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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Proposition Every morphism f: See also the Wikipedia article for the idea of the proof. Abdlian 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. 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 cateories functors from a small category to an Abelian category are Abelian as well.

Embedding of abelian categories into Ab is discussed in.

Context Enriched category theory enriched aabelian theory Background category theory monoidal categoryclosed monoidal category cosmosmulticategorybicategorydouble categoryvirtual double category Basic concepts enriched category enriched functorprofunctor enriched actegories 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.

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. The notion of abelian category is self-dual: Given any pair AB of objects in an abelian category, there is a special zero morphism from A to B.

By using this site, you agree to the Terms of Use and Privacy Policy. Retrieved from ” https: For a Noetherian ring R R the category of finitely generated R R -modules is an abelian category that lacks these properties. The essential image of I is a full, additive subcategory, but I is not exact.

See for instance remark 2. The last point is of relevance in particular for higher categorical generalizations of additive categories. They are what make an additive category abelian.

## Abelian categories

The motivating prototype example of an abelian category is the category of abelian groupsAb. Views Read Edit View history. Going still further one should categoriew able to obtain a nice theorem describing the image of the embedding of the weak 2-category of. For example, the poset of subobjects of any given object A is a bounded lattice. The first part of this theorem can also be found as Prop. Let A be an abelian category, C a full, additive subcategory, and I the inclusion functor.

### abelian category in nLab

There are numerous types of full, additive categoriez of abelian categories that occur in nature, as well as some conflicting terminology. While additive categories differ significantly from toposesthere is an intimate relation between abelian xbelian and toposes. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum.

An abelian category is a pre-abelian category satisfying the following equivalent conditions.

Every small abelian category admits a fullfaithful and exact functor to the category R Mod R Mod for some ring R R. A discussion about to which extent abelian categories are a general context for homological algebra is archived at nForum here.

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

However, in most examples, the Ab Ab -enrichment is evident from categoriss start and does not need to be constructed in this way. See AT category for more on that. Monographs 3Academic Press The concept of exact sequence arises naturally in this setting, and it turns out that exact functorsi.

Important theorems categoriss 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.

## Abelian category

These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometrycohomology and pure category theory.

We can also characterize which abelian categories are equivalent to a category of R R -modules:. 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. All of the constructions used categorids that field are relevant, such as exact sequences, and especially short exact sequencesand derived functors.

For more discussion abelisn the n n -Cafe. Abelian categories are very stable categories, for example they are regular and they satisfy the snake aelian. It follows that every abelian category is a balanced category. This is the celebrated Freyd-Mitchell embedding theorem discussed below.

### Abelian category – Wikipedia

Axioms AB1 and AB2 were also given. Since by remark every monic is regularhence strongit categoriss that epimono epi, mono is an orthogonal factorization system in an abelian category; see at epi, mono factorization system. Deligne tensor product of abelian categories. 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.

This highlights the foundational relevance of the category of Abelian groups in the theory and categoties canonical nature.

Abelian categories were introduced by Buchsbaum under the name of “exact category” and Grothendieck in order to unify various cohomology theories. The two were defined differently, but they had similar properties. Remark By the second formulation of the definitionin an abelian category every monomorphism is a regular monomorphism ; every epimorphism is a regular epimorphism. From Wikipedia, categorie free encyclopedia.