higher monadic descent



Locality and descent



Special and general types

Special notions


Extra structure



Higher algebra



Higher monadic descent is the generalization of the notion of monadic descent from category theory to higher category theory. It relates to the descent of ∞-stacks as ordinary monad descent relates to stacks.

See also cohomological descent.


Amitsur complex, Sweedler corings, Hopf algebroids

For ϕ:BA\phi \colon B \longrightarrow A a homomorphism of suitable monoids, there is the corresponding pull-push adjunction (extension of scalars \dashv restriction of scalars) on categories of modules

(() BAϕ *):Mod Aϕ *() BAMod B. ((- )\otimes_B A \dashv \phi^\ast ) \;\colon\; Mod_A \stackrel{\overset{(-)\otimes_B A}{\leftarrow}}{\underset{\phi^\ast}{\longrightarrow}} Mod_B \,.

The bar construction of the corresponding monad is the corresponding Amitsur complex.

(e.g. Hess 10, section 6)


2-categorical monadic descent

A monadicity theorem for pseudomonads is proved in

  • I. J. Le Creurer, F. Marmolejo, E. M. Vitale, Beck’s theorem for pseudo-monads, J. Pure Appl. Algebra 173 (2002), no. 3, 293–313.

In 2-category-theory, a notion of 2-fibered categories are defined in Gray’s work and then again introduced and discussed at length by Claudio Hermida, who has good ideas on higher n (and I will be trilled to hear once that he found the time to return to the topic and give us good answers). An appendix in

  • Claudio Hermida, Descent on 2-fibrations and strongly 2-regular 2-categories, Appl. Categ. Structures 12 (2004), no. 5-6, 427–459.

is discussing a 2-categorical version of Beck-Chevalley condition needed to compare it with the 2-monadic descent; there is also a version of Beck’s theorem sketched there.

(,1)(\infty,1)-categorical monadic descent

A comprehensive treatment in the context of (∞,1)-category-theory, general theory of (∞,1)-monads and their monadicity theorem is in

later absorbed as

Unfortunately, Kontsevich‘s monadicity theorem (July 2004) which is in the setup of A-∞-categories, still remains unpublished. The triangulated version is in Rosenberg’s lectures

  • A. L. Rosenberg, Topics in noncommutative algebraic geometry, homological algebra and K-theory, preprint MPIM Bonn 2008-57 pdf, page 36-37.

Its proof is based on Verdier's abelianization functor.

See also, for another point of view,

Discussion of monadic descent for simplicially enriched categories is in

and for quasi-categories in

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