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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Embedding of abelian categories into Ab is discussed in.
The category of sheaves of abelian groups on any categgories is abelian. Retrieved from ” https: This exactness concept has been axiomatized in the theory of exact categoriesforming a very special case of regular categories. Monographs 3Academic Press The concept of abelian categories is one in a sequence of notions of additive and abelian categories. Every monomorphism is a kernel and every epimorphism is a cokernel.
Not every abelian category is a concrete category such as Ab or R R Mod. The exactness properties of abelian categories have many features in common with exactness properties of toposes or of pretoposes.
This epimorphism is called the coimage of fwhile the monomorphism is called the image of f. However, in most examples, the Ab Ab -enrichment is evident from the start and does not need to be constructed in this way.
Deligne tensor product of abelian categories. Recall the following fact about pre-abelian categories from this propositiondiscussed there:. The concept of exact sequence arises naturally in this setting, and it turns out that exact functorsi. Important theorems that apply in all abelian categories include the five lemma and the short five lemma as a special caseas well as the snake lemma and the nine lemma as a special case.
This highlights the foundational relevance of the category of Abelian groups in the theory and its canonical nature. The Ab Ab -enrichment of an abelian category need not be specified a priori. 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.
See also the Wikipedia article for the idea of the proof. The motivating prototype example of an abelian category is the category of abelian groupsAb. Remark The notion of abelian category is self-dual: Let A be an abelian category, C a full, additive subcategory, and I the inclusion functor.
They are the following:. 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. Abelian categories are very stable categories, for example they are regular and they satisfy the snake lemma. 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.
A similar statement is true for additive categoriesalthough the most natural result in that case gives only enrichment over abelian monoids ; see semiadditive category. While additive categories differ significantly from toposesthere is an intimate relation between abelian categories and toposes. Subobjects and quotient objects are well-behaved in abelian categories.
The theory originated in an effort to categoreis several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. So 1 implies 2.
Note that the enriched structure on hom-sets is a consequence of the first three axioms of feeyd first definition.
Abelian category – Wikipedia
If A is completethen we can remove the requirement that G be finitely generated; most generally, we can form finitary enriched limits in A. Abelian categories are the most general setting for homological algebra. Grothendieck unified the two theories: But under suitable conditions this comes down to working subject to an embedding into Ab Absee the discussion at Embedding into Ab below.
For more see at Freyd-Mitchell embedding theorem. All of the constructions used in that field are relevant, such as exact sequences, and especially short exact sequencesand derived functors. Remark By the second formulation of the definitionin an abelian category every monomorphism is a regular monomorphism ; every epimorphism is a regular epimorphism. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometrycohomology and pure category theory.
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. For more discussion see the n n -Cafe.
The abelian category is also feeyd comodule ; Hom GA can be interpreted as an object of A. See AT category for more on that. Theorem Let C C be an abelian category. Given any pair AB of objects in an abelian fregd, there is a special zero morphism from A to B. 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.
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.
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.