# presheaf of a topological basis

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 $\mathcal{C}$ be a complete category, we now define the presheaf of $\mathcal{C}$-objects over the basis $\mathcal{B}$ of the topology of $X$ to be a contravariant functor

 $\mathcal{P}:\mathcal{B}\rightarrow\mathcal{C}$
Title presheaf of a topological basis PresheafOfATopologicalBasis 2013-03-22 16:22:36 2013-03-22 16:22:36 jocaps (12118) jocaps (12118) 14 jocaps (12118) Definition msc 14F05 msc 54B40 msc 18F20 site