Riehl and Verity have introduced an "∞-cosmic'' framework in which they redevelop the category theory of ∞-categories using 2-categorical arguments. In this paper, we begin with a self-contained review of the parts of their theory needed to discuss adjunctions and monadicity. This is applied in order to extend to the ∞-categorical context the classical criterion for fully faithfulness of the comparison functor induced by an adjunction. We discuss the relation with previous work in the literature--which primarily uses model-categorical techniques--and indicate applications to descent theory.

Sulyma was supported in part by NSF grant DMS-1564289