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

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

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

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

is exact for every collection ${\displaystyle \{U_{\alpha }\}_{\alpha }}$ of open sets, where ${\displaystyle U:=\bigcup _{\alpha }U_{\alpha }}$ and ${\displaystyle U_{\alpha ,\beta }:=U_{\alpha }\cap U_{\beta }}$. (Notice that this is dual to the sheaf condition.) Approximately, exactness at ${\displaystyle F(U)}$ means that every element over ${\displaystyle U}$ can be represented as a finite sum of elements that live over the smaller opens ${\displaystyle U_{\alpha }}$, while exactness at ${\displaystyle \bigoplus _{\alpha }F(U_{\alpha })}$ 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 ${\displaystyle U_{\alpha ,\beta }}$.

Equivalently, ${\displaystyle F}$ is a cosheaf if

• for all open sets ${\displaystyle U}$ and ${\displaystyle V}$, ${\displaystyle F(U\cup V)}$ is the pushout of ${\displaystyle F(U\cap V)\to F(U)}$ and ${\displaystyle F(U\cap V)\to F(V)}$, and
• for any upward-directed family ${\displaystyle \{U_{\alpha }\}_{\alpha }}$ of open sets, the canonical morphism ${\displaystyle \varinjlim F(U_{\alpha })\to F(\bigcup _{\alpha }U_{\alpha })}$ 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 ${\displaystyle {\mathcal {C}}}$ is not an abelian category.

Examples

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

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

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 ${\displaystyle U}$ to ${\displaystyle {\overline {C}}_{k}(U;\mathbb {Z} )}$ is in fact a cosheaf.

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

Notes

References