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 Ab Ab -enrichment of an abelian category need not be specified a priori. Every monomorphism is a kernel and every epimorphism is a cokernel. This epimorphism is called the coimage of fwhile the monomorphism frehd called the image of f. Here is an explicit example of a full, additive subcategory of an abelian category which is itself abelian but the inclusion functor is not exact.
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 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. Categoriees first part of this theorem can also be found as Prop.
Recall the following fact about pre-abelian categories from this propositiondiscussed there:. For more discussion of the Freyd-Mitchell embedding theorem see there. The motivating prototype example of an abelian category is the category of abelian groupsAb. The reason is that R Mod R Mod has all small category limits and xategories.
In an abelian category, every morphism f can be written as the composition of an epimorphism followed by a monomorphism. Some references claim that this property characterizes abelian categories among pre-abelian ones, but it is not clear to categroies authors of this page why this should be so, although we do not currently have a qbelian see this discussion.
Abelian categories are the most general setting for homological algebra. Proposition Every morphism f: In mathematicsan abelian category is a category in which morphisms and objects can be abwlian and in which kernels and cokernels exist and have desirable properties.
Axioms AB1 and AB2 were also given. Abelian categories are very stable categories, for example they are regular and they satisfy the snake lemma. The concept of exact sequence arises naturally in this setting, and it turns out that exact functorsi.
These axioms are still in common use to this day. This exactness concept has been axiomatized in the theory of exact categoriesforming a very special case of regular 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.
This definition is equivalent  to the following “piecemeal” definition:. Subobjects and quotient objects are well-behaved in abelian categories. Context Enriched category theory enriched category theory Background category theory monoidal categoryclosed monoidal category cosmosdategoriesbicategorydouble 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 Cattegories enrichment enriched homotopical category enriched model category model structure on homotopical presheaves Edit this sidebar.
The proof that R Mod R Mod is an abelian category is spelled out for instance in. They are the following:.
While additive categories differ significantly from toposesthere is an intimate relation between abelian categories and toposes. See also the Wikipedia article for the idea of the proof. Additive categories Homological algebra Niels Henrik Abel. This result can be found as Theorem 7. See also the catlist discussion on comparison between abelian categories and topoi AT categories. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometrycohomology and pure category theory.
The last point is of relevance in particular for higher categorical generalizations of additive categories. 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. This is the celebrated Freyd-Mitchell embedding theorem discussed below.
See AT category for more on that. 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.
The exactness properties of abelian categories have many features in common freeyd exactness properties of toposes or of pretoposes. At the time, there was a cohomology theory for sheavesabeliian a cohomology theory for groups. An abelian category is a pre-abelian category satisfying the following equivalent conditions.
Abelian category – Wikipedia
For more see at Freyd-Mitchell embedding theorem. See for instance remark 2. 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.
The concept of abelian categories is one in a sequence of notions of additive and abelian categories. From Wikipedia, the free encyclopedia. By the second formulation of the definitionin an abelian category. For example, the poset of subobjects of any given object A is a bounded lattice. Let C C be an abelian category. For a Noetherian ring R R the category of finitely generated R R -modules is an abelian category that lacks these properties.
Popescu, Abelian categories with applications to rings and modulesLondon Math. Going still further one should be able to obtain a nice theorem describing the image of the embedding of the weak 2-category of.
Note that the enriched structure on hom-sets is a consequence of the first three axioms of the first definition. 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 frehd underlying the objects. Deligne tensor product of abelian categories. Theorem Let C C be an abelian category.
Let A be an abelian category, C a full, additive subcategory, and I the inclusion functor.