presheaf of a topological basis

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

$$𝒫:ℬ\to 𝒞$$

Title	presheaf of a topological basis
Canonical name	PresheafOfATopologicalBasis
Date of creation	2013-03-22 16:22:36
Last modified on	2013-03-22 16:22:36
Owner	jocaps (12118)
Last modified by	jocaps (12118)
Numerical id	14
Author	jocaps (12118)
Entry type	Definition
Classification	msc 14F05
Classification	msc 54B40
Classification	msc 18F20
Related topic	site