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charset="utf-8" Content-Transfer-Encoding: quoted-printable Hi all, So here's the thing: my setup is fibrations (non-cloven in general) over a = given model of algebraic set theory, with anafunctors over the base built u= sing functors preserving cartesian morphisms in the usual way. So I'm avoid= ing choice both internal to the model of AST, as well as externally, so tha= t eso + ff faithful functors aren't always equivalences in the 2-category o= f fibrations over the given base. I presume B=C3=A9nabou thought at some point about distributors between fib= rations, but I'd have to dig through those old scanned documents to see if = there is a paper record. My approach would presumably be a special case of = that, as he so forcefully pointed out to me a bit over a decade ago on this= list. If anyone recalls anything along these lines, I'd appreciate a point= er. All the best, David David Roberts Webpage: https://ncatlab.org/nlab/show/David+Roberts Blog: https://thehighergeometer.wordpress.com On Sat, 10 Feb 2024 at 06:25, Dusko Pavlovic > wrote: we are all departing, of course. there is no power law there. the way i understand plack's statement about the paradigm changes when peop= le depart is that some people leave behind houses for other people to live = in, whereas others leave behind fortresses with nothing to defend. there is= a paper by thomas kuhn about "The Function of Dogma in Scientific Research= " which i think speaks to us more clearly than his famous "Scientific Revol= utions". -- dusko On Thu, Feb 8, 2024 at 3:48=E2=80=AFPM Michael Barr, Prof. > wrote: We're departing, we're departing. Just give us a few more years. Michael ________________________________ From: Dusko Pavlovic > Sent: Thursday, February 8, 2024 7:02 PM To: David Roberts > Cc: categories@mq.edu.au > Subject: Re: Fibrewise opposite fibration FWIW, the history of problems around choosing representatives of equivalenc= e classes may very well be a completely self-inflicted artifact of academic= politics. if it were up to young cantor, sets could be taken with well-ord= ers and any equivalence class would have a minimal representative. in the w= orld where math is done in collaboration with computers, everything *is* we= ll-ordered, and cantor was right. if a fibration E is computable, the equiv= alence classes of spans that define E^op have minimal representatives and a= re effectively defined. in lawvere's words: if we don't transition to cardi= nalities but stay in a boolean topos with cantor-bernstein, all surjections= split effectively. but dedekind convinced cantor that he should yield to the great professors = and not assume that the reals can be well-ordered. then dedekind used canto= r's basic construction in Zahlenbericht (with a reference to cantor in the = manuscript, no reference in the published version), whereas cantor spent 10= years trying to *prove* that all sets can be well-ordered. enter zermelo t= o edit cantor's collected works, to criticize cantor for well-ordering, and= to packages this problematic idea behind the second-order quantifier in hi= s own contribution: the axiom of choice. so that we could happily choose th= e representatives of equivalence classes ever after from behind the second-= order quantifier =3D& but it will be like max planck said: the new paradigm will win when the gen= erations suppressing it depart. in the world where math is computed, zermel= o was wrong, cantor was right, and equivalence classes have minimal represe= ntatives. OR if we develop category theory categorically, the dual hom-sets= will be effectively definable... :) -- dusko On Sat, Jan 27, 2024 at 7:21=E2=80=AFPM David Roberts > wrote: Hi all what with all the discussion of B=C3=A9nabou and fibrations, I have a question about what happens at a foundational level when one takes the opposite of a fibration E --> B fibrewise. Call this (E/B)^op --> B For reference, one can see section 5 of Streicher's https://arxiv.org/abs/1801.02927 for the construction. The point is that the morphisms are defined to be equivalence classes of certain data. However, in a setting where one cannot necessarily form equivalence classes, it's less clear how to proceed. The point is that I don't want to be assuming any particular foundations here, just working at the level of a first-order theory (in the way that ETCS is a first-order theory of sets, say) The only thing I can think of is that the construction actually describes a category weakly enriched in 0-truncated groupoids (or whatever you want to call the first-order description of such a thing). You still get a functor down to the base 1-category, and perhaps one has to now think about what it means for such a thing to be a fibration, without passing to the plan 1-category quotient. That is probably fine for my purposes, but then you have to worry that taking the fibrewise opposite again should return the original fibration, at least up to equivalence. The original construction with the equivalence classes gives back the original thing up to *isomorphism*: ((E/B)^op/B)^op \simeq E, over B. So now one has to think about what the fiberwise opposite construction looks like for these slightly generalised fibrations (enriched with 0-truncated groupoids), and one would hope that this gives back the original thing after two applications (again, up to the appropriate notion of equivalence). Note that the construction in the literature (eg Streicher's notes, or Jacob's book) has the fibres (E/B)^op_b of the fibrewise opposite be *isomorphic* to the opposite of the original fibres E_b. In this fancier setting, one might also only get equivalence, but I haven't checked that. Has anyone thought about something like this before? Or any pointers to anything related? Best wishes, David Roberts Webpage: https://ncatlab.org/nlab/show/David+Roberts Blog: https://thehighergeometer.wordpress.com You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | Learn = more about Microsoft 365 Groups You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | Learn = more about Microsoft 365 Groups You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | = Learn more about Microsoft 365 Groups --00000000000075372c061101286d Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable
Hi all,

So here's the thing: my setup is fibrations (non-cloven in general) ov= er a given model of algebraic set theory, with anafunctors over the base bu= ilt using functors preserving cartesian morphisms in the usual way. So I'm = avoiding choice both internal to the model of AST, as well as externally, so that eso + ff faithful functor= s aren't always equivalences in the 2-category of fibrations over the given= base.

I presume B=C3=A9nabou thought at some point about distributors betwee= n fibrations, but I'd have to dig through those old scanned documents to se= e if there is a paper record. My approach would presumably be a special cas= e of that, as he so forcefully pointed out to me a bit over a decade ago on this list. If anyone recalls anything= along these lines, I'd appreciate a pointer.

All the best,
David




On Sat, 10 Feb 2024 at 06:25, Dusko P= avlovic <duskgoo@gmail.com> = wrote:
we are all departing, of course. there is no power law there.

the way i understand plack's statement about the paradigm changes when= people depart is that some people leave behind houses for other people to = live in, whereas others leave behind fortresses with nothing to defend. the= re is a paper by thomas kuhn about "The Function of Dogma in Scientific Research" which i think spe= aks to us more clearly than his famous "Scientific Revolutions".<= /div>

-- dusko

On Thu, Feb 8, 2024 at 3:48=E2=80=AFP= M Michael Barr, Prof. <barr.michael@mcgill.ca> wrote:
We're departing, we're departing.  Just give us a few more years.

Michael
From: Dusko Pavlovic <duskgoo@gmail.com>
Sent: Thursday, February 8, 2024 7:02 PM
To: David Roberts <droberts.65537@gmail.com>
Cc: catego= ries@mq.edu.au <categories@mq.edu.au>
Subject: Re: Fibrewise opposite fibration
 
FWIW, the history of problems around choosing representatives of equiv= alence classes may very well be a completely self-inflicted artifact of aca= demic politics. if it were up to young cantor, sets could be taken with wel= l-orders and any equivalence class would have a minimal representative. in the world where math is done in co= llaboration with computers, everything *is* well-ordered, and cantor was ri= ght. if a fibration E is computable, the equivalence classes of spans that = define E^op have minimal representatives and are effectively defined. in lawvere's words: if we don't transition to= cardinalities but stay in a boolean topos with cantor-bernstein, all surje= ctions split effectively.

but dedekind convinced cantor that he should yield to the great profes= sors and not assume that the reals can be well-ordered. then dedekind used = cantor's basic construction in Zahlenbericht (with a reference to cantor in= the manuscript, no reference in the published version), whereas cantor spent 10 years trying to *prove* th= at all sets can be well-ordered. enter zermelo to edit cantor's collected w= orks, to criticize cantor for well-ordering, and to packages this problemat= ic idea behind the second-order quantifier in his own contribution: the axiom of choice. so that we could = happily choose the representatives of equivalence classes ever after from b= ehind the second-order quantifier =3D&

but it will be like max planck said: the new paradigm will win when th= e generations suppressing it depart. in the world where math is computed, z= ermelo was wrong, cantor was right, and equivalence classes have minimal re= presentatives. OR if we develop category theory categorically, the dual hom-sets will be effectively defin= able...

:)
-- dusko

On Sat, Jan 27, 2024 at 7:21=E2=80=AFPM David Roberts <= droberts.6553= 7@gmail.com> wrote:
Hi all

what with all the discussion of B=C3=A9nabou and fibrations, I have a
question about what happens at a foundational level when one takes the
opposite of a fibration E --> B fibrewise. Call this (E/B)^op --> B
For reference, one can see section 5 of Streicher's
https://arxiv.org/abs/1801.02927 for th= e construction. The point is
that the morphisms are defined to be equivalence classes of certain
data. However, in a setting where one cannot necessarily form
equivalence classes, it's less clear how to proceed. The point is that
I don't want to be assuming any particular foundations here, just
working at the level of a first-order theory (in the way that ETCS is
a first-order theory of sets, say)

The only thing I can think of is that the construction actually
describes a category weakly enriched in 0-truncated groupoids (or
whatever you want to call the first-order description of such a
thing). You still get a functor down to the base 1-category, and
perhaps one has to now think about what it means for such a thing to
be a fibration, without passing to the plan 1-category quotient.

That is probably fine for my purposes, but then you have to worry that
taking the fibrewise opposite again should return the original
fibration, at least up to equivalence. The original construction with
the equivalence classes gives back the original thing up to
*isomorphism*: ((E/B)^op/B)^op \simeq E, over B. So now one has to
think about what the fiberwise opposite construction looks like for
these slightly generalised fibrations (enriched with 0-truncated
groupoids), and one would hope that this gives back the original thing
after two applications (again, up to the appropriate notion of
equivalence).

Note that the construction in the literature (eg Streicher's notes, or
Jacob's book) has the fibres (E/B)^op_b of the fibrewise opposite be
*isomorphic* to the opposite of the original fibres E_b. In this
fancier setting, one might also only get equivalence, but I haven't
checked that.

Has anyone thought about something like this before? Or any pointers
to anything related?

Best wishes,

David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts
Blog: https://thehighergeometer.wordpress.com
 
 
You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message.
 
View g= roup files &nbs= p; |   Leave group   | &n= bsp; Learn more about Microsoft 365 Groups
 
 
 
You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message.
 
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You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message.
 
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