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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Recall the following fact about pre-abelian categories from this propositiondiscussed there:.
All of the constructions used in that field are relevant, such as exact sequences, and especially freydd exact sequencesand derived functors. 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. It is such that much of the homological algebra of chain complexes can be developed inside every abelian category. However, in most actegories, the Ab Ab -enrichment is evident from the start and does not need to be constructed in this way.
For more discussion of the Freyd-Mitchell embedding theorem see there.
abelian category in nLab
They are what make an additive category abelian. For example, the poset of subobjects of any given object A is a bounded lattice. The essential image of I is a full, additive subcategory, but I is not exact.
The reason is that R Mod R Mod has all small category limits and colimits. Embedding of abelian categories into Ab is discussed in. In frreyd, much of category theory was developed as a language to study these similarities.
Abelian categories are very stable categories, for example they are regular and they satisfy the snake lemma.
Every small abelian category admits a fullfaithful and exact functor to the category R Mod R Mod for some ring R R. The last point is of relevance in particular for higher categorical generalizations of additive categories.
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. At the time, there was a cohomology theory for sheavesand a cohomology theory for groups. The first part of this theorem can also be found as Prop.
Given any pair AB of objects in an abelian category, there is a special zero morphism from A to B. Retrieved from ” https: The exactness properties of abelian categories have many features freyf common with exactness properties of toposes or of pretoposes.
Abelian categories were introduced by Buchsbaum under the name of “exact category” and Grothendieck in order to unify various cohomology theories. The concept of exact sequence arises naturally in this setting, and it turns out that exact functorsi. Every monomorphism aelian a kernel and every epimorphism is a cokernel. 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 theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. This definition is equivalent abelixn to the following “piecemeal” abelisn. Grothendieck unified the two theories: The notion of abelian category is self-dual: 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.
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 categoriex abelian groups and of modules.
These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometrycohomology and pure category theory. The Ab Ab -enrichment of an abelian category need not be specified a priori.
Remark By the second formulation of the definitionin an abelian categorkes every monomorphism is a regular monomorphism ; every epimorphism is a regular epimorphism.