The coherator $\Theta^{\infty}_W$ of cubical weak $\infty$-categories with connections

Document Type : Research Paper

Author

Laboratoire de Math\'ematiques d'Orsay, UMR 8628, Universit\'e de Paris-Saclay and CNRS, B\^atiment 307, Facult\'e des Sciences d'Orsay, 94015 ORSAY Cedex, France.

Abstract

This work exhibits two applications of the combinatorial approach in [12] of the small category $\Theta_0$ which objects are cubical pasting diagrams. First we provide an accurate description of the monad $\mathbb{S}=(S,\lambda,\mu)$ acting on the category ${\mathbb{C}\mathbb{S}\text{ets}}$ of cubical sets (without degeneracies and connections), which algebras are cubical strict $\infty$-categories with connections, and show that this monad is cartesian, which solve a conjecture in \cite{camark-cub}. Secondly we give a precise construction of the cubical coherator $\Theta^{\infty}_W$ which set-models are cubical weak $\infty$-categories with connections, and we also give a precise construction of the cubical coherator $\Theta^{\infty}_{W^{0}}$ which set-models are cubical weak $\infty$-groupoids with connections.  

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References

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Categories and General Algebraic Structures with Applications
Volume 21, Issue 1
July 2024
Pages 69-126

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