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.

10.48308/cgasa.2023.104127

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. 

Keywords

Main Subjects

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
[3] Kachour, C., Algebraic models of cubical weak ∞-categories with connections, Categ. General Alg. Structures Appl. 16(1) (2022), 143-187.
[4] Kachour, C., Algebraic models of cubical weak higher structures, Categ. General Alg. Structures Appl. 16(1) (2022), 189-220.
[5] Kachour, C., The coherator Θ∞ W of cubical weak ∞-categories with connections, To appears in Categ. General Alg. Structures Appl. (2023).
[6] Camell Kachour, Introduction to Higher Cubical Operads. Pr´epublication de l’IHES, republished in HAL: https://hal.science/hal-04306455
[7] Lair, C., Trames et s´emantiques cat´egoriques des syst`emes de trames, Journal Diagrammes, 18 (1987), 47 pages.
Categories and General Algebraic Structures with Applications

Articles in Press, Accepted Manuscript
Available Online from 24 December 2023

Files

Share

How to cite

Statistics