Holmstrom Abelian category

An introduction by Daniel Murfet.

Borceaux vol 2.

http://ncatlab.org/nlab/show/additive+and+abelian+categories

P. Freyd, Abelian categories, Harper, 1966.

[Ga] P. Gabriel, Des categoryégories abéliennes, Bull. Soc. Math. France 90 (1962), 323-448. ONline version exists

Grothendieck’s Tohoku paper

An abelian category is an additive category in which every map:

• admits a kernel and a cokernel (Ab1)
• is strict (Ab2)

Recall that a kernel of a morphism $f: X \to Y$ is the fiber product $X \times_Y 0$. Also recall that a morphism is strict if the natural map from its coimage to its image is an isomorphism.

Some properties of abelian cats:

• mono + epi implies iso
• finite limits and colimits exist
• any functor category from a small to an abelian category is abelian
• the opposite of an abelian category is abelian

An abelian category is called semisimple if all short exact sequences split.

A full subcategory of an abelian category is called

• thick if it is closed under kernels, cokernels and extensions.
• generating if “its objects can surject to anything”
• cogenerating if “its objects can ”eat“ anything”, i.e. can receive an injection from any object.

Recall that Hom is left exact in both variables. An object is said to be injective if it is exact as a contravariant functor. An abelian category has enough injectives if the injectives are cogenerating. An object is projective if it is exact as a covariant functor. Enough projectives means that the projectives are generating.

Properties of injective objects: “Any morphism from a subobject to I can be extended to the whole object”. (Probably enough for generators). Three facts about short exact sequences: In any short exact sequence, if A, B inj, then C inj. Also, if A inj, then the sequence splits. In a split short exact sequence, B is inj iff A & C are inj.

Freyd-Mitchell thm: Let C be a small abelian category. Then there exists a ring R and an exact fully faithful functor from C to R-mod. (See Kashiwara-Schapira chapter 9)

Some further “axioms” on abelian cats that are sometimes useful to consider (these are not satisfied for all abelian cats):

• All direct sums exist (Ab3)
• (assuming Ab3) “A compatible family of morphisms on an increasingly filtered family of subobjects induce a unique morphism on the sum subobject”. (Ab5)
• Also Ab4. See Tohoku

Thm: Let $C$ be abelian, satisfying Ab3. Then direct limits exist in $C$, and the functor $\lim_{\to} : Hom(I, C) \to C$ is additive and right exact. (What is the significance of this result??)

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A category is said to be $F$-linear if the Hom sets are $F$-modules, composition is $F$-linear, and if the category admits finite sums. A functor which respects the $F$-module structure will also respect direct sums, up to a natural isomorphism.

nLab page on Abelian category