Difference between revisions of "Pre-sheaf"
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| − | A contravariant [[Functor|functor]] | + | A ''pre-sheaf on a topological space $X$ with values in a category $\def\cK{ {\mathcal K}}\cK$'' (e.g., the category of sets, groups, modules, rings, etc.) |
| + | is a contravariant | ||
| + | [[Functor|functor]] $F$ from the | ||
| + | [[Category|category]] of open sets of $X$ and their natural inclusion mappings into $\cK$. Depending on $\cK$, the functor $F$ is called a pre-sheaf of sets, groups, modules, rings, etc. The morphisms $F(U)\to F(V)$ corresponding to the inclusions $V\subseteq U$ are called restriction homomorphisms. | ||
| − | Every pre-sheaf generates a [[Sheaf|sheaf]] on | + | Every pre-sheaf generates a |
| + | [[Sheaf|sheaf]] on $X$ (cf. | ||
| + | [[Sheaf theory|Sheaf theory]]). | ||
| − | ==== | + | ====Comment==== |
| − | More generally, if | + | More generally, if $\def\cC{ {\mathcal C}}\cC$ is any |
| + | [[Small category|small category]], the term "pre-sheaf on $\cC$" is used to denote a contravariant (usually set-valued) functor defined on $\cC$ (cf. | ||
| + | [[Site|Site]]). | ||
Latest revision as of 16:46, 24 November 2013
2020 Mathematics Subject Classification: Primary: 14-XX [MSN][ZBL]
A pre-sheaf on a topological space $X$ with values in a category $\def\cK{ {\mathcal K}}\cK$ (e.g., the category of sets, groups, modules, rings, etc.) is a contravariant functor $F$ from the category of open sets of $X$ and their natural inclusion mappings into $\cK$. Depending on $\cK$, the functor $F$ is called a pre-sheaf of sets, groups, modules, rings, etc. The morphisms $F(U)\to F(V)$ corresponding to the inclusions $V\subseteq U$ are called restriction homomorphisms.
Every pre-sheaf generates a sheaf on $X$ (cf. Sheaf theory).
Comment
More generally, if $\def\cC{ {\mathcal C}}\cC$ is any small category, the term "pre-sheaf on $\cC$" is used to denote a contravariant (usually set-valued) functor defined on $\cC$ (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