Locally constant sheaf

Last updated April 27, 2024

In algebraic topology, a locally constant sheaf on a topological space X is a sheaf ${\mathcal {F}}$ on X such that for each x in X, there is an open neighborhood U of x such that the restriction ${\mathcal {F}}|_{U}$ 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 $X=\mathbb {C}$ , ${\mathcal {O}}_{X}$ be the sheaf of holomorphic functions on X and $P:{\mathcal {O}}_{X}\to {\mathcal {O}}_{X}$ 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).^[1]

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

\Pi _{1}X\to \mathbf {Set} ,\,x\mapsto {\mathcal {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 $\Pi _{1}X\to \mathbf {Set}$ is of the above form; i.e., the functor category $\mathbf {Fct} (\Pi _{1}X,\mathbf {Set} )$ 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.^[2]^[3]

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References

↑ 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

Locally constant sheaf at the nLab
https://golem.ph.utexas.edu/category/2010/11/locally_constant_sheaves.html (recommended)

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[1] Kashiwara & Schapira 2002 , Example 2.9.14.

[2] Szamuely, Tamás (2009). "Fundamental Groups in Topology". Galois Groups and Fundamental Groups. Cambridge University Press. p. 57. ISBN 9780511627064.

[3] 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.

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