Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585321120240701The coherator $\Theta^{\infty}_W$ of cubical weak $\infty$-categories with connections6912610413910.48308/cgasa.2023.104139ENCamell KachourLaboratoire 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.0009-0004-9550-1648Journal Article20230914This 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. https://cgasa.sbu.ac.ir/article_104139_624697552a25e8bee0393e08ecf9ddf6.pdf