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Difference between revisions of "Pre-sheaf"

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''on a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074340/p0743401.png" /> with values in a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074340/p0743402.png" /> (e.g. the category of sets, groups, modules, rings, etc.)''
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{{MSC|14}}
  
A contravariant [[Functor|functor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074340/p0743403.png" /> from the [[Category|category]] of open sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074340/p0743404.png" /> and their natural inclusion mappings into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074340/p0743405.png" />. Depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074340/p0743406.png" />, the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074340/p0743407.png" /> is called a pre-sheaf of sets, groups, modules, rings, etc. The morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074340/p0743408.png" /> corresponding to the inclusions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074340/p0743409.png" /> are called restriction homomorphisms.
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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.)
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is a contravariant
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[[Functor|functor]] $F$ from the
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[[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074340/p07434010.png" /> (cf. [[Sheaf theory|Sheaf theory]]).
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Every pre-sheaf generates a
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[[Sheaf|sheaf]] on $X$ (cf.
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[[Sheaf theory|Sheaf theory]]).
  
  
  
====Comments====
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====Comment====
More generally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074340/p07434011.png" /> is any [[Small category|small category]], the term  "pre-sheaf on C"  is used to denote a contravariant (usually set-valued) functor defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074340/p07434012.png" /> (cf. [[Site|Site]]).
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More generally, if $\def\cC{ {\mathcal C}}\cC$ is any
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[[Small category|small category]], the term  "pre-sheaf on $\cC$"  is used to denote a contravariant (usually set-valued) functor defined on $\cC$ (cf.
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[[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=30766
This article was adapted from an original article by E.G. Sklyarenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article