Symmetry in the cubical Joyal model structure

We study properties of the cubical Joyal model structures on cubical sets by means of a combinatorial construction which allows for convenient comparisons between categories of cubical sets with and without symmetries. In particular, we prove that the cubical Joyal model structures on categories of cubical sets with connections are cartesian monoidal. Our techniques also allow us to prove that the geometric product of cubical sets (with or without connections) is symmetric up to natural weak equivalence in the cubical Joyal model structure, and to obtain induced model structures for $(\infty,1)$-categories on cubical sets with symmetries.

💡 Research Summary

The paper investigates symmetry properties of the cubical Joyal model structures on various categories of cubical sets. After recalling the basic cube category □ S generated by faces, projections, diagonals, connections, transpositions, and reversals, the author introduces sub‑categories □ A obtained by selecting a subset A ⊆ {∧, ∨, Σ, ρ, δ}. The corresponding presheaf categories cSet A serve as the ambient settings for minimal cubical sets (□ ∅), cubical sets with connections (□ ∧∨), and cubical sets with symmetries (□ Σ), among others.

A monoidal product ⊗ on each cSet A is defined via Day convolution, yielding the “geometric product”. This product satisfies the desirable dimension‑addition property □ m ⊗ □ n ≅ □ (m + n), which the ordinary cartesian product lacks in most cube categories. However, ⊗ is not symmetric in general; symmetry holds only when the transposition generator Σ is present.

The central technical tool is the notion of “standard decomposition cubes”. These are explicit combinatorial objects that allow certain inclusions of cubical sets to be expressed as transfinite compositions of open‑box fillings. Using these, the paper constructs three adjoint triples i ! ⊣ i* ⊣ i* relating categories with and without additional structure maps, and shows that the unit of the left adjoint i ! is a natural trivial cofibration in the cubical Joyal model structure.

From this analysis two main theorems follow. First (Theorem 4.3), for any cubical sets X and Y, the geometric products X ⊗ Y and Y ⊗ X are naturally weakly equivalent; thus the geometric product is symmetric up to weak equivalence even when the underlying cube category lacks symmetries. Second (Theorem 5.12), when connections are present (cSet ∧∨), the cubical Joyal model structure is cartesian monoidal, and the geometric product is weakly equivalent to the cartesian product. This resolves the known homotopical deficiencies of the cartesian product in connection‑free settings.

Moreover, by applying the same adjunction analysis to categories that include symmetries, the author obtains induced model structures for (∞, 1)‑categories on symmetric cubical sets (Theorem 4.7). These induced structures are Quillen equivalent to the original cubical Joyal model via the forgetful functor, providing alternative models of (∞, 1)‑categories that incorporate symmetry.

Overall, the paper offers a concise combinatorial framework—standard decomposition cubes—that replaces more intricate factorisation arguments in previous work. This framework yields clean proofs of symmetry up to weak equivalence, cartesian monoidality, and the existence of symmetric induced model structures, thereby strengthening the foundations for using cubical sets in higher‑category theory, homotopy type theory, and related areas.