Enriched quasicategories and the templicial homotopy coherent nerve

Lowen, Wendy; Mertens, Arne

doi:10.2140/agt.2025.25.1029

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Enriched quasicategories and the templicial homotopy coherent nerve

Wendy Lowen and Arne Mertens

Algebraic & Geometric Topology 25 (2025) 1029–1074

DOI: 10.2140/agt.2025.25.1029

Abstract

We lay the foundations for a theory of quasicategories in a monoidal category $𝒱$ replacing $Set$ , aimed at realising weak enrichment in the category $S 𝒱$ of simplicial objects in $𝒱$ . To accommodate noncartesian monoidal products, we make use of an ambient category $S_{\otimes} 𝒱$ of templicial, or “tensor-simplicial”, objects in $𝒱$ , 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 $S 𝒱$ -enriched categories to $S_{\otimes} 𝒱$ . We show that an $S 𝒱$ -enriched category whose underlying simplicial category is locally Kan is turned into a quasicategory in $𝒱$ by this nerve functor.

Keywords

quasicategory, enriched category, necklace, templicial object, homotopy coherent nerve

Mathematical Subject Classification

Primary: 18D20, 18N60

Secondary: 18M05, 18N50

References

Publication

Received: 15 February 2023

Revised: 22 December 2023

Accepted: 5 March 2024

Published: 16 May 2025

Authors

Wendy Lowen
Departement Wiskunde Universiteit Antwerpen Antwerpen Belgium

Arne Mertens
Departement Wiskunde Universiteit Antwerpen Antwerpen Belgium

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Volume 25, issue 2 (2025)