Re: [HoTT] Joyal's definition of elementary higher topos

[Michael Shulman <shulman@sandiego.edu>]

Actually let me modify that: (G3) holds in any type-theoretic model
topos.  And since coproducts are disjoint, (G2) is implied by (G1) and
(G3); while since the initial object is strict, (G1) merely asserts
that it is fibrant.  So if the initial object is fibrant, then all of
(G1-G3) hold.  The initial object isn't always fibrant, but I wouldn't
be surprised if every Grothendieck (∞,1)-topos could be presented by a
model category of this sort in which the initial object is fibrant, at
least in a classical metatheory, for the same reason that every
Grothendieck 1-topos is overt classically: we can simply remove from
the site all objects that are covered by the empty family.  But I
haven't checked the details in the ∞-case.

And I still don't know any way to make the NNO fibrant.

On Fri, Feb 21, 2020 at 2:13 PM Michael Shulman <shulman@sandiego.edu> wrote:
>
> I believe the best that's known is that (assuming an inaccessible
> cardinal) any Grothendieck (∞,1)-topos can be presented by a model
> category -- namely, a left exact localization of an injective model
> structure on simplicial presheaves -- satisfying all of Joyal's axioms
> except those involving coproducts (G1-G3) and fibrancy of the NNO
> (A2).  Most of the properties are easy to show from the definitions;
> G6 and G7 follow from the fact that it presents a Grothendieck
> (∞,1)-topos; L2 follows from an adjoint pushout-product calculation;
> and I showed L6 myself most recently in
> https://arxiv.org/abs/1904.07004.
>
> The extra axioms (G1-G3) and (A2) hold in many examples -- e.g. the
> injective model structure itself, which presents a presheaf
> (∞,1)-topos, and probably also other examples such as sheaves on
> locally connected sites.  But in other cases even the initial object
> may not be fibrant.  Personally, my current opinion (subject to
> change) is that (G1-G3) and (A2) are unreasonably strong, and
> unnecessary for most purposes.
>
>
> On Fri, Feb 21, 2020 at 5:23 AM Bas Spitters <b.a.w.spitters@gmail.com> wrote:
> >
> > In 2014, Andra Joyal proposed a definition of an elementary higher topos.
> >
> > "This lecture contains a proposed definition that is not an
> > (∞,1)-category but a presentation of one by a model category-like
> > structure; this is closer to the type theory, but further from the
> > intended examples. In particular, there are unresolved coherence
> > questions even as to whether every Grothendieck (∞,1)-topos can be
> > presented by a model in Joyal’s sense (in particular, how strict can a
> > universe be made, and can the natural numbers object be made
> > fibrant)."
> > https://ncatlab.org/nlab/show/elementary+%28infinity%2C1%29-topos
> >
> > Has there been any progress on these coherence questions?
> >
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