# A homotopical Dold-Kan correspondence for Joyal's category $\Theta$ and other test categories

Léo Hubert

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Léo Hubert. A homotopical Dold-Kan correspondence for Joyal's category $\Theta$ and other test categories.
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# A HOMOTOPICAL DOLD-KAN CORRESPONDENCE FOR JOYAL'S CATEGORY Θ AND OTHER TEST CATEGORIES

LÉO HUBERT

**ABSTRACT.** We prove that for any test category $A$, in the sense of Grothendieck, satisfying a compatibility condition between homology equivalences and weak equivalences of presheaves, the homotopy category of abelian presheaves on $A$ is equivalent to the non-negative derived category of abelian groups. This provides a homotopical generalization of the Dold-Kan correspondence for presheaves of abelian groups over a wide range of test categories. This equivalence of homotopy categories comes from a Quillen equivalence for a model structure on abelian presheaves that we introduce under these conditions. We then show that this result applies to Joyal's category $\Theta$.

## INTRODUCTION

In the first chapters of his 1983 manuscript *Pursuing Stacks* [15], Grothendieck introduced the notion of *test categories* to get an axiomatic characterization of small categories whose category of presheaves modelizes, in a canonical way, the category *Hot* of topological spaces up to weak homotopy equivalences. This theory has since been described and developed further, notably in [24], [11], [19], [12], [3] and [1].

However, Chapters V to VII of *Pursuing Stacks* have not met the same enthusiasm yet. In these chapters, Grothendieck introduced a formalism to study the homology theory of abelian presheaves on small categories, different from the one used in functor homology (see for example [26]). He also conjectured a homotopical generalization of the Dold-Kan correspondence for abelian presheaves on test categories. The present article aims to expose the basics of this formalism and to prove Grothendieck's conjecture, under additional conditions applying notably to Joyal's category $\Theta$ [20].

**A brief overview of test categories.** In this context, the category *Hot* is realized, using Illusie-Quillen's theorem [18], as the Gabriel-Zisman localization of the category *Cat* of small categories by the class $W_\infty$ of functors whose image by the nerve functor is a simplicial weak equivalence. Given any small category $A$, the functor

$$i_A : \hat{A} \to \mathrm{Cat}, \quad X \mapsto A/X$$

sending every presheaf to its category of elements allows to define a class of weak equivalences of presheaves denoted by $W_A$, consisting of the natural transformations in $\hat{A}$ whose image by the functor $i_A$ is in $W_\infty$. Grothendieck denotes by $\mathrm{Hot}_A$ the localization of $\hat{A}$ by $W_A$, and defines a *pseudo-test category* to be a category $A$ such that the functor $\mathrm{Hot}_A \to \mathrm{Hot}$ induced by $i_A$ is an equivalence of categories.

However, this notion is too general for Grothendieck's purpose of finding elegant characterizations, and he gradually restricts his scope. First, he defines *weak test categories* by asking, in addition, for the right adjoint $i_A^*$: $\mathrm{Cat} \to \hat{A}$ of $i_A$ to preserve weak equivalences. He then manages to find very simple characterizations of categories $A$ whose slices $A/a$ are weak test categories for every object $a$ of $A$, that he calls *local test categories*. Moreover, any local test category with trivial homotopy groups is also a weak test category: *test categories* are then defined to be categories which are both local and weak test categories. Finally, *strict test categories* are test categories $A$ such that the functor $i_A$ commutes, up to weak equivalences, with the product of presheaves.

Important examples of test categories include the category $\Delta$ of non-empty finite ordinals, the categorical category $\square$ and its variations with connections [25], and the dendroidal category $\Omega$ [3]. A particularly important example is given by the category $\Theta$ who, besides being a strict test category [12], is deeply linked to both strict $\omega$-categories ([22],[7]) and weak higher categories ([20],[5],[29],[2]).

<footnote n="2020">2020 Mathematics Subject Classification. 18F20, 18G31, 18G90, 18N40, 55U35.</footnote>

<footnote>Key words and phrases. abelian presheaves, test categories, homology, Dold-Kan correspondence.</footnote>
**Linearization of homotopy types.** The homology of a small category is traditionally defined as the homology of its simplicial nerve, and we can therefore define the homology of a presheaf on any small category as the homology of its category of elements. However, this process is rather complex, and gives a preeminent role to the category $\Delta$. In the context of test categories, Grothendieck asks in [15] a handful of questions regarding what he calls the linearization of homotopy types.

Denote by Hotab the non-negative derived category of abelian groups, and by $\hat{A}^{ab}$ the category of abelian presheaves on $A$.

(1) Given a small category $A$, can we define a canonical functor

$$H_A : \hat{A}^{ab} \to \text{Hotab}$$

so that the following diagram

<figure><img src="image_2.png" /></figure>

where $C_N$ is the normalized chain complex functor and $N$ is the simplicial nerve, is commutative up to a natural isomorphism?

We introduce in this paper such a functor, defined, following Grothendieck's ideas, as the left derived functor of the colimit functor.

This allows to define a class of weak equivalences of abelian presheaves on $A$ that we denote by $W_A^{ab}$, whose elements are the morphisms sent to isomorphisms by the functor $H_A$. Note that, in the case of simplicial abelian groups, one usually takes as weak equivalences morphisms whose underlying morphism of simplicial sets is a weak equivalence. A very important observation in our context is the fact that the functor $C_N : \hat{\Delta}^{ab} \to \text{Ch}_+(\text{Ab})$ sends these morphisms to quasi-isomorphisms.

(2) Let $U : \hat{A}^{ab} \to \hat{A}$ be the forgetful functor. Can we exhibit conditions on $A$ to ensure that the functor $H_A$ sends morphisms of $U^{-1}W_A$ to isomorphisms in Hotab?

We will focus on categories, such as $\Delta$, for which the classes $W_A^{ab}$ and $U^{-1}W_A$ coincide. We call this the strong Whitehead condition. We won't study in this paper categories satisfying only the inclusion $U^{-1}W_A \subset W_A^{ab}$.

Finally, the functor $C_N : \hat{\Delta}^{ab} \to \text{Ch}_+(\text{Ab})$ has another important property, expressed by the Dold-Kan theorem: it is an equivalence of categories ([14],[21]). In general, abelian presheaves on test categories are not equivalent to non-negative chain complexes of abelian groups. For example, the category of bisimplicial abelian groups is equivalent to the category of double chain complexes of the first quadrant. However, a reasonable generalization of the Dold-Kan correspondence would be an answer to the following question:

(3) If the answer to question (2) is positive, we get a functor

$$\overline{H_A} : \text{Hotab}_A^U \to \text{Hotab}$$

where $\text{Hotab}_A^U$ is the localization of $\hat{A}^{ab}$ by the class $U^{-1}W_A$. Is it true that if $A$ is a test category, then the functor $H_A$ is an equivalence of categories?

The present article shows that the answer to this last question is positive for categories satisfying the strong Whitehead condition.

**Theorem (5.14).** Let $A$ be a test category satisfying the strong Whitehead condition. Then the functor $\overline{H_A} : \text{Hotab}_A^U \to \text{Hotab}$ is an equivalence of categories.

A major ingredient of the proof comes from functors satisfying the condition of Quillen's Theorem A [27, Theorem A], that are called aspherical functors in the context of Grothendieck's homotopy theory. Given any aspherical functor $u : A \to B$, the induced restriction functor $u^* : \hat{B} \to \hat{A}$ is homotopy cofinal, meaning that it induces an isomorphism between homotopy colimits.

We show that if $u : A \to B$ is an aspherical functor and $A$ satisfies the strong Whitehead condition, then the same holds for $B$. We also study the class $W_\infty^{ab}$ of functors between small categories which induce isomorphisms in homology, and the corresponding notion of $W_\infty^{ab}$-aspherical
morphisms. We show that the restriction functor $(u^*)^{ab} : \hat{B}^{ab} \to \hat{A}^{ab}$ induced by a $W_\infty^{ab}$-aspherical functor commutes with the homology functor. Moreover, using the Grothendieck-Cisinski model structure on the category of presheaves over a local test category [11, Corollary 4.2.18], we prove the following theorem:

**Theorem.** Let $A$ be a local test category satisfying the strong Whitehead condition.

(5.9) The category $\hat{A}^{ab}$ can be given the structure of a cofibrantly generated model category whose weak equivalences are the elements of the class $W_A^{ab} = U^{-1}W_A$.

(5.11) If $u : A \to B$ is an aspherical functor and $B$ is also a local test category, then the functor $(u^*)^{ab} : \hat{B}^{ab} \to \hat{A}^{ab}$ is a left Quillen equivalence.

In particular, the equivalence of localized categories of Theorem 5.14 can be promoted to a Quillen equivalence with the projective model structure on non-negative chain complexes of abelian groups.

As an application of Theorem 5.14, we prove that Joyal’s category $\Theta$ satisfies the homotopical Dold-Kan correspondence. Our contribution to this proof consists mainly in the study of a variation on the notion of aspherical functors for functors whose codomain is a presheaf category. We prove that if $A$ satisfies the strong Whitehead condition, and if $j : B \to \hat{A}$ is an aspherical functor in this sense, then $B$ also satisfies the strong Whitehead condition. We then use previous work from Ara and Maltsiniotis [4] showing that the functor $\Delta \to \hat{\Theta}$ given by the cellular nerve ([6], [7]) and Street’s oriental functor [30] is an aspherical functor. This then implies that $\Theta$ satisfies the strong Whitehead condition, and therefore satisfies the conditions of Theorem 5.14.

We would like to point out that one can also try to characterize small categories $A$ for which the functor $H_A$ induces an equivalence of categories between the localization of the pair $(\hat{A}^{ab}, W_A^{ab})$ and Hotab, without requiring the condition of question (2). This is partly done in the author’s PhD thesis [17] from which this article is extracted, with still more investigations to be held. For example, the category $\Delta/\mathbb{N}(\mathbb{G}_r)$, where $\mathbb{G}_r$ is the reflexive globular category, is a test category that doesn’t satisfy the strong Whitehead condition. However, one can prove using [17, Proposition 4.3.6] that it satisfies this version of the homotopical Dold-Kan correspondence.

**Structure of the paper.** Section 1 consists of a brief recollection of definitions and results concerning test categories that we will use throughout this paper. In Section 2, we introduce the homology functor and show that, when restricted to presheaves of sets, it computes the homology of the category of elements. In Section 3, we expose Grothendieck’s definition of the tensor product of functors and use it to produce explicit computations of the homology functor. In Section 4, we introduce the class $W_\infty^{ab}$ of functors inducing isomorphisms in homology and we prove that the restriction functor induced by a $W_\infty^{ab}$-aspherical morphism commutes with homology. Section 5 introduces the strong Whitehead condition, and contains the two main theorems stated in the introduction. In Section 6, we focus on Joyal’s category $\Theta$ and show, using Ara and Maltsiniotis’s results [4], that $\Theta_n$, for any integer $n \ge 1$, and $\Theta$ are strong Whitehead categories, and thus verify the homotopical Dold-Kan correspondence.

**Acknowledgment.** I am grateful to Georges Maltsiniotis, whose project to edit *Pursuing Stacks* made the present work possible, and to Dimitri Ara for his guidance during my PhD, from which this article is extracted.

All definitions in Sections 1 to 3, except for the term *homologically pseudo-test*, are due to Grothendieck [15].

**Notation.** If $\mathcal{C}$ is a category, we denote by $\text{Ob}(\mathcal{C})$ its class of objects and by $\text{Arr}(\mathcal{C})$ its class of morphisms. Given another category $\mathcal{D}$, we denote by $\text{Hom}(\mathcal{C}, \mathcal{D})$ the category of functors $\mathcal{C} \to \mathcal{D}$ and natural transformations between them. We denote by $\text{Set}$ the category of sets, $\text{Ab}$ the category of abelian groups and $\text{Cat}$ the category of small categories.

If $A$ is a small category, $\hat{A}$ is the category $\text{Hom}(A^{\text{op}}, \text{Set})$ of set-valued presheaves on $A$. Given an object $a$ of $A$, we denote by $a$ the associated representable presheaf on $A$. Given a functor $u : A \to B$ between small categories, we denote by $u^* : \hat{B} \to \hat{A}$ the restriction functor, and by $u_!$ (resp. $u_*$) its left (resp. right) adjoint.

We denote by $e$ the final object of $\text{Cat}$, and by $e_{\hat{A}}$ the final object of $\hat{A}$ for any small category $A$.
# 1. PRELIMINARIES ON TEST CATEGORIES

We begin by a brief recollection of definitions and facts concerning the theory of test categories that will be used throughout this paper. We refer to [24] for a detailed exposition of this theory.

**1.1.** If $u : A \to B$ is a functor and $b$ is an object of $B$, we denote by $A/b$ the category whose objects are pairs $(a, f)$ where $a$ is an object of $A$ and $f : ua \to b$ is a morphism in $B$, and whose morphisms from an object $(a, f)$ to an object $(a', f')$ are morphisms $g : a \to a'$ such that $f'u(g) = f$. In particular, if $A$ is a small category, $u : A \to \tilde{A}$ is the Yoneda embedding and $X$ is a presheaf on $A$, the category $A/X$ is the category of elements of $X$. This defines a functor

$$i_A : \tilde{A} \to \mathrm{Cat}, \quad A \mapsto A/X$$

which admits as a right adjoint the functor

$$i_A^* : \mathrm{Cat} \to \tilde{A}, \quad C \mapsto \mathrm{Hom}(A/a, C).$$

**1.2.** A class $W \subset \mathrm{Arr}(\mathrm{Cat})$ is weakly saturated if it satisfies the following conditions:

(1) identities are in $W$;

(2) if two out of three arrows in a commutative triangle in $\mathrm{Cat}$ are in $W$, then so is the third;

(3) if $i : X' \to X$ and $r : X \to X'$ are two functors between small categories such that $ri = \mathrm{id}_X$, and $ir$ is in $W$, then $r$ is also in $W$;

**1.3.** A class $W \subset \mathrm{Arr}(\mathrm{Cat})$ is a basic localizer if it is weakly saturated and satisfies the following conditions (recall that $e$ is the final object of $\mathrm{Cat}$):

(4) if $A$ is a small category with a final object, then the morphism $A \to e$ is in $W$;

(5) if $u : A \to B$ is a functor in $\mathrm{Cat}$ such that for every object $b$ of $B$, the functor

$$u/b : A/b \to B/b, \quad (a, u(a) \to b) \mapsto (u(a), u(a) \to b)$$

is in $W$, then $u$ is in $W$.

**1.4.** Let $W$ be a basic localizer in $\mathrm{Cat}$. We say that a small category is $W$-aspherical if the unique morphism $A \to e$ is in $W$. We say that a functor $u : A \to B$ is $W$-aspherical if for every object $b$ of $B$, the morphism $u/b$ is in $W$. Finally, we say that a presheaf $X$ on $A$ is $W$-aspherical if $i_A(X)$ is a $W$-aspherical category. One can prove that any representable presheaf is $W$-aspherical, since for any object $a$ of $A$, the category $A/a$ has a final object.

**Proposition 1.5** (Grothendieck). Let $u : A \to B$ be a functor between small categories. The following conditions are equivalent:

(1) $u$ is $W$-aspherical;

(2) a presheaf $X$ on $B$ is $W$-aspherical if and only if $u^*(X)$ is $W$-aspherical;

(3) for every object $b$ of $B$, the presheaf $u^*(b)$ is $W$-aspherical;

(4) for every presheaf $X$ on $B$, the functor

$$u/X : A/X \to B/X, \quad (a, u(a) \to X) \mapsto (u(a), u(a) \to X)$$

is in $W$;

Moreover, these conditions imply the following condition:

(5) for any morphism $f : X \to Y$ in $\tilde{B}$, the functor $i_B(f)$ is in $W$ if and only if $i_A(u^*(f))$ is in $W$.

*Proof.* See [24, Proposition 1.2.9]. □

<figure><img src="image_3.png" /></figure>

**1.6.** We denote by $W_\infty$ the class of morphisms in $\mathrm{Cat}$ sent to simplicial weak equivalences by the simplicial nerve functor. Using [27, Theorem A], it is easy to prove that $W_\infty$ is a basic localizer. We call *Thomason equivalences*, or simply *weak equivalences of categories*, its elements.

**1.7.** We denote by **Hot** the localization of $\mathrm{Cat}$ by the class $W_\infty$. Recall ([18, VI, Theorem 3.3]) that the simplicial nerve functor induces an equivalence of categories between **Hot** and the homotopy category of simplicial sets.

In the rest of this paper, we will simply use the adjective *aspherical* to refer to $W_\infty$-asphericity.
**1.8.** Let $A$ be a small category. A morphism $f : X \to Y$ of $\hat{A}$ is a weak equivalence if its image by the functor $i_A$ is in $W_\infty$. We denote by $W_A$ the class of weak equivalences of presheaves on $A$, and by $\text{Hot}_A$ the localization of $\hat{A}$ by the class $W_A$.

The functor $i_A$ induces a functor $\text{Hot}_A \to \text{Hot}$, and we say that $A$ is a *pseudo-test category* if this functor is an equivalence of categories.

**1.9.** We say that $A$ is a *weak test category* if the following conditions are satisfied:

(1) $i_A^*(W_\infty) \subset W_A$;

(2) for every presheaf $X$, the unit morphism $\eta_X : X \to i_A^*i_A(X)$ is in $W_A$;

(3) for every small category $X$, the counit morphism $\varepsilon_C : i_A i_A^*(C) \to C$ is in $W_\infty$.

In particular, every weak test category is a pseudo-test category.

**1.10.** A small category $A$ is a *local test category* if for any object $a$ of $A$, the category $A/a$ is a weak test category. If $A$ is both a weak test category and a local test category, we say that $A$ is a *test category*.

**Theorem 1.11 (Grothendieck-Cisinski).** Let $A$ be a local test category. The category $\hat{A}$ can be given the structure of a model category where the weak equivalences are the elements of $W_A$ and the cofibrations are the monomorphisms.

*Proof.* See [11, Corollary 4.2.18]. □

<figure><img src="image_4.png" /></figure>

**Proposition 1.12 (Cisinski).** Let $u : A \to B$ be an aspherical functor between local test categories. Then $(u^*, u_*)$ is a Quillen equivalence for the Grothendieck-Cisinski model structure on $\hat{A}$ and $\hat{B}$.

*Proof.* See [11, Proposition 6.4.29]. □

<figure><img src="image_5.png" /></figure>

**1.13.** Let $A$ be a small category. A presheaf $X$ on $A$ is locally $W$-aspherical if the canonical functor $i_A(X) \to A$ is $W$-aspherical. We say that $A$ is totally aspherical if it is aspherical, and if for any presheaves $X$ and $Y$ on $A$, the canonical morphism $i_A(X \times Y) \to i_A(X) \times i_A(Y)$ is aspherical.

**Proposition 1.14 (Grothendieck).** Let $A$ be a small category. The following conditions are equivalent:

(1) $A$ is totally aspherical;

(2) the diagonal functor $A \to A \times A$ is aspherical;

(3) every aspherical presheaf on $A$ is locally aspherical.

*Proof.* See [24, Proposition 1.6.1]. □

<figure><img src="image_6.png" /></figure>

**Proposition 1.15 (Grothendieck).** If $A$ is a totally aspherical category and $u : A \to B$ is an aspherical functor, then $B$ is also totally aspherical.

*Proof.* See [24, Proposition 1.6.5]. □

<figure><img src="image_7.png" /></figure>

**1.16.** A *strict test category* is a test category which is also totally aspherical.

**Proposition 1.17 (Grothendieck).** Let $A$ be a totally aspherical category. The following conditions are equivalent:

(1) $A$ is a weak test category;

(2) $A$ is a local test category;

(3) $A$ is a test category;

(4) $A$ is a strict test category.

*Proof.* See [24, Proposition 1.6.6]. □

<figure><img src="image_8.png" /></figure>

**Proposition 1.18 (Grothendieck).** Let $u : A \to B$ be a functor. If $A$ is totally aspherical and if $B$ is a local test category, then $A$ and $B$ are both strict test categories.

*Proof.* See [24, Corollary 1.6.11]. □

<figure><img src="image_9.png" /></figure>
**1.19.** Let $A$ be a small category and $i: A \to Cat$ a functor. Consider the functor

$$i^*: Cat \to \hat{A}, \quad C \mapsto \operatorname{Hom}_{Cat}(i(a), C).$$

Observe that if $i$ is the inclusion functor from the simplex category $\Delta$ to $Cat$, then $i^*$ is the simplicial nerve functor. We say that $i$ is an *aspherical functor* if it satisfies the following conditions:

(1) for every object $a$ of $A$, the category $i(a)$ is aspherical;

(2) a small category $C$ is aspherical if and only if the presheaf $i^*(C)$ is aspherical.

**Proposition 1.20 (Grothendieck).** Let $A$ be a small category. Then $A$ is a weak test category if and only if $i_A$ is an aspherical functor.

*Proof.* See [24, Remark 1.7.2]. □

**Proposition 1.21 (Grothendieck).** Let $u: A \to B$ be a functor between small categories, and $j: B \to Cat$ a functor such that for every object $b$ of $B$, the category $j(b)$ is aspherical.

(1) if $u$ is aspherical, then $ju: A \to Cat$ is aspherical if and only if $j$ is aspherical;

(2) if $j$ is fully faithful and $ju: A \to Cat$ is an aspherical functor, then $u$ and $j$ are aspherical.

*Proof.* See [15, Lemma 1.7.4]. □

**1.22.** Let $A$ be a small category. We say that a functor $i: A \to Cat$ is a *weak test functor* if $A$ is a weak test category and if $i$ is aspherical. We say that $i$ is a *local test functor* if for every object $a$ of $A$, the functor $A/a \to Cat$ induced by $i$ is a weak test functor. Finally, $i$ is a *test functor* if it is both a weak test functor and a local test functor.

**Theorem 1.23 (Grothendieck).** Let $A$ be a small category and $i: A \to Cat$ a functor such that for every object $a$ of $A$, the category $i(a)$ is aspherical. The following conditions are equivalent:

(1) $i$ is a local test functor (and $A$ is a local test category);

(2) for every small aspherical category $C$, the presheaf $i^*(C)$ is locally aspherical.

*Proof.* See [24, Theorem 1.7.13]. □

## 2. HOMOLOGY, HOMOTOPY COLIMITS AND THE CATEGORY OF ELEMENTS

In the context of test categories, the homology of a presheaf is the homology of its category of elements. We will now introduce, following Grothendieck's ideas in [15], a homology functor for *abelian* presheaves that generalizes this idea. We show that it also generalizes the homology of groups and of simplicial abelian groups.

**2.1.** If $A$ is a small category, we denote by $\hat{A}^{ab} = \operatorname{Hom}(A^{op}, Ab)$ the category of abelian presheaves on $A$. Given a presheaf $X$ on $A$, we denote by $\mathbb{Z}_A^{(X)}$ the abelian presheaf on $A$ sending every object $a$ of $A$ to the free abelian group on the set $Xa$. Following Grothendieck's terminology [15], we will call the *Whitehead functor* the functor

$$\mathrm{Wh}_A: \hat{A} \to \hat{A}^{ab}, \quad X \mapsto \mathbb{Z}_A^{(X)},$$

or simply the *abelianization functor*. If $a$ is an object of $A$, we will simply write $\mathbb{Z}_A^{(a)}$ for the image of the representable presheaf $a$ by the functor $\mathrm{Wh}_A$, and we will refer to it as the abelian presheaf represented by $a$.

**2.2.** We denote by **Hotab** the derived category $\mathcal{D}_+(Ab)$, obtained by localizing non-negative chain complexes of abelian groups by quasi-isomorphisms. We define a functor

$$\mathrm{H}_A: \hat{A}^{ab} \to \mathrm{Hotab}, \quad X \mapsto \mathrm{H}(A, X)$$

where $\mathrm{H}(A, X)$ is the homotopy colimit of the diagram

$$A^{op} \xrightarrow{X} Ab \xrightarrow{in_0} Ch_+(Ab),$$

$in_0$ being the inclusion of abelian groups into chain complexes concentrated in degree 0. We say that $\mathrm{H}(A, X)$ is the *homology of A with coefficients in X*, or simply the *homology of X*.
that is commutative up to a natural isomorphism.

**Definition 2.13.** A small category $A$ is a homologically pseudo-test category if the functor $\overline{H_A}$ is an equivalence of categories.

*Remark 2.14.* Inspired by the terminology of test categories, the prefix «pseudo» underlines the fact that we do not ask for any conditions on the quasi-inverse of the functor $\overline{H_A}$. We introduce in [17] a notion of homologically test categories and homologically test functors that we won't mention in this paper.

*Example 2.15.* The prime example of a homologically pseudo-test category is given by the simplex category $\Delta$. Recall that the Dold-Kan correspondence ([21, Theorem 8.1], [14, Theorem 1.9]) states that the normalized chain complex functor $C_N : \hat{\Delta}^{ab} \to Ch_+(Ab)$ is an equivalence of categories. Moreover, as explained in Example 2.4, this functor computes the homology functor $H_\Delta$, which implies that the induced functor between the localized categories is also an equivalence of categories.

*Example 2.16.* We will prove in this paper that the categories $\Theta_n$ for every integer $n > 0$, and $\Theta$ are homologically pseudo-test categories. Using techniques that we do not need to introduce in this paper, we provide in [17, Chapter 5] numerous other examples, including the subcategory $\Delta'$ of monomorphisms of $\Delta$ and the cubical category $\square$ without connections. We also show in [17], using materials exposed here, that the category $\Delta^n$ for every integer $n > 0$, the cubical category with connections and the reflexive globular category $\mathbb{G}_r$ are homologically pseudo-test categories.

### 3. INTEGRATORS

Grothendieck describes in [15] explicit methods to compute the homology functor without the need of simplicial techniques and the Bousfield-Kan formula [9], that we expose in this section. While we wont do an extensive study of integrators in this article (this has been done in [17]), we cannot solely rely on the Bousfield-Kan formula to study $\mathcal{W}_\infty^{ab}$-asphericity in Section 4.

**3.1.** If $A$ is a small category, its *additive envelope*, denoted $Add(A)$, is the small category defined by the following universal property: for every additive category $\mathcal{M}$ (that is, an Ab-enriched category with finite biproducts and a zero object), there is a natural isomorphism

$$ \underline{\mathrm{Hom}}(A, \mathcal{M}) \simeq \underline{\mathrm{Hom}}_{\mathrm{add}}(\mathrm{Add}(A), \mathcal{M}) $$

where $\underline{\mathrm{Hom}}_{\mathrm{add}}$ denotes the subcategory of functors commuting to finite sums. For any functor $F : A \to \mathcal{M}$, we denote by $\mathrm{Add}(F)$ the functor making the following diagram commutative:

<figure><img src="image_10.png" /></figure>

**3.2.** For any small category $A$, the additive envelope $\mathrm{Add}(A)$ can be realized as the full subcategory of abelian presheaves on $A$ of the form

$$ \bigoplus_{i \in I} \mathbb{Z}_A^{(a_i)} : a \mapsto \bigoplus_{i \in I} \mathbb{Z}^{(\mathrm{Hom}_A(a, a_i))} $$

where $I$ is a *finite set* and $a_i$ is an object of $A$ for every $i$ in $I$. Given any additive category $\mathcal{M}$ and any functor $F : A \to \mathcal{M}$, the value of $\mathrm{Add}(F)$ on an object $\bigoplus_{i \in I} \mathbb{Z}_A^{(a_i)}$ is then given by the formula

$$ \mathrm{Add}(F) \left( \bigoplus_{i \in I} \mathbb{Z}_A^{(a_i)} \right) = \bigoplus_{i \in I} F(a_i) . $$

**Proposition 3.3.** For any small category $A$, $\mathrm{Add}(A)$ is a dense subcategory of $\hat{A}^{ab}$.

*Proof.* We need to show that for any abelian presheaf $X$ on $A$, the canonical morphism

$$ \lim_{L \to \mathrm{Add}(A)/X} \hat{A}^{ab} \to L $$
is an isomorphism. The proof is similar to the proof of the density of representable presheaves in the Set-valued case, with an additional use of the diagonal morphisms $(\mathbb{Z}_A^{(a)}, X) \to (\mathbb{Z}_A^{(a)} \oplus \mathbb{Z}_A^{(a)}, X)$ of Add$(A)/X$ to check that cocones with vertex $Y$ correspond to morphisms of abelian presheaves from $X$ to $Y$. We leave the details to the reader, which can also be found in [17, Proposition 2.2.4]. $\square$

**3.4.** Given a small category $A$, a cocomplete additive category $\mathcal{M}$ and a functor $F : A \to \mathcal{M}$, we get, using Proposition 3.3, a functor $F_I^{ab} : \hat{\mathcal{A}}^{ab} \to \mathcal{M}$ by setting, for any abelian presheaf $X$ on $A$,

$$F_I^{ab}(X) = \lim_{L \in \mathrm{Add}(A)/X} \mathrm{Add}(F)(L).$$

**Proposition 3.5.** Let $\mathcal{M}$ be a cocomplete additive category and $A$ a small category. For any functor $F : A \to \mathcal{M}$, the functor

$$F_I^{ab} : \hat{\mathcal{A}}^{ab} \to \mathcal{M}$$

is left adjoint to the functor

$$(F^*)^{ab} : \mathcal{M} \to \hat{\mathcal{A}}^{ab}$$

defined for every object $x$ of $\mathcal{M}$ by

$$(F^*)^{ab}(x) : A^{\mathrm{op}} \to \mathcal{M} \\ a \mapsto \mathrm{Hom}_M(F(a), x).$$

*Proof.* For any object $x$ in $\mathcal{M}$ and any abelian presheaf $X$ on $A$, we have the following chain of natural isomorphisms:

$$
\begin{align*}
\operatorname{Hom}_M(F_I^{ab} X, x) &\approx \operatorname{Hom}_M\left(\lim_{L \in \operatorname{Add}(A)/X}^M \operatorname{Add}(F)(L), x\right) \\
&\approx \lim_{L \in \operatorname{Add}(A)/X}^{\operatorname{Ab}} \operatorname{Hom}_M\left(\bigoplus_i F(a_i), x\right) \\
&\approx \lim_{L \in \operatorname{Add}(A)/X}^{\operatorname{Ab}} \prod_i (F^*)^{ab}(x)(a_i) \\
&\approx \lim_{L \in \operatorname{Add}(A)/X}^{\operatorname{Ab}} \prod_i \operatorname{Hom}_{\hat{\mathcal{A}}^{ab}}(\mathbb{Z}_A^{(a_i)}, (F^*)^{ab}(x)) \\
&\approx \operatorname{Hom}_{\hat{\mathcal{A}}^{ab}}\left(\lim_{L \in \operatorname{Add}(A)/X}^{\hat{\mathcal{A}}^{ab}} L, (F^*)^{ab}(x)\right) \\
&\approx \operatorname{Hom}_{\hat{\mathcal{A}}^{ab}}(X, (F^*)^{ab}(x)). \quad \square
\end{align*}
$$

**Corollary 3.6.** Let $\mathcal{M}$ be a cocomplete additive category and $A$ a small category. For any functor $F : A \to \mathcal{M}$, the functor $F_I^{ab} : \hat{\mathcal{A}}^{ab} \to \mathcal{M}$ is cocontinuous.

**Proposition 3.7.** The restriction functor

$$\underline{\operatorname{Hom}}_I(\hat{\mathcal{A}}^{ab}, \mathcal{M}) \to \underline{\operatorname{Hom}}(A, \mathcal{M}), \quad F \mapsto F \circ \operatorname{Wh}_A$$

is an equivalence of categories, and a quasi-inverse is given by the functor

$$\underline{\operatorname{Hom}}(A, \mathcal{M}) \to \underline{\operatorname{Hom}}_I(\hat{\mathcal{A}}^{ab}, \mathcal{M}), \quad F \mapsto F_I^{ab},$$

where $\underline{\operatorname{Hom}}_I$ denotes the subcategory of cocontinuous functors.

*Proof.* By construction, if $a$ is an object of $A$, we have a natural isomorphism $F_I^{ab}(\mathbb{Z}^{(a)}) \simeq F(a)$. Moreover, if $F : \hat{\mathcal{A}}^{ab} \to \mathcal{M}$ is a cocontinuous functor, we can check that $F$ and $(F \circ \operatorname{Wh}_A)_I^{ab}$ agree by checking on representables, which is straightforward. $\square$

**3.8.** Let $A$ be a small category. Denote by

$$- \odot_A - : \hat{\mathcal{A}}^{ab} \times \widehat{\mathcal{A}}^{\mathrm{op}}^{ab} \to \mathrm{Ab}$$
the bifunctor defined by the commutative diagram

<figure><img src="image_11.png" /></figure>

where we used the fact that $\widehat{A}^{\text{op}} = \text{Hom}(A, \text{Ab})$. In other words, for every abelian presheaf $X$ on $A$ and every abelian presheaf $Y$ on $A^{\text{op}}$, we have the following identity:

$$X \odot_A Y = Y_I^{\text{ab}}(X) .$$

**Proposition 3.9.** Let $A$ be an object of a small category $A$ and $Y$ an abelian presheaf on $A^{\text{op}}$. There is a canonical isomorphism

$$\mathbb{Z}_A^{(a)} \odot_A Y \simeq Y(a)$$

in Ab.

*Proof.* This is already contained in Proposition 3.7. □

*Remark 3.10.* The latter proposition, together with cocontinuity, is enough to see that the bifunctor $- \odot_A -$ coincide with the tensor product of functors: for every abelian presheaf $X$ on $A$ and every presheaf $Y$ on $A^{\text{op}}$, we have a natural isomorphism of abelian groups

$$X \odot_A Y \simeq \int^{a \in A} X_a \otimes Y_a .$$

While we could have used this description in this article, we chose to expose the formalism introduced by Grothendieck in [15]. We will still, however, refer to this bifunctor as the tensor product of functors.

**Proposition 3.11.** Let $X$ be an abelian presheaf on $A$ and $Y$ an abelian presheaf on $A^{\text{op}}$. There is a natural isomorphism of abelian groups

$$X \odot_A Y \simeq Y \odot_{A^{\text{op}}} X .$$

*Proof.* The functor $- \odot_A -$ is cocontinuous on the first variable by Corollary 3.6. Furthermore, for any abelian presheaf $X$ on $A$, the functor

$$X \odot_A - : \widehat{A}^{\text{op}} \to \text{Ab}$$

is the composite of the equivalence of categories given by Proposition 3.7 and the evaluation functor, both of which are also cocontinuous. The situation is identical for the functor $- \odot_{A^{\text{op}}} -$, therefore we only have to check the isomorphism with representables on both arguments. For any object $a$ in $A$ and any object $a'$ in $A^{\text{op}}$, we have

$$\mathbb{Z}_A^{(a)} \odot_A \mathbb{Z}_{A^{\text{op}}}^{(a')} \simeq \mathbb{Z}(\text{Hom}_{A^{\text{op}}}(a, a'))$$

and

$$\mathbb{Z}_{A^{\text{op}}}^{(a')} \odot_{A^{\text{op}}} \mathbb{Z}_A^{(a)} \simeq \mathbb{Z}(\text{Hom}_A(a', a))$$

which concludes the proof. □

In the rest of this section, we will show how the tensor product of functors relates to the homology functor we introduced in Section 2.

**3.12.** Recall that for any category $A$, the category $\widehat{A}^{\text{ab}}$ is an abelian category with enough projective objects. To describe projective objects, we use the *infinite additive envelope*, denoted $\text{Addinf}(A)$, defined by the following universal property: for every additive category $\mathcal{M}$, there is a natural isomorphism

$$\text{Hom}(A, \mathcal{M}) \simeq \text{Hom}_{\text{addinf}}(\text{Addinf}(A), \mathcal{M})$$

where $\text{Hom}_{\text{addinf}}$ denotes the subcategory of functors commuting to arbitrary sums. It can be realized as the full subcategory of $\widehat{A}^{\text{ab}}$ whose objects are arbitrary sums of representables.

**Proposition 3.13.** Let $A$ be a small category.
(1) Every object of Addinf(A) is a projective object in $\hat{A}^{ab}$.

(2) Every object in $\hat{A}^{ab}$ admits a projective resolution by objects of Addinf(A).

(3) Projective objects in $\hat{A}^{ab}$ are retracts of objects of Addinf A.

**Proof.** The proof of (1) is straightforward since coproducts of projectives are projectives. For (2), given an abelian presheaf $X$ on $A$, we set

$$L_X = \bigoplus_{a \in \text{Ob}(A)} \bigoplus_{x \in X_a} \mathbb{Z}_A^{(a)}$$

and we define an epimorphism $\varphi : L_X \to X$ on each summand $(a, x)$ by defining $\varphi_{(a,x)}$ to be the element of $X_a$ represented by the morphism $\mathbb{Z}_A^{(a)} \xrightarrow{x} X$. The proof of the last point then follows from the fact that if $X$ is a projective object in $\hat{A}^{ab}$, the morphism $\varphi : L_X \to X$ admits a section, exhibiting $X$ as a retract of $L_X$. $\square$

**Proposition 3.14.** Let $X$ be an abelian presheaf on $A$ and $Y$ an abelian presheaf on $A^{\text{op}}$. The functors

$$X \odot_A - : \widehat{A^{\text{op}}}^{ab} \to \text{Ab} \\ - \odot_A Y : \hat{A}^{ab} \to \text{Ab}$$

both admit left derived functors

$$L(X \odot_A -) : \mathcal{D}_+(\widehat{A^{\text{op}}}^{ab}) \to \text{Hotab}, \\ L(- \odot_A Y) : \mathcal{D}_+(\hat{A}^{ab}) \to \text{Hotab},$$

and there is a natural isomorphism

$$L(X \odot_A -)(Y) \simeq L(- \odot_A Y)(X)$$

in Hotab.

**Proof.** Consider two projective resolutions $M_\bullet \xrightarrow{\epsilon} X$ and $L_\bullet \xrightarrow{\eta} Y$ by objects of Addinf(A) and Addinf(A$^{\text{op}}$) respectively. We can then form the chain complexes of abelian groups $M_\bullet \odot_A Y$, $X \odot_A L_\bullet$, as well as the double complex $M_\bullet \odot_A L_\bullet$ obtained by applying the bifunctor $\odot_A$ term by term. Considering the complexes $M_\bullet \odot_A Y$ and $X \odot_A L_\bullet$ as double complexes concentrated in the first column or in the first row, the resolutions $\epsilon$ and $\eta$ provide morphisms of double complexes which in turn induce morphisms of complexes

$$\mathrm{Tot}(M_\bullet \odot_A Y) \xleftarrow{M_\bullet \odot_A \eta} \mathrm{Tot}(M_\bullet \odot_A L_\bullet) \xrightarrow{\epsilon \odot_A L_\bullet} \mathrm{Tot}(X \odot_A L_\bullet).$$

We will use a classical argument from homological algebra (see for example [32, 2.7]) to show that these morphisms are quasi-isomorphisms. For this, we only need to show that the functors

$$- \odot_A L_j, \quad M_i \odot_A -, \quad i, j \in \mathbb{N}$$

are exact. By symmetry (Proposition 3.11), it is enough to show that $N \odot_A -$ is exact for every object $N = \bigoplus_{i \in I} \mathbb{Z}_A^{(a_i)}$ in Addinf(A). Since for every abelian presheaf $Y$ on $B$ we have

$$(\bigoplus_{i \in I} \mathbb{Z}_A^{(a_i)}) \odot_A Y = \bigoplus_{i \in I} Y(a_i),$$

and since the evaluation and direct sum functors are exact, we can conclude. $\square$

**Lemma 3.15.** Let $A$ be a small category. If $X$ is an abelian presheaf on $A$, there is a natural isomorphism of abelian groups

$$X \odot_A \mathbb{Z}_{A^{\text{op}}} \simeq \lim_{\longrightarrow} X.$$
**Proof.** Given an abelian presheaf $X$ on $A$, we have the following chain of natural isomorphisms in Ab:

$$
\begin{align*}
X \odot_A \mathbb{Z}_{A^{\text{op}}} &\approx X \odot_A \mathbb{Z}_{A^{\text{op}}} \stackrel{(e_{A^{\text{op}}})}{\rightleftharpoons} \\
&\approx X \odot_A \mathbb{Z}_{A^{\text{op}}} \stackrel{\left(\lim_{\underset{a \in A^{\text{op}}} \to^{A^{\text{op}}} a}\right)}{\rightleftharpoons} \\
&\approx X \odot_A \lim_{\underset{a \in A^{\text{op}}} \to^{A^{\text{op}}} a} \mathbb{Z}_{A^{\text{op}}}^{(a)} \\
&\approx \lim_{\underset{a \in A^{\text{op}}} \to^{A^{\text{op}}} a} \mathbb{Z}_{A^{\text{op}}}^{(a)} \\
&\approx \lim_{\underset{a \in A^{\text{op}}} \to^{A^{\text{op}}} a} X(a) . \quad \square
\end{align*}
$$

**Corollary 3.16.** Let $A$ be a small category, and let $L_\bullet$ be a projective resolution of $\mathbb{Z}_{A^{\text{op}}}$ in the category $\widehat{A^{\text{op}}}^{ab}$. For any abelian presheaf $X$ on $A$, there is a canonical natural isomorphism

$$
\mathrm{H}(A, X) \simeq X \odot_A L_{\bullet}.
$$

in Hotab.

*Proof.* Since we defined the functor $H_A : \widehat{A}^{ab} \to \text{Hotab}$ to be the left derived functor of the colimit functor (2.2), there is nothing else to prove. $\square$

**Definition 3.17.** An integrator on $A$ is a projective resolution

$$
L = \left( L_0 \leftarrow^{d_0} L_1 \leftarrow^{d_1} L_2 \leftarrow^{d_2} \dots \right) , \quad L \xrightarrow{\varepsilon} \mathbb{Z}_{A^{\text{op}}}
$$

of the constant abelian presheaf $\mathbb{Z}_{A^{\text{op}}}$ of value $\mathbb{Z}$ on $A^{\text{op}}$. This means that each $L_n$ is a projective presheaf on $A^{\text{op}}$, and that $\varepsilon_0 : L_0 \to \mathbb{Z}$ is the cokernel of $d_0$.

**3.18.** We say that $L$ is a free integrator if it is an integrator whose components $L_n$ are in $\text{Addinf}(A^{\text{op}})$ for every integer $n \ge 0$. This allows for very simple computations thanks to the following proposition.

**Proposition 3.19.** If $X$ is an abelian presheaf on $A$ and if $L = \bigoplus_{i \in I} \mathbb{Z}_{A^{\text{op}}}^{(a_i)}$ is an object of $\text{Addinf}(A^{\text{op}})$, there is a natural isomorphism

$$
X \odot_A L \approx \bigoplus_{i \in I} X(a_i)
$$

in Ab.

*Proof.* We can use the symmetry proved in proposition 3.11 together with proposition 3.9. $\square$

*Remark 3.20.* Since $\mathrm{Ch}_+(\widehat{A^{\text{op}}}^{ab}) = \mathrm{Hom}(A, \mathrm{Ch}_+(\mathrm{Ab}))$, an integrator $L$ on a small category $A$ can be seen either as a complex in $\mathrm{Add}(A^{\text{op}})$, or as a functor $A \to \mathrm{Ch}_+(\mathrm{Ab})$. We will use interchangeably the two points of view.

*Example 3.21.* Taking for $A$ the category $\Delta$, let

$$
L_{\Delta} = (\mathbb{Z}_{\Delta^{\text{op}}}^{(\Delta_0)} \leftarrow \mathbb{Z}_{\Delta^{\text{op}}}^{(\Delta_1)} \leftarrow \mathbb{Z}_{\Delta^{\text{op}}}^{(\Delta_2)} \leftarrow \dots)
$$

be the complex in $\mathrm{Add}(\Delta^{\text{op}})$ whose differential is given by the alternate sum of cofaces. For every integer $n \ge 0$, $\mathbb{Z}_{\Delta^{\text{op}}}^{(\Delta_n)}$ is the abelian presheaf on $\Delta^{\text{op}}$

$$
\mathbb{Z}_{\Delta^{\text{op}}}^{(\Delta_n)} : \Delta_k \mapsto \mathbb{Z}^{(\text{Hom}_{\Delta}(\Delta_n, \Delta_k))},
$$

hence the direction of the arrows. Given any abelian simplicial group $X$, the complex $X \odot (L_\Delta)_\bullet$ is, by Proposition 3.19, the non-normalized chain complex associated to $X$

$$
X \odot (L_{\Delta})_{\bullet} \approx (X_0 \leftarrow X_1 \leftarrow X_2 \leftarrow \dots)
$$

with differential given by the alternative sum of faces. We claim that the complex $L_\Delta$ is a free integrator on $\Delta$. Proving this amounts to compute the homology of representables, which is a classical exercise. Details can be found for example in [17, Example 2.5.7]. Thus, we get another proof of the fact that the non-normalized chain complex computes homology, as defined
in paragraph 2.2. Since the normalized and non-normalized chain complex functors are pointwise quasi-isomorphic, this also gives another proof of the fact that $C_N : \hat{\Delta}^{ab} \to Ch_+(Ab)$ computes the homology functor.

**3.22.** Any small category $A$ can be equipped with a free integrator that we call the *Bousfield-Kan integrator*, defined as the composite

$$ \ell_A : A \hookrightarrow \hat{A} \xrightarrow{i_A} \mathrm{Cat} \xrightarrow{N} \hat{\Delta} \xrightarrow{\mathrm{Wh}_{\Delta}} \hat{\Delta}^{ab} \xrightarrow{L_{\Delta!}^{ab}} \mathrm{Ch}_+(Ab) . $$

Indeed, for any integer $n \ge 0$, we have

$$ (\ell_A)_n = \bigoplus_{\Delta_n \xrightarrow{u} A} \mathbb{Z}_{A^{\mathrm{op}}}^{(\varphi n)} : a \mapsto \bigoplus_{\Delta_n \xrightarrow{u} A} \mathbb{Z}^{(\mathrm{Hom}_A(\varphi n, a))} $$

which shows that $\ell_A$ is a complex with components in $\mathrm{Add}(A^{\mathrm{op}})$. Moreover, given any object $a$ of $A$, the category $A/a$ is aspherical, which means that the functor $A/a \to e$ is in $W_\infty$. Applying the nerve and the functor $L_{\Delta!}^{ab}$ then gives a natural quasi-isomorphism $\varepsilon : \ell_A(a) \to L_{\Delta!}^{ab} \mathbb{Z}_{\Delta^{\mathrm{op}}}^{(N(e))}$. The codomain of this morphism is the complex of abelian groups

$$ L_{\Delta!}^{ab} \mathbb{Z}_{\Delta^{\mathrm{op}}}^{(N(e))} = (\mathbb{Z} \stackrel{0}{\leftarrow} \mathbb{Z} \stackrel{\mathrm{id}}{\leftarrow} \mathbb{Z} \stackrel{0}{\leftarrow} \mathbb{Z} \stackrel{\mathrm{id}}{\leftarrow} \dots) , $$

which we equip with the quasi-isomorphism to $\mathbb{Z}[0]$ given by the identity in degree 0. Composing these morphisms, we get a natural quasi-isomorphism $\ell_A(a) \to \mathbb{Z}[0]$ which is given by the augmentation morphism $\ell_A(a)_0 \to \mathbb{Z}$. This proves that $\ell_A$ is a free integrator on $A$.

Given any abelian presheaf $X$ on $A$, the complex $(\ell_A)_!^{ab}(X)$ is then the non-normalized complex of the simplicial replacement [9] of $X$:

$$ (\ell_A)_!^{ab}(X) \simeq (\dots \leftarrow \bigoplus_{\Delta_n \xrightarrow{u} A} Xu(n) \leftarrow \bigoplus_{\Delta_{n+1} \xrightarrow{u} A} Xu(n+1) \leftarrow \dots) $$

whose differential is given on generators by

$$ d\langle\Delta_{n+1} \xrightarrow{u} A, x\rangle = \sum_{0 \le i \le n+1} (-1)^i \langle\Delta_n \xrightarrow{\delta_i} \Delta_{n+1} \xrightarrow{u} A, d_i(x)\rangle $$

where we denoted by $\delta_i$ the $i$-th coface maps and by $d_i$ its image by $X$. Using this integrator amounts to use the classical Bousfield-Kan formula given in [9, XII, 5.5].

A detailed study of integrators is not needed for the purpose of this paper, since we will mainly (but not exclusively) use the Bousfield-Kan integrator. More details are available in [17] where we give, for example, integrators on the cubical category $\square$ and the globular category $\mathcal{G}$. We also use this notion to prove several homotopical Dold-Kan correspondences for categories that do not satisfy the strong Whitehead condition mentioned in the introduction.

### 4. ASPHERICAL MORPHISMS FOR HOMOLOGY

It is fairly easy to see from Proposition 1.5 that if $u : A \to B$ is an aspherical functor, the restriction functor $u^* : \hat{B} \to \hat{A}$ induces an isomorphism in homology for presheaves of sets. In this section, we show that it is enough to ask for $u$ to be aspherical with respect to the class of homology isomorphisms in $\mathrm{Cat}$, and that the restriction functor also induces an isomorphism in homology for arbitrary abelian presheaves.

**4.1.** We denote by $W_\infty^{ab}$ the class of functors in $\mathrm{Cat}$ inducing an isomorphism in homology. More precisely, a functor $u : A \to B$ between small categories is an element of $W_\infty^{ab}$ if and only if the induced morphism

$$ H(A, \mathbb{Z}) \xrightarrow{H(u, \mathbb{Z})} H(B, \mathbb{Z}) $$

is an isomorphism in $\mathrm{Hotab}$. Using the standard way of computing homology of categories with the simplicial nerve functor, we get the following equivalence:

$$ u \in W_\infty^{ab} \iff C_N \mathbb{Z}_\Delta^{(N(u))} \text{ is a quasi-isomorphism.} $$

**Proposition 4.2.** The class $W_\infty^{ab}$ is a basic localizer. Moreover, we have the inclusion $W_\infty \subset W_\infty^{ab}$, and every aspherical functor is $W_\infty^{ab}$-aspherical.
Proof. This follows (see [23, Corollary 1.5.14]) from the fact that $W_{\infty}^{ab}$ is the class of equivalences for the derivator induced by the localizer $(Ch_+(Ab), W_{qis})$ where $W_{qis}$ is the class of quasi-isomorphisms of non-negative chain complexes of abelian groups. More details (in French) can be found in [17, annexe A]. The inclusion $W_{\infty} \subset W_{\infty}^{ab}$ can be seen directly from the equivalence above, or, without using the simplicial description, by the minimality of $W_{\infty}$ among basic localizers [10]. □

4.3. Let $u : A \to B$ be a functor between small categories. Recall that the functor $u^* : \hat{B} \to \hat{A}$ is a right adjoint. In particular, it preserves abelian group objects and induces a functor that we denote

$$ (u^*)^{ab} : \hat{B}^{ab} \to \hat{A}^{ab}. $$

The functor $u^*$ also admits a right adjoint $u_* : \hat{A} \to \hat{B}$, which, by the same argument, induces a functor

$$ u_*^{ab} : \hat{A}^{ab} \to \hat{B}^{ab} $$

which is right adjoint to the functor $(u^*)^{ab}$. The two following squares are then commutative up to a natural isomorphism:

$$ (4.3.1) \qquad \begin{tikzcd} \hat{B}^{ab} \arrow[r, "(u^*)^{ab}", "u^*"'] & \hat{A}^{ab} \arrow[d, "u"] \arrow[d, "u"] & \hat{A}^{ab} \arrow[r, "u_*^{ab}", "u_*"'] & \hat{B}^{ab} \arrow[d, "u"] \arrow[d, "u"] \\ \hat{B} \arrow[r, "u^*", "u_*"'] & \hat{A} \arrow[d, "u_*"'] & \hat{A} \arrow[r, "u_*^{ab}", "u_*"'] & \hat{B} \arrow[d, "u_*"'] \arrow[d, "u_*"'] \end{tikzcd} $$

The left adjoint to $u^*$, however, does not preserve abelian group objects in general. But since $\hat{B}^{ab}$ is a cocomplete additive category, Proposition 3.5 allows to extend the functor $A \xrightarrow{u} B \xrightarrow{\mathrm{Wh}_B} \hat{B}^{ab}$ to a cocontinuous functor

$$ (\mathrm{Wh}_B \circ u)_!^{ab} : \hat{A}^{ab} \to \hat{B}^{ab} $$

that we will simply denote by $u_!^{ab}$. Proposition 3.5 guarantees that this functor is a left adjoint to the functor

$$ ((\mathrm{Wh}_B \circ u)^*)^{ab} : \hat{B}^{ab} \to \hat{A}^{ab}, \quad Y \mapsto (a \mapsto \mathrm{Hom}_{\hat{B}^{ab}}(\mathbb{Z}_B^{(u(a))}, Y)) $$

which is nothing but the functor $(u^*)^{ab}$. Commutativity of the squares in diagram 4.3.1 imply that the following two squares

$$ (4.3.2) \qquad \begin{tikzcd} \hat{B} \arrow[r, "u^*", "u_*"'] & \hat{A} \arrow[d, "\mathrm{Wh}_B"'] \arrow[d, "\mathrm{Wh}_A"'] & \hat{A} \arrow[r, "u_!", "u_*"'] & \hat{B} \arrow[d, "\mathrm{Wh}_B"'] \arrow[d, "\mathrm{Wh}_A"'] \\ \hat{B}^{ab} \arrow[r, "(u^*)^{ab}", "u_*^{ab}"'] & \hat{A}^{ab} \arrow[d, "u_*^{ab}"'] & \hat{A}^{ab} \arrow[r, "u_*^{ab}", "u_*^{ab}"'] & \hat{B}^{ab} \arrow[d, "u_*^{ab}"'] \arrow[d, "u_*^{ab}"'] \end{tikzcd} $$

are also commutative up to a natural isomorphism.

4.4. Let $A$ and $B$ be two small categories, and $u : A \to B$ a $W_{\infty}^{ab}$-aspherical functor. Recall from Proposition 1.5 that the functor

$$ u/X : A/X \to B/X, \quad (a, u(a) \to X) \mapsto (u(a), u(a) \to X) $$

is then in $W_{\infty}^{ab}$. Thus, taking homology with integer coefficients, the morphism

$$ \mathrm{H}(u/X, \mathbb{Z}) : \mathrm{H}(A/X, \mathbb{Z}) \to \mathrm{H}(B/X, \mathbb{Z}) $$

is an isomorphism in Hotab. Using Remark 2.11 together with the commutativity of diagram 4.3.2, this means that there is a natural isomorphism

$$ \mathrm{H}(A, (u^*)^{ab} \mathbb{Z}_B^{(X)}) \simeq \mathrm{H}(B, \mathbb{Z}_B^{(X)}) $$

in Hotab. In other words, $(u^*)^{ab}$ induces an isomorphism in homology for free abelian presheaves. We will now show that this still holds for arbitrary abelian presheaves.
4.5. For any object $b$ of $\mathcal{B}$, we denote by $\ell_A^B : \mathcal{B} \to \mathrm{Ch}_+(\mathcal{Ab})$ the functor making the following diagram

<figure><img src="image_12.png" /></figure>

commutative, where $\ell_A$ denotes the Bousfield-Kan integrator introduced in paragraph 3.22. Concretely, for any object $b$ of $\mathcal{B}$, $\ell_A^B(b)$ is the complex which is given in degree $n \ge 0$ by

$$ (u^*)^{ab} \mathbb{Z}_B^{(b)} \odot_A (\ell_A)_n = \bigoplus_{\Delta_n \xrightarrow{\varphi} A} \mathbb{Z}^{(\mathrm{Hom}_B(u\varphi(n),b))} . $$

In particular, $\ell_A^B$ is a complex in $\mathrm{Add}(\mathcal{B}^{\mathrm{op}})$. Moreover, we get the equality

$$ \ell_A^B(b) = (L_\Delta)_!^{ab} \mathbb{Z}_\Delta^{(N(A/b))} . $$

Therefore, applying the functor

$$ \mathrm{Cat} \xrightarrow{N} \hat{\Delta} \xrightarrow{\mathrm{Wh}_\Delta} \hat{\Delta}^{ab} \xrightarrow{(L_\Delta)_!^{ab}} \mathrm{Ch}_+(\mathcal{Ab}) $$

to $u/b : A/b \to B/b$ gives a morphism of chain complexes $\lambda_{u,b} : \ell_A^B(b) \to \ell_B(b)$ which is natural in $b$, since for any morphism $f : b \to b'$ in $\mathcal{B}$, the square

$$ \begin{tikzcd} A/b \arrow[r, "u/b"] \arrow[d, "A/f"] & B/b \arrow[d, "B/f"] \\ A/b' \arrow[r, "u/b'"'] & B/b' \end{tikzcd} $$

is commutative. Given any abelian presheaf $X$ on $\mathcal{B}$, we can then define a morphism

$$ \lambda_{u,X} = X \odot_B \lambda_u : X \odot_B \ell_A^B \to X \odot_B \ell_B, $$

which fits in the following diagram:

<figure><img src="image_13.png" /></figure>

and which is described in the following way: for any abelian presheaf $X$ on $\mathcal{B}$ and any integer $n \ge 0$, we have

$$ X \odot_B (\ell_A^B)_n = \bigoplus_{\Delta_n \xrightarrow{\varphi} A} X(u\varphi(n)), \quad X \odot_B (\ell_B)_n = \bigoplus_{\Delta_n \xrightarrow{\varphi} B} X(\varphi(n)), $$

and $(\lambda_{u,X})_n$ is given on generators by the formula $(\lambda_{u,X})_n \langle \varphi, x \rangle = \langle u \circ \varphi, x \rangle$.

**Proposition 4.6.** Let $u : A \to B$ be a functor between small categories. The following conditions are equivalent:

(1) $u$ is $\mathcal{W}_\infty^{ab}$-aspherical;

(2) for any object $b$ in $\mathcal{B}$, the morphism $\lambda_{u,b}$ is a quasi-isomorphism;

(3) for any abelian presheaf $X$ on $\mathcal{B}$, the morphism $\lambda_{u,X}$ is a quasi-isomorphism;

(4) the diagram

<figure><img src="image_14.png" /></figure>

is commutative up to a natural isomorphism.

Moreover, these conditions imply the following condition
(5) for any morphism $f$ in $\hat{B}^{ab}$, we have the equivalence

$$f \in \mathcal{W}_B^{ab} \iff (u^*)^{ab} f \in \mathcal{W}_A^{ab}.$$

*Proof.* The implication (4) $\implies$ (5) is immediate. To prove (1) $\implies$ (2), recall that if $u$ is $\mathcal{W}_\infty^{ab}$-aspherical, Proposition 1.5 implies that for any object $b$ in $B$, the morphism $u/b$ is in $\mathcal{W}_\infty^{ab}$. By construction, $\lambda_{u,b}$ is then a quasi-isomorphism. For the implication (2) $\implies$ (3), recall that $\ell_A^B$ is a complex in $\mathrm{Add}(B^{\mathrm{op}})$ which implies, with the hypothesis, that $\lambda_u : \ell_A^B \to \ell_B$ is a quasi-isomorphisms between complexes of projective abelian presheaves on $B^{\mathrm{op}}$. Since for any abelian presheaf $X$ on $B$, the functor $X \odot_B -$ is a left adjoint, it is right exact, hence preserves quasi-isomorphisms between complexes of projective objects. Therefore, $\lambda_{u,X} = X \odot_B \lambda_u$ is a quasi-isomorphism. Implication (3) $\implies$ (4) comes from the fact that $\ell_B$ is an integrator on $B$, so that with this hypothesis, $\ell_A^B$ is also an integrator on $B$. Finally, to prove (4) $\implies$ (1), we use Proposition 1.5 and prove that for any object $b$ of $B$, the category $A/b$ is $\mathcal{W}_\infty^{ab}$-aspherical: this comes from the isomorphisms

$$\mathrm{H}(A/b, \mathbb{Z}) \simeq \mathrm{H}(A, \mathbb{Z}_A^{(u^*(b))}) \simeq \mathrm{H}(B, \mathbb{Z}_B^{(b)}) \simeq \mathrm{H}(B/b, \mathbb{Z})$$

in Hotab, and from the fact that $B/b$ is a $\mathcal{W}_\infty^{ab}$-aspherical category.

□

## 5. STRONG WHITEHEAD CATEGORIES

There are two natural candidates to the role of weak equivalences of abelian presheaves on a small category: morphisms inducing isomorphisms in homology, and morphisms whose underlying morphism of presheaves of sets is a weak equivalence. We show in this section that any test category for which these two classes coincide is a homologically pseudo-test category, as defined in Definition 2.13.

**Definition 5.1.** We say that a small category $A$ is a strong Whitehead category if the two classes of morphisms $\mathcal{W}_A^{ab}$ and $U^{-1}\mathcal{W}_A$ in $\hat{A}^{ab}$ coincide, where $U : \hat{A}^{ab} \to \hat{A}$ denotes the forgetful functor. This means that for any morphism $f$ of abelian presheaves on $A$, we have the equivalence

$$f \in \mathcal{W}_A^{ab} \iff U(f) \in \mathcal{W}_A.$$

*Example 5.2.* Recall that for any morphism of simplicial abelian groups $f : X \to Y$ and any integer $n > 0$, there is a commutative square

$$\begin{tikzcd}
& & \pi_n(U X, 0) \arrow[r, "\pi_n(U f)"'] & \pi_n(U Y, 0) \\
& & \coloneqq \arrow[d] & \coloneqq \arrow[d] \\
\mathcal{H}_n(\Delta, \mathbb{Z}^{(X)}) & \mathcal{H}_n(\Delta, \mathbb{Z}^{(Y)}) & &
\end{tikzcd}$$

and similarly in degree 0. This shows that $f$ is in $\mathcal{W}_A^{ab}$ if and only if $U f$ induces an isomorphism between homotopy groups. Illusie-Quillen Theorem [27, VI, Theorem 3.3] then allows to conclude that $\Delta$ is a strong Whitehead category.

**Proposition 5.3.** Let $u : A \to B$ be an aspherical functor. If $A$ is a strong Whitehead category, then $B$ is also a strong Whitehead category.

*Proof.* If $u$ is aspherical, then it is also $\mathcal{W}_\infty^{ab}$-aspherical by Proposition 4.2, and we get the sequence of equivalences

$$\begin{align*}
f \in \mathcal{W}_B^{ab} &\iff (u^*)^{ab} f \in \mathcal{W}_A^{ab} && (\mathcal{W}_\infty^{ab}\text{-asphericity}) \\
&\iff U(u^*)^{ab} f \in \mathcal{W}_A && (\text{strong Whitehead condition}) \\
&\iff u^* U f \in \mathcal{W}_A && (\text{commutativity of diagram 4.3.1}) \\
&\iff U f \in \mathcal{W}_B && (\text{asphericity})
\end{align*}$$

which shows that $B$ is a strong Whitehead category.

□

*Example 5.4.* Since $\Delta$ is totally aspherical [24, Proposition 1.6.14], the diagonal functor $\Delta \to \Delta \times \Delta$ is aspherical. This implies that for any integer $n > 0$, the category $\Delta^n$ is also a strong Whitehead category. We will see in Proposition 6.7 that there exists an aspherical functor $\Delta^n \to \Theta_n$, ensuring that $\Theta_n$ is also a strong Whitehead category for any integer $n > 0$.
**5.5.** If $\mathcal{M}$ is a locally presentable category and $I$ is a class of morphisms in $\mathcal{M}$, we denote by $l(I)$ (resp. $r(I)$) the class of morphisms satisfying the left (resp. right) lifting property with respect to $I$. We recall the following theorem:

**Theorem 5.6 (Crans).** Let $(\mathcal{M}, W_{\mathcal{M}}, \text{Cof}_{\mathcal{M}}, \text{Fib}_{\mathcal{M}})$ be a closed model category cofibrantly generated by the pair $(I, J)$. Let $\mathcal{M}'$ be a locally presentable category and $G : \mathcal{M} \to \mathcal{M}'$ a functor admitting a right adjoint $D$. Suppose that $D(l(r(GJ))) \subset W_{\mathcal{M}}$. Then $\mathcal{M}'$ admits a closed model category structure generated by the pair $(GI, GJ)$, whose weak equivalences are the elements of $D^{-1}(W_{\mathcal{M}})$ and fibrations are the elements of $D^{-1}(\text{Fib}_{\mathcal{M}})$.

*Proof.* See [13, Theorem 3.3]. □

In what follows, we will use Theorem 5.6 to transfer the Grothendieck-Cisinski model structure (1.11) on $\hat{A}$ to the category $\hat{A}^{ab}$ in the case where $A$ is both a strong Whitehead and a local test category.

**Proposition 5.7.** Let $A$ be a small category and $i : X \hookrightarrow Y$ a monomorphism of presheaves on $A$. If $L : A \to \text{Ch}_+(\text{Ab})$ is a free integrator on $A$, then the morphism

$$L_i^{ab}(\mathbb{Z}_A^{(i)}) : L_i^{ab}(\mathbb{Z}_A^{(X)}) \to L_i^{ab}(\mathbb{Z}_A^{(Y)})$$

is a monomorphism with projective cokernel.

*Proof.* Since $L$ is a free integrator, we can write, for any integer $n \ge 0$,

$$L_n = \bigoplus_{i \in I_n} \mathbb{Z}_{A^{op}}^{(a_i)}$$

and, by additivity, we only have to show that if $a$ is an object of $A$, then the morphism of abelian groups

$$\mathbb{Z}_A^{(i)} \odot \mathbb{Z}_{A^{op}}^{(a)} : \mathbb{Z}_A^{(X)} \odot_A \mathbb{Z}_{A^{op}}^{(a)} \to \mathbb{Z}_A^{(Y)} \odot_A \mathbb{Z}_{A^{op}}^{(a)}$$

is a monomorphism with projective cokernel. But this is the free abelian group morphism

$$\mathbb{Z}^{(i_a)} : \mathbb{Z}^{(Xa)} \to \mathbb{Z}^{(Ya)}$$

which is a monomorphism with cokernel the free abelian group on the set $Ya \setminus \text{Im} i_a$. □

**Corollary 5.8.** Let $A$ be a local test category, $j : X \to Y$ a trivial cofibration for the Grothendieck-Cisinski model category structure on $\hat{A}$, and $L : A \to \text{Ch}_+(\text{Ab})$ a free integrator on $A$. Then the morphism $L_i^{ab}(\mathbb{Z}_A^{(j)})$ is a trivial cofibration in the projective model structure on $\text{Ch}_+(\text{Ab})$.

*Proof.* Since we proved in Proposition 2.7 that the functor $L_i^{ab} \text{Wh}_A$ sends weak equivalences of presheaves on $A$ to quasi-isomorphisms, we can conclude using the proposition above. □

**Theorem 5.9.** Let $A$ be a strong Whitehead and local test category. The category $\hat{A}^{ab}$ can be given the structure of a model category where weak equivalences are the elements of $W_{\hat{A}}^{ab}$ and fibrations are the morphisms of abelian presheaves whose underlying morphism of presheaves is a fibration in the Grothendieck-Cisinski model structure on $\hat{A}$.

*Proof.* We will use Theorem 5.6 taking for $G$ the functor

$$\text{Wh}_A : \hat{A} \to \hat{A}^{ab}.$$

Since $\hat{A}^{ab}$ is locally presentable, we only need to show that if $J$ is a set of generating trivial cofibrations for $\hat{A}$, then we have the inclusion $l(r(\mathbb{Z}J)) \subset \text{U}^{-1}W_A$. Let $L$ be a free integrator on $A$. Since $A$ is a strong Whitehead category, we have

$$\text{U}^{-1}W_A = (L_i^{ab})^{-1}W_{\text{qis}}$$

and, therefore, we only need to show that the functor $L_i^{ab} : \hat{A}^{ab} \to \text{Ch}_+(\text{Ab})$ sends elements of $l(r(\mathbb{Z}J))$ to quasi-isomorphisms. Since the functor $L_i^{ab}$ is a left adjoint, the small object argument (see for example [16, Corollary 10.5.22]) implies that

$$L_i^{ab}(l(r(\mathbb{Z}J))) \subset l(r(L_i^{ab}\mathbb{Z}(J))) .$$

Since we just showed that elements of $L_i^{ab}(\mathbb{Z}J)$ are trivial cofibrations in $\text{Ch}_+(\text{Ab})$, and since trivial cofibrations are stable under pushouts, transfinite compositions and retracts, we can conclude that every element of $l(r(\mathbb{Z}J))$ is also in $W_{\hat{A}}^{ab}$. □
Remark 5.10. The cofibrations in this model structure are generated by the image of monomorphisms by the functor $\mathrm{Wh}_A : \hat{A} \to \hat{A}^{ab}$. In particular, every free abelian presheaf is a cofibrant object.

**Proposition 5.11.** Let $A$ be a strong Whitehead and local test category, $B$ a local test category and $u : A \to B$ an aspherical functor, Then:

(1) $B$ is a strong Whitehead category, and can be endowed with the model structure defined in Theorem 5.9;

(2) the functors

$$ (u^*)^{ab} : \hat{B}^{ab} \to \hat{A}^{ab}, \quad u_*^{ab} : \hat{A}^{ab} \to \hat{B}^{ab} $$

forms a Quillen equivalence for the model structure defined in Theorem 5.9.

*Proof.* We have already proved the first point in Proposition 5.3. For the second point, recall that cofibrations and trivial cofibrations in $\hat{B}^{ab}$ are generated by the image under $\mathrm{Wh}_B : \hat{B} \to \hat{B}^{ab}$ of cofibrations and trivial cofibrations in the Grothendieck-Cisinski structure on $\hat{B}$. To prove that $(u^*)^{ab}$ is a left Quillen functor, we only have to check that it sends morphisms of the form $\mathbb{Z}_B^{(i)}$, where $i$ is a cofibration in $\hat{B}$, to cofibrations of $\hat{A}^{ab}$ (since we already know that it preserves weak equivalences). This follows from the fact that $\mathrm{Wh}_A$ and $u^*$ are left Quillen functors, and by the commutativity of diagram 4.3.1.

Let's prove that $((u^*)^{ab}, u_*^{ab})$ is a Quillen equivalence. Let $X$ be an abelian presheaf on $B$ and $Y$ a fibrant object of $\hat{A}^{ab}$. Then $\mathrm{U}(Y)$ is a fibrant object of $\hat{A}$, and $\mathrm{U}(X)$ is a cofibrant object of $\hat{B}$. Using Proposition 1.12, we can then conclude from the following chain of equivalences:

$$ \begin{align*} f : (u^*)^{ab} X &\to Y \in \mathcal{W}_A^{ab} & (\text{strong Whitehead condition for A}) \\ \Leftrightarrow & U f : U (u^*)^{ab} X \to U Y \in \mathcal{W}_A & \\ \Leftrightarrow & U f : u^* U X \to U Y \in \mathcal{W}_A & (\text{commutativity of 4.3.1}) \\ \Leftrightarrow & (U f)^{\sharp} : U X \to u_* U Y \in \mathcal{W}_B & (\text{Proposition 1.12}) \\ \Leftrightarrow & U (f^{\sharp}) : U X \to U u_*^{ab} Y \in \mathcal{W}_B & (\text{commutativity of 4.3.1}) \\ \Leftrightarrow & f^{\sharp} : X \to u_*^{ab} Y \in \mathcal{W}_B^{ab} & (\text{strong Whitehead condition for B}) \end{align*} $$

where we used the symbol $(-)^{\sharp}$ to denote the transpose of morphisms. $\square$

**Corollary 5.12.** Let $A$ be a strong Whitehead and local test category, $B$ a local test category, and $u : A \to B$ an aspherical functor. Then $A$ is a homologically pseudo-test category (2.13) if and only if the same holds for $B$.

*Proof.* By Proposition 5.3, since $u$ is aspherical and $A$ is a strong Whitehead category, then $B$ is a strong Whitehead category. Thus, we can endow $\hat{B}^{ab}$ with the model category structure described in Theorem 5.9. Since aspherical morphisms are $\mathcal{W}_\infty^{ab}$-aspherical, the diagram

<figure><img src="image_15.png" /></figure>

is commutative up to a natural isomorphism, and Proposition 5.11 shows that the horizontal arrow is an equivalence of categories. We can then conclude that $H_A$ is an equivalence of categories if and only if $H_B$ is an equivalence of categories. $\square$

*Example 5.13.* As a direct application of this corollary, the category $\Delta^n$ for $n > 0$ is a homologically pseudo-test category, as explained in example 5.4.

**Theorem 5.14.** If $A$ is a strong Whitehead and test category, then $A$ is a homologically pseudo-test category.

*Proof.* Let $A_0$ be a homologically pseudo-test, strong Whitehead and strict test category (for example, we can take the category $\Delta$ of simplices). Let $B$ be the full subcategory of $\mathrm{Cat}$ whose
objects are the images by $i_A$ of objects of $A$ and images by $i_{A_0}$ of objects of $A_0$. We then get the following commutative diagram

<figure><img src="image_16.png" /></figure>

in $\mathcal{C}at$. We will show that $u$ and $v$ are aspherical functors, and that $B$ is a strong Whitehead and local test category, before applying Corollary 5.12. Since $A_0$ is a local test category, the functor $i_{A_0}$ is aspherical by Proposition 1.20. Moreover $i$ is full and faithful, so Proposition 1.21 implies that $i$ and $v$ are aspherical. Since $A$ is a local test category, the same argument shows that $i_A$ and $u$ are also aspherical. Since $A_0$ is a strong Whitehead category and $v$ is aspherical, Proposition 5.3 implies that $B$ is a strong Whitehead category,

Let's show that $B$ is a local test category. Consider a small aspherical category $C$. Since $i$ is aspherical, $i^*(C)$ is an aspherical presheaf on $B$ by Theorem 1.23. But since $v$ is aspherical and $A_0$ is totally aspherical, Proposition 1.15 implies that $B$ is also totally aspherical. This implies by Proposition 1.14 that $i^*(C)$ is a locally aspherical presheaf. By Theorem 1.23, this implies that $B$ is a local test category and that $i$ is a local test functor. Finally, since $A$, $A_0$ and $B$ are strong Whitehead and local test categories, and since there is a zig-zag of aspherical functors

$$A \xrightarrow{u} B \leftarrow^{v} A_0,$$

we can conclude using Corollary 5.12 that $A$ is a homologically pseudo-test category, since the same holds for $A_0$. $\square$

*Remark 5.15.* We showed in Proposition 5.7 that if $L$ is a free integrator on a small category $A$, the functor

$$L_i^{ab} : \hat{A}^{ab} \to \mathcal{Ch}_+(\mathbf{Ab})$$

sends monomorphisms of free abelian presheaves to monomorphisms with projective cokernels. Moreover, any small category can be equipped a free integrator, as explained in paragraph 3.22. In particular, if $A$ is a strong Whitehead and test category, we actually get a Quillen equivalence between the model structure on $\hat{A}^{ab}$ given in Theorem 5.9 and the projective model structure on $\mathcal{Ch}_+(\mathbf{Ab})$.

## 6. JOYAL'S CATEGORY $\Theta$ IS A HOMOLOGICALLY PSEUDO-TEST CATEGORY

We first recall the construction of $\Theta$ using wreath products, due to Berger [8], which makes it easy to define an aspherical functor $\Delta^n \to \Theta_n$ for every integer $n \ge 0$, allowing to apply directly Corollary 5.12. For $\Theta$, we will use a variation on the notion of asphericity for functors whose codomain is a presheaf category and show the analogue of Proposition 5.3 in this context. This will prove, using previous results from Ara and Maltsiniotis [4], that $\Theta$ is a strong Whitehead category.

**6.1.** Let $A$ be a small category. We denote by $\Delta \circled{A}$ the small category whose objects are pairs $[\Delta_n; (a_i)_{1 \le i \le n}]$ where $n$ is a non-negative integer and $(a_i)_{1 \le i \le n}$ is a family of object of $A$, and whose morphisms

$$[\Delta_n; (a_i)_{1 \le i \le n}] \to [\Delta_m; (a'_j)_{1 \le j \le m}]$$

are pairs $[\varphi, \mathbf{f}]$ where $\varphi : \Delta_n \to \Delta_m$ is a morphism of $\Delta$ and

$$\mathbf{f} = (f_{ji} : a_i \to a'_j)_{1 \le i \le n, \varphi(i-1) < j \le \varphi(i)}$$

is a family of morphisms of $A$. Given two composable morphisms

$$[\Delta_n; (a_i)_{1 \le i \le n}] \xrightarrow{[\varphi, \mathbf{f}]} [\Delta_{n'}; (a'_{i'})_{1 \le i' \le n'}] \xrightarrow{[\varphi', \mathbf{f}']}] [\Delta_{n''}; (a''_{i''})_{1 \le i'' \le n''}]$$

their composite is defined by the pair $[\varphi'', \mathbf{f}'']$ with $\varphi'' = \varphi'\varphi$ and

$$\mathbf{f}'' = (f''_{i''i})_{1 \le i \le n}, \quad \varphi''(i-1) < i'' \le \varphi''(i), \quad f''_{i''i} = f'_{i''i'} f'_{i'i}$$

where $i'$ is the only integer such that $\varphi(i-1) < i' \le \varphi(i)$ and $\varphi'(i' - 1) < i'' \le \varphi'(i')$.
**6.2.** Given a functor $F: A \to B$ between small categories, we denote by

$$ \Delta \lrcorner F : \Delta \lrcorner A \to \Delta \lrcorner B $$

the functor sending objects $[\Delta_n; (a_i)_i]$ of $\Delta \lrcorner A$ to $[\Delta_n; (F(a_i))_i]$, and morphisms $[\varphi, \mathbf{f} = (f_{ji})_{i,j}]$ to

$$ (\Delta \lrcorner F)[\varphi, \mathbf{f}] = [\varphi, F(\mathbf{f}) = (F(f_{ji}))_{i,j}] . $$

Note that if $F$ is a faithful (resp. fully faithful) functor, then the same is true for the functor $\Delta \lrcorner F$. In particular, if $a$ is an object of $A$, we get a faithful functor

$$ I_a : \Delta \lrcorner e \simeq \Delta \xrightarrow{\Delta \lrcorner a} \Delta \lrcorner A , $$

where we also denoted by $a$ the corresponding functor from the terminal category $e$ to $A$. This functor is fully faithful if $a$ doesn't admit non-trivial endomorphisms. This construction is also functorial in $a$, and we get a functor denoted

$$ \mu_A : \Delta \times A \to \Delta \lrcorner A, \quad (\Delta_n, a) \mapsto I_a(\Delta_n) . $$

**6.3.** For any integer $n \ge 0$, we denote by $\Theta_n$ the category defined inductively in the following way:

* $\Theta_0 = e$ is the final small category;

* $\Theta_{n+1} = \Delta \lrcorner \Theta_n$ for $n \ge 0$.

Notice that since $\Delta_0$ is the final object of $\Delta$, we get a fully faithful functor $\Theta_n \hookrightarrow \Theta_{n+1}$ for any integer $n \ge 0$. We denote by $\Theta$ the colimit of the diagram

$$ \Theta_0 \hookrightarrow \Theta_1 \hookrightarrow \Theta_2 \hookrightarrow \dots $$

in $Cat$.

*Remark 6.4.* This category is equivalent to the category introduced by Joyal in [20]. A description of $\Theta$ as a full subcategory of the category $\omega Cat$ was conjectured in [6] and proven independently in [22] and [8].

**Theorem 6.5** (Cisinski-Maltsiniotis). For any integer $n \ge 1$, the category $\Theta_n$ is a strict test category. Moreover, the category $\Theta$ is also a strict test category.

*Proof.* See [12]. □

**6.6.** For any integer $n \ge 0$, we define a functor

$$ m_n : \Delta^n \to \Theta_n $$

inductively in the following way:

* $m_0$ is the identity on $e = \Delta^0 = \Theta_0$;

* $m_{n+1}$ is the composite

$$ \Delta^{n+1} = \Delta \times \Delta^n \xrightarrow{\Delta \times m_n} \Delta \times \Theta_n \xrightarrow{\mu_{\Theta_n}} \Theta_{n+1} . $$

**Proposition 6.7** (Ara-Maltsiniotis). For any integer $n \ge 0$,

$$ m_n : \Delta^n \to \Theta_n $$

is an aspherical functor.

*Proof.* See [4, Corollary 5.5]. □

**Corollary 6.8.** For any integer $n \ge 1$, $\Theta_n$ is a homologically pseudo-test category.

*Proof.* Since $\Delta$ is totally aspherical, the diagonal functor $\Delta \to \Delta^n$ is aspherical. The composite functor $\Delta \to \Delta^n \to \Theta_n$ is then also aspherical. Moreover, $\Delta$ is a strong Whitehead category, and Proposition 5.3 implies that $\Theta_n$ is a strong Whitehead category. Since, by Proposition 6.5, $\Theta_n$ is also a test category, this is a particular case of Theorem 5.14. □

We will now prove that $\Theta$ is also a homologically pseudo-test category. For this, we will use a variation on the notion of aspherical functors.

**6.9.** Let $\mathcal{W} \subset \mathrm{Arr}(\mathrm{Cat})$ be a basic localizer. If $j : B \to \hat{A}$ is a functor, we get a functor

$$ j^* : \hat{B} \to \hat{A}, \quad X \mapsto (b \mapsto \mathrm{Hom}_{\hat{A}}(j(b), X)) . $$

We say that $j$ is aspherical if the following conditions are satisfied:
(1) for every object $b$ of $B$, the presheaf $j(b)$ on $A$ is aspherical;

(2) for every object $b$ of $B$, the presheaf $j^*j(b)$ on $B$ is aspherical;

(3) for every object $a$ of $A$, the presheaf $j^*(a)$ on $B$ is aspherical.

**6.10.** Let $j : B \to \hat{A}$ be a functor. Consider the full subcategory $C$ of $\hat{A}$ whose objects are representables presheaves, and images by $j$ of objects of $B$. By construction, the diagram

<figure><img src="image_17.png" /></figure>

where we denoted by $h$ the Yoneda embedding, is commutative. This implies that the diagram

<figure><img src="image_18.png" /></figure>

is also commutative.

**Proposition 6.11.** In diagram 6.10.1, if $j$ is an aspherical functor, then $j'$ and $h'$ are aspherical.

*Proof.* Let $c$ be an object of $C$. Recall from Proposition 1.5 that we only need to show that $j'^*(c)$ and $h'^*(c)$ are aspherical presheaves. Since $i$ is fully faithful, there is a natural isomorphism $i^*i(c) \simeq c$ in $\hat{C}$, which induces a natural isomorphism

$$h'^*(c) \simeq h'^*i^*i(c) \simeq i(c) .$$

We distinguish two cases: if $c$ is an object of $A$, then $i(c)$ is a representable presheaf, which implies that it is aspherical. If $c$ is a $j'(b)$ for an object $b$ of $B$, then $i(c) = j(b)$ is an aspherical functor by the first condition of paragraph 6.9. In both cases, $h'^*(c)$ is an aspherical presheaf on $A$, which proves that $h'$ is aspherical.

We proceed in the same way to show that $j$ is aspherical: we have a natural isomorphism

$$j'^*(c) \simeq j'^*i^*i(c) \simeq j^*i(c)$$

and we distinguish two cases: either $c$ is an object of $A$, in which case $j^*i(c)$ is aspherical by condition 3 of paragraph 6.9, or $c$ is a $j'(b)$ for an object $b$ of $B$, in which case $j^*ij'(b) = j^*j(b)$ is aspherical by condition 2. We have proved that $j'$ is also an aspherical functor. $\square$

**Proposition 6.12.** Let $A$ be a small category and $B$ a strong Whitehead category. If $j : B \to \hat{A}$ is an aspherical functor, then $A$ is a strong Whitehead category.

*Proof.* We keep the notation of paragraph 6.10. Since $i^*$ and $h'^*$ are right adjoint functors, they preserve abelian group objects and induce functors

$$(i^*)^{ab} : \hat{A}^{ab} \to \hat{C}^{ab}, \quad (h'^*)^{ab} : \hat{C}^{ab} \to \hat{A}^{ab}$$

such that, in diagram

<figure><img src="image_19.png" /></figure>

every square except the top-left one are commutative. Since $h'^*i^* = \text{id}$, the exterior square is also commutative. Notice also that we have $(h'^*)^{ab}(i^*)^{ab} = \text{id}$.
By Proposition 5.3, we know that $C$ is a strong Whitehead category, since we proved that $j'$ is aspherical. Moreover, $h'$ is also an aspherical morphism. We can then conclude by the following chain of equivalences, for any morphism $f$ of $\hat{A}$:

$$ \begin{align*} f \in \mathcal{W}_A^{ab} &\iff (h'^*)^{ab}(i^*)^{ab} f \in \mathcal{W}_A^{ab} \\ &\iff (i^*)^{ab} f \in \mathcal{W}_C^{ab} && (\text{asphericity for } h') \\ &\iff U_C(i^*)^{ab} f \in \mathcal{W}_C && (\text{strong Whitehead condition for } C) \\ &\iff i^* U_A f \in \mathcal{W}_C && (\text{commutativity of bottom-left square}) \\ &\iff h'^* i^* U_A f \in \mathcal{W}_A && (\text{asphericity for } h') \\ &\iff U_A(f) \in \mathcal{W}_A . && \square \end{align*} $$

We will now show that $\Theta$ is a strong Whitehead category. For this, we just point out the existence of an aspherical functor $\Delta \to \hat{\Theta}$ and refer to the literature for an explicit description, as our purpose is only to apply proposition 6.12.

**6.13.** Denote by $\omega Cat$ the category of strict $\omega$-categories and strict $\omega$-functors. There is an inclusion $\Theta \hookrightarrow \omega Cat$ (see [22], [7]) that allows to define a fully faithful functor called the *cellular nerve*

$$ N_{\Theta} : \omega Cat \to \hat{\Theta} . $$

Moreover, Street introduced in [30] a functor $\mathcal{O} : \Delta \to \omega Cat$, which induces a functor

$$ N : \omega Cat \to \hat{\Delta} $$

called the *Street nerve*. In [4], Ara and Maltsiniotis prove that these two nerve functors are equivalent, meaning that the square

<figure><img src="image_20.png" /></figure>

is commutative up to a natural isomorphism. One of the steps of their proof is the following:

**Theorem 6.14 (Ara-Maltsiniotis).** *The functor*

$$ j : \Delta \xrightarrow{\mathcal{O}} \omega Cat \xrightarrow{N_{\Theta}} \hat{\Theta} $$

*is an aspherical functor, within the meaning of paragraph 6.9.*

*Proof.* See [4, Theorem 5.8]. □

**Theorem 6.15.** *The category $\Theta$ is a strong Whitehead and homologically pseudo-test category.*

*Proof.* By Proposition 6.12, the asphericity of $j$ implies that $\Theta$ is a strong Whitehead category. Since it is also a test category, we can conclude with Theorem 5.14. □

## REFERENCES

1. Dimitri Ara, *The groupoidal analogue $\tilde{\Theta}$ to Joyal’s category $\Theta$ is a test category*, Applied Categorical Structures **20** (2012), no. 6, 603–649.

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LÉO HUBERT, AIX-MARSEILLE UNIV, CNRS, I2M, MARSEILLE, FRANCE

E-mail: leo.hubert.math@mailo.com

URL: https://leohubert.pages.math.cnrs.fr