model structure on cubical sets


Model category theory

model category



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Presentation of (,1)(\infty,1)-categories

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Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



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Homotopy groups

Basic facts




There is a model category structure on the category [ op,Set][\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.


There is an evident simplicial set-valued functor

sSet \Box \to sSet

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

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

Similarly there is a canonical Top-valued functor

Top \Box \to Top
1 n(Δ Top 1) n. \mathbf{1}^n \mapsto (\Delta^1_{Top})^n \,.

The corresponding nerve and realization adjunction

(||Sing ):TopSing ||Set 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.


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

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

  • cofibrations are the monomorphisms.

This is (Jardine, section 3).


Homotopy theory

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


The unit of the adjunction

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

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

The counit of the adjunction

|Sing X|X |Sing_\Box X| \to X

is a weak equivalence in TopTop for every topological space XX.

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

Ho(Top)Ho(Set op). 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).


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

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

There is also the old work

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

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.

Revised on March 10, 2016 04:37:33 by Urs Schreiber (