Abstract. Higher inductive types are a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a typetheoretic style. They have proven very fruitful for the "synthetic" development of homotopy theory within type theory, as well as in formalizing ordinary set-level mathematics in type theory. In this article, we construct models of a wide range of higher inductive types in a fairly wide range of settings. We introduce the notion of cell monad with parameters: a semanticallydefined scheme for specifying homotopically well-behaved notions of structure. We then show that any suitable model category has weakly stable typal initial algebras for any cell monad with parameters. When combined with the local universes construction to obtain strict stability, this specializes to give models of specific higher inductive types, including spheres, the torus, pushout types, truncations, the James construction, and general localisations. Our results apply in any sufficiently nice Quillen model category, including any right proper simplicial Cisinski model category (such as simplicial sets) and any locally presentable locally cartesian closed category (such as sets) with its trivial model structure. In particular, any locally presentable locally cartesian closed (∞, 1)-category is presented by some model category to which our results apply.
On the other hand, Lumsdaine and Shulman also study the semantics of higher inductive types using a semantic scheme (cell monads with parameters) and they prove existence in sufficiently nice Quillen model categories REF .
Semantics of higher inductive types
arXiv: Logic
Mathematics