nLab
test category

Contents

Context

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

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

The notion of a test category (Grothendieck 83) is meant to axiomatize common features of categories of shapes used to model homotopy types in homotopy theory, such as the categories of simplicial sets, cubical sets or cellular sets.

Definition

Given any small category 𝒞\mathcal{C}, one considers 𝒞\mathcal{C}-sets, hence presheaves on 𝒞\mathcal{C}, hence contravariant functors from 𝒞\mathcal{C} to Set.

Given an object cCc\in C, one considers the representable functor Hom 𝒞(,c)=:Δ cHom_{\mathcal{C}}(-,c)=:\Delta^c. If X:𝒞 opSetX \colon \mathcal{C}^{op} \to Set is a 𝒞\mathcal{C}-set, the elements of X(c)X(c) are called the cc-cells. By the Yoneda lemma, they correspond to the natural transformations Δ cX\Delta^c\to X.

Let the cell category of XX, denoted i 𝒞Xi_{\mathcal{C}} X, be the full subcategory of the overcategory 𝒞Set/X\mathcal{C}Set/X whose objects are the transformations of the form Δ cX\Delta^c\to X. (This is another name for the category of elements of XX.)

The correspondence Xi 𝒞XX\mapsto i_{\mathcal{C}}X extends to a functor i 𝒞:𝒞Seti_{\mathcal{C}} \colon\mathcal{C}Set \to Cat, which has a right adjoint i 𝒞 *:Cat𝒞Seti_{\mathcal{C}}^* \colon Cat\to\mathcal{C}Set whose object part is given by the formula

i 𝒞 *(D)(c)Hom Cat(𝒞/c,D). i_{\mathcal{C}}^*(D)(c) \coloneqq Hom_{Cat}(\mathcal{C}/c,D) \,.

Denote the counit of the adjunction ϵ:i 𝒞i 𝒞 *Id Cat\epsilon : i_{\mathcal{C}}i_{\mathcal{C}}^*\to Id_{Cat}.

Two 𝒞\mathcal{C}-sets XX and YY are called weakly equivalent if there is a morphism f:XYf \colon X\to Y inducing an equivalence f *:i 𝒞Xi 𝒞Yf_* \colon i_{\mathcal{C}} X\to i_{\mathcal{C}} Y of their cell categories, i.e., the induced map of nerves (“classifying spaces”) B(i 𝒞X)B(i 𝒞Y)B(i_{\mathcal{C}} X)\to B(i_{\mathcal{C}} Y) is a weak equivalence of simplicial sets. The functor i 𝒞:𝒞SetCati_{\mathcal{C}}:\mathcal{C}Set\to Cat induces a functor i 𝒞*:Ho(𝒞Set)Ho(Cat)i_{\mathcal{C}*}:Ho(\mathcal{C}Set)\to Ho(Cat) of the homotopy categories.

A 𝒞\mathcal{C}-set XX is called aspherical if the category i 𝒞(X)i_{\mathcal{C}}(X) is weakly contractible, i.e. the nerve B(i 𝒞(X))B(i_{\mathcal{C}}(X)) is a weakly contractible simplicial set. Note that if 𝒞\mathcal{C} is a weakly contractible category, then this is equivalent to the condition that the map X1X \to 1 to the terminal presheaf is a weak equivalence of 𝒞\mathcal{C}-sets.

A weak test category is a small category 𝒞\mathcal{C} such that, for any category DD in CatCat which has a terminal object, the 𝒞\mathcal{C}-set i 𝒞 *(D)i_{\mathcal{C}}^\ast(D) is aspherical.

A test category is any small category 𝒜\mathcal{A} such that

  • (𝒜\mathcal{A} is aspherical) its (geometric realization of the) nerve (“classifying space”) |𝒜|\vert \mathcal{A}\vert is contractible

  • (𝒜\mathcal{A} is a “local test category”) for every object aa in 𝒜\mathcal{A} require the overcategory 𝒜/a\mathcal{A}/a to be a weak test category. Thus for each a𝒜a \in \mathcal{A} and any category DD with a terminal object, we require that B(i 𝒜/a(i 𝒜/a *(D)))B(i_{\mathcal{A}/a}(i_{\mathcal{A}/a}^\ast(D))) be a weakly contractible simplicial set.

A strict test category is a test category 𝒜\mathcal{A} such that

  • i 𝒞:𝒞SetCati_{\mathcal{C}} : \mathcal{C}Set \to Cat preserves finite products up to weak equivalence,

or equivalently, such that

  • the induced functor i 𝒞*:Ho(𝒞Set)Ho(Cat)i_{\mathcal{C}*}:Ho(\mathcal{C}Set)\to Ho(Cat) preserves finite products.

Then one proceeds with 𝒜\mathcal{A}-sets.

If 𝒜\mathcal{A} is a test category and 𝒞\mathcal{C} any small category whose classifying space is contractible (which may or may not be a test category itself), then their cartesian product 𝒜×𝒞\mathcal{A}\times\mathcal{C} is a test category.

Properties

Homotopy 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).

Examples

Apart from the archeytpical example of the simplex category we have the following

References

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)

  • J. F. 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

That fact that the tree category is a test category was proved in

A short introduction can be found in

Last revised on February 9, 2021 at 04:02:32. See the history of this page for a list of all contributions to it.