Discussion of Homotopy Type Theory and Univalent Foundations
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From: Michael Shulman <shulman@sandiego.edu>
To: Thomas Streicher <streicher@mathematik.tu-darmstadt.de>
Cc: "HomotopyTypeTheory@googlegroups.com"
	<homotopytypetheory@googlegroups.com>
Subject: Re: [HoTT] "type-theoretic model structures"
Date: Mon, 18 Feb 2019 13:07:19 -0800	[thread overview]
Message-ID: <CAOvivQzwY_kLRHcFP=vK9jMUNrjSQVwOsFwq=N-mf=FwSLChCg@mail.gmail.com> (raw)
In-Reply-To: <20190218205750.GE24000@mathematik.tu-darmstadt.de>

Identity types do not by themselves require all objects to be fibrant.
Sorry, my parenthetical was intended to mention *another* way in which
tribes are more specific than display map categories, since you seemed
to be conflating the two.  I think a clan is just a display map
category in which all objects are fibrant.  A tribe adds to this a
weak factorization system suitable for modeling identity types.

On Mon, Feb 18, 2019 at 12:57 PM Thomas Streicher
<streicher@mathematik.tu-darmstadt.de> wrote:
>
> I don't see why identity types require every object to be fibrant.
> But Andr'e has this requirement also for clans which really are
> display map cats with a few more additional requirements.
>
> Thomas
>
> > Every object in a tribe is fibrant.  (A tribe is not just a display
> > map category; it also has the categorical structure corresponding to
> > identity types.)  For purposes of modeling type theory, the
> > non-fibrant objects are of course irrelevant, since every concrete
> > context does have a chain of display maps to 1.  And yes, of course,
> > one doesn't need infinitary structure to model type theory; as I said,
> > that's one of the differences between a tribe and a type-theoretic
> > model category, that the latter has infinitary structure but the
> > former doesn't.
>
> --
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      reply	other threads:[~2019-02-18 21:07 UTC|newest]

Thread overview: 6+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-02-18 10:25 Thomas Streicher
2019-02-18 14:32 ` Michael Shulman
2019-02-18 20:30   ` Thomas Streicher
2019-02-18 20:44     ` Michael Shulman
2019-02-18 20:57       ` Thomas Streicher
2019-02-18 21:07         ` Michael Shulman [this message]

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