Abelian Category - Grothendieck's Axioms

Grothendieck's Axioms

In his Tôhoku article, Grothendieck listed four additional axioms (and their duals) that an abelian category A might satisfy. These axioms are still in common use to this day. They are the following:

• AB3) For every set {A_i} of objects of A, the coproduct *A_i exists in A (i.e. A is cocomplete).
• AB4) A satisfies AB3), and the coproduct of a family of monomorphisms is a monomorphism.
• AB5) A satisfies AB3), and filtered colimits of exact sequences are exact.

and their duals

• AB3*) For every set {A_i} of objects of A, the product PA_i exists in A (i.e. A is complete).
• AB4*) A satisfies AB3*), and the product of a family of epimorphisms is an epimorphism.
• AB5*) A satisfies AB3*), and filtered limits of exact sequences are exact.

Axioms AB1) and AB2) were also given. They are what make an additive category abelian. Specifically:

• AB1) Every morphism has a kernel and a cokernel.
• AB2) For every morphism f, the canonical morphism from coim f to im f is an isomorphism.

Grothendieck also gave axioms AB6) and AB6*).