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Somehow I missed that Mike’s 3 talks were going ahead: Yesterday (28 April), 5 May and 12 May, details here.
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?
[…] Question answered in the talk.
Is there a way to define dependent/homogeneous identity types without using universes?
I meant heterogoeneous identity types, rather than “homogeneous” identity types.
@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.
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.
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:
In all cases one needs extra niceness properties, but this is the basic distinguishing structure. Does that go any way towards answering the question?
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 $U$.
It works fine with universes a la Tarski as long as every type belongs to some universe.
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.
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.
Looks like Mike Shulman gave primitive rules for dependent identity types in his second talk yesterday.
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.
The video of the 2nd lecture has been posted to Youtube here.
Thanks, Madeleine!
For anybody interested, the third lecture has been posted to Youtube here.
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