An algebraic definition of (∞,n)-categories

In this paper we define a sequence of monads T(∞, n)(n ε N) on the category ∞-Gr of ∞-graphs. We conjecture that algebras for T(∞, 0), which are defined in a purely algebraic setting, are models of ∞-groupoids. More generally, we conjecture that T(∞;n)-algebras are models for (∞, n)-categories. We prove that our (∞, 0)-categories are bigroupoids when truncated at level 2.