Idea
The category of small additive presheaves of abelian groups on an additive category contains a subcategory of finitely presented (that is compact) objects. This subcategory has a nice universal property.
Definition
Let TT be an additive subcategory. An additive presheaf is finitely presented if it is a cokernel of a morphism of representables. The full subcategory of object parts of these cokernels is A(T)A(T), the Freyd’s abelianization of TT.
Thus, an object xx of A(T)A(T) is a presheaf such that there exist an exact sequence of presheaves of abelian groups of the form
Hom(−,a)→Hom(−,b)→x→0 Hom(-,a)\to Hom(-,b) \to x \to 0
where a,ba,b are objects in AA.
Last revised on January 21, 2026 at 17:04:45...
Idea
The category of small additive presheaves of abelian groups on an additive category contains a subcategory of finitely presented (that is compact) objects. This subcategory has a nice universal property.
Definition
Let TT be an additive subcategory. An additive presheaf is finitely presented if it is a cokernel of a morphism of representables. The full subcategory of object parts of these cokernels is A(T)A(T), the Freyd’s abelianization of TT.
Thus, an object xx of A(T)A(T) is a presheaf such that there exist an exact sequence of presheaves of abelian groups of the form
Hom(−,a)→Hom(−,b)→x→0 Hom(-,a)\to Hom(-,b) \to x \to 0
where a,ba,b are objects in AA.
Last revised on January 21, 2026 at 17:04:45. See the history of this page for a list of all contributions to it.