From: Michael Shulman <firstname.lastname@example.org> To: Bas Spitters <email@example.com> Cc: homotopytypetheory <firstname.lastname@example.org> 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 <email@example.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? > > -- > You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. > To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheoryfirstname.lastname@example.org. > To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOoPQuQc317E8qUEOWJ2JQD_iXAFyE%3DWcttruqd5A8tRpHqttg%40mail.gmail.com. -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheoryemail@example.com. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOvivQya7JthjksQC-XQ%2Bjuc3N_-O3JJibDjRM8F1hyML4CMAQ%40mail.gmail.com.
next prev parent reply index 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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