Semantics of higher inductive types

TL;DR

This paper develops models for higher inductive types within various homotopical and categorical settings, enabling their use in synthetic homotopy theory and formalized mathematics in type theory.

Contribution

It introduces the notion of cell monads with parameters and proves the existence of weakly stable typal initial algebras in suitable model categories, facilitating models of higher inductive types.

Findings

Models of higher inductive types like spheres, torus, and truncations are constructed.

Applicable in a wide range of model categories including simplicial sets and locally presentable categories.

Provides a unifying framework for higher inductive types in homotopy-theoretic and categorical contexts.

Abstract

Higher inductive types are a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic 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 semantically-defined 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, simplicially locally cartesian closed, 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 $(\infty,1)$-category is presented by some model category to which our results apply.