Re: [HoTT] Canonical forms for initiality

[Andrea Vezzosi <sanz...@gmail.com>]

On Mon, Oct 16, 2017 at 7:01 PM, Michael Shulman <shu...@sandiego.edu> wrote:
> [...]
> However, when doing this intrinsically (which is the direct way to get
> a useful induction principle) and without h-level restriction, I found
> that there seems to be a new problem: you end up needing to substitute
> hereditarily not just one term but a whole context morphism, and then
> in defining the clauses for the graph of that hereditary substitution
> you need to know already that it is associative.  So you add another
> judgment for the graph of its associativity, but in defining its
> clauses you need to know that the associativity is coherent, and so
> on.

This is interesting but I can't quite see it, what would be the type
of the "graph of associativity" judgment?
And how would you use it in the term/type judgments?

> The result is an inductive-inductive definition with infinite
> dependency, which we don't know how to make sense of internally.  But
> if we can make sense of it (or make it finite by capping the h-level
> somewhere), then this tower of dependent inductive-inductive types
> ought to present the entire (semi)simplicial nerve of the syntactic
> category of the type theory (probably in the form of something like a
> "comprehension quasicategory").
>
> The induction principle of this inductive-inductive definition should
> then let us interpret it into an arbitrary sufficiently structured
> quasicategory, or even an internally defined "complete Segal space" if
> we do it in a homotopy type theory, such as the canonical universe,
> thereby solving the autophagy problem.  On the other hand, since it
> has no path-constructors, an encode-decode argument should prove that
> it actually consists of h-sets, which should then allow us to
> interpret untyped syntax into it.
>
> Of course this is a VERY rough sketch and there are a lot of ways it
> could go wrong, plenty of which I expect people will point out.  (-:
>
> Mike
>
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