Locally constant sheaf: Difference between revisions
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*{{Citation | last1=Kashiwara | first1=Masaki | last2=Schapira | first2=Pierre | author1-link=Masaki Kashiwara | title=Sheaves on Manifolds | isbn=3540518614 | year=2002 | publisher=Springer | location=Berlin}} |
*{{Citation | last1=Kashiwara | first1=Masaki | last2=Schapira | first2=Pierre | author1-link=Masaki Kashiwara | title=Sheaves on Manifolds | isbn=3540518614 | year=2002 | publisher=Springer | location=Berlin}} |
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*{{cite book | last1=Kashiwara | first1=Masaki | last2=Schapira | first2=Pierre | author1-link=Masaki Kashiwara|doi=10.1007/978-3-662-02661-8|title=Sheaves on Manifolds publisher=Springer | location=Berlin |year=2002 |volume=292 |isbn=978-3-662-02661-8| url={{Google books|qfWcUSQRsX4C|page=131|plainurl=yes}}}} |
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*{{cite web |last1=Lurie's |first1=J. |title=§ A.1. of Higher Algebra (Last update: September 2017)|url=https://www.math.ias.edu/~lurie/papers/HA.pdf}} |
*{{cite web |last1=Lurie's |first1=J. |title=§ A.1. of Higher Algebra (Last update: September 2017)|url=https://www.math.ias.edu/~lurie/papers/HA.pdf}} |
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Revision as of 06:53, 3 December 2022
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]
References
- ^ Kashiwara–Schapira, Example 2.9.14.
- ^ 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, Berlin: Springer, ISBN 3540518614
- Kashiwara, Masaki; Schapira, Pierre (2002). Sheaves on Manifolds publisher=Springer. Vol. 292. Berlin. doi:10.1007/978-3-662-02661-8. ISBN 978-3-662-02661-8.
{{cite book}}
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(help)CS1 maint: location missing publisher (link) - Lurie's, J. "§ A.1. of Higher Algebra (Last update: September 2017)" (PDF).
External links
- https://ncatlab.org/nlab/show/locally+constant+sheaf
- https://golem.ph.utexas.edu/category/2010/11/locally_constant_sheaves.html (recommended)