Re: Descent for fibred monads

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

> Thank you, Richard, for the clarification. But you say
>
>> This deficiency also applies to the example I started with, of a locally
>> connected topos E---->S.
>
> and this not "also" but "therefore". I mean, my example of G-sets is the
> simplest non-trivial (in the sense of Galois theory) special case of your
> locally connected topos example, as you surely know.

Yes indeed!

> And another comment: to say that HI is left exact is the same as to say
> that
> I is left exact - so, your story is a localization story, right?

That's right. In this case, in fact, things are a bit boring; to say
that the reflection IH is a fibred reflection, equivalently a
localisation, means that the E-maps are stable under pullback, which in
turn means---by Section 6 of CJKP---that the M-maps already descend
along effective descent morphisms. So "trivial = locally trivial".

Thinking about this further, the situation here is actually completely
typical: if p: D ---> C is a fibration, and T is a fibred monad on D,
then any effective descent morphism for the fibration p will also be one
for the fibration T-Alg ----> C. Indeed, to say that a fibration sees a
particular map as an effective descent morphism is expressible as a
(bicategorical) orthogonality property in the 2-category Fib(C). Thus
the class of fibrations with this property is closed under bilimits; in
particular, under Eilenberg-Moore objects of monads. In short: if
objects descend, then for any fibred monad T, also T-algebras descend.

So, to summarise:

(a) If p: D ---> C is a fibration with sums, and T a fibred monad on p,
and f: x--->y in C, then the category of descent data of T-algebras
w.r.t. f is isomorphic to the category of f^* T_y f_!-algebras.

(b) If, in the same situation, f is an effective descent morphism for p,
then the category of descent data of T-algebras w.r.t. f is equivalent
to the category of T_y algebras

(c) Thus, in this situation, the category of f^* T_y f_! algebras is
equivalent to the category of T_y-algebras.

And unfortunately, the requirement of T's fibredness rules out all the
interesting examples, such as those coming from Galois theory. So
perhaps this is why (a) above does not appear in the literature; the
monadic treatment it promises for "local" structure is in fact, by (c),
only valid for local structure that already descends.

Thanks for the careful readings, George - I think I understand what is
going on here much better now!

Richard

>
> George
>
> --------------------------------------------------
> From: "Richard Garner" <richard.garner@mq.edu.au>
> Sent: Saturday, May 17, 2014 9:16 AM
> To: "George Janelidze" <janelg@telkomsa.net>; "Categories list"
> <categories@mta.ca>
> Subject: Re: categories: Re:  Descent for fibred monads
>
>> 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
>>

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