Resurrecting this thread from many years ago, because I was thinking about it again recently, it seems to me that although { f & isTrackedByRelEquiv(f) } satisfies the rule sym (sym e) = e judgmentally, it doesn't satisfy the rule that sym id = id judgmentally.  (In particular, I am not sure what relational equivalence to use for the identity equivalence which does not change judgmentally when I flip the order of its arguments.)  Is there a version of equivalence which simultaneously satisfies that the inverse of the identity is judgmentally the identity, and that inverting an equivalence twice judgmentally gives you what you started with?

-Jason

On Thu, Nov 13, 2014 at 12:59 PM Vladimir Voevodsky <vladimir@ias.edu> wrote:
In general no. But their model is essentially syntactic and more or less complete. Or, to be more precise, I would expect it to 
be more or less complete. 

V.


On Nov 13, 2014, at 9:55 PM, Peter LeFanu Lumsdaine <p.l.lumsdaine@gmail.com> wrote:

On Thu, Nov 13, 2014 at 12:04 PM, Vladimir Voevodsky <vladimir@ias.edu> wrote:
The question is about how you interpret this operation for the univalent universe. If there is an interpretation of such an operation then there is a way to define equivalences between types in an involutionary way.

I don’t follow why this should be the case.  This shows that there is some notion of equivalence *in the model* (i.e. constructed in the meta-theory) which is strictly involutive.  But there is no reason that this notion need be definable in the syntax of the object theory, is there?

–p. 

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