on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
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
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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.
There is an evident simplicial set-valued functor
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
The corresponding nerve and realization adjunction
is the cubical analogue of the simplicial nerve and realization discussed above.
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).
The following theorem establishes a form of the homotopy hypothesis for cubical sets.
is a weak equivalence in $Set^{{\Box}^{op}}$ for every cubical set $A$.
The counit of the adjunction
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
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
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.