I'm learning about sheaves and sheaf cohomology with the eventually goal of using these tools to study Riemann surfaces. My references are Forster's Lectures on Riemann Surfaces and Iverson's Cohomology of Sheaves. Both sources identify a presheaf $F$ as a functor from a category of ordered open sets in a topological space $X$ to the category of abelian groups. They also mention one can define a presheaf mapping to commutative rings and vector spaces among other objects.
I was wondering: is there a characterization of what categories a presheaf $F$ can map to? Abelian categories or concrete categories seem like a reasonable guess; however it would be nice if there's a specific characterization.
A presheaf is just a functor, with no particular properties. So it's possible to define one with values in an arbitrary category. Sets, spaces, chain complexes, and spectra are other common examples not listed there. But for defining sheaves, you need to be able to describe the sheaf condition, which requires the existence of certain products and equalizers in the target category. To be able to define sheaves on an arbitrary space with values in $\mathcal{C}$, then, $\mathcal{C}$ should have all limits.