Aspects of Cubical Higher Category Theory

Abstract

In this article we show how to build main aspects of our paper on globular weak $(\infty ,n)$-categories, but now for the cubical geometry. Thus we define a monad on the category $\mathbb{CS}ets$ of cubical sets which algebras are models of cubical weak $\infty$-categories. Also for each $n\in \mathbb{N}$ we define a monad on $\mathbb{CS}ets$ which algebras are models of cubical weak $(\infty ,n)$-categories. And finally we define a monad on the category $\mathbb{CS}et{s}^2$ which algebras are models of cubical weak $\infty$-functors, and a monad on the category $\mathbb{CS}et{s}^4$ which algebras are models of cubical weak natural $\infty$-transformations.

40 pages; An improved version of the previous version; for the paragraph referring to $(\infty ,m)$-categories, we treated only the case $m=0$. An even more improved version dealing with all $m\in \mathbb{N}$ will be proposed later

References in corpus (4)

• On the Classification of Topological Field Theories
• Grothendieck $\infty$-groupoids, and still another definition of $\infty$-categories
• Thin fillers in the cubical nerves of omega-categories
• Cubical $(\omega ,p)$-categories

Cited by in corpus (1)

• The algebra of entanglement and the geometry of composition