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charset="utf-8" Content-Transfer-Encoding: quoted-printable I think there is a problem with this analogy. A basis for a vector space is= actual (higher) data, but a cleavage for a fibration is property-like data= . Morphisms of fibrations that weakly preserve the cleavage are just morphism= s of fibrations. Morphisms of vector spaces that preserve a chosen basis are very different = from just morphisms of vector spaces. On Thu, 1 Feb 2024, 12:07 Thomas Streicher, > wrote: Dear Jon, for constructing the opposite of a fibration you do not need at all that the chosen cleavage is split. It is easy to see that fixing the cartesian arrow also fixes the vertical arrow. If everybody is happy with this solution I am fine. It is in the same spiri= t as chosen pullbacks or chosen finite limits. The reason for my reluctance is that nobody would consider as a natural notion a vector space with a chosen basis. The virtue of strong choice is that then such additional structure can be pulled out of the hat at demand. Thomas 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 --00000000000049a78f06105044c8 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable
I think there is a problem with this analogy. A basis for= a vector space is actual (higher) data, but a cleavage for a fibration is = property-like data.

Morphisms of fibrations that weakly preserve the cleavage= are just morphisms of fibrations.

Morphisms of vector spaces that preserve a chosen basis a= re very different from just morphisms of vector spaces.

On Thu, 1 Feb 2024, 12:07 Thomas Stre= icher, <streiche= r@mathematik.tu-darmstadt.de> wrote:
Dear Jon,

for constructing the opposite of a fibration you do not need at all
that the chosen cleavage is split. It is easy to see that fixing the
cartesian arrow also fixes the vertical arrow.

If everybody is happy with this solution I am fine. It is in the same spiri= t
as chosen pullbacks or chosen finite limits.
The reason for my reluctance is that nobody would consider as a
natural notion a vector space with a chosen basis.
The virtue of strong choice is that then such additional structure can be pulled out of the hat at demand.


Thomas
 
 
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