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 of open subsets with union U, the canonical map is an equivalence. * (3) is the pushout of and . The basic example is 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 is a cosheaf.