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This is fairly obscure but the problem he highlights can be overcome easily by localising at the saturation of S_f, for D(f)=D(g) if and only if the saturations are equal, and localising at saturation is an isomorphism

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It seems though that we would want the computer to be able to do this kind of reasoning itself, and not rely on humans "pre-resolving" all such problems.

I understood the presentation to be on the distinction between equality and isomorphism and when an isomorphism "is an equality". From the slide on homotopy type theory I get the impression that he finds it unsatisfactory to simply consider all isomorphisms as equalities.

But I might have misunderstood your objection?

So many times it’s necessary to ‘identify’ two more objects which are isomorphic, and the ‘canonical’ is supposed to justify why this doesn’t cause a problem.

The reason it is necessary here is that the mapping D(f) -> A_f is a priori multi valued, and we need it to be single valued.

However it’s fairly easy to make a definition which is trivially single valued , so I don’t think it’s a very instructive example of the phenomenon.

Probably more pertinent is an n-fold tensor product with different bracket ordering

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