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.