[TYPES] Equivalence invariance for MLTT (reference request)

[andrej.bauer at andrej.com]

Dear Jon,

I must have seen an early version of Per’s Nagel lecture somewhere. I definitely wasn’t at CMU in 2013. It looks like the thing I was looking for. I’ll try to cajole some young minds here at TYPES 2023 to formalize the lectures.

With kind regards,

Andrej

> On 13 Jun 2023, at 15:50, Jon Sterling <jon at jonmsterling.com> wrote:
> 
> Dear Andrej,
> 
> I am not completely sure because it is a while since I had watched this, but I think this might be related to the topic of Per Martin-Löf's 2013 Earnest Nagel Lecture “Invariance Under Isomorphism and Definability”, which you can find here: https://urldefense.com/v3/__https://www.cmu.edu/dietrich/philosophy/events/nagel-lectures/past-lectures.html__;!!IBzWLUs!WXYpkNUV5tYJlVXuiZQI97C8hXEYZVcohkLYzo-pn_4VNL1D6ez6rV1xf8vsge4MN36Af4fD_ieivgawrmCzHwwZ5MXYlZ03oes$ . But I don't recall if Per proves the exact theorem you want; it might be a little different, judging from the abstract.
> 
> Best,
> Jon
> 
> 
> On 12 Jun 2023, at 20:56, andrej.bauer at andrej.com <mailto:andrej.bauer at andrej.com> wrote:
> 
>> [ The Types Forum, http://lists.seas.upenn.edu/mailman/listinfo/types-list ]
>> 
>> Dear all,
>> 
>> in preparation for my TYPES 2023 talk I realized I don’t actually know of anyone having proved the following about MLTT (Σ + Π + Id + Nat).
>> 
>> EQUIVALENCE INVARIANCE: Let P be a well-formed type expression with a type meta-variable X. If A and B are closed type expressions and e : A ≃ B an equivalence between them, then the type of equivalences P[A/X] ≃ P[B/X] is inhabited.
>> 
>> There are many possible variants, of course, and I’d be interested in learning about any results in this direction, especially ones that don’t throw in any axioms.
>> 
>> I am vaguely remebering that it has been done for Church’s simple type theory, which actually sounds, well, simple. Does anyone know a reference?
>> 
>> I think there might have been some work by Bob Harper & Dan Licata (https://urldefense.com/v3/__https://www.cs.cmu.edu/*drl/pubs/lh112tt/lh122tt-final.pdf__;fg!!IBzWLUs!Ww4XrFxjdSUVFGjTgD3MToleD85TRwLRGFIy_xjkQqvSw3nuqa9fWxMNxAWYPT5FyS0Rr9hCcWv8p05bA6Xc3ZdFibwiK0ulYns$ ), and another by Nicolas Tabareau & Matthieu Sozeau (https://urldefense.com/v3/__https://doi.org/10.1145/3236787__;!!IBzWLUs!Ww4XrFxjdSUVFGjTgD3MToleD85TRwLRGFIy_xjkQqvSw3nuqa9fWxMNxAWYPT5FyS0Rr9hCcWv8p05bA6Xc3ZdFibwin1ZSLzI$ ), which cuts thing off at the groupoid level. I am not even sure if they really prove an analogue of the principle stated above.
>> 
>> But how about pure MLTT, has anyone done it?
>> 
>> With kind regards,
>> 
>> Andrej