nLab model structure on cubical sets

Idea

There is a model category structure on the category $[\Box^{op},Set]$ of cubical sets whose homotopy theory is that of the classical 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

$$\Box \to sSet$$

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

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

Similarly there is a canonical Top-valued functor

$$\Box \to Top$$

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

The corresponding nerve and realization adjunction

$$(|-| \dashv Sing_\Box) : Top \stackrel{\overset{|-|}{\leftarrow}}{\underset{Sing_\Box}{\to}} Set^{\Box^{op}}$$

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

Theorem

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

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

• cofibrations are the monomorphisms.

This is (Jardine, section 3).

Explicitly, a set of generating cofibrations is given by the boundary inclusions $\partial \Box^n \to \Box^n$, and a set of generating acyclic cofibrations is given by the horn inclusions $\sqcap_{k,\epsilon}^n \to \Box^n$. This is (Cisinski, Thm 8.4.38). Thus, as a consequence of Cisinski’s work, the fibrations are exactly cubical Kan fibrations.

Properties

Homotopy theory

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

Theorem

The unit of the adjunction

$$A \to Sing_\Box(|A|)$$

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

The counit of the adjunction

$$|Sing_\Box X| \to X$$

is a weak equivalence in $Top$ for every topological space $X$.

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

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

This is (Jardine, theorem 29, corollary 30).

In fact, by the discussion at adjoint (∞,1)-functor it follow that the derived functors of the adjunction exhibit the simplicial localizations of cubical sets equivalent to that of simplicial sets, hence makes their (∞,1)-categories equivalent (hence equivalent to ∞Grpd).

Other kinds of cubical sets

In cubical type theory one uses more structured notions of cubical set (symmetric, cartesian, De Morgan, etc.) In most cases such categories have both a test model structure, which is equivalent to spaces, and a cubical-type model structure that corresponds to the interpretation of type theory. In many cases these are not equivalent, and the cubical-type model structure does not model classical homotopy types. See cubical-type model structure for more discussion.

Joyal-type model structures

In complete analogy to simplicial sets, there is also an analogue of the Joyal model structure on cubical sets, with or without connection. See the article model structures for cubical quasicategories.

Related concepts

• cubical set

• cubical set with connections

• test category, strict test category

• Cisinski model structure

• model structures for cubical quasicategories

References

Cubical sets without connections

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

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

Cisinski also derives explicit generating cofibrations and generating acyclic cofibrations using his theory of generalized Reedy category, or categories skelettiques. See Section 8.4.

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

• J. F. Jardine, Cubical homotopy theory: a beginning (2002) (pdf)

Cubical sets with max-connections

The following paper proves that cubical sets with connections (more specifically, max-connections) form a strict test category and therefore admit a cartesian model structure that is Quillen equivalent to the Kan–Quillen model structure on simplicial sets:

• Georges Maltsiniotis, La catégorie cubique avec connexions est une catégorie test stricte, Homology, Homotopy and Applications 11:2 (2009), 309-326. doi.

Cubical sets with max-connections and min-connections

The case of cubical sets with both max-connections and min-connections largely follows the case of cubical sets with max-connections, the corresponding category of cubes again being a strict test category. The relevant results are stated explicitly as Corollary 3 and Theorem 3 of

• Ulrik Buchholtz, Edward Morehouse, Varieties of Cubical Sets, arXiv.

Cartesian cubical sets

The case of the category of presheaves over the free finite product category with a bipointed object is treated in:

• Steve Awodey, Cartesian cubical model categories (arXiv:2305.00893)