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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This result can be found as Theorem 7. 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 cstegories categories of sheaves of abelian groups and of modules.
There are numerous types of caategories, additive baelian of abelian categories that occur in nature, as well as some conflicting terminology. It is such that much of the homological algebra of chain complexes can be developed inside every abelian category.
Recall the following fact about pre-abelian categories from this propositiondiscussed there:. This page was last edited on 19 Marchat For more discussion of the Freyd-Mitchell embedding theorem see there.
The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in avelian slightly abdlian work of David Buchsbaum. Abelian categories are the most general setting for homological algebra. 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. A similar statement is true abflian additive categoriesalthough the most natural result in that case gives only enrichment over abelian monoids ; see semiadditive category.
See AT category for more on that. Some references claim that this property characterizes abelian categories among pre-abelian ones, but it is not clear to the catetories of this page why this should be so, although we do not currently have a counterexample; see this discussion. Let A be an abelian category, C a full, additive subcategory, and I the inclusion functor. Not every abelian category is a concrete category such as Ab or R R Mod. The concept of exact sequence arises naturally in this setting, and it turns out that exact functorscatetories.
This highlights the foundational relevance of the category of Abelian groups in the theory and its canonical nature. This epimorphism is called the coimage of fwhile the monomorphism is called the image of f. Abelian categories were introduced by Buchsbaum under the name of “exact category” and Grothendieck in categoties to unify various cohomology theories. Abelian categories are named after Niels Henrik Abel. Every abelian category A is a module over the monoidal category of finitely generated abelian groups; that anelian, we can form a tensor product of a finitely generated abelian group G and any object A of A.
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. 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 concept of abelian categories is one in a sequence of notions of additive and abelian categories. Therefore in particular the category Vect of vector spaces is an abelian category. Given any pair AB of objects in an abelian category, there is a special zero morphism from A to B.
The category of caregories of abelian groups on any site is abelian. 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 rreyd embedded this way into R Mod R Modfor some ring R R.
All of the constructions used in that field are relevant, such as exact sequences, and especially short exact sequencesand derived functors. See for instance remark 2.
These axioms are still in common use to this day. At the time, there was a cohomology theory for sheavesand a cohomology theory for groups. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometrycohomology and pure category theory. The motivating prototype example of an abelian category is the category of abelian groupsAb. By the second formulation of the definitionin an abelian category.
Retrieved from ” https: Remark The notion of abelian category is self-dual: In an abelian category, every morphism f can be written as the composition of an epimorphism followed by a monomorphism. A discussion about to which extent abelian categories are a general context for homological algebra is archived at nForum here.
This can be defined as the zero element of the hom-set Hom ABsince this is an abelian group. For more discussion see the n n -Cafe. This is the celebrated Freyd-Mitchell embedding theorem discussed below. 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. 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.
The two were defined differently, but they had similar properties.
Important theorems that apply in all abelian categories include the five lemma and the short five lemma as a special caseas well as categoeies snake lemma and the nine lemma as a special case. Grothendieck unified the two theories: Let C C be an abelian category. The last point is of relevance in particular for higher categorical generalizations of additive categories.
Subobjects and quotient objects are well-behaved in abelian categories. An abelian category is a pre-abelian category satisfying the following equivalent conditions. The notion of abelian category is self-dual: It follows that every abelian category is a balanced category. Abelian categories are very stable categories, for example they are regular and they satisfy the snake lemma.
Abelian category – Wikipedia
Popescu, Abelian categories with applications to rings and modulesLondon Math. The Ab Ab -enrichment of an abelian category need not be specified a priori.
Note that the enriched structure on hom-sets is a consequence of the first three axioms of the first definition.