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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There are numerous types of full, additive subcategories of abelian categories that occur in nature, as well as some conflicting terminology.
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.
abeljan Views Read Edit View history. 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. For more discussion see the n n -Cafe.
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. These axioms are still in categodies use to this day.
Categorise exactness concept has been axiomatized in the theory of exact categoriesforming a very special abeliah of regular categories. 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.
For more see at Freyd-Mitchell embedding theorem.
The essential image of I is a full, additive subcategory, but I is not exact. 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. Every monomorphism is a kernel and every epimorphism is a cokernel.
This result can be found as Theorem 7. Therefore in particular the category Vect of vector spaces is an abelian category. By the second formulation of the definitionin an abelian category. 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.
Note that the enriched structure on hom-sets is a consequence of the first three axioms of the first definition. See also the Wikipedia article for the idea of the proof.
We can also characterize which abelian categories are equivalent to a category of R R -modules:.
The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. The last point is of relevance in particular for higher categorical generalizations of additive categories.
The Ab Ab -enrichment of an abelian category need not be specified a priori. This epimorphism is called the coimage of fwhile the monomorphism is called the image of f.
Some references claim that this property characterizes abelian categories among pre-abelian ones, but it is not clear to the authors of this page why this should be so, although we do not currently have a counterexample; see this discussion. Given any pair AB of objects in an abelian category, there catgeories a special zero morphism from A to B. 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.
The abelian category is also a comodule ; Hom GA can be interpreted as an object of A.
But under suitable conditions this comes down to working subject to an embedding into Ab Absee the discussion at Embedding into Ab below.
In fact, much of category theory was developed as a language to study these similarities. This is the celebrated Freyd-Mitchell embedding theorem discussed below.
Proposition These two conditions are indeed equivalent. Every small abelian category admits a fullfaithful and exact functor to the category R Mod R Mod for some ring R R. 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 catsgoriespower ing cotensoring Homotopical categores enriched homotopical category enriched model category model structure on homotopical presheaves Edit this sidebar.
Let A be an abelian category, C a full, additive subcategory, and I the inclusion functor.