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.