[HoTT] Re: [Agda] Higher inductive-recursive definitions?

[Altenkirch Thorsten]

Thank you for pointing this out Guillaume. I always fall for this one :-)

We often want to say that a function on paths is an omega functor..

Thorsten

From: Guillaume Brunerie <guillaume.brunerie at gmail.com<mailto:guillaume.brunerie at gmail.com>>
Date: Friday, 23 May 2014 16:27
To: Thorsten Altenkirch <psztxa at exmail.nottingham.ac.uk<mailto:psztxa at exmail.nottingham.ac.uk>>
Cc: agda list <agda at lists.chalmers.se<mailto:agda at lists.chalmers.se>>, "HomotopyTypeTheory at googlegroups.com<mailto:HomotopyTypeTheory at googlegroups.com>" <homotopytypetheory at googlegroups.com<mailto:homotopytypetheory at googlegroups.com>>
Subject: [HoTT] Re: [Agda] Higher inductive-recursive definitions?


Hi Thorsten,

I don't know about the elimination rule, but I just wanted to point out that you won't get a univalent universe in this way because eqU refl and refl won't be identified, and similarly for concat (eqU p) (eqU q) and eqU (concat p q).
And the obvious way to fix requires something like defining semi-simplicial types internally...

Guillaume

Le 23 mai 2014 15:23, "Altenkirch Thorsten" <psztxa at exmail.nottingham.ac.uk<mailto:psztxa at exmail.nottingham.ac.uk>> a écrit :
I would like to combine higher inductive definitions (I.e. have path constructors) with induction-recursion. One application would be to define a closed universe which is univalent. However, I cannot see any reasonable way to define an eliminator.

Ok, I start with a simple universe(using Agda) - an inductive recursive definition

data U : Set
El : U -> Set

data U where
  nat : U
  pi : (a : U)(b : El a -> U) -> U

El nat = Nat
El (pi a b) = (x : El a) -> El (b x)

Now I can define an eliminator for the universe which allows me to define dependent functions by recursion over type codes:

ElimU : (X : U -> Set)
       -> (X nat)
       -> ((a : U) -> X a -> (b : El a -> U) -> ((x : El a) -> X (b x)) -> X (pi a b))
       -> (a : U) -> X a
ElimU X n p nat = n
ElimU X n p (pi a b) = p a (ElimU X n p a) b (? x -> ElimU X n p (b x))

However, I also would like to add a path constructor, which identifies codes if the have the same semantics:

postulate
  eqU : forall {a b} -> El a == El b -> a == b

But I don't see a good way to modify the Eliminator. It seems that this corresponds to a condition on the eliminator as a whole.

An alternative is to quotient the universe afterwards. However, the problem is that in this case I cannot lift the pi constructor to the quotiented universe – the usual problem when quotienting infinitary constructors. This can usually be overcome by defining the path constructors mutually..

Any ideas? Maybe the whole thing doesn't make sense semantically?

Thorsten


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