Discussion of Homotopy Type Theory and Univalent Foundations
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From: Michael Shulman <shulman@sandiego.edu>
To: Bas Spitters <b.a.w.spitters@gmail.com>
Cc: homotopytypetheory <homotopytypetheory@googlegroups.com>
Subject: Re: [HoTT] Joyal's definition of elementary higher topos
Date: Fri, 21 Feb 2020 14:13:24 -0800
Message-ID: <CAOvivQya7JthjksQC-XQ+juc3N_-O3JJibDjRM8F1hyML4CMAQ@mail.gmail.com> (raw)
In-Reply-To: <CAOoPQuQc317E8qUEOWJ2JQD_iXAFyE=Wcttruqd5A8tRpHqttg@mail.gmail.com>

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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Thread overview: 3+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2020-02-21 13:23 Bas Spitters
2020-02-21 22:13 ` Michael Shulman [this message]
2020-02-23 23:56   ` Michael Shulman

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