A-infinity-space

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## Definition

An **$A_\infty$-space** is a [[homotopy type]] $X$ that is equipped with the structure of a [[monoid]] up to [[coherence|coherent]] higher [[homotopy]]:

that means it is equipped with

1. a binary product operation $\cdot \colon X \times X \to X$

1. a choice of [[associativity]] [[homotopy]]; $\eta_{x,y,z} :  (x\cdot y) \cdot z \to x \cdot (y \cdot z)$;

1. a choice of [[pentagon law]] homotopy between five such $\eta$s;

1. and so ever on, as controlled by the [[associahedron|associahedra]].

In short one may say: an **$A_\infty$-space** is an [[A-∞ algebra]]/[[monoid in an (∞,1)-category]] in the [[(∞,1)-category]] [[∞Grpd]]/[[Top]]. See there for more details.

## Properties

### Relation to H-monoids

If in the definition of an $A_\infty$-space one discards all the higher homotopies and retains only the _existence_ of an [[associativity]]-[[homotopy]], then one has the notion of _[[H-monoid]]_. Put another way, An $A_\infty$-space in the [[(∞,1)-category]] [[∞Grpd]]/[[Top]] becomes an [[H-monoid]] in the [[homotopy category of an (∞,1)-category|homotopy]] [[Ho(Top)]]. And lifting an [[H-monoid]] structure to an $A_\infty$-space structure means lifting a monoid structure through the projection from the [[(∞,1)-category]] [[∞Grpd]]/[[Top]] to [[Ho(Top)]].

### Relation to $A_\infty$-categories

The [[delooping]] of an $A_\infty$-space is an [[A-∞ category]]/[[(∞,1)-category]] with a single object. (Beware that in standard literature "$A_\infty$-category" is often, but not necessarily, reserved for a _[[stable (∞,1)-category]]_).

There is an [[equivalence of (∞,1)-categories]] between pointed connected [[A-∞ categories]]/[[(∞,1)-categories]] and $A_\infty$-spaces.

## Related concepts

[[!include k-monoidal table]]

* [[model structure for dendroidal left fibrations]]

* [[H-space]], [[A-n space]]

* [[monoid]], [[monoidal groupoid]]

* [[k-tuply associative n-groupoid]]

* [[A-infinity category]]

## References

$A_\infty$-spaces were introduced by [[Jim Stasheff]] as a refinement of an [[H-group]] taking into account higher [[coherence|coherences]].

* [[Jim Stasheff]], _Homotopy associative H-spaces I_, _II_, Trans. Amer. Math. Soc. __108__ (1963), 275--312 [MR158400](http://www.ams.org/mathscinet-getitem?mr=158400)

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A-model

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## Idea

What is called the _A-model_ [[topological string]] is the 2-dimensional [[topological conformal field theory]] corresponding to the [[Calabi–Yau category]] called the [[Fukaya category]] of a [[symplectic manifold]] $(X,\omega)$. This is the [[Poisson sigma-model]] of the underlying [[Poisson manifold]] after appropriate [[gauge fixing]] ([AKSZ 97, p 19](#AKSZ)). The A-model on $X$ is effectively the [[Gromov–Witten theory]] of $X$.

The A-model arose in formal [[physics]] from considerations of [[string theory|superstring]]-propagation on [[Calabi-Yau spaces]]: it may be motivated by considering the [[vertex operator algebra]] of the 2d[[CFT|SCFT]] given by the [[supersymmetric sigma-model]] with [[target space]] $X$ and then deforming it such that one of the super-[[Virasoro algebra|Virasoro]] generators squares to $0$. The resulting "topologically twisted" algebra may then be read as being the [[BRST complex]] of a [[TCFT]].

One can also define an A-model for [[Landau-Ginzburg model|Landau–Ginzburg models]]. The category of [[D-brane|D-branes]] for the corresponding [[open string]] theory is given by the [[Fukaya–Seidel category]].

By [[homological mirror symmetry]], the A-model is dual to the  [[B-model]].

## Properties

### Lagrangian

The [[action functional]] of the A-model is that associated by [[AKSZ theory]] to a Lagrangian [[submanifold]] in a [[target space|target]] [[symplectic Lie n-algebroid]] which is the [[Poisson Lie algebroid]] of a [[symplectic manifold]].

See the _[references on Lagrangian formulation](#LagrangianLit)_.

### Boundary theory / holography

On [[coisotropic submanifold|coisotropic]] [[branes]] in symplectic target manifolds that arise by complexification of [[phase spaces]], the boundary [[path integral]] of the A-model computes the [[quantization]] of that phase space. For details see

* [[quantization via the A-model]].

and

* [[holographic principle]].

### Second quantization / effective background field theory
 {#SecondQuantization}

The [[second quantization]] [[effective field theory|effective]] background field theory defined by the [[perturbation series]] of the A-model string has been argued to be [[Chern-Simons theory]]. ([Witten 92](#Witten92), [Costello 06](#Costello06))

For more on this see at _[TCFT -- Worldsheet and effective background theories](http://ncatlab.org/nlab/show/TCFT#ActionFunctionals)_. A related mechanism is that of _[[world sheets for world sheets]]_.

## Related concepts

* [[schreiber:∞-Chern-Simons theory]]

* [[sigma-model]]

  * [[AKSZ sigma-model]]

    * [[Poisson sigma-model]]

      * **A-model**, [[B-model]]

      * [[half-twisted model]]

    * [[Courant sigma-model]]

      * [[Chern-Simons theory]]

  * [[topological membrane]]

* [[topologically twisted D=4 super Yang-Mills theory]]

* [[Landau-Ginzburg model]]

[[!include gauge theory from AdS-CFT -- table]]

## References

### General

The A-model was first conceived in

* [[Edward Witten]], _Topological sigma models_, Commun. Math. Phys. __118__ (1988) 411--449, [euclid](http://projecteuclid.org/euclid.cmp/1104162092), [MR90b:81080](http://www.ams.org/mathscinet-getitem?mr=0958805)

An early review is in

* [[Edward Witten]]. _Mirror manifolds and topological field theory_, in: Essays on mirror manifolds, pp. 120–-158. Int. Press, Hong Kong, 1992. ([arXiv:hep-th/9112056](http://arxiv.org/abs/hep-th/9112056)).

The motivation from the point of view of [[string theory]] is reviewed for instance in

* [[Paul Aspinwall]], _D-Branes on Calabi-Yau Manifolds_ ([arXiv:hep-th/0403166](http://arxiv.org/abs/hep-th/0403166))

A summary of these two reviews is in

* H. Lee, _Review of topological field theory and homological mirror symmetry_ ([pdf](http://people.maths.ox.ac.uk/leeh/files/CYMSmini.pdf))

### Action functional {#LagrangianLit}

That the A-model [[Lagrangian]] arises in [[AKSZ theory]] by [[gauge fixing]] the [[Poisson sigma-model]] was observed in

* {#AKSZ} M. Alexandrov, [[Maxim Kontsevich|M. Kontsevich]], [[Albert Schwarz|A. Schwarz]], O. Zaboronsky, around page 19 in _The geometry of the master equation and topological quantum field theory_, Int. J. Modern Phys. A 12(7):1405--1429, 1997

with more details in

* {#BonechiCattaneoIraso16} Francesco Bonechi, [[Alberto Cattaneo]], Riccardo Iraso, _Comparing Poisson Sigma Model with A-model_ ([arXiv:1607.03411](http://arxiv.org/abs/1607.03411))

Review and further discussion includes

* Francesco Bonechi, [[Maxim Zabzine]], section 5.3 of _Poisson sigma model on the sphere_ ([arXiv:0706.3164](http://arxiv.org/abs/0706.3164))

Also

* [[Noriaki Ikeda]], _Deformation of graded (Batalin-Volkvisky) Structures_, in Dito, Lu, Maeda, [[Alan Weinstein]] (eds.) _Poisson geometry in mathematics and physics_ Contemp. Math. 450, AMS (2008)

Discussion of how the [[second quantization]] [[effective field theory]] given by the A-model [[perturbation series]] is [[Chern-Simons theory]] is in

* [[Edward Witten]], _Chern-Simons Gauge Theory As A String Theory_, Prog.Math. 133 (1995) 637-678 ([arXiv:hep-th/9207094](http://arxiv.org/abs/hep-th/9207094))
 {#Witten92}

* [[Kevin Costello]], _Topological conformal field theories and gauge theories_ ([arXiv:math/0605647](http://arxiv.org/abs/math/0605647))
 {#Costello06}

formalizing at least aspects of the observations in

[[!redirects A-model]]
[[!redirects A-models]]


A-n space

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## Definition

An **$A_n$-space** or **$A_n$-algebra in spaces** is a space (in the sense of an [[homotopy type]]/[[infinity-groupoid|$\infty$-groupoid]], usually presented by a [[topological space]] or a [[simplicial set]]) with a multiplication that is associative up to higher homotopies involving up to $n$ variables.

* An $A_0$-space is a pointed space.

* Same with an $A_1$-space.

* An $A_2$-space is an [[H-space]].

* An [[A_3-space|$A_3$-space]] is a homotopy associative H-space (but no coherence is required of the associator).

* An $A_4$-space has an [[associativity]] homotopy that satisfies the [[pentagon identity]] up to [[homotopy]], but no further [[coherence law|coherence]].

* ...

* An [[A-infinity space|$A_\infty$-space]] has all [[coherence|coherent]] higher associativity homotopies.

## Related concepts

* [[A-infinity space|$A_\infty$-space]]

* [[E-n algebra]]

* [[A2-groupoid]]

* [[A3-groupoid]]

* [[k-tuply associative n-groupoid]]

## References

The notion is originally due to:

* {#Stasheff63a} [[Jim Stasheff]], *Homotopy associativity of H-spaces I*, Trans. Amer. Math. Soc. **108** 2 (1963) 275-292 [[doi:10.2307/1993608](https://doi.org/10.2307/1993608)]

* {#Stasheff63b} [[Jim Stasheff]], *Homotopy associativity of H-spaces II* **108** 2 (1963) 293-312 [[doi:10.2307/1993609](https://doi.org/10.2307/1993609), [doi:10.1090/S0002-9947-1963-0158400-5](https://doi.org/10.1090/S0002-9947-1963-0158400-5)]

Early review:

* {#Adams78} [[John Adams]], §2.2 of: _Infinite loop spaces_, Hermann Weyl lectures at IAS 1974, Annals of Mathematics Studies **90**, Princeton University Press (1978) [[ISBN:9780691082066](https://press.princeton.edu/books/paperback/9780691082066/infinite-loop-spaces-am-90-volume-90), [doi:10.1515/9781400821259](https://doi.org/10.1515/9781400821259)]

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A-theory

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## Idea

Waldhausen's _A-theory_ ([Waldhausen 85](#Waldhausen85)) of a connected [[homotopy type]] $X$ is the [[algebraic K-theory]] of the [[suspension spectrum]] $\Sigma^\infty_+ (\Omega X)$ of the [[loop space]] $\Omega X$, hence of the [[∞-group ∞-rings]] $\mathbb{S}[\Omega X]$ of the [[looping]] [[∞-group]] $\Omega X$, hence the K-theory of the [[parametrized spectra]] over $X$ ([Hess-Shipley 14](#HessShipley14)).

## Related concepts

* [[iterated algebraic K-theory]]

* [[coring spectrum]]

## References

The definition is originally due to

* {#Waldhausen85} [[Friedhelm Waldhausen]], _Algebraic K-theory of spaces_ Algebraic and geometric topology (Ne Brunswick, N. J., 1983), 318–419, Lecture Notes in Math., 1126, Springer, Berlin, 1985

The interpretation in terms of certain [[module spectra]] over the [[Spanier-Whitehead dual]] of $X$ is due to

* [[Andrew Blumberg]], [[Michael Mandell]], _Derived Koszul Duality and Involutions in the Algebraic K-Theory of Spaces_, J. Topology 4 (2011), no. 2, 327-342 ([arXiv:0912.1670](https://arxiv.org/abs/0912.1670))

and the interpretation in terms of $\mathbb{S}[\Omega X]$-[[module spectra]]  and [[Koszul duality|Koszul dually]] in terms of $\mathbb{S}[X]$-[[comodule spectra]] (using the canonical [[coring spectrum]]-[[structure]] of [[suspension spectra]]) is due to:

* {#HessShipley14} [[Kathryn Hess]], [[Brooke Shipley]], _Waldhausen K-theory of spaces via comodules_, Advances in Mathematics **290** (2016) 1079-1137 [[arXiv:1402.4719](https://arxiv.org/abs/1402.4719), [doi:10.1016/j.aim.2015.12.019](https://doi.org/10.1016/j.aim.2015.12.019)]

[[!redirects Waldhausen a-theory]]


A. Ch. Ganchev

* [arXiv page](https://arxiv.org/search/hep-th?searchtype=author&query=Ganchev%2C+A)

## Selected writings

On [[WZW models]] at [[rational number|fractional]] [[level (Chern-Simons theory)|level]]:

* P. Furlan, [[A. Ch. Ganchev]], R. Paunov, [[Valentina B. Petkova]], *Solutions of the Knizhnik-Zamolodchikov Equation with Rational Isospins and the Reduction to the Minimal Models*, Nucl. Phys. B394 (1993) 665-706 ([arXiv:hep-th/9201080](https://arxiv.org/abs/hep-th/9201080), doi:10.1016/0550-3213(93)90227-G)

* [[A. Ch. Ganchev]], [[Valentina B. Petkova]], G. M. T. Watts, *A note on decoupling conditions for generic level $\widehat{s l}(3)_k$ and fusion rules*, Nucl. Phys. B **571** (2000) 457-478 $[$doi:10.1016/S0550-3213(99)00745-2$]$

category: people


A. G. Walker

Arthur Geoffrey Walker was a professor at the University of Liverpool,
working in [[differential geometry]], [[general relativity]], and [[cosmology]].

In particular, he is responsible for the [[Friedmann–Lemaître–Robertson–Walker metric]] in [[cosmology]].

## Selected writings

On [[vector fields as derivations]]:

* [[W. F. Newns]], [[A. G. Walker]],  _Tangent Planes To a Differentiable Manifold_.  Journal of the London Mathematical Society s1-31:4 (1956), 400–407 ([doi:10.1112/jlms/s1-31.4.400](https://doi.org/10.1112/jlms/s1-31.4.400))

category: people

[[!redirects Arthur Geoffrey Walker]]
[[!redirects Arthur G. Walker]]
[[!redirects Arthur Walker]]


A. J. Tolland > history

< [[A. J. Tolland]]


A. P. Balachandran

__A. P. Balachandran__ is an indian-american theoretical physicist, retired from the Syracuse University.

* [Wikipedia entry](https://en.wikipedia.org/wiki/A._P._Balachandran)

## Selected writings

On the notion of [[entropy]] in [[quantum probability theory]]:

* [[A. P. Balachandran]], T. R. Govindarajan, Amilcar R. de Queiroz, A. F. Reyes-Lega, _Algebraic approach to entanglement and entropy_, Phys. Rev. A 88, 022301 (2013) ([arXiv:1301.1300](http://arxiv.org/abs/1301.1300))

* [[A. P. Balachandran]], A. R. de Queiroz, S. Vaidya, *Entropy of Quantum States: Ambiguities*, Eur. Phys. J. Plus 128, 112 (2013) ([arXiv:1212.1239](https://arxiv.org/abs/1212.1239), [doi:10.1140/epjp/i2013-13112-3](https://doi.org/10.1140/epjp/i2013-13112-3))

On [[fiber bundles in physics]]:

* [[A. P. Balachandran]], G. Marmo, B.-S. Skagerstam, A. Stern, _Gauge Symmetries and Fibre Bundles_, Lect. Notes in Physics 188, Springer-Verlag, Berlin, 1983 ([arXiv:1702.08910](https://arxiv.org/abs/1702.08910))

## Related $n$Lab pages

* [[fiber bundles in physics]]

* [[entanglement]],

* [[entropy]]

* [[Hopf fibration]]

category: people


A. Yung

## Selected writings

On [[vortex strings]] in $\mathcal{N}=2$ super QCD identified as [[superstrings]] in [[type II string theory]] [[KK-compactification|compactified]] on a [[conifold]]:

* [[Mikhail Shifman]], [[A. Yung]], *Supersymmetric Solitons and How They Help Us Understand Non-Abelian Gauge Theories*, Rev. Mod. Phys. **79** 1139 (2007) [[doi:10.1103/RevModPhys.79.1139](https://doi.org/10.1103/RevModPhys.79.1139), [arXiv:hep-th/0703267](https://arxiv.org/abs/hep-th/0703267)

* [[Mikhail Shifman]], [[A. Yung]], *Supersymmetric Solitons*, Cambridge University Press (2009) [[doi:10.1017/CBO9780511575693](https://doi.org/10.1017/CBO9780511575693)]

* [[Mikhail Shifman]], [[A. Yung]], *Critical String from Non-Abelian Vortex in Four Dimensions*,
Physics Letters B **750** (2015) 416-419
[[arXiv:1502.00683](https://arxiv.org/abs/1502.00683), [doi:10.1016/j.physletb.2015.09.045](https://doi.org/10.1016/j.physletb.2015.09.045)
]

* P. Koroteev, [[Mikhail Shifman]], [[A. Yung]], *Studying Critical String Emerging from Non-Abelian Vortex in Four Dimensions*, Phys.Lett. B **759** (2016) 154-158
[[arXiv:1605.01472](https://arxiv.org/abs/1605.01472), [doi:10.1016/j.physletb.2016.05.075](https://doi.org/10.1016/j.physletb.2016.05.075)
]

and interpreted in [[AdS/QCD]], in the presence of the [[Kalb-Ramond field]]:

* [[A. Yung]], *NS Three-form Flux Deformation for the Critical Non-Abelian Vortex String* [[arXiv:2209.08118](https://arxiv.org/abs/2209.08118)]

category: people


A.A. Sharapov

* [publications](http://arxiv.org/find/hep-th/1/au:+Sharapov_A/0/1/0/all/0/1)

## Selected writings

On [[non-Lagrangian field theory]] such as [[self-dual higher gauge theory]]:

* S. L. Lyakhovich, [[A. A. Sharapov]], _Quantizing non-Lagrangian gauge theories: an augmentation method_, JHEP 0701 047 (2007) [[arXiv:hep-th/0612086](http://arxiv.org/abs/hep-th/0612086)]

category: people

[[!redirects A. A. Sharapov]]
[[!redirects A. Sharapov]]


A1

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## Idea

In the [[ADE-classification]], the items labeled $A1$ include the following:

1. as [[finite subgroups of SO(3)]]:

   the [[cyclic group of order 2]]

   $\mathbb{Z}/2$

1. as [[finite subgroups of SU(2)]]:

   the [[cyclic group of order 2]]

   $\mathbb{Z}/2$

1. as [[simple Lie groups]]: the [[special unitary group]] in 2 complex dimensions

   [[SU(2)]]

1. as a [[Dynkin diagram]]/[[Dynkin quiver]]:

\begin{tikzpicture}
  \node (center) at (0,0) {};

  \draw[fill=black] (center) circle (.1);

\end{tikzpicture}

\linebreak

## Related concepts

[[!include ADE -- table]]


A1-cohesive homotopy type theory

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## Idea

**$\mathbb{A}^1$-cohesive homotopy type theory**,  **affine-cohesive homotopy type theory**, or **algebraic-cohesive homotopy type theory** is a version of [[cohesive homotopy type theory]] which has an [[affine line]] [[type]] $\mathbb{A}^1$ and the [[axiom of A1-cohesion|axiom of $\mathbb{A}^1$-cohesion]]. $\mathbb{A}^1$-cohesive homotopy type theory provides a [[synthetic mathematics|synthetic]] [[foundation]] for [[A1-homotopy theory|$\mathbb{A}^1$-homotopy theory]], where the affine line is the standard [[affine line]] in the [[Nisnevich site]] of [[smooth schemes]] of [[finite type]] over a [[Noetherian scheme]].

### Motivic homotopy type theory
 {#MotivicHomotopyTypeTheory}

[[Mitchell Riley]]'s [[bunched logic|bunched]] [[linear homotopy type theory]] [[Riley (2022)](dependent+linear+type+theory#Riley22)] is a [[synthetic mathematics|synthetic]] foundations for [[parameterized stable homotopy theory|parameterized]] [[stable homotopy theory]]. This means that there should be an $\mathbb{A}^1$-cohesive [[bunched logic|bunched]] [[linear homotopy type theory]] which behaves as a synthetic foundations for [[motivic homotopy theory]].

## Presentation

Similarly to [[real-cohesive homotopy type theory]], we assume a [[spatial type theory]] presented with crisp term judgments $a::A$. In addition, we also assume the spatial type theory has an [[affine line]] [[type]] $\mathbb{A}^1$, and $\mathbb{A}^1$-[[localization of a type at a family of functions|localizations]] $\mathcal{L}_{\mathbb{A}^1}(-)$.

Given a type $A$, let us define $\mathrm{const}_{A, \mathbb{A}^1}:A \to (\mathbb{A}^1 \to A)$ to be the type of all constant functions in the [[affine line]] $\mathbb{A}^1$:
$$\delta_{\mathrm{const}_{A, \mathbb{A}^1}}(a, r):\mathrm{const}_{A, \mathbb{A}^1}(a)(r) =_A a$$
There is an equivalence $\mathrm{const}_{A, 1}:A \simeq (1 \to A)$ between the type $A$ and the type of functions from the [[unit type]] $1$ to $A$.
Given types $B$ and $C$ and a function $F:(B \to A) \to (C \to A)$, type $A$ is **$F$-[[localization of a type at a family of functions|local]]** if the function $F:(B \to A) \to (C \to A)$ is an [[equivalence of types]].

A crisp type $\Xi \vert () \vdash A$ is **discrete** if the function $(-)_\flat:\flat A \to A$ is an [[equivalence of types]].

The **axiom of $\mathbb{A}^1$-cohesion** states that for the crisp affine line $\Xi \vert () \vdash \mathbb{A}^1 \; \mathrm{type}$, given any crisp type $\Xi \vert () \vdash A \; \mathrm{type}$, $A$ is discrete if and only if $A$ is $(\mathrm{const}_{A, 1}^{-1} \circ \mathrm{const}_{A, \mathbb{A}^1})$-local.

This allows us to define discreteness for non-crisp types: a type $A$ is **discrete** if $A$ is $(\mathrm{const}_{A, 1}^{-1} \circ \mathrm{const}_{A, \mathbb{A}^1})$-local.

The [[shape modality]] in $\mathbb{A}^1$-cohesive homotopy type theory is then defined as the $\mathbb{A}^1$-localization $\esh(A) \coloneqq \mathcal{L}_{\mathbb{A}^1}(A)$, which ensures that the shape of $\mathbb{A}^1$ itself is a [[contractible type]].

## See also

* [[cohesive homotopy type theory]]
* [[A1-homotopy theory|$\mathbb{A}^1$-homotopy theory]]
* [[axiom of A1-cohesion|axiom of $\mathbb{A}^1$-cohesion]]

## References

For the presentation of the underlying [[spatial type theory]] used for $\mathbb{A}^1$-cohesive homotopy type theory, see:

* [[Mike Shulman]], *Brouwer's fixed-point theorem in real-cohesive homotopy type theory*, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))

[[!redirects A1-cohesive homotopy type theory]]
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[[!redirects affine-cohesive homotopy type theory]]
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[[!redirects algebraic-cohesive homotopy type theory]]
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a1-meson

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## Idea

The $a_1$-meson is the [[light meson]] which (in the [[Wigner classification]]) is the [[Lorentz group|Lorentz]] [[spin group]] [[pseudo-vector representation]] as well as the [[isospin]] [[vector representation]].

This is the [[chiral partner]] of the [[rho-meson]].

\begin{imagefromfile}
    "file_name": "LightAndChiralPartnerMesonFieldsPinIII.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": 0,
        "right": 10,
        "bottom": 0,
        "left": 20
    }
\end{imagefromfile}

## Related concepts

* [[b1-meson]]

## References

* [[Stefan Leupold]], Markus Wagner, _Chiral Partners in a Chirally Broken World_, International Journal of Modern Physics A Vol. 24, No. 02n03, pp. 229-236 (2009) ([arXiv:0807.2389](https://arxiv.org/abs/0807.2389))

* A. E. Dorokhova, N. I. Kochelevba. A. P. Martynenkoc. F. A. Martynenkoc. A. E. Radzhabovb, _The contribution of axial-vector mesons to hyperfine structure of muonic hydrogen_, Physics Letters B Volume 776, 10 January 2018, Pages 105-110 Physics Letters B ([doi:10.1016/j.physletb.2017.11.027](https://doi.org/10.1016/j.physletb.2017.11.027))

[[!redirects a1-mesons]]


A2

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

#Contents#
* table of contents
{:toc}

## Idea

In the [[ADE-classification]], the items labeled $A2$ include the following:

1. as [[finite subgroups of SO(3)]]:

   the [[cyclic group of order 3]]

   $\mathbb{Z}/3$

1. as [[finite subgroups of SU(2)]]:

   the [[cyclic group of order 3]]

   $\mathbb{Z}/3$

1. as [[simple Lie groups]]: the [[special unitary group]] in 3 complex dimensions

   [[SU(3)]]

1. as a [[Dynkin diagram]]/[[Dynkin quiver]]:

\begin{tikzpicture}
  \node (left) at (-.5,0) {};
  \node (right) at (.5,0) {};

  \draw[fill=black] (left) circle (.1);
  \draw[fill=black] (right) circle (.1);

  \draw (left) to (right);
\end{tikzpicture}

\linebreak

## Related concepts

[[!include ADE -- table]]


A3

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

#Contents#
* table of contents
{:toc}

## Idea

In the [[ADE-classification]], the items labeled $A3$ include the following:

1. as [[finite subgroups of SO(3)]]:

   the [[cyclic group of order 4]]

   $\mathbb{Z}/4$

1. as [[finite subgroups of SU(2)]]:

   the [[dicyclic group]]

   $Dic_1 \simeq \mathbb{Z}/4$

1. as [[simple Lie groups]]: the [[special unitary group]] in 4 complex dimensions

   [[SU(4)]]

1. as a [[Dynkin diagram]]/[[Dynkin quiver]]:

\begin{tikzpicture}
  \node (center) at (0,0) {};
  \node (left) at (-1,0) {};
  \node (right) at (1,0) {};

  \draw[fill=black] (center) circle (.1);
  \draw[fill=black] (left) circle (.1);
  \draw[fill=black] (right) circle (.1);

  \draw (center) to (left);
  \draw (center) to (right);
\end{tikzpicture}

\linebreak

## Properties

### Exceptional isomorphism
  {#ExceptionalIsomorphism}

The A3 [[Dynkin diagram]] coincides with the [[D3]] diagram, hence with the result of removing one node from the [[D4]] diagram. In the [[classification of simple Lie groups]] this coincidence reflects the exceptional [[isomorphism]] between [[SU(4)]] and [[Spin(6)]]:

\begin{tikzpicture}

  \node (Spin6) at (0,1.4)  {$\mathrm{Spin}(6)$};
  \node (SU4) at (3.4,1.4)  {$\mathrm{SU}(4)$};

  \node at (1.7,1.4) {$\simeq$};

  \node (center) at (0,0) {};
  \node (topright) at (30:1) {};
  \node (left) at (180-30:1) {};
  \node (botright) at (0,-1) {};

  \draw[fill=black] (center) circle (.1);
  \draw[fill=black] (topright) circle (.1);
  \draw[draw=lightgray, fill=lightgray] (botright) circle (.1);
  \draw[fill=black] (left) circle (.1);

  \draw (center) to (topright);
  \draw[lightgray] (center) to (botright);
  \draw (center) to (left);

  \begin{scope}[shift={(3.4,0)}]
  \node (center) at (0,0) {};
  \node (left) at (-1,0) {};
  \node (right) at (+1,0) {};

  \draw[fill=black] (center) circle (.1);
  \draw[fill=black] (left) circle (.1);
  \draw[fill=black] (right) circle (.1);

  \draw (center) to (left);
  \draw (center) to (right);
  \end{scope}
\end{tikzpicture}

## Related concepts

[[!include ADE -- table]]

[[!redirects D3]]


A4-spatial groupoid > history

[[!redirects A4-spatial groupoid]]

see discussion [here](https://nforum.ncatlab.org/discussion/14220/ktuply-associative-ngroupoid/?Focus=100308#Comment_100308)


Aarne Ranta

[webpage](http://www.cse.chalmers.se/~aarne/)

## Selected writings

On [[dependent type theory]] and natural language (parsing and [[syntactic sugar|sugaring]]):

* {#Ranta93} [[Aarne Ranta]], _Type theory and the informal language of mathematics_, TYPES '93: Proceedings of the international workshop on Types for proofs and programs, pp. 352–365, ([pdf](/nlab/files/InfLang.pdf))

* {#Ranta94} [[Aarne Ranta]], §1.6, §1.7, §9 in: *Type-theoretical grammar*, Oxford University Press (1994) [[ISBN:9780198538578](https://global.oup.com/academic/product/type-theoretical-grammar-9780198538578)]

## Related pages

* [[dependent type theoretic methods in natural language semantics]]

category: people


Aaron Bergman

* [arXiv entry](http://arxiv.org/find/hep-th/1/au:+Bergman_A/0/1/0/all/0/1)

## Selected writings

On [[Bridgeland stability conditions]] for [[B-branes]] of the [[B-model]] [[topological string]]:

* [[Aaron Bergman]], _Stability Conditions and Branes at Singularities_,  Journal of High Energy Physics 2008.10 (2008): 07 ([arXiv:hep-th/0702092](http://arxiv.org/abs/hep-th/0702092))

Application of [Block 2005](Lie+infinity-algebroid+representation#Block05) to [[D-branes]] in [[topological string theory]]:

* {#Bergman08} [[Aaron Bergman]], _Topological D-branes from Descent_ ([arXiv:0808.0168](http://arxiv.org/abs/0808.0168))

On [[non-geometric string vacua]] ([[T-folds]]):

* [[Aaron Bergman]], Daniel Robbins, _Ramond-Ramond Fields, Cohomology and Non-Geometric Fluxes_ ([arXiv:0710.5158](http://arxiv.org/abs/0710.5158))

On [[T-duality]]:

* [[Katrin Becker]], [[Aaron Bergman]], _Geometric Aspects of D-branes and T-duality_ ([arXiv:0908.2249](http://arxiv.org/abs/0908.2249))

category: people


Aaron F

### September 2012 ###

Wow, time flies! I'm now a Ph.D. student at UT Austin, and you can find some of my writing and stuff on my [department home page](http://www.ma.utexas.edu/users/afenyes). Things I'm thinking about these days include...
* Quadratic differentials, [spectral networks](http://arxiv.org/abs/1204.4824), and $SL_2\mathbb{C}$ connections on Riemann surfaces (for a reading course).
* Classical and quantum $\mathbb{Z}/2\mathbb{Z}$ gauge theories on compact surfaces (for a [math club](http://www.ma.utexas.edu/mathclub/) talk).
* What the $\clubsuit\ast\mho\heartsuit$ is going on in [this paper](http://arxiv.org/abs/1105.4170) (for a graduate geometry seminar).
* Markov algorithms (maybe for a middle school summer program somewhere, someday).

### October 2009 ###

I'm a master's student interested in mathematical physics. Or is it physical mathematics? Here are some things I've been wondering about lately...
* The [general boundary formulation](http://arxiv.org/abs/hep-th/0509122) of quantum field theory.
* Whether or not it would be pedagogically useful to introduce topology as a special case of [[pretopological space|pretopology]].

+-- {: .query}
Welcome, Aaron!  It might be good if you used your full name, although I don\'t think anybody will try to force you.

It also helps if you log your changes in the [latest changes](http://www.math.ntnu.no/~stacey/Mathforge/nForum/?CategoryID=5) category of the Forum.  In particular, people will be much more likely to look at your work on [[induced representation]] if you do that.

I have also thought that pretopological spaces are pedagogically simpler than topological spaces.  I also like to motivate [[uniform spaces]] as a uniform version of pretopological rather than topological spaces (although we do have the theorem that any pretopological uniform space is topological).  On the other hand, now that I\'m getting to [[locale]] theory, I\'m not as interested in pretopological spaces ....

---[[Toby Bartels]]
=--

category: people


Aaron Greicius

* [webpage](http://greicius.wordpress.com/about-3/)

category: people


Aaron Lauda

* [website](https://sites.google.com/view/lauda-home/home)

## Selected writings

On [[Frobenius monads]] and [[ambidextrous adjunctions]] in [[2-categories]] more general than [[Cat]], with motivation from [[2d TQFT]]:

* {#Lauda05} [[Aaron Lauda]], *Frobenius algebras and ambidextrous adjunctions*, Theory and Applications of Categories **16** 4 (2006) 84-122 [[arXiv:math/0502550](http://arxiv.org/abs/math/0502550), [tac:16-04](http://www.tac.mta.ca/tac/volumes/16/4/16-04abs.html)]

## Videos

On [[knot theory]]:

* [[Aaron Lauda]], _Knot theory explained to newbies in 1 min 24 sec lightning idea_,  USC Dornsife College of Letters, Arts and Sciences ([video](https://www.youtube.com/embed/kxaWqKM5JyQ)).

category: people
[[!redirects A. Lauda]]


Aaron M. Smith

* [GoogleScholar page](https://scholar.google.com/citations?user=e0yBkScAAAAJ)

## Selected writings

On [[infinity-local systems|$\infty$-local systems]] in the sense of [[(infinity,1)-vector bundles|$(\infty,1)$-vector bundles]] with [[flat infinity-connections|flat $\infty$-connections]]:

* [[Jonathan Block]], [[Aaron M. Smith]], *The higher Riemann--Hilbert correspondence*, Advances in Mathematics
**252** (2014) 382-405 [[arXiv:0908.2843](https://arxiv.org/abs/0908.2843), [doi:10.1016/j.aim.2013.11.001](https://doi.org/10.1016/j.aim.2013.11.001)]

category: people

[[!redirects Aaron Smith]]


Aaron Mazel-Gee

* [webpage](http://etale.site/)

## Selected writing

On [[tmf]]:

* _You could've invented $tmf$_, April 2013 ([pdf slides](http://math.berkeley.edu/~aaron/writing/ustars-tmf-beamer.pdf))

Introducing [[model (infinity,1)-category|model $\infty$-category]]-theory:

* [[Aaron Mazel-Gee]], *Model ∞-categories I: some pleasant properties of the ∞-category of simplicial spaces* ([arXiv:1412.8411](https://arxiv.org/abs/1412.8411))

* [[Aaron Mazel-Gee]], *Model ∞-categories II: Quillen adjunctions*, New York Journal of Mathematics **27** (2021) 508-550.  ([arXiv:1510.04392](https://arxiv.org/abs/1510.04392), [nyjm:27-21](http://nyjm.albany.edu/j/2021/27-21.html))

* [[Aaron Mazel-Gee]], *Model ∞-categories III: the fundamental theorem*, New York Journal of Mathematics **27** (2021) 551-599 ([arXiv:1510.04777](https://arxiv.org/abs/1510.04777), [nyjm:27-22](http://nyjm.albany.edu/j/2021/27-22.html))

## Related entries

* [[tmf]]

* [[homotopy Kan fibration]]

category: people

[[!redirects amg]]


Aaron Stump

* [personal page](https://homepage.divms.uiowa.edu/~astump/)

## Selected writings

On [[Agda]] as a [[certified programming|verified]] [[functional programming language]]:

* [[Aaron Stump]], *Verified Functional Programming in Agda*, Association for Computing Machinery and Morgan & Claypool (2016) [[doi:10.1145/2841316](https://doi.org/10.1145/2841316), ISBN:978-1-970001-27-3]

category: people


Ab

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

# Contents
* table of contents
{: toc}

## Idea

_$Ab$_ denotes the [[category]] of [[abelian group|abelian groups]]: it has abelian groups as [[object|objects]] and [[group homomorphisms]] between these as [[morphism|morphisms]].

The archetypical example of an abelian group is the group $\mathbb{Z}$ of integers, and for many purposes it is useful to think of $Ab$ equivalently as the [[category of modules]] over $\mathbb{Z}$

$$
  Ab \simeq \mathbb{Z} Mod
 \,.
$$

The category $Ab$ serves as the basic [[enriched category theory|enriching]] category in [[homological algebra]]. There [[Ab-enriched categories]] play much the same role as [[Set]]-enriched categories ([[locally small categories]]) play in general.

In this vein, the analog of $Ab$ in [[homotopy theory]] -- or rather in _[[stable homotopy theory]]_ -- is the category of [[spectra]], either regarded as the [[stable homotopy category]] or rather refined to the [[stable (infinity,1)-category of spectra]]. A spectrum is much like an abelian group up to [[coherence law|coherent]] [[homotopy]] and the role of the archetypical abelian group $\mathbb{Z}$ is the played by the [[sphere spectrum]] $\mathbb{S}$.

## Properties

### Free abelian groups

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

The category $Ab$ is a [[concrete category]], the [[forgetful functor]]

$$
  U : Ab \to Set
$$

to [[Set]] sends a group, regarded as a [[set]] $A$ equipped with the [[structure]] $(+,0)$ of a chosen element $0 \in A$ and a binary, associative and 0-unital operation $+$ to its underlying set

$$
  (A, +, 0) \mapsto A
  \,.
$$

This functor has a [[left adjoint]] $F : Set \to Ab$ which sends a set $S$ to the [[free abelian group]] $\mathbb{Z}[S]$ on this set: the group of [[formal linear combinations]] of elements in $S$ with [[coefficients]] in $\mathbb{Z}$.

=--

### Direct sum, direct product and tensor product
 {#DirectSumEtc}

We discuss basic properties of binary operations on the category of abelian groups: [[direct product]], [[direct sum]] and [[tensor product]]. Below in _[Monoidal and bimonoidal structure](#MonoidalAndBipermutativeStructure)_ we put these structures into a more abstract context.

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

For $A, B \in Ab$ two [[abelian groups]], their
[[direct product]] $A \times B$ is the abelian group whose elements are pairs $(a, b)$ with $a \in A$ and $b \in B$, whose 0-element is $(0,0)$ and whose addition operation is the componentwise addition

$$
  (a_1, b_1) + (a_2, b_2) = (a_1 + a_2, b_1 + b_2)
  \,.
$$

This is at the same time [[generalized the|the]] [[direct sum]] $A \oplus B$.

Similarly for $I \in $[[FinSet]]$\hookrightarrow$ [[Set]] a [[finite set]], we have

$$
  \oplus_{i \in I} A_i \simeq \prod_i A_{i}
  \,.
$$

But for $I \in Set$ a set which is not finite, there is a difference: the [[direct sum]] $\oplus_{i \in I} A_i$ of an $I$-indexed family ${A_i}_{i \in I}$ of abelian groups is the sub-group of the direct product on those elements for which only finitely many components are non-0

$$
  \oplus_{i \in I} A_i \hookrightarrow \prod_i A_i
  \,.
$$

=--

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

The [[trivial group]] $0 \in Ab$ (the group with a single element) is a [[unit]] for the direct sum: for every abelian group we have

$$
  A \oplus 0 \simeq 0 \oplus A \simeq A
  \,.
$$

=--

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

In view of remark \ref{FreeForgetful} this means that the direct sum of ${\vert I \vert}$ copies of the additive group of [[integers]] with themselves is equivalently the [[free abelian group]] on $I$:

$$
  \oplus_{i \in I} \mathbb{Z} \simeq \mathbb{Z}[I]
  \,.
$$

=--

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

For $A$ and $B$ two [[abelian groups]], their [[tensor product of abelian groups]] is the group $A \otimes B$ with the property that a group homomorphism $A \otimes B \to C$ is equivalently a [[bilinear map]] out of the _set_ $A \times B$.

=--

See at _[[tensor product of abelian groups]]_ for details.

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

The [[unit]] for the tensor product of abelian groups is the additive group of [[integers]]:

$$
  A \otimes \mathbb{Z} \simeq \mathbb{Z} \otimes A \simeq A
  \,.
$$

=--

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

The [[tensor product of abelian groups]] [[distributivity law|distributes]] over arbitrary [[direct sums]]:

$$
  A \otimes (\oplus_{i \in I} B_i) \simeq \oplus_{i \in I} A \otimes B_i
  \,.
$$

=--

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

For $I \in Set$ and $A \in Ab$, the [[direct sum]] of ${\vert I\vert}$ copies of $A$ with itself is equivalently the [[tensor product of abelian groups]] of the [[free abelian group]] on $I$ with $A$:

$$
  \oplus_{i \in I} A \simeq (\oplus_{i \in I} \mathbb{Z}) \otimes A
  \simeq (\mathbb{Z}[I]) \otimes A
  \,.
$$

=--

### Symmetric monoidal and bimonoidal structure
 {#MonoidalAndBipermutativeStructure}

With the definitions and properties discussed above in _[Direct sum, etc.](#DirectSumEtc)_ we have the following

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

The category $Ab$ becomes a [[monoidal category]]

1. under [[direct sum]] $(Ab, \oplus, 0)$;

1. under [[tensor product of abelian groups]] $(Ab, \otimes, \mathbb{Z})$.

Indeed with both structures combined we have

* $(Ab, \oplus, \otimes, 0, \mathbb{Z})$

is a [[bimonoidal category]] (and can be made a [[bipermutative category]]).

=--

It's also easy to see that under direct sum or tensor product, Ab can be turned into a [[symmetric monoidal category]] by equipping it with the appropriate braiding map. For example, under $\oplus$, the braiding is $\sigma_{A, B}(a, b) = (b, a)$.

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

A [[monoid]] [[internalization|internal to]] $(Ab, \otimes, \mathbb{Z})$ is equivalently a [[ring]].

=--

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

A [[monoid]] in $(Ab, \oplus, 0)$ is equivalently just an abelian group again (since $\oplus$ is the [[coproduct]] in $Ab$, so every object has a unique monoid structure with respect to it).

=--

### Pointed objects

$Ab$ is a [[monoidal category]] with [[tensor unit]] $\mathbb{Z}$, so the [[pointed objects in a monoidal category|pointed objects]] in $Ab$ are the objects $A$ with a [[group homomorphism]] $\mathbb{Z} \to A$.

### Closed monoidal structure

Abelian groups are equivalently $\mathbb{Z}$-modules. Because the [[Mod|category of $R$-modules]] $Mod_R$ is closed monoidal for all [[commutative rings]] $R$, $Ab = Mod_{\mathbb{Z}}$ is also closed monoidal.

### Natural numbers object

The [[natural numbers object]] in $Ab$ is the [[free abelian group]] $\mathbb{Z}[\mathbb{N}] = \bigoplus_{n \in \mathbb{N}} \mathbb{Z}$ on the [[natural numbers]], and comes with group homomorphisms $z_0:\mathbb{Z} \to \mathbb{Z}[\mathbb{N}]$ and $z_s:\mathbb{Z}[\mathbb{N}] \to \mathbb{Z}[\mathbb{N}]$ such that for all abelian groups $A$ and group homomorphisms $f:\mathbb{Z} \to A$, $g: A \to A$, there is a unique group homomorphism $\phi_{f, g}:\mathbb{Z}[\mathbb{N}] \to A$ making the following diagram commute:

$$\array{
\mathbb{Z} & \stackrel{z_0}{\to} & \mathbb{Z}[\mathbb{N}] & \stackrel{z_s}{\leftarrow} & \mathbb{Z}[\mathbb{N}] \\
 & \mathllap{f} \searrow & \downarrow \mathrlap{\phi_{f, g}} & & \downarrow \mathrlap{\phi_{f, g}} \\
 & & A & \underset{g}{\leftarrow} & A
}$$

Abelian groups are $\mathbb{Z}$-[[modules]], so the free $\mathbb{Z}$-module $\mathbb{Z}[\mathbb{N}]$ has a function $v:\mathbb{N} \to \mathbb{Z}[\mathbb{N}]$ representing the [[basis of a free module|basis]] of $\mathbb{Z}[\mathbb{N}]$; it has the property that for all integers $m \in \mathbb{Z}$, $m \cdot v(0) = z_0(m)$ and for all $n \in \mathbb{N}$, $m \cdot v(s(n)) = z_s(m \cdot v(n))$, where $m \cdot v$ is the scalar multiplication of an element $v$ by an integer $m$ in a $\mathbb{Z}$-module.

The [[ring]] structure on $\mathbb{Z}[\mathbb{N}]$ is defined by double [[induction]] on $\mathbb{Z}[\mathbb{N}]$, we define

$$(-)(-):\mathbb{Z}[\mathbb{N}] \times \mathbb{Z}[\mathbb{N}] \to \mathbb{Z}[\mathbb{N}] \otimes \mathbb{Z}[\mathbb{N}] \to \mathbb{Z}[\mathbb{N}]$$

by

$$z_0(m)z_0(n) = z_0(m \cdot n) \qquad z_s(v)z_0(n) = z_s(v z_0(n))$$
$$z_0(m)z_s(w) = z_s(z_0(m) w) \qquad z_s(v)z_s(w) = z_s(z_s(vw))$$

for all $m, n \in \mathbb{Z}$ and $v, w \in \mathbb{Z}[\mathbb{N}]$ (recall the definition of [[addition]] in the [[natural numbers]], inductively defined by $0(p) + 0(q) = 0(p \cdot q)$, $s(m) + 0(p) = s(m + 0(p))$, $0(p) + s(n) = s(0(p) + n)$, and $s(m) + s(n) = s(s(m + n))$ for all $p, q \in \mathbb{1}$ and $m, n \in \mathbb{N}$). It is a [[commutative ring]] and represents multiplication in the [[polynomial ring]] $\mathbb{Z}[X]$; the group homomorphism $z_0$ represents the function which takes integers to constant polynomials, and $z_s$ represents the function which takes a polynomial and multiplies it by the indeterminate $X$.

### Enrichment over $Ab$

Categories [[enriched category|enriched]] over $Ab$ are called [[additive category|pre-additive categories]] or sometimes just additive categories.  If they satisfy an extra exactness condition they are called [[abelian category|abelian categories]]. See at _[[additive and abelian categories]]_.

## Related concepts

* [[CMon]]

* [[Mod]]

* [[sAb]]

category: category

[[!redirects AbGrp]]

[[!redirects category of abelian groups]]
[[!redirects categories of abelian groups]]

[[!redirects AbelianGroups]]

[[!redirects ZMod]]
[[!redirects Z Mod]]
[[!redirects Z-Mod]]

[[!redirects category of Z-modules]]
[[!redirects categories of Z-modules]]

[[!redirects category of integer modules]]
[[!redirects categories of integer modules]]

[[!redirects ZModules]]
[[!redirects Z Modules]]
[[!redirects Z-Modules]]

[[!redirects IntegerModules]]


Ab-enriched category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
#### Additive and abelian categories
+--{: .hide}
[[!include additive and abelian categories - contents]]
=--
=--
=--

# $Ab$-enriched categories
* tic
{: toc}

## Idea

An _$Ab$-enriched category_  (or, if small, _ringoid_) is a [[category enriched]] over the [[monoidal category]] [[Ab]] of [[abelian groups]] with its usual [[tensor product]].

Sometimes they are called [[pre-additive category|pre-additive categories]], but sometimes that term also implies the existence of a [[zero object]].

## Definition

Explicitly, an __$Ab$-enriched category__ is a [[category]] $C$ such that
for all objects $a,b$ the [[hom-set]] $Hom_C(a,b)$
is equipped with the structure of an [[abelian group]]; and
such that for all triples $a,b,c$ of objects the
[[composition]] operation
$
  \circ_{a,b,c} : Hom_C(a,b) \times Hom_C(b,c) \to Hom_C(a,c)
$
is bilinear. A __ringoid__ is small $Ab$-enriched category.

## Remarks

*  $Ab$-enriched categories are called ringoids since the concept is a [[horizontal categorification]] (or 'oidification') of the concept of a [[ring]].

*  There is a canonical forgetful functor $Ab \to Set_*$ from
abelian groups to [[pointed set]]s, which sends each group to its underlying set with point being the neutral element. Using this functor, every $Ab$-enriched category $C$ is in particular also a category that is enriched over pointed sets (that is, a category with [[zero morphisms]]). This is sufficient for there to be a notion of [[kernel]] and [[cokernel]] in $C$.

*  In general, [[abelian category|abelian categories]] are the most important examples of $Ab$-enriched categories.  See [[additive and abelian categories]].

## Finite products are absolute

One of the remarkable facts about $Ab$-enriched categories is that finite [[products]] (and [[coproducts]]) are [[absolute limit|absolute limits]].  This implies that finite products coincide with finite coproducts, and are preserved by _any_ $Ab$-enriched functor.

### Zero objects

In an $Ab$-enriched category $C$, any [[initial object]] is also a [[terminal object]], hence a [[zero object]], and dually.  An object $a\in C$ is a zero object just when its identity $1_a$ is equal to the zero morphism $0:a\to a$ (that is, the identity element of the abelian group $\hom_C(a,a)$).  Expressed in this way, it is easy to see that any $Ab$-enriched functor preserves zero objects.

### Biproducts

For $c_1, c_2 \in C$ two objects in an $Ab$-enriched category $C$,  [[generalized the|the]] [[product]] $c_1 \times c_2$ coincides  with [[generalized the|the]] [[coproduct]] $c_1 \sqcup c_2$ when either exists.

For example, if $c_1 \times c_2$ exists, with [[projection]] maps $p_i\colon c_1 \times c_2 \to c_i$, then according to the [[universal property]] of products, there are unique maps
$$
  q_i\colon c_i \to c_1 \times c_2
$$
such that $p_i q_i = 1_{c_i}$ and $p_j q_i = 0$ for $j \neq i$, and these maps $q_i$ are the coproduct [[coprojections]], i.e., they realize $c_1 \times c_2$ as the coproduct of $c_1$ and $c_2$. Indeed, for any maps $r_1\colon c_1 \to e$ and $r_2\colon c_2 \to e$, it is easily checked that

$$r = r_1 p_1 + r_2 p_2\colon c_1 \times c_2 \to e$$

satisfies $r q_1 = r_1$ and $r q_2 = r_2$, and is the unique map satisfying these equations. The full argument is spelled out at *[[additive category]]*.

By a [[formal duality|dual]] argument, if the coproduct $c_1 \sqcup c_2$ exists, then it may also be realized as the product of $c_1$ and $c_2$. Either way, the product or coproduct is called a *[[biproduct]]* or (sometimes) a *[[direct sum]]* and is generally denoted
$$
  c_1 \oplus c_2.
$$
It can be characterized diagrammatically as an object $c_1\oplus c_2$ equipped with morphisms $q_i \colon c_i\to c_1\oplus c_2$ and $p_i \colon c_1\oplus c_2 \to c_i$ such that $p_i q_j = \delta_{i j}$ and $q_1 p_1 + q_2 p_2 = 1_{c_1\oplus c_2}$.  Expressed in this form, it is clear that any $Ab$-[[enriched functor]] [[preserved limit|preserves]] biproducts.

## As a generalisation of rings

When using the term 'ringoid', one often assumes a ringoid to be [[small category|small]].

Ringoids share many of the properties of (noncommutative) [[rings]].  For instance, we can talk about (left and right) [[modules]] over a ringoid $R$, which can be defined as $Ab$-enriched [[functors]] $R\to Ab$ and $R^{op}\to Ab$.  [[bimodule|Bimodules]] over ringoids have a tensor product (the enriched [[tensor product of functors]]) under which they form a [[bicategory]], also known as the bicategory $Ab Prof$ of $Ab$-enriched [[profunctors]].  Modules over a ringoid also form an [[abelian category]] and thus have a [[derived category]].

One interesting operation on ringoids is the ($Ab$-enriched) [[Cauchy completion]], which is the completion under finite [[direct sums]] and [[split idempotents]].  In particular, the Cauchy completion of a ring $R$ is the category of [[finitely generated object|finitely generated]] [[projective object|projective]] $R$-modules (aka [[split monomorphism|split]] [[subobjects]] of finite-rank [[free object|free]] modules).  Every ringoid is [[equivalence|equivalent]] to its Cauchy completion in the bicategory $Ab Prof$, and two ringoids are equivalent in $Ab Prof$ if and only if their Cauchy completions are [[equivalence of categories|equivalent]] as $Ab$-enriched categories.  This sort of equivalence is naturally called [[Morita equivalence]].

See also [[dg-category]].

## Examples

* The category [[Ab]] is [[closed monoidal category|closed monoidal]] and hence canonically enriched over itself.

* An $Ab$-enriched category with one object is precisely a [[ring]].

* For any small $Ab$-enriched category $R$, the enriched [[presheaf category]] $[R^{op},Ab]$ is, of course, $Ab$-enriched.  If $R$ is a ring, as above, then $[R^{op},Ab]$ is the category of $R$-modules.

## Related concepts

* [[hom-group]]

* [[additive category]]

## References

* John Baez, [Ringoids](http://golem.ph.utexas.edu/category/2006/09/ringoids.html), blog
* C. Weibel, [[An Introduction to Homological Algebra]], Cambridge Univ. Press
* [[Daniel Murfet]], _Localisation of ringoids_, [pdf](http://therisingsea.org/notes/LocalisationOfRingoids.pdf) 2006 notes
* N. Popescu, _Abelian categories with applications to rings and modules_, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. MR0340375

[[!redirects Ab-enriched category]]
[[!redirects Ab-enriched categories]]
[[!redirects Ab-enriched]]
[[!redirects Ab-category]]
[[!redirects Ab-categories]]
[[!redirects ringoid]]
[[!redirects ringoids]]


abc conjecture

#Contents#
* table of contents
{:toc}

## Idea

The _abc conjecture_ (or _ABC conjecture_) is a [[number theory|number theoretic]] conjecture due to ([Oesterlé-Masser 1985](#OesterléMasser)), which says that there are only [[finite set|finitely many]] integer solutions to the [[equation]]

$$
  a + b = c   \;\;\; for \;\;\;  a,b,c \geq 1 \in \mathbb{N}
$$

(or instead $a+b+c= 0$)
if one requires the [[integer numbers]] $a,b,c$ to have no common factor as well as having "joint power" greater than a given bound.

Here the _power_ of $a,b,c$ is

$$
  P(a,b,c) \coloneqq \frac{log|c|}{log(rad(a\cdot b\cdot c))}
  \,,
$$

where the [[radical]] $rad(n)$ of an [[integer]] $n$ is the product of all its distinct [[prime factors]].

The precise form of the conjecture is:

+-- {: .un_theorem}
###### Conjecture
**(abc conjecture)**

For any number $\epsilon\gt 0$ there are only finitely many positive relatively prime ([[coprime integers|coprime]]) [[integer]] solutions $(a,b,c)$ to the [[equation]] $a + b = c$   with power $P(a,b,c)\geq 1+\epsilon$.

=--

According to ([Mazur](#Matzur)):

> The beauty of such a Conjecture is that it captures the intuitive sense that triples of numbers which satisfy a linear relation, and which are divisible by high perfect powers, are rare; the precision of the Conjecture goads one to investigate this rarity quantitatively. Its very statement makes an attractive appeal to perform a range of numerical experiments that would test the empirical waters. On a theoretical level, it is enlightening to understand its relationship to the constellation of standard arithmetic theorems, conjectures, questions, etc., and we shall give some indications of this below.

## Mason's Theorem

According to [Lang](#Lang), one important antecedent of the abc conjecture is a simple but at the time unexpected relation for the function field case, published in 1984. Consider polynomials $f \in k[t]$ over an algebraically closed field $k$ of characteristic $0$, and define $n_0(f)$ to be the number of _distinct_ roots of $f$, counted _without_ regard to multiplicity.

+-- {: .un_theorem}
###### Theorem ([Mason](#Mason))
Let $a, b, c \in k[t]$ be relatively prime polynomials, not all constant, such that $a + b = c$. Then $\max \{deg(a), deg(b), deg(c)\} \leq n_0(a b c) - 1$.
=--

+-- {: .proof}
###### Proof
Let $f = a/c$, $g = b/c$, so that $f + g = 1$. Taking the derivative, we obtain

$$\frac{f'}{f} f + \frac{g'}{g} g = 0$$

whence

$$b/a = g/f = -\frac{f'/f}{g'/g}.$$

Put

$$a(t) = c_1 \prod (t - \alpha_i)^{m_i}, \qquad b(t) = c_2 \prod (t - \beta_j)^{n_j}, \qquad c(t) = c_3 \prod (t - \gamma_k)^{p_k}.$$

Then

$$\frac{b}{a} = -\frac{f'/f}{g'/g} = -\frac{\sum \frac{m_i}{t - \alpha_i} - \sum \frac{p_k}{t - \gamma_k}}{\sum \frac{n_j}{t - \beta_j} - \sum \frac{p_k}{t - \gamma_k}} .$$

A common denominator for $f'/f$ and $g'/g$ is given by

$$N_0 = \prod (t - \alpha_i) \prod (t - \beta_j) \prod (t - \gamma_k)$$

whose degree is $n_0(a b c)$. We then have

$$\frac{b}{a} = -\frac{N_0 f'/f}{N_0 g'/g}$$

where the numerator and denominator on the right are polynomials. However, since $b$ and $a$ are relatively prime, the fraction $b/a$ is already in lowest terms. From this we conclude that $deg(b) \leq deg(N_0 f'/f) \leq n_0(a b c) - 1$, and similarly $deg(a) \leq deg(N_0 g'/g) \leq n_0(a b c) - 1$, which completes the proof.
=--

+-- {: .un_cor}
###### Corollary (FLT for polynomials)
Assume $x, y, z \in k[t]$ are relatively prime polynomials, not all constant, and suppose $x^n + y^n = z^n$. Then $n \leq 2$.
=--

+-- {: .proof}
###### Proof
From Mason's theorem, we conclude $n deg(x) = deg(x^n) \leq deg(x) + deg(y) + deg(z) - 1$, and similarly upon replacing $x$ by $y$ and $z$ on the left. Adding the results, we have

$$n(deg(x) + deg(y) + deg(z)) \leq 3(deg(x) + deg(y) + deg(z)) - 3$$

which is impossible if $n \geq 3$.
=--

Guided by analogies between the ring of integers and the ring of polynomials in one variable, and building on insights of Mason, Frey, Szpiro, and others, Masser and Oesterlé were led to formulate the abc conjecture for integers as follows. Again define $N_0(m)$ for $m$ a non-zero integer to be the number of distinct primes dividing $m$.

* Conjecture: For all $\epsilon \lt 0$ there exists $C(\epsilon) \lt 0$ such that for relatively prime integers $a, b, c$ satisfying $a + b = c$, we have
$$\max({|a|}, {|b|}, {|c|}) \leq C(\epsilon)N_0(a b c)^{1+\epsilon}.$$

Of course, this differs from the polynomial case because of the presence of $1+ \epsilon$ in the exponent, but this is a necessary evil. For example, for any $C \gt 0$, we can find relatively prime $a$, $b$, $c$ with $a + b = c$ and ${|a|} \gt C N_0(a b c)$: take $a = 3^{2^n}$, $b = -1$, and observe by repeated application of $x^2 - y^2 = (x-y)(x+y)$ that $c = a + b$ is of the form $2^n d$ for some integer $d$. Taking $n$ sufficiently large, we can easily derive the claimed inequality.

## Relation to other statements

The abc conjecture implies the [[Mordell conjecture]] ([Elkies](#Elkies)).

It is equivalent to the general form of [[Szpiro's conjecture]].

* [[Vojta's conjecture]]

## References

### General

The abc conjecture was stated in

* {#OesterléMasser} Joseph Oesterlé, David Masser (1985)

Mason's theorem was presented in

* {#Mason}  R. C. Mason, Equations over function fields. In _Number Theory, Proceedings of the Noordwijkerhout_, Springer Lecture Notes 1068 (1984), 149-157.

Material on Mason's theorem and its relation to the abc conjecture was taken from

* {#Lang} Serge Lang, Algebra ($3^{rd}$ Edition), Addison-Wesley (1993), 194-196.

The relation to the [[Mordell conjecture]] is discussed in

* {#Elkies} [[Noam Elkies]], _ABC conjecture implies Mordell_, Int. Math. Research Notices 7 (1991) 99-109

The relation to [[Szpiro's conjecture]] is discussed in

* Matt Baker (notes taken by William Stein), _Elliptic curves, the ABC conjecture, and points of small canonical height_ ([pdf](http://modular.math.washington.edu/mcs/archive/Fall2001/notes/12-10-01/12-10-01.pdf))

See also

* [[Barry Mazur]], _Questions about number_, [pdf](http://www.math.harvard.edu/~mazur/papers/scanQuest.pdf) scan
 {#Mazur}

* PolyMath, _[ABC conjecture](http://michaelnielsen.org/polymath1/index.php?title=ABC_conjecture)_

*  Abderrahmane Nitaj, _[The ABC Conjecture Homepage](http://www.math.unicaen.fr/~nitaj/abc.html)_

* Wikipedia, _[abc conjecture](http://en.wikipedia.org/wiki/Abc_conjecture)_

### Alleged proof via IUT

[[Shinichi Mochizuki]] anounced the proof which the mathematical community perceives as a serious but unchecked claim. See the references at _[[inter-universal Teichmüller theory]]_.

Comments on the proof are at

*  _[[Mochizuki's proof of abc]]_.

* MathOverflow, _[Philosophy behind Mochizuki's work on the ABC conjecture](http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/106658#106658)_

A popular account of the problem of the math community checking the proof is in

* Caroline Chen, _The Paradox of the Proof_ ([web](http://projectwordsworth.com/the-paradox-of-the-proof/))

[[!redirects abc-conjecture]]


Abdelmalek Abdesselam

* [webpage](http://people.virginia.edu/~aa4cr/)

## related $n$Lab entries

* [[renormalization group]]

category: people


abductive reasoning

#Contents#
* table of contents
{:toc}

##General idea

**Abductive** [[reasoning]] is a process whereby one reasons to the truth of an explanation from its ability to account for what is observed. It is therefore sometimes also known as **inference to the best explanation**.

[[Charles Peirce]], the originator of the term, illustrated the differences between [[deductive reasoning|deduction]], [[inductive reasoning|induction]], and abduction by the following example.

* **Deduction**
  * All beans in that bag are white.
  * These beans are from that bag.
  * Therefore, these beans are white.

* **Induction**
  * These beans are from that bag.
  * These beans are white.
  * Therefore, all beans in that bag are white.

* **Abduction**
  * These beans are white.
  * All beans in that bag are white.
  * Therefore, these beans are from that bag.

It is not completely clear what Peirce meant by abduction, which he also termed **retroduction**. Clearly the inference cannot be to just any possible explanation, e.g., in the case above, there might have been many other bags full of white beans. But before we decide what constitutes a **best** explanation, we had been inquire into the nature of explanation itself.

There is an extensive literature about explanation in the Philosophy of Science, for example, ([FourDecades](#FourDecades)). Clearly it is not merely a matter of devising propositions, perhaps a general law and a particular statement, which have the thing to be explained (explicandum) as a consequence. We want the proposed explanation to 'give the reason for' the observation. A thorough account of what constitutes the 'reason' for something is notoriously difficult to formulate. For some, it is a matter of subsuming the observation under a general covering law, while for others, it is a matter of giving a causal or mechanistic story with the observation as the outcome.

Note also that there is a growing literature now on the concept of 'explanatory proofs' in mathematics, it being felt that one may have proved a mathematical fact without understanding 'why' it is true.

For some, abduction also signifies the creative process of coming up with a good explanation. Otherwise, if it is merely a case of assessing a range of existing rival hypotheses as explanations, it may be possible to employ [[Bayesian reasoning]], generally taken to be a form of [[inductive reasoning]].

> If you really can find an explanation having sufficient probability to be worth consideration, you escape in great measure from reposing upon retroduction [abduction] and make your inference inductive. (Peirce, [Harvard Lectures, p. 193](#Harvard))

## Abduction as lifting

In Peirce's [Harvard lectures, p. 315](#Harvard), he describes the triad -- _deduction_, _induction_, _abduction_ -- in terms of the logical relations between three concepts, $M$, $P$ and $S$.

* Deduction strings together, say, $M$ is $P$ and $P$ is $S$ to give $M$ is $S$.

* Induction looks to generalise from $M$ is $S$, taking $M$ as a sample of $P$, to conclude that $P$ is $S$.

* Abduction looks to explain why $M$ is $S$, having noted that $P$ is $S$, by hypothesising that $M$ is $P$.

Seen from the point of view of category theory, this would seem rather like: composition, [[extension]], and [[lifting]]. Induction as a kind of extension seems quite reasonable. Abduction may account for an instance of some concept, $E$, by lifting to a concept, $C$, through a law connecting cause, $C$, to effect, $E$.

## References

* Wesley Salmon, 1989, _Four Decades of Scientific Explanation_, University of Pittsburgh Press. {#FourDecades}

* Proposed formalization as a functor between [categories of structures](structure+in+model+theory#categories_of_structures) can be found in Fernando Tohmé, Gianluca Caterina, Rocco Gangle, [_Abduction: A categorical characterization_](http://doi.org/10.1016/j.jal.2014.12.004), Journal of Applied Logic, Volume 13, Issue 1, March 2015, Pages 78-90

* Gerhard Schurz, 2008, _Patterns of abduction_, Synthese 164:201–234.

* {#Harvard} [[Charles Peirce]], 1992, _Reasoning and the logic of things_, Harvard University Press (lectures from 1898, [book](http://www.hup.harvard.edu/catalog.php?isbn=9780674749672))

* Peter Krause, _Presupposition and abduction in type theory_, In Working Notes of. CLNLP-95: Computational Logic and Natural Language Processing.

[[!redirects abduction]]
[[!redirects abductive reasoning]]
[[!redirects abductive inference]]
[[!redirects abductive logic]]


Abdus Salam

* [Wikipedia entry](http://en.wikipedia.org/wiki/Abdus_Salam)

## Selected writings

Co-introducing the theory of the [[electroweak field]]:

* [[Abdus Salam]], [[John Clive Ward]], *Weak and electromagnetic interactions*, Nuovo Cimento. **11** 4 (1959) 568–577 [[doi:10.1007/BF02726525](https://doi.org/10.1007/BF02726525)]

Introducing [[super Yang-Mills theory]]:

* [[Abdus Salam]], [[John Strathdee]], _Super-symmetry and non-Abelian gauges_, Physics Letters B Volume 51, Issue 4, 19 August 1974, Pages 353-355 (doi:10.1016/0370-2693(74)90226-3)

Introducing [[superspace]] and [[superfields]] for [[supersymmetry|supersymmetric]] [[quantum field theory]]:

* [[Abdus Salam]], [[John Strathdee]], _Supergauge Transformations_, Nucl.Phys. B76 (1974) 477-482 ([spire:89208](http://inspirehep.net/record/89208))

* [[Abdus Salam]], [[John Strathdee]], _Superfields and Fermi-Bose symmetry_, Physical Review D11, 1521-1535 (1975) ([doi:10.1142/9789812795915_0051](https://doi.org/10.1142/9789812795915_0051))

On [[supergravity]]:

* [[Abdus Salam]], [[Ergin Sezgin]] (eds.), *Supergravities in Diverse Dimensions*, Elsevier & World Scientific (1990) [[doi:10.1142/0277](https://doi.org/10.1142/0277)]

## Related entries

* [[superspace]], [[superfield]]

* [[8-dimensional supergravity]]

* [[grand unified theory]]

* [[E11]]

category: people


Abel-Jacobi map

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

#Contents#
* table of contents
{:toc}

## Idea

In [[algebraic geometry|algebraic]]/[[complex geometry]] The term *Abel-Jacobi map* refers to various [[group homomorphisms]] from certain [[groups]] of [[algebraic cycles]] to some sorts of [[Jacobians]] or generalized Jacobians. Such maps generalize the classical Abel-Jacobi map from points of a complex algebraic curve to its [[Jacobian]], which answers the question of which divisors of degree zero arise from [[meromorphic functions]].

## Definition

### for curves

Let $X$ be a smooth projective [[complex curve]]. Recall that a [[Weil divisor]] on $X$ is a [[formal linear combination]] of [[closed points]]. Classically, the Abel-Jacobi map
$$
  \alpha
  \;\colon\;
  \Div^0(X)
   \longrightarrow
  J(X)
  ,\,
$$
on the group of [[Weil divisors]] of degree zero, is defined by [[integration]]. According to [[Abel's theorem]], its [[kernel]] consists of the [[principal divisors]], i.e. the ones coming from [[meromorphic functions]].

### on Deligne cohomology

The cycle map to [[de Rham cohomology]] due to [Zein & Zucker (1981)](#ZeinZucker81) is discussed in [Esnault & Viehweg (1988), section 6](#EsnaultViehweg88), the refinement to [[Deligne cohomology]] in [Esnault & Viehweg (1988), section 6](#EsnaultViehweg88). By the characterization of [[intermediate Jacobians]] as subgroups of the [[Deligne complex]] (see *[intermediate Jacobian -- characterization as Hodge-trivial Deligne cohomology](intermediate+Jacobian#CharacterizationAsHodgeTrivialDeligneCohomology)*) this induces a map from cycles to [[intermediate Jacobians]]. This is the Abel-Jacobi map ([Esnault & Viehweg (1988), theorem 7.11](#EsnaultViehweg88)).

### on higher Chow groups

An Abel-Jacobi map on [[higher Chow groups]] is discussed in [K-L-MS 04](#KLMS04).

### via extensions of mixed Hodge structures

An alternate construction of the Abel-Jacobi map, via [[Hodge theory]], is due to Arapura-Oh.
By a theorem of Carlson, the [[Jacobian]] is identified with the following group of [[extensions]] in the [[abelian category]] of [[mixed Hodge structures]]:
  $$ J(X) = Ext^1_{MHS}(\mathbf{Z}(-1), H^1(X, \mathbf{Z})) $$
where $\mathbf{Z}(-1)$ is the Tate Hodge structure.
Given a [[divisor]] $D$ of degree zero, one can associate to it a certain class in the above extension group.
This gives a map
  $$ \alpha : Div^0(X) \longrightarrow J(X) $$
which is called the Abel-Jacobi map.
The Abel theorem says that its [[kernel]] is precisely the subgroup of [[principal divisors]], i.e. divisors which come from invertible rational functions.
See ([Arapura-Oh, 1997](#ArapuraOh97)) for details of this construction.

## Related concepts

* [[Hodge-filtered differential cohomology]]

## References

* {#ZeinZucker81} Fouad El Zein and Steven Zucker, _Extendability of normal functions associated to algebraic cycles_, Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982), Ann. of Math. Stud., vol. 106, Princeton Univ. Press, Princeton, NJ, 1984, pp. 269–288. [MR 756857](http://www.ams.org/mathscinet-getitem?mr=756857)

* {#EsnaultViehweg88} [[Hélène Esnault]], [[Eckart Viehweg]], _Deligne-Beilinson cohomology_, in: [[Michael Rapoport]], [[Norbert Schappacher]], [[Peter Schneider]] (eds.), _[[Beilinson's Conjectures on Special Values of L-Functions]]_, Perspectives in Mathematics **4**, Academic Press, Inc. (1988) [ISBN:978-0-12-581120-0, [[EsnaultViehweg-DeligneBeilinsonCohomology.pdf:file]]]

* {#Voisin02} [[Claire Voisin]], section 12 of _[[Hodge theory and Complex algebraic geometry]] I,II_,  Cambridge Stud. in Adv. Math. __76, 77__, 2002/3

* {#ArapuraOh97} Donu Arapura, Kyungho Oh. _On the Abel-Jacobi map for non-compact varieties_. Osaka Journal of Mathematics 34 (1997), no. 4, 769--781. [Project Euclid](http://projecteuclid.org/euclid.ojm/1200787781).

* {#KLMS04} Matt Kerr, James Lewis, Stefan Müller-Stach, _The Abel-Jacobi map for higher Chow groups_, 2004, [arXiv:0409116](http://arxiv.org/abs/math/0409116).

* Wikipedia, _[Abel-Jacobi map](http://en.wikipedia.org/wiki/Abel–Jacobi_map)_

Remarks on generalization to the more general context of [[anabelian geometry]] are in

* [[Minhyong Kim]], _Galois Theory and Diophantine geometry, 2009 ([pdf](http://www.ucl.ac.uk/~ucahmki/cambridgews.pdf))

Refinement of the [[Abel-Jacobi map]] to [[Hodge filtration|Hodge filtered]] [[differential cobordism cohomology theory|differential]] [[MU]]-[[cobordism cohomology theory]]:

* [[Gereon Quick]], *An Abel-Jacobi invariant for cobordant cycles*,
Documenta Mathematica **21** (2016) 1645–1668 [[arXiv:1503.08449](https://arxiv.org/abs/1503.08449)]

* [[Knut B. Haus]], [[Gereon Quick]], *Geometric Hodge filtered complex cobordism* [[arXiv:2210.13259](https://arxiv.org/abs/2210.13259)]

Introduction and survey:

* [[Gereon Quick]], *Geometric Hodge filtered complex cobordism*, [talk at](Center+for+Quantum+and+Topological+Systems#QuickMar2023) *[[CQTS]]* (March 2023) [video:[YT](https://www.youtube.com/watch?v=pMu0gT5kIBo)]

[[!redirects Abel-Jacobi maps]]

[[!redirects Abel-Jacobi theorem]]
[[!redirects Abel-Jacobi theorems]]


abelian 2-category

[[abelian category|Abelian categories]] are a common generalization of the category of abelian groups, categories of $R$-modules where $R$ is a ring, and the categories of sheaves of $\mathcal{O}_X$-modules where $\mathcal{O}_X$ is a sheaf of rings on a topological space $X$. Similarly, in the categorical dimension 2, one wants to define an __abelian 2-category__ as a generalization of symmetric 2-groups and possibly some 2-related categories. There are a few different proposals in the literature.

* Mathieu Dupont, _Abelian categories in dimension 2_, [arxiv/0809.1760](http://arxiv.org/abs/0809.1760)
* Hiroyuki Nakaoka, _Comparison of the definitions of Abelian 2-categories_, [arxiv/0904.0078](http://arxiv.org/abs/0904.0078)

[[!redirects Abelian 2-category]]


abelian 3-group

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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

An _abelian 3-group_ is an [[abelian infinity-group]] which is a [[3-group]].

## Examples

If $(\mathcal{C}, \otimes)$ is a [[symmetric monoidal 2-category]] then its [[Picard 3-group]] is an abelian 3-group.

## Related concepts

* [[∞-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**

[[!redirects abelian 3-group]]


abelian 4-group

+-- {: .rightHandSide}
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+-- {: .hide}
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#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

An _abelian 4-group_ is an [[abelian ∞-group]] which is a [[4-group]].

## Related concepts

* [[∞-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**

[[!redirects abelian 4-groups]]


abelian 7-group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

An _abelian 7-group_ is an [[abelian ∞-group]] which is a [[7-group]].

## Related concepts

* [[∞-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**

[[!redirects abelian 7-groups]]


This page compiles pointers related to:

* [[Nicolae Popescu]]:

  \linebreak

  __Abelian categories with applications to rings and modules__

  \linebreak

  London Mathematical Society Monographs **3*

  Academic Press (1973)

  [MR0340375](http://www.ams.org/mathscinet-getitem?mr=0340375)

on the [[algebra]] of [[rings]] and [[modules]] via [[abelian categories]].

\linebreak

Contents:

__Note to the reader__

Some terminology, notations and conventions used throughout the book.

__Categories and functors__

The notion of a category; examples. Special objects and morphisms. Functors. Inductive and
projective limits. Adjoint functors; equivalence of categories.

__Abelian categories__

Preadditive and additive category. The canonical factorization of a morphism; preabelian categories.
Abelian categories. Fibred products and fibred sums. Basic lemmas on abelian categories. The
isomorphism theorems. Direct sums of subobjects. Inductive limits; the conditions Ab.

__Additive functors__

Additive functors. Exactness of functors; injective and projective objects. The injective and
projective objects in the category Ab. Categories of additive functors; modules. Special objects in
abelian categories. Tensor products; a characterization of functor categories. Tensor products of
modules. Flat modules. Some remarks on projective modules. Injective envelopes. Semisimple
rings.

__Localization__

Categories of fractions; calculus of fractions. The spectral category of an abelian category. The
quotient category of an abelian category relative to a dense subcategory. The section functor.
Localization in categories with injective envelopes. Localization in Ab 3 categories. Sheaves over a
topological space. Torsion theories. Localizing subcategories in categories of modules. Localization
in categories of modules. Left exact functors; the embedding theorem. The study of the localization
ring of a ring. The complete ring of quotients. Some remarks on Grothendieck categories.
Finiteness conditions on localizing systems. Flat epimorphism of rings. Left quasi-orders of a ring.
Rings of fractions. Left orders. Left spectrum of a ring. Bilocalizing subcategories.

__The Krull-Remak-Schmidt theorem and decomposition theories__

The classical Krull-Remak-Schmidt theorem. The structure of spectral categories. Local
coirreducible categories. Decomposition theory. Semi-noetherian categories. Semi-artinian
categories. Noetherian and artinian categories. Locally noetherian categories. Noetherian and
artinian rings. Primary decomposition theory. Decomposition theories on locally noetherian categories.

__Duality__

Linearly compact subcategories. Topologically linearly compact rings. The duality theorem for
Grothendieck categories. Duality theory for l.n.- and l.f.-categories. Colocalization.

__Bibliography__

__Index__

category: reference


abelian category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
#### Additive and abelian categories
+--{: .hide}
[[!include additive and abelian categories - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

##Idea

The notion of _abelian category_ is an abstraction of basic properties of the category [[Ab]] of [[abelian groups]], more generally of the category $R$[[Mod]] of [[modules]] over some [[ring]], and still more generally of categories of [[sheaves]] of abelian groups and of modules. It is such that much of the [[homological algebra]] of [[chain complexes]] can be developed inside every abelian category.

The concept of abelian categories is one in a sequence of notions of [[additive and abelian categories]].

While additive categories differ significantly from [[toposes]], there is an intimate relation between abelian categories and toposes. See _[[AT category]]_ for more on that.

##Definition

Recall the following fact about [[pre-abelian categories]] from [this proposition](pre-abelian+category#DecompositionOfMorphisms), discussed there:

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

Every [[morphism]] $f \colon A\to B$ in a [[pre-abelian category]] has a canonical de-[[composition]]

\[
  \label{PreImageFactorization}
  A
  \overset{\; p \;}{\twoheadrightarrow}
  \coker\big(\ker f\big)
  \xrightarrow{\; \overline{f} \;}
  \ker\big(\coker(f)\big)
  \xhookrightarrow{\; i \;}
  B
\]

where:

* $p$ is a [[cokernel]], hence  an [[epimorphism]],

* $i$ is a [[kernel]], hence a [[monomorphism]].

=--

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

An **abelian category** is a [[pre-abelian category]] satisfying the following equivalent conditions.

1. For every [[morphism]] $f$, the canonical morphism $\bar{f} \colon coker(ker(f)) \to ker(coker(f))$ (eq:PreImageFactorization) from prop. \ref{DecompositionOfMorphisms} is an [[isomorphism]] (hence providing an [[image]] factorization $A \twoheadrightarrow im(f) \hookrightarrow B$).

1. Every [[monomorphism]] is a [[kernel]] and every [[epimorphism]] is a [[cokernel]].

=--

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

These two conditions are indeed equivalent.

=--

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

The first condition implies that if $f$ is a [[monomorphism]] then $f \cong \ker(\coker(f))$ (in the category of objects over $B$) so $f$ is a kernel. Dually if $f$ is an [[epimorphism]] it follows that $f \cong coker(ker(f))$. So (1) implies (2).

The converse can be found in, among other places, Chapter VIII of ([MacLane](#MacLane)).

=--

## Properties

### General
 {#PropertiesGeneral}

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

The notion of abelian category is self-dual: [[opposite category|opposite]] of any abelian category is abelian.

=--

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

By the second formulation of the definition \ref{AbelianCategory},  in an abelian category

* every [[monomorphism]] is a [[regular monomorphism]];

* every [[epimorphism]] is a [[regular epimorphism]].

It follows that every abelian category is a _[[balanced category]]_.

=--

\begin{prop}\label{PullbackPreservesEpimorphisms}
  In an [[abelian category]], [[pullback]] preserves [[epimorphisms]] and [[pushout]] preserves [[monomorphisms]].
\end{prop}
Because every abelian category is a [[regular category]]. For an explicit proof see, e.g., [Selick, Prop. 1.3.13](#Selick).

### Factorization of morphisms
 {#FactorizationOfMorphisms}

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

In an abelian category every morphism decomposes [[generalized the|uniquely up to a unique isomorphism]] into the composition of an [[epimorphism]] and a [[monomorphism]], via prop \ref{DecompositionOfMorphisms} combined with def. \ref{AbelianCategory}.

Since by remark \ref{RegularEpisAndMonos} every monic is [[regular monomorphism|regular]], hence [[strong monomorphism|strong]], it follows that $(epi, mono)$ is an [[orthogonal factorization system]] in an abelian category; see at _[[(epi, mono) factorization system]]_.

=--

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

Some references claim that this property characterizes abelian categories among pre-abelian ones, but it is not clear to the authors of this page why this should be so, although we do not currently have a counterexample; see [this discussion](http://nforum.mathforge.org/discussion/4094/?Focus=33415#Comment_33415).

=--

### Canonical $Ab$-enrichment
 {#CanonicalAbEnrichment}

The $Ab$-enrichment of an abelian category need not be specified a priori.  If an arbitrary (not necessarily pre-additive) [[locally small category|locally small]] category $C$ has a [[zero object]], binary products and coproducts, kernels, cokernels and the property that every monic is a kernel arrow and every epi is a cokernel arrow (so that all monos and epis are [[normal monomorphism|normal]]), then it can be equipped with a unique addition on the morphism sets such that composition is bilinear and $C$ is abelian with respect to this structure.  However, in most examples, the $Ab$-enrichment is evident from the start and does not need to be constructed in this way.  (A similar statement is true for [[additive categories]], although the most natural result in that case gives only enrichment over abelian [[monoids]]; see [[semiadditive category]].)

The last point is of relevance in particular for [[infinity-category|higher categorical]] generalizations of additive categories. See for instance [remark 2.14, p. 5](http://www.math.harvard.edu/~lurie/papers/DAG-I.pdf#page=5) of [[Jacob Lurie]]'s [[Stable Infinity-Categories]].

### Relation to exactness properties of toposes
 {#RelationToToposes}

The [[exactness properties]] of abelian categories have many features in common with exactness properties of [[toposes]] or of [[pretoposes]].

[Freyd (1999)](AT+category#Freyd99) gave a sharp description of the properties shared by these categories, introducing a new concept called _[[AT categories]]_ (for "abelian-topos"), and showing convincingly that the difference between the A and the T can be concentrated precisely in the difference of the behavior of the initial object.

### Embedding theorems
 {#EmbeddingTheorems}

Not every [[abelian category]] is a [[concrete category]] such as [[Ab]] or $R$[[Mod]]. But for many proofs in [[homological algebra]] it is very convenient to have a concrete abelian category, for that allows one to check the behaviour of morphisms on actual _elements_ of the sets underlying the [[objects]].

The following _embedding theorems_, however, show that under good conditions an abelian category can be _embedded_ into [[Ab]] as a [[full subcategory]] by an [[exact functor]], and generally can be embedded this way into $R Mod$, for some ring $R$. This is the celebrated _[[Freyd-Mitchell embedding theorem]]_  discussed [below](#FreydMitchellEmbedding).

This implies for instance that proofs about [[exact sequence|exactness of sequences]] in an abelian category can always be obtained by a naive argument on elements -- called a "[[diagram chasing|diagram chase]]" -- because that does hold true after such an embedding, and the exactness of the embedding means that the notion of exact sequences is preserved by it.

Alternatively, one can reason with [[generalized elements]] in an abelian category, without explicitly embedding it into a larger concrete category, see at _[[element in an abelian category]]_. But under suitable conditions this comes down to working subject to an embedding into $Ab$, see the discussion at _[Embedding into Ab](#EmbeddingIntoAb)_ below.

#### Counterexamples

First of all, it's easy to see that not every abelian category is equivalent to $R$[[Mod]] for some ring $R$.  The reason is that $R Mod$ has all [[small category]] [[limits]] and [[colimits]].  For a [[Noetherian ring]] $R$ the category of [[finitely generated module|finitely generated]] $R$-modules is an abelian category that lacks these properties.

#### Embedding into $Ab$
 {#EmbeddingIntoAb}

(...)

([Bergman 1974](#Bergman))

(...)

#### Freyd-Mitchell embedding into $R Mod$
 {#FreydMitchellEmbedding}

+-- {: .num_thm}
###### Mitchell's Embedding Theorem

Every small abelian category admits a [[full functor|full]], [[faithful functor|faithful]] and [[exact functor|exact]] functor to the category $R Mod$ for some ring $R$.

=--

+-- {: .proof}
###### Proof
This result can be found as Theorem 7.34 on page 150 of Peter Freyd's book [Abelian Categories](http://www.emis.de/journals/TAC/reprints/articles/3/tr3.pdf#page=176).  His terminology is a bit outdated, in that he calls an abelian category "fully abelian" if admits a full and faithful exact functor to a category of $R$-modules.  See also the [Wikipedia article](http://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem) for the idea of the proof.
=--

For more see at _[[Freyd-Mitchell embedding theorem]]_.

We can also characterize which abelian categories _are_ equivalent to a category of $R$-modules:

+-- {: .num_theorem}
###### Theorem
Let $C$ be an abelian category.  If $C$ has all [[small category|small]] [[coproducts]] and has a [[compact object|compact]] [[projective object|projective]] [[generator]], then $C \simeq R Mod$ for some ring $R$.  In fact, in this situation we can take $R = C(x,x)^{op}$ where $x$ is any compact projective generator.   Conversely, if $C \simeq R Mod$, then $C$ has all small coproducts and $x = R$ is a compact projective generator.
=--

+-- {: .proof}
###### Proof
This theorem, minus the explicit description of $R$, can be found as Exercise F on page 103 of Peter Freyd's book [Abelian Categories](http://www.emis.de/journals/TAC/reprints/articles/3/tr3.pdf#page=132).
The first part of this theorem can also be found as Prop. 2.1.7. of Victor Ginzburg's [Lectures on noncommutative geometry](http://arxiv.org/PS_cache/math/pdf/0506/0506603v1.pdf#page=4).  Conversely, it is easy to see that $R$ is a compact projective generator of $R Mod$.
=--

One can characterize functors between categories of $R$-modules that are either (isomorphic) to functors of the form $B \otimes_R -$ where $B$ is a bimodule or those which look as Hom-modules. For the characterization of the tensoring functors see [[Eilenberg-Watts theorem]].

Going still further one should be able to obtain a nice theorem describing the image of the embedding of the weak 2-category of

* rings
* bimodules
* bimodule homomorphisms

into the strict 2-category of

* abelian categories
* right exact functors
* natural transformations.

For more discussion see the [$n$-Cafe](http://golem.ph.utexas.edu/category/2007/08/questions_about_modules.html).

## Examples

* Of course, [[Ab]] is abelian,

* the [[category of modules|category $R$Mod of (left) modules]] over any [[ring]] $R$ is abelian

* Therefore in particular the category [[Vect]] of vector spaces over any field is an abelian category

* The full subcategory of $R$Mod whose objects are the Noetherian left $R$-modules is abelian, since it contains any submodule or quotient module of any of its objects (see Theorem 2.3.8 p.103 of Berrick and Keating in the textbook references below).

* Similarly, the full subcategory of $R$Mod whose objects are the Artinian left $R$-modules is abelian, since it contains any submodule or quotient module of any of its objects (loc. cit.).

* Also similarly, the full subcategory of $R$Mod whose objects are the Artinian semisimple modules is abelian, since it contains any submodule or quotient module of any of its objects (loc. cit.) .

* as is the [[category of representations]] of a [[group]] (e.g. [here](https://unapologetic.wordpress.com/2008/12/15/the-category-of-representations-is-abelian/))

* The [[category of sheaves|category of]] [[sheaves of abelian groups]] on any [[site]] is abelian.

Counter-examples:

* The category of [[torsion subgroup|torsion-free]] abelian groups is pre-abelian, but not abelian: the monomorphism $2:\mathbb{Z}\to\mathbb{Z}$ is not a kernel.

## Related concepts

* [[additive and abelian categories]]

* [[abelian subcategory]]

* [[Deligne tensor product of abelian categories]]

* [[pseudo-abelian category]]

* [[quasi-abelian category]]

* [[semi-abelian category]]

* [[length of an object]]

## References

Maybe the first reference on abelian categories, then still called _exact categories_ is

* D. A. Buchsbaum, _Exact categories and duality_, Transactions of the American Mathematical Society Vol. 80, No. 1 (1955), pp. 1-34 ([JSTOR](http://www.jstor.org/stable/1993003))

Further foundations of the theory were then laid in

* [[Alexander Grothendieck]], _[[Tohoku|Sur quelques points d'algèbre homologique]], Tôhoku Math. J. vol 9, n.2, 3, 1957.

Other classic references:

* [[Pierre Gabriel]], *[[Des Catégories Abéliennes]]*, Bulletin de la Société Mathématique de France **90** (1962) 323-448 [[numdam:BSMF_1962__90__323_0](http://www.numdam.org/item?id=BSMF_1962__90__323_0)]

* {#Freyd64} [[Peter Freyd]], _Abelian Categories -- An Introduction to the theory of functors_, originally published by Harper and Row, New York(1964), Reprints in Theory and Applications of Categories **3** (2003)  [[tac:tr3](http://www.emis.de/journals/TAC/reprints/articles/3/tr3abs.html), [pdf](https://www.emis.de/journals/TAC/reprints/articles/3/tr3.pdf)]

Textbook accounts:

* [[Nicolae Popescu]], _[[Abelian categories with applications to rings and modules]]_, London Math. Soc. Monographs __3__, Academic Press (1973) [[MR0340375](http://www.ams.org/mathscinet-getitem?mr=0340375)]

* [[Saunders MacLane]], Chapter VIII of: _[[Categories for the Working Mathematician]]_ (1978)

* A. J. Berrick and M. E. Keating, *Categories and Modules, with K-theory in View*, Cambridge Studies in Advanced Mathematics **67**, Cambridge University Press (2000

* [[Masaki Kashiwara]], [[Pierre Schapira]], Section 8 of: *[[Categories and Sheaves]]*, Grundlehren der Mathematischen Wissenschaften **332**, Springer (2006) [[doi:10.1007/3-540-27950-4](https://link.springer.com/book/10.1007/3-540-27950-4), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/kashiwara2.pdf)]

* {#EGNO15} [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], Chapter 1 of: *Tensor Categories*, AMS Mathematical Surveys and Monographs **205** (2015) [[ISBN:978-1-4704-3441-0](https://bookstore.ams.org/surv-205), [pdf](http://www-math.mit.edu/~etingof/egnobookfinal.pdf)]

Further review:

* Rankey Datta, _An introduction to abelian categories_  (2010) ([pdf](http://www-bcf.usc.edu/~lauda/teaching/rankeya.pdf))

* {#Selick} Paul Selick, *Homological Algebra Notes* ([pdf](www.math.toronto.edu/selick/mat1352/1350notes.pdf),[[Selick_HomologicalAlgebra.pdf:file]])

On [[diagram chasing]] and [[elements in an abelian category]]:

* {#Bergman} [[George Bergman]], _A note on abelian categories -- translating element-chasing proofs, and exact embedding in abelian groups_ (1974) [[pdf](http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf), [[Bergman-ElementChasing.pdf:file]]]

For more discussion of the _[[Freyd-Mitchell embedding theorem]]_ see there.

The proof that $R Mod$ is an abelian category is spelled out for instance in

* Rankeya Datta, _The category of modules over a commutative ring and abelian categories_ ([pdf](http://www.math.columbia.edu/~ums/pdf/Rankeya_R-mod_and_Abelian_Categories.pdf))

A discussion about to which extent abelian categories are a general context for [[homological algebra]] is archived at nForum [here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=2052&Focus=17680#Comment_17680).

See also the [catlist 1999 discussion](http://www.mta.ca/~cat-dist/catlist/1999/atcat) on comparison between abelian categories and topoi ([[AT categories]]).

Formalization of abelian categories as [[univalent categories]] in [[univalent foundations of mathematics]] ([[homotopy type theory]]):

* [[unimath]], *Abelian Categories* [[UniMath.CategoryTheory.Abelian](https://unimath.github.io/doc/UniMath/4dd5c17/UniMath.CategoryTheory.Abelian.html)]

[[!redirects abelian categories]]
[[!redirects Abelian category]]


abelian differential graded Lie algebra

An abelian [[differential graded Lie algebra]] is a [[cochain complex]] seen as a differential graded Lie algebra with trivial [[Lie algebra|Lie bracket]]. In the particular case when the cochain complex is concentrated in degree zero, one recovers the notion of [[abelian Lie algebra]].


abelian group

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

#Contents#
* table of contents
{:toc}

## Definition

An _abelian group_ (named after [[Niels Henrik Abel]]) is a [[group]] $A$ where the multiplication satisfies the commutative law: for all elements $x, y\in A$ we have

$$
  x y = y x
  \,.
$$

The [[category]] with abelian groups as [[object|objects]] and group homomorphisms as [[morphism|morphisms]] is called [[Ab]].

Every abelian group has the canonical structure of a [[module]] over the [[commutative ring]] $\mathbf{Z}$.  That is, [[Ab]] = $\mathbf{Z}$-[[Mod]].

### With subtraction and unit only

This definition of abelian group is based upon [[Toby Bartels]]'s definition of an [[associative quasigroup]]:

An **abelian group** is a [[pointed set]] $(A, 0)$ with a binary operation $(-)-(-):A \times A \to A$ called **subtraction** such that

* for all $a \in A$, $a - a = 0$

* for all $a \in A$, $0 - (0 - a) = a$

* for all $a \in A$ and $b \in A$, $a - (0 - b) = b - (0 - a)$

* for all $a \in A$, $b \in A$, and $c \in A$, $a - (b - c) = (a - (0 - c)) - b$

For every element $a \in A$, the inverse element is defined as $-a \coloneqq 0 - a$ and addition is defined as $a + b \coloneqq a - (-b)$.

Addition is commutative:

$$a + b = a - (0 - b) = b - (0 - a) = b + a$$

and associative

$$(a + b) + c = (a - (0 - b)) - (0 - c)$$
$$(a + b) + c = (b - (0 - a)) - (0 - c)$$
$$(a + b) + c = b - ((0 - c) - a)$$
$$(a + b) + c = b - ((0 - c) - (0 - (0 - a)))$$
$$(a + b) + c = b - ((0 - a) - (0 - (0 - c)))$$
$$(a + b) + c = b - ((0 - a) - c)$$
$$(a + b) + c = (b - (0 - c)) - (0 - a)$$
$$(a + b) + c = a - (0 - (b - (0 - c)))$$
$$(a + b) + c = a + (b + c)$$

and has left identities

$$0 + a = 0 - (0 - a) = a$$

and right identities

$$a + 0 = 0 + a = a$$

and has left inverses

$$-a + a = (0 - a) - (0 - a) = 0$$

and right identities

$$a + (-a) = -a + a = 0$$

Thus, these axioms form an abelian group.

## Properties

### In homotopy theory

From the [[nPOV]], just as a [[group]] $G$ may be thought of as a ([[pointed object|pointed]]) [[groupoid]] $\mathbf{B}G$ with a single object -- as discussed at [[delooping]] -- an abelian group $A$ may be understood as a (pointed) [[2-groupoid]] $\mathbf{B}^2 A$ with a single object and a single morphism: the delooping of the delooping of $A$.

$$
  \mathbf{B}^2 A
  =
  \left\{
    \array{
      & \nearrow \searrow^{\mathrlap{Id}}
      \\
      \bullet
      &\Downarrow^{a \in A}&
      \bullet
      \\
      & \searrow \nearrow_{\mathrlap{Id}}
    }
  \right\}
  \,.
$$

The [[exchange law]] for the composition of [[2-morphisms]] in a [[2-category]] forces the product on the $a \in A$ here to be commutative. This reasoning is known as the [[Eckmann-Hilton argument]] and is the same as the reasoning that finds that the second [[homotopy group]] of a space has to be abelian.

So the identitfication of abelian groups with one-object, one-morphism 2-groupoids may also be thought of as an identification with 2-[[truncated]] and 2-[[connected]] [[homotopy types]].

### Relation to other concepts

A [[monoid]] in [[Ab]] with its standard [[monoidal category]] structure, equivalently a ([[pointed object|pointed]]) [[Ab]]-[[enriched category]] with a single object, is a [[ring]].

## Generalizations

Generalizations of the notion of abelian group in [[higher category theory]] include

* abelian [[group object in an (infinity,1)-category|group objects in an (∞,1)-category]]

* notably abelian [[simplicial groups]]

* and [[spectrum|spectra]].

An abelian group may also be seen as a [[discrete category|discrete]] [[compact closed category]].

## Related entries

* [[commutative magma]]

* [[commutative invertible semigroup]]

* [[tensor product of abelian groups]], [[direct sum of abelian groups]]

* [[free abelian group]], [[finite abelian group]]

* [[theory of abelian groups]]

* [[abelianization]]

* [[anabelian group]]

* [[module]], [[ring]]

* [[commutative ring]], [[commutative algebra]]

* [[super abelian group]], [[super module]]

* [[quadratic abelian group]]

* [[tensor ring]], [[Clifford ring]]

* [[Ab]]

* [[sheaf of abelian groups]]

* [[Ab-enriched category]], [[abelian category]]

* [[abelian ∞-group]]

  * [[symmetric 2-group]]

  * [[symmetric 3-group]]

* [[halving group]]

* [[symmetric midpoint group]]

## References

Textbook account:

* [[László Fuchs]], *Abelian Groups*, Springer (2015) [[doi:10.1007/978-3-319-19422-6](https://doi.org/10.1007/978-3-319-19422-6)]

Formalization of abelian groups in [[univalent foundations of mathematics]] ([[homotopy type theory]] with the [[univalence axiom]]):

* [[Univalent Foundations Project]], Section 6.11 of: *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]], Section 4.12 of: **[[Symmetry]]** (2021)

[[!redirects abelian groups]]
[[!redirects Abelian group]]
[[!redirects Abelian groups]]