(3) Suppose that the isomorphism (73) is known to exist whenever $f$ is either a smooth morphism or a closed immersion of codimension 1 between smooth connected varieties. Then the natural isomorphism (73) exists for every morphism $f: T \to S$ in $\mathfrak{S}m_k$.

*Proof.* (1) The uniqueness of $K_f^{\mathcal{M},\dagger}$ follows from the fact that $\iota_T : \mathcal{M}(T) \to \mathrm{Perv}(T)$ is faithful, while its invertibility follows from the fact that $\iota_T$ is also conservative (being exact).

(2) We only need to check that the isomorphisms (73) are compatible with composition: given two composable morphisms $f: T \to S$ and $g: S \to V$ in $\mathfrak{S}m_k$, we have to show that the diagram in $\mathcal{M}(T)$

$$ \begin{array}{ccc} f^\dagger g^\dagger K_V^{\mathcal{M}} & \xrightarrow{K_g^{\mathcal{M},\dagger}} & f^\dagger K_S^{\mathcal{M}} \\ \llap{\left.\vphantom{\begin{array}{@{}c@{}}\strut\\\strut\end{array}}\right]} & & \llap{\left.\vphantom{\begin{array}{@{}c@{}}\strut\\\strut\end{array}}\right]} \\ (gf)^\dagger K_V^{\mathcal{M}} & \xrightarrow{K_{gf}^{\mathcal{M},\dagger}} & K_T^{\mathcal{M}} \end{array} $$

is commutative or, equivalently, that its image under the faithful functor $\iota_T : \mathcal{M}(T) \to \mathrm{Perv}(T)$

$$ \begin{array}{ccc} \iota_T(f^\dagger g^\dagger K_V^{\mathcal{M}}) & \xrightarrow{K_g^{\mathcal{M},\dagger}} & \iota_T(f^\dagger K_S^{\mathcal{M}}) \\ \llap{\left.\vphantom{\begin{array}{@{}c@{}}\strut\\\strut\end{array}}\right]} & & \llap{\left.\vphantom{\begin{array}{@{}c@{}}\strut\\\strut\end{array}}\right]} \\ \iota_T((gf)^\dagger K_V^{\mathcal{M}}) & \xrightarrow{K_{gf}^{\mathcal{M},\dagger}} & \iota_T(K_T^{\mathcal{M}}). \end{array} $$

is commutative. But the latter coincides with the outer rectangle of the diagram

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

where the three five-term pieces are commutative by construction while all the other inner pieces are commutative by naturality.

(3) As one can easily deduce from the proof of the previous point, for every two given composable morphisms $f: T \to S$ and $g: S \to V$ in $\mathfrak{S}m_k$ such that the isomorphisms $K_f^{\mathcal{M},\dagger}$ and $K_g^{\mathcal{M},\dagger}$ are already known to exist, the natural isomorphism $K_{gf}^{\mathcal{M},\dagger}$ exists as well and equals the composite

$$ (gf)^\dagger K_V^{\mathcal{M}} = f^\dagger g^\dagger K_V^{\mathcal{M}} \xrightarrow{K_g^{\mathcal{M},\dagger}} f^\dagger K_S^{\mathcal{M}} \xrightarrow{K_f^{\mathcal{M},\dagger}} K_T^{\mathcal{M}}. $$

This implies that it suffices to know that the isomorphism $K_f^{\mathcal{M},\dagger}$ exists whenever $f$ is a smooth morphism or a closed immersion in $\mathfrak{S}m_k$. Even better, once one knows that $K_p^{\mathcal{M},\dagger}$ exists for every smooth morphism $p$ (in particular, for every open immersion), it suffices to show that the isomorphism $K_z^{\mathcal{M},\dagger}$ exists for every closed immersion $z$ between smooth connected varieties.
In order to conclude, we now claim that, in order to know the existence of $K_z^{\mathcal{M},\dagger}$ for every closed immersion between smooth varieties $z: Z \hookrightarrow S$, it suffices to know that it exists whenever $z$ is a closed immersion of pure codimension 1. To see this, recall that, given a closed immersion $z: Z \hookrightarrow S$ in $\mathfrak{Sm}_k$, it is always possible to find a finite open covering $\{S_i \subset S\}_{i \in I}$ such that each closed immersion $z_i: z^{-1}(S_i) \hookrightarrow S_i$ can be written as a composite of closed immersions of pure codimension 1. In this situation, for each index $i \in I$ form the Cartesian square

$$ \begin{array}{ccc} Z_i & \xrightarrow{z_i} & S_i \\ \Big|_{u_{Z,i}} & & \Big|_{u_i} \\ Z & \xrightarrow{z} & S. \end{array} $$

Arguing as in the proof of the previous point one easily sees that, if the four isomorphism $K^{\mathcal{M},\dagger}$, $K^{\mathcal{M},\dagger}$, $K^{\mathcal{M},\dagger}$, $K^{\mathcal{M},\dagger}$ all exist, then they make the diagram in $\mathcal{M}(Z_i)$

$$ \begin{array}{ccc} z_i^\dagger u_i^\dagger K_S & \xrightarrow{K_{u_i}^{\mathcal{M},\dagger}} & z_i^\dagger K_{S_i} \\ \Big|_{u_{Z,i}} & & \Big|_{K_{Z_i}^{\mathcal{M},\dagger}} \\ u_{Z,i}^\dagger z^\dagger K_S & \xrightarrow{K_z^{\mathcal{M},\dagger}} & u_{Z,i}^\dagger K_Z \end{array} $$

commute. Conversely, since the $\mathfrak{Sm}\text{Op}(S)$-fibered categories $\mathcal{M}(\cdot)$ and $\text{Perv}(\cdot)$ are stacks for the Zariski topology, the existence of the isomorphisms $K_{z_i}^{\mathcal{M},\dagger}$, $K_{u_i}^{\mathcal{M},\dagger}$ and $K_{u_{Z,i}}^{\mathcal{M},\dagger}$ for all $i \in I$ implies the existence of $K_z^{\mathcal{M},\dagger}$. This proves the claim and concludes the proof. $\square$

**Proof of Proposition 4.3.** By [Ter23E, Prop. 1.7, Lemma 2.7], it suffices to construct the motivic unit section $\mathbf{1}^{\mathcal{M}}$ and the isomorphism $w: \iota(\mathbf{1}^{\mathcal{M}}) \xrightarrow{\sim} \mathbf{1}$ over $\mathfrak{Sm}_k$. To this end, we apply the criterion of Lemma 4.4 taking $K$ to be the shifted unit section of the $\mathfrak{Sm}_k$-fibered category $D^b(\text{Perv}(\cdot))$: for every connected $S \in \mathfrak{Sm}_k$ we set $K_S := \mathbf{1}_S[\dim(S)]$, and we extend the definition to general $S \in \mathfrak{Sm}_k$ in the obvious way. In a similar manner, we define the shifted motivic unit objects: for every connected $S \in \mathfrak{Sm}_k$ we set $K_S^{\mathcal{M}} := \pi_S(\mathbf{1}_S[\dim(S)])$, and we extend the definition to general $S \in \mathfrak{Sm}_k$ in the obvious way. Note that we have a canonical isomorphism in $\text{Perv}(S)$

$$ (75) \qquad \tilde{w}_S : \iota_S(K_S^{\mathcal{M}}) \xrightarrow{\sim} K_S, $$

which is defined as the composite

$$ \iota_S(K_S^{\mathcal{M}}) := \iota_S \pi_S(\mathbf{1}_S[\dim(S)]) = \text{Bti}_S^*(\mathbf{1}_S[\dim(S)]) = \mathbf{1}_S[\dim(S)] =: K_S $$

in the case when $S$ is connected and by the obvious extension in the general case. Finally, we define the unshifted motivic unit objects by setting $\mathbf{1}_S^{\mathcal{M},\dagger} := K_S^{\mathcal{M}}[-\dim(S)]$ when $S \in \mathfrak{Sm}_k$ is connected, and extending the definition to general $S \in \mathfrak{Sm}_k$ in the obvious way. As above, we have a canonical isomorphism in $D^b(\text{Perv}(S))$

$$ (76) \qquad \iota_S(\mathbf{1}_S^{\mathcal{M}}) \xrightarrow{\sim} \mathbf{1}_S, $$

defined as the composite

$$ \iota_S(\mathbf{1}_S^{\mathcal{M}}) = \iota_S(K_S^{\mathcal{M}}[-\dim(S)]) = \iota_S(K_S^{\mathcal{M}})[-\dim(S)] \xrightarrow{\sim} K_S[-\dim(S)] = \mathbf{1}_S $$
when $S \in \mathbf{Sm}_k$ is connected and by the obvious extension in the general case. Clearly, turning the collection $(\mathbf{1}_S^{\mathcal{M}})_{S \in \mathbf{Sm}_k}$ into a section of $D^b(\mathcal{M}(\cdot))$ with respect to the usual inverse image functors $f^*$ in such a way that the isomorphisms (76) define an isomorphism of sections is the same as turning the collection $(K_S^{\mathcal{M}})_{S \in \mathbf{Sm}_k}$ into a section of $D^b(\mathcal{M}(\cdot))$ with respect to the shifted inverse image functors $f^\dagger$ in such a way that the isomorphisms (75) define an isomorphism of sections.

By Lemma 4.4(3), in order to achieve the latter task, it suffices to construct the isomorphisms $K_f^{\mathcal{M},\dagger}$ in the cases where $f$ is either a smooth morphism or a closed immersion of codimension 1. We treat these two cases separately as follows:

* Let $p: P \to S$ be a smooth morphism in $\mathbf{Sm}_k$. We define the isomorphism in $\mathcal{M}(P)$

$$K_p^{\mathcal{M},\dagger} : p^\dagger K_S^{\mathcal{M}} \xrightarrow{\sim} K_P^{\mathcal{M}}$$

by taking the composite

$$p^\dagger K_S^{\mathcal{M}} := p^\dagger \pi_S(\mathbf{1}_S[\dim(S)]) = \pi_P(p^\dagger \mathbf{1}_S[\dim(S)]) \xrightarrow{\mathbf{1}_P^\dagger} \pi_P(\mathbf{1}_P[\dim(P)]) =: K_P^{\mathcal{M}}.$$

The associated diagram in $\mathbf{Perv}(P)$

$$\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
  p^\dagger \i_S(K_S^{\mathcal{M}}) \arrow[r, "\tilde{\theta}_P"] & \i_P(p^\dagger K_S^{\mathcal{M}}) \arrow[r, "K_P^{\mathcal{M},\dagger}"] & \i_P(K_P^{\mathcal{M}}) \\
  \coloneqq \coloneqq & \coloneqq & \coloneqq \\
  p^\dagger K_S \arrow[r, "K_P^\dagger"] & & K_P
\end{tikzcd}$$

which can be more explicitly written in the form

$$\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
  p^\dagger \mathrm{Bti}_S^*(\mathbf{1}_S[\dim(S)]) \arrow[r, "\theta_P"] & \mathrm{Bti}_P^*(p^\dagger \mathbf{1}_S[\dim(S)]) \arrow[r, "\mathbf{1}_P^\dagger"] & \mathrm{Bti}_P^*(\mathbf{1}_P[\dim(P)]) \\
  \coloneqq \coloneqq & \coloneqq & \coloneqq \\
  \coloneqq & \coloneqq & \coloneqq \\
  p^\dagger \mathbf{1}_S[\dim(S)] \arrow[r, "\mathbf{1}_P^\dagger"] & & \mathbf{1}_P[\dim(P)]
\end{tikzcd}$$

is commutative since $w: \mathrm{Bti}^*(\mathbf{1}) \xrightarrow{\sim} \mathbf{1}$ is a morphism of sections.

* Let $z: Z \hookrightarrow S$ be a closed immersion of codimension 1 in $\mathbf{Sm}_k$. In order to define the isomorphism in $\mathcal{M}(Z)$

$$\mathbf{1}_z^{\mathcal{M},*} : z^\dagger K_S^{\mathcal{M}} \xrightarrow{\sim} K_Z^{\mathcal{M}}$$

we describe the morphism in $D^b(\mathcal{M}(S))$

$$K_S^{\mathcal{M}} \to z_* K_Z^{\mathcal{M}} [1] \qquad (77)$$

corresponding to it under adjunction. To this end, let $u: U \hookrightarrow S$ denote the open immersion complementary to $z$. Consider the distinguished triangle in $\mathbf{DA}_{ct}^{\hat{\epsilon}t}(S, \mathbb{Q})$

$$z_* \mathbf{1}_Z[\dim(Z)] \to u_* u^* \mathbf{1}_S[\dim(S)] \to \mathbf{1}_S[\dim(S)] \xrightarrow{+} \qquad (78)$$

obtained from the usual localization triangle

$$u_* u^* \mathbf{1}_S[\dim(S)] \to \mathbf{1}_S[\dim(S)] \to z_* z^* \mathbf{1}_S[\dim(S)] \xrightarrow{+}$$

after inserting the isomorphism $\mathbf{1}_z^*: z^* \mathbf{1}_S \xrightarrow{\sim} \mathbf{1}_Z$ and rotating backwards. Note that all the terms in (78) belong to $\mathcal{D}^0(S)$ by construction. Hence, applying the functor $\pi_S: \mathcal{D}^0(S) \to \mathcal{M}(S)$ and using the canonical isomorphisms $\beta_S \circ z_* = z_* \circ \beta_Z$ and $\beta_S \circ u_* u^* = u_* \circ \beta_U \circ u^* = u_* u^* \circ \beta_S$, we obtain a short exact sequence in $\mathcal{M}(S)$

$$0 \to z_* K_Z^{\mathcal{M}} \to u_* u^* K_S^{\mathcal{M}} \to K_S^{\mathcal{M}} \to 0.$$
in a form which only involves t-exact functors by replacing inverse images under closed immersions by the corresponding direct images.

For sake of brevity, we formulate our preliminary results only for the right unit action; of course, the same conclusions apply to the left unit action as well.

**Lemma 4.5.** Regard $D^b(\mathcal{M}(\cdot))$ as a $\mathfrak{Sm}_k$-fibered category, and consider the motivic unit section $\mathbb{1}^{\mathcal{M}} = (\mathbb{1}_{S}^{\mathcal{M}})_{S \in \mathfrak{Sm}_k}$ constructed in Proposition 4.3. Suppose that we are given, for every $S \in \mathfrak{Sm}_k$, a natural isomorphism of functors $D^b(\mathcal{M}(S)) \to D^b(\mathcal{M}(S))$

$$u_r = u_{r,S} : M^{\bullet} \otimes \mathbb{1}_S^{\mathcal{M}} \xrightarrow{\sim} M^{\bullet}.$$

Then the following statements hold:

(1) For every morphism $f : T \to S$ in $\mathfrak{Sm}_k$, let $D_f$ denote the diagram of functors $D^b(\mathcal{M}(S)) \to D^b(\mathcal{M}(T))$

$$ \begin{array}{ccc} f^* M^{\bullet} \otimes f^* \mathbb{1}_S^{\mathcal{M}} & \xrightarrow{m} & f^*(M^{\bullet} \otimes \mathbb{1}_S^{\mathcal{M}}) \\ \bigdownarrow \mathbb{1}_f^* & & \bigdownarrow u_r \\ f^* M^{\bullet} \otimes \mathbb{1}_T^{\mathcal{M}} & \xrightarrow{u_r} & f^* M^{\bullet}. \end{array} $$

Then, in order for $D_f$ to be commutative for every morphism $f$ in $\mathfrak{Sm}_k$, it suffices that it be commutative for every smooth morphism and for every closed immersion.

(2) Let $z : Z \hookrightarrow S$ be a closed immersion in $\mathfrak{Sm}_k$, and let $D'_z$ denote the diagram $D'_z$ of functors $D^b(\mathcal{M}(Z)) \to D^b(\mathcal{M}(S))$

$$ \begin{array}{ccc} z_* N^{\bullet} \otimes z_* z^* \mathbb{1}_S^{\mathcal{M}} & \xleftarrow{\eta} & z_* N^{\bullet} \otimes \mathbb{1}_S^{\mathcal{M}} \\ \bigdownarrow \mathbb{1}_z^* & & \bigdownarrow u_r \\ z_* N^{\bullet} \otimes z_* \mathbb{1}_Z^{\mathcal{M}} & \xrightarrow{\bar{m}} & z_* (N^{\bullet} \otimes \mathbb{1}_Z^{\mathcal{M}}) \xrightarrow{u_r} & z_* N^{\bullet}. \end{array} $$

Then $D_z$ is commutative if and only if $D'_z$ is commutative.

*Proof.* Both assertions follow by a standard application of the factorization method of [Ter23F]. Since unit constraints are not discussed in [Ter23F], for the reader's convenience we include the argument:

(1) Given a morphism $f : T \to S$ in $\mathfrak{Sm}_k$, choose a factorization of it in $\mathrm{Var}_k$ of the form

$$f : T \xrightarrow{t} P \xrightarrow{p} S$$

with $t$ a closed immersion and $p$ a smooth morphism; note that $P$ is smooth since so is $S$ as in the statement, so this factorization exists in $\mathfrak{Sm}_k$. Then, by construction, the diagram $D_f$ coincides
with the outer part of the diagram

<figure><img src="image_9.png" /></figure>

where the two central trapezoids are essentially the diagrams $D_p$ and $D_z$. Since all the remaining pieces of the diagram are already known to be commutative, the thesis follows.

(2) Let $D_z''$ denote the diagram of functors $D^b(M(Z)) \to D^b(M(Z))$

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

We claim that the commutativity of $D_z$ and of $D_z'$ are both equivalent to the commutativity of $D_z''$. We prove each equivalence directly as follows:

* It is clear that the commutativity of $D_z$ implies that of $D_z''$. Conversely, $D_z$ can be written as the outer part of the diagram

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

where the central square is essentially $D_z''$ and the lateral pieces are already known to be commutative. This shows that $D_z$ is commutative if and only if so is $D_z''$.
• The diagram $D'_z$ can be written as the outer part of the diagram

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

where all arrows are invertible by construction and all pieces except possibly the square

$$ \begin{array}{ccc} z_*(z^*z_*N^\bullet \otimes z^*1_S^{\mathcal{M}}) & \xrightarrow{m} & z_*z^*(z_*N^\bullet \otimes 1_S^{\mathcal{M}}) \\ \begin{tikzpicture}[baseline={([yshift=-0.5ex]current bounding box.center)}] \node[circle, fill, inner sep=1.2pt] (u) at (0,0) {}; \node[above=1pt of u] {$1_S^{\mathcal{M},*}$}; \draw[->] (u) -- (-1,0) node[left] {$\xrightarrow{u_r}$}; \end{tikzpicture} & & \begin{tikzpicture}[baseline={([yshift=-0.5ex]current bounding box.center)}] \node[circle, fill, inner sep=1.2pt] (u) at (0,0) {}; \node[above=1pt of u] {$1_S^{\mathcal{M},*}$}; \draw[->] (u) -- (-1,0) node[left] {$\xrightarrow{u_r}$}; \end{tikzpicture} \\ z_*(z^*z_*N^\bullet \otimes 1_Z^{\mathcal{M}}) & \xrightarrow{u_r} & z_*z^*z_*N^\bullet \end{array} $$

are commutative by naturality. Hence $D'_z$ is commutative if and only if the above square is commutative; since the functor $z_* : D^b(\mathcal{M}(Z)) \to D^b(\mathcal{M}(S))$ is fully faithful, this is the case if and only if $D''_z$ is commutative.

This proves the claim and concludes the proof.

□

In order to be able to apply Lemma 4.5(2), we need to use the following variant of Proposition 3.6:

**Proposition 4.6.** Let $z : Z \hookrightarrow S$ be a closed immersion in $\mathbf{Sm}_k$. Then the bi-exact functor

$$ z_{*}(-) \otimes z_{*}(-[\dim(Z)]) = (z_{*}(-) \otimes z_{*}(-))[\dim(Z)] : \mathcal{M}^{0}(Z) \times \mathcal{M}^{0,\mathrm{loc}}(Z) \to \mathcal{M}^{0}(S) $$

is canonically isomorphic to the functor obtained by applying [Ter23U, Prop. 4.3] to the diagram

$$ \begin{array}{ccc} \mathcal{D}^0(Z) \times \mathcal{D}^{0,\mathrm{loc}}(Z) & \xrightarrow{(z_*(-) \otimes z_*(-))[\dim(Z)]} & \mathcal{D}^0(S) \\ \begin{tikzpicture}[baseline={([yshift=-0.5ex]current bounding box.center)}] \node[circle, fill, inner sep=1.2pt] (u) at (0,0) {}; \node[above=1pt of u] {$\beta_Z^0 \times \beta_Z^{0,\mathrm{loc}}$}; \draw[->] (u) -- (-1,0) node[left] {$\xrightarrow{\beta_S^0}$}; \end{tikzpicture} & & \begin{tikzpicture}[baseline={([yshift=-0.5ex]current bounding box.center)}] \node[circle, fill, inner sep=1.2pt] (u) at (0,0) {}; \node[above=1pt of u] {$\mathrm{Perv}(Z) \times \mathrm{Perv}(Z)$}; \draw[->] (u) -- (-1,0) node[left] {$\xrightarrow{\mathrm{Perv}(Z)[\dim(Z)])} \mathrm{Perv}(S)$}; \end{tikzpicture} \end{array} $$

*Proof.* As the reader can check, the argument used for the proof of Proposition 3.6 goes through modulo the obvious changes; we omit the details. □

*Proof of Proposition 4.1.* By [Ter23E, Prop. 3.9, Lemma 4.3(3)], it suffices to construct the motivic unit constraint and check the unitarity of $\iota : D^b(\mathcal{M}(\cdot)) \to D^b(\mathrm{Perv}(\cdot))$ (via the isomorphism $w : \iota(1_S^{\mathcal{M}}) \xrightarrow{\sim} 1$ constructed in Proposition 4.3) over $\mathbf{Sm}_k$. To this end, we apply the criterion of Lemma 4.5.
In the first place, fix $S \in \mathbf{Sm}_k$ and consider the two diagrams

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

The shifted right (resp. left) unit isomorphisms of functors $\mathcal{D}^0(S) \to \mathcal{D}^0(S)$

$$ u_r = u_{r,S} : A \otimes^\dagger \mathbf{1}_S[\dim(S)] \xrightarrow{\sim} A, \quad u_l = u_{l,S} : \mathbf{1}_S[\dim(S)] \otimes^\dagger A \xrightarrow{\sim} A $$

and the shifted right (resp. left) unit isomorphism of functors $\mathbf{Perv}(S) \to \mathbf{Perv}(S)$

$$ u_r = u_{r,S} : K \otimes^\dagger \mathbf{1}_S[\dim(S)] \xrightarrow{\sim} K, \quad u_l = u_{l,S} : \mathbf{1}_S[\dim(S)] \otimes^\dagger K \xrightarrow{\sim} K $$

satisfy the compatibility condition of [Ter23U, Defn. 3.1], because the same holds for the corresponding natural isomorphisms on $\mathbf{DA}_{ct}^{\mathcal{E}t}(\cdot, \mathbb{Q})$ and on $\mathcal{D}_c^b(\cdot, \mathbb{Q})$ - this is just a way of rephrasing the unitarity of $\mathbf{Bti}^*$: $\mathbf{DA}_{ct}^{\mathcal{E}t}(\cdot, \mathbb{Q}) \to \mathcal{D}_c^b(\cdot, \mathbb{Q})$. Moreover, in both cases the shifted left and right unit isomorphisms coincide up to a sign (up to a sign $(-1)^{\dim(S)}$) when evaluated on the shifted object $\mathbf{1}_S[\dim(S)]$. Therefore, applying [Ter23U, Prop. 3.4], we obtain two canonical natural isomorphisms between functors $\mathcal{M}(S) \to \mathcal{M}^0(S)$

$$ (79) \qquad \tilde{u}_r = \tilde{u}_{r,S} : M \otimes^\dagger \mathbf{1}_S^{\mathcal{M}}[\dim(S)] \xrightarrow{\sim} M, \qquad \tilde{u}_l = \tilde{u}_{l,S} : \mathbf{1}_S^{\mathcal{M}}[\dim(S)] \otimes^\dagger M \xrightarrow{\sim} M $$

compatible with the corresponding shifted unit isomorphisms on $\mathbf{Perv}(\cdot)$; they coincide (up to a sign $(-1)^{\dim(S)}$) when evaluated on the shifted object $\mathbf{1}_S^{\mathcal{M}}[\dim(S)]$. Applying Lemma 2.5, unshifting the unit objects and the tensor product functors, and applying Proposition 3.6, we obtain two unit natural isomorphisms of functors $\mathcal{D}^b(\mathcal{M}^0(S)) \to \mathcal{D}^b(\mathcal{M}^0(S))$

$$ (80) \qquad \tilde{u}_r = \tilde{u}_{r,S} : M^{\bullet} \otimes \mathbf{1}_S^{\mathcal{M}} \xrightarrow{\sim} M^{\bullet}, \qquad \tilde{u}_l = \tilde{u}_{l,S} : \mathbf{1}_S^{\mathcal{M}} \otimes M^{\bullet} \xrightarrow{\sim} M^{\bullet} $$

again compatible with the corresponding unit isomorphisms on $\mathcal{D}^b(\mathbf{Perv}(\cdot))$; they coincide (with no sign) when evaluated on the unshifted object $\mathbf{1}_S^{\mathcal{M}}$, thereby satisfying condition (uITS-1) of [Ter23T, Defn. 5.1].

It remains to show that the natural isomorphisms (80) satisfy condition (uITS-2) of [Ter23T, Defn. 5.1]. For sake of simplicity, we only write the proof for the right unit action; of course, the same argument will apply to the left unit action as well. What we have to show in the case of the right unit action is that, for every morphism $f : T \to S$ in $\mathbf{Sm}_k$, the diagram $D_f$ associated to $f$ as in the statement of Lemma 4.5 is commutative. To this end, applying Lemma 4.5(1) we reduce to considering the cases where $f$ is either a smooth morphism or a closed immersion; applying Lemma 4.5(2), we can also rephrase the case of a closed immersion $z$ in terms of the commutativity of the alternative diagram $D'_z$ introduced there. Here are the details:
• Let $p: P \to S$ be a smooth morphism in $\mathfrak{Sm}_k$. The diagram of functors $D^b(\mathcal{M}^0(S)) \to D^b(\mathcal{M}^0(P))$

$$ \begin{array}{ccc} p^*M^\bullet \otimes p^*\mathbf{1}_S^{\mathcal{M}} & \xrightarrow{\tilde{m}} & p^*(M^\bullet \otimes \mathbf{1}_S^{\mathcal{M}}) \\ \bigdownarrow_{\mathbf{1}_P^{\mathcal{M},*}} & & \bigdownarrow_{\tilde{u}_r} \\ p^*M^\bullet \otimes \mathbf{1}_P^{\mathcal{M}} & \xrightarrow{\tilde{u}_r} & p^*M^\bullet \end{array} $$

is commutative if and only if so is the diagram

$$ \begin{array}{ccc} p^\dagger M^\bullet \otimes p^\dagger K_S^{\mathcal{M}} & \xrightarrow{\tilde{m}} & p^\dagger(M^\bullet \otimes K_S^{\mathcal{M}}) \\ \bigdownarrow_{\mathbf{1}_P^{\mathcal{M},*}} & & \bigdownarrow_{\tilde{u}_r} \\ p^\dagger M^\bullet \otimes K_P^{\mathcal{M}} & \xrightarrow{\tilde{u}_r} & p^\dagger M^\bullet. \end{array} $$

Since all the functors involved in the latter diagram are both dg-enhanced and $t$-exact for the obvious $t$-structures, by Proposition 2.6(1) they coincide with the trivial derived functors of the corresponding exact functors $\mathcal{M}^0(S) \to \mathcal{M}^0(P)$. Hence (see Remark 2.7) the above diagram is commutative if and only if the corresponding diagram of functors $\mathcal{M}^0(S) \to \mathcal{M}^0(P)$

$$ \begin{array}{ccc} p^\dagger M \otimes p^\dagger \mathbf{1}_S^{\mathcal{M}} & \xrightarrow{\tilde{m}} & p^\dagger(M \otimes \mathbf{1}_S^{\mathcal{M}}) \\ \bigdownarrow_{\mathbf{1}_P^{\mathcal{M},*}} & & \bigdownarrow_{\tilde{u}_r} \\ p^\dagger M \otimes \mathbf{1}_P^{\mathcal{M}} & \xrightarrow{\tilde{u}_r} & p^\dagger M \end{array} $$

is commutative. Since all the functors and natural transformations involved in the latter diagram come from the lifting principles of universal abelian factorizations, it suffices to show that the corresponding diagram of functors in $\mathbf{DA}_{ct}^{\acute{e}t}(\cdot, \mathbb{Q})$ is commutative, which follows from the validity of axiom (uITS-2) there.

• Let $z: Z \hookrightarrow S$ be a closed immersion in $\mathfrak{Sm}_k$. By Lemma 4.5(2), it suffices to show that the diagram of triangulated functors $D^b(\mathcal{M}^0(Z)) \to D^b(\mathcal{M}^0(S))$

$$ \begin{array}{ccc} z_* M^\bullet \otimes z_* z^* \mathbf{1}_S^{\mathcal{M}} & \xleftarrow{\eta} & z_* M^\bullet \otimes \mathbf{1}_S^{\mathcal{M}} \\ \bigdownarrow_{\mathbf{1}_Z^{\mathcal{M},*}} & & \bigdownarrow_{\tilde{u}_r} \\ z_* M^\bullet \otimes z_* \mathbf{1}_Z^{\mathcal{M}} & \xrightarrow{\tilde{m}} & z_*(M^\bullet \otimes \mathbf{1}_Z^{\mathcal{M}}) \xrightarrow{\tilde{u}_r} & z_* M^\bullet \end{array} $$

is commutative. Since all the functors involved in this diagram are dg-enhanced and $t$-exact for the obvious $t$-structures, by Proposition 2.6(1) they coincide with the trivial derived functors of the corresponding exact functors $\mathcal{M}^0(Z) \to \mathcal{M}^0(S)$. Hence, applying Proposition 2.6(2), we see that the above diagram is commutative if and only if the corresponding diagram of functors $\mathcal{M}^0(Z) \to \mathcal{M}^0(S)$

$$ \begin{array}{ccc} z_* M \otimes z_* z^* \mathbf{1}_S^{\mathcal{M}} & \xleftarrow{\eta} & z_* M \otimes \mathbf{1}_S^{\mathcal{M}} \\ \bigdownarrow_{\mathbf{1}_Z^{\mathcal{M},*}} & & \bigdownarrow_{\tilde{u}_r} \\ z_* M \otimes z_* \mathbf{1}_Z^{\mathcal{M}} & \xrightarrow{\tilde{m}} & z_*(M \otimes \mathbf{1}_Z^{\mathcal{M}}) \xrightarrow{\tilde{u}_r} & z_* M \end{array} $$

is commutative. As a consequence of Proposition 3.6 and Proposition 4.6, all the functors and natural transformations appearing in the latter diagram come from the lifting principles of universal abelian factorizations. Thus it suffices to show that the corresponding diagram of functors in $\mathbf{DA}_{ct}^{\acute{e}t}(\cdot, \mathbb{Q})$ is commutative, which follows from the validity of axiom (uITS-2) there.

This concludes the proof. $\square$

<figure><img src="image_14.png" /></figure>
### 4.3. Compatibility with the associativity and commutativity constraints.
In order to make the picture complete, it remains to study the compatibility between the unit constraint of Proposition 4.1 and the associativity and commutativity constraints constructed in Lemma 2.11 and Lemma 2.12. Since the unit constraint has been defined in the language of internal tensor structures, whereas the associativity and commutativity constraints have been defined in the language of external tensor structures, we need to decide whether to check the two compatibility conditions in the internal or in the external setting. Working in the external setting turns out to be easier, since everything can be more directly related to the lifting principles of universal abelian factorizations. In order to explain why this is the case, we start with the following observation:

**Remark 4.7.** For every $S_1, S_2 \in \mathfrak{Sm}_k$, consider the diagram

$$
\begin{tikzcd}
\mathcal{D}^0(S_1) \arrow[r, "-\boxtimes K_{S_2}"] \arrow[d, "\beta_{S_1}^0"] & \mathcal{D}^0(S_1 \times S_2) \arrow[d, "\beta_{S_1 \times S_2}^0"] \\
\mathrm{Perv}(S_1) \arrow[r, "-\boxtimes K_{S_2}"] & \mathrm{Perv}(S_1 \times S_2)
\arrow[dl, "\downarrow"]
\end{tikzcd}
$$
$$
\begin{tikzcd}
\mathcal{D}^0(S_1) \arrow[r, "pr_1^{12,\dagger}"] \arrow[d, "\beta_{S_1}^0"] & \mathcal{D}^0(S_1 \times S_2) \arrow[d, "\beta_{S_1 \times S_2}^0"] \\
\mathrm{Perv}(S_1) \arrow[r, "pr_1^{12,\dagger}"] & \mathrm{Perv}(S_1 \times S_2).
\arrow[dl, "\downarrow"]
\end{tikzcd}
$$

Therefore, applying [Ter23U, Prop. 3.4], we obtain a canonical natural isomorphism between functors $\mathcal{M}^0(S_1) \to \mathcal{M}^0(S_1 \times S_2)$

$$M_1 \boxtimes K_{S_2}^{\mathcal{M}} \xrightarrow{\sim} pr_1^{12,\dagger} M_1.$$

As a consequence of Proposition 3.6 we see that, for every $S \in \mathfrak{Sm}_k$, the motivic right unit isomorphism in (79) coincides with the composite isomorphism

$$M_1 \otimes_M K_S^{\mathcal{M}} = M_1 \otimes \mathbf{1}_S^{\mathcal{M}} := \Delta_S^*(M_1 \boxtimes \mathbf{1}_S^{\mathcal{M}}) = \Delta_S^\dagger(M_1 \boxtimes K_S^{\mathcal{M}}) \xrightarrow{\sim} \Delta_S^\dagger pr_1^{12,\dagger} M_1 = M_1.$$

Of course, the analogue result holds for the motivic left unit isomorphism.

We can now easily deduce the sought-after compatibility properties as follows:

**Lemma 4.8.** The external unit constraint obtained from Proposition 4.1 and the external associativity constraint of Lemma 2.11 are compatible in the sense of [Ter23T, Defn. 6.7].

*Proof.* We have to show that the associativity and unit constraints satisfy condition (auETS) from [Ter23T, Defn. 6.7]. For sake of simplicity, we only check the commutativity of the third family of diagrams mentioned there; the commutativity of the other two families follows by a similar argument. Using the analogue of [Ter23E, Lemma 3.11(2)] for external tensor structures, we see that it suffices to consider smooth varieties only. In the end, what we have to show is that, for every $S_1, S_2, S_3 \in \mathfrak{Sm}_k$, the diagram of functors $D^b(\mathcal{M}^0(S_1)) \times D^b(\mathcal{M}^0(S_2)) \to D^b(\mathcal{M}^0(S_1 \times S_2 \times S_3))$

$$
\begin{tikzcd}
(M_1 \boxtimes M_2) \boxtimes \mathbf{1}_{S_3}^{\mathcal{M}} \arrow[r, "a"'] & M_1 \boxtimes (M_2 \boxtimes \mathbf{1}_{S_3}^{\mathcal{M}}) \arrow[d, "u_r"] \\
\mathrm{pr}_{23}^{123,*}(M_1 \boxtimes M_2) \arrow[r, "m"'] & M_1 \boxtimes \mathrm{pr}_2^{12,*} M_2
\arrow[d, "u_r"]
\end{tikzcd}
$$
is commutative. After shifting the unit objects, using Remark 4.7, we reduce to showing that the diagram of functors $\mathcal{M}^0(S_1) \times \mathcal{M}^0(S_2) \to \mathcal{M}^0(S_1 \times S_2 \times S_3)$

$$ \begin{array}{ccc} (M_1 \boxtimes M_2) \boxtimes K_{S_3}^{\mathcal{M}} & \xrightarrow{a} & M_1 \boxtimes (M_2 \boxtimes K_{S_3}^{\mathcal{M}}) \\ \begin{array}{c} \searrow \\ \downarrow \end{array} & & \begin{array}{c} \searrow \\ \downarrow \end{array} \\ pr_{23}^{123,\dagger}(M_1 \boxtimes M_2) & \leftarrow_{m} & M_1 \boxtimes pr_2^{12,\dagger}M_2 \end{array} $$

is commutative. Since all the functors and natural transformations appearing in the latter diagram come from the lifting principles of universal abelian factorizations, it suffices to check that the corresponding diagram on $\mathbf{DA}_{\text{ct}}^{\text{\'et}}(\cdot, \mathbb{Q})$ is commutative, which (after unshifting the unit objects) follows from the validity of axiom (auETS) there. $\square$

**Lemma 4.9.** *The internal unit constraint of Proposition 4.1 and the internal commutativity constraint obtained from Lemma 2.12 are compatible in the sense of [Ter23T, Defn. 6.11].*

*Proof.* We have to show that the commutativity and unit constraints satisfy condition (cuETS) from [Ter23T, Defn. 6.11]. Using the analogue of [Ter23E, Lemma 3.11(3)] for external tensor structure, we see that it suffices to consider smooth varieties only. In the end, what we have to show is that, for every $S_1, S_2 \in \mathbf{Sm}_k$, the diagram of functors $D^b(\mathcal{M}(S_1)) \to D^b(\mathcal{M}(S_1 \times S_2))$

$$ \begin{array}{ccc} M_1^{\bullet} \boxtimes 1_{S_2}^{\mathcal{M}} & \xrightarrow{c} & \tau^*(1_{S_2}^{\mathcal{M}} \boxtimes M_1^{\bullet}) \\ \begin{array}{c} \searrow \\ \downarrow \end{array} & & \begin{array}{c} \searrow \\ \downarrow \end{array} \\ pr_1^{12,*} M_1^{\bullet} & = & \tau^* pr_1^{21,*} M_1^{\bullet} \end{array} $$

is commutative. After shifting the unit object, using Remark 4.7, we reduce to showing that the diagram of functors $\mathcal{M}^0(S_1) \to \mathcal{M}^0(S_1 \times S_2)$

$$ \begin{array}{ccc} M_1 \boxtimes K_{S_2}^{\mathcal{M}} & \xrightarrow{c} & \tau^*(K_{S_2}^{\mathcal{M}} \boxtimes M_1) \\ \begin{array}{c} \searrow \\ \downarrow \end{array} & & \begin{array}{c} \searrow \\ \downarrow \end{array} \\ pr_1^{12,\dagger} M_1 & = & \tau^* pr_1^{21,\dagger} M_1 \end{array} $$

is commutative. Since all the functors and natural transformations appearing in the latter diagram come from the lifting principles of universal abelian factorizations, it suffices to check that the corresponding diagram on $\mathbf{DA}_{\text{ct}}^{\text{\'et}}(\cdot, \mathbb{Q})$ is commutative, which (after unshifting the unit object) follows from the validity of axiom (cuETS) there. $\square$

*Proof of Theorem 2.1.* Combine the results of Proposition 2.8, Lemma 2.11, Lemma 2.12 and Lemma 2.13 obtained previously with the results of Proposition 4.1, Lemma 4.8 and Lemma 4.9 obtained in the present section. $\square$

## 5. INTERNAL HOMOMORPHISMS

In this section, we complete the construction of the six functor formalism on perverse motives by showing the existence of internal homomorphisms and their compatibility with those on the underlying categories of perverse sheaves. As usual, we state the main result in the language of stable homotopy 2-functors:
**Theorem 5.1.** The monoidal stable homotopy 2-functor $D^b(M(\cdot))$ is closed in the sense of [Ayo07a, Defn. 2.3.50]. Moreover, for every $S \in \text{Var}_k$, the canonical natural transformation between functors $D^b(M(S))^{op} \times D^b(M(S)) \to D^b(\text{Perv}(S))$

$$ \iota_S \underline{\text{Hom}}_S(M_1^\bullet, M_2^\bullet) \to \underline{\text{Hom}}_S(\iota_S(M_1^\bullet), \iota_S(M_2^\bullet)) $$

is invertible.

The strategy of the proof consists of two main steps: on the one hand, an abstract construction of internal homomorphisms as an ind-adjoint of the internal tensor product; on the other side, the adjunction for distinguished motivic local systems obtained in Corollary 3.13. As usual, following the conventions of Section 2, we write the categories of perverse motives as "$\mathcal{M}^0$" in all constructions and proofs specifically involving the tensor product, and simply as "$\mathcal{M}$" otherwise.

The abstract construction of internal homomorphism is inspired by Arapura's construction of motivic direct images in [Ara13, § 5.1], based on a suitable adjoint functor theorem for derived categories of Grothendieck abelian categories from [Fra01]. Compared to Arapura's case, our construction requires a longer preparation in order to extend the internal tensor product to the level of derived ind-categories.

**5.1. Recollections on ind-categories.** We start by reviewing some facts about ind-categories of (Noetherian) abelian categories and their derived categories. Recall from [SGA4, Exp. 1, § 8.2] that, for every small category $\mathcal{C}$, the ind-category Ind$\mathcal{C}$ is the full subcategory of $\text{Fun}(\mathcal{C}^{\text{op}}, \text{Sets})$ consisting of all those presheaves which are filtered colimits of representables; the Yoneda embedding $\mathcal{C} \hookrightarrow \text{Fun}(\mathcal{C}^{\text{op}}, \text{Sets})$ can be refined to a fully faithful functor $\mathcal{C} \hookrightarrow \text{Ind}\mathcal{C}$.

**Proposition 5.2.** Let $\mathcal{A}$ be an essentially small abelian category. Then:

(1) The category Ind$\mathcal{A}$ is a Grothendieck abelian category, and the canonical embedding $\mathcal{A} \hookrightarrow \text{Ind}\mathcal{A}$ is exact.

(2) If $\mathcal{A}$ is Noetherian, then its essential image in Ind$\mathcal{A}$ consists of the subcategory of Noetherian objects inside Ind$\mathcal{A}$.

(3) The canonical triangulated functor $D^b(\mathcal{A}) \to D(\text{Ind}\mathcal{A})$ is fully faithful and induces an equivalence

$$ D^b(\mathcal{A}) \xrightarrow{\sim} D^b_{\mathcal{A}}(\text{Ind}\mathcal{A}) := \{A^\bullet \in D(\text{Ind}\mathcal{A}) \mid H^n(A^\bullet) = 0 \ \forall |n| \gg 0, H^n(A^\bullet) \in \mathcal{A} \ \forall n \in \mathbb{Z}\}. $$

*Proof.* The first statement is proved in [KS06, Thm. 8.6.5]; the second one is an easy consequence of [Hub93, Lemma 1.3]; the third one follows from [KS06, Prop. 13.1.12(i), Thm. 15.3.1(i)]. □

We need to extend certain additive functors of abelian categories to the level of ind-categories and to their derived categories. The following result describes this in detail:

**Proposition 5.3.** Let $F : \mathcal{A}_1 \to \mathcal{A}_2$ be a functor between abelian categories. Then:

(1) Up to canonical natural isomorphism, there exists a unique functor

$$ \text{Ind}(F) : \text{Ind}_{\mathcal{A}_1} \to \text{Ind}_{\mathcal{A}_2} $$

extending $F$ and commuting with arbitrary small filtered colimits.

(2) If $F$ is additive, then so it $\text{Ind}(F)$.

(3) If $F$ is faithful (resp. full) then so is $\text{Ind}(F)$.

(4) If $F$ is left-exact (resp. right-exact), then so is $\text{Ind}(F)$.

Moreover, the assignment $F \mapsto \text{Ind}(F)$ is canonically compatible with composition of functors.

*Proof.* The first statement is proved in [SGA4, Exp. 1, § 8.6] and does not require the assumption that $\mathcal{A}_1$ and $\mathcal{A}_2$ are abelian; the second statement follows directly from the construction of $\text{Ind}(F)$; the third statement is [KS06, Prop. 6.1.10], and the fourth statement is [KS06, Cor. 8.6.8]. Finally, the compatibility with composition follows directly from the construction. □
Similarly, we need to extend certain natural transformations to the level of ind-categories. This can be
done as follows:

**Proposition 5.4.** Let $\mathcal{A}_1$ and $\mathcal{A}_2$ be two abelian categories. Then:

(1) Given two functors $F, G : \mathcal{A}_1 \to \mathcal{A}_2$, and a natural transformation $\alpha : F \to G$, the following hold:

(i) There exists a unique natural transformation of functors $\mathrm{Ind}\mathcal{A}_1 \to \mathrm{Ind}\mathcal{A}_2$

$$ \mathrm{Ind}(\alpha) : \mathrm{Ind}(F) \to \mathrm{Ind}(G) $$

such that, for every object $A = \lim_{\to i \in I} A_i$ of $\mathrm{Ind}\mathcal{A}_1$, the diagram in $\mathrm{Ind}\mathcal{A}_2$

$$ (81) \qquad \begin{tikzcd} \lim_{\to i \in I} F(A_i) \arrow[r, "\lim_{\to i \in I} \alpha(A_i)"'] & \lim_{\to i \in I} G(A_i) \arrow[d, "\sim"] \\ \mathrm{Ind}(F)(\lim_{\to i \in I} A_i) \arrow[r, "\mathrm{Ind}(\alpha)(A)"'] & \mathrm{Ind}(G)(\lim_{\to i \in I} A_i) \arrow[d, "\sim"] \end{tikzcd} $$

is commutative.

(ii) The natural transformation $\mathrm{Ind}(\alpha)$ is invertible if and only if $\alpha$ is invertible.

(2) Given three functors $F, G, H : \mathcal{A}_1 \to \mathcal{A}_2$, for every choice of natural transformations $\alpha : F \to G$ and $\beta : G \to H$, we have the equality

$$ \mathrm{Ind}(\beta \circ \alpha) = \mathrm{Ind}(\beta) \circ \mathrm{Ind}(\alpha). $$

*Proof.* In the first statement, (i) follows from the fact that the commutativity of the diagram (81) uniquely determines $\mathrm{Ind}(\alpha)$, while (ii) is a consequence of the second statement of the proposition; the latter follows from the explicit definition of $\mathrm{Ind}(\alpha)$ and $\mathrm{Ind}(\beta)$ via commutative diagrams of the form (81). $\square$

The main case of interest for us concerns the unit and co-unit transformations associated to an adjoint
pair, in which case we can apply the following criterion:

**Corollary 5.5.** Let $\mathcal{A}_1$ and $\mathcal{A}_2$ be abelian categories, and let

$$ F : \mathcal{A}_1 \stackrel{\sim}{\to} \mathcal{A}_2 : G $$

be an adjoint pair of functors. Then the functors $\mathrm{Ind}(F)$ and $\mathrm{Ind}(G)$ fit into an adjunction

$$ \mathrm{Ind}(F) : \mathrm{Ind}\mathcal{A}_1 \stackrel{\sim}{\to} \mathrm{Ind}\mathcal{A}_2 : \mathrm{Ind}(G) $$

compatible with the original one.

*Proof.* Indeed, one can use Proposition 5.4 to extend the unit and co-unit transformations to the level of ind-categories, as well as to check that the extended natural transformations still satisfy the triangular identities of adjunctions. $\square$

We also need to extend some short exact sequences of exact functors to the level of ind-categories. To
this end, we note the following result:

**Lemma 5.6.** Let $\mathcal{A}_1$ and $\mathcal{A}_2$ be essentially small abelian categories. Then, for every short exact sequence of functors $\mathcal{A}_1 \to \mathcal{A}_2$

$$ 0 \to F' \to F \to F'' \to 0, $$

the induced sequence of functors $\mathrm{Ind}\mathcal{A}_1 \to \mathrm{Ind}\mathcal{A}_2$

$$ 0 \to \mathrm{Ind}(F') \to \mathrm{Ind}(F) \to \mathrm{Ind}(F'') \to 0 $$

is exact as well.

*Proof.* Since $\mathrm{Ind}\mathcal{A}_2$ is a Grothendieck abelian category by Proposition 5.2(1), the thesis follows from the fact that filtered colimits of exact sequences in Grothendieck abelian categories are exact by definition. $\square$
5.2. **Indization of perverse motives.** In order to construct motivic internal homomorphisms, we first need to extend part of the functoriality of the stable homotopy 2-functor $D^b(\mathcal{M}(\cdot))$ to the assignment $S \mapsto D(\mathrm{Ind.}\mathcal{M}(S))$.

Specifically, we are interested in direct and inverse images under open and closed immersions. By Proposition 5.3, for every closed immersion $i: Z \hookrightarrow S$, the fully faithful exact functor $i_*: \mathcal{M}(Z) \to \mathcal{M}(S)$ extends to a fully faithful exact functor

$$i_* : \mathrm{Ind.}\mathcal{M}(Z) \to \mathrm{Ind.}\mathcal{M}(S).$$

Similarly, for every open immersion $j: U \hookrightarrow S$, the exact functor $j^*: \mathcal{M}(S) \to \mathcal{M}(U)$ extends to an exact functor

$$j^* : \mathrm{Ind.}\mathcal{M}(S) \to \mathrm{Ind.}\mathcal{M}(U).$$

Moreover, if $j$ is affine, the fully faithful exact functors $j_!: \mathcal{M}(U) \to \mathcal{M}(S)$ extend to fully faithful exact functors

$$j_!: j_* : \mathrm{Ind.}\mathcal{M}(U) \to \mathrm{Ind.}\mathcal{M}(S)$$

and, using Corollary 5.5, we obtain canonical adjunction

$$j_!: \mathrm{Ind.}\mathcal{M}(U) \rightleftharpoons \mathrm{Ind.}\mathcal{M}(S) : j^*, \quad j^*: \mathrm{Ind.}\mathcal{M}(S) \rightleftharpoons \mathrm{Ind.}\mathcal{M}(U) : j_*.$$

The result of [IM19, Prop. 4.2] can be generalized to the setting of ind-categories as follows:

**Proposition 5.7.** For every closed immersion $i: Z \hookrightarrow S$, the triangulated functor $i_*: D(\mathrm{Ind.}\mathcal{M}(Z)) \to D(\mathrm{Ind.}\mathcal{M}(S))$ admits a canonical left adjoint

$$i^*: D(\mathrm{Ind.}\mathcal{M}(S)) \to D(\mathrm{Ind.}\mathcal{M}(Z))$$

sending $D^b(\mathcal{M}(S))$ to $D^b(\mathcal{M}(Z))$.

Following the argument for the proof of [IM19, Prop. 4.2], we start by establishing an auxiliary result:

**Lemma 5.8.** Given a closed immersion $i: Z \hookrightarrow S$ with complementary open immersion $j: U \hookrightarrow S$, the triangulated functor $i_*: D(\mathrm{Ind.}\mathcal{M}(Z)) \to D(\mathrm{Ind.}\mathcal{M}(S))$ is fully faithful and induces an equivalence

$$D(\mathrm{Ind.}\mathcal{M}(Z)) \xrightarrow{\sim} D_Z(\mathrm{Ind.}\mathcal{M}(S)) := \ker \{j^*: D(\mathrm{Ind.}\mathcal{M}(S)) \to D(\mathrm{Ind.}\mathcal{M}(U))\}.$$

*Proof.* As in the proof of [IM19, Thm. 4.1], one reduces formally to the case where $S \in \mathrm{Aff}_k$ and $Z = V(f)$ for some $f \in \mathcal{O}(S)$. In this case, consider the exact functor

$$\Phi_f : \mathrm{Ind.}\mathcal{M}(S) \to \mathrm{Ind.}\mathcal{M}(Z).$$

We claim that the restriction to $D_Z(\mathrm{Ind.}\mathcal{M}(S))$ of its trivial derived functor

$$\Phi_f : D(\mathrm{Ind.}\mathcal{M}(S)) \to D(\mathrm{Ind.}\mathcal{M}(Z))$$

provides a quasi-inverse to $i_*$. The natural isomorphism of functors $D(\mathrm{Ind.}\mathcal{M}(Z)) \to D(\mathrm{Ind.}\mathcal{M}(Z))$

$$\Phi_f(i_*M^\bullet) = M^\bullet$$

is induced by the natural isomorphism (20) after passing to the ind-completions and deriving. The natural isomorphisms of functors $D_Z(\mathrm{Ind.}\mathcal{M}(S)) \to D_Z(\mathrm{Ind.}\mathcal{M}(S))$

$$i_* \Phi_f(M^\bullet) = M^\bullet$$

is obtained from the distinguished triangles of functors $D(\mathrm{Ind.}\mathcal{M}(S)) \to D(\mathrm{Ind.}\mathcal{M}(S))$

$$i_* \Psi_f (j^* M^\bullet) \to \Omega_f (M^\bullet) \to M^\bullet \xrightarrow{\pm} j_* j^* M^\bullet \to \Omega_f (M^\bullet) \to i_* \Phi_f (M^\bullet) \xrightarrow{\pm}$$

induced from the short exact sequences of exact functors $\mathrm{Ind.}\mathcal{M}(S) \to \mathrm{Ind.}\mathcal{M}(S)$ extending (15) and (16), taking into account Lemma 5.6 above. $\square$
Proof of Proposition 5.7. We have to show that the functor $i_* : D(\mathrm{Ind.}\mathcal{M}(Z)) \to D(\mathrm{Ind.}\mathcal{M}(S))$ admits a left adjoint sending $D^b(\mathcal{M}(S))$ to $D^b(\mathcal{M}(Z))$. As a consequence of Lemma 5.8, this is equivalent to showing that the inclusion functor $D_Z(\mathrm{Ind.}\mathcal{M}(S)) \subset D(\mathrm{Ind.}\mathcal{M}(S))$ admits a left adjoint sending $D^b(\mathcal{M}(S))$ to $D_Z^b(\mathcal{M}(S))$. The latter can be constructed using Čech complexes in the same way as in the proof of [IM19, Prop. 4.2]; see also Construction 3.7. $\square$

Remark 5.9. (1) For every pair of composable closed immersions $z' : Z' \hookrightarrow Z$ and $z : Z \hookrightarrow S$ in $\mathrm{Var}_k$, the natural isomorphism of exact functors $\mathcal{M}(Z') \to \mathcal{M}(Z)$

$$ \overline{\mathrm{conn}}_{z',z} : (zz')_* M \xrightarrow{\sim} z_* z'_* M $$

induces an analogous isomorphism of the corresponding exact functors $\mathrm{Ind.}\mathcal{M}(Z') \to \mathrm{Ind.}\mathcal{M}(S)$ via Proposition 5.4, whence an analogous natural isomorphism of their trivial derived functors $D(\mathrm{Ind.}\mathcal{M}(Z')) \to D(\mathrm{Ind.}\mathcal{M}(S))$. By adjunction, we obtain a canonical isomorphism of functors $D(\mathrm{Ind.}\mathcal{M}(S)) \to D(\mathrm{Ind.}\mathcal{M}(Z'))$

$$ \mathrm{conn}_{z',z} : (zz')^* M^\bullet \xrightarrow{\sim} z'^* z^* M^\bullet. $$

(2) Consider a Cartesian square in $\mathrm{Var}_k$

$$ \begin{array}{ccc} V_Z & \xrightarrow{z_V} & V \\ \big|_{i_Z} & & \big|_i \\ Z & \xrightarrow{z} & S \end{array} $$

where both $z$ and $i$ are closed immersions. Then the natural transformation of functors $D(\mathrm{Ind.}\mathcal{M}(V)) \to D(\mathrm{Ind.}\mathcal{M}(Z))$

$$ z^* i_* M^\bullet \xrightarrow{\eta} z^* i_{Z,*} z_V^* M^\bullet = z^* z_* i_{Z,*} z_V^* M^\bullet \xrightarrow{\epsilon} i_{Z,*} z_V^* M^\bullet $$

is invertible. This is equivalent to the invertibility of the unit arrow, which can be shown via the associated localization triangle; the existence of the latter follows directly from the construction of [IM19, Prop. 4.2].

(3) Consider a Cartesian square in $\mathrm{Var}_k$

$$ \begin{array}{ccc} V_Z & \xrightarrow{z_V} & V \\ \big|_{v_Z} & & \big|_v \\ Z & \xrightarrow{z} & S \end{array} $$

where $z$ is a closed immersion and $v$ is an affine open immersion. Since the natural transformation of exact functors $\mathcal{M}(Z) \to \mathcal{M}(V)$

$$ v^* z_* M \xrightarrow{\eta} z_{V,*} z_V^* v^* z_* M = z_{V,*} v_Z^* z^* z_* M \xrightarrow{\epsilon} z_{V,*} v_Z^* M $$

is invertible by proper base-change, by Proposition 5.4 the same is true for the induced natural transformation of exact functors $\mathrm{Ind.}\mathcal{M}(Z) \to \mathrm{Ind.}\mathcal{M}(V)$, and hence also for the natural transformation of their trivial derived functors $D(\mathrm{Ind.}\mathcal{M}(Z)) \to D(\mathrm{Ind.}\mathcal{M}(V))$. We deduce that the corresponding natural transformation between left adjoints $D(\mathrm{Ind.}\mathcal{M}(V)) \to D(\mathrm{Ind.}\mathcal{M}(Z))$

$$ v_{Z, !} z_V^* M^\bullet \xrightarrow{\eta} v_{Z, !} z_V^* v^* v_! M^\bullet = v_{Z, !} v_Z^* z^* v_! M^\bullet \xrightarrow{\epsilon} z^* v_! M^\bullet $$

is invertible as well. Moreover, it is clear from the construction that the natural transformation of functors $D(\mathrm{Ind.}\mathcal{M}(S)) \to D(\mathrm{Ind.}\mathcal{M}(V_Z))$

$$ z_V^* v^* M^\bullet \xrightarrow{\eta} z_V^* v^* z_* z^* M^\bullet \xrightarrow{\sim} z_V^* z_{V,*} v_Z^* z^* M^\bullet \xrightarrow{\epsilon} v_Z^* z^* M^\bullet $$
is invertible.

(4) In fact, following [IM19, § 4.2], one could turn the assignment $S \mapsto D(\mathrm{Ind.}\mathcal{M}(S))$ into a $\mathrm{Var}_k$-fibered category. We omit the details.

**5.3. Abstract construction of internal homomorphisms.** After the above preliminaries, we are ready to describe our abstrac construction of internal homomorphisms on the derived ind-categories of perverse motives. Here are the details:

**Construction 5.10.** Fix $S \in \mathrm{Var}_k$. For every object $M^\bullet \in C^b(\mathcal{M}^0(S))$, we construct a canonical triangulated functor

$$ (82) \qquad - \otimes M^\bullet : D(\mathrm{Ind.}\mathcal{M}^0(S)) \to D(\mathrm{Ind.}\mathcal{M}^0(S)) $$

extending the functor $- \otimes M^\bullet : D^b(\mathcal{M}^0(S)) \to D^b(\mathcal{M}^0(S))$. Since the latter is, by definition, the composite

$$ D^b(\mathcal{M}^0(S)) \xrightarrow{-\boxtimes M^\bullet} D^b(\mathcal{M}^0(S \times S)) \xrightarrow{\Delta_S^*} D^b(\mathcal{M}^0(S)), $$

the task reduces to constructing the two single extensions

$$ (83) \qquad - \boxtimes M^\bullet : D(\mathrm{Ind.}\mathcal{M}^0(S)) \to D(\mathrm{Ind.}\mathcal{M}^0(S \times S)) $$

and

$$ (84) \qquad \Delta_S^* : D(\mathrm{Ind.}\mathcal{M}^0(S \times S)) \to D(\mathrm{Ind.}\mathcal{M}^0(S)) $$

and, since the second extension has already been defined in Proposition 5.7, it remains to construct the first one. To this end, choose $a, b \in \mathbb{Z}$ such that $M^\bullet \in C^{[a,b]}(\mathcal{M}^0(S))$. Recall from [KS06, Lemma 15.4.1] that we have a canonical identification

$$ \mathrm{Ind}C^{[a,b]}(\mathcal{M}^0(S \times S)) = C^{[a,b]}(\mathrm{Ind.}\mathcal{M}^0(S \times S)). $$

Consider the exact functor

$$ - \boxtimes M^\bullet : \mathrm{Ind.}\mathcal{M}^0(S) \to \mathrm{Ind}C^{[a,b]}(\mathcal{M}^0(S \times S)) = C^{[a,b]}(\mathrm{Ind.}\mathcal{M}^0(S \times S)) $$

extending the functor $- \boxtimes M^\bullet : \mathcal{M}^0(S) \to C^{[a,b]}(\mathcal{M}^0(S \times S))$. By construction, the induced functor

$$ - \boxtimes M^\bullet : D(\mathrm{Ind.}\mathcal{M}^0(S)) \to D(C^{[a,b]}(\mathrm{Ind.}\mathcal{M}^0(S \times S))) \xrightarrow{\mathrm{Tot}^\bullet} D(\mathrm{Ind.}\mathcal{M}^0(S \times S)) $$

does not depend on the choice of the integers $a, b$ and defines the sought-after extension (83). Using the independence from $a, b$ of the construction for each single $M^\bullet$, one can easily see that the construction of the functor (82) is covariantly functorial with respect to $M^\bullet \in D^b(\mathcal{M}^0(S))$.

Note that both functors (83) and (84) commute with small direct sums: for the first one, this follows from the construction (as the functor $\mathrm{Tot}^\bullet$ commutes with small direct sums); for the second one, this follows from the fact that it is a left adjoint. Therefore the functor (82) commutes with small direct sums as well. Since, by [Fra01, Thm. 3.1] the category $D(\mathrm{Ind.}\mathcal{M}^0(S))$ satisfies Brown representability, we deduce that, for every fixed $B^\bullet \in D(\mathrm{Ind.}\mathcal{M}^0(S))$, the functor

$$ D(\mathrm{Ind.}\mathcal{M}^0(S)) \to D(\mathrm{Ind.}\mathcal{M}^0(S)), \quad A^\bullet \rightsquigarrow \mathrm{Hom}_{D(\mathrm{Ind.}\mathcal{M}^0(S))}(A^\bullet \otimes M^\bullet, B^\bullet) $$

is representable by an object $\mathrm{Hom}_S(M^\bullet, B^\bullet) \in D(\mathrm{Ind.}\mathcal{M}^0(S))$. By construction, the assignment $B^\bullet \rightsquigarrow \mathrm{Hom}_S(M^\bullet, B^\bullet)$ is functorial in $B^\bullet \in D(\mathrm{Ind.}\mathcal{M}^0(S))$, and so defines a functor

$$ (85) \qquad \mathrm{Hom}_S(M^\bullet, -) : D(\mathrm{Ind.}\mathcal{M}^0(S)) \to D(\mathrm{Ind.}\mathcal{M}^0(S)) $$

which is a triangulated right adjoint to (82). Since, as already noted above, the construction of (82) is covariantly functorial in $M^\bullet \in D^b(\mathcal{M}^0(S))$, we deduce that the construction of (85) is contravariantly functorial in $M^\bullet \in D^b(\mathcal{M}^0(S))$.
Note that it is not clear a priori whether (85) also sends $D^b(\mathcal{M}^0(S))$ to itself; in fact, showing this is one of the main points in the proof of Theorem 5.1.

In order to understand the abstract internal homomorphisms functors just defined better, we now study their compatibility with open and closed immersions.

**Construction 5.11.** For every open immersion $j: U \hookrightarrow S$, we have a canonical natural isomorphism of functors $D(\mathrm{Ind.}\mathcal{M}^0(S)) \times D^b(\mathcal{M}^0(S)) \to D(\mathrm{Ind.}\mathcal{M}^0(U))$

$$ \tilde{m} = \tilde{m}_j : j^*A^\bullet \otimes j^*M^\bullet \xrightarrow{\sim} j^*(A^\bullet \otimes M^\bullet) $$

extending the given isomorphism of functors $D^b(\mathcal{M}^0(S)) \times D^b(\mathcal{M}^0(S)) \to D^b(\mathcal{M}^0(S))$. It is obtained as the composite

$$ j^*A^\bullet \otimes j^*M^\bullet := \Delta_U^*(j^*A^\bullet \boxtimes j^*M^\bullet) \xrightarrow{\tilde{m}} \Delta_U^*(j \times j)^*(A^\bullet \boxtimes M^\bullet) = j^*\Delta_U^*(A^\bullet \boxtimes M^\bullet) =: j^*(A^\bullet \otimes M^\bullet) $$

where the first arrow is obtained from the natural isomorphism of bi-exact functors $\mathcal{M}^0(S) \times \mathcal{M}^0(S) \to \mathcal{M}^0(U \times U)$

$$ \tilde{m}_{j,j} : j^*A^\bullet \boxtimes j^*M^\bullet \xrightarrow{\sim} (j \times j)^*(A^\bullet \boxtimes M^\bullet) $$

after passing to the ind-completion in the first variable and then to the derived categories, while the second arrow is described in Remark 5.9(3).

Similarly, for every closed immersion $i: Z \hookrightarrow S$, we have a canonical natural isomorphism of functors $D(\mathrm{Ind.}\mathcal{M}^0(Z)) \times D^b(\mathcal{M}^0(Z)) \to D(\mathrm{Ind.}\mathcal{M}^0(S))$

$$ \tilde{m} = \tilde{m}_i : i_*A^\bullet \otimes i_*M^\bullet \xrightarrow{\sim} i_*(A^\bullet \otimes M^\bullet) $$

extending the given isomorphism of functors $D^b(\mathcal{M}^0(Z)) \times D^b(\mathcal{M}^0(Z)) \to D^b(\mathcal{M}^0(S))$. It is obtained as the composite

$$ i_*A^\bullet \otimes i_*M^\bullet = \Delta_S^*(i_*A^\bullet \boxtimes i_*M^\bullet) \xrightarrow{\tilde{m}} \Delta_S^*(i \times i)_*(A^\bullet \boxtimes M^\bullet) \xrightarrow{\sim} i_*\Delta_Z^*(A^\bullet \boxtimes M^\bullet) =: i_*(A^\bullet \otimes M^\bullet). $$

where the first arrow is obtained from the natural isomorphism of bi-exact functors $\mathcal{M}^0(Z) \times \mathcal{M}^0(Z) \to \mathcal{M}^0(S \times S)$

$$ \tilde{m}_{i,i} : i_*A \boxtimes i_*M \xrightarrow{\sim} (i \times i)_*(A \boxtimes M) $$

after passing to the ind-completion in the first variable and then to the derived categories, while the second arrow is described in Remark 5.9(2).

**Lemma 5.12.** Let $j: U \hookrightarrow S$ be an affine open immersion, and let $i: Z \hookrightarrow S$ denote the complementary closed immersion. Then the natural transformations of functors $D(\mathrm{Ind.}\mathcal{M}^0(S)) \times D^b(\mathcal{M}^0(S)) \to D(\mathrm{Ind.}\mathcal{M}^0(S))$

$$ (86) \qquad j_!(j^*A^\bullet \otimes j^*M^\bullet) \xrightarrow{\eta} j_!(j^*A \otimes j^*j_!j^*M) \xrightarrow{\tilde{m}} j_!j^*(A \otimes j_!j^*M) \xrightarrow{\epsilon} A^\bullet \otimes j_!j^*M^\bullet $$

and

$$ (87) \qquad A^\bullet \otimes i_*i^*M^\bullet \xrightarrow{\eta} i_*i^*A^\bullet \otimes i_*i^*M^\bullet \xrightarrow{\tilde{m}} i_*(i^*A^\bullet \otimes i^*M^\bullet) $$

are invertible.

*Proof.* Showing that the natural transformation (86) is invertible is equivalent to showing the invertibility of the co-unit transformation

$$ \epsilon : j_!j^*(A^\bullet \otimes j_!j^*M^\bullet) \to A^\bullet \otimes j_!j^*M^\bullet. $$
*Proof.* We have to show that, for every $S \in \mathrm{Var}_k$ and every $M^\bullet \in D^b(\mathcal{M}^0(S))$, the functor

$$ \underline{\mathrm{Hom}}_S(M^\bullet, -) : D(\mathrm{Ind.}\,\mathcal{M}^0(S)) \to D(\mathrm{Ind.}\,\mathcal{M}^0(S)) $$

sends $D^b(\mathcal{M}^0(S))$ to itself. We argue by Noetherian induction on $S$. If $\dim(S) = 0$, the result holds by hypothesis if $M^\bullet$ is concentrated in degree 0; the general case follows immediately by dévissage, since $D^b(\mathcal{M}^0(S))$ is a triangulated subcategory of $D(\mathrm{Ind.}\,\mathcal{M}^0(S))$ and the functor (88) is triangulated in the first variable.

For the inductive step, suppose that $\dim(S) > 0$ and that the result is known to hold for all proper closed subvarieties of $S$. As in the previous paragraph, we reduce immediately to the case where the complex $M^\bullet$ is concentrated in degree 0, so that we may identify it with an object $M \in \mathcal{M}^0(S)$. In this case, choose a dense affine open immersion $j : U \hookrightarrow S$ satisfying the hypothesis in the statement; let $i : Z \hookrightarrow S$ denote the complementary closed immersion. For every $B^\bullet \in D^b(\mathcal{M}^0(S))$, applying the functor $\underline{\mathrm{Hom}}_S(-, B^\bullet)$ to the localization triangle

$$ j_!j^*M \to M \to i_*i^*M \xrightarrow{+} $$

we obtain the distinguished triangle in $D(\mathrm{Ind.}\,\mathcal{M}^0(S))$

$$ \underline{\mathrm{Hom}}_S(i_*i^*M, B^\bullet) \to \underline{\mathrm{Hom}}_S(M, B^\bullet) \to \underline{\mathrm{Hom}}_S(j_!j^*M, B^\bullet) \xrightarrow{+} $$

Since $D^b(\mathcal{M}^0(S))$ is a triangulated subcategory of $D(\mathrm{Ind.}\,\mathcal{M}^0(S))$, it suffices to show that the two lateral terms in the latter triangle belong to $D^b(\mathcal{M}^0(S))$ in order to conclude the same for the term in the middle. By Corollary 5.13 we have canonical isomorphisms

$$ \underline{\mathrm{Hom}}_S(i_*i^*M, B^\bullet) = i_*\underline{\mathrm{Hom}}_Z(i^*M, i^!B^\bullet), \quad \underline{\mathrm{Hom}}_S(j_!j^*M, B^\bullet) = j_*\underline{\mathrm{Hom}}_U(j^*M, j^!B^\bullet). $$

Since we know that $i^*M, i^!B^\bullet \in D^b(\mathcal{M}^0(S))$, applying the inductive hypothesis we deduce that $\underline{\mathrm{Hom}}_Z(i^*M, i^!B^\bullet) \in D^b(\mathcal{M}^0(Z))$, and therefore that $i_*\underline{\mathrm{Hom}}_Z(i^*M, i^!B^\bullet) \in D^b(\mathcal{M}^0(S))$. Similarly, since we know that $j^!B^\bullet \in D^b(\mathcal{M}^0(U))$, applying the hypothesis on $j^*M$ we deduce that $\underline{\mathrm{Hom}}_U(j^*M, j^!B^\bullet) \in D^b(\mathcal{M}^0(U))$. This concludes the proof. $\square$

In the second place, we need a criterion ensuring that the monoidal morphism $\iota : D^b(\mathcal{M}^0(\cdot)) \to D^b(\mathrm{Perv}(\cdot))$ is compatible with internal homomorphisms.

**Proposition 5.15.** Suppose that the hypotheses (and hence the conclusions) of Proposition 5.14 are satisfied. In addition, suppose that, for every $S \in \mathrm{Var}_k$ and every $M \in \mathcal{M}^0(S)$, there exists a dense affine open immersion $j : U \hookrightarrow S$ such that the canonical natural transformation of functors $D^b(\mathcal{M}^0(U)) \to D^b(\mathrm{Perv}(U))$

$$ \iota_U \underline{\mathrm{Hom}}_U(j^*M, B^\bullet) \to \underline{\mathrm{Hom}}_S(\iota_U(j^*M), \iota_U(B^\bullet)) $$

is invertible. Then, for every $S \in \mathrm{Var}_k$, the canonical natural transformation of functors $D^b(\mathcal{M}^0(S)) \to D^b(\mathrm{Perv}(S))$

$$ \iota_S \underline{\mathrm{Hom}}_S(M^\bullet, B^\bullet) \to \underline{\mathrm{Hom}}_S(\iota_S(M^\bullet), \iota_S(B^\bullet)) $$

is invertible.

As a preparation for the proof of Proposition 5.15, we need to describe the relations among the various natural isomorphisms witnessing the compatibility of internal homomorphisms with open and closed immersions. For simplicity, we state these auxiliary results in the setting of (closed) monoidal stable homotopy 2-functors of [Ayo07a, § 2.3].

**Lemma 5.16.** Let $R : \mathbb{H}_1 \to \mathbb{H}_2$ be a monoidal morphism of monoidal stable homotopy 2-functors. Then the following assertions hold:
(1) For every closed immersion $i: Z \hookrightarrow S$, and every $M \in \mathbb{H}_1(Z)$, the diagram of functors $\mathbb{H}_1(S) \to \mathbb{H}_2(S)$

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

is commutative.

(2) For every open immersion $j: U \hookrightarrow S$, and every $M \in \mathbb{H}_1(U)$, the diagram of functors $\mathbb{H}_1(S) \to \mathbb{H}_2(S)$

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

is commutative.

*Proof.* We only prove the second assertion; the first one can be proved in a similar way. Developing the definitions of the two vertical isomorphisms in the second diagram, we obtain the more explicit diagram

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

By inserting, on the top side of the latter, the diagram

<figure><img src="image_18.png" /></figure>
and, on its bottom side, the diagram

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

we are reduced to considering the outer part of the diagram

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

where the two lateral pieces are commutative by naturality while the central rectangle is commutative by definition. $\square$

**Corollary 5.17.** Let $R: \mathbb{H}_1 \to \mathbb{H}_2$ be a monoidal morphism of monoidal stable homotopy 2-functors. Then the following assertions hold:

(1) For every closed immersion $i: Z \hookrightarrow S$, the diagram of functors $\mathbb{H}_1(S) \to \mathbb{H}_2(S)$

<figure><img src="image_21.png" /></figure>

is commutative.

(2) For every open immersion $j: U \hookrightarrow S$, and every $K \in \mathbb{H}_1(U)$ the diagram of functors $\mathbb{H}_1(S) \to \mathbb{H}_2(S)$

<figure><img src="image_22.png" /></figure>

is commutative.

*Proof.* By adapting the argument in the proof of [Ayo07a, Prop. 2.3.57], one sees that the two statements are formal consequences of the corresponding statements of Lemma 5.16. We omit the details. $\square$
Proof of Proposition 5.15. We have to show that, for every $S \in \mathrm{Var}_k$ and every $M^\bullet \in D^b(\mathcal{M}^0(S))$, the
natural transformation of functors $D^b(\mathcal{M}^0(S)) \to D^b(\mathrm{Perv}(S))$

$$
\iota_S \mathrm{Hom}_S(M^\bullet, B^\bullet) \to \mathrm{Hom}_S(\iota_S(M^\bullet), \iota_S(B^\bullet))
$$

is invertible. We argue by Noetherian induction on $S$. If $\dim(S) = 0$, the result holds by hypothesis if $M^\bullet$
is concentrated in degree 0; the general case follows immediately by dévissage.

For the inductive step, assume that $\dim(S) > 0$ and that the result is known to hold for all proper
closed subvarieties of $S$. As in the previous paragraph, we may reduce immediately to the case where the
complex $M^\bullet$ is concentrated in degree 0, so that we may identify it with an object $M \in \mathcal{M}^0(S)$. In this
case, choose a Zariski-dense affine open immersion $j : U \hookrightarrow S$ satisfying the hypothesis in the statement;
let $i : Z \hookrightarrow S$ denote the complementary closed immersion. For every $B^\bullet \in D^b(\mathcal{M}^0(S))$, we obtain the
morphism of distinguished triangles in $D^b(\mathcal{M}^0(S))$

$$
\begin{equation} \tag{89}
\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
  \iota_S \Hom_S(i_* i^* M, B^\bullet) \arrow[r] \arrow[d] & \iota_S \Hom_S(M, B^\bullet) \arrow[r] \arrow[d] & \iota_S \Hom_S(j_* j^* M, B^\bullet) \arrow[r, "+1"] \arrow[d] \\
  \Hom_S(\iota_S(i_* i^* M), \iota_S(B^\bullet)) \arrow[r] \arrow[d] & \Hom_S(\iota_S(M), \iota_S(B^\bullet)) \arrow[r] \arrow[d] & \Hom_S(\iota_S(j_* j^* M), \iota_S(B^\bullet)) \arrow[r, "+1"] \arrow[d]
\end{tikzcd}
\end{equation}
$$

In order to show that the central vertical arrow is an isomorphism, it suffices to show that so are the two
lateral vertical arrows. But the two arrows in question are the two marked arrows in the diagrams in
$D^b(\mathrm{Perv}(S))$

$$
\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
  \iota_S i_* \underbrace{\operatorname{Hom}_Z(i^*M, i^!B^\bullet)}_{\substack{\uparrow \\ \iota}} \arrow[r] \arrow[d] & i_* \iota_Z \underbrace{\operatorname{Hom}_Z(i^*M, i^!B^\bullet)}_{\substack{\downarrow \\ i_* \iota_Z(i^*M), i^! \iota_S(B^\bullet)}} \arrow[r] \arrow[d] & i_* \underbrace{\operatorname{Hom}_Z(\iota_Z(i^*M), \iota_Z(i^!B^\bullet))}_{\substack{\uparrow \\ \iota}} \\
  \iota_S \underbrace{\operatorname{Hom}_S(i_* i^* M, B^\bullet)}_{\substack{\uparrow \\ \iota}} \arrow[r, "*"'] \arrow[d] & \underbrace{\operatorname{Hom}_S(\iota_S(i_* i^* M), \iota_S(B^\bullet))}_{\substack{\downarrow \\ \iota}} \arrow[r] \arrow[d] & \underbrace{\operatorname{Hom}_S(i_* \iota_Z(i^* M), \iota_S(B^\bullet))}_{\substack{\uparrow \\ \iota}}
\end{tikzcd}
$$

and

$$
\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
  \iota_S j_* \underbrace{\operatorname{Hom}_U(j^*M, j^!B^\bullet)}_{\substack{\uparrow \\ \iota}} \arrow[r] \arrow[d] & j_* \iota_U \underbrace{\operatorname{Hom}_U(j^*M, j^!B^\bullet)}_{\substack{\downarrow \\ j_* \iota_U(j^*M), j^! \iota_S(B^\bullet)}} \arrow[r] \arrow[d] & j_* \underbrace{\operatorname{Hom}_U(\iota_U(j^*M), \iota_U(j^!B^\bullet))}_{\substack{\uparrow \\ \iota}} \\
  \iota_S \underbrace{\operatorname{Hom}_S(j_* j^* M, B^\bullet)}_{\substack{\uparrow \\ \iota}} \arrow[r, "*"'] \arrow[d] & \underbrace{\operatorname{Hom}_S(\iota_S(j_* j^* M), \iota_S(B^\bullet))}_{\substack{\downarrow \\ \iota}} \arrow[r] \arrow[d] & \underbrace{\operatorname{Hom}_S(j_* \iota_U(j^*M), \iota_S(B^\bullet))}_{\substack{\uparrow \\ \iota}}
\end{tikzcd}
$$

respectively. Now, taking into account Remark 5.9(4), we see that the result of Lemma 5.16 applies to
our setting as well, and thus the same holds for Corollary 5.17. Therefore the two natural diagrams above
are commutative. Moreover, in each of them, the vertical arrow in the lower-right corner is invertible by
Corollary 5.13, while the horizontal arrow in the upper-right corner is invertible by assumption. We deduce
that the marked arrow is invertible as well. This concludes the proof. $\square$

Proof of Theorem 5.1. We want to apply the criteria of Proposition 5.14 and Proposition 5.15. To this end, fix an object $M \in \mathcal{M}^0(S)$. Choose a smooth Zariski-dense affine open immersion $j: U \hookrightarrow S$ such that $j^*M \in \mathcal{M}^{0,\mathrm{loc}}(U)$: this is possible since the $\mathrm{SmOp}(S)$-fibered subcategory $\mathcal{M}^{0,\mathrm{loc}}$ of $\mathcal{M}^0$ is weakly cofinal by Lemma 3.3. Up to writing $U$ as the disjoint union of its connected components and treating each component separately, we may assume that $U$ is connected of dimension $d \ge 0$. We divide the proof into two
main steps: first of all, we check that the hypotheses are satisfied if $j^*M$ lies in the image of $\mathbf{R}(\mathcal{D}^{0,loc}(U))$; then, we explain how to deduce the general case. Here are the details:

(Step 1) Suppose first that the object $j^*M$ lies in the image of $\mathbf{R}(\mathcal{D}^{0,loc}(U))$ under the quotient functor $\pi_U : \mathbf{A}(\mathcal{D}^{0,loc}(U)) \to \mathcal{M}^{0,loc}(U)$. In this case, by Corollary 3.13, we have a canonical adjunction of exact functors

$$ (90) \qquad F : \mathcal{M}^0(U) \stackrel{\sim}{\rightleftarrows} \mathcal{M}^0(U) : G, $$

where $F := - \otimes j^*M[-d]$ and $G := \mathbb{D}_U(j^*M[-d] \otimes \mathbb{D}_U(-))$. In order to deduce the thesis, the main point is to make sure that the induced exact functor $\mathrm{Ind}(F) : \mathrm{Ind}(\mathcal{M}^0(U)) \to \mathrm{Ind}(\mathcal{M}^0(U))$ is naturally isomorphic to (the co-restriction of) the functor

$$ - \otimes j^*M[-d] : \mathrm{Ind}(\mathcal{M}^0(U)) \to D(\mathrm{Ind}(\mathcal{M}^0(U))) $$

obtained from Construction 5.10.

In order to establish this claim, we only need to analyze carefully the various constructions involved. Since the functor $\Delta_{U,*} : \mathrm{Ind}(\mathcal{M}^0(U)) \to \mathrm{Ind}(\mathcal{M}^0(U \times U))$ is exact and fully faithful by Proposition 5.3, this is equivalent to saying that the composite functor

$$ \mathrm{Ind}(\mathcal{M}^0(U)) \xrightarrow{\mathrm{Ind}(F)} \mathrm{Ind}(\mathcal{M}^0(U)) \xrightarrow{\Delta_{U,*}} \mathrm{Ind}(\mathcal{M}^0(U \times U)) $$

is naturally isomorphic to (the co-restriction of) the composite functor

$$ \mathrm{Ind}(\mathcal{M}^0(U)) \xrightarrow{-\otimes j^*M[-d]} D(\mathrm{Ind}(\mathcal{M}^0(U))) \xrightarrow{\Delta_{U,*}} D(\mathrm{Ind}(\mathcal{M}^0(U \times U))). $$

By construction, the latter can be written as the composite

$$ \mathrm{Ind}(\mathcal{M}^0(U)) \xrightarrow{-\boxtimes j^*M[-d]} \mathrm{Ind}(\mathcal{M}^0(U \times U)[-d]) \xrightarrow{\Delta_{U,*} \Delta_U^*} D(\mathrm{Ind}(\mathcal{M}^0(U \times U))). $$

Moreover, following the idea of Construction 3.7, the functor $\Delta_{U,*} \Delta_U^*$ : $\mathrm{Ind}(\mathcal{M}^0(U \times U)[-d] \to D(\mathrm{Ind}(\mathcal{M}^0(U \times U)))$ can be written as

$$ \mathrm{Ind}(\mathcal{M}^0(U \times U)[-d]) \xrightarrow{\tilde{C}_{U \times U, \mathcal{U}}^*} C^{[-n, d-n]}(\mathrm{Ind}(\mathcal{M}^0(U \times U))) \to D^{[-n, d-n]}(\mathrm{Ind}(\mathcal{M}^0(U \times U))) \\ \subset D(\mathrm{Ind}(\mathcal{M}^0(U \times U))), $$

where $\mathcal{U}$ denotes some fixed affine open covering of $U \times U \setminus \Delta_U(U)$ with $n$ elements. Now, for every Grothendieck abelian category $\mathcal{A}$, the additive functors $H^i : \mathcal{C}^b(\mathcal{A}) \to \mathcal{A}$ commute with filtered colimits. Since the category $\mathrm{Ind}(\mathcal{M}^0(U \times U))$ is Grothendieck abelian by Proposition 5.2(1), we deduce that the composite functors

$$ \mathrm{Ind}(\mathcal{M}^0(U)) \xrightarrow{-\boxtimes j^*M[-d]} \mathrm{Ind}(\mathcal{M}^0(U \times U)[-d]) \xrightarrow{\tilde{C}_{U \times U, \mathcal{U}}^*} C^{[-n, d-n]}(\mathrm{Ind}(\mathcal{M}^0(U \times U))) \xrightarrow{H^i} \mathrm{Ind}(\mathcal{M}^0(U \times U)) $$

vanish for every $i \neq 0$, and that the composite functor

$$ \mathrm{Ind}(\mathcal{M}^0(U)) \xrightarrow{-\boxtimes j^*M[-d]} \mathrm{Ind}(\mathcal{M}^0(U \times U)[-d]) \xrightarrow{\tilde{C}_{U \times U, \mathcal{U}}^*} C^{[-n, d-n]}(\mathrm{Ind}(\mathcal{M}^0(U \times U))) \xrightarrow{H^0} \mathrm{Ind}(\mathcal{M}^0(U \times U)) $$

commutes with filtered colimits. Hence the two functors considered in the claim both satisfy the conditions of Proposition 5.3(1). By the essential uniqueness in the statement of Proposition 5.3(1), we conclude that they are naturally isomorphic as claimed.

Now we can consider the adjunction

$$ \mathrm{Ind}(F) : D(\mathrm{Ind}(\mathcal{M}^0(U))) \stackrel{\sim}{\rightleftarrows} D(\mathrm{Ind}(\mathcal{M}^0(U))) : \mathrm{Ind}(G) $$
obtained from (90) by applying Corollary 5.5 and then passing to the derived categories. As a con-
sequence of the claim just proved, the triangulated functor Ind($F$) : $D(\text{Ind.}\mathcal{M}^0(U)) \to D(\text{Ind.}\mathcal{M}^0(U))$
is naturally isomorphic to the functor

$$
- \otimes j^* M[-d] : \mathrm{D}(\mathrm{Ind.}\mathcal{M}^0(U)) \to \mathrm{D}(\mathrm{Ind.}\mathcal{M}^0(U))
$$

obtained from Construction 5.10: indeed, since the latter triangulated functor is dg-enhanced by
construction and t-exact for the obvious t-structures by the claim, by Proposition 2.6(1) it is natur-
ally isomorphic to the trivial derived functor of the induced exact functor between the hearts. Taking
the right adjoints, we deduce that the functor Ind($G$) : $D(\text{Ind.}\mathcal{M}^0(U)) \to D(\text{Ind.}\mathcal{M}^0(U))$ is natur-
ally isomorphic to the triangulated functor $\operatorname{Hom}_U(j^*M[-d], -)$ obtained from Construction 5.10.
But Ind($G$) sends $D^b(\mathcal{M}^0(U))$ to itself by construction, and the induced natural transformation of
functors $D^b(\mathcal{M}^0(U)) \to D^b(\operatorname{Perv}(U))$

$$
\iota_U G(B^\bullet) \to \operatorname{Hom}_U(\iota_U(j^*M)[-d], \iota_U(B^\bullet))
$$

is invertible as a consequence of Corollary 3.13. We deduce that the triangulated functor

$$
\underline{\operatorname{Hom}}_U(j^*M, -) = \underline{\operatorname{Hom}}_U(j^*M[-d], -)[-d] : \operatorname{D}(\operatorname{Ind.}\mathcal{M}^0(U)) \to \operatorname{D}(\operatorname{Ind.}\mathcal{M}^0(U))
$$

sends $D^b(\mathcal{M}^0(U))$ to itself, and that the induced natural transformation of functors $D^b(\mathcal{M}^0(U)) \to D^b(\mathrm{Perv}(U))$

$$
\iota_U \underline{\operatorname{Hom}}_U(j^*M, B^\bullet) \to \underline{\operatorname{Hom}}_U(\iota_U(j^*M), \iota_U(B^\bullet))
$$

is invertible. This concludes the proof in the case where $j^*M$ lies in the image of $\mathbf{R}(\mathcal{D}^{0,\mathrm{loc}}(U))$.

(Step 2) For a general choice of $M \in \mathcal{M}^0(S)$, the object $j^*M \in \mathcal{M}^{0,\text{loc}}(U)$ does not necessarily satisfy the previous condition. However, by construction, every object of $\mathbf{A}(\mathcal{D}^{0,\text{loc}}(U))$ can be written as the kernel of some morphism in $\mathbf{R}(\mathcal{D}^{0,\text{loc}}(U))$. Hence we can always choose a co-presentation of the form

$$
j^*M = \ker \{ \pi_U(C_1) \to \pi_U(C_2) \}
$$

for some morphism $C_1 \to C_2$ in $\mathbf{R}(\mathcal{D}^{0,loc}(U))$. Write $N_1 := \pi_U(C_1)$, $N_2 := \pi_U(C_2)$ for convenience, and set $N_3 := \text{coker}\{\pi_U(C_1) \to \pi_U(C_2)\}$. Since the image of $\mathbf{R}(\mathcal{D}^{0,loc}(U))$ in $\mathbf{A}(\mathcal{D}^{0,loc}(U))$ is closed under cokernels (as every cokernel of presheaves of finite presentation is itself of finite presentation), the object $N_3$ belongs to the image of $\mathbf{R}(\mathcal{D}^{0,loc}(U))$. By construction, the object $j^*M$ is canonically isomorphic to the complex $[N_1 \to N_2 \to N_3]$ in $D^{[0,3]}(\mathcal{M}^0(U))$. Moreover, by the previous step of the proof, each of the three functors

$$
\underline{\mathrm{Hom}}_U(N_i, -) : \mathrm{D}(\mathrm{Ind.}\mathcal{M}^0(U)) \to \mathrm{D}(\mathrm{Ind.}\mathcal{M}^0(U)) \quad (i = 1, 2, 3)
$$

sends $D^b(\mathcal{M}^0(U))$ to itself, and each of the three induced natural transformations of functors
$D^b(\mathcal{M}^0(U)) \to D^b(\mathrm{Perv}(U))$

$$
\iota_U \underline{\mathrm{Hom}}_S(N_i, B^\bullet) \to \underline{\mathrm{Hom}}_U(\iota_U(N_i), \iota_U(B^\bullet)) \quad (i = 1, 2, 3)
$$

is invertible. Since both of these properties are triangulated with respect to the first variable, we
conclude that they hold for *M* as well. This concludes the proof.

□

Remark 5.18. As explained in [IM19, § 6], there exists a theory of weights for perverse motives compatible with the analogous notion of weight for perverse sheaves of geometric origin. By construction, the operations $f^*$, $f_*$, $f_!$ and $f^!$ constructed in [IM19] on the level of perverse motives enjoy the same weight-exactness properties as the corresponding operations on perverse sheaves as described in [BBD82, § 5.1.14].

The same observation applies to the tensor product and internal homomorphisms functors constructed
here. Namely, given $S \in \mathrm{Var}_k$ and letting $D^b_{\le w}(\mathcal{M}(S))$ (resp. $D^b_{\ge w}(\mathcal{M}(S))$) denote the full subcategory
of $D^b(\mathcal{M}(S))$ collecting the complexes of weights at most $w$ (resp. at least $w$), we have that the internal tensor product functor restricts to

$$- \otimes - : D_{\le w_1}^b(\mathcal{M}(S)) \times D_{\le w_2}^b(\mathcal{M}(S)) \to D_{w_1+w_2}^b(\mathcal{M}(S)),$$

while the internal homomorphisms functor restricts to

$$\underline{\mathrm{Hom}}_S(-, -) : D_{\le w_1}^b(\mathcal{M}(S)) \times D_{\ge w_2}^b(\mathcal{M}(S)) \to D_{-w_1-w_2}^b(\mathcal{M}(S)).$$

**5.5. Motivic Verdier duality.** We close this section with some useful remarks about Verdier duality for perverse motives. Recall that, for every $S \in \mathrm{Var}_k$, the motivic Verdier duality functor

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

was constructed in [IM19, § 2.8] as an application of the general lifting principles of universal abelian factorizations. The functor $\mathbb{D}_S$ canonically defines an involutive anti-equivalence of $\mathcal{M}(S)$: there exists a canonical natural isomorphism between functors $\mathcal{M}(S) \to \mathcal{M}(S)$

$$M \xrightarrow{\sim} \mathbb{D}_S(\mathbb{D}_S(M))$$

which is induced by the usual bi-duality isomorphisms on $\mathbf{DA}_{ct}^{\mathcal{E}t}(\cdot, \mathbb{Q})$ and on $\mathrm{Perv}(\cdot)$.

What is still lacking at the moment is a precise link between the motivic Verdier duality and the internal homomorphisms of perverse motives: Corollary 3.13 given a partial result in this direction; however, not much else could be said without an additional ingredient that was only recently obtained in Tubach's work [Tub23]. Namely, by [Tub23, Thm. 4.43] there exists a canonical unitary symmetric monoidal morphism of stable homotopy 2-functors

$$(91) \qquad \mathrm{Nri}^* : \mathbf{DA}_{ct}^{\mathcal{E}t}(\cdot, \mathbb{Q}) \to D^b(\mathcal{M}(\cdot)),$$

with the property that the composite morphism

$$\mathbf{DA}_{ct}^{\mathcal{E}t}(\cdot, \mathbb{Q}) \xrightarrow{\mathrm{Nri}^*} D^b(\mathcal{M}(\cdot)) \xrightarrow{\iota} D^b(\mathrm{Perv}(\cdot)) = D_c^b(\cdot, \mathbb{Q})$$

recover the Betti realization $\mathrm{Bti}^* : \mathbf{DA}_{ct}^{\mathcal{E}t}(\cdot, \mathbb{Q}) \to D_c^b(\cdot, \mathbb{Q})$ up to canonical equivalence. The proof is an application of Drew and Gallauer's universality theorem [DG22, Thm. 7.14], which requires to construct an $\infty$-categorical enhancement of the six operations on perverse motives in advance. The construction of such an enhancement is based on a motivic adaptation of Nori's classical argument for the derived category of constructible sheaves from [Nor18].

Using the existence of the morphism (91), we can prove the following result:

**Proposition 5.19.** Fix a variety $S \in \mathrm{Var}_k$ with structural morphism $a_S : S \to \mathrm{Spec}(k)$. Then:

(1) There exists a canonical natural isomorphism of functors $D^b(\mathcal{M}(S)) \to D^b(\mathcal{M}(S))^{op}$

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

(2) There exists a canonical natural isomorphism of functors $D^b(\mathcal{M}(S))^{op} \times D^b(\mathcal{M}(S)) \to D^b(\mathcal{M}(S))$

$$\underline{\mathrm{Hom}}_S(M^\bullet, N^\bullet) \xrightarrow{\sim} \mathbb{D}_S(M^\bullet \otimes \mathbb{D}_S(N^\bullet)).$$

*Proof.* (1) Using the conservativity of $\iota_S : D^b(\mathcal{M}(S)) \to D^b(\mathrm{Perv}(S)) = D_c^b(S, \mathbb{Q})$, one easily sees that the canonical natural transformation of functors $\mathbf{DA}_{ct}^{\mathcal{E}t}(S, \mathbb{Q})^{op} \times \mathbf{DA}_{ct}^{\mathcal{E}t}(S, \mathbb{Q}) \to \mathbf{DA}_{ct}^{\mathcal{E}t}(S, \mathbb{Q})$

$$\mathrm{Nri}_S^* \underline{\mathrm{Hom}}_S(A, B) \to \underline{\mathrm{Hom}}_S(\mathrm{Nri}_S^*(A), \mathrm{Nri}_S^*(B))$$

induced by the monoidality of $\mathrm{Nri}_S^*$ is necessarily invertible, since the same holds true for the Betti realization $\mathrm{Bti}_S^* : \mathbf{DA}_{ct}^{\mathcal{E}t}(S, \mathbb{Q}) \to D_c^b(S, \mathbb{Q})$. In particular, we obtain a canonical isomorphism of functors $\mathbf{DA}_{ct}^{\mathcal{E}t}(S, \mathbb{Q})^{op} \to D^b(\mathcal{M}(S))$

$$\mathrm{Nri}_S^* (\underline{\mathrm{Hom}}_S(A, a_S^! \mathbf{1}_{\mathrm{Spec}(k)})) \xrightarrow{\sim} \underline{\mathrm{Hom}}_S(\mathrm{Nri}_S^*(A), a_S^! \mathbf{1}_{\mathrm{Spec}(k)})$$
which, by construction, satisfies the hypotheses of [Ter23U, Cor. 2.9]. Applying the latter result, we get the thesis.

(2) We have a canonical natural transformation between functors $D^b(\mathcal{M}(S))^{op} \times D^b(\mathcal{M}(S)) \to D^b(\mathcal{M}(S))$

$$ (92) \qquad \underline{\mathrm{Hom}}_S(M^{\bullet}, N^{\bullet}) \to \underline{\mathrm{Hom}}_S(M^{\bullet} \otimes \underline{\mathrm{Hom}}_S(N^{\bullet}, K_S^{\mathcal{M}}), K_S^{\mathcal{M}}) $$

whose value on the pair $(M^{\bullet}, N^{\bullet})$ is the morphism in $D^b(\mathcal{M}(S))$ corresponding under adjunction to the composite

$$ A \otimes \underline{\mathrm{Hom}}_S(B, a_S^! 1_{\mathrm{Spec}(k)}) \otimes \underline{\mathrm{Hom}}_S(A, B) $$

$\|\|$

$$ \underline{\mathrm{Hom}}_S(B, a_S^! 1_{\mathrm{Spec}(k)}) \otimes A \otimes \underline{\mathrm{Hom}}_S(A, B) $$

$\|\|$

$$ \underline{\mathrm{Hom}}_S(B, a_S^! 1_{\mathrm{Spec}(k)}) \otimes \underline{\mathrm{Hom}}_S(A, B) \otimes A \xrightarrow{\mathrm{ev}} \underline{\mathrm{Hom}}_S(B, a_S^! 1_{\mathrm{Spec}(k)}) \otimes B \xrightarrow{\mathrm{ev}} a_S^! 1_{\mathrm{Spec}(k)}. $$

Since the analogous natural transformation on $D^b(\mathrm{Perv}(\cdot))$ is known to be invertible, it follows from Theorem 5.1 that (92) is invertible as well. Combining this with the previous point, we obtain the result. $\square$

*Remark 5.20.* Clearly, the natural isomorphism of Proposition 5.19 extends the one constructed in Corollary 3.13.

## 6. MOTIVIC LOCAL SYSTEMS

In this short final section, we define Tannakian categories of motivic local systems over smooth, geometrically connected varieties and we establish their basic properties. As an application, we show that the derived categories of perverse motives satisfy Gallauer’s axiomatic of *constructible systems* from [Gal22] and we deduce a topological reconstruction theorem for perverse motives.

**6.1. Tannakian categories of motivic local systems.** As explained in [HMS17, § II.9], the abelian category $\mathcal{M}(k)$ of $\mathbb{Q}$-linear Nori motives over $k$ is a neutral Tannakian category over $\mathbb{Q}$ with fibre functor given by the canonical faithful exact functor $\iota_k : \mathcal{M}(k) \hookrightarrow \mathrm{vect}_{\mathbb{Q}}$. The Tannaka dual of $\mathcal{M}(k)$ is a pro-affine algebraic group $\mathcal{G}_{\mathrm{mot}}(k)$ called the *motivic Galois group* of $k$.

In the setting of motivic sheaves, a candidate category of motivic local system over general bases was proposed by Arapura in [Ara13, § 6] within his own theory of constructible motivic sheaves; see also [Ara19, § 2]. While internal homomorphisms are not available in full generality in Arapura’s theory, one can still show that these categories of motivic local systems are indeed Tannakian by generalizing the approach for Nori motives, where the main ingredients are provided by the Basic Lemma and Poincaré duality for $k$-varieties. A technical defect of Arapura’s construction is that his categories of motivic local systems are not visibly full subcategories inside the corresponding categories of constructible motivic sheaves.

Here we propose an alternative approach that exploits the six functor formalism on perverse motives in its full strength. By definition, our categories of motivic local systems will be full subcategories inside the corresponding categories of perverse motives. The existence of internal homomorphisms on perverse motives allows us to obtain duality as a straightforward consequence of the rigidity of ordinary local systems.

**Definition 6.1.** Let $S \in \mathrm{Var}_k$ be smooth and geometrically connected. We introduce the full abelian subcategory

$$ \mathcal{M}\mathrm{Loc}_p(S) := \{M \in \mathcal{M}(S) \mid \iota_S(M) \in \mathrm{Loc}_p(S)\} \subset \mathcal{M}(S), $$
is commutative. Hence, for every object $M \in \mathcal{M}(S)$, it is natural to define its dual $M^{\vee}$ by the analogous formula

$$M^{\vee} := \underline{\mathrm{Hom}}_{S}(M[-d], \mathbf{1}_{S}^{\mathcal{M}})[d].$$

As a consequence of Theorem 5.1, the latter formula defines a functor $\mathcal{M}\mathrm{Loc}_p(S) \to \mathcal{M}\mathrm{Loc}_p(S)^{op}$ and we have a canonical isomorphisms of functors $\mathcal{M}\mathrm{Loc}_p(S) \to \mathrm{Loc}_p(S)^{op}$

$$\iota_S(M^{\vee}) \xrightarrow{\sim} (\iota_S M)^{\vee}.$$

In order to conclude that the construction $M \rightsquigarrow M^{\vee}$ defines the strong dual in $\mathcal{M}\mathrm{Loc}_p(S)$, we have to check the following two conditions:

(i) The canonical natural transformation of functors $\mathcal{M}\mathrm{Loc}_p(S)^{op} \times \mathcal{M}\mathrm{Loc}_p(S) \to \mathcal{M}\mathrm{Loc}_p(S)$

$$M^{\vee}[-d] \otimes N[-d] := \underline{\mathrm{Hom}}_{S}(M[-d], \mathbf{1}_{S}^{\mathcal{M}}) \otimes N[-d] \to \underline{\mathrm{Hom}}_{S}(M[-d], N[-d])$$

corresponding under adjunction to

$$\begin{array}{ccc}
\underline{\mathrm{Hom}}_{S}(M[-d], \mathbf{1}_{S}^{\mathcal{M}}) & \xrightarrow{N[-d] \otimes -} & \underline{\mathrm{Hom}}_{S}(N[-d] \otimes M[-d], N[-d]) \\
& \parallel & \\
\underline{\mathrm{Hom}}_{S}(N[-d], \underline{\mathrm{Hom}}_{S}(M[-d], N[-d])) & &
\end{array}$$

is invertible.

(ii) The canonical natural transformation of functors $\mathcal{M}\mathrm{Loc}_p(S) \to \mathcal{M}\mathrm{Loc}_p(S)$

$$M \to \underline{\mathrm{Hom}}_{S}(\underline{\mathrm{Hom}}_{S}(M[-d], \mathbf{1}_{S}^{\mathcal{M}})[d)[-d], \mathbf{1}_{S}^{\mathcal{M}})[d] := (M^{\vee})^{\vee}$$

defined as the composite

$$M = M[-d][d] \to \underline{\mathrm{Hom}}_{S}(\underline{\mathrm{Hom}}_{S}(M[-d], \mathbf{1}_{S}^{\mathcal{M}}), \mathbf{1}_{S}^{\mathcal{M}})[d] = \underline{\mathrm{Hom}}_{S}(\underline{\mathrm{Hom}}_{S}(M[-d], \mathbf{1}_{S}^{\mathcal{M}})[d)[-d], \mathbf{1}_{S}^{\mathcal{M}})[d]$$

is invertible.

As usual, the validity of each of these two conditions follows from the validity of the corresponding condition on $\mathrm{Loc}_p(S)$ and the conservativity of $\iota_S : \mathcal{M}\mathrm{Loc}_p(S) \to \mathrm{Loc}_p(S)$. This concludes the proof. $\square$

**Remark 6.3.** The definition of the shifted tensor product in (93) does not coincide with the one given in Section 3. Of course, the two functors in question are naturally isomorphic via any of the two composites in (the shift of) the diagram of functors $\mathcal{M}\mathrm{Loc}_p(S) \times \mathcal{M}\mathrm{Loc}_p(S) \to \mathcal{M}\mathrm{Loc}_p(S)[-d]$

$$M[-d] \otimes N[-d] \xrightarrow{\sim} (M \otimes N[-d])[-d]$$

$\searrow^2$

$\searrow^2$

$$(M[-d] \otimes N)[-d] \xrightarrow{\sim} (M \otimes N)[-2d].$$

However, the latter diagram is $(-1)^d$-commutative. Hence, if $d$ is odd, the two versions of the shifted tensor product are not canonically isomorphic. The definition of Section 3 is adapted to the setting of Proposition 3.6, but it is not well-suited for the task of defining compatible associativity, commutativity and unit constraints by shifting those of perverse motives.

**Definition 6.4.** For every smooth connected $S \in \mathrm{Var}_k$, we let $\mathcal{G}_{\mathrm{mot}}(S)$ denote the Tannaka dual of the Tannakian category $\mathcal{M}\mathrm{Loc}_p(S)$, and we call it the *motivic Galois group* of $S$.
By construction, the motivic Galois group $\mathcal{G}_{mot}(S)$ is a pro-algebraic group over $\mathbb{Q}$. We plan to investigate the structure of these groups systematically in future work. One basic question is to compute the difference between $\mathcal{G}_{mot}(S)$ and Nori's original motivic Galois group $\mathcal{G}_{mot}(k)$. Note that the structural morphism $a_S : S \to \text{Spec}(k)$ induces an exact tensor functor

$$a_S^\dagger := a_S^*[\dim(S)] : \mathcal{M}(k) \to \mathcal{M}\text{Loc}_p(S),$$

which, under Tannaka duality, corresponds to a morphism of pro-algebraic groups

$$a_{S, \dagger} := \mathcal{G}_{mot}(S) \to \mathcal{G}_{mot}(k).$$

Intuitively, the difference between the two groups should be measured the usual Tannakian category of topological local systems on the complex-analytic space $S^\sigma$; more precisely, by the category of local systems of geometric origin. At least over the generic point of $S$, this intuition is made precise by E. Jacobsen in [Jac23, Thm. 8.4].

**6.2. Reconstruction theorem.** In his recent paper [Gal22], M. Gallauer obtained general reconstruction results in the framework of tensor-triangulated geometry. The general question is how to recover topological information about a variety $S \in \text{Var}_k$ from the structure of a tensor-triangulated category attached to it. Gallauer's approach relies on the notion of *constructible systems*, which allows him to study the theory of supports using the general machinery of six functor formalisms.

The starting point for our applications is [Gal22, Prop. 3.1], which asserts (as a particular case) that the $\text{Var}_k$-fibered category $D^b(\text{Perv}(\cdot))$ defines a constructible system in the sense of [Gal22, Defn. 2.3]; moreover, if $k$ is algebraically closed, this constructible system is *generically simple* in the sense of [Gal22, Defn. 2.13]. In the latter case, by [Gal22, Thm. 5.21] it follows that, for every $S \in \text{Var}_k$, the underlying Zariski topological space of $S$ can be canonically recovered as the smashing spectrum of the tensor-triangulated category $D^b(\text{Perv}(S))$. Analogous reconstruction results hold in the setting of $\ell$-adic sheaves and mixed Hodge modules. As an application of our results, we obtain the following motivic refinement:

**Theorem 6.5.** Suppose that $k$ is algebraically closed. Then, for every $S \in \text{Var}_k$, the underlying topological space of $S$ is canonically homeomorphic to the smashing spectrum of the tensor-triangulated category $D^b(\mathcal{M}(S))$.

We start by proving the following result:

**Lemma 6.6.** For every choice of $k$, the $\text{Var}_k$-fibered category $D^b(\mathcal{M}(\cdot))$ is a constructible system in the sense of [Gal22, Defn. 2.3]; moreover, if $k$ is algebraically closed, this constructible system is generically simple in the sense of [Gal22, Defn. 2.13].

*Proof.* For every $S \in \text{Sm}_k$, we define the subcategory of lisse objects in $D^b(\mathcal{M}(S))$ to be

$$D_{ls}^b(\mathcal{M}(S)) := \{M^\bullet \in D^b(\mathcal{M}(S)) \mid H^i(M^\bullet) \in \mathcal{M}\text{Loc}_p(S) \ \forall i \in \mathbb{Z}\} \subset D^b(\mathcal{M}(S)).$$

By construction, a complex $M^\bullet \in D^b(\mathcal{M}(S))$ belongs to $D_{ls}^b(\mathcal{M}(S))$ if and only if the ordinary cohomology objects of the underlying complex $\iota_S(M^\bullet) \in D^b(\text{Perv}(S)) = D_c^b(S, \mathbb{Q})$ are local systems: in other words, the category $D_{ls}^b(\mathcal{M}(S))$ is nothing but the inverse image of the subcategory

$$D_{ls}^b(S, \mathbb{Q}) := \{K^\bullet \in D_c^b(S, \mathbb{Q}) \mid H^i(K^\bullet) \in \text{Loc}(S) \ \forall i \in \mathbb{Z}\}$$

under $\iota_S : D^b(\mathcal{M}(S)) \to D^b(\text{Perv}(S)) = D_c^b(S, \mathbb{Q})$. Since $D_{ls}^b(S, \mathbb{Q})$ is a tensor-triangulated subcategory of $D_c^b(S, \mathbb{Q})$, we deduce that $D_{ls}^b(\mathcal{M}(S))$ is a tensor-triangulated subcategory of $D^b(\mathcal{M}(S))$. The fact that, as $S$ varies in $\text{Sm}_k$, the subcategories $D_{ls}^b(S, \mathbb{Q}) \subset D_c^b(S, \mathbb{Q})$ satisfy the conditions in [Gal22, Defn. 2.3], easily implies the validity of the same conditions for the subcategories $D_{ls}^b(\mathcal{M}(S)) \subset D^b(\mathcal{M}(S))$.
Remark 6.7. As remarked in [Gal22, Cor. 5.2], in certain cases the scheme $S$ is completely determined by the knowledge of its underlying topological space. In these cases, it follows that the scheme $S$ is determined by the tensor-triangulated category $D^b(\mathcal{M}(S))$.

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LUCA TERENZI
UMPA, ENS DE LYON
46 ALLÉE D'ITALIE, 69007 LYON (FRANCE)

Email address: luca.terenzi@ens-lyon.fr