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3-Crossed modules, Quasi-categories, and the Moore complex

Masaki Fukuda, Tommy Shu

Abstract

The established equivalence between 2-crossed modules and Gray 3-groups [M. Sarikaya and E. Ulualan, 2024] serves as a benchmark for higher-dimensional algebraic models. However, to the best of our knowledge, the established definitions of 3-crossed modules [Z. Arvasi, T. S. Kuzpinari, and E. Ö. Uslu, 2009] are not clearly suited for extending this equivalence. In this paper, we propose an alternative formulation of a 3-crossed module, equipped with a new type of lifting, which is specifically designed to serve as a foundation for this higher-order categorical correspondence. As the primary results of this paper, we validate this new structure. We prove that the simplicial set induced by our 3-crossed module forms a quasi-category. Furthermore, we show that the Moore complex of length 3 associated with a simplicial group naturally admits the structure of our 3-crossed module. This work establishes our definition as a robust candidate for modeling the next level in this algebraic-categorical program.

3-Crossed modules, Quasi-categories, and the Moore complex

Abstract

The established equivalence between 2-crossed modules and Gray 3-groups [M. Sarikaya and E. Ulualan, 2024] serves as a benchmark for higher-dimensional algebraic models. However, to the best of our knowledge, the established definitions of 3-crossed modules [Z. Arvasi, T. S. Kuzpinari, and E. Ö. Uslu, 2009] are not clearly suited for extending this equivalence. In this paper, we propose an alternative formulation of a 3-crossed module, equipped with a new type of lifting, which is specifically designed to serve as a foundation for this higher-order categorical correspondence. As the primary results of this paper, we validate this new structure. We prove that the simplicial set induced by our 3-crossed module forms a quasi-category. Furthermore, we show that the Moore complex of length 3 associated with a simplicial group naturally admits the structure of our 3-crossed module. This work establishes our definition as a robust candidate for modeling the next level in this algebraic-categorical program.
Paper Structure (15 sections, 9 theorems, 72 equations, 39 figures)

This paper contains 15 sections, 9 theorems, 72 equations, 39 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simplicial sets and simplicial groups.
  4. Quasi-categories
  5. 2-crossed modules
  6. Remark.
  7. 2-crossed modules and simplicial sets
  8. 3-crossed modules and simplicial sets
  9. Remark.
  10. Example of 3-crossed module
  11. Construction from a Moore complex
  12. Construction from a 2-crossed module
  13. Conclusion and Future Work
  14. 3-crossed module and diagram
  15. Cocycle condition

Key Result

Theorem 1

Let $W \coloneqq (L \xrightarrow{\partial} H \xrightarrow{\partial} G,\{-,-\} )$ be a 2-crossed module. Then the simplicial set $M_ W$ is a quasi-category.

Figures (39)

  • Figure 2: Compositon of $G$
  • Figure 3: Two types of $H$
  • Figure 4: Unit element of $G$
  • Figure 5: Vertical compositon of $H$
  • Figure 6: Unit element of $H$
  • ...and 34 more figures

Theorems & Definitions (19)

  • Theorem 1
  • Lemma 1
  • proof
  • Corollary 1
  • Lemma 2
  • proof
  • proof : of Theorem \ref{['2cm and qusi']}
  • Definition 1
  • Lemma 3
  • proof
  • ...and 9 more