Thank you, Richard, for the clarification. But you say

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This deficiency also applies to the example I started with, of a locally connected topos E---->S.

and this not "also" but "therefore". I mean, my example of G-sets is the simplest non-trivial (in the sense of Galois theory) special case of your locally connected topos example, as you surely know.

Yes indeed!

And another comment: to say that HI is left exact is the same as to say that I is left exact - so, your story is a localization story, right?

That's right. In this case, in fact, things are a bit boring; to say that the reflection IH is a fibred reflection, equivalently a localisation, means that the E-maps are stable under pullback, which in turn means---by Section 6 of CJKP---that the M-maps already descend along effective descent morphisms. So "trivial = locally trivial". Thinking about this further, the situation here is actually completely typical: if p: D ---> C is a fibration, and T is a fibred monad on D, then any effective descent morphism for the fibration p will also be one for the fibration T-Alg ----> C. Indeed, to say that a fibration sees a particular map as an effective descent morphism is expressible as a (bicategorical) orthogonality property in the 2-category Fib(C). Thus the class of fibrations with this property is closed under bilimits; in particular, under Eilenberg-Moore objects of monads. In short: if objects descend, then for any fibred monad T, also T-algebras descend. So, to summarise: (a) If p: D ---> C is a fibration with sums, and T a fibred monad on p, and f: x--->y in C, then the category of descent data of T-algebras w.r.t. f is isomorphic to the category of f^* T_y f_!-algebras. (b) If, in the same situation, f is an effective descent morphism for p, then the category of descent data of T-algebras w.r.t. f is equivalent to the category of T_y algebras (c) Thus, in this situation, the category of f^* T_y f_! algebras is equivalent to the category of T_y-algebras. And unfortunately, the requirement of T's fibredness rules out all the interesting examples, such as those coming from Galois theory. So perhaps this is why (a) above does not appear in the literature; the monadic treatment it promises for "local" structure is in fact, by (c), only valid for local structure that already descends. Thanks for the careful readings, George - I think I understand what is going on here much better now! Richard

George

-------------------------------------------------- From: "Richard Garner" <richard.garner@mq.edu.au> Sent: Saturday, May 17, 2014 9:16 AM To: "George Janelidze" <janelg@telkomsa.net>; "Categories list" <categories@mta.ca> Subject: Re: categories: Re: Descent for fibred monads

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Ah! You are quite correct. I was hasty in saying that the Galois theory situation is an example of the result I am interested in. The reason it does not work is that the reflection HI does not induce a fibred monad on C. The semi-left-exactness ensures the simple formula for the reflection: A--->B goes to the pullback of HIA ----> HIB along B---->IHB. What it does not ensure is that pullback commutes with reflection---which would be to ask that HI be left exact.

This deficiency also applies to the example I started with, of a locally connected topos E---->S. The "fibred monad" Delta pi_0 is only fibred over S, whereas I need it to be fibred over E. So in fact it seems that a correct example is given by a topos with totally connected components --- meaning that the left adjoint pi_0 of Delta preserves pullbacks. In this case, then, the analogue of (2) does hold.

Richard

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