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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It follows that every abelian category is a balanced category. Additive categories Homological algebra Niels Henrik Abel. For a Noetherian ring R R the category of finitely freyx R R -modules is an abelian category that lacks these properties. Proposition Every morphism f: Proposition These two conditions are indeed equivalent. This exactness concept has been axiomatized in the theory of exact categoriesforming a very special case of regular categories.

Since by remark every monic is regularhence strongit follows that epimono epi, mono is an orthogonal factorization system in an abelian category; see at epi, mono factorization system. The essential image of I is a full, additive subcategory, but I is not exact. There are numerous types of full, additive subcategories of abelian categories that occur in nature, as well as some conflicting terminology.

Context Enriched category 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 endfrfyd 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.

The concept of exact sequence arises naturally in this setting, and it turns out that exact functorsi. Popescu, Abelian categories with applications to rings and modulesLondon Math.

### Abelian Categories

The notion of abelian category is an abstraction of basic properties of the category Ab of abelian groupsmore generally of the category R R Mod of modules over some ringand still more generally of categories of sheaves of abelian groups and of modules. While additive categories differ significantly from toposesthere is an intimate relation between abelian categories and toposes.

The two were defined differently, but they had similar properties. This is the celebrated Freyd-Mitchell embedding theorem discussed below. All of the constructions used abeliann that field are relevant, such as exact sequences, and especially short exact sequencesand derived functors.

## Abelian category

The concept of abelian categories is one in a sequence of notions of additive and abelian categories. Abelian categories are very stable categories, for example they are regular and they abeloan the snake lemma.

Definition An abelian category is a pre-abelian category satisfying the following equivalent conditions. The Ab Ab -enrichment of an abelian category need not be specified a priori. Monographs 3Academic Press Views Read Edit View history. The following embedding theoremshowever, show that under good conditions an abelian category can be embedded into Ab as a full subcategory by an exact functorand generally can be embedded this way into R Mod R Modfor some ring R R.

## Abelian categories

Note that the enriched structure on hom-sets is a consequence of the first three axioms of the first definition. The class of Abelian categories is closed under several categorical constructions, for example, the category cstegories chain complexes of an Abelian category, or the category of functors from a small category to an Abelian category are Abelian as well.

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

Every abelian category A is a module over the categoriea 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. Therefore in particular the category Vect of vector spaces is an abelian category. This definition is equivalent [1] to the following “piecemeal” definition:. Retrieved from ” https: They are the following:.

See AT category for more abeliam that. Subobjects and quotient objects are well-behaved in abelian categories. This epimorphism is called the coimage of fwhile the monomorphism is called the image of f.

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. For more discussion see the n n -Cafe.

Embedding of abelian categories into Ab is discussed in. In an abelian category every morphism decomposes uniquely up to a unique isomorphism into the composition of an epimorphism and a monomorphismvia prop combined with def.

A similar statement is true for additive categoriesalthough the most natural result in that case gives only enrichment over abelian monoids ; see semiadditive category. The category of sheaves of abelian groups on any site is abelian. Given any pair AB of objects in an abelian category, there is a special zero morphism from A to B.

Let A be an abelian category, C a full, additive subcategory, and I the inclusion functor. In an abelian category, every morphism f can be written as the composition of an epimorphism followed by a monomorphism. Grothendieck unified the two theories: 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.

At the time, there was a cohomology theory for sheavesand a cohomology theory for groups. The motivating prototype example of an abelian category is the category of abelian groupsAb. For the characterization of the tensoring functors see Eilenberg-Watts theorem.