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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For the characterization of the tensoring functors see Eilenberg-Watts theorem. Abelian categories are the most general setting for homological algebra. There are numerous types of full, additive subcategories of abelian cateegories that occur in nature, as well as some conflicting terminology. Proposition These two conditions are indeed equivalent.

Alternatively, one can reason with generalized elements in an abelian category, without explicitly embedding it into a larger concrete category, see at element in abe,ian abelian category. These axioms are still in common use to this day. 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 essential image of Abeliaj is a full, additive subcategory, but I is not exact.

While additive categories differ significantly from toposesthere is an intimate relation between abelian categories and toposes.

An abelian category is a pre-abelian category satisfying the following equivalent conditions. 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 notion of abelian category is self-dual: All of the constructions used in that field are relevant, such as exact sequences, and especially short exact sequencesand derived functors. So 1 implies 2. The category of sheaves of abelian groups on any site is abelian.

Proof This result can be found as Theorem 7.

## Abelian categories

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. Definition An abelian category is a pre-abelian category satisfying the following categorles conditions. In an abelian category every morphism decomposes uniquely up to a unique isomorphism into the composition rreyd an epimorphism and a monomorphismvia prop combined with def.

These stability categpries make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometrycohomology and pure category theory. In fact, much of category theory was developed as a language to study these similarities.

Abelian categories are named after Niels Henrik Abel. The proof that R Mod R Mod is an abelian category is spelled out for instance in.

### Abelian category – Wikipedia

Recall the following fact about pre-abelian categories from this propositiondiscussed there:. This highlights the xbelian relevance of the category of Abelian groups in the theory and its canonical nature.

Categorjes 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.

By the second formulation of the definitionin an abelian category. Note that the enriched structure on hom-sets is a consequence of the first three axioms of the first definition.

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 are very stable categories, for example they are regular and they satisfy the snake lemma.

For more see at Freyd-Mitchell embedding theorem.

## Abelian category

Retrieved from ” https: The reason is that R Mod R Mod has all small category limits and colimits. However, in most examples, the Ab Ab -enrichment is evident from the start and does not need to be constructed in this way.

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.

The Ab Ab -enrichment of an abelian category need not be specified a priori. Remark The notion of abelian category is self-dual:

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