$\mathrm{\infty }$-categorical monadicity and descent

Riehl and Verity have introduced an “$\mathrm{\infty }$-cosmic” framework in which they redevelop the category theory of $\mathrm{\infty }$-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 $\mathrm{\infty }$-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.