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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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Basic technical things about simplicial sets to have a good understanding of quasicategories
Simplicial sets are absolutely fundamental to the study of quasi-categories. There's no easy list of things you have to know and other things you don't have to know. Therefore, you should try to gain …