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 exactness properties of abelian categories have many features in common with exactness properties of toposes or of pretoposes. See for instance remark 2. The reason is that R Mod R Mod has all small category limits and colimits. Proof This result can be found as Theorem 7. Therefore in particular the category Vect of vector spaces is an abelian category. In an abelian category, every morphism f can be written as the composition of an epimorphism followed by a monomorphism.

Remark By the second formulation of the definitionin an abelian category every monomorphism is a regular monomorphism ; every epimorphism is a regular epimorphism. This definition is equivalent [1] to the following “piecemeal” definition:. This is the celebrated Freyd-Mitchell embedding theorem discussed below. Categoroes unified the two theories: All of the constructions used feyd that field are relevant, such as categoires sequences, and especially short exact sequencesand derived functors.

Going still further one ctaegories be able to obtain a nice theorem describing the image of the embedding of the weak 2-category of. The essential image of I is a full, additive subcategory, but I is not exact.

At the time, there was a cohomology theory for sheavesand a cohomology theory for groups. Given any pair AB of objects in an abelian category, there is a special zero morphism from A to B. Proposition In an abelian category every morphism decomposes categorids up to a unique isomorphism into the composition of an epimorphism and a monomorphismvia prop combined with def.

They are the following:. Recall the following fact about pre-abelian categories from this propositiondiscussed there:.

## Abelian categories

The notion of abelian category is an abstraction of basic properties of the category Ab of abelian frdydmore 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.

The abelian category is also a comodule ; Hom GA can be interpreted as an object of A.

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. In fact, much of category theory was developed as a language to study these similarities. For the characterization of the tensoring functors see Eilenberg-Watts theorem. The first part of this theorem can also be found as Prop.

Here is an explicit example of a full, additive subcategory of an abelian category which is frey abelian but the inclusion functor is not exact. See also the Wikipedia article for the idea of the proof. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly cxtegories work of David Buchsbaum.

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.

Theorem Let C C be an abelian category.

This result can be found as Theorem 7. This page was last edited on 19 Marchat 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.

Every small abelian category admits a fullfaithful and exact functor to the catdgories R Mod R Mod for some ring R R. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometrycohomology and ahelian category theory.

### Abelian Categories

This highlights the foundational relevance of the category of Abelian groups in the theory and its canonical nature. Let A be an abelian category, C a full, additive subcategory, and I the inclusion functor. By the second formulation of the definitionin an abelian category. However, in most examples, the Ab Ab -enrichment is evident from the start and does not need to be constructed in this way.

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.

Abelian categories are very stable categories, for example they are regular and they satisfy the snake lemma. Views Read Edit View history. They are what make an additive category abelian.

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.

Definition An abelian category is a pre-abelian category satisfying the following equivalent conditions. The last point is of relevance in particular for higher categorical generalizations of additive categories. Let C C be an abelian category.

## Abelian category

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.

It is such that much of the homological algebra of chain complexes can be developed inside every abelian category. 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. Proposition Every morphism f: The Ab Ab -enrichment of an abelian category need not be specified a priori.

For more discussion of the Freyd-Mitchell embedding theorem see there.

Remark The notion of abelian category is self-dual: