#
Monads of effective descent type and comonadicity

##
Bachuki Mesablishvili

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.

http://www.tac.mta.ca/tac/volumes/16/1/16-01.dvi

http://www.tac.mta.ca/tac/volumes/16/1/16-01.ps

http://www.tac.mta.ca/tac/volumes/16/1/16-01.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/1/16-01.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/1/16-01.ps

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