Combinatorial approach of the category \Theta_۰ of cubical pasting diagrams

In globular higher category theory the small category \Theta_۰ of finite rooted trees plays an important role: for example the objects of \Theta_۰ are the arities of the operations inside the free globular \omega-operad \mathbb{B}^۰ of Batanin, which \mathbb{B}^۰-algebras are models of globular weak \infty-categories; also this globular \Theta_۰ is an important tool to build the coherator \Theta^{\infty}_{W^۰} of Grothendieck which {\mathbb{S}\text{ets}}-models are globular weak \infty-groupoids. Cubical higher category needs similarly its \Theta_۰. In this work we describe, combinatorially, the small category \Theta_۰ which objects are cubical pasting diagrams and which morphisms are morphisms of cubical sets.

Keywords:

Pasting diagrams , pasting schemes , sketch theory , higher order terms