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A new model structure on dendroidal spaces for the theory of $\infty$-operads

João Candeias, Javier J. Gutiérrez

TL;DR

The paper develops a new cofibrantly generated model structure $\mathbf{sSet}^{\Omega^{op}}_{\rm Op}$ on dendroidal spaces, providing a robust model for the homotopy theory of $\infty$-operads. Its fibrant objects are dendroidal Segal spaces, with Dwyer–Kan equivalences as weak equivalences and isofibrations as fibrations, while cofibrations are normal monomorphisms extended at the object level; this model is proven to be Quillen equivalent to the Cisinski–Moerdijk dendroidal models, thereby connecting operadic and higher-categorical viewpoints. The authors establish a rich web of comparisons with existing models (simplicial Segal spaces, complete dendroidal Segal spaces, and operads) via Quillen adjunctions and equivalences, and show that the new framework is homotopically enriched over the Cat-model. They further demonstrate practical benefits by showing that, for well-behaved diagrams, homotopy limits and colimits can be computed without the completion step, enabling simpler computations in many cases. Overall, the work extends the theory of $\infty$-categories to operadic contexts, providing a versatile, compatible model for $\infty$-operads and their homotopy-theoretic properties.

Abstract

We introduce a new model structure on the category of dendroidal spaces, designed to provide a further model for the homotopy theory of $\infty$-operads. This model is directly analogous to a recent construction on the category of simplicial spaces by Moser and Nuiten, and can be seen as its dendroidal counterpart. In our new model structure, the fibrant objects are the dendroidal Segal spaces, while the cofibrations form a subclass of those in the Segal space model structure. The weak equivalences between fibrant objects are precisely the Dwyer--Kan equivalences, and the fibrations between them are the isofibrations. We prove that this model structure is Quillen equivalent to the Cisinski--Moerdijk model structures on dendroidal sets and dendroidal spaces, thereby establishing a compatible extension of the theory of $\infty$-categories to the operadic setting.

A new model structure on dendroidal spaces for the theory of $\infty$-operads

TL;DR

The paper develops a new cofibrantly generated model structure on dendroidal spaces, providing a robust model for the homotopy theory of -operads. Its fibrant objects are dendroidal Segal spaces, with Dwyer–Kan equivalences as weak equivalences and isofibrations as fibrations, while cofibrations are normal monomorphisms extended at the object level; this model is proven to be Quillen equivalent to the Cisinski–Moerdijk dendroidal models, thereby connecting operadic and higher-categorical viewpoints. The authors establish a rich web of comparisons with existing models (simplicial Segal spaces, complete dendroidal Segal spaces, and operads) via Quillen adjunctions and equivalences, and show that the new framework is homotopically enriched over the Cat-model. They further demonstrate practical benefits by showing that, for well-behaved diagrams, homotopy limits and colimits can be computed without the completion step, enabling simpler computations in many cases. Overall, the work extends the theory of -categories to operadic contexts, providing a versatile, compatible model for -operads and their homotopy-theoretic properties.

Abstract

We introduce a new model structure on the category of dendroidal spaces, designed to provide a further model for the homotopy theory of -operads. This model is directly analogous to a recent construction on the category of simplicial spaces by Moser and Nuiten, and can be seen as its dendroidal counterpart. In our new model structure, the fibrant objects are the dendroidal Segal spaces, while the cofibrations form a subclass of those in the Segal space model structure. The weak equivalences between fibrant objects are precisely the Dwyer--Kan equivalences, and the fibrations between them are the isofibrations. We prove that this model structure is Quillen equivalent to the Cisinski--Moerdijk model structures on dendroidal sets and dendroidal spaces, thereby establishing a compatible extension of the theory of -categories to the operadic setting.
Paper Structure (29 sections, 43 theorems, 107 equations)

This paper contains 29 sections, 43 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Trees
  4. Simplicial and dendroidal sets and spaces
  5. Vertical and horizontal directions
  6. Discrete and constant dendroidal spaces
  7. Representables and simplicial enrichment
  8. Skeleta and coskeleta
  9. Normal monomorphisms
  10. Categories, operads and their nerves
  11. Tensor product of dendroidal sets and spaces
  12. Monoidality and weak enrichment
  13. Segal pre-operads
  14. Generalized Reedy categories and model structures
  15. Dendroidal Segal spaces
  16. ...and 14 more sections

Key Result

Theorem 1.1

There exists a cofibrantly generated model structure on $\mathbf{sSet}^{\Omega^\mathrm{op}}$, which we denote by $\mathbf{sSet}^{\Omega^\mathrm{op}}_{\rm Op}$, such that: Moreover, this model structure is Quillen equivalent to the Cisinski--Moerdijk model structures on dendroidal sets and dendroidal spaces, thereby providing yet another model for the theory of $\infty$-operads.

Theorems & Definitions (96)

  • Theorem 1.1
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4
  • Remark 3.5
  • Proposition 3.6
  • proof
  • Definition 3.7
  • Proposition 3.8
  • ...and 86 more