presheaf

Noun

 * 1)  An abstract mathematical construct which associates data to the open sets of a topological space,  generalizing the situation of functions, fiber bundles, manifold structure, etc. on a topological space (but not necessarily in such a way as to make the local and global data compatible, as in a sheaf). Formally, A contravariant functor $$\mathcal{F}$$ whose domain is a category whose objects are open sets of a topological space (called the base space or underlying space) and whose morphisms are inclusion mappings. The image of each open set under $$\mathcal{F}$$ is an object whose elements are called sections, and are which are said to be over the given open set; the image of each inclusion map $$A \to B $$ under $$\mathcal{F}$$ is a morphism $$\mathcal{F}(B) \to \mathcal{F}(A)$$, called the restriction from $$B$$ to $$A$$ and denoted $$\operatorname{res}_{B,A}$$ or $$|_{B,A}$$.

Usage notes

 * If the base space is denoted as X and the presheaf's codomain is denoted A, then the presheaf is said to be "on X, with values in A".

Hyponyms

 * sheaf