Re: Descent for fibred monads

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

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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>


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