From mboxrd@z Thu Jan 1 00:00:00 1970 Date: Thu, 12 Oct 2017 14:55:11 -0700 (PDT) From: =?UTF-8?Q?Mart=C3=ADn_H=C3=B6tzel_Escard=C3=B3?= To: Homotopy Type Theory Message-Id: <164e6589-5d92-4642-add6-fdbe7439d164@googlegroups.com> In-Reply-To: <82f74559-e82f-4773-bd7d-2886bd9b38fb@googlegroups.com> References: <69c716dc-7fbf-4c07-a128-21c75fc996da@googlegroups.com> <82f74559-e82f-4773-bd7d-2886bd9b38fb@googlegroups.com> Subject: Re: Vladimir Voevodsky MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="----=_Part_15732_252028612.1507845311783" ------=_Part_15732_252028612.1507845311783 Content-Type: multipart/alternative; boundary="----=_Part_15733_463399984.1507845311784" ------=_Part_15733_463399984.1507845311784 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 7bit 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. ------=_Part_15733_463399984.1507845311784 Content-Type: text/html; charset=utf-8 Content-Transfer-Encoding: quoted-printable
On Thursday, 12 October 2017 20:24:26 UTC+1, Daniel R. Gra= yson wrote:
P= S: it is the oldest email I have from him with the word "univalent&quo= t; or "univalence" in it.

= =C2=A0In the vein of bringing to public record=C2=A0 things that Vladimir s= aid, 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>

<= /div>

> On Oct 22, 2015, at 3:32 PM, Martin Escardo &= lt;m.e...@cs.bham.ac.uk> wrote:
>=C2=A0
> Hi V= ladimir,
>=C2=A0
> (0) When did you formulate hle= vels in type theory?

Probably in early 2010
<= div>
> (1) When did you formulate the univalence axiom?

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

= >=C2=A0
>=C2=A0 =C2=A0 (And when did you give your model fo= r 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 oth= er properties to require from them I did not know.

>=C2=A0
> (2) When did you prove that univalence implies f= unction extensionality?

July 2010.

<= /div>
>=C2=A0
> I am giving a talk next week trying to = rigorously explain the univalence
> axiom to classical mathema= ticians. This will involve, of course, trying
> to first expla= in Martin-Loef type theory, particularly the identity type.
>= =C2=A0
> One thing between you and Martin-Loef is Hofmann-Stre= icher's groupoid
> model, in which they have a proto-form = of univalence. Were you inspired
> by that, or were your thoug= hts independent of that?

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

>=C2=A0
> (Also: what was = your first reaction when you saw the identity type for
> the f= irst 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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