Re: [HoTT] Identity versus equality

[Michael Shulman <shu...@sandiego.edu>]

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