Aspects of Cubical Higher Category Theory

TL;DR

This paper develops a cubical geometric framework for weak higher categories, defining monads on cubical sets that model various levels of cubical weak $ $-categories, functors, and transformations.

Contribution

It introduces monads on cubical sets for modeling cubical weak $ $-categories, functors, and transformations, extending globular theories to cubical geometry.

Findings

Defined monads for cubical weak $ $-categories

Constructed monads for cubical weak $ $-functors

Established models for cubical weak natural $ $-transformations

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 $\mathbb{C}\mathbb{S}ets$ of cubical sets which algebras are models of cubical weak $\infty$-categories. Also for each $n\in\mathbb{N}$ we define a monad on $\mathbb{C}\mathbb{S}ets$ which algebras are models of cubical weak $(\infty,n)$-categories. And finally we define a monad on the category $\mathbb{C}\mathbb{S}ets^2$ which algebras are models of cubical weak $\infty$-functors, and a monad on the category $\mathbb{C}\mathbb{S}ets^4$ which algebras are models of cubical weak natural $\infty$-transformations.