Re: Vladimir Voevodsky

["Martín Hötzel Escardó" <"escardo..."@gmail.com>]

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On Thursday, 12 October 2017 20:24:26 UTC+1, Daniel R. Grayson wrote:
>
> PS: it is the oldest email I have from him with the word "univalent" or 
> "univalence" in it.
>

 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 <vlad...@ias.edu>
To: Martin Escardo <m.es...@cs.bham.ac.uk>
CC: Prof. Vladimir Voevodsky <vlad...@ias.edu>


> On Oct 22, 2015, at 3:32 PM, Martin Escardo <m.es...@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.



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