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Suppose that $F$ is a sheaf of categories (on a Grothendieck site or even a topological space). By this, I mean a sheaf in the naive 1-categorical sense, so it can equivalently be viewed as a category object in sheaves of sets. By taking the nerve, one can view $F$ as a simplicial sheaf. $F$ will not take values in Kan complexes however (unless $F$ takes values in groupoids).

Question: Are there checkable conditions for $F$ to satisfy homotopy descent (besides the Cech diagram consisting of fibrations), in the sense that if $RF$ is the fibrant replacement of $F$ in the Joyal model structure on simplicial sheaves (so I'm modelling $\infty$-sheaves of $\infty$-groupoids here), then the canonical map $$F \to RF$$ is object-wise a weak equivalence of simplicial sets?

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  • $\begingroup$ Have you looked at the old Joyal–Tierney paper on "strong stacks"? There they describe strict sheaves of groupoids but I am informed it also extends to strict sheaves of categories. $\endgroup$ – Zhen Lin Jul 3 '14 at 19:00
  • $\begingroup$ Thanks Zhen. I took a quick look, but it appears as they are considering stacks of categories, whereas, I want to consider stacks of infinity groupoids, but I want to represent the homotopy type by the nerve of a category, if you follow what I mean. $\endgroup$ – David Carchedi Jul 3 '14 at 19:25
  • $\begingroup$ Ah, I see. I thought you were talking about the other kind of descent condition. So the problem is to compute some homotopy limits... I don't know of any theorems in that direction. $\endgroup$ – Zhen Lin Jul 3 '14 at 22:55

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