The four operations on perverse motives

Let $k$ be a field of characteristic zero with a fixed embedding $\sigma:k\hookrightarrow \mathbb{C}$ into the field of complex numbers. Given a $k$-variety $X$, we use the triangulated category of \'etale motives with rational coefficients on $X$ to construct an abelian category $\mathscr{M}(X)$ of perverse mixed motives. We show that over $\mathrm{Spec}(k)$ the category obtained is canonically equivalent to the usual category of Nori motives and that the derived categories $\mathrm{D}^{\mathrm{b}}(\mathscr{M}(X))$ are equipped with the four operations of Grothendieck (for morphisms of quasi-projective $k$-varieties) as well as nearby and vanishing cycles functors and a formalism of weights. In particular, as an application, we show that many classical constructions done with perverse sheaves, such as intersection cohomology groups or Leray spectral sequences, are motivic and therefore compatible with Hodge theory. This recovers and strengthens work by Zucker, Saito, Arapura and de Cataldo-Migliorini and provide an arithmetic proof of the pureness of intersection cohomology with coefficients in a geometric variation of Hodge structures.


Introduction
Let k be a field of characteristic zero with a fixed embedding σ : k ֒→ C into the field of complex numbers. A k-variety is a separated k-scheme of finite type.
In present work we construct the four operations on the derived categories of perverse Nori motives. In order to combine the tools available from [6,7] and [17,18] in a most efficient way, we define the abelian category of perverse Nori motives on a given k-variety as a byproduct of the triangulated category of constructible étale motives on the same variety. Over the base field the category obtained still coincides with the usual category of Nori motives but now, as we show, it is possible to equip the derived categories of these abelian categories with the four operations of Grothendieck as well as nearby and vanishing cycles functors.
In particular, as an application, we show that many classical constructions done with perverse sheaves, such that intersection cohomology groups or Leray spectral sequences, are motivic and therefore compatible with Hodge theory. This recover and strenghten works by Zucker [68], Saito [59], Arapura [4] and de Cataldo-Migliorini [26]. Moreover it provides an arithmetic proof via reduction to positive characteristic and the Weil conjecture of the pureness of the Hodge structure on intersection cohomology with coefficients in a geometric variation of Hodge structures.

Conjectural picture and some earlier works
Before getting into more depth, let us discuss perverse motives from the perspective of perverse sheaves and recall some parts of the conjectural picture and re lated earlier works.
For someone interested in perverse sheaves, perverse motives can be thought of as perverses sheaves of geometric origin. However, the classical definition of these perverse sheaves as a full subcategory of the category of all perverse sheaves is not entirely satisfactory. Indeed, it contains too many morphisms and consequently, as we take kernels and cokernels of morphisms which shouldn't be considered, too many objects. For example, perverse sheaves of geometric origin should define mixed Hodge modules and therefore any morphism between them should also be a morphism of mixed Hodge modules. Therefore, one expects the category of perverse motives/perverse sheaves of geometric origins to be an abelian category endowed with a faithful exact functor into the category of perverse sheaves.
According to Grothendieck, there should exists a Q-linear abelian category MM(k) whose objects are called mixed motives. Given an embedding σ : k ֒→ C, the category MM(k) should comes with a faithful exact functor

MM(k) → MHS
to the category of (polarizable) mixed Hodge structures MHS, called the realization functor. The mixed Hodge structure on the i-th Betti cohomology group H i (X) of a given a k-variety X should come via the realization functor from a mixed motive H i M (X). The appealing beauty lies in the expected properties of this category, in particular, the conjectural relations between extension groups and algebraic cycles (see e.g. [46]), or the relation with periods rings and motivic Galois groups (see e.g. for a survey [12]).
As part of Grothendieck's more general cohomological program, the category MM(k) should underly a system of coefficients. For any k-variety X, there should exist an abelian category MM(X) of mixed motives and their derived categories should satisfy a formalism of (adjoint) triangulated functors a formalism which has been at the heart of Grothendieck's approach to every cohomology theory. The motives H i M (X) would then be given as the i-th cohomology of the image under a M * of a complex of mixed motives Q M X that should realizes to the standard constant sheaf Q X on the associated analytic space X an . Grothendieck was looking for abelian categories modeled after the categories of constructible sheaves, but as pointed out by Beȋlinson and Deligne one could/should also look for categories modeled after perverse sheaves (see e.g. [28]).
Many attemps have been made to carry out at least partially but unconditionnally Grothendieck's program. The most succesful attempt in constructing the triangulated category of mixed motives stems from Morel-Voevodsky's stable homotopy theory of schemes. The best candidate so far is the triangulated category DA ct (X) of constructible étale étale motivic sheaves (with rational coefficients) extensively studied by Ayoub in [6,7,9]. The theory developed in [6,7] provides these categories with the Grothendieck four operations and, as shown by Voevodsky in [66], Chow groups of smooth algebraic k-varieties can be computed as extension groups in the category DA ct (k).
On the abelian side, Nori has constructed a candidate for the abelian category of mixed motives over k. The construction of Nori's abelian category HM(k) is Tannakian in essence and, since it is a category of comodules over some Hopf algebra, it comes with a built-in motivic Galois group. Moreover any Nori motive has a canonical weight filtration and Arapura has shown in [5, Theorem 6.4.1] that the full subcategory of pure motives coincides with the semi-simple abelian category defined by André in [3] using motivated algebraic cycles (see also [41, Theorem 10.2.5]). More generally, attempts have been made to define Nori motives over k-varieties. Arapura has defined a constructible variant in [5] and the first author a perverse variant in [44]. However, the Grothendieck four operations have not been constructed (at least in their full extent) in those contexts. For example in [5], the direct image functor is only available for structural morphisms or projective morphisms and no extraordinary inverse image is defined.
Note that the two different attempts should not be unrelated. One expects the triangulated DA ct (k) to possess a special t-structure (called the motivic t-structure) whose heart should be the abelian category of mixed motives. This is a very deep conjecture which implies for example the Lefschetz and Künneth type standard conjectures (see [16]). As of now, the extension groups in Nori's abelian category of mixed motives are known to be related with algebraic cycles only very poorly.
However, striking unconditional relations between the two different approaches have still been obtained. In particular, in [22], Gallauer-Choudhury have shown that the motivic Galois group constructed by Ayoub in [10, 11] using the triangulated category of étale motives is isomorphic to the motivic Galois group obtained by Nori's construction.

Contents of this paper
Let us now describe more precisely the contents of our paper. Given a k-variety X, consider the bounded derived category D b c (X, Q) of sheaves of Q-vector spaces with algebraically constructible cohomology on the analytic space X an associated with the base change of X along σ and the category of perverse sheaves P(X) which is the heart of the perverse t-structure on D b c (X, Q) introduced in [19]. Let DA ct (X) be the triangulated category of constructible étale motivic sheaves (with rational coefficients) which is a full triangulated subcategory of the Q-linear counterpart of the stable homotopy category of schemes SH(X) introduced by Morel and Voevodsky (see [47,52,65]). This category has been extensively studied by Ayoub in [6,7,9] and comes with a realization functor Bti * X : DA ct (X) → D b c (X, Q) (see in [8]) and thus, by composing with the perverse cohomology functor, with a homological functor p H 0 P with values in P(X). The category of perverse motives considered in the present paper is defined (see Section 2) as the universal factorization of p H 0 P , where M (X) is an abelian category, p H 0 M is a homological functor and rat M X is a faithful exact functor. This kind of universal construction goes back to Freyd and is recalled in Section 1. As we see in Section 6, ℓ-adic perverse sheaves can also be used to defined the category of perverse motives (see Definition 6.3 and Proposition 6.10) Given a morphism of k-varieties f : X → Y , the four functors where developed by Verdier [63] (see also [48]) on the model of the theory developed by Grothendieck for étale and ℓ-adic sheaves [2]. The nearby and vanishing cycles functors k were constructed by Grothendieck in [1] (here X η denotes the generic fiber and X σ the special fiber). By a theorem of Gabber, the functors ψ g := Ψ g [−1] and φ g := Φ g [−1] are t-exact for the perverse t-structures and thus induce exact functors

Categorical preliminaries
Let us recall in this section a few universal constructions related to abelian and triangulated categories. They date back to Freyd's construction of the abelian hull of an additive category [32] and have been considered in many different forms in various works (see e.g. [64,50,55,15]).
Let S be an additive category. Let Mod(S) be the category of right S-modules, that is, the category of additive functors from S op to the category Ab of abelian groups. The category Mod(S) is abelian and a sequence of right S-modules is exact if and only if for every s ∈ S the sequence of abelian groups A right S-module F is said to be of finite presentation if there exist objects s, t in S and an exact sequence Definition 1.1. Let S be an additive category. We denote by R(S) the full subcategory of Mod(S) consisting of right S-modules of finite presentation.
The category R(S) is an additive category with cokernels (the cokernel of a morphism of right S-modules of finite presentation is of finite presentation) and the Yoneda functor h S : S → R(S) is a fully faithful additive functor. Recall that, given a morphism t → s in S, a morphism r → t is called a pseudo-kernel if the sequence Note that the construction can be dualized so that there is a universal way to add kernels to an additive category. One simply set L(S) := R(S op ) op . The two constructions can be combined to add both cokernels and kernels at the same time. Let S be an additive category and let A ad (S) := L(R(S)).
Then the functor h : S → A ad (S) is a fully faithful additive functor and A ad (S) is an abelian category which enjoys the following universal property (this is Freyd's abelian hull). Proposition 1.3. Let A be an abelian category and F : S → A be an additive functor, then there exists, up to a natural isomorphism, a unique exact functor Note also that the category A ad (S) is canonically equivalent to R(L(S)). This construction can be used to provide an alternative description of Nori's category (see [15]). Let Q be a quiver, A be an abelian category and T : Q → A be a representation. Let P(Q) be the path category and P(Q) ⊕ be its additive completion obtained by adding finite direct sums. Then, up to natural isomorphisms, we have a commutative diagram where ̺ T is an additive functor and ρ T an exact functor. The kernel of ρ T is a thick subcategory of A qv (Q) and we define the abelian category A qv (Q, T ) to be the quotient of A qv (Q) by this kernel. By construction, the functor ρ T has a canonical factorisation where π T is an exact functor and r T is a faithful exact functor. If we denote by T the composition of the representation Q → A qv (Q) and the functor π T : where T is a representation and r T is a faithful exact functor. It is easy to see that the above factorization is universal among all factorizations of T of the form where B is an abelian category, R is a representation and s is a faithful exact functor. In particular, whenever Nori's construction is available, e.g. if A is Noetherian, Artinian and has finite dimensional Hom-groups over Q (see [44]), then the category A qv (Q, T ) is equivalent to Nori's abelian category associated with the quiver representation T .
Let us consider the case when Q is an additive category and T is an additive functor. Then, up to natural isomorphisms, we have a commutative diagram where T * is an exact functor. The kernel of T * is a thick subcategory of A ad (Q) and we define the abelian category A ad (Q, T ) to be the quotient of A ad (Q) by this kernel.
satisfies the universal property that defines A qv (Q, T ). Consider a factorization of the representation T of the quiver Q where B is an abelian category, R is a representation and s is a faithful exact functor. Since s is faithful, R must be an additive functor. Therefore, we get a commutative diagram (up to natural isomorphisms) The exactness and the faithfulness of s implies that the above diagram can be further completed into a commutative diagram (up to natural isomorphisms) This shows the desired universal property.

Triangulated case
Let us finally consider the special case when S is a triangulated category. In that case the additive category S has pseudo-kernels and pseudo-cokernels, in particular, the category A tr (S) := R(S) 1 is an abelian category. The Yoneda embedding h S : S → A tr (S) is a homological functor and is universal for this property (see [54,Theorem 5.1.18]). In particular, if A is an abelian category, any homological functor H : S → A admits a canonical factorization where ρ H is an exact functor. This factorization of H is universal among all such factorizations. The kernel of ρ H is a thick subcategory of A tr (S) and we define the abelian category A tr (S, H) to be the quotient of A tr (S) by this kernel. By construction, the functor ρ H has a canonical factorisation where π H is an exact functor and r H is a faithful exact functor. Setting H S := π H • h S , it provides a canonical factorization of H where H S is a homological functor and r H a faithful exact functor. It is easy to see that the above factorization is universal among all factorizations of H of the form where L is a homological functor and s is a faithful exact functor.
If S is a triangulated category, we can also see S simply as a quiver (resp. an additive category) and the homological functor H : S → A simply as a representation (resp. an additive functor). In particular, we have at our disposal the universal factorizations of the representation H where the arrows on the right are exact and faithful functors.
where B is an abelian category, R is an additive functor and s is a faithful exact functor. Since s is faithful, R must be homological. Therefore, we get a commutative diagram (up to natural isomorphisms) The exactness and the faithfulness of s imply that the above diagram can be further completed into a commutative diagram (up to natural isomorphisms) This shows the desired universal property.

Definition
To construct the abelian category of perverse motives M (X) used in the present work, we take S to be the triangulated category of constructible étale motives with rational coefficients DA ct (X) and H to be the homological functor p H 0 P obtained by composing of the Betti realization where rat M X is a faithful exact functor and p H 0 M is a homological functor. Let us recall the two consequences (denoted by P1 and P2 below) of the universal property of the factorization where F is a triangulated functor, G is an exact functor and α is an invertible natural transformation, there exists a commutative diagram where E is an exact functor and β, γ are invertible natural transformations such that the diagram be a commutative diagram in which F 1 , F 2 are triangulated functors, G 1 , G 2 are exact functors, α 1 , α 2 are invertible natural transformations and λ, µ are natural transformations. Let be commutative diagrams given in the property P1, then there exists a unique natural transformation θ : Let (Sch/k) be the category of quasi-projective k-varieties and C be a subcategory of (Sch/k). The properties P1 and P2 can be used to lift (covariant or contravariant) 2-functors. Indeed, let F : C → TR be a 2-functor (let's say covariant to fix the notation), where TR is the 2-category of triangulated categories, such that F(X) = DA ct (X) for every k-variety X in C. Similarly, let Ab be the 2-category of abelian categories, and let G : C → Ab be a 2-functor such that G(X) = P(X) for every k-variety X in C and that G(f ) is exact for every morphism f in C. Assume that (Θ, α) : F → G is a 1-morphism of 2-functors such that Θ X = p H 0 P for every X ∈ C and that α f is invertible for every morphism f in C.
Let f : X → Y be a morphism in C. By applying P1 to the square DA ct (X) we get a commutative diagram where E(f ) is an exact functor and β f , γ f are invertible natural transformations such that the diagram can be seen as functors between the derived category of perverse sheaves (for their construction in terms of perverse sheaves see [18]).
Let Proposition 2.4]). Let θ be the collection of these natural transformations, then (Bti * , θ) is a morphism of stable homotopical 2-functors in the sense of [8, Definition 3.1]. Following the notation in [8], we denote by

Direct images under affine and quasi-finite morphisms
Let QAff (Sch/k) be the subcategory of (Sch/k) with the same objects but in which we only retain the morphisms that are quasi-finite and affine. By [19,Corollaire 4.1.3], for such a morphism f : X → Y , the functors are t-exact for the perverse t-structures. In particular, they induce exact functors between categories of perverse sheaves and by applying the property P1 to the canonical transformation compatible with the composition of morphisms.

Inverse image along a smooth morphism
Let f : X → Y be a smooth morphism of k-varieties. Then, Ω f is a locally free O X -module of finite rank. Let d f its rank (which is constant on each connected component of X). Then, d f is the relative dimension of f (see [36, (17.10.2)]) and if g : Y → Z is a smooth morphism, then d g•f = d g + d f with the obvious abuse of notation (see [36, (17.10.3)]). By [19, 4.2.4], the functor is t-exact for the perverse t-structures. In particular, it induces an exact functor between the categories of perverse sheaves and by applying the property P1 to the canonical transformation where the functor in the middle f * M [d f ] is an exact functor and θ DA f , θ M f are invertible natural transformations such that the diagram Proof. As for Proposition 2.3, the proof is a simple application of property P2. The details are left to the reader.

Duality
The result in this subsection will be used in the proof of Proposition 5.3. Let D P X be the duality functor for perverse sheaves and ε P X : Id → D P X • D P X be the canonical 2-isomorphism. Recall that, given a smooth morphism f : (1) There exist a contravariant exact functor D M (2) There exists a 2-isomorphism Proof. Again, the proof is a simple application of property P2. The details are left to the reader.

Perverse motives as a stack
Let S be a k-variety. Let us denote by AffEt/S be the category of affine étale schemes over S endowed with the étale topology. As in [62, Tag 02XU], the 2functor can be turned into a fibered category M → AffEt/S such that the fibre over an object U of AffEt/X is the category M (U ).
Proposition 2.7. The fibered category M → AffEt/S is a stack for the étale topology.
Proof. Let U be a k-variety, I be a finite set and U = (u i : U i → U ) i∈I be a covering of U by affine and étale morphisms. If J ⊆ I is a nonempty subset of I, we denote by U J the fiber product of the U j , j ∈ J, over U and by u J : U J → U the induced morphism. Given an object A ∈ M (U ), and k ∈ Z, we set into a complex using the alternating sum of the maps obtained from the unit of the adjunction in Proposition 2.5. The unit of this adjunction also provides a canonical morphism . This morphism induces a quasi-isomorphism on the underlying complex of perverse sheaves and so is a quasi-isomorphism itself since the forgetful functor to the derived category of perverse sheaves is conservative.
By [62,Tag 0268], to prove the proposition we have to show the following: (1) if U is an object in AffEt/S and A, B are objects in M (U ), then the presheaf on AffEt/U is a sheaf for the étale topology; (2) for any covering U = (u i : U i → U ) i∈I of the site AffEt/S, any descent datum is effective.
We already now that the similar assertions are true for perverse sheaves by [19, Let us first prove (1). Let A, B be objects in M (U ) and K, L be their underlying perverse sheaves. Consider the canonical commutative diagram The lower row is exact and the vertical arrows are injective. We only have to check that the upper row is exact at the middle term. Let c be an element in which belongs to the equalizer of the two maps on the right-hand side. Then, it defines (by adjunction) a morphism c 0 and a morphism c 1 such that the square A is the kernel of the upper map in (3) and B is the kernel of the lower map. Hence, Now we prove (2). Consider a descent datum. In other words, consider, for every i ∈ I, an object A i in M (U i ) and, for every i, j ∈ I, an isomorphism φ ij : given on (u i ) M * A i by the difference of the maps obtained by composing the morphism induced by adjunction either the identity or the isomorphism φ ij . Using the fact that descent data on perverse sheaves are effective, it is easy to see that A makes the given descent datum effective.

A simpler generating quiver
Let X be a k-variety. Consider the quiver Pairs eff • For every vertex (a : Y → X, Z, i) in Pairs eff X and every closed subscheme W ⊆ Z, we have an edge where z : Z ֒→ Y is the closed immersion.
The quiver Pairs eff X admits a natural representation in D b c (X, Q). If c = (a : Y → X, Z, i) is a vertex in the quiver Pairs eff X and u : U ֒→ Y is the inclusion of the complement of Z in Y , then we set (4). The morphism f maps Z 1 to Z 2 and therefore U : where the arrow in the middle is given by the exchange morphism. By taking the image of this morphism under a 2! [−i], we get a morphism  [6,7] and their compatibility with their topological counterpart on the triangulated categories D b c (X, Q) shown in [8], imply that the quiver representation B can be lifted via the realization functor Bti * X to a quiver representation B : Pairs eff X → DA ct (X). Recall the definition of the category of perverse Nori motives considered in [44]. Definition 2.8. Let X be a k-variety. The category of effective perverse Nori motives is the abelian category The category N (X) is then obtained by inverting the Tate twist (see [44, 7.6] for details). By construction, the category of Nori motives HM(k) of [29] coincides with N (k). Remark 2.9. There is a difference between the representation p H 0 •B used here and the representation used in [44, 7.2-7.4] (see [44, Remark 7.8]). In loc.cit. the relative dualizing complex u ! P a ! P Q X is used instead of the absolute dualizing complex K U . If X is smooth, then the two different choices lead to equivalent categories.
Recall that the category M (X) can also be obtained by considering DA ct (X) simply as a quiver, that is it is canonically equivalent to the abelian category In particular, since the diagram is commutative (up to natural isomorphisms), there exists a canonical faithful exact functor Lemma 2.10. The functor (6) extends to a faithful exact Proof. By construction (see [44, 7.6]), the Tate twist in N eff (X) is the exact functor induced by the morphism of quivers Q : To prove the lemma it is enough to observe that there is a natural isomorphism in DA ct (X) between B(Q(Y, Z, n)) and B(Y, Z, n)(−1). Proposition 2.11. The category M (k) is canonically equivalent to the abelian category of Nori motives HM(k). More precisely the functor (7) is an equivalence when X = Spec(k) Proof. (See also [14,Proposition 4.12]) Consider the triangulated functor R N,s : DA ct (X) → D b (HM(k)) constructed in [22,Proposition 7.12]. Up to a natural isomorphism, the diagram is commutative. In particular, it provides a factorization of the cohomological This implies the existence of a canonical faithful exact functor M (k) → HM(k) such that is commutative up to a natural isomorphism. Using the universal properties, it is easy to see that it is a quasi-inverse to (7).
The following conjecture seems reasonable and reachable via our current technology. Conjecture 2.12. Let X be a smooth k-variety. Let N (X) be the category of perverse motives constructed in [44] and ) be the triangulated functor constructed in [43]. Then, the Betti realization Bti * X is isomorphic to the composition c (X, Q) If Conjecture 2.12 holds then the same proof as the one of Proposition 2.11 implies the following Conjecture 2.13. Let X be a smooth k-variety. Then, the functor (7) is an equivalence.

Unipotent nearby and vanishing cycles
In [17], Beȋlinson has given an alternate construction of unipotent vanishing cycles functors for perverse sheaves and has used it to explain a glueing procedure for perverse sheaves (see [17,Proposition 3.1]). In this section, our main goal is to obtain similar results for perverse Nori motives. Later on, the vanishing cycles functor for perverse Nori motives will play a crucial role in the construction of the inverse image functor (see Section 4).
Given the way the abelian categories of perverse Nori motives are constructed from the triangulated categories of étale motives, our first step is to carry out Beȋlinson's constructions for perverse sheaves within the categories of étale motives or analytic motives (these categories being equivalent to the classical unbounded derived categories of sheaves of Q-vector spaces on the associated analytic spaces). This is done in Subsection 3.2 and Subsection 3.4. Our starting point is the logarithmic specialization system constructed by Ayoub in [7]. However, by working in triangulated categories instead of abelian categories as Beȋlinson did, one has to face the classical functoriality issues, one of the major drawback of triangulated categories. To avoid these problems and ensure that all our constructions are functorial we will rely heavily on the fact that the triangulated categories of motives underlie a triangulated derivator.
Only then, using the compatibility with the Betti realization, will we be able to obtain in Subsection 3.5 the desired functors for perverse Nori motives.

Reminder on derivators
Let us recall some features of triangulated (a.k.a. stable) derivators D needed in the construction of the motivic unipotent vanishing cycles functor and the related exact triangles. For the general theory, originally introduced by Grothendieck [37], we refer to [6,7,24,34,35,51].
We will assume that our derivator D is defined over all small categories. In our applications, the derivators considered will be of the form D := DA(S, −) for some k-variety S. Given a functor ρ : A → B, we denote by the structural functor and its right and left adjoint.
Notation: We let e be the punctual category reduced to one object and one morphism. Given a small category A, we denote by p A : A → e the projection functor and, if a is an object in A, we denote by a : e → A the functor that maps the unique object of e to a. Given n ∈ N, we let n be the category n ← · · · ← 1 ← 0.
If one thinks of functors in Hom(A op , D(e)) as diagrams of shape A, then an object in D(A) can be thought as a "coherent diagram" of shape A. Indeed, every object M in D(A) has an underlying diagram of shape A called its A-skeleton and defined to be the functor A op → D(e) which maps an object a in A to the object a * M of D(e). This construction gives the A-skeleton functor which is not an equivalence in general (coherent diagrams are richer than simple diagrams). We say that M ∈ D(A) is a coherent lifting of a given diagram of shape A if its A-skeleton is isomorphic to the given diagram.
We will not give here the definition of a stable derivator (see e.g. [   This functor is conservative. Moreover if A is directed, it is full and essentially surjective. Note that the property (Der3) is usually not part of the definition of a stable derivator but is satisfied for derivators coming from stable model categories (so in particular for DA(S, −)). We denote by = 1 × 1 the category We denote by the full subcategory of that does not contain the object (0, 0) and by i : → the inclusion functor. We denote by (−, 1) : 1 → the fully faithful functor which maps 0 to (0, 1) and 1 to (1, 1). Similarly we denote by the full subcategory of that does not contain the object (1, 1) and i : → the inclusion functor. We denote by (0, −) : 1 → the fully faithful functor that maps 0 and 1 respectively to (0, 0) and (0, 1) An object M in D( ) is said to be cocartesian (resp. cartesian) if and only if the canonical morphism Let be the category There are three natural ways to embed in and an object M ∈ D( ) is said to be cocartesian if the squares in D( ) obtained by pullback along those embeddings are cocartesian. A coherent triangle is a cocartesian object M ∈ D( ) such that (0, 1) * M and (2, 0) * M are zero. For such an object, we have a canonical isomorphism (0, 0) * M ≃ (2, 1) * M [1] and the induced sequence is an exact triangle in D(e).
One of the main advantage of working in a stable derivator is the possibility to associate with a coherent morphism M ∈ D(1) functorially a coherent triangle. Let us briefly recall the construction of this triangle. Let U be the full subcategory of that does not contain (0, 0) and (1, 0). Denote by v : 1 → U the functor that maps 0 and 1 respectively to (1, 1) and (2, 1) and by u : U → the inclusion functor. The image under the functor is a coherent triangle. Using (Der1-2), we see that (10) provides an exact triangle where the cofiber functor Cof is defined by Using Der1-3 it is easy to see that this functor is also given by 1 → is the fully faithful functor that maps 0 and 1 respectively to (1, 0) and (1, 1). Note that we have an isomorphism is the fully faithful functor that maps 0 and 1 respectively to (0, 0) and (1, 0).
The construction of the cofiber functor Cof and the cofiber triangle (11) can be dualized to get a fiber functor Fib and a fiber triangle. Let us recall the following lemma (see e.g. [35, Proposition 15.1.10] for a proof).
is an isomorphism.
There is also a functorial version of the octahedron axiom in D (see e.g. [35, Proof of Theorem 9.44]), that is, there is a functor D(2) → D(O) which associates to a coherent sequence of morphisms a coherent octahedron diagram. Here the category O ⊆ 4 × 2 is the full subcategory that does not contain the objects (4, 0) and (0, 2). In other words, O is the category  We have the following lemma.
In particular, it follows from Lemma 3.2 that Since the inverse image of w ♯ ω * along the fully faithful functor → O that maps the square (8) to the square Let us recall [6, Lemma 1.4.8].
is commutative. Moreover the whole diagram is commutative.
Note that in loc.cit. the lemma is stated only in the case I = e. However its proof works in the more general situation considered here.
We will need the following technical lemma.
Proof. Let M be the category of (unbounded) complexes of Q-vector spaces with its projective model structure. If J is a small category and S a separated k-scheme of finite type, we let Spec Σ T (PreShv(Sm/(S, J))) be the category of symmetric T -spectra of presheaves on Sm/(S, J) with values in M endowed with its semiprojective A 1 -stable model structure (see [ Remark 3.7. Assume I = e. Given M in DA(X), as in Remark 3.5, we have an exact triangle

Motivic unipotent vanishing cycles functor
Let Log f be the logarithmic specialization system constructed in [7, 3.6] (see also [9, p.103-109]). It is defined by where L og ∨ is the commutative associative unitary algebra in DA(G m,k ) constructed in [7, Définition 3.6.29] (see also [9, Définition 11.6]). The monodromy triangle To construct the motivic unipotent vanishing cycles functor, we shall use the fact that the 1-skeleton functor ) is full and essentially surjective. This allows to choose an object L in DA(A 1 k , 1) that lifts the morphism Q(0) → j * L og ∨ obtained as the composition of the adjunction morphism Q(0) → j * Q(0) and the image under j * of the unit Q(0) → L og ∨ of the commutative associative unitary algebra L og ∨ . Moreover, using the monodromy triangle (15), we can fix an isomorphism between L og ∨ (−1) and the cofiber of j * L such that the diagram Let be the full subcategory of that does not contain (0, 1). Denote by i : → the inclusion and by p , : → the unique functor which is the identity on and maps (0, 1) to (0, 0). Consider the functor . By pulling back along the closed immersion i : X σ ֒→ X we get the functor which is a coherent lifting of the commutative square Let (−, 1) : 1 → be the fully faithful functor that maps 0 and 1 respectively to (0, 1) and (1, 1). In particular, the 1-skeleton of (−, 1) Definition 3.8. The motivic unipotent vanishing cycles functor Φ f : DA(X) → DA(X σ ) is defined as the composition of (−, 1) * i * Θ f (−) and the cofiber functor : By construction, we get a natural transformation can : and an exact triangle We also get a natural transformation such that var • can = N . Indeed, let (−, 0) : 1 → be the fully faithful functor that maps 0 and 1 respectively to (0, 0) and (1, 0).
The chosen isomorphism between L og ∨ (−1) and the cofiber of j * L induces an isomorphism between Log f (j * (−))(−1) and the cofiber of (−, 0) By applying the coherent triangle functor u ♯ v * to the object i * Θ f (−) of the category DA(X σ , ) = DA(X σ , 1 × 1), we get a functor which is a coherent lifting of the commutative diagram The category 1 × is given by  inside 1 × . In the next subsection, we will be mainly focusing on the functor which is a coherent lifting of the commutative square

Maximal extension functor
Let us now construct Beȋlinson's maximal extension functor Ξ f (see [17]) and the related exact triangles in the triangulated categories of étale motives. This will be essential to prove Theorem 3.15 and for glueing perverse motives. By applying the coherent triangle functor u ♯ v * to the object Θ f (−) in DA(X, ) = DA(X, 1 × 1), we get a functor u ♯ v * Θ f : DA(X) → DA(X, 1 × ) which is a coherent lifting of the commutative diagram (Here • is some motive which we do not need to specify). The category 1 × is given by We denote by α : 1 × → × 1 the functor which maps (17) to  1, 1, 0).

By construction, we have an exact triangle
Since the canonical morphisms are isomorphisms, the exact triangle (18) can be rewritten as On the other hand, we have an exact triangle Proposition 3.11. There are exact triangles and Proof. Let us first construct (20) using the functorial version of the octahedron axiom (see Subsection 3.1). Recall that by definition be the fully faithful functor that maps 0, 1 and 2 respectively to (0, 0, 1, 0), (0, 1, 1, 0) and (1, 1, 1, 0). Recall that cm : 1 → 2 is the fully faithful functor that maps 0 and 1 respectively to 0 and 2. Then, β • (−, 1, 0) = γ • cm. In particular, we get that Using the exact triangle (13) given by the functorial octahedron axiom, we get an exact triangle − − → . However, by construction, we have an exact triangle Using Remark 3.7, we see that . This constructs the exact triangle (20).
Consider now the localization triangle To obtain (21) it is enough to check that j * Ξ f (−) is isomorphic to j * (−) [1] and that i * Ξ f (−) is isomorphic to Log f (j * (−)). The first isomorphism is obtained by applying j * to (20) and the second isomorphism is obtained by applying i * to (19). and Proof. Using (20), the exact triangle (22) is obtained by applying Lemma 3.1 to the cartesian square (i ) * Cof(Σ f (−)).
Since j * i * = 0, (22) provides an isomorphism between j * Ω f (−) and j * (−) [1]. Now, consider the localization triangle To construct (23), it is enough to obtain an isomorphism between i * Ω f (−) and Φ f (−). By definition However since i * j ! = 0, the canonical morphism is an isomorphism. Given that By Remark 3.9, the canonical morphism is an isomorphism. This concludes the proof.

Betti realization
Let X be a complex algebraic variety. Let AnDA(X) be the triangulated category of analytic motives. This category is obtained as the special case of the category SH an M (X) considered in [8] when the stable model category M is taken to be the category of unbounded complexes of Q-vector spaces with its projective model structure. Recall that the canonical triangulated functor i * X : D(X) → AnDA(X) (24) is an equivalence of categories (see [8, Théorème 1.8]). Here D(X) denotes the (unbounded) derived category of sheaves of Q-vector spaces on the associated analytic space X an . The functor An X : Sm/X → AnSm/X an which maps a smooth X-scheme Y to the associated X an -analytic space Y an induces a triangulated functor The Betti realization Bti * X of [8] is obtained as the composition of (25) and a quasiinverse to (24).
Let L og ∨ P be the image under the Betti realization of the motive L og ∨ and consider the specialization system it defines

Recall that in
and the two triangles

Moreover, we have canonical natural transformations
are isomorphisms when applied to constructible motives (see [8, Théorème 3.9]) and are also compatible with the various exact triangles.
As proved in [8, Théorème 4.9], the Betti realization is compatible with the (total) nearby cycles functors for constructible motives. In this subsection, we will need the compatibility of the Betti realization with the unipotent nearby cycles functors.
Lemma 3.13. The functor Log P f (−) is isomorphic to the unipotent nearby cycles functor ψ un f (−). Let e : C → C × ; z → exp(z) be the universal cover of the punctured complex plane C × . The group of deck transformations is identified with Z by mapping the integer k ∈ Z to the deck transformation z → z + 2iπk.
Let E n be the unipotent rational local system on C × of rank n+1 with (nilpotent) monodromy given by one Jordan block of maximal size. It underlies a variation of Q-mixed Hodge structures described e.g. in [58, §1.1].
Let us recall the description of this local system and relate it to Ayoub's logarithmic motive L og ∨ n . The following description is given in [57,2.3. Remark]. Let E n be the subsheaf of e * Q C annihilated by (T − Id) n+1 where T is the automorphism of e * Q C induced by the deck transformation corresponding to 1 ∈ Z. The restriction of T to E n is unipotent and we denote by N = log T the associated nilpotent endomorphism.
The sheaf E n is a local system on C × of rank n + 1. Let (E n ) 1 be its fiber over 1. We have an inclusion Note that the automorphism T acts by mapping a sequence (a k ) k∈Z to (a k+1 ) k∈Z . Let τ n be the element in (E n ) 1 given by τ n = (k n /n!) k∈Z . The family (1, τ 1 , . . . , τ n ) is a basis of (E n ) 1 such that T (τ r ) = r k=0 τ k /(r − k)! for every r ∈ [ [1, n]]. The matrix with respect to the basis (1, τ 1 , . . . , τ n ) of the unipotent endomorphism T of (E n ) 1 is thus given by n k=0 (J n ) k /k! where J n is the nilpotent Jordan block of size n + 1 and therefore N is given by the Jordan block J n in the basis (1, τ 1 , . . . , τ n ).
The multiplication e * Q C ⊗ e * Q C → e * Q C induces a morphism of local systems E k ⊗E ℓ → E k+ℓ . In particular, for n ∈ N * , we have a canonical morphism E ⊗n 1 → E n which defines a morphism Sym n E 1 → E n . (26) If τ := τ 1 , then τ n = τ n /n! and the above description of E n implies that (26) is an isomorphism.
Let us consider the Kummer natural transform e K : Id(−)(−1)[−1] → Id(−) in Betti cohomology (see [7,Définition 3.6.22]). By [60, 5.1 Lemma], the local system E 1 fits into an exact triangle By [8, Théorème 3.19] the Betti realization is compatible with the Kummer transform (for constructible motives). In particular, we have a natural isomorphism Bti * K → E 1 where K ∈ DA(G m ) is the motivic Kummer extension, that is, the cone of the Kummer natural transform for étale motives (see [7,Lemme 3.6.28]). Since the Betti realization Bti * is a symmetric monoidal functor, it induces an isomorphism for every integer n ∈ N. Therefore, we get an isomorphism where E is the ind-local system given by E = colim n∈N × E n . Let K ∈ D b c (X, Q), the unipotent nearby cycles functor ψ un f is given by [57, (2.3.3)] or [17,56]). With this description, Lemma 3.13 is an immediate consequence of (27). Proof. Since the functor ψ un f (−)[−1] is t-exact for the perverse t-structure, the corollary is an immediate consequence of Lemma 3.13 and the exact triangles relating the various functors.

Application to perverse motives
Now, we can apply the universal property of the categories of perverse motives to obtain four exact functors . Moreover we have four canonical exact sequences obtained from the exact triangles relating the four functors used in the construction. Two exact sequences As well as two exact sequences  We first consider the case of the immersion of a special fiber. Lemma 3.16. Let X be a k-variety and f : X → A 1 k be a morphism. Let i : X σ ֒→ X be the closed immersion of the special fiber in X and Z be a closed subscheme of X σ . Then, the exact functor Proof. We may assume Z = X σ . Indeed, let u : X \ Z ֒→ X and v : X σ \ Z ֒→ X σ be the open immersion. By Proposition 2.3 applied to cartesian square Since the functor i ′M * is conservative (it is faithful exact), we see that an object A in M (X σ ) belongs to Ker v * if and only if i M * A belongs to Ker u * . Hence, it is enough to show that Let us show that the functor p Φ M f is a quasi-inverse. Let X η be the generic fiber and j : X η ֒→ X be the open immersion. The exact triangle (16), provides an isomorphism of endomorphisms of DA(X σ ) between i * i * and Φ f [−1]i * . By composing with the isomorphism of functors i * i * → Id, we get an isomorphism of functors between the identity of DA(X σ ) and Similarly, we get an isomorphism between the identity of D(X σ , Q) and the func- Since these isomorphisms are compatible with the Betti realization, the property P2, ensures that p Φ M f i M * is isomorphic to the identity functor of the category M (X σ ).
An isomorphism between the identity of M Xσ (X) := Ker j * M and i * M p Φ f is provided by the exact sequences − − → (the first terms vanish for objects in the kernel of j * M ). This concludes the proof.
Proof of Theorem 3.15. Using Proposition 2.7, we may assume that X is an affine scheme. Let U be the open complement of Z in X and let f 1 , . . . , f r be elements in By Lemma 3.16, all these functors are equivalences. This concludes the proof.

Inverse images
The purpose of this section is to extend the (contravariant) 2-functor Liss H * M constructed in Subsection 2.5 into a (contravariant) 2-functor Suppose that j : V ֒→ X is an affine open immersion, that A is an object of M (X) and that B is an object of M (V ). Let i 1. Then, we have corresponding to u is also 0. Applying this to an open cover j 1 : U 1 ֒→ X, . . . , j n : U n ֒→ X of X by affine subsets and using the fact the canonical map B → n r=1 (j r ) M * (j r ) * M B given by Proposition 2.5 is a monomorphism for every object B of M (X), we reduce to the case where X is affine.
If X is affine, then, as in the proof of Theorem 3.15, we write i = i r • · · · • i 1 , where Z 1 = Z, Z r+1 = X, and, for every k ∈ {1, . . . , r}, i k : Z k → Z k+1 is the immersion of the complement of an open set of the form is an equivalence of categories. So we may assume that there exists f ∈ O(X) such that i is the immersion of the complement of D(f ). In that case, we showed in the proof of , which shows that C • takes its values in the full subcategory D b Z (M (X)). Let B ∈ D b Z (M (X)). Using the long exact sequence associated with this triangle and Proposition 2.5 which ensures that as desired.
In the general case, the adjoint C • can be constructed by considering a finite set I and an affine open covering U = (j i : U i → U ) i∈I . For every J ⊆ I, let j J be the inclusion i∈J U i ֒→ X. We define an exact functor C • : M (X) → C b (M (X)) in the following way. Let A be an object of M (X). We set : Let Z be a closed immersion such that the open immersion j : U ֒→ X of the complement of Z in X is affine. It follows from the proof of Proposition 4.2 that we have a canonical exact triangle Moreover the diagram of 2-isomorphisms between exact functors from M (S) to M (X) is commutative.
is commutative for every object A in D b (M (X)). Since all the entries of the above diagram are t-exact functors up to a shift by the relative dimension d of f , using [67,Theorem 1], it is enough to check the commutativity of the diagram when A belongs to M (X). Then, it amounts to the commutativity of a diagram of perverse motives on Y ′ and this can be checked on the underlying perverse sheaves.

Lemma 4.7. Consider a commutative diagram
is commutative. We can refine (35) into the following commutative diagrams The desired compatibility is now a consequence of Proposition 2.3, Lemma 4.7 and Lemma 4.5.

Main theorem
In Subsection 3.5, we have shown that the unipotent nearby and vanishing cycles functors can be defined at the level of perverse Nori motives.
Our goal is to prove that the four operations (1) can be lifted to the derived categories of perverse Nori motives. To obtain these various functors (and their compatibility relations) with the least amount of effort, we have chosen to follow Ayoub's approach developed in [6] around the notion of stable homotopical 2-functor, which encompasses in a small package all the ingredients needed to build the rest of the formalism.

Statement of the theorem
As before, (Sch/k) denotes the category of quasi-projective k-varieties. Recall that a contravariant 2-functor is an open immersion in (Sch/k) and i : Z → X is the closed immersion of the complement, then the pair (j * , i * ) is conservative. (5) If p : A 1 X → X is the canonical projection, then the unit morphism Id → p * p * is invertible. (6) If s is the zero section of the canonical projection p : A 1 X → X, then p ♯ s * : H(X) → H(X) is an equivalence of categories. The main theorem of [6] says that these data can be expanded into a complete formalism of the four operations (see [6,Scholie 1.4.2]). In particular, we can apply [6, Scholie 1.4.2] to get the functors (36). The next subsection is devoted to the proof of Theorem 5.1, and the reader will find some applications of the main theorem in Subsection 5.3.

Proof of the main theorem (Theorem 5.1)
We start by showing the existence of the direct image functor. The most important step is the proof of the existence of the direct image along the projection of the affine line A 1 Y onto its base Y . Proposition 5.2. For every morphism f : X → Y in (Sch/k), the functor In the proof, all products are fiber products over the base field k and A 1 is the affine line over k.
Step 1 : Suppose first that f is a closed immersion. Then f * M admits f M * as a right adjoint by construction of f * M , we know point (2) by Lemma 4.3, and the last statement of (1) is true by (2) and by conservativity of rat M X .
Step 2 : Now we consider the case where f is the projection morphism p : As before, if we can prove that p * M admits a right adjoint satisfying (2), then the last part of (3) will follow automatically.
We consider the following commutative diagram : where q 1 = id A 1 × p, q 2 is the product of the projection A 1 → Spec k and of id A 1 ×Y , i is the product of the diagonal morphism of A 1 and of id Y , and j is the complementary open inclusion. We also denote by s : Y → A 1 × Y the zero section of p. By the smooth base change theorem (or a direct calculation), the base change map p * P p P * → q P 1 * q * 2P is an isomorphism, so we get a functorial isomorphism p P * ≃ s * P p * P p P * → s * P q P 1 * q * 2P . Let K be a perverse sheaf on Y . Then L := q * 2P K [1] is perverse, and we have i * P L = K [1], so we get an exact sequence of perverse sheaves on A 1 × Y : Applying the functor q P 1 * and using the fact that q 1 • i = id A 1 ×Y , we get an exact triangle : Finally, we get an exact sequence of perverse sheaves on A 1 × Y : . We have just proved that these functors are t-exact (of course, this is obvious for the first one) and that there is a functorial exact triangle − − → . The functors F P and G P are defined in terms of the four operations. The existence of these operations in the categories DA ct (−) and the compatibility of the Betti realization with the four operations (see [8, Théorème 3.19]), imply by the universal property of the categories of perverse motives that there exist : • two exact functors , and the Betti realization of H M is isomorphic to q P 1 * q * P 2 . We now define a functor Note also the following useful fact. We denote by f : A 1 × Y → A 1 the first projection and by a : G m × Y → A 1 × Y the inclusion. Then applying s * M to the connecting map in the third exact sequence of Subsection 3.5, we get a natural transformation , whose composition with the functor H M is invertible. Indeed, we can check this last statement after applying the functors rat M Y , and then this follows from the exact triangle q P 1 * q * P 2 → F P → G P

+1
− − → and the fact that the composition of the natural transformation s * P → p Log B f a * P [1] and of the functor q P 1 * q * P 2 ≃ p * P p P * ≃ Q A 1 ⊠ p P * is invertible. As the functor p Log M f a * M is exact, we get an isomorphism from • . That is, we consider the commutative diagram are texact and we have a natural transformation F ′ P → G ′ P . As before, we can lift these functors and transformation to endofunctors the second projection and by a ′ the injection of A 1 × G m × Y into A 1 × A 1 × Y , we get as above an invertible natural transformation from q M 1• to the mapping cone of the morphism of exact functors Let's show that the base change isomorphism p * P p P * ∼ − → q P 1 * q * 2P lifts to a morphism p * M p M • → q M 1• q * 2M (which will automatically be an isomorphism). We have invertible natural transformations F ′ P • q * 2P ≃ q * 2P • F P and G ′ P • q * 2P ≃ q * 2P • G P . As all the functors involved are t-exact up to the same shift, the transformations lift to natural transformations Composing on the left with t * M and using the connection isomorphism t * . Composing this isomorphism with the unit of the adjunction (i * M , i M * ) and using the connection and are t-exact and the counit of the adjunction (J P ! , J * P ) induces a natural transformation from the second one to the first one. Hence, the functor (37) induces an exact endofunctor H ′′ But we also have an invertible natural transformation of t-exact functors can be written as f = pji, where i is a closed immersion, j is an open immersion, and p is a second projection Z × Y → Y with Z smooth equidimensional. We already know the result for closed immersions, so we just need to prove it for j and p. But the proof in these two cases is exactly the same as in [ where γ M f is the invertible natural transformation of Proposition 5.2. Using the expressions of M δ * ♯ and M η * ♯ given in (38) and (39), this follows directly from Proposition 2.6 (2) and Proposition 2.6 (1), which ensure that the diagram is invertible since all the other morphisms are. Therefore ξ M i is also invertible.

Some consequences
In this subsection, we draw some immediate consequences of the main theorem (Theorem 5.1). A Q-local system L on a k-variety X will be called geometric if there exists a smooth proper morphism g : Z → X such that L = R i g * Q for some integer i ∈ Z.

Intersection cohomology
The four operations formalism allows the definition of a motivic avatar of intersection complexes. In particular, intersection cohomology groups with coefficients in geometric systems are motivic. More precisely: ). (Note that B := j M ! * A with the notation of Definition 6.18).) This implies that IH i (X, L ) : where π : X → Spec k is the structural morphism. This shows, in particular, that intersection cohomology groups carry a natural Hodge structure. If X is a smooth projective curve, and L underlies a polarizable variation of Hodge structure, then the Hodge structure on the intersection cohomology groups was constructed by Zucker in [68, (7.2) Theorem, (11.6) Theorem]. In general, it follows from Saito's work on mixed Hodge modules [59] and a different proof has been given in [25]. We consider the weights in the next section (see Theorem 6.24 and Corollary 6.25).

Leray spectral sequences
Let f : X → Y be a morphism of quasi-projective k-varieties and L be a Q-local system on X. Then, we can associate with it two Leray spectral sequences in Betti cohomology: the classical one and the perverse one The main theorem of [4] shows that, if L = Q X is the constant local system on X and the morphism f is projective, then the classical Leray sequence is motivic, that is, it is the realization of a spectral sequence in the abelian category of Nori motives over k (see precisely [4,Theorem 3.1]).
This property is still true without the projectivity assumption and also more generally if the local system L is geometric: Corollary 5.6. If the local system L is geometric, then the classical Leray spectral sequence and the perverse Leray spectral sequence are spectral sequences of Nori motives over k.
Proof. If the local system L is geometric, there exists a smooth proper morphism g : Z → X such that L = R i g * Q Z for some integer i ∈ Z. Then L is the image under the functor rat M X of the perverse motive c H i (g M * Q M Z ), where c H i is the cohomological functor associated with the constructible t-structure (see below). So the result follows from the functoriality of the direct image functors.
In particular, the Leray spectral sequences are spectral sequences of (polarizable) mixed Hodge structures. The compatibility of the classical Leray spectral sequence result in Hodge theory was already proved by Zucker in [68] when X is a curve and more generally, for both spectral sequences, by Saito if L underlies an admissible variation of mixed Hodge structures (see [59]). This result has been recovered by de Cataldo and Migliorini with different technics in [26].

Nearby cycles
The theory developed here also shows that nearby cycles functors applied to perverse motives produce Nori motives.
Corollary 5.7. Let X be a k-variety, f : X → A 1 k a flat morphism with smooth generic fiber X η and L be a Q-local system on X η . If L is geometric, then, for every point x ∈ X σ (k) and every integer i ∈ Z, the Betti cohomology H i (Ψ f (L ) x ) of the nearby fiber is canonically a Nori motive over k.
Proof. The nearby cycles functor ψ f := Ψ f [−1] is t-exact for the perverse tstructure. Since it exists in the triangulated category of constructible étale motives (see [7]) and the Betti realization is compatible with the nearby cycles func-

Exponential motives
The perverse motives introduced in the present paper and their stability under the four operations can be used also in the study of exponential motives as introduced in [31]. Indeed, recall that Kontsevich and Soibelman define an exponential mixed Hodge structure as a mixed Hodge module A on the complex affine line A 1 C such that p * A = 0, where p : A 1 C → Spec(C) is the projection (see [49]). Their definition can be mimicked in the motivic context and the abelian category of exponential Nori motives can be defined as the full subcategory of M (A 1 k ) formed by the objects which have no global cohomology.

Constructible t-structure
Let us conclude by a possible comparison with Arapura's construction from [5]. Let X be a k-variety and consider the following full subcategories of D b (M (X)) As in [59, 4.6. Remarks] (see also [5, Theorem C.0.12]), we can check that these categories define a t-structure on D b (M (X)).
Let ct M (X) be the heart of this t-structure. Then, the functor rat M X induces a faithful exact functor from ct M (X) into the abelian category of constructible sheaves of Q-vector spaces on X. Then, using the universal property of the category of constructible motives M(X, Q) constructed by Arapura in [5], we get a faithful exact functor M(X, Q) → ct M (X). Is this functor an equivalence? If X = Spec k, then both categories are equivalent to the abelian category of Nori motives, so this functor is an equivalence.

Weights
In this section, we will use results on motives and weight structures from [21,38]. To apply these references directly in our context, we will make use of the fact that, if S is a Noetherian scheme of finite dimension, then Ayoub's category DA ct (S) is canonically isomorphic to the category of Beȋlinson motives defined in Cisinski and Déglise's book [23]. This is [23, Theorem 16.2.18] and will henceforth be used without further comment. (Note also that, though the authors of [21,38] have chosen to use Beȋlinson's motives, étale motives could have been used.)

Continuity of the abelian hull
Remember that, in chapter 5 of Neeman's book [54], there are four constructions of the abelian hull of a triangulated category. The first one gives a lax 2-functor from the 2-category of triangulated categories to that of abelian categories, but the other three constructions give strict 2-functors. If we use the fourth construction, which Neeman calls D(S) (see [ Then the canonical functor A tr (S) → lim − →i∈I A tr (S i ) is an equivalence of abelian categories.

Étale realization and ℓ-adic perverse Nori motives
Let S be a Noetherian excellent scheme finite-dimensional scheme, let ℓ be a prime number invertible over S; we assume that ℓ is odd and that S is a Q-scheme. (By Exposé XVIII-A of [42], the hypotheses above imply Hypothesis 5.1 in Ayoub's paper [9].) Under this hypothesis, Ayoub has constructed an étale ℓ-adic realization functor on DA ct (S), compatible with pullbacks.
Using results of Gabber (see [33], and also sections 4 and 5 of Fargues's article [30]), we can construct an abelian category P(S, Q ℓ ) of ℓ-adic perverse sheaves on S, satisfying all the usual properties. In particular, we get a perverse cohomology functor p H 0 ℓ := p H 0 • R ét S : DA ct (S) → P(S, Q ℓ ). where rat M S,ℓ is a faithful exact functor and p H 0 M is a homological functor. By the universal property of M (S) ℓ , we also get pullback functors between these categories as soon as the pullback functor between the categories of ℓ-adic complexes preserves the category of perverse sheaves.
We will use the following important fact : If we fix a base field k of characteristic 0 and only consider schemes that are quasi-projective over k, then the main theorem (stated in Subsection 5.1) stays true for the categories M (S) ℓ , for any odd prime number ℓ. Of course, we have to replace D b ct (S) and the Betti realization functor by D b c (S, Q ℓ ) and the étale realization functor in all the statements. Indeed, the proof of the main theorem, of of the statements that it uses, still work if we use the ℓ-adic étale realization instead of the Betti realization. The only result that requires a slightly different proof is Lemma 3.13 : we have to show that the ℓadic realization of Ayoub's logarithmic motive L og ∨ n is the local system used in Beilison's construction of the unipotent nearby cycle functor (see 1.1 and 1.2 of [20] or Definition 5.2.1 of [53]). As in the proof of Lemma 3.13, it suffices to check this for n = 1, and then it follows from Lemma 11.22 of [9].

Mixed horizontal perverse sheaves
Let k be a field and S be a k-scheme of finite type. We also fix a prime number ℓ invertible over S. If k is finitely generated over its prime field, then the category D b m (S, Q ℓ ) of mixed horizontal Q ℓ -complexes and its perverse t-structure with heart P m (S, Q ℓ ) (the category of mixed horizontal ℓ-adic perverse sheaves on S) were constructed in Huber's article [39] (see also [53, section 2]). We can actually use exactly the same process to define D b m (S, Q ℓ ) and P m (S, Q ℓ ) for arbitrary k. The only result in [53, section 2] that used the fact that k is finitely generated is the fact that the obvious exact functor from P m (S, Q ℓ ) to the usual category P(S, Q ℓ ) of ℓ-adic perverse sheaves on S is fully faithful. (This is not true for a general field k, for example it is obviously false if S = Spec k and k is algebraically closed.) All the other results of [39] about the 6 operations and their interaction with weights stay true for the same reasons as in that paper.
Remark 6.4. We could also have defined D b m (S, Q ℓ ) and P m (S, Q ℓ ) as direct 2limits of the similar categories over models of S over finitely generated subfields of k. This gives the same result because, for any field extension k ⊂ k ′ , the pullbacks functor D b m (S, Q ℓ ) → D b m (S k ′ , Q ℓ ) is exact and t-exact. Now we want to show that the realization functor rat M S,ℓ : M (S) ℓ → P(S, Q ℓ ) factors through the subcategory P m (S, Q ℓ ).
We have a continuity theorem for the categories of étale motives, proved in [13, Corollaire 1. Using the definition of mixed horizontal ℓ-adic complexes, we immediately get the following corollary. C and from Proposition 6.8, applied to the family of subfields of k that can be embedded in C.
Now we treat the case of a general k-scheme S. As in the first case, (i) follows from (ii) and from Proposition 6.8. So suppose that we have an embedding σ : k → C. We prove the desired result by induction on the dimension of S. The case dim S = 0 has already been treated, so we may assume that dim S > 0 and that the result is known for all the schemes of lower dimension. We denote by M → [M ] the canonical functor DA ct (S) → A tr (DA ct (S)); as DA ct (S) is a triangulated category, this is a fully faithful functor. Let X be an object of A tr (DA ct (S)). By construction of A tr (DA ct (S)), there exists a morphism N → M in DA ct (S) such that X is the cokernel of

Definition of weights
We fix a field k of characteristic zero and a quasi-projective scheme S over k. We first define weights via the ℓ-adic realizations. Definition 6.12. Let w ∈ Z. Let K be an object of M (S). We say that K is of weight ≤ w (resp. ≥ w) if rat M S,ℓ (K) ∈ Ob(P m (S, Q ℓ )) is of weight ≤ w (resp. ≥ w) for every odd prime number ℓ. We say that K is pure of weight w if it is both of weight ≤ w and of weight ≥ w.
In Proposition 6.17, we will a more intrisic definition of weights that does not use the realization functors. Definition 6.13. A weight filtration on an object K of M (S) is an increasing filtration W • K on K such that W i K = 0 for i small enough, W i K = K for i big enough, and W i K/W i−1 K is pure of weight i for every i ∈ Z.
The next result follows immediately from the similar result in the categories of mixed horizontal perverse sheaves (see Proposition 3.4 and Lemma 3.8 of [39]). Proposition 6.14. Let K, L be objects of M (S), and let w ∈ Z. a commutative diagram (up to isomorphisms of functors)