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A sheaf is a [[Pre-sheaf|pre-sheaf]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s0848301.png" /> (cf. also [[Sheaf theory|Sheaf theory]]) on a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s0848302.png" /> such that for every union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s0848303.png" /> of open subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s0848304.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s0848305.png" /> the following conditions are satisfied:
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{{MSC|14}}
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{{TEX|done}}
  
a) if on every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s0848306.png" /> the restrictions of two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s0848307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s0848308.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s0848309.png" /> coincide, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s08483010.png" />;
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A sheaf is a
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[[Pre-sheaf|pre-sheaf]] $F$ (cf. also
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[[Sheaf theory|Sheaf theory]]) on a topological space $X$ such that for every union $\def\l{\lambda} U=\bigcup_\l U_\l$ of open subsets $U_\l$ of $X$ the following conditions are satisfied:
  
b) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s08483011.png" /> are such that for any pair of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s08483012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s08483013.png" /> the restrictions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s08483014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s08483015.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s08483016.png" /> coincide, then there exists an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s08483017.png" /> which on each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s08483018.png" /> has restriction coinciding with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s08483019.png" />.
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a) if on every $U_\l$ the restrictions of two elements $s$ and $s'$ in $F(U)$ coincide, then $s'=s$;
  
Every sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s08483020.png" /> is isomorphic to the sheaf of continuous sections of a certain covering space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s08483021.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s08483022.png" />, which is determined uniquely up to an isomorphism (by a covering space one means a continuous mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s08483023.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s08483024.png" /> which is a local homeomorphism), therefore a sheaf is also commonly understood to be the covering space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084830/s08483025.png" /> itself (see [[Sheaf theory|Sheaf theory]]).
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b) if $s_\l\in F(U_\l)$ are such that for any pair of indices $\l$ and $\mu$ the restrictions of $s_\l$ and $s_\mu$ to $U_\l\cap U_\mu$ coincide, then there exists an element $s\in F(U)$ which on each $U_\l$ has restriction coinciding with $s_\l$.
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Every sheaf on $X$ is isomorphic to the sheaf of continuous [[section]]s of a certain [[covering]] space $p:E\to X$ over $X$, which is determined uniquely up to an isomorphism (by a covering space one means a continuous mapping from $E$ onto $X$ which is a local homeomorphism), therefore a sheaf is also commonly understood to be the covering space $p:E\to X$ itself (see
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[[Sheaf theory|Sheaf theory]]).
  
  
  
 
====Comments====
 
====Comments====
Generalizing the above notion of a sheaf on a topological space, it is also possible to define sheaves on an arbitrary [[Site|site]]. Cf. also [[Topos|Topos]].
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Generalizing the above notion of a sheaf on a topological space, it is also possible to define sheaves on an arbitrary
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[[Site|site]]. Cf. also
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[[Topos|Topos]].
  
For a more detailed treatment of sheaves, and additional references, see [[Sheaf theory|Sheaf theory]].
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For a more detailed treatment of sheaves, and additional references, see
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[[Sheaf theory|Sheaf theory]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G.E. Bredon, "Sheaf theory" , McGraw-Hill (1967) {{MR|0221500}} {{ZBL|0158.20505}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Godement, "Topologie algébrique et théorie des faisceaux" , Hermann (1958) {{MR|0102797}} {{ZBL|0080.16201}} </TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|Br}}||valign="top"| G.E. Bredon, "Sheaf theory", McGraw-Hill (1967) {{MR|0221500}} {{ZBL|0158.20505}}
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|-
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|valign="top"|{{Ref|Go}}||valign="top"| R. Godement, "Topologie algébrique et théorie des faisceaux", Hermann (1958) {{MR|0102797}} {{ZBL|0080.16201}}
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|-
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|}

Revision as of 14:43, 24 November 2013

2020 Mathematics Subject Classification: Primary: 14-XX [MSN][ZBL]

A sheaf is a pre-sheaf $F$ (cf. also Sheaf theory) on a topological space $X$ such that for every union $\def\l{\lambda} U=\bigcup_\l U_\l$ of open subsets $U_\l$ of $X$ the following conditions are satisfied:

a) if on every $U_\l$ the restrictions of two elements $s$ and $s'$ in $F(U)$ coincide, then $s'=s$;

b) if $s_\l\in F(U_\l)$ are such that for any pair of indices $\l$ and $\mu$ the restrictions of $s_\l$ and $s_\mu$ to $U_\l\cap U_\mu$ coincide, then there exists an element $s\in F(U)$ which on each $U_\l$ has restriction coinciding with $s_\l$.

Every sheaf on $X$ is isomorphic to the sheaf of continuous sections of a certain covering space $p:E\to X$ over $X$, which is determined uniquely up to an isomorphism (by a covering space one means a continuous mapping from $E$ onto $X$ which is a local homeomorphism), therefore a sheaf is also commonly understood to be the covering space $p:E\to X$ itself (see Sheaf theory).


Comments

Generalizing the above notion of a sheaf on a topological space, it is also possible to define sheaves on an arbitrary site. Cf. also Topos.

For a more detailed treatment of sheaves, and additional references, see Sheaf theory.

References

[Br] G.E. Bredon, "Sheaf theory", McGraw-Hill (1967) MR0221500 Zbl 0158.20505
[Go] R. Godement, "Topologie algébrique et théorie des faisceaux", Hermann (1958) MR0102797 Zbl 0080.16201
How to Cite This Entry:
Sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sheaf&oldid=23571
This article was adapted from an original article by E.G. Sklyarenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article