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 concept of abelian categories is one in a sequence of notions of additive and abelian categories. For the characterization of the tensoring functors see Eilenberg-Watts theorem. See also the catlist discussion on comparison between abelian categories and topoi AT categories. 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.
These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometrycohomology and pure category theory. 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.
Additive categories Homological algebra Niels Henrik Abel. This definition is equivalent  to the following “piecemeal” definition:.
The motivating prototype example of an abelian category is the category of abelian groupsAb. See AT category for more on that. Abelian categories are the most general setting for homological algebra. Deligne tensor product of abelian categories. Retrieved from ” https: If A is completethen we can remove the requirement that G be finitely generated; most generally, we can form finitary enriched limits in A.
Theorem Let C C be an abelian category. Monographs 3Academic Press Therefore in particular the category Vect of vector spaces is an abelian category.
In fact, much of category theory was developed as a language to study these similarities. This is the celebrated Freyd-Mitchell embedding theorem discussed below. The first part of this theorem can also be found as Prop. The abelian category is also a comodule ; Hom GA can be interpreted as an object of A. This highlights the foundational relevance of the category of Abelian groups in the theory and its canonical nature.
See for instance remark 2. Alternatively, one can reason with rfeyd elements in an abelian category, without explicitly embedding it into a larger concrete category, see at element in an abelian category.
Subobjects and quotient objects are well-behaved in abelian categories. Abelian categories are named after Niels Henrik Abel. So 1 implies 2. The essential image of I is a full, additive subcategory, but I is not exact.
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 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 cstegories ring R R.
These axioms are still in common use to this day. From Wikipedia, the free encyclopedia. It is cqtegories that much of the homological algebra of chain complexes can be freyyd inside every abelian 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. In an abelian category, every morphism f can be written as the composition of an epimorphism followed by a monomorphism.
The concept of exact sequence arises naturally in this setting, and it turns out that exact functorsi. Embedding of abelian categories into Ab is discussed in. Since by remark every monic is regularhence strongit follows that epimono epi, mono is an orthogonal factorization system in an aelian category; see at epi, mono factorization system. While additive categories differ significantly from toposesthere is an intimate fregd between abelian categories and toposes.
abelian category in nLab
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But for many proofs in homological algebra it is abelixn 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. Every monomorphism is a kernel and every epimorphism is a cokernel. Going still further one should be able to obtain a nice theorem describing the image of the embedding of the weak 2-category of.
The categkries were defined differently, but they had similar properties. Proposition These two conditions are indeed equivalent. 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.