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 HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOoPQuQc317E8qUEOWJ2JQD_iXAFyE%3DWcttruqd5A8tRpHqttg%40mail.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? > > -- > 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 HomotopyTypeTheory+unsubscribe@googlegroups.com. > 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 HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOvivQya7JthjksQC-XQ%2Bjuc3N_-O3JJibDjRM8F1hyML4CMAQ%40mail.gmail.com.

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? > > > > -- > > 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 HomotopyTypeTheory+unsubscribe@googlegroups.com. > > 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 HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOvivQwXyK0QMj-F0%2BCcEV1VjQoFTDh9P7P9630chx1V9Hpapg%40mail.gmail.com.