Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the equivalence of Brantner's and Chu--Haugseng's approaches to enriched $\infty$-operads

Kensuke Arakawa

Abstract

We prove that two models of (monochromatic) enriched $\infty$-operads, due to Brantner and Chu--Haugseng, are equivalent. We show this as a consequence of the equivalence of two models of monoidal $\infty$-categories of symmetric sequences and the composition product, due to Brantner and Haugseng. As a consequence, constructions and results formulated in either framework, such as notions of algebra and Koszul duality, are also shown to be equivalent.

On the equivalence of Brantner's and Chu--Haugseng's approaches to enriched $\infty$-operads

Abstract

We prove that two models of (monochromatic) enriched -operads, due to Brantner and Chu--Haugseng, are equivalent. We show this as a consequence of the equivalence of two models of monoidal -categories of symmetric sequences and the composition product, due to Brantner and Haugseng. As a consequence, constructions and results formulated in either framework, such as notions of algebra and Koszul duality, are also shown to be equivalent.
Paper Structure (17 sections, 34 theorems, 127 equations, 1 table)

This paper contains 17 sections, 34 theorems, 127 equations, 1 table.

Table of Contents

  1. Stating the main result
  2. Brantner's model
  3. Haugseng's model
  4. Main Result
  5. Review on Localization
  6. Composition product and Day convolution from model-categorical viewpoint
  7. Day convolution
  8. Composition Product
  9. Closer look at Brantner's model
  10. Closer look at Haugseng's model
  11. Proof of the main result
  12. Results on $\left(\infty,2\right)$-categories
  13. Definitions
  14. Mapping $\infty$-categories
  15. Cartesian fibrations
  16. ...and 2 more sections

Key Result

Theorem A

There is a natural equivalence of monoidal $\infty$-categories

Theorems & Definitions (113)

  • Theorem A: Theorem \ref{['thm:main']}
  • Proposition 1.1
  • proof
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.6
  • Remark 1.9
  • Definition 1.10
  • Remark 1.11
  • ...and 103 more