Presheaf

English

Etymology

From Template:Prefix.

Noun

1. Template:Lb 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 $A\to B$ under $ℱ$ is a morphism $ℱ(B)\to ℱ(A)$, called the restriction from $B$ to $A$ and denoted ${\mathrm{res}}_{B,A}$ or ${|}_{B,A}$.
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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