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Enriched quasi-categories and the templicial homotopy coherent nerve

Wendy Lowen, Arne Mertens

TL;DR

This work develops a foundations-level framework for enriched $(\infty,1)$-categories in a general monoidal setting by introducing templicial objects and the templicial nerve, enabling weak enrichment to be realized inside the simplicial objects of $\mathcal{V}$. The authors construct a templicial analogue of the homotopy coherent nerve, prove a central result that locally Kan underlying data yields quasi-categories in $\mathcal{V}$, and establish a rich necklace-categorical apparatus that underpins these constructions. They show that the templicial nerve embeds strictly, relate the templicial and classical nerves, and provide a simplification of categorification via flagged necklace data. The framework recovers the classical quasi-category theory when $\mathcal{V}=\mathrm{Set}$, while offering a natural path toward enriched $\infty$-categories and potential model-categorical equivalences in the non-cartesian setting. Overall, the paper lays the structural groundwork for quasi-categories in a monoidal category and sets the stage for future homotopy-theoretic developments and rectification results.

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.

Enriched quasi-categories and the templicial homotopy coherent nerve

TL;DR

This work develops a foundations-level framework for enriched -categories in a general monoidal setting by introducing templicial objects and the templicial nerve, enabling weak enrichment to be realized inside the simplicial objects of . The authors construct a templicial analogue of the homotopy coherent nerve, prove a central result that locally Kan underlying data yields quasi-categories in , and establish a rich necklace-categorical apparatus that underpins these constructions. They show that the templicial nerve embeds strictly, relate the templicial and classical nerves, and provide a simplification of categorification via flagged necklace data. The framework recovers the classical quasi-category theory when , while offering a natural path toward enriched -categories and potential model-categorical equivalences in the non-cartesian setting. Overall, the paper lays the structural groundwork for quasi-categories in a monoidal category and sets the stage for future homotopy-theoretic developments and rectification results.

Abstract

We lay the foundations for a theory of quasi-categories in a monoidal category replacing , aimed at realising weak enrichment in the category of simplicial objects in . To accomodate non-cartesian monoidal products, we make use of an ambient category of templicial - or 'tensor-simplicial' - objects in , 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 -enriched categories to templicial objects. We show that an -enriched category whose underlying simplicial category is locally Kan, is turned into a quasi-category in by this nerve functor.
Paper Structure (22 sections, 37 theorems, 131 equations)

This paper contains 22 sections, 37 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. The main goal
  3. Templicial objects and necklace categories
  4. Overview of the paper
  5. Notations and conventions
  6. Templicial objects
  7. Simplicial versus templicial objects
  8. The templicial nerve
  9. The Eilenberg-Zilber lemma
  10. Necklaces and necklace categories
  11. Necklaces
  12. Necklace categories
  13. Some constructions revisited
  14. Enriching the homotopy coherent nerve
  15. The classical homotopy coherent nerve
  16. ...and 7 more sections

Key Result

Theorem 1

There is a right-adjoint functor from the category of small $S\mathcal{V}$-enriched categories to the category of templicial objects in $\mathcal{V}$ with the following properties.

Theorems & Definitions (105)

  • Theorem
  • Proposition 2.1: leinster2000homotopy, Proposition 3.1.7
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Proposition 2.8
  • ...and 95 more