Joyal's definition of elementary higher topos

[Bas Spitters]

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?

[Michael Shulman]

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.

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[Michael Shulman]

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.