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charset="utf-8" Content-Transfer-Encoding: quoted-printable 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 --MCBoundary=_12401281151410701 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=UTF-8 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. Th= e 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
 
 
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