Let $C$ be a site, $\mathbf{S}$ some ($\infty$-? homotopy?) category of spaces. 



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> **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?
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Have such things been studied? References are welcome.



Homotopy cosheaf?
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.


