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| − | ''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083740/s0837401.png" />'' | + | ''$p:X \rightarrow Y$'' |
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| − | A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083740/s0837402.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083740/s0837403.png" />. In a wider sense, a section of any morphism in an arbitrary category is a right-inverse morphism. | + | A mapping $s : Y \rightarrow X$ for which $p \circ s = \mathrm{id}_Y$. In a wider sense, a section of any morphism in an arbitrary category is a right-inverse morphism. |
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| | + | ====Comments==== |
| | + | If $U \subset Y$ is a subset of $Y$, a section over $U$ of $p$ is a mapping $s : U \rightarrow X$ such that $p(s(u)) = u$ for all $u \in U$. |
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| | + | For a vector bundle $E \stackrel{p}{\rightarrow} Y$, where the mapping $p$ is part of the structure defined, one speaks of a section of the vector bundle $E$ rather than of a section of $p$. This applies, e.g., also to sheaves and fibrations. A standard notation for the set of sections in such a case is $\Gamma(E)$, or $\Gamma(U,E)$ for the set of sections of $E$ over $U$. |
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| − | ====Comments====
| + | {{TEX|done}} |
| − | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083740/s0837404.png" /> is a subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083740/s0837405.png" />, a section over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083740/s0837406.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083740/s0837407.png" /> is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083740/s0837408.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083740/s0837409.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083740/s08374010.png" />. For a vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083740/s08374011.png" />, where the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083740/s08374012.png" /> is part of the structure defined, one speaks of a section of the vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083740/s08374013.png" /> rather than of a section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083740/s08374014.png" />. This applies, e.g., also to sheaves and fibrations. A standard notation for the set of sections in such a case is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083740/s08374015.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083740/s08374016.png" /> for the set of sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083740/s08374017.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083740/s08374018.png" />.
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Revision as of 18:10, 15 November 2014
$p:X \rightarrow Y$
A mapping $s : Y \rightarrow X$ for which $p \circ s = \mathrm{id}_Y$. In a wider sense, a section of any morphism in an arbitrary category is a right-inverse morphism.
If $U \subset Y$ is a subset of $Y$, a section over $U$ of $p$ is a mapping $s : U \rightarrow X$ such that $p(s(u)) = u$ for all $u \in U$.
For a vector bundle $E \stackrel{p}{\rightarrow} Y$, where the mapping $p$ is part of the structure defined, one speaks of a section of the vector bundle $E$ rather than of a section of $p$. This applies, e.g., also to sheaves and fibrations. A standard notation for the set of sections in such a case is $\Gamma(E)$, or $\Gamma(U,E)$ for the set of sections of $E$ over $U$.
How to Cite This Entry:
Section of a mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Section_of_a_mapping&oldid=18495
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article