Semantic factorization and descent (Q2105683)

The principal objective in this paper is to present a counterpart account to the Bénabou-Roubaud theorem [\textit{J. Benabou} and \textit{J. Roubaud}, C. R. Acad. Sci., Paris, Sér. A 270, 96--98 (1970; Zbl 0287.18007)] in the setting of two-dimensional category theory or in the so-called formal category theory, giving the semantic factorization via descent, hence giving, in particular, a characterization of monodicity via descent. The paper aims \begin{itemize} \item[1.] to get a formal monadicity theorem given by a \(2\)-dimensional exact condition, and \item[2.] to better understand the relation between descent and monadicity in a given \(2\)-category and, together with [\textit{F. Lucatelli Nunes}, Theory Appl. Categ. 33, 390--444 (2018; Zbl 1405.18002)], get alternative guiding templates for the development of higher descent theory and monadicity. \end{itemize}