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Generally, a section is a right inverse. On the other hand, if $F$ is a ($\mathsf{Set}$-valued) sheaf, then the elements of $FU$ are usually also called sections.

Why is this terminology justified and how is it related to right inverses?

Please do not rely on any theory of bundles (I don't know any).

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    $\begingroup$ Are you familiar with sheaves as spaces with a projection (almost like covering spaces)? $\endgroup$ May 13, 2015 at 9:19
  • $\begingroup$ @DanielFischer no :( $\endgroup$
    – user153312
    May 13, 2015 at 10:01
  • $\begingroup$ @DanielFischer I now understand you were talking about étalé spaces. I assume your answer would have been in the same spirit as the one given by N.H below. $\endgroup$
    – user153312
    Jun 2, 2015 at 19:26
  • $\begingroup$ You assume correctly. Except that I'm used to calling these things sheaves rather than étalé spaces, so I'd have phrased it differently. $\endgroup$ Jun 2, 2015 at 19:34

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All this is very classic, you can find it in the book of Hartshorne for example.

Assume we have a space $X$ and an local homeomorphism $\pi : Y \to X$. For every open $U \subset X$ we can look at the continuous maps $s : U \to Y$ such that $\pi \circ s = id_U$. These maps are "sections" in the classical sens, and of course you can take the restriction. Therefore, this space gives you a sheaf $\mathscr S(U) := \{s:U \to Y \mid \pi \circ s = id_U, \text{s is continous} \}$. The canonical examples are the covering spaces (or vector bundles,etc...).

But in fact every sheaf can be obtained in this way. Now assume you have a sheaf $\mathscr F$ on some space $X$. You can construct a space $Y$, called the étalé space of $\mathscr F$. As a set, $Y = \cup_{p \in X}\mathscr F_x$. We obtain a covering map $\pi :Y \to X$. We define the topology on $Y$ in the following way : if $f \in \mathscr F(U)$ then $[U,f]$ be all the germs $s_y$ such that $s_y = f_y$ and $y \in U$. Then, all the $[U,f]$ form a basis for a topology, and $\pi$ is now a local homeomorphism. We just have to see that we recover the original sheaf but the stalk $\mathscr F_x \cong \mathscr S_x$ are naturally isomorphic so every sheaf arise as the sheaf of sections of an étalé space.

So a sheaf could be seen exactly as a map $\pi : Y \to X $ which is a local homeomorphism. In fact, this is the definition taken in the book of Godement, or FAC for example.

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    $\begingroup$ In other words, the justification for the name 'section' is simply the fact each sheaf is the sheaf of sections over its étalé space? $\endgroup$
    – user153312
    Jun 2, 2015 at 19:24
  • $\begingroup$ Yes, that's the explanation ! $\endgroup$
    – user171326
    Jun 2, 2015 at 19:40
  • $\begingroup$ Relevant. $\endgroup$ Jul 19, 2023 at 11:22

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