presheaf.

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 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 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→B under ℱ is a morphism ℱ(B)→ℱ(A), called the restriction from B to A and denoted operatorname res_(B,A) or |_(B,A).