abelian infinity-group

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#### Higher algebra
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#Contents#
* table of contents
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## Idea

An ordinary [[group]] is either an [[abelian group]] or not. For an [[∞-group]] there is an infinite tower of notions ranging from completely general non-abelian [[∞-groups]], over [[braided infinity-group|braided $\infty$-groups]], [[sylleptic infinity-groups|sylleptic $\infty$-groups]] ..., to ever more abelian groups  By an _abelian ∞-group_ (not an established term) one may want to mean an [[∞-group]] which is maximally abelian, in this sense.

Technically, the level of abelianness may be encoded (see at *[[May recognition theorem]]*) by the [[En-operads|$E_n$-operads]] as $n$ ranges from 1 to $\infty$: On the non-abelian end, a general [[∞-group]] is equivalently a groupal [[algebra over an operad|algebra]] over $E_1$, also known as the [[associative operad]], hence is a  groupal [[A-∞ algebra]]; while at the abelian end a groupal [[E-infinity space|$E_\infty$-space]] is an *[[infinite loop space]]* or *[[connective spectrum]]*. See also the *[periodic table](k-tuply+groupal+n-groupoid#PeriodicTable)* of [[k-tuply monoidal n-groupoid|$k$-tuply monoidal $n$-groupoids]].

Notice that referring to [[connective spectra]] as "abelian $\infty$-groups" (which is not standard) matches the established terminology for *[[non-abelian cohomology]]* (which is standard): The qualifier "non-abelian" in [[non-abelian cohomology]] is in contrast to [[Whitehead-generalized cohomology theories]] which are [[Brown representability theorem|represented]] by [[spectra]].

In a more restrictive sense one may say that plain abelian [[cohomology]] is just *[[ordinary cohomology|ordinary]]* cohomology theory, subsuming only those [[Whitehead-generalized cohomology theories]] which are [[Brown representability theorem|represented]] specifically by [[Eilenberg-MacLane spectra]]. Under the [[Dold-Kan correspondence]] these are [[equivalence of (infinity,1)-categories|equivalently]] [[chain complexes]] of [[abelian groups]]. One may think of these as being yet more commutative than general spectra and might want to reserve the term "abelian $\infty$-group" for them.

## Proposition

### Relation to commutative $\infty$-rings

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

Write

$$
  gl_1 \; \colon \;  CRing_\infty \to AbGrp_\infty
$$

for the [[(∞,1)-functor]] which sends a [[E-∞ ring|commutative ∞-ring]] to its [[∞-group of units]].

=--

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

The [[∞-group of units]] [[(∞,1)-functor]] of def. \ref{GroupOfUnitsFunctor} is a  right-[[adjoint (∞,1)-functor]] (or at least a [[right adjoint]] on [[homotopy category of an (∞,1)-category|homotopy categories]])

$$
  CRing_\infty
  \stackrel{\overset{\mathbb{S}[-]}{\leftarrow}}{\underset{gl_1}{\to}}
  AbGrp_\infty
  \,.
$$

=--

This is ([ABGHR 08, theorem 2.1](#ABGHR08)).

## Examples

* A [[0-truncated]] abelian $\infty$-group is equivalently an [[abelian group]].

* A [[1-truncated]] abelian $\infty$-group is equivalently a [[symmetric 2-group]].

* A [[2-truncated]] abelian $\infty$-group is equivalently a [[symmetric 3-group]].

## Related concepts

[[!include k-monoidal table]]

* [[∞-group]], [[k-tuply groupal n-groupoid]]

* [[braided ∞-group]],

  * [[braided 2-group]]

  * [[braided 3-group]]

* [[sylleptic ∞-group]]

  * [[sylleptic 3-group]]

* **abelian ∞-group**

  * [[abelian group]]

  * [[abelian 2-group]]

  * [[abelian 3-group]]

## References

General discussion is in section 5 of

* [[Jacob Lurie]], _[[Higher Algebra]]_

Discussion in the context of [[E-∞ rings]] and [[twisted cohomology]] is in

* {#ABGHR08} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], _Units of ring spectra and Thom spectra_ ([arXiv:0810.4535](http://arxiv.org/abs/0810.4535))

[[!redirects abelian infinity-groups]]

[[!redirects k-monoidal ∞-group]]

[[!redirects abelian ∞-group]]
[[!redirects abelian ∞-groups]]


abelian Lie algebra

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## Definition

A [[Lie algebra]] $\mathfrak{g}$ is __abelian__ if its bracket is identically 0, in that for all $x,y \in \mathfrak{g}$ we have

$$
  [x,y] = 0
  \,.
$$

## Examples

Every [[vector space]] has a (necessarily unique) abelian Lie algebra structure.  As such, we may identify an abelian Lie algebra with its underlying vector space.

A $0$-dimensional or $1$-dimensional Lie algebra must be abelian.  The $0$-dimensional Lie algebra is the [[trivial Lie algebra]].  The $1$-dimensional Lie algebra is a [[simple object]] in [[LieAlg]], but it is traditionally *not* considered a [[simple Lie algebra]].

## Lie integration

Under [[Lie integration]] abelian Lie algebras integrate to [[abelian Lie group]]s.

[[!redirects abelian Lie algebra]]
[[!redirects abelian Lie algebras]]
[[!redirects Abelian Lie algebra]]
[[!redirects Abelian Lie algebras]]


abelian model category

## Idea

An abelian model category is an [[abelian category]] with a compatible [[model structure]].

## Definition

An **abelian model category** is an [[abelian category]] $\mathcal{A}$ that is [[complete category|complete]] and [[cocomplete category|cocomplete]], together with a [[model structure]] such that

* (AMC1) A [[morphism]] $i : A \to B$ is a [[cofibration]] if and only if it is a [[monomorphism]] with [[cofibrant object|cofibrant]] [[cokernel]].
* (AMC2) A [[morphism]] $p : X \to Y$ is a [[fibration]] if and only if it is an [[epimorphism]] with [[fibrant object|fibrant]] [[kernel]].

## Relation to cotorsion pairs

[Hovey](#Hovey) has shown that, roughly speaking, [[model structures]] on [[abelian categories]] correspond to [[cotorsion pairs]].

In one direction we have

+-- {: .num_prop}
###### Proposition
Let $\mathcal{A}$ be an [[abelian model category]], i.e. an [[abelian category]] with a compatible [[model structure]].  Let $\mathcal{C}$, $\mathcal{F}$, and $\mathcal{W}$ denote the classes of cofibrant, fibrant, and trivial objects, respectively.

Then $(\mathcal{C} \cap \mathcal{W}, \mathcal{F})$ and $(\mathcal{C}, \mathcal{F} \cap \mathcal{W})$ are complete [[cotorsion pairs]].
=--

And under some more assumptions we have a converse

+-- {: .num_theorem}
###### Theorem
Let $\mathcal{A}$ be an [[abelian category]] that is [[complete category|complete]] and [[cocomplete category|cocomplete]].  Let $\mathcal{C}$, $\mathcal{F}$, and $\mathcal{W}$ denote three classes of objects in $\mathcal{A}$, such that $\mathcal{W}$ is a [[thick subcategory]], and $(\mathcal{C} \cap \mathcal{W}, \mathcal{F})$ and $(\mathcal{C}, \mathcal{F} \cap \mathcal{W})$ are complete [[cotorsion pairs]].

Then there exists a unique [[abelian model structure]] on $\mathcal{A}$ such that $\mathcal{C}$, $\mathcal{F}$, $\mathcal{W}$ are the classes of [[cofibrant]], [[fibrant]], and trivial objects, respectively.
=--

Under certain assumptions on the cotorsion pair we can further guarantee that the associated model structure is [[monoidal model category|monoidal]].

+-- {: .num_theorem}
###### Theorem
Under the assumptions of the previous theorem, suppose further that $\mathcal{A}$ is [[closed monoidal category|closed]] [[symmetric monoidal category|symmetric monoidal]], and that

* Every object in the class $\mathcal{C}$ is [[flat]] ($X \otimes \cdot$ is an [[exact functor]]).
* For any two objects $X$ and $Y$ in $\mathcal{C}$, the tensor product $X \otimes Y$ is also in $\mathcal{C}$.  If one of $X$ and $Y$ is further in $\mathcal{W}$ then $X \otimes Y$ is also in $\mathcal{W}$.
* The unit is in $\mathcal{C}$.

Then $\mathcal{A}$ is a [[monoidal model category]] (with the model structure given by the previous theorem).
=--

## References

* [[Mark Hovey]], _Cotorsion pairs, model category structures, and representation theory_, Math. Z. 241 (2002), no. 3, 553–592. MR 2003m:55027
{#Hovey}

An overview is in

* [[Mark Hovey]], _Cotorsion pairs and model categories_, 2006 ([pdf](http://homepages.math.uic.edu/~bshipley/hovey.pdf))

[[!redirects abelian model categories]]
[[!redirects abelian model structure]]
[[!redirects abelian model structures]]


abelian sheaf

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#### Additive and abelian categories
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#### Topos Theory
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> see also _[[sheaf of abelian groups]]_

#Contents#
* table of contents
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## Idea

An **abelian sheaf** is a [[sheaf]] with values in an [[abelian category]] which usually is itself, or is taken to be embedded in, a category of [[complex]]es in an [[abelian category]].

In light of the [[Dold-Kan correspondence]] this means that abelian sheaves can usefully be regarded as special cases of [[simplicial presheaf|simplicial presheaves]] and in particular the corresponding [[derived category]] of abelian sheaves, traditionally mainly investigated in terms of [[sheaf cohomology]], is analogous to the [[homotopy category]] of abelian [[infinity-stack homotopically|infinity-stacks]]. In this way, via [[Dold-Kan correspondence|Dold-Kan]], plain abelian sheaves already go a long way towards (abelian) [[Higher Topos Theory]], which is one way of understanding the relevance of the concept of abelian sheaves.

For instance [[Deligne cohomology]], which classifies higher abelian [[gerbe]]s (certain [[infinity-stack]]s) with [[connection]]), is the [[sheaf cohomology]] of a certain class of sheaves with values in abelian complexes. This is understood conceptually by realizing that after embedding complexes of abelian sheaves -- via [[Dold-Kan correspondence|Dold-Kan]] -- into  general [[simplicial presheaf|simplicial sheaves]], a complex of abelian sheaves becomes an abelian $\infty$-prestack and the computation of its [[sheaf cohomology]] corresponds to passing to its [[infinity-stackification]].

## Properties

### Projective objects

See at _[[projective object]]_ the section _[Existence of enough projectives](projective+object#ExistenceOfEnoughProjectives)_.

## Related concepts

* [[sheaf on a topological space]]

* [[abelian group]]

* [[abelian sheaf cohomology]]

* an abelian sheaf of [[torsion groups]] is called a _[[torsion sheaf]]_

## References

A basic textbook introduction begins for instance around Definition 1.5.6 of

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

A detailed textbook discussion is in section 18 of

* [[Masaki Kashiwara]], [[Pierre Schapira]], _Categories and Sheaves_, Grundlehren der Mathematischen Wissenschaften __332__, Springer (2006)

category: sheaf theory
[[!redirects abelian sheaves]]


abelian sheaf cohomology

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#### $(\infty,1)$-Topos Theory
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[[!include (infinity,1)-topos - contents]]
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#Contents#
* table of contents
{:toc}

##Idea##

### General

The [[cohomology]] $H^n(X,F)$ of a [[topological space]] $X$ with values in a [[sheaf of abelian groups]] / [[abelian sheaf]] $F$ was originally defined as the value of the right [[derived functor]] of the [[global section]] functor, the [[derived direct image]] functor.

But by embedding sheaves with values in abelian groups as special cases of [[simplicial presheaf|simplicial sheaves]] into the more general context of [[(∞,1)-sheaves|∞-groupoid-valued sheaves]] via the [[Dold-Kan correspondence]] and thus the abelian sheaf cohomology into the more general context of the intrinsic [[nonabelian cohomology]] of an [[(∞,1)-topos]] $\mathbf{H} = Sh_{(\infty,1)}(C)$, this definition becomes equivalent to a special case of the general notion of [[nonabelian cohomology]] defined simply as the set of homotopy classes of maps

$$
  H^n(X,F) = \pi_0 \mathbf{H}(X,\mathbf{B}^n F)
$$

from the space $X$ regarded a ("nonabelian"!) sheaf, to the [[Eilenberg-MacLane object]] in degree $n$, defined by $F$.

The relation of this more conceptual and more general point of view on abelian sheaf cohomology to the original definition was originally clarified in

* [[Kenneth Brown]], [[BrownAHT|Abstract Homotopy Theory and Generalized Sheaf Cohomology]]

(whose proof is reproduced below).

Brown constructed effectively the [[homotopy category of an (∞,1)-category|homotopy category]] of $\mathbf{H}$ using a model of a [[category of fibrant objects]] paralleling the  [[model structure on simplicial presheaves]] as a [[presentable (∞,1)-category|presentation]] of the [[(∞,1)-category of (∞,1)-sheaves]]. This says that ordinary abelian sheaf cohomology in fact computes the equivalence classes of the [[∞-stackification]] of a sheaf with values in [[chain complexes]] of [[abelian groups]].

The general [[(∞,1)-topos]]-theoreric perspective on cohomology is described in more detail at [[cohomology]]. The details on how to realize abelian sheaf cohomology as an example of this are discussed below.

### More details on this idea ###

Using the [[Dold-Kan correspondence]] in [[higher topos theory]], [[complex]]es of [[abelian sheaf|abelian sheaves]] can be understood as a generalization of [[topological space]]s, in a precise sense recalled below.
This induces a notion of cohomology of [[complex]]es of [[abelian sheaf|abelian sheaves]] from the familiar notion of cohomology of spaces.

Which is a simple one: recall that the [[cohomology]] of one [[topological space]] $X$ with coefficients in another space $A$ is nothing but the space of morphisms (continuous maps) $H(X,A) := [X,A]$ from $X$ to $A$ -- or, in a more restrictive sense traditionally adopted, the set $\Pi_0[X,A]$ of connected path components of this space.

Similarly, when considering [[chain complexes]] of [[abelian sheaves]] in their natural [[higher topos theory|higher topos theoretic]] home, the cohomology of a complex of sheaves $A$ on a space $X$ is nothing but the hom-space $H(X,A) = [X,A]$ -- where the space $X$ itself is regarded as a special case of a sheaf.

One reason this conceptually simple picture is not usually presented is that the space $X$ is typically not represented by an _abelian_ complex of sheaves, so that the full simplicity of the story becomes manifest only in general [[nonabelian cohomology]].

More precisely, via the [[Dold-Kan correspondence]] (non-negatively graded) [[complex]]es of [[abelian sheaf|abelian sheaves]] -- which are equivalently [[sheaf|sheaves]] with values in (non-negatively graded) [[category of chain complexes|categories of chain complexes]] -- can be regarded as special cases of [[simplicial presheaf|simplicial sheaves]]. But thanks to the [[model structure on simplicial presheaves|model category structure]] on the category of [[simplicial presheaf|simplicial sheaves]], this in turn is a model for the [[(infinity,1)-topos]] of  [[space and quantity|generalized spaces]] called [[infinity-stack]]s. The very point of $(\infty,1)$-[[(infinity,1)-topos|topoi]] is that they are [[(infinity,1)-category|(infintiy,1)-categories]] which behave in all structural aspects relevant for [[homotopy theory]] as the archetypical example [[Top]] does. In particular, as in [[Top]], the notion of [[cohomology]] in any [[(infinity,1)-topos]] simply coincides with that of [[hom-space]]s.

In the 1-categorical [[model category|model theoretic models]] these hom-spaces are computed technically by [[derived functor]]s. More precisely, the Hom-space $[X,A]$ for $X$ an ordinary space computes the [[global section]]s $\Gamma(X,A)$ of the complex of [[abelian sheaf|abelian sheaves]] $A$ which is computed by the right [[derived functor]] of the [[global section]] $R \Gamma(X,-)$ of the [[global section functor]] $\Gamma(X,-)$, which does exist entirely within the abelian context.

This, then, is the definition of sheaf cohomology as usually presented: the cohomology of the complex $R \Gamma(X,A)$.

## Properties

### As intrinsic (∞,1)-topos cohomology
 {#AsIntrinsicInfinityToposCohomology}

Under the ([[stable Dold-Kan correspondence|stable]]) [[Dold-Kan correspondence]] we have the following
identification of sheaves taking values in [[chain complexes]]
with sheaves taking values in [[infinity-groupoids]] and
[[spectrum|spectra]], crucial for a conceptual understanding of
abelian sheaf cohomology:

let $X$ be a [[site]]

* the category $Sh(X, Ch_+(Ab))$ of non-negatively graded
[[chain complex]]es of [[abelian sheaf|abelian sheaves]] is
[[homotopical functor|homotopically]] [[equivalence|equivalent]]
to the category $Sh(X, sAb)$ of [[sheaf|sheaves]] with values in
simplicial abelian groups (i.e. [[Kan complex]]es with strict abelian group structure);

* the category $Sh(X, Ch(Ab))$ of unbounded
[[chain complex]]es of [[abelian sheaf|abelian sheaves]] is
[[equivalence of categories|equivalent]] to $Sh(X, Sp(Ab))$, the category
of sheaves with values in [[combinatorial spectrum|combinatorial spectra]]
internal to abelian groups.

Let how $F \in Sh(X,Ab)$ be a [[sheaf]] on a [[site]] $X$ with values in the category [[Ab]] of abelian groups.

For $n \in \mathbb{N}$ write $B^n F \in Sh(X, Ch_+(Ab))$ for the [[complex]] of [[sheaf|sheaves]] with values in abelian groups which is trivial everywhere except in degree $n$, where it is given by $F$.

By the [[Dold-Kan correspondence]] we can regard $B^n F$ equivalently as a complex of sheaves of abelian groups as well as sheaf with values in [[infinity-groupoid]]s.

Write $H$ for the [[(infinity,1)-category]] of
[[simplicial presheaf|simplicial sheaves]] on $X$ and $H_{ab}$ for the
[[(infinity,1)-category]] of complexes of [[abelian sheaf|abelian sheaves]] on $X$.

Write $X$ for the terminal sheaf of $X$, i.e. for the [[sheaf]] that corresponds to the space $X$ itself.

Then

$$
  H^n(X,A) := \pi_0 H(X,\mathbf{B}^n F)
$$

is the degree $n$ [[cohomology]] class of $X$ with values in $F$, regarded as computed in [[nonabelian cohomology]].

Now write $\mathbb{Z}[X]$ for the free abelianization of the sheaf $X$. This is the sheaf constant on the abelian group $\mathbb{Z}$ of integers. Then the above cohomology set, which of course happens to be a cohomology group here, due to the abelianness of $F$, is canonically isomorphic to the cohomology set

$$
  \cdots \simeq \pi_0 H_{ab}(\mathbb{Z}[X], \mathbf{B}^n F)
$$

which can be regarded as the [[hom-set]] in the [[derived category]] of [[complex]]es of [[abelian sheaf|abelian sheaves]]. This, in turn, is the same as the traditional expression

$$
  \cdots \simeq R^n \Gamma(X,F)
$$

giving the $n$th [[derived functor]] of the [[global section functor]] of the [[abelian sheaf]] $F$.

This, finally, is the same group as obtained by choosing any [[complex]] $I_F$ of [[abelian sheaf|abelian sheaves]] that is [[injective complex|injective]] and [[quasi-isomorphism|quasi-isomorphic]] to $F$ regarded as a complex concentrated in degree 0 and then computing the $n$ [[homology]] group of the complex $\Gamma(X,I_F)$ of global sections of $F$:

$$
  \cdots \simeq H_n(\Gamma(X,I_F))
  \,.
$$

Historically the development of abelian sheaf cohomology was precisely in reverse order to this derivation from the general $(\infty,1)$-[[(infinity,1)-category|categorical]] [[cohomology]].

+-- {: .num_theorem}
###### Theorem (K. Brown, 1973)

Let $X$ be a [[topological space]], $F$ a [[sheaf]] on (the [[category of open subsets]] of) $X$ with values in abelian groups, and $\mathbf{B}^n F = K(F,n)$ the image of the complex of abelian sheaves $F[n]$ ($F$ in degree $n$, trivial elsewhere) under the [[Dold-Kan correspondence]] in sheaves with values in [[Kan complex]]es
$$
  \Gamma \circ (-) : Sh(X,Ch_+(Ab)) \to Sh(X, AbSimpGrp)
$$
$$
  F[n] \mapsto K(F,n) =: \mathbf{B}^n F
  \,.
$$

Then we have the following natural isomorphism of cohomologies:

$$
  H^n(X,F) \simeq H(X, \mathbf{B}^n F)
$$

where

* on the left we have ordinary abelian sheaf cohomology defined as the right [[derived functor]] of the global sections functor
  $$
    H^n(X,F) := (R \Gamma_X)(F)
    \,;
  $$

* on the right we have [[nonabelian cohomology]], namely the hom-set in the [[homotopy category]] of Kan complex valued [[model structure on simplicial presheaves|simplicial sheaves]]
  $$
    H(X, \mathbf{B}^n F) := Ho_{Sh(X,\infty Grpd)}(X,\mathbf{B}^n X)
   \,.
  $$

=--

+-- {: .proof}
###### Proof

This is the first four steps in the proof of theorem 2 in [[BrownAHT]].

The proof proceeds along the following four steps, which we describe in more detail below:

$$
  \begin{aligned}
    H^n(X,F) & \simeq
     Ho_{Sh(X,Ch(Ab))}(\mathbb{Z}, F[n])
     \\
     & \simeq
     Ho_{Sh(X,Ch_+(Ab))}(\mathbb{Z}, F[n])
     \\
     & \simeq
     Ho_{Sh(X,AbSimpGrp)}(\mathbb{Z}X, K(F,n))
     \\
     & \simeq
     Ho_{Sh(X,\infty Grpd)}(X, K(F,n))
  \end{aligned}
$$

1. By the [[derived functor]] definition of sheaf cohomology, $H^n(X,F)$ is the cohomology of any complex of sheaves $I^\bullet \in Sh(X,Ch(Ab))$ that is [[injective object|injective]] and weakly equivalent to $F[n]$, $F[n] \stackrel{\simeq}{\to} I^\bullet$:
   $$
     H^n(X,F) \simeq H^0(I^\bullet(X))
     \,.
   $$
   On the other hand, by the general formula for hom-sets in [[homotopy category|homtotopy categories]] obtained by localizing at the [[calculus of fractions|multiplicative system]] given by [[quasi-isomorphism]]s of complexes (e.g. def. 13.1.2 in [[Categories and Sheaves|CaS]]) we have
   $$
     Ho_{Sh(X,Ch(Ab))}(\mathbb{Z}, F[n])
     \simeq
     colim_{Y^\bullet \stackrel{\simeq}{\to} \mathbb{Z}} Hom_{K(Sh(X,Ab))}(Y, I^\bullet)
     \,.
   $$
   But due to the injectiveness of $I^\bullet$, the integrand on the right is constant (lemma 14.1.5 in [[Categories and Sheaves|CaS]]) and hence the colimit is isomorphic to $\cdots \simeq Hom_{K(Sh(X,Ab))}(\mathbb{Z}, I^\bullet) \simeq H^0(I^\bullet(X))$, as desired.

1. The second step uses that the inclusion functor
   $$
     Ho_{Sh(X,Ch_+(Ab))} \hookrightarrow Ho_{Sh(X,Ch(Ab))}
   $$
   is [[full and faithful functor|full and faithful]].
   This in turn follows from

   * first observing that the inclusion
     $S : Sh(X,Ch_+(Ab)) \hookrightarrow Sh(X, Ch(Ab))$ of
     chain complexes concentrated in non-negative degree
     into all complexes of sheaves is
     [[full and faithful functor|full and faithful]] and
     has the obvious [[right adjoint]]
     $T : Sh(X,Ch(Ab)) \to Sh(X, Ch_+(Ab))$ obtained by
     **t**runcating a complex.

    * By inspection, or else by the general properties
     of [[adjoint functor]]s (see the list of properties
     given there) this implies that $Id \to T \circ S$
     is an [[isomorphism]]. This implies that also
     $Id \to Ho T \circ Ho S$ is an [[isomorphism]].
    * But by the adjoint functor lemma for homotopical
     categories, $Ho S$ is also left adjoint to $Ho T$
     (since both preserve weak equivalences). So that
     once again with the general properties of
     [[adjoint functor]]s it follows that
     $Ho S$ is
     [[full and faithful functor|full and faithful]].

1. The third step uses that the [[normalized chain complex]] functor $Sh(X,AbSimpGrp) \to Sh(X, Ch_+(Ab))$ is an [[equivalence of categories]] that preserves the respective weak equivalences and homotopies.

1. The fourth step finally uses that the [[stuff, structure, property|forgetful functor]] $Sh(X, SimpAbGrp) \to Sh(X, \infty Grpd)$ that only remembers the [[Kan complex]] underlying a [[simplicial group]] has a [[left adjoint]], the free abelian group functor $\mathbb{Z} : Sh(X,\infty Grpd) \to Sh(X, AbSimpGrp)$ (see [[Dold-Kan correspondence]] for details), and that preserves weak equivalences (see the discussion at [[simplicial group]] for more on that).

=--

### Relation to derived direct images

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

Let $f^{-1} \colon Y \to X$ be a [[morphism of sites]].
Then the $q$th [[derived functor]] $R^q f_\ast$ of the induced [[direct image]] functor sends $\mathcal{F} \in Ab(Sh(X_{et}))$ to the [[sheafification]] of the [[presheaf]]

$$
  U_Y
  \mapsto
  H^q(f^{-1}(U_Y), \mathcal{F})
  \,,
$$

where on the right we have the degree $q$ [[abelian sheaf cohomology]] [[cohomology group|group]] with [[coefficients]] in the given $\mathcal{F}$.

=--

(e.g. [Tamme, I (3.7.1), II (1.3.4)](#Tamme), [Milne, 12.1](#Milne)).

+-- {: .proof}
###### Proof

We have a [[commuting diagram]]

$$
  \array{
    Ab(PSh(X)) &\stackrel{(-)\circ f^{-1}}{\longrightarrow}& Ab(PSh(Y))
    \\
    \uparrow^{\mathrlap{inc}} && \downarrow^{L}
    \\
    Ab(Sh(X)) &\stackrel{f_\ast}{\longrightarrow}& Ab(Sh(Y))
  }
  \,,
$$

where the right vertical morphism is [[sheafification]]. Because $(-) \circ f^{-1}$ and $L$ are both [[exact functors]] it follows that for $I^\bullet \to \mathcal{F}$ an [[injective resolution]] that

$$
  \begin{aligned}
     R^p f_\ast(\mathcal{F})
     & :\simeq
     H^p( f_\ast I)
     \\
     & = H^p(L I^\bullet(f^{-1}(-)))
     \\
     & = L (H^p(I^\bullet)(f^{-1}(-)))
  \end{aligned}
$$

=--

## Examples

* [[de Rham cohomology]]

* [[Dolbeault cohomology]]

* [[Spencer cohomology]]

* [[Deligne cohomology]]

* [[etale cohomology]]

* [[crystalline cohomology]]

* [[syntomic cohomology]]

## Related concepts

* [[sheaf]], [[cohomology]]

* **abelian sheaf cohomology**

  * [[model structure on chain complexes]]

  * [[resolutions]]: [[soft sheaf]], [[fine sheaf]], [[flabby sheaf]]

  * [[triangulated category of sheaves]]

  * [[Verdier duality]]

## References

Textbook account:

* [[Jean Gallier]], [[Jocelyn Quaintance]], *Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry*, World Scientific (2022) [[doi:10.1142/12495](https://doi.org/10.1142/12495), [webpage](https://www.cis.upenn.edu/~jean/gbooks/sheaf-coho.html)]

Original texts:

* {#Godement58} [[Roger Godement]], *Topologie algébrique et theorie des faisceaux*, Hermann, Paris (1958) [[webpage](https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement), [[Godement-TopologieAlgebrique.pdf:file]]]

The traditional definition of sheaf cohomology in terms of the right [[derived functor]] of the [[global sections]] functor:

* [[Ugo Bruzzo]], _Derived Functors and Sheaf Cohomology_, Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes: Volume 2 ([doi:10.1142/11473](https://doi.org/10.1142/11473))

* Chênevert, Kassaei, _Sheaf Cohomology_ ([pdf](http://www.math.mcgill.ca/goren/SeminarOnCohomology/Sheaf_Cohomology.pdf))

* Cibotaru, _Sheaf cohomology_ ([pdf](http://www.nd.edu/~lnicolae/sheaves_coh.pdf))

* [[Patrick Morandi]], _Sheaf cohomology_ ([pdf](http://sierra.nmsu.edu/morandi/notes/SheafCohomology.pdf))

Its discussion in the more general [[nonabelian cohomology]] and [[infinity-stack]] context emphasized above is due to

* [[Kenneth Brown]], _[[BrownAHT|Abstract Homotopy Theory and Generalized Sheaf Cohomology]]_

This uses homotopical structures of a [[category of fibrant objects]] on complexes of abelian sheaves. Discussion of actual [[model structure on chain complexes]] of abelian sheaves is in

* [[Mark Hovey]], _Model category structures on chain complexes of sheaves_, Trans. Amer. Math. Soc. 353, 6 ([pdf](http://www.mathaware.org/tran/2001-353-06/S0002-9947-01-02721-0/S0002-9947-01-02721-0.pdf))

A discussion of the [[Čech cohomology]] description of sheaf cohomology along the above lines is in

* [[Tibor Beke]], _Higher Čech Theory_ ([web](http://www.math.uiuc.edu/K-theory/0646/), [pdf](http://www.math.uiuc.edu/K-theory/0646/cech.pdf))

See also:

* {#Duskin79} [[John Duskin]], _Higher-dimensional torsors and the cohomology of topoi: the abelian theory_, p. 255-279 in: _Applications of sheaves_,  Lecture Notes in Mathematics **753**, Springer (1979) [[doi:10.1007/BFb0061822](https://doi.org/10.1007/BFb0061822)]

* {#Tamme} [[Günter Tamme]], section II 1 of _[[Introduction to Étale Cohomology]]_

* {#Milne} [[James Milne]], section 7 of _[[Lectures on Étale Cohomology]]_

[[!redirects chain complex of sheaves]]
[[!redirects chain complexes of sheaves]]

[[!redirects sheaf of chain complexes]]
[[!redirects sheaves of chain complexes]]

[[!redirects abelian presheaf]]
[[!redirects abelian presheaves]]


abelian subcategory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Notions of subcategory
+-- {: .hide}
[[!include notions of subcategory]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

For $\mathcal{A}$ an [[abelian category]] an _abelian subcategory_ is a [[subcategory]] $\mathcal{B} \to \mathcal{A}$ such that $\mathbf{B}$ is also [[abelian category|abelian]] and such that every [[exact sequence]] in $\mathcal{B}$ is also exact in $\mathcal{A}$.

## References

Section 1.2 of

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

[[!redirects abelian subcategories]]


abelian variety

#Contents#
* table of contents
{:toc}

## Idea

Abelian varieties are higher dimensional analogues of [[elliptic curve]]s (which are included) -- they are [[varieties]] equipped with a structure of an [[abelian group]], hence abelian [[group schemes]], whose multiplication and inverse are regular maps.

## Definition

In his book _Abelian Varieties_, David Mumford defines an **abelian variety** over an algebraically closed field $k$ to be a [[complete algebraic variety|complete]] [[algebraic group]] over $k$.  Remarkably, any such thing is an _abelian_ algebraic group.  The assumption of connectedness is necessary for that conclusion.

##  Automatic abelianness

David Mumford gives at two proofs that every complete algebraic group over an algebraically closed field is automatically abelian.  One of them uses a 'rigidity lemma' which has an interesting category-theoretic interpretation.  We outline this here:

In simple terms, the rigidity lemma says that under certain circumstances "a 2-variable function $f(x,y)$ that is independent of $x$ for one value of $y$ is independent of $x$ for all values of $y$."

More precisely, in a category with products, say a morphism $f: X \to Y$ is constant if it factors through the unique morphism $X \to 1$.  Say a morphism $f : X \times Y \to Z$ is independent of $X$ if it factors through the projection $X \times Y \to Y$.

Say a point of $Y$ is a morphism $p: 1 \to Y$.  Say a morphism $f: X \times Y \to Z$ is independent of $X$ at some point $p$ of $Y$ if $f \circ (1_X \times p) : X \to Z$ is constant.

**Definition.** A category with finite products **obeys the rigidity lemma** if any morphism $f: X \times Y \to Z$ that is independent of $X$ at some point of $Y$ is in fact independent of $X$.

**Theorem 1.** The category of complete algebraic varieties over an algebraically complete field $k$ has finite products and obeys the rigidity lemma.

+-- {: .proof}
######Proof
The hard part, the rigidity lemma, is proved for complete algebraic varieties on page 43 of Mumford's _Abelian Varieties_.  Mumford mentions in a footnote that complete algebraic varieties are automatically irreducible, and he later seems to assume without much explanation that they are connected: these points could use some clarification, at least for amateurs.
=--

**Theorem 2.** Suppose $G,H$ are group objects in a category $C$ with finite products obeying the rigidity lemma.  Suppose $f : G \to H$ is any morphism in $C$ preserving the identity.  Then $f$ is a homomorphism.

+-- {: .proof}
######Proof
The idea is this: suppose $C$ is a [[concrete category]] and look at the function $k : G \times G \to H$ given by

$$k(g,g') = f(g\cdot g')\cdot (f(g) \cdot f(g'))^{-1} $$

Assume $f(1) = 1$.  Then $k(1,g') = 1$ for all $g' \in G$, so by the rigidity lemma $k(g,g')$ is independent of $g'$ and we can write $k(g,g') = r(g)$.  Furthermore $k(g,1) = 1$ for all $g \in G$ so $k(g,g')$ is independent of $g$.  This means that $r(g)$ is independent of $g$, but $r(1) = k(1,1) = 1$ so $r(g) = 1$ for all $g$.  This says that $f(g \cdot g') = f(g) \cdot f(g')$, so $f$ preserves multiplication. This in turn implies that $f$ preserves inverses, so $f$ is a group homomorphism.

In fact a version of this argument works in any category with finite products obeying the rigidity lemma.  The expression $f(g\cdot g')\cdot (f(g) \cdot f(g'))^{-1}$ compiles to a particular morphism $G\times G \xrightarrow{k} H$.  The fact that $f$ preserves the identity $e_G:1\to G$ implies that both composites $G \xrightarrow{(id,e_G)} G\times G \xrightarrow{k} H$ and $G \xrightarrow{(e_G,id)} G\times G \xrightarrow{k} H$ are constant at the identity $e_H:1\to H$.  (This is a straightforward calculation in the internal logic of a category with products, or alternatively a slightly tedious diagram chase.)

In particular, $k$ is independent of the first $G$ in its domain at the point $e_G : 1\to G$ of the second $G$ in its domain.  So by the rigidity lemma, there exists a morphism $r : G\to H$ such that the composite $G\times G \xrightarrow{\pi_2} G \xrightarrow{r} H$ is equal to $k$.  Now precompose both of these with $G \xrightarrow{(e_G,id)} G\times G$: the first gives $r \circ \pi_2 \circ (e_G,id) = r$ and the second gives $k \circ (e_G,id) = e_H \circ !$.  Thus, $r$ is constant at the identity of $H$, and hence so is $k$.  This implies $f$ preserves multiplication, and thus also inverses (again, by a calculation in internal logic or a diagram chase).
=--

**Corollary 1.** If $C$ is a category with finite products obeying the rigidity lemma, any group object in $C$ is abelian.

+-- {: .proof}
######Proof
If $G$ is a group object in $C$, the inverse map $inv: G \to G$ preserves the identity, so by the above theorem it is a group homomorphism.  This in turn implies that $G$ is abelian.
=--

**Corollary 2.** Let $Var_*$ be the category of pointed complete varieties over an algebraically closed field $k$, and let $AbVar$ be the category of abelian varieties over $k$.  Then the forgetful functor $U : AbVar \to Var_*$ is full.

+-- {: .proof}
######Proof
This follows immediately from the two theorems above.
=--

A consequence of Corollary 2 is that if $Alb : Var_* \to AbVar$ is the left adjoint to $U$, sending any connected pointed projective variety to its Albanese variety, the monad $T = U \circ Alb$ is an [[idempotent monad]].  For more on this see [[Albanese variety]].

## Related concepts

* [[theta function]]

* [[Cartier duality]]

* [[dual abelian group scheme]]

* [[Albanese variety]]

## Literature

* C. Bartocci, [[Ugo Bruzzo]], D. Hernandez Ruiperez, _Fourier-Mukai and Nahm transforms in geometry and mathematical physics_, Progress in Mathematics 276, Birkhauser 2009.

* [[M. Demazure]], [[P. Gabriel]], _Groupes algebriques_, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970 – has functor of points point of view; for review see Bull. London Math. Soc. (1980) 12 (6): 476-478, [doi](http://dx.doi.org/10.1112/blms/12.6.476b)

* [[Daniel Huybrechts]], _Fourier-Mukai transforms in algebraic geometry_, Oxford Mathematical Monographs. 2006. 307 pages.

* J. S. Milne, _Abelian varieties_, course notes, [pdf](http://www.jmilne.org/math/CourseNotes/AV.pdf)

* David Mumford, _Abelian varieties_, Oxford Univ. Press 1970.

* [[Alexander Polishchuk]], _Abelian varieties, theta functions and the Fourier transform_, Cambridge Univ. Press 2003.

* Goro Shimura, _Abelian varieties with complex multiplication and modular functions_, Princeton Univ. Press 1997.

* [[André Weil]], _Courbes algébriques et variétés abéliennes_, Paris: Hermann 1971

In [[E-infinity geometry]]:

* {#LurieI} [[Jacob Lurie]], _Elliptic Cohomology I: Spectral Abelian Varieties_ ([pdf](http://www.math.harvard.edu/~lurie/papers/Elliptic-I.pdf))

For a discussion of how the rigidity lemma gives 'automatic abelianness' see:

* [Two miracles in algebraic geometry](https://golem.ph.utexas.edu/category/2016/08/the_magic_of_algebraic_geometr.html), _The n-Category Caf&eacute_.

[[!redirects abelian varieties]]
[[!redirects abelian group scheme]]
[[!redirects abelian group schemes]]


abelianization

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

# Contents
* table of contents
{: toc}

## Idea

Abelianisation is the process of [[free functor|freely]] making an [[algebraic structure]] 'abelian'. There are several notions of abelianizations for various algebraic structures, notably there is the abelianization of [[non-abelian groups]] to [[abelian groups]].

There is also [[Verdier's abelianization functor]] from a [[triangulated category]] to an [[abelian category]] with some universal property; this is treated in a separate entry.

## For groups

### Definition

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

For $G$ a [[group]], its **abelianization** $G^{ab} \in $ [[Grp]] is the [[quotient]] of $G$ by its [[commutator subgroup]]:

$$
  G^{ab} \coloneqq G/[G,G]
  \,,
$$

=--

A group whose abelianization is the [[trivial group]] is called a _[[perfect group]]_.

The abelianization is an [[abelian group]]. Indeed, it is the [[universal construction|universal]] abelian group induced by $G$, in the following sense:

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

Abelianization extends to a [[functor]] $(-)^{ab} \colon $ [[Grp]] $\to$ [[Ab]] and this functor is [[left adjoint]] to the [[forgetful functor]] $U \colon Ab \to Grp$ from abelian groups to group.

=--

Hence abelianization is the _[[free construction]]_ of an abelian group from a group.

### Examples

#### Homotopy groups

+-- {: .num_example }
###### Example

Given a [[pointed space|pointed]] [[connected topological space]] $(X,a)$, its first [[singular homology|singular]] [[homology group]] with coefficients in the [[integers]] is the abelianisation of its [[fundamental group]]:
$$ H_1(X,\mathbb{Z}) \cong \pi_1(X,a)^{ab} .$$
This is a [[natural isomorphism]] filling the following diagram of [[functors]]:
$$ \array {
   Top_{\geq 1}^{*/} & \overset{\pi_1}\longrightarrow & Grp \\
   \llap{U}\downarrow & & \downarrow\rlap{ab} \\
   Top & \underset{H_1({-},\mathbb{Z})}\longrightarrow & Ab Grp
} $$
(where $U$ [[forgetful functor|forgets]] the point).

This example can also be done starting with an arbitrary pointed topological space and letting $U$ take the [[connected component]] of the point.  Of course, this example really lives in [[∞ Grpd]] and so we could start with a (pointed, maybe connected) [[simplicial set]], [[Kan complex]], etc.
=--

For more discussion of this see at _[[singular homology]]_ the section _[Relation to homotopy groups](singular%20homology#RelationToHomotopyGroups)_.

#### Free abelian groups

A [[free abelian group]] on a set $S$ is the abelianization of the [[free group]] on $S$.

In other words, if $F \colon Set \to Grp$ is the [[free group]]-functor and $F_{Ab} \colon Set \to Ab$ is the [[free abelian group]] functor, then

$$
  \array{
    Set &&\stackrel{F_{ab}}{\to}&& Ab
    \\
    & {}_{\mathllap{F_{grp}}}\searrow && \nearrow_{\mathrlap{(-)^{ab}}}
    \\
    && Grp
  }
$$

commutes up to a canonical isomorphism. This is because we have a corresponding commutative diagram of forgetful functors

$$
  \array{
    Set &&\stackrel{U_{ab}}{\leftarrow}&& Ab
    \\
    & {}_{\mathllap{U_{grp}}}\nwarrow && \swarrow_{\mathrlap{U}}
    \\
    && Grp
  }
$$

and $(-)^{ab} \circ F_{grp}$ is left adjoint to $U_{grp} \circ U$.

## For monoids etc

Abelianisation of monoids works pretty much like abelianisation of groups.

We can also do abelianisation of [[monoid objects]] in many [[monoidal categories]] (or [[closed categories]] or more generally [[multicategories]]).  For example, we can form abelianisations of [[rings]], which are monoid objects in [[Ab]].

We can even form abelianisations of [[semigroups]] or [[magmas]].

## For Lie algebras

Lie algebras are not monoid objects in any category, but one still considers [[abelian Lie algebras]], which may be identified with their underlying [[vector spaces]].  These are so called because they correspond to abelian [[Lie groups]].  Lie algebras also can be abelianised.

[[!redirects abelianization]]
[[!redirects abelianizations]]
[[!redirects abelianisation]]
[[!redirects abelianisations]]


Abhijit Champanerkar

* [webpage](https://www.math.csi.cuny.edu/abhijit/)

## Selected writings

On [[knot theory]], [[knot complements]], etc.:

* [[Abhijit Champanerkar]], _The geometry of knot complements_ ([pdf](https://www.math.csi.cuny.edu/~abhijit/talks/knot-geometry-h.pdf), [[ChampanerkarKnotComplements.pdf:file]])

category: people


Abhijit Gadde

* [Institute page](http://qst.theory.tifr.res.in/StringTheoryStud/faculty1.php)

* [InSpire page](https://inspirehep.net/authors/1065230)

## Selected writings

On [[KK-compactification]] of [[D=6 N=(2,0) SCFT]] on [[4-manifolds]] to [[2d CFTs]]:

* [[Abhijit Gadde]], [[Sergei Gukov]], [[Pavel Putrov]], *Fivebranes and 4-manifolds*, in: *Arbeitstagung Bonn 2013*, Progress in Mathematics **319**, Birkhäuser (2016) [[arXiv:1306.4320](https://arxiv.org/abs/1306.4320), [doi:10.1007/978-3-319-43648-7_7](https://doi.org/10.1007/978-3-319-43648-7_7)]

On [[E-strings]] and emergence of [[SU(2)]]  [[flavor (particle physics)|flavor]]-[[chiral symmetry|symmetry]] on [[M5-branes]] in [[heterotic M-theory]] (in the [[D=6 N=(1,0) SCFT]] on [[small instantons]] in [[heterotic string theory]]):

* {#GHKKLV15} [[Abhijit Gadde]], [[Babak Haghighat]], [[Joonho Kim]], [[Seok Kim]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _6d String Chains_, JHEP **1802** (2018) 143 [[arXiv:1504.04614](https://arxiv.org/abs/1504.04614)]

category: people


Abhishek Banerjee

* [personal page](https://sites.google.com/site/abhishekb1313/)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=132991)

## Selected writings

Discussion of [[opposite categories]] of [[commutative monoids in a symmetric monoidal category]] regarded as categories of generalized [[affine schemes]]:

* [[Abhishek Banerjee]], *The relative Picard functor on schemes over a symmetric monoidal category*, Bulletin des Sciences Mathématiques **135** 4 (2011) 400-419 [[doi:10.1016/j.bulsci.2011.02.001](https://doi.org/10.1016/j.bulsci.2011.02.001)]

* [[Abhishek Banerjee]], *On integral schemes over symmetric monoidal categories* [[arXiv:1506.04890](https://arxiv.org/abs/1506.04890)]

* [[Abhishek Banerjee]], *Noetherian Schemes over abelian symmetric monoidal categories*, International Journal of Mathematics **28** 07 (2017) 1750051 [[doi:1410.3212](https://arxiv.org/abs/1410.3212), [doi:10.1142/S0129167X17500513](https://doi.org/10.1142/S0129167X17500513)]

category: people


ABJM theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _ABJM model_ ([ABJM 08](#ABJM08)) is an $\mathcal{N} = 6$ [[3d superconformal gauge field theory]] involving [[Chern-Simons theory]] with [[gauge group]] [[special unitary group|SU(N)]] and [[minimal coupling|coupled]] to [[matter fields]]. For [[level (Chern-Simons theory)|Chern-Simons level]] $k$ it is supposed to describe the [[worldvolume]] theory of $N$ coincident [[black brane|black]] [[M2-branes]] at an $\mathbb{Z}/k$-[[cyclic group]] [[orbifold]] [[singularity]] with [[near-horizon geometry]] $AdS_4 \times S^7/(\mathbb{Z}/k)$ (see at _[M2-branes -- As a black brane](M2-brane#AsABlackBrane)_).

[[!include superconformal symmetry -- table]]

For $k = 2$ the supersymmetry of the ABJM model increases to $\mathcal{N} = 8$. For $k = 2$ and $N = 2$ the ABJM model reduces to the [[BLG model]] ([ABJM 08, section 2.6](#ABJM08)).

Due to the matter coupling, the ABJM model is no longer a [[topological field theory]] as pure Chern-Simons is, but it is still a [[conformal field theory]]. As such it is thought to correspond under [[AdS-CFT]] duality to [[M-theory]] on [[anti-de Sitter spacetime|AdS4]] $\times$ [[7-sphere|S7]]/$\mathbb{Z}/k$ (see also [MFFGME 09](#MFFGME09)).

Notice that the worldvolume $SU(N)$ gauge group enhancement at an $\mathbb{Z}_k$-[[ADE singularity]] is akin to the [[enhanced gauge symmetry|gauge symmetry enhancement]] of the [[effective field theory]] for [[M-theory on G2-manifolds]] at the same kind of singularities (see at _[M-theory on G2-manifolds -- Nonabelian gauge groups](M-theory+on+G2-manifolds#EnhancedGaugeGroups)_).

More generally, classification of the [[near horizon geometry]] 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]] shows that these are the [[Cartesian product]] $AdS_4 \times (S^7/G)$ of 4-[[dimension|dimensional]] [[anti de Sitter spacetime]] with a 7-[[dimension|dimensional]] [[spherical space form]] $S^7/{\widehat{G}}$ with [[spin structure]] and $N \geq 4$, for $\widehat{G}$ a [[finite subgroup of SU(2)]] ([MFFGME 09](#MFFGME09), see [here](spherical+space+form#7DSphericalSpaceFormsWithSpinStructure)).

[[!include 7d spherical space forms -- table]]

## Properties

### AdS/CFT duality

Under  [[holographic duality]] supposed to be related to [[11-dimensional supergravity|M-theory]] on $AdS_4 \times S^7 / \mathbb{Z}_k$.

### Boundary conditions

Discussion of [[boundary conditions]] of the BLG model, leading to [[brane intersection]] with [[M-wave]], [[M5-brane]] and [[MO9-brane]] is in ([Chu-Smith 09](#ChuSmith09), [BPST 09](#BPST09)).

## Related concepts

* [[super Chern-Simons theory]]

* [[BLG model]]

* [[membrane matrix model]]

[[!include superconformal symmetry -- table]]

## References

### Precursors

Precursor considerations in

* [[John Schwarz]], _Superconformal Chern-Simons Theories_ ([arXiv:arXiv:hep-th/0411077](https://arxiv.org/abs/hep-th/0411077)

The lift of [[Dp-D(p+2)-brane bound states]] in [[string theory]] to [[M2-M5-brane bound states]]/[[E-strings]] in  [[M-theory]], under [[duality between M-theory and type IIA string theory]]+[[T-duality]], via generalization of [[Nahm's equation]] (this eventually motivated the [[BLG-model]]/[[ABJM model]]):

* [[Anirban Basu]], [[Jeffrey Harvey]], _The M2-M5 Brane System and a Generalized Nahm's Equation_, Nucl.Phys. B713 (2005) 136-150 ([arXiv:hep-th/0412310](https://arxiv.org/abs/hep-th/0412310))

* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], Section 2.2.1 of _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 ([arXiv:1203.3546](http://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))

This inspired the [[BLG model]]:

* {#BaggerLambert06} [[Jonathan Bagger]], [[Neil Lambert]], _Modeling Multiple M2's_, Phys. Rev. D75, 045020 (2007). ([hep-th/0611108](http://arxiv.org/abs/hep-th/0611108)).

* [[Jonathan Bagger]], [[Neil Lambert]], _Gauge Symmetry and Supersymmetry of Multiple M2-Branes_, Phys. Rev. D77, 065008 (2008). ([arXiv:0711.0955](http://arXiv.org/abs/0711.0955)).

### General

The original article on the $N=6$-case is

* {#ABJM08} [[Ofer Aharony]], [[Oren Bergman]], [[Daniel Jafferis]], [[Juan Maldacena]], _$N=6$ superconformal Chern-Simons-matter theories, M2-branes and their gravity duals_, JHEP 0810:091,2008, [DOI:10.1088/1126-6708/2008/10/091](http://iopscience.iop.org/article/10.1088/1126-6708/2008/10/091/meta;jsessionid=FCE6764D4E19F3038C9530E50B057A56.c3.iopscience.cld.iop.org) ([arXiv:0806.1218](http://arxiv.org/abs/0806.1218))

and for [[discrete torsion]] in the [[supergravity C-field]] in

* [[Ofer Aharony]], [[Oren Bergman]], [[Daniel Jafferis]], _Fractional M2-branes_, JHEP 0811:043, 2008 ([arXiv:0807.4924](https://arxiv.org/abs/0807.4924))

  (on [[fractional M2-brane|fractional M2-branes]])

inspired by the $N=8$-case of the [[BLG model]] ([Bagger-Lambert 06](#BaggerLambert06))

The $N=5$-case is discussed in

* {#HLLLP08} Kazuo Hosomichi, Ki-Myeong Lee, Sangmin Lee, Sungjay Lee, [[Jaemo Park]], _$\mathcal{N}=5,6$ Superconformal Chern-Simons Theories and M2-branes on Orbifolds_, JHEP 0809:002, 2008 ([arXiv:0806.4977](https://arxiv.org/abs/0806.4977))

* {#BHRSS08} [[Eric Bergshoeff]], [[Olaf Hohm]], Diederik Roest, [[Henning Samtleben]], [[Ergin Sezgin]], _The Superconformal Gaugings in Three Dimensions_, JHEP0809:101, 2008 ([arXiv:0807.2841](https://arxiv.org/abs/0807.2841))

* {#AharonyBergmanJafferis08} [[Ofer Aharony]], [[Oren Bergman]], [[Daniel Jafferis]], _Fractional M2-branes_, JHEP 0811:043, 2008 ([arXiv:0807.4924](https://arxiv.org/abs/0807.4924))

The $N=4$-case is discussed in

* {#HLLLP08b} Kazuo Hosomichi, Ki-Myeong Lee, Sangmin Lee, Sungjay Lee, [[Jaemo Park]], _\mathcal{N}=4 Superconformal Chern-Simons Theories with Hyper and Twisted Hyper Multiplets_, JHEP 0807:091,2008 ([arXiv:0805.3662](https://arxiv.org/abs/0805.3662))

* {#ChenWu10} Fa-Min Chen, Yong-Shi Wu, _Superspace Formulation in a Three-Algebra Approach to D=3, N=4,5 Superconformal Chern-Simons Matter Theories_, Phys.Rev.D82:106012, 2010 ([arXiv:1007.5157](https://arxiv.org/abs/1007.5157))

More on the role of [[discrete torsion]] in the [[supergravity C-field]] is in

* [[Mauricio Romo]], _Aspects of ABJM orbifolds with discrete torsion_, J. High Energ. Phys. (2011) 2011 ([arXiv:1011.4733](https://arxiv.org/abs/1011.4733))

Discussion of [[boundary conditions]] leading to [[brane intersection]] laws with the [[M-wave]], [[black brane|black]] [[M5-brane]] and [[MO9]] is in

* {#ChuSmith09} [[Chong-Sun Chu]], Douglas J. Smith, _Multiple Self-Dual Strings on M5-Branes_, JHEP 1001:001, 2010 ([arXiv:0909.2333](https://arxiv.org/abs/0909.2333))

* {#BPST09} [[David Berman]], [[Malcolm J. Perry]], [[Ergin Sezgin]], [[Daniel C. Thompson]], *Boundary Conditions for Interacting Membranes*, JHEP 1004:025, 2010 [[arXiv:0912.3504](https://arxiv.org/abs/0912.3504), doi:10.1007/JHEP04(2010)025]

As a [[matrix model]],:

* {#MohammedMuruganNastase10} Asadig Mohammed, Jeff Murugan, [[Horatiu Nastase]], _Looking for a Matrix model of ABJM_, Phys. Rev. D82:086004, 2010 ([arXiv:1003.2599](https://arxiv.org/abs/1003.2599))

Review includes

* {#KlebanovTorri10} [[Igor Klebanov]], Giuseppe Torri, _M2-branes and AdS/CFT_, Int.J.Mod.Phys.A25:332-350, 2010 ([arXiv;0909.1580](https://arxiv.org/abs/0909.1580))

* Neil B. Copland, _Introductory Lectures on Multiple Membranes_ ([arXiv:1012.0459](https://arxiv.org/abs/1012.0459))

* {#Lambert12} [[Neil Lambert]], _M-Theory and Maximally Supersymmetric Gauge Theories_, Annual Review of Nuclear and Particle Science, Vol. 62:285-313 ([arXiv:1203.4244](https://arxiv.org/abs/1203.4244), [doi:10.1146/annurev-nucl-102010-130248](https://doi.org/10.1146/annurev-nucl-102010-130248))

* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 2013 ([arXiv:1203.3546](https://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))

* {#Lambert18}  [[Neil Lambert]], _Lessons from M2's and Hopes for M5's_, _Proceedings of the [LMS-EPSRC Durham Symposium](http://www.maths.dur.ac.uk/lms/):_ _[[Higher Structures in M-Theory 2018]]_ Fortschritte der Physik, 2019 ([arXiv:1903.02825](https://arxiv.org/abs/1903.02825), [slides pdf](http://www.maths.dur.ac.uk/lms/109/talks/1877lambert.pdf), [video recording](http://www.maths.dur.ac.uk/lms/109/movies/1877lamb.mp4))

Discussion of [[Montonen-Olive duality]] in [[D=4 super Yang-Mills theory]] via [[ABJM-model]] as [[D3-brane]] model after [[double dimensional reduction]] followed by [[T-duality]]:

* [[Koji Hashimoto]], Ta-Sheng Tai, [[Seiji Terashima]], _Toward a Proof of Montonen-Olive Duality via Multiple M2-branes_, JHEP 0904:025, 2009 ([arxiv:0809.2137](https://arxiv.org/abs/0809.2137))

Discussion of extension to [[boundary field theory]] (describing [[M2-branes]] ending on [[M5-branes]]) includes

* {#BermanThomson09} [[David Berman]], Daniel Thompson, _Membranes with a boundary_, Nucl.Phys.B820:503-533,2009 ([arXiv:0904.0241](http://arxiv.org/abs/0904.0241))

A kind of [[double dimensional reduction]] of the ABJM model to something related to [[type II superstrings]] and [[D1-branes]] is discussed in

* [[Horatiu Nastase]], Constantinos Papageorgakis, _Dimensional reduction of the ABJM model_,  JHEP 1103:094,2011 ([arXiv:1010.3808](http://arxiv.org/abs/1010.3808))

Discussion of the ABJM model in [[Horava-Witten theory]] and reducing to [[heterotic strings]] is in

* {#Lambert15} [[Neil Lambert]], _Heterotic M2-branes_, Physics Letters B Volume 749, 7 October 2015, Pages 363-367 ([arXiv:1507.07931](http://arxiv.org/abs/1507.07931))

Discussion of the model as a [[higher gauge theory]] (due to its coupling to the [[supergravity C-field]]) is in

* Sam Palmer, [[Christian Saemann]], section 2 of _M-brane Models from Non-Abelian Gerbes_, JHEP 1207:010, 2012 ([arXiv:1203.5757](http://arxiv.org/abs/1203.5757))

* Sam Palmer, [[Christian Saemann]], _The ABJM Model is a Higher Gauge Theory_, IJGMMP 11 (2014) 1450075 ([arXiv:1311.1997](http://arxiv.org/abs/1311.1997))

Classification of the possible [[superpotentials]] via [[representation theory]] is due to

* {#MFFME09} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Elena Méndez-Escobar]], _Superpotentials for superconformal Chern-Simons theories from representation theory_, J. Phys. A 42:485204,2009 ([arXiv:0908.2125](http://arxiv.org/abs/0908.2125))

and derived from this a classification of the possible [[orbifold|orbifolding]] (see at _[[spherical space form]]: [7d with spin structure](spherical+space+form#7DSphericalSpaceFormsWithSpinStructure)_) 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))

* Paul de Medeiros, [[José Figueroa-O'Farrill]], _Half-BPS M2-brane orbifolds_, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. ([arXiv:1007.4761](http://arxiv.org/abs/1007.4761), [eujclid:atmp/1408561553](https://projecteuclid.org/euclid.atmp/1408561553))

* [[José Figueroa-O'Farrill]], _M2-branes, ADE and Lie superalgebras_, talk at IPMU 2009 ([pdf](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/Hongo.pdf))

Discussion via the [[conformal bootstrap]]:

* Nathan B. Agmon, [[Shai Chester]], Silviu S. Pufu, _The M-theory Archipelago_ ([arXiv:1907.13222](https://arxiv.org/abs/1907.13222))

* Damon J. Binder, [[Shai Chester]], Max Jerdee, Silviu S. Pufu, _The 3d $\mathcal{N}=6$ Bootstrap: From Higher Spins to Strings to Membranes_ ([arXiv:2011.05728](https://arxiv.org/abs/2011.05728))

See also

* Nadav Drukker, [[Marcos Marino]], Pavel Putrov, _From weak to strong coupling in ABJM theory_ ([arXiv:1007.3837](http://arxiv.org/abs/1007.3837))

* [[Shai Chester]], Silviu S. Pufu, Xi Yin, _The M-Theory S-Matrix From ABJM: Beyond 11D Supergravity_ ([arXiv:1804.00949](https://arxiv.org/abs/1804.00949))

Computation of [[black hole entropy]] in 4d via [[AdS4-CFT3 duality]] from [[holographic entanglement entropy]] in the ABJM theory for the [[M2-brane]] is discussed in

* Jun Nian, Xinyu Zhang, _Entanglement Entropy of ABJM Theory and Entropy of Topological Black Hole_ ([arXiv:1705.01896](https://arxiv.org/abs/1705.01896))

Discussion of [[higher curvature corrections]] in the abelian case:

* Shin Sasaki, _On Non-linear Action for Gauged M2-brane_, JHEP 1002:039, 2010 ([arxiv:0912.0903](https://arxiv.org/abs/0912.0903))

### Mass deformation

The [[Myers effect]] in [[M-theory]] for [[M2-branes]] polarizing into [[M5-branes]] of ([[fuzzy 3-sphere|fuzzy]]) [[3-sphere]]-shape ([[M2-M5 brane bound states]]):

* [[Iosif Bena]], _The M-theory dual of a 3 dimensional theory with reduced supersymmetry_, Phys. Rev. D62:126006, 2000 ([arXiv:hep-th/0004142](https://arxiv.org/abs/hep-th/0004142))

* Masato Arai, [[Claus Montonen]], Shin Sasaki, _Vortices, Q-balls and Domain Walls on Dielectric M2-branes_, JHEP 0903:119, 2009 ([arXiv:0812.4437](https://arxiv.org/abs/0812.4437))

* [[Iosif Bena]], [[Mariana Graña]], Stanislav Kuperstein, Stefano Massai, _Tachyonic Anti-M2 Branes_, JHEP 1406:173, 2014 ([arXiv:1402.2294](https://arxiv.org/abs/1402.2294))

With emphasis on the role of the [[Page charge]]/[[Hopf WZ term]]:

* [[Krzysztof Pilch]], Alexander Tyukov, [[Nicholas Warner]], _Flowing to Higher Dimensions: A New Strongly-Coupled Phase on M2 Branes_, JHEP11 (2015) 170 ([arXiv:1506.01045](https://arxiv.org/abs/1506.01045))

Via the [[mass-deformed ABJM model]]:

* [[Jaume Gomis]], Diego Rodriguez-Gomez, [[Mark Van Raamsdonk]], [[Herman Verlinde]], _A Massive Study of M2-brane Proposals_, JHEP 0809:113, 2008 ([arXiv:0807.1074](https://arxiv.org/abs/0807.1074))

* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], Section 6.4 of: _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 ([arXiv:1203.3546](http://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))

The corresponding D2-NS5 bound state under [[duality between M-theory and type IIA string theory]]:

* [[Iosif Bena]], Aleksey Nudelman, _Warping and vacua of $(S)YM_{3+1}$_, Phys. Rev. D62 (2000) 086008 ([arXiv:hep-th/0005163](https://arxiv.org/abs/hep-th/0005163))

On [[Wilson line]]-[[quantum observables]] and [[bosonization]] in the ABJM model:

* [[Amit Sever]], _Line Operators in Chern-Simons-Matter Theories and Bosonization in Three Dimensions_, talk at *[[Strings 2022]]* [[indico:4940843](https://indico.cern.ch/event/1085701/contributions/4940843), [slides](https://indico.cern.ch/event/1085701/contributions/4940843/attachments/2482058/4261226/Slides_Sever.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=a9f7872c-1603-4954-9030-192a72e4f456)]

[[!redirects ABJM model]]
[[!redirects ABJM-model]]

[[!redirects D=3 SCFT]]

[[!redirects mass-deformed ABJM model]]


AbMon > history

[[!redirects AbMon]]

The [[category]] of commutative (abelian) [[monoids]].

category: category

[[!redirects Ab Mon]]


Abner Shimony

* [Wikipedia entry](https://en.wikipedia.org/wiki/Abner_Shimony)

* Alisa Bokulich and Don Howard: *[Abner Shimony (1928-2015)](https://www.bu.edu/cphs/about/abner-shimony/)*

## Selected writings

On [[Bell's theorem]] in [[quantum physics]]:

* [[John F. Clauser]], [[Abner Shimony]], *Bell's theorem. Experimental tests and implications*, Rep. Prog. Phys. **41** (1978) 1881 [[doi:10.1088/0034-4885/41/12/002](https://iopscience.iop.org/article/10.1088/0034-4885/41/12/002)]

category: people


About

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## History
 {#History}

The [[HomePage|nLab]] is a _collaborative [wiki](http://en.wikipedia.org/wiki/Wiki)_.  It grew out of the desire ([I](http://golem.ph.utexas.edu/category/2007/09/towards_a_higherdimensional_wi.html), [II](http://golem.ph.utexas.edu/category/2008/11/beyond_the_blog.html)) to have a place for development (the "Lab" in "$n$Lab") and indexed archives of the ideas and concepts surrounding the discussions at the [The $n$-Category Café](http://golem.ph.utexas.edu/category/).  These discussions primarily are about [[mathematics]], [[physics]] and [[philosophy]] from the [[higher structures]] [[n-point of view|point of view]] of [[homotopy theory]]/[[algebraic topology]], [[homotopy type theory|homotopy]] [[type theory]], [[higher category theory|higher]] [[category theory]] and [[higher algebra|higher]] [[categorical algebra]] (the "$n$" in "$n$Lab" and "[[nPOV]]"), as well as from the perspective of [[string theory]] for physics related articles.

The $n$Lab was [created](http://golem.ph.utexas.edu/category/2008/11/nlab.html) in November 2008 by [[Urs Schreiber]], using software provided and set up by [[Jacques Distler]].  The name "$n$Lab" (short for "$n$-Category Lab", but perhaps less restrictive) was dreamt up by Lisa Raphals ([[John Baez]]'s wife).

Discussion of and around edits to the $n$Lab happen on the [$n$Forum](https://nforum.ncatlab.org/) which roughly plays the role of the "talk"-pages at Wikipedia.

## What the $n$Lab is

The most apt analogy for the $n$Lab is of a **group lab book**.  A lab book for a research scientist is a place where they write down anything that they consider relevant for their work; a group lab book is one that several researchers with common interests use.  The material recorded in a lab book can include:

* Notes from seminars
* Notes from papers and books
* Summaries of known work
* Observations and results from experiments
* Ideas for future work

Apart from not having actual physical experiments, this matches the content of the $n$Lab fairly well.  The key difference between the $n$Lab and an "ordinary" lab book is that it is public.  By making it public we hope to achieve two things:

1. To enable others to benefit from our work while it is still being done.
2. To benefit from the work of others while we are doing it.

There is much more to a research project than that which appears in the final paper and we believe that since our results and our final proofs will be public, then both we and those who might read such a paper can benefit from beginning the interaction much earlier in its development.

Thus by making our lab book public, we hope that casual passers-by will stop, read, and scribble something on it that will help us do our research.

There are three things to say about this that are worth making very clear.

1. Our reasons for having the $n$Lab are ultimately selfish.  We have set it up and run it to make it easier for us to do our research.
2. But to make it _work_ properly in public, we need to make it attractive to others so that they will stop by, read our jottings, and scribble something of their own.
3. We aim to achieve a system whereby those who put most in are also those who get most out.

For more recent impressions of the nature of this wiki see also at: *[[schreiber:What is... the nLab]]*.

## Who Are We?

The $n$Lab is not a community project set up for the education and improvement of Humanity.  It originally grew out of the [The $n$-Category Café](http://golem.ph.utexas.edu/category/) and the loose-knit community surrounding that blog.  It is not possible to give a precise description of who is in that community, and the community involved in the $n$Lab has itself evolved from that starting point.  Moreover, even if it were possible to describe that community we would not wish to.  We are not a closed group, we are always keen for others to join us.  The two simple rules are:

1. To join, you need to be willing to interact with the group and material already present, from the [[n-point of view]] or the [[string theory]] point of view for material on [[physics]].  Specifically: is there something already on the $n$Lab that you are interested in and is there someone already working in the $n$Lab that you would like to talk to about it?
2. We will ask you to leave if what you are doing has little connection with what the rest of the group is doing and if it is interfering with others trying to do their work. This is of course unpleasant, since most people who come to contribute do so with honorable intentions. To help minimize possible areas of friction, it may help to read [[writing in the nLab]].

##In Practice?

The way that this is being worked out in practice is as follows.  Currently it seems that common motivations for contributing to the $n$Lab are

 * to **assemble information** on [[mathematics]], [[physics]] and [[philosophy]] in a modern unified way, from the perspective of [[higher algebra]], [[homotopy theory]], [[type theory]], [[category theory]], and [[higher category theory]], as well as from the perspective of [[string theory]] for physics related articles

* to **provide exposition** of this information which is useful to a wide range of readers of differing expertise;

 * to jointly **develop ideas** and research on applying higher algebraic, homotopical, type theoretic, categorical, and higher categorical concepts and tools to issues in [[mathematics]], [[physics]] and [[philosophy]], and [[string theory]] concepts and tools to issues in [[physics]].

To some extent this involves

* collecting in principle "well known" definitions and facts in an encyclopaedic fashion, to the extent that these are not readily available, or not in the desired form, for instance on [Wikipedia](http://en.wikipedia.org/wiki/Main_Page). Depending on the enthusiasm of those who decide to contribute, this _could_ eventually develop into something like a modern version of a [Bourbaki](http://en.wikipedia.org/wiki/Bourbaki)-like project. The potential for this is conceivably there, but of course this is an ambitious idea. Time will show if the $n$Lab can live up to this goal.

But on top of this encyclopaedic function, and hopefully in parallel to that, the $n$Lab is

* intended as providing a _laboratory_ for collaborative development of ideas -- for research. Within the community out of which the $n$Lab grew exists the feeling that there is considerable potential for the fruitful application and development of higher algebraic, homotopical, type theoretic,  categorical, and higher categorical concepts and tools to various areas in [[mathematics]], [[physics]] and [[philosophy]], as well as the fruitful application and development of [[string theory]] concepts and tools to various areas in [[physics]]. Several contributors to the $n$Lab are actively involved in research along these lines.
The $n$Lab is meant as a place to collect, develop and present such research.

These points together imply that

* on the $n$Lab we do not hesitate to provide non-traditional perspectives, definitions and explanations of terms and phenomena for mathematics, physics, and philosophy, if we feel that these are the _right_ perspectives, definitions and explanations from a modern unified higher algebraic, homotopical, type theoretic, categorical, and higher categorical perspective, and we also do not hesitate to provide non-traditional perspectives, definitions and explanations of terms and phenomena for physics if we feel that these are the _right_ perspectives, definitions and explanations from a string theory perspective;

* at the same time we want to indicate clearly which part of an entry is traditional common material, which is a modern but widely-accepted reformulation and which is genuinely the result of original research by a contributor or by several contributors;

* we will intersperse possibly controversial but always constructive **discussion** into entries if we feel the need. While in general the [nforum](https://nforum.ncatlab.org/) is where we have chat and discussion and the $n$Lab is where we compile material, few entries on the $n$Lab are or can be meant as representing a final truth. The $n$Lab will be the better the more we discuss its contents. For more on how to inject discussion into entries see the [[HowTo]] page.

##What the $n$Lab is not

Most importantly the $n$Lab is

* **not complete** and **not meant to be complete**. Neither its general structure nor each single entry are meant to be optimal in their current state. Many existing entries, possibly all of them, deserve to be and are meant to be eventually improved and expanded on.
Many entries are [stubs](http://en.wikipedia.org/wiki/Wikipedia:Stub)!

**If you find yourself annoyed by the state any given entry is in**, for whatever reason, please feel encouraged to edit it in order to improve the situation.

**Notice**: an entry being in a pitiful state is usually more a sign of nobody having spared the time and energy to work on it, than of our joint incompetence to write a decent entry if we were being paid for doing it. So if you find your eyebrows raised by some entry, don't turn away to be the next one _not_ to work on it. Instead, improve it. We all do this voluntarily. Most of us have other duties to attend. So don't be annoyed with "us", help us.

So if you feel existing material needs to be changed, you can do so. If you feel further material needs to be added, different perspectives need to be amplified, you can add a new paragraph, headed by a suitable headline. **Be bold**: The $n$Lab will be the better the more people decide to contribute to it.

See the [[HowTo]] page for information on how to contribute to the $n$Lab and edit and/or create its entries.

##Organization of the $n$Lab

* see [[organization of the nLab]]

## Technology and Support

Historically, the nlab was powered by [Instiki](https://golem.ph.utexas.edu/wiki/instiki/show/HomePage), but now runs on a [custom fork](https://github.com/ncatlab/nlab). All bug reports or other software issues/requests for the nLab are currently best raised in the category [nLab Technical Matters](https://nforum.ncatlab.org/21/) at the nForum, but can also be posted [at github](https://github.com/ncatlab/nlab/issues).

The physical server running the nLab lives at [Carnegie Mellon University](http://www.cmu.edu), and is supported by the [[homotopy type theory]] MURI award FA9550-15-1-0053 from the United States Air Force Office of Scientific Research.  Any opinions, findings and conclusions or recommendations expressed on the nLab are those of the authors and do not necessarily reflect the views of the AFOSR.

category: meta


Abraham Adrian Albert

* [Wikipedia entry](https://en.wikipedia.org/wiki/Abraham_Adrian_Albert)

## related $n$Lab entries

* [[Albert algebra]]

category:people

[[!redirects Abraham Albert]]
[[!redirects Adrian Albert]]


Abraham Fraenkel

Abraham Fraenkel was a mathematician who worked on [[set theory]] and [[foundations]]. He is responsible for the 'F' in [[ZFC]], but also did work using [[ZFA]].

Among other things, he also introduced the [[basic Fraenkel model|basic]] and [[second Fraenkel model|second Fraenkel models]], early examples of [[forcing]] using what is now known as a [[permutation model]].

[[!redirects Fraenkel]]

category:people


Abraham Robinson

Abraham Robinson was a German born mathematician, who studied in Jerusalem, before taking up various positions in France then in Cranfield (the staff college of the RAF in the UK, during the second world war), Israel, Canada and finally in the USA at UCLA and Yale. He is the 'father' of Nonstandard Analysis, the first full formalisation of the intuitive infinitesimal calculus, used historically in mathematics, and still in use in physics.  He died of cancer in 1974.

* [Abraham Robinson in Wikipedia](http://en.wikipedia.org/wiki/Abraham_Robinson)

* [Abraham Robinson's biography](http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Robinson.html) on the St. Andrews History website.

category: people


Abraham Westerbaan

* [personal page](https://bram.westerbaan.name/home/about.html)

* [GoogleScholar page](https://scholar.google.com/citations?user=6Hz9H7sAAAAJ&hl=en)

## Selected writings

On [[categorical semantics]] of [[quantum programming languages]] on [[von Neumann algebras]] for the [[quantum lambda-calculus]]:

* [[Kenta Cho]], [[Abraham Westerbaan]], *Von Neumann Algebras form a Model for the Quantum Lambda Calculus*, QPL 2016 [[arXiv:1603.02133](https://arxiv.org/abs/1603.02133), [pdf](https://www.cs.ru.nl/K.Cho/papers/model-qlc.pdf), slides:[pdf](http://qpl2016.cis.strath.ac.uk/pdfs/2cho-final.pdf), [[Cho-vNQuantumCalculus.pdf:file]]]

category: people


abrupt category

Each [[category with star-morphisms]] gives rise to a category (abrupt
category, see a remark below why I call it "abrupt"), as described below.
Below for simplicity I assume that the set $M$ and the set of our indexed
families of functions are disjoint. The general case (when they are not
necessarily disjoint) may be easily elaborated by the reader.

% Objects are indexed (by $\operatorname{arity}m$ for some
$m \in M$) families of objects of the category $C$ and an (arbitrarily
choosen) object $\operatorname{None}$ not in this set

% There are the following disjoint sets of morphisms:

* indexed (by $\operatorname{arity} m$ for some $m \in M$) families of morphisms of $C$

* elements of $M$

* the identity morphism $\operatorname{id}_{\operatorname{None}}$ on $\operatorname{None}$

% Source and destination of morphims are defined by the formulas:

* $\operatorname{Src}f = \lambda i \in
    \operatorname{dom}f : \operatorname{Src}f_i$;

* $\operatorname{Dst}f = \lambda i \in
\operatorname{dom}f : \operatorname{Dst}f_i$

* $\operatorname{Src}m =\operatorname{None}$

* $\operatorname{Dst}m
    =\operatorname{Obj}_m$.

% Compositions of morphisms are defined by the formulas:

* $g \circ f = \lambda i \in \operatorname{dom}f : g_i
    \circ f_i$

* $f \circ m =\operatorname{StarProd} \left( m ; f
\right)$

* $m \circ \operatorname{id}_{\operatorname{None}} = m$

* $\operatorname{id}_{\operatorname{None}} \circ \operatorname{id}_{\operatorname{None}} = \operatorname{id}_{\operatorname{None}}$

% Identity morphisms for an object $X$ are:

* $\lambda i \in X : \operatorname{id}_{X_i}$ if $X \neq
\operatorname{None}$

* $\operatorname{id}_{\operatorname{None}}$
if $X =\operatorname{None}$

We need to prove it is really a category.

**Proof** We need to prove:

* Composition is associative

* Composition with identities complies with the identity law.

Really:

* $\left( h \circ g \right) \circ f = \lambda i \in \operatorname{dom} f :
\left( h_i \circ g_i \right) \circ f_i = \lambda i \in \operatorname{dom} f : h_i
\circ \left( g_i \circ f_i \right) = h \circ \left( g \circ f \right)$;

$g \circ \left( f \circ m \right) = \operatorname{StarComp} \left( \operatorname{StarComp}
\left( m ; f \right) ; g \right) = \operatorname{StarComp} \left( m ; \lambda i \in
\operatorname{arity} m : g_i \circ f_i \right) = \operatorname{StarComp} \left( m ; g \circ
f \right) = \left( g \circ f \right) \circ m$;

$f \circ \left( m \circ \operatorname{id}_{\operatorname{None}} \right) = f \circ m = \left(
f \circ m \right) \circ \operatorname{id}_{\operatorname{None}}$.

* $m \circ \operatorname{id}_{\operatorname{None}} = m$; $\operatorname{id}_{\operatorname{Dst} m} \circ
m = \operatorname{StarComp} \left( m ; \lambda i \in \operatorname{arity} m :
\operatorname{id}_{\operatorname{Obj}_m i} \right) = m$.

**Remark** I call the above defined category abrupt category because (excluding identity morphisms) it allows composition with an $m \in M$ only on the left (not on the right) so that the morphism $m$ is "abrupt" on the right.

[[category with star-morphisms|Categories with star-morphisms]] and **abrupt categories** arise in research of [[cross-composition product]].


absolute coequalizer

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## Definition

An **absolute coequalizer** in a [[category]] $C$ is a [[coequalizer]] which is preserved by *any* [[functor]] $F\colon C \to D$.  This is a special case of an [[absolute colimit]].

## Characterization

Intuitively, an absolute coequalizer is a diagram that is a coequalizer "purely for diagrammatic reasons."  The most common example is a [[split coequalizer]].  A trivial example of an absolute coequalizer that is not split is a diagram of the form
$$ X \; \underoverset{f}{f}{\rightrightarrows}\; Y \overset{1_Y}{\to} Y $$
whenever $f$ is not a [[split epimorphism]].

In fact, split coequalizers and "trivial" absolute coequalizers are the cases $n=1$ and $0$ of a general characterization of absolute coequalizers, which we now describe.  Suppose that
$$ X\; \underoverset{f_1}{f_0}{\rightrightarrows}\; Y \overset{e}{\to} Z $$
is an absolute coequalizer.  Then it must be preserved, in particular, by the hom-functor $hom(Z,-)\colon C \to Set$; that is, we have a coequalizer diagram
$$ hom(Z,X)\; \underoverset{f_1\circ -}{f_0\circ -}{\rightrightarrows}\; hom(Z,Y) \overset{e\circ -}{\to} hom(Z,Z)$$
in $Set$.  In particular, that means that $e\circ -$ is surjective, and so in particular there exists some $s\colon Z\to Y$ such that $e s = 1_Z$.  In other words, $e$ is [[split epimorphism|split epic]].

Now the given coequalizer must also be preserved by the hom-functor $hom(Y,-)$, so we have another coequalizer diagram
$$ hom(Y,X)\; \underoverset{f_1\circ -}{f_0\circ -}{\rightrightarrows}\; hom(Y,Y) \overset{e\circ -}{\to} hom(Y,Z)$$
in $Set$.  We also have two elements $1_Y$ and $s e$ in $hom(Y,Y)$ with the property that $e \circ 1_Y = e = e \circ s e$ (since $e s = 1_Z$).

However, a coequalizer of two functions $h_0,h_1\colon P\to Q$ in $Set$ is constructed as the [[quotient set]] of $Q$ by the [[equivalence relation]] generated by the image of $(h_0,h_1)\colon P\to Q\times Q$.  That means that we set
$q\sim q'$ iff there is a finite sequence $p_1,\dots, p_n$ of elements of $P$ and a finite sequence $\varepsilon_0,\dots,\varepsilon_n$ with $\varepsilon_i\in\{0,1\}$, such that $h_{\varepsilon_1}(p_1)=q$,
$h_{1-\varepsilon_i}(p_i)=h_{\varepsilon_{i+1}}(p_{i+1})$, and $h_{1-\varepsilon_n}(p_n)=q'$.  We consider $q=q'$ as the case $n=0$.

Therefore, since $1_Y$ and $s e$ are in the same class of the equivalence relation on $hom(Y,Y)$ generated by $f_0$ and $f_1$, they must be related by such a finite chain of elements of $hom(Y,X)$.  That is, we must have morphisms $t_1,\dots, t_n\colon Y\to X$ and a sequence of binary digits $\varepsilon_1,\dots,\varepsilon_n$ such that $f_{\varepsilon_1} t_1=1_Y$, $f_{1-\varepsilon_i} t_i = f_{\varepsilon_i} t_{i+1}$, and $f_{1-\varepsilon_n}t_n=s e$.  (Note that if $n=1$ then this says precisely that we have a split coequalizer, and if $n=0$ it is the trivial case above.)  Conversely, it is easy to check that given $s$ and $t_1,\dots, t_n$ satisfying these equations, the given fork must be a coequalizer, for essentially the same reason that any split coequalizer is a coequalizer.  Thus we have a complete characterization of absolute coequalizers.

This characterization is essentially a special case of the characterization of [[absolute colimits]] (in unenriched categories).

## Examples

* [[Beck coequalizer]]

## References

* [[Robert Pare]], *Absolute coequalizers*, Lecture Notes in Math. 86 (1969), 132-145.

[[!redirects absolute coequalizers]]


absolute cohomology

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## Contents
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## Idea

Roughly something like a combination of a "[[geometry|geometric]]" and an [[arithmetic geometry|arithmetic]] concept of [[cohomology]].

## Examples

* [[absolute Hodge cohomology]]

* [[syntomic cohomology]]

* [[motivic cohomology]]

## References

* Roy Joshua, _K-Theory and Absolute Cohomology for algebraic stacks_ ([KTheory](http://www.math.uiuc.edu/K-theory/0732/))

* [[Cohomology Theory Database]], _[Absolute cohomology](http://notes.andreasholmstrom.org/ct.php?n=Absolute+cohomology)_

* Jan Nekovar, section 3 of _Beilinson's Conjectures_ ([pdf](http://people.math.jussieu.fr/~nekovar/pu/mot.pdf))


absolute colimit

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## Idea

An **absolute colimit** is a [[colimit]] which is [[preserved limit|preserved]] by any [[functor]] whatsoever.  In general this happens because the colimit is a colimit for purely "diagrammatic" reasons.  The notion is most important in [[enriched category theory]].

Of course, there is a dual notion of **absolute limit**, but it is used less frequently.

## Definitions

The term "absolute colimit" is actually used for two closely related, but distinct, notions.

### Particular absolute colimits

+-- {: .num_defn #DefParticular}
###### Definition
A particular [[colimit]] diagram in a particular [[category]] $C$ is an **absolute colimit** if it is [[preserved limit|preserved]] by every [[functor]] with domain $C$.
=--

This definition makes sense also in [[enriched category theory]]: for any $V$, a [[weighted colimit]] in a particular $V$-[[enriched category]] $C$ is an **absolute colimit** if it is preserved by every $V$-functor with domain $C$.

Note, however, that a [[conical colimit]] in a $V$-category $C$ may be preserved by all $V$-functors without being preserved by all unenriched functors on the underlying ordinary category $C_o$.

Generalizing in a different direction, absolute colimits in $Set$-enriched categories can be regarded as the particular case of [[postulated colimit]]s in [[sites]] where the site has the [[trivial topology]].

### Weights for absolute colimits

+-- {: .num_defn #DefWeight}
###### Definition
For a given $V$, a [[weighted colimit|weight]] $\Phi\colon D^{op} \to V$ for colimits is an **absolute weight**, or a **weight for absolute colimits**, if $\Phi$-weighted colimits in *all* $V$-categories are preserved by all $V$-functors.
=--

Absolute colimits of this sort are also called **Cauchy colimits**.  A $V$-[[category]] which admits all absolute colimits --- that is, all $V$-weighted colimits whose weights are absolute -- is called [[Cauchy complete category|Cauchy complete]].  By the characterization below, it is equivalent to admit *limits* weighted by all weights for absolute limits.

## Characterizations

Both types of absolute colimits admit pleasant characterizations.

### Particular absolute colimits

For a particular [[cocone]] $\mu \colon F \to \Delta A$ under a functor $F\colon I\to C$ (all in the Set-enriched world), the following are equivalent:

* $\mu$ is an absolute colimiting cocone.

* $\mu$ is a colimiting cocone which is is preserved by the [[Yoneda embedding]] $C \hookrightarrow [C^{op},Set]$.

* $\mu$ is a colimiting cocone which is preserved by the representable functors $C(F(i),-)\colon C\to Set$ (for all $i\in I$) and $C(A,-)\colon C\to Set$.

* There exists $i_0\in I$ and $d_0\colon A\to F(i_0)$ such that
  1. For every $i\in I$, $d_0 \circ \mu_i$ and $1_{F(i)}$ are in the same connected component of the [[comma category]] $(F(i) / F)$.
  1. $\mu_{i_0} \circ d_0 = 1_{A}$.

The equivalence of the first two is basically because the Yoneda embedding is the [[free cocompletion]] of $C$.  The third clearly follows from the second.  The fourth follows from the third by inspecting exactly what preservation by those representables means in terms of colimits in [[Set]] (as is explained in more detail in the special case of [[absolute coequalizers]]).  Finally, it is straightforward to check that the fourth implies that $\mu$ is colimiting, and it is clearly a property preserved by any functor.

Note that in particular, the fourth condition implies that $A$ is a [[retract]] of $F(i_0)$.  Also, the first half of the fourth condition by itself characterizes absolute [[weak colimits]].

It is also possible to prove directly that the third condition implies the first two, without extracting the fourth condition along the way.  Namely, Let $B$ be the full subcategory of $C$ consisting of the objects $F(i)$ and $A$.  Then $F$ defines a functor $I\to B$; call it $F'$.  Note that $A$ is also the colimit of $F'$ in $B$.  Moreover, by the equivalence of the first two conditions, $A$ is an *absolute* colimit of $F'$, since by hypothesis it is preserved by all representable functors out of $B$.  Therefore, this colimit is in particular preserved by the inclusion $B\hookrightarrow C$, along with its composite with any functor out of $C$; so $A$ is an absolute colimit of $F$.

### Weights for absolute colimits
 {#CharacterizationsWeightsForAbsoluteColimits}

Let $V$ be a Bénabou [[cosmos]] and $J\colon K ⇸ A$ a $V$-[[profunctor]].  Then the following are equivalent:

* $J$ is a weight for absolute colimits (i.e. $J$-weighted colimits in any $V$-category are preserved by all $V$-functors)

* $J$ has a [[right adjoint]] $J^*$ in the [[bicategory]] $V$-[[Prof]].

* There is a weight $J^*\colon A ⇸ K$ such that $J$-weighted colimits coincide naturally with $J^*$-weighted limits.

## Examples
 {#Examples}

### Particular absolute colimits

Of course, every colimit weighted by a weight for absolute colimits is itself a particular absolute colimit.  But it may also happen that a particular colimit may be absolute without all colimits of that shape being absolute.  For example (in ordinary category theory, with $V=Set$):

* [[split coequalizers]] are absolute, and figure in Beck's [[monadicity theorem]].
* More generally (of course), [[absolute coequalizers]] are absolute.
* [[absolute pushouts]] appear in [[elegant Reedy categories]].

We can also say something about non-examples.

* Initial objects (in $Set$-enriched categories) are *never* absolute.  If $0$ is an initial object, then it is never preserved by the representable functor $C(0,-)\colon C \to Set$.

* Similarly, coproducts in $Set$-enriched categories are never absolute.

### Weights for absolute colimits
 {#WExampleseightsForAbsoluteColimits}

In ordinary $Set$-enriched category theory there are not very many weights for absolute colimits, but we have

* [[split idempotents]].

In fact, this example is "universal," in that an ordinary category is Cauchy complete iff it has split idempotents, although not every absolute colimit "is" the splitting of an idempotent.  More precisely, the class of absolute $Set$-limits is the [[saturated class of limits|saturation]] of idempotent-splittings.

In [[enriched category theory]] there can be more types of absolute colimits.  For instance:

* in categories with [[zero morphisms]] (that is, enriched over [[pointed sets]]), [[initial objects]] are absolute.

* in [[Ab-enriched categories]] (or, more generally, categories enriched over commutative [[monoids]]), finite [[biproducts]] are absolute.  Finite biproducts and splitting of idempotents together are "universal" absolute colimits for Ab-enrichment.

* in [[SupLat]]-enriched categories, arbitrary small biproducts are absolute, and together with splitting of idempotents these generate all absolute colimits.

* in [[dg-categories]] (or more generally, categories enriched over [[graded set|graded sets]]), *shifts/suspensions* and [[mapping cones]] are absolute.

* in Lawvere [[metric spaces]], limits of Cauchy sequences are absolute.  This is the origin of the name "Cauchy colimit."

* in [[posets]], suprema of subsets with a greatest element are absolute.

* in categories [[bicategory-enriched category|enriched]] over the [[bicategory]] (or [[double category]]) of relations in a [[site]], *gluings* are absolute.  In this case the enriched categories can roughly be identified with [[separated presheaves]] and the Cauchy-complete ones with [[sheaves]].

* in categories enriched over [[rational number|rational]] [[vector spaces]], quotients by finite [[group actions]] are absolute.

New kinds of absolute (co)limits also arise in [[higher category theory]].

* for [[(∞,1)-categories]] (enriched over $\infty$-groupoids), splitting of idempotents is a universal absolute colimit.

* in [[stable (∞,1)-categories]] (which are enriched, in the $(\infty,1)$-categorical sense, over the $(\infty,1)$-category of [[spectra]]), initial objects and [[pushouts]] are absolute, and therefore so are all finite colimits.

## Generalisations

If $K : A \to C$ is a functor, then a colimit in $A$ is **$K$-absolute** if it is preserved by the [[nerve]] $N_K : C \to [A^{\mathrm{op}}, \mathscr{V}]$ of $K$. This generalises absoluteness, taking $K$ to be the [[identity functor]].

## References

* [[Ross Street]], *Absolute colimits in enriched categories*, [Cahiers 1983](http://www.numdam.org/item/CTGDC_1983__24_4_377_0/)
* [[Robert Pare]], *On absolute colimits*, J. Alg. 19 (1971), 80-95.
* [[Max Kelly]], _Basic Concepts of Enriched Category Theory_, Cambridge University Press, Lecture Notes in Mathematics 64 (1982)
* [[Mike Shulman]]'s answer to [*What are _all_ of the exactness properties enjoyed by stable ∞-categories?*](https://mathoverflow.net/a/267324)

[[!redirects absolute colimits]]
[[!redirects absolute limit]]
[[!redirects absolute limits]]
[[!redirects Cauchy colimit]]
[[!redirects Cauchy colimits]]


absolute conclusion

[[!redirects agar.io]]
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# Contents
* table of contents
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[[!redirects Absolute conclusion]]
[[!redirects the absolute conclusion]]
[[!redirects Absoluter Schluss]]
[[!redirects der absolute Schluss]]

## Idea
The **absolute conclusion**, or in German, _der absolute Schluss_, is a key concept in the speculative philosophy of the German philosopher [[Georg Hegel|G. W. F. Hegel]].

The name of the concepts plays on the triple sense of _Schluss_ in German evoking _end_ , as well as 'Zusammenschluss' - _union_ and _logical deduction_. The primary sense is that the absolute conclusion unites three deductions into a deductive relation to conclude the development of a concept or part of the philosophical system. It is called _absolute_ because its ' premises' and 'conclusion' are themselves deductions that relate to other deductions, whence, it is a _deduction between deductions_.

The main idea is that this deductive pattern improves on ordinary 'relative' deductions that hinge on unmediated premises for their mediation, by elevating each premise into conclusive positions and vice versa, thereby solving a 'contradiction' well known in logic since Aristotle's _Analytica Posteriora_, namely that the desired proofs require unproved axioms.[^nat]

[^nat]: That it is also possible to proceed from (retractable) assumptions instead of (true) axioms was shown by the end of the 1920s by G. Gentzen and S. Jaskowski (cf. [[natural deduction]]).

Hegel's model for a logical deduction is an Aristotelian [[syllogism]] consisting of two premises and a conclusion that involve a subject notion $S$ and a predicate notion $P$ and a middle term $M$. Hegel first identifies then  $S$ with a _general_ notion, $P$ with a _particular_ notion, and $M$ with a _singular_ notion. The absolute conclusion then consists of three syllogisms where the general, the particular and the singular change position in the syllogistic figure such that each takes the place thereby expressing the deep-structural identity between the three notions.

The _objectivity_ and _reality_ of an entity is viewed as the **absolute mediation** of the general, particular and singular parts involved in its notion as expressed by such an absolute conclusion. E.g. in the philosophy of religion Hegel ([Enzyklopädie III](#Hegel10), pp.374ff) interprets the Christian trinity as an absolute conclusion where the conceptions of god as father (generality), is united with the conception of god as a son (particularity) and the conception as holy spirit (singularity). Once the identity of these concepts is expressed, the trinitarian concept of god realizes itself as communital spirit in the church - i.e. once rightly understood god lives in and as the spirit of community in human society.

This might seem to be a somewhat speculative use of the absolute conclusion but a similar passage from a completed mediation to objectivity occurs in the subjective logic of "Wissenschaft der Logik" when the subjective notion transforms into 'objectivity'. That a mediation of moments yields to objectivity sounds rather strange but the background here is Hegel's reinterpretation of Kant's concept of 'transcendental apperception' where the integration of perceptions under the notion of an object is accompanied by the 'I think' i.e. the unity of perception gets correlated with the unity of a self. Hegel equates the unity of the self with the complete mediation of the concept's moments from which the thought of an object springs: the self that flows around such a circle of mediations is viewed as the generic invariant that imparts its invariance to objects.

Beyond all metaphysical speculation one can view Hegel's concept as a general proposal for the **architecture of foundational theories** that 'escapes' the Münchhausen problem of a foundational (vicious) circle by entangling three circles into a circle. Under this interpretation one would view it as the suggestion that foundations should ideally consists of three fundamental systems that interpret each other and thereby corroborate each other.

To get an intuition one might think of the three different notions of [[computability]] as Turing machines, recursive functions and lambda calculus whose mutual equivalence back the Church-Turing hypothesis resulting in an 'absolute' concept of algorithm. In a similar vein is [[Bob Harper|R. Harper's]] [[computational trinitarianism]]. Also relevant is the possibility in [[categorical logic]] to interpret classes of categories via an external 'objective' description, then as models of a 'subjective' internal language and as models of appropriate [[sketch|sketches]] at an intermediate level.

Notice the proximity of the triads involved in such foundations to the mathematical rendering of the triadic structure involving the imaginary, the real, and the symbolic in Lacan's theory of subjectivity as proposed in [[René Guitart|Guitart's]] theory of _Borromean objects_ ([2009](#Guitart09)).

To sum up: the concept of the absolute conclusion enjoys currently little popularity among philosophers not even among Hegelian philophers despite the importance Hegel attached to it. It might nevertheless be worthwhile to probe into this concept from the perspective of contemporary foundational theories in the mathematical sciences.

## Some background

Hegel apparently introduced the concept in his philosophy around 1800 in or shortly before the Jena period. He purloined it from Plato's [[Timaeus dialogue|Timaios dialogue]] (**7**, 32c):

>But it isn't possible to combine two things well all by themselves, without a third; there has to be some bond between the two that unites them. Now the best bond is one that really and truly makes a unity of itself together with the things bonded by it, and this in the nature of things is accomplished by proportion. For whenever of three numbers which are either solids or squares the middle term between any two of them is such that what the first term is to it, it is to the first, then, since the middle term turns out to be both first and and last, and the last and first likewise both turn out to be middle terms, they will all of necessity turn out to have the same relationship to each other, and, given this, will all be unified.[^cite]

[^cite]: Translation by D. J. Zeyl, p.1237 in [Cooper (ed.) 1997](#Plato). Note that 'most harmonious bond' is closer to the Greek text than 'best bond'. Hegel quotes it as 'der Bande schönstes' in German.

Hegel cites this passage already in _'Differenz des Fichteschen und Schellingschen Systems der Philosophie'_ ([1801](#Hegel2), p.97). A similar configuration with triangles instead of circles occurs also in his (lost) fragment on the _'divine triangle'_ as reported and dated to 1804 by [[Karl Rosenkranz|Rosenkranz]] (cf. [Hegel Werke 2](#Hegel2), pp.534-539) which is somewhat reminiscent of an adjoint triple $L\dashv M\dashv R$ as an [[adjoint triple|adjunction between adjunctions]].

The conception seems to have been the subject of the discussions between Hegel and Schelling during their collaboration in Jena, since the latter alludes to the display of the absolute by the three judgements in two texts of the epoch, namely the dialogue _'Bruno oder Über das göttliche und natürliche Prinzip der Dinge'_ ([1802](#Bruno)) and _'Philosophie und Religion'_ ([1804](#SchellSchrift)).

The mature Hegel comments the Timaios passage extensively and with emphatic approval in his [[Lectures on the History of Philosophy]] (cf. [Hegel Werke 19](#Hegel19), pp.89-91).

## Related entries

* [[Aufhebung]]

* [[Science of Logic]]

* [[infinite judgement]]

* [[construction in philosophy]]

* [[computational trinitarianism]]

* [[syllogism]]

* [[Timaios]]

## References

* {#Plato} J. M. Cooper (ed.), _Plato - Complete Works_ , Hackett Indianapolis 1997.

* N. Füzesi, _Hegels drei Schlüsse_ , Alber München 2004.

* {#Guitart09}[[René Guitart| R. Guitart]], _Klein's Group as a Borromean Object_ , Cah. Top. Géo. Diff. Cat. **L** no.2 (2009) pp.144-155. ([pdf](http://rene.guitart.pagesperso-orange.fr/textespublications/Guitart%202009%20G168CTGDC.pdf))

* {#Hegel2} [[Georg Hegel|G. W. F. Hegel]], _Jenaer Schriften 1801-1807 - Werke 2_ , Suhrkamp Frankfurt a. M 1986.

* {#Hegel8} [[Georg Hegel|G. W. F. Hegel]], _Enzyklopädie der philosophischen Wissenschaften I - Werke 8_ , Suhrkamp Frankfurt a. M 1986[1817, rev. 1830]. (§ 187 Zusatz, p.339f.)

* {#Hegel10} [[Georg Hegel|G. W. F. Hegel]], _Enzyklopädie der philosophischen Wissenschaften III - Werke 10_ , Suhrkamp Frankfurt a. M 1986[1817, rev. 1830]. (§§ 569-571, pp.376f; §§ 575-77, pp.393f)

* {#Hegel19} [[Georg Hegel|G. W. F. Hegel]], _Vorlesungen über die Geschichte der Philosophie II - Werke 19_ , Suhrkamp Frankfurt a. M 1986. (ch. III A. 2, pp.89-91)

* W. Jaeschke, _Die Schlüsse der Philosophie (§§ 574-577)_ , pp.479-486 in Drüe et al. (eds.), _Hegels 'Enzyklopädie der philosophischen Wissenschaften'  (1830) - Ein Kommentar zum Systemgrundriß_ , Suhrkamp Frankfurt a. M. 2000.

* G. Sans, _Die Realisierung des Begriffs - Eine Untersuchung zu Hegels Schlusslehre_ , Akademie Verlag Berlin 2004.

* {#SchellSchrift}F. W. J. Schelling, _Schriften 1804-1812_ , Union Berlin 1982. (sect. 1, p.48)

* {#Bruno}F. W. J. Schelling, _Bruno oder Über das göttliche und natürliche Prinzip der Dinge_ , Reclam Leipzig 1986. (sect. IIB 7b, p.88)

* G. Werckmeister, _Hegels absoluter Schluss als logische Grundstruktur der Objektivität_ , PhD TU Kaiserslautern 2009. ([link](https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2160))