# 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 $\varinjlim F(U_{\alpha })\to F\left(\bigcup _{\alpha }U_{\alpha }\right)$ 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 ${\mathcal {C}}$ 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.