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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Every monomorphism is a kernel and every epimorphism is a cokernel. Let A be an abelian category, C a full, additive subcategory, and I the inclusion functor. This highlights the foundational relevance of the category of Abelian groups in the theory and its canonical nature.

However, in most examples, the Ab Ab -enrichment is evident from the start and does not need to be constructed in this way. Axioms AB1 and AB2 were also given.

For more see at Freyd-Mitchell embedding theorem. See for instance remark 2.

Abelian categories

Retrieved from ” https: See also the Wikipedia article for the idea of the proof. 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.


Not every abelian category is a concrete category such as Ab or R R Mod. Every small abelian category admits a fullfaithful and exact functor to the category R Mod R Mod for some ring R R. See AT category for more on that. The Ab Ab -enrichment of an abelian category need not be specified a priori. Recall the following fact about pre-abelian categories from this propositiondiscussed there:.

Abelian Categories

Theorem Let C C be an freydd category. The proof that R Mod R Mod is an abelian category is spelled out for instance in. Aeblian epimorphism is called the coimage of fwhile the monomorphism is called the image of f.

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. This page was last edited on 19 Marchat Subobjects and quotient objects are well-behaved in abelian categories.

Abelian categories are very stable categories, baelian example they are regular and they satisfy the snake lemma. While additive categories differ significantly from toposesthere is an intimate relation between abelian categories and toposes. In an abelian category, every morphism f can be written as the composition of an epimorphism followed by a monomorphism. Deligne tensor product of abelian categories.

Abelian category – Wikipedia

By the second formulation of the definitionin an abelian category. This exactness concept has been axiomatized in the theory of exact categoriesforming a very special case of regular categories. In fact, much of category theory was developed as a language to study these similarities. Going still further one should be able to obtain a nice theorem describing the image of the embedding of the weak 2-category of.


This is the celebrated Freyd-Mitchell embedding theorem discussed below. These axioms are still in common use to this day. 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. Note that the enriched structure on hom-sets is a consequence of the first three axioms of the first definition.

For example, the poset of subobjects of any given object A is a bounded lattice. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum.

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