Monads of effective descent type and comonadicity

Bachuki Mesablishvili

Theory and applications of categories, Tome 16 (2006), pp. 1-45.

Résumé

We show, for an arbitrary adjunction $F \dashv U : \cal B \to \cal A$ with $\cal B$ Cauchy complete, that the functor $F$ is comonadic if and only if the monad $T$ on $\cal A$ induced by the adjunction is of effective descent type, meaning that the free $T$-algebra functor $F^{T}: \cal A \to \cal A^{T}$ 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, or Set$_{\star}$, 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.

Détail

Classification : 18A40, 18C15, 18C20, 16W30
Keywords: Monad of effective descent type, (co)monadicity, separable functor, coring, descent data