Derived categories of functors and quiver sheaves

Abstract

Let [Formula: see text] be small category and [Formula: see text] an arbitrary category. Consider the category [Formula: see text] whose objects are functors from [Formula: see text] to [Formula: see text] and whose morphisms are natural transformations. Let [Formula: see text] be another category, and again, consider the category [Formula: see text]. Now, given a functor [Formula: see text] we construct the induced functor [Formula: see text]. Assuming [Formula: see text] and [Formula: see text] to be abelian categories, it follows that the categories [Formula: see text] and [Formula: see text] are also abelian. We have two main goals: first, to find a relationship between the derived category [Formula: see text] and the category [Formula: see text]; second relate the functors [Formula: see text] and [Formula: see text]. We apply the general results obtained to the special case of quiver sheaves.