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Let $\mathcal{A}$ be a locally constant sheaf on a topological space $X$ and let $\sigma:\Delta_p\to X$ denote a singular $p$-simplex. Writing the pullback of $\mathcal A$ by $\sigma$ as $\sigma^\ast(\mathcal A)$, Bredon's book on sheaf theory (page 26 in the second edition) says:

Since $\mathcal{A}$ is locally constant and $\Delta_p$ is simply connected, $\sigma^{\ast}(\mathcal{A})$ is a constant sheaf on $\Delta_p$.

(Emphasis mine).

I realize this is pretty basic, but I can't seem to figure out why the simply-connectedness of $\Delta_p$ enters into this. Is it in general true that the pullback of a locally constant sheaf by a continuous function whose domain is simply connected is a constant sheaf on said domain? Could someone hint at a proof for this?

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To get some intuition, you might try to find, say, a map $f:S^1 \rightarrow X$ and a locally constant sheaf $\mathscr{F}$ over $X$ such that $f^*\mathscr{F}$ isn't constant. – Aaron Mazel-Gee May 16 '12 at 19:25
up vote 4 down vote accepted

The espace étalé of a locally constant sheaf on a space $X$ is a covering space of $X$. If $X$ is simply connected, any covering space is trivial, and thus the sheaf you started with must be constant.

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I'm sorry gspr, did my minor edit (removing the word "additionally") make your comment disappear? You asked how to see this without using espaces étalé (is that the right plural?) since Bredon uses this fact before he discusses the espace étalé. I don't know the answer to that question. – Omar Antolín-Camarena May 16 '12 at 13:13
No, no problem. I realized my comment was nonsensical. Everything you say is essentially direct. – gspr May 16 '12 at 13:14
Oh, OK. I thought for a second that the StackExchange software deleted all comments on edited answers to avoid the situation depicted in this comic. – Omar Antolín-Camarena May 16 '12 at 13:21
Dear Omar, the plural is espaces étalés. – Georges Elencwajg May 16 '12 at 15:14
Thanks @GeorgesElencwajg. – Omar Antolín-Camarena May 16 '12 at 15:55

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