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1. starting stub on path types

Anonymous

Anonymous

3. added rules for path types

Anonymous

• CommentRowNumber4.
• CommentAuthorGuest
• CommentTimeSep 3rd 2022

The rules for path types currently do not include any coercion or composition rules. Do the path types actually behave like paths/identity types without such coercion and composition rules?

Also, I think the given rules are for Cartesian cubical type theories. Are the rules the same for other flavors of cubical type theories such as De Morgan cubical type theory and the BCH model?

• CommentRowNumber5.
• CommentAuthorGuest
• CommentTimeSep 3rd 2022

This page should be renamed to “cubical path type”, since path type is also used as another name for Martin-Löf’s identity types.

4. path type -> cubical path type, identity type -> Martin-Löf identity type

Anonymous

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeSep 4th 2022

Re 4: Yes, you need the coercion/composition rules to complete the definition. Those are the ones that depend on the flavor of cubical type theory: I think the ones written on the page so far are the same in all cases.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeSep 4th 2022

I’m a little confused by what is written in the “regularity” section at the moment. Could whoever wrote it please provide a reference?

• CommentRowNumber9.
• CommentAuthorjonsterling
• CommentTimeSep 6th 2022

Hi Mike, I didn’t write this, but the regularity rule appears to be an equivalent reformulation of the usual regularity rule — these are two ways to make A constant. Does this make sense?

• CommentRowNumber10.
• CommentAuthorjonsterling
• CommentTimeSep 6th 2022

I think that section is a little weird, though, since if you don’t have coercion at all, there’s no chance of having J and regularity won’t be the problem. Indeed, you need heterogeneous composition, I think just coercion wouldn’t be enough.

• CommentRowNumber11.
• CommentAuthorGuest
• CommentTimeSep 6th 2022

The coercion and regularity rules seem to be lifted from the second XTT article listed in the references but whoever wrote that section on regularity seems to have forgotten that defining the J-eliminator requires composition, which was explicitly stated in that XTT article

Example 2.10 (Identity type). Using composition, we may define a combinator with the same type as Martin-Löf’s J eliminator for the identity type.

• CommentRowNumber12.
• CommentAuthorMike Shulman
• CommentTimeSep 6th 2022

One thing that confuses me is that I wouldn’t expect “adding regularity” to entail adding new operations, only new axioms satisfied by the existing operations. Is that similar to what you’re saying?

• CommentRowNumber13.
• CommentAuthorjonsterling
• CommentTimeSep 7th 2022

Yes, I think the changes to this section could be:

1) Remove the comment about adding coercion, keep the regularity rule 2) If coercion shall be discussed, discuss it elsewhere in the context of composition.

Adding regularity does not add any new operation. (Though, tangentially, I suppose one way to add regularity is to add a new “cofibration-former” that corresponds to the reflexivity map.)

5. moving coercion to Rules section

Anonymous