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.

References

It is more generally unclear how to judge whether a homotopy theory is “the homotopy theory of spaces” constructively.

— The equivariant model structure on cartesian cubical sets  (2406.18497 - Awodey et al., 2024) in Subsection “Constructivity” (Section 1.6)

The interesting question remains how rich the theory is, namely: ``Does every phenomenon observed in the classical topology have a counterpart in $A$-homotopy theory?''. This question is formalized in the {discrete homotopy hypothesis}, which states that the localization of the category of graphs along those maps which induce isomorphisms on $A$-homotopy groups is equivalent to the classical homotopy category, that is the category of CW-complexes with morphisms being homotopy classes of continuous maps. Recently, a major advancement towards this conjecture has been made in , where it was shown that the localization along those graph maps, which induce isomorphisms on the first $n$ homotopy groups is equivalent to the category of CW-complexes localized at those maps that induce isomorphisms on the first $n$ classical homotopy groups.

— Eilenberg-MacLane spaces in A-homotopy theory  (2610.03497 - Wille, 2 Oct 2026) in Section 1, Introduction