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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The Ab Ab -enrichment of an abelian category need not be specified a priori. This definition is equivalent [1] to the following “piecemeal” definition:. Here is an explicit example of a full, additive ableian of an abelian category which is itself abelian but the inclusion functor is not exact.

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

## Abelian categories

See for instance remark 2. This page was last edited on 19 Marchat Abelian categories are very stable categories, for example they are regular and they satisfy the snake lemma. Let C C be an abelian category. There are numerous types of full, additive subcategories of abelian categories that occur in nature, as well as some conflicting terminology.

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.

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. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum.

In an abelian category, every morphism f can be written as the composition of an epimorphism followed by a monomorphism. Axioms AB1 and AB2 were also given. Given any pair AB of objects in an abelian category, there is a special zero morphism from A to B.

These axioms are still in common use to this day. Grothendieck unified the two theories: This epimorphism is called the coimage of fwhile the monomorphism is called the image of f. See AT category for more on that. They are the following:.

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

### Abelian category – Wikipedia

Cateogries The notion of abelian category is self-dual: 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.

For more see at Freyd-Mitchell embedding theorem. Subobjects and quotient objects are well-behaved in abelian categories. The notion of abelian category is self-dual: Abelian categories are the most general setting for homological algebra. Proposition These two conditions are indeed equivalent.

By cztegories this site, you agree to the Terms of Use and Privacy Policy. Note that the enriched structure on hom-sets is a consequence of the first three axioms of the first definition.

The concept of abelian categories is one in a sequence of notions abdlian additive and abelian categories.

### Abelian Categories

Proposition Every morphism f: Monographs 3Academic Press Abelian categories were introduced by Buchsbaum under the name of “exact category” and Grothendieck in order to unify various cohomology theories. By the second formulation of the definitionin an abelian category. It is such that much of the homological algebra of chain complexes can be developed inside every abelian category.

From Wikipedia, the free encyclopedia. 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 last point is of relevance in particular for higher categorical generalizations of additive categories. Proof This result categorjes be found as Theorem 7. Views Read Edit View history.

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

These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometrycohomology and pure category theory. The reason is that R Mod R Mod has all small category limits abelisn colimits. This can be defined as the zero element of the hom-set Hom ABsince this is an abelian group.

This result can be found as Theorem 7.

We can also characterize which abelian categories are equivalent to a category of R R -modules:. Every monomorphism is a kernel and every epimorphism is a cokernel.