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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Views Read Edit View history. Axioms AB1 and AB2 were also given.

Abelian Categories

Remark By the second formulation of the definitionin an abelian category every monomorphism is a regular monomorphism ; every epimorphism is a regular epimorphism.

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

Remark The notion of abelian category is self-dual: Therefore in particular the category Vect of vector spaces is an abelian category.

See also the Wikipedia article for the idea of the proof.

Abelian categories

It follows that every abelian category is a balanced category. Let C C be an abelian category.

Grothendieck unified the two theories: Note that the enriched structure on hom-sets is a consequence of the first three axioms of the first definition. But under suitable conditions this comes down to working subject to an embedding into Ab Absee the discussion at Embedding into Ab frey. This page was last edited on 19 Marchat It is such that much of the homological caategories of chain complexes can be developed inside every abelian category.

Abelian category

Proposition Every morphism f: Every monomorphism is a kernel and every epimorphism is a cokernel. The concept of exact sequence arises naturally in this setting, and it turns out that exact functorsi. Proof This result can be found as Theorem 7. 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. For a Noetherian ring R R the category abepian finitely generated R R -modules is an abelian category that lacks these properties.


Given any pair ACtegories of objects in an abelian category, there is a special zero morphism from A to B. These stability properties make them inevitable in homological algebra and beyond; the theory has major freydd in algebraic geometrycohomology and fdeyd category theory.

Alternatively, one can reason with generalized elements in an abelian category, without explicitly embedding it into a larger concrete category, see at element fteyd an abelian category. They are what make an additive category abelian. These axioms are still in common use to this day. The class of Abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an Abelian category, or the category of functors from a small category to an Abelian category are Abelian as well.

Popescu, Abelian categories with applications to rings and modulesLondon Math. Deligne tensor product of abelian categories. This can be defined as cwtegories zero element of the hom-set Hom ABsince this is an abelian group. From Wikipedia, the free encyclopedia. For more discussion see the n n -Cafe. We can also characterize which abelian categories are equivalent to a category of R R -modules:.

Abelian categories are very stable categories, for example they are regular and they satisfy the snake lemma. Context Enriched category theory enriched category categogies Background category theory monoidal categoryclosed monoidal category cosmosmulticategorybicategorydouble categoryvirtual double category Basic concepts enriched category enriched functorprofunctor enriched functor category 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.

The exactness properties of abelian categories have many features in common with cayegories properties of toposes or of pretoposes. All of the constructions used in that field are relevant, such as exact sequences, and especially categoeies exact sequencesand derived functors. By using this site, you agree to the Terms of Use and Privacy Policy.


For the characterization of the tensoring functors see Eilenberg-Watts theorem.

The notion of abelian category is self-dual: The essential image of I is a full, additive subcategory, but I is not exact. Important theorems that apply in all abelian categories include the abellian lemma and the short five lemma as a special caseas well as the snake lemma and the nine lemma as a special case.

Abelian category – Wikipedia

Abelian categories were introduced by Buchsbaum under the name of “exact category” and Grothendieck in order to unify various cohomology theories.

Proposition These two conditions are indeed equivalent. By the second formulation of the definitionin an abelian category.

This is the celebrated Freyd-Mitchell embedding theorem discussed below. Abelian categories are named after Niels Henrik Abel. This highlights the foundational relevance of abwlian category of Aeblian groups in the theory and its canonical nature. Let A be an abelian category, C a full, additive subcategory, and I the inclusion functor. At the time, there was a cohomology theory for sheavesand a cohomology theory for groups.

For more see at Freyd-Mitchell embedding theorem.

The two were defined differently, but they had similar properties. The motivating prototype example of an abelian category is the category of abelian groupsAb. This definition is equivalent [1] to the following “piecemeal” definition:. The concept of abelian categories is one in a sequence of notions of additive and abelian categories.