Discussion of Homotopy Type Theory and Univalent Foundations
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From: "Martín Hötzel Escardó" <"escardo..."@gmail.com>
To: Homotopy Type Theory <HomotopyT...@googlegroups.com>
Subject: Univalence <-> equivalence induction
Date: Thu, 17 May 2018 23:36:44 -0700 (PDT)	[thread overview]
Message-ID: <4f23688f-af7c-43cb-946c-988f9d476848@googlegroups.com> (raw)


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Equivalence induction says that in order to prove something for all 
equivalences, it is enough to prove it for all identity equivalences for 
all types.

This follows from univalence. But also, conversely, univalence follows from 
it:

   http://www.cs.bham.ac.uk/~mhe/agda-new/UF-Univalence.html#JEq

Is this known? Some years ago it was claimed in this list that equivalence 
induction would be strictly weaker than univalence. 

To prove the above, I apply a technique I learned from Peter Lumsdaine, 
that given an abstract identity system (Id, refl , J) with no given 
"computation rule" for J, produces another identity system (Id, refl , J' , 
J'-comp) with
a "propositional computation rule" J'-comp for J'.

   http://www.cs.bham.ac.uk/~mhe/agda-new/Lumsdaine.html

Martin


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             reply	other threads:[~2018-05-18  6:36 UTC|newest]

Thread overview: 6+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2018-05-18  6:36 Martín Hötzel Escardó [this message]
2018-05-18 13:04 ` [HoTT] " Egbert Rijke
2018-05-18 15:40   ` Michael Shulman
2018-05-18 21:03     ` Martín Hötzel Escardó
2018-05-19 18:09 ` Nicolai Kraus
2018-05-19 19:38   ` Thierry Coquand

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