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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.

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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.

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  • $\begingroup$ Thank you! I'll digest this and see if I have any follow-up questions... $\endgroup$
    – Piotr Achinger
    Commented Feb 22, 2018 at 9:55
  • $\begingroup$ P.S. I am surprised that this doesn't seem to be a classical notion, since such functors appear in nature quite often. For example, if $f:X\to Y$ is a sufficiently nice map of topological spaces, then $(U\subseteq Y) \mapsto f^{-1}(U)$ should be such a "homotopy cosheaf" on $Y$. $\endgroup$
    – Piotr Achinger
    Commented Feb 22, 2018 at 18:24
  • $\begingroup$ @PiotrAchinger Maybe I was unclear. This is a classical notion, it is just not significantly different from sheaves with values in any other (∞-)category. I am a bit uncertain of what kind of results you expect, do you have some statement in mind? $\endgroup$
    – Denis Nardin
    Commented Feb 22, 2018 at 18:34
  • $\begingroup$ For example: you can give exactly the same definition for sets: a cosheaf of sets is just a sheaf valued in $\mathrm{Set}^{op}$. These things do occasionally appear, but they do not seem to me to be special in any way. Cosheaves valued in stable ∞-categories or abelian categories are more interesting: for example they are the natural setting for Verdier duality (since sections of a sheaf with proper support form naturally a cosheaf). $\endgroup$
    – Denis Nardin
    Commented Feb 22, 2018 at 18:44

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