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Partial groups as symmetric simplicial sets

Philip Hackney, Justin Lynd

TL;DR

The paper presents a new characterization of partial groups as a reflective subcategory of symmetric sets, via the introduction of spiny symmetric sets and their equivalence with partial groupoids (and reduced spiny symmetric sets with partial groups). It constructs explicit reflections onto reduced and spiny symmetric sets, enabling concrete computation of limits and colimits in the partial-group setting by first forming them in symmetric sets and then applying the reflection. Central technical tools include matrix encodings of simplices, the L operation, and the spine-based notion of spiny symmetry, along with word classifiers $\mathfrak{F}^m$ that give free partial groups on length $m$. The results unify partial groups and partial groupoids within the symmetric-set framework, provide a complete and cocomplete theory, and supply practical methods for colimit constructions and pushouts through explicit reflections.

Abstract

We give a new characterization of partial groups as a subcategory of symmetric (simplicial) sets. This subcategory has an explicit reflection, which permits one to compute colimits in the category of partial groups. We also introduce the notion of a partial groupoid, which encompasses both groupoids and partial groups.

Partial groups as symmetric simplicial sets

TL;DR

The paper presents a new characterization of partial groups as a reflective subcategory of symmetric sets, via the introduction of spiny symmetric sets and their equivalence with partial groupoids (and reduced spiny symmetric sets with partial groups). It constructs explicit reflections onto reduced and spiny symmetric sets, enabling concrete computation of limits and colimits in the partial-group setting by first forming them in symmetric sets and then applying the reflection. Central technical tools include matrix encodings of simplices, the L operation, and the spine-based notion of spiny symmetry, along with word classifiers that give free partial groups on length . The results unify partial groups and partial groupoids within the symmetric-set framework, provide a complete and cocomplete theory, and supply practical methods for colimit constructions and pushouts through explicit reflections.

Abstract

We give a new characterization of partial groups as a subcategory of symmetric (simplicial) sets. This subcategory has an explicit reflection, which permits one to compute colimits in the category of partial groups. We also introduce the notion of a partial groupoid, which encompasses both groupoids and partial groups.
Paper Structure (10 sections, 28 theorems, 34 equations, 1 figure)

This paper contains 10 sections, 28 theorems, 34 equations, 1 figure.

Table of Contents

  1. Simplicial sets, partial groups, and partial groupoids
  2. Simplicial sets and symmetric sets
  3. Simplicial sets and partial group(oid)s
  4. Elements as matrices
  5. Spiny symmetric sets
  6. Partial groupoids as spiny symmetric sets
  7. Reflections onto reduced and spiny symmetric sets
  8. The reduction of a symmetric set
  9. The reflection onto spiny symmetric sets
  10. Consequences

Key Result

Theorem A

The category of spiny symmetric sets is equivalent to the category of partial groupoids. The category of reduced spiny symmetric sets is equivalent to the category of partial groups.

Figures (1)

  • Figure 1: Flattening a symmetric set

Theorems & Definitions (74)

  • Theorem A
  • Theorem B
  • Corollary C
  • Example C
  • Remark C: Groupoids and cyclic sets
  • Definition C
  • Definition C
  • Definition C
  • Definition C: Partial Groupoids
  • Theorem C: González
  • ...and 64 more