New 3-Crossed Module Formulation for Higher-Order Categorical Correspondence
Research Paper#Mathematics (Category Theory, Higher-Dimensional Algebra)🔬 Research|Analyzed: Jan 3, 2026 19:35•
Published: Dec 28, 2025 05:56
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•ArXivAnalysis
This paper addresses a key challenge in higher-dimensional algebra: finding a suitable definition of 3-crossed modules that aligns with the established equivalence between 2-crossed modules and Gray 3-groups. The authors propose a novel formulation of 3-crossed modules, incorporating a new lifting mechanism, and demonstrate its validity by showing its connection to quasi-categories and the Moore complex. This work is significant because it provides a potential foundation for extending the algebraic-categorical program to higher dimensions, which is crucial for understanding and modeling complex mathematical structures.
Key Takeaways
- •Proposes a new formulation of 3-crossed modules with a novel lifting mechanism.
- •Demonstrates the validity of the new structure by connecting it to quasi-categories.
- •Shows that the Moore complex of length 3 of a simplicial group admits the proposed 3-crossed module structure.
- •Aims to provide a foundation for extending the algebraic-categorical program to higher dimensions.
Reference / Citation
View Original"The paper validates the new 3-crossed module structure by proving that the induced simplicial set forms a quasi-category and that the Moore complex of length 3 associated with a simplicial group naturally admits the structure of the proposed 3-crossed module."