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From: Michael Shulman <shulman@sandiego.edu>
To: Nicola Gambino <N.Gambino@leeds.ac.uk>
Cc: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] Papers on constructive simplicial homotopy theory
Date: Thu, 18 Jul 2019 05:21:23 -0700 [thread overview]
Message-ID: <CAOvivQySjeDQXj_wWt9m1oQbAJj0+Dn-NQ0yBX__DO1N3NBC_g@mail.gmail.com> (raw)
In-Reply-To: <D49A1FEA-4CE1-448F-97A8-46065AF9E7B6@leeds.ac.uk>
For something that ought to work in the internal language of a
category with finite limits, like the first paper on weak model
categories, it should technically suffice to have Sigma- and Id-types
with UIP (since those are a version of that internal language). If we
wanted to internalize more of the abusive external statements like "f
is an acyclic fibration if it has right lifting for every
cofibration", it should be enough to add Pi-types and a universe.
To enhance this to the internal language of an elementary
(predicative) 1-topos with NNO or an analogue of CZF, coproducts,
propositional truncations, pushouts, and a natural numbers type should
be enough. I'm very curious to hear where propositional truncations
and pushouts are used (if ever), since in so many places the "for all
x, exists y" is actually an untruncated Sigma. Certainly pushouts
appear in the small object argument, but I wonder whether those
pushouts could be implemented with coproducts in the case when they
are pushouts of cofibrations that are sufficiently "complemented
inclusions" (the pushout of A -> X along a complemented inclusion A ->
A+B is just X+B).
On Thu, Jul 18, 2019 at 12:55 AM Nicola Gambino <N.Gambino@leeds.ac.uk> wrote:
>
> Dear Mike,
>
> On 17 Jul 2019, at 18:56, Michael Shulman <shulman@sandiego.edu> wrote:
>
> Most of these papers describe the situation with phrases like "we are
> working in the internal language of a category with finite limits" or
> an elementary topos with NNO, or in CZF, and by an "abuse of language"
> we interpret "for all x there exists a y" as referring to the giving
> of a function assigning a y to each x. But wouldn't it be more
> precise and less abusive to just work in dependent type theory with
> Sigma and Id types, and sometimes Pi and Nat, and use the untruncated
> propositions-as-types logic where "for all x there exists a y"
> literally means Pi(x) Sigma(y) and therefore (by the "type-theoretic
> principle of non-choice") automatically induces a function assigning a
> y to each x? That would also allow asking and answering the question
>
> of how much UIP is required -- do these model structures exist in
> HoTT?
>
>
> Thank you for your email.
>
> Your suggestion of working in a dependent type theory is interesting. I am not sure what kind of dependent type theory would be sufficient to develop these papers and what would be the best approach to the formalization (e.g. via sets-as-hsets or via sets-as-setoids).
>
> Regarding the dependent type theory, apart from basic rules, I guess one would need:
>
> - some extensionality,
> - propositional truncations,
> - pushouts,
> - some inductive types (for the instances of the small object argument)
> - at least one universe (cf. quantification over all Kan complexes).
>
> One could then keep track explicitly of which existential quantifies are to be left untruncated and which ones can be truncated, and then see if everything can be done in HoTT.
>
> Is this the kind of thing you had in mind?
>
> Another approach to avoiding the abuse of language, suggested by Andre’ Joyal, is to develop a theory of “split” weak factorisation systems, i.e. weak factorisation systems in which one has a given choice of fillers, and work with them. This would be a variant of the theory of algebraic weak factorisation systems. We are working on that.
>
> With best wishes,
> Nicola
>
> PS The first link in my email was incorrect. Simon Henry’s paper "Weak model categories in classical and constructive mathematics” is available at https://arxiv.org/abs/1807.02650.
>
>
>
>
>
> Dr Nicola Gambino
> Associate Professor in Pure Mathematics
> School of Mathematics, University of Leeds
> http://www1.maths.leeds.ac.uk/~pmtng/
>
>
>
>
> --
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```

prev parent reply other threads:[~2019-07-18 12:21 UTC|newest]Thread overview:5+ messages / expand[flat|nested] mbox.gz Atom feed top 2019-07-15 10:18 Nicola Gambino 2019-07-17 17:56 ` Michael Shulman 2019-07-18 7:55 ` Nicola Gambino 2019-07-18 8:15 ` Bas Spitters2019-07-18 12:21 ` Michael Shulman [this message]

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