Enriched quasi-categories and the templicial homotopy coherent nerve

TL;DR

This paper develops a foundational theory of quasi-categories within a monoidal category setting, introducing templicial objects and a homotopy coherent nerve to extend enrichment concepts beyond classical frameworks.

Contribution

It introduces a new framework for quasi-categories in monoidal categories using templicial objects and constructs a homotopy coherent nerve functor for enriched categories.

Findings

The homotopy coherent nerve converts locally Kan enriched categories into quasi-categories.

The framework accommodates non-cartesian monoidal products.

Establishes foundational tools for weak enrichment in monoidal categories.

Abstract

We lay the foundations for a theory of quasi-categories in a monoidal category $\mathcal{V}$ replacing $\mathrm{Set}$, aimed at realising weak enrichment in the category $S\mathcal{V}$ of simplicial objects in $\mathcal{V}$. To accomodate non-cartesian monoidal products, we make use of an ambient category $S_{\otimes}\mathcal{V}$ of templicial - or 'tensor-simplicial' - objects in $\mathcal{V}$, which are certain colax monoidal functors following Leinster. Inspired by the description of the categorification functor due to Dugger and Spivak, we construct a templicial analogue of the homotopy coherent nerve functor which goes from $S\mathcal{V}$-enriched categories to templicial objects. We show that an $S\mathcal{V}$-enriched category whose underlying simplicial category is locally Kan, is turned into a quasi-category in $\mathcal{V}$ by this nerve functor.