Re: Descent for fibred monads

[Richard Garner <richard.garner@mq.edu.au>]

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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