Constructive criteria to recognize the homotopy theory of spaces

Develop constructive criteria and diagnostic methods to ascertain when a constructively defined Quillen model structure presents the homotopy theory of spaces. In particular, formulate a constructive procedure to determine whether a given homotopy theory is classically equivalent to the Kan–Quillen model structure on simplicial sets or otherwise identify constructive invariants that characterize such equivalence.

Background

The authors provide a constructively definable model structure on cartesian cubical sets and discuss issues surrounding constructive verification of equivalence to the classical homotopy theory of spaces. While classical tools (e.g., Reedy cell complex presentations) facilitate such comparisons, they are not generally constructive and can interfere with coherence needed for type-theoretic models.

A general constructive framework to judge when a homotopy theory is "the homotopy theory of spaces" would aid in assessing models that are classically equivalent but potentially distinct constructively, and would support broader constructive developments analogous to Shulman’s classical results for Grothendieck ∞-toposes.