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• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeApr 29th 2022

Somehow I missed that Mike’s 3 talks were going ahead: Yesterday (28 April), 5 May and 12 May, details here.

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeApr 29th 2022

Is there any ’Trinitarian’ story to tell about the advantages of HOTT over the other generations of HoTT? Do the differences correspond to anything in category theory?

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeApr 29th 2022
• (edited Apr 29th 2022)

[…] Question answered in the talk.

1. Is there a way to define dependent/homogeneous identity types without using universes?

2. I meant heterogoeneous identity types, rather than “homogeneous” identity types.

• CommentRowNumber6.
• CommentAuthorUlrik
• CommentTimeApr 30th 2022

@4,5: There are many equivalent ways of defining dependent identity types, several of which don’t need universes. The simplest is perhaps as an identity type in the right (or left) fiber, using transport. (I.e., $(a =^B_p b) :\equiv (\mathrm{trp}^P\,p\,a =_{B(b)} b$, for $a,b:A$, $p:a=_A b$, and $B$ a type family over $A$.) But you can also express them as an indexed inductive type.

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeApr 30th 2022

I assume Madeleine’s question was about the specific heterogeneous identity types in H.O.T.T., since the latter was the subject of this thread. One can simply postulate all the n-ary heterogeneous identity types, along with all their structure, rather than deriving them from n-ary ap and the universe. Semantically, one may want to do that anyway, and then separately assert equalities saying that they happen to coincide with the underlying correspondences of ap into the universe. But for a syntactic introduction, I find it clean and nicely motivating to derive them from the latter.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeApr 30th 2022

Is there any ’Trinitarian’ story to tell about the advantages of HOTT over the other generations of HoTT? Do the differences correspond to anything in category theory?

Roughly I would say that:

1. Book HoTT corresponds to Quillen model categories.
2. Cubical type theories correspond to Quillen model categories containing an interval object.
3. H.O.T.T. corresponds to Quillen model categories enriched over semicartesian cubical sets. (This will become clearer after the second and third talks.)

In all cases one needs extra niceness properties, but this is the basic distinguishing structure. Does that go any way towards answering the question?

3. My question was about the heterogeneous identity types in Mike’s presentation of higher observational type theory. Mike’s presentation implicitly uses universes a la Russell, where every type is a term of a universe. But Mike’s definition of heterogeneous identity types through universes presents some problems in a type theory which uses universes a la Tarski: since it is not true that every type is a term of a universe a la Tarski, the given definition is only valid (after translating appropriately) for those types that are some coercion $El(A)$ of a term $A:U$ of a universe $U$.

• CommentRowNumber10.
• CommentAuthorMike Shulman
• CommentTimeMay 1st 2022

It works fine with universes a la Tarski as long as every type belongs to some universe.

• CommentRowNumber11.
• CommentAuthorMike Shulman
• CommentTimeMay 1st 2022

If you have types that don’t belong to any universe, then you’d have to give primitive rules for the heterogeneous equality types, as I suggested in #7.

• CommentRowNumber12.
• CommentAuthorDavid_Corfield
• CommentTimeMay 3rd 2022

Re #8, thanks Mike! Since you seem to be convinced that H.O.T.T. represents a philosophical improvement, perhaps I expected this to be clear in the direction of change of categorical semantics, say to something simpler. Maybe I shouldn’t expect this though.

4. Looks like Mike Shulman gave primitive rules for dependent identity types in his second talk yesterday.

• CommentRowNumber14.
• CommentAuthorDavidRoberts
• CommentTimeMay 6th 2022
• (edited May 6th 2022)

Is there any reason the video is not yet available on YouTube? I think the first one was up quite promptly.

Edit Oh, I see Jonas Frey gave a temporary “link for the impatient” here: https://u.pcloud.link/publink/show?code=XZXBjpVZ8cOAFR7oWoRbobS6NWsNE82pz3Ny, the official upload will happen presumably before the day is out.

5. The video of the 2nd lecture has been posted to Youtube here.

• CommentRowNumber16.
• CommentAuthorDavidRoberts
• CommentTimeMay 6th 2022