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 that these give a Quillen equivalence, in particular recovering the classical one in the process. We then demonstrate that this equivalence descends to any Cartesian closed left Bousfield localization in a natural way. As an application, we demonstrate a version of Yoneda's lemma for quasicategories enriched in any such Cartesian closed localization.
Keywords: enriched categories, higher category theory, homotopy theory
2020 MSC: 18A30, 18D05, 18E35, 18G55, 55P10, 55U35
Theory and Applications of Categories, Vol. 37, 2021, No. 23, pp 709-817.
Published 2021-06-21.
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