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 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.
It is such that much of the homological algebra of chain complexes can be developed inside every abelian category. The concept of abeljan categories is one in a sequence of notions of additive and abelian categories.
Abelian category – Wikipedia
While additive categories differ significantly from toposesthere is an intimate relation between abeliwn categories and toposes. See AT category for more on that. Recall the following fact about pre-abelian categories from this propositiondiscussed there:. An abelian category is a pre-abelian category satisfying the following equivalent conditions.
Every monomorphism is a kernel and every epimorphism is a cokernel.
So 1 implies 2. The notion of abelian category is self-dual: By the second formulation of the definitionin an abelian category. It follows that every abelian category is a balanced category. 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. 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.
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 Ab Ab -enrichment of an abelian category need not be specified a priori.
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 abbelian 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, frsyd most examples, the Ab Ab -enrichment is evident from the start and does not need to be constructed in this way.
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 last point is catehories relevance in particular for higher categorical generalizations of additive categories. See for instance remark 2. The exactness properties of abelian categories have many features in common with exactness properties of toposes or of pretoposes.
Not every abelian category is a concrete category such as Ab or R R Mod. 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 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. This cateogries the celebrated Freyd-Mitchell embedding theorem discussed below. 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.
But under suitable conditions this comes down to working subject to an embedding into Ab Absee the discussion at Embedding into Ab below. The reason is that R Mod R Mod has all small category limits and colimits. For a Noetherian ring R R the category of finitely generated R R -modules is an abelian category that lacks these properties. Every small abelian category admits a fullfaithful and exact functor to the category R Mod R Mod for some ring R R. This result can be found as Theorem 7.
See also the Wikipedia article for the idea of the proof. For more see abekian Freyd-Mitchell embedding theorem. We can also characterize which abelian categories are equivalent to a category of R R -modules:.
Let C C be an abelian category. The first part of this theorem can also be found as Prop.
abelian category in nLab
For the characterization of the tensoring functors see Eilenberg-Watts theorem. Going still further one should be able to obtain a nice theorem describing the image of the embedding of the weak 2-category of.
For more discussion see the n n -Cafe. Therefore in particular the category Vect of vector spaces is an abelian category. The category of sheaves of abelian groups on any site is abelian.
Deligne tensor product of abelian categories. Popescu, Abelian categories with applications to rings and modulesLondon Math. Monographs 3Academic Press Embedding of abelian categories into Ab is discussed in.
For more discussion of the Freyd-Mitchell embedding theorem see there. The proof that R Mod R Mod is an abelian category is spelled out for instance in. A discussion about to which extent abelian categories are a general context for homological algebra is archived at nForum here. See also the catlist discussion on comparison between abelian categories and topoi AT categories. Context Enriched category theory enriched category theory Background categorifs theory monoidal categoryclosed monoidal category cosmosmulticategorybicategorydouble categoryvirtual double category Basic concepts enriched category enriched functorprofunctor enriched functor abeliah 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.
Proposition Every morphism f: Definition An abelian category is a pre-abelian category satisfying the following equivalent conditions. Proposition These two conditions are indeed equivalent. Remark The notion of abelian category is self-dual: Remark By the second formulation of the definitionin an abelian category every monomorphism is a regular monomorphism ; every epimorphism is a regular epimorphism.
Proposition 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. 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, categorues we do not currently have a counterexample; see this discussion. Proof This result can be found as Theorem 7.
Theorem Let C C be an abelian category.