Re: [HoTT] Identity versus equality

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

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Yes, this is nice. We should add it to the next revision of the paper
(arXiv:1705.03307).

On Sun, May 10, 2020 at 5:29 PM Michael Shulman <shu...@sandiego.edu>
wrote:

> Okay.  But the implication works in the other way, doesn't it?  A
> product indexed by exo-Nat is the exo-limit of a tower of finite
> products.  So maybe the tower axiom is the best one.
>
> On Sun, May 10, 2020 at 9:13 AM Nicolai Kraus <nicola...@gmail.com>
> wrote:
> >
> > 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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