In topology, a branch of mathematics, a cosheaf with values in an ∞-category C that admits colimits 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 ${\displaystyle U_{i}}$ of open subsets with union U, the canonical map ${\displaystyle \varinjlim F(U_{i})\to F(U)}$ is an equivalence.
• (3) ${\displaystyle F(U\cup V)}$ is the pushout of ${\displaystyle F(U\cap V)\to F(U)}$ and ${\displaystyle F(U\cap V)\to F(V)}$.

The basic example is ${\displaystyle U\mapsto C_{*}(U;A)}$ where on the right is the singular chain complex of U with coefficients in an abelian group A.

Example:^[1] If f is a continuous map, then ${\displaystyle U\mapsto f^{-1}(U)}$ is a cosheaf.