TY - JOUR

T1 - Hom weak ω-categories of a weak ω-category

AU - Cottrell, Thomas

AU - Fujii, Soichiro

N1 - Funding Information:
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.

PY - 2022/4/12

Y1 - 2022/4/12

N2 - 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 -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 -category based on an earlier definition by Batanin and construct, for each weak -category, an underlying (weak -category)-enriched graph consisting of the same objects and for each pair of objects x and y, a hom weak -category. We also show that our construction is functorial with respect to weak -functors introduced by Garner.

AB - 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 -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 -category based on an earlier definition by Batanin and construct, for each weak -category, an underlying (weak -category)-enriched graph consisting of the same objects and for each pair of objects x and y, a hom weak -category. We also show that our construction is functorial with respect to weak -functors introduced by Garner.

KW - Weak ω-category

KW - identity type

KW - intensional Martin-Löf type theory

KW - operad

KW - weak ω-functor

KW - weak ω-groupoid

UR - http://www.scopus.com/inward/record.url?scp=85144349487&partnerID=8YFLogxK

U2 - 10.1017/S0960129522000111

DO - 10.1017/S0960129522000111

M3 - Article

VL - 32

SP - 420

EP - 441

JO - Mathematical Structures in Computer Science

JF - Mathematical Structures in Computer Science

SN - 0960-1295

IS - 4

ER -