Cosheaf

Definition

We associate to a topological space $X$ its category of open sets $\mathrm{Op}(X)$, whose objects are the open sets of $X$, with a (unique) morphism from $U$ to $V$ whenever $U\subset V$. Fix a category ${𝒞}$. Then a precosheaf (with values in ${𝒞}$) is a covariant functor $F:\mathrm{Op}X\to {𝒞}$, i.e., $F$ consists of

• for each open set $U$ of $X$, an object $F(U)$ in ${𝒞}$, and
• for each inclusion of open sets $U\subset V$, a morphism ${\iota }_{U,V}:F(U)\to F(V)$ in ${𝒞}$ such that
  • ${\iota }_{U,U}={\mathrm{i}\mathrm{d}}_{F(U)}$ for all $U$ and
  • ${\iota }_{U,V}\circ {\iota }_{V,W}={\iota }_{U,W}$ whenever $U\subset V\subset W$.

Suppose now that ${𝒞}$ is an abelian category that admits small colimits. Then a cosheaf is a precosheaf $F$ for which the sequence

$$\underset{(\alpha ,\beta )}{⨁}F(U_{\alpha ,\beta })\stackrel{\sum _{(\alpha ,\beta )}({\iota }_{U_{\alpha ,\beta },U_{\alpha }}-{\iota }_{U_{\alpha ,\beta },U_{\beta }})}{\to }\underset{\alpha }{⨁}F(U_{\alpha })\stackrel{\sum _{\alpha }{\iota }_{U_{\alpha },U}}{\to }F(U)\to 0$$

is exact for every collection $\{U_{\alpha }{\}}_{\alpha }$ of open sets, where $U:=\bigcup _{\alpha }{U}_{\alpha }$ and $U_{\alpha ,\beta }:=U_{\alpha }\cap U_{\beta }$. (Notice that this is dual to the sheaf condition.) Approximately, exactness at $F(U)$ means that every element over $U$ can be represented as a finite sum of elements that live over the smaller opens $U_{\alpha }$, while exactness at $\underset{\alpha }{⨁}F(U_{\alpha })$ means that, when we compare two such representations of the same element, their difference must be captured by a finite collection of elements living over the intersections $U_{\alpha ,\beta }$.

Equivalently, $F$ is a cosheaf if

• for all open sets $U$ and $V$, $F(U\cup V)$ is the pushout of $F(U\cap V)\to F(U)$ and $F(U\cap V)\to F(V)$, and
• for any upward-directed family $\{U_{\alpha }{\}}_{\alpha }$ of open sets, the canonical morphism $\underrightarrow{\lim }F(U_{\alpha })\to F\left(\bigcup _{\alpha }{U}_{\alpha }\right)$ is an isomorphism. One can show that this definition agrees with the previous one. This one, however, has the benefit of making sense even when ${𝒞}$ is not an abelian category.

Examples

A motivating example of a precosheaf of abelian groups is the singular precosheaf, sending an open set $U$ to $C_k(U;\mathbb{\mathbb{Z}})$, the free abelian group of singular $k$-chains on $U$. In particular, there is a natural inclusion ${\iota }_{U,V}:C_k(U;\mathbb{\mathbb{Z}})\to C_k(V;\mathbb{\mathbb{Z}})$ whenever $U\subset V$. However, this fails to be a cosheaf because a singular simplex cannot be broken up into smaller pieces. To fix this, we let $s:C_k(U;\mathbb{\mathbb{Z}})\to C_k(U;\mathbb{\mathbb{Z}})$ be the barycentric subdivision homomorphism and define ${\bar{C}}_k(U;\mathbb{\mathbb{Z}})$ to be the colimit of the diagram

$$C_k(U;\mathbb{\mathbb{Z}})\stackrel{s}{\to }{C}_k(U;\mathbb{\mathbb{Z}})\stackrel{s}{\to }{C}_k(U;\mathbb{\mathbb{Z}})\stackrel{s}{\to }\dots .$$

In the colimit, a simplex is identified with all of its barycentric subdivisions. One can show using the Lebesgue number lemma that the precosheaf sending $U$ to ${\bar{C}}_k(U;\mathbb{\mathbb{Z}})$ is in fact a cosheaf.

Fix a continuous map $f:Y\to X$ of topological spaces. Then the precosheaf (on $X$) of topological spaces sending $U$ to $f^{-1}(U)$ is a cosheaf.

References

• 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. 
• 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 (2014). "§ 3, in particular Thm 3.10". Sheaves, cosheaves and applications (Doctoral dissertation). University of Pennsylvania. p. 34. ProQuest 1553207954.