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Enriched quasicategories and the templicial homotopy coherent nerve

Wendy Lowen and Arne Mertens

Algebraic & Geometric Topology 25 (2025) 1029–1074
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 S𝒱 of simplicial objects in 𝒱. To accommodate noncartesian monoidal products, we make use of an ambient category S𝒱 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 S𝒱-enriched categories to S𝒱. We show that an S𝒱-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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