Triangulation functor creating weak equivalences for induced cubical model structures

Determine whether the triangulation functor T: cSet_B → sSet, with sSet equipped with the Quillen or Joyal model structure as appropriate, creates the weak equivalences of the induced model structures on cSet_B arising from the forgetful functor i^*: cSet_B → cSet_∅ (namely, the left- and right-induced Grothendieck model structures and, when B does not contain reversals, the left- and right-induced cubical Joyal model structures). In particular, for B = P (the poset cube category with faces, connections, symmetries, and diagonals), ascertain whether T creates the weak equivalences of the left-induced cubical Joyal model structure on cSet_P, which would imply that this model structure coincides with Hackney–Rovelli’s cSet_(m,1).

Background

The paper constructs left- and right-induced model structures on categories of cubical sets with symmetries and connections via the forgetful functor i^*: cSet_B → cSet_∅, yielding Grothendieck and cubical Joyal model structures that are Quillen equivalent to their counterparts on minimal cubical sets. A key tool in relating these cubical models to simplicial models is the triangulation functor T: cSet_B → sSet, which in several classical cases (e.g., for A ⊆ ∅) is known to create weak equivalences.

For certain cube categories without reversals, such as B = {∨, Σ, δ} (and the isomorphic case with ∧) and B = P, existing results show that T creates weak equivalences for the test model structures. The unresolved issue is whether T likewise creates the weak equivalences for the induced model structures introduced in this paper. In the specific case B = P, establishing this for the left-induced cubical Joyal model structure would identify it with Hackney–Rovelli’s model cSet_(m,1), which is defined to have monomorphisms as cofibrations and weak equivalences created by triangulation.