Some extensions of the fundamental theorem of identity types

[fredrik-bakke]

So my recent work on π₀-trivial types got me looking for other applications of this concept, and here's one. Essentially, this PR gives an alternative phrasing of the extended fundamental theorem of identity types such that the assumption of inhabitedness/pointedness on the base type A is not needed.

Edit: the scope of this PR has grown since the above description was written.

Summary

• Unbased version of the extended fundamental theorem of identity types
• Structured equality duality
• Strong preunivalence
• Strengthen definition of preunivalent categories
• Weaken assumptions on the type theoretic Yoneda lemma

[fredrik-bakke]

Wait a second, I'm doing a stupid

[EgbertRijke]

Do you use "structure" in the same way here as in the structure identity principle?

[fredrik-bakke]

No, here structure is used differently. Here structure is what the theorem is used to construct, not to characterize.

[EgbertRijke]

No, here structure is used differently. Here structure is what the theorem is used to construct, not to characterize.

Probably you should be very careful with terminology then. The title "fundamental theorem of identity types for structures" is suggestive of a theorem related to the structure identity principle. Could you think about phrasing your work in a less ambiguous way?

[fredrik-bakke]

No, here structure is used differently. Here structure is what the theorem is used to construct, not to characterize.

Probably you should be very careful with terminology then. The title "fundamental theorem of identity types for structures" is suggestive of a theorem related to the structure identity principle. Could you think about phrasing your work in a less ambiguous way?

Thanks, that is good to keep in mind! Sorry, this PR was originally something different, but then I realized I was thinking about things wrong so I need to think a little more before it is ready

[fredrik-bakke]

As an aside, this PR made me realize there's a canonical strengthening of preunivalence one might ought to consider, so stay tuned for that :)

[fredrik-bakke]

I figured out what the proper statement for the extended fundamental theorem should be, so we're back in business 😁

[fredrik-bakke]

Okay, now I think this PR is ready to be looked at. I renamed the previous "fundamental theorem of identity types for structures" to "structured equality duality", but that's just a name I'm making up so let me know what you feel about it. I also did some other things, see the PR main body.

[fredrik-bakke]

I've tried writing some explanations as you asked for. If you are not happy with them, I would prefer to change terminology and or remove formalizations until the explanations are no longer needed.

[EgbertRijke]

Thank you so much! I think your extra note is very informative, I like it!

I think (or presume) that more people are familiar with the ordinary preunivalence principle than with the stronger version. Having the preunivalence principle in the library with this note that there is also a stronger common generalization is very worthwhile!

So, thank you for going through the extra effort, and for this cool new insight about identity types! I'll merge this PR now.