Contents

# 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 $\mathcal{C}$, 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 $\epsilon : i_{\mathcal{C}}i_{\mathcal{C}}^*\to Id_{Cat}$.

A morphism of $\mathcal{C}$-sets $f\colon X\to Y$ is a weak equivalence is the induced map $i_{\mathcal{C}}(f)\colon i_{\mathcal{C}} X\to i_{\mathcal{C}} Y$ 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. Two $\mathcal{C}$-sets $X$ and $Y$ are called weakly equivalent if there is a weak equivalence $f \colon X\to Y$. 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.

###### Definition

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

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

or equivalently, such that

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

One can show that a small category $\mathcal{C}$ is a weak test category if and only if both the unit $\epsilon : i_{\mathcal{C}}i_{\mathcal{C}}^*\to Id_{Cat}$ and the co-unit $\eta : Id_{\mathcal{C}Set/X}\to i^*_{\mathcal{C}}i_{\mathcal{C}}$ are weak equivalence; see Lemme 1.3.8 and Proposition 1.3.9 in (Maltsiniotis). In particular, one then has an equivalence of categories

$Ho(\mathcal{C}Set)\cong Ho(Cat)\, .$

An example of weak test category which is not a test category is the category $\Delta_+$ (so that a presheaf on $\Delta_+$ is a semi-simplicial set). This comes from a more general family of examples studied in (Cisinski, Maltsiniotis).

## Properties

### Homotopy category

The homotopy category of a category of presheaves over a (weak) 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 (weak) test category are models for homotopy types of ∞-groupoids. There is a relative analogue of this situation for local test categories (see below).

### 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). In fact the possibility of such model structures characterizes local test categories:

###### Theorem

A small category $\mathcal{A}$ is a local test category if and only if the category of presheaves of sets on $\mathcal{A}$ is equipped with a model category structure whose cofibrations are the monomorphisms with the weak equivalences defined as above. Such a model category structure is always proper. Furthermore, if $\mathcal{A}$ is a local test category, the assignment $X\mapsto (N(i_{\mathcal{A}}(X))\to N(\mathcal{A}))$ is a Quillen equivalence

$\mathcal{A}Set\to SSet/N(\mathcal{A})$

(where $SSet/\mathcal{C}$ is equipped with the sliced Kan-Quillen model structure), and thus induces equivalences of homotopy categories

$Ho(\mathcal{A}Set)\cong Ho(Cat/\mathcal{A})\cong Ho(SSet/N(\mathcal{A}))\, .$
###### Proof

See Théorème 1.4.3 and Corollaire 4.2.18 in (Cisinski) for the first and second assertion. The Quillen equivalence is a particular case of Proposition 4.4.28 in loc. cit. The properness of the model structure is a particular case of Théorème 4.4.30 in loc. cit.

###### Remark

If $\mathcal{A}$ is a local test category, one can describe the model structure on $\mathcal{A}$-sets independently from the functor $i_\mathcal{A}$ as follows: it is the minimal Cisinski model structure such that any morphism between representable presheaves is a weak equivalence and such that any presheaf on $\mathcal{A}$ is canonically an homotopy colimit of representable presheaves. This is Proposition 6.4.26 in (Cisinski).

## Examples

Apart from the archetypical 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 connections is even a strict test category (Maltsiniotis, 2008).

• The Theta category and its groupoid-analog $\tilde \Theta$ are strict test categories (Cisinski, Maltsiniotis and 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.

• The cycle category of Connes is a local test category. The corresponding model structure models homotopy types over $K(\mathbb{Z},2)$; see paragraph 8.5.14 and Proposition 8.5.19 in (Cisinski).

• Given a local test category $\mathcal{A}$ and a small category $\mathcal{C}$, the product $\mathcal{A}\times\mathcal{C}$ is a local test category. It is a test category if both $\mathcal{A}$ and $\mathcal{C}$ have weakly contractible nerves. This is a special case of a more general fact: if $\mathcal{A}$ is a local test category, for any a Grothendieck fibration $p:\mathcal{X}\to\mathcal{A}$ (with small fibers), the category $\mathcal{X}$ is a local test category. This follows from Corollaire 1.7.15, Exemple 3.2.2 and the dual version of Proposition 3.2.9 in (Maltsiniotis, 2005). In particular, given any presheaf of groups $G$ on a local test category $\mathcal{A}$, one gets a nice homotopy theory of representations of $G$; see Scholie 7.2.15 in Cisinski).

Any small category $\mathcal{A}$ which is closed under finite products and which contains an interval is a strict test category (where an interval is an object $I$ equipped with two morphisms from the terminal object $d^e:*\to I$, $e=0,1$, such that there is no maps from an object of $\mathcal{A}$ which factors through both $d^0$ and $d^1$); see Corollaire 1.5.7 in (Maltsiniotis). Particular cases include the following examples.

• The category of non-empty finite sets is a strict test category (hence symmetric simplicial sets are a model of homotopy types of CW-complexes).

• The full subcategory of the category of topological spaces, whose objects are closed balls in euclidian spaces, is a strict test category.

• The category of open balls in euclidian spaces, with $C^\infty$-maps as morphisms, is a strict test category.

• The category of contractible Stein manifolds, with holomorphic maps as morphisms, is a strict test category.

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

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

and the Theta category itself is discussed in

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

A short introduction can be found in