In the vein of bringing to public record  things that Vladimir said, here is a short interview.

-------- Forwarded Message --------

Subject: Re: historical question

Date: Thu, 22 Oct 2015 16:08:14 -0400

From: Vladimir Voevodsky <vl...@ias.edu>

To: Martin Escardo <m.e...@cs.bham.ac.uk>

CC: Prof. Vladimir Voevodsky <vl...@ias.edu>

> On Oct 22, 2015, at 3:32 PM, Martin Escardo <m.e...@cs.bham.ac.uk> wrote:

> 

> Hi Vladimir,

> 

> (0) When did you formulate hlevels in type theory?

Probably in early 2010

> (1) When did you formulate the univalence axiom?

Originally in 2005 as a property of morphisms (fibrations). In late 2009 as a formula in type theory.

> 

>    (And when did you give your model for it?)

In a sense in 2006, only I did not know how to model the Martin-Lof identity types and thought that different identity types will need to be introduced that will satisfy univalence but what other properties to require from them I did not know.

> 

> (2) When did you prove that univalence implies function extensionality?

July 2010.

> 

> I am giving a talk next week trying to rigorously explain the univalence

> axiom to classical mathematicians. This will involve, of course, trying

> to first explain Martin-Loef type theory, particularly the identity type.

> 

> One thing between you and Martin-Loef is Hofmann-Streicher's groupoid

> model, in which they have a proto-form of univalence. Were you inspired

> by that, or were your thoughts independent of that?

I was not inspired by it. In fact I tried several times to understand what they are saying and never could.

> 

> (Also: what was your first reaction when you saw the identity type for

> the first time? Did you immediately connect it with path spaces?)

Not at all. I did not make this connection until late 2009. All the time before it I was hypnotized by the mantra that the only inhabitant of the Id type is reflexivity which made then useless from my point of view.

Vladimir.