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\begin{document}

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\section*{abelian category}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{enriched_category_theory}{}\paragraph*{{Enriched category theory}}\label{enriched_category_theory}

[[!include enriched category theory contents]]

\hypertarget{additive_and_abelian_categories}{}\paragraph*{{Additive and abelian categories}}\label{additive_and_abelian_categories}

[[!include additive and abelian categories - contents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{PropertiesGeneral}{General}\dotfill \pageref*{PropertiesGeneral} \linebreak
\noindent\hyperlink{FactorizationOfMorphisms}{Factorization of morphisms}\dotfill \pageref*{FactorizationOfMorphisms} \linebreak
\noindent\hyperlink{CanonicalAbEnrichment}{Canonical $Ab$-enrichment}\dotfill \pageref*{CanonicalAbEnrichment} \linebreak
\noindent\hyperlink{RelationToToposes}{Relation to exactness properties of toposes}\dotfill \pageref*{RelationToToposes} \linebreak
\noindent\hyperlink{EmbeddingTheorems}{Embedding theorems}\dotfill \pageref*{EmbeddingTheorems} \linebreak
\noindent\hyperlink{counterexamples}{Counterexamples}\dotfill \pageref*{counterexamples} \linebreak
\noindent\hyperlink{EmbeddingIntoAb}{Embedding into $Ab$}\dotfill \pageref*{EmbeddingIntoAb} \linebreak
\noindent\hyperlink{FreydMitchellEmbedding}{Freyd-Mitchell embedding into $R Mod$}\dotfill \pageref*{FreydMitchellEmbedding} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The notion of \emph{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 \emph{[[AT category]]} for more on that.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Recall the following fact about [[pre-abelian categories]] from \href{pre-abelian+category#DecompositionOfMorphisms}{this proposition}, discussed there:

\begin{prop}
\label{DecompositionOfMorphisms}\hypertarget{DecompositionOfMorphisms}{}
Every [[morphism]] $f \colon A\to B$ in a [[pre-abelian category]] has a canonical decomposition

\begin{displaymath}
A\stackrel{p}\to \coker(\ker f)\stackrel{\bar{f}}\to\ker(\coker f)\stackrel{i}\to B
\end{displaymath}
where $p$ is a [[cokernel]], hence an [[epimorphism|epi]], and $i$ is a [[kernel]], and hence [[monomorphism|monic]].

\end{prop}
\begin{defn}
\label{AbelianCategory}\hypertarget{AbelianCategory}{}
An \textbf{abelian category} is a [[pre-abelian category]] satisfying the following equivalent conditions.

\begin{enumerate}%
\item For every [[morphism]] $f$, the canonical morphism $\bar{f} \colon coker(ker(f)) \to ker(coker(f))$ of prop. \ref{DecompositionOfMorphisms} is an [[isomorphism]] (hence providing an [[image]] factorization $A \to im(f) \to B$).


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



\end{enumerate}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
These two conditions are indeed equivalent.

\end{prop}
\begin{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 (\hyperlink{MacLane}{MacLane}).

\end{proof}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{PropertiesGeneral}{}\subsubsection*{{General}}\label{PropertiesGeneral}

\begin{remark}
\label{}\hypertarget{}{}
The notion of abelian category is self-dual: [[opposite category|opposite]] of any abelian category is abelian.

\end{remark}
\begin{remark}
\label{RegularEpisAndMonos}\hypertarget{RegularEpisAndMonos}{}
By the second formulation of the definition \ref{AbelianCategory}, in an abelian category

\begin{itemize}%
\item every [[monomorphism]] is a [[regular monomorphism]];


\item every [[epimorphism]] is a [[regular epimorphism]].



\end{itemize}
It follows that every abelian category is a \emph{[[balanced category]]}.

\end{remark}
\hypertarget{FactorizationOfMorphisms}{}\subsubsection*{{Factorization of morphisms}}\label{FactorizationOfMorphisms}

\begin{prop}
\label{}\hypertarget{}{}
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 \emph{[[(epi, mono) factorization system]]}.

\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
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 \href{http://nforum.mathforge.org/discussion/4094/?Focus=33415#Comment_33415}{this discussion}.

\end{remark}
\hypertarget{CanonicalAbEnrichment}{}\subsubsection*{{Canonical $Ab$-enrichment}}\label{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 \href{http://www.math.harvard.edu/~lurie/papers/DAG-I.pdf#page=5}{remark 2.14, p. 5} of [[Jacob Lurie]]`s [[Stable Infinity-Categories]].

\hypertarget{RelationToToposes}{}\subsubsection*{{Relation to exactness properties of toposes}}\label{RelationToToposes}

The [[exactness properties]] of abelian categories have many features in common with exactness properties of [[toposes]] or of [[pretoposes]]. In a fascinating post to the categories mailing list, [[Peter Freyd]] gave a sharp description of the properties shared by these categories, introducing a new concept called \emph{[[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.

\hypertarget{EmbeddingTheorems}{}\subsubsection*{{Embedding theorems}}\label{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 \emph{elements} of the sets underlying the [[objects]].

The following \emph{embedding theorems}, however, show that under good conditions an abelian category can be \emph{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 \emph{[[Freyd-Mitchell embedding theorem]]} discussed \hyperlink{FreydMitchellEmbedding}{below}.

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 \emph{[[element in an abelian category]]}. But under suitable conditions this comes down to working subject to an embedding into $Ab$, see the discussion at \emph{\hyperlink{EmbeddingIntoAb}{Embedding into Ab}} below.

\hypertarget{counterexamples}{}\paragraph*{{Counterexamples}}\label{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.

\hypertarget{EmbeddingIntoAb}{}\paragraph*{{Embedding into $Ab$}}\label{EmbeddingIntoAb}

(\ldots{})

(\hyperlink{Bergman}{Bergman 1974})

(\ldots{})

\hypertarget{FreydMitchellEmbedding}{}\paragraph*{{Freyd-Mitchell embedding into $R Mod$}}\label{FreydMitchellEmbedding}

\begin{thm}
\label{}\hypertarget{}{}
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 $R$.

\end{thm}
\begin{proof}
This result can be found as Theorem 7.34 on page 150 of Peter Freyd's book \href{http://www.emis.de/journals/TAC/reprints/articles/3/tr3.pdf#page=176}{Abelian Categories}. 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 \href{http://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem}{Wikipedia article} for the idea of the proof.

\end{proof}
For more see at \emph{[[Freyd-Mitchell embedding theorem]]}.

We can also characterize which abelian categories \emph{are} equivalent to a category of $R$-modules:

\begin{theorem}
\label{}\hypertarget{}{}
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.

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

\end{proof}
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

\begin{itemize}%
\item rings
\item bimodules
\item bimodule homomorphisms

\end{itemize}
into the strict 2-category of

\begin{itemize}%
\item abelian categories
\item right exact functors
\item natural transformations.

\end{itemize}
For more discussion see the \href{http://golem.ph.utexas.edu/category/2007/08/questions_about_modules.html}{$n$-Cafe}.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item Of course, [[Ab]] is abelian,


\item the [[category of modules]] over any [[ring]] is abelian


\item Therefore in particular the category [[Vect]] of vector spaces is an abelian category


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


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



\end{itemize}
Counter-examples:

\begin{itemize}%
\item 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.

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[additive and abelian categories]]


\item [[abelian subcategory]]


\item [[Deligne tensor product of abelian categories]]


\item [[pseudo-abelian category]]


\item [[quasi-abelian category]]


\item [[length of an object]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Maybe the first reference on abelian categories, then still called \emph{exact categories} is

\begin{itemize}%
\item D. A. Buchsbaum, \emph{Exact categories and duality}, Transactions of the American Mathematical Society Vol. 80, No. 1 (1955), pp. 1-34 (\href{http://www.jstor.org/stable/1993003}{JSTOR})

\end{itemize}
Further foundations of the theory were then laid in

\begin{itemize}%
\item [[Alexander Grothendieck]], \emph{[[Tohoku|Sur quelques points d'algèbre homologique]], T\^o{}hoku Math. J. vol 9, n.2, 3, 1957.}

\end{itemize}
Other classic references, now available online, include:

\begin{itemize}%
\item [[Pierre Gabriel]], \emph{[[Des Catégories Abéliennes]]}


\item [[Peter Freyd]], \emph{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, No. 3, 2003 (\href{http://www.emis.de/journals/TAC/reprints/articles/3/tr3abs.html}{TAC}, \href{http://emis.maths.adelaide.edu.au/journals/TAC/reprints/articles/3/tr3.pdf}{pdf})



\end{itemize}
Textbook accounts include

\begin{itemize}%
\item [[Saunders MacLane]], \emph{[[Categories for the Working Mathematician]]} .


\item N. Popescu, \emph{[[Abelian categories with applications to rings and modules]]}, London Math. Soc. Monographs \textbf{3}, Academic Press 1973. xii+467 pp. \href{http://www.ams.org/mathscinet-getitem?mr=0340375}{MR0340375}


\item [[nLab:Pavel Etingof]], Shlomo Gelaki, Dmitri Nikshych, [[nLab:Victor Ostrik]], chapter 1 of \emph{Tensor categories}, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 (\href{http://www-math.mit.edu/~etingof/egnobookfinal.pdf
}{pdf})



\end{itemize}
Reviews include

\begin{itemize}%
\item Rankey Datta, \emph{An introduction to abelian categories} (2010) (\href{http://www-bcf.usc.edu/~lauda/teaching/rankeya.pdf}{pdf})

\end{itemize}
Embedding of abelian categories into [[Ab]] is discussed in

\begin{itemize}%
\item [[George Bergman]], \emph{A note on abelian categories -- translating element-chasing proofs, and exact embedding in abelian groups} (1974) (\href{http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf}{pdf})

\end{itemize}
For more discussion of the \emph{[[Freyd-Mitchell embedding theorem]]} see there.

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

\begin{itemize}%
\item Rankeya Datta, \emph{The category of modules over a commutative ring and abelian categories} (\href{http://www.math.columbia.edu/~ums/pdf/Rankeya_R-mod_and_Abelian_Categories.pdf}{pdf})

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

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

[[!redirects abelian categories]]



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