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

Document Type : Research Paper

Author

Laboratoire 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.

Abstract

In 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. 

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References

[1] Brown, R., Higgins, P.J., and Sivera, R., “Nonabelian Algebraic Topology”, European Mathematical Society, Tracts in Mathematics Volume 15, 2011.
[2] Kachour, C., Combinatorial approach to the category Θ0 of cubical pasting diagrams. https://arxiv.org/pdf/2102.09787.pdf
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Categories and General Algebraic Structures with Applications
Volume 21, Issue 1
July 2024
Pages 19-68

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