Hi Alexander,
I think what you are asking is "what is the internal language of an
oo-topos". This is doesn't really make much sense as stated since we don't
have a definition for "internal language" in an oo-topos. But we expect
that once we can define such a notion, then it ought to be HoTT. This will
require further work, but recently there have been some big strides towards
this direction with Mike Shulman's work .
The Kripke-Joyal semantics of a topos of sheaves is essentially a
dictionary between the "internal language" of this 1-topos and what it
means externally. I have quotation marks here because I am not being very
precise and phrases like "internal language" do have precise meanings which
I don't care to look up right now.
You are correct that these have been used in practice. Just look at Ingo
Blechschmidt's work
. He discusses
on pg. 199 briefly the thing that you consider.
The thing to note with Mike's work is theorem 11.1, which shows that every
oo-topos can be presented by a "type-theoretic model topos". Now you can
think of Kripke-Joyal semantics as a "machine" that can translate between
internal and external statements. Here is a page on the nlab
that
shows what this will probably end up looking like.
To get an idea what such a machine might do, have a look at Charles Rezk's
"translation"
of a HoTT proof of Blakers-Massey into higher topos theory. The key point
here is that this proof was not known in higher topos theory before. It is
a very natural argument in HoTT however. I would say this is pretty
fruitful.
The reason I keep saying "probably" is because nobody has actually formally
written down these semantics that I know of. Kripke-Joyal semantics are
fairly technical already, just look at Ingo's thesis. And you are correct
that with HoTT there are lot's of subtleties involved.
So to answer you final question: yes it is being investigated and yes it
seems to be fruitful.
I will add however that apart from Ingo's work, I don't know of many people
using Kripke-Joyal semantics to actually do algebraic geometry. It's true
that Ingo discovered some new arguments (generic freeness) but these have
yet to catch on with "regular" algebraic geometers. This is because even
though it is a 200 page thesis, it only covers foundational aspects of the
subject, i.e. the basics of scheme theory. There is much more work to be
done before it gets mainstream.
We will probably see algebraic topologists using the internal language of
(oo-)toposes well before algebraic geometers do. Since we already have
examples of these almost being done.
These are just my thoughts on your question and is no means the word of an
expert.
Best,
Ali Caglayan
On Tue, Oct 8, 2019 at 4:07 PM Alexander Gietelink Oldenziel <
a.f.d.a.gietelinkoldenziel@gmail.com> wrote:
> Dear all,
>
> It is my understanding that the interpretation of HoTT inside higher topoi
> is an ongoing field of research. Much of this business involves subtle
> strictness properties and things like contextual and syntactic categories.
> In the 1-topos case there is the Kripke-Joyal/Stack - semantics. These
> semantics are easy to use in practice. Naively, it seems one could lift
> the Kripke-Joyal semantics by not asking for truth but simply an inhabitant
> of a type.
>
> Has this been investigated at all? Is it ultimately fruitless?
>
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