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 On Sat, May 17, 2014, at 04:53 AM, George Janelidze wrote:

Dear Richard,

I am sorry, but, unless I completely misunderstood what you are saying, what you call "(2)" is simply wrong. Moreover, this can be seen in the 'very first" example of Galois theory. For, take:

(a) C to be the category of G-sets, where G is any fixed non-trivial group;

(b) X to be the category of sets;

(c) I -| H to be what you called "pi_0 -| Delta" in your first message (that is, for A in C, I(A) is the set of orbits of A, while for S in X, H(S) is the set S equipped with the trivial action of G);

(d) B = 1, the one-element G-set;

(e) E = G, considered as a G-set, on which G acts via its multiplication.

Then C / E is equivalent to the category of sets, and in particular each of its objects is a coproduct of copies of its terminal object G=G; and let us calculate your monad, which is sufficient to do for G=G:

(g) Your C / E --Sum_p--> C / B sends G=G to G-->1;

(h) Then I^B sends G-->1 to 1=1, the terminal object of X / I(B) = X / 1;

(i) H^B and p^* preserves the terminal object;

(j) that is, your monad sends G=G to G=G, and so it is the identity monad.

But the right monad is the free G-set monad (if we identify C / E with the category of sets).

Please either confirm or explain what have I misunderstood in your message.

George

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