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On the other hand, if your fibration comes cloven, then it is easy = to define the opposite in terms of this cleaving, and this construction is = clearly involutive.=20 There is plenty of reason to think that cloven fibrations are the correct n= otion to work with in settings that do not satisfy the axiom of choice (of = course, we would not ask that morphisms of fibrations preserve cleavings). = Evidence for this viewpoint is provided by univalent foundations, in which = fibrations of univalent categories automatically have (unique!) cleavings = =E2=80=94=C2=A0but, here we mean unique up to homotopy. As we can always ta= ke a Rezk completion, there is nothing really lost by restricting attention= to fibrations of univalent categories, which by virtue of their (unique) c= leaving admit a very simple description of fiberwise opposites. Best, Jon On Sun, Jan 28, 2024, at 12:51 AM, 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://protect-au.mimecast.com/s/yc6pC5QP8ySGoRWrSzfuzE?domain=3Darxiv.o= rg=20 > =20 > 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://protect-au.mimecast.com/s/zq_BC6XQ68fKEMG4t6fSnX?domain= =3Dncatlab.org=20 > > Blog: https://protect-au.mimecast.com/s/pBrRC71R63CkvyWBtBluYx?domain=3Dt= hehighergeometer.wordpress.com=20 > >=20 >=20 > You're receiving this message because you're a member of the Categories= =20 > mailing list group from Macquarie University. To take part in this=20 > conversation, reply all to this message.=20 >=20 > View group files=20 > =20 > | Leave group=20 > =20 > | Learn more about Microsoft 365 Groups --MCBoundary=_12401282255270201 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=utf-8 Dear David,

I did also find the quotienting to be kind of a hack, but it is indeed a bi= t unclear what one should do in the context of a "weak fibration"= in the sense of Ahrens and Lumsdaine, i.e. one that does not come equipped= with a cleaving. On the other hand, if your fibration comes cloven, then i= t is easy to define the opposite in terms of this cleaving, and this constr= uction is clearly involutive.

There is plenty of reason to think that cloven fibrations are the correct n= otion to work with in settings that do not satisfy the axiom of choice (of = course, we would not ask that morphisms of fibrations preserve cleavings). = Evidence for this viewpoint is provided by univalent foundations, in which = fibrations of univalent categories automatically have (unique!) cleavings = =E2=80=94 but, here we mean unique up to homotopy. As we can always ta= ke a Rezk completion, there is nothing really lost by restricting attention= to fibrations of univalent categories, which by virtue of their (unique) c= leaving admit a very simple description of fiberwise opposites.

Best,
Jon


On Sun, Jan 28, 2024, at 12:51 AM, David Roberts 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
> <https://protect-au.mimecast.com/s/ULb4CgZ05Jf2rlDBs3s= P1k?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/ygpkCjZ12Rfq2jVOS= 1CfGU?domain=3Dncatlab.org>
> Blog: https://thehighergeometer.wor= dpress.com
> <https://protect-au.mimecast.com= /s/bSqFCk815RCQ7nPwt8C6hN?domain=3Dthehighergeometer.wordpress.com><= BR> >
>
> 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 > conversation, reply all to this message.
>
> View group files
> <https://outlook.office365.com/owa/categor= ies@mq.edu.au/groupsubscription.ashx?source=3DEscalatedMessage&action= =3Dfiles&GuestId=3D3d33085c-21ea-4036-99e4-8fc96e8c25b8>
> | Leave group
> <https://outlook.office365.com/owa/categor= ies@mq.edu.au/groupsubscription.ashx?source=3DEscalatedMessage&action= =3Dleave&GuestId=3D3d33085c-21ea-4036-99e4-8fc96e8c25b8>
> | Learn more about Microsoft 365 Groups <https://aka.= ms/o365g>
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