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What should we say about this topic from the perspective of Higher Observational TT where laws are definitional, e.g., identity of pairs and pairs of identities (slide 3 and 13 of Mike’s slides) and this goes for dependent pair and function too (slide 40)? Might one say definitional univalence (slide 26) leads to structural definitional equality?
My understanding is that this is the whole motivation and point of HiObsTT: To make all identification types be definitionally respectful of their type’s structure.
One wouldn’t be able to tell this from what “Anonymous” has written at higher observational type theory (this entry would deserve a complete overhaul), and it is not immediate to extract this from writings available elsewhere. But that’s my impression.
Thanks. It’s a shame we don’t get to hear more of the motivation:
Homotopy theory is emergent rather than explicit; all rules have a convincing philosophical justification. (slide 8)
To decompose this: I suppose that
the whole motivation and point of (non-higher) “observational type theory” is to make equality of types be the structure-preserving equalities,
the “higher” analog is meant to do just the same but with equality replaced by identification types.
So your main question is whether point (1.) is the case, and if so whether it is admitted anywhere.
So let’s see:
In the original article “Towards observational type theory” the motivation for the approach and terminology is:
two functions are equal, if there are equal pointwise or to put it differently, if all observations about them agree.
It seems that “observations” is not the best word here, since it is specific to the case of function types.
The actual rule for equality of (a) dependent function types, (b) dependent pair types and (c) well-founded inductive types are stated in the very bottom left of p. 3.
After realizing that what looks like norms are meant to be equalities (!, deduced from inside two paragraphs above in that left column) one sees that this bottom left formula says that equality of all such types is to imply component-wise equaities — and the converse is declared in section 2.3.
In summary, this is to declare that the equalities of compound types are componentwise, hence what the algebraist would (and did) call: homo-morphically, i.e. structure-preservingly.
added pointer to:
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