nLab
test category

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 $\mathcal{C}$, 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 $X \colon \mathcal{C}^{op} \to Set$ 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 $\Delta^c\to X$. (This is another name for the category of elements of $X$.)

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

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

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

Two $\mathcal{C}$-sets $X$ and $Y$ are called weakly equivalent if there is a morphism $f \colon X\to Y$ inducing an equivalence $f_* \colon i_{\mathcal{C}} X\to i_{\mathcal{C}} Y$ of their cell categories, i.e., the induced map of nerves (“classifying spaces”) $B(i_{\mathcal{C}} X)\to B(i_{\mathcal{C}} Y)$ 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 $\mathcal{C}$-set $X$ is called aspherical if the category $i_{\mathcal{C}}(X)$ is weakly contractible, i.e. the nerve $B(i_{\mathcal{C}}(X))$ is a weakly contractible simplicial set. Note that if $\mathcal{C}$ is a weakly contractible category, then this is equivalent to the condition that the map $X \to 1$ to the terminal presheaf is a weak equivalence of $\mathcal{C}$-sets.

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

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 each $a \in \mathcal{A}$ and any category $D$ with a terminal object, we require that $B(i_{\mathcal{A}/a}(i_{\mathcal{A}/a}^\ast(D)))$ be a weakly contractible simplicial set.

A strict test category is a test category $\mathcal{A}$ 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, and later appeared as Ara, Cisinski, Moerdijk, 2019.

Related concepts

• direct category

• basic localizer

• test topos

References

The notion of test category was introduced in

• Alexandre Grothendieck, Pursuing Stacks, 1983 (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)

• 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

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

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

• Dimitri Ara, Denis-Charles Cisinski, Ieke Moerdijk, The dendroidal category is a test category, Mathematical Proceedings of the Cambridge Philosophical Society, 167(1), 107-121, 2019, (doi:10.1017/S030500411800021X, arXiv:1703.07098).

A short introduction can be found in

• Chris Kapulkin, Introduction to Test Categories PDF.