nLab
higher monadic descent

Idea

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.

Examples

Amitsur complex, Sweedler corings, Hopf algebroids

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

$$((- )\otimes_B A \dashv \phi^\ast ) \;\colon\; Mod_A \stackrel{\overset{(-)\otimes_B A}{\leftarrow}}{\underset{\phi^\ast}{\longrightarrow}} Mod_B \,.$$

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

(e.g. Hess 10, section 6)

Related concepts

• descent

  • cover

  • cohomological descent

  • monadic descent,

    • Sweedler coring

    • higher monadic descent

    • descent in noncommutative algebraic geometry

  • descent spectral sequence

• sheaf, (2,1)-sheaf, 2-sheaf (∞,1)-sheaf

References

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.

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

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

• Jacob Lurie, Derived algebraic geometry II: Noncommutative algebra, math.CT/0702299, Section 3.

later absorned as

• Jacob Lurie, section 6.2.4 of Higher Algebra

Unfortunately, Kontsevich’s monadicity theorem (July 2004) which is in the setup of A-∞-categories, still remains unpublished.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

• Paul Balmer, Descent in triangulated categories, pdf

Discussion of monadic descent for simplicially enriched categories is in

• Kathryn Hess, A general framework for homotopic descent and codescent, arXiv/1001.1556

• Kathryn Hess, Brooke Shipley, The homotopy theory of coalgebras over a comonad (arXiv:1205.3979)

and for quasi-categories in

• Emily Riehl, Dominic Verity, Homotopy coherent adjunctions and the formal theory of monads (arXiv:1310.8279)