Aspects of Cubical Higher Category Theory

In this article we show how to build main aspects of our paper on globular weak $(\mathrm{\infty },n)$-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 $\mathrm{\infty }$-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 $(\mathrm{\infty },n)$-categories. And finally we define a monad on the category $\mathbb{C}\mathbb{S}et{s}^2$ which algebras are models of cubical weak $\mathrm{\infty }$-functors, and a monad on the category $\mathbb{C}\mathbb{S}et{s}^4$ which algebras are models of cubical weak natural $\mathrm{\infty }$-transformations.