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 To: Martin Escardo CC: Prof. Vladimir Voevodsky > On Oct 22, 2015, at 3:32 PM, Martin Escardo 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.