Let $X$ be a topological space and let $\mathcal B$ be a basis of its topology. We can regard $\mathcal B$ as a category with objects being the open sets in $\mathcal B$ and arrows/morphisms between $U,V\in\mathcal B$ to exists only if $U\subset V$ , and where the only element of $\mathcal B(U,V)$ is the injection map $U\hookrightarrow V$ . Let now $\Ccal$ be a complete category, we now define the presheaf of $\Ccal$ -objects over the basis $\mathcal B$ of the topology of $X$ to be a contravariant functor $$\P:\mathcal B\rightarrow \Ccal$$