In mathematics, an **abelian category** is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abelian groups, **Ab**. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very *stable* categories; for example they are regular and they satisfy the snake lemma. The class 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.

A category is **abelian** if it is * preadditive * and

- it has a zero object,
- it has all binary biproducts,
- it has all kernels and cokernels, and
- all monomorphisms and epimorphisms are normal.

This definition is equivalent^{ [1] } to the following "piecemeal" definition:

- A category is
*preadditive*if it is 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. - A preadditive category is
*additive*if every finite set of objects has a biproduct. This means that we can form finite direct sums and direct products. In^{ [2] }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. 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 categories, forming a very special case of regular categories.

- 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 vector spaces over a fixed field
*k*is abelian, as is the category of finite-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 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 category of all functors from**C**to**A**forms an abelian category. If**C**is small and 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.

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 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 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 and maps , we have , where lim denotes the filtered colimit. - AB6*)
**A**satisfies AB3*), and given a family of cofiltered categories and maps , we have , where lim denotes the cofiltered limit.

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*.

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 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 enriched limits in **A**.

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 (and the nine lemma as a special case).

An abelian category is called **semi-simple** if there is a collection of objects called **simple objects** (meaning the only sub-objects of any are the zero object and itself) such that an object can be decomposed as a direct sum (denoting the coproduct of the abelian category)

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 are not semi-simple; in fact, this is the case if and only if is a semisimple ring.

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

- Category of vector spaces over a fixed field
- By Maschke's theorem the category of representations of a finite group over a field whose characteristic does not divide is a semi-simple abelian category.
- The category of coherent sheaves on a Noetherian scheme is semi-simple if and only if 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 groups vanish, meaning the cohomological dimension is 0. This only happens when the skyscraper sheaves at a point have Zariski tangent space equal to zero, which is isomorphic to using local algebra for such a scheme.
^{ [3] }

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 has the representation

which only has one subrepresentation of dimension . In fact, this is true for any unipotent group ^{ [4] }^{pg 112}.

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 is a short exact sequence in**A**such that two of 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 Serre subcategory if, for all short exact sequences in**A**we have*M*in**C**if and only if both 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 admits a 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 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, the algebra of upper-triangular matrices over *k*, and the category of finite-dimensional -modules. Then each is an abelian category and we have an inclusion functor 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.

Abelian categories were introduced by Buchsbaum (1955) (under the name of "exact category") and Grothendieck (1957) 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 derived functors on abelian categories; the abelian category of sheaves of abelian groups on a topological space, and the abelian category of *G*-modules for a given group *G*.

In mathematics, the **inverse limit** is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can be defined in any category although their existence depends of the category that is considered. They are a special case of the concept of limit in category theory.

In mathematics, a **category** is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions.

In mathematics, specifically in category theory, a **preadditive category** is another name for an **Ab-category**, i.e., a category that is enriched over the category of abelian groups, **Ab**. That is, an **Ab-category****C** is a category such that every hom-set Hom(*A*,*B*) in **C** has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas:

In mathematics, specifically in category theory, an **additive category** is a preadditive category **C** admitting all finitary biproducts.

In mathematics, specifically in category theory, a **pre-abelian category** is an additive category that has all kernels and cokernels.

An **exact sequence** is a sequence of morphisms between objects such that the image of one morphism equals the kernel of the next.

**Homological algebra** is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.

The following outline is provided as an overview of and guide to category theory, the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of *objects* and *arrows*, where these collections satisfy certain basic conditions. Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.

In category theory, a branch of abstract mathematics, an **equivalence of categories** is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.

**Mitchell's embedding theorem**, also known as the **Freyd–Mitchell theorem** or the **full embedding theorem**, is a result about abelian categories; it essentially states that these categories, while rather abstractly defined, are in fact concrete categories of modules. This allows one to use element-wise diagram chasing proofs in these categories. The theorem is named after Barry Mitchell and Peter Freyd.

In mathematics, particularly homological algebra, an **exact functor** is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much of the work in homological algebra is designed to cope with functors that *fail* to be exact, but in ways that can still be controlled.

In mathematics, the category **Ab** has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in **Ab**.

In mathematics, especially in the field of category theory, the concept of **injective object** is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories. The dual notion is that of a projective object.

In mathematics, the category **Grp** has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.

This is a glossary of properties and concepts in category theory in mathematics.

In category theory, a **regular category** is a category with finite limits and coequalizers of a pair of morphisms called **kernel pairs**, satisfying certain *exactness* conditions. In that way, regular categories recapture many properties of abelian categories, like the existence of *images*, without requiring additivity. At the same time, regular categories provide a foundation for the study of a fragment of first-order logic, known as regular logic.

In mathematics, an **exact category** is a concept of category theory due to Daniel Quillen which is designed to encapsulate the properties of short exact sequences in abelian categories without requiring that morphisms actually possess kernels and cokernels, which is necessary for the usual definition of such a sequence.

In mathematics, the **category of rings**, denoted by **Ring**, is the category whose objects are rings and whose morphisms are ring homomorphisms. Like many categories in mathematics, the category of rings is large, meaning that the class of all rings is proper.

In mathematics, a **Grothendieck category** is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957 in order to develop the machinery of homological algebra for modules and for sheaves in a unified manner. The theory of these categories was further developed in Pierre Gabriel's seminal thesis in 1962.

In mathematics, the **Gabriel–Popescu theorem** is an embedding theorem for certain abelian categories, introduced by Pierre Gabriel and Nicolae Popescu (1964). It characterizes certain abelian categories as quotients of module categories.

- ↑ Peter Freyd, Abelian Categories
- ↑ Handbook of categorical algebra, vol. 2, F. Borceux
- ↑ "algebraic geometry - Tangent space in a point and First Ext group".
*Mathematics Stack Exchange*. Retrieved 2020-08-23. - ↑ Humphreys, James E. (2004).
*Linear algebraic groups*. Springer. ISBN 0-387-90108-6. OCLC 77625833.

- Buchsbaum, David A. (1955), "Exact categories and duality",
*Transactions of the American Mathematical Society*,**80**(1): 1–34, doi: 10.1090/S0002-9947-1955-0074407-6 , ISSN 0002-9947, JSTOR 1993003, MR 0074407 - Freyd, Peter (1964),
*Abelian Categories*, New York: Harper and Row - Grothendieck, Alexander (1957), "Sur quelques points d'algèbre homologique",
*Tohoku Mathematical Journal*, Second Series,**9**: 119–221, doi: 10.2748/tmj/1178244839 , ISSN 0040-8735, MR 0102537 - Mitchell, Barry (1965),
*Theory of Categories*, Boston, MA: Academic Press - Popescu, Nicolae (1973),
*Abelian categories with applications to rings and modules*, Boston, MA: Academic Press

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.