Ordinary inductive types is initial algebras for free monads. However, HITs are not initial algebras for endofunctors but presented monads. 


From nLab, [initial algebra of a presentable (infinity,1)-monad](https://ncatlab.org/nlab/show/initial+algebra+of+a+presentable+%28infinity%2C1%29-monad) and [blog comment](http://golem.ph.utexas.edu/category/2011/04/homotopy_type_theory_vi.html#c041358), Peter and Mike constructed models for HITs in their paper. 


Where is the reference or link for this paper?



Models for Higher Inductive Types in Homotopy Type Theory
<https://arxiv.org/abs/1705.07088> . I've updated the nLab page.


There are also some slides available at my [web page](http://home.sandiego.edu/~shulman/papers/).


