Locally constant sheaf
In algebraic topology, a locally constant sheaf on a topological space X is a sheaf on X such that for each x in X, there is an open neighborhood U of x such that the restriction 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 , be the sheaf of holomorphic functions on X and given by . Then the kernel of P is a locally constant sheaf on but not constant there (since it has no nonzero global section).[1]
If is a locally constant sheaf of sets on a space X, then each path in X determines a bijection Moreover, two homotopic paths determine the same bijection. Hence, there is the well-defined functor
where 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 is of the above form; i.e., the functor category 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.[2][3]
References
[edit]- ^ Kashiwara & Schapira 2002, Example 2.9.14.
- ^ Szamuely, Tamás (2009). "Fundamental Groups in Topology". Galois Groups and Fundamental Groups. Cambridge University Press. p. 57. ISBN 9780511627064.
- ^ Mac Lane, Saunders (1992). "Sheaves of sets". Sheaves in geometry and logic : a first introduction to topos theory. Ieke Moerdijk. New York: Springer-Verlag. p. 104. ISBN 0-387-97710-4. OCLC 24428855.
- Kashiwara, Masaki; Schapira, Pierre (2002). Sheaves on Manifolds. Vol. 292. Berlin: Springer. doi:10.1007/978-3-662-02661-8. ISBN 978-3-662-02661-8.
- Lurie's, J. "§ A.1. of Higher Algebra (Last update: September 2017)" (PDF).
External links
[edit]- Locally constant sheaf at the nLab
- https://golem.ph.utexas.edu/category/2010/11/locally_constant_sheaves.html (recommended)