Narrow Results

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. Let ${ℬ}$ be another category, and again, consider the category ${𝒞}({ℬ})$. Now, given a functor $F:{𝒜}\to {ℬ}$ we construct the induced functor $F_{{𝒞}}:{𝒞}({𝒜})\to {𝒞}({ℬ})$. Assuming ${𝒜}$ and ${ℬ}$ to be abelian categories, it follows that the categories ${𝒞}({𝒜})$ and ${𝒞}({ℬ})$ are also abelian. We have two main goals: first, to find a relationship between the derived category $D({𝒞}({𝒜}))$ and the category ${𝒞}(D({𝒜}))$; second relate the functors $R(F_{{𝒞}})$ and ${(RF)}_{{𝒞}}:{𝒞}(D({𝒜}))\to {𝒞}(D({ℬ}))$. We apply the general results obtained to the special case of quiver sheaves.