Rigidification and the Coherent Nerve for Enriched Quasicategories

TL;DR

This paper develops a model category framework for enriched quasicategories over a regular Cartesian Reedy category, introducing a coherent nerve and realization that establish a Quillen equivalence with enriched categories, extending to localizations and yielding a Yoneda lemma.

Contribution

It introduces a new model structure for enriched quasicategories, along with coherent nerve and realization functors, and proves their Quillen equivalence to enriched categories, extending to localizations.

Findings

Established a model category for enriched quasicategories.

Proved the coherent nerve and realization form a Quillen equivalence.

Extended results to localizations and derived a Yoneda lemma.

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.