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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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Descent properties of spaces

For the first problem, as I wrote in the comments, the author is implicitly using the language of $(\infty,1)$-categories, where it does make sense to speak of such a functor. To understand why, you c …
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