Locally constant sheaf

{{Short description|Sheaf theory}}

In algebraic topology, a locally constant sheaf on a topological space X is a sheaf $\backslash mathcal\{F\}$ on X such that for each x in X, there is an open neighborhood U of x such that the restriction $\backslash mathcal\{F\}|\_U$ is a constant sheaf on U. It is also called a local system. When X is a stratified space, a constructible sheaf is roughly a sheaf that is locally constant on each member of the stratification.

A basic example is the orientation sheaf on a manifold since each point of the manifold admits an orientable open neighborhood (while the manifold itself may not be orientable).

For another example, let $X\; =\; \backslash mathbb\{C\}$, $\backslash mathcal\{O\}\_X$ be the sheaf of holomorphic functions on X and $P:\; \backslash mathcal\{O\}\_X\; \backslash to\; \backslash mathcal\{O\}\_X$ given by $P\; =\; z\; \{\backslash partial\; \backslash over\; \backslash partial\; z\}\; -\; \{1\; \backslash over\; 2\}$. Then the kernel of P is a locally constant sheaf on $X\; -\; \backslash \{0\backslash \}$ but not constant there (since it has no nonzero global section).{{harvnb|Kashiwara|Schapira|2002|loc=Example 2.9.14.}}

If $\backslash mathcal\{F\}$ is a locally constant sheaf of sets on a space X, then each path $p:\; [0,\; 1]\; \backslash to\; X$ in X determines a bijection $\backslash mathcal\{F\}\_\{p(0)\}\; \backslash overset\{\backslash sim\}\backslash to\; \backslash mathcal\{F\}\_\{p(1)\}.$ Moreover, two homotopic paths determine the same bijection. Hence, there is the well-defined functor

:$\backslash Pi\_1\; X\; \backslash to\; \backslash mathbf\{Set\},\; \backslash ,\; x\; \backslash mapsto\; \backslash mathcal\{F\}\_x$

where $\backslash Pi\_1\; X$ is the fundamental groupoid of X: the category whose objects are points of X and whose morphisms are homotopy classes of paths. Moreover, if X is path-connected, locally path-connected and semi-locally simply connected (so X has a universal cover), then every functor $\backslash Pi\_1\; X\; \backslash to\; \backslash mathbf\{Set\}$ is of the above form; i.e., the functor category $\backslash mathbf\{Fct\}(\backslash Pi\_1\; X,\; \backslash mathbf\{Set\})$ is equivalent to the category of locally constant sheaves on X.

If X is locally connected, the adjunction between the category of presheaves and bundles restricts to an equivalence between the category of locally constant sheaves and the category of covering spaces of X.{{Cite book|last=Szamuely|first=Tamás|url=https://www.cambridge.org/core/books/galois-groups-and-fundamental-groups/2511B1C10ACF174A0F444A045D9C1F89#|title=Galois Groups and Fundamental Groups|chapter=Fundamental Groups in Topology| date=2009|publisher=Cambridge University Press|isbn=9780511627064|pages=57}}{{Cite book|last=Mac Lane|first=Saunders|url=https://www.worldcat.org/oclc/24428855|title=Sheaves in geometry and logic : a first introduction to topos theory|chapter=Sheaves of sets|chapter-url={{Google books|SGwwDerbEowC|page=104|plainurl=yes}}| date=1992|publisher=Springer-Verlag|others=Ieke Moerdijk|isbn=0-387-97710-4|location=New York|pages=104|oclc=24428855}}