Paths and cylinders
In the 1980-s, Grothendieck in Pursuing Stacks introduced test categories to make the variants of the homotopy theory based on the usage of combinatorial models with some kind of cell structure (e.g., simplicial sets, cubical sets and cellular sets) independent of a particular combinatorial model.
Given any small category , one considers -sets, hence presheaves on , hence contravariant functors from to .
Given an object , one considers the representable functor . If is a -set, the elements of are called the -cells. By the Yoneda lemma, they correspond to the natural transformations .
Let the cell category of , denoted , be the full subcategory of the overcategory whose objects are the transformations of the form . The correspondence extends to a functor which has a right adjoint whose object part is given by the formula
Denote the counit of the adjunction .
Two -sets and are weakly equivalent if there is a map inducing an equivalence of their cell categories, i.e., the induced map of nerves (“classifying spaces”) is a weak equivalence of simplicial sets. Functor induces a functor of the homotopy categories.
A weak test category is a small category such that, for any category in , the component of the counit is an equivalence of categories.
A test category is any small category such that
A strict test category is a test category such that
- preserves finite products up to weak equivalence,
or equivalently, such that
- the induced functor preserves finite products.
Then one proceeds with -sets.
If is a test category and any small category whose classifying space is contractible (which may or may not be a test category itself), then their cartesian product is a test category.
The homotopy category of a category of presheaves over a test category, as a category with weak equivalences is equivalent to the standard homotopy category of homotopy theory: that of the category of simplicial sets/topological spaces with weak equivalences being weak homotopy equivalences.
In other words, presheaves over a test category are models for homotopy types of ∞-groupoids.
Model category structure
The presheaf category over a test category with the above weak equivalences admits a model category structure: the model structure on presheaves over a test category. This is due to (Cisinski) with further developments due to (Jardine).
Apart from the archeytpical example of the simplex category we have the following
The notion of test category was introduced in
Various conjectures made there are proven in
which moreover develops the main toolset and establishes the model structure on presheaves over a test category.
General surveys include
Georges Maltsiniotis, La théorie de l’homotopie de Grothendieck, Astérisque, 301, pp. 1-140, (2005) (see html)
Rick Jardine, Categorical homotopy theory, Homot. Homol. Appl. 8 (1), 2006, pp.71–144, (HHA, pdf)
That the cube category is a test category is asserted without proof in (Grothendieck). A proof is spelled out in (Cisinski)
That it is not a strict test category is implicitly already in
- Dan Kan, Abstract homotopy. I , Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 1092–1096. (pdf)
and led to the preference for simplicial sets over cubical sets.
That the category of cubes equipped with connection on a cubical set forms a strict test category is shown in
- Georges Maltsiniotis, La catégorie cubique avec connexions est une catégorie test stricte . (French. English summary) Homology, Homotopy Appl. 11 (2009), no. 2, 309–326. (web)
The test category nature of the groupoidal Theta category is discussed in
A short introduction can be found in
- Chris Kapulkin, Introduction to Test Categories PDF.