Nonabelian cosheaf homotopy is a notion in the context of nonabelian cohomology:
Given an infinity-category-valued (pseudo)copresheaf $\mathbf{B} : Spaces \to \infty-Cat$, its homotopy $\pi(-,\mathbf{B})$ is the $\infty$-category valued (pseudo)copresheaf which assigns to each space the limit over all codescent data:
Here the limit is over all hypercovers of $X$.
Dual to nonabelian cosheaf homotopy is nonabelian sheaf cohomology.
A co-presheaf whose value on each space is equivalent to its codescent $\infty$-category for any cover of that space is an infinity-co-stack.