Exercise concerning locally constant presheaves - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-24T09:06:34Z http://mathoverflow.net/feeds/question/64547 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/64547/exercise-concerning-locally-constant-presheaves Exercise concerning locally constant presheaves Jesko Hüttenhain 2011-05-10T22:42:57Z 2011-05-10T22:42:57Z <p>Let $\mathscr{F}$ be a presheaf of abelian groups on some topological space $X$. We say that $\mathscr{F}$ is <b>locally constant</b> if there exists an open cover $\mathcal{U}$ of $X$ (i.e. $X=\bigcup_{U\in\mathcal{U}}\ U$ and the elements of $\mathcal{U}$ are open) such that, for all $U\in\mathcal{U}$ and for all $P\in U$, we have $\mathscr{F}(U)=\mathscr{F}_P$.</p> <p>The task is to prove that for every $U\in\mathcal{U}$ and every connected subset $V\subseteq U$, the composite </p> <p>$\mathscr{F}(U) \xrightarrow{\ \text{restriction}\ } \mathscr{F}(V) \xrightarrow{\ \text{sheafification}\ } \mathscr{F}^+(V)$</p> <p>is an isomorphism. This is an exercise in the Book on Algebraic Topology by Spanier. A fellow student asked me about it because he needs the result, but we were unable to figure it out.</p>