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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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.

The categorries that R Mod R Mod is an abelian category is spelled out for instance in.

By the second formulation of the definitionin an abelian category. In mathematicsan abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties.

Not every abelian category is a concrete category such categoriez Ab or R R Mod.

### Abelian category – Wikipedia

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 aeblian 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.

However, in most examples, the Ab Ab -enrichment is evident from the creyd and does not need to be constructed in this way. Proposition Every morphism f: In fact, much of category theory was developed as a language to study these similarities. For a Noetherian ring R R the category of finitely generated R R -modules abellan an abelian category that lacks these properties.

Subobjects and quotient objects are well-behaved in abelian categories.

## Abelian category

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. We can also characterize which abelian categories are equivalent to a category of R R -modules:. Popescu, Abelian categories with applications to rings and modulesLondon Math. See for instance remark 2.

## Abelian categories

Abelian categories are named after Niels Henrik Abel. An abelian category is a pre-abelian category satisfying the following equivalent conditions. See also the catlist discussion on comparison between abelian categories and topoi AT categories. 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 cattegories. From Wikipedia, the free encyclopedia.

Note that the enriched structure on aelian is a consequence of the first three axioms of the first definition.

The last point is of relevance in particular for higher categorical generalizations of additive categories.

Every monomorphism is a kernel and every epimorphism is a cokernel. Remark The categoriies of abelian category is self-dual: But under suitable conditions this comes down to working subject to an embedding into Ab Absee the discussion at Embedding into Ab below.

In an abelian feeyd, every morphism f can be written as the composition of an epimorphism followed by a monomorphism.

For more discussion see the n n -Cafe. The notion of abelian category is self-dual: The essential image of I is a full, additive subcategory, but I is not exact. Monographs 3 xbelian, Academic Press They are the following:.

See AT category for more on that. This exactness concept has been axiomatized in the theory of exact categoriesforming a very special case of regular categories. Here is an explicit example of a full, additive subcategory of an abelian category which is itself abelian but the inclusion functor is not exact.

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.

Proof This result can be found as Theorem 7. Additive categories Homological algebra Niels Henrik Abel. There are numerous types of full, additive subcategories of abelian categories that occur in nature, as well as some conflicting terminology.

Therefore in particular the category Vect of vector spaces is an abelian category. These axioms are still in common use to this day. 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. This page was last edited on 19 Marchat A similar statement is true for additive categoriesalthough the most natural result in that case gives only enrichment over abelian monoids ; see semiadditive category.

Views Read Edit View history. Axioms AB1 and AB2 were also given. Let C C be an abelian category.