# Pre-sheaf

From Encyclopedia of Mathematics

*on a topological space with values in a category (e.g. the category of sets, groups, modules, rings, etc.)*

A contravariant functor from the category of open sets of and their natural inclusion mappings into . Depending on , the functor is called a pre-sheaf of sets, groups, modules, rings, etc. The morphisms corresponding to the inclusions are called restriction homomorphisms.

Every pre-sheaf generates a sheaf on (cf. Sheaf theory).

#### Comments

More generally, if is any small category, the term "pre-sheaf on C" is used to denote a contravariant (usually set-valued) functor defined on (cf. Site).

**How to Cite This Entry:**

Pre-sheaf.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Pre-sheaf&oldid=16592

This article was adapted from an original article by E.G. Sklyarenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article