(pre)sheaves on path connected neigh

Question

Presheaves are defined by using neighborhoods of a point. Is there a way to restrict this construction to path connected neighborhoods of points? What is the name of the object which assigns other objects to path connected nbds. of points? Thank you. (Note: I am not interested in restricting presheaves to path connected open sets)

Answer 1

A presheaf can be defined on any category: it is just another name for a contravariant functor from that category to any other category. In particular, if you have a topological space $X$ you can associate to it a category $\mathcal Open (X)$ whose objects are the open subsets $U\subset X$ and the morphisms the inclusions $U\subset V$. A presheaf in the usual sense is then indeed a functor from $\mathcal Open (X)$ to another category. Now you have a full subcategory $\mathcal Pathconopen (X)\subset \mathcal Open (X)$ obtained by taking only path connected open subsets of $X$ and their inclusions, and you can call a conravariant functor from it a presheaf . All fine and well. However it is impossible to say when a presheaf is a sheaf in that context: the problem is that if a path connected open set $U\subset X$ is covered by path connected open subsets $U_i$, the intersections $U_i\cap U_j$ will in general not be path connected and the sheaf conditions thus don't make sense.