nLab
test category

Idea

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.

Definition

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

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

Let the cell category of $X$, denoted $i_{\mathcal{C}} X$, be the full subcategory of the overcategory $\mathcal{C}Set/X$ whose objects are the transformations of the form $\Delta^c\to X$. The correspondence $X\mapsto i_{\mathcal{C}}X$ extends to a functor $i_{\mathcal{C}}:\mathcal{C}Set\to Cat$ which has a right adjoint $i_{\mathcal{C}}^*:Cat\to\mathcal{C}Set$ whose object part is given by the formula

$$i_{\mathcal{C}}^*(D)(c):= Hom_{Cat}(\mathcal{C}/c,D).$$

Denote the counit of the adjunction $\epsilon : i_{\mathcal{C}}i_{\mathcal{C}}^*\to Id_{Cat}$.

Two $\mathcal{C}$-sets $X$ and $Y$ are weakly equivalent if there is a map $f:X\to Y$ inducing an equivalence $f_* : i_{\mathcal{C}} X\to i_{\mathcal{C}} Y$ of their cell categories, i.e., the induced map of nerves (“classifying spaces”) $B(i_{\mathcal{C}} X)\to B(i_{\mathcal{C}} Y)$ is a weak equivalence of simplicial sets. The functor $i_{\mathcal{C}}:\mathcal{C}Set\to Cat$ induces a functor $i_{\mathcal{C}*}:Ho(\mathcal{C}Set)\to Ho(Cat)$ of the homotopy categories.

A weak test category is a small category $\mathcal{C}$ such that, for any category $D$ in $Cat$, the component of the counit $\epsilon_D : i_{\mathcal{C}}i_{\mathcal{C}}^* D \to D$ is an equivalence of categories.

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

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

• ($\mathcal{A}$ is a “local test category”) for every object $a$ in $\mathcal{A}$ require the overcategory $\mathcal{A}/a$ to be a weak test category. Thus for any category $D$ in $Cat$, $\epsilon_D : i_{\mathcal{A}/a}i_{\mathcal{A}/a}^* D \to D$ is an equivalence of categories.

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

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

or equivalently, such that

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

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

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

• The cube category is a test category (Grothendieck, Cisinski), however not a strict one (Kan). (The corresponding model category is discussed at model structure on cubical sets.=)

  The category of cubes equipped with connection on a cubical set is even a strict test category (Maltsiniotis, 2008).

• The groupoid-analog $\tilde \Theta$ of the Theta category is a test category (Ara).

• The tree category $\Omega$ is a test category. This was proven in an unpublished note of Cisinski.

Related concepts

• direct category

• basic localizer

References

The notion of test category was introduced in

• Alexandre Grothendieck, Pursuing Stacks, djvu

Various conjectures made there are proven in

• Denis-Charles Cisinski, Les préfaisceaux comme modèles des types d’homotopie, Asterisque 308, pdf.

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

• Dimitri Ara, The groupoidal analogue $\tilde \Theta$ to Joyal’s category $\Theta$ is a test category (arXiv:1012.4319)

A short introduction can be found in

• Chris Kapulkin, Introduction to Test Categories PDF.