The coherator Θ∞W of cubical weak∞-categories with connections

This work exhibits two applications of the combinatorial approach in [12] of the small category Θ0 which objects are cubical pasting diagrams. First we provide an accurate description of the monad S = (S, λ, µ) acting on the category CSets of cubical sets (without degeneracies and connections), which algebras are cubical strict ∞-categories with connections, and show that this monad is cartesian, which solve a conjecture in [16]. Secondly we give a precise construction of the cubical coherator Θ∞ W which set-models are cubical weak ∞-categories with connections, and we also give a precise construction of the cubical coherator Θ∞ W0 which set-models are cubical weak ∞-groupoids with connections.