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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, an abelian category is a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
in which
morphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ... s and
objects Object may refer to: General meanings * Object (philosophy) An object is a philosophy, philosophical term often used in contrast to the term ''Subject (philosophy), subject''. A subject is an observer and an object is a thing observed. For mo ...
can be added and in which
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
s and
cokernel The cokernel of a linear mapping In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ... s exist and have desirable properties. The motivating prototypical example of an abelian category is the
category of abelian groups In mathematics, the category theory, category Ab has the abelian groups as object (category theory), objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every Small category, small abelian category can ...
, Ab. The theory originated in an effort to unify several
cohomology theories In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of as ...
by
Alexander Grothendieck Alexander Grothendieck (; ; ; 28 March 1928 – 13 November 2014) was a mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative ... and independently in the slightly earlier work of
David Buchsbaum David Alvin Buchsbaum (November 6, 1929 – January 8, 2021) was a mathematician at Brandeis University , mottoeng = Truth even unto its innermost parts , established = , type = Private research university , president = Ronald D. Liebowitz ...
. Abelian categories are very ''stable'' categories; for example they are
regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * Regular (Badfinger song), "Regular" (Badfinger song) * Regular tunin ...
and they satisfy the
snake lemma The snake lemma is a tool used in mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their cha ...
. The
class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differently f ...
of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors 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. Abelian categories are named after Niels Henrik Abel.

# Definitions

A category is abelian if it is ''Preadditive category, preadditive'' and *it has a zero object, *it has all binary biproducts, *it has all Kernel (category theory), kernels and
cokernel The cokernel of a linear mapping In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ... s, and *all monomorphisms and epimorphisms are Normal morphism, normal. This definition is equivalent to the following "piecemeal" definition: * A category is ''Preadditive category, preadditive'' if it is enriched category, enriched over the monoidal category Ab of abelian groups. This means that all hom-sets are abelian groups and the composition of morphisms is bilinear operator, bilinear. * A preadditive category is ''Additive category, additive'' if every finite set of objects has a biproduct. This means that we can form finite direct sum of modules, direct sums and direct products. In Def. 1.2.6, it is required that an additive category has a zero object (empty biproduct). * An additive category is ''preabelian category, preabelian'' if every morphism has both a kernel (category theory), kernel and a
cokernel The cokernel of a linear mapping In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ... . * Finally, a preabelian category is abelian if every monomorphism and every epimorphism is normal morphism, normal. 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 hom-sets is a ''consequence'' of the first three axioms of the first definition. This highlights the foundational relevance of the category of Abelian groups in the theory and its canonical nature. The concept of exact sequence arises naturally in this setting, and it turns out that exact functors, 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, exact categories, forming a very special case of regular category, regular categories.

# 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 (mathematics), ring, then the category of all left (or right) module (mathematics), 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 module, 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 vector spaces over a fixed field (mathematics), field ''k'' is abelian, as is the category of finite-Hamel dimension, dimensional 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 sheaf (mathematics), sheaves of abelian groups on ''X'' is an abelian category. More generally, the category of sheaves of abelian groups on a Grothendieck topology, Grothendieck site is an abelian category. In this way, abelian categories show up in algebraic topology and algebraic geometry. * If C is a Category (mathematics)#Small and large categories, small category and A is an abelian category, then the functor category, category of all functors from C to A forms an abelian category. If C is small and preadditive category, preadditive, then the category of all additive functors 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.

# 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 filtered colimits of exact sequences are exact. and their duals * AB3*) For every indexed family (''A''''i'') of objects of A, the Product (category theory), product P''A''''i'' exists in A (i.e. A is Complete category, complete). * AB4*) A satisfies AB3*), and the product of a family of epimorphisms is an epimorphism. * AB5*) A satisfies AB3*), and filtered limits 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 $I_j, j\in J$ and maps $A_j : I_j \to A$, we have $\prod_ \lim_ A_j = \lim_ \prod_ A_j$, where lim denotes the filtered colimit. * AB6*) A satisfies AB3*), and given a family of cofiltered categories $I_j, j\in J$ and maps $A_j : I_j \to A$, we have $\sum_ \lim_ A_j = \lim_ \sum_ A_j$, where lim denotes the cofiltered limit.

# 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 0 (number), 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 (category theory), image'' of ''f''. Subobjects and quotient objects 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 (mathematics), 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 category, complete, then we can remove the requirement that ''G'' be finitely generated; most generally, we can form finitary enriched limits in A.

# 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 derived functors. 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 The snake lemma is a tool used in mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their cha ...
(and the nine lemma as a special case).

## Semi-simple Abelian categories

An abelian category $\mathbf$ is called semi-simple if there is a collection of objects $\_ \in \text\left(\mathbf\right)$ called simple objects (meaning the only sub-objects of any $X_i$ are the zero object $0$ and itself) such that an object $X \in \text\left(\mathbf\right)$ can be decomposed as a direct sum (denoting the coproduct of the abelian category)
$X \cong \bigoplus_ X_i$
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.

### Examples

Some Abelian categories found in nature are semi-simple, such as * Category of vector spaces $\text\left(k\right)$ over a fixed field $k$ * By Maschke's theorem the category of representations $\text_k\left(G\right)$ 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, coherent sheaves on a Noetherian scheme, Noetherian Scheme (mathematics), 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 $\text^1$ groups vanish, meaning the cohomological dimension is 0. This only happens when the skyscraper sheaves $k_x$ at a point $x \in X$ have Zariski tangent space equal to zero, which is isomorphic to $\text^1\left(k_x,k_x\right)$ using local algebra for such a scheme.

### Non-examples

There do exist some natural counter-examples of abelian categories which are not semi-simple, such as certain categories of Representation theory, representations. For example, the category of representations of the Lie group $\left(\mathbb,+\right)$ has the representation
$a \mapsto \begin 1 & a \\ 0 & 1 \end$
which only has one subrepresentation of dimension $1$. In fact, this is true for any unipotent grouppg 112.

# 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 Pullback (category theory), pullbacks of epimorphisms and Pushout (category theory), 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 $0 \to M\text{'} \to M \to M\text{'}\text{'} \to 0$ is a short exact sequence in A such that two of $M\text{'},M,M\text{'}\text{'}$ 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 subquotients. * C is a localizing subcategory, Serre subcategory if, for all short exact sequences $0 \to M\text{'} \to M \to M\text{'}\text{'} \to 0$ in A we have ''M'' in C if and only if both $M\text{'},M\text{'}\text{'}$ are in C. In other words, C is closed under extensions and subquotients. 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 $Q\colon\mathbf A \to \mathbf A/\mathbf C$ admits a Adjoint functors, right adjoint. * 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 subcategory#Types of subcategories, 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 $n\times n$ matrices over ''k'', and $\mathbf A_n$ the category of finite-dimensional $T_n$-modules. Then each $\mathbf A_n$ is an abelian category and we have an inclusion functor $I\colon\mathbf A_2 \to \mathbf A_3$ 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.

# 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 sheaf (mathematics), sheaves, and a cohomology theory for Group (mathematics), 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 derived functors on abelian categories; the abelian category of sheaves of abelian groups on a topological space, and the abelian category of G-module, ''G''-modules for a given group ''G''.