cosmos

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###Context###
#### Category theory
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[[!include category theory - contents]]
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#### Enriched category theory
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[[!include enriched category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In ([[enriched category theory|enriched]]) [[category theory]] by a *cosmos* one means a "good place in which to do (enriched) category theory". Typically what is meant is a [[base of enrichment]] $V$ (being by default a [[monoidal category]] or more generally a [[bicategory]] or similar) which carries enough extra structure for intended [[category theory|category-theoretic]]-constructions (such as [[enriched functor categories]], etc.) to exist in $V$-[[enriched category theory]].

The word is chosen by analogy with the notion of *[[toposes]]* which may be regarded as "a good places to do [[set theory]]".

## Definitions

There are a number of different, inequivalent, definitions of "cosmos" in the literature.

### Bénabou cosmoi

A [[Bénabou cosmos]] (see [Street 74, p. 1](#Street74)) is a [[complete category|complete]] and [[cocomplete category|cocomplete]] (hence [[bicomplete category|bicomplete]]) [[closed monoidal category|closed]] [[symmetric monoidal category]].

This is the ideal situation for $V$-[[enriched category theory]].

### Indexed Bénabou cosmoi

[Shulman (2013)](#Shulman13) introduced an [[indexed category|indexed]] generalization of [[Bénabou cosmoi]], appropriate for studying enriched [[indexed categories]] over a base category.  Notably, while the definition of [[Bénabou cosmoi]] is not "[[elementary topos|elementary]]" (it involves infinitary (non-[[finite limit|finite]]) [[limits]] and [[colimits]]), the indexed version is elementary; the infinitary structure is folded into the indexing base category.  The notion of [[Bénabou cosmoi]] is recovered as particular cosmoi indexed over [[Set]].

### Street's "fibrational cosmoi"

[[Ross Street]] has taken a different tack, defining a "cosmos" to be the collection of (enriched) categories and relevant structure for doing category theory, rather than the "base" category $V$ over which the enrichment occurs.

[Street (1974)](#Street74) defined a **(fibrational) cosmos** to be a [[2-category]] in which internal [[Grothendieck fibrations|fibrations]] are well-behaved and representable by a structure of "presheaf objects" (later realized to be a special sort of [[Yoneda structure]]).  Note that while this includes [[Cat]], it does *not* include [[VCat]] for non-[[cartesian monoidal category|cartesian]] $V$, since [[internal fibrations]] are poorly behaved there. The definition is given in [Street (1980)](#Street80):

A *fibrational cosmos* is a [[2-category]] $K$ such that

* [[finite limits]] exist in $K$;

* there is a [[2-adjunction]] $P^* \colon K \leftrightarrows K^{\text{coop}} : P$;

* each object $A$ admits a [[discrete fibration]] from $P A$ satisfying two technical properties.

The objects $P A$ are the "presheaf objects" that represent fibrations.

### Street's second definition

[Street (1981)](#Street81) instead defines a cosmos to be a [[2-category]] that "behaves like the 2-category $V$-$Mod$ of [[enriched categories]] and [[profunctors]]".  The precise definition:

A cosmos is a 2-category (or [[bicategory]]) such that:

* Small ([[2-limit|weak, or bi-]]) [[coproducts]] exist.

* Each [[monad]] admits a [[Kleisli construction]] (analogous to the [[exact category|exactness]] of a topos).

* It is locally small-[[cocomplete category|cocomplete]], i.e. its hom-categories have small colimits that are preserved by composition on each side.

* There exists a small "Cauchy generator".

These hypotheses imply that it is equivalent to the bicategory of categories and [[profunctors]] [[enriched category|enriched]] over some "base" *bicategory*.  (Note the generalization from enrichment over a monoidal category to enrichment over a bicategory.)

Defined in this way, cosmoi are closed under [[opposite category|dualization]], parametrization and [[localization]], suitably defined.

## Related notions

* [[Bénabou cosmos]]

* [[base of enrichment]]

* An [[infinity-cosmos]] is a "good place in which to do higher category theory" as axiomatized by Riehl and Verity in their work on the foundations of $(\infty,1)$- and $(\infty,n)$-category theory.

## References

Apparently there is no written account by [[Jean Bénabou]] of his definition of cosmos. One finds it recounted in [Street 74, p. 1](#Street74):

> "to J. Bénabou the word means "bicomplete symmetric monoidal category", such categories $\mathcal{V}$ being rich enough so that the theory of categories enriched in  $\mathcal{V}$ develops to a large extent just as the theory of ordinary categories."

* {#Street74} [[Ross Street]],  *Elementary cosmoi I.* in *Category Seminar*, Lecture Notes in Mathematics **420**, Springer (1974) [[doi:10.1007/BFb0063103](https://doi.org/10.1007/BFb0063103)]

* {#Street80} [[Ross Street]], *Cosmoi of internal categories*, Transactions of the American Mathematical Society **258** 2 (1980) 271-318 [[doi:10.2307/1998059](https://doi.org/10.2307/1998059)]

* {#Street81} [[Ross Street]], *Cauchy characterization of enriched categories*, Rend. Sem. Mat. Fis. Milano 51 (1981): 217-233, Reprints in Theory and Applications of Categories, **4** (2004) 1-16 [[tac:tr4](http://www.tac.mta.ca/tac/reprints/articles/4/tr4abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/4/tr4.pdf)]

* {#Shulman13} [[Mike Shulman]], *Enriched indexed categories*, Theory and Applications of Categories, **28** 21 (2013) 616-695 ([tac:28-21](http://www.tac.mta.ca/tac/volumes/28/21/28-21abs.html))

[[!redirects cosmoi]]

[[!redirects cosmos for enrichment]]


cosocle

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[Specht module]]

## References

See also:

* Wikipedia, *[Cosocle](https://en.wikipedia.org/wiki/Cosocle)*

[[!redirects cosocle]]


cospan

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## Definition

In any [[category]], a **cospan** is a diagram like this:
$$
  \array{
     && a &&&& b
      \\
      &
      && {}_{f}\searrow
       &
       & \swarrow_g
      &&
     \\
     &&&&
     c
     &&&&
  }
$$

A cospan in the category $C$ is the same as a [[span]] in the [[opposite category]] $C^{op}$.  So, all general facts about cospans in $C$ are general facts about spans in $C^{op}$, and the reader may turn to the entry on [[span|spans]] to learn more.

A cospan that admits a [[cone]] is called a [[quadrable cospan]].

Cospans in a category $V$ with small colimits form a [[bicategory]] whose objects are the objects of $V$, whose morphisms are cospans between two objects, and whose 2-morphisms $\eta$ are commuting diagrams of the form

$$
  \array{
    && S
    \\
    & {}^{\sigma_{S}}\nearrow && \nwarrow^{\tau_S}
    \\
    a &&\downarrow^\eta&& b
    \\
    & {}_{\sigma_T}\searrow
    && \swarrow_{\tau_T}
    \\
    && T
  }
  \,.
$$

The category of cospans from $a$ to $b$ is naturally a [[enriched category|category enriched in]] $V$: for

$$
  \array{
    && S
    \\
    & {}^{\sigma_{S}}\nearrow && \nwarrow^{\tau_S}
    \\
    a &&&& b
    \\
    & {}_{\sigma_T}\searrow
    && \swarrow_{\tau_T}
    \\
    && T
  }
$$

two parallel cospans in $V$, the $V$-object
${}_a[S,T]_b$
of morphisms between them is the pullback

$$
  \array{
    {}_a[S,T]_b
     &\to&
     pt
    \\
    \downarrow && \downarrow^{\sigma_T \times \tau_T}
    \\
    [S,T] &\stackrel{\sigma_S^* \times \sigma_T^*}{\to}&
    [a \sqcup b, T]
  }
$$

formed in analogy to the enriched hom of [[pointed object]]s. The [[initial object]] of $V$ is the [[coproduct]] of $a$ and $b$.

If $V$ has a terminal object, $pt$, then cospans from $pt$ to itself are [[bi-pointed object]]s in $V$.

##Related concepts

* [[cospan cotrace]]
* [[multi-cospan]]
* [[Cospans in Algebraic Topology]]
* [[multispan]]
* the [cograph of a function](http://ncatlab.org/nlab/show/graph+of+a+function#cograph)
* the [[cograph of a functor]]

* [[sink]], [[cosink]]

* [[product]], [[coproduct]]

##References

Topological cospans and their role as models for [[cobordism]]s are discussed in

* Marco Grandis, _Collared cospans, cohomotopy and TQFT (Cospans in algebraic topology, II)_ ([pdf](http://www.dima.unige.it/~grandis/wCub2.pdf))

[[!redirects cospan]]
[[!redirects cospans]]
[[!redirects opspan]]
[[!redirects opspans]]
[[!redirects co-span]]
[[!redirects co-spans]]


cospan cotrace

[[!redirects co-span co-trace]]
#Idea#

One can naturally think of a [[cospan]] as the abstraction of a [[cobordism]]. For instance an [[interval object]] cospan models the standard topological interval $[0,1]$ regarded as a cobordism from pt to pt. The _cospan cotrace_ on the interval glues the two ends of the interval together to produce a _circle_ regarded as a cospan from $\emptyset$ to itself.

The [[duality|concrete dual]] of a cospan, obtained by mapping it into some target object, is a [[span]], which in the context of [[groupoidification]] and [[geometric function theory]] can be interpreted as a generalized linear map. On such a generalized linear map, there is a notion of _trace_, the [[span trace]].

The _cospan cotrace_ is the concept dual to that: the image of the cotrace of a cospan under mapping it into a target object is the span trace of the result of mapping the original cospan to that target object.

#Definition#

For

$$
  \array{
    && T
    \\
    & {}^{in}\nearrow && \nwarrow^{out}
    \\
    \Sigma &&&& \Sigma
  }
$$

a [[cospan]] with identical left and right index object $\Sigma$, its **cospan cotrace** $cotr(T)$ is the composite of the result

$$
  \array{
    && T
    \\
    & {}^{in \sqcup out}\nearrow && \nwarrow
    \\
    \Sigma \sqcup \Sigma &&&& \emptyset
  }
$$

of dualizing one leg of the cospan with the cospan

$$
  \array{
    && \Sigma
    \\
    & {}^{}\nearrow && \nwarrow^{Id \sqcup Id}
    \\
    \emptyset &&&& \Sigma \sqcup \Sigma
  }
$$

i.e. the [[pushout]]

$$
  \array{
    &&&&
    \mathrm{cotr}T
    \\
    &&& \nearrow && \nwarrow
    \\
    && \Sigma &&&& T
    \\
    & {}^{}\nearrow && \nwarrow^{Id \sqcup Id}
    && {}^{in \sqcup out}\nearrow && \nwarrow
    \\
    \emptyset &&&& \Sigma \sqcup \Sigma &&&& \emptyset
  }
$$

regarded as a cospan from the initial object
$\emptyset$ to $\emptyset$

$$
  \array{
    && cotr(T)
    \\
    & {}^{}\nearrow && \nwarrow
    \\
    \emptyset &&&& \emptyset
  }
  \,.
$$

## Definition for multi-cospans

More generally, the trace of a [[multi-cospan]] over $n$ identical of its index objects $\Sigma$ is the composite with the multi-cospan

$$
  \array{
    & \Sigma
    \\
    & {}^{Id}\nearrow \uparrow^{Id} & \cdots
    \\
    \Sigma & \Sigma & \cdots & \Sigma & \cdots
  }
$$

#Examples

## Cotracing topological interval to circle

Let the ambient category be [[Top]], let $I = [0,1]$ be the standard topological interval and let $e := [0,\epsilon]$ be a small interval, for some $0 \lt \epsilon \lt 1/2$ -- to be thought here as a _[[collar]]_ of the point $pt$.

Let

$$
  \array{
     && I
     \\
     & {}\nearrow && \nwarrow^{1-\epsilon+(-)}
     \\
     e &&&& e
  }
$$

be the interval regarded as a collared cobordisms from the point to the point. Its cotrace, the pushout

$$
  \array{
     cotr(I) &\leftarrow& I
     \\
     \uparrow && \uparrow^{in \sqcup out}
     \\
     e &\stackrel{Id \sqcup Id}{\leftarrow}& e \sqcup e
  }
$$

is the result of gluing the ends of the interval to each other, i.e. the circle

$$
  cotr(I) = S^1
  \,.
$$

+--{.query}
[[Urs Schreiber|Urs]]: This may require a bit more care
with the topology involved. I still need to check the reference below for more details.
=--

See also

* Marco Grandis, _Collared cospans, cohomotopy and TQFT (Cospans in Algebraic Topology II)_ ([pdf](http://www.dima.unige.it/~grandis/wCub2.pdf))

## Cotracing category interval object to the natural numbers

Let the ambient category be [[Cat]], let $I = \{a \to b\}$ be the standard [[interval object]] in [[Cat]] and let
$pt = \{\bullet\}$ be the terminal category.

Let

$$
  \array{
     && I
     \\
     & {}^{pt \mapsto a}\nearrow && \nwarrow^{pt \mapsto b}
     \\
     pt &&&& pt
  }
$$

be the standard [[interval object]] in [[Cat]]
regarded in the standard way as a cospan from the
point to the point.

Dualizing it to

$$
  \array{
     && I
     \\
     & {}^{in \sqcup out}\nearrow && \nwarrow^{}
     \\
     pt \sqcup pt &&&& \emptyset
  }
$$

corresponds to thinking of it as a "bent interval"

$$
  \array{
    pt
    \\
    & \searrow
    \\
    && \downarrow
    \\
    & \swarrow
    \\
    pt
  }
  \,.
$$

Accordingly, the co-span

$$
  \array{
     && pt
     \\
     & {}^{}\nearrow && \nwarrow^{Id \sqcup Id}
     \\
     \emptyset &&&& pt \sqcup pt
  }
$$

can be thought of as

$$
  \array{
    & pt
    \\
    \nearrow
    \\
    \nwarrow
    \\
    & pt
  }
  \,.
$$

Gluing these two arcs together yields the
cotrace, the pushout

$$
  \array{
     cotr(I) &\leftarrow& I
     \\
     \uparrow && \uparrow^{in \sqcup out}
     \\
     pt &\stackrel{Id \sqcup Id}{\leftarrow}& pt \sqcup pt
  }
  \,,
$$

which is the result of gluing the ends of the interval object to each other, which here is the [[skeleton]] of the [[fundamental category]] of the [[directed space|directed circle]]

$$
  \array{
    && \rightarrow
    \\
    & \nearrow && \searrow
    \\
    \uparrow &&&& \downarrow
    \\
    & \nwarrow && \swarrow
    \\
    && \leftarrow
  }
  \,,
$$

namely the [[monoid]] of natural numbers, regarded as a one-object category:

$$
  cotr(I) = \mathbf{B} \mathbb{N} = \{\bullet \stackrel{n}{\to} \bullet | n \in \mathbb{N}\}
  \,.
$$

If instead we start with the standard interval object in groupoids, $I_{inv} = \{a \stackrel{\simeq}{\to} b\}$ with the nontrivial morphism from $a$ to $b$ being an [[isomorphism]], then the co-trace in question is the [[skeleton]] of the [[fundamental groupoid]] of the ordinary [[topological space|topological circle]]

$$
  cotr(I_{inv}) = \mathbf{B} \mathbb{Z} = \{\bullet \stackrel{n}{\to} \bullet | n \in \mathbb{Z}\}
  \,.
$$

#Remarks#

* The dual notion is that of [[span trace]]

#References#

While the concept is obvious, it is apparently (?) not discussed yet in the (young) literature on the subject. On the blog the concept was mentioned in

* Urs Schreiber, [(co)-traces](http://golem.ph.utexas.edu/category/2008/05/hopkinslurie_on_baezdolan.html#c021537)


Cospans in Algebraic Topology

A series of articles by [[Marco Grandis]] on describing [[extended cobordism]]s in terms of [[multi-cospan]]s in [[Top]]

* I, Higher cospans and weak cubical categories ([pdf](http://www.tac.mta.ca/tac/volumes/18/12/18-12.pdf))

* II, Collared cospans, cohomotopy and TQFT ([pdf](http://www.dima.unige.it/~grandis/wCub2.pdf))

* III: Cubical cospans and higher cobordisms ([arXiv](http://arxiv.org/abs/0806.2359))

Some of the crucial ideas appearing here are

* a notion of cubical [[multispan|multi-cospans]] in some category;

* the observation that [[extended cobordism]]s should be modeled by such multi-cospans (in particular in [[Top]]);

* the observation that such cospans naturally map into [[span]]s by homming them into target objects, and that such spans can be linearized such as to yield [[FQFT]]s.

  * (Grandis seems not to mention (?) the fact that [[groupoidification]] makes this interpretation of spans as linear maps very natural)

* the relevance of evaluating functors which preserve (homotopy) [[pushout]]s on (multi)-cospans.

category: reference


cospectrum

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[[!include stable homotopy theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The concept of a _cospectrum_ is the [[formal dual]] to the concept of a ([[sequential spectrum|sequential]]) [[spectrum object]]. Where a [[sequential spectrum]] is a [[sequence]] of [[objects]] $(X_n)$ equipped with a sequence of [[morphisms]] of the form $\Sigma X_n \to X_{n+1}$ (for $\Sigma$ a [[reduced suspension]] operation), a _cospectrum_ is such a sequence of objects, but with the direction of the structure maps reversed: $X_{n+1} \to \Sigma X_{n}$.

## References

* Elon L. Lima, _The Spanier-Whitehead duality in new homotopy categories_, Summa Brasil. Math. __4__ (1959) 91–148 (1959) [MR0116332](http://www.ams.org/mathscinet-getitem?mr=116332)

* A. Jankowski, _Remarks on generalized homology theories_, Comment. Math. Prace Mat. __15__ (1971) 185–199 [MR405404](http://www.ams.org/mathscinet-getitem?mr=405404)

* {#Hikida81} Mizuho Hikida, _On CW cospectra_, Hiroshima Math. J. Volume 11, Number 2 (1981), 347-368 ([Euclid](https://projecteuclid.org/euclid.hmj/1206134106))

[[!redirects cospectra]]

[[!redirects co-spectrum]]
[[!redirects co-spectra]]

[[!redirects cospectrum object]]
[[!redirects cospectrum objects]]

[[!redirects co-spectrum object]]
[[!redirects co-spectrum objects]]


Costas Kounnas

* [webpage](http://www.theorie.physik.uni-muenchen.de/MATH/members/asc/professors/kounnas/index.html)

## related $n$Lab entries

* [[Starobinsky model of cosmic inflation]]

category: people


cotangent bundle

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### Context
#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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# Contents
* table of contents
{: toc}

## Idea

Given a [[differentiable manifold]] $X$, the __cotangent bundle__ $T^*(X)$ of $X$ is the [[dual vector bundle]] over $X$ dual to the [[tangent bundle]] $T x$ of $X$.

A __cotangent vector__ or __covector__ on $X$ is an element of $T^*(X)$.  The __cotangent space__ of $X$ at a point $a$ is the [[fiber]] $T^*_a(X)$ of $T^*(X)$ over $a$; it is a [[vector space]].  A __covector field__ on $X$ is a [[section]] of $T^*(X)$.  (More generally, a [[differential form]] on $X$ is a section of the [[exterior algebra]] of $T^*(X)$; a covector field is a __[[differential 1-form]]__.)

Given a covector $\omega$ at $a$ and a [[tangent vector]] $v$ at $a$, the pairing $\langle{\omega,v}\rangle$ is a [[scalar]] (a [[real number]], usually).  This (with some details about linearity and universality) is basically what it means for $T^*(X)$ to be the [[dual vector bundle]] to $T_*(X)$.  More globally, given a covector field $\omega$ and a [[tangent vector field]] $v$, the paring $\langle{\omega,v}\rangle$ is a scalar [[function]] on $X$.

Given a point $a$ in $X$ and a differentiable (real-valued) [[partial function]] $f$ defined near $a$, the __differential__ $\mathrm{d}_a f$ of $f$ at $a$ is a covector on $X$ at $a$; given a tangent vector $v$ at $a$, the pairing is given by
$$ \langle{\mathrm{d}_a f, v}\rangle = v[f] ,$$
thinking of $v$ as a [[derivation]] on [[differentiable functions]] defined near $a$.  (It is really the [[germ]] at $a$ of $f$ that matters here.)  More globally, given a differentiable function $f$, the __[[de Rham differential]]__ $\mathrm{d}f$ of $f$ is a covector field on $X$; given a vector field $v$, the pairing is given by
$$ \langle{\mathrm{d}f, v}\rangle = v[f] ,$$
thinking of $v$ as a derivation on differentiable functions.

One can also *define* covectors at $a$ to be germs of differentiable functions at $a$, modulo the [[equivalence relation]] that $\mathrm{d}_a f = \mathrm{d}_a g$ if $f - g$ is constant on some neighbourhood of $a$.  In general, a covector field won\'t be of the form $\mathrm{d}f$, but it will be a sum of terms of the form $h \mathrm{d}f$.  More specifically, a covector field $\omega$ on a coordinate patch can be written
$$ \omega = \sum_i \omega_i\, \mathrm{d}x^i $$
in local coordinates $(x^1,\ldots,x^n)$.
This fact can also be used as the basis of a definition of the cotangent bundle.

## Properties

### Symplectic structure
 {#SymplecticStructure}

Every cotangent bundle $T^\ast X$ carries itself a canonical [[differential 1-form]]

$$
  \theta
   \in
  \Omega^1(T^* X)
$$

with the property that under the [[isomorphism]]

$$
  j \;\colon\; \Gamma(T^* X) \stackrel{\simeq}{\to} \Omega^1(X)
$$

between [[differential 1-forms]] and smooth [[sections]] of the [[cotangent bundle]] we have for every smooth section $\sigma \in \Gamma(T^* X)$ the identification

$$
  \sigma^* \theta = j(\sigma)
$$

between the [[pullback of a differential form|pullback]] of $\theta$ along $\sigma$ and the 1-form corresponding to $\sigma$ under $j$.

This unique differential 1-form $\theta \in \Omega^1(T^* X)$ is called the **[[Liouville-Poincaré 1-form]]** or **canonical form** or **tautological form** on the cotangent bundle.

The [[de Rham differential]] $\omega \coloneqq d \theta$ is a [[symplectic form]]. Hence every cotangent bundle is canonically a [[symplectic manifold]].

On a [[coordinate chart]] $\mathbb{R}^n$ of $X$ with canonical [[coordinate functions]] denoted $(x^i)$, the cotangent bundle over the chart is $T^\ast \mathbb{R}^n \simeq \mathbb{R}^n \times \mathbb{R}^n$ with canonical coordinates $((x^i), (p_j))$. In these coordinates the canonical 1-form is (using [[Einstein summation convention]])

$$
  \theta = p_i d x^i
$$

and hence the [[symplectic form]] is

$$
  \omega = d p_i \wedge d q^i
  \,.
$$

## Related concepts

* [[tangent bundle]]

* [[differential form]]

* [[exterior bundle]]

* [[Liouville-Poincaré 1-form]]

* [[principal symbol]], [[bicharacteristic flow]]

* [[wavefront set]], [[microlocal analysis]]

* [[microsupport]], [[microlocal sheaf theory]]

[[!redirects cotangent vector]]
[[!redirects cotangent vectors]]
[[!redirects covector]]
[[!redirects covectors]]

[[!redirects cotangent space]]
[[!redirects cotangent spaces]]
[[!redirects cotangent vector space]]
[[!redirects cotangent vector spaces]]

[[!redirects covector field]]
[[!redirects covector fields]]
[[!redirects cotangent vector field]]
[[!redirects cotangent vector fields]]
[[!redirects differential 1-form]]
[[!redirects differential 1-forms]]
[[!redirects smooth differential 1-form]]
[[!redirects smooth differential 1-forms]]
[[!redirects 1-form]]
[[!redirects 1-forms]]

[[!redirects cotangent bundle]]
[[!redirects cotangent bundles]]
[[!redirects Cotangent bundle]]
[[!redirects Cotangent bundles]]


cotangent complex

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###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of **cotangent complex** is a _derived_  or [[(∞,1)-category|(∞,1)-categorical]] refinement of the notion of [[Kähler differential]]s.

Traditionally this has been conceived in terms of [[model category]] presentations. This we discuss in the section

* [Model category presentation](#ModCat).

From the [[nPOV]], the notion of cotangent complex has a more intrinsic description as being the [[left adjoint]] to the [[tangent (∞,1)-category]] projection. This we discuss in the section

* [(∞,1)-Categorical description](#InfCat)

### Motivation

The construction of [[moduli space]]s depends strongly on local properties like smoothness and [[transversal map|transversality]] of intersections when trying to [[representable functor|represent]] the [[functor]] of assigning families of objects to the varying base of family. When passing to the classes of equivalent objects, one faces the problem of having nontrivial [[automorphism]]s.

At the [[infinitesimal object|infinitesimal]] level [[automorphism]]s correspond to the [[derivation]]s. Taking derivations is represented by the [[module]] of relative [[Kähler differential]]s which sufficies in good cases. Its correct [[derived algebraic geometry|derived]] replacement is the **cotangent complex** of Grothendieck-Illusie. One can typically split the information about a map of higher rings into its "discrete part" and infinitesimal obstruction theory governed by the cotangent complex.

## Model category presentation {#ModCat}

### Quillen's definition: left derived functor of Kähler differentials
 {#LeftDerivedFunctorOfKaehlerDifferentials}

The **cotangent complex** functor is effectively the left [[derived functor]] of the [[Kähler differential]]s assignment.

To talk about the nonabelian derived functors, Quillen introduced a [[simplicial ring|model structure on the category of simplicial commutative rings]].

Given a morphism $f: A\to B$ in [[CRing]], which makes $B$ an $A$-algebra, the fiber of the [[tangent category]] $AbGr(A Alg/B)$ of abelian group objects in the slice category $A Alg/B$ of $A$-algebras over $B$ is equivalent both to the category of $B$-modules and the trivial (= square zero) extensions of $A$ by $B$-modules.

In particular we can consider the forgetful functor $AbGr(A Alg/B)\to A Alg/B$ which has a [[left adjoint]] $\Omega_{A/B} Ab_{B/A} : A Alg/B\to AbGr(A Alg/B)\cong {}_B Mod$. This is the [[Kähler differential]]s functor.

All said is true for [[simplicial ring|simplicial commutative rings]] as well.

+-- {: .num_defn}
###### Definition

The **relative cotangent complex** functor is the left [[derived functor]]

$$
  \mathbb{L} \Omega_{B/A} :
   s A Alg/B\to s AbGr(A Alg/B)\cong s B Mod
$$

Its value on $B$ is the relative cotangent complex $L_{B/A}$

=--

The [[André-Quillen cohomology]] of $R$ is the [[cohomology]] of $\mathbb{L}\Omega(R)$.

### Explicit resolutions

Here is one way to compute the required cofibrant [[resolution]]
for the construction of the left [[derived functor]] for the case that $A = k$ is a [[field]].

Let $P : CAlg \to CAlg$ be the [[comonad]] induced by the [[adjunction]]

$$
  U : CAlg \to Set : k[-]
$$

that sends a commutative $k$-algebra $R$ to the polynomial algebra on its underlying set.

Let $P_\bullet R$ be the corresponding [[bar construction]] [[simplicial ring|simplicial algebra]]. The canonical morphism $P_\bullet R \to R$ with $R$ on the right regarded as a constant simplicial object is a resolution of $R$.

Forming degreewise the module of [[Kähler differential]]s on this yields the simplicial object $\Omega_{k}(P_\bullet R)$, which is a $P_\bullet R$-module.

+-- {: .num_prop}
###### Proposition

The cotangent complex of $R$ is equivalent to

$$
  (\mathbb{L} \Omega_{/k}) (R) \simeq
  R \otimes_{P_\bullet R} \Omega_k(P_\bullet R)
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

Notice that the universal property of the [[Kähler differential]]s is that for $R$ a ring and $N$ an $R$-module, we have

$$
  Hom(\Omega(R), N) \simeq Der(R,N)
  \,.
$$

Accordingly, it follows that the [[André-Quillen cohomology]] of $R$ with values in $N$, which is the cohomology of the cosimplicial object

$$
  Hom((\mathbb{L}\Omega)(R), N)
$$

is equivalently the cohomology of the object

$$
  \begin{aligned}
    \cdots & \simeq
    Hom(R \otimes_{P_\bullet R} \Omega_k(P_\bullet R), N)
    \\
    & \simeq
    Hom_{P_\bullet R}( \Omega_k(P_\bullet R), N)
    \\
    & \simeq
    Der(P_\bullet R, N)
  \end{aligned}
  \,.
$$

In particular we have the the degree-0 cohomology of this complex is the module of ordinary derivations

$$
  H^0(Der(P_\bullet R, N)) \simeq Der(R,N)
  \,.
$$

=--

+-- {: .num_prop}
###### Proposition

If in the above $k$ is [[field]] of characteristic 0, then
[[André-Quillen cohomology]] of the $k$-algebra $R$ with coefficients in a [[module]] $N$ is a [[direct sum]]mand of the corresponding [[Hochschild cohomology]]:

$$
  H^q(Hom(\mathbb{L} \Omega (R)), N)
  \simeq
  HH^{q+1}_{(1)}(R,N)
  \,,
$$

where the subscript refers to [[Hodge decomposition]] of [[Hochschild cohomology]].

=--

This is in section 8.8 of

* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_

## $(\infty,1)$-categorical description {#InfCat}

The _cotangent complex_ is a generalization to [[higher category theory]] and [[higher algebra]] of the notion of [[cotangent bundle]] in the sense of [[Kähler differential]]s.

Recall from above that for $C =$ [[CRing]] the ordinary category of commutative rings, the cotangent complex functor is the [[section]]

$$
  \Omega_K : Ring \to Mod
$$

of the canonical [[bifibration]] $Mod \to Ring$ of [[module]]s over [[ring]]s that is on objects given by forming the module of [[Kähler differential]]s.

This generalizes to the case where [[CRing]] is replaced by any [[(∞,1)-category]] $C$: the cotangent complex functor for $C$ is here the [[left adjoint]] [[section]]

$$
  \Omega : C \to T_C
$$

of the [[tangent (∞,1)-category]] projection $dom : T_C \to C$.

In particular, when $C = ...$, then the cotangent complex assigns ... .

## Further properties and applications

### Characterization via universal properties

See [[cotangent complex in derived geometry]]

### Application: lifting properties of classical rings to derived rings

For more background see [[deformation theory]].

Apart from simplicial rings we can consider $E_\infty$-rings. A map of connective $E_\infty$-rings is an equivalence, if it induces an isomorphism at the level of $\pi_0$ plus a condition on the relative cotangent complex.
Similarly, one can express the descent properties of higher stacks via the usual gluing at the bottom level plus the obstruction theory for relative cotangent complex. Study of an appropriate version of the [[Postnikov tower]] is a systematic way to do this.

## Related concepts

* [[deformation theory]], [[derived deformation theory]]

* [[tangent complex]], [[André-Quillen cohomology]], [[Hochschild cohomology]]

* **cotangent complex**, [[André-Quillen homology]], [[Hochschild homology]]

* [[topological André-Quillen homology]], [[topological Hochschild homology]]

## References

See also [[deformation theory]] and references therein.

* [[Alexander Grothendieck]], _Catégories cofibrées additives et complexe cotangent relatif_, Lec. Notes in Math. __79__

* [[Luc Illusie]], _Complexe cotangent et déformations I_, Lec. Notes Math. __239__, Springer 1971, xv+355 pp.; _II_, LNM __283__, Springer 1972. vii+304 xv+355 pp.

* [[Kai Behrend]], [[Barbara Fantechi]], _The intrinsic normal cone_, Invent. Math. __128__ (1997), no. 1, 45--88, MR1437495 (98e:14022) [arXiv:alg-geom/9601010](http://arxiv.org/abs/alg-geom/9601010)

* [[Barbara Fantechi]], M. Manetti, _Obstruction calculus for functors of Artin rings I_, J. Algebra __202__ (1998), no. 2, 541--576, MR1617687 (99f:14004).

* [[Jacob Lurie]], [[Deformation Theory]], ([arXiv:0709.3091](http://arxiv.org/abs/0709.3091))

* [[Stefan Schwede]], _Spectra in model categories and applications to the algebraic cotangent complex_, J. Pure Appl. Alg. __120__, 77--104 (1997) (doi)

* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_ section 8.8.

* [[Gabriele Vezzosi]], _A note on the cotangent complex in derived algebraic geometry_, [arxiv/1008.0601](http://arxiv.org/abs/1008.0601)

* [[The Stacks Project]], _The cotangent complex_ ([pdf](http://stacks.math.columbia.edu/download/cotangent.pdf))

A short exposition (from the point of view of [[formal schemes]]) is in

* chapter 5 (5.29-5.31) in Luc Illusie, _Grothendieck's existence theorem in formal geometry_, in [[FGA explained]] (179--233) MR2223409; (draft version [pdf](http://cdsagenda5.ictp.it//askArchive.php?categ=a0255&id=a0255s3t3&ifd=15021&down=1&type=lecture_notes))

The cotangent complex for a general [[algebra over an operad]] in [[chain complexes]] is discussed in section 7 of

* {#Hinich} [[Vladimir Hinich]],  _Homological algebra of homotopy algebras_ Communications in algebra, 25(10). 3291-3323 (1997)([arXiv:q-alg/9702015](http://arxiv.org/abs/q-alg/9702015), _Erratum_ ([arXiv:math/0309453](http://arxiv.org/abs/math/0309453)))

In terms of [[model category]] presentations for [[tangent (infinity,1)-categories]]:

* [[Yonatan Harpaz]], [[Joost Nuiten]], [[Matan Prasma]], _Tangent categories of algebras over operads_ ([arXiv:1612.02607](https://arxiv.org/abs/1612.02607))

* _The abstract cotangent complex and Quillen cohomology of enriched categories_ ([arXiv:1612.02608](https://arxiv.org/abs/1612.02608))

[[!redirects cotangent complexes]]


cotangent complex in derived geometry

#Contents#
* table of contents
{:toc}

## Idea

A _cotangent complex_ is a certain [[spectrum object]] which exerts full control of the linear-order [[deformation theory|deformation]] & [[obstruction]] theory in a [[moduli problem]]. Consequently, a construction of a cotangent complex constitutes a complete understanding of the [[deformation theory]] of the situation.

There's both a *local* and *global* version of this idea. Let $X$ be a derived prestack. One can seek the following objects:

- $\mathbb{L}_{X, x} \in QCoh(U)$, for  a "point" $U \xrightarrow{x} X \in Aff_{/X}$. The local/point-wise cotangent complex.

- $\mathbb{L}_X \in QCoh(X)$. The global cotangent complex.

In the case that $X$ is a nice geometric object (e.g. an n-[[geometric stack]]), $\mathbb {L}_{X, x}$ can be viewed as the ([[derived geometry|derived]]) [[cotangent space]] at a point, and $\mathbb {L}_X$ the global [[cotangent bundle]].

That being said, the global cotangent complex exist for a much broader class of prestacks $X$ than just the geometric ones, allowing one to talk about the "existence of a good deformation theory everywhere" even when the moduli problem is not representable.

## Universal Properties

### Ideas & Summary

Before diving into the various variants of cotangent complexes, we summarize their relationships here.

- Local cotangent complex can be obtained from global tangent complex by restriction (when the latter exists)
- The absolute tangent complexes can be obtained from the relative ones by considering the morphism $X \to pt$.

We've already explained the global/local distinction above, we briefly discuss the absolute/relative distinction now.

In [[algebraic geometry]], relative notions are notions about families. e.g. [[flat morphism|flatness]]/[[formally smooth morphism|smoothness]]/[[formally etale morphism|etaleness]] over a base [[scheme]]/[[algebraic stack|stack]]. Relative cotangent complexes encode deformation theory of families. Of course these are analogous to [[vertical tangent bundle|relative]] cotangent bundles in [[differential geometry]].

We also note that even if one is only interested in finding an absolute cotangent complex, some of the most useful computational lemmas require you to compute relative cotangent complexes. Hence the notion is truly ubiquitous in deformation theory.

### Local Cotangent Complex

We begin by discussing what it means for the cotangent complex to control deformation/obstruction theory.

Let

- $x: U \rightarrow X \in Aff_{/X}$
- $\iota_M: U \hookrightarrow U^M$ the *trivial* square-0 thickening by a $M \in QCoh(U)$.

The  [[$\infty$-groupoid]] encoding all extensions of $x$ along $\iota_M$ is called the space of derivations. This can be written as a (homotopy) fiber diagram in the $\infty$-category of spaces:

\begin{definition}
$$Der_{X, x}(M)  := Map_{pSt}(U^M, X)  \times_{Map_{pSt}(U, X)}\{x\}$$
\end{definition}

Formation of these homotopy fibers is functorial in $M \in QCoh(U)$, hence determines a functor

$$Der_{X, x}: QCoh(U) \rightarrow Spaces$$

\begin{definition}
A cotangent complex for $X$ and $x$ is a object in $QCoh(U)$ corepresenting the functor $Der_{X, x}$. Such an object is denoted $\mathbb {L}_{X, x}$
\end{definition}

Hence, a cotangent complex controls, up to homotopy, the deformation theory of $x$ in $X$ by the following mechanism.

$$
[\text{extensions of x along }  U\hookrightarrow U^M] \simeq [\mathbb {L}_{X, x} \to M]
$$

Hence this construction translates the geometric situation on the left to the *linear* situation on the right.

### Global Cotangent Complex

\begin{definition}
An object $\mathbb{L}_X \in QCoh(X)$ is said to be a *global cotangent complex* if for all $x: U \rightarrow X \in Aff_{/X}$, $x^*\mathbb {L}_X \in QCoh(U)$ is a cotangent local complex at $x$.
\end{definition}

Hence, one can informally write the condition as existence of a compatible system of equivalences $x^*\mathbb {L}_X \simeq \mathbb {L}_{X, x}$.

### Relative Cotangent Complexes

Both the local and global notions above can be relativized...

## Basic Properties

### Cotangent Exact Sequence

\begin{lemma}
Let $X \xrightarrow{a} Y \xrightarrow{b} Z \in Fun(\Delta^2, pSt)$ such that each arrow has a global relative cotangent complex, the following diagram

$$
a^*\mathbb{L}_{Y/Z} \to \mathbb{L}_{X/Z} \to \mathbb{L}_{X/Y}
$$

is a fiber (and hence cofiber) diagram in the stable $\infty$ category $QCoh(X)$.
\end{lemma}

For example, taking $Z \simeq pt$ exhibits the $\mathbb{L}_{X/Y}$ as the cofiber of $f^*\mathbb{L}_{Y} \to \mathbb{L}_{X}$

### Excision

## Constructions

There are several ways to demonstrate the existence of cotangent complexes via explicit constructions:

- **Derived functors**: in the setting of dg-algebras, one can take left derived functors of the classical Kahler differentials functor $L\Omega_{(-)}$.  For example, concretely one can take a quasi-free resolution and apply $\Omega$ degree-wise.
- **Stabilization**: when $X$ is geometric (e.g. a spectral Deligne Mumford stack), there is a construction via the "tangent $\infty$-category", which is a sort of relative stabilization procedure.

Some elaboration of these ideas are found in the article  [[cotangent complex]].


cotangent function

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Trigonometry
+-- {: .hide}
[[!include trigonometry -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[trigonometry]] the _cotangent function_ is one of the basic [[trigonometric functions]].

## Definition

The contangent is the [[ratio]] of the [[cosine]] and the [[sine]]:

$$
  \cot(x) = \frac{\cos(x)}{\sin(x)}
$$

## Properties

### Relations with tangent function

* The cotangent and [[tangent function|tangent]] are reciprocal to each other: $\cot x = \frac1{\tan x}$.

* The cotangent and tangent are complementary to each other: $\cot x = \tan (\frac{\pi}{2} - x)$.

* Double angle formula: $\tan x = \cot x - 2\cot 2x$.

### Relation to Bernoulli numbers

The [[hyperbolic functions|hyperbolic]] analog $\coth x = \frac{e^x + e^{-x}}{e^x - e^{-x}}$ is related to $\cot x$ via the formula

$$\cot x = i\coth i x.$$

Meanwhile $\coth x$ is related to the [[Bernoulli numbers]] $B_n$, defined by the [[exponential generating function]]

$$\frac{x}{e^x - 1} = \sum_{n \geq 0} \frac{B_n x^n}{n!},$$

through a series of equations

$$\frac{x}{e^x - 1} + \frac{x}{2} = \frac{x(2 + e^x - 1)}{2(e^x - 1)} = \frac{x}{2} \frac{e^x + 1}{e^x - 1} = \frac{x}{2} \frac{e^{x/2} + e^{-x/2}}{e^{x/2} - e^{-x/2}} = \frac{x}{2}\coth \frac{x}{2}.$$

Notice the right side defines an even function. Therefore

$$\frac{x}{2}\coth \frac{x}{2} = \sum_{n \geq 0} \frac{B_{2n} x^{2n}}{(2n)!}$$

and so

$$x \cot x = i x \coth i x = \sum_{n \geq 0} \frac{B_{2n} (2i x)^{2n}}{(2n)!} = \sum_{n \geq 0} (-1)^n \frac{B_{2n} 2^{2n} x^{2n}}{(2n)!}.$$

### "Eisenstein" series expansion

The cotangent is the [[logarithmic derivative]] of the [[sine function]]:

$$\cot x = (\log (\sin x))'.$$

Applying this observation to the Euler-Weierstrass [[product formula for the sine function]] (see there for a proof):

$$\sin (\pi x) = \pi x \prod_{n=1}^\infty \left(1 - \frac{x^2}{n^2}\right)$$

one obtains the following summation formula for the cotangent:

$$\pi\, \cot (\pi x) = \frac1{x} + \sum_{n=1}^\infty \left(\frac1{x + n} + \frac1{x - n}\right)$$

This expansion was used by Eisenstein as a starting point for developing the theory of trigonometric functions; Eisenstein's account of elliptic functions (cf. the eponymous [[Eisenstein series]]), developed further by Weierstrass, Kronecker, and others, runs parallel to his trigonometric theory, as explained later by Weil. For some more details, see these [notes](#vsv) by Varadarajan.

### Relation to zeta function values

\begin{proposition}
The power series identity

$$\pi x \cot \pi x = 1 - 2\sum_{k \geq 1} \zeta(2k) x^{2k}$$
holds over an open domain where the series converges, ${|x|} \lt 1$.
\end{proposition}

\begin{proof}
From the Eisenstein expansion, we have
$$\pi x\, \cot (\pi x) = x\, \left(\frac1{x} + \sum_{n=1}^\infty \left(\frac1{x + n} + \frac1{x - n}\right)\right) = 1 + \sum_{n \geq 0} \frac{2x^2}{x^2 - n^2} = 1 - 2\sum_{n \geq 1} \frac{x^2/n^2}{1 - x^2/n^2}.$$
By a geometric series expansion, the last expression is

$$1 - 2\sum_{n \geq 1} \sum_{k \geq 1} \left(\frac{x^2}{n^2}\right)^k = 1 - 2\sum_{k \geq 1} x^{2k} \sum_{n \geq 1} \frac1{n^{2k}}$$
which is the same as $1 - 2\sum_{k \geq 1} \zeta(2k)x^{2k}$.
\end{proof}

## Related concepts

* [[tangent function]]

* [[trigonometric identity]]

* [[Eisenstein series]]

* [[product formula for the sine function]]

## References

* Wikipedia, _[Trigonometric functions -- tan](https://en.wikipedia.org/wiki/Trigonometric_functions#tan)_

* {#vsv} [[Veeravalli Varadarajan]], *Circular Functions* ([pdf](http://www.math.ucla.edu/~vsv/fullnotes.pdf)).

[[!redirects cotangent functions]]
[[!redirects cotan]]
[[!redirects cot]]
[[!redirects cotangent]]


cotensor product

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

> This page is about cotensor products of comodules over coalgebras.  The word "cotensor" is also used for [[powers]] in enriched categories, and for the [[multiplicative disjunction]] in a [[polycategory]]/[[linearly distributive category]].

***

#Contents#
* table of contents
{:toc}

## Definition

Given a [[monoidal category]] $\mathcal{M}$ and a [[coalgebra]] $C$ in $\mathcal{M}$ denote by $\mathcal{M}^{C}$ ($ resp. {}^{C}\mathcal{M}$)
the category of right (resp. left) ${C}$-[[comodules]]; similarly for an algebra $E$, denote by ${}_E\mathcal{M}$ (resp. $\mathcal{M}_E$) the [[category of modules|category of left E-modules]]
(right $E$-modules). If the monoidal category is [[symmetric monoidal category|symmetric]] or there is instead an appropriate [[distributive law]], then there are extensions of this notation to [[bimodules]], [[bicomodules]], relative [[Hopf modules]], entwined modules etc. e.g. Write ${}_E\mathcal{M}^B$ for left-right relative $(E,B)$-Hopf modules where $E$ is a $B$-comodule algebra over a bialgebra $B$.

Let $k$ be a commutative unital [[ring]], and let $\mathcal{M}$ be $k$-linear (in particular it has [[zero morphisms]]).

+-- {: .num_defn }
###### Definition

Given a [[coalgebra]] $C$ in $\mathcal{M}$, a left $C$-[[comodule]] $(N,\rho_N \colon N\to N\otimes C)$, a right $C$-comodule $(M,\rho_M \colon M\to C\otimes M)$, their __cotensor product__ is an object in $\mathcal{M}$ given by the [[kernel]]

$$
  N \Box M
  \coloneqq
  \mathrm{ker} (\rho_N \otimes \mathrm{id}_M - \mathrm{id}_N \otimes \rho_M ).
$$

=--

If [[equalizers]] exist in $\mathcal{M}$, this formula extends to a [[bifunctor]]

$$
   {}\Box = \Box^{C}
  \colon
\mathcal{M}^{C} \times {}^{C}\mathcal{M} \rightarrow \mathcal{M}
  \,.
$$

If $B$ is a [[bialgebra]] in $\mathcal{M}$ and $E$ is a right $B$-comodule algebra then the same formula defines a bifunctor

$$
  \Box
    \colon
  {}_{E}\mathcal{M}^{B} \times {}^{B}\mathcal{M} \rightarrow
{}_{E}\mathcal{M}
  \,.
$$

## Properties

### Induced comodules and a relation to tensor product

Let now $\mathcal{M}=({}_k\mathrm{Mod},\otimes_k)$ be the symmetric monoidal category of $k$-modules where $k$ is a commutative unital ring.

Let $D$ be another $k$-coalgebra, with coproduct $\Delta_C$.
If $D$ is [[flat module|flat]] as a $k$-module
(e.g. $k$ is a field), and $N$ a left $D$- right $C$-bicomodule,
then the cotensor product $N \Box M$ is a $D$-subcomodule of the [[tensor product]] $N \otimes_k M$. In particular, under the flatness assumption, if $\pi \colon D \rightarrow C$ is a homomorphism of coalgebras (usually surjective in applications), then $D$ is a left $D$-right $C$-bicomodule via $\Delta_D$ and $(\id \otimes \pi) \circ \Delta_D$
respectively, hence
$$
\mathrm{Ind}^D_C : C-Comod\to D-comod: \coloneqq D \Box^C -
$$
is a functor from left $C$- to left
$D$-comodules called the [[induced comodule|induction]] (corepresentation) functor for left comodules from $C$ to $D$.

### For comodules over commutative Hopf coalgebroids
 {#ForComodulesOverCommutativeHopfCoalgebroids}

+-- {: .num_prop #LeftComodulesToRightComodules}
###### Proposition

Consider a [[commutative Hopf algebroid]] $\Gamma$ over $A$ ([def.](commutative+Hopf+algebroid#CommutativeHopfAlgebroidDefinitionInExplicitComponents)). Any left comodule $N$ over $\Gamma$ ([def.](commutative+Hopf+algebroid#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.](commutative+Hopf+algebroid#CommutativeHopfAlgebroidDefinitionInExplicitComponents)), and given $N_1$ a right $\Gamma$-comodule and $N_2$ a left comodule ([def.](commutative+Hopf+algebroid#CommutativeHopfAlgebroidComodule)), then their **cotensor product** $N_1 \Box_\Gamma N_2$ is the [[kernel]] of the difference of the two coaction morphisms:

$$
  N_1 \Box_\Gamma N_2
    \;\coloneqq\;
  ker
  \left(
    N_1 \otimes_A N_2
    \overset{\Psi_{N_1}\otimes_{A} id - id \otimes_A \Psi_{N_2} }{\longrightarrow}
  \right)
  \,.
$$

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 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 $A$,
and given $N_1, N_2$ two left $\Gamma$-comodules , 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 $A$,
and given $N_1, N_2$ two left $\Gamma$-comodules, 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]].

=--

## Related concepts

* a special case of the cotensor product as a functor of one variable is the [[induced comodule]] construction
* The [[derived functor]] of cotensoring is called _[[Cotor]]_.

## References

Cotensor products in [[noncommutative geometry]] appear in the role of [[space of sections]] of [[associated bundle|associated]] [[vector bundles]] of [[quantum principal bundle]]s (which in affine case correspond to [[Hopf-Galois extensions]]). See e.g.

* [[Shahn Majid]], _Foundations of quantum groups theory_, 2nd extended edition, paperback, Cambridge Univ. Press 2000.

Basic homological algebra of cotensor products for coalgebras over a field is advanced in

* Yukio Doi, _Homological coalgebra_, J. Math. Soc. Japan __33__:1 (1981) 31–50 [MR597479](https://mathscinet.ams.org/mathscinet-getitem?mr=597479) [euclid](https://projecteuclid.org/euclid.jmsj/1239888030)

(Co)flatness conditions play major role for study of cotensor products when coalgebras are over commutative rings, or in the categories of bimodules ([[coring]]s),

* Leonid Positselski, _Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures_, [arxiv/0708.3398](https://arxiv.org/abs/0708.3398)
* [[T. Brzeziński]], R. Wisbauer, _Corings and comodules_, London Math. Soc. Lec. Note Series 309, Cambridge 2003.

For a nonaffine extension of the sections of associated quantum vector bundle, using [[localization]] theory see

* [[Zoran Škoda]], _Coherent states for Hopf algebras_,
Lett. Math. Phys. 81 (2007), no. 1, 1--17. ([arXiv:math.QA/0303357](http://front.math.ucdavis.edu/0303.5357))

In [[Hopf algebra]] theory motivated by algebraic topology, cotensor products appear as early as in

* [[John Milnor]], [[John Moore]], _On the structure of Hopf algebras_, Ann. of Math. (2)  81  1965 211--264

and along with its derived functor Cotor used in computation of $E^2$-term of a [[spectral sequence]] of a [[fibration]] in

* [[Samuel Eilenberg]], John. C Moore, _Homology and fibrations. I. Coalgebras, cotensor product and its derived functors_, Comment. Math. Helv. __40__ (1966) 199–236 [MR203730](https://mathscinet.ams.org/mathscinet-getitem?mr=203730) [doi](http://doi.org/10.1007/BF02564371)

The authors mention that they learned the notion from Mac Lane who knew it earlier in more general contexts.

A textbook account is in

* [[Doug Ravenel]], section A1.1 of _[[Complex cobordism and stable homotopy groups of spheres]]_, 1986/2003

An important problem is that the cotensor product of bicomodules is in general (even for $\mathcal{M}={}_k\mathrm{Mod}$) *not associative*, even up to an isomorphism.

Cotensor products play a prominent role in various treatments of [[Galois theory]] in [[noncommutative geometry]]; a particularly geometric approach is within a version of [[noncommutative algebraic geometry]] based on usage of monoidal categories, as sketched in

* [[Tomasz Maszczyk]], Noncommutative geometry through monoidal categories I, [arXiv:0707.1542](http://front.math.ucdavis.edu/0707.1542)

[[!redirects cotensor products]]


cotopological localization

##Idea

Let $\mathcal{X}$ be an $\infty$-topos and let $\mathcal{Y} \subseteq \mathcal{X}$ be an [[accessible (∞,1)-category|accessible]] left exact localization of $\mathcal{X}$. We say that $\mathcal{Y}$ is a **cotopological localization** of $\mathcal{X}$ if the left adjoint $L : \mathcal{X} \to \mathcal{Y}$ to the inclusion of $\mathcal{Y}$ in $\mathcal{X}$ satisfies either of the following equivalent conditions:

1. For every monomorphism $u$ in $\mathcal{X}$, if $L^{\ast}u$ is an equivalence in $\mathcal{Y}$, then $u$ is an equivalence in $\mathcal{X}$.

1. For every morphism $u \in \mathcal{X}$, if $L^{\ast}u$ is an equivalence in $\mathcal{Y}$, then $f$ is ∞-connective.

The [[hypercompletion]] $\mathcal{X}^{\wedge}$ of an ∞-topos $\mathcal{X}$ can be characterized as the maximal cotopological localization of $\mathcal{X}$ (that is, the cotopological localization which is obtained by inverting as many morphisms as possible).

Every [[localization of an (∞,1)-category|localization]] can be obtained by combining [[topological localization|topological]] and cotopological localizations.

##Examples

In the context of the [[Goodwillie calculus]], the whole [[Goodwillie-Taylor tower]] consists of cotopological localizations of the classifying ∞-topos for pointed ∞-connective objects.

##Related concepts

* [Whitehead theorem in $(\infty,1)$-toposes](Whitehead+theorem#in_general_toposes)

##References

* 6.5.2.17 of [[Higher Topos Theory]]


cotopology

## Idea

Formally, a cotopology of a topological space $(X, \mathcal{T})$ is a coarser topology $\mathcal{T}^*$ on the same set $X$ that does not forget too many sets.
It can be used to generalize the [[Baire category theorem]] and to characterize [[topologically complete space|topological completeness]]. The pair of both topologies, the original and the coarser one, constitute an example of a [[bitopological space]].

## Definitions

\begin{definition}
Let $(X, \mathcal{T})$ be a topological space.
A topology $\mathcal{T}^*$ on $X$ is called a __cotopology__ of $\mathcal{T}$---and $(X, \mathcal{T}^*)$ is called a __cospace__ of $(X, \mathcal{T})$---if

1. $\mathcal{T}^*$ is coarser than $\mathcal{T}$, i.e., $\mathcal{T}^* \subseteq \mathcal{T}$;

1. for each point $x$ and each $\mathcal{T}$-closed $\mathcal{T}$-neighborhood $V$ of $x$ in $X$ there exists a $\mathcal{T}^*$-closed $\mathcal{T}$-neighborhood $U$ of $x$ in $X$ such that $U$ is contained in $V$.

\end{definition}

If $\mathcal{T}$ is [[regular space|regular]], the last condition can be replaced by other conditions, see [[bitopological space#implications|this proposition]].

\begin{definition}
A topological space $X$ is called __cocompact__ if there is a cotopology $\mathcal{T}^*$ on $X$ which is [[compact space|compact]].
\end{definition}

## Properties

Every cocompact [[regular space]] is a [[Baire space]]. A [[metrisable topological space|metrisable space]] is [[topologically complete space|topologically complete]] if and only if it is cocompact.

A space that admits only Hausdorff cospaces is equivalently an H-closed space, i.e. it (is Hausdorff and) is not a proper dense subspace of another space.

## Related Concepts

* [[bitopological space]]

## References

The concept principally appeared in

* J. D. Weston, _On the comparison of topologies_ 1956, Journal of the London Mathematical Society, vol. s1-32 no. 3, pp. 342-354.

De Groot made cotopologies popular by giving a unifying and generalizing version of the [[Baire category theorem]]

* de Groot, _Subcompactness and the Baire category theorem_ 1963, Nederl. Akad. Wetensch. Proc., Ser. A66 = Indag. Math., vol. 25, pp. 761-767;

unfortunately, his proof contained a gap that was later closed by

* Isidore Fleischer, _On 'Subcompactness and the Baire category theorem'_ 1979, Nederl. Akad. Wetensch. Proc. Ser. A 82 = Indag. Math. 41, pp. 9-11.

In this context some

* _Colloquium co-topologie_ at the _Mathematisch Centrum_, today _[Centrum Wiskunde & Informatica](https://www.cwi.nl/)_, in Amsterdam in 1964

is mentioned.
Further developments include

* Aarts, de Groot, McDowell, _Cotopology for metrizable spaces_ 1970, Duke Mathematical Journal vol. 37.

* [[George Strecker]] and G. Viglino, _Cotopology and Minimal Hausdorff Spaces_ 1969, Proceedings of the American Mathematical Society Vol. 21 No. 3.

[[!redirects cocompact space]]


cotopos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

The notion of *cotopos* is [[abstract duality|dual]] to that of a [[topos]].

A co-elementary cotopos should be the 1-categorical version of a [[co-Heyting algebra]] (which model [[dual intuitionistic logic]]), and a co-Grothendieck cotopos should be the 1-categorical version of a [[coframe]] (which model [[closed sublocales]] in [[topology]]).

## Definition

### Co-elementary cotopos

A co-elementary cotopos is a [[finitely cocomplete category|finitely cocomplete]] [[cocartesian coclosed category|cocartesian coclosed]] [[category]] with a [[quotient object coclassifier]].

### Co-Grothendieck cotopos

A co-Grothendieck cotopos is a [[locally small category|locally small]] [[finitely cocomplete category]] with a small [[cogenerator]], with all small [[products]] which are codisjoint and pushout-stable, and where all internal [[bisimulations]] have effective [[subobjects]], which are also pushout-stable.

## Properties

Every [[cotopos]] is a [[protomodular category]].

## Examples

A [[binary relation]] $R$ between sets $A$ and $B$ is *injective* if for all $a \in A$, $b \in A$, and $c \in B$, if $R(a, c)$ and $R(b, c)$ then $a = b$. A binary relation $R$ between sets $A$ and $B$ is *onto* if for all $b \in B$ there exists an element $a \in A$ such that $R(a, b)$. The category of sets and injective onto binary relations is a cotopos.

## See also

* [[coclosed category]]

* [[coclosed monoidal category]]

* [[cocartesian monoidal category]]

* [[cocartesian coclosed category]]

* [[locally cocartesian coclosed category]]

* [[coslice category]]

* [[protomodular category]]

* [[M-cotopos]]

* [[co-well-pointed cotopos]]

* [[quotient object coclassifier]]

* [[co-Heyting algebra]]

* [[coframe]]

* [[topos]]

## References

* Mamuka Jibladze, "Cosheaves, coframes, cotoposes: some new facts, some old questions" ([web archive](https://web.archive.org/web/20130602214748/http://www.wra1th.plus.com/gcw/rants/math/PSSL/1997.html))

[[!redirects cotopoi]]
[[!redirects cotoposes]]


Cotor

#Contents#
* table of contents
{:toc}

## Idea

The [[derived functor]] of a [[cotensor product]] functor of [[comodules]] is often called "Cotor", in analogy with the notation "[[Tor]]" for the derived functor of a [[tensor product]] functor.

## References

* [[Doug Ravenel]], section A1.2 of _[[Complex cobordism and stable homotopy groups of spheres]]_, 1986/2003


cotorsion pair

## Idea

Cotorsion pairs in [[abelian categories]] are a notion generalizing both [[projective object|projective]] and [[injective objects]].

## Definition

Let $\mathcal{A}$ be an [[abelian category]].

+-- {: .num_defn}
###### Definition
A **cotorsion pair** in $\mathcal{A}$ is a pair $(\mathcal{D},\mathcal{E})$ of classes
of objects of $\mathcal{A}$ such that

* $D \in \mathcal{D}$ if and only if $Ext^1(D, E) = 0$ for all $E \in \mathcal{E}$
* $E \in \mathcal{E}$ if and only if $Ext^1(D, E) = 0$ for all $D \in \mathcal{D}$.
=--

+-- {: .num_example}
###### Example
Let $\mathcal{I}$ denote the class of [[injective objects]] of $\mathcal{A}$.  Then $(\mathcal{A}, \mathcal{I})$ is a cotorsion pair, where we write $\mathcal{A}$ for the class of all objects of $\mathcal{A}$.

Dually, there is a cotorsion pair $(\mathcal{P}, \mathcal{A})$ where $\mathcal{P}$ is the class of [[projective objects]] of $\mathcal{A}$.
=--

+-- {: .num_example #FlatCotorsionPair}
###### Example
The **flat cotorsion pair** on the category of $R$-[[modules]] (for some [[ring]] $R$) is defined by letting $\mathcal{D}$ be the class of [[flat module|flat]] $R$-modules and $\mathcal{E}$ the class of [[cotorsion modules]], i.e. $R$-modules $E$ such that $Ext^1_R(D,E)=0$ for all flat $D$.
=--

+-- {: .num_defn}
###### Definition
We say that a cotorsion pair $(\mathcal{D},\mathcal{E})$ **has enough injectives** if for every [[object]] $X$ of $\mathcal{A}$, there exist objects $D \in \mathcal{D}$ and $E \in \mathcal{E}$ such that there is an exact sequence
\[ 0 \to X \to E \to D \to 0. \]
We say that it **has enough projectives** if for every [[object]] $X$ of $\mathcal{A}$, there exist objects $D \in \mathcal{D}$ and $E \in \mathcal{E}$ such that there is an exact sequence
\[ 0 \to E \to D \to X \to 0. \]
=--

Note that the category $\mathcal{A}$ has enough injectives if and only if the cotorsion pair $(\mathcal{A}, \mathcal{I})$ has enough injectives, and $\mathcal{A}$ has enough projectives if and only if $(\mathcal{P}, \mathcal{A})$ has enough projectives.

+-- {: .num_defn}
###### Definition
A cotorsion pair $(\mathcal{D},\mathcal{E})$ is called **complete** if it has enough injectives and enough projectives.
=--

## References

* [[Mark Hovey]], _Cotorsion pairs and model categories_, Contemp. Math. __436__ (2007) 277-296 [pdf](http://homepages.math.uic.edu/~bshipley/hovey.pdf) [gBooks](https://books.google.hr/books?id=UZ8bCAAAQBAJ) [doi](https://doi.org/10.1090/conm/436/08413)

[[!redirects cotorsion pairs]]


Coulomb branch

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For [[N=2 D=4 super Yang-Mills theory]] the [[moduli space]] of [[vacuum expectation values]] (VEVs) of the theory is meant to locally be a [[Cartesian product]] of spaces of

1. [[moduli]] of the [[vector multiplet]] (the [[gauge field]] sector)

   this is called the *Coulomb branch*

1. moduli of the [[hypermultiplet]] (the scalar matter field sector)

   this is called the *[[Higgs branch]]*.

These are thought to be [[duality in physics|dual]] to each other under a version of [[mirror symmetry]]. This is largely the topic of [[Seiberg-Witten theory]].

Definitions of the Coulomb and Higgs branches have been extended to [[N=4 D=3 super Yang-Mills theory]].

## Definition

### D=4

### D=3

Given $G$ a complex reductive group and $M$ a complex [[symplectic representation]] of the form $M=N \bigoplus N^*$, the Coulomb branch is defined as the spectrum of the ring given by equivariant Borel-Moore homology $H_\bullet^{G_\mathcal{O}} ( \mathcal{R} )$ with convolution product. $\mathcal{R}$ is a space of triples of a $G$ principle bundle over the formal disk $\mathcal{P}$, a trivialization over the punctured formal disk $\phi$ and a section $s$ of $\mathcal{P} \times_G N$ with the restriction that $\phi (s)$ extends over the puncture.

By including $\mathbb{C}^*$ equivariance as well, this gives a non-commutative deformation. This gives a Poisson structure at the classical level.

#### Examples

* If $N=0$, this gives the Bezrukavnikov-Finkelberg-Mirkovic (BFM) space for the Langlands dual $^L G$.

## Related concepts

* [[symplectic duality]]

* [[fractional D-brane]], [[McKay correspondence]]

## References

### General

The terminology "Coulomb branch" and "Higgs branch" first appears in

* {#SeibergWitten94} [[Nathan Seiberg]], [[Edward Witten]], _Monopoles, duality and chiral symmetry breaking in $N=2$ supersymmetric QCD_, Nucl.Phys. __B431__ (1994) 484--550 ([doi](https://doi.org/10.1016/0550-3213%2894%2990214-3) [arXiv:hep-th/9408099](https://arxiv.org/abs/hep-th/9408099))

The definition is summarized (specifically for [[super QCD]]) in [Assel-Cremoni 17, Section 2.1](#AsselCremoni17).

Quick exposition of the basic idea includes

* {#Albertsson03} Cecilia Albertsson, around p. 31 of _Superconformal D-branes and moduli spaces_ ([arXiv:hep-th/0305188](https://arxiv.org/abs/hep-th/0305188))

Mathematical discussion in the case of [[D=3 N=4 super Yang-Mills theory]]:

* [[Hiraku Nakajima]], _Introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional $\mathcal{N} = 4$ gauge theories_ ([arXiv:1706.05154](http://arxiv.org/abs/1706.05154))

* [[Hiraku Nakajima]], _Towards a mathematical definition of Coulomb branches of 3-dimensional $\mathcal{N} = 4$ gauge theories, I_ ([arXiv:1503.03676](https://arxiv.org/abs/1503.03676))

* [[Alexander Braverman]], [[Michael Finkelberg]], [[Hiraku Nakajima]], _Towards a mathematical definition of Coulomb branches of 3-dimensional $\mathcal{N} = 4$ gauge theories, II_, Adv. Theor. Math. Phys. __22__ (2018) 1071-1147 ([arXiv:1601.03586](http://arxiv.org/abs/1601.03586))

* [[Alexander Braverman]], [[Michael Finkelberg]], [[Hiraku Nakajima]], *Coulomb branches of 3d $\mathcal{N}=4$ quiver gauge theories and slices in the affine Grassmannian* (with appendices by Alexander Braverman, Michael Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Hiraku Nakajima, Ben Webster, and Alex Weekes), Advances in Theoretical and Mathematical Physics __23__:1 (2019) ([arXiv:1604.03625](https://arxiv.org/abs/1604.03625))

  > (relation to the [[moduli space of Yang-Mills monopoles]])

* [[Ben Webster]], _Koszul duality between Higgs and Coulomb categories $\mathcal{O}$_, ([arXiv:1611.06541](https://arxiv.org/abs/1611.06541))

See also:

* [[Constantin Teleman]], _The role of Coulomb branches in 2D gauge theory_, ([arXiv:1801.10124](https://arxiv.org/abs/1801.10124))

* Justin Hilburn, Joel Kamnitzer, Alex Weekes, _BFN Springer Theory_, Commun. Math. Phys. __402__ (2023) 765--832 (2023) ([doi](https://doi.org/10.1007/s00220-023-04735-4) [arXiv:2004.14998](https://arxiv.org/abs/2004.14998))

* Zijun Zhou, *Virtual Coulomb branch and vertex function* ([arXiv:2107.06135](https://arxiv.org/abs/2107.06135))

On [[mirror symmetry]] between [[Higgs branches]]/[[Coulomb branches]] of [[D=3 N=4 super Yang-Mills theory]] (with emphasis of [[Hilbert schemes of points]]):

* [[Jan de Boer]], [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], _Mirror symmetry in three-dimensional gauge theories, quivers and D-branes_, Nucl. Phys. __B493__:101--147, 1997 ([arXiv:hep-th/9611063](https://arxiv.org/abs/hep-th/9611063))

* [[Alexander Braverman]], [[Michael Finkelberg]], [[Hiraku Nakajima]], _Line bundles over Coulomb branches_ ([arXiv:1805.11826](https://arxiv.org/abs/1805.11826))

### Singularities

Discussion of Coulomb branch [[singularities]]:

* [[Mathew Bullimore]], [[Tudor Dimofte]], [[Davide Gaiotto]], _The Coulomb branch of 3d $\mathcal{N}=4$ theories_, Commun. Math. Phys. (2017) 354: 671 ([arXiv:1503.04817](https://arxiv.org/abs/1503.04817))

  (relation to [[Nahm's equations]])

* {#AsselCremoni17} [[Benjamin Assel]], [[Stefano Cremonesi]], _The infrared physics of bad theories_, SciPost Phys. 3, 024 (2017) ([arXiv:1707.03403](https://arxiv.org/abs/1707.03403))

[[!redirects Coulomb branches]]


Coulomb gauge

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A special kind of [[gauge fixing]] condition on a [[connection on a bundle]].

## Definition

Let $A_0$ and $A$ be two [[connection on a bundle|connections on a bundle]]. Then $A$ is said to be in _Coulomb gauge_ relative to $A_0$ if

$$
  d_{A_0}^*(A-A_0) = 0
  \,.
$$

## Related concepts

* [[gauge group]], [[gauge transformation]]

* [[gauge fixing]]

  * [[temporal gauge]]

  * **Coulomb gauge**


countable

* [[countable set]]

* [[countable ordinal]]

[[!redirects countably]]


countable chain condition

# Contents #
* table of contents
{:toc}

## Idea

The countable chain condition, appearing in various guises (for [[topological spaces]], for [[posets]], for [[Boolean algebras]]), frequently recurs in discussions on [[forcing]] in [[set theory]], particularly in discussions about preservation of [[cardinals]] and [[cofinalities]] in forcing extensions.

It is something of a misnomer because it is really a condition about [[antichains]]. But in the topological context it is equivalent to a condition on chains, and the name has stuck.

## Definitions

+-- {: .num_defn}
###### Definition
Suppose $P$ is a poset with a [[minimum|bottom]] element $0$. A *(strong) antichain* in $P$ is a subset $A \subseteq P$ such that for all $a, b \in A$, we have either $a = b$ or $0$ is their [[meet]] (we need not assume general meets exist in $P$). If $P$ is a poset *without* a bottom, we define $A \subseteq P$ to be an antichain if it becomes an antichain in $P^+$, the poset obtained by freely adjoining a bottom to $P$; this is equivalent to saying $a, b$ have no lower bound in $P$.
=--

+-- {: .num_defn}
###### Definition
A poset $P$ satisfies the *countable chain condition* if every antichain in $P$ is a [[countable set]] (i.e., at most denumerable).

A topological space $X$ satisfies the countable chain condition if its [[frame]] of open sets $Op(X)$ does (as a poset).
=--

For future reference, we record a result on the countable chain condition in [[Heyting algebras]] $H$. Let $Reg(H)$ denote the Boolean algebra of regular elements ($x \in H$ such that $x = \neg \neg x$).

+-- {: .num_lemma #reg}
###### Lemma
A Heyting algebra $H$ satisfies the countable chain condition iff $Reg(H)$ does.
=--

+-- {: .proof}
###### Proof
The inclusion $Reg(H) \hookrightarrow H$ preserves meets and $0$, so if every antichain in $H$ is countable, then the same is true in $Reg(H)$.

The quotient map $\neg \neg: H \to Reg(H)$ also preserves meets and $0$ (details are at [[Heyting algebra]]), and moreover the restriction of this map to an antichain $A \subseteq H$ is an injection (if $\neg \neg a = \neg \neg b$ for $a, b \in A$, then $a = b$ or $a \wedge b = 0$; in the latter case $a \leq \neg \neg a = \neg \neg a \wedge \neg \neg b = \neg \neg (a \wedge b) = 0$ and similarly $b = 0$, so $a = b$). So if every antichain in $Reg(H)$ is countable, the same is true in $H$.
=--

### Chains, not antichains

+-- {: .num_prop}
###### Proposition
Let $B$ be a complete Boolean algebra. For cardinals $\kappa$, there is a bijective correspondence between

* maps $g: \kappa \to B \setminus \{0\}$ such that $\{g(\alpha): \alpha \lt \kappa\}$ is an antichain in $B$;

* injective [[cocontinuous functor|colimit-preserving]] poset maps $f: \kappa \to B$,

here regarding $\kappa$ as an [[ordinal]].
=--

+-- {: .proof}
###### Proof
Given an antichain $g: \kappa \to B \setminus \{0\}$, define a colimit-preserving chain $f: \kappa \to B$ by $f(\beta) = \sum_{\alpha \lt \beta} g(\alpha)$ (so $f(0) = 0$, as mandated by colimit-preservation). Given an injective colimit-preserving chain $f: \kappa \to B$, define an antichain $g: \kappa \to B \setminus \{0\}$ by $g(\alpha) = f(\alpha + 1) \setminus f(\alpha) \coloneqq f(\alpha + 1) \wedge \neg f(\alpha)$. It is straightforward to verify that these assignments are inverse to one another.
=--

+-- {: .num_cor}
###### Corollary
$B$ satisfies the countable chain condition iff every chain $b_0 \lt b_1 \lt \ldots$ in $B$ is countable.
=--

This corollary is stated in terms of ascending chains, but by self-duality of Boolean algebras, it could equally well be stated in terms of descending chains.

For topological spaces $X$, the frame $H = Op(X)$ satisfies the countable chain condition iff $Reg(H)$, the complete Boolean algebra of regular open sets, satisfies the countable chain condition (Lemma \ref{reg}). Now a regular element is just an element of the form $\neg U$, which for an open set $U$ is the complement of its closure. Thus for the topological case we deduce the following formulation in terms of chains:

+-- {: .num_prop}
###### Proposition
A space $X$ satisfies the countable chain condition if and only if every collection of open sets $\{U_\alpha\}_{\alpha \lt \kappa}$ that satisfies the condition

* $\widebar{U_\alpha} \subset \widebar{U_\beta}$ (strict inclusion between closures) whenever $\alpha \lt \beta$

is countable.
=--

## Products of separable spaces

We show in this section that arbitrary products of separable spaces satisfy the countable chain condition.

+-- {: .num_prop #sep}
###### Proposition
A [[separable space]] satisfies the countable chain condition.
=--

+-- {: .proof}
###### Proof
Let $D$ be a countable dense set, and suppose given a collection of pairwise disjoint inhabited open sets $\{U_\alpha: \alpha \in A\}$. Each $U_\alpha$ contains at least one $d \in D$. Thus the map $D \to A$ sending $d \in D$ to the unique $\alpha \in A$ such that $d \in U_\alpha$ is surjective, and so $A$ is at most countable.
=--

Next we will need a result in combinatorial set theory. A *$\Delta$-system*, also called a *sunflower*, is a collection of finite sets $A_i$ such that any two intersect in the same set: there is some "common core" $S$ such that $A_i \cap A_j = S$ when $i \neq j$.

+-- {: .num_lemma}
###### Lemma
**(Sunflower lemma)**
Let $\mathcal{A}$ be an *uncountable* collection of finite sets. Then there is some uncountable subset $\mathcal{B} \subseteq \mathcal{A}$ that is a sunflower.
=--

+-- {: .proof}
###### Proof
Without loss of generality, we may suppose all $A \in \mathcal{A}$ have the same cardinality $n$. We argue by induction on $n$. The case $n = 0$ is trivial. Assume the result holds for $n$, and suppose ${|A|} = n + 1$ for all $A \in \mathcal{A}$. If some $a \in X = \bigcup \mathcal{A}$ belongs to uncountably many $A \in \mathcal{A}$, then these $A$ form a collection $\mathcal{A}'$, and by inductive hypothesis the collection $\{A \setminus \{a\}: A \in \mathcal{A}'\}$ has a sunflower $\mathcal{B}'$. Then induction goes through by forming the sunflower $\{B + \{a\}: B \in \mathcal{B}'\}$.

Otherwise, each $a \in X$ belongs to at most countably many $A \in \mathcal{A}$. In this case we may form an uncountable pairwise disjoint sequence $\{A_\alpha \in \mathcal{A}: \alpha \in \omega_1\}$ by transfinite induction: if $A_\alpha$ have been chosen for all $\alpha: \alpha \lt \beta \lt \omega_1$, then only countably many $A$ have an inhabited intersection with $\bigcup_{\alpha \lt \beta} A_\alpha$, and we simply choose an $A_\beta$ belonging to the uncountable collection that remains after removing such $A$. (So here the common core $S$ of the sunflower is the empty set.)
=--

+-- {: .num_prop #prod}
###### Proposition
If $\{X_i\}_{i \in I}$ is an arbitrary family of spaces such that any finite product of the $X_i$ satisfies the countable chain condition, then the full product $X = \prod_{i \in I} X_i$ also satisfies the countable chain condition.
=--

+-- {: .proof}
###### Proof
Suppose otherwise, that $\{U_\alpha\}_{\alpha \in A}$ is an uncountable collection of pairwise disjoint inhabited open sets of $X$. Shrinking them if necessary, we may suppose each $U_\alpha$ is a basic open of the form $\prod_{i \in F_\alpha} U_{\alpha, i} \times \prod_{i \notin F_\alpha} X_i$, where $F_\alpha \subseteq I$ is a finite set. The collection $\{F_\alpha\}$ has a sunflower by the Sunflower Lemma, say with common core $S$, and without loss of generality we may suppose $\{F_\alpha\}$ *is* that sunflower (i.e., that the $\alpha$\'s used to index the sunflower are just the $\alpha \in A$: throw away any other $\alpha \in A$). Let $\pi_S: X \to \prod_{i \in S} X_i$ be the obvious projection map. We first claim that the sets $\pi_S(U_\alpha)$ are pairwise disjoint, a purely set-theoretic matter.

Suppose instead $u \in \pi_S(U_\alpha) \cap \pi_S(U_{\alpha'})$. Regard $u \in \prod_{i \in S} X_i$ as a section of the canonical projection $\sum_{i \in S} X_i \to S$. That it belongs to $\pi_S(U_\alpha)$ simply means it extends to some section $\sigma$ of the canonical map $\sum_{i \in A} X_i \to A$ such that the restriction of $\sigma$ to $F_\alpha$ factors through the inclusion $\sum_{i \in F_\alpha} U_{\alpha, i} \hookrightarrow \sum_{i \in A} X_i$, as on the left half of the following diagram:

$$\array{
\sum_{i \in F_\alpha} U_{\alpha, i} & \hookrightarrow & \sum_{i \in A} X_i & \hookleftarrow & \sum_{i \in F_{\alpha'}} U_{\alpha', i} \\
\mathllap{\exists v} \uparrow & & \mathllap{\sigma} \uparrow \uparrow \mathrlap{\sigma'} & & \uparrow \mathrlap{\exists w} \\
F_\alpha & \hookrightarrow & A & \hookleftarrow & F_{\alpha'}\\
 & \nwarrow & & \nearrow & \\
 & & S & &
}$$

Similarly there is a section $\sigma': A \to \sum_{i \in A} X_i$ and a partial section $w$, as shown on the right half. With the partial sections $v$ and $w$ in hand, amalgamate them to a partial section over the union $F_\alpha \cup F_{\alpha'}$ (we can do this, as $v: F_\alpha \to \sum_{i \in A}$ and $w: F_{\alpha'} \to \sum_{i \in A} X_i$ both restrict to the same map $u$ on the intersection $S = F_\alpha \cap F_{\alpha'}$). Then extend the amalgamation $v \cup w$ however you please to a full section $\sigma'': A \to \sum_{i \in A} X_i$. This $\sigma''$ belongs to both $U_\alpha$ and $U_{\alpha'}$, contradicting their disjointness, and proving the claim.

Thus the $\pi_S(U_\alpha)$ form an uncountable collection of open, inhabited, pairwise disjoint sets, contradicting the countable chain condition hypothesis on the finite product $\prod_{i \in S} X_i$. With this the proof is complete.
=--

+-- {: .num_cor}
###### Corollary
An arbitrary product of separable spaces $X_i$ satisfies the countable chain condition.
=--

+-- {: .proof}
###### Proof
Any finite product of separable spaces $X_i$ is separable and thus satisfies the countable chain condition by Proposition \ref{sep}, so the full product does as well by Proposition \ref{prod}.
=--

## Related countability properties of topological spaces

[[!include topology - countability axioms]]

[[!redirects countable chain conditions]]


countable choice

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### Context
#### Foundations
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[[!include foundations - contents]]
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# Countable choice
* table of contents
{: toc}

## Idea

The axiom of __countable choice__ ($CC$), also called $AC_\omega$ or $AC_N$, is a weak form of the [[axiom of choice]]; it says that the set of [[natural number]]s is a [[projective object]] in [[Set]].  (Recall that the full axiom of choice states that *every* set is projective.)

In [[classical mathematics]], countable choice is usually accepted because the full axiom of choice is accepted.  In [[constructive mathematics]] the situation is more subtle.  For varying reasons, some schools of constructive mathematics accept countable choice (though they reject the full axiom of choice).  On the other hand, countable choice is not valid in the [[internal logic]] of a general topos, so if one desires this level of generality then it should not be assumed.  There are also philosophical constructivist arguments against it.  [[Fred Richman]] [(RichmanFTA)](#RichmanFTA) has said that

> Countable choice is a blind spot for constructive mathematicians in much the same way as excluded middle is for classical mathematicians.

All the reasoning in this page is [[constructive mathematics|constructive]].

## Definition

### In set theory

More explicitly, let $X$ be any [[set]] and let $p\colon X \to \mathbf{N}$ be a [[surjection]].  Then the axiom of countable choice states that $p$ has a [[section]].  If you phrase the axiom of choice in terms of [[entire relations]], then countable choice states that any entire relation from $\mathbf{N}$ to any other set contains (in the [[2-poset]] [[Rel]]) a [[functional relation|functional]] entire relation.

### In dependent type theory

In [[dependent type theory]], **countable choice** says that the [[dependent product type|dependent product]] of a family of inhabited sets indexed by the [[natural numbers]] is itself inhabited:

$$\frac{\Gamma, n:\mathbb{N} \vdash A(n) \; \mathrm{type}}{\Gamma \vdash \mathrm{AC}_\mathbb{N}^A:\left(\prod_{n:\mathbb{N}} \mathrm{isSet}(A(n))\right) \to \left(\prod_{n:\mathbb{N}} [A(n)]\right) \to \left[\prod_{n:\mathbb{N}} A(n)\right]}$$

Alternatively, the axiom of countable choice states that given a family of sets $A(n)$ indexed by natural numbers $n$ and a family of propositions $P(n, x)$ indexed by natural numbers $n$ and element $x:A(n)$, if for all natural numbers $n$ there merely exists an element $x:A(n)$ such that $P(n, x)$, then there merely exists a dependent function $g:\prod_{n:\mathbb{N}} A(n)$ such that for all natural numbers $n$, $P(n, g(n))$:

$$\frac{\Gamma, n:\mathbb{N} \vdash A(n) \; \mathrm{type} \quad \Gamma, n:\mathbb{N}, x:A(n) \vdash P(n, x) \; \mathrm{type}}{\Gamma \vdash \mathrm{AC}_\mathbb{N}^{A, P}:\left(\prod_{n:\mathbb{N}} \mathrm{isSet}(A(n)) \times \prod_{x:A(n)} \mathrm{isProp}(P(n, x))\right) \to \left(\prod_{n:\mathbb{N}} \exists x:A(n).P(n, x)\right) \to \exists g:\prod_{n:\mathbb{N}} A(n).\prod_{n:\mathbb{N}} P(n, g(n))}$$

## Consequences

Here we collect some consequences of the countable axiom of choice.

* The [[Cauchy real numbers]] and [[Dedekind real numbers]] coincide (the [[law of excluded middle]] also separately implies this).

* The [[Rosolini dominance]] is a [[dominance]] (the [[law of excluded middle]] also separately implies this).

* [[countable unions of countable sets are countable]]

## Variations

### $COSHEP$ & $DC$

Unlike the full axiom of choice, countable choice is often considered to be a [[constructive mathematics|constructively]] acceptable principle.  In particular, it does not imply the principle of [[excluded middle]].  It is a consequence of [[COSHEP]].  A stronger version of countable choice, also a consequence of $COSHEP$, is the axiom of [[dependent choice]] ($DC$).  In general, $DC$ is enough to justify results in [[analysis]] involving [[sequences]].

### $AC_{00}$

Sometimes in foundations it is useful to consider a weaker version of countable choice, called __$AC_{00}$__.  This states that any entire relation from $\mathbf{N}$ to itself contains a functional entire relation.  In terms of surjections, this states that any surjection $p\colon X \to \mathbf{N}$ has a section if $X$ is a [[subset]] of $\mathbf{N} \times \mathbf{N}$ and $p$ is the [[restriction]] to $X$ of a product projection.  $AC_{00}$ is enough to prove that every [[Dedekind real number]] is a [[Cauchy real number]] (the converse is always true).

### Weak countable choice {#WCC}

The axiom of __weak countable choice__ ($WCC$) states that a surjection $p\colon X \to \mathbf{N}$ has a section if, whenever $m \neq n$, at least one of the [[preimages]] $p^*(m)$ and $p^*(n)$ is a [[singleton]].  $WCC$ follows (for different reasons) from either $CC$ or [[excluded middle]].  On the other hand, $WCC$ is enough to prove that the [[fundamental theorem of algebra]] in the sense that every non-constant complex polynomial has a root; see [Bridges et al (1998)](#BRS).

### Another weak countable choice {#ACN2}

An axiom variously called $AC_{weak}$ and $AC_{N2}$ is countable choice for subsets of $\{0,1\}$; that is, every $\mathbb{N}$-indexed sequence of inhabited subsets of $\{0,1\}$ has a choice function.  Like $WCC$ above, this also follows from either $CC$ or excluded middle.  It is enough to prove the equivalence of the [[Dedekind reals]] and [[Cauchy reals]].  See [Saving et al (2021)](#Saving2021).

### Very weak countable choice

An even weaker form of countable choice was proposed by [Martin Escardo](#EscardoCN); it states that any surjection of the form $A \sqcup (\mathbf{N}\times B) \to \mathbf{N}$ has a section, where $A\to \mathbf{N}$ is a [[decidable subset]] and $B$ is an arbitrary set with $\mathbf{N}\times B \to \mathbf{N}$ the projection.  This follows from WCC and also from the [[limited principle of omniscience]]; see the [constructivenews discussion](#EscardoCN).

### Topos violating the CAC

One easy example is the category $Sh([0,1])$, the sheaves on the unit interval, since in that topos the [[real numbers object#classical_dedekind_real_numbers|Dedekind real numbers]] and the [[real numbers object#cauchy_real_numbers|Cauchy real numbers]] are not isomorphic, and this is a consequence of the internal countable choice.

Discussion [here](https://mathoverflow.net/questions/79807/example-of-a-topos-that-violates-countable-choice).

### In higher category theory

To formulate a version of the axiom of countable choice in a higher category, one has to make an appropriate choice of the meaning of "epimorphism". In most cases, it is best to choose [[effective epimorphisms|effective epimorphism in an (infinity,1)-category]] or a related notion such as [[eso morphisms]].

There are multiple version of the axiom of countable choice, depending on what [[truncated object|truncation]] requirements, if any, are applied to the domain.

* An $(\infty,1)$-category satisfies the **axiom of countable $n$-choice**, or $CC_n$, if every $n$-[[truncated morphism]] into the [[natural numbers object]] has a section. We write $CC_\infty$ for the **axiom of countable infinity-choice**: the statement that *every* morphism with codomain the [[natural numbers object]] has a section.

These are stronger axioms as $n$ increases.

There are also "internal" versions of these axioms.

* In [[homotopy type theory]] (the internal logic of an $(\infty,1)$-topos), the internal version of $CC_n$ is "every surjection onto the natural numbers type with $n$-type fibers has a section", or equivalently
  $$\prod_{(Y:\mathbb{N}\to n Type)} \Big( \prod_{(x:\mathbb{N})} \Vert Y(x)\Vert \to \Big\Vert \prod_{(x:\mathbb{N})} Y(x) \Big\Vert \Big) $$

* More generally, we can replace the $(-1)$-truncation by the $k$-truncation to obtain a family of axioms $CC_{k,n}$.

* We can also replace the $(-1)$-truncation by the assertion of $k$-connectedness, obtaining the **axiom of countable $k$-connected choice**.

## References

* {#BRS} [[Fred Richman]], [[Douglas Bridges]], Peter Schuster, *A weak countable choice principle*. Proceedings of the American Mathematical Society 128(9):2749-2752, March 2000. [[doi:10.1090/S0002-9939-00-05327-2](http://dx.doi.org/10.1090/S0002-9939-00-05327-2)]

*  {#EscardoCN}Martin Escardo et. al., *Special case of countable choice*, message and discussion to the constructivenews list, [google groups](https://groups.google.com/d/msg/constructivenews/PeLsQWDFJNg/7piOmUHbAAAJ)

* {#RichmanFTA} [[Fred Richman]], *The fundamental theorem of algebra: a constructive development without choice*. Pacific Journal of Mathematics **196** 1 (2000) 213–230 [[doi:10.2140/pjm.2000.196.213](http://dx.doi.org/10.2140/pjm.2000.196.213), [pdf](https://msp.org/pjm/2000/196-1/pjm-v196-n1-p10-p.pdf)]

* {#Saving2021} Mark Saving et al (2021). *A weak form of countable choice*. [MathOverflow](https://mathoverflow.net/questions/385805/a-weak-form-of-countable-choice/).

See also

* Wikipedia, _[Axiom of countable choice](https://en.wikipedia.org/wiki/Axiom_of_countable_choice)_

category: foundational axiom

[[!redirects countable choice]]
[[!redirects countable axiom of choice]]
[[!redirects axiom of countable choice]]

[[!redirects countable n-choice]]
[[!redirects countable axiom of n-choice]]
[[!redirects axiom of countable n-choice]]

[[!redirects countable infinity-choice]]
[[!redirects countable axiom of infinity-choice]]
[[!redirects axiom of countable infinity-choice]]

[[!redirects AC00]]

[[!redirects weak countable choice]]
[[!redirects WCC]]


countable cover

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### Context
#### Topology
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[[!include topology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A _countable cover_ is a [[cover]] by a [[countable set]] of patches.

Accordingly a _countable open cover_ is an [[open cover]] with a countable set of patches.

A [[topological space]] for which every [[open cover]] has a countable subcover is called a _[[Lindelöf space]]_.

## Related concepts

* [[finite cover]]

* [[locally finite cover]]

* [[countably compact topological space]]

[[!redirects countable covers]]

[[!redirects countable open cover]]
[[!redirects countable open covers]]