nLab
model structure on cubical sets

Idea

There is a model category structure on the category $[{\square }^{\mathrm{op}},\mathrm{Set}]$ of cubical sets whose homotopy theory is that of the standard model structure on simplicial sets.

Using this version of the homotopy hypothesis-theorem, cubical sets are a way to describe the homotopy type of ∞-groupoids using of all the geometric shapes for higher structures the cube.

Definition

There is an evident simplicial set-valued functor

$$\square \to \mathrm{sSet}$$\Box \to sSet

from the cube category to sSet, which sends the cubical $n$-cube to the simplicial $n$-cube

$$1^n\mapsto (\Delta [1]{)}^{\times n}.$$\mathbf{1}^n \mapsto (\Delta[1])^{\times n} \,.

Similarly there is a canonical Top-valued functor

$$\square \to \mathrm{Top}$$\Box \to Top

$$1^n\mapsto ({\Delta }_{\mathrm{Top}}^1{)}^n.$$\mathbf{1}^n \mapsto (\Delta^1_{Top})^n \,.

The corresponding nerve and realization adjunction

$$(\mid -\mid ⊣{\mathrm{Sing}}_{\square }):\mathrm{Top}\stackrel{\stackrel{\mid -\mid }{\leftarrow }}{\underset{{\mathrm{Sing}}_{\square }}{\to }}{\mathrm{Set}}^{{\square }^{\mathrm{op}}}$$(|-| \dashv Sing_\Box) : Top \stackrel{\overset{|-|}{\leftarrow}}{\underset{Sing_\Box}{\to}} Set^{\Box^{op}}

is the cubical analogue of the simplicial nerve and realization discussed above.

Theorem

There is a model structure on cubical sets ${\mathrm{Set}}^{{\square }^{\mathrm{op}}}$ whose

• weak equivalences are the morphisms that become weak equivalences under geometric realization $\mid -\mid$;

• cofibrations are the monomorphisms.

This is (Jardine, section 3).

The following theorem establishes a form of the homotopy hypothesis for cubical sets.

Theorem

The unit of the adjunction

$$A\to {\mathrm{Sing}}_{\square }(\mid A\mid )$$A \to Sing_\Box(|A|)

is a weak equivalence in ${\mathrm{Set}}^{{\square }^{\mathrm{op}}}$ for every cubical set $A$.

The counit of the adjunction

$$\mid {\mathrm{Sing}}_{\square }X\mid \to X$$|Sing_\Box X| \to X

is a weak equivalence in $\mathrm{Top}$ for every topological space $X$.

It follows that we have an equivalence of categories induced on the homotopy categories

$$\mathrm{Ho}(\mathrm{Top})\simeq \mathrm{Ho}({\mathrm{Set}}^{{\square }^{\mathrm{op}}}).$$Ho(Top) \simeq Ho(Set^{\Box^{op}}) \,.

This is Jardine, theorem 29, corollary 30.

References

Using that the cube category is a test category a model structure on cubical sets follows as a spcial case of the model structure on presheaves over a test category, due to

• Denis-Charles Cisinski, Les préfaisceaux comme type d'homotopie, Asterisque 308.

The model structure on cubical sets as above is given in detail in

• Rick Jardine, Model structure on cubical sets (2002) (pdf)

There is also the old work

• Victor Gugenheim, On supercomplexes Trans. Amer. Math. Soc. 85 (1957), 35–51

in which “supercomplexes” are discussed, that combine simplicial sets and cubical sets (def 5). There are functors from simplicial sets to supercomplexes (after Defn 5) and, implicitly, from supercomplexes to cubical sets (in Appendix II). This was written in 1956, long before people were thinking as formally as nowadays and long before Quillen model theory, but a comparison of the homotopy categories might be in there.