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A simplicial presheaf is a presheaf on a site (e.g., the category of topological spaces) taking values in simplicial sets (i.e., a contravariant functor from the site to the category of simplicial sets).
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Matching objects and hypercovers in topology
This follows from the construction of $(\operatorname{cosk}_m(X_\bullet))_n$ as a limit indexed by $(\Delta/[n])^{\operatorname{op}}_{\leq m}$ (in the non-augmented case) or $(\Delta_a/[n])^{\operator …