nLab model structure on cubical sets

Idea

Various (∞,1)-categories can be presented by model structures on the category $[\Box^{op},Set]$ of cubical sets, i.e., presheaves on a category of cubes $\Box$.

For example, the theory of test categories produces model structures on categories 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.

Cubical sets can also be used to build model structures presenting other homotopy theories, such the model structures for cubical quasicategories presenting (∞,1)-categories.

Presentations of infinity-groupoids

Ordered cubical sets

Write $\Box$ for the ordered cube category defined at category of cubes. There is an evident simplicial set-valued functor

$$\begin{array}{rcl} \Box &\to& sSet \\ I^n &\mapsto& (\Delta[1])^{\times n} \end{array}$$

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

Similarly, there is a canonical Top-valued functor

$$\begin{array}{rcl} \Box &\to& Top \\ I^n &\mapsto& [0,1]^{n} \end{array}$$

The corresponding nerve and realization adjunction

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

is the cubical analogue of the simplicial nerve and realization adjunction.

Theorem

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

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

• cofibrations are the monomorphisms.

This is (Jardine, section 3).

Proposition

An explicit 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 2006, Thm 8.4.38).

As a consequence, the fibrations are exactly cubical Kan fibrations. 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 $[\Box^{op},Set]$ 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

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

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

In fact, by the discussion at adjoint (∞,1)-functor it follows 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 cubical sets

The model structure from Theorem and the equivalence of its homotopy theory to that of Top can be derived using the theory of test categories (see (Cisinski 2006, Section 8.4)). Test category theory can be used more generally to derive model structures presenting ∞Grpd on $[\Cube^{op},Set]$ for other cube categories $\Box$.

The weak equivalences are not in general induced from the weak homotopy equivalences by the geometric realization functor $|-| \colon [\Box^{op},Set] \to Top$ sending $I^n$ to $[0,1]^{n}$, but rather from the weak equivalences of the Thomason model structure by the realization $[\Box^{op},Set] \to Cat$ sending $I^n$ to the slice category $\Box/I^n$. It is also not always true that the trivial cofibrations are generated by cubical horn inclusions $\sqcap_{k,\epsilon}^n \to \Box^n$, as is the case for the minimal cube category; such characterizations rely on Reedy category structure on the cube category.

A case of particular interest is the ordered cube category with a connection. Unlike the minimal cube category, this cube category is a strict test category (Maltsiniotis 2009), meaning its weak equivalences are closed under product. This is one motivation for using cubes with connections as opposed to the minimal cube category; see connection on a cubical set for more details.

Buchholtz and Morehouse (2017) outline a class of cube categories and investigate which are test and/or strict test categories.

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.

Cubical-type model structures

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 presents ∞Grpd, 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.

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, Relational and Algebraic Methods in Computer Science (RAMICS 2017), Lecture Notes in Computer Science 10226 (2017) 77–92 [arXiv:1701.08189, doi:10.1007/978-3-319-57418-9_5]

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)