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.
Keywords: Monad of effective descent type, (co)monadicity, separable functor, coring, descent data
2000 MSC: 18A40, 18C15, 18C20, 16W30
Theory and Applications of Categories,
Vol. 16, 2006,
No. 1, pp 1-45.
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