$\infty$-categorical monadicity and descent

TL;DR

This paper extends classical monadicity and descent criteria to the realm of $ abla$-categories using an $ abla$-cosmic framework, bridging 2-categorical and $ abla$-categorical approaches.

Contribution

It develops an $ abla$-categorical approach to monadicity and descent, providing new criteria and connecting with existing model-categorical methods.

Findings

Extended classical monadicity criteria to $ abla$-categories.

Connected $ abla$-categorical and model-categorical approaches.

Indicated applications to descent theory.

Abstract

Riehl and Verity have introduced an "$\infty$-cosmic" framework in which they redevelop the category theory of $\infty$-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 $\infty$-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.