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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Grothendieck unified the two theories: This is the celebrated Freyd-Mitchell embedding theorem discussed below. In fact, much of category theory was developed as a language to study these similarities. So 1 implies 2. The exactness properties of abelian categories have many features in common with exactness properties of toposes or of pretoposes.
See for instance remark 2. 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.
For example, the poset of subobjects of any given object A is a bounded lattice. Proposition These two conditions are indeed equivalent. A discussion about to which extent abelian categories are a general context for homological algebra is archived at nForum here. Note that the enriched structure on hom-sets is a consequence of the first three axioms of the first definition. 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.
Remark The notion of abelian category is self-dual: The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. While additive categories differ significantly from toposesthere is an intimate relation between abelian categories and toposes.
Abelian categories are very stable categories, for example they are regular categkries 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.
Let C C be an abelian category.
The motivating prototype example of an abelian category is the category of abelian groupsAb. Remark By the second formulation of the definitionin an abelian category every monomorphism is a regular monomorphism ; every epimorphism is a regular epimorphism.
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. This result can be found as Theorem 7. Additive categories Homological algebra Niels Henrik Abel. The abelian category is also a comodule ; Hom GA can be interpreted as an object of A.
The proof that R Mod R Mod is an abelian category is spelled out for instance in. Theorem Let C C be an abelian category. Context Enriched category theory enriched category theory Background category theory monoidal categoryclosed monoidal category cosmosmulticategorybicategorydouble 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 Homotopical enrichment enriched homotopical category enriched model category model structure on homotopical presheaves Edit this sidebar.
Going still further one should be able to obtain a nice theorem describing the image of the embedding of the weak 2-category of. For the characterization of the tensoring functors see Eilenberg-Watts theorem.
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 for additive categoriesalthough the most natural result in that case gives only enrichment over abelian monoids ; see semiadditive category.
abelian category in nLab
Abelian categories are the most general setting for homological algebra. It is such that much of the homological algebra of chain complexes can be developed inside every abelian category. 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. See also the catlist discussion on comparison between abelian categories and topoi AT categories.
The essential image of I is a full, additive subcategory, but I is not exact. 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. Definition An abelian category is a pre-abelian category satisfying the following equivalent conditions.
The notion of abelian category is self-dual: From Wikipedia, the free encyclopedia. The two were defined differently, but they had similar properties.
The concept of exact sequence arises naturally in this setting, and it turns out that exact functorsi. There are numerous types of full, additive subcategories of abelian categories that occur in nature, as well as some conflicting terminology.
Every monomorphism is a kernel and every epimorphism is a cokernel. These axioms are still in common use to this day.
For more discussion see the n n -Cafe. At the time, there was a cohomology theory for sheavesand a cohomology theory for groups.
Subobjects and quotient objects are well-behaved in abelian categories. See AT category for more on that.