Derived categories of functors and Fourier--Mukai transform for quiver sheaves

TL;DR

This paper explores the relationship between derived categories of functor categories and quiver sheaves, establishing connections and a version of Mukai's Theorem for these structures.

Contribution

It introduces new relationships between derived categories of functor categories and relates functors in this context, applying results to quiver sheaves.

Findings

Established a relationship between D(C(A)) and C(D(A))

Related functors R(F_C) and (RF)_C

Proved a version of Mukai's Theorem for quiver sheaves

Abstract

Let C be small category and A an arbitrary category. Consider the category C(A) whose objects are functors from C to A, and whose morphisms are natural transformations. Given a functor F : A --> B one obtains an induced functor F_C : C(A) --> C(B) . If A and B are abelian categories, we have that C(A) and C(B) are also abelian, and one has two functors R(F_C) : D(C(A)) --> D(C(B)) and (RF)_ C : C(D(A)) --> C(D (B)). The goals of this paper are: 1) to find a relationship between D (C(A)) and C(D(A)); 2) to relate the functors R(F_C) and (RF)_C. As an application, we prove a version of Mukai's Theorem for quiver sheaves.