Locally constant sheaf
In algebraic topology, a locally constant sheaf on a topological space X is a sheaf [math]\displaystyle{ \mathcal{F} }[/math] on X such that for each x in X, there is an open neighborhood U of x such that the restriction [math]\displaystyle{ \mathcal{F}|_U }[/math] 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 [math]\displaystyle{ X = \mathbb{C} }[/math], [math]\displaystyle{ \mathcal{O}_X }[/math] be the sheaf of holomorphic functions on X and [math]\displaystyle{ P: \mathcal{O}_X \to \mathcal{O}_X }[/math] given by [math]\displaystyle{ P = z {\partial \over \partial z} - {1 \over 2} }[/math]. Then the kernel of P is a locally constant sheaf on [math]\displaystyle{ X - \{0\} }[/math] but not constant there (since it has no nonzero global section).[1]
If [math]\displaystyle{ \mathcal{F} }[/math] is a locally constant sheaf of sets on a space X, then each path [math]\displaystyle{ p: [0, 1] \to X }[/math] in X determines a bijection [math]\displaystyle{ \mathcal{F}_{p(0)} \overset{\sim}\to \mathcal{F}_{p(1)}. }[/math] Moreover, two homotopic paths determine the same bijection. Hence, there is the well-defined functor
- [math]\displaystyle{ \Pi_1 X \to \mathbf{Set}, \, x \mapsto \mathcal{F}_x }[/math]
where [math]\displaystyle{ \Pi_1 X }[/math] 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 [math]\displaystyle{ \Pi_1 X \to \mathbf{Set} }[/math] is of the above form; i.e., the functor category [math]\displaystyle{ \mathbf{Fct}(\Pi_1 X, \mathbf{Set}) }[/math] 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
- ↑ Kashiwara & Schapira 2002, Example 2.9.14.
- ↑ Szamuely, Tamás (2009). "Fundamental Groups in Topology". Galois Groups and Fundamental Groups. Cambridge University Press. pp. 57. ISBN 9780511627064. https://www.cambridge.org/core/books/galois-groups-and-fundamental-groups/2511B1C10ACF174A0F444A045D9C1F89#.
- ↑ Mac Lane, Saunders (1992). "Sheaves of sets". Sheaves in geometry and logic : a first introduction to topos theory. Ieke Moerdijk. New York: Springer-Verlag. pp. 104. ISBN 0-387-97710-4. OCLC 24428855. https://books.google.com/books?id=SGwwDerbEowC&pg=PA104.
- Kashiwara, Masaki; Schapira, Pierre (2002). Sheaves on Manifolds. 292. Berlin: Springer. doi:10.1007/978-3-662-02661-8. ISBN 978-3-662-02661-8. https://books.google.com/books?id=qfWcUSQRsX4C&pg=PA131.
- Lurie's, J.. "§ A.1. of Higher Algebra (Last update: September 2017)". https://www.math.ias.edu/~lurie/papers/HA.pdf.
External links
- Locally constant sheaf in nLab
- https://golem.ph.utexas.edu/category/2010/11/locally_constant_sheaves.html (recommended)
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