Finitary abelian category

A cocomplete abelian category is finitary in the directed-union sense if every cocone under a diagram indexed by a nonempty directed poset, with all its legs monomorphisms, induces a monomorphism from the colimit to the cocone vertex. The category of abelian groups has this property: two colimit representatives mapped to the same vertex become equal at a common upper index because that cocone leg is injective. This use of finitary concerns directed unions, rather than the preservation-of-filtered-colimits meaning used for finitary monads.

In a complete finitary abelian category, the comparison $j$ with diagonal matrix entries the identity and off-diagonal entries zero is monic. Express its source as the directed colimit of finite biproducts. Their maps into the product have finite-coordinate retractions and are therefore split monomorphisms; the finitary axiom applies to this cocone. If the category is also cofinitary abelian category, duality makes $j$ epic and hence invertible. For countably many copies of any object, the shift-and-fold argument in countable biproducts in an additive category force triviality then forces the identity to be zero, so every object is a zero object.