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Combinatorial and homotopical aspects of $E_n$-operads

Christian Schlichtkrull

TL;DR

The paper addresses the problem of realizing $E_n$-operads from categorical (pre)operads by purely combinatorial means. It develops and analyzes edge-label/orientation operads $ ext{G}_n$, $ ext{K}_n$, $ ext{K}_n^e$, and the $n$-fold monoidal operad $ ext{M}_n$, along with restricted preoperads $ ext{M}_n^{ ightarrow}$ and $ ext{M}_n^{ ightarrow}$, proving that inclusions into $ ext{K}_n^e$ are homotopy initial/final. This yields that $| ext{P}|$ is an $E_n$-(pre)operad whenever $ ext{P}$ is a categorical sub(pre)operad of $ ext{K}_n^e$ containing one of the preoperads, and it provides a new, purely combinatorial route to the $E_n$-property, reducing to the $n=2$ case and connecting to the little $n$-cubes via known BFV07 arguments. The results extend the repertoire of $E_n$-operads arising from categorical structures, clarify dualities with Getzler–Jones and Milgram preoperads, and circumvent potential Reedy cofibrancy obstructions highlighted in prior work.

Abstract

We show that a certain class of categorical operads give rise to $E_n$-operads after geometric realization. The main arguments are purely combinatorial and avoid the technical topological assumptions otherwise found in the literature.

Combinatorial and homotopical aspects of $E_n$-operads

TL;DR

The paper addresses the problem of realizing -operads from categorical (pre)operads by purely combinatorial means. It develops and analyzes edge-label/orientation operads , , , and the -fold monoidal operad , along with restricted preoperads and , proving that inclusions into are homotopy initial/final. This yields that is an -(pre)operad whenever is a categorical sub(pre)operad of containing one of the preoperads, and it provides a new, purely combinatorial route to the -property, reducing to the case and connecting to the little -cubes via known BFV07 arguments. The results extend the repertoire of -operads arising from categorical structures, clarify dualities with Getzler–Jones and Milgram preoperads, and circumvent potential Reedy cofibrancy obstructions highlighted in prior work.

Abstract

We show that a certain class of categorical operads give rise to -operads after geometric realization. The main arguments are purely combinatorial and avoid the technical topological assumptions otherwise found in the literature.
Paper Structure (12 sections, 21 theorems, 58 equations)

This paper contains 12 sections, 21 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Notation and conventions
  4. Operads arising from edge labels and orientations
  5. The complete graph operad $\mathscr{K}_n$
  6. The extended complete graph operad $\mathscr{K}^{e}_n$
  7. The operad $\mathscr{M}_n$ governing $n$-fold monoidal categories
  8. The restricted preoperads $\mathscr{M}_n^{\shortuparrow}$ and $\mathscr{M}_n^{\shortdownarrow}$
  9. Duality properties
  10. Homotopy initial and homotopy final inclusions
  11. The proof of Theorem \ref{['thm:homotopy-final-initial']} for $n=2$
  12. The relation to little $n$-cubes

Key Result

Theorem 1.1

Consider an inclusion $j\colon \mathcal{A}\to \mathcal{B}$ of partially ordered subsets of $\mathscr{K}_n^e(S)$. If $\mathcal{A}$ contains $\mathscr{M}_n^{\shortdownarrow}(S)$, then $j$ is homotopy initial and if $\mathcal{A}$ contains $\mathscr{M}_n^{\shortuparrow}(S)$, then $j$ is homotopy final.

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.5
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['thm:intro-homotopy-initial-final']}
  • Example 3.2
  • Remark 3.3
  • Corollary 3.4
  • Theorem 3.5
  • Remark 3.6
  • ...and 36 more