nLab abelian category

\tableofcontents

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, discussed there:

Properties

General

\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.

Factorization of morphisms

Canonical $Ab$-enrichment

The Ab-enrichment of an abelian category need not be specified a priori, this is Prop. \ref{UniquenessOfAbEnrichment} below.

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 commutative monoids., cf. at semiadditive category.

The last point is of relevance in particular for higher categorical generalizations of additive categories. See for instance remark 2.14, p. 5 of Jacob Lurie’s Stable Infinity-Categories.

\begin{proposition} \label{UniquenessOfAbEnrichment} If a (not necessarily pre-additive) category $C$ has

1. a zero object,

2. binary products or binary coproducts,

3. all kernels and cokernels,

4. the property that every monomorphism is a kernel and every epimorphism is a cokernel

   (so that all monos and epis are normal),

then there is a unique Ab-enrichment of $C$ and with it $C$ is abelian.

\end{proposition} (cf. Freyd & Scedrov 1990 §1.598, and Lean formalization in Mathlib.CategoryTheory.Abelian.NonPreadditive) \begin{proof} We give a streamlined version of the argument in Freyd & Scedrov 1990 §1.597-§1.598, which proceeds via the notion of exact categories (where the analog of Step 3 (iii) is immediate). The construction of the subtraction morphism $\sigma_B$ (Step 4) and of the addition from it (Step 6) is that of loc. cit. §1.597 (beware of misprints in the proof on p. 93 there: “$x \colon C \to \mathrm{Cok}(\theta)$” should be “$z \colon \cdots$”, “$y$ is monic” should be “$y$ is epic”, and “forces $y$ to be an isomorphism” should be “forces $\theta$ to be an isomorphism”).

We write $0$ also for the zero morphisms $A \to 0 \to B$; these form a two-sided ideal under composition. For binary products we write $p_1, p_2$ for the projections, $\langle f, g \rangle$ for the pairing of morphisms, $\Delta_B \coloneqq \langle \mathrm{id}_B, \mathrm{id}_B \rangle \colon B \to B \times B$ for the diagonal morphism, and

Step 0 (reduction to binary products). All hypotheses except the second are invariant under passage to the opposite category, while the second turns binary products into binary coproducts and vice versa. The conclusion is invariant as well: $Ab$-enrichments of $C$ are the same as those of $C^{op}$, and $C$ is abelian iff $C^{op}$ is (see below). Hence it suffices to consider the case that $C$ has binary products, which we assume from now on.

Step 1 (normal monos and epis).

(i) If $m$ is the kernel of some morphism $g$, then $m$ is the kernel of its own cokernel $c \coloneqq \mathrm{coker}(m)$: Since $g \circ m = 0$, we have $g = g' \circ c$ for some $g'$, so that $c \circ z = 0$ implies $g \circ z = 0$, whence $z$ factors uniquely through $m$. Dually, every epimorphism, being a cokernel by assumption, is the cokernel of its kernel.

(ii) Every split monomorphism is a monomorphism, hence by assumption a kernel, hence by (i) the kernel of its cokernel. Dually for split epimorphisms. This applies in particular to $\Delta_B$ and $\iota_1$ (split by $p_1$), to $\iota_2$ (split by $p_2$), and to $p_1$, $p_2$ (split by $\iota_1$, $\iota_2$, respectively).

Step 2. The morphism $\iota_1 \colon A \to A \times B$ is a kernel of $p_2$, and $p_2$ is a cokernel of $\iota_1$: First, $p_2 \circ \iota_1 = 0$, and if $p_2 \circ z = 0$ then $z = \langle p_1 \circ z, 0 \rangle = \iota_1 \circ p_1 \circ z$, uniquely so since $\iota_1$ is mono. Hence $\iota_1 = \mathrm{ker}(p_2)$, and so $p_2 = \mathrm{coker}(\iota_1)$ by Step 1. Symmetrically, $\iota_2 = \mathrm{ker}(p_1)$ and $p_1 = \mathrm{coker}(\iota_2)$.

Step 3 (equalizers, and a criterion for isomorphisms).

(i) $C$ has equalizers: For $f, g \colon A \to B$ let $e \coloneqq \mathrm{ker}\big( \mathrm{coker}(\Delta_B) \circ \langle f, g \rangle \big)$. Since $\Delta_B = \mathrm{ker}\big(\mathrm{coker}(\Delta_B)\big)$ by Step 1, a morphism $z$ into $A$ factors (necessarily uniquely) through $e$ iff $\langle f \circ z, g \circ z \rangle$ factors through $\Delta_B$, hence iff $f \circ z = g \circ z$. So $e$ is an equalizer of $f$ and $g$.

(ii) If $f \colon A \to B$ has vanishing kernel, in that $f \circ z = 0$ implies $z = 0$, then $f$ is a monomorphism: Let $k \colon P \to A \times A$ be the equalizer of $f \circ p_1$ and $f \circ p_2$ (the kernel pair of $f$), and write $q_i \coloneqq p_i \circ k$. Since $f \circ p_1 \circ \Delta_A = f \circ p_2 \circ \Delta_A$, there is $s \colon A \to P$ with $k \circ s = \Delta_A$, hence with $q_1 \circ s = \mathrm{id}_A = q_2 \circ s$. Now if $q_1 \circ z = 0$, then $f \circ q_2 \circ z = f \circ q_1 \circ z = 0$, hence $q_2 \circ z = 0$, hence $k \circ z = \langle q_1 \circ z, q_2 \circ z \rangle = 0$, hence $z = 0$, since $k$ is mono. Thus the kernel of the split epimorphism $q_1$ is $0 \to P$, so that, by Step 1, $q_1$ is a cokernel of $0 \to P$, hence an isomorphism, with inverse $s$. Therefore $q_2 = q_2 \circ s \circ q_1 = q_1$. Finally, if $f \circ x = f \circ y$ then $\langle x, y \rangle = k \circ u$ for some $u$, and so $x = q_1 \circ u = q_2 \circ u = y$.

(iii) If $f \colon A \to B$ has vanishing kernel and vanishing cokernel (in that also $t \circ f = 0$ implies $t = 0$), then $f$ is an isomorphism: By (ii), $f$ is mono, hence by Step 1 a kernel of $\mathrm{coker}(f)$. But $\mathrm{coker}(f) = 0$, and the kernel of a zero morphism out of $B$ is $\mathrm{id}_B$. Since kernels are unique up to unique isomorphism, $f$ is an isomorphism.

Step 4 (the subtraction morphism). For any object $B$, let $q \colon B \times B \to Q$ be a cokernel of $\Delta_B$, and consider

• $r$ has vanishing kernel: If $r \circ z = 0$, then $\iota_1 \circ z$ factors through $\Delta_B = \mathrm{ker}(q)$ (by Step 1), say as $\iota_1 \circ z = \Delta_B \circ w$. Applying $p_2$ to this equation gives $0 = w$, and then applying $p_1$ gives $z = w = 0$.

• $r$ has vanishing cokernel: If $t \circ r = 0$, then $t \circ q$ vanishes on $\iota_1$ and hence factors through $p_2 = \mathrm{coker}(\iota_1)$ (by Step 2), say as $t \circ q = s \circ p_2$. Then $s = s \circ p_2 \circ \Delta_B = t \circ q \circ \Delta_B = 0$, so that $t \circ q = 0$, and hence $t = 0$, since $q$ is epi.

By Step 3 (iii), $r$ is an isomorphism, and we set

This is a cokernel of $\Delta_B$ which satisfies

(For $C =$ Ab this is the map $(b_1, b_2) \mapsto b_1 - b_2$.) Moreover, $\sigma$ is natural: For $f \colon A \to B$ we have $\sigma_B \circ (f \times f) = f \circ \sigma_A$. Namely, $\sigma_B \circ (f \times f) \circ \Delta_A = \sigma_B \circ \Delta_B \circ f = 0$, so that $\sigma_B \circ (f \times f) = f' \circ \sigma_A$ for some $f'$, and then

Step 5 (properties of subtraction). For $x, y \colon A \to B$ define

(the notation is justified by Step 6). Then, for all composable morphisms $x, y, z, w$:

1. $x - 0 = x$, since $\langle x, 0 \rangle = \iota_1 \circ x$;

2. $x - x = 0$, since $\langle x, x \rangle = \Delta_B \circ x$;

3. $z \circ (x - y) = z \circ x - z \circ y$, by naturality of $\sigma$ (Step 4) and since $(z \times z) \circ \langle x, y \rangle = \langle z \circ x, z \circ y \rangle$;

4. $(x - y) \circ w = x \circ w - y \circ w$, since $\langle x, y \rangle \circ w = \langle x \circ w, y \circ w \rangle$.

From this follows the middle-two interchange law: For $u, v, x, y \colon A \to B$ we have

To see this, observe first that for $a, b \colon A \to D$ and $c, d \colon D \to B$, items 3 and 4 give

and then specialize to $D = B \times B$, $a = \langle u, v \rangle$, $b = \langle x, y \rangle$, $c = p_1$, $d = p_2$.

Step 6 (the abelian group structure). Define

Then $x + 0 = x - 0 = x$, by items 2 and 1 of Step 5, and

by items 1, 2 and the interchange law. Applying the interchange law three times gives

Setting here $u = y = 0$ shows that $+$ is commutative, and setting $v = 0$ shows that it is associative (this is the Eckmann-Hilton argument). Moreover, $(0 - x) + x = (0 - x) - (0 - x) = 0$, so that every $x$ has the inverse $-x \coloneqq 0 - x$ (and $x + (-y) = x - y$, by the formula for $0 + x$ above). Hence each hom-set $C(A,B)$ is an abelian group, whose neutral element is the zero morphism. Finally, composition is bilinear, since by items 3 and 4 of Step 5

and similarly $(x + y) \circ w = x \circ w + y \circ w$. This is an $Ab$-enrichment of $C$.

Step 7 (abelianness). With respect to this enrichment, $\iota_1 \circ p_1 + \iota_2 \circ p_2 = \mathrm{id}_{A \times B}$, as seen by composing with $p_1$ and with $p_2$. Together with $p_1 \circ \iota_1 = \mathrm{id}_A$, $p_2 \circ \iota_2 = \mathrm{id}_B$ and $p_2 \circ \iota_1 = 0$, $p_1 \circ \iota_2 = 0$, this means that binary products are biproducts, so that $C$ is an additive category, hence (having kernels and cokernels) a pre-abelian category in which, by assumption, every monomorphism is a kernel and every epimorphism is a cokernel. Therefore $C$ is abelian, by the second condition in the definition above.

Step 8 (uniqueness). Let $+'$ be any $Ab$-enrichment of $C$. Its neutral elements are the zero morphisms, because $C(A,0)$ is a singleton, so that $A \to 0$ is neutral, and composition is bilinear. Consider

By bilinearity of composition we have, for $x, y \colon A \to B$,

as well as $\nabla' \circ \iota_1 = \mathrm{id}_B +' 0 = \mathrm{id}_B$ and similarly $\nabla' \circ \iota_2 = \mathrm{id}_B$. Hence, using Step 7,

and therefore $x +' y = x + y$. \end{proof}

Relation to exactness properties of toposes

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

Freyd (1999) 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

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.

This implies for instance that proofs about exactness of sequences in an abelian category can always be obtained by a naive argument on elements – called a “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 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 $R$-modules is an abelian category that lacks these properties.

Embedding into $Ab$

(…)

(Bergman 1974)

(…)

Freyd-Mitchell embedding into $R Mod$

For more see at Freyd-Mitchell embedding theorem.

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

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.

Examples

• Of course, Ab is abelian,

• the 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)

• The category of sheaves of abelian groups on any site is abelian.

Counter-examples:

• The category of 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

• Grothendieck category

• pseudo-abelian category

• quasi-abelian category

• semi-abelian category

• length of an object

References

Possibly the first reference on abelian categories, then still called exact categories:

• D. A. Buchsbaum: Exact categories and duality, Transactions of the American Mathematical Society 80 1 (1955) 1–34 [[jstor:1993003](http://www.jstor.org/stable/1993003)]

Further foundations of the theory were then laid in

• Alexander Grothendieck: Sur quelques points d'algèbre homologique, Tôhoku Math. J. 9 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)]

• Peter Freyd: Abelian Categories – An Introduction to the theory of functors, 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]

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)

• Peter Freyd, Andre Scedrov: Categories, chapter one of: Categories, Allegories, Mathematical Library 39, North-Holland (1990) [[ISBN 978-0-444-70368-2](https://www.elsevier.com/books/categories-allegories/freyd/978-0-444-70368-2), math-library:vol/39/suppl/C, ark:/13960/s2mnkcjwggx]

• 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]

• 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]

Further review:

• Rankey Datta, An introduction to abelian categories (2010) (pdf)

• Paul Selick, Homological Algebra Notes (pdf,pdf)

On diagram chasing and elements in an abelian category:

• 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), pdf]

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

A proof that $R Mod$ is an abelian category:

• Rankeya Datta, The category of modules over a commutative ring and abelian categories (pdf)

A 2-monadic characterization of abelian categories:

• Martin Markl, Dominik Trnka: Lax Distributivity and a Characterization of Abelian Categories [[arXiv:2607.25740](https://arxiv.org/abs/2607.25740)]

A discussion about to which extent abelian categories are a general context for homological algebra is archived at nForum here.

See also the catlist 1999 discussion 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)]

Generalization of abelian categories to (n,1)-categories:

• Nao Mochizuki, Higher-dimensional generalization of abelian categories via DG Categories [[arXiv:2501.06955](https://arxiv.org/abs/2501.06955)]