Discussion of Homotopy Type Theory and Univalent Foundations
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From: Michael Shulman <shulman@sandiego.edu>
To: Martin Escardo <escardo.martin@googlemail.com>
Cc: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] All (∞,1)-toposes have strict univalent universes
Date: Wed, 17 Apr 2019 16:44:06 -0700	[thread overview]
Message-ID: <CAOvivQzW1tiBE509QcoHe_4Yr-UH_e8DXXEDAE29Jw_sTxWMxw@mail.gmail.com> (raw)
In-Reply-To: <7724b923-fc14-0bd2-57b6-ba341f5641bd@googlemail.com>

Here's a brief summary of the steps:

1. Every Grothendieck (∞, 1)-topos can be presented by an accessible
left exact left Bousfield localization of an injective model structure
on a category of enriched simplicial presheaves.

2. Such presentations are right proper Cisinski model categories,
hence (as we already knew) model MLTT, even with many HITs, but not
previously known to model universes.

3. The fibrations in an injective model structure have no explicit
"cofibrantly generated" description in terms of a lifting property,
but it turns out they do have a fairly explicit "algebraic"
description in terms of the rectifiability of homotopy coherent
natural transformations (a homotopical version of "coflexible
algebras").

4. Now define a presheaf U by U(x) = the set of injective fibrations
over the representable C(-,x) *equipped with* such fibrancy structure
(suitably rectified to become a strict presheaf), and we get a fibrant
and univalent universe.

5. Finally, any accessible left exact localization can be presented
internally by a lex modality (using the recent characterization
thereof by Anel-Biedermann-Finster-Joyal), and the universe of modal
types for a lex modality is modal, giving a universe for the localized
model structure.

For more detail, you can read the introduction to the paper (or the
whole thing, of course), or look at these slides from last month's
Midwest HoTT seminar:
http://home.sandiego.edu/~shulman/papers/injmodel-talk.pdf

On Wed, Apr 17, 2019 at 3:59 PM 'Martin Escardo' via Homotopy Type
Theory <HomotopyTypeTheory@googlegroups.com> wrote:
>
> On 16/04/2019 13:06, Ali Caglayan wrote:
> > Mike has released this new preprint on the arXiv:
> >
> > All (∞,1)-toposes have strict univalent universes
> > <https://arxiv.org/abs/1904.07004>
> >
> > Quoting the abstract:
> >
> > Thus, homotopy type theory can be used as a formal language for
> > reasoning internally to (∞, 1)-toposes, just as higher-order logic is
> > used for 1-toposes.
>
> This is awesome.
>
> Perhaps it deserves a short explanation (or at least listing) of the
> crucial steps gere in this list.
>
> Best,
> Martin
>
> --
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  reply	other threads:[~2019-04-17 23:44 UTC|newest]

Thread overview: 5+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-04-16 12:06 Ali Caglayan
2019-04-17 22:59 ` 'Martin Escardo' via Homotopy Type Theory
2019-04-17 23:44   ` Michael Shulman [this message]
2019-04-18 10:09     ` [HoTT] All (???,1)-toposes " Thomas Streicher
2019-04-18 10:16       ` Michael Shulman

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