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Answer by David Carchedi for Site dependance of the Cech weak equivalences on...
It is independent, provided the two sites of definition, also yield the same infinity topos (so e.g. when they both have finite limits, or the infinity-topos is hypercomplete). Given two simplicial...
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Let $\mathcal{T}= sh(C,J)$ a Grothendieck topos of sheaves over a (ordinary) site.One endows the category of simplicial presheaves over $C$ with the "Rezk-Lurie" model structure: that is we start with...
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