Pseudo-Kan Extensions and Descent Theory

Abstract: There are two main constructions in classical descent theory: the category of algebras and the descent category, which are known to be examples of weighted bilimits. We give a formal approach to descent theory, employing formal consequences of commuting properties of bilimits to prove classical and new theorems in the context of Janelidze-Tholen "Facets of Descent II", such as B\'{e}nabou-Roubaud Theorems, a Galois Theorem, embedding results and formal ways of getting effective descent morphisms. In order to do this, we develop the formal part of the theory on commuting bilimits via pseudomonad theory, studying idempotent pseudomonads and proving a $2$-dimensional version of the adjoint triangle theorem. Also, we work out the concept of pointwise pseudo-Kan extension, used as a framework to talk about bilimits, commutativity and the descent object. As a subproduct, this formal approach can be an alternative perspective/guiding template for the development of higher descent theory.