Combinatorial approach of the category Θ0 of cubical pasting diagrams

In globular higher category theory the small category Θ0 of finite rooted trees plays an important role: for example the objects of Θ0 are the arities of the operations inside the free globular ω-operad B 0 of Batanin, which B 0 -algebras are models of globular weak ∞-categories; also this globular Θ0 is an important tool to build the coherator Θ∞ W0 of Grothendieck which Sets-models are globular weak ∞-groupoids. Cubical higher category needs similarly its Θ0. In this work we describe, combinatorially, the small category Θ0 which objects are cubical pasting diagrams and which morphisms are morphisms of cubical sets.