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charset=UTF-8 Content-Transfer-Encoding: quoted-printable >in the world where math is done in collaboration with computers, everythin= g *is* well-ordered, and cantor was right.=20 I think, only if we record each run of each programme. The ordering is not = stable in this sense. Sergei Soloviev Le Vendredi, F=C3=A9vrier 09, 2024 01:02 CET, Dusko Pavlovic a =C3=A9crit: > FWIW, the history of problems around choosing representatives of equivale= nce classes may very well be a completely self-inflicted artifact of academ= ic politics. if it were up to young cantor, sets could be taken with well-o= rders and any equivalence class would have a minimal representative. in the= world where math is done in collaboration with computers, everything *is* = well-ordered, and cantor was right. if a fibration E is computable, the equ= ivalence classes of spans that define E^op have minimal representatives and= are effectively defined. in lawvere's words: if we don't transition to car= dinalities but stay in a boolean topos with cantor-bernstein, all surjectio= ns split effectively. >=20 > but dedekind convinced cantor that he should yield to the great professor= s and not assume that the reals can be well-ordered. then dedekind used can= tor's basic construction in Zahlenbericht (with a reference to cantor in th= e manuscript, no reference in the published version), whereas cantor spent = 10 years trying to *prove* that all sets can be well-ordered. enter zermelo= to edit cantor's collected works, to criticize cantor for well-ordering, a= nd to packages this problematic 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 behind the secon= d-order quantifier =3D& >=20 > but it will be like max planck said: the new paradigm will win when the g= enerations suppressing it depart. in the world where math is computed, zerm= elo was wrong, cantor was right, and equivalence classes have minimal repre= sentatives. OR if we develop category theory categorically, the dual hom-se= ts will be effectively definable... >=20 > :) > -- dusko >=20 > On Sat, Jan 27, 2024 at 7:21=E2=80=AFPM David Roberts > wrote: > Hi all >=20 > 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 >=20 > For reference, one can see section 5 of Streicher's > https://protect-au.mimecast.com/s/pw9JCk815RCQBglBI2ERjy?domain=3Darxiv.o= rg 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) >=20 > 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. >=20 > 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). >=20 > 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. >=20 > Has anyone thought about something like this before? Or any pointers > to anything related? >=20 > Best wishes, >=20 > David Roberts > Webpage: https://protect-au.mimecast.com/s/FuxkClx1OYUZELGEt9Uog4?domain= =3Dncatlab.org > Blog: https://protect-au.mimecast.com/s/qblGCmO5wZsnlvXlhB28KI?domain=3Dt= hehighergeometer.wordpress.com >=20 >=20 > You're receiving this message because you're a member of the Categories m= ailing list group from Macquarie University. To take part in this conversat= ion, reply all to this message. >=20 > View group files | Leave group | Lear= n more about Microsoft 365 Groups >=20 >=20 >=20 > You're receiving this message because you're a member of the Categories m= ailing list group from Macquarie University. To take part in this conversat= ion, reply all to this message. >=20 > View group files | Leave group | Lear= n more about Microsoft 365 Groups > --MCBoundary=_12402092225320081 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=UTF-8 >in the world where math is done in collaboration with compu= ters, everything *is* well-ordered, and cantor was right.

I think, only if we record each run of each programme. The ordering is not = stable
in this sense.

Sergei Soloviev

Le Vendredi, Février 09, 2024 01:02 CET, Dusko Pavlovic <duskgoo@= gmail.com> a écrit:

> 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 collaboration with computers, everything *i= s* well-ordered, and cantor was right. 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 surjec= tions 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 spe= nt 10 years trying to *prove* that all sets can be well-ordered. enter zerm= elo to edit cantor's collected works, to criticize cantor for well-ordering= , and to packages this problematic idea behind the second-order quantifier = in his own contribution: the axiom of choice. so that we could happily choo= se the representatives of equivalence classes ever after from behind the se= cond-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 definable...
>
> :)
> -- dusko
>
> On Sat, Jan 27, 2024 at 7:21=E2=80=AFPM David Roberts <droberts.655= 37@gmail.com<mailto:droberts.65537@gmail.com>> wrote:
> Hi all
>
> what with all the discussion of Bénabou 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<http= s://protect-au.mimecast.com/s/JQQrCzvkmpf2ooA0c4GaAU?domain=3Darxiv.org= > 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<= BR> > 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<= BR> > 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<https://protect-au.mimecast.com/s/DNIaCANpnDCWAAzvC8EeE= A?domain=3Dncatlab.org>
> Blog: https://thehighergeometer.wor= dpress.com<https://protect-au.mim= ecast.com/s/JbslCBNqgBC455OghNfAGM?domain=3Dthehighergeometer.wordpress.com= >
>
>
> You're receiving this message because you're a member of the Categorie= s mailing list group from Macquarie University. To take part in this conver= sation, reply all to this message.
>
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>
>
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