Rigidification and the Coherent Nerve for Enriched Quasicategories

Abstract: We introduce, for (\C) a regular Cartesian Reedy category a model category whose fibrant objects are an analogue of quasicategories enriched in simplicial presheaves on (C). We then develop a coherent realization and nerve for this model structure and demonstrate using an enriched version of the necklaces of Dugger and Spivak that our model category is Quillen-equivalent to the category of categories enriched in simplicial presheaves on (\C). We then show that for any Cartesian-closed left-Bousfield localization of the category of simplicial presheaves on (\C), the coherent nerve and realization descend to a Quillen equivalence on the localizations of these model categories. As an application, we demonstrate a version of Yoneda's lemma for these enriched quasicategories.