Abelian category

 ( Additive Categories ) 

In mathematics, an abelian category is a category in which and objects can be added and in which kernels and exist and have desirable properties.

The motivating prototypical example of an abelian category is the category of abelian groups, .

Abelian categories are very stable categories; for example they are regular category and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of of an abelian category, or the category of from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory.

Mac Lane

says Alexander Grothendieck defined abelian categories in 1957, but there is a reference that says Samuel Eilenberg's disciple, David Buchsbaum, had proposed the concept in his 1955 PhD thesis, and Grothendieck popularized it under the name "abelian category".

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Definitions A category is abelian if it is preadditive and

• it has a zero object,
• it has all binary ,
• it has all kernels and , and
• all and are Normal morphism.

This definition is equivalentPeter Freyd, Abelian Categories to the following "piecemeal" definition:

• A category is preadditive if it is enriched over the monoidal category of . This means that all are abelian groups and the composition of morphisms is bilinear.
• A preadditive category is additive if every finite set of objects has a biproduct. This means that we can form finite direct sums and . In Handbook of categorical algebra, vol. 2, F. Borceux Def. 1.2.6, it is required that an additive category have a zero object (empty biproduct).
• An additive category is preabelian if every morphism has both a kernel and a cokernel.
• Finally, a preabelian category is abelian if every monomorphism and every epimorphism is normal morphism. This means that every monomorphism is a kernel of some morphism, and every epimorphism is a cokernel of some morphism.

Note that the enriched structure on is a consequence of the first three of the first definition. This highlights the foundational relevance of the category of in the theory and its canonical nature.

The concept of exact sequence arises naturally in this setting, and it turns out that , i.e. the functors preserving exact sequences in various senses, are the relevant functors between abelian categories. This exactness concept has been axiomatized in the theory of Exact category, forming a very special case of regular category.

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Examples

• As mentioned above, the category of all abelian groups is an abelian category. The category of all finitely generated abelian groups is also an abelian category, as is the category of all finite abelian groups.
• If R is a ring, then the category of all left (or right) modules over R is an abelian category. In fact, it can be shown that any small abelian category is equivalent to a full subcategory of such a category of modules ( Mitchell's embedding theorem).
• If R is a left-noetherian ring, then the category of finitely generated left modules over R is abelian. In particular, the category of finitely generated modules over a noetherian commutative ring is abelian; in this way, abelian categories show up in commutative algebra.
• As special cases of the two previous examples: the category of over a fixed field k is abelian, as is the category of finite-Hamel dimension vector spaces over k.
• If X is a topological space, then the category of all (real or complex) vector bundles on X is not usually an abelian category, as there can be monomorphisms that are not kernels.
• If X is a topological space, then the category of all sheaves of abelian groups on X is an abelian category. More generally, the category of sheaves of abelian groups on a Grothendieck site is an abelian category. In this way, abelian categories show up in algebraic topology and algebraic geometry.
• If C is a small category and A is an abelian category, then the functor category from C to A forms an abelian category. If C is small and preadditive, then the category of all from C to A also forms an abelian category. The latter is a generalization of the R-module example, since a ring can be understood as a preadditive category with a single object.

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Grothendieck's axioms In his Tōhoku article, Grothendieck listed four additional axioms (and their duals) that an abelian category A might satisfy. These axioms are still in common use to this day. They are the following:

• AB3) For every indexed family ( A _i) of objects of A, the coproduct * A_i exists in A (i.e. A is cocomplete).
• AB4) A satisfies AB3), and the coproduct of a family of monomorphisms is a monomorphism.
• AB5) A satisfies AB3), and of are exact.

and their duals

• AB3*) For every indexed family ( A _i) of objects of A, the product P A _i exists in A (i.e. A is complete).
• AB4*) A satisfies AB3*), and the product of a family of epimorphisms is an epimorphism.
• AB5*) A satisfies AB3*), and of exact sequences are exact.

Axioms AB1) and AB2) were also given. They are what make an additive category abelian. Specifically:

• AB1) Every morphism has a kernel and a cokernel.
• AB2) For every morphism f, the canonical morphism from coim f to im f is an isomorphism.

Grothendieck also gave axioms AB6) and AB6*).

• AB6) A satisfies AB3), and given a family of filtered categories $I\_j,\; j\backslash in\; J$ and maps $A\_j\; :\; I\_j\; \backslash to\; A$, we have $\backslash prod\_\{j\backslash in\; J\}\; \backslash lim\_\{I\_j\}\; A\_j\; =\; \backslash lim\_\{I\_j,\; \backslash forall\; j\backslash in\; J\}\; \backslash prod\_\{j\backslash in\; J\}\; A\_j$, where lim denotes the filtered colimit.
• AB6*) A satisfies AB3*), and given a family of cofiltered categories $I\_j,\; j\backslash in\; J$ and maps $A\_j\; :\; I\_j\; \backslash to\; A$, we have $\backslash sum\_\{j\backslash in\; J\}\; \backslash lim\_\{I\_j\}\; A\_j\; =\; \backslash lim\_\{I\_j,\; \backslash forall\; j\backslash in\; J\}\; \backslash sum\_\{j\backslash in\; J\}\; A\_j$, where lim denotes the cofiltered limit.

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Elementary properties Given any pair A, B of objects in an abelian category, there is a special zero morphism from A to B. This can be defined as the zero element of the hom-set Hom( A, B), since this is an abelian group. Alternatively, it can be defined as the unique composition A → 0 → B, where 0 is the zero object of the abelian category.

In an abelian category, every morphism f can be written as the composition of an epimorphism followed by a monomorphism. This epimorphism is called the coimage of f, while the monomorphism is called the image of f.

and are well-behaved in abelian categories. For example, the poset of subobjects of any given object A is a bounded lattice.

Every abelian category A is a module over the monoidal category of finitely generated abelian groups; that is, we can form a tensor product of a finitely generated abelian group G and any object A of A. The abelian category is also a comodule; Hom( G, A) can be interpreted as an object of A. If A is complete, then we can remove the requirement that G be finitely generated; most generally, we can form finitary in A.

Given an object $A$ in an abelian category, flatness refers to the idea that $-\; \backslash otimes\; A$ is an exact functor. See flat module or, for more generality, flat morphism.

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Related concepts Abelian categories are the most general setting for homological algebra. All of the constructions used in that field are relevant, such as exact sequences, and especially short exact sequences, and . Important theorems that apply in all abelian categories include the five lemma (and the short five lemma as a special case), as well as the snake lemma (and the nine lemma as a special case).

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Semi-simple Abelian categories An abelian category $\backslash mathbf\{A\}$ is called semi-simple if there is a collection of objects $\backslash \{X\_i\backslash \}\_\{i\; \backslash in\; I\}\; \backslash in\; \backslash text\{Ob\}(\backslash mathbf\{A\})$ called simple objects (meaning the only sub-objects of any $X\_i$ are the zero object $0$ and itself) such that an object $X\; \backslash in\; \backslash text\{Ob\}(\backslash mathbf\{A\})$ can be decomposed as a direct sum (denoting the coproduct of the abelian category)

$X\; \backslash cong\; \backslash bigoplus\_\{i\; \backslash in\; I\}\; X\_i$

This technical condition is rather strong and excludes many natural examples of abelian categories found in nature. For example, most module categories over a ring $R$ are not semi-simple; in fact, this is the case if and only if $R$ is a semisimple ring.

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Examples Some abelian categories found in nature are semi-simple, such as

• The category of $\backslash text\{Vect\}(k)$ over a fixed field $k$.
• By Maschke's theorem the category of representations $\backslash text\{Rep\}\_k(G)$ of a finite group $G$ over a field $k$ whose characteristic does not divide $|G|$ is a semi-simple abelian category.
• The category of Coherent sheaf on a Noetherian scheme is semi-simple if and only if $X$ is a finite disjoint union of irreducible points. This is equivalent to a finite coproduct of categories of vector spaces over different fields. Showing this is true in the forward direction is equivalent to showing all $\backslash text\{Ext\}^1$ groups vanish, meaning the cohomological dimension is 0. This only happens when the skyscraper sheaves $k\_x$ at a point $x\; \backslash in\; X$ have Zariski tangent space equal to zero, which is isomorphic to $\backslash text\{Ext\}^1(k\_x,k\_x)$ using local algebra for such a scheme.

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Non-examples There do exist some natural counter-examples of abelian categories which are not semi-simple, such as certain categories of representations. For example, the category of representations of the Lie group $(\backslash mathbb\{R\},+)$ has the representation

which only has one subrepresentation of dimension $1$. In fact, this is true for any unipotent group

Humphreys, James E. (2025). 9780387901084, Springer. . ISBN 9780387901084

^{pg 112}.

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Subcategories of abelian categories There are numerous types of (full, additive) subcategories of abelian categories that occur in nature, as well as some conflicting terminology.

Let A be an abelian category, C a full, additive subcategory, and I the inclusion functor.

• C is an exact subcategory if it is itself an exact category and the inclusion I is an exact functor. This occurs if and only if C is closed under pullbacks of epimorphisms and pushouts of monomorphisms. The exact sequences in C are thus the exact sequences in A for which all objects lie in C.
• C is an abelian subcategory if it is itself an abelian category and the inclusion I is an exact functor. This occurs if and only if C is closed under taking kernels and cokernels. Note that there are examples of full subcategories of an abelian category that are themselves abelian but where the inclusion functor is not exact, so they are not abelian subcategories (see below).
• C is a thick subcategory if it is closed under taking direct summands and satisfies the 2-out-of-3 property on short exact sequences; that is, if $0\; \backslash to\; M\text{'}\; \backslash to\; M\; \backslash to\; M$ \to 0 is a short exact sequence in A such that two of $M\text{'},M,M$ lie in C, then so does the third. In other words, C is closed under kernels of epimorphisms, cokernels of monomorphisms, and extensions. Note that P. Gabriel used the term thick subcategory to describe what we here call a Serre subcategory.
• C is a topologizing subcategory if it is closed under .
• C is a Serre subcategory if, for all short exact sequences $0\; \backslash to\; M\text{'}\; \backslash to\; M\; \backslash to\; M$ \to 0 in A we have M in C if and only if both $M\text{'},M$ are in C. In other words, C is closed under extensions and . These subcategories are precisely the kernels of exact functors from A to another abelian category.
• C is a localizing subcategory if it is a Serre subcategory such that the quotient functor $Q\backslash colon\backslash mathbf\; A\; \backslash to\; \backslash mathbf\; A/\backslash mathbf\; C$ admits a Adjoint functors.
• There are two competing notions of a wide subcategory. One version is that C contains every object of A (up to isomorphism); for a full subcategory this is obviously not interesting. (This is also called a lluf subcategory.) The other version is that C is closed under extensions.

Here is an explicit example of a full, additive subcategory of an abelian category that is itself abelian but the inclusion functor is not exact. Let k be a field, $T\_n$ the algebra of upper-triangular $n\backslash times\; n$ matrices over k, and $\backslash mathbf\; A\_n$ the category of finite-dimensional $T\_n$-modules. Then each $\backslash mathbf\; A\_n$ is an abelian category and we have an inclusion functor $I\backslash colon\backslash mathbf\; A\_2\; \backslash to\; \backslash mathbf\; A\_3$ identifying the simple projective, simple injective and indecomposable projective-injective modules. The essential image of I is a full, additive subcategory, but I is not exact.

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History Abelian categories were introduced by (under the name of "exact category") and in order to unify various cohomology theories. At the time, there was a cohomology theory for sheaves, and a cohomology theory for groups. The two were defined differently, but they had similar properties. In fact, much of category theory was developed as a language to study these similarities. Grothendieck unified the two theories: they both arise as on abelian categories; the abelian category of sheaves of abelian groups on a topological space, and the abelian category of G-module for a given group G.

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See also

• Triangulated category

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