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.