Cosheaf

In topology, a branch of mathematics, a cosheaf is a dual notion to that of a sheaf that is useful in studying Borel-Moore homology.

Definition

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

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

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

$$\bigoplus _{(\alpha ,\beta )}F(U_{\alpha ,\beta })\xrightarrow {\sum _{(\alpha ,\beta )}(\iota _{U_{\alpha ,\beta },U_{\alpha }}-\iota _{U_{\alpha ,\beta },U_{\beta }})} \bigoplus _{\alpha }F(U_{\alpha })\xrightarrow {\sum _{\alpha }\iota _{U_{\alpha },U}} F(U)\to 0$$

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 $\bigoplus _{\alpha }F(U_{\alpha })$ 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 $\varinjlim F(U_{\alpha })\to F(\bigcup _{\alpha }U_{\alpha })$ 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 ${\mathcal {C}}$ 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 $C_{k}(U;\mathbb {Z} )$, the free abelian group of singular $k$-chains on $U$. In particular, there is a natural inclusion $\iota _{U,V}:C_{k}(U;\mathbb {Z} )\to C_{k}(V;\mathbb {Z} )$ 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 $s:C_{k}(U;\mathbb {Z} )\to C_{k}(U;\mathbb {Z} )$ be the barycentric subdivision homomorphism and define ${\overline {C}}_{k}(U;\mathbb {Z} )$ to be the colimit of the diagram

$$C_{k}(U;\mathbb {Z} )\xrightarrow {s} C_{k}(U;\mathbb {Z} )\xrightarrow {s} C_{k}(U;\mathbb {Z} )\xrightarrow {s} \ldots .$$

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 ${\overline {C}}_{k}(U;\mathbb {Z} )$ 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.
• 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.
• Curry, Justin Michael (2015). "Topological data analysis and cosheaves". Japan Journal of Industrial and Applied Mathematics. 32 (2): 333–371. arXiv:1411.0613. doi:10.1007/s13160-015-0173-9. S2CID 256048254.
• Positselski, Leonid (2012). "Contraherent cosheaves". arXiv:1209.2995 [math.CT].
• Rosiak, Daniel (25 October 2022). Sheaf Theory through Examples. MIT Press. ISBN 9780262362375.
• Lurie, Jacob. "Tamagawa Numbers via Nonabelian Poincare Duality, Lecture 8: Nonabelian Poincare Duality in Topology" (PDF). School of Mathematics, Institute for Advanced Study.
• Curry, Justin (2014). "§ 3, in particular Thm 3.10". Sheaves, cosheaves and applications (Doctoral dissertation). University of Pennsylvania. p. 34. arXiv:1303.3255. ProQuest 1553207954.