• Corpus ID: 173990720

Descent Data and Absolute Kan Extensions

@article{Nunes2019DescentDA,
  title={Descent Data and Absolute Kan Extensions},
  author={Fernando Lucatelli Nunes},
  journal={arXiv: Category Theory},
  year={2019}
}

• Fernando Lucatelli Nunes
• Published 3 June 2019
• Mathematics
• arXiv: Category Theory

The fundamental construction underlying descent theory, the lax descent category, comes with a functor that forgets the \textit{descent data}. We prove that, in any $2$-category with lax descent objects, the forgetful morphisms create all absolute Kan extensions. As a consequence, we get a monadicity theorem which says that a right adjoint functor is monadic if and only if it is, up to the composition with an equivalence, a functor that forgets descent data. In particular, within the classical context of descent theory, we show that, in a fibred category, the forgetful functor between the category of internal actions of a precategory and the category of internal actions of the underlying discrete precategory is monadic if and only if it has a left adjoint. More particularly, this shows that one of the implications of the celebrated Benabou-Roubaud theorem does not depend on the so called Beck-Chevalley condition. Namely, we prove that, in a fibred category with pullbacks, whenever an effective descent morphism induces a right adjoint functor, the functor is monadic.