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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.
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