Discussion of Homotopy Type Theory and Univalent Foundations
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From: Thorsten Altenkirch <Thorsten....@nottingham.ac.uk>
To: nicolai.kraus <nicola...@gmail.com>,
	"homotopyt...@googlegroups.com" <homotopyt...@googlegroups.com>
Subject: Re: [HoTT] "Identifications" ?
Date: Mon, 4 May 2020 15:57:53 +0000	[thread overview]
Message-ID: <B0040E66-5205-412A-9BCA-6B61A1D09CC9@nottingham.ac.uk> (raw)
In-Reply-To: <b1b554b1-1c70-0bd6-5c09-cd5f0f2e48f9@gmail.com>

Ok, I apologize discussions on names and syntax are at the same time very attractive and are also scoffed at because they seem not to have much content. 

I am quite happy with Mike's explanation to use the word "identifications" for inhabitants of equality types. "Paths" or "equality proofs" or "evidence for equality" are also good alternatives.

However, I prefer to use the word "equality" even equality is not propositional. Two mathematical objects which share all properties we can talk about should be considered equal even if there is more than one reason / proof / identification that this is the case. Moreover, there is no other "equality" we can talk about in Type Theory.


On 04/05/2020, 16:47, "homotopyt...@googlegroups.com on behalf of Nicolai Kraus" <homotopyt...@googlegroups.com on behalf of nicola...@gmail.com> wrote:

    For what it's worth, "identification" is much closer to Per Martin-Löf's 
    original terminology ("identity type") than "equality" or "path" are.

    On 04/05/2020 15:48, Stefan Monnier wrote:
    >> I don't think using "identification" necessarily implies any
    >> difference between "identification" and "equality".  I don't think of
    >> it that way.  For me the point is just to have a word that refers to
    >> an *element* of an identity type.  Calling it "an equality" can have
    >> the wrong connotation because classically, an equality is just a
    >> proposition (or a true proposition), whereas an element of an identity
    >> type carries information.  Calling it "an identification" suggests
    >> exactly the information that it carries: a way of identifying two
    >> things.
    > I thought that's what "path" was for?
    >          Stefan "who really doesn't know what he's talking about"

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  reply	other threads:[~2020-05-04 15:57 UTC|newest]

Thread overview: 26+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2020-05-04  9:35 Thorsten Altenkirch
2020-05-04 10:59 ` [HoTT] " stre...
2020-05-04 11:04   ` Steve Awodey
2020-05-04 11:17   ` Thorsten Altenkirch
2020-05-04 11:42     ` Nicolai Kraus
2020-05-04 12:04       ` Thorsten Altenkirch
2020-05-04 12:06     ` Thomas Streicher
2020-05-04 12:12       ` Thorsten Altenkirch
2020-05-04 12:39         ` Thomas Streicher
2020-05-04 13:16 ` Michael Shulman
2020-05-04 14:17   ` Thorsten Altenkirch
2020-05-04 14:48   ` Stefan Monnier
2020-05-04 15:46     ` Nicolai Kraus
2020-05-04 15:57       ` Thorsten Altenkirch [this message]
2020-05-04 15:59     ` Michael Shulman
2020-05-04 16:07       ` Steve Awodey
2020-05-04 16:17         ` Thorsten Altenkirch
2020-05-04 16:53           ` Steve Awodey
2020-05-04 17:25             ` Thorsten Altenkirch
2020-05-04 17:43               ` Michael Shulman
2020-05-04 17:55               ` Steve Awodey
2020-05-04 16:21         ` Peter LeFanu Lumsdaine
2020-05-04 16:16       ` Joyal, André
2020-05-04 20:38         ` Joyal, André
2020-05-07 19:43 ` Martín Hötzel Escardó
2020-05-08 10:41   ` [HoTT] " Thorsten Altenkirch

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