Additive functors to abelian groups: "additional structure" and functors induced by "additive pseudo-functors"; references?

Question

Let $\mathcal{A}$ be a small additive category. Consider the category $PreSh(\mathcal{A})$ of all additive functors from $\mathcal{A}^{op}$ into abelian groups; note that this category is abelian and any object of $\mathcal{A}$ becomes projective in it (if we apply the Yoneda embedding for this setting). Now note that $PreSh(\mathcal{A})$ does not fully determine $\mathcal{A}$; so there can exist "natural" functors from $PreSh(\mathcal{A})$ into $PreSh(\mathcal{B})$ that do not come from additive functors from $B$ into $A$. Did anybody study functors of this sort?

Besides, any "formal" direct sum of $\mathcal{A}$-morphisms satisfying obvious finiteness conditions "acts" on objects of $PreSh(\mathcal{A})$. One can form a certain (associative unital) ring $R$ of formal direct sums; one can possibly describe $PreSh(\mathcal{A})$ as a certain modules over this ring. Now, I would like to localize this $R$ and to factorize it by certain (two-sided and non-homogeneous) ideals. So, I wonder whether one can "categorify" the corresponding full subcategories of $PreSh(\mathcal{A})$.

Any references or hints would be very welcome!

Answer 1

You want to look up Morita theory for enriched categories.

By a "linear category" I will mean an $\text{Ab}$-enriched category. Write $\widehat{A}$ for the category of presheaves of abelian groups on $A$. It has a universal property: it is universal for maps $A \to \widehat{A}$ into a cocomplete linear category. Hence cocontinuous functors $\widehat{A} \to \widehat{B}$ correspond to functors $A \to \widehat{B}$, which in turn correspond to profunctors or bimodules $A \times B^{op} \to \text{Ab}$. This is a many-object version of the Eilenberg-Watts theorem.