model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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).
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.
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).
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.
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
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
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:
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