Re: Are Joyal--Tierney fibrations exponentiable?

[Zhen Lin Low <zll22@cam.ac.uk>]

Dear Thomas,

Thank you for your reply. Michael Warren also pointed out to me (in a
private reply) that one can construct dependent products for split
fibrations of internal groupoids. Unfortunately, this doesn't quite
answer my question.

As far as I know, Joyal--Tierney fibrations are cloven (or rather,
cleavable), but they do not have to split. After all, in the case
where the Grothendieck topos we start with is Set, the Joyal--Tierney
model structure coincides with the standard model structure on Grpd,
in which the fibrations really are just isofibrations in the usual
sense. (I assume the axiom of choice here; the theory of cofibrantly
generated model categories breaks down otherwise.)

Incidentally, since you bring up universes, perhaps I should explain
why I am focusing on Joyal--Tierney fibrations: I am wondering about
the strength of propositional resizing. As you say, the groupoid
interpretation makes sense constructively; but it is not hard to see
that propositional resizing in the groupoid interpretation implies a
weak form of the axiom of choice (namely, WISC) in the ambient set
theory. This essentially boils down to the difference between weak
equivalences (= fully faithful and essentially surjective on objects)
and equivalences. The difference disappears when one restricts to
Joyal--Tierney fibrations: this is a special case of the so-called
"Whitehead theorem" in model category theory.

--
Zhen Lin


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