Descent for fibred monads

Descent for fibred monads

[Richard Garner]

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


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

Re: Descent for fibred monads

[George Janelidze]

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

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

Re: Descent for fibred monads

[Richard Garner]

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


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

Re: Descent for fibred monads

[George Janelidze]

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/ ]

Re: Descent for fibred monads

[Richard Garner]

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


On Sat, May 17, 2014, at 04:53 AM, George Janelidze wrote:
> 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
>

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

Re: Descent for fibred monads

[Richard Garner]

> 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
>>

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

Re: Descent for fibred monads

[Richard Garner]

Actually

> 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

This bit is clear rubbish. But the rest of my question remains.

Richard


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