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Hom weak ω-categories of a weak ω-category

Published online by Cambridge University Press:  12 May 2022

Thomas Cottrell  and
Soichiro Fujii Open the ORCID record for Soichiro Fujii [Opens in a new window]
Thomas Cottrell
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
Soichiro Fujii*
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan
*
*Corresponding author. Email: s.fujii.math@gmail.com
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Abstract

Classical definitions of weak higher-dimensional categories are given inductively, for example, a bicategory has a set of objects and hom categories, and a tricategory has a set of objects and hom bicategories. However, more recent definitions of weak n-categories for all natural numbers n, or of weak $\omega$ -categories, take more sophisticated approaches, and the nature of the ‘hom is often not immediate from the definitions’. In this paper, we focus on Leinster’s definition of weak $\omega$ -category based on an earlier definition by Batanin and construct, for each weak $\omega$ -category $\mathcal{A}$ , an underlying (weak $\omega$ -category)-enriched graph consisting of the same objects and for each pair of objects x and y, a hom weak $\omega$ -category $\mathcal{A}(x,y)$ . We also show that our construction is functorial with respect to weak $\omega$ -functors introduced by Garner.

Keywords

Weak ω-categoryweak ω-groupoidweak ω-functoroperadintensional Martin-Löf type theoryidentity type

Information

Type
Special Issue: The Power Festschrift
Information
Mathematical Structures in Computer Science , Volume 32 , Special Issue 4: The Power Festschrift , April 2022 , pp. 420 - 441
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

We gratefully acknowledge the support of Royal Society grant IE160402. The second author is supported by ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603), JST. No data were generated in association with this paper.

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