Monadic descent

In mathematics, especially category theory, a monadic descent is roughly an idea to encode descent data using a monad.

Bénabou-Roubaud theorem

The Bénabou-Roubaud theorem says that (roughly) given a bifibration satisfying the Beck–Chevalley condition for p, the category of descent data is canonically equivalent to the category of algebras of the monad induced by $p_{!},p^{*}$.

See also

• Beck's monadicity theorem

References

• Bénabou, Jean; Roubaud, Jacques (1970). "Monades et descente". C. R. Acad. Sci. Paris Sér. A 270: 96–98. https://gallica.bnf.fr/ark:/12148/bpt6k480298g/f100. 
• Kahn, Bruno (2025). "On the Bénabou-Roubaud theorem". Cahiers de topologie et géométrie différentielle catégoriques LXVI (2): 3–12. https://cahierstgdc.com/wp-content/uploads/2025/04/KAHN_LXVI-2.pdf. 
• Janelidze, George; Tholen, Walter (1994). "Facets of descent, I". Applied Categorical Structures 2 (3): 245–281. doi:10.1007/BF00878100. https://doi.org/10.1007/BF00878100. 
• Janelidze, G.; Tholen, W. (1997). "Facets of Descent, II". Applied Categorical Structures 5 (3): 229–248. doi:10.1023/A:1008697013769. https://doi.org/10.1023/A:1008697013769. 
• Nunes, Fernando Lucatelli (2018). "Pseudo-Kan Extensions and Descent Theory". Theory and Applications of Categories 33: 390–444. doi:10.70930/tac/ncck98gb. 

Further reading

• "Monadic descent". https://ncatlab.org/nlab/show/monadic+descent. 
• "Bénabou-Roubaud theorem". https://ncatlab.org/nlab/show/B%C3%A9nabou-Roubaud+theorem. 
• "English Reference for the Bénabou-Roubaud theorem". https://mathoverflow.net/questions/279097/english-reference-for-the-b%C3%A9nabou-roubaud-theorem.  -English translation of Bénabou&Roubaud(1970).