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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Popescu, Abelian categories with applications to rings and modulesLondon Math. There are numerous types of full, additive subcategories of abelian categories that occur in nature, categoriss well as some conflicting terminology.
For the characterization cstegories the tensoring functors see Eilenberg-Watts theorem. Given any pair AB of objects in an abelian category, there is a special zero morphism from A to B. But under suitable conditions this comes down to working subject to an embedding into Ab Absee the discussion at Embedding into Ab below.
It is such that much of the homological algebra of chain complexes can be developed inside every abelian category. The category of sheaves of abelian groups on any site is abelian. The concept of exact sequence arises naturally in this setting, and it turns out that exact functorsi.
A discussion about to which extent abelian categories are a general context for homological algebra is archived at nForum here. The reason is catehories R Mod R Mod has all small category limits and colimits.
Every abelian category A is a module over the monoidal category of finitely generated abelian groups; that is, we can form a tensor product of a finitely generated abelian group G and any object A of A.
These axioms are still in common use to this day. Embedding of abelian categories into Ab abeliah discussed in. A similar statement is true for additive categoriesalthough the most natural result in that case gives only enrichment over abelian monoids ; see semiadditive category.
abelian category in nLab
It follows that every abelian category is a balanced category. The first part of this theorem can also be found as Prop.
Abelian category – Wikipedia
At the time, there was a cohomology theory for sheavesand a cohomology theory for groups. Definition An abelian category is a pre-abelian category satisfying the following equivalent conditions.
An abelian category is a pre-abelian category satisfying the following equivalent conditions. The two were defined differently, but they had similar properties.
Going still further one should be able to obtain a nice theorem describing the image of the embedding of the weak 2-category of. See also the Wikipedia article for the idea of the proof. So 1 implies 2. Recall the following fact about pre-abelian categories from this propositiondiscussed there:.
See for instance remark 2. Every small abelian category admits a fullfaithful and exact functor to the category R Mod R Mod for some ring R R.
These stability properties make them inevitable in catdgories algebra and beyond; the theory has major applications in algebraic geometrycohomology and pure category theory.
This is the celebrated Freyd-Mitchell embedding theorem discussed below. Retrieved from ” https: Not every abelian category is a concrete category such as Ab or R R Mod. The exactness properties of abelian categories dategories many features in common with exactness properties of toposes or of pretoposes. The notion of abelian category is self-dual: 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.
Therefore abeloan particular the category Vect of vector spaces is an abelian category. For more discussion see the n n -Cafe.
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. Abelian categories were introduced by Buchsbaum under the name of “exact category” and Grothendieck in order to unify various cohomology theories. Theorem Let C C be an abelian category. We can also characterize which abelian categories are equivalent to a category catfgories R R -modules:.
Deligne tensor product of abelian categories. This can be defined as the zero element of the hom-set Hom ABsince this is an abelian group. In fact, much of category theory was developed as a language to study these similarities.
See AT category for more on that.
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. Proposition In an abelian category every morphism decomposes uniquely up to a unique isomorphism into zbelian composition of an epimorphism and a monomorphismvia prop combined with def.
Remark The notion of abelian category greyd self-dual: Axioms AB1 and AB2 were also given. However, in most examples, the Ab Ab -enrichment is evident from the start and does not need to be constructed in this way.
Proposition These two conditions are indeed equivalent.