Cosheaf

In topology, a branch of mathematics, a cosheaf with values in an ∞-category C that admits colimits is a functor F from the category of open subsets of a topological space X (more precisely its nerve) to C such that

• (1) The F of the empty set is the initial object.
• (2) For any increasing sequence [math]\displaystyle{ U_i }[/math] of open subsets with union U, the canonical map [math]\displaystyle{ \varinjlim F(U_i) \to F(U) }[/math] is an equivalence.
• (3) [math]\displaystyle{ F(U \cup V) }[/math] is the pushout of [math]\displaystyle{ F(U \cap V) \to F(U) }[/math] and [math]\displaystyle{ F(U \cap V) \to F(V) }[/math].

The basic example is [math]\displaystyle{ U \mapsto C_*(U; A) }[/math] where on the right is the singular chain complex of U with coefficients in an abelian group A.

Example: If f is a continuous map, then [math]\displaystyle{ U \mapsto f^{-1}(U) }[/math] is a cosheaf.

See also

• sheaf (mathematics)

References

• Lurie, Jacob. "Tamagawa Numbers via Nonabelian Poincare Duality, Lecture 8: Nonabelian Poincare Duality in Topology". School of Mathematics, Institute for Advanced Study.. http://www.math.harvard.edu/~lurie/282ynotes/LectureVIII-Poincare.pdf. 
• Curry, Justin (2013). Sheaves, cosheaves and applications, § 3, in particular Thm 3.10 p. 34. ProQuest 18750. 
• Bredon, Glen E. (24 January 1997). Sheaf Theory. Springer. ISBN 9780387949055. https://books.google.com/books?id=zGdqWepiT1QC&q=cosheaf. 
• Bredon, Glen (1968). "Cosheaves and homology". Pacific Journal of Mathematics 25: 1–32. doi:10.2140/pjm.1968.25.1. 
• Funk, J. (1995). "The display locale of a cosheaf". Cahiers de Topologie et Géométrie Différentielle Catégoriques 36 (1): 53–93. http://eudml.org/doc/91560. 
• Curry, Justin Michael (2015). "Topological data analysis and cosheaves". Japan Journal of Industrial and Applied Mathematics 32 (2): 333–371. doi:10.1007/s13160-015-0173-9. 
• Positselski, Leonid (2012). "Contraherent cosheaves". arXiv:1209.2995 [math.CT].
• Rosiak, Daniel (25 October 2022). Sheaf Theory through Examples. MIT Press. ISBN 9780262362375. https://books.google.com/books?id=HudaEAAAQBAJ&pg=PA306.