What is "cosheaf"

n. (context math English) The dual of a sheaf.

In topology, a branch of mathematics, a cosheaf with values in an ∞-category C that admits colimit is a functor F from the category of open subsets of a topological space X (more precisely its nerve) to C such that

• (1) The F of the empty set is the initial object.
• (2) For any increasing sequence U of open subsets with union U, the canonical map $\varinjlim F(U_i) \to F(U)$ is an equivalence.
• (3) F(U ∪ V) is the pushout of F(U ∩ V) → F(U) and F(U ∩ V) → F(V).

The basic example is U ↦ C(U; A) where on the right is the singular chain complex of U with coefficients in an abelian group A.

Example: If f is a continuous map, then U ↦ f(U) is a cosheaf.