. 345-414 ([pdf](http://www.maths.ed.ac.uk/cheltsov/quotient/pdf/reid3.pdf))
(The last formula on page 409 has a typo: there should be no $r$ in the [[denominator]].)
Discussion of [[resolution of singularities|resolution]] of ADE-singularities in terms of [[hyper-Kähler geometry]]:
* {#Kronheimer89a} [[Peter Kronheimer]], _The construction of ALE spaces as hyper-Kähler quotients_, J. Differential Geom. Volume 29, Number 3 (1989), 665-683. ([euclid:1214443066](https://projecteuclid.org/euclid.jdg/1214443066))
* {#Kronheimer89b} [[Peter Kronheimer]], _A Torelli-type theorem for gravitational instantons_, J. Differential Geom. Volume 29, Number 3 (1989), 685-697 ([euclid:1214443067](https://projecteuclid.org/euclid.jdg/1214443067))
and in terms of preprojective algebras:
* {#CBH98} William Crawley-Boevey, Martin P. Holland, _Noncommutative deformations of Kleinian singularities_, Duke Math. J. Volume 92, Number 3 (1998), 605-635 ([euclid:1077231679](https://projecteuclid.org/euclid.dmj/1077231679))
Reviews and lecture notes:
* _[Classification of singularities](http://www.mathematik.uni-kl.de/~zca/Reports_on_ca/29/paper_html/node10.html)_
* {#Burban} Igor Burban, _Du Val Singularities_ ([pdf](http://www.mi.uni-koeln.de/~burban/singul.pdf))
* [[Miles Reid]], _The Du Val Singularities $A_n$, $D_n$, $E_6$, $E_7$, $E_8$_ ([pdf](http://homepages.warwick.ac.uk/~masda/surf/more/DuVal.pdf))
* Anda Degeratu, _Crepant Resolutions of Calabi-Yau Orbifolds_, 2004 ([pdf](https://home.mathematik.uni-freiburg.de/degeratu/crepant.pdf))
* {#Siegel14} Kyler Siegel, section 6 of _The Ubiquity of the ADE classification in Nature_ , 2014 ([pdf](http://math.stanford.edu/~ksiegel/ADEClassifications.pdf))
* MathOverflow, _[Resolving ADE singularities by blowing up](http://mathoverflow.net/q/186368/381)_
On the [[Chen-Ruan cohomology|Chen-Ruan]] [[orbifold cohomology]] of ADE-singularities:
* [[Fabio Perroni]], *Orbifold Cohomology of ADE-singularities* ([arXiv:0510528](https://arxiv.org/abs/math/0510528))
* [[Fabio Perroni]], *Chen-Ruan cohomology of ADE-singularities*, International Journal of Mathematics, Vol. 18, No. 9 (2007) 1009-1059 ([arXiv:math/0605207](https://arxiv.org/abs/math/0605207), [doi:10.1142/S0129167X07004436](https://doi.org/10.1142/S0129167X07004436))
Families of examples of [[G2 manifolds|G2 orbifolds]] with ADE singularities are constructed in
* {#Reidegeld15} [[Frank Reidegeld]], _$G_2$-orbifolds from K3 surfaces with ADE-singularities_ ([arXiv:1512.05114](http://arxiv.org/abs/1512.05114))
* [[Frank Reidegeld]], _$G_2$-orbifolds with ADE-singularities_ ([pdf](https://core.ac.uk/download/pdf/159317626.pdf))
[[Riemannian geometry]] of manifolds with ADE singularities is discussed in
* Boris Botvinnik, [[Jonathan Rosenberg]], _Positive scalar curvature on manifolds with fibered singularities_ ([arXiv:1808.06007](https://arxiv.org/abs/1808.06007))
See also
* Wikipedia, _[du Val singularity](https://en.wikipedia.org/wiki/Du_Val_singularity)_
### Via Bridgeland stability
Discussion of [[Bridgeland stability conditions]] for ([[resolution of singularities|resolutions of]]) ADE singularities includes:
* {#Bridgeland05} [[Tom Bridgeland]], _Stability conditions and Kleinian singularities_, International Mathematics Research Notices 2009.21 (2009): 4142-4157 ([arXiv:0508257](https://arxiv.org/abs/math/0508257))
* {#IshiiUedaUehara10} Akira Ishii, Kazushi Ueda, Hokuto Uehara, _Stability conditions on $A_n$-singularities_, Journal of Differential Geometry 84 (2010) 87-126 ([arXiv:math/0609551](https://arxiv.org/abs/math/0609551))
and specifically over [[Dynkin quivers]]
* {#Qiu15} [[Yu Qiu]], Def. 2.1 _Stability conditions and quantum dilogarithm identities for Dynkin quivers_, Adv. Math., 269 (2015), pp 220-264 ([arXiv:1111.1010](https://arxiv.org/abs/1111.1010))
* [[Tom Bridgeland]], [[Yu Qiu]], Tom Sutherland, _Stability conditions and the $A_2$ quiver_ ([arXiv:1406.2566](https://arxiv.org/abs/1406.2566))
### In string theory
{#ReferencesInStringTheory}
Discussion of [[ADE-singularities]] in [[string theory]] on [[orbifolds]]:
#### M-theory on ADE-orbifolds reducing to D6-branes in type II
[[M-theory lift of gauge enhancement on D6-branes]]:
* {#Sen97} [[Ashoke Sen]], _A Note on Enhanced Gauge Symmetries in M- and String Theory_, JHEP 9709:001,1997 ([arXiv:hep-th/9707123](http://arxiv.org/abs/hep-th/9707123))
* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], Section 6.3.3 of: _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012
#### Heterotic M-theory on ADE-orbifolds
[[heterotic M-theory on ADE-orbifolds]]:
* {#Sen97} [[Ashoke Sen]], _A Note on Enhanced Gauge Symmetries in M- and String Theory_, JHEP 9709:001,1997 ([arXiv:hep-th/9707123](http://arxiv.org/abs/hep-th/9707123))
* {#FLO99} Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], _Intersecting Orbifold Planes and Local Anomaly Cancellation in M-Theory_, Nucl. Phys. B554: 437-483, 1999 ([arXiv:hep-th/9903028](https://arxiv.org/abs/hep-th/9903028))
* {#FLO00a} Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], _Local Anomaly Cancellation, M-Theory Orbifolds and Phase-Transitions_, Nucl. Phys. B589: 269-291, 2000 ([arXiv:hep-th/0005251](https://arxiv.org/abs/hep-th/0005251))
* {#FLO00b} Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], _An M-Theory Perspective on Heterotic K3 Orbifold Compactifications_, Int. J. Mod. Phys. A18:3273-3314, 2003 ([arXiv:hep-th/0010087](https://arxiv.org/abs/hep-th/0010087))
* {#FLO00c} Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], _Twisted Sectors and Chern-Simons Terms in M-Theory Orbifolds_, Int. J. Mod. Phys. A18: 2995-3014, 2003 ([arXiv:hep-th/0011031](https://arxiv.org/abs/hep-th/0011031))
* {#KSTY99} [[Vadim Kaplunovsky]], J. Sonnenschein, [[Stefan Theisen]], S. Yankielowicz, _On the Duality between Perturbative Heterotic Orbifolds and M-Theory on $T^4/Z_N$_, Nuclear Physics B Volume 590, Issues 1–2, 4 December 2000, Pages 123-160 Nuclear Physics B ([arXiv:hep-th/9912144](https://arxiv.org/abs/hep-th/9912144), <a href="https://doi.org/10.1016/S0550-3213(00)00460-0">doi:10.1016/S0550-3213(00)00460-0</a>)
* {#GKSTY02} E. Gorbatov, [[Vadim Kaplunovsky]], J. Sonnenschein, [[Stefan Theisen]], S. Yankielowicz, _On Heterotic Orbifolds, M Theory and Type I' Brane Engineering_, JHEP 0205:015, 2002 ([arXiv:hep-th/0108135](https://arxiv.org/abs/hep-th/0108135))
* {#HSS18} [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_, CMP (2019) ([arXiv:1805.05987](https://arxiv.org/abs/1805.05987), [doi:10.1007/s00220-019-03442-3](http://link.springer.com/article/10.1007/s00220-019-03442-3))
#### Heterotic string theory on ADE-orbifolds
[[heterotic string theory]] on ADE-orbifolds:
* [[Paul Aspinwall]], [[David Morrison]], _Point-like Instantons on K3 Orbifolds_, Nucl. Phys. B503 (1997) 533-564 ([arXiv:hep-th/9705104](https://arxiv.org/abs/hep-th/9705104))
* {#Witten99} [[Edward Witten]], _Heterotic String Conformal Field Theory And A-D-E Singularities_, JHEP 0002:025, 2000 ([arXiv:hep-th/9909229](https://arxiv.org/abs/hep-th/9909229))
#### Type II-string theory on ADE-orbifolds
{#ReferencesTypeIIStringsOnADE}
The observation that the [[worldsheet]] [[2d CFT]] correspoding to a string probing (a [[D-brane]] on) an $A_{\kappa-1}$-type singularity $\mathbb{H}/_{C_{\kappa + 2}}$ is the chiral [[WZW model]] for the [[affine Lie algebra]] [[su(2)]] at level $\kappa - 2$ (plus some trivial summands):
* [[Hirosi Ooguri]] [[Cumrun Vafa]], p. 10-12 of: *Two-Dimensional Black Hole and Singularities of CY Manifolds*, Nucl. Phys. B **463** (1996) 55-72 ([arXiv:hep-th/9511164](https://arxiv.org/abs/hep-th/9511164), [doi:10.1016/0550-3213%2896%2900008-9](https://doi.org/10.1016/0550-3213%2896%2900008-9))
* [[Wolfgang Lerche]], [[Carsten Andrew Lütken]], [[Christoph Schweigert]], p. 4 of: *D-Branes on ALE Spaces and the ADE Classification of Conformal Field Theories*, Nucl.Phys. B **622** (2002) 269-278 ([doi:10.1016/S0550-3213%2801%2900613-7](https://doi.org/10.1016/S0550-3213%2801%2900613-7), [arXiv:hep-th/0006247](https://arxiv.org/abs/hep-th/0006247))
On how this $\widehat{\mathfrak{su}(2)}^{\kappa-2}$-CFT encodes the [[BPS states]] of [[special unitary group|$SU(\kappa)$]]-[[SYM]] on [[D3-branes]] transverse to the singularity:
* [[Wolfgang Lerche]], *On a Boundary CFT Description of Nonperturbative $\mathcal{N} = 2$ Yang-Mills Theory* ([arXiv:hep-th/0006100](https://arxiv.org/abs/hep-th/0006100))
An interpretation of this phenomenon, under the expected [[K-theory classification of D-brane charge]], as due to the (somewhat neglected) sector of [[twisted equivariant K-theory]] where the twist is by an [[inner local system]] which may appear inside an A-type singularity:
* [[Hisham Sati]], [[Urs Schreiber]], (around p. 23 of) *[[schreiber:Anyonic defect branes in TED-K-theory]]* ([arXiv:2203.11838](https://arxiv.org/abs/2203.11838))
#### Type $I'$-string theory on ADE-orbifolds
[[type I' string theory]] on ADE-orbifolds
* {#BergmanRodriguezGomez12} [[Oren Bergman]], Diego Rodriguez-Gomez, Section 3 of: _5d quivers and their $AdS_6$ duals_, JHEP07 (2012) 171 ([arxiv:1206.3503](https://arxiv.org/abs/1206.3503))
#### Type $I$-string theory on ADE-orbifolds
[[type I string theory]] on ADE-orbifolds
* [[Kenneth Intriligator]], _New String Theories in Six Dimensions via Branes at Orbifold Singularities_, Adv. Theor. Math. Phys.1:271-282, 1998 ([arXiv:hep-th/9708117](https://arxiv.org/abs/hep-th/9708117))
#### M-theory on $G_2$-orbifolds with ADE-singularities
[[M-theory on G2-manifolds]] $\,$ [with ADE-singularities](M-theory+on+G2-manifolds#EnhancedGaugeGroups):
* {#Acharya98} [[Bobby Acharya]], _M theory, Joyce Orbifolds and Super Yang-Mills_, Adv. Theor. Math. Phys. 3 (1999) 227-248 ([arXiv:hep-th/9812205](http://arxiv.org/abs/hep-th/9812205))
* {#Acharya00} [[Bobby Acharya]], _On Realising $N=1$ Super Yang-Mills in M theory_ ([arXiv:hep-th/0011089](http://arxiv.org/abs/hep-th/0011089))
* {#AcharyaSpence00} [[Bobby Acharya]], B. Spence, _Flux, Supersymmetry and M theory on 7-manifolds_ ([arXiv:hep-th/0007213](http://arxiv.org/abs/hep-th/0007213))
* {#Acharya02} [[Bobby Acharya]], _M Theory, $G_2$-manifolds and Four Dimensional Physics_, Classical and Quantum Gravity Volume 19 Number 22, 2002 ([pdf](http://users.ictp.it/~pub_off/lectures/lns013/Acharya/Acharya_Final.pdf))
* {#AtijayMaldacenaVafa00} [[Michael Atiyah]], [[Juan Maldacena]], [[Cumrun Vafa]], _An M-theory Flop as a Large N Duality_, J. Math. Phys.42:3209-3220, 2001 ([arXiv:hep-th/0011256](https://arxiv.org/abs/hep-th/0011256))
* {#BeasleyWitten02} [[Chris Beasley]], [[Edward Witten]], *A Note on Fluxes and Superpotentials in M-theory Compactifications on Manifolds of $G_2$ Holonomy*, JHEP 0207:046 (2002) ([arXiv:hep-th/0203061](http://arxiv.org/abs/hep-th/0203061))
* {#AtiyahWitten01} [[Michael Atiyah]], [[Edward Witten]] *M-Theory dynamics on a manifold of $G_2$-holonomy*, Adv. Theor. Math. Phys. 6 (2001) ([arXiv:hep-th/0107177](http://arxiv.org/abs/hep-th/0107177))
* {#Witten01} [[Edward Witten]], _Anomaly Cancellation On Manifolds Of $G_2$ Holonomy_ ([arXiv:hep-th/0108165](http://arxiv.org/abs/hep-th/0108165))
* {#AcharyaWitten01} [[Bobby Acharya]], [[Edward Witten]], _Chiral Fermions from Manifolds of $G_2$ Holonomy_ ([arXiv:hep-th/0109152](http://arxiv.org/abs/hep-th/0109152))
#### F-theory with ADE-singularities
[[F-theory]] with ADE-singularities
* [[Paul Aspinwall]], [[David Morrison]], _Point-like Instantons on K3 Orbifolds_, Nucl. Phys. B503 (1997) 533-564 ([arXiv:hep-th/9705104](https://arxiv.org/abs/hep-th/9705104))
See also at _[[F-branes -- table]]_
[[!redirects ADE singularities]]
[[!redirects ADE-singularity]]
[[!redirects ADE-singularities]]
[[!redirects ADE orbifold]]
[[!redirects ADE orbifolds]]
[[!redirects ADE-orbifold]]
[[!redirects ADE-orbifolds]]
[[!redirects du Val singularity]]
[[!redirects du Val singularities]]
|
Adeel Khan | https://ncatlab.org/nlab/source/Adeel+Khan | * [website](https://www.preschema.com)
Master's thesis:
* [[Adeel Khan]], _On derived categories and noncommutative motives of varieties_, [arXiv](http://arxiv.org/abs/1401.7222).
## Related entries
* [[Chow motive]]
* [[noncommutative motive]]
* [[derived algebraic geometry]]
* [[algebraic stacks]]
[[!redirects Adeel Khan Yusufzai]]
[[!redirects Adeel Ahmad Khan]] |
Adel Bilal | https://ncatlab.org/nlab/source/Adel+Bilal |
* [webpage](http://www.lpt.ens.fr/?page=membre&id=25&lang=fr)
## Selected writings
On the [[discrete light cone quantization]] formulation of [[M-theory]] in view of the [[BFSS matrix model]]:
* [[Adel Bilal]], *DLCQ of M-Theory as the Light-Like Limit*, Phys. Lett. B **435** (1998) 312-318 [[arXiv:hep-th/9805070](https://arxiv.org/abs/hep-th/9805070), <a href="https://doi.org/10.1016/S0370-2693(98)00811-9">doi:10.1016/S0370-2693(98)00811-9</a>]
Review of the [[BFSS matrix model]]:
* [[Adel Bilal]], *M(atrix) Theory : a Pedagogical Introduction*, Fortsch. Phys. **47** (1999) 5-28 [[arXiv:hep-th/9710136](https://arxiv.org/abs/hep-th/9710136), <a href="https://doi.org/10.1002/(SICI)1521-3978(199901)47:1/3%3C5::AID-PROP5%3E3.0.CO;2-B">doi:10.1002/(SICI)1521-3978(199901)47:1/3<5::AID-PROP5>3.0.CO;2-B</a>]
On the [[Green-Schwarz mechanism]] in [[heterotic M-theory]]:
* [[Adel Bilal]], Jean-Pierre Derendinger, Roger Sauser, _M-Theory on $S^1/\mathbb{Z}_2$ : New Facts from a Careful Analysis_, Nucl. Phys. B576 (2000) 347-374 ([arXiv:hep-th/9912150](https://arxiv.org/abs/hep-th/9912150))
On the [[non-abelian DBI action]]:
* [[Eric Bergshoeff]], [[Adel Bilal]], [[Mees de Roo]], A. Sevrin, _Supersymmetric non-abelian Born-Infeld revisited_, JHEP 0107, 029 (2001) [[arXiv:hep-th/0105274](https://arxiv.org/abs/hep-th/0105274)]
category: people |
adelic integral | https://ncatlab.org/nlab/source/adelic+integral |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Integration theory
+--{: .hide}
[[!include integration theory - contents]]
=--
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The concept of _adelic integration_ is a concept of [[integration]] in [[arithmetic geometry]] over [[rings of adeles]].
This was introduced for [[arithmetic curves]] in ([Tate 50](#Tate50), [Iwasawa 52](#Iwasawa52), [Iwasawa 92](#Iwasawa92)) and used to establish [[analytic continuation]], [[functional equations]] and [[Euler product]]-form of [[L-functions]] and [[zeta functions]], see [below](#ZetaFunctions). Here one speaks of _[[Iwasawa-Tate theory]]_. In ([Fesenko 08, chapter 3](#Fesenko08)) adelic integration is generalized to [[higher arithmetic geometry]]. A textbook account is in ([Goldfeld-Hundley 11, section 2,2](#GoldfeldHundley11))).
## Examples
### Zeta functions and L-functions
{#ZetaFunctions}
* The [completed Riemann zeta function](Riemann+zeta+function#TheCompletedZetaFunction) (see there) is naturally an adelic integral (reviewed e.g. in [Goldfeld-Hundley 11, (2.2.6)](#GoldfeldHundley11)).
For L-functions: [Wikipedia Rankin–Selberg method -- Modern adelic theory](http://en.wikipedia.org/wiki/Rankin–Selberg_method#Modern_adelic_theory)
### $p$-Adic string scattering amplitudes
In the context of [[p-adic string theory]] (see there) one considers adelic integral version of the [[Veneziano amplitude]] and of some further [[string scattering amplitudes]] for the [[open string]].
### Adelic path integral
Analogies with the [[Feynman path integral]] are pointed out in ([Fesenko 06, section 18](#Fesenko06))...
## Related entries
* [[Iwasawa-Tate theory]]
* [[Mellin transform]]
## References
The original articles are
* {#Tate50} [[John Tate]], _[[Fourier analysis in number fields, and Hecke's zeta-functions]]_, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 305–34 1950
* {#Iwasawa52} {#Iwasawa92} [[Kenkichi Iwasawa]], _A note on functions_, Proc. ICM 1950, [link](http://www.mathunion.org/ICM/ICM1950.1) [MR0044534](http://www.ams.org/mathscinet-getitem?mr=0044534); _On the rings of valuation vectors_, Ann. Math. (II) 57:2 (Mar., 1953), pp. 331-356 [jstor](http://www.jstor.org/stable/1969863); Letter to [[Jean Dieudonne]] [link](http://books.google.com/books?ei=jyALTq-_L4nkiAL6zrHXAQ)
The generalization to [[higher arithmetic geometry]] and hence to [[arithmetic zeta functions]] is due to
* {#Fesenko06} [[Ivan Fesenko]], _Measure, integration and elements of harmonic analysis on generalized loop spaces_, Proc. of
St Petersburg Math. Soc. 12(2005), 179–199; English transl. in AMS Transl. Series 2, 219 (2006), 149–164 ([pdf](http://www.maths.nott.ac.uk/personal/ibf/aoh.pdf))
* {#Fesenko08} [[Ivan Fesenko]], chaper 3 of _Adelic approch to the zeta function of arithmetic schemes in dimension two_, Moscow Math. J. 8 (2008), 273–317 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/ada.pdf))
* {#FesenkoAnalysisI} [[Ivan Fesenko]], _Analysis on arithmetic schemes I_ ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/a1.pdf))
* {#Fesenko10} [[Ivan Fesenko]], _Analysis on arithmetic schemes II_, Journal of K-theory 5: 437–557, 2010 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/a2.pdf))
A textbook account is in
* {#GoldfeldHundley11} [[Dorian Goldfeld]], [[Joseph Hundley]], chapter 2 of _Automorphic representations and L-functions for the general linear group_, Cambridge Studies in Advanced Mathematics 129, 2011 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/text/gl2.pdf))
[[!redirects adelic integrals]]
[[!redirects adelic integration]]
[[!redirects idelic integral]]
[[!redirects idelic integrals]]
[[!redirects idelic integration]] |
adequate equivalence relation | https://ncatlab.org/nlab/source/adequate+equivalence+relation | An [[algebraic cycle]] on a [[scheme]] $X$ of finite type over a [[field]] $k$ is a finite linear combination $\sum_{i=1}^r n_i Z_i$ of integral closed subschemes $Z_i\subset X$ with integral coefficients $n_i$. The algebraic cycles form a group $\mathcal{Z}= \mathcal{Z}_*$ of algebraic cycles on $X$ which is graded by the dimensions of the cycles. Sometimes (for equidimensional $X$) one looks at the grading by codimension $\mathcal{Z}_*$.
Let $\mathrm{SmProj}_k$ be the category of smooth projective [[algebraic variety|varieties]] over $k$. A rule giving an [[equivalence relation]] $\mathcal{Z}^*(X)$ for every $X$ in $\mathrm{SmProj}_k$, and which is compatible with grading is an **adequate equivalence relation** such that
1. For every pair of cycles $a,b\in \mathcal{Z}^*(X)$, there exists $a'\sim a$ such that $a'$ is transversal to $b$.
2. Consider a product $X\times Y$ in $\mathrm{SmProj}_k$, denote by $p_X,p_Y$ its projections. Consider cycles $a\in \mathcal{Z}^*(X)$ and $b\in \mathcal{Z}^*(X\times Y)$ such that $b$ and $p_X^*(a)$ intersect properly. Then $a\sim 0$ implies $p_Y_*(p_X^*(a)\cdot b) \sim 0$ where $p_X^*(a)\cdot b$ denotes the intersection product.
The [[intersection product]], which is associative but only partially defined on $\mathcal{Z}^*$, then becomes globally defined on $\mathcal{Z}^*/{\sim}$.
Typical choices are rational, algebraic and numerical adequate equivalence relations. The rational is the finest and the numerical is the coarsest nonzero adequate equivalence relation. |
adequate subcategory > history | https://ncatlab.org/nlab/source/adequate+subcategory+%3E+history | See [[dense functor]]. |
adhesive category | https://ncatlab.org/nlab/source/adhesive+category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--
# Adhesive categories
* table of contents
{: toc}
## Idea
An *adhesive category* is a [[category]] in which [[pushouts]] of [[monomorphisms]] exist and "behave more or less as they do in the category of [[Set|sets]]", or equivalently in any [[topos]].
## Definition
+-- {: .num_defn}
###### Definition
The following conditions on a [[category]] $C$ are equivalent. When they are satisfied, we say that $C$ is **adhesive**.
1. $C$ has [[pullbacks]] and [[pushouts]] of [[monomorphisms]], and pushout squares of monomorphisms are also pullback squares and are stable under pullback.
1. $C$ has pullbacks, and pushouts of monomorphisms, and the latter are also [[2-colimit|(bicategorical)]] pushouts in the [[bicategory]] of [[spans]] in $C$.
1. (If $C$ is small) $C$ has pullbacks and pushouts of monomorphisms, and admits a [[fully faithful functor|full embedding]] into a [[Grothendieck topos]] preserving pullbacks and pushouts of monomorphisms.
1. $C$ has pullbacks and pushouts of monomorphisms, and in any cubical diagram:
if $X\to Y$ is a monomorphism, the bottom square is a pushout, and the left and back faces are pullbacks, then the top face is a pushout if and only if the front and right face are pullbacks. In other words, pushouts of monomorphisms are [[van Kampen colimits]].
=--
## Properties
Notice that generally [[monomorphisms]] are preserved by [[pullback]] (see [there](monomorphism#MonomorphismsArePreservedByPullback)) but in a general category they may not need to be preserved by [[pushout]]. In an adhesive category, however, they are:
+-- {: .num_prop}
###### Proposition
In an adhesive category, suppose given a [[pushout]] square
$$
\array{
C
& \overset{m}{\longrightarrow}
&
A
\\
\mathllap{{}^f}\big\downarrow
&&
\big\downarrow\mathrlap{{}^g}
\\
B
&
\underset{n}{\longrightarrow}
&
D
}
$$
such that $m$ is a [[monomorphism]]. Then:
1. $n$ is also a monomorphism.
2. The square is also a [[pullback]] square.
3. The square is also a [[distributivity pullback]] around $(g,m)$; hence in particular $n = \forall_g m$ is the [[universal quantification]].
=--
For a proof of the above proposition, see ([Lack, prop. 2.1](#Lack)) and ([Lack-Sobocinski, Lemmas 2.3 and 2.8](#LS1)). The latter Lemma 2.8 states only that $n = \forall_g m$ (a weaker universal property since it refers only to other *monomorphisms* into $D$), but the proof applies more generally.
+-- {: .num_prop}
###### Proposition
An adhesive category with a [[strict initial object]] is automatically an [[extensive category]].
=--
We define a [[pushout complement]] of $m:C\to A$ and $g:A\to D$ to be a pair of arrows $f:C\to B$ and $n:B\to D$ such that $n f = g m$ and this [[commutative square]] is a [[pushout]]. The following proposition is crucial in [[double pushout graph rewriting]].
+-- {: .num_prop}
###### Proposition
In an adhesive category, if $m:C\to A$ is mono and $g:A\to D$ is any morphism, then if a pushout complement exists, it is unique up to unique isomorphism.
=--
+-- {: .proof}
###### Proof
We give only a sketch; details are in [(LS, Lemma 4.5)](#LS). If $(f,n)$ and $(f',n')$ are two pushout complements, consider the two pushout squares as morphisms in the [[arrow category]] with target $g$, and take their pullback. The resulting commutative cube can be viewed as a morphism in the category of commutative squares from the pullback square of $m$ against itself (which is again $m$, since $m$ is mono) to the pullback square of $n$ against $n'$. Denote the vertex of the latter pullback square by $U$. Applying the van Kampen property in two directions, we find that the maps $U\to B$ and $U\to B'$ are both pushouts of $1_C$, hence isomorphisms. This gives an isomorphism between the pushout complements; it is unique since $n$ and $n'$ are mono (being pushouts of the mono $m$).
=--
+-- {: .num_prop}
###### Proposition
In an adhesive category, every monomorphism is [[regular monomorphism|regular]].
In particular, every adhesive category is [[balanced category|balanced]].
=--
+-- {: .proof}
###### Proof
Let $m: A \to B$ be a monomorphism.
By adhesiveness, the cokernel pair (pushout)
$$
\array{
A
& \overset{m}{\longrightarrow}
&
B
\\
\mathllap{{}^m}\big\downarrow
&&
\big\downarrow\mathrlap{{}^u}
\\
B
&
\underset{v}{\longrightarrow}
&
P
\rlap{.}
}
$$
is a pullback.
This exhibits $m$ as an equalizer of $u$ and $v$.
=--
## Examples
{#Examples}
\begin{example}
Any [[topos]] is adhesive ([Lack-Sobocisnki](#LSToposes)). For [[Grothendieck toposes]] this is easy, because $Set$ is adhesive and adhesivity is a condition on colimits and finite limits, hence preserved by functor categories and left-exact localizations. For [[elementary toposes]] it is a theorem of [Lack and Sobocinski](#LSToposes).
The fact that [[monomorphisms]] are preserved by [[pushouts]] in toposes plays a central role for [[Cisinski model structures]] such as notably the [[classical model structure on simplicial sets]], where the [[class]] of monomorphisms is identified with the class of [[cofibrations]] and as such required to be closed under pushout (in particular).
\end{example}
\begin{example}
Some [[counter-examples]]:
* The [[1-category]] [[Cat]] of [[strict categories]] and [[functors]] is _not_ adhesive.
* Neither are the categories of [[posets]], [[topological spaces]], and [[groupoids]].
\end{example}
([Lack and Sobocinski, Counterexample 7](#LS)).
## Related concepts
Adhesiveness is an [[exactness property]], similar to being a [[regular category]], an [[exact category]], or an [[extensive category]]. In particular, it can be phrased in the language of "lex colimits".
## References
* {#LS1} [[Steve Lack]] and [[Pawel Sobocinski]], *Adhesive and quasiadhesive categories*, RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 39 (2005) no. 3, pp. 511-545, [Numdam](http://www.numdam.org/item/ITA_2005__39_3_511_0/),
[author PDF](https://www.ioc.ee/~pawel/papers/adhesivejournal.pdf)
* {#LS} [[Steve Lack]] and [[Pawel Sobocinski]], *Adhesive categories*,
In: Walukiewicz I. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2004. Lecture Notes in Computer Science, vol 2987. Springer, Berlin, Heidelberg. doi:[10.1007/978-3-540-24727-2_20](https://doi.org/10.1007/978-3-540-24727-2_20),
[author PDF](https://www.ioc.ee/~pawel/papers/adhesive.pdf)
* {#LSToposes} [[Steve Lack]] and [[Pawel Sobocinski]], *Toposes are adhesive*, In: Corradini A., Ehrig H., Montanari U., Ribeiro L., Rozenberg G. (eds) Graph Transformations. ICGT 2006. Lecture Notes in Computer Science, vol 4178. Springer, Berlin, Heidelberg. doi:[10.1007/11841883_14](https://doi.org/10.1007/11841883_14),
[author PDF](https://www.ioc.ee/~pawel/papers/toposesAdhesive.pdf)
* {#Lack} [[Steve Lack]], *An embedding theorem for adhesive categories*, Theory and Applications of Categories, Vol. 25, 2011, No. 7, pp 180-188. [journal page](http://www.tac.mta.ca/tac/volumes/25/7/25-07abs.html), arXiv:[1103.0600](https://arxiv.org/abs/1103.0600)
* {#GarnerLack12} [[Richard Garner]], [[Steve Lack]]: *On the axioms for adhesive and quasiadhesive categories*, Theory and Applications of Categories, **27** 3 (2012) 27-46 [[arXiv:1108.2934](http://arxiv.org/abs/1108.2934), [tac:27-03](http://www.tac.mta.ca/tac/volumes/27/3/27-03abs.html)]
[[!redirects adhesive categories]]
|
ADHM construction | https://ncatlab.org/nlab/source/ADHM+construction |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A method for constructing [[Yang-Mills instanton]] solutions. ([ADHM](#ADHM))
## Related concepts
* [[Nahm transform]]
* [[Skyrmion]]
## References
The original articles:
* {#ADHM} [[Michael Atiyah]], [[Nigel Hitchin]], [[Vladimir Drinfeld]], [[Yuri Manin]], *Construction of instantons*, Physics Letters A **65** 3 (1978) 185-187 [<a href="https://doi.org/10.1016/0375-9601(78)90141-X">doi:10.1016/0375-9601(78)90141-X</a>]
* Norman H. Christ, Erick J. Weinberg, and Nancy K. Stanton, *General self-dual Yang-Mills solutions*, Phys. Rev. D **18** 6 (1978) [[doi:10.1103/PhysRevD.18.2013](https://doi.org/10.1103/PhysRevD.18.2013)]
> (emphasis on well-separated instanton solutions)
Surveys:
* [[Bert Lindenhovius]], *Instantons and the ADHM construction*, Amsterdam (2011) [[[Lindenhovius-Instantons.pdf:file]]]
* {#DoreyHollowoodKhozeMattis02} Nick Dorey, Timothy J. Hollowood, Valentin V. Khoze, Michael P. Mattis, *The Calculus of Many Instantons*, Phys. Rept. **371** (2002) 231-459 [[arXiv:hep-th/0206063](https://arxiv.org/abs/hep-th/0206063), <a href="https://doi.org/10.1016/S0370-1573(02)00301-0">doi:10.1016/S0370-1573(02)00301-0</a>]
In 8 dimensions:
* Atsushi Nakamula, Shin Sasaki, Koki Takesue, *ADHM construction of (anti-)self-dual instantons in eight dimensions*, Nuclear Physics B **910** (2016) 199-224 [[arXiv:1604.01893](https://arxiv.org/abs/1604.01893), [doi:10.1016/j.nuclphysb.2016.06.026](https://doi.org/10.1016/j.nuclphysb.2016.06.026)]
For the relation to [[Skyrmions]] see there.
[[!redirects ADHM instanton]]
[[!redirects ADHM instantons]]
|
adiabatic | https://ncatlab.org/nlab/source/adiabatic |
For the moment there is:
* [[adiabatic switching]]
* [[quantum adiabatic theorem]]
## References
See also
* Wikipedia, *[List of adiabatic concepts](https://en.wikipedia.org/wiki/List_of_adiabatic_concepts)*
category: disambiguation
|