Locally constant sheaf

In algebraic topology, a locally constant sheaf on a topological space X is a sheaf ${\displaystyle {\mathcal {F}}}$ on X such that for each x in X, there is an open neighborhood U of x such that the restriction ${\displaystyle {\mathcal {F}}|_{U}}$ 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

open neighborhood (while the manifold itself may not be orientable).

For another example, let ${\displaystyle X=\mathbb {C} }$, ${\displaystyle {\mathcal {O}}_{X}}$ be the sheaf of holomorphic functions on X and ${\displaystyle P:{\mathcal {O}}_{X}\to {\mathcal {O}}_{X}}$ given by ${\displaystyle P=z{\partial \over \partial z}-{1 \over 2}}$. Then the kernel of P is a locally constant sheaf on ${\displaystyle X-\{0\}}$ but not constant there (since it has no nonzero global section).^[1]

If ${\displaystyle {\mathcal {F}}}$ is a locally constant sheaf of sets on a space X, then each

${\displaystyle p:[0,1]\to X}$ in X determines a bijection ${\displaystyle {\mathcal {F}}_{p(0)}{\overset {\sim }{\to }}{\mathcal {F}}_{p(1)}.}$ Moreover, two

${\displaystyle \Pi _{1}X\to \mathbf {Set} ,\,x\mapsto {\mathcal {F}}_{x}}$

where ${\displaystyle \Pi _{1}X}$ is the

universal cover

), then every functor ${\displaystyle \Pi _{1}X\to \mathbf {Set} }$ is of the above form; i.e., the functor category ${\displaystyle \mathbf {Fct} (\Pi _{1}X,\mathbf {Set} )}$ is equivalent to the category of locally constant sheaves on X.

If X is

References

• 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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