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A moperadic approach to cyclotomic associators

Damien Calaque, Martin Gonzalez

Abstract

This is a companion paper to "Ellipsitomic associators". We provide a (m)operadic description of Enriquez's torsor of cyclotomic associators, as well as of its associated cyclotomic Grothendieck-Teichmüller groups.

A moperadic approach to cyclotomic associators

Abstract

This is a companion paper to "Ellipsitomic associators". We provide a (m)operadic description of Enriquez's torsor of cyclotomic associators, as well as of its associated cyclotomic Grothendieck-Teichmüller groups.

Paper Structure

This paper contains 29 sections, 13 theorems, 120 equations.

Table of Contents

  1. Moperads
  2. Moperads over an operad
  3. Example of a moperad over an operad: coloured Stasheff polytopes
  4. Pointing
  5. Group actions
  6. Reminders on associators and G(R)T
  7. Compactified configuration space of the plane
  8. The operad of parenthesized braids
  9. The operad of (parenthesized) chord diagrams
  10. Drinfeld associators
  11. The Grothendieck--Teichmüller group
  12. The graded Grothendieck--Teichmüller group
  13. Bitorsor structure
  14. Parenthesized braids with a frozen strand
  15. Compactified configuration space of the annulus
  16. ...and 14 more sections

Key Result

Theorem 1

The moperad in groupoid $\mathbf{PaB}^1$ is generated by an arity $1$ arrow $E$ and an arity $2$ arrow $\Psi$, with relations eqn:cU, eqn:MP, eqn:RP, and eqn:O.

Theorems & Definitions (46)

  • Theorem : Theorem \ref{['PaB1']}
  • Theorem : Theorem \ref{['Ass:cyc:iso']}
  • Remark 1.1
  • Example 1.2
  • Example 2.1: of arrows in small arity
  • Theorem 2.2
  • Remark 3.1
  • Example 3.2: Description of $\mathbf{PaB}^1(1)$
  • Example 3.3: Notable arrow in $\mathbf{PaB}^1(2)$
  • Remark 3.4
  • ...and 36 more