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# Tensor structure on perverse Nori motives

Luca Terenzi

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Luca Terenzi. Tensor structure on perverse Nori motives. 2024. hal-04817307

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Distributed under a Creative Commons Attribution 4.0 International License
# TENSOR STRUCTURE ON PERVERSE NORI MOTIVES

LUCA TERENZI

ABSTRACT. Let $k$ be a field of characteristic 0 endowed with a fixed field embedding $\sigma : k \to \mathbb{C}$. In this paper we complete the construction of the six functor formalism on perverse Nori motives over quasi-projective $k$-varieties initiated by F. Ivorra and S. Morel. Our main contribution is the construction of a canonical closed unitary symmetric monoidal structure on the bounded derived categories of perverse Nori motives compatible with the analogous structure on the underlying constructible derived categories; along the way, we extend some of Nori's original results to perverse Nori motives. As a consequence, we obtain well-behaved Tannakian categories of motivic local systems over smooth, geometrically connected $k$-varieties. Our constructions do not depend on the chosen complex embedding of $k$ and, in fact, our results generalize to arbitrary base fields of characteristic 0.

## CONTENTS

<table><tr><td>Introduction</td><td>1</td></tr><tr><td>Notation and conventions</td><td>9</td></tr><tr><td>1. Beilinson gluing for perverse motives</td><td>10</td></tr><tr><td>2. Construction of the external tensor structure</td><td>28</td></tr><tr><td>3. Tensoring with distinguished motivic local systems</td><td>40</td></tr><tr><td>4. Construction of the unit constraint</td><td>56</td></tr><tr><td>5. Internal homomorphisms</td><td>68</td></tr><tr><td>6. Motivic local systems</td><td>83</td></tr><tr><td>References</td><td>88</td></tr></table>

## INTRODUCTION

**Motivation and goal of the paper.** Let $k$ be a field of characteristic 0 endowed with a fixed field embedding into the complex numbers $\sigma : k \to \mathbb{C}$. In the late 1990's, M. Nori made a spectacular proposal for an unconditional construction of the conjectural $\mathbb{Q}$-linear abelian category of mixed motives over $k$. By design, Nori's category $\mathcal{M}(k)$ is the target of the universal cohomology theory on $k$-varieties comparable to $\mathbb{Q}$-linear singular cohomology of the underlying complex-analytic spaces; in particular, it comes equipped with a canonical faithful exact functor into finite-dimensional $\mathbb{Q}$-vector spaces

$$ (1) \qquad \iota_k : \mathcal{M}(k) \to \mathrm{vect}_{\mathbb{Q}} $$

<footnote n="2020">2020 Mathematics Subject Classification. 14F42, 14F43, 14C15.</footnote>

<footnote>Key words and phrases. Motives, perverse sheaves, sheaf operations.</footnote>

<footnote>The author acknowledges support by the GK 1821 "Cohomological Methods in Geometry" at the University of Freiburg and by the Labex Milyon at the ENS Lyon.</footnote>
which recovers singular cohomology groups when applied to the cohomological motives of k-varieties. Through the classical comparison isomorphisms relating singular cohomology with algebraic de Rham co-homology and with the various ℓ-adic cohomologies, Nori's category $\mathcal{M}(k)$ also acquires faithful exact func-tors into the associated coefficient categories recovering the corresponding cohomology groups of k-varieties.

According to the conjectures summarized by Beilinson in [Bei87H, § 5.10], the theory of mixed motives should admit an enhancement to a theory of mixed motivic sheaves: for every (quasi-projective) k-variety $S$ there should exist an abelian category of mixed motivic sheaves over $S$ providing a universal enhancement of the classical analytic and $\ell$-adic categories of perverse sheaves; as $S$ varies, their bounded derived categories should admit a complete six functor formalism analogously to the classical analytic and $\ell$-adic derived categories, as well as a compatible theory of weights. The theory of *perverse Nori motives*, introduced by F. Ivorra and S. Morel in [IM19], provides an unconditional candidate for the conjectural abelian categories of mixed motivic sheaves predicted by Beilinson: for every (quasi-projective) k-variety $S$, they define a $\mathbb{Q}$-linear abelian category $\mathcal{M}(S)$ which comes equipped with a canonical faithful exact functor into algebraically constructible $\mathbb{Q}$-linear perverse sheaves

$$ (2) \qquad \iota_S : \mathcal{M}(S) \to \mathrm{Perv}(S). $$

Their main result [IM19, Thm. 5.1] asserts that, as $S$ varies, the bounded derived categories $D^b(\mathcal{M}(S))$ are canonically endowed with Grothendieck's operations $f^*$, $f_*$, $f_!$ and $f^!$, compatibly with the analogous operations on the underlying derived categories $D^b(\mathrm{Perv}(S))$ along the triangulated functors

$$ (3) \qquad \iota_S : D^b(\mathcal{M}(S)) \to D^b(\mathrm{Perv}(S)) $$

induced by the exact functors (2). The main goal of the present paper is to complete the construction of the six functor formalism on perverse Nori motives by defining a canonical closed unitary symmetric monoidal structure on the categories $D^b(\mathcal{M}(S))$ in such a way that the functors (3) are unitary symmetric monoidal and compatible with internal homomorphisms.

**History and previous work.** Before explaining our results in more detail, let us review the construction of perverse Nori motives. In order to put it into perspective, it is convenient to start from the very beginning of the theory. Nori's original definition of the abelian category $\mathcal{M}(k)$ is based on an abstract category-theoretic result about representations of quivers into vector spaces. The idea is to introduce a quiver $\mathrm{Pairs}_k$ encoding the basic features of singular cohomology of (pairs of) k-varieties: functoriality along morphisms of pairs, long exact cohomology sequences of triples, and Tate twists. Vertices of $\mathrm{Pairs}_k$ are typically written as quadruples $(X, Y; n, w)$ consisting of a k-variety $X$, a closed subvariety $Y$, a cohomological degree $n$ and a Tate degree $w$. Singular cohomology can be regarded as a quiver representation

$$ (4) \qquad b_k : \mathrm{Pairs}_k \to \mathrm{vect}_{\mathbb{Q}}, $$

and the conjectural category of mixed motives over $k$ should be the smallest non-full abelian subcategory of $\mathrm{vect}_{\mathbb{Q}}$ containing the image of $b_k$. Nori was able to construct such a category explicitly in terms of the representation $b_k$, thus obtaining the abelian category $\mathcal{M}(k)$ equipped with the forgetful functor (1). While Nori's construction was tailored to representation of quivers into vector spaces (or, slightly more generally, into finitely generated modules over a Noetherian ring), it was later generalized to representations of quivers into arbitrary abelian categories in [BVCL18] using the language of categorical logic; the latter result was then translated into the familiar language of categories in [BVP18], based on P. Freyd's classical construction of the abelian hull of an additive category from [Fre65]. In its final form, the result asserts that, for every quiver representation into an abelian category

$$ \beta : \mathcal{D} \to \mathcal{A}, $$

there exists a smallest non-full abelian subcategory of $\mathcal{A}$ containing the image of $\beta$, called the associated *universal abelian factorization*. However, the simplicity of Nori's construction and of its later variants hides
a serious drawback: it does not allow one to compute morphisms or extension groups in $\mathcal{M}(k)$ explicitly, except for a few trivial cases. Indeed, since universal abelian factorizations are defined by purely categorical methods, it is typically impossible to describe their inner shape directly.

One of the most surprising achievements of Nori's work is the construction of a canonical tensor product on $\mathcal{M}(k)$ making it a neutral Tannakian category over $\mathbb{Q}$, with canonical fibre functor provided precisely by the forgetful functor (1). The Tannaka dual of $\mathcal{M}(k)$, denoted $\mathcal{G}_{mot}(k)$, is called the *motivic Galois group* of $k$. Even if our knowledge of the structure of this group is extremely poor, it allows for an elegant motivic interpretation of the Kontsevich-Zagier conjecture about periods of algebraic varieties; in fact, this interpretation was arguably Nori's original motivation for introducing $\mathcal{M}(k)$ (see [HMS17, §§ II, III] for a detailed account of this story).

In contrast with the existence of the abelian category $\mathcal{M}(k)$, the existence of a tensor structure on it is a highly non-formal result. The intuitive idea is to get a tensor product on $\mathcal{M}(k)$ by defining a suitable pairing on the quiver $\mathbf{Pairs}_k$ compatible with tensor product of singular cohomology groups and show that this descends to a tensor product on the universal abelian factorization. However, the only reasonable guess for such a pairing on $\mathbf{Pairs}_k$, given by the rule

$$ (X_1, Y_1; n_1, w_1) \otimes (X_2, Y_2, n_2, w_2) := (X_1 \times X_2, Y_1 \times X_2 \cup X_1 \times Y_2; n_1 + n_2, w_1 + w_2), $$

is not always compatible with tensor product of singular cohomology groups: using Künneth's formula, one sees that the pairing is well-behaved only when restricted to the subquiver $\mathbf{Pairs}_k^0 \subset \mathbf{Pairs}_k$ collecting all those vertices $(X, Y; n, w)$ such that the singular cohomology of the pair $(X, Y)$ (twisted by $w$) is concentrated precisely in degree $n$. Applying Nori's categorical construction to the restricted representation

$$ b_k^0 := b_k|_{\mathbf{Pairs}_k^0} : \mathbf{Pairs}_k^0 \to \mathrm{vect}_{\mathbb{Q}}, $$

one obtains an abelian subcategory $\mathcal{M}^0(k)$ of $\mathcal{M}(k)$ which, by design, admits a well-defined tensor product

$$ (5) \qquad - \otimes - : \mathcal{M}^0(k) \times \mathcal{M}^0(k) \to \mathcal{M}^0(k) $$

turning the restriction of (1) into a tensor functor. Now the problem becomes to show that the natural inclusion $\mathcal{M}^0(k) \subset \mathcal{M}(k)$ is in fact an equivalence. This was proven by Nori as a consequence of Beilinson's result [Bei87D, Lemma 3.3], quoted by Nori in [Nor18] as the Basic Lemma. From Nori's perspective, the Basic Lemma is a result about the existence of algebraic skeletal filtrations on affine $k$-varieties; in fact, it is the fundamental ingredient in the geometric interpretation of the cohomological perverse filtration described by M.A. de Cataldo and L. Migliorini in [DCM10].

The Basic Lemma turns out to be a crucial input into another major result of Nori's theory: namely, the existence of a canonical triangulated realization

$$ (6) \qquad R_k : \mathbf{DM}_{gm}(k, \mathbb{Q}) \to \mathcal{D}^b(\mathcal{M}^0(k)). $$

Here, $\mathbf{DM}_{gm}(k, \mathbb{Q})$ denotes Voevodsky's triangulated category of $\mathbb{Q}$-linear geometric motives over $k$, which so far represents the best known candidate for the bounded derived category of the conjectural abelian category of mixed motives over $k$; it is endowed with a triangulated functor

$$ (7) \qquad \mathrm{Bti}_k^* : \mathbf{DM}_{gm}(k, \mathbb{Q}) \to \mathcal{D}^b(\mathrm{vect}_{\mathbb{Q}}) $$

called the Betti realization, which should be regarded as a triangulated analogue of (1). Nori's realization functor (6) has the distinctive property that the composite functor

$$ \mathbf{DM}_{gm}(k, \mathbb{Q}) \xrightarrow{R_k} \mathcal{D}^b(\mathcal{M}^0(k)) \xrightarrow{\iota_k} \mathcal{D}^b(\mathrm{Perv}(S)) \xrightarrow{\mathrm{real}_k} \mathcal{D}^b(\mathrm{vect}_{\mathbb{Q}}) $$

recovers the Betti realization (7) up to canonical natural isomorphism. Since, in addition, the singular representation (4) factors as the composite

$$ \mathbf{Pairs}_k \xrightarrow{M_k} \mathbf{DM}_{gm}(k, \mathbb{Q}) \xrightarrow{\mathrm{Bti}_k^*} \mathcal{D}^b(\mathrm{vect}_{\mathbb{Q}}) \xrightarrow{H^0} \mathrm{vect}_{\mathbb{Q}} $$
for a suitable map of quivers $M_k : \text{Pairs}_k \to \mathbf{DM}_{gm}(k, \mathbb{Q})$, the equivalence $\mathcal{M}^0(k) = \mathcal{M}(k)$ follows formally. Furthermore, it follows that Nori's category $\mathcal{M}(k)$ can be reconstructed directly from Voevodsky's category $\mathbf{DM}_{gm}(k, \mathbb{Q})$: it suffices to apply Nori's category-theoretic construction to the composite functor

$$ \beta_k : \mathbf{DM}_{gm}(k, \mathbb{Q}) \xrightarrow{\text{Bti}_k^*} \mathcal{D}^b(\text{vect}_{\mathbb{Q}}) \xrightarrow{\mathcal{H}^0} \text{vect}_{\mathbb{Q}}, $$

regarded as a quiver representation. As pointed out in [BVHP20, Thm. 4.10], Nori's tensor product (5) can be also constructed starting from the latter presentation of $\mathcal{M}(k)$, and the realization (6) is a tensor-triangulated functor. The existence of a canonical functor relating Voevodsky's triangulated category with the derived category of Nori's abelian category is a strong indication that Nori's construction gives the correct definition of mixed motives over $k$. The triangulated category $\mathbf{DM}_{gm}(k, \mathbb{Q})$ is expected to admit a *motivic t-structure* rendering the Betti realization (7) t-exact; its heart should be the true conjectural abelian category of mixed motives (see [Ayo17, § 3.3]). A consequence of the existence of the realization (6) is that, if the motivic t-structure on $\mathbf{DM}_{gm}(k, \mathbb{Q})$ exists, then its heart is necessarily equivalent to $\mathcal{M}(k)$. Let us mention that an alternative construction of the motivic Galois group $\mathcal{G}_{mot}(k)$ avoiding Nori motives was provided by Ayoub in [Ayo14H1, Ayo14H2] based on his weak Tannakian formalism for monoidal triangulated categories. The fact that the motivic Galois group arising from Ayoub's construction coincides with Nori's Galois group $\mathcal{G}_{mot}(k)$ was proved by U. Chudhurry and M. Gallauer in [CG17] and, up to now, represents the strongest piece of evidence to the general expectations just mentioned; Nori's realization (6) plays a crucial role in the proof.

The theory of Voevodsky motives has been already expanded to a theory of motivic sheaves satisfying a large part of Beilinson's predictions: for every (quasi-projective) $k$-variety $S$ there exists a triangulated category $\mathbf{DM}_{gm}(S, \mathbb{Q})$ of $\mathbb{Q}$-linear geometric motives over $S$; as $S$ varies, they enjoy a complete six functor formalism and admit a compatible weight structure in the sense of Bondarko's paper [Bon14]. The construction of the six functors in the motivic world was sketched by V. Voevodsky himself in his lecture [Del01], developed systematically in J. Ayoub's thesis work [Ayo07a, Ayo07b], and further refined in D.-C. Cisinski and F. Déglise's book [CD19]. By [Ayo14E, Thm. B.1], Voevodsky's category $\mathbf{DM}_{gm}(S, \mathbb{Q})$ is canonically equivalent to Ayoub's category $\mathbf{DA}_{ct}^{\text{ét}}(S, \mathbb{Q})$ of constructible étale motives over $S$ without transfers defined in [Ayo14E, § 3]; by [CD19, Thm. 16.2.18], the latter is also equivalent to the category of constructible Beilinson motives defined in [CD19, § IV]. The theory of [IM19] is developed using mostly Ayoub's categories $\mathbf{DA}_{ct}^{\text{ét}}(S, \mathbb{Q})$ and adopting the language of *stable homotopy 2-functors* developed in [Ayo07a, Ayo07b], [Ayo10] and [Ayo14E]. For every (quasi-projective) $k$-variety $S$, Ayoub constructed in [Ayo10] a triangulated Betti realization

$$ (8) \qquad \text{Bti}_S^* : \mathbf{DA}_{ct}^{\text{ét}}(S, \mathbb{Q}) \to \mathcal{D}_c^b(S, \mathbb{Q}) $$

into the usual algebraically constructible derived category $\mathcal{D}_c^b(S, \mathbb{Q})$ associated to the complex-analytic space $S^\sigma$. By [Ayo10, Thm. 3.19], as $S$ varies the functors (8) are canonically compatible with the six operations on the two sides, thereby defining a morphism of stable homotopy 2-functors

$$ (9) \qquad \text{Bti}^* : \mathbf{DA}_{ct}^{\text{ét}}(\cdot, \mathbb{Q}) \to \mathcal{D}_c^b(\cdot, \mathbb{Q}). $$

Moreover, by [Ayo10, Prop. 4.7], the realization functors (8) are also compatible with nearby and vanishing cycles functors on the two sides.

The $\mathbb{Q}$-linear abelian category of perverse Nori motives $\mathcal{M}(S)$ is defined in [IM19, § 2.1] by taking the universal abelian factorization of the homological functor

$$ (10) \qquad \beta_S : \mathbf{DA}_{ct}^{\text{ét}}(S, \mathbb{Q}) \xrightarrow{\text{Bti}_S^*} \mathcal{D}_c^b(S, \mathbb{Q}) \xrightarrow{\mathcal{P}H^0} \text{Perv}(S), $$
which may be also regarded as a mere quiver representation. The abelian category $\mathcal{M}(S)$ comes equipped with the faithful exact functor (2), which extends to the conservative triangulated functor (3). As a consequence of Nori's results mentioned above, the category $\mathcal{M}(S)$ recovers Nori's original category $\mathcal{M}(k)$ when $S = \text{Spec}(k)$. By [IM19, Thm. 5.1], the triangulated categories $D^b(\mathcal{M}(S))$ canonically assemble into a stable homotopy 2-functor in the sense of [Ayo07a], and the triangulated functors (3) canonically assemble into a morphism of stable homotopy 2-functors

$$ (11) \qquad \iota : D^b(\mathcal{M}(\cdot)) \to D^b(\mathrm{Perv}(\cdot)) $$

in the sense of [Ayo10]. Furthermore, as proved in [IM19, § 3], the theory of perverse Nori motives is endowed with the analogue of Beilinson's gluing functors from [Bei87G]. These functors, whose definition builds upon the theory of specialization systems developed in [Ayo07b], play a crucial role in the construction of motivic inverse and direct images obtained in [IM19, §§ 4, 5].

**Main results.** Our main result asserts that the four functor formalism on perverse Nori motives constructed in [IM19] admits a canonical enhancement to a complete six functor formalism. In the language of stable homotopy 2-functors, it can be conveniently summarized as follows:

**Theorem (Theorem 2.1, Theorem 5.1).** The stable homotopy 2-functor $D^b(\mathcal{M}(\cdot))$ obtained in [IM19, Thm. 5.1] is canonically closed unitary symmetric monoidal. Moreover, the morphism of stable homotopy 2-functors (11) is canonically a unitary symmetric monoidal morphism which also respects internal homomorphisms.

The first intermediate result is the construction of the monoidal structure. The basic principle - as always, in the theory of Nori motives - is that the universal property of the categories $\mathcal{M}(S)$ can be exploited to lift functors from the underlying constructible derived categories, provided these functors can be expressed in terms of the six functor formalism and are t-exact for the perverse t-structures. One basic technical issue is that the internal tensor product

$$ - \otimes - : D^b(\mathcal{M}(S)) \times D^b(\mathcal{M}(S)) \to D^b(\mathcal{M}(S)) $$

is not t-exact for the perverse t-structures unless $\dim(S) = 0$. The abstract theory developed in [Ter23T] allows us to overcome this issue by working with the external tensor product

$$ - \boxtimes - : D^b(\mathcal{M}(S_1)) \times D^b(\mathcal{M}(S_2)) \to D^b(\mathcal{M}(S_1 \times S_2)) $$

which is t-exact in each variable by [BBD82, § 4.2.7]. But this is still not enough: as explained in the previous paragraph, lifting the external tensor product directly to $\mathcal{M}(S)$ is not allowed even over $S = \text{Spec}(k)$, for the simple reason that, given two complexes $K_1^\bullet \in D^b(\text{Perv}(S_1))$ and $K_2^\bullet \in D^b(\text{Perv}(S_2))$, the equality

$$ {}^pH^0(K_1^\bullet) \boxtimes {}^pH^0(K_2^\bullet) = {}^pH^0(K_1^\bullet \boxtimes K_2^\bullet) $$

does not hold unless one among $K_1^\bullet$ and $K_2^\bullet$ is concentrated in perverse degree 0. In fact, this is just a different incarnation of the Künneth formula issue already encountered for the tensor product on $\mathcal{M}(k)$; of course, defining Nori motives starting from the quiver of Voevodsky motives $\mathbf{DM}_{gm}(k, \mathbb{Q})$ instead than from Nori's original quiver $\mathrm{Pairs}_k$ does not remove this technical issue. By mimicking Nori's strategy, for every k-variety $S$ we introduce a convenient abelian subcategory $\mathcal{M}^0(S)$ of $\mathcal{M}(S)$ on which the external tensor product can be defined directly, and we then show that the natural inclusion $\mathcal{M}^0(S) \subset \mathcal{M}(S)$ is in fact an equivalence. Here is the precise statement:

**Theorem (Theorem 1.12).** For every (quasi-projective) k-variety $S$, consider the full additive subcategory

$$ \mathbf{DA}_{ct}^{\acute{e}t, 0}(S, \mathbb{Q}) := \{ A \in \mathbf{DA}_{ct}^{\acute{e}t}(S, \mathbb{Q}) \mid \mathrm{Bti}_S^*(A) \in \mathrm{Perv}(S) \} \subset \mathbf{DA}_{ct}^{\acute{e}t}(S, \mathbb{Q}), $$
and let $\mathcal{M}^0(S)$ denote the universal abelian factorization of the additive functor

$$ \beta_S^0 := \beta_S|_{\mathcal{D}\mathcal{A}_{ct}^{\text{et},0}(S, \mathbb{Q})} : \mathcal{D}\mathcal{A}_{ct}^{\text{et},0}(S, \mathbb{Q}) \to \text{Perv}(S). $$

Then the canonical faithful exact functor $\mathcal{M}^0(S) \hookrightarrow \mathcal{M}(S)$ is in fact an equivalence.

Note that our definition of $\mathcal{D}\mathcal{A}_{ct}^{\text{et},0}(S, \mathbb{Q})$ is just the natural analogue of Nori's definition of the subquiver Pairs$_k^0$ in the setting of Voevodsky motives. In order to make our constructions uniform, it is convenient to view the original category $\mathcal{M}(S)$ as the universal abelian factorization of (10) regarded as an additive functor. Unfortunately, we are not able prove our comparison result by generalizing Nori's geometric ideas over arbitrary base varieties - although we believe that this should be possible. Instead, we use Nori's equivalence over fields as the starting step of a Noetherian induction argument, the inductive step of which exploits the motivic version of Beilinson's gluing functors obtained in [IM19, § 3].

Once the single external tensor product functors are constructed, turning their collection into the sought-after monoidal structure amounts to constructing monoidality isomorphisms along morphisms of k-varieties as well as compatible associativity, commutativity and unit constraints. Using the abstract theory developed in [Ter23F] and [Ter23E], most of the required constructions can be reduced to the level of abelian categories, where the universal property of perverse Nori motives can be exploited to lift natural transformations between exact functors. The most delicate point is constructing the unit constraint and checking its compatibility with the rest of the structure: this requires a closer analysis of the tensor product with certain distinguished motivic local systems. The consequences of this analysis play a crucial role also in the construction of internal homomorphisms. The latter result is quite lengthy and technical, but it is based on a simple strategy (inspired to a construction of D. Arapura from [Ara13, § 5]): in a first step, we extend part of the functoriality of perverse Nori motives to the derived ind-categories $D(\text{Ind.}\mathcal{M}(S))$, which allows us to define internal homomorphisms functors abstractly as formal right adjoints of the tensor product; in a second step, we show that these formal internal homomorphisms respect the bounded derived categories $D^b(\mathcal{M}(S))$ and moreover are compatible with the usual internal homomorphisms on the underlying categories of perverse sheaves. Our analysis of the tensor product with distinguished motivic local systems, combined with repeated use of localization triangles, is the key ingredient for the second step.

One of the most interesting consequences of the six functor formalism on perverse Nori motives is the existence of well-behaved Tannakian categories of motivic local systems over suitable bases. It can be stated as follows:

**Theorem (Theorem 6.2).** For every smooth, geometrically connected k-variety $S$ of dimension $d$, the full subcategory

$$ \mathcal{M}\text{Loc}_p(S) := \{M \in \mathcal{M}(S) \mid \iota_S(M) \in \text{Loc}(S)[d]\} \subset \mathcal{M}(S) $$

is canonically a neutral Tannakian category over $\mathbb{Q}$.

The smoothness assumption in the statement is necessary to ensure that ordinary local systems over $S$ are perverse sheaves up to shifting by $d$.

Our categories of motivic local systems provide the ambient categories for E. Jacobsen's motivic version of Hain's theorem described in [Jac23, Thm. 8.4]. We plan to study the categories $\mathcal{M}\text{Loc}_p(S)$ and their associated motivic Galois groups systematically in future work.

**Related and future work.** In the same period when the results of the present paper were completed, significant progress in the theory of perverse Nori was achieved in a complementary direction by work of S. Tubach. Building upon Nori's classical argument for the derived category of constructible sheaves from [Nor18], he shows in [Tub23] that the triangulated category $D^b(\mathcal{M}(S))$ is canonically equivalent to the bounded derived category of its so-called constructible heart (see [IM19, § 5.4] for the definition of the latter). The main consequence of this result is that the six functors on perverse Nori motives can be
canonically enhanced to $\infty$-functors. In addition to allowing for an extension of the theory of perverse Nori motives to not necessarily quasi-projective $k$-varieties, this implies the existence of a canonical morphism of stable homotopy 2-functors

$$ (12) \qquad \mathrm{Nri}^* : \mathbf{D}\mathbf{A}_{ct}^{\mathrm{et}}(\cdot, \mathbb{Q}) \to \mathcal{D}^b(\mathcal{M}(\cdot)) $$

lifting the Betti realization (9); in fact, with the $\infty$-categorical lifting of the six operations at hand, both the existence and the essential uniqueness of (12) follow immediately from B. Drew and M. Gallauer's universality theorem for triangulated motives from [DG22].

Let us stress that even constructing the single realization functors $\mathrm{Nri}_S^*$ over general $S$ directly seems a hard task. To the author's knowledge, the only plausible candidate construction available to date is the one proposed by Ivorra in [Ivo17], which only works over smooth bases. Ivorra's construction is based on cellular complexes of perverse sheaves over $S$ attached to relative smooth affine $S$-schemes; it factors trough another abelian category of perverse Nori motives introduced by Ivorra in [Ivo17], which is conjectured to be equivalent to $\mathcal{M}(S)$ although, a priori, it is only a non-full subcategory of the latter. The main problem with Ivorra's realization functors is that neither their compatibility with the Betti realization nor their compatibility with the six operations are fully understood yet (mostly due to concrete technical issues concerning filtrations of varieties over a base). The expectation is that Ivorra's realization functors should recover the functors (12) over smooth bases; in fact, this would be a consequence of the expected compatibility of Ivorra's functors with the Betti realization and with the six operations.

The results of [Tub23] are related to our results in two ways. On the one hand, the existence of a canonical closed monoidal structure on perverse Nori motives is used crucially in Tubach's variation of Nori's argument as described in [Tub23, § 2]. On the other hand, the existence of the triangulated realization (12) allows us to derive some pleasant complementary results concerning the motivic Verdier duality functors: we show that these functors, which were defined in [IM19, § 2.8] purely in terms of the universal property of perverse Nori motives, admit the usual description in terms of internal homomorphisms to dualizing complexes; and that, as a consequence, internal homomorphisms of perverse Nori motives admit the classical reformulation in terms of tensor product and Verdier duality. Curiously enough, we were able to establish the latter result directly in some cases: this is Corollary 3.13, which in turn was a crucial ingredient in our construction of internal homomorphisms!

In order to make the whole picture terser, it would be useful to have an alternative, more direct construction of perverse Nori motives at our disposal. In his recent preprint [Ayo21], J. Ayoub constructed a new stable homotopy 2-functor enhancing the category $\mathcal{D}^b(\mathcal{M}(k))$; while Ayoub's six functor formalism is obtained all at once and comes equipped with a canonical $\infty$-categorical enhancement, it does not manifestly enjoy a universal property analogous to that of Ivorra and Morel's categories. In an ongoing project (joint with E. Jacobsen) we aim at showing that Ayoub's triangulated categories are equivalent to Ivorra and Morel's categories $\mathcal{D}^b(\mathcal{M}(S))$ compatibly with the six functor formalism. A significant step towards this goal consists in giving a precise meaning to the intuition that the Tannakian categories $\mathcal{M}\mathrm{Loc}_p(S)$ should be thought of as categories of $\mathcal{M}(k)$-valued local systems of geometric origin.

**Structure of the paper.** Most constructions of the present paper are based on the abstract properties of perverse motives as universal abelian factorizations. The basic functoriality results for universal abelian factorizations used in the setting of perverse motives are quickly discussed in [IM19, § 2.1], based on the detailed results of [Ivo17]. A more systematic treatment, which also includes the extension of the above discussion to multi-linear functors, is given in our note [Ter23U], which we will often refer to.

In Section 1 we recall and extend the discussion of [IM19, §§ 3, 4] concerning Beilinson's gluing functors in the setting of perverse motives. Our first goal is to obtain a gluing result for the abelian categories of perverse motives analogous to Beilinson's classical result for perverse sheaves (Proposition 1.7). We deduce an abstract comparison criterion asserting that any system of abelian subcategories $\mathcal{M}^0(S) \subset \mathcal{M}(S)$ which
is stable under the gluing functors and induces equivalences over the generic points of all $k$-varieties must be in fact a system of equivalences (Proposition 1.9). We then construct a system of abelian subcategories $\mathcal{M}^0(S)$ (Notation 3.1) which, by design, are more adapted to defining the external tensor product functors than the original categories $\mathcal{M}(S)$, and we show that they satisfy the assumptions of our abstract comparison criterion (Theorem 1.12). This allows us to construct the tensor structure on perverse motives by working with the categories $\mathcal{M}^0(S)$ in place of the categories $\mathcal{M}(S)$. Along the way, we study the behavior of Nori motives over function fields of $k$-varieties in some detail (Proposition 1.15 and Corollary 1.17).

In Section 2 we state our first main result concerning the existence of the monoidal structure on perverse motives together with its associativity, commutativity and unit constraints (Theorem 2.1). After some general categorical preliminaries (Proposition 2.6), we construct the whole monoidal structure except for the unit constraint: in the first place, we construct the external monoidality isomorphisms (Proposition 2.8) by exploiting the reduction results of [Ter23F]; we then endow the external tensor product with compatible associativity and commutativity constraints (Lemma 2.11, Lemma 2.12 and Lemma 2.13).

Section 3 is a crucial technical interlude needed both for the unit constraint and for internal homomorphisms of perverse motives. We introduce, for every smooth $k$-variety $S$, a further abelian subcategory $\mathcal{M}^{0,loc}(S)$ of $\mathcal{M}^0(S)$ (Notation 3.1) and we study the restriction of the internal tensor product to $\mathcal{M}^0(S) \times \mathcal{M}^{0,loc}(S)$: the main result is that this functor, which a priori is only defined indirectly from the external tensor product, can be actually described in terms of the universal property of perverse motives (Proposition 3.6). As a consequence, we readily deduce the existence of internal homomorphisms attached to certain 'distinguished motivic local systems' (Corollary 3.13).

In Section 4 we finally complete the proof of Theorem 2.1 by constructing the unit constraint for perverse motives (Proposition 4.1). In the first place, we construct the motivic unit section (Proposition 4.3) by exploiting the reduction results of [Ter23E]. We then use it to construct the motivic unit constraint, and we check that the latter is compatible with the associativity and commutativity constraints defined previously (Lemma 4.8 and Lemma 4.9).

In Section 5 we prove the existence of internal homomorphisms for perverse motives (Theorem 5.1). The strategy consists of two main steps: firstly, we construct an abstract ind-adjoint to the tensor product functors (Construction 5.10); secondly, using the partial results obtained at the end of Section 3, we check that these functors respect the bounded derived categories and that they are compatible with the usual internal homomorphisms on the underlying constructible derived categories by applying an abstract criterion (Proposition 5.14 and Proposition 5.15). To conclude, we explain how our results, combined with Tubach's results from [Tub23], imply the expected relations between Verdier duality and internal homomorphisms in the setting of perverse motives (Proposition 5.19).

In the final Section 6 we introduce abelian categories of motivic local systems over smooth, geometrically connected $k$-varieties (Definition 6.1) and we show that they are canonically neutral Tannakian over $\mathbb{Q}$ (Theorem 6.2). As an application in the direction tensor-triangulated geometry, we describe a topological reconstruction result for perverse motives (Theorem 6.5).

**Acknowledgments.** The contents of this paper correspond to the fourth and central chapter of my Ph.D. thesis, written at the University of Freiburg under the supervision of Annette Huber-Klawitter. It is a pleasure to thank her for introducing me to the subject of Nori motives and suggesting to me the problem of constructing the tensor structure on perverse motives, for following my progresses on it closely, as well as for her constant support and encouragement. I would also like to thank Joseph Ayoub, Denis-Charles Cisinski, Martin Gallauer, Fritz Hörmann, Florian Ivorra, Emil Jacobsen, Sophie Morel, Wolfgang Soergel, and Swann Tubach for useful discussions around the subject of this article.
## NOTATION AND CONVENTIONS

### Fibered categories.

• We follow the notation and conventions on fibered categories used in [Ayo07a] and in [CD19], as displayed explicitly in [Ter23T, § 1].

### Algebraic Geometry.

• We fix a field $k$ of characteristic 0 endowed with a fixed field embedding $\sigma: k \hookrightarrow \mathbb{C}$.

• $\mathbf{Var}_k$ denotes the category of algebraic varieties over $k$: objects are reduced, separated, finite type $k$-schemes; morphisms are $k$-morphisms.

• By the closed immersion *complementary* to a given open immersion $j: U \hookrightarrow S$ in $\mathbf{Var}_k$ we mean the closed immersion $i: Z \hookrightarrow S$ where $Z$ denotes the closed complement of $U$ in $S$ endowed with its reduced scheme structure.

• We consider the following full subcategories of $\mathbf{Var}_k$:

– $\mathbf{Sm}_k$: the subcategory of smooth $k$-varieties;

– $\mathbf{Aff}_k$: the subcategory of affine $k$-varieties.

• For every $S \in \mathbf{Var}_k$ we write:

– $a_S: S \to \mathrm{Spec}(k)$ for the structural morphism;

– $\mathcal{O}(S)$ for the $k$-algebra of global regular functions on $S$;

– $\mathbf{Op}(S)$ for the full subcategory of the slice category $\mathbf{Sm}_k/S$ whose objects are the Zariski-dense open immersions into $S$;

– $\mathbf{AffOp}(S)$ for the full subcategory $\mathbf{Aff}_k \subset \mathbf{Op}(S)$;

– $\mathbf{SmOp}(S)$ for the full subcategory $\mathbf{Sm}_k \subset \mathbf{Op}(S)$.

### Analytic sheaves.

• For every $S \in \mathbf{Var}_k$, we use the following notation:

– $S^\sigma$ denotes the complex analytic space whose points are the $\mathbb{C}$-points of $S$ over $k$ via $\sigma$.

– $\mathcal{C}t(S)$ denotes the abelian category of sheaves of $\mathbb{Q}$-vector spaces on $S^\sigma$ which are algebraically constructible.

– $\mathrm{Loc}(S)$ denotes the full abelian subcategory of $\mathcal{C}t(S)$ consisting of all locally constant sheaves.

– $D_c^b(S, \mathbb{Q})$ denotes the derived category of complexes of sheaves of $\mathbb{Q}$-vector spaces on $S^\sigma$ with bounded algebraically constructible cohomology.

– $\mathrm{Perv}(S)$ denotes the heart of the self-dual perverse $t$-structure on $D_c^b(S, \mathbb{Q})$ as defined in [BBD82], and ${}^pH^0: D_c^b(S, \mathbb{Q}) \to \mathrm{Perv}(S)$ denotes the corresponding homological functor.

### $\ell$-adic sheaves.

• Let $\ell$ be a fixed prime number. For every $S \in \mathbf{Var}_k$, we use the following notation:

– $\mathcal{C}t(S)_\ell$ denotes the abelian category of $\ell$-adic algebraically constructible sheaves on the small étale site $S_{\ell t}$.

– $\mathrm{Loc}(S)_\ell$ denotes the full abelian subcategory of $\mathcal{C}t(S)_\ell$ consisting of all lisse sheaves.

– $D_{c,\ell}^b(S, \mathbb{Q}_\ell)$ denotes the constructible $\ell$-adic derived category as defined by Ekedahl in [Eke90] or equivalently by Bhatt-Scholze in [BS15].

– $\mathrm{Perv}(S)_\ell$ denotes the heart of the self-dual perverse $t$-structure on $D_{c,\ell}^b(S, \mathbb{Q}_\ell)$ as defined in [Mor19, § 2.2], and ${}^pH^0: D_{c,\ell}^b(S, \mathbb{Q}_\ell) \to \mathrm{Perv}(S)_\ell$ denotes the corresponding homological functor.
## Triangulated motives and realizations.

* For every $S \in \mathrm{Var}_k$, we use the following notation:

– $\mathbf{DA}^{\mathrm{\acute{e}t}}_{ct}(S, \mathbb{Q})$ denotes the triangulated category of constructible étale motives with $\mathbb{Q}$-coefficients constructed in [Ayo07b, § 4.5.2] (see also [Ayo14E, § 3]), implicitly identified with the category of constructible Beilinson motives defined in [CD19, § IV] via the equivalence of [CD19, Thm. 16.2.18].

– We let

$$ \mathrm{Bti}_S^* : \mathbf{DA}^{\mathrm{\acute{e}t}}_{ct}(S, \mathbb{Q}) \to D_{c,\ell}^b(S, \mathbb{Q}) $$

denote the *Betti realization* over $S$ associated to $\sigma$ constructed in [Ayo10].

– Let $\ell$ be a fixed prime number. We let

$$ R_{\ell,S} : \mathbf{DA}^{\mathrm{\acute{e}t}}_{ct}(S, \mathbb{Q}) \to D_{c,\ell}^b(S, \mathbb{Q}) $$

denote the $\ell$-adic realization over $S$ constructed in [Ayo14E].

## Universal abelian factorizations.

* For every additive category $\mathcal{D}$, we write:

– $\mathbf{R}(\mathcal{D})$ for the full subcategory of $\mathrm{Fun}(\mathcal{D}^{op}, \mathbf{Ab})$ consisting of all presheaves of finite presentation (i.e. cokernels of representable presheaves).

– $\mathbf{L}(\mathcal{D}) := \mathbf{R}(\mathcal{D}^{op})^{op}$.

– $\mathbf{A}(\mathcal{D}) := \mathbf{L}(\mathbf{R}(\mathcal{D}))$ for Freyd's abelian hull of $\mathcal{D}$ introduced in [Fre65] (see also [Ter23U, § 1]).

* For every additive functor $\beta : \mathcal{D} \to \mathcal{A}$ from an additive category $\mathcal{D}$ to an abelian category $\mathcal{A}$, we write:

– $\beta^+ : \mathbf{A}(\mathcal{D}) \to \mathcal{A}$ for the unique exact functor extending $\beta$;

– $\mathbf{A}(\beta) : \mathbf{A}(\mathcal{D}) / \ker(\beta^+)$ for the associated universal abelian factorization.

## Perverse Nori motives.

* For every $S \in \mathrm{Var}_k$, we write:

– $\mathcal{D}(S)$ for the underlying additive category of $\mathbf{DA}^{\mathrm{\acute{e}t}}_{ct}(S, \mathbb{Q})$;

– $\beta_S : \mathcal{D}(S) \to \mathrm{Perv}(S)$ for the homological functor

$$ \mathbf{DA}^{\mathrm{\acute{e}t}}_{ct}(S, \mathbb{Q}) \xrightarrow{\mathrm{Bti}_S^*} D_c^b(S, \mathbb{Q}) \xrightarrow{\nu H^0} \mathrm{Perv}(S) $$

regarded as an additive functor.

– $\mathcal{M}(S)$ for the category of *perverse Nori motives* over $S$, defined as the universal abelian factorization $\mathbf{A}(\beta_S)$ of the additive functor $\beta_S$.

– $\pi_S$ for the quotient functor $\mathbf{A}(\mathcal{D}(S)) \to \mathcal{M}(S)$, and also for the composite functor $\mathcal{D}(S) \to \mathbf{A}(\mathcal{D}(S)) \to \mathcal{M}(S)$,

– $\iota_S$ for the induced faithful exact functor $\mathcal{M}(S) \hookrightarrow \mathrm{Perv}(S)$.

# 1. BEILINSON GLUING FOR PERVERSE MOTIVES

Our first main goal is to obtain an alternative description of perverse Nori motives which is more adapted to defining the external tensor product. The precise result is Theorem 1.12 below, which can be regarded as a generalization of Nori's philosophy of *good pairs* (see [HMS17, § II.9]) to arbitrary base varieties. The proof relies on an abstract comparison criterion formalized as Proposition 1.9; this in turn is based on Proposition 1.7, a motivic analogue of the classical gluing formalism for perverse sheaves introduced by Beilinson in [Bei87G].

Before stating the main results, we review the construction of [IM19, § 3.5] which provides all the essential ingredients. We then obtain the motivic gluing formalism by simply adapting Beilinson's classical argument
to the motivic setting; our presentation is modeled on Morel's note [Mor18]. The proof of the final result is preceded by a detailed discussion about Nori motives over function fields, which leads to the auxiliary results Proposition 1.15 and Corollary 1.17. In our opinion, these results are of independent interest and represent a useful complement to [IM19, § 6.5].

1.1. Review of the motivic gluing functors. We start by recalling the relevant results from [IM19, § 3.5]. To begin with, fix a variety $S \in \mathrm{Var}_k$ and a regular function $f \in \mathcal{O}(S)$ with vanishing locus $Z$; let $i: Z \hookrightarrow S$ denote the corresponding closed immersion, and let $j: U \hookrightarrow S$ denote the complementary open immersion. Since $j$ is affine by construction, by [IM19, § 2.4] there exist canonical exact functors

$$j_!, j_*: \mathcal{M}(U) \to \mathcal{M}(S)$$

which mimic the formal properties of the underlying functors on perverse sheaves, namely: they are left and right adjoint to the inverse image functor $j^*: \mathcal{M}(S) \to \mathcal{M}(U)$, respectively; they are fully faithful; they are intertwined under the motivic Verdier duality functors of [IM19, § 2.8].

As shown in [IM19, § 3.5], the classical gluing functors for perverse sheaves defined in [Bei87G] can be canonically lifted to exact functors

$$^p\Psi_f: \mathcal{M}(U) \to \mathcal{M}(Z), \quad ^p\Phi_f: \mathcal{M}(S) \to \mathcal{M}(Z),$$

$$^p\Xi_f: \mathcal{M}(S) \to \mathcal{M}(S), \quad ^p\Omega_f: \mathcal{M}(S) \to \mathcal{M}(S).$$

In the setting of perverse sheaves, the functors $^p\Psi_f$ and $^p\Phi_f$ are the classical unipotent nearby cycles and vanishing cycles functors (shifted by $-1$, so that they become $t$-exact for the perverse $t$-structures), while the functor $^p\Xi_f$ is Beilinson's maximal extension functor (or rather, the composite of the latter with $j^*$). The functor $\Omega_f$ first appeared in Saito's work on mixed Hodge modules [Sai90].

Remark 1.1. In [IM19] the functor $^p\Psi_f: \mathcal{M}(U) \to \mathcal{M}(Z)$ is denoted $^p\mathrm{Log}_f$, since it comes from Ayoub's logarithmic specialization system constructed in [Ayo07b, § 3.6].

The functors $^p\Psi_f$ and $^p\Xi_f$ fit into short exact sequences of functors $\mathcal{M}(S) \to \mathcal{M}(S)$

$$(13) \qquad 0 \to j_*j^*M \to {}^p\Xi_f(M) \to i_*{}^p\Psi_f(j^*M)(-1) \to 0$$

and

$$(14) \qquad 0 \to i_*{}^p\Psi_f(j^*M) \to {}^p\Xi_f(M) \to j_*j^*M \to 0$$

which are intertwined under motivic Verdier duality; thus the composite $j_*j^*M \to {}^p\Xi_f(M) \to j_*j^*M$ coincides with the composite $j_*j^*M \xrightarrow{\epsilon} M \xrightarrow{\eta} j_*j^*M$. There are two further relevant short exact sequences of functors $\mathcal{M}(S) \to \mathcal{M}(S)$

$$(15) \qquad 0 \to i_*{}^p\Psi_f(j^*M) \to {}^p\Omega_f(M) \to M \to 0$$

and

$$(16) \qquad 0 \to j_*j^*M \to {}^p\Omega_f(M) \to i_*{}^p\Phi_f(M) \to 0.$$

The functors $^p\Omega_f$ and $^p\Xi_f$ are also related by a canonical natural transformation of functors $\mathcal{M}(S) \to \mathcal{M}(S)$

$${}^p\Omega_f(M) \to {}^p\Xi_f(M)$$

which allows one to combine the exact sequences (14) and (15) into a commutative diagram of the form

$$(17) \qquad \begin{array}{ccc} 0 & \to & i_*{}^p\Psi_f(j^*M) \to {}^p\Xi_f(M) \to j_*j^*M \to 0 \\ \parallel & & \uparrow \\ 0 & \to & i_*{}^p\Psi_f(j^*M) \to {}^p\Omega_f(M) \to M \to 0. \end{array}$$
All these fact were noted in [IM19, § 3.5]. Our first task is to deduce further information about the mutual relations among the gluing functors.

In the proof of the following results, we often need to use the fully faithfulness of the exact functor $i_* : \mathcal{M}(Z) \to \mathcal{M}(S)$ proven in [IM19, Thm. 4.1].

**Lemma 1.2.** With the above notation, there exist a canonical natural isomorphism between functors $\mathcal{M}(S) \to \mathcal{M}(S)$

$$ (18) \qquad \mathcal{P}\Omega_f(M) = \ker \{ \mathcal{P}\Xi_f(M) \oplus M \twoheadrightarrow j_*j^*M \} $$

and a canonical natural isomorphism between functors $\mathcal{M}(S) \to \mathcal{M}(Z)$

$$ (19) \qquad \mathcal{P}\Phi_f(M) = i^*H^0[j_*j^*M \xrightarrow{\pm} \mathcal{P}\Xi_f(M) \oplus M \twoheadrightarrow j_*j^*M]. $$

*Proof.* Regard the commutative diagram (17) as a double complex. Since both rows are exact, we deduce that its total complex is exact as well. Hence the total complex of the truncated double complex

$$ \begin{array}{ccc} 0 & \to & \mathcal{P}\Xi_f(M) \to j_*j^*M \to 0 \\ \parallel & & \uparrow \\ 0 & \to & \mathcal{P}\Omega_f(M) \end{array} \qquad \begin{array}{ccc} \eta & \to & \mathcal{P}\Xi_f(M) \oplus M \end{array} $$

is exact as well. This yields the natural isomorphism (18). Moreover, since the arrow $j_*j^*M \to \mathcal{P}\Xi_f(M)$ in the short exact sequence (13) is a monomorphism, the canonical arrow

$$ \mathrm{im} \{ j_*j^*M \xrightarrow{\pm} \mathcal{P}\Xi_f(M) \oplus M \} \to \mathrm{im} \{ j_*j^*M \to \mathcal{P}\Xi_f(M) \} $$

is an isomorphism. This yields a natural isomorphism

$$ i_* \mathcal{P}\Phi_f(M) = \mathrm{coker} \{ j_*j^*M \to \mathcal{P}\Omega_f(M) \} \qquad \text{by (16)} $$

$$ = \mathrm{coker} \{ j_*j^*M \xrightarrow{\pm} \mathrm{ker} \{ \mathcal{P}\Xi_f(M) \oplus M \twoheadrightarrow j_*j^*M \} \} \qquad \text{by (18)} $$

$$ = H^0[j_*j^*M \xrightarrow{\pm} \mathcal{P}\Xi_f(M) \oplus M \twoheadrightarrow j_*j^*M]. $$

Thus we obtain the natural isomorphism (19) as the composite

$$ \mathcal{P}\Phi_f(M) = i^*i_* \mathcal{P}\Phi_f(M) = i^*H^0[j_*j^*M \xrightarrow{\pm} \mathcal{P}\Xi_f(M) \oplus M \twoheadrightarrow j_*j^*M], $$

where the first passage follows from the fully faithfulness of $i_*$. $\square$

The following consequence plays a fundamental role in the construction of motivic inverse images under closed immersions described in [IM19, § 4.1]. We spell it out since it is relevant for our constructions as well.

**Corollary 1.3.** With the above notation, there exists a canonical isomorphism of functors $\mathcal{M}(Z) \to \mathcal{M}(Z)$

$$ (20) \qquad \mathcal{P}\Phi_f(i_*M) = M. $$

In other words, the functor $\mathcal{P}\Phi_f$ is canonically left-inverse to $i_*$.

*Proof.* Combining any of the two short exact sequences (13) and (14) with the vanishing $j^* \circ i_* = 0$, we deduce the vanishing $\mathcal{P}\Xi_f \circ i_* = 0$. Hence we obtain the sought-after isomorphism (20) as the composite

$$ \begin{align*} \mathcal{P}\Phi_f(i_*M) &= i^*H^0[j_*j^*i_*M \xrightarrow{\pm} \mathcal{P}\Xi_f(i_*M) \oplus i_*M \twoheadrightarrow j_*j^*i_*M] && \text{by (19)} \\ &= i^*H^0[0 \to 0 \oplus i_*M \oplus 0] && \text{as } j^* \circ i_* = 0 \\ &= i^*i_*M = M, \end{align*} $$
where the last passage follows again from the fully faithfulness of $i_*$. $\square$

With these results at our disposal, we can lift the classical nilpotent monodromy operator $N_f$, together with its canonical factorization (see [Mor18, §§ 2, 6]) to the setting of perverse motives.

**Construction 1.4.** Keep the above notation. Consider the composite natural transformation of functors $\mathcal{M}(S) \to \mathcal{M}(S)$

$$ (21) \qquad i_*{}^p \Psi_f(j^*M) \hookrightarrow {}^p \Xi_f(M) \to i_*{}^p \Psi_f(j^*M)(-1), $$

where the two arrows come from the short exact sequences (13) and (14), respectively. Since the functor $i_* : \mathcal{M}(Z) \to \mathcal{M}(S)$ is fully faithful, there is a unique natural transformation of functors $\mathcal{M}(S) \to \mathcal{M}(Z)$

$$ (22) \qquad N_f : {}^p \Psi_f(j^*M) \to {}^p \Psi_f(j^*M)(-1) $$

whose image under $i_*$ coincides with (21). We look for a canonical factorization of $N_f$ through the functor ${}^p \Phi_f$. To this end, note that the arrow $i_*{}^p \Psi_f(j^*M) \hookrightarrow {}^p \Xi_f(M)$ factors through ${}^p \Omega_f(M) = \ker \{ {}^p \Xi_f(M) \oplus M \twoheadrightarrow j_*j^*M \}$, because we have

$$ \operatorname{Hom}_{\mathcal{M}(S)}(i_*{}^p \Psi_f(j^*M), j_*j^*M) = \operatorname{Hom}_{\mathcal{M}(U)}(j^*i_*{}^p \Psi_f(j^*M), j^*M) = 0 $$

as $j^* \circ i_* = 0$. Hence we obtain a canonical natural transformation of functors $\mathcal{M}(S) \to \mathcal{M}(Z)$

$$ (23) \qquad \operatorname{can}_f : {}^p \Psi_f(j^*M) \to {}^p \Phi_f(M) $$

by declaring that its image under the fully faithful functor $i_* : \mathcal{M}(Z) \to \mathcal{M}(S)$ be the induced arrow

$$ \begin{tikzcd} i_*{}^p \Psi_f(j^*M) \arrow[r] \arrow[d] & i_*{}^p \Phi_f(M) \arrow[d] \\ \ker \{ {}^p \Xi_f(M) \oplus M \twoheadrightarrow j_*j^*M \} \arrow[r, "\simeq"] & H^0[j_*j^*M \xrightarrow{\pm} {}^p \Xi_f(M) \oplus M \twoheadrightarrow j_*j^*M] \arrow[l] \arrow[r] \arrow[d] \end{tikzcd} $$

Dually, note that the arrow ${}^p \Xi_f(M) \to i_*{}^p \Psi_f(M)(-1)$ is trivial on $\operatorname{im} \{ j_*j^*M \xrightarrow{\pm} {}^p \Xi_f(M) \oplus M \}$, because we have

$$ \operatorname{Hom}_{\mathcal{M}(S)}(j_*j^*M, i_*{}^p \Psi_f(M)) = \operatorname{Hom}_{\mathcal{M}(U)}(j^*M, j^*i_*{}^p \Psi_f(M)) = 0 $$

as $j^* \circ i_* = 0$. Hence we obtain a canonical natural transformation of functors $\mathcal{M}(S) \to \mathcal{M}(Z)$

$$ (24) \qquad \operatorname{var}_f : {}^p \Phi_f(M) \to i_*{}^p \Psi_f(j^*M)(-1) $$

by declaring that its image under the fully faithful functor $i_* : \mathcal{M}(Z) \to \mathcal{M}(S)$ be the induced arrow

$$ \begin{tikzcd} i_*{}^p \Phi_f(M) \arrow[r] & i_*{}^p \Psi_f(j^*M) \arrow[r, "\simeq"] \arrow[d] & H^0[j_*j^*M \xrightarrow{\pm} {}^p \Xi_f(M) \oplus M \twoheadrightarrow j_*j^*M] \arrow[d] \\ & \operatorname{coker} \{ j_*j^*M \xrightarrow{\pm} {}^p \Xi_f(M) \oplus M \} \arrow[r, "\twoheadleftarrow"] & \end{tikzcd} $$

By construction, we have the equality of natural transformations

$$ (25) \qquad \operatorname{var}_f \circ \operatorname{can}_f = N_f : {}^p \Psi_f(j^*M) \to {}^p \Psi_f(j^*M)(-1). $$

Before stating the main result about gluing of perverse motives, it is convenient to define one last short exact sequence of gluing functors.
Proof. The fact that $\delta_{S,f}$ is well-defined follows from the relation (25). Let us construct a canonical isomorphism $\delta_{S,f} \circ \gamma_{S,f} \simeq \text{id}_{\mathcal{M}(S,f)}$. Given an object $(M_U, M_Z; u, v) \in \mathcal{M}(S, f)$, write the object $\delta_{S,f}\gamma_{S,f}(M_U, M_Z; u, v) \in \mathcal{M}(S, f)$ as a quadruple $(M'_U, M'_Z; u', v')$. We have a canonical isomorphism

$$
\begin{align*}
M'_U &:= j^* \gamma_{S,f}(M_U, M_Z; u, v) \\
&:= j^* H^0 [i_*{}^p \Psi_f(M_U) \xrightarrow{\pm} {}^p \Xi_f(j!M_U) \oplus i_* M_Z \xrightarrow{\bar{\phantom{\pm}}} i_*{}^p \Psi_f(M_U)(-1)] \\
&= H^0 [j^* i_*{}^p \Psi_f(M_U) \xrightarrow{\pm} j^{*p} \Xi_f(j!M_U) \oplus j^* i_* M_Z \xrightarrow{\bar{\phantom{\pm}}} j^* i_*{}^p \Psi_f(M_U)(-1)] \\
&= H^0 [0 \to j^{*p} \Xi_f(j!M_U) \oplus 0 \to 0] \qquad \text{as } j^* \circ i_* = 0 \\
&= j^{*p} \Xi_f(j!M_U) \\
&= j^* j! M_U \qquad \text{by (13)} \\
&= M_U,
\end{align*}
$$

where the last passage follows from the fully faithfulness of $j!$. Similarly, we have a canonical isomorphism

$$
\begin{align*}
M'_Z &:= {}^p\Phi_f \gamma_{S,f}(M_U, M_Z; u, v) \\
&:= {}^p\Phi_f H^0 [i_*{}^p \Psi_f(M_U) \xrightarrow{\pm} {}^p\Xi_f(j!M_U) \oplus M_Z \xrightarrow{\bar{\phantom{\pm}}} i_*{}^p \Psi_f(M_U)(-1)] \\
&= H^0 [{}^p\Phi_f i_*{}^p \Psi_f(M_U) \xrightarrow{\pm} {}^p\Phi_f {}^p\Xi_f(j!M_U) \oplus {}^p\Phi_f i_* M_Z \xrightarrow{\bar{\phantom{\pm}}} {}^p\Phi_f i_*{}^p \Psi_f(M_U)(-1)] \\
&= H^0 [{}^p\Psi_f(M_U) \xrightarrow{\pm} {}^p\Phi_f {}^p\Xi_f(j!M_U) \oplus M_Z \xrightarrow{\bar{\phantom{\pm}}} {}^p\Psi_f(M_U)(-1)] \quad \text{by (20)} \\
&= M_Z. \quad \text{by (27)}
\end{align*}
$$

The fact that these two isomorphisms glue to an isomorphism in $\mathcal{M}(S, f)$ follows formally from the construction; we omit the details.

Conversely, let us construct a canonical isomorphism $\gamma_{S,f} \circ \delta_{S,f} \simeq \text{id}_{\mathcal{M}(S)}$. Given an object $M \in \mathcal{M}(S)$, we have

$$
\begin{align*}
\delta_{S,f} \gamma_{S,f}(M) &:= H^0 [i_*{}^p \Psi_f(j^*M) \xrightarrow{\pm} {}^p \Xi_f(M) \oplus i_*{}^p \Phi_f(M) \xrightarrow{\bar{\phantom{\pm}}} i_*{}^p \Psi_f(j^*M)(-1)] \\
&= \operatorname{coker} \{i_*{}^p \Psi_f(j^*M) \to O_f(M)\},
\end{align*}
$$

where, for notational convenience, we have set

$$
O_f(M) := \ker \{ {}^p\Xi_f(M) \oplus i_*{}^p\Phi_f(M) \xrightarrow{\bar{\phantom{\pm}}} i_*{}^p\Psi_f(j^*M)(-1) \}.
$$

Applying the Snake Lemma to the commutative diagram of functors $\mathcal{M}(S) \to \mathcal{M}(S)$ with exact rows

$$
\begin{tikzcd}
0 \arrow[r] \arrow[d] & {}^p\Omega_f(M) \arrow[r, "+"'] \arrow[d] & {}^p\Xi_f(M) \oplus {}^p\Omega_f(M) \arrow[d] \arrow[r, "-"'] & {}^p\Xi_f(M) \arrow[d] \arrow[r] & 0 \\
0 \arrow[r] & O_f(M) \arrow[r, "+"'] & {}^p\Xi_f(M) \oplus i_*{}^p\Phi_f(M) \arrow[r, "-"'] & i_*{}^p\Psi_f(j^*M)(-1) \arrow[r] & 0
\end{tikzcd}
$$

and using the short exact sequences (13) and (16), we obtain the vanishing

$$
\ker \{ {}^p\Omega_f(M) \to O_f(M) \} = 0 = \operatorname{coker} \{ {}^p\Omega_f(M) \to O_f(M) \},
$$

from which it follows that the canonical arrow ${}^p\Omega_f(M) \to O_f(M)$ is an isomorphism. Hence we obtain a canonical isomorphism

$$
\delta_{S,f} \gamma_{S,f}(M) = \operatorname{coker} \{i_*{}^p \Psi_f(j^*M) \to O_f(M)\} = \operatorname{coker} \{i_*{}^p \Psi_f(j^*M) \to {}^p\Omega_f(M)\} = M
$$

where the last passage follows from the short exact sequence (15). $\square$
As a consequence of the motivic gluing formalism, we can now state an abstract comparison criterion that we will later apply to obtain the sought-after alternative description of perverse motives.

For sake of clarity, it is convenient to introduce a suitable notion of cofinality for inclusions of fibered categories over a cofiltered small category:

**Definition 1.8.** Let $\mathcal{W}$ be a cofiltered small category. Given a $\mathcal{W}$-fibered category $\mathcal{C}$ and a $\mathcal{W}$-fibered subcategory $\mathcal{C}^0$, we say that the inclusion of $\mathcal{W}$-fibered categories $\mathcal{C}^0 \subset \mathcal{C}$ is weakly cofinal if the canonical functor

$$2 - \varinjlim_{U \in \mathcal{W}^{\mathrm{op}}} \mathcal{C}^0(U) \to 2 - \varinjlim_{U \in \mathcal{W}^{\mathrm{op}}} \mathcal{C}(U)$$

is an equivalence.

Recall that, for every $S \in \mathrm{Var}_k$, we let $\mathrm{SmOp}(S)$ denote the poset of smooth Zariski-dense open subsets of $S$. We can state our comparison criterion as follows:

**Proposition 1.9.** Suppose that we are given, for every $S \in \mathrm{Var}_k$, a (not necessarily full) abelian subcategory $\mathcal{M}^0(S)$ of $\mathcal{M}(S)$, in such a way that the following conditions are satisfied:

(i) As $S$ varies in $\mathrm{Var}_k$, the categories $\mathcal{M}^0(S)$ are stable under the following exact functors:

(a) for every affine open immersion $j: U \hookrightarrow S$, the inverse image functor $j^*$ as well as the direct image functors $j_1$ and $j_*$;

(b) for every closed immersion $i: Z \hookrightarrow S$, the direct image functor $i_*$;

(c) for every $S \in \mathrm{Var}_k$, the Tate twist functor $(-)(-)(-1)$;

(d) for every $S \in \mathrm{Var}_k$ and every $f \in \mathcal{O}(S)$, the functors ${}^p\Psi_f$, ${}^p\Phi_f$, ${}^p\Xi_f$ and ${}^p\Omega_f$.

(ii) For every $S \in \mathrm{Aff}_k$ and every $f \in \mathcal{O}(S)$, the natural transformations in the exact sequences (13), (14), (15) and (16) restrict to natural transformations of functors $\mathcal{M}^0(S) \to \mathcal{M}^0(S)$.

(iii) For every $S \in \mathrm{Aff}_k$, the natural inclusion of $\mathrm{SmOp}(S)$-fibered categories $\mathcal{M}^0 \subset \mathcal{M}$ is weakly cofinal in the sense of Definition 1.8.

Then, for every $S \in \mathrm{Var}_k$, the inclusion $\mathcal{M}^0(S) \subset \mathcal{M}(S)$ is in fact an equivalence.

*Proof.* The proof consists of two main steps: in the first place we reduce the proof to the case of affine varieties, while in the second step we prove the thesis in the affine case. As the reader will see, the first step is a standard Zariski-descent argument which only uses the stability of the categories $\mathcal{M}(S)$ under inverse images and direct images along affine open immersions; on the other hand, the second step exploits the full power of the gluing formalism.

(Step 1) Assume for a moment that the inclusion $\mathcal{M}^0(U) \subset \mathcal{M}(U)$ is known to be an equivalence for every $U \in \mathrm{Aff}_k$, and let us show that the same conclusion holds for every $S \in \mathrm{Var}_k$. Fix one variety $S \in \mathrm{Var}_k$, and choose a finite affine open covering $\{u_r : U_r \hookrightarrow S\}_{r \in I}$; for every $r, s \in I$, form the Cartesian diagram

$$\begin{tikzcd}
U_r \cap U_s \arrow[r, "u_s^{(r)}"] \arrow[d, "u_r^{(s)}"] & U_r \arrow[d, "u_r"] \\
U_s \arrow[r, "u_s"] & S
\end{tikzcd}$$

and let $u_{r,s}$ denote the composite open immersion $U_r \cap U_s \hookrightarrow S$. Note that each $u_r$, each $u_r^{(s)}$ and each $u_{r,s}$ is affine, because $S$ is separated over $k$ by definition. Consider the usual exact sequence of exact functors $\mathcal{M}(S) \to \mathcal{M}(S)$

$$\bigoplus_{r,s \in I} u_{r,s}! u_{r,s}^* M \to \bigoplus_{r \in I} u_{r,s}! u_r^* M \to M \to 0.$$
Note that the functors $\bigoplus_{r,s \in I} u_{r,s,!} u_{r,s}^*$ and $\bigoplus_{r \in I} u_{r,!} u_r^*$ take values in the essential image of $\mathcal{M}^0(S)$ by condition (i) in the statement, since, by our assumption, the functors $u_{r,s}^*$ and $u_r^*$ take values in the essential image of $\mathcal{M}^0(U_r \cap U_s)$ and $\mathcal{M}^0(U_r)$, respectively. We claim that the natural transformation of functors $\mathcal{M}(S) \to \mathcal{M}(S)$

$$ \bigoplus_{r,s \in I} u_{r,s,!} u_{r,s}^* M \to \bigoplus_{r \in I} u_{r,!} u_r^* M $$

takes values in $\mathcal{M}^0(S)$ as well; this will immediately imply that the inclusion of abelian categories $\mathcal{M}^0(S) \subset \mathcal{M}(S)$ is both essentially surjective and fully faithful, thus an equivalence. To prove the claim, it suffices to recall that, for every $r, s \in I$, the arrow $u_{r,s,!} u_{r,s}^* M \to u_{r,!} u_r^* M$ is nothing but the composite

$$ u_{r,s,!} u_{r,s}^* M = u_{r,!} u_{s,!}^{(r)} u_s^{(r),*} u_r^* M \xrightarrow{\epsilon} u_{r,!} u_r^* M, $$

and to note the second arrow belongs to $\mathcal{M}^0(S)$ by condition (i), since the arrow $u_{s,!}^{(r)} u_s^{(r),*} u_r^* M \xrightarrow{\epsilon} u_r^* M$ belongs to $\mathcal{M}^0(U_r)$ by our assumption.

(Step 2) We show that, for every $S \in \text{Aff}_k$, the inclusion $\mathcal{M}^0(S) \subset \mathcal{M}(S)$ is an equivalence. We argue by Noetherian induction on $S$, starting from the case where $\dim(S) = 0$. The base case follows directly from hypothesis (iii), since in this case the only object of $\text{SmOp}(S)$ is $S$ itself. For the inductive step, assume that $\dim(S) > 0$ and that the result is known to hold for all proper closed subvarieties $Z$ of $S$ (this is legitimate, since every such $Z$ is itself affine).

Let us first show that the inclusion $\mathcal{M}^0(S) \subset \mathcal{M}(S)$ is essentially surjective. Given an object $M \in \mathcal{M}(S)$, choose any regular function $f \in \mathcal{O}(S)$ with smooth Zariski-dense non-vanishing locus $D(f)$ such that the object $M|_{D(f)} \in \mathcal{M}(D(f))$ belongs to $\mathcal{M}^0(D(f))$: the existence of such a function follows from condition (iii) in the statement (using the fact that, as $f$ varies in $\mathcal{O}(S)$, the distinguished open subsets $D(f)$ form a basis for the Zariski topology of the affine variety $S$). Let $j : D(f) \hookrightarrow S$ and $i : V(f) \hookrightarrow S$ denote the corresponding open and closed immersions. By Proposition 1.7, we have a canonical isomorphism in $\mathcal{M}(S)$

$$ M = \gamma_{S,f} \delta_{S,f}(M) := H^0[i_*{}^p \Psi_f(j^*M) \xrightarrow{\pm} {}^p \Xi_f(M) \oplus i_*{}^p \Phi_f(M) \to i_*{}^p \Psi_f(j^*M)(-1)]. $$

We claim that the complex on the right is isomorphic to a complex in $\mathcal{M}^0(S)$; this will imply that $M$ belongs to the essential image of the abelian subcategory $\mathcal{M}^0(S)$ inside $\mathcal{M}(S)$. In order to prove the claim, it suffices to note that:

- The objects $i_*{}^p \Psi_f(j^*M)$, ${}^p \Xi_f(M)$, $i_*{}^p \Psi_f(j^*M)(-1)$ belong to the essential image of $\mathcal{M}^0(S)$ by condition (i), since $j^*M$ lies in the essential image of $\mathcal{M}^0(D(f))$ by construction.

- The object $i_*{}^p \Phi_f(M)$ belongs to $\mathcal{M}^0(S)$ by condition (i), since the inclusion $\mathcal{M}^0(V(f)) \subset \mathcal{M}(V(f))$ is essentially surjective by inductive hypothesis.

- The arrows $i_*{}^p \Psi_f(j^*M) \to {}^p \Xi_f(M)$ and ${}^p \Xi_f(M) \to i_*{}^p \Psi_f(j^*M)(-1)$ belong to $\mathcal{M}^0(S)$ by condition (ii).

- The arrows $i_*{}^p \Psi_f(j^*M) \to i_*{}^p \Phi_f(M)$ and $i_*{}^p \Phi_f(M) \to i_*{}^p \Psi_f(j^*M)(-1)$, being in the image of $i_*$, belong to $\mathcal{M}^0(S)$ by condition (i), since the inclusion $\mathcal{M}^0(V(f)) \subset \mathcal{M}(V(f))$ is fully faithful by inductive hypothesis.

This shows the essential surjectivity.

Let us now show that the inclusion $\mathcal{M}^0(S) \subset \mathcal{M}(S)$ is full. Given a morphism $\alpha : M_1 \to M_2$ in $\mathcal{M}(S)$, choose any regular function $f \in \mathcal{O}(S)$ with smooth, Zariski-dense non-vanishing locus $D(f)$ such that the objects $M_1|_{D(f)}$, $M_2|_{D(f)} \in \mathcal{M}(D(f))$ belong to the essential image of $\mathcal{M}^0(D(f))$ and the morphism $\alpha|_{D(f)} : M_1|_{D(f)} \to M_2|_{D(f)}$ also lies in $\mathcal{M}^0(D(f))$: as in the previous case, the existence of such a function follows from condition (iii) in the statement. As before, let $j :$
$D(f) \hookrightarrow S$ and $i: V(f) \hookrightarrow S$ denote the corresponding open and closed immersions. Under the canonical isomorphisms $M_r = \gamma_{S,f}\delta_{S,f}(M_r)$, $r = 1, 2$, constructed in the proof of Proposition 1.7, the morphism $\alpha$ is induced by the morphism of complexes in $\mathcal{M}(S)$

$$ \begin{aligned} i_*{}^p\Psi_f(j^*M_1) &\xrightarrow{+} {}^p\Xi_f(M_1) \oplus i_*{}^p\Phi_f(M_1) \xrightarrow{-} i_*{}^p\Psi_f(j^*M_1)(-1) \\ \begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)}, scale=0.7] \draw (-0.5,0) -- (0.5,0); \node[above] at (-0.2,0) {$i_*{}^p\Psi_f(j^*\alpha)$}; \node[below] at (0.2,0) {$i_*{}^p\Psi_f(j^*M_2)$}; \draw (-0.5,0.2) -- (-0.2,0.2); \draw (0.5,0.2) -- (0.2,0.2); \node[right] at (0.5,0.2) {$i_*{}^p\Psi_f(j^*M_2)$}; \node[left] at (-0.5,0.2) {$i_*{}^p\Psi_f(j^*M_1)$}; \end{tikzpicture} &\xrightarrow{+} {}^p\Xi_f(M_2) \oplus i_*{}^p\Phi_f(M_2) &\xrightarrow{-} i_*{}^p\Psi_f(j^*M_2)(-1). \end{aligned} $$

By the same argument as in the previous paragraph, both complexes are isomorphic to complexes in $\mathcal{M}^0(S)$. We claim that the three vertical arrows lie in $\mathcal{M}(S)$ as well; this will imply that the morphism $\alpha$ belongs to the abelian subcategory $\mathcal{M}^0(S)$ of $\mathcal{M}(S)$. In order to prove the claim, it suffices to note that:

- The arrows $i_*{}^p\Psi_f(j^*\alpha)$, ${}^p\Xi_f(\alpha)$ and $i_*{}^p\Psi_f(j^*\alpha)(-1)$ lie in $\mathcal{M}^0(S)$ by condition (i), since the arrow $j^*\alpha$ lies in $\mathcal{M}^0(D(f))$ by construction.

- The arrow $i_*{}^p\Phi_f(\alpha)$, being in the essential image of the fully faithful functor $i_*$, lies in $\mathcal{M}^0(S)$ by condition (i), since the inclusion $\mathcal{M}^0(V(f)) \subset \mathcal{M}(V(f))$ is full by inductive hypothesis.

This shows the fully faithfulness.

The proof is now complete.

□

At first sight, the amount of conditions to check in order to apply Proposition 1.9 looks massive. However, the stability conditions at points (i) and (ii) are automatically satisfied as long as the categories $\mathcal{M}^0(S)$ are defined by t-exactness conditions. In fact, all the actual work in the proof of Theorem 1.12 below goes into checking the weak cofinality property required in condition (iii).

### 1.3. Application to perverse motives.
In the final part of this section, we apply the criterion of Proposition 1.9 to obtain the promised alternative description of perverse motives. Let us first fix the relevant notation:

**Notation 1.10.** Let $S \in \mathrm{Var}_k$.

* We introduce the full additive subcategory

$$ \mathcal{D}^0(S) := \{ A \in \mathbf{D}\mathbf{A}_{ct}^{\text{et}}(S, \mathbb{Q}) \mid \mathrm{Bti}_S^*(A) \in \mathrm{Perv}(S) \} \subset \mathcal{D}(S). $$

* We let $\mathcal{M}^0(S)$ denote the universal abelian factorization of the additive functor

$$ \beta_S^0 := \beta_S|_{\mathcal{D}^0(S)} : \mathcal{D}^0(S) \to \mathrm{Perv}(S). $$

We implicitly regard it as a possibly non-full abelian subcategory of $\mathcal{M}(S)$ via the canonical faithful exact functor

$$ \mathcal{M}^0(S) \hookrightarrow \mathcal{M}(S) $$

obtained by applying [Ter23U, Prop. 2.5] to the diagram

$$ \mathcal{D}^0(S) \xrightarrow{\text{id}} \mathcal{D}(S) $$

$$ \begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)}, scale=0.7] \draw (-0.5,0) -- (0.5,0); \node[above] at (-0.2,0) {$\beta_S^0$}; \node[below] at (0.2,0) {$\beta_S$}; \draw (-0.5,0.2) -- (-0.2,0.2); \draw (0.5,0.2) -- (0.2,0.2); \node[right] at (0.5,0.2) {$\mathrm{Perv}(S)$}; \node[left] at (-0.5,0.2) {$\mathrm{Perv}(S)$}; \end{tikzpicture} \xrightarrow{\text{id}} \mathrm{Perv}(S). $$

* For simplicity, we still write

- $\pi_S$ for the quotient functor $\mathbf{A}(\mathcal{D}^0(S)) \to \mathcal{M}^0(S)$, and also for the composite functor $\mathcal{D}^0(S) \to \mathbf{A}(\mathcal{D}^0(S)) \to \mathcal{M}^0(S)$,

- $\iota_S$ for the induced faithful exact functor $\mathcal{M}^0(S) \hookrightarrow \mathrm{Perv}(S)$.
This convention will not create any ambiguity.

Even if the definition of the additive categories $\mathcal{D}^0(S)$ (and, therefore, also that of the abelian categories $\mathcal{M}^0(S)$) depends a priori on the field embedding $\sigma : k \hookrightarrow \mathbb{C}$, we purposely omit any mention of it from the notation. This choice is justified by the following result:

**Lemma 1.11.** Let $\sigma' : k \hookrightarrow \mathbb{C}$ be another complex embedding of $k$; for every $S \in \mathrm{Var}_k$, write $\mathrm{Bti}^*_{S,\sigma'} : \mathrm{DA}^{\mathrm{et}}_{ct}(S, \mathbb{Q}) \to \mathrm{D}^b_c(S^{\sigma'}, \mathbb{Q})$ for the associated Betti realization functor. Then the additive subcategory

$$ \mathcal{D}^0_{\sigma'}(S) := \{ A \in \mathrm{DA}^{\mathrm{et}}_{ct}(S, \mathbb{Q}) \mid \mathrm{Bti}^*_{S,\sigma'}(A) \in \mathrm{Perv}(S^{\sigma'}) \} \subset \mathcal{D}(S) $$

coincides with the category $\mathcal{D}^0(S)$ defined above.

*Proof.* We have to show that, for every object $A \in \mathrm{DA}^{\mathrm{et}}_{ct}(S, \mathbb{Q})$, the equivalence

$$ \mathrm{Bti}^*_{S,\sigma}(A) \in \mathrm{Perv}(S) \iff \mathrm{Bti}_{S,\sigma'}(A) \in \mathrm{Perv}(S)_{\ell} $$

holds. We claim that, in fact, for every fixed $n \in \mathbb{Z}$, the equivalence

$$ {}^pH^n(\mathrm{Bti}^*_{S}(A)) = 0 \iff {}^pH^n(R_{\ell,S}(A)) = 0 $$

holds; the claim clearly implies the thesis. The claim can be proved by induction on $\dim(S)$ using the method of the proof of [IM19, Prop. 6.11]. We omit the details. $\square$

The rest of this section is devoted to the proof of the following result:

**Theorem 1.12.** For every $S \in \mathrm{Var}_k$, the canonical inclusion $\mathcal{M}^0(S) \subset \mathcal{M}(S)$ is in fact an equivalence.

The proof of this result, given at the end of the present section, is essentially a direct application of Proposition 1.9. However, this requires some careful preparation: firstly, we need to make sense of perverse Nori motives over function fields of $k$-varieties by a colimit construction; secondly, we need to reduce ourselves to the case of function fields over finitely generated subfields of $k$, so that we are allowed to embed these function fields into $\mathbb{C}$; thirdly, in such a situation, we need to interpret the categories of perverse Nori motives over function fields as categories of motives in Nori's sense, so that we are allowed to exploit Nori's geometric results over the function fields in question.

In order to achieve all these steps, our original approach was to exploit the alternative definition of perverse motives via the $\ell$-adic realization, as done in [IM19, § 6.5]: by the comparison result [IM19, Prop. 6.11], for every $S \in \mathrm{Var}_k$, the abelian category $\mathcal{M}(S)$ can be recovered as the universal abelian factorization associated to the homological functor

$$ (28) \qquad \mathrm{DA}^{\mathrm{et}}_{ct}(S, \mathbb{Q}) \xrightarrow{R_{\ell,S}} \mathrm{D}^b_{c,\ell}(S, \mathbb{Q}_{\ell}) \xrightarrow{{}^pH^0} \mathrm{Perv}(S)_{\ell}. $$

Here, $\mathrm{D}^b_{c,\ell}(S, \mathbb{Q}_{\ell})$ denotes the constructible $\ell$-adic derived category (as defined in [Eke90] or equivalently in [BS15]) and $\mathrm{Perv}(S)_{\ell}$ denotes the heart of the self-dual perverse $t$-structure on $\mathrm{D}^b_{c,\ell}(S, \mathbb{Q}_{\ell})$ (as defined, for example, in [Mor19]). The triangulated functor

$$ R_{\ell,S} : \mathrm{DA}^{\mathrm{et}}_{ct}(S, \mathbb{Q}) \to \mathrm{D}^b_{c,\ell}(S, \mathbb{Q}_{\ell}) $$

is the $\ell$-adic realization constructed by Ayoub in [Ayo14E]; as $S$ varies in $\mathrm{Var}_k$, these functors assemble into a morphism of monoidal stable homotopy $2$-functors. Compared to the Betti realization, the $\ell$-adic realization has the key advantage of being absolute and independent of any auxiliary choices (such as the choice of a complex embedding $\sigma : k \hookrightarrow \mathbb{C}$).
Remark 1.13. In fact, as far as we can see, the proof of [IM19, Prop. 6.11] tacitly passes through the $\ell$-adic derived categories $D_{c,\ell}^b(S_{\bar{k}}, \mathbb{Q}_{\ell})$ over the base-changed varieties $S_{\bar{k}} := S \times_{\text{Spec}(k)} \text{Spec}(\bar{k})$. Indeed, the proof of [IM19, Prop. 6.11] in the case where $S = \text{Spec}(k)$ uses the existence of Huber’s mixed realizations from [Hub00], which yields a comparison isomorphism between the Betti and $\ell$-adic realizations of $\mathbf{D}\mathbf{A}_{ct}^{\ell t}(k, \mathbb{Q})$ extending the classical comparison isomorphism between Betti and $\ell$-adic cohomology. Recall, however, that the latter takes the form

$$H^n(X^\sigma, \mathbb{Q}) \otimes_{\mathbb{Q}} \mathbb{Q}_\ell = H_\ell^n(X_{\bar{k}}, \mathbb{Q}_\ell)$$

for $X \in \text{Var}_k$. Therefore, the existence of the comparison isomorphism alone does not imply directly a comparison between the Betti realization and the absolute $\ell$-adic realization. In order to get the desired comparison, one has to use the fact that the inverse image functors

$$D_{c,\ell}^b(S, \mathbb{Q}_\ell) \to D_{c,\ell}^b(S_{\bar{k}}, \mathbb{Q}_\ell)$$

are conservative and commute with the six operations. Indeed, this fact implies formally that the universal abelian factorization associated to the homological functor (28) does not change if one composes further with the exact inverse image functor $\text{Perv}(S)_\ell \to \text{Perv}(S_{\bar{k}})_\ell$, because the latter is faithful.

In terms of the absolute $\ell$-adic realization, both the definition of Nori motives over function fields and the reduction to finitely generated fields alluded to above can be easily obtained from the continuity property [IM19, Prop. 6.9]. However, the precise form of Nori’s geometric results that we want to use in the proof of Theorem 1.12 is only available with respect to the Betti realization. This would force us to use the comparison result [IM19, Prop. 6.11] twice: the first time to pass from the Betti to the $\ell$-adic version, the second time to go back from the $\ell$-adic to the Betti version.

In order to avoid such an artificial-sounding argument, here we follow a different approach which does not rely on the $\ell$-adic realization at all. Instead, we define abelian categories of Nori motives over function fields of $k$-varieties by a direct colimit presentation, and we study them from the perspective of the Betti realization. The crucial result is Proposition 1.15 below, where these abelian categories are identified as categories of motives in Nori’s sense, at least when the function field in question admits a complex embedding extending the original embedding of $k$. This is, in our opinion, a very pleasant result, which provides a clear geometric explanation behind the seemingly miraculous independence results of [IM19, Prop. 6.11]. Together with Corollary 1.17, this allows us to reduce the proof of Theorem 1.12 to the finitely generated case as promised.

In the first place, we need to recall how Ivorra and Morel’s definition of perverse motives can be extended to function fields of $k$-varieties; we need a similar extension for the categories introduced in Notation 3.1.

**Construction 1.14.** Let $L$ be a finitely generated extension field of $k$, and fix one $S \in \text{Var}_k$ with function field $L := k(S)$: this means that $S$ is irreducible and the sheaf of $\mathcal{O}_S$-modules

$$U \mapsto \text{Frac}(\mathcal{O}_S(U))$$

is identified with the constant sheaf with value $L$ over the Zariski site of $S$. We can write

$$\text{Spec}(L) = \lim_{\underset{U \in \text{AffOp}(S)}{\leftarrow}} U$$

as the inverse limit of all affine open subsets of $S$ in the category of $k$-schemes. By the continuity property of triangulated étale motives proved in [Ayo07a], the canonical functor

$$2 - \lim_{\underset{U \in \text{AffOp}(S)^{op}}{\leftarrow}} \mathbf{D}\mathbf{A}_{ct}^{\ell t}(U, \mathbb{Q}) \to \mathbf{D}\mathbf{A}_{ct}^{\ell t}(\text{Spec}(L), \mathbb{Q})$$
induced by the inverse image functors $\mathbf{DA}_{ct}^{\text{et}}(U, \mathbb{Q}) \to \mathbf{DA}_{ct}^{\text{et}}(\text{Spec}(L), \mathbb{Q})$ is an equivalence. Therefore, by [Ter23U, Lemma 5.12(2)], the filtered 2-colimit of abelian categories

$$ (29) \qquad 2 - \lim_{U \in \text{AffOp}(S)} \mathcal{M}(U) $$

can be canonically identified with the universal abelian factorization of the additive functor

$$ \beta_{L,\sigma} : \mathcal{D}(\text{Spec}(L)) = 2 - \lim_{U \subset S} \mathcal{D}(U) \xrightarrow{\beta_U} 2 - \lim_{U \subset S} \text{Perv}(U), $$

where, in analogy to the notation $\mathcal{D}(U)$ for $\mathbf{DA}_{ct}^{\text{et}}(U, \mathbb{Q})$ employed for $U \in \text{Var}_k$, we write $\mathcal{D}(\text{Spec}(L))$ for the underlying additive category of $\mathbf{DA}_{ct}^{\text{et}}(\text{Spec}(L), \mathbb{Q})$. Let us suggestively write $\mathcal{M}(\text{Spec}(L))$ for the abelian category (29); note that, in the case where the extension $L/k$ is finite (so that the $k$-scheme $\text{Spec}(L)$ belongs to $\text{Var}_k$), this agrees with the definition of perverse motives over $\text{Spec}(L)$ given in [IM19]. Clearly, in the definition of the 2-colimit (29), one can replace the poset $\text{AffOp}(S)$ by any cofinal subposet: for instance, we can use the poset $\text{SmOp}(S) \cap \text{AffOp}(S)$ consisting of all smooth affine open subsets of $S$. It is also easy to see that the definition of $\mathcal{M}(\text{Spec}(L))$ is independent of the chosen model $S$ of $L$; we leave the details to the interested reader.

Keeping the above notation, let us also introduce the additive subcategory

$$ \mathcal{D}^0(\text{Spec}(L)) := 2 - \lim_{U \in \text{AffOp}(S)^{op}} \mathcal{D}^0(U) \subset \mathcal{D}(\text{Spec}(L)). $$

Again, this definition does not depend on the chosen model $S$ of $L$ and does not change if one replaces the poset $\text{AffOp}(S)$ by any cofinal subposet such as $\text{SmOp}(S) \cap \text{AffOp}(S)$. Moreover, let $\mathcal{M}^0(\text{Spec}(L))$ denote the universal abelian factorization of the additive functor

$$ \beta_{L,\sigma}^0 := \beta_{L,\Sigma} |_{\mathcal{D}^0(\text{Spec}(L))} : \mathcal{D}^0(\text{Spec}(L)) = 2 - \lim_{U \subset S} \mathcal{D}^0(U) \xrightarrow{\beta_U} 2 - \lim_{U \subset S} \text{Perv}(U), $$

which is independent of the chosen presentation as well; if the extension $L/k$ is finite, this is compatible with the original definition given in Notation 3.1. Applying [Ter23U, Lemma 5.12] as above, we see that $\mathcal{M}^0(\text{Spec}(L))$ can be canonically identified with the filtered 2-colimit of abelian categories

$$ 2 - \lim_{U \in \text{AffOp}(S)^{op}} \mathcal{M}^0(U), $$

or with the analogous 2-colimit over the cofinal subposet $\text{SmOp}(S) \cap \text{AffOp}(S)$.

A priori, the categories of perverse Nori motives over function fields just introduced are not categories of motives over a field in Nori's sense; this might give rise to some confusion. As a concrete example, suppose that the field extension $L/k$ is finite, so that the $k$-scheme $\text{Spec}(L)$ belongs to $\text{Var}_k$; fix a field embedding $\Sigma : L \hookrightarrow \mathbb{C}$ extending the original embedding $\sigma : k \hookrightarrow \mathbb{C}$. Then we have two distinct ways to define an abelian category of perverse Nori motives out of $\mathbf{DA}_{ct}^{\text{et}}(\text{Spec}(L), \mathbb{Q})$: on the one side, as the category $\mathcal{M}(\text{Spec}(L))$ obtained via $\sigma$ (by considering $\text{Spec}(L)$ as a $k$-variety); on the other side, as the category $\mathcal{M}(L)$ obtained via $\Sigma$ (by considering $L$ as the base field). It is natural to expect these two recipes to define the same abelian category.

In the following proposition, we check that this in indeed the case; in fact, the same independence result holds for arbitrary finitely generated field extensions of $k$.

**Proposition 1.15.** Let $L/k$ be a finitely generated field extension, and assume that there exists a field embedding $\Sigma : L \hookrightarrow \mathbb{C}$ extending the original embedding $\sigma : k \hookrightarrow \mathbb{C}$. Then the two exact functors

$$ \beta_{\text{Spec}(L),\sigma}^{+} : \mathbf{A}(\mathcal{D}(L)) \to \text{vect}_{\mathbb{Q}}, \qquad \beta_{L,\Sigma}^{+} : \mathbf{A}(\mathcal{D}(L)) \to \text{vect}_{\mathbb{Q}} $$
$$ (\text{resp. } \beta_{\text{Spec}(L),\sigma}^{0,+} : \mathbf{A}(\mathcal{D}^0(L)) \to \text{vect}_{\mathbb{Q}}, \quad \beta_{L,\Sigma}^{0,+} : \mathbf{A}(\mathcal{D}^0(L)) \to \text{vect}_{\mathbb{Q}}) $$

have the same kernel. In particular, the category $\mathcal{M}(\text{Spec}(L))$ (resp. $\mathcal{M}^0(\text{Spec}(L))$) is canonically equivalent to $\mathcal{M}(L)$ (resp. $\mathcal{M}^0(L)$).

*Proof.* For sake of simplicity, we only write down the argument for the categories $\mathcal{M}(\text{Spec}(L))$ and $\mathcal{M}(L)$; the argument for the categories $\mathcal{M}^0(\text{Spec}(L))$ and $\mathcal{M}^0(L)$ can be obtained by easily adapting the former, and we leave it to the interested reader.

In the first place, we describe the key geometric construction behind the proof. Fix one variety $S \in \text{Var}_k$ with function field $L$, and write

$$ \text{Spec}(L) = \varinjlim_{U \in \text{SmOp}(S)} U $$

as the limit in the category of $k$-schemes over all smooth non-empty open subsets of $S$. For every $U \in \text{SmOp}(S)$, the canonical morphism of $k$-schemes $\eta_U : \text{Spec}(L) \to U$ is the inclusion of the generic point; the induced monomorphism of sheaves of $k$-algebras

$$ \mathcal{O}_U \xrightarrow{\eta_U^*} L \xrightarrow{\Sigma} \mathbb{C} $$

determines a complex point $z_{U,\Sigma} \in U^\sigma$ which, by construction, does not lie on any proper algebraic $k$-subvariety of $U$. In fact, for every inclusion $V \subset U$ in $\text{SmOp}(S)$, the induced inclusion of complex-analytic spaces $V^\sigma \subset U^\sigma$ identifies $z_{V,\Sigma}$ with $z_{U,\Sigma}$. By construction, for every $U \in \text{SmOp}(S)$ the diagram

$$ \begin{array}{ccc} \mathbf{D}\mathbf{A}_{ct}^{\acute{e}t}(U, \mathbb{Q}) & \xrightarrow{\eta_U^*} & \mathbf{D}\mathbf{A}_{ct}^{\acute{e}t}(L, \mathbb{Q}) \\ \Big\downarrow \mathrm{Bti}_{U,\sigma}^* & & \Big\downarrow \mathrm{Bti}_{L,\Sigma}^* \\ D_c^b(U, \mathbb{Q}) & \xrightarrow{z_{U,\Sigma}^*} & D^b(\mathrm{vect}_{\mathbb{Q}}) \end{array} $$

is commutative up to natural isomorphism. In fact, we have a canonical natural isomorphism

$$ z_{U,\Sigma}^* \circ \mathrm{Bti}_{U,\sigma}^* = \mathrm{Bti}_{L,\Sigma}^* \circ \eta_U^* $$

as functors on the entire category of étale motives $\mathbf{D}\mathbf{A}^{\acute{e}t}(U, \mathbb{Q})$: this extends the identification of complex-analytic spaces

$$ X^{\sigma} \times_{U^{\sigma}} z_{U,\Sigma} = (X \times_U \text{Spec}(L))^{\Sigma}, $$

which holds functorially in $X \in \text{Sm}/U$; this fact was observed in the proof of [Ayo14H2, Prop. 2.20]. From the compatibility of the complex points $z_{U,\Sigma}$ under restriction, we deduce that the diagram

$$ \begin{array}{ccc} 2 - \varinjlim_{\underset{\sim}{\to} U \in \text{SmOp}(S)^{op}} \mathbf{D}\mathbf{A}_{ct}^{\acute{e}t}(U, \mathbb{Q}) & \xrightarrow{\sim} & \mathbf{D}\mathbf{A}_{ct}^{\acute{e}t}(L, \mathbb{Q}) \\ \Big\downarrow \mathrm{Bti}_{U,\sigma}^* \Big\downarrow & & \Big\downarrow \mathrm{Bti}_{L,\Sigma}^* \\ 2 - \varinjlim_{\underset{\sim}{\to} U \in \text{SmOp}(S)^{op}} D_c^b(U, \mathbb{Q}) & \xrightarrow{(z_{U,\Sigma}^*)_U} & D^b(\mathrm{vect}_{\mathbb{Q}}) \end{array} $$
is commutative up to canonical natural isomorphism as well. Applying the perverse truncation functors, we deduce a commutative diagram of the form

$$
\begin{align*}
2 - \lim_{\substack{\longrightarrow \\ \left\lfloor \mathrm{Bt}_{U,\sigma}^* \right\rfloor_{U} \\ \longrightarrow}} \mathrm{DA}_{ct}^{\acute{e}t}(U, \mathbb{Q}) &\xrightarrow{\sim} \mathrm{DA}_{ct}^{\acute{e}t}(L, \mathbb{Q}) \\
2 - \lim_{\substack{\longrightarrow \\ \left\lfloor \mathrm{P} H^0 \right\rfloor_{U} \\ \longrightarrow}} \mathrm{D}_c^b(U, \mathbb{Q}) &\xrightarrow{(z_U^*_U)_U} \mathrm{D}^b(\mathrm{vect}_{\mathbb{Q}}) \\
2 - \lim_{\substack{\longrightarrow \\ \left\lfloor \mathrm{P} H^0 \right\rfloor_{U} \\ \longrightarrow}} \mathrm{Perv}(U) &\leftarrow\xrightarrow{\sim} 2 - \lim_{\substack{\longrightarrow \\ \left\lfloor \mathrm{Loc}_p}(U) \\ \longrightarrow}} \mathrm{Loc}_p(U) \xrightarrow{(z_{U,\Sigma}^*)_U} \mathrm{vect}_{\mathbb{Q}}
\end{align*}
$$

where the lower-left horizontal arrow is the equivalence induced by the inclusions $\mathrm{Loc}_p(U) \subset \mathrm{Perv}(U)$ for $U \in \mathrm{SmOp}(S)$, while the lower-right horizontal arrow is defined by the exact shifted inverse image functors

$$ (30) \qquad z_{U,\Sigma}^{\dagger} := z_{U,\Sigma}^{*}[-\dim(U)] : \mathrm{Loc}_p(U) \to \mathrm{vect}_{\mathbb{Q}}. $$

Passing to the abelian hulls, we obtain a commutative diagram of the form

$$
\begin{align}
& \mathbf{A}(2 - \lim_{\substack{\longrightarrow \\ \left\lfloor \beta_{\mathrm{Spec}(L),\sigma}^{+} \right\rfloor_{U} \\ \longrightarrow}} \mathbf{DA}_{ct}^{\acute{e}t}(U, \mathbb{Q})) \xrightarrow{\sim} \mathbf{A}(\mathbf{DA}_{ct}^{\acute{e}t}(L, \mathbb{Q})) \nonumber \\
(31) \qquad & 2 - \lim_{\substack{\longrightarrow \\ \left\lfloor \mathrm{Loc}_p}(U) \\ \longrightarrow}} \mathrm{Loc}_p(U) \xrightarrow{(z_{U,\Sigma}^{\dagger})_U} \mathrm{vect}_{\mathbb{Q}}. \nonumber
\end{align}
$$

Proving the thesis amounts to showing that the equivalence in the upper horizontal arrow identifies the kernels of the two vertical exact functors.

Let us first explain how to prove this in the simple case where the field extension $L/k$ is geometrically integral. In this case, each $U \in \mathrm{SmOp}(S)$ is geometrically connected, and so the associated complex-analytic space $U^\sigma$ is connected as well. This implies that the exact functors (30) are all faithful. Therefore the same holds for the lower horizontal arrow in (31). But then it follows formally that the two vertical exact functors in (31) have the same kernel; compare with the proof of [Ter23U, Prop. 1.10].

If the field extension $L/k$ is not assumed to be geometrically integral, the argument is similar but slightly more involved: the issue is that the exact functors (30) are no longer necessarily faithful, since the complex-analytic spaces $U^\sigma$ are no longer necessarily connected. To remedy this issue, the rough idea is to consider multiple complex embeddings of $L$ at once so that the corresponding points $z_{U,\Sigma}$ hit all connected components of $U^\sigma$. In order to make this idea work, we need to describe the construction of the points $z_{U,\Sigma}$ in more explicit geometric terms. To this end, choose a transcendence basis $t_1, \dots, t_n$ for $L/k$ with the property that we can write $L = k(t_1, \dots, t_n)[x]/P(x)$ where $P(x) \in k(t_1, \dots, t_n)[x]$ is a monic polynomial in $x$ with coefficients in $k[t_1, \dots, t_n]$ defining a smooth subvariety inside the affine space $\mathbf{A}_k^{n+1}$ with coordinates $t_1, \dots, t_n, x$. To achieve this, it suffices to apply the strong form of Noether's Normalization Lemma to some chosen affine $U \in \mathrm{SmOp}(S)$ (this is legitimate, since the base field $k$ is infinite). In this way, we get an identification

$$ U = \mathrm{Spec}(k[t_1, \dots, t_n, x]/P(x)) $$

as closed subvarieties of $\mathbf{A}_k^{n+1}$. Note that, by construction, $P(x)$ is irreducible in $k(t_1, \dots, t_n)[x]$.
For sake of brevity, write $L_0 := k(t_1, \dots, t_n)$; write also $\Sigma_0 := \Sigma|_{L_0}: L_0 \hookrightarrow \mathbb{C}$, and let $\Sigma_1, \dots, \Sigma_{\deg(P)}$ denote the $\deg(P)$ distinct complex embeddings of $L$ extending $\Sigma_0$ (among which figures the original embedding $\Sigma$). Moreover, write $P^\sigma(x)$ for the image of $P(x)$ in $\mathbb{C}(t_1, \dots, t_n)[x]$ via $\sigma$, and let

$$P^{\sigma}(x) = \prod_{i=1}^{r} Q_i(x)$$

be its factorization into irreducible polynomials. Note that, since the coefficients of $P^\sigma(X)$ belong to the unique factorization domain $\mathbb{C}[t_1, \dots, t_n]$, the same holds for the coefficients of each factor $Q_i(x)$. Thus each $Q_i(x)$ defines a connected closed subvariety $U_i$ of $\mathbb{A}_{\mathbb{C}}^{n+1}$, and the complex-analytic space $U^\sigma$ breaks into connected components as

$$U^{\sigma} = U_1 \amalg \dots \amalg U_{\deg(P)}.$$

This implies, in particular, that each complex variety $U_i$ is smooth, since so is $U^\sigma$. Moreover, write $P^{\Sigma_0}(x)$ for the image of $P(x)$ in $\mathbb{C}[x]$ via $\Sigma_0$, and consider the factorization

$$P^{\Sigma_0}(x) = \prod_{i=1}^{r} \bar{Q}_i(x),$$

where $\bar{Q}_i(x)$ denotes the image of $Q_i(x)$ in $\mathbb{C}[x]$ via the $\mathbb{C}$-algebra homomorphism

$$\mathbb{C}[t_1, \dots, t_n] \to \mathbb{C}, \quad t_i \mapsto \Sigma_0(t_i).$$

The complex embeddings $\Sigma_1, \dots, \Sigma_{\deg(P)}$ extending $\Sigma_0$ are in natural bijection with the complex roots of $P^{\Sigma_0}(x)$: giving such an embedding $\Sigma'$ amounts to choosing a factor $Q_i(x)$ of $P^\sigma(x)$ and specifying a complex root of $\bar{Q}_i(x)$; by construction, the corresponding complex point $z_{U,\Sigma'} \in U^\sigma$ belongs to the connected component $U_i$. Thus the complex points

$$z_{U, \Sigma_1}, \dots, z_{U, \Sigma_{\deg(P)}}$$

jointly hit every connected component of $U^\sigma$. Similarly, for each dense open subset $V \subset U$, the points $z_{U,\Sigma_i}$ restrict to a family of complex points $z_{V,\Sigma_i} \in V^\sigma$ which jointly hit every connected component of $V^\sigma$.

Finally, we are ready to prove the thesis. Consider the commutative diagram

$$\begin{array}{ccc} \mathbf{A}(2 - \lim_{\underset{\sim}{\longrightarrow}}_{U \in \mathrm{SmOp}(S)^{op}} \mathbf{D}\mathbf{A}_{ct}^{\mathrm{\acute{e}t}}(U, \mathbb{Q})) & \xrightarrow{\sim} & \mathbf{A}(\mathbf{D}\mathbf{A}_{ct}^{\mathrm{\acute{e}t}}(L, \mathbb{Q})) \\ 2 - \lim_{\underset{\sim}{\longrightarrow}}_{U \in \mathrm{SmOp}(S)^{op}} \mathrm{Loc}_p(U) & \xrightarrow{(z_{U,\Sigma_i}^\dagger)_{U,i}} & \prod_{i=1}^{\deg(P)} \mathrm{vect}_{\mathbb{Q}} \end{array} \quad (32)$$

obtained by tying together the individual diagrams of shape (31) for the embeddings $\Sigma_1, \dots, \Sigma_{\deg(P)}$. For every dense open subset $V \subset U$, the collection of exact shifted inverse image functors

$$z_{V, \Sigma_1}^{\dagger}, \dots, z_{V, \Sigma_{\deg(P)}}^{\dagger} : \mathrm{Loc}_p(V) \to \mathrm{vect}_{\mathbb{Q}}$$

is jointly faithful, precisely because the points $z_{V,\Sigma_i}$ hit every connected component of $V^\sigma$. Therefore the lower horizontal arrow in (32) is a faithful exact functor as well. As in the geometrically connected case, this implies formally that the two vertical exact functors in (32) have the same kernel. In order to conclude, it now suffices to show that the right-most vertical arrow in the two diagrams (31) and (32) have the same kernel. Equivalently, we need to show that the various exact functors

$$\beta_{L, \Sigma_1}^{+}, \dots, \beta_{L, \Sigma_{\deg(P)}}^{+} : \mathbf{A}(\mathbf{D}\mathbf{A}_{ct}^{\mathrm{\acute{e}t}}(L, \mathbb{Q})) \to \mathrm{vect}_{\mathbb{Q}}$$

all have the same kernel. This follows from the fact that the induced exact functors

$$\beta_{L, \Sigma_1}^{+}, \dots, \beta_{L, \Sigma_{\deg(P)}}^{+} : \mathbf{A}(\mathbf{D}\mathbf{A}_{ct}^{\mathrm{\acute{e}t}}(L, \mathbb{Q})) \to \mathrm{vect}_{\mathbb{Q}} \xrightarrow{-\otimes_{\mathbb{Q}} L} \mathrm{vect}_L$$
are all isomorphic to each other: indeed, the homological functors

$$ \beta_{L,\Sigma_1, \dots, \Sigma_{\deg(L)}} : \mathbf{D}\mathbf{A}_{ct}^{\ell t}(L, \mathbb{Q}) \to \mathrm{vect}_{\mathbb{Q}} \xrightarrow{-\otimes_{\mathbb{Q}} L} \mathrm{vect}_L $$

are all isomorphic to the algebraic de Rham realization of Voevodsky motives via Huber's construction of mixed realizations from [Hub00]. □

**Remark 1.16.** Note that both the proof of Lemma 1.11 above and the final part of the proof of Proposition 1.15 use crucially the existence of comparison isomorphisms relating the various Betti cohomology theories associated to different complex embeddings of the base fields up to extension of scalars. To the author's knowledge, the only way to obtain such a comparison is via some other cohomology theory, defined independently of any such embedding, which compares to all Betti cohomology theories at once. As far as the argument is concerned, it is essentially irrelevant which particular auxiliary cohomology is chosen.

In the proof of Proposition 1.15, we chose to use de Rham cohomology instead of $\ell$-adic cohomology since both the definition and the comparison with Betti cohomology are much deeper results for the latter than for the former. The proof of Lemma 1.11 is based on the same argument as [IM19, Prop. 6.11], where the authors use $\ell$-adic cohomology and not de Rham cohomology. Note, however, that if one is only interested in the comparison results of [IM19, Prop. 6.11] for different Betti realizations, one can use the de Rham realization in place of the $\ell$-adic realization, since the latter is actually used as a comparison tool only over $k$-varieties of dimension 0.

Note that, in general, it might be simply impossible to extend the original embedding $\sigma : k \hookrightarrow \mathbb{C}$ to a given function field $L/k$ (for instance if $k = \mathbb{C}$ and $\sigma = \mathrm{id}_{\mathbb{C}}$). In order to exploit Proposition 1.15 to study the abelian category $\mathcal{M}(\mathrm{Spec}(L))$ as a genuine category of Nori motives over $L$, we need to make sure that we can always reduce to the case where such an embedding extension is possible.

In fact, as we explain now, we can always reduce to the case where $L$ (or equivalently, $k$) is finitely generated over $\mathbb{Q}$. The next result is a consequence of Proposition 1.15, applied not directly to the original field extension $L/k$ but rather to the induced extensions between suitable finitely generated subfields thereof.

**Corollary 1.17.** Let $L/k$ be a finitely generated field extension. Let $\mathcal{S}_L$ denote the filtered poset of all subfields of $L$ which are finitely generated over $\mathbb{Q}$; for each $F \in \mathcal{S}_L$, choose an embedding $\Sigma_F : F \hookrightarrow \mathbb{C}$ extending the restriction $\sigma|_{F \cap k} : F \cap k \subset k \hookrightarrow \mathbb{C}$.

Then, as $F$ varies in $\mathcal{S}_L$, the abelian categories $\mathcal{M}(F)$ (resp. $\mathcal{M}^0(F)$) defined via the chosen embeddings $\Sigma_F$ canonically assemble into an abelian fibered category over $\mathcal{S}_L^{\mathrm{op}}$, and we have a canonical equivalence

$$ \mathcal{M}(L) = 2 - \lim_{F \in \mathcal{S}_L} \mathcal{M}(F) \\ (\mathrm{resp.} \ \mathcal{M}^0(L) = 2 - \lim_{F \in \mathcal{S}_L} \mathcal{M}^0(F)). $$

**Proof.** Again, we only give the argument for the categories $\mathcal{M}(L)$ and $\mathcal{M}(F)$; the argument for the categories $\mathcal{M}^0(L)$ and $\mathcal{M}^0(F)$ can be obtained by easily adapting the former and is left to the interested reader.

For any fixed $F \in \mathcal{S}_L$, and any choice of two complex embeddings $\Sigma_F^{(1)}, \Sigma_F^{(2)} : F \hookrightarrow \mathbb{C}$, the abelian categories $\mathcal{M}(F_1, \Sigma_F^{(1)})$ and $\mathcal{M}(F_2, \Sigma_F^{(2)})$ coincide, as a consequence of Huber's construction of mixed realizations; this was already noted in the proof of [IM19, Prop. 6.11]. Therefore, for every inclusion $F_1 \subset F_2$ in $\mathcal{S}_L$, we have a well-defined exact functor

$$ \mathcal{M}(F_1) := \mathcal{M}(F_1, \Sigma_{F_1}) = \mathcal{M}(F_1, \Sigma_{F_2}|_{F_1}) \to \mathcal{M}(\mathrm{Spec}(F_2), \Sigma_{F_2}|_{F_1}) = \mathcal{M}(F_2, \Sigma_{F_2}) =: \mathcal{M}(F_2), $$

where the last equivalence is given by Proposition 1.15. It is easy to see that these functors are compatible with composition of inclusions in $\mathcal{S}_L$, thereby defining an abelian fibered category as stated.
The proof of the last assertion is not difficult but a bit lengthy, since it requires us to describe everything in more explicit geometric terms, similarly to what done in the proof of Proposition 1.15. To begin with, fix some $S \in \text{Aff}_k$ with function field $L$, as done at the beginning of Construction 1.14; for the present argument it is not relevant whether $S$ is smooth or not. Applying the strong form of Noether's Normalization Lemma, we can write

$$S = \text{Spec}(k[t_1, \dots, t_n][x]/P(x))$$

where $t_1, \dots, t_n$ form a transcendence basis for $L/k$ and $P(x) \in k[t_1, \dots, t_n][x]$ is an irreducible monic polynomial in $x$. From now on, we regard $S$ as a closed subvariety of the affine space $\mathbf{A}_k^{n+1}$ with coordinates $t_1, \dots, t_n, x$, and we tacitly identify the function field $L := k(S)$ with $k(t_1, \dots, t_n)[x]/P(x)$.

For notational convenience, let $k_0$ denote the smallest subfield of $k$ such that $P(x) \in k_0[t_1, \dots, t_n][x]$; by construction, $k_0$ is finitely generated over $\mathbb{Q}$. Consider the filtered poset $\mathcal{S}_{k/k_0}$ consisting of all subfields of $k$ which are finitely generated over $\mathbb{Q}$ and contain $k_0$, and let $\mathcal{S}_{L/k_0} \subset \mathcal{S}_L$ denote the cofinal subposet consisting of fields of the form $k'(t_1, \dots, t_n)/P(x)$ with $k' \in \mathcal{S}_{k/k_0}$; we thus have an obvious map of posets $\mathcal{S}_{k/k_0} \to \mathcal{S}_{L/k_0}$, and we write the image of a typical element $k' \in \mathcal{S}_{k/k_0}$ under it as $F'$.

For each $k' \in \mathcal{S}_{k/k_0}$, let $\text{AffOp}_{k'}(S) \subset \text{AffOp}(S)$ denote the subposet consisting of all affine open subsets of $S$ which are defined over $k'$: by definition, an open subset $U \in \text{AffOp}(S)$ belongs to $\text{AffOp}_{k'}(S)$ if and only if the closed subvariety $S \setminus U \subset \mathbf{A}_k^{n+1}$ is defined over $k'$, which means that the associated ideal $\mathbb{I}(S \setminus U) \subset k[x_1, \dots, x_n]$ satisfies the equality

$$\mathbb{I}(S \setminus U) = (\mathbb{I}(S \setminus U) \cap k'[x_1, \dots, x_n]) \otimes_{k'} k.$$

Note that the definition of $\text{AffOp}_{k'}(S)$ depends on the chosen presentation of $S$ as a closed subvariety of $\mathbf{A}_k^{n+1}$ (for instance, it is not stable under $k$-automorphisms of $\mathbf{A}_k^{n+1}$). Nevertheless, since every closed subvariety of $\mathbf{A}_k^{n+1}$ is defined over some field finitely generated over $\mathbb{Q}$, we have the filtered union

$$(33) \qquad \text{AffOp}(S) = \bigcup_{k' \in \mathcal{S}_{k'/k_0}} \text{AffOp}_{k'}(S).$$

For each $k' \in \mathcal{S}_{k/k_0}$, we let $S_{(k')}$ denote the closed subvariety of $\mathbf{A}_{k'}^{n+1}$ defined by the polynomial $P(x) \in k'[t_1, \dots, t_n][x]$. Similarly, for every $U \in \text{AffOp}_{k'}(S)$, we let $U_{(k')}$ denote the corresponding affine $k'$-variety: its closed complement $S_{(k')} \setminus U_{(k')}$ is the closed subvariety of $\mathbf{A}_{k'}^n$ with associated ideal $\mathbb{I}(S \setminus U) \cap k'[x_1, \dots, x_n]$. Note that, by construction, every $U \in \text{AffOp}(S)$ can be written as a filtered inverse limit with affine transition maps

$$U = \varinjlim_{k' \in \mathcal{S}_{k/k_0}^{op}} U_{(k')}$$

in the category of schemes. Therefore, by [Ayo14E, Cor. 3.22], we have a canonical equivalence

$$2 - \lim_{k' \in \mathcal{S}_{k'/k_0}^{op}} \mathbf{D} \mathbf{A}_{ct}^{\text{et}}(U_{(k')}, \mathbb{Q}) \xrightarrow{\sim} \mathbf{D} \mathbf{A}_{ct}^{\text{et}}(U, \mathbb{Q}).$$

Note also that, for $k' \in \mathcal{S}_{k/k_0}$ and $U \in \text{AffOp}_{k'}(S)$, we have a canonical identification of complex-analytic spaces

$$U_{(k')}^{\sigma} = U^{\sigma},$$

and the resulting diagram

$$\begin{tikzcd}
\mathbf{D} \mathbf{A}_{ct}^{\text{et}}(U_{(k')}, \mathbb{Q}) \arrow[r] & \mathbf{D} \mathbf{A}_{ct}^{\text{et}}(U, \mathbb{Q}) \\
\big|_{\mathcal{S}_{k'/k_0}^{op}} \arrow[u] \arrow[r] & \big|_{\mathcal{S}_{U, \sigma}} \arrow[u] \\
D_c^b(U_{(k')}^{\sigma}, \mathbb{Q}) \arrow[r] & D_c^b(U^{\sigma}, \mathbb{Q})
\end{tikzcd}$$
is commutative up to natural isomorphism. In fact, we have a canonical natural isomorphism

$$ \mathrm{Bti}^*_{U(k'), \sigma_{k'}} = \mathrm{Bti}^*_{U, \sigma} $$

as functors on the entire category of étale motives $\mathbf{DA}^{\text{ét}}(U_{(k')}, \mathbb{Q})$: this extends the identification of complex-analytic spaces

$$ X^{\sigma_{k'}} = (X \otimes_{k'} k)^{\sigma}, $$

which holds functorially in $X \in \mathbf{Sm}/U_{(k')}}}$. Passing to the colimit over $\mathcal{S}_{k/k_0}^{\text{op}}$, and applying [Ter23U, Lemma 5.12], this yields a canonical equivalence

$$ (34) \qquad 2 - \varinjlim_{k' \in \mathcal{S}_{k/k_0}} \mathcal{M}(U_{(k')}, \sigma|_{k'}) = \mathcal{M}(U, \sigma), $$

which is clearly compatible with restrictions under inclusions in $\mathbf{AffOp}(S)$. Lastly, note that, for every $k' \in \mathcal{S}_{k/k_0}$, the obvious inclusion of posets $\mathbf{AffOp}_{k'}(S) \subset \mathbf{AffOp}(S_{(k')})$ is cofinal.

Putting all these observation together, we obtain the chain of equivalences

$$
\begin{align*}
\mathcal{M}(L) &= \mathcal{M}(\mathrm{Spec}(L), \sigma) & \text{by Proposition 1.15} \\
&= 2 - \varinjlim_{U \in \mathrm{AffOp}(S)^{\mathrm{op}}} \mathcal{M}(U, \sigma) & \text{by definition} \\
&= 2 - \varinjlim_{k' \in \mathcal{S}_{k/k_0}} 2 - \varinjlim_{U \in \mathrm{AffOp}_{k'}(S)} \mathcal{M}(U, \sigma) & \text{by (33)} \\
&= 2 - \varinjlim_{k' \in \mathcal{S}_{k/k_0}} 2 - \varinjlim_{U \in \mathrm{AffOp}_{k'}(S)} \mathcal{M}(U_{(k')}, \sigma|_{k'}) & \text{by (34)} \\
&= 2 - \varinjlim_{k \in \mathcal{S}_{k/k_0}} 2 - \varinjlim_{V \in \mathrm{AffOp}(S_{(k')})} \mathcal{M}(V, \sigma|_{k'}) & \text{by cofinality} \\
&= 2 - \varinjlim_{k' \in \mathcal{S}_{k/k_0}} \mathcal{M}(\mathrm{Spec}(F'), \sigma|_{k'}) & \text{by definition} \\
&= 2 - \varinjlim_{k' \in \mathcal{S}_{k/k_0}} \mathcal{M}(F', \Sigma_{F'}) & \text{by Proposition 1.15} \\
&= 2 - \varinjlim_{F \in \mathcal{S}_L} \mathcal{M}(F, \Sigma_F) & \text{by cofinality.}
\end{align*}
$$

This concludes the proof. $\square$

*Proof of Theorem 1.12.* We claim that, as $S$ varies in $\mathbf{Var}_k$, the abelian subcategories $\mathcal{M}^0(S)$ of $\mathcal{M}(S)$ satisfy the conditions in the statement of Proposition 1.9. This will imply the thesis.

The validity of condition (i) follows from the fact that all the exact functors involved are constructed via the method described in [Ter23U, Prop. 2.5]. Similarly, the validity of condition (ii) follows from the fact that all the natural transformations involved are constructed via the method described in [Ter23U, Prop. 3.4].

Let us prove the validity of condition (iii): we have to show that, for every $S \in \mathbf{Var}_k$, the induced faithful exact functor

$$ (35) \qquad 2 - \varinjlim_{U \in \mathbf{SmOp}(S)^{\mathrm{op}}} \mathcal{M}^0(U) \hookrightarrow 2 - \varinjlim_{U \in \mathbf{SmOp}(S)^{\mathrm{op}}} \mathcal{M}(U) $$

is an equivalence. Note that $\mathbf{SmOp}(S)$-fibered categories $\mathcal{M}$ and $\mathcal{M}^0$ are stacks for the Zariski topology: for $\mathcal{M}$ this result is proved in [IM19, Prop. 2.7], while for $\mathcal{M}^0$ the same result follows using condition (i). As a consequence, in order to prove that (35) is an equivalence for a general $S \in \mathbf{Var}_k$, it suffices to treat
each irreducible component of S separately; in other words, without loss of generality we may assume S irreducible.

In this situation, let $L := k(S)$ denote the function field of $S$. Then, by the discussion of Construction 1.14, the functor (35) can be identified with the canonical inclusion

$$ \mathcal{M}^0(\mathrm{Spec}(L)) \hookrightarrow \mathcal{M}(\mathrm{Spec}(L)) $$

and, by Corollary 1.17, the latter can be identified with the functor

$$ 2 - \lim_{F \in \mathcal{F}_L^{\mathrm{op}}} \mathcal{M}^0(F) \hookrightarrow 2 - \lim_{F \in \mathcal{F}_L^{\mathrm{op}}} \mathcal{M}(F) $$

induced by the inclusions of categories of Nori motives

$$ (36) \qquad \mathcal{M}^0(F) \hookrightarrow \mathcal{M}(F) $$

over every finitely generated subfield $F \subset L$; here, the category $\mathcal{M}(F)$ is defined in terms of a complex embedding $\Sigma_F : F \hookrightarrow \mathbb{C}$ extending $\sigma|_{F \cap k}$. Therefore, in order to conclude, it suffices to show that the functors (36) are all equivalences. So fix a field $F$ finitely generated over $\mathbb{Q}$. By [BVHP20, Prop. 3.8], in order to prove that (36) is an equivalence, it suffices to show that, up to natural isomorphism, the Betti realization

$$ \mathrm{Bti}_{F, \Sigma_F}^* : \mathbf{D} \mathcal{A}_{ct}^{\mathrm{et}}(F, \mathbb{Q}) \to \mathcal{D}^b(\mathrm{vect}_{\mathbb{Q}}) $$

factors through the functor

$$ \iota_{F, \Sigma_F}^0 : \mathcal{D}^b(\mathcal{M}^0(F)) \to \mathcal{D}^b(\mathrm{vect}_{\mathbb{Q}}). $$

The latter claim was established by Nori; see [CG17, Prop. 7.1] or [Har16, Thm. 7.4.17] for a detailed proof. $\square$

## 2. CONSTRUCTION OF THE EXTERNAL TENSOR STRUCTURE

In this section we start constructing and studying the tensor structure on perverse motives. The final outcome of our effort will be the following theorem:

**Theorem 2.1.** The stable homotopy 2-functor $\mathcal{D}^b(\mathcal{M}(\cdot))$ on $\mathrm{Var}_k$ is canonically unitary symmetric monoidal in the sense of [Ayo07a, Defn. 2.3.1]. Moreover, the morphism of stable homotopy 2-functors

$$ \iota : \mathcal{D}^b(\mathcal{M}(\cdot)) \to \mathcal{D}^b(\mathrm{Perv}(\cdot)) $$

is canonically unitary symmetric monoidal in the sense of [Ayo10, Defn. 3.2].

The complete proof of this result will be obtained only at the very end of Section 4, as a combination of several partial results that we will collect along the way. More precisely, the results of the present section prove everything except for the existence of the motivic unit constraint and its compatibility with the classical unit constraint on $\mathcal{D}^b(\mathrm{Perv}(\cdot))$; this is the subject of Section 4 and requires the technical discussion developed in Section 3.

After discussing some general preliminaries about multi-linear functors and natural transformations thereof, the central part of the present section is devoted to constructing the single external tensor product functors on perverse motives as well as the monoidality isomorphisms for inverse image functors. In the last part, we will construct the associativity and commutativity constraints and show that they are compatible in the expected way.
**2.1. Categorical preliminaries.** Several constructions in Ivorra and Morel's paper [IM19] are based on a technical recognition principle for functors and natural transformations in the constructible derived categories: given two varieties $T, S \in \mathrm{Var}_k$ and a triangulated functor $F : D_c^b(S, \mathbb{Q}) \to D_c^b(T, \mathbb{Q})$ which is t-exact for the perverse t-structures, one wants to have a criterion guaranteeing that $F$ coincides with the trivial derived functor of its restriction to the perverse hearts

$$F : \mathrm{Perv}(S) \to \mathrm{Perv}(T)$$

under Beilinson's equivalences. More abstractly, given two abelian categories $\mathcal{A}$ and $\mathcal{B}$ and a triangulated functor $F : D^b(\mathcal{A}) \to D^b(\mathcal{B})$ which is t-exact for the obvious t-structures, one wants to have a criterion to decide whether $F$ coincides with the trivial derived functor of its restriction to the hearts

$$F : \mathcal{A} \to \mathcal{B}.$$

This is not always the case, due to the technical defects of triangulated categories. However, by [Vol10, Thm. 1], in order for this to hold it suffices that $F$ admit a dg-enhancement; a similar result holds for triangulated natural transformations between t-exact triangulated functors. This is summarized in [IM19, Prop. 4.4].

We are going to apply the same recognition principle to the case of multi-linear functors and natural transformations thereof. Before stating the precise result, let us spell out what we mean by multi-triangulated functors and natural transformations thereof.

**Definition 2.2.** Let $\mathcal{D}_1, \dots, \mathcal{D}_n$ and $\mathcal{E}$ be triangulated categories.

(1) A *multi-triangulated functor* from $\mathcal{D}_1 \times \dots \times \mathcal{D}_n$ to $\mathcal{E}$ is a functor

$$F : \mathcal{D}_1 \times \dots \times \mathcal{D}_n \to \mathcal{E}$$

which is functorially triangulated separately in each variable in such a way that, for every choice of $i, j \in \{1, \dots, n\}$ with $i < j$, the diagram of functors $\mathcal{D}_1 \times \dots \times \mathcal{D}_n \to \mathcal{E}$

$$F(A_1^\bullet, \dots, A_i^\bullet[1], \dots, A_j^\bullet[1], \dots, A_n^\bullet) \xrightarrow{\sim} F(A_1^\bullet, \dots, A_i^\bullet, \dots, A_j^\bullet[1], \dots, A_n^\bullet)[1]$$

$\downarrow$

$\downarrow$

$$F(A_1^\bullet, \dots, A_i^\bullet[1], \dots, A_j^\bullet, \dots, A_n^\bullet)[1] \xrightarrow{\sim} F(A_1^\bullet, \dots, A_i^\bullet, \dots, A_j^\bullet, \dots, A_n^\bullet)[2]$$

is anti-commutative.

(2) A *multi-triangulated natural transformation* between $F$ and $G$ is a natural transformation of functors $\mathcal{D}_1 \times \dots \times \mathcal{D}_n \to \mathcal{E}$

$$\alpha : F \to G$$

which is triangulated separately in each variable in the sense that, for every choice of $i \in \{1, \dots, n\}$, the diagram of functors $\mathcal{D}_1 \times \dots \times \mathcal{D}_n \to \mathcal{E}$

$$F(A_1^\bullet, \dots, A_i^\bullet[1], \dots, A_n^\bullet) \xrightarrow{\alpha} G(A_1^\bullet, \dots, A_i^\bullet[1], \dots, A_n^\bullet)$$

$\downarrow$

$\downarrow$

$$F(A_1^\bullet, \dots, A_i^\bullet, \dots, A_n^\bullet)[1] \xrightarrow{\alpha} G(A_1^\bullet, \dots, A_i^\bullet, \dots, A_n^\bullet)[1]$$

is commutative.

In order to motivate the definition, we quickly describe the fundamental example given by derived functors of multi-exact functors. In order to introduce it, we need some ad hoc notation about multi-dimensional complexes:

**Notation 2.3.** Let $\mathcal{A}$ be an abelian category. For every integer $n \ge 1$, we use the following notation:
• We regard $\mathbb{Z}^n$ as a poset with the order relation defined by

$$ (i_1, \dots, i_n) \le (i'_1, \dots, i'_n) \iff i_r \le i'_r \quad \forall r = 1, \dots, n. $$

• We let $\square_n^b(\mathcal{A})$ denote the category of $n$-dimensional bounded complexes with values in $\mathcal{A}$: it is the full subcategory of the functor category $\mathbf{Fun}(\mathbb{Z}^n, \mathcal{A})$ consisting of those functors $A^\square = (A^{i_1, \dots, i_n})_{i_1, \dots, i_n}$ such that:

(i) For every $(i_1, \dots, i_n) \in \mathbb{Z}^n$ and every $r = 1, \dots, n$, the composite morphism

$$ A^{i_1, \dots, i_r, \dots, i_n} \to A^{i_1, \dots, i_r+1, \dots, i_n} \to A^{i_1, \dots, i_r+2, \dots, i_n} $$

vanishes.

(ii) There exists $N \ge 0$ (depending on $A^\bullet$) such that $A^{i_1, \dots, i_n} = 0$ whenever $|i_1| + \dots + |i_n| > N$.

• We let

$$ \mathrm{Tot}^\square : \square_n^b(\mathcal{A}) \to \square_1^b(\mathcal{A}) = C^b(\mathcal{A}) $$

denote the total complex functor with respect to a fixed choice of signs.

We are now able to state the fundamental example concisely as follows:

**Lemma 2.4.** Let $\mathcal{A}_1, \dots, \mathcal{A}_n$ and $\mathcal{B}$ be abelian categories, and let $F : \mathcal{A}_1 \times \dots \times \mathcal{A}_n \to \mathcal{B}$ be a multi-exact functor. Then the formula

$$ F(A_1^\bullet, \dots, A_n^\bullet) := \mathrm{Tot}^\square(F(A_n^{i_1}, \dots, A_n^{i_n})_{i_1, \dots, i_n}) $$

canonically defines a multi-triangulated functor

$$ F : D^b(\mathcal{A}_1) \times \dots \times D^b(\mathcal{A}_n) \to D^b(\mathcal{B}). $$

*Proof.* If $n=2$, the classical computation in the case where $F$ is the tensor product functor in the category of vector spaces over some field carries over to the general framework described in the statement. The same computation also works for $n > 2$. We omit the details. $\square$

**Lemma 2.5.** Let $\mathcal{A}_1, \dots, \mathcal{A}_n$ and $\mathcal{B}$ be abelian categories, and let $F, G : \mathcal{A}_1 \times \dots \times \mathcal{A}_n \to \mathcal{B}$ be two multi-exact functors. Suppose that we are given a natural transformation of functors $\mathcal{A}_1 \times \dots \mathcal{A}_n \to \mathcal{B}$

$$ \alpha : F(A_1, \dots, A_n) \to G(A_1, \dots, A_n). $$

Then the formula

$$ \alpha(A_1^\bullet, \dots, A_n^\bullet) : \begin{tikzcd} & F(A_1^\bullet, \dots, A_n^\bullet) \arrow[d, "G(A_1^\bullet, \dots, A_n^\bullet)"'] \\ \mathrm{Tot}^\square(F(A_1^{i_1}, \dots, A_n^{i_n})_{i_1, \dots, i_n}) & (\kappa(A_1^{i_1}, \dots, A_n^{i_n})_{i_1, \dots, i_n}) \arrow[d, "\sim"'] \arrow[r, "\mathrm{Tot}^\square(G(A_1^{i_1}, \dots, A_n^{i_n})_{i_1, \dots, i_n})"'] \end{tikzcd} $$

defines a multi-triangulated natural transformation between functors $D^b(\mathcal{A}_1) \times \dots \times D^b(\mathcal{A}_n) \to D^b(\mathcal{B})$

$$ \alpha : F(A_1^\bullet, \dots, A_n^\bullet) \to G(A_1^\bullet, \dots, A_n^\bullet) $$

Moreover, this construction is compatible with identities of functors as well as with composition of natural transformations; in particular, it preserves invertibility of natural transformations.

*Proof.* The standard computation in the case where $F$ and $G$ are suitable combinations of tensor product functors in the category of vector spaces over some field carries over to the general setting described in the statement. We omit the details. $\square$

Finally, we can state the aforementioned generalization of [Vol10, Thm. 1] to the multi-linear setting as follows:

**Proposition 2.6.** Let $\mathcal{A}_1, \dots, \mathcal{A}_n$ and $\mathcal{B}$ be abelian categories. The following statements hold:
(1) Let $F: D^b(\mathcal{A}_1) \times \dots \times D^b(\mathcal{A}_n) \to D^b(\mathcal{B})$ be a multi-triangulated functor which is $t$-exact in each variable for the obvious $t$-structures, and suppose that it can be enhanced to a dg-quasifunctor. Then $F$ is canonically isomorphic to the functor obtained from the induced multi-exact functor

$$F: \mathcal{A}_1 \times \dots \times \mathcal{A}_n \to \mathcal{B}$$

via the construction of Lemma 2.4.

(2) Let $F, G: D^b(\mathcal{A}_1) \times \dots \times D^b(\mathcal{A}_n) \to D^b(\mathcal{B})$ be two multi-triangulated functors which are $t$-exact in each variable for the obvious $t$-structures, and suppose that they can be enhanced to dg-quasifunctors; fix one such enhancement. Moreover, let $\alpha: F \to G$ be a multi-triangulated natural transformation, and suppose that it can be enhanced to a morphism of dg-quasifunctors. Then $\alpha$ coincides with the canonical natural transformation obtained from the induced natural transformation of functors $\mathcal{A}_1 \times \dots \times \mathcal{A}_n \to \mathcal{B}$

$$\alpha: F \to G$$

via the construction of Lemma 2.5.

*Proof.* For sake of simplicity, we only sketch the argument in the case $n=2$.

(1) For every fixed object $A_1 \in \mathcal{A}_1$, the induced triangulated functor

$$F(A_1, -): D^b(\mathcal{A}_2) \to D^b(\mathcal{B})$$

it $t$-exact by hypothesis and inherits a dg-enhancement from $F$. Applying [Vol10, Thm. 1] to it, we deduce that it coincides with the trivial derived functor of the induced exact functor

$$F(A_1, -): \mathcal{A}_2 \to \mathcal{B}.$$

Now, arguing by dévissage in the first variable, one obtains the analogous conclusion for $F$.

(2) For every fixed object $A_1 \in \mathcal{A}_1$, the induced natural transformation of triangulated functors $\mathcal{A}_2 \to \mathcal{B}$

$$\alpha(A_1, -): F(A_1, -) \to G(A_1, -)$$

is triangulated by hypothesis and inherits a dg-enhancement from $\alpha$. Applying [Vol10, Thm. 1] to it, we deduce that it coincides with the canonical natural transformation induced by the natural transformation between functors $\mathcal{A}_2 \to \mathcal{B}$

$$\alpha(A_1, -): F(A_1, -) \to G(A_1, -).$$

Again, arguing by dévissage in the first variable, one obtains the analogous conclusion for $\alpha$.

□

*Remark 2.7.* In particular, since the construction of Lemma 2.5 is compatible with composition of natural transformations, the result of Proposition 2.6(2) can be used to show the commutativity of natural diagrams of $t$-exact multi-triangulated functors by checking the commutativity of the induced diagram of multi-exact functors on the hearts of the various $t$-structures.

In the setting of perverse motives, Proposition 2.6(1) asserts that, given varieties $S_1, \dots, S_n, T \in \mathrm{Var}_k$, every dg-enhanced multi-triangulated functor $F: D_c^b(S_1, \mathbb{Q}) \times \dots \times D_c^b(S_n, \mathbb{Q}) \to D_c^b(T, \mathbb{Q})$ which is $t$-exact for the perverse $t$-structures separately in each variable is canonically isomorphic to the multi-derived functor of its restriction

$$F: \mathrm{Perv}(S_1) \times \dots \times \mathrm{Perv}(S_n) \to \mathrm{Perv}(T)$$

under Beilinson’s equivalences; Proposition 2.6(2) asserts that the same holds for natural transformations between $t$-exact multi-triangulated functors.
All the multi-triangulated functors between constructible derived categories that we will need to consider are suitable combinations of three types of functors: shifted inverse images under smooth equidimensional morphisms, direct images under closed immersions, and external tensor product functors.

## 2.2. External tensor structure.
We can now start constructing the monoidal structure on perverse motives. As explained in the introduction of the paper, the lack of t-exactness of the usual internal tensor product functors forces one to work systematically with the external tensor product functors. The legitimacy of this approach is completely justified by the results of [Ter23T], where we show that the entire theory of monoidal fibered categories can be canonically expressed in terms of the external tensor product: there exists a canonical dictionary relating the structure and properties of the internal tensor product to the structure and properties of the external tensor product. In the rest of this section we systematically adopt the language of external tensor structures introduced in [Ter23T], which allows us to express our main constructions in a concise and precise form.

In fact, all the results of this section are proved for the abelian categories $\mathcal{M}^0(S)$ defined in Notation 3.1; the corresponding results for the categories $\mathcal{M}(S)$ will be simply deduced by transport of structure via the equivalences of Theorem 1.12. This operation is legitimate because, as pointed out at the beginning of the proof of Theorem 1.12, the categories $\mathcal{M}^0(S)$ are stable under all the exact functors listed in the statement of Proposition 1.9, and the structure of stable homotopy 2-functor of $D^b(\mathcal{M}(\cdot))$ obtained in [IM19] is constructed using only these functors (together with the motivic Verdier duality functors $\mathbb{D}_S$, which also preserve the subcategories $\mathcal{M}^0(S)$). Mostly for psychological reasons, we will use the more precise notation "$\mathcal{M}^0$" in place of "$\mathcal{M}$" whenever developing construction or proofs where it is necessary to use the alternative description of perverse motives. However, as in the case of Theorem 2.1 above, we will adopt the usual notation "$\mathcal{M}$" in the formulation of our main results.

Our first task is to construct the single external tensor product functors together with external monoidality isomorphisms witnessing their compatibility with inverse image functors of perverse motives, in such a way that everything is compatible with the corresponding structure on the underlying categories of perverse sheaves. As always in the theory of Nori motives, the bulk of the construction is based on the general lifting principles for universal abelian factorizations; in this case, we will apply the specific lifting results for external tensor structures on abelian fibered categories as summarized in [Ter23U, § 6]. Even if general inverse image functors of perverse motives are definitely not exact, the results of [Ter23F] can be used to reduce the sought-after structure to a minimal amount of data involving only inverse images under smooth morphisms and direct images under closed immersions, which are always exact on perverse motives (up to dividing into connected components and shifting appropriately). Our first intermediate result towards Theorem 2.1 can be stated as follows:

**Proposition 2.8.** The triangulated $\mathbf{Var}_k$-fibered category $D^b(\mathcal{M}(\cdot))$ carries a canonical triangulated external tensor structure in the sense of [Ter23T, Defn. 2.6]. Moreover, the morphism of $\mathbf{Var}_k$-fibered categories $\iota : D^b(\mathcal{M}(\cdot)) \to D^b(\mathrm{Perv}(\cdot))$ carries a canonical external tensor structure in the sense of [Ter23T, Defn. 8.5].

The basic input from the theory of triangulated motives is the existence of a canonical external tensor structure on the $\mathbf{Var}_k$-fibered categories $\mathbf{DA}_{ct}^{\mathcal{E}t}(\cdot, \mathbb{Q})$ and $D_c^b(\cdot, \mathbb{Q})$ related by a canonical external tensor structure on the Betti realization

$$ \mathrm{Bti}^* : \mathbf{DA}_{ct}^{\mathcal{E}t}(\cdot, \mathbb{Q}) \to D_c^b(\cdot, \mathbb{Q}). $$

Under the canonical dictionary of [Ter23T], they correspond to the usual monoidal structures constructed as part of the six functor formalism in [Ayo07a, Ayo07b] and [Ayo10].

**Notation 2.9.** From now on, we adopt the following notation:

* For every morphism $f : T \to S$ in $\mathbf{Var}_k$, we use the following notation:
- We let $\theta = \theta_f : f^* \circ \text{Bti}_S^* \xrightarrow{\sim} \text{Bti}_T \circ f^*$ denote the Bti*-transition isomorphism along $f$ constructed in [Ayo10].

- We let $\tilde{\theta} = \tilde{\theta}_f : f^* \circ \iota_S \xrightarrow{\sim} \iota_T \circ f^*$ denote the $\iota$-transition isomorphism along $f$ constructed in [IM19, § 2.1].

* For every closed immersion $z: Z \hookrightarrow S$ in $\text{Var}_k$, we use the following notation:

- We let $\tilde{\theta} = \tilde{\theta}_z : \text{Bti}_S^* \circ z_* \xrightarrow{\sim} z_* \circ \text{Bti}_Z^*$ denote the natural isomorphism obtained as in [Ter23F, § 1] from $\theta_z$.

- We let $\tilde{\tilde{\theta}} = \tilde{\tilde{\theta}}_z : \iota_S \circ z_* \xrightarrow{\sim} z_* \circ \iota_Z$ denote the natural isomorphism obtained as in [Ter23F, § 1] from $\tilde{\theta}_z$.

* We use the symbol $(\boxtimes, m; a, c)$ for the canonical symmetric associative triangulated external tensor structures on $\mathbf{DA}_{ct}^{\text{\'et}}(\cdot, \mathbb{Q})$ and $\mathbf{D}_c^b(\cdot, \mathbb{Q})$.

* We let $\rho$ denote the canonical triangulated external tensor structure on the morphism of triangulated $\text{Var}_k$-fibered categories $\text{Bti}^* : \mathbf{DA}_{ct}^{\text{\'et}}(\cdot, \mathbb{Q}) \to \mathbf{D}_c^b(\cdot, \mathbb{Q})$.

The proof of Proposition 2.8 crucially exploits the general factorization techniques introduced in [Ter23F]: these are applied to the setting of fibered categories over $\text{Var}_k$ with respect to the subcategories $\text{Var}_k^{sm}$ and $\text{Var}_k^{cl}$ collecting smooth morphisms and the closed immersions, respectively (see [Ter23F, Ex. 2.2]).

In order to simplify the notation in some parts of the proofs, it is convenient to adopt the following notational convention:

**Notation 2.10.** Let $\mathbb{H}$ be a triangulated $\text{Var}_k$-fibered category which is localic in the sense of [Ter23F, Defn. 3.4].

* Let $f: T \to S$ be a morphism in $\text{Var}_k$ such that the number $e := \dim(\mathcal{O}_{T,t}) - \dim(\mathcal{O}_{S,f(t)})$ is independent of $t \in T$. We define the shifted inverse image functor

$$f^{\dagger} := f^*[e] : \mathbb{H}(S) \to \mathbb{H}(T).$$

The two cases of interest for us are the following:

(i) when $f$ is a smooth morphism of constant relative dimension $e$;

(ii) when $f$ is a regular closed immersion of constant codimension $e$.

* More generally, if $f: T \to S$ is a morphism whose restriction to each connected component of $S$ has the property described in the previous point, we define the shifted inverse image functor $f^{\dagger}$ by extending the previous definition in the obvious way.

This notation is compatible with composition and interacts with connection isomorphisms and $R$-transition isomorphisms in the expected way.

**Proof of Proposition 2.8.** In the first place, fix two varieties $S_1, S_2 \in \text{Var}_k$ and consider the diagram

$$ \begin{array}{ccc} \mathcal{D}^0(S_1) \times \mathcal{D}^0(S_2) & \xrightarrow{-\boxtimes-} & \mathcal{D}^0(S_1 \times S_2) \\ \bigdownarrow_{\beta_{S_1}^0 \times \beta_{S_2}^0} & & \bigdownarrow_{\beta_{S_1 \times S_2}^0} \\ \text{Perv}(S_1) \times \text{Perv}(S_2) & \xrightarrow{-\boxtimes-} & \text{Perv}(S_1 \times S_2). \end{array} $$

The natural isomorphism of functors $\mathbf{DA}_{ct}^{\text{\'et}}(S_1, \mathbb{Q}) \times \mathbf{DA}_{ct}^{\text{\'et}}(S_2, \mathbb{Q}) \to \mathbf{D}_c^b(S_1 \times S_2, \mathbb{Q})$

$$ \text{Bti}_{S_1}^*(A_1) \boxtimes \text{Bti}_{S_2}^*(A_2) \xrightarrow{\sim} \text{Bti}_{S_1 \times S_2}^*(A_1 \boxtimes A_2) $$

restricts to a natural isomorphism between the two composite functors $\mathcal{D}^0(S_1) \times \mathcal{D}^0(S_2) \to \text{Perv}(S_1 \times S_2)$. Applying [Ter23U, Prop. 3.3], we obtain a canonical bi-exact functor

$$ (37) \qquad - \boxtimes - = - \boxtimes_{S_1, S_2} - : \mathcal{M}^0(S_1) \times \mathcal{M}^0(S_2) \to \mathcal{M}^0(S_1 \times S_2) $$
together with a canonical natural isomorphism of functors $\mathcal{M}^0(S_1) \times \mathcal{M}^0(S_2) \to \mathrm{Perv}(S_1 \times S_2)$

$$ (38) \qquad \tilde{\rho} = \tilde{\rho}_{S_1, S_2} : \iota_{S_1}(M_1) \boxtimes \iota_{S_2}(M_2) \xrightarrow{\sim} \iota_{S_1 \times S_2}(M_1 \boxtimes M_2). $$

By Lemma 2.4 and Lemma 2.5, these extend to a bi-triangulated functor

$$ (39) \qquad - \boxtimes - : D^b(\mathcal{M}^0(S_1)) \times D^b(\mathcal{M}^0(S_2)) \to D^b(\mathcal{M}^0(S_1 \times S_2)) $$

and a bi-triangulated natural isomorphism of functors $D^b(\mathcal{M}^0(S_1)) \times D^b(\mathcal{M}^0(S_2)) \to D^b(\mathrm{Perv}(S_1 \times S_2))$

$$ (40) \qquad \tilde{\rho} = \tilde{\rho}_{S_1, S_2} : \iota_{S_1}(M_1^\bullet) \boxtimes \iota_{S_2}(M_2^\bullet) \xrightarrow{\sim} \iota_{S_1 \times S_2}(M_1^\bullet \boxtimes M_2^\bullet). $$

In order to turn the collection of functors (39) into an external tensor structure on $D^b(\mathcal{M}(\cdot))$, we have to construct the external monoidality isomorphisms with respect to inverse image functors in $\mathrm{Var}_k$ compatibly with composition. Once this is done, in order to check that the collection of natural transformations (40) defines an external tensor structure on the morphism $\iota : D^b(\mathcal{M}(\cdot)) \to D^b(\mathrm{Perv}(\cdot))$, we have to show that they are compatible with the external monoidality isomorphisms on the two sides. If this holds, then the conservativity of $\iota$ implies the validity of the projection formulae on $D^b(\mathcal{M}(\cdot))$.

Therefore, as a consequence of the discussion in [Ter23T, §§ 3, 5], we see that the sought-after external tensor structures on $D^b(\mathcal{M}(\cdot))$ and on $\iota$ are completely determined by the underlying external tensor cores, as defined in [Ter23T, Defn. 5.2, Defn. 5.6], and so we are reduced to the task of constructing the latter. To this end, we apply the results of [Ter23U, § 6] to the underlying triangulated $\mathrm{Var}_k^{sm}$-fibered and $\mathrm{Var}_k^{cl,op}$-fibered categories as follows:

* Regard $\mathrm{Bti}^*$: $\mathrm{DA}_{ct}^{\dot{\epsilon}t}(\cdot, \mathbb{Q}) \to D_c^b(\cdot, \mathbb{Q})$ as a morphism of $\mathrm{Var}_k^{sm}$-fibered categories with respect to the shifted inverse image functors $p^\dagger$ introduced in Notation 2.10. Using the additivity of relative dimensions with respect to direct products of morphisms, we see that the canonical external tensor structures on $\mathrm{DA}_{ct}^{\dot{\epsilon}t}(\cdot, \mathbb{Q})$, $D_c^b(\cdot, \mathbb{Q})$ and $\mathrm{Bti}^*$ induce external tensor structures on the $\mathrm{Var}_k^{sm}$-fibered categories considered. By restriction, these in turn induce analogous external tensor structures on the $\mathrm{Var}_k^{sm}$-fibered categories $\mathcal{D}^0(\cdot)$ and $\mathrm{Perv}(\cdot)$ as well as on the morphism of $\mathrm{Var}_k^{sm}$-fibered categories $\beta^0: \mathcal{D}^0(\cdot) \to \mathrm{Perv}(\cdot)$. Applying [Ter23U, Prop. 6.4], we obtain canonical external tensor structures on the abelian $\mathrm{Var}_k^{sm}$-fibered category $\mathcal{M}^0(\cdot)$ as well as on the morphism of abelian $\mathrm{Var}_k^{sm}$-fibered categories $\iota: \mathcal{M}^0(\cdot) \to \mathrm{Perv}(\cdot)$. Finally, applying Lemma 2.5 and shifting the inverse image functors back, the latter define triangulated external tensor structures on the triangulated $\mathrm{Var}_k^{sm}$-fibered category $D^b(\mathcal{M}(\cdot))$ as well as on the triangulated morphism $\iota: D^b(\mathcal{M}(\cdot)) \to D^b(\mathrm{Perv}(\cdot))$.

* Regard $\mathrm{Bti}^*$: $\mathrm{DA}_{ct}^{\dot{\epsilon}t}(\cdot, \mathbb{Q}) \to D_c^b(\cdot, \mathbb{Q})$ as a morphism of $\mathrm{Var}_k^{cl,op}$-fibered categories with respect to the direct image functors $z_*$.$\mathrm{Using}$ [Ter23F, Lemma 1.9, Lemma 1.13], we see that the canonical external tensor structures on $\mathrm{DA}_{ct}^{\dot{\epsilon}t}(\cdot, \mathbb{Q})$, $D_c^b(\cdot, \mathbb{Q})$ and $\mathrm{Bti}^*$ induce external tensor structures on the $\mathrm{Var}_k^{cl,op}$-fibered categories considered. By restriction, these in turn induce analogous external tensor structures on the $\mathrm{Var}_k^{cl,op}$-fibered categories $\mathcal{D}^0(\cdot)$ and $\mathrm{Perv}(\cdot)$ as well as on the morphism of $\mathrm{Var}_k^{cl,op}$-fibered categories $\beta^0: \mathcal{D}^0(\cdot) \to \mathrm{Perv}(\cdot)$. Applying [Ter23U, Prop. 6.4], we obtain canonical external tensor structures on the abelian $\mathrm{Var}_k^{cl,op}$-fibered category $\mathcal{M}^0(\cdot)$ as well as on the morphism of abelian $\mathrm{Var}_k^{cl,op}$-fibered categories $\iota: \mathcal{M}^0(\cdot) \to \mathrm{Perv}(\cdot)$. Finally, applying Lemma 2.5 and using again [Ter23F, Lemma 1.9, Lemma 1.13], the latter define triangulated external tensor structures on the triangulated $\mathrm{Var}_k^{cl,op}$-fibered category $D^b(\mathcal{M}(\cdot))$ as well as on the triangulated morphism $\iota: D^b(\mathcal{M}(\cdot)) \to D^b(\mathrm{Perv}(\cdot))$.

* We have to show that the constructions of the previous two points satisfy the compatibility condition of [Ter23F, Defn. 5.2] and thus define an external tensor core on the $\mathrm{Var}_k$-fibered category $D^b(\mathcal{M}(\cdot))$; since the validity of the conditions of [Ter23F, Defn. 8.5] has been already justified in the previous two points, we will be able to conclude that these two constructions also define an external tensor core on the morphism of $\mathrm{Var}_k$-fibered categories $\iota: D^b(\mathcal{M}(\cdot)) \to D^b(\mathrm{Perv}(\cdot))$.
Let us then check the two conditions (C'-ETC) and (T-ETC) of [Ter23F, Defn. 5.2] are satisfied: (C'-ETC) Given two Cartesian squares in Var_k

$$ 
\begin{array}{ccc}
& z'_i & P_i \\
P_i, Z_i & \xrightarrow{z'_i} & P_i \\
\downarrow p'_i & \xrightarrow{\downarrow} & \downarrow p_i \\
Z_i & \xrightarrow{z_i} & S_i
\end{array}
\qquad (i = 1, 2)
$$

where $p_i$ (and hence $p'_i$) is smooth while $z_i$ (and hence $z'_i$) is a closed immersion, we have to show that the diagram of functors $D^b(\mathcal{M}^0(Z_1)) \times D^b(\mathcal{M}^0(Z_2)) \to D^b(\mathcal{M}^0(P_1 \times P_2))$

$$ 
\begin{array}{ccc}
& \bar{m} & (p_1 \times p_2)^*(z_{1,*} M_1 \boxtimes z_{2,*} M_2) \\
p_1^* z_{1,*} M_1 \boxtimes p_2^* z_{2,*} M_2 & \xrightarrow{\bar{m}} & (p_1 \times p_2)^*(z_1 \times z_2)_*(M_1^\bullet \boxtimes M_2^\bullet) \\
\downarrow & \xrightarrow{\downarrow} & \downarrow \\
z_{1,*} p'_1 M_1 \boxtimes z'_{2,*} p'^*_2 M_2 & \xrightarrow{\bar{m}} & (z'_1 \times z'_2)_*(p_1^* M_1 \boxtimes p_2^* M_2^\bullet) \xrightarrow{\bar{m}} (z'_1 \times z'_2)_*(p_1 \times p_2)^*(M_1^\bullet \boxtimes M_2^\bullet)
\end{array}
$$

is commutative. After shifting the inverse image functors under smooth morphisms as in Notation 2.10, all the triangulated functors involved become t-exact for the obvious t-structures. Since all of them are obviously dg-enhanced, we are allowed to invoke Lemma 2.5 (see Remark 2.7) which reduces us to showing that the diagram of exact functors $\mathcal{M}^0(Z_1) \times \mathcal{M}^0(Z_2) \to \mathcal{M}^0(P_1 \times P_2)$

$$ 
\begin{array}{ccc}
& \bar{m} & (p_1 \times p_2)^{\dagger}(z_{1,*} M_1 \boxtimes z_{2,*} M_2) \\
p_1^{\dagger} z_{1,*} M_1 \boxtimes p_2^{\dagger} z_{2,*} M_2 & \xrightarrow{\bar{m}} & (p_1 \times p_2)^{\dagger}(z_1 \times z_2)_*(M_1 \boxtimes M_2) \\
\downarrow & \xrightarrow{\downarrow} & \downarrow \\
z'_{1,*} p_1^{\dagger} M_1 \boxtimes z'_{2,*} p_2^{\dagger} M_2 & \xrightarrow{\bar{m}} & (z'_1 \times z'_2)_*(p_1^{\dagger} M_1 \boxtimes p_2^{\dagger} M_2) \xrightarrow{\bar{m}} (z'_1 \times z'_2)_*(p_1 \times p_2)^{\dagger}(M_1 \boxtimes M_2)
\end{array}
$$

is commutative. Since all the functors and natural transformations figuring in the latter diagram are obtained via the lifting principles for universal abelian factorizations, using [Ter23U, Rem. 4.7(1)(2)] we see that the sought-after commutativity reduces to the commutativity of the analogous diagram of functors on $\mathbf{DA}_{ct}^{\hat{\text{et}}}(\cdot, \mathbb{Q})$. After shifting back, the latter in turn reduces to the validity of axiom (C'-ETC) on $\mathbf{DA}_{ct}^{\hat{\text{et}}}(\cdot, \mathbb{Q})$, which we already know.

(T-ETC) Given two commutative triangles in Var_k

$$ 
\begin{array}{ccc}
& h_i & P_i \\
Q_i & \xrightarrow{h_i} & P_i \\
\downarrow q_i & \xrightarrow{\downarrow} & \downarrow p_i \\
S_i & \xrightarrow{\downarrow} & S_i
\end{array}
\qquad (i = 1, 2)
$$

where $p_i$ and $q_i$ are smooth while $h_i$ is a closed immersion (necessarily regular), we have to show that the diagram of functors $D^b(\mathcal{M}^0(S_1)) \times D^b(\mathcal{M}^0(S_2)) \to D^b(\mathcal{M}^0(Q_1 \times Q_2))$

$$ 
\begin{array}{ccc}
& \bar{m} & (q_1 \times q_2)^*(M_1^\bullet \boxtimes M_2^\bullet) \\
q_1^* M_1^\bullet \boxtimes q_2^* M_2^\bullet & \xrightarrow{\bar{m}} & (q_1 \times q_2)^*(M_1^\bullet \boxtimes M_2^\bullet) \\
\downarrow & \xrightarrow{\downarrow} & \downarrow \\
h_1^* p_1^* M_1^\bullet \boxtimes h_2^* p_2^* M_2^\bullet & \xrightarrow{\bar{m}} & (h_1 \times h_2)^*(p_1^* M_1^\bullet \boxtimes p_2^* M_2^\bullet) \xrightarrow{\bar{m}} (h_1 \times h_2)^*(p_1 \times p_2)^*(M_1^\bullet \boxtimes M_2^\bullet)
\end{array}
$$

is commutative. As before, we may assume that $p_1, p_2$ and $q_1, q_2$ are equidimensional. In this case, after shifting appropriately so that all the functors involved become t-exact for the obvious
t-structures, we are reduced to showing that the diagram of exact functors $\mathcal{M}^0(S_1) \times \mathcal{M}^0(S_2) \to \mathcal{M}^0(Q_1 \times Q_2)$

$$
\begin{tikzcd}
& & q_1^\dagger M_1 \boxtimes q_2^\dagger M_2 \arrow[r, "\tilde{m}"] \arrow[d, "\|"'] & (q_1 \times q_2)^\dagger (M_1 \boxtimes M_2) \arrow[d, "\|"'] \\
& h_1^\dagger p_1^\dagger M_1 \boxtimes h_2^\dagger p_2^\dagger M_2 \arrow[r, "\tilde{m}"] & (h_1 \times h_2)^\dagger (p_1^\dagger M_1 \boxtimes p_2^\dagger M_2) \arrow[r, "\tilde{m}"] & (h_1 \times h_2)^\dagger (p_1 \times p_2)^\dagger (M_1 \boxtimes M_2)
\end{tikzcd}
$$

is commutative. Unfortunately, this cannot be deduced directly from general properties of
universal abelian factorizations, since the inverse images under closed immersions on perverse
motives constructed in [IM19, § 4.1] are not defined via the lifting principles of universal abelian
factorizations. In any case, in order to prove the above diagram is commutative, it suffices to
show that its image under the faithful exact functor $\iota_{Q_1 \times Q_2}$

$$
\begin{tikzcd}
& & \iota_{Q_1 \times Q_2} (q_1^\dagger M_1 \boxtimes q_2^\dagger M_2) \arrow[r, "\tilde{m}"] \arrow[d, "\|"'] & \iota_{Q_1 \times Q_2} ((q_1 \times q_2)^\dagger (M_1 \boxtimes M_2)) \arrow[d, "\|"'] \\
\iota_{Q_1 \times Q_2} (h_1^\dagger p_1^\dagger M_1 \boxtimes h_2^\dagger p_2^\dagger M_2) \arrow[r, "\tilde{m}"] & \iota_{Q_1 \times Q_2} ((h_1 \times h_2)^\dagger (p_1^\dagger M_1 \boxtimes p_2^\dagger M_2)) \arrow[r, "\tilde{m}"] & \iota_{Q_1 \times Q_2} ((h_1 \times h_2)^\dagger (p_1 \times p_2)^\dagger (M_1 \boxtimes M_2))
\end{tikzcd}
$$

is commutative, and this can be checked directly. Indeed, by inserting, on the left side of the
latter diagram, the diagram

$$
\begin{tikzcd}
& & \iota_{Q_1 \times Q_2} (q_1^\dagger M_1 \boxtimes q_2^\dagger M_2) \arrow[r, "\tilde{\rho}"] \arrow[d, "\|"'] & \iota_{Q_1} (q_1^\dagger M_1) \boxtimes \iota_{Q_2} (q_2^\dagger M_2) \arrow[r, "\tilde{\theta}"] \arrow[d, "\|"'] & q_1^\dagger \iota_{S_1} (M_1) \boxtimes q_2^\dagger \iota_{S_2} (M_2) \arrow[d, "\|"'] \\
& h_1^\dagger p_1^\dagger \iota_{S_1} (M_1) \boxtimes h_2^\dagger p_2^\dagger \iota_{S_2} (M_2) \arrow[r, "\tilde{\theta}", "\downarrow"'] & \iota_{Q_1} (h_1^\dagger p_1^\dagger M_1) \boxtimes \iota_{Q_2} (h_2^\dagger p_2^\dagger M_2) \arrow[r, "\tilde{\theta}", "\downarrow"'] & h_1^\dagger \iota_{P_1} (p_1^\dagger M_1) \boxtimes h_2^\dagger \iota_{P_2} (p_2^\dagger M_2)
\end{tikzcd}
$$

and, on its right side, the diagram

$$
\begin{tikzcd}
& & (q_1 \times q_2)^{\dagger} (\iota_{S_1}(M_1) \boxtimes \iota_{S_2}(M_2)) \arrow[r, "\tilde{\rho}"] \arrow[d, "\|"'] & (q_1 \times q_2)^{\dagger} (\iota_{S_1} \times_{S_2} (M_1 \boxtimes M_2)) \arrow[r, "\tilde{\theta}"] \arrow[d, "\|"'] & \iota_{Q_1 \times Q_2} ((q_1 \times q_2)^{\dagger} (M_1 \boxtimes M_2)) \arrow[d, "\|"'] \\
& h_1 \times h_2)^{\dagger} (p_1 \times p_2)^{\dagger} (\iota_{S_1}(M_1) \boxtimes \iota_{S_2}(M_2)) \arrow[r, "\tilde{\rho}"] \arrow[d, "\|"'] & h_1 \times h_2)^{\dagger} (p_1 \times p_2)^{\dagger} \iota_{S_1} \times_{S_2} (M_1 \boxtimes M_2) \arrow[r, "\tilde{\theta}", "\uparrow"'] & h_1 \times h_2)^{\dagger} \iota_{P_1} \times_{P_2} ((p_1 \times p_2)^{\dagger} (M_1 \boxtimes M_2))
\end{tikzcd}
$$
both of which are commutative by naturality and by axiom (mor-Var<sub>k</sub>-fib) of [Ter23T, Defn. 8.5],
we are reduced to considering the outer part of the diagram

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

Here, the two four-term pieces are commutative by naturality, the three six-term pieces are commutative by axiom (mor-ETS) of [Ter23T, Defn. 3.6], and the central five-term piece (after shifting back) is commutative by the validity of axiom (T-ETC) on $D^b(\mathrm{Perv}(\cdot))$.

This concludes the proof.

□

2.3. **External associativity and commutativity constraints.** We go on by equipping the external tensor core of Proposition 2.8 with compatible external associativity and commutativity constraints; for a general reference about these notions, see [Ter23T, §§ 3–5] and [Ter23F, § 4]. For sake of clarity, we first construct the two constraints separately and only check their compatibility in a second moment.

**Lemma 2.11.** The external tensor structure on $D^b(\mathcal{M}(\cdot))$ carries a canonical external associativity constraint in the sense of [Ter23T, Defn. 3.2], with respect to which the morphism of $\mathrm{Var}_k$-fibered categories $\iota: D^b(\mathcal{M}(\cdot)) \to D^b(\mathrm{Perv}(\cdot))$ is associative in the sense of [Ter23T, Defn. 8.13].
Proof. In the first place, fix three varieties $S_1, S_2, S_3 \in \mathrm{Var}_k$ and consider the diagram

$$
\begin{tikzcd}
& & \mathcal{D}^0(S_1) \times \mathcal{D}^0(S_2) \times \mathcal{D}^0(S_3) \arrow[r, "(-\boxminus-) \times \mathrm{id}"] \arrow[d, "\beta_{S_1}^0 \times \beta_{S_2}^0 \times \beta_{S_3}^0"] & \mathcal{D}^0(S_1 \times S_2) \times \mathcal{D}^0(S_3) \arrow[r, "(-\boxminus-) \times \mathrm{id}"] \arrow[d, "\beta_{S_1}^0 \times S_2 \times \beta_{S_3}^0"] & \mathcal{D}^0(S_1 \times S_2 \times S_3) \arrow[r, "(-\boxminus-) \times \mathrm{id}"] \arrow[d, "\beta_{S_1}^0 \times S_2 \times S_3"] & \mathcal{D}^0(S_1 \times S_2 \times S_3) \\
& \mathcal{D}^0(S_1) \times \mathcal{D}^0(S_2) \times \mathcal{D}^0(S_3) & \mathcal{D}^0(S_1) \times \mathcal{D}^0(S_2 \times S_3) & \mathcal{D}^0(S_1 \times S_2 \times S_3) & \mathcal{D}^0(S_1 \times S_2 \times S_3) \arrow[r, "(-\boxminus-) \times \mathrm{id}"] \arrow[d, "\beta_{S_1}^0 \times \beta_{S_2}^0 \times \beta_{S_3}^0"] & \mathcal{D}^0(S_1 \times S_2 \times S_3) \arrow[r, "(-\boxminus-) \times \mathrm{id}"] \arrow[d, "\beta_{S_1}^0 \times S_2 \times S_3"] & \mathcal{D}^0(S_1 \times S_2 \times S_3) \arrow[r, "(-\boxminus-) \times \mathrm{id}"] \arrow[d, "\beta_{S_1}^0 \times S_2 \times S_3"] & \mathcal{D}^0(S_1 \times S_2 \times S_3)
\end{tikzcd}
$$

$$
\begin{tikzcd}
& & \mathcal{D}^0(S_1) \times \mathcal{D}^0(S_2) \times \mathcal{D}^0(S_3) \arrow[r, "\mathrm{id} \times (-\boxminus-)"] & \mathcal{D}^0(S_1) \times \mathcal{D}^0(S_2 \times S_3) \arrow[r, "(-\boxminus-)"] & \mathcal{D}^0(S_1 \times S_2 \times S_3) \arrow[r, "(-\boxminus-)"] \arrow[d, "\beta_{S_1}^0 \times \beta_{S_2}^0 \times \beta_{S_3}^0"] & \mathcal{D}^0(S_1 \times S_2 \times S_3) \arrow[d, "\beta_{S_1}^0 \times S_2 \times S_3"] & \mathcal{D}^0(S_1 \times S_2 \times S_3) \arrow[r, "(-\boxminus-)"] \arrow[d, "\beta_{S_1}^0 \times S_2 \times S_3"] & \mathcal{D}^0(S_1 \times S_2 \times S_3).
\end{tikzcd}
$$

$$
\begin{array}{l}
\text{The associativity isomorphism of functors } \mathcal{D}^0(S_1) \times \mathcal{D}^0(S_2) \times \mathcal{D}^0(S_3) \to \mathcal{D}^0(S_1 \times S_2 \times S_3) \\
\phantom{\text{The associativity isomorphism of functors }} (A_1 \boxtimes A_2) \boxtimes A_3 \xrightarrow{\sim} A_1 \boxtimes (A_2 \boxtimes A_3)
\end{array}
$$

$$
\begin{array}{l}
\text{and the associativity isomorphisms of functors } \mathrm{Perv}(S_1) \times \mathrm{Perv}(S_2) \times \mathrm{Perv}(S_3) \to \mathrm{Perv}(S_1 \times S_2 \times S_3) \\
\phantom{\text{and the associativity isomorphisms of functors }} (K_1 \boxtimes K_2) \boxtimes K_3 \xrightarrow{\sim} K_1 \boxtimes (K_2 \boxtimes K_3)
\end{array}
$$

satisfy the compatibility condition of [Ter23U, Defn. 6.7(1)], because the same holds for the corresponding
natural isomorphisms on $\mathbf{DA}_{ct}^{\acute{e}t}(\cdot, \mathbb{Q})$ and on $\mathcal{D}^b(\cdot, \mathbb{Q})$ - this is just a way of rephrasing the associativity
of $\mathrm{Bti}^* : \mathbf{DA}_{ct}^{\acute{e}t}(\cdot, \mathbb{Q}) \to \mathcal{D}_c^b(\cdot, \mathbb{Q})$. Therefore, applying [Ter23U, Prop. 3.6], we obtain a canonical natural
isomorphism between multi-exact functors $\mathcal{M}^0(S_1) \times \mathcal{M}^0(S_2) \times \mathcal{M}^0(S_3) \to \mathcal{M}^0(S_1 \times S_2 \times S_3)$

$$
(41) \qquad (M_1 \boxtimes M_2) \boxtimes M_3 \xrightarrow{\sim} M_1 \boxtimes (M_2 \boxtimes M_3)
$$

compatible with the corresponding associativity isomorphism on $\mathrm{Perv}(\cdot)$. By Lemma 2.5, this extends to
a multi-triangulated natural isomorphism between functors $\mathcal{D}^b(\mathcal{M}^0(S_1)) \times \mathcal{D}^b(\mathcal{M}^0(S_2)) \times \mathcal{D}^b(\mathcal{M}^0(S_3)) \to$
$\mathcal{D}^b(\mathcal{M}^0(S_1 \times S_2 \times S_3))$

$$
(42) \qquad (M_1^\bullet \boxtimes M_2^\bullet) \boxtimes M_3^\bullet \xrightarrow{\sim} M_1^\bullet \boxtimes (M_2^\bullet \boxtimes M_3^\bullet)
$$

again compatible with the corresponding associativity isomorphism on $\mathcal{D}^b(\mathrm{Perv}(\cdot))$.

As $S_1, S_2, S_3$ vary in $\mathrm{Var}_k$, the natural isomorphisms (42) satisfy condition (aETS-1) of [Ter23T, Defn. 3.2]:
to see this it suffices to check that the natural isomorphisms (41) satisfy the same condition, which in turn
follows from the validity of axiom (aETS-1) on $\mathbf{DA}_{ct}^{\acute{e}t}(\cdot, \mathbb{Q})$.

By [Ter23F, Lemma 5.5(1)], in order to conclude that the natural isomorphisms (42) define an associ-
ativity constraint for the $\mathrm{Var}_k$-fibered category $\mathcal{D}^b(\mathcal{M}(\cdot))$, it suffices to show that they define associativity
constraints for the underlying $\mathrm{Var}_k^{sm}$-fibered and $\mathrm{Var}_k^{cl,op}$-fibered categories. By construction, the latter
task reduces to checking that the natural isomorphisms (41) define associativity constraints for the abelian
$\mathrm{Var}_k^{sm}$-fibered and $\mathrm{Var}_k^{cl,op}$-fibered category $\mathcal{M}(\cdot)$, which is indeed the case by [Ter23U, Lemma 6.8(1)].

The associativity of the morphism $\iota$ amounts to the compatibility between the natural isomorphisms (42)
and the corresponding natural isomorphisms on $\mathcal{D}^b(\mathrm{Perv}(\cdot))$ that we have already mentioned above. $\square$

**Lemma 2.12.** The external tensor structure on $\mathcal{D}^b(\mathcal{M}(\cdot))$ carries a canonical external commutativity constraint in the sense of [Ter23T, Defn. 4.2], with respect to which the morphism of $\mathrm{Var}_k$-fibered categories $\iota: \mathcal{D}^b(\mathcal{M}(\cdot)) \to \mathcal{D}^b(\mathrm{Perv}(\cdot))$ is symmetric in the sense of [Ter23T, Defn. 8.17].
compatible. By [Ter23U, Lemma 6.10], this reduces to checking that the external associativity and com-mutativity constraints on $\mathbf{DA}_{ct}^{\dot{e}t}(\cdot, \mathbb{Q})$ are compatible, which is indeed the case. $\square$

### 3. TENSORING WITH DISTINGUISHED MOTIVIC LOCAL SYSTEMS

Before completing the construction of the tensor structure on perverse motives, we need to analyze the tensor product functors defined in the previous section more closely with the aim of deducing more information about the motivic internal tensor product.

Recall that, using the canonical dictionary of [Ter23T], the symmetric associative external tensor structure on $D^b(\mathcal{M}(\cdot))$ obtained in Section 2 can be translated canonically into an internal tensor structure: explicitly, for every $S \in \mathrm{Var}_k$ the internal tensor product on $D^b(\mathcal{M}(S))$ is defined via the classical formula

$$ (45) \qquad M_1^\bullet \otimes M_2^\bullet := \Delta_S^*(M_1^\bullet \boxtimes M_2^\bullet), $$

where $\Delta_S : S \hookrightarrow S \times S$ denotes the diagonal embedding. Hence we can canonically view $D^b(\mathcal{M}(\cdot))$ as a (non-unitary, for the moment) symmetric monoidal $\mathrm{Var}_k$-fibered category. The two parts of the promised monoidal structure that it remains to obtain are the motivic unit constraint and the motivic internal homomorphisms. In both cases, trying to work with the external tensor product does not seem to lead one very far: we definitely need to describe our next constructions in terms of the internal tensor product. The problem is that, at this stage, the internal tensor product functors on perverse motives remains quite obscure, since these are deduced from the external tensor product functors rather than obtained directly from the lifting principles of universal abelian factorizations.

The main goal of this section is to provide a more direct description of the internal tensor product functor (45) in the case where the base $S$ is smooth and one of the two complexes involved has a certain distinguished form: this is Proposition 3.6, the technical proof of which occupies most part of the present section. Roughly speaking, we will show that, over any $S \in \mathfrak{Sm}_k$, a suitable restriction of the internal tensor product functor on $D^b(\mathcal{M}(S))$ can also be obtained directly from the lifting principles of universal abelian factorizations. This result will play a crucial role both in Section 4, where we construct the motivic unit constraint, and in Section 5, where we construct internal homomorphisms of perverse motives.

### 3.1. Distinguished motivic local systems.
In order to formalize our idea, we need to introduce another abelian subcategory $\mathcal{M}^{0,loc}(S)$ of $\mathcal{M}^0(S)$ obtained as the universal abelian factorization of a further additive subquiver $\mathcal{D}^{0,loc}(S)$ of $\mathcal{D}^0(S)$. Recall that we let $\mathrm{Loc}(S)$ denote the usual $\mathbb{Q}$-linear abelian category of local systems over $S$.

**Notation 3.1.** For every $S \in \mathfrak{Sm}_k$, we use the following notation:

* We define an abelian subcategory $\mathrm{Loc}_p(S)$ of $\mathrm{Perv}(S)$ by setting

$$ \mathrm{Loc}_p(S) = \{ K \in \mathrm{Perv}(S) \mid K[-\dim(S)] \in \mathrm{Loc}(S) \} $$

if $S$ is connected, and extending the definition in the obvious way for general smooth $S$.

* We introduce the full additive subcategory

$$ \mathcal{D}^{0,loc}(S) := \{ A \in \mathcal{D}^0(S) \mid \mathrm{Bti}_S^*(A) \in \mathrm{Loc}_p(S) \} \subset \mathcal{D}^0(S), $$

and we let

$$ \beta_S^{0,loc} : \mathcal{D}^{0,loc}(S) \to \mathrm{Loc}_p(S) $$

denote the restriction of $\beta_S^0$ to $\mathcal{D}^{0,loc}(S)$.

* We let $\mathcal{M}^{0,loc}(S)$ denote the universal abelian factorization of the additive functor

$$ \beta_S^{0,loc} := \beta_S|_{\mathcal{D}^{0,loc}(S)} : \mathcal{D}^0(S) \to \mathrm{Perv}(S). $$
We implicitly regard it as a (not necessarily full) abelian subcategory of $\mathcal{M}^0(S)$ via the canonical
faithful exact functor

$$
\mathcal{M}^{0, \mathrm{loc}}(S) \hookrightarrow \mathcal{M}^{0}(S)
$$

obtained by applying [Ter23U, Prop. 2.5] to the diagram

$$
\begin{tikzcd}
\mathcal{D}^{0, \mathrm{loc}}(S) \arrow[r] \arrow[d] & \mathcal{D}^0(S) \arrow[d] \\
\mathrm{Loc}_p(S) \arrow[r] \arrow[d] & \mathrm{Perv}(S).
\end{tikzcd}
$$

• For simplicity, we still write

- $\pi_S$ for the quotient functor $\mathbf{A}(\mathcal{D}^{0,\mathrm{loc}}(S)) \to \mathcal{M}^{0,\mathrm{loc}}(S)$, and also for the composite functor $\mathcal{D}^{0,\mathrm{loc}}(S) \to \mathbf{A}(\mathcal{D}^{0,\mathrm{loc}}(S)) \to \mathcal{M}^{0,\mathrm{loc}}(S)$,

- $\iota_S$ for the induced faithful exact functor $\mathcal{M}^{0,\mathrm{loc}}(S) \hookrightarrow \mathrm{Loc}_p(S) \subset \mathrm{Perv}(S)$.

This convention will not create any ambiguity.

Remark 3.2. By construction, for every smooth (equidimensional) morphism $p: P \to S$ in $\mathbf{Sm}_k$, the functor $p^\dagger: \mathcal{D}^0(S) \to \mathcal{D}^0(P)$ introduced in Notation 2.10 sends $\mathcal{D}^{0,\mathrm{loc}}(S)$ to $\mathcal{D}^{0,\mathrm{loc}}(P)$, and therefore the corresponding exact functor $p^\dagger: \mathcal{M}^0(S) \to \mathcal{M}^0(P)$ sends $\mathcal{M}^{0,\mathrm{loc}}(S)$ to $\mathcal{M}^{0,\mathrm{loc}}(P)$. For every $S \in \mathrm{Var}_k$, we can restrict the picture to the cofiltered category $\mathbf{SmOp}(S)$ introduced before Proposition 1.9: as $U$ varies in $\mathbf{SmOp}(S)$, the categories $\mathcal{M}^{0,\mathrm{loc}}(U)$ canonically assemble into an abelian $\mathbf{SmOp}(S)$-fibered subcategory $\mathcal{M}^{0,\mathrm{loc}}(\cdot)$ of $\mathcal{M}^0(\cdot)$.

In view of the applications to Section 5, it is important to note the following fact:

**Lemma 3.3.** For every $S \in \mathrm{Var}_k$, the $\mathbf{SmOp}(S)$-fibered category $\mathcal{M}^{0,\mathrm{loc}}$ is weakly cofinal in $\mathcal{M}^0$ in the sense of Definition 1.8.

*Proof.* By [Ter23U, Cor. 5.15], it suffices to show that the $\mathbf{SmOp}(S)$-fibered subcategory $\mathcal{D}^{0,\mathrm{loc}}$ of $\mathcal{D}^0$ is weakly cofinal. Since $\mathcal{D}^{0,\mathrm{loc}}$ is a full $\mathbf{SmOp}(S)$-fibered subcategory of $\mathcal{D}^0$, it suffices to check the condition on objects. To this end, using the definition of $\mathcal{D}^{0,\mathrm{loc}}$, it suffices to show that the inclusion of $\mathbf{SmOp}(S)$-fibered categories $\mathrm{Loc}_p(\cdot) \subset \mathrm{Perv}(\cdot)$ is weakly cofinal, which is obvious. $\square$

Remark 3.4. The existence of a perverse motivic *t*-structure on $\mathbf{DA}^{\dot{e}_t}_{ct}(S, \mathbb{Q})$ would imply the fullness of the inclusion $\mathcal{M}^{0,\mathrm{loc}}(S) \subset \mathcal{M}(S)$. Unfortunately, we have no idea how to show this unconditionally; in any case, Lemma 3.3 suffices for all our applications here.

We now describe the auxiliary construction on which the main results of the present section are based:

**Construction 3.5.** Fix $S \in \mathbf{Sm}_k$. We define a shifted tensor product functor

$$
(46) \qquad - \otimes^\dagger - : \mathbf{DA}^{\dot{e}_t}_{ct}(S, \mathbb{Q}) \times \mathbf{DA}^{\dot{e}_t}_{ct}(S, \mathbb{Q}) \to \mathbf{DA}^{\dot{e}_t}_{ct}(S, \mathbb{Q})
$$

by the formula

$$
A_1 \otimes^\dagger A_2 := (A_1 \otimes A_2)[-\dim(S)]
$$

in the case where $S$ is connected, and extending it in the obvious way for general $S$. Similarly, we define a
shifted tensor product functor

$$
(47) \qquad - \otimes^{\dagger} - : \mathcal{D}^{b}(\mathrm{Perv}(S)) \times \mathcal{D}^{b}(\mathrm{Perv}(S)) \to \mathcal{D}^{b}(\mathrm{Perv}(S))
$$

by the formula

$$
K_1 \bullet \otimes^\dagger K_2 \bullet := (K_1 \bullet \otimes K_2 \bullet)[-\dim(S)]
$$
in the case where $S$ is connected, and extending it in the obvious way for general $S$. Note that the functor (47) sends $\mathrm{Perv}(S) \times \mathrm{Loc}_p(S)$ to $\mathrm{Perv}(S)$, and therefore the functor (46) sends $\mathcal{D}^0(S) \times \mathcal{D}^{0,\mathrm{loc}}(S)$ to $\mathcal{D}^0(S)$. Applying [Ter23U, Prop. 4.3] to the diagram

$$ \begin{array}{ccc} \mathcal{D}^0(S) \times \mathcal{D}^{0,\mathrm{loc}}(S) & \xrightarrow{-\otimes^\dagger -} & \mathcal{D}^0(S) \\ \bigdownarrow_{\beta_S^0 \times \beta_S^{0,\mathrm{loc}}} & & \bigdownarrow_{\beta_S^0} \\ \mathrm{Perv}(S) \times \mathrm{Loc}_p(S) & \xrightarrow{-\otimes^\dagger -} & \mathrm{Perv}(S) \end{array} $$

we obtain a bi-exact functor

$$ (48) \qquad - \otimes^\dagger - : \mathcal{M}^0(S) \times \mathcal{M}^{0,\mathrm{loc}}(S) \to \mathcal{M}^0(S) $$

together with a canonical natural isomorphism of functors $\mathcal{M}^0(S) \times \mathcal{M}^{0,\mathrm{loc}}(S) \to \mathrm{Perv}(S)$

$$ \hat{\rho}^{\dagger} = \hat{\rho}_{S}^{\dagger} : \iota_{S}(M) \otimes^{\dagger} \iota_{S}(L) \xrightarrow{\sim} \iota_{S}(M \otimes^{\dagger} L). $$

It is a priori unclear whether the functor (48) is related to (the restriction of) the internal tensor product of perverse motives defined via the formula (45). It is a pleasant surprise that the two functors are indeed related in the way one would expect:

**Proposition 3.6.** Let $S \in \mathrm{Sm}_k$ be connected of dimension $d$. Then there exists a canonical isomorphism of functors $\mathcal{M}^0(S) \times \mathcal{M}^{0,\mathrm{loc}}(S) \to \mathcal{M}^0(S)$

$$ (49) \qquad M \otimes^\dagger L \xrightarrow{\sim} (M \otimes L)[-d] $$

rendering the diagram of functors $\mathcal{M}^0(S) \times \mathcal{M}^{0,\mathrm{loc}}(S) \to \mathrm{Perv}(S)$

$$ \begin{array}{ccc} \iota_S(M \otimes^\dagger L) & \xrightarrow{\sim} & \iota_S((M \otimes L)[-d]) \\ \hat{\rho}^\dagger & \xrightarrow{\sim} & \hat{\rho} \\ \iota_S(M) \otimes^\dagger \iota_S(L) & \xrightarrow{\sim} & (\iota_S(M) \otimes \iota_S(L)[-d]) \end{array} $$

commutative.

Most of the rest of the section is devoted to the technical proof of this result. Before discussing the details, let us briefly explain what is the general strategy behind the proof and why implementing it correctly turns out to be quite subtle. Consider the diagram

$$ \begin{array}{ccc} \mathcal{D}^0(S) \times \mathcal{D}^{0,\mathrm{loc}}(S) & \xrightarrow{-\boxtimes -} & \mathcal{D}^0(S \times S) \xrightarrow{\Delta_S^*(-d]} & \mathcal{D}(S) \\ \bigdownarrow_{\beta_S^0 \times \beta_S^{0,\mathrm{loc}}} & & \bigdownarrow_{\beta_S^0 \times S} & \\ \mathrm{Perv}(S) \times \mathrm{Loc}_p(S) & \xrightarrow{-\boxtimes -} & \mathrm{Perv}(S \times S) & \xrightarrow{\Delta_S^*(-d]} & \mathrm{D}^b(\mathrm{Perv}(S)) \end{array} $$

where both squares are commutative up to canonical natural isomorphism. By construction, the vertical and horizontal composites recover (the restrictions of) the shifted tensor product functors (46) and (47), respectively; in particular they take values in $\mathcal{D}^0(S)$ and $\mathrm{Perv}(S)$, respectively. If the functor

$$ (51) \qquad \Delta_S^*(-d] : \mathrm{D}^b(\mathrm{Perv}(S \times S)) \to \mathrm{D}^b(\mathrm{Perv}(S)) $$
was t-exact for the obvious t-structures, then the corresponding functor $\Delta_S^*(-d) : \mathcal{D}(S \times S) \to \mathcal{D}(S)$ would send $\mathcal{D}^0(S \times S)$ to $\mathcal{D}^0(S)$. Hence one could refine the previous diagram to a diagram of the form

$$ \begin{array}{ccc} \mathcal{D}^0(S) \times \mathcal{D}^{0, loc}(S) & \xrightarrow{-\boxtimes -} & \mathcal{D}^0(S \times S) \xrightarrow{\Delta_S^*(-d)} \mathcal{D}^0(S) \\ \Big\downarrow_{\beta_S^0 \times \beta_S^{0, loc}} & & \Big\downarrow_{\beta_S^0 \times S} \\ \text{Perv}(S) \times \text{Loc}_p(S) & \xrightarrow{-\boxtimes -} & \text{Perv}(S \times S) \xrightarrow{\Delta_S^*(-d)} \text{Perv}(S) \end{array} $$

and one could try to deduce that the functor (48) is naturally isomorphic to the composite

$$ \mathcal{M}^0(S) \times \mathcal{M}^{0, loc}(S) \xrightarrow{-\boxtimes -} \mathcal{M}^0(S \times S) \xrightarrow{\Delta_S^*(-d)} \mathcal{M}^0(S) $$

by using the general functoriality of universal abelian factorizations from [Ter23U]. If this was possible, then the conclusion of Proposition 3.6 would follow trivially from the fact that lifting exact functors to universal abelian factorizations is compatible with composition.

Needless to say, the functor (51) is not t-exact, hence the argument sketched here simply does not work. On the other hand, it is well-known that the composite functor

$$ \Delta_{S,*} \circ \Delta_S^*(-d) : \mathcal{D}^b(\text{Perv}(S \times S)) \to \mathcal{D}^b(\text{Perv}(S \times S)) $$

can be modeled as an exact functor $\mathcal{C}^b(\text{Perv}(S \times S)) \to \mathcal{C}^b(\text{Perv}(S \times S))$ by means of a suitable Čech complex construction. For instance, in the simple case where $S$ is the affine line $\mathbb{A}_k^1$, one can use the total complex associated to the exact functor $\mathcal{C}^b(\text{Perv}(\mathbb{A}_k^2)) \to \mathcal{C}^{[0,1]}(\mathcal{C}^b(\text{Perv}(\mathbb{A}_k^2)))$

$$ (52) \qquad K^{\bullet} \rightsquigarrow [j_1, j^* K^{\bullet} \to K^{\bullet}], $$

where $j$ denotes the affine open immersion of the complement of the diagonal in $\mathbb{A}_k^2$. The key advantage is that the complex in (52) lifts canonically to the level of perverse motives. In general, the model for $\Delta_{S,*} \circ \Delta_S^*(-d)$ depends on the choice of a finite affine covering of the open complement of $\Delta_S(S)$ in $S \times S$, but the derived functor of the latter is essentially independent of such a choice. In fact, this idea is exploited in the proof of [IM19, Prop. 4.2] to construct the inverse image functor $\Delta_S^*$ - or, more generally, inverse image functors along closed immersions - in the setting of perverse motives. We are going to prove Proposition 3.6 by a similar method.

**3.2. Some Čech complex computations.** For the reader's convenience, we start by recalling the Čech complex construction just mentioned in the setting of perverse sheaves and perverse motives.

**Construction 3.7.** Let $s: S \hookrightarrow Y$ be a closed immersion in $\text{Var}_k$, and let $u: U \hookrightarrow Y$ denote the complementary open immersion. Suppose that we are given a finite affine open covering $\mathcal{U} = \{U_i \subset U\}_{i=1}^d$ of $U$. For every subset $I \subset \{1, \dots, d\}$, set

$$ U_I := \bigcap_{i \in I} U_i $$

and let $j_I: U_I \hookrightarrow Y$ denote the corresponding open immersion (with the agreement that $U_\emptyset = Y$ and $j_\emptyset = \text{id}_Y$). Note that each $j_I$ is affine, because $Y$ is separated over $k$ by definition. For every choice of subsets $I, I' \subset \{1, \dots, d\}$ satisfying $I' \subset I$, there exists a canonical natural transformation of functors $\text{Perv}(Y) \to \text{Perv}(Y)$

$$ \alpha_{I,I'}^\mathcal{U} : j_{I,\cdot}j_I^*K \to j_{I',\cdot}j_{I'}^*K $$

induced by the co-unit along the inclusion of $U_I$ into $U_{I'}$. For each $r = 0, \dots, d$, set

$$ U_{\mathcal{U}}^{(r)} := \coprod_{\substack{I \subset \{1, \dots, d\} \\ \#I = r}} U_I $$
**Construction 3.8.** Keep the notation and assumptions of Construction 3.7. For every choice of subsets $I, I' \subset \{1, \dots, d\}$ satisfying $I' \subset I$, consider the canonical natural transformation of functors $\mathbf{DA}_{ct}^{\text{ét}}(Y, \mathbb{Q}) \to \mathbf{DA}_{ct}^{\text{ét}}(Y, \mathbb{Q})$

$$ \alpha_{I,I'}^{\mathcal{Y}} : j_{I,\downarrow j_I^*} A \to j_{I',\downarrow j_I^*} A $$

induced by the co-unit along the inclusion of $U_I$ into $U_{I'}$. By construction, the diagram of functors $\mathbf{DA}_{ct}^{\text{ét}}(Y, \mathbb{Q}) \to D_c^b(Y, \mathbb{Q})$

$$ \begin{array}{ccc} j_{I,\downarrow j_I^*} \mathbf{Bti}_Y^*(A) & \xrightarrow{\alpha_{I,I'}^{\mathcal{Y}}} & j_{I',\downarrow j_I^*} \mathbf{Bti}_Y^*(A) \\ \downarrow & & \downarrow \\ j_{I,\downarrow} \mathbf{Bti}_{U_I}^*(j_I^* A) & j_{I',\downarrow} \mathbf{Bti}_{U_{I'}}^*(j_{I'}^* A) & \downarrow \\ \downarrow & & \downarrow \\ \mathbf{Bti}_Y^*(j_{I,\downarrow} j_I^* A) & \xrightarrow{\alpha_{I,I'}^{\mathcal{Y}}} & \mathbf{Bti}_Y^*(j_{I',\downarrow} j_{I'}^* A) \end{array} $$

is commutative under Beilinson's equivalence $D^b(\text{Perv}(Y)) \xrightarrow{\sim} D_c^b(Y, \mathbb{Q})$. Let $[-1,0]$ denote the partially ordered set $\{-1 < 0\}$; for every integer $n \ge 0$ we regard the product category $[-1,0]^n$ as a small diagram (with the convention that $[-1,0]^0$ is just the singleton category). We are particularly interested in the diagram $[-1,0]^d$: in this case, for every subset $I \subset \{0, \dots, d\}$, we let $\underline{I}$ denote the object $(\epsilon_1, \dots, \epsilon_d) \in [-1,0]^d$ whose $i$-th coordinate $\epsilon_i$ equals $-1$ if $i \in I$ and $0$ otherwise. Note that the assignment $I \rightsquigarrow \underline{I}$ defines an anti-isomorphism between the powerset of $\{1, \dots, d\}$ (regarded as a poset with the usual inclusion ordering) and the category $[-1,0]^d$. Given two subsets $I, I' \subset \{1, \dots, d\}$ satisfying $I' \subset I$, we let $e_{I,I'} : \underline{I} \to \underline{I}'$ denote the corresponding morphism in $[-1,0]^d$. Following the idea of [IM19, Lemma 3.4, Lemma 3.6], we can construct a triangulated functor

$$ \check{\mathcal{C}}_{Y, \mathcal{Y}}^{\square} : \mathbf{DA}_{ct}^{\text{ét}}(Y, \mathbb{Q}) \to \mathbf{DA}_{ct}^{\text{ét}}((Y, [-1, 0]^d), \mathbb{Q}) $$

providing a coherent lifting of the $[-1,0]^d$-indexed diagram of functors $\mathbf{DA}_{ct}^{\text{ét}}(Y, \mathbb{Q}) \to \mathbf{DA}_{ct}^{\text{ét}}(Y, \mathbb{Q})$

$$ \underline{I}(A) := j_{I,\downarrow} j_I^* A, \quad e_{I,I'}(A) := \alpha_{I,I'}^{\mathcal{Y}}(A) : j_{I,\downarrow} j_I^* A \to j_{I',\downarrow} j_{I'}^* A. $$

Concretely, one can consider the two obvious morphisms of diagram of $k$-varieties

$$ \begin{array}{ccc} (\underline{I} \mapsto U_I)_{\underline{I} \in [-1,0]^d} & \xrightarrow{\xi} & (\underline{I} \mapsto Y)_{\underline{I} \in [-1,0]^d} \\ \downarrow & & \downarrow \\ Y & & \end{array} $$

and define the functor $\check{\mathcal{C}}_{Y, \mathcal{Y}}^{\square}$ by the formula

$$ \check{\mathcal{C}}_{Y, \mathcal{Y}}^{\square} := \xi_! \circ \nu^*. $$

Now, for every integer $n \ge 1$, let

$$ \mathrm{Cof}_n : \mathbf{DA}_{ct}^{\text{ét}}((Y, [-1, 0]^n), \mathbb{Q}) \to \mathbf{DA}_{ct}^{\text{ét}}((Y, [-1, 0]^{n-1}), \mathbb{Q}) $$

denote the co-fiber with respect to the last coordinate of $[-1,0]^n = [-1,0]^{n-1} \times [-1,0]$. We let

$$ \mathrm{Tot}^{\square} := \mathrm{Cof}_1 \circ \dots \circ \mathrm{Cof}_d : \mathbf{DA}_{ct}^{\text{ét}}((Y, [-1, 0]^d), \mathbb{Q}) \to \mathbf{DA}_{ct}^{\text{ét}}(Y, \mathbb{Q}) $$

denote the associated derived total complex functor, and we set

$$ \check{\mathcal{C}}_{Y, \mathcal{Y}}^{\square} := \mathrm{Tot}^{\square} \circ \check{\mathcal{C}}_{Y, \mathcal{Y}}^{\square} : \mathbf{DA}_{ct}^{\text{ét}}(Y, \mathbb{Q}) \to \mathbf{DA}_{ct}^{\text{ét}}(Y, \mathbb{Q}). $$
Similarly, we can define a functor

$$
\sigma_{\le (-1, \dots, -1)} \check{C}_{Y, \mathcal{U}}^{\square} : \mathbf{D}\mathbf{A}_{ct}^{\acute{e}t}(Y, \mathbb{Q}) \to \mathbf{D}\mathbf{A}_{ct}^{\acute{e}t}((Y, [-1, 0]^d), \mathbb{Q})
$$

providing a coherent lifting of the $[-1,0]^d$-indexed diagram of functors $\mathbf{D}\mathbf{A}_{ct}^{\acute{e}t}(Y, \mathbb{Q}) \to \mathbf{D}\mathbf{A}_{ct}^{\acute{e}t}(Y, \mathbb{Q})$

$$
\underline{I}(A) := \begin{cases} J_d! J_d^* A, & I = \{1, \dots, d\} \\ 0, & I \neq \{1, \dots, d\}. \end{cases}
$$

Note that there is an obvious natural transformation of functors $\mathbf{D}\mathbf{A}_{ct}^{\acute{e}t}(Y, \mathbb{Q}) \to \mathbf{D}\mathbf{A}_{ct}^{\acute{e}t}((Y, [-1, 0]^d), \mathbb{Q})$

$$
(54) \qquad \check{C}_{Y, \mathcal{U}}^{\square}(A) \to \sigma_{\le (-1, \dots, -1)} \check{C}_{Y, \mathcal{U}}^{\square}(A).
$$

One can repeat the entire construction replacing $\mathbf{D}\mathbf{A}_{ct}^{\acute{e}t}(\cdot, \mathbb{Q})$ with any stable homotopy algebraic derivator in the sense of [Ayo07a, § 2.4.2]. As a consequence of [Ayo10, Thm. 1.8, Lemme 1.10], the stable homotopy 2-functor $D_c^b(\cdot, \mathbb{Q})$ canonically underlies a stable homotopy algebraic derivator, and the Betti realization

$$
\mathrm{Bti}^* : \mathbf{D}\mathbf{A}_{ct}^{\acute{e}t}(\cdot, \mathbb{Q}) \to D_c^b(\cdot, \mathbb{Q})
$$

can be canonically extended over diagrams of $k$-varieties. Thus we can define a functor

$$
\check{C}_{Y, \mathcal{U}}^{\square} : D_c^b(Y, \mathbb{Q}) \to D_c^b((Y, [-1, 0]^d), \mathbb{Q})
$$

providing a coherent lifting of the $[-1,0]^d$-indexed diagram of functors $D_c^b(Y, \mathbb{Q}) \to D_c^b(Y, \mathbb{Q})$

$$
\underline{I}(K^\bullet) := j_{I, !} j_I^* K^\bullet, \quad e_{I, I'}(A) := \alpha_{I, I'}^\mathcal{U}(K^\bullet) : j_{I, !} j_I^* K^\bullet \to j_{I', !} j_{I'}^* K^\bullet,
$$

as well as a functor

$$
\sigma_{\le (-1, \dots, -1)} \check{C}_{Y, \mathcal{U}}^{\square} : D_c^b(Y, \mathbb{Q}) \to D_c^b((Y, [-1, 0]^d), \mathbb{Q})
$$

providing a coherent lifting of the $[-1,0]^d$-indexed diagram of functors $D_c^b(Y, \mathbb{Q}) \to D_c^b(Y, \mathbb{Q})$

$$
\underline{I}(K^{\bullet}) := \begin{cases} J_d! J_d^* K^{\bullet}, & I = \{1, \dots, d\} \\ 0, & I \neq \{1, \dots, d\}. \end{cases}
$$

As in the previous case, they are related by a canonical natural transformation of functors $D_c^b(Y, \mathbb{Q}) \to D_c^b((Y, [-1, 0]^d), \mathbb{Q})$

$$
\check{C}_{Y, \mathcal{U}}^{\square}(K^{\bullet}) \to \sigma_{\le (-1, \dots, -1)} \check{C}_{Y, \mathcal{U}}^{\square}(K^{\bullet}).
$$

Note that, moreover, there exist canonical natural isomorphisms of functors $\mathbf{D}\mathbf{A}_{ct}^{\acute{e}t}(Y, \mathbb{Q}) \to D_c^b((Y, [-1, 0]^d), \mathbb{Q})$

$$
\check{C}_{Y, \mathcal{U}}^{\square} \mathbf{Bti}_{Y}^{*}(A) \xrightarrow{\sim} \mathbf{Bti}_{(Y, [-1, 0]^d)}^{*}(\check{C}_{Y, \mathcal{U}}^{\square}(A))
$$

and

$$
\sigma_{\le (-1, \dots, -1)} \check{C}_{Y, \mathcal{U}}^{\square} \mathbf{Bti}_{Y}^{*}(A) \xrightarrow{\sim} \mathbf{Bti}_{(Y, [-1, 0]^d)}^{*}(\sigma_{\le (-1, \dots, -1)} \check{C}_{Y, \mathcal{U}}^{\square}(A))
$$

rendering the diagram of functors $\mathbf{D}\mathbf{A}_{ct}^{\acute{e}t}(Y, \mathbb{Q}) \to D_c^b((Y, [-1, 0]^d), \mathbb{Q})$

$$
\begin{tikzcd}
& \check{C}_{Y, \mathcal{U}}^{\square} (\mathrm{Bti}_Y^*(A)) \arrow[r, "\sim"] & \mathrm{Bti}_{(Y, [-1, 0]^d)}^*(\check{C}_{Y, \mathcal{U}}^{\square}(A)) \\
\sigma_{\le (-1, \dots, -1)} \check{C}_{Y, \mathcal{U}}^{\square} (\mathrm{Bti}_Y^*(A)) \arrow[r, "\sim"] & \mathrm{Bti}_{(Y, [-1, 0]^d)}^*(\sigma_{\le (-1, \dots, -1)} \check{C}_{Y, \mathcal{U}}^{\square}(A)) &
\end{tikzcd}
$$

commutative. We can also define a total complex functor

$$
\mathrm{Tot}^{\square} := \mathrm{Cof}_1 \circ \dots \circ \mathrm{Cof}_d : D_c^b((Y, [-1, 0]^d), \mathbb{Q}) \to D_c^b(Y, \mathbb{Q}).
$$
By construction, the composite

$$ \check{\mathcal{C}}_{Y, \mathcal{U}} := \mathrm{Tot}^{\square} \circ \check{\mathcal{C}}_{Y, \mathcal{U}}^{\square} : D_c^b(Y, \mathbb{Q}) \to D_c^b(Y, \mathbb{Q}) $$

recover the functor (53) under Beilinson's equivalence $D^b(\mathrm{Perv}(Y)) \xrightarrow{\sim} D_c^b(Y, \mathbb{Q})$. Therefore we obtain a canonical natural isomorphism of functors $\mathbf{DA}_{ct}^{\dot{\mathrm{e}}t}(Y, \mathbb{Q}) \to D_c^b(Y, \mathbb{Q})$

$$ \check{\mathcal{C}}_{Y, \mathcal{U}} \mathrm{Bti}_Y(A) \xrightarrow{\sim} \mathrm{Bti}_Y^*(\check{\mathcal{C}}_{Y, \mathcal{U}}(A)) $$

by taking the composite

$$ \begin{tikzcd} \check{\mathcal{C}}_{Y, \mathcal{U}} \mathrm{Bti}_Y(A) \arrow[r] \arrow[d] & \mathrm{Bti}_Y^*(\check{\mathcal{C}}_{Y, \mathcal{U}}(A)) \arrow[d] \\ \mathrm{Tot}^{\square} \check{\mathcal{C}}_{Y, \mathcal{U}}^\square (\mathrm{Bti}_Y^*(A)) \arrow[r, "\sim"] & \mathrm{Tot}^{\square} \mathrm{Bti}_{Y, [-1,0]^d}^*(\check{\mathcal{C}}_{Y, \mathcal{U}}^\square(A)) \arrow[r, "\cong"] & \mathrm{Bti}_Y^*(\mathrm{Tot}^{\square} \check{\mathcal{C}}_{Y, \mathcal{U}}^\square(A)) \end{tikzcd} $$

By construction, the diagram of functors $\mathbf{DA}_{ct}^{\dot{\mathrm{e}}t}(Y, \mathbb{Q}) \to D_c^b(Y, \mathbb{Q})$

$$ \begin{tikzcd} \check{\mathcal{C}}_{Y, \mathcal{U}} \mathrm{Bti}_Y(A) \arrow[r, "\sim"] \arrow[d] & \mathrm{Bti}_Y^*(\check{\mathcal{C}}_{Y, \mathcal{U}}(A)) \arrow[d] \\ s_* s^* \mathrm{Bti}_Y^*(A) \arrow[r, "\theta"] \arrow[d] & s_* \mathrm{Bti}_S^*(s^* A) \arrow[r, "\cong"'] & \mathrm{Bti}_Y^*(s_* s^* A) \end{tikzcd} \qquad (55) $$

is commutative.

We need to compare the total Čech complex functor on the level of perverse motives with the analogous functor on triangulated motives. This turns out to be quite subtle, since the two functors are constructed in different ways. In order to proceed, it is convenient to introduce yet a further subdiagram of $\mathcal{D}^0(S)$:

**Notation 3.9.** Given a closed immersion $s : S \hookrightarrow Y$ in $\mathbf{Sm}_k$, we introduce the full additive subcategory

$$ \mathcal{D}^{0,s}(Y) := \{B \in \mathcal{D}^0(Y) \mid s^\dagger B \in \mathcal{D}^0(S)\} \subset \mathcal{D}^0(Y). $$

We can state the promised result about Čech complex functors as follows:

**Lemma 3.10.** Let $s : S \hookrightarrow Y$ be a closed immersion of constant codimension $d$ in $\mathbf{Sm}_k$. Suppose that the open complement $U$ of $S$ in $Y$ admits a finite affine open covering $\mathcal{U}$ with exactly $d$ elements. Then there exist a canonical natural isomorphism of functors $\mathcal{D}^{0,s}(Y) \to \mathcal{M}^0(Y)$

$$ (56) \qquad \pi_Y(\check{\mathcal{C}}_{Y, \mathcal{U}}[-d](B)) \xrightarrow{\sim} \check{\mathcal{C}}_{Y, \mathcal{U}}[-d](\pi_Y(B)) $$

such that the diagram of functors $\mathcal{D}^{0,s}(Y) \to \mathrm{Perv}(Y)$

$$ \begin{tikzcd} \mathrm{Bti}_Y^*(\check{\mathcal{C}}_{Y, \mathcal{U}}[-d]B) \arrow[r, "\sim"] \arrow[d] & \check{\mathcal{C}}_{Y, \mathcal{U}}[-d]\mathrm{Bti}_Y^*(B) \arrow[d] \\ \iota_Y \pi_Y(\check{\mathcal{C}}_{Y, \mathcal{U}}[-d]B) \arrow[r, "\sim"'] \arrow[d] & \iota_Y(\check{\mathcal{C}}_{Y, \mathcal{U}}[-d]\pi_Y(B)) \arrow[d] \end{tikzcd} \qquad (57) $$

is commutative.

*Proof.* First of all, let us check that the composite functor

$$ \check{\mathcal{C}}_{Y, \mathcal{U}}[-d] \circ \pi_Y : \mathcal{D}^{0,s}(Y) \to \mathcal{M}^0(Y) \to D^b(\mathcal{M}^0(Y)) $$
really takes values in $\mathcal{M}^0(Y)$ - or, equivalently, that its composition with the conservative $t$-exact functor $\iota_Y : D^b(\mathcal{M}^0(Y)) \to D^b(\mathrm{Perv}(Y))$ takes values in $\mathrm{Perv}(Y)$. To this end, for every object $B \in \mathcal{D}^{0,s}(Y)$ we compute

$$
\begin{align*}
\iota_Y(\check{C}_{Y,\mathcal{U}}[-d](\pi_Y(B))) &= \check{C}_{Y,\mathcal{U}}[-d](\iota_Y \pi_Y(B)) \\
&= \check{C}_{Y,\mathcal{U}}[-d](\beta_Y(B)) \\
&= s_* s^*(-[d]\beta_Y(B)) \\
&= s_* s^*(-[d]\mathrm{Bti}_Y^*(B)) && \text{as } B \in \mathcal{D}^0(Y) \\
&= s_* \mathrm{Bti}_S^*(s^*(-[d]B)) \\
&= s_* \beta_S(s^*(-[d]B)) && \text{as } B \in \mathcal{D}^{0,s}(Y).
\end{align*}
$$

The latter object belongs indeed to Perv(Y) in view of the definition of β_S and by the t-exactness of s*: D_c^b(S, Q) → D_c^b(Y, Q).

In order to define the sought-after natural isomorphism (56), we start by constructing a natural trans-
formation between exact functors $\mathcal{D}^0(Y) \to C^{[-d,0]}(\mathcal{M}^0(Y))$

$$
\pi_Y(\check{C}_{Y,\mathcal{U}}[-d](B))[d] \to \check{C}_{Y,\mathcal{U}}^{\bullet}(\pi_Y(B)).
$$

With the notation of Construction 3.7 and Construction 3.8, this amounts to giving a natural transformation
of functors $\mathcal{D}^0(Y) \to \mathcal{M}^0(Y)$

$$
(58) \qquad \pi_Y(\check{C}_{Y,\mathcal{U}}[-d](B)) \to J_{d,!} J_d^* \pi_Y(B)
$$

such that the composite

$$
(59) \quad \pi_Y(\check{C}_{Y,\mathcal{U}}[-d](B)) \to J_{d,!} J_d^* \pi_Y(B) \to J_{d-1,!} J_{d-1}^* \pi_Y(B) = \bigoplus_{\substack{I \subset \{1, \dots, d\} \\ \#I = d-1}} j_{I,!} j_I^* \pi_Y(B)
$$

vanishes. We define (58) by taking the composite

$$
\[
\begin{tikzcd}
& \pi_Y(\check{C}_{Y,\mathcal{U}}[-d](B)) \arrow[r] \arrow[d] & J_{d,!} J_d^* \pi_Y(B), \arrow[d] \\
\pi_Y(\mathrm{Tot}^{\square}[-d]\check{C}_{Y,\mathcal{U}}^{\square}(B)) \arrow[r] & \pi_Y(\mathrm{Tot}^{\square}[-d]\sigma_{\le (-1,\dots,-1)}\check{C}_{Y,\mathcal{U}}^{\square}(B)) \arrow[r] & \pi_Y(J_{d,!}J_d^*B)
\end{tikzcd}
\]
$$

where the first horizontal arrow is induced by (54). The vanishing of (59) follows from the existence of a
commutative diagram of the form

<figure><img src="image_4.png" /></figure>
where the upper horizontal composite vanishes by construction. We deduce a natural transformation of
functors $\mathcal{D}^0(Y) \to D^{[0,d]}(\mathcal{M}^0(Y))$

$$
(60) \qquad \begin{tikzcd}
\pi_Y(\check{C}_{Y,\mathcal{U}}[-d]B) & \check{C}_{Y,\mathcal{U}}[-d](\pi_Y(B)) \\
\llap{$\text{Tot}^{\bullet} \pi_Y(\check{C}_{Y,\mathcal{U}}[-d](B))$} \arrow[r] & \llap{$\text{Tot}^{\bullet} \check{C}_{Y,\mathcal{U}}^{\bullet}[-d](\pi_Y(B))$}
\end{tikzcd}
$$

In an analogous way, we obtain a natural transformation of functors $\mathcal{D}^0(Y) \to D^{[-d,0]}(\mathrm{Perv}(Y))$

$$
(61) \qquad \beta_Y(\check{C}_{Y,\mathcal{U}}[-d](B)) \to \check{C}_{Y,\mathcal{U}}[-d](\beta_Y(B))
$$

making the diagram of functors $\mathcal{D}^0(Y) \to D^{[0,d]}(\mathrm{Perv}(Y))$

$$
(62) \qquad \begin{tikzcd}[column sep=2.8em, row sep=2.8em]
\beta_Y(\check{C}_{Y,\mathcal{U}}[-d](B)) \arrow[r] \arrow[d] & \check{C}_{Y,\mathcal{U}}[-d](\beta_Y(B)) \\
\check{C}_{Y,\mathcal{U}}[-d](\iota_Y \pi_Y(B)) \arrow[u] \arrow[r] \arrow[d] & \\
\iota_Y \pi_Y(\check{C}_{Y,\mathcal{U}}[-d](B)) \arrow[u] \arrow[r] \arrow[d] & \iota_Y(\check{C}_{Y,\mathcal{U}}[-d](\pi_Y(B)))
\end{tikzcd}
$$

commutative. We claim that the value of (60) on a given object $B \in \mathcal{D}^0(Y)$ is an isomorphism if and only if $B \in \mathcal{D}^{0,s}(Y)$. Using the commutativity of the diagram (62) and the conservativity of $\iota_Y : D^{[0,d]}(\mathcal{M}^0(Y)) \to D^{[0,d]}(\mathrm{Perv}(Y))$, we see that this is equivalent to showing that the value of (61) on a given object $B \in \mathcal{D}^0(Y)$ is an isomorphism if and only if $B \in \mathcal{D}^{0,s}(Y)$. To this end, taking the image under ${}^pH^0$ of the commutative diagram (55), the arrow (61) can be identified with the canonical arrow in $\mathrm{Perv}(Y)$

$$
(63) \qquad {}^p H^0 \mathrm{Bti}_Y^*(s_* s^\dagger B) = {}^p H^0 s_* \mathrm{Bti}_S^*(s^\dagger B) = s_* {}^p H^0 \mathrm{Bti}_S^*(s^\dagger B) \to s_* \mathrm{Bti}_S^*(s^\dagger B).
$$

Here, the latter arrow in the composite can be also written as

$$
s_* {}^p H^0 s^\dagger \mathrm{Bti}_Y^*(B) \to s_* s^\dagger \mathrm{Bti}_Y^*(B)
$$

and is induced by the left-t-exactness of the functor $s^\dagger = s^*(-d] : D_c^b(Y, \mathbb{Q}) \to D_c^b(S, \mathbb{Q})$ (which follows precisely from the fact that $U$ admits an affine open covering with $d$ elements). Using the conservativity of $s_* : D_c^b(S, \mathbb{Q}) \to D_c^b(Y, \mathbb{Q})$, we see that the arrow (63) is invertible if and only if $\mathrm{Bti}_S^*(s^\dagger B) \in \mathrm{Perv}(S)$.

We define the natural isomorphism (56) by taking the restriction of (60) to $\mathcal{D}^{0,s}(Y)$. By construction,
the diagram (57) coincides with the outer part of the diagram

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

where the two triangles are commutative by definition and the bottom piece is already known to be com-
mutative. Hence, in order to conclude the proof it suffices to show that the top piece is commutative as well.
Unwinding the various definitions, the commutativity of the upper piece follows from the commutativity of
the outer part of the diagram
<figure><img src="image_6.png" /></figure>

which in turn follows from the commutativity of the four inner pieces, which holds by construction. This
concludes the proof. □

**Corollary 3.11.** Let $s: S \hookrightarrow Y$ be a closed immersion of constant codimension $d$ in $\mathfrak{S}m_k$. Then there exists a natural isomorphism of functors $\mathcal{D}^{0,s}(Y) \to \mathcal{M}^0(S)$

$$
(64) \qquad \zeta_s : \pi_S(s^\dagger B) \xrightarrow{\sim} s^\dagger \pi_Y(B)
$$

such that the diagram of functors $\mathcal{D}^{0,s}(Y) \to \operatorname{Perv}(S)$

$$
(65) \qquad
\begin{tikzcd}
Bti_S^*(s^\dagger B) \arrow[r, "\theta"] & s^\dagger Bti_Y^*(B) \arrow[d, "\simeq"] \\
\vline & \vline \arrow[d, "\simeq"] \\
\iota_S \pi_S(s^\dagger B) & s^\dagger \iota_Y \pi_Y(B) \arrow[r, "\zeta_s"] \arrow[d, "\tilde{\theta}"] & J_{d, !} J_d^* Bti_Y^*(B)
\end{tikzcd}
$$

is commutative.

*Proof.* Note that, for every closed immersion $s: S \hookrightarrow Y$ as in the statement, the faithfulness and conserv-
vativity of the functor $\iota_S: \mathcal{M}^0(S) \hookrightarrow \operatorname{Perv}(S)$ implies that a natural transformation as in (64) rendering the
diagram (65) commutative is necessarily uniquely determined and invertible, provided it exists.

We start with a preliminary observation. Let $s: S \hookrightarrow Y$ be as in the statement; let $v: V \hookrightarrow Y$ be an
open immersion, and form the Cartesian square in $\mathfrak{S}m_k$

$$
\begin{tikzcd}
U \arrow[r, "s_U"] \arrow[d, "u"] & V \arrow[d, "v"] \\
S \arrow[r, "s"] & Y.
\end{tikzcd}
$$

Suppose for a moment that the thesis is known to hold both for $s$ and for $s_U$. We claim that, in this
situation, the diagram of functors $\mathcal{D}^{0,s}(Y) \to \mathcal{M}^0(U)$

$$
\begin{tikzcd}
u^* \pi_S(s^\dagger B) \arrow[r, "\zeta_s"] & u^* s^\dagger \pi_Y(B) \arrow[d, "\cong"] \\
\vline & \vline \arrow[d, "\cong"] \\
\pi_U(u^* s^\dagger B) & s^\dagger v^* \pi_Y(B) \arrow[r, "\zeta_{sU}"] \arrow[d, "\cong"] & s^\dagger U \pi_V(v^* B)
\end{tikzcd}
$$
is commutative. In order to show this, it suffices to check the commutativity of its image under the faithful functor $\iota_U : \mathcal{M}^0(U) \hookrightarrow \mathrm{Perv}(U)$

$$ \begin{array}{ccc} \iota_U(u^*\pi_S(s^\dagger B)) & \xrightarrow{\zeta_s} & \iota_U(u^*s^\dagger\pi_Y(B)) \\ \llap{$\vrule width 0.5pt height 0.5pt width 0.5pt$} \\ \iota_U\pi_U(u^*s^\dagger B) & & \iota_U(s_U^\dagger v^*\pi_Y(B)) \\ \llap{$\vrule width 0.5pt height 0.5pt width 0.5pt$} \\ \iota_U\pi_U(s_U^\dagger v^*B) & \xrightarrow{\zeta_{s_U}} & \iota_U(s_U^\dagger\pi_V(v^*B)) \end{array} $$

that we can decompose as

$$ \begin{array}{ccc} \iota_U(u^*\pi_S(s^\dagger B)) & \xrightarrow{\zeta_s} & \iota_U(u^*s^\dagger\pi_Y(B)) \\ \llap{$\vrule width 0.5pt height 0.5pt width 0.5pt$} \\ \xleftarrow{\tilde{\theta}} & & \xrightarrow{\tilde{\theta}} \\ u^*\iota_S\pi_S(s^\dagger B) & \xrightarrow{\zeta_s} & u^*\iota_S(s^\dagger\pi_Y(B)) \\ \llap{$\vrule width 0.5pt height 0.5pt width 0.5pt$} \\ u^*\mathrm{Bti}_S^*(s^\dagger B) & \leftarrow_{\theta} & u^*s^\dagger\mathrm{Bti}_Y^*(B) = u^*s^\dagger\iota_Y\pi_Y(B) \\ \llap{$\vrule width 0.5pt height 0.5pt width 0.5pt$} \\ \theta & & \llap{$\vrule width 0.5pt height 0.5pt width 0.5pt$} \\ \iota_U\pi_U(u^*s^\dagger B) = \mathrm{Bti}_U^*(u^*s^\dagger B) & \xrightarrow{\zeta_s} & s_U^\dagger v^*\mathrm{Bti}_Y^*(B) = s_U^\dagger v^*\iota_V\pi_V(B) \xrightarrow{\zeta_s} \iota_U(s_U^\dagger v^*\pi_Y(B)) \\ \llap{$\vrule width 0.5pt height 0.5pt width 0.5pt$} \\ \mathrm{Bti}_U^*(s_U^\dagger v^*B) & \leftarrow_{\theta} & s_U^\dagger\mathrm{Bti}_V^*(v^*B) = s_U^\dagger\iota_V(v^*\pi_Y(B)) \\ \llap{$\vrule width 0.5pt height 0.5pt width 0.5pt$} \\ & & \xrightarrow{\tilde{\theta}} \\ s_U^\dagger\iota_V\pi_V(v^*B) & \xrightarrow{\zeta_{s_U}} & \xrightarrow{\tilde{\theta}} \\ \llap{$\vrule width 0.5pt height 0.5pt width 0.5pt$} \\ \iota_U\pi_U(s_U^\dagger v^*B) & \xrightarrow{\zeta_{s_U}} & \iota_U(s_U^\dagger\pi_V(v^*B)) \end{array} $$

Here, the two five-term pieces involving $\zeta_s$ and $\zeta_{s_U}$ are commutative by assumption, the six-term piece in the upper-right part as well as the two five-term pieces in the upper-left part and in the central part are commutative by [Ter23U, Prop. 2.5(2)], the central six-term piece is commutative by construction, and the remaining three pieces are commutative by naturality.

As a consequence of this observation we deduce that, in order to show the existence of the natural isomorphism $\zeta_s$ along a given closed immersion $s : S \hookrightarrow Y$, it suffices to show that, for some finite open covering $\{v_i : V_i \hookrightarrow Y\}_{i \in I}$, the isomorphisms $\zeta_{s_i}$ along all restrictions $s_i : s^{-1}(V_i) \hookrightarrow V_i$ exist: this follows from the fact that both $\mathcal{M}^0(\cdot)$ and $\mathrm{Perv}(\cdot)$ are stacks for the Zariski topology. Therefore, since both $Y$ and $S$ are smooth by assumption (hence, in particular, the closed immersion $s$ is regular of codimension $d$), we may reduce to the case where $Y$ is affine and connected and the open complement $U$ of $S$ in $Y$ admits a finite affine open covering $\mathcal{U}$ with exactly $d$ elements: for example, for each point of $S$ one can take the distinguished open subsets associated to the elements of some regular sequence of length $d$ defining $S$ inside $Y$ locally around the chosen point.
Remark 3.12. Of course, the result of Proposition 3.6 admits a symmetric version where the roles of the two variables are exchanged: there exists a canonical isomorphism of functors $\mathcal{M}^{0,loc}(S) \times \mathcal{M}^0(S) \to \mathcal{M}^0(S)$

$$L \otimes^\dagger M \xrightarrow{\sim} (L \otimes M)[-d]$$

such that the diagram of functors $\mathcal{M}^{0,loc}(S) \times \mathcal{M}^0(S) \to \mathrm{Perv}(S)$

$$\iota_S(L \otimes^\dagger M) \xrightarrow{\sim} \iota_S((L \otimes M)[-d]) \xrightarrow{\hspace{2.5cm}} \iota_S(L \otimes M)[-d]$$

$$\bar{\rho}^\dagger \uparrow$$

$$\rho \uparrow$$

$$\iota_S(L) \otimes^\dagger \iota_S(M) \xrightarrow{\hspace{2.5cm}} (\iota_S(L) \otimes \iota_S(M))[-d]$$

is commutative.

Recall that both stable homotopy 2-functors $\mathbf{DA}^{\mathcal{E}\mathcal{t}}_{ct}(\cdot, \mathbb{Q})$ and $D_c^b(\cdot, \mathbb{Q})$ are endowed with Verdier duality functors defined, over any given $S \in \mathrm{Var}_k$, by the formula

$$\mathbb{D}_S(A) := \underline{\mathrm{Hom}}_S(A, a_S^! \mathbf{1}_{\mathrm{Spec}(k)}),$$

where $a_S : S \to \mathrm{Spec}(k)$ denotes the structural projection. Since, as $S$ varies in $\mathrm{Var}_k$, the Betti realization functors $\mathrm{Bti}_S^* : \mathbf{DA}^{\mathcal{E}\mathcal{t}}_{ct}(S, \mathbb{Q}) \to D^b(S, \mathbb{Q})$ commute with the six operations on the two sides, they also commute with Verdier duality in a canonical way. The fact that $\mathbb{D}_S$ defines a duality means that, in both cases, the canonical arrow

$$A \to \mathbb{D}_S(\mathbb{D}_S(A))$$

is invertible. Using this, one can deduce a canonical identification

$$(69) \qquad \underline{\mathrm{Hom}}_S(A, B) \xrightarrow{\sim} \mathbb{D}_S(A \otimes \mathbb{D}_S(B))$$

in both cases; again, these are compatible with one another under the Betti realization. Since, for every $S \in \mathrm{Var}_k$, the equivalence

$$\mathbb{D}_S : D_c^b(S, \mathbb{Q}) \xrightarrow{\sim} D_c^b(S, \mathbb{Q})^{op}$$

is $t$-exact for the perverse $t$-structures, it restricts to an equivalence

$$\mathbb{D}_S : \mathrm{Perv}(S) \xrightarrow{\sim} \mathrm{Perv}(S)^{op}$$

and so, by [IM19, § 2.8], it canonically lifts to an equivalence

$$\mathbb{D}_S : \mathcal{M}(S) \xrightarrow{\sim} \mathcal{M}(S)^{op}$$

that one might still call Verdier duality. Note that these functors are obtained for free without even the need to know that internal homomorphisms of perverse motives actually exist! In fact, one might try to use the above formula (69) in the setting of perverse motives as a way of defining internal homomorphisms there. In this way one would obtain, over every $S \in \mathrm{Var}_k$, a bi-triangulated functor

$$(70) \qquad \underline{\mathrm{Hom}}_S(-, -) : D^b(\mathcal{M}(S))^{op} \times D^b(\mathcal{M}(S)) \to D^b(\mathcal{M}(S))$$

endowed with a canonical natural isomorphism of functors $D^b(\mathcal{M}(S))^{op} \times D^b(\mathcal{M}(S)) \to D^b(\mathrm{Perv}(S))$

$$\iota_S \underline{\mathrm{Hom}}_S(M_1^\bullet, M_2^\bullet) \xrightarrow{\sim} \underline{\mathrm{Hom}}_S(\iota_S(M_1^\bullet), \iota_S(M_2^\bullet)).$$

However, with this approach it would remain unclear whether the functor (70) defines a right adjoint to the internal tensor product as expected: the issue is that, for a general object $M^\bullet \in D^b(\mathcal{M}(S))$, none of the two functors

$$- \otimes M^\bullet, \underline{\mathrm{Hom}}_S(M^\bullet, -) : D^b(\mathcal{M}(S)) \to D^b(\mathcal{M}(S))$$

is exact - and, even worse, none of them is obtained directly from the lifting principles of universal abelian factorizations. However, the result of Proposition 3.6 puts us in the desired situation at least for some choices of $M^\bullet$. This is what we want to prove now.
In order to state the final result of this section, recall that, for any additive category $\mathcal{D}$, we let $\mathbf{R}(\mathcal{D})$ denote the category of finitely presented abelian presheaves on $\mathcal{D}$ (i.e. cokernels of representable presheaves) and we let $\mathbf{A}(\mathcal{D})$ denote Freyd's abelian hull of $\mathcal{D}$. By construction, every object of $\mathbf{A}(\mathcal{D})$ can be written as the kernel of an arrow in $\mathbf{R}(\mathcal{D})$. Moreover, for every additive functor $\beta: \mathcal{D} \to \mathcal{A}$ into an abelian category $\mathcal{A}$, the corresponding universal abelian factorization $\mathbf{A}(\beta)$ is defined as a Serre quotient of $\mathbf{A}(\mathcal{D})$, and so the image of $\mathbf{R}(\mathcal{D})$ in $\mathbf{A}(\beta)$ generates the whole of $\mathbf{A}(\beta)$ under kernels.

**Corollary 3.13.** Let $S \in \mathbf{Sm}_k$ be connected of dimension $d$, and let $\mathbb{D}_S$ denote the motivic Verdier duality functor defined in [IM19, § 2.8]. Let $L \in \mathcal{M}^{0,loc}(S)$, and suppose that $L$ lies in the image of $\mathbf{R}(\mathcal{D}^{0,loc}(S))$ under the quotient functor $\pi_S: \mathbf{A}(\mathcal{D}^{0,loc}(S)) \to \mathcal{M}^{0,loc}(S)$. Then there is a canonical adjunction

$$ - \otimes L[-d] : \mathcal{M}^0(S) \rightleftarrows \mathcal{M}^0(S) : \mathbb{D}_S(L[-d] \otimes \mathbb{D}_S(-)) $$

compatible with the corresponding adjunction on the underlying categories of perverse sheaves in the sense of [Ter23U, Defn. 3.6].

*Proof.* As a consequence of Proposition 3.6 and Remark 3.12, it suffices to construct, for every $L$ as in the statement, an adjunction

$$ (71) \qquad - \otimes^\dagger L : \mathcal{M}^0(S) \rightleftarrows \mathcal{M}^0(S) : \mathbb{D}_S(L \otimes^\dagger \mathbb{D}_S(-)) $$

compatible with the corresponding adjunction on the underlying categories of perverse sheaves in the sense of [Ter23U, Defn. 3.6]. As a consequence of the faithfulness of the functor $\iota_S: \mathcal{M}^0(S) \hookrightarrow \mathrm{Perv}(S)$, we see that such an adjunction (or, more precisely, its associated unit and co-unit transformations) is uniquely determined, provided it exists. We construct the adjoint pair in the statement in two steps of increasing generality:

(Step 1) Suppose first that $L = \pi_S(A)$ for some $A \in \mathcal{D}^{0,loc}(S)$. In this case, we obtain the sought-after adjunction by applying [Ter23U, Cor. 3.9] to the diagrams

$$ \begin{tikzcd} \mathcal{D}^0(S) \arrow[r, "- \otimes^\dagger A"] & \mathcal{D}^0(S) \arrow[d, "\beta_S^0"] \\ \mathrm{Perv}(S) \arrow[r, "- \otimes^\dagger \beta_S^0(A)"'] & \mathrm{Perv}(S) \arrow[d, "\beta_S^0"] \end{tikzcd} $$

and

$$ \begin{tikzcd} \mathcal{D}^0(S) \arrow[r, "\mathbb{D}_S"] & \mathcal{D}^0(S) \arrow[d, "\beta_S^0"] \\ \mathrm{Perv}(S) \arrow[r, "\mathbb{D}_S"] & \mathrm{Perv}(S) \arrow[d, "\beta_S^0(A) \otimes^\dagger -"] \arrow[r, "\beta_S^0(A) \otimes^\dagger -"] & \mathrm{Perv}(S) \arrow[r, "\mathbb{D}_S"] & \mathrm{Perv}(S) \arrow[d, "\beta_S^0"] \end{tikzcd} $$

Note that, as a consequence of [Ter23U, Rem. 3.5(2)], the natural bijections

$$ \mathrm{Hom}_{\mathcal{M}^0(S)}(M \otimes^\dagger \pi_S(A), N) = \mathrm{Hom}_{\mathcal{M}^0(S)}(M, \mathbb{D}_S(\pi_S(A) \otimes^\dagger \mathbb{D}_S(N))) $$

are functorial with respect to $A \in \mathcal{D}^{0,loc}(S)$.

(Step 2) Let $L \in \mathbf{R}(\mathcal{D}^{0,loc}(S))$. Choose a presentation of the form $L = \mathrm{coker}\{\pi_S(A) \to \pi_S(B)\}$ for some morphism $A \to B$ in $\mathcal{D}^{0,loc}(S)$. Then we obtain a natural bijection

$$ \mathrm{Hom}_{\mathcal{M}^0(S)}(M \otimes^\dagger L, N) = \mathrm{Hom}_{\mathcal{M}^0(S)}(M, \mathbb{D}_S(L \otimes^\dagger \mathbb{D}_S(N))) $$
by combining the two chains of bijections

$$
\begin{align*}
\operatorname{Hom}_{\mathcal{M}^0(S)}(M \otimes^\dagger L, N) \\
&= \operatorname{Hom}_{\mathcal{M}^0(S)}(M \otimes^\dagger \operatorname{coker}\{\pi_S(A) \to \pi_S(B)\}, N) \\
&= \operatorname{Hom}_{\mathcal{M}^0(S)}(\operatorname{coker}\{M \otimes^\dagger \pi_S(A) \to M \otimes^\dagger \pi_S(B)\}, N) \\
&= \ker\{\operatorname{Hom}_{\mathcal{M}^0(S)}(M \otimes^\dagger \pi_S(B), N) \to \operatorname{Hom}_{\mathcal{M}^0(S)}(M \otimes^\dagger \pi_S(A), N)\}
\end{align*}
$$

and

$$
\begin{align*}
\operatorname{Hom}_{\mathcal{M}^0(S)}(M, \mathbb{D}_S(L \otimes^\dagger \mathbb{D}_S(N))) \\
&= \operatorname{Hom}_{\mathcal{M}^0(S)}(M, \mathbb{D}_S(\operatorname{coker}\{\pi_S(A) \to \pi_S(B)\} \otimes^\dagger \mathbb{D}_S(N))) \\
&= \operatorname{Hom}_{\mathcal{M}^0(S)}(M, \mathbb{D}_S(\operatorname{coker}\{\pi_S(A) \otimes^\dagger \mathbb{D}_S(N) \to \pi_S(B) \otimes^\dagger \mathbb{D}_S(N)\})) \\
&= \operatorname{Hom}_{\mathcal{M}^0(S)}(M, \ker\{\mathbb{D}_S(\pi_S(B) \otimes^\dagger \mathbb{D}_S(N)) \to \mathbb{D}_S(\pi_S(A) \otimes^\dagger \mathbb{D}_S(N))\}) \\
&= \ker\{\operatorname{Hom}_{\mathcal{M}^0(S)}(M, \mathbb{D}_S(\pi_S(B) \otimes^\dagger \mathbb{D}_S(N))) \to \operatorname{Hom}_{\mathcal{M}^0(S)}(M, \mathbb{D}_S(\pi_S(A) \otimes^\dagger \mathbb{D}_S(N)))\}
\end{align*}
$$

via the previous case. It is clear from the construction that the adjunction obtained in this way is
compatible with the corresponding adjunction on perverse sheaves.

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

In Section 6 we will explain how to prove that the same result holds in full generality, by combining our
own results with certain related recent advances in the theory of Nori motives. We stress, however, that the
results just mentioned rely crucially on the construction of internal homomorphisms obtained in Section 5,
where, in turn, the essential ingredient is provided precisely by the tiny-looking Corollary 3.13.

4. CONSTRUCTION OF THE UNIT CONSTRAINT

By Proposition 2.8 we have a canonical monoidal structure on perverse motives endowed with mutually compatible associativity and commutativity constraints. In order to complete the proof of Theorem 2.1, we need to construct a unit constraint compatible with the rest of the structure. This is achieved in the present section building on the auxiliary results of Section 3. The extension techniques developed in [Ter23E] allow us to complete our construction by working with smooth varieties only: this will often prove convenient in order to avoid technical issues related to singular varieties. Over smooth varieties, we will often make use of shifted inverse image functors and shifted tensor products as defined in Notation 2.10 and Construction 3.5, respectively.

As usual, we write perverse Nori motives with the symbol $\mathcal{M}^0$ rather than as $\mathcal{M}$ in all those passages
that depend closely on the construction of the tensor product, but not in the statement of the main result:

**Proposition 4.1.** The internal tensor structure on $D^b(\mathcal{M}(\cdot))$ constructed in Proposition 2.8 carries a canonical internal unit constraint in the sense of [Ter23T, Defn. 5.1], with respect to which the morphism of $\text{Var}_k$-fibered categories $\iota: D^b(\mathcal{M}(\cdot)) \to D^b(\text{Perv}(\cdot))$ is canonically unitary in the sense of [Ter23T, Defn. 8.20].

In the first place, we need to construct the unit section for perverse motives.

**Notation 4.2.** We adopt the following notation:

* We use the symbol $\mathbf{1} = (\mathbf{1}_S)_{S \in \text{Var}_k}$ to denote the canonical unit sections of the monoidal $\text{Var}_k$-fibered categories $\mathbf{DA}^{\text{et}}_{ct}(\cdot, \mathbb{Q})$ and $\mathbf{D}^b_c(\cdot, \mathbb{Q})$.

• We let $w : \mathrm{Bti}^*(\mathbf{1}) \xrightarrow{\sim} \mathbf{1}$ denote the canonical isomorphism of unit sections defined in [Ayo10].

* We use the symbol $u$ to denote the internal unit constraints of $\mathbf{DA}^{\text{et}}_{ct}(\cdot, \mathbb{Q})$ and $\mathbf{D}^b_c(\cdot, \mathbb{Q})$.
4.1. **Construction of the unit section.** In the first place, we need to define the unit section of the $\mathrm{Var}_k$-fibered category $D^b(\mathcal{M}(\cdot))$.

**Proposition 4.3.** The $\mathrm{Var}_k$-fibered category $D^b(\mathcal{M}(\cdot))$ possesses a canonical unit section $1_{\mathcal{M}} = (1_{S}^{\mathcal{M}})_{S \in \mathrm{Var}_k}$, and the morphism of $\mathrm{Var}_k$-fibered categories $\iota : D^b(\mathcal{M}(\cdot)) \to D^b(\mathrm{Perv}(\cdot))$ induces a canonical isomorphism of unit sections

$$ \tilde{w} : \iota(1_{\mathcal{M}}) \xrightarrow{\sim} 1. $$

The basic technical issue is that, if a variety $S \in \mathrm{Var}_k$ is not smooth, the unit sheaf $1_S \in D_c^b(S, \mathbb{Q})$ is not a perverse sheaf (not even up to decomposing $S$ into connected components and shifting appropriately); in fact, a priori it is not obvious at all that the unit sheaf belongs to the essential image of the triangulated functor $\iota : D^b(\mathcal{M}(S)) \to D^b(\mathrm{Perv}(S))$. However, the structure of stable homotopy 2-functor on $D^b(\mathcal{M}(\cdot))$ allows one to easily overcome this issue by working directly over smooth varieties only: this idea can be made rigorous via the embedding method of [Ter23E, § 1], which we apply to the setting of fibered categories over $\mathrm{Var}_k$ with respect to the full subcategory $\mathrm{Sm}_k$ of smooth varieties.

The core of the proof of Proposition 4.3 is constructing the isomorphisms witnessing the compatibility of the various motivic unit objects with respect to inverse image functors. Using a factorization method analogous to that of [Ter23F], we easily reduce to the cases of smooth morphisms and closed immersions, which we treat separately. While the case of smooth morphisms is quite straightforward, the case of closed immersions is slightly tricky and requires some preliminary discussion. Our approach consists in reducing to the case of closed immersions of codimension 1, which can be treated directly by means of suitable localization triangles; it is plausible that the Čech complex constructions described in Section 3 could lead to an alternative approach, but we have not tried to follow this direction.

Our preliminary results can be summarized as follows:

**Lemma 4.4.** Regard $D^b(\mathcal{M}(\cdot))$ and $D^b(\mathrm{Perv}(\cdot))$ as triangulated $\mathrm{Sm}_k$-fibered categories with inverse image functors $f^\dagger$ as in Notation 2.10.

Let $K$ be a section of the $\mathrm{Sm}_k$-fibered category $D^b(\mathrm{Perv}(\cdot))$ such that, for every connected $S \in \mathrm{Sm}_k$, the object $K_S \in D^b(\mathrm{Perv}(S))$ lies in degree 0. Suppose that we are given, for every $S \in \mathrm{Sm}_k$, and object $K_S^{\mathcal{M}} \in \mathcal{M}(S)$ together with an isomorphism in $D^b(\mathrm{Perv}(S))$

$$ (72) \qquad w_S : \iota_S(K_S^{\mathcal{M}}) \xrightarrow{\sim} K_S. $$

Then the following statements hold:

$$ (1) \quad \text{Fix a morphism } f : T \to S \text{ in } \mathrm{Sm}_k. \text{ Then there exists at most one morphism in } D^b(\mathcal{M}(T)) \\ (73) \qquad K_f^{\mathcal{M},\dagger} : f^\dagger K_S^{\mathcal{M}} \to K_T^{\mathcal{M}} $$

making the diagram in $\mathrm{Perv}(T)$

$$ \begin{tikzcd} f^\dagger \iota_S(K_S^{\mathcal{M}}) \arrow[r, "\tilde{\theta}_f"] & \iota_T(f^\dagger K_S^{\mathcal{M}}) \arrow[r, "K_f^{\mathcal{M},\dagger}"] & \iota_T(K_T^{\mathcal{M}}) \arrow[d, "\tilde{w}_S"] \\ f^\dagger K_S \arrow[r, "K_f^{\dagger}"] & \arrow[d, "\tilde{w}_T"'] & \end{tikzcd} $$

commute; if this is the case, it is automatically an isomorphism.

$$ (2) \quad \text{Suppose that, for every morphism } f : T \to S \text{ in } \mathrm{Sm}_k, \text{ the natural isomorphism (73) exists. Then} \\ \text{the natural isomorphisms (73) turn the collection of objects } \{K_S^{\mathcal{M}}\}_{S \in \mathrm{Sm}_k} \text{ into a section } K \text{ of the} \\ \mathrm{Sm}_k\text{-fibered category } D^b(\mathcal{M}(\cdot)), \text{ and the morphisms (72) define an isomorphism between sections of} \\ D^b(\mathrm{Perv}(\cdot)) \\ (74) \qquad w : \iota(K^{\mathcal{M}}) \xrightarrow{\sim} K. $$