Locally constant sheaf explained

l{F} on X such that for each x in X, there is an open neighborhood U of x such that the restriction l{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 = C , l{O} _{X} be the sheaf of holomorphic functions on X and P: l{O} _{X} \to l{O} _{X} given by P = z {\partial \over \partial z} - { 1 \over 2 } . Then the kernel of P is a locally constant sheaf on X - \{ 0 \} but not constant there (since it has no nonzero global section).

If

l{F} is a locally constant sheaf of sets on a space X, then each path p: [ 0 , 1 ] \to X in X determines a bijection l{F} _{p(0)} \overset{\sim}\to l{F} _{p(1)} . Moreover, two homotopic paths determine the same bijection. Hence, there is the well-defined functor\Pi _{1} X \to Set , x \mapsto l{F} _{x} where \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 \Pi _{1} X \to Set is of the above form; i.e., the functor category Fct (\Pi _{1} X, 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.^{[1] [2]}

References

• Book: Kashiwara . Masaki . Schapira . Pierre . Masaki Kashiwara. 10.1007/978-3-662-02661-8. [{{Google books|qfWcUSQRsX4C|page=131|plainurl=yes}} Sheaves on Manifolds]. Springer . Berlin . 2002 . 292 . 978-3-662-02661-8.
• Web site: Lurie's . J. . § A.1. of Higher Algebra (Last update: September 2017).

External links

• • https://golem.ph.utexas.edu/category/2010/11/locally_constant_sheaves.html (recommended)

Notes and References

1. Book: Szamuely, Tamás. Galois Groups and Fundamental Groups. Fundamental Groups in Topology. 2009. Cambridge University Press. 9780511627064. 57.
2. Book: Mac Lane, Saunders. Sheaves in geometry and logic : a first introduction to topos theory. Sheaves of sets. . 1992. Springer-Verlag. Ieke Moerdijk. 0-387-97710-4. New York. 104. 24428855.