Abelian categories are the most general category in which one can The idea and the name “abelian category” were first introduced by. In mathematics, an abelian category is a category in which morphisms and objects can be .. Peter Freyd, Abelian Categories; ^ Handbook of categorical algebra, vol. 2, F. Borceux. Buchsbaum, D. A. (), “Exact categories and duality”. BOOK REVIEWS. Abelian categories. An introduction to the theory of functors. By Peter. Freyd. (Harper’s Series in Modern Mathematics.) Harper & Row.

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Theorem Let C C be an abelian category. This result can be found as Theorem 7. This exactness concept has been axiomatized categkries the theory of exact categoriesforming a very special case of regular categories.

A discussion about to which extent abelian categories are a general context for homological algebra is archived at nForum here. Therefore in particular the category Vect of vector spaces is an abelian category. In an abelian category every morphism decomposes uniquely up to a categorjes isomorphism into the composition of an epimorphism and a monomorphismvia prop combined with def.

Abelian categories

This can be defined as the zero element of the hom-set Hom ABsince this is an abelian group. For the characterization of the tensoring functors see Eilenberg-Watts theorem. If an arbitrary not necessarily pre-additive locally small category C C has a zero objectbinary 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 normalthen it can be equipped with a unique addition on the morphism sets such that composition is bilinear and C C is abelian with respect to this structure.


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. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometrycohomology and pure category theory. The essential image of I is a full, additive subcategory, but I is not exact.

An abelian category is a pre-abelian category satisfying the following equivalent conditions.

Abelian Categories

The concept of abelian categories is one in a sequence of notions of additive and abelian categories. In fact, much of category theory was developed as a language to study these similarities. It follows that every abelian category is a balanced category.

They are the following:. Abelian categories are very stable categories, for example they are regular and they satisfy the snake lemma. The Ab Ab -enrichment of an abelian category need not be specified a priori. Every small abelian category admits a fullfaithful and exact functor to the category R Mod R Mod for some ring R Xbelian.

Remark The notion of abelian category is self-dual: In an abelian category, every morphism f can be written as the composition of an epimorphism followed by a monomorphism. All of the constructions used in that field are relevant, such as exact sequences, categoried especially short exact sequencesand derived functors.

See for instance remark 2. This is the celebrated Freyd-Mitchell embedding theorem discussed below. Context Enriched ctegories theory enriched category theory Background category theory monoidal categoryclosed monoidal category cosmosmulticategorybicategorydouble categoryvirtual double category Basic concepts enriched category enriched functorprofunctor enriched functor category Universal constructions weighted limit endcoend Extra stuff, structure, property copower ing tensoringpower ing cotensoring Homotopical enrichment enriched homotopical category enriched model category model structure on homotopical presheaves Edit this sidebar.


At the time, there was a cohomology theory for sheavesand a cohomology theory for groups. The proof that R Mod R Mod is an abelian categoriies is spelled out for instance in.

Abelian category – Wikipedia

Remark 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 this discussion. Proposition In an abelian category every morphism decomposes uniquely up to a unique isomorphism into the composition of an epimorphism and a monomorphismfreys prop combined with def.

The two were defined differently, but they had similar properties. This definition is equivalent [1] to the following “piecemeal” definition:. Abelizn categories Homological algebra Niels Henrik Abel.

Proof This result can be found as Theorem 7. The notion of abelian category is self-dual: A similar statement is true for additive categoriesalthough the most natural result in that case gives only enrichment over abelian monoids ; see semiadditive category.

In mathematicsan abelian category is a categoires in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties.

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