Monads of effective descent type and comonadicity

By Bachuki Mesablishvili. Theory and Applications of Categories, 2006

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We show, for an arbitrary adjunction FU : B→ Awith B Cauchy complete, that the functor F is comonadic if and only if the monad T on A induced by the adjunction is of effective descent type, meaning that the free T-algebra functor F T : A→A T is comonadic. This result is applied to several situations: In Section 4 to give a sufficient condition for an exponential functor on a cartesian closed category to be monadic, in Sections 5 and 6 to settle the question of the comonadicity of those functors whose domain is Set ,o rSet, or the category of modules over a semisimple ring, in Section 7 to study the effectiveness of (co)monads on module categories. Our final application is a descent theorem for noncommutative rings from which we deduce an important result of A. Joyal and M. Tierney and of J.-P. Olivier, asserting that the effective descent morphisms in the opposite of the category of commutative unital rings are precisely the pure monomorphisms.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/12fb8858a1dde44cd3823b086c82e6e910c36028

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Citations in the corpus

Stacks of group representations (2013, background): “In algebraic geometry, descent has been systematically studied by Grothendieck and the Diadochi and applies to many frameworks, including monads; see Mesablishvili [Mes06].”
Bimonads and Hopf monads on categories (2007, background): “17 of [20] that, when A is Cauchy complete, the functor φ is monadic.”