Categorially a *higher inductive type* is an initial algebra of an endofunctor. In particular $W$[[W-type|-type]]s correspond to initial algebras of [[nLab:polynomial functor]]s. * Steve Awodey, Nicola gambino, Kristina Sojakova, inductive types in homotopy type theory, [arXiv:1201.3898](http://arxiv.org/abs/1201.3898)