Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
Paths and cylinders
There is a model category structure on the category 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.
There is an evident simplicial set-valued functor
from the cube category to sSet, which sends the cubical -cube to the simplicial -cube
Similarly there is a canonical Top-valued functor
The corresponding nerve and realization adjunction
is the cubical analogue of the simplicial nerve and realization discussed above.
This is (Jardine, section 3).
The following theorem establishes a form of the homotopy hypothesis for cubical sets.
The unit of the adjunction
is a weak equivalence in for every cubical set .
The counit of the adjunction
is a weak equivalence in for every topological space .
It follows that we have an equivalence of categories induced on the homotopy categories
This is Jardine, theorem 29, corollary 30.
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
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
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.