Aspects of Cubical Higher Category Theory

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.