If we are only interested in providing one -particular- model of HITs, the paper

on cubical type theory describes a way to interpret HIT together with a univalent

universe which is stable by HIT operations. This gives in particular the consistency

and the proof theoretic power of this extension of type theory.

The approach uses an operation of “flattening an open box”, which solves in

this case the issue of interpreting HIT with parameters (such as propositional

truncation or suspension) without any coherence issue.

Since the syntax used in this paper is so close to the semantics, we limited

ourselves to a syntactical presentation of this interpretation. But it can directly

be transformed to a semantical interpretation, as explained in the following note

(which also incorporates a nice simplification of the operation of flattering

an open box noticed by my coauthors). I also try to make more explicit in the note

what is the problem solved by the “flattening boxes” method.

Only the cases of the spheres and propositional truncation are described, but one

would expect the method to generalise to other HITs covered e.g. in the HoTT book.

On 25 May 2017, at 20:25, Michael Shulman <shu...@sandiego.edu> wrote:

The following long-awaited paper is now available:

Semantics of higher inductive types
Peter LeFanu Lumsdaine, Mike Shulman
https://arxiv.org/abs/1705.07088

From the abstract:

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 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.

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