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.Template:Further explanation needed

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 $C_k(U;\mathbb{Z})$, 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 ${\bar{C}}_k(U;\mathbb{Z})$ to be the colimit of the diagram

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

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 ${\bar{C}}_k(U;\mathbb{Z})$ 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

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