Combinatorial approach to the category $\Theta_0$ of cubical pasting diagrams

In these notes we describe models of globular weak $(\infty,m)$-categories ($m\in\mathbb{N}$) in the Grothendieck style, i.e for each $m\in\mathbb{N}$ we define a globular coherator $\Theta^{\infty}_{\mathbb{M}^m}$ whose set-models are globular weak $(\infty,m)$-categories. Then we describe the combinatorics of the small category $\Theta_0$ whose objects are cubical pasting diagrams and whose morphisms are morphisms of cubical sets. This provides an accurate description of the monad, on the category of cubical sets (without degeneracies and connections), of cubical strict $\infty$-categories with connections. We prove that it is a cartesian monad, solving a conjecture in \cite{camark-cub-1}. This puts us in a position to describe the cubical coherator $\Theta^{\infty}_W$ whose set-models are cubical weak $\infty$-categories with connections and the cubical coherator $\Theta^{\infty}_{W^{0}}$ whose set-models are cubical weak $\infty$-groupoids with connections.