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Cloning systems and action operads

Javier Aramayona, Federico Cantero Morán, Víctor Carmona, Javier J. Gutiérrez

TL;DR

The paper establishes a precise dictionary between action operads and restricted operadic cloning systems, showing that these two frameworks for assembling Thompson-like groups are categorically equivalent. It then extends the correspondence to operadic cloning systems via general action operads, and introduces bilateral cloning systems, detailing how they capture left-extensions and interact with right-extensions through cloning maps. By investigating braid groups as a principal example, the authors illustrate forward and reverse constructions between action operads and cloning systems, and generalize to broader settings including signed permutations, crossed interval groups, and PROs. The work thus unifies operadic and cloning-theoretic approaches to Thompson-like groups, and provides multiple equivalent perspectives (through crossed groups and PROs) that facilitate further applications and future work on fundamental groups of operads and related Thompson-type groups.

Abstract

Action operads and cloning systems are, respectively, the main ingredients in two approaches for axiomatically constructing Thompson-like groups due to Thumann and Witzel-Zaremsky. In this paper, we prove that action operads are equivalent to cloning systems that admit a certain extra structure, and which we call bilateral cloning systems. In addition, we describe their relation with crossed interval groups and product categories.

Cloning systems and action operads

TL;DR

The paper establishes a precise dictionary between action operads and restricted operadic cloning systems, showing that these two frameworks for assembling Thompson-like groups are categorically equivalent. It then extends the correspondence to operadic cloning systems via general action operads, and introduces bilateral cloning systems, detailing how they capture left-extensions and interact with right-extensions through cloning maps. By investigating braid groups as a principal example, the authors illustrate forward and reverse constructions between action operads and cloning systems, and generalize to broader settings including signed permutations, crossed interval groups, and PROs. The work thus unifies operadic and cloning-theoretic approaches to Thompson-like groups, and provides multiple equivalent perspectives (through crossed groups and PROs) that facilitate further applications and future work on fundamental groups of operads and related Thompson-type groups.

Abstract

Action operads and cloning systems are, respectively, the main ingredients in two approaches for axiomatically constructing Thompson-like groups due to Thumann and Witzel-Zaremsky. In this paper, we prove that action operads are equivalent to cloning systems that admit a certain extra structure, and which we call bilateral cloning systems. In addition, we describe their relation with crossed interval groups and product categories.
Paper Structure (15 sections, 12 theorems, 85 equations, 17 figures)

This paper contains 15 sections, 12 theorems, 85 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Cloning systems
  3. Action operads
  4. Action operads
  5. General action operads
  6. Fundamental groups of $\mathop{\mathrm{\mathsf{G}}}\nolimits$-operads and unoriented ribbon braids
  7. A case study: the correspondence for braid groups
  8. From the action operad to the (bilateral) cloning system.
  9. From the (operadic) bilateral cloning system to the action operad
  10. From action operads to cloning systems
  11. From cloning systems to action operads
  12. Cloning systems as crossed groups
  13. The demi-simplicial set of symmetric groups
  14. Crossed simplicial groups and crossed interval groups
  15. Cloning systems and PROs

Key Result

Lemma 3.10

Condition (2') in the definition of general action operad $\mathop{\mathrm{\mathsf{G}}}\nolimits$ can be replaced by the following: Moreover, Condition (ii) in the definition of a morphism of general action operads $\phi\colon \mathop{\mathrm{\mathsf{G}}}\nolimits\to \widehat{\mathop{\mathrm{\mathsf{G}}}\nolimits}$ is equivalent to requiring:

Figures (17)

  • Figure 1: Cloning map $\kappa_2^4$ on $\mathop{\mathrm{\mathsf{Br}}}\nolimits_4$.
  • Figure 2: Condition (vii), $\kappa_3^3\circ \iota=\iota\circ\iota$.
  • Figure 3: Condition (vi), $\iota\circ \zeta=\zeta\circ \iota$.
  • Figure 4: Condition (viii).
  • Figure 5: Condition (ix).
  • ...and 12 more figures

Theorems & Definitions (63)

  • Example 2.1: Cloning system for symmetric groups
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.7
  • Example 2.8: Braid groups
  • Example 2.9
  • Example 2.10: Direct powers
  • Example 2.11: Upper triangular matrices
  • ...and 53 more