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Homotopy theory of simplicial parametrized operads

Gregoire Marc

TL;DR

The paper develops a unifying framework for operads using ${\mathbf{F}}$-multicategories and double-categorical methods, introduces a convolution-based composition product and a monad that recover ${\mathbb{M}}$-operads as algebras, and constructs model structures for simplicial (coloured) ${\mathbb{M}}$-operads via transfer from projective models. It shows that this setup encompasses coloured symmetric and non-symmetric operads and recovers Nardin–Shah’s simplicial parametrized operads, providing a foundation for comparing all models of equivariant operads. Furthermore, the theory of parametrized operads via orbital pairs connects to equivariant operad frameworks and suggests a pathway to complete Segal-object descriptions and ∞-operad comparisons in this parametrized context. The outlined program aims to unify diverse approaches to equivariant and parametrized operads, enabling Quillen equivalences and ∞-categorical comparisons across models.

Abstract

We define a generalization of (coloured) operads based on double lax functors and we construct a model structure on the associated category of generalized simplicial (coloured) operads. In particular, we obtain a model structure on the category of simplicial (coloured) O-operads of Nardin and Shah.

Homotopy theory of simplicial parametrized operads

TL;DR

The paper develops a unifying framework for operads using -multicategories and double-categorical methods, introduces a convolution-based composition product and a monad that recover -operads as algebras, and constructs model structures for simplicial (coloured) -operads via transfer from projective models. It shows that this setup encompasses coloured symmetric and non-symmetric operads and recovers Nardin–Shah’s simplicial parametrized operads, providing a foundation for comparing all models of equivariant operads. Furthermore, the theory of parametrized operads via orbital pairs connects to equivariant operad frameworks and suggests a pathway to complete Segal-object descriptions and ∞-operad comparisons in this parametrized context. The outlined program aims to unify diverse approaches to equivariant and parametrized operads, enabling Quillen equivalences and ∞-categorical comparisons across models.

Abstract

We define a generalization of (coloured) operads based on double lax functors and we construct a model structure on the associated category of generalized simplicial (coloured) operads. In particular, we obtain a model structure on the category of simplicial (coloured) O-operads of Nardin and Shah.

Paper Structure

This paper contains 22 sections, 61 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Structure of the paper
  4. Related work
  5. Notations and conventions
  6. Acknowledgements
  7. $\mathbb{M}$-operads
  8. ${\mathbf{F}}$-multicategories
  9. $\mathbb{M}$-operads
  10. Coloured $\mathbb{M}$-operads
  11. Parametrized operads
  12. Convolution product for double categories
  13. Monoidal structure on spans
  14. Left Kan extension along morphisms of cocartesian fibrations
  15. Convolution product for double categories
  16. ...and 7 more sections

Key Result

Theorem A

(corocompprod) For every ${\mathbf{F}}$-multicategory $\mathbb{M}$ that is target right fibrant in the sense of deftrf and for every cartesian closed category ${\mathcal{V}}$, the category $\operatorname{Coll}_{\mathbb{M}}({\mathcal{V}})$ of $\mathbb{M}$-collections in ${\mathcal{V}}$ is endowed wit

Theorems & Definitions (158)

  • Theorem A
  • Theorem B
  • Theorem C
  • Corollary 1
  • Definition 1: Ben1
  • Proposition 1
  • proof
  • Remark 1
  • Proposition 2
  • proof
  • ...and 148 more