Locally constant sheaf

Short description: Sheaf theory

In algebraic topology, a locally constant sheaf on a topological space X is a sheaf ${ℱ}$ on X such that for each x in X, there is an open neighborhood U of x such that the restriction ${ℱ}{|}_U$ 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 $X=\mathbb{ℂ}$, ${{𝒪}}_X$ be the sheaf of holomorphic functions on X and $P:{{𝒪}}_X\to {{𝒪}}_X$ given by $P=z\frac{\partial }{\partial z}-\frac{1}{2}$. 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 ${ℱ}$ is a locally constant sheaf of sets on a space X, then each path $p:[0,1]\to X$ in X determines a bijection ${{ℱ}}_{p(0)}\stackrel{\sim }{\to }{{ℱ}}_{p(1)}.$ Moreover, two homotopic paths determine the same bijection. Hence, there is the well-defined functor

${\mathrm{\Pi }}_{1}X\to \mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭},x\mapsto {{ℱ}}_x$

where ${\mathrm{\Pi }}_{1}X$ 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 ${\mathrm{\Pi }}_{1}X\to \mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭}$ is of the above form; i.e., the functor category $\mathbf{𝐅}\mathbf{𝐜}\mathbf{𝐭}({\mathrm{\Pi }}_{1}X,\mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭})$ 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.

References

• 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)