TITLE: Homotopy cosheaf?
QUESTION [6 upvotes]: Let $C$ be a site, $\mathbf{S}$ some ($\infty$-? homotopy?) category of spaces. 

Question. What do you call a (covariant!) functor $F:C\to \mathbf{S}$ enjoying the following property: for every hypercovering $a\colon V_\bullet \to U$ in $C$, the induced map $$ {\rm hocolim}\, F(V_\bullet)\longrightarrow F(U) $$ is a homotopy equivalence?

Have such things been studied? References are welcome.

REPLY [6 votes]: This is just a sheaf valued in the ∞-category $\mathrm{Space}^{op}$. It is usually called a cosheaf. A place where this kind of thing shows up is in factorization algebras, that can be described as particular cosheaves over the Ran space. 
I don't think they behave  significantly differently from sheaves in any other complete ∞-category, so I do not believe there are specific references for them.