Discussion of Homotopy Type Theory and Univalent Foundations
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From: Matt Oliveri <atmacen@gmail.com>
To: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] Why do we need judgmental equality?
Date: Sun, 10 Feb 2019 23:01:42 -0800 (PST)	[thread overview]
Message-ID: <b901140c-c578-4b41-9d58-753120e76fde@googlegroups.com> (raw)
In-Reply-To: <730FBE36-8E4F-45F5-9DB9-3B3A04E708FA@nottingham.ac.uk>


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I think you're right. From discussions about autophagy, it seems like no 
one knows how to match judgmental equality using equality types, unless 
that equality type family is propositionally truncated in some way.

Consequently, my guess is that Valery's Q transformation actually yields 
something rather like a 2-level system.

On Saturday, February 9, 2019 at 7:30:07 AM UTC-5, Thorsten Altenkirch 
wrote:
>
> Hi,
>
>  
>
> what we need is a strict equality on all types. If we would state the laws 
> of type theory just using the equality type we would also need to add 
> coherence laws. Since I would include the laws for substitution (never 
> understood why substitution is different from application) this would 
> include the laws for infinity categories and this would make even basic 
> type theory certainly much more complicated if not unusable. Instead one 
> introduces a 2-level system with strict equality on one level and weak 
> equality on another. For historic and pragmatic reasons this is combined 
> with the computational aspects of type theory which is expressed as 
> judgemental equality. However, there are reasons to separate these 
> concerns, e.g. to work with higher dimensional constructions in type theory 
> such as semi-simplicial types it is helpful to work with hypothetical 
> strict equalities (see our paper (
> http://www.cs.nott.ac.uk/~psztxa/publ/csl16.pdf). 
>
>  
>
> I do think that the computational behaviour of type theory is important 
> too. However, this can be expressed by demandic a form of computational 
> adequacy, that is for every term there is a strictly equal normal form. It 
> is not necessary that strict equality in general is decidable (indeed 
> different applications of type theory may demand different decision 
> procedures).
>
>  
>
> Thorsten
>
>  
>
>  
>
> *From: *<homotopyt...@googlegroups.com <javascript:>> on behalf of Felix 
> Rech <s9fe...@gmail.com <javascript:>>
> *Date: *Wednesday, 30 January 2019 at 11:55
> *To: *Homotopy Type Theory <homotopyt...@googlegroups.com <javascript:>>
> *Subject: *[HoTT] Why do we need judgmental equality?
>
>  
>
> In section 1.1 of the HoTT book it says "In type theory there is also a 
> need for an equality judgment." Currently it seems to me like one could, in 
> principle, replace substitution along judgmental equality with explicit 
> transports if one added a few sensible rules to the type theory. Is there a 
> fundamental reason why the equality judgment is still necessary?
>
>  
>
> Thanks,
>
> Felix Rech
>

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  reply	other threads:[~2019-02-11  7:01 UTC|newest]

Thread overview: 71+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-01-30 11:54 Felix Rech
2019-02-05 23:00 ` [HoTT] " Matt Oliveri
2019-02-06  4:13   ` Anders Mörtberg
2019-02-09 11:55     ` Felix Rech
2019-02-16 15:59     ` Thorsten Altenkirch
2019-02-17  1:25       ` Michael Shulman
2019-02-17  7:56         ` Thorsten Altenkirch
2019-02-17  9:14           ` Matt Oliveri
2019-02-17  9:18           ` Michael Shulman
2019-02-17 10:52             ` Thorsten Altenkirch
2019-02-17 11:35               ` streicher
2019-02-17 11:44                 ` Thorsten Altenkirch
2019-02-17 14:24                   ` Bas Spitters
2019-02-17 19:36                   ` Thomas Streicher
2019-02-17 21:41                     ` Thorsten Altenkirch
2019-02-17 12:08             ` Matt Oliveri
2019-02-17 12:13               ` Matt Oliveri
2019-02-20  0:22               ` Michael Shulman
2019-02-17 14:22           ` [Agda] " Andreas Abel
2019-02-17  9:05         ` Matt Oliveri
2019-02-17 13:29         ` Nicolai Kraus
2019-02-08 21:19 ` Martín Hötzel Escardó
2019-02-08 23:31   ` Valery Isaev
2019-02-09  1:41     ` Nicolai Kraus
2019-02-09  8:04       ` Valery Isaev
2019-02-09  1:58     ` Jon Sterling
2019-02-09  8:16       ` Valery Isaev
2019-02-09  1:30   ` Nicolai Kraus
2019-02-09 11:38   ` Thomas Streicher
2019-02-09 13:29     ` Thorsten Altenkirch
2019-02-09 13:40       ` Théo Winterhalter
2019-02-09 11:57   ` Felix Rech
2019-02-09 12:39     ` Martín Hötzel Escardó
2019-02-11  6:58     ` Matt Oliveri
2019-02-18 17:37   ` Martín Hötzel Escardó
2019-02-18 19:22     ` Licata, Dan
2019-02-18 20:23       ` Martín Hötzel Escardó
2019-02-09 11:53 ` Felix Rech
2019-02-09 14:04   ` Nicolai Kraus
2019-02-09 14:26     ` Gabriel Scherer
2019-02-09 14:44     ` Jon Sterling
2019-02-09 20:34       ` Michael Shulman
2019-02-11 12:17         ` Matt Oliveri
2019-02-11 13:04           ` Michael Shulman
2019-02-11 15:09             ` Matt Oliveri
2019-02-11 17:20               ` Michael Shulman
2019-02-11 18:17                 ` Thorsten Altenkirch
2019-02-11 18:45                   ` Alexander Kurz
2019-02-11 22:58                     ` Thorsten Altenkirch
2019-02-12  2:09                       ` Jacques Carette
2019-02-12 11:03                   ` Matt Oliveri
2019-02-12 15:36                     ` Thorsten Altenkirch
2019-02-12 15:59                       ` Matt Oliveri
2019-02-11 19:27                 ` Matt Oliveri
2019-02-11 21:49                   ` Michael Shulman
2019-02-12  9:01                     ` Matt Oliveri
2019-02-12 17:54                       ` Michael Shulman
2019-02-13  6:37                         ` Matt Oliveri
2019-02-13 10:01                           ` Ansten Mørch Klev
2019-02-11 20:11                 ` Matt Oliveri
2019-02-11  8:23       ` Matt Oliveri
2019-02-11 13:03         ` Jon Sterling
2019-02-11 13:22           ` Matt Oliveri
2019-02-11 13:37             ` Jon Sterling
2019-02-11  6:51   ` Matt Oliveri
2019-02-09 12:30 ` [HoTT] " Thorsten Altenkirch
2019-02-11  7:01   ` Matt Oliveri [this message]
2019-02-11  8:04     ` Valery Isaev
2019-02-11  8:28       ` Matt Oliveri
2019-02-11  8:37         ` Matt Oliveri
2019-02-11  9:32           ` Rafaël Bocquet

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