Re: Descent for fibred monads

["George Janelidze" <janelg@telkomsa.net>]

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

--------------------------------------------------
From: "Richard Garner" <richard.garner@mq.edu.au>
Sent: Friday, May 16, 2014 10:29 AM
To: "George Janelidze" <janelg@telkomsa.net>; "Categories list"
<categories@mta.ca>
Subject: categories: Re:  Descent for fibred monads

> Dear George,
>
> Thanks for your message. I should say that the locally connected topos
> example was just intended to be a sample application of the modified
> monadic descent theorem quoted at the start of my message. But as you
> point out, one could also apply it in the setting of categorical Galois
> theory that you refer to. In the terminology of [CJKP] this would say
> something like:
>
> Let I -| H : X ----> C be an admissible reflection, and p: E --> B an
> effective descent map in C. Then Spl(E,p) is isomorphic to the category
> of algebras for the monad
>
> C / E --Sum_p--> C / B --H^B.I^B--> C / B --p^*--> C / E
>
> Which decomposes into the two statements:
>
> (1) Let I -| H : X ----> C be an admissible reflection, and p: E --> B
> an effective descent map in C. Then Spl(E,p) is isomorphic to the
> category of M-on-objects discrete fibrations over the kernel-pair \bar B
> of p --- i.e. C^{\bar B} /\ M / {\bar B} in the terminology of [CJKP]
>
> -- which is (part of) the Theorem on p.26 of ibid.; and
>
> (2) Let I -| H : X ----> C be an admissible reflection, and p: E --> B
> _any_ map in C. Then C^{\bar B} /\ M / {\bar B} is isomorphic to the
> category of algebras for the monad
>
> C / E --Sum_p--> C / B --H^B.I^B--> C / B --p^*--> C / E
>
> --- and it is really this (2) which I am interested in. Does this come
> up in the categorical Galois theory literature?
>
> The other example I attempted in my original message, but botched rather
> badly, involving vector bundles, was an attempt to give some application
> of this modified descent theorem to a fibred monad which is not a fibred
> reflection.  The categorical Galois theory example is compelling because
> one has a fibred monad whose algebras do not descend along effective
> descent morphisms (although, of course, the underlying objects do). The
> point is that the algebras for lots of fibred monads DO descend along
> effective descent morphisms, e.g., any fibred monad on a topos E ----> S
> induced by a finitary algebraic theory in S. So I guess a subsidiary
> question is whether there are any compelling examples of non-idempotent
> fibred monads whose algebras do not descend.
>
> Richard
>
>
>

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