Algebraic Definition of weak ($\infty$; n)-Categories

Abstract

In this paper we define a sequence of monads $\mathbb{T}^(\infty;n)$ $(n\in\mathbb{N})$ on $\infty$-$\mathbb{G}\text{r}$, the category of the $\infty$-graphs. We conjecture that algebras for $\mathbb{T}^(0;n)$ which are defined in a purely algebraic setting, are models of weak $\infty$-groupoids. And for all $n>1$ we conjecture that algebras for $\mathbb{T}^(\infty;n)$ which are defined in a purely algebraic setting, are models of weak $(\infty; n)$-categories.