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.^{[further explanation needed]}

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 $\varinjlim F(U_{\alpha })\to F\left(\bigcup _{\alpha }U_{\alpha }\right)$ is an isomorphism. One can show that this definition agrees with the previous one.^[1] 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.^[2]

Notes

^ Bredon, Glen E. (24 January 1997). Sheaf Theory. Springer. ISBN 9780387949055.
^ Lurie, Jacob. "Tamagawa Numbers via Nonabelian Poincare Duality, Lecture 9: Nonabelian Poincare Duality in Algebraic Geometry" (PDF). School of Mathematics, Institute for Advanced Study.

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.

[1] Bredon, Glen E. (24 January 1997). Sheaf Theory. Springer. ISBN 9780387949055.

[2] Lurie, Jacob. "Tamagawa Numbers via Nonabelian Poincare Duality, Lecture 9: Nonabelian Poincare Duality in Algebraic Geometry" (PDF). School of Mathematics, Institute for Advanced Study.

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