Pre-sheaf
From Encyclopedia of Mathematics
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
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
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