Sheaf as a functor

Question

Let $X$ be any topological space, $S$ - any category (e.g. of sets). Consider a new category $C$: its objects are only open subsets of $X$ and a set of morphisms from $U$ to $V$ is nonempty if and only if $U\subset V$. In this case we define the only one morphism $\phi_{UV}:U\rightarrow V$, that includes naturally $U$ into $V$. Thus we can define a presheaf of sets on $X$ as a contravariant functor $\Phi$ from $C$ to $S$ (that functor turns inclusion $U\subset V$ into restriction $res_{VU}$ - a morphism in $S$).
Perhaps, the following question is too popular, but i cannot get the answer myself.
Is it possible to define a sheaf in the same way? I mean some simple categorical terms.

Answer 1

Yes, it is. If the target category is complete, the gluing axiom translates into $F(U) \rightarrow \prod_i F(U_i) \rightrightarrows \prod_{i,j} F(U_i \cap U_j)$ being a difference kernel diagram. This is actually a very common definition of sheafs, especially since it is no longer dependend on the starting category being the category of open sets of a topological space. On can (and does) define (pre)sheaves on any category in this way. This is especially useful when working with Grothendieck topologies etc.

Greetings.