higher inductive type (Rev #2)

Categorially a *higher inductive type* in an extensional type theory is an initial algebra of an endofunctor. In intensional type theory this analogy fails.

In particular $W$-types in an extensional type theory correspond to initial algebras of polynomial functors. Also this is not true for intensional type theories.

This failure can be (at least for $W$-types and some more general cases) remedied by replacing “initial” by “homotopy initial”. This is the main result (to be found in §2) of

- Steve Awodey, Nicola gambino, Kristina Sojakova, inductive types in homotopy type theory, arXiv:1201.3898