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. Zbl 0287.18007.
• Kahn, Bruno (2025). "On the Bénabou-Roubaud theorem" (PDF). Cahiers de topologie et géométrie différentielle catégoriques. LXVI (2): 3–12. arXiv:2404.00868.
• Janelidze, George; Tholen, Walter (1994). "Facets of descent, I". Applied Categorical Structures. 2 (3): 245–281. doi:10.1007/BF00878100.
• Janelidze, G.; Tholen, W. (1997). "Facets of Descent, II". Applied Categorical Structures. 5 (3): 229–248. doi: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". ncatlab.org.
• "Bénabou-Roubaud theorem". ncatlab.org.
• "English Reference for the Bénabou-Roubaud theorem". -English translation of Bénabou&Roubaud(1970).