Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585321120240701Combinatorial approach of the category $\Theta_0$ of cubical pasting diagrams196810412710.48308/cgasa.2023.104127ENCamell KachourLaboratoire de Math\'ematiques d'Orsay, UMR 8628,
Universit\'e de Paris-Saclay and CNRS,
B\^atiment 307, Facult\'e des Sciences d'Orsay,
94015 ORSAY Cedex, France.0009-0004-9550-1648Journal Article20231224In globular higher category theory the small category $\Theta_0$ of finite rooted trees plays an important role: for example the objects of $\Theta_0$ are the arities of the operations inside the free globular $\omega$-operad $\mathbb{B}^0$ of Batanin, which $\mathbb{B}^0$-algebras are models of globular weak $\infty$-categories; also this globular $\Theta_0$ is an important tool to build the coherator $\Theta^{\infty}_{W^0}$ of Grothendieck which ${\mathbb{S}\text{ets}}$-models are globular weak $\infty$-groupoids. Cubical higher category needs similarly its $\Theta_0$. In this work we describe, combinatorially, the small category $\Theta_0$ which objects are cubical pasting diagrams and which morphisms are morphisms of cubical sets. https://cgasa.sbu.ac.ir/article_104127_f77d5847a666cf26ad9963292d77126e.pdf