Re: [HoTT] Identity versus equality

[Nicolai Kraus <nicola...@gmail.com>]

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I have to correct what I said an hour ago (thanks, Mike). We don't know
whether "exo-Nat is cofibrant" implies that exo-limits of towers are
fibrant. (And probably it doesn't.)
In other words, we don't know the connection between axioms (A2) and (A3)
in arXiv:1705.03307.
 -- Nicolai

On Sun, May 10, 2020 at 4:23 PM Nicolai Kraus <nicola...@gmail.com>
wrote:

> Yes, I think that is one main motivation for this axiom (that you've
> suggested in this form :-) and I also believe that it was Vladimir's
> main motivation for his axiom "exo-Nat is fibrant". I think the two
> axioms really serve the same purpose, but the "cofibrant" version is
> much more harmless.
>
> On 10/05/2020 16:16, Michael Shulman wrote:
> > I forget -- does "exo-Nat is cofibrant" imply that exo-limits of
> > towers of fibrations are fibrant?  That's another useful axiom that
> > holds in models and might make it easier to construct coinductive
> > types with judgmental computation rules.
> >
> > On Sun, May 10, 2020 at 7:52 AM Nicolai Kraus <nicola...@gmail.com>
> wrote:
> >> I would guess that "exo-Nat is cofibrant" justifies the coinductive
> type in question, but not its judgmental computation rules. And the
> judgmental computation rules are probably the very reason why one would
> want this coinductive type. But this is just a guess.
> >> -- Nicolai
> >>
> >> On Sun, May 10, 2020 at 3:35 PM Michael Shulman <shu...@sandiego.edu>
> wrote:
> >>> Many or all coinductive types can be constructed, at least up to
> >>> equivalence, using Pi-types and (some kind of) Nat.  Is there any
> >>> chance that "exo-Nat is cofibrant" could be used to justify the
> >>> existence/fibrancy of the coinductive types you want?
> >>>
> >>> On Sun, May 10, 2020 at 7:20 AM Nicolai Kraus <nicola...@gmail.com>
> wrote:
> >>>> On 10/05/2020 15:01, Michael Shulman wrote:
> >>>>> On Sun, May 10, 2020 at 4:46 AM Thorsten Altenkirch
> >>>>> <Thorsten....@nottingham.ac.uk> wrote:
> >>>>>> Defining simplicial types isn't entirely straightforward (as you
> know I suppose), because Delta is not directed. We can do semi-simplicial
> types as a Reedy limit, i.e. an infinite Sigma type
> >>>>> We can certainly *talk* about simplicial types in 2LTT as exofunctors
> >>>>> from the exocategory Delta to the exocategory Type.  I assume the
> >>>>> point you're making is that we don't have a (fibrant) *type of*
> >>>>> simplicial types, whereas we do have a fibrant type of semisimplicial
> >>>>> types (under appropriate axioms)?
> >>>> Judging from the rest of his message, I believe that Thorsten was
> >>>> talking about the direct replacement construction in Christian's and
> my
> >>>> abstract arXiv:1704.04543. With the assumption "exo-Nat is cofibrant",
> >>>> this gives us a fibrant type that one could call "simplicial types"
> (and
> >>>> Thorsten does). But of course it's an encoding. If we decide to use
> such
> >>>> encodings, I fear we may lose the main advantage that the "axiomatic"
> >>>> representations in HoTT have, namely avoiding encodings. (I mean the
> >>>> "main advantage" of HoTT compared to traditional approaches, e.g.
> taking
> >>>> bisimplicial sets.)
> >>>>
> >>>>>> You need some extra principles, e.g. that strict Nat is fibrant or
> maybe better that certain coinductive types exist.
> >>>>> Personally, I think the best axiom to use here is that exo-Nat is
> >>>>> *cofibrant*, i.e. Pi-types with domain exo-Nat preserve fibrancy.  We
> >>>>> don't know how to model "exo-Nat is fibrant" in all higher toposes,
> >>>>> but it's easy to interpret "exo-Nat is cofibrant" in such models,
> >>>>> since Pi-types with domain exo-Nat are just externally-infinite
> >>>>> products.
> >>>> I completely agree with your preference for this axiom :-)
> >>>> But Thorsten does has a point if we consider the "engineering level"
> >>>> that was discussed earlier in this thread. Allowing coinductive types
> >>>> with exo-Nat as an index makes it possible to use your paper (Higher
> >>>> Stucture Identity Principle, arXiv:2004.06572) and get a construction
> of
> >>>> semi-simplicial types which is more convenient to use in a proof
> assistant.
> >>>>
> >>>> -- Nicolai
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> .
>
>

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