\(\infty\)-categorical monadicity and descent (Q2360802)

This work gives a contribution to the study of monadic adjunctions and descent in a very general and abstract framework, from higher category theory. Its setup is \textit{\(\infty\)-cosmoi} (see Definitions 2.1 and 2.7), which come from the works by \textit{E. Riehl} and \textit{D. Verity} [Adv. Math. 280, 549--642 (2015; Zbl 1319.18005); Adv. Math. 286, 802--888 (2016; Zbl 1329.18020); J. Pure Appl. Algebra 221, No. 3, 499--564 (2017; Zbl 1378.18007)]. Given a \textit{homotopy coherent monad} in this context, one may define the \(\infty\)-category of \textit{homotopy coherent algebras} over it (see Definition 2.34). Every \textit{homotopy coherent adjunction} gives rise to a homotopy coherent monad and to a comparison functor from the source of the right adjoint to the category of homotopy coherent algebras over it. The main result of the article (Theorem of its Introduction) gives a necessary and sufficient condition so that this comparison is fully faithfull (see Definition 3.2 for the meaning of this notion in the framework of \(\infty\)-cosmoi). It is applied to a very general notion of \textit{descent} (see \S 3.3), which is the main motivation of this work, in particular to some kind of \textit{descent spectral sequences} (see \S 3.4).