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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]