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 On Fri, May 16, 2014, at 05:22 PM, George Janelidze wrote:
Dear Richard,
I would like to see your question formulated more precisely, and showing general (admissible) Galois theory example instead of the locally connected topos example. Some days ago you recommended Carboni-Janelidze-Kelly-Pare paper as one of references for factorization systems (thank you for that!), and now please look at Section 5 of that paper. Note that "admissible"="semi-left-exact" can be replaced with "fibration".
Best regards, George Janelidze
-------------------------------------------------- From: "Richard Garner" <richard.garner@mq.edu.au> Sent: Thursday, May 15, 2014 1:15 PM To: "Categories list" <categories@mta.ca> Subject: categories: Descent for fibred monads
Dear categorists,
Does the following variant of the Benabou-Roubaud/Beck monadic descent theorem appear anywhere?
Let p:E--->B be a fibration with sums and let T:E--->E be a fibred monad over B. Let q: E^T ----> B be the induced fibration of T-algebras. Let f: x--->y in B. Then to give T-algebra descent data for f---that is, a diagram over the kernel-pair of f valued in E^T---is equally to give an algebra for the composite monad
E_x ----f_!----> E_y ----T_y---> E_y ---f^*----> E_x
This doesn't seem to be an application of the usual monadic descent theorem to q: E^T ---> B; that would identify T-algebra descent data for f with algebras for a monad on (E^T)_x, not on E_x.
For example, take E ----> S a connected topos with pi_0 -| Delta -| Gamma. Let T be the monad for constant objects on E induced by the fibred adjunction pi_0 -| Delta. Given f: U --->> 1 in E, to give T-algebra descent data for f is to give a locally constant object split by U. So such objects are equally the algebras for the monad
E/U -----> E/U (A--->U) |----> (Delta pi_0 A) x U ----> U
In the same situation, take T to be the monad for free vector spaces E ---pi_0---> S ---Fv---> S ---Delta---> E induced by the free vector space monad Fv on S. Then T-algebra descent data over U --->> 1 is a vector bundle split by U; so such objects are equally algebras for the monad (A--->U) |----> (Delta Fv pi_0 A) x U ---> U
Richard
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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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Richard Garner