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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeApr 17th 2018

Mention that geometric homotopy type theory is not well-defined yet.

• CommentRowNumber2.
• CommentAuthorspitters
• CommentTimeApr 19th 2019

Link to HITs

• CommentRowNumber3.
• CommentAuthormbid
• CommentTimeApr 21st 2019

I’m not sure higher inductive types, in the sense of a generalization of W-types, would make sense in a geometric homotopy type theory. 1-categorically, W-types are defined as initial algebras of polynomial endofunctors, and these make sense only if there are dependent products, which are, however, not stable under inverse image. Arithmetic universes, which should be precisely the models of whatever geometric type theory is, have free models for all (finite) finite limit theories. I’ve been told that it does not follow from this that an arithmetic universe U has all W-types even if U happens to be lcc. Having just free models to finite limit theories is thus strictly weaker than having W-types. So I think the infinity categorical version of geometric type theory should similarly have free models for finite hlimit theories instead of W-types.

• CommentRowNumber4.
• CommentAuthormbid
• CommentTimeApr 21st 2019

Mention essentially algebraic (infinity,1) theories instead of higher inductive types

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeApr 22nd 2019

Clarified further the relationship.

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