Since $\mathbf{Perf}_{/\mathbb{A}^1}(X_{\mathrm{Del}})$ and $\mathbf{Perf}(X) \times \mathbb{A}^1$ (resp. $\mathbf{Perf}_{/[\mathbb{A}^1/\mathbb{G}_m]}(X_{\mathrm{Del}}, \mathbb{G}_m)$ and $\mathbf{Perf}(X) \times [\mathbb{A}^1/\mathbb{G}_m]$) are geometric as derived stacks over $\mathbb{A}^1$ (resp. $[\mathbb{A}^1/\mathbb{G}_m]$) by [50, §6.1.1] and [79, Corollary 3.29] respectively, and $\mathbf{Coh}(X) \times \mathbb{A}^1$ (resp. $\mathbf{Coh}(X) \times [\mathbb{A}^1/\mathbb{G}_m]$) is geometric in $\mathrm{dSt}_{/\mathbb{A}^1}$ (resp. $\mathrm{dSt}_{/[\mathbb{A}^1/\mathbb{G}_m]}$) because of Proposition 2.24, we conclude that $\mathbf{Coh}_{/\mathbb{A}^1}(X_{\mathrm{Del}})$ and $\mathbf{Coh}_{/[\mathbb{A}^1/\mathbb{G}_m]}(X_{\mathrm{Del}}, \mathbb{G}_m)$ are geometric as well. ■

We have canonical maps $\mathbf{Coh}_{/\mathbb{A}^1}(X_{\mathrm{Del}}) \to \mathbb{A}^1$ and $\mathbf{Coh}_{/[\mathbb{A}^1/\mathbb{G}_m]}(X_{\mathrm{Del}}, \mathbb{G}_m) \to [\mathbb{A}^1/\mathbb{G}_m]$. Unraveling the definitions, we see that

$$ \mathbf{Coh}_{/\mathbb{A}^1}(X_{\mathrm{Del}}) \times_{\mathbb{A}^1} \{0\} \simeq \mathbf{Coh}(X_{\mathrm{Dol}}) \quad \text{and} \quad \mathbf{Coh}_{/\mathbb{A}^1}(X_{\mathrm{Del}}) \times_{\mathbb{A}^1} \{1\} \simeq \mathbf{Coh}(X_{\mathrm{dR}}), $$

while

$$ \begin{aligned} \mathbf{Coh}_{/[\mathbb{A}^1/\mathbb{G}_m]}(X_{\mathrm{Del}}, \mathbb{G}_m) \times_{[\mathbb{A}^1/\mathbb{G}_m]} \mathbb{B}\mathbb{G}_m &\simeq \mathbf{Coh}_{\mathbb{C}^*}(X_{\mathrm{Dol}}) \\ \mathbf{Coh}(X_{\mathrm{Del}}, \mathbb{G}_m) \times_{[\mathbb{A}^1/\mathbb{G}_m]} [\mathbb{G}_m/\mathbb{G}_m] &\simeq \mathbf{Coh}(X_{\mathrm{dR}}) \times \mathbb{B}\mathbb{G}_m. \end{aligned} \quad (6.1) $$

We also consider the open substack $\mathbf{Coh}^*_{/\mathbb{A}^1}(X_{\mathrm{Del}}) \subset \mathbf{Coh}_{/\mathbb{A}^1}(X_{\mathrm{Del}})$ for which the fiber at zero is the derived moduli stack $\mathbf{Coh}^{\mathrm{ss}, 0}(X_{\mathrm{Dol}})$ of semistable Higgs bundles on $X$ of degree zero (cf. [74, §7]).

Similarly, we can define the derived moduli stacks of extensions of Deligne's $\lambda$-connections. Thus, we have the convolution diagram in $\mathrm{dSt}_{/\mathbb{A}^1}$:

<figure><img src="image_31.png" /></figure>

and the convolution diagram in $\mathrm{dSt}_{/[\mathbb{A}^1/\mathbb{G}_m]}$:

<figure><img src="image_32.png" /></figure>

Because of Corollaries 3.16 and 3.18, it follows that the map $p$ above is derived lci. A similar result holds when we restrict to the open substack $\mathbf{Coh}^*_{/\mathbb{A}^1}(X_{\mathrm{Del}})$ and the corresponding open substack of extensions. Following the same arguments as in §4, we can encode such convolution diagrams into 2-Segal objects, and obtain the following:

**Proposition 6.4.** Let $X$ be a smooth projective complex curve. Then

* there is a 2-Segal object $\mathcal{S} \cdot \mathbf{Coh}_{/\mathbb{A}^1}(X_{\mathrm{Del}})$ which endows $\mathbf{Coh}_{/\mathbb{A}^1}(X_{\mathrm{Del}})$ with the structure of an $\mathbb{E}_1$-monoid object in $\mathrm{Corr}^\times(\mathrm{dGeom}_{/\mathbb{A}^1})_{\mathrm{lci}, \mathrm{rps}}$;

* there is a 2-Segal object $\mathcal{S} \cdot \mathbf{Coh}_{/[\mathbb{A}^1/\mathbb{G}_m]}(X_{\mathrm{Del}}, \mathbb{G}_m)$ which endows $\mathbf{Coh}_{/[\mathbb{A}^1/\mathbb{G}_m]}(X_{\mathrm{Del}}, \mathbb{G}_m)$ with the structure of an $\mathbb{E}_1$-monoid object in $\mathrm{Corr}^\times(\mathrm{dGeom}_{/[\mathbb{A}^1/\mathbb{G}_m]})_{\mathrm{lci}, \mathrm{rps}}$.

A similar result holds for $\mathbf{Coh}^*_{/\mathbb{A}^1}(X_{\mathrm{Del}})$ and $\mathbf{Coh}^*_{/[\mathbb{A}^1/\mathbb{G}_m]}(X_{\mathrm{Del}}, \mathbb{G}_m)$.
**Corollary 6.5.** Coh<sub>pro</sub><sup>b</sup>(Coh<sub>/A<sup>1</sup>/X<sub>Del</sub></sub>) and Coh<sub>pro</sub><sup>b</sup>(Coh<sub>[[A<sup>1</sup>]/G<sub>m</sub>](X<sub>Del</sub>, G<sub>m</sub>) are E<sub>1</sub>-monoid objects in Pro(Cat<sub>∞</sub><sup>st</sup>). A similar result holds for Coh<sub>/A<sup>1</sup>/X<sub>Del</sub></sub><sup>*</sup> and Coh<sub>[[A<sup>1</sup>]/G<sub>m</sub>](X<sub>Del</sub>, G<sub>m</sub>).

By combining the results above with (6.1), we get

**Theorem 6.6.** Let X be a smooth projective complex curve. Then

$$ \mathrm{Coh}_{\mathrm{pro}, \mathbb{C}^*}^{\mathrm{b}}(\mathrm{Coh}_{/\mathbb{A}^1}^{(*)}(X_{\mathrm{Del}})) \simeq \mathrm{Coh}_{\mathrm{pro}}^{\mathrm{b}}(\mathrm{Coh}_{/[\mathbb{A}^1/G_m]}^{(*)}(X_{\mathrm{Del}}, G_m)) $$

is a module over Perf<sub>ilt</sub> and we have E<sub>1</sub>-monoidal functors

$$ \Phi^{(*)}: \mathrm{Coh}_{\mathrm{pro}, \mathbb{C}^*}^{\mathrm{b}}(\mathrm{Coh}_{/\mathbb{A}^1}^{(*)}(X_{\mathrm{Del}})) \otimes_{\mathrm{Perf}_{ilt}} \mathrm{Perf}_{\mathbb{C}} \to \mathrm{Coh}^{\mathbb{b}}(\mathrm{Coh}(X_{\mathrm{dR}})) $$

$$ \Psi^{(*)}: \mathrm{Coh}_{\mathrm{pro}, \mathbb{C}^*}^{\mathrm{b}}(\mathrm{Coh}_{/\mathbb{A}^1}^{(*)}(X_{\mathrm{Del}})) \otimes_{\mathrm{Perf}_{ilt}} \mathrm{Perf}_{\mathrm{gr}} \to \mathrm{Coh}_{\mathbb{C}^*}^{\mathrm{b}}(\mathrm{Coh}^{(\mathrm{ss}, 0)}(X_{\mathrm{Dol}})). $$

Following Simpson [74, §7], we expect the following to be true:

**Conjecture 6.7** (Cat-HA version of the non-abelian Hodge correspondence). The morphisms $\Phi^*$ and $\Psi^*$ are equivalences, i.e., Coh<sub>pro</sub><sup>b</sup>(Coh<sub>(X<sub>dR</sub>)</sub>) is filtered by Coh<sub>pro, C*</sub><sup>b</sup>(Coh<sub>/A<sup>1</sup>/X<sub>Del</sub></sub>) with associated graded Coh<sub>pro, C*</sub><sup>b</sup>(Coh<sub>ss, 0</sub><sup>*</sup>(X<sub>Dol</sub>)).

# 7. A Cat-HA version of the Riemann-Hilbert correspondence

In this section we briefly consider a complex analytic analogue of the theory developed so far. Thanks to the foundational work on derived analytic geometry [22, 34, 48, 51] most of the constructions and results obtained so far carry over in the analytic setting. After sketching how to define the derived analytic stack of coherent sheaves, we focus on two main results. The first is the construction of a monoidal functor between the algebraic and the analytic categorified Hall algebras coming from non-abelian Hodge theory. The second is to provide an equivalence between the analytic categorified Betti algebra and the de Rham one. This equivalence is an instance of the Riemann-Hilbert correspondence, and it is indeed induced by the main results of [22, 49].

## 7.1. The analytic stack of coherent sheaves

We refer to [22, §2] for a review of derived analytic geometry. Using the notations introduced there, we denote by AnPerf the complex analytic stack of perfect complexes (see §4 in loc. cit.). Similarly, given derived analytic stacks X and Y, we let AnMap(X, Y) be the derived analytic stack of morphisms between them.

Fix a derived geometric analytic stack X. We wish to define a substack of AnPerf(X) := AnMap(X, AnPerf) classifying families of coherent sheaves on X. The same ideas of §2 apply, but as usual some extra care to deal with the notion of flatness in analytic geometry is needed.

**Definition 7.1.** Let S be a derived Stein space and let $f: X \to S$ be a morphism of derived analytic stacks. We say that an almost perfect complex $\mathcal{F} \in \mathrm{APerf}(X)$ has tor-
amplitude within $[a, b]$ relative to $S$ (resp. tor-amplitude $\le n$ relative to $S$) if for every $\mathcal{G} \in \mathrm{APerf}^{\mathcal{G}}(S)$ one has

$$ \pi_i(\mathcal{F} \otimes f^*\mathcal{G}) \simeq 0, \quad i \notin [a, b] \quad (\text{resp. } i > n). $$

We let $\mathrm{APerf}_{\mathcal{S}}^{\le n}(X)$ denote the full subcategory of $\mathrm{APerf}(X)$ spanned by those sheaves of almost perfect modules $\mathcal{F}$ on $X$ having tor-amplitude $\le n$ relative to $S$. We write

$$ \mathrm{Coh}_{\mathcal{S}}(X) := \mathrm{APerf}_{\mathcal{S}}^{\le 0}(X), $$

and we refer to it as the $\infty$-category of flat families of coherent sheaves on $X$ relative to $Y$.

The above definition differs from [53, Definitions 7.1 & 7.2]. We prove in Lemma 7.3 that they are equivalent.

**Lemma 7.2.** Let $X$ be a derived analytic stack, let $S \in \mathrm{dStn}_C$ and let $f: X \to S$ be a morphism in $\mathrm{dAnSt}$. Assume that there exists a flat$^{28}$ effective epimorphism $u: U \to X$. Then $\mathcal{F} \in \mathrm{APerf}(X)$ has tor-amplitude within $[a, b]$ relative to $S$ if and only if $u^*(\mathcal{F})$ has tor-amplitude within $[a, b]$ relative to $S$.

*Proof.* Let $\mathcal{G} \in \mathrm{APerf}^{\mathcal{G}}(S)$. Then since $u$ is a flat effective epimorphism, we see that the pullback functor $u^*: \mathrm{APerf}(X) \to \mathrm{APerf}(U)$ is $t$-exact and conservative. Therefore $\pi_i(\mathcal{F} \otimes f^*\mathcal{G}) \simeq 0$ if and only if

$$ u^*(\pi_i(\mathcal{F} \otimes f^*\mathcal{G})) \simeq \pi_i(u^*(\mathcal{F}) \otimes u^*f^*\mathcal{G}) \simeq 0. $$

The conclusion follows.

<figure><img src="image_33.png" /></figure>

**Lemma 7.3.** Let $f: X \to S$ be a morphism of derived analytic stacks. Assume that $X$ is geometric and that $S$ is a derived Stein space. Then $\mathcal{F} \in \mathrm{QCoh}(X)$ has tor-amplitude within $[a, b]$ relative to $S$ if and only if there exists a smooth Stein covering $\{u_i: U_i \to X\}$ such that $\Gamma(U_i; u_i^*\mathcal{F})$ has tor-amplitude within $[a, b]$ as $\Gamma(S; \mathcal{O}_S^{\mathrm{alg}})$-module.

*Proof.* Using Lemma 7.2, we can reduce the problem to the case where $X$ is a derived Stein space. Notice that $\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G} \in \mathrm{APerf}(X)$. Therefore, Cartan's theorem B applies and shows that $\pi_i(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}) = 0$ if and only if $\pi_i(f_*(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G})) = 0$. Observe now that there is a canonical morphism

$$ \eta_{\mathcal{F}, \mathcal{G}}: f_*(\mathcal{F}) \otimes_{\mathcal{O}_S} \mathcal{G} \to f_*(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}). $$

When $\mathcal{G} = \mathcal{O}_S$ this morphism is obviously an equivalence. We claim that it is an equivalence for any $\mathcal{G} \in \mathrm{APerf}(S)$.

<footnote>$^{28}$A morphism $f: U \to X$ of derived analytic stacks is said to be flat if the pullback functor $f^*: \mathrm{APerf}(X) \to \mathrm{APerf}(U)$ is $t$-exact.</footnote>
This question is local on $S$. Write $A_S := \Gamma(S; \mathcal{O}_S^{\text{alg}})$. Using [22, Lemma 4.12] we can restrict ourselves to the case where $\mathcal{G} \simeq \varepsilon_S^*(M)$ for some $M \in \text{APerf}(A_S)$. Here $\varepsilon_S^*: A_S$-Mod $\to \mathcal{O}_S$-Mod is the functor introduced in [22, §4.2]. In this case, we see that since $\eta_{\mathcal{F},\mathcal{G}}$ is an equivalence when $\mathcal{G} = \mathcal{O}_S$, it is also an equivalence whenever $M$ (and hence $\mathcal{G}$) is perfect. In the general case, we use [35, 7.2.4.11 (5)] to find a simplicial object $P^\bullet \in \text{Fun}(\Delta^{\text{op}}, \text{APerf}(A_S))$ such that

$$|P^\bullet| \simeq M.$$

Write $\mathcal{P}^\bullet := \varepsilon_S^*(P^\bullet)$. Reasoning as in [53, Corollary 3.5], we deduce that

$$|\mathcal{P}^\bullet| \simeq \varepsilon_S^*(M) \simeq \mathcal{G}.$$

It immediately follows that

$$\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G} \simeq |\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{P}^\bullet|,$$

and proving that $\eta_{\mathcal{F},\mathcal{G}}$ is an equivalence is reduced to checking that $f_*$ preserves the above colimit. Since the above diagram as well as its colimit takes values in $\text{APerf}(X)$, we can apply Cartan's theorem B. The descent spectral sequence degenerates, and therefore the conclusion follows. ■

**Corollary 7.4.** Let $f: X \to S$ be a morphism as in the previous lemma. Let $j: \text{cl}S \to S$ be the canonical morphism and consider the pullback diagram

<figure><img src="image_34.png" /></figure>

Then an almost perfect complex $\mathcal{F} \in \text{APerf}(X)$ has tor-amplitude within $[a, b]$ relative to $S$ if and only if $i^*\mathcal{F}$ has tor-amplitude within $[a, b]$ relative to $\text{cl}S$.

*Proof.* The map $j$ is a closed immersion and therefore so is $i$. In particular, for any $\mathcal{G} \in \text{APerf}(\text{cl}S)$ the canonical map

$$f^*j_*(\mathcal{G}) \to i_*f_0^*(\mathcal{G})$$

is an equivalence.$^{29}$ Moreover, the projection formula holds for $i$, and $i_*$ is $t$-exact. Suppose that $\mathcal{F}$ has tor-amplitude within $[a, b]$ relative to $S$. Let $\mathcal{G} \in \text{APerf}^\diamond(\text{cl}S)$. Then

$$i_*(i^*\mathcal{F} \otimes_{\mathcal{O}_{X_0}} f_0^*\mathcal{G}) \simeq \mathcal{F} \otimes_{\mathcal{O}_X} i_*f_0^*\mathcal{G} \simeq \mathcal{F} \otimes_{\mathcal{O}_X} f^*j_*\mathcal{G}.$$

Since $j_*$ is $t$-exact, we have $j_*\mathcal{G} \in \text{APerf}^\diamond(S)$, and therefore the above tensor product is concentrated in homological degree $[a, b]$. In other words, $i^*\mathcal{F}$ has tor-amplitude within $[a, b]$ relative to $\text{cl}S$. For the converse, it is enough to observe that $j_*$ induces an equivalence $\text{APerf}^\diamond(\text{cl}S) \simeq \text{APerf}^\diamond(S)$. ■

<footnote>$^{29}$Ultimately, this can be traced back to the unramifiedness of the analytic pregeometry $\mathcal{T}_{\text{an}}(\mathbb{C})$. See [53, Lemma 6.1] for an argument in the non-archimedean case.</footnote>
**Definition 7.5.** Let $S \in \mathrm{dSt}_C$ and let $f: X \to S$ be a morphism in $\mathrm{dAnSt}$. A morphism $u: U \to X$ is said to be universally flat relative to $S$ if for every derived Stein space $S' \in \mathrm{dSt}$ and every morphism $S' \to S$ the induced map $S' \times_S U \to S' \times_S X$ is flat. We say that a morphism $u: U \to X$ is universally flat if it is universally flat relative to $\mathrm{Spec}(\mathbb{C})$.

**Remark 7.6.** Let $S$ be an affine derived scheme and let $f: X \to S$ and $u: U \to X$ be morphisms of derived stacks. If $f$ is flat, then for every morphism $S' \to S$ of affine derived schemes, the morphism $S' \times_S U \to S' \times_S X$ is flat; see [50, Proposition 2.3.16]. In the analytic setting, it is difficult to prove a similar result, because it essentially relies on base change for maps between derived affine schemes (see [50, Proposition 2.3.4]), which is not available in the analytic setting.

**Corollary 7.7.** Let $X$ be a derived analytic stack and let $S$ be a derived Stein space. Assume that there exists a universally flat effective epimorphism $u: U \to X$ where $U$ is geometric and underived. Let $f: S' \to S$ be a morphism of derived Stein spaces and consider the pullback

$$ \begin{array}{ccc} X \times S' & \xrightarrow{g} & X \times S \\ \downarrow^{q} & & \downarrow^{p} \\ S' & \xrightarrow{f} & S \end{array} $$

If $\mathcal{F} \in \mathrm{APerf}(X \times S)$ has tor-amplitude within $[0, 0]$ relative to $S$, then $g^*\mathcal{F}$ has tor-amplitude within $[0, 0]$ relative to $S'$.

*Proof.* Since $u: U \to X$ is universally flat, the morphisms $U \times S \to X \times S$ and $U \times S' \to X \times S'$ are flat. Therefore Lemma 7.2 shows that we can restrict ourselves to the case $X = U$. Using Corollary 7.4, we can reduce the problem to the case where $S$ and $S'$ are underived. Since the question is local on $X$, we can furthermore assume that $X$ is a Stein space. At this point, the conclusion follows directly from [10, §8.3, Proposition 3]. ■

Using the above corollary, we can therefore define a derived analytic stack **AnCoh**(X), which is a substack of **AnPerf**(X).

In what follows, we will often restrict ourselves to the study of **AnCoh**(X<sup>an</sup>), where now $X$ is an algebraic variety. Combining [22, Proposition 5.2 & Theorem 5.5] we see that if $X$ is a proper complex scheme, then there is a natural equivalence<sup>30</sup>

$$ \mathrm{Perf}(X)^{\mathrm{an}} \simeq \mathrm{AnPerf}(X^{\mathrm{an}}). \qquad (7.1) $$

We wish to extend this result to **Coh**(X)<sup>an</sup> and **AnCoh**(X<sup>an</sup>). Let us start by constructing the map between them. The map **Perf**(X)<sup>an</sup> → **AnPerf**(X<sup>an</sup>) is obtained by adjunction from the map

$$ \mathrm{Perf}(X) \to \mathrm{AnPerf}(X^{\mathrm{an}}) \circ (-)^{\mathrm{an}}, $$

<footnote n="30"><sup>30</sup>The derived analytification functor has been first introduced in [34, Remark 12.26] and studied extensively in [48, §4]. For a review, see [22, §3.1].</footnote>
which, for $S \in d\mathrm{Aff}^{\mathrm{afp}}$, is induced by applying $(-)^{\sim} : \mathrm{Cat}_{\infty} \to \mathcal{S}$ to the analytification functor

$$ \mathrm{Perf}(X \times S) \to \mathrm{Perf}(X^{\mathrm{an}} \times S^{\mathrm{an}}). $$

It is therefore enough to check that this functor respects the two subcategories of families of coherent sheaves relative to $S$ and $S^{\mathrm{an}}$, respectively.

**Lemma 7.8.** Let $f: X \to S$ be a morphism of derived complex stacks locally almost of finite presentation. Suppose that $X$ is geometric and that $S$ is affine. Then $\mathcal{F} \in \mathrm{APerf}(S)$ has tor-amplitude within $[a, b]$ relative to $S$ if and only if $\mathcal{F}^{\mathrm{an}} \in \mathrm{APerf}(X^{\mathrm{an}})$ has tor-amplitude within $[a, b]$ relative to $S^{\mathrm{an}}$.

*Proof.* Suppose first that $\mathcal{F}^{\mathrm{an}}$ has tor-amplitude within $[a, b]$ relative to $S^{\mathrm{an}}$. Let $\mathcal{G} \in \mathrm{APerf}^{\mathcal{G}}(S)$. Then we have to check that $\pi_i(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}) = 0$ for $i \notin [a, b]$. As the analytification functor $(-)^{\mathrm{an}}$ is $t$-exact and conservative, this is equivalent to checking that we have $\pi_i((\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G})^{\mathrm{an}}) = 0$. But

$$ (\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G})^{\mathrm{an}} \simeq \mathcal{F}^{\mathrm{an}} \otimes_{\mathcal{O}_{X^{\mathrm{an}}}} f^{\mathrm{an}*}(\mathcal{G}^{\mathrm{an}}), \qquad (7.2) $$

and the conclusion follows from the fact that $\mathcal{G}^{\mathrm{an}} \in \mathrm{APerf}^{\mathcal{G}}(S^{\mathrm{an}})$.

Suppose now that $\mathcal{F}$ has tor-amplitude within $[a, b]$ relative to $S = \mathrm{Spec}(A)$. We can check that $\mathcal{F}^{\mathrm{an}}$ has tor-amplitude within $[a, b]$ relative to $S^{\mathrm{an}}$ locally on $S^{\mathrm{an}}$. For every derived Stein open subspace $j_U: U \subset S^{\mathrm{an}}$, write

$$ A_U := \Gamma(U; \mathcal{O}_{S^{\mathrm{an}}}^{\mathrm{alg}}|_U). $$

Write $a_U: \mathrm{Spec}(A_U) \to S$ for the morphism induced by the canonical map $A \to A_U$. Consider the two pullback squares

$$ \begin{tikzcd} X_U \arrow[r, "b_U"] \arrow[d, "f_U"] & X \arrow[d, "f"'] \\ \mathrm{Spec}(A_U) \arrow[r, "a_U"] \arrow[d, "S"'] & S \arrow[d, "f^{\mathrm{an}}"] \arrow[r, "j_U"] & S^{\mathrm{an}} \end{tikzcd} $$

There is a natural analytification functor relative to $U$,

$$ (-)^{\mathrm{an}}_U : \mathrm{APerf}(X_U) \to \mathrm{APerf}(X_U^{\mathrm{an}}). $$

Moreover, the canonical map

$$ i_U^*(\mathcal{H}^{\mathrm{an}}) \to (b_U^*(\mathcal{H}))_U^{\mathrm{an}} $$

is an equivalence for every $\mathcal{H} \in \mathrm{APerf}(X)$.

Fix now $\mathcal{G} \in \mathrm{APerf}(S^{\mathrm{an}})$. If $\mathcal{G} \simeq (\tilde{\mathcal{G}})^{\mathrm{an}}$ for some $\tilde{\mathcal{G}} \in \mathrm{APerf}^{\mathcal{G}}(S)$, then the equivalence (7.2) shows that

$$ \pi_i(\mathcal{F}^{\mathrm{an}} \otimes_{\mathcal{O}_X^{\mathrm{an}}} f^{\mathrm{an}*}(\mathcal{G})) = 0 $$
for $i \notin [a, b]$. In the general case, we choose a double covering $\{V_i \in U_i \in S^{\text{an}}\}$ by relatively compact derived Stein open subspaces of $S^{\text{an}}$. Using [22, Lemma 4.12] we can find $\tilde{\mathcal{G}}_i \in \text{APerf}(A_{V_i})$ such that $\mathcal{G}|_{V_i} \simeq \varepsilon_{V_i}^*(\tilde{\mathcal{G}}_i)$. Here $\varepsilon_{V_i}^*$ is the functor introduced in [22, §4.2]. At this point, we observe that Lemma 2.4 guarantees that $b_U^*(\mathcal{F})$ has tor-amplitude within $[a, b]$ relative to $\text{Spec}(A_U)$. The conclusion then follows from the argument given in the first case. ■

As a consequence, we find a morphism

$$ \mathrm{Coh}(X) \to \mathrm{AnCoh}(X^{\mathrm{an}}) \circ (-)^{\mathrm{an}}, $$

which by adjunction induces

$$ \mu_X: \mathrm{Coh}(X)^{\mathrm{an}} \to \mathrm{AnCoh}(X^{\mathrm{an}}), $$

which is compatible with the morphism $\mathrm{Perf}(X)^{\mathrm{an}} \to \mathrm{AnPerf}(X^{\mathrm{an}})$.

**Proposition 7.9.** If $X$ is a proper complex scheme, the natural transformation

$$ \mu_X: \mathrm{Coh}(X)^{\mathrm{an}} \to \mathrm{AnCoh}(X^{\mathrm{an}}) $$

is an equivalence.

*Proof.* Reasoning as in the proof of the equivalence (7.1) in [22, Proposition 5.2], we restrict ourselves to checking that for every derived Stein space $U \in \mathrm{dStn}_\mathbb{C}$ and every compact derived Stein subspace $K$ of $U$, the natural morphism

$$ \underset{K \subset V \subset U}{\text{“colim”}} \mathrm{Coh}_{\mathrm{Spec}(A_V)}(\mathrm{Spec}(A_V) \times X) \to \underset{K \subset V \subset U}{\text{“colim”}} \mathrm{Coh}_V(V \times X^{\mathrm{an}}) $$

is an equivalence in $\mathrm{Ind}(\mathrm{Cat}_\infty^{\mathrm{st}})$. Here the colimit is taken over the family of open Stein neighborhoods $V$ of $K$ inside $U$. Using [22, Lemma 5.13] we see that for every $V$, the functor

$$ \mathrm{Coh}_{\mathrm{Spec}(A_V)}(\mathrm{Spec}(A_V) \times X) \to \mathrm{Coh}_V(V \times X^{\mathrm{an}}) $$

is fully faithful. The conclusion now follows by combining [22, Proposition 5.15] and the “only if” direction of Lemma 7.8. ■

## 7.2. Categorical Hall algebras in the $\mathbb{C}$-analytic setting

Let $X \in \mathrm{dAnSt}$ be a derived analytic stack. In the previous section, we have introduced the analytic stack $\mathrm{AnCoh}(X)$ parameterizing families of sheaves of almost perfect modules over $X$ of tor-amplitude $\le 0$ relative to the base. Similarly, we can define the derived analytic stacks $\mathrm{AnPerf}^{\mathrm{ext}}$, $\mathrm{AnPerf}^{\mathrm{ext}}(X)$, and $\mathrm{AnCoh}^{\mathrm{ext}}(X)$. We deal directly with the Waldhausen construction.

We define the simplicial derived analytic stack

$$ S_{\bullet} \mathrm{AnPerf}: \mathrm{dStn}_{\mathbb{C}}^{\mathrm{op}} \to \mathrm{Fun}(\Delta^{\mathrm{op}}, S) $$
by sending an object $[n] \in \Delta$ and a derived Stein space $S$ to the full subcategory of$^{31}$

$$ \mathcal{S}_n \mathrm{Perf}(S) \hookrightarrow \mathrm{Fun}(T_n, \mathrm{Perf}(S)). $$

Since each $T_n$ is a finite category, [22, Corollary 7.2] and the flatness of the relative analytification proven in [52, Proposition 4.17] imply that the natural map

$$ (\mathcal{S} \cdot \mathrm{Perf})^{\mathrm{an}} \to \mathcal{S} \cdot \mathrm{AnPerf} $$

is an equivalence. Moreover, [22, Proposition 7.3] implies that the analytification commutes with the limits appearing in the 2-Segal condition. We can therefore deduce that $\mathcal{S} \cdot \mathrm{AnPerf}$ is a 2-Segal object in $\mathrm{dAnSt}$. From this, we deduce immediately that for every derived analytic stack $X$, $\mathrm{AnMap}(X, \mathcal{S} \cdot \mathrm{AnPerf})$ is again a 2-Segal object. At this point, the same reasoning as for Lemma 4.1 yields

**Proposition 7.10.** Let $X \in \mathrm{dAnSt}$ be a derived geometric analytic stack. Then $\mathcal{S} \cdot \mathrm{AnCoh}(X)$ is a 2-Segal object in $\mathrm{dAnSt}$, and therefore it endows the derived analytic stack $\mathrm{AnCoh}(X)$ with the structure of an $\mathbb{E}_1$-monoid object.

The morphism (7.2) can be naturally upgraded to a natural transformation

$$ \mathcal{S} \cdot \mathrm{Coh}(X) \to \mathcal{S} \cdot \mathrm{AnCoh}(X^{\mathrm{an}}) \circ (-)^{\mathrm{an}} $$

in $\mathrm{Fun}(\Delta^{\mathrm{op}}, \mathrm{dSt})$. By adjunction, we therefore find a morphism of simplicial objects

$$ (\mathcal{S} \cdot \mathrm{Coh}(X))^{\mathrm{an}} \to \mathcal{S} \cdot \mathrm{AnCoh}(X^{\mathrm{an}}). $$

**Remark 7.11.** Suppose that $X$ is such that each $\mathcal{S}_n \mathrm{Coh}(X)$ is geometric. Then [22, Proposition 7.3] implies that $(\mathcal{S} \cdot \mathrm{Coh}(X))^{\mathrm{an}}$ is a 2-Segal object in $\mathrm{dAnSt}$.

Let $Y \in \mathrm{dAnSt}$ be a derived analytic stack and let $u: U \to Y$ be a flat effective epimorphism from an underived geometric analytic stack $U$. As above, we are able to define the derived stack $\mathrm{AnCoh}(Y)$. Notice that $\mathrm{AnCoh}(Y)$ only depends on $Y$ and not on $U$. However, as in the algebraic case, the proof of the functoriality of $\mathrm{AnCoh}(Y)$ relies on the existence of $U$ and on Lemma 7.7. In addition, we have

$$ \mathrm{AnCoh}(Y) \simeq \mathrm{AnPerf}(Y) \times_{\mathrm{AnPerf}(U)} \mathrm{AnCoh}(U). $$

This is the analytic counterpart of Lemma 2.11.

Similarly, we can define $\mathrm{AnCoh}^{\mathrm{ext}}(Y)$ and $\mathrm{AnBun}^{\mathrm{ext}}(Y)$ and more generally their Waldhausen analogues $\mathcal{S} \cdot \mathrm{AnCoh}(Y)$ and $\mathcal{S} \cdot \mathrm{AnBun}(Y)$. We immediately obtain

**Proposition 7.12.** Let $Y \in \mathrm{dAnSt}$ be a derived analytic stack and let $u: U \to Y$ be a flat effective epimorphism from an underived geometric analytic stack $U$. Then $\mathcal{S} \cdot \mathrm{AnCoh}(Y)$ is a 2-Segal object and it endows $\mathrm{AnCoh}(Y)$ with the structure of an $\mathbb{E}_1$-monoid object in $\mathrm{dAnSt}$.

<footnote>$^{31}$See §4.1 for the notations used here.</footnote>
As a particular case, let $X$ be a smooth proper connected analytic space. Simpson's shapes $X_B$, $X_{dR}$, $X_{Dol}$, and $X_{Del}$ also exist in derived analytic geometry (as introduced e.g. in [22, §5.2]). We have the following analytic analog of Proposition 4.3.

**Corollary 7.13.** Let $X \in \mathrm{dAnSt}$ be a derived geometric analytic stack and let $Y$ be one of the following stacks: $X_B$, $X_{dR}$, or $X_{Dol}$. Then $\mathcal{S}\cdot\mathrm{AnCoh}(Y)$ is a 2-Segal object in $\mathrm{dAnSt}$, and therefore it endows the derived analytic stack $\mathrm{AnCoh}(Y)$ with the structure of an $\mathbb{E}_1$-monoid object.

Our next step is to construct the categorified Hall algebras in the analytic setting. The lack of quasi-coherent sheaves in analytic geometry forces us to consider a variation of the construction considered in §4.2. We start with the following construction:

**Construction 7.14.** Let $\mathcal{T}_{\mathrm{disc}}(\mathbb{C})$ be the full subcategory of $\mathrm{Sch}_{\mathbb{C}}$ spanned by finite-dimensional affine spaces $A_{\mathbb{C}}^n$. Given an $\infty$-topos $\mathcal{X}$, sheaves on $\mathcal{X}$ with values in $\mathrm{Alg}_{\mathbb{C}}$ can be canonically identified with product preserving functors $\mathcal{T}_{\mathrm{disc}}(\mathbb{C}) \to \mathcal{X}$. We let $^{\mathbb{R}}\mathrm{Top}(\mathcal{T}_{\mathrm{disc}}(\mathbb{C}))$ denote the $\infty$-category of $\infty$-topoi equipped with a sheaf of derived commutative $\mathbb{C}$-algebras. The construction performed in [33, Notation 2.2.1] provides us with a functor

$$ \Gamma: (^{\mathbb{R}}\mathrm{Top}(\mathcal{T}_{\mathrm{disc}}(\mathbb{C})))^{\mathrm{op}} \to \mathrm{Alg}_{\mathbb{C}}. $$

Equipping both $\infty$-categories with the cocartesian monoidal structure, we see that $\Gamma$ can be upgraded to a right-lax symmetric monoidal structure. Composing with the symmetric monoidal functor $\mathrm{QCoh}: \mathrm{Alg}_{\mathbb{C}} \to \mathrm{Cat}_{\infty}^{\mathrm{st}}$ we therefore obtain a right-lax symmetric monoidal functor

$$ (^{\mathbb{R}}\mathrm{Top}(\mathcal{T}_{\mathrm{disc}}(\mathbb{C})))^{\mathrm{op}} \to \mathrm{Cat}_{\infty}^{\mathrm{st}}. $$

We denote the sheafification of this functor with respect to the étale topology on $^{\mathbb{R}}\mathrm{Top}(\mathcal{T}_{\mathrm{disc}}(\mathbb{C}))$ (see [33, Definition 2.3.1]) by

$$ \mathcal{O}\text{-Mod}: (^{\mathbb{R}}\mathrm{Top}(\mathcal{T}_{\mathrm{disc}}(\mathbb{C})))^{\mathrm{op}} \to \mathrm{Cat}_{\infty}^{\mathrm{st}}. $$

Observe that $\mathcal{O}\text{-Mod}$ is canonically endowed with a right-lax symmetric monoidal structure.

Consider the natural forgetful functor

$$ (-)^{\mathrm{alg}}: \mathrm{dAn}_{\mathbb{C}} \to ^{\mathbb{R}}\mathrm{Top}(\mathcal{T}_{\mathrm{disc}}(\mathbb{C})). $$

Equipping both $\infty$-categories with the cartesian monoidal structure, we see that $(-)^{\mathrm{alg}}$ can be upgraded to a left-lax monoidal functor. We still denote by $\mathcal{O}\text{-Mod}$ the composition

$$ (\mathrm{dAn}_{\mathbb{C}})^{\mathrm{op}} \xrightarrow{(-)^{\mathrm{alg}}} (^{\mathbb{R}}\mathrm{Top}(\mathcal{T}_{\mathrm{disc}}(\mathbb{C})))^{\mathrm{op}} \xrightarrow{\mathcal{O}\text{-Mod}} \mathrm{Cat}_{\infty}^{\mathrm{st}}, $$

which canonically inherits the structure of a right-lax monoidal functor. Given $X \in \mathrm{dAn}_{\mathbb{C}}$, we denote by $\mathcal{O}_X\text{-Mod}$ its image via this functor.
This functor admits a canonical subfunctor

$$ \mathrm{APerf}: \mathrm{dAn}_{\mathbb{C}}^{\mathrm{op}} \to \mathrm{Cat}_{\infty}^{\mathrm{st}}, $$

which sends a derived $\mathbb{C}$-analytic space to the full subcategory of $\mathcal{O}_X$-Mod spanned by sheaves of almost perfect modules. Observe that sheaves of almost perfect modules are closed under exterior product, and therefore $\mathrm{APerf}$ inherits the structure of a right-lax monoidal functor. Moreover, if $f: X \to Y$ is proper, then [48, Theorem 6.5] implies that the functor

$$ f_*: \mathcal{O}_X\text{-Mod} \to \mathcal{O}_Y\text{-Mod} $$

restricts to a functor

$$ f_*: \mathrm{APerf}(X) \to \mathrm{APerf}(Y), $$

which is right adjoint to $f^*$.

**Lemma 7.15.** Let

$$ \begin{array}{ccc} X' & \xrightarrow{u} & X \\ \downarrow g & & \downarrow f \\ Y' & \xrightarrow{v} & Y \end{array} $$

be a pullback square in $\mathrm{dAn}_\mathbb{C}$. Assume that the truncations of $X, X'$, $Y$ and $Y'$ are separated analytic spaces. If $f$ is proper then the commutative diagram

$$ \begin{array}{ccc} \mathrm{APerf}(Y) & \xrightarrow{v^*} & \mathrm{APerf}(Y') \\ \downarrow f^* & & \downarrow g^* \\ \mathrm{APerf}(X) & \xrightarrow{u^*} & \mathrm{APerf}(X') \end{array} $$

is vertically right adjointable.

*Proof.* We adapt the proof of [53, Theorem 6.8] to the complex analytic setting. The key input is unramifiedness for the pregeometry $\mathcal{T}_{\mathrm{an}}(\mathbb{C})$, proven in [34, Proposition 11.6], which has as a consequence Proposition 11.12 (3) in loc. cit. In turn, this implies that the statement of this lemma holds true when $g$ is a closed immersion. Knowing this, Steps 1 and 2 of the proof of [53, Theorem 6.8] apply without changes. Step 3 applies as well, with the difference that in the $\mathbb{C}$-analytic setting we can reduce to the case where $Y' = \mathrm{Sp}(\mathbb{C})$ is the $\mathbb{C}$-analytic space associated to a point. In particular, the map $Y' = \mathrm{Sp}(\mathbb{C}) \to Y$ is now automatically a closed immersion, and therefore the conclusion follows. ■

Let $\mathrm{dAn}_{\mathbb{C}}^{\mathrm{sep}}$ denote the full subcategory of $\mathrm{dAn}_\mathbb{C}$ spanned by derived $\mathbb{C}$-analytic spaces whose truncation is a separated analytic space. Lemma 7.15 shows that the assumptions of [12, Theorem 3.2.2(b)] are satisfied with $\mathrm{horiz} = \mathrm{all}$ and $\mathrm{vert} = \mathrm{proper}$. As a consequence, we can extend $\mathrm{APerf}$ to a functor

$$ \mathrm{APerf}: \mathrm{Corr}(\mathrm{dAn}_{\mathbb{C}}^{\mathrm{sep}})_{\mathrm{proper}, \mathrm{all}}^{\mathrm{isom}} \to \mathrm{Cat}_{\infty}^{\mathrm{st}}. $$
Moreover, the considerations in [12, §3.3.1] show that this functor inherits a canonical right-lax monoidal structure. Using [12, Theorem 8.6.1.5] we obtain (via right Kan extension) a right-lax monoidal functor

$$ \mathrm{APerf}: \mathrm{Corr}(\mathrm{dAnSt})_{\mathrm{all},\mathrm{rps}}^{\mathrm{isom}} \to \mathrm{Cat}_{\infty}^{\mathrm{st}}. $$

Here rps denotes the class of 1-morphisms representable by proper derived C-analytic spaces.

Given a derived C-analytic space $X$, we denote by $\mathrm{Coh}^b(X)$ the full subcategory of $\mathrm{APerf}(X)$ spanned by locally cohomologically bounded sheaves of almost perfect modules.

**Lemma 7.16.** Let $f: X \to Y$ be a morphism of derived geometric analytic stacks. If $f$ is lci$^{32}$ then it has finite tor-amplitude and in particular it induces a functor

$$ f^*: \mathrm{Coh}^b(Y) \to \mathrm{Coh}^b(X). $$

*Proof.* The argument of [54, Corollary 2.9] applies. ■

As a consequence, we obtain a right-lax monoidal functor

$$ \mathrm{Coh}^b: \mathrm{Corr}^\times(\mathrm{dAnSt})_{\mathrm{rps,lci}} \to \mathrm{Cat}_\infty^{\mathrm{st}}. $$

Finally, we want to restrict ourselves to derived geometric analytic stacks. In particular, we need that $\mathrm{AnCoh}(Y)$ and the corresponding 2-Segal space to be geometric. So, first note that if $Y \in \mathrm{dSt}$ is a derived stack, then we obtain as before a natural transformation

$$ S_{\bullet} \mathrm{Coh}(Y)^{\mathrm{an}} \to S_{\bullet} \mathrm{AnCoh}(Y) \qquad (7.3) $$

in $\mathrm{Fun}(\Delta^{\mathrm{op}}, \mathrm{dAnSt})$.

Let $X$ be a smooth and proper complex scheme. By [22, Proposition 5.2], $\mathrm{AnPerf}(X)$ is equivalent to the analytification $\mathrm{Perf}(X)^{\mathrm{an}}$ of the derived stack $\mathrm{Perf}(X) = \mathrm{Map}(X, \mathrm{Perf})$. Thus, $\mathrm{AnPerf}(X)$ is a locally geometric derived stack, locally of finite presentation.

**Lemma 7.17.** The map (7.3) induces an equivalence $(S_{\bullet} \mathrm{Coh}(X))^{\mathrm{an}} \simeq S_{\bullet} \mathrm{AnCoh}(X^{\mathrm{an}})$. In particular, for each $n \ge 0$ the derived analytic stack $S_n \mathrm{AnCoh}(X^{\mathrm{an}})$ is locally geometric and locally of finite presentation.

*Proof.* When $n = 1$, this is exactly the statement of Proposition 7.9. The proof of the general case is similar, and there are no additional subtleties. ■

Let $X$ be a smooth proper connected complex scheme. As proved in [22, §5.2], the analytification functor commutes with Simpson's shape functor, i.e., we have the following canonical equivalences:

$$ (X_{\mathrm{dR}})^{\mathrm{an}} \simeq (X^{\mathrm{an}})_{\mathrm{dR}}, \quad (X_B)^{\mathrm{an}} \simeq (X_B)^{\mathrm{an}}, \quad (X_{\mathrm{Dol}})^{\mathrm{an}} \simeq (X^{\mathrm{an}})_{\mathrm{Dol}}. $$

<footnote>$^{32}$In this setting, it means that the analytic cotangent complex $\mathbb{L}_{X/Y}^{\mathrm{an}}$ introduced in [52] is perfect and has tor-amplitude within $[0, 1]$.</footnote>
**Lemma 7.18.** Let $* \in \{\text{B, dR, Dol}\}$. Then (7.3) induces an equivalence $(\mathcal{S}_{\bullet}\mathcal{Coh}(X_*))^{\text{an}} \simeq \mathcal{S}_{\bullet}\mathcal{AnCoh}((X^{\text{an}})_*)$. In particular, for each $n \ge 0$ the derived analytic stack $\mathcal{S}_n\mathcal{AnCoh}((X^{\text{an}})_*)$ is locally geometric and locally of finite presentation.

*Proof.* The proof of Proposition 7.9 applies, with the following caveat: rather than invoking [22, Lemma 5.13 & Proposition 5.15], we instead use Propositions 5.26 (for the de Rham case), 5.28 (for the Betti case) and 5.32 (for the Dolbeault case) in [22]. ■

Finally, we are able to give the analytic counterpart of Theorem 4.9:

**Theorem 7.19.** Let $Y$ be one of the following derived stacks:

(1) a smooth proper complex scheme of dimension either one or two;

(2) the Betti, de Rham or Dolbeault stack of a smooth projective curve.

Then the composition

$$ \begin{array}{ccc} \mathrm{Coh}^b(\mathrm{AnCoh}(Y^{\mathrm{an}})) \times \mathrm{Coh}^b(\mathrm{AnCoh}(Y^{\mathrm{an}})) & \xrightarrow{\boxtimes} & \mathrm{Coh}^b(\mathrm{AnCoh}(Y^{\mathrm{an}}) \times \mathrm{AnCoh}(Y^{\mathrm{an}})) \\ \xrightarrow{q_* \circ p^*} & & \mathrm{Coh}^b(\mathrm{AnCoh}(Y^{\mathrm{an}})), \end{array} $$

where the map on the right-hand side is induced by the 1-morphism in correspondences

<figure><img src="image_35.png" /></figure>

endows $\mathrm{Coh}^b(\mathrm{AnCoh}(Y))$ with the structure of an $\mathbb{E}_1$-monoidal stable $\infty$-category.

*Proof.* The only main point to emphasize is how to use the tor-amplitude estimates for the map $p$ in the algebraic case (i.e., Proposition 3.10 and Corollaries 3.14, 3.16, and 3.18) in the analytic setting. First of all, we use Lemmas 7.17 and 7.18 to identify the 2-Segal object $\mathcal{S}_{\bullet}\mathcal{AnCoh}(Y^{\text{an}})$ with $(\mathcal{S}_{\bullet}\mathcal{Coh}(Y))^{\text{an}}$. Then it remains to check that $p^{\text{an}}$ is derived lci, where now $p$ is the map appearing in (4.2). This follows by combining Lemma 7.8 and [52, Theorem 5.21]. ■

**Corollary 7.20.** Let $Y$ be as in Theorem 7.19. Then the derived analytification functor induces a morphism in $\mathrm{Alg}_{\mathbb{E}_1}(\mathrm{Cat}^{\mathrm{st}})$,

$$ \mathrm{Coh}^b(\mathrm{Coh}(Y)) \to \mathrm{Coh}^b(\mathrm{AnCoh}(Y^{\mathrm{an}})). $$

*Proof.* By using Lemmas 7.17 and 7.18, we have $\mathrm{Coh}^b(\mathrm{Coh}(Y)^{\mathrm{an}}) \simeq \mathrm{Coh}^b(\mathrm{AnCoh}(Y^{\mathrm{an}}))$ as $\mathbb{E}_1$-monoid objects. The analytification functor $(-)^{\mathrm{an}}$ promotes to a symmetric monoidal functor

$$ (-)^{\mathrm{an}}: \mathrm{Corr}^x(d\mathrm{St}) \to \mathrm{Corr}^x(d\mathrm{AnSt}). $$

Combining Lemma 7.8 and [52, Theorem 5.21], we conclude that $(-)^{\mathrm{an}}$ preserves lci morphisms. Moreover, [52, Lemma 3.1 (3)] and [51, Proposition 6.3], we see that $(-)^{\mathrm{an}}$ also
preserves proper morphisms. Finally, using the derived GAGA theorems [48, Theorems 7.1 & 7.2] we see that $(-)^{\text{an}}$ takes morphisms which are representable by proper schemes to morphisms which are representable by proper analytic spaces.<fnref n="33" /> Therefore, it restricts to a symmetric monoidal functor

$$(-)^{\text{an}}: \text{Corr}^{\times}(\text{dSt})_{\text{rps,lci}} \to \text{Corr}^{\times}(\text{dAnSt})_{\text{rps,lci}}.$$

The analytification functor for coherent sheaves induces a natural transformation of right-lax symmetric monoidal functors

$$\text{Coh}^{\text{b}} \to \text{Coh}^{\text{b}} \circ (-)^{\text{an}}.$$

Here both functors are considered as functors $\text{dSt} \to \text{Cat}_{\infty}$. Using the universal property of the category of correspondences, we can extend this natural transformation of right-lax symmetric monoidal functors defined over the category of correspondences. The key point is to verify that if $p: X \to Y$ is a proper morphism of geometric derived stacks locally almost of finite presentation, then the diagram

$$\begin{array}{ccc} \text{Coh}^{\text{b}}(X) & \xrightarrow{(-)^{\text{an}}} & \text{Coh}^{\text{b}}(X^{\text{an}}) \\ \Big\downarrow_{p_*} & & \Big\downarrow_{p_*^{\text{an}}} \\ \text{Coh}^{\text{b}}(Y) & \xrightarrow{(-)^{\text{an}}} & \text{Coh}^{\text{b}}(Y^{\text{an}}) \end{array}$$

commutes. This is a particular case of [48, Theorem 7.1]. The conclusion follows. ■

### 7.3. The derived Riemann-Hilbert correspondence

Let $X$ be a smooth proper connected complex scheme. In [49, §3] there is constructed a natural transformation

$$\eta_{\text{RH}}: X_{\text{dR}}^{\text{an}} \to X_{\text{B}}^{\text{an}},$$

which induces for every derived analytic stack $Y \in \text{dAnSt}$ a morphism

$$\eta_{\text{RH}}^{*}: \text{AnMap}(X_{\text{dR}}^{\text{an}}, Y) \to \text{AnMap}(X_{\text{B}}^{\text{an}}, Y).$$

It is then shown in [49, Theorem 6.11] that this map is an equivalence when $Y = \text{AnPerf}$.<fnref n="34" />

Taking $Y = S \cdot \text{AnPerf}$, we see that $\eta_{\text{RH}}$ induces a morphism of 2-Segal objects

$$\eta_{\text{RH}}^{*}: S \cdot \text{AnPerf}(X_{\text{dR}}^{\text{an}}) \to S \cdot \text{AnPerf}(X_{\text{B}}^{\text{an}}).$$

By applying the functor $\text{2-Seg}(\text{dAnSt}) \to \text{Alg}_{\mathbb{E}_1}(\text{Corr}^{\times}(\text{dAnSt}))$, we therefore conclude that

$$\eta_{\text{RH}}^{*}: \text{AnPerf}(X_{\text{dR}}^{\text{an}}) \to \text{AnPerf}(X_{\text{B}}^{\text{an}})$$

acquires a natural structure of morphism between $\mathbb{E}_1$-monoid objects:

<footnote n="33">³³Using [51, Proposition 6.3] it is enough to prove that the analytification takes representable morphisms with geometric target to representable morphisms. This immediately follows from [51, Proposition 2.25].</footnote>

<footnote n="34">³⁴See [22, Corollary 7.6] for a discussion of which other derived analytic stacks $Y$ see $\eta_{\text{RH}}$ as an equivalence.</footnote>
**Proposition 7.21.** *The morphism*

$$ \eta_{\mathrm{RH}}^*: S \bullet \mathrm{AnPerf}(X_{\mathrm{dR}}^{\mathrm{an}}) \to S \bullet \mathrm{AnPerf}(X_{\mathrm{B}}^{\mathrm{an}}) $$

is an equivalence. Moreover, it restricts to an equivalence

$$ \eta_{\mathrm{RH}}^*: S \bullet \mathrm{AnCoh}_{\mathrm{dR}}(X) \to S \mathrm{AnCoh}_{\mathrm{B}}(X). $$

*Proof.* Fix a derived Stein space $S \in \mathrm{dStn}_\mathbb{C}$. Then [49, Theorem 6.11] provides an equivalence of stable $\infty$-categories

$$ \mathrm{Perf}(X_{\mathrm{dR}}^{\mathrm{an}} \times S) \simeq \mathrm{Perf}(X_{\mathrm{B}}^{\mathrm{an}} \times S). $$

Therefore, for every $n \ge 0$ we obtain an equivalence

$$ \begin{aligned} S_n \mathbf{AnPerf}(X_{\mathrm{dR}}^{\mathrm{an}})(S) &\simeq \mathrm{Fun}(T_n, \mathrm{Perf}(X_{\mathrm{dR}}^{\mathrm{an}} \times S)) \simeq \mathrm{Fun}(T_n, \mathrm{Perf}(X_{\mathrm{B}}^{\mathrm{an}} \times S)) \\ &\simeq S_n \mathbf{AnPerf}(X_{\mathrm{B}}^{\mathrm{an}})(S). \end{aligned} $$

The first statement follows at once. The second statement follows automatically given the commutativity of the natural diagram

<figure><img src="image_36.png" /></figure>

**Theorem 7.22** (CoHA version of the derived Riemann–Hilbert correspondence). There is an equivalence of stable $\mathbb{E}_1$-monoidal $\infty$-categories

$$ (\mathrm{Coh}^b(\mathrm{AnCoh}_{\mathrm{dR}}(X)), \otimes_{\mathrm{dR}}^{\mathrm{an}}) \simeq (\mathrm{Coh}^b(\mathrm{AnCoh}_{\mathrm{B}}(X)), \otimes_{\mathrm{B}}^{\mathrm{an}}) $$

**Remark 7.23.** In the algebraic setting we considered the finer invariant $\mathrm{Coh}_{\mathrm{pro}}^b$, which is more adapted to the study of non-quasi-compact stacks. Among its features, there is the fact that for every derived stack $Y$ there is a canonical equivalence (cf. Proposition A.5)

$$ K(\mathrm{Coh}_{\mathrm{pro}}^b(Y)) \simeq K(\mathrm{Coh}_{\mathrm{pro}}^b(\mathrm{cl}Y)). $$

In the $\mathbb{C}$-analytic setting, a similar treatment is possible, but it is more technically involved. In the algebraic setting, the construction of $\mathrm{Coh}_{\mathrm{pro}}^b$ relies on the machinery developed in §A, which provides a canonical way of organizing exhaustion by quasi-compact substackles into a canonical ind-object. In the $\mathbb{C}$-analytic setting, one cannot proceed verbatim, because quasi-compact $\mathbb{C}$-analytic substackles are extremely rare and it is not true that every geometric derived analytic stack admits an open exhaustion by quasi-compact ones. Rather, one would have to use compact Stein subsets; see [22, Definition 2.14]. Combining [36, Corollary 4.5.1.10] and [22, Theorem 4.13], it would then be possible to compare the $K$-theory of the resulting pro-category of bounded coherent sheaves on a derived analytic stack $Y$ with the one of the classical truncation $\mathrm{cl}Y$. We will not develop the full details here.
## Appendix A. Ind quasi-compact stacks

The main object of study of the paper is the derived stack **Coh**(S) of coherent sheaves on S, where S is a smooth and proper scheme or one of Simpson's shapes of a smooth and proper scheme. This stack is typically not quasi-compact, and this requires some care when studying its invariants, such as the G-theory. For example, when X is a quasi-compact geometric derived stack, the inclusion $i: \mathrm{cl}^X \hookrightarrow X$ induces a canonical equivalence

$$i_*: G(\mathrm{cl}^X) \xrightarrow{\sim} G(X).$$

This relies on Quillen's theorem of the heart and the equivalence $\mathrm{Coh}^{\heartsuit}(\mathrm{cl}^X) \simeq \mathrm{Coh}^{\heartsuit}(X)$ induced by $i_*$. In particular, one needs quasi-compactness of X to ensure that the t-structure on $\mathrm{Coh}^b(X)$ is (globally) bounded. In this appendix, we set up a general framework to deal with geometric derived stacks that are not necessarily quasi-compact.

### A.1. Open exhaustions

Let $j: \mathrm{dGeom}^{qc} \hookrightarrow \mathrm{dSt}$ be the inclusion of the full subcategory of $\mathrm{dSt}$ spanned by quasi-compact geometric derived stacks. Left Kan extension along $j$ induces a functor

$$\Psi: \mathrm{dSt} \to \mathrm{PSh}(\mathrm{dGeom}^{qc}).$$

**Lemma A.1.** Let $X \in \mathrm{dGeom}$ be a locally geometric derived stack.

(1) There exists a (possibly transfinite) sequence

$$\emptyset = U_0 \hookrightarrow U_1 \hookrightarrow \dots \hookrightarrow U_\alpha \hookrightarrow U_{\alpha+1} \hookrightarrow \dots$$

of quasi-compact Zariski open substackles of X whose colimit is X.

(2) Let $Y \in \mathrm{dGeom}^{qc}$ be a quasi-compact geometric derived stack. For any exhaustion of X by quasi-compact Zariski open substackles of X as in the previous point, the canonical morphism

$$\underset{\alpha}{\mathrm{colim}} \mathrm{Map}_{\mathrm{dSt}}(Y, U_\alpha) \to \mathrm{Map}_{\mathrm{dSt}}(Y, X)$$

is an equivalence.

(3) The object $\Psi(X)$ belongs to the full subcategory $\mathrm{Ind}(\mathrm{dGeom}^{qc})$ of $\mathrm{PSh}(\mathrm{dGeom}^{qc})$.

*Proof.* Let $V \to X$ be a smooth atlas, where $V$ is a scheme. Let $V' \hookrightarrow V$ be the inclusion of a quasi-compact open Zariski subset. Let

$$V' := \check{\mathcal{C}}(V' \to X)$$

be the Čech nerve of $V' \to X$ and set

$$U' := |V'|.$$
The canonical map $U' \to X$ is representable by open Zariski immersions, and $U'$ is a quasi-compact stack. Since this construction is obviously functorial in $V'$, we see that any exhaustion of $V$ by quasi-compact Zariski open subschemes induces a similar exhaustion of $X$, thus completing the proof of (1).

We now prove (2). Fix an exhaustion of $X$ by quasi-compact Zariski open substackles of $X$ as in (1). For every index $\alpha$, the map $U_\alpha \to U_{\alpha+1}$ is an open immersion and therefore it is $(-1)$-truncated in dSt. Using [32, Proposition 5.5.6.16], we see that

$$ \mathrm{Map}_{\mathrm{dSt}}(Y, U_\alpha) \to \mathrm{Map}_{\mathrm{dSt}}(Y, U_{\alpha+1}) $$

is $(-1)$-truncated as well. The same holds for the maps $\mathrm{Map}_{\mathrm{dSt}}(Y, U_\alpha) \to \mathrm{Map}_{\mathrm{dSt}}(Y, X)$. As a consequence, the map

$$ \mathrm{colim}_{\alpha} \mathrm{Map}_{\mathrm{dSt}}(Y, U_\alpha) \to \mathrm{Map}_{\mathrm{dSt}}(Y, X) $$

is $(-1)$-truncated. To prove that it is an equivalence, we are left to check that it is surjective on $\pi_0$. Let $f: Y \to X$ be a morphism. Write $Y_\alpha := U_\alpha \times_X Y$. Then the sequence $\{Y_\alpha\}$ is an open exhaustion of $Y$, and since $Y$ is quasi-compact there must exist an index $\alpha$ such that $Y_\alpha = Y$. This implies that $f$ factors through $U_\alpha$, and therefore the proof of (2) is achieved.

As for (3), this immediately follows from (2) and [32, Corollary 5.3.5.4 (1)].

**Corollary A.2.** *The functor $\Psi$ restricts to a limit preserving functor*

$$ (-)_{\mathrm{ind}}: \mathrm{dGeom} \to \mathrm{Ind}(\mathrm{dGeom}^{\mathrm{qc}}). $$

## A.2. G-theory of non-quasi-compact stacks

As a consequence, when $X$ is a locally geometric derived stack, we can canonically promote $\mathrm{Coh}^b(X)$ to a pro-category

$$ \mathrm{Coh}_{\mathrm{pro}}^{b}(X) := \mathrm{Coh}^{b}(X_{\mathrm{ind}}) \simeq \text{``lim''} \mathrm{Coh}^{b}(U_{\alpha}) \in \mathrm{Pro}(\mathrm{Cat}_{\infty}^{\mathrm{st}}). $$

In particular, we can give the following definition:

**Definition A.3.** Let $X \in \mathrm{dGeom}$ be a locally geometric derived stack. The pro-spectrum of G-theory of $X$ is

$$ G_{\mathrm{pro}}(X) := K(\mathrm{Coh}_{\mathrm{pro}}^{b}(X)) \in \mathrm{Pro}(\mathrm{Sp}). $$

The spectrum of G-theory of $X$ is the realization of $G_{\mathrm{pro}}(X)$:

$$ G(X) := \lim_{\alpha} G_{\mathrm{pro}}(X) \in \mathrm{Sp}. $$

**Remark A.4.** If $X$ is quasi-compact, then $X_{\mathrm{ind}}$ is equivalent to a constant ind-object. As a consequence, both $\mathrm{Coh}_{\mathrm{pro}}^b(X)$ and $G_{\mathrm{pro}}(X)$ are equivalent to constant pro-objects and $G(X)$ simply coincides with the spectrum $K(\mathrm{Coh}^b(X))$.
**Proposition A.5.** Let $X \in \mathrm{dGeom}$ be a locally geometric derived stack. The inclusion $i: {}^\mathrm{cl}X \hookrightarrow X$ induces a canonical equivalence

$$i_*: G_{\mathrm{pro}}({}^\mathrm{cl}X) \xrightarrow{\sim} G_{\mathrm{pro}}(X),$$

and therefore an equivalence

$$i_*: G({}^\mathrm{cl}X) \xrightarrow{\sim} G(X).$$

*Proof.* Choose an exhaustion $\{U_\alpha\}$ of $X$ by quasi-compact open Zariski subsets as in Lemma A.1 (1). Then $\{{}^\mathrm{cl}U_\alpha\}$ is an exhaustion of ${}^\mathrm{cl}X$, and the map $i_*: G_{\mathrm{pro}}({}^\mathrm{cl}X) \to G_{\mathrm{pro}}(X)$ can be computed as

$$\lim_{\alpha} K(\mathrm{Coh}^b({}^\mathrm{cl}U_\alpha)) \to \lim_{\alpha} K(\mathrm{Coh}^b(U_\alpha)).$$

Since each $U_\alpha$ is quasi-compact, this is a levelwise equivalence. Therefore, it is also an equivalence at the level of pro-objects. The second statement follows by passing to realizations. ■

**Definition A.6.** Let $X \in \mathrm{dGeom}$ be a locally geometric derived stack. We define

$$G_0(X) := \pi_0 G(X).$$

**Remark A.7.** In [25, 60], the authors defined $G_0$ of a non-quasi-compact geometric classical stack $Y$ as the limit of the $G_0(V_\alpha)$ for an exhaustion $\{V_\alpha\}$ of $Y$ by quasi-compact Zariski open substackles.

The relation between the above two definitions is as follows. Let $X \in \mathrm{dGeom}$ be a locally geometric derived stack and let $\{U_\alpha\}$ be an exhaustion of $X$ by quasi-compact Zariski open substackles. Then there exists a short exact sequence

$$0 \to \lim_{\alpha}^{1} \pi_1 G(U_{\alpha}) \to G_0(X) \to \lim_{\alpha} G_0(U_{\alpha}) \to 0$$

in the abelian category of abelian groups.

**Remark A.8.** Now, we discuss the quasi-compactness of the moduli stacks of coherent sheaves we deal with in the main body of the paper.

Let $Y$ be a smooth projective complex variety. First, recall that the classical stack ${}^\mathrm{cl}\mathrm{Coh}(Y)$ decomposes into the disjoint union

$${}^\mathrm{cl}\mathrm{Coh}(Y) = \bigsqcup_P {}^\mathrm{cl}\mathrm{Coh}^P(Y),$$

with respect to the Hilbert polynomials of coherent sheaves. Here, the stacks on the right-hand side are introduced in §2.3.2. We have a corresponding decomposition at the level of derived enhancements

$$\mathrm{Coh}(Y) = \bigsqcup_P \mathrm{Coh}^P(Y). \qquad (\mathrm{A.1})$$
Now, fix $P(m) \in \mathbb{Q}[m]$. As shown e.g. in the proof of [31, Théorème 4.6.2.1], there exists an exhaustion $\{U_n^P\}_{n \in N}$ of $\mathrm{Coh}^P(Y)$ by quasi-compact open substackles such that the truncation $^{cl}U_n^P$ is a quotient stack by a certain open subset of a Quot scheme for any $n$. By [25, Proposition 4.1.1], a similar description holds for the moduli stack $\mathrm{Coh}_{\mathrm{prop}}(Y)$ of coherent sheaves with proper support on a smooth quasi-projective complex variety $Y$. Note that the decomposition (A.1) for the stack $\mathrm{Coh}_0(Y) := \mathrm{Coh}_{\mathrm{prop}}^{\le 0}(Y)$ reduces to

$$ \mathrm{Coh}_0(Y) = \bigsqcup_{k \in \mathbb{Z}_{\ge 0}} \mathrm{Coh}_0^k(Y). $$

By using the explicit description of the $U_n^k$'s, one can show that $^{cl}\mathrm{Coh}_0^k(Y)$ is a quasi-compact quotient stack, hence the stack $\mathrm{Coh}_0^k(Y)$ is quasi-compact.

Now, let $Y$ be a smooth projective complex curve. The moduli stack $\mathrm{Coh}_{\mathrm{Dol}}(Y)$ is not quasi-compact. On the other hand, the moduli stacks $\mathrm{Bun}_B^n(Y)$ and $\mathrm{Bun}_{\mathrm{DR}}^n(Y)$ are quasi-compact quotient stacks. The truncations of these stacks are quotients by the Betti and de Rham representation spaces respectively (cf. [72]). The derived stacks are quotients by the derived versions of these representation spaces (see e.g. [47, §1.2]).

### A.3. Correspondences

We finish this section by providing a formal extension of Gaitsgory–Rozenblyum correspondence machine in the setting of not necessarily quasi-compact stacks. Let $\mathcal{S}$ be an $(\infty, 2)$-category, seen as an $(\infty, 1)$-category weakly enriched in $\mathrm{Cat}_\infty$, in the sense of [14, 21]. We write $\mathrm{Cat}_\infty^{(2)}$ for $\mathrm{Cat}_\infty$ thought as weakly enriched over itself in the natural way (i.e. for the $(\infty, 2)$-category of $(\infty, 1)$-categories). Consider the 2-categorical Yoneda embedding

$$ y: \mathcal{S} \to 2\text{-Fun}(\mathbb{S}^{1\text{-op}}, \mathrm{Cat}_\infty^{(2)}). $$

Then [21, Corollary 6.2.7] guarantees that is 2-fully faithful. We let $2\text{-Ind}(\mathcal{S})$ be the full 2-subcategory of $2\text{-Fun}(\mathbb{S}^{1\text{-op}}, \mathrm{Cat}_\infty^{(2)})$ spanned by those functors that commute with finite $\mathrm{Cat}_\infty$-limits. The fully faithful functor $\mathcal{S} \to \mathrm{Cat}_\infty$ induces a fully faithful embedding

$$ \mathrm{Ind}(\mathbb{S}^{1\text{-cat}}) \to (2\text{-Fun}(\mathbb{S}^{1\text{-op}}, \mathrm{Cat}_\infty^{(2)}))^{1\text{-cat}}. $$

We let $2\text{-Ind}(\mathcal{S})$ be the full 2-subcategory of $2\text{-Fun}(\mathbb{S}^{1\text{-op}}, \mathrm{Cat}_\infty^{(2)})$ spanned by the essential image of the above functor.<fnref n="35" /> By construction, the 1-category underlying $2\text{-Ind}(\mathcal{S})$ coincides with $\mathrm{Ind}(\mathbb{S}^{1\text{-cat}})$.

**Remark A.9.** We are not sure about the intrinsic meaning of the above definition of $2\text{-Ind}(\mathcal{S})$ from a 2-categorical perspective. It is probably too little to be the correct 2-Ind construction, and we are not aware whether it satisfies some 2-categorical universal property. If “colim”$_\alpha x_\alpha$ and “colim”$_\beta y_\beta$ are two objects in $2\text{-Ind}(\mathcal{S})$ in the above sense, the

<footnote n="35">³⁵We thank Andrea Gagna and Ivan Di Liberti for helping us conceiving this definition.</footnote>
mapping category is given by the formula

$$ \mathrm{Fun}_{2-\mathrm{Ind}(\mathcal{C})}(\text{``colim''}_{\alpha} x_{\alpha}, \text{``colim''}_{\beta} y_{\beta}) = \lim_{\alpha} \mathrm{colim}_{\beta} \mathrm{Fun}_{\mathcal{S}}(x_{\alpha}, y_{\beta}). $$

This is all that is needed in what follows.

We now equip $\mathcal{C}$ with three markings $\mathcal{C}_{\text{horiz}}$, $\mathcal{C}_{\text{vert}}$ and $\mathcal{C}_{\text{adm}}$ satisfying the conditions of [12, §7.1.1.1]. We define three markings $\mathrm{Ind}(\mathcal{C})_{\text{horiz}}$, $\mathrm{Ind}(\mathcal{C})_{\text{vert}}$ and $\mathrm{Ind}(\mathcal{C})_{\text{adm}}$ on $\mathrm{Ind}(\mathcal{C})$ by declaring that a morphism $f: X \to Y$ in $\mathrm{Ind}(\mathcal{C})$ belongs to $\mathrm{Ind}(\mathcal{C})_{\text{horiz}}$ (resp. $\mathrm{Ind}(\mathcal{C})_{\text{vert}}$, $\mathrm{Ind}(\mathcal{C})_{\text{adm}}$) if it is representable by morphisms in $\mathcal{C}_{\text{horiz}}$ (resp. $\mathcal{C}_{\text{vert}}$, $\mathcal{C}_{\text{adm}}$). It is then straightforward to check that the conditions of [12, §7.1.1.1] are again satisfied. Next, we let

$$ \Phi: \mathcal{C}^{\mathrm{op}} \to \mathrm{Cat}_{\infty} $$

be a functor and let $\Phi_{\text{horiz}}$ be its restriction to $(\mathcal{C}_{\text{horiz}})^{\mathrm{op}}$. Passing to ind-objects, we obtain a functor

$$ \Phi^{\mathrm{ind}}: \mathrm{Ind}(\mathcal{C})^{\mathrm{op}} \to \mathrm{Pro}(\mathrm{Cat}_{\infty}). $$

We let $\Phi_{\text{horiz}}^{\mathrm{ind}}$ be its restriction to $(\mathrm{Ind}(\mathcal{C})_{\text{horiz}})^{\mathrm{op}}$. Before stating the next key proposition, we recall the definition of bivariance:

**Definition A.10.** A functor $\Phi: \mathcal{C}^{\mathrm{op}} \to \mathrm{Cat}_{\infty}$ is said to be a $\mathcal{C}_{\text{vert}}$-right *bivariant* if for every morphism $f: X \to Y$ in $\mathcal{C}_{\text{vert}}$, the functor $\Phi(f): \Phi(Y) \to \Phi(X)$ admit a right adjoint $\Phi_*(f)$. A $\mathcal{C}_{\text{vert}}$-right bivariant functor is said to have *base change with respect to* $\mathcal{C}_{\text{horiz}}$ if for every pullback diagram

$$ \begin{tikzcd} W \arrow[r, "g'"] \arrow[d, "f'"] & X \arrow[d, "f"'] \\ Z \arrow[r, "g"] & Y \end{tikzcd} $$

where $f \in \mathcal{C}_{\text{vert}}$ and $g \in \mathcal{C}_{\text{horiz}}$, the square

$$ \begin{tikzcd} \Phi(Y) \arrow[r, "\Phi(g)"'] & \Phi(Z) \\ \Phi(f) \arrow[d] & \Phi(f') \\ \Phi(X) \arrow[r, "\Phi(g')"'] & \Phi(W) \end{tikzcd} $$

is vertically right adjointable.

**Remark A.11.** In [12, Chapter 7], the above property is not directly considered. It rather corresponds to the right Beck–Chevalley property (see Definition 7.3.2.2 in loc. cit.) for functors with values in $(\mathrm{Cat}_{\infty}^{(2)})^{2-\mathrm{op}}$.

**Proposition A.12.** *Keeping the above notation and assumptions, suppose furthermore that*

(1) $\Phi$ is $\mathcal{C}_{\text{vert}}$-right bivariant and has base change with respect to $\mathcal{C}_{\text{horiz}}$.
(2) Every object in $\mathrm{Ind}(\mathcal{C})$ can be represented as a filtered colimit whose transition maps belong to $\mathcal{C}_{\mathrm{horiz}}$.

Then $\Phi_{\mathrm{horiz}}^{\mathrm{ind}}$ is $\mathrm{Ind}(\mathcal{C})_{\mathrm{vert}}$-right bivariant and has base change with respect to $\mathrm{Ind}(\mathcal{C})_{\mathrm{horiz}}$.

*Proof.* Let $f: X \to Y$ be a morphism in $\mathrm{Ind}(\mathcal{C})$. Choose a representation $Y \simeq$ “colim”$\alpha$ $Y_\alpha$, where the transition maps belong to $\mathcal{C}_{\mathrm{horiz}}$. For every index $\alpha$, we let

$$X_{\alpha} := Y_{\alpha} \times_{Y} X$$

and we let $f_{\alpha}: X_{\alpha} \to Y_{\alpha}$ be the induced morphism. By definition of $\mathrm{Ind}(\mathcal{C})_{\mathrm{vert}}$, $X_{\alpha}$ belongs to $\mathcal{C}$ and $f_{\alpha}$ is a morphism in $\mathcal{C}_{\mathrm{vert}}$. The morphism

$$\Phi(f_{\alpha}): \Phi(Y_{\alpha}) \to \Phi(X_{\alpha})$$

admits a left adjoint $\Phi_!(f_\alpha)$. Since $\Phi_{\mathrm{horiz}}$ satisfies the right Beck–Chevalley property with respect to $\mathcal{C}_{\mathrm{vert}}$, the morphisms $\Phi_!(f_\alpha)$ assemble into a morphism

$$\Phi_!(f): \text{``lim''} \quad \Phi(X_\alpha) \to \text{``lim''} \quad \Phi(Y_\alpha)$$

in 2-Pro($\mathrm{Cat}_\infty^{(2)}$). The triangular identities for the adjunction $\Phi_!(f_\alpha) \dashv \Phi(f_\alpha)$ induce tri- angular identities exhibiting $\Phi_!(f)$ as a left adjoint to $\Phi(f)$ in the $(\infty, 2)$-category 2-Pro($\mathrm{Cat}_\infty^{(2)}$). For every morphism $Z \to Y$ in $\mathrm{Ind}(\mathcal{C})_{\mathrm{horiz}}$, we let $Z_\alpha := Y_\alpha \times_Y Z$. The induced morphism $Z_\alpha \to Y_\alpha$ belongs to $\mathcal{C}_{\mathrm{horiz}}$ by definition. In this way, we can describe the Beck–Chevalley transformation for the diagram

$$\begin{tikzcd}
\Phi(Y) \arrow[r] \arrow[d] & \Phi(X) \arrow[d] \\
\Phi(Z) \arrow[r] & \Phi(X \times_Y Z)
\end{tikzcd}$$

in terms of the Beck–Chevalley transformation for the diagram

$$\begin{tikzcd}
\Phi(Y_\alpha) \arrow[r] \arrow[d] & \Phi(X_\alpha) \arrow[d] \\
\Phi(Z_\alpha) \arrow[r] & \Phi(X_\alpha \times_{Y_\alpha} Z_\alpha)
\end{tikzcd}$$

which holds by assumption.

■

Seeing $\mathrm{Pro}(\mathrm{Cat}_\infty)$ as the underlying 1-category of 2-Pro($\mathrm{Cat}_\infty^{(2)}$), we obtain

**Corollary A.13.** Keeping the above notation, assume $\mathcal{C}_{\mathrm{vert}} \subset \mathcal{C}_{\mathrm{horiz}}$ and $\mathcal{C}_{\mathrm{vert}} = \mathcal{C}_{\mathrm{adm}}$. Then under the assumptions of Proposition A.12, the functor

$$\Phi^{\mathrm{ind}}: \mathrm{Ind}(\mathcal{C})^{\mathrm{op}} \to \mathrm{Pro}(\mathrm{Cat}_{\infty})$$

uniquely extends to a functor

$$\Phi_{\mathrm{corr}}^{\mathrm{ind}}: \mathrm{Corr}(\mathrm{Ind}(\mathcal{C}))_{\mathrm{vert}, \mathrm{horiz}}^{\mathrm{vert}} \to 2\mathrm{Pro}(\mathrm{Cat}_{\infty}^{(2)}),$$
and its restriction to $\text{Corr}(\text{Ind}(\mathcal{C}))_{\text{vert,horiz}}$ factors through the maximal $(\infty, 1)$-category $\text{Pro}(\text{Cat}_{\infty})$ of 2-Pro($\text{Cat}_{\infty}^{(2)}$).

*Proof.* It is enough to apply [12, Theorem 7.3.2.2 (b)] to the $(\infty, 2)$-category

$$S = 2-\text{Pro}(\text{Cat}_{\infty}^{(2)})^{2-\text{op}}.$$

See also [39, Theorem 4.2.6].

<figure><img src="image_37.png" /></figure>

**Corollary A.14.** There exists a uniquely defined right-lax symmetric monoidal functor

$$\text{QCoh}_{\text{pro}}: \text{Corr}^{\times}(\text{Ind}(\text{dGeom}^{\text{qc}}))_{\text{rps,all}} \to \text{Pro}(\text{Cat}_{\infty}).$$

*Proof.* Take $\mathcal{C} = \text{dSch}^{\text{qc}}$ and $\mathcal{D} = \text{dGeom}^{\text{qc}}$. For $\mathcal{C}$, we take horiz = all and adm = vert = proper. Observe that condition (5) in [12, §7.1.1.1] is satisfied. Consider the functor

$$\text{QCoh}: (\text{dSch}^{\text{qc}})^{\text{op}} \to \text{Cat}_{\infty}.$$

Applying [12, Theorem 7.3.2.2], we obtain a functor

$$\text{QCoh}: \text{Corr}(\text{dSch}^{\text{qc}})_{\text{proper,all}}^{\text{proper}} \to \text{Cat}_{\infty}^{(2)}.$$

For $\mathcal{C}$, $\mathcal{D}$ we now take horiz = all and vert = rps (morphisms that are representable by proper schemes) and adm = isom. Then [12, Theorem 8.6.1.5] supplies an extension of $\text{QCoh}$,

$$\text{QCoh}: \text{Corr}(\text{dGeom}^{\text{qc}})_{\text{rps,all}}^{\text{rps}} \to \text{Cat}_{\infty}^{(2)}.$$

Thanks again to [12, Theorem 7.3.2.2], the above functor is uniquely determined by its restriction

$$\text{QCoh}: (\text{dGeom}^{\text{qc}})^{\text{op}} \to \text{Cat}_{\infty}.$$

This is our $\Phi$. It satisfies the hypothesis of Proposition A.12. Thus, applying Corollary A.13, we obtain a functor

$$\text{QCoh}: \text{Corr}(\text{Ind}(\text{dGeom}^{\text{qc}}))_{\text{rps,all}}^{\text{rps}} \to 2-\text{Pro}(\text{Cat}_{\infty}^{(2)}),$$

which we restrict to a functor

$$\text{QCoh}_{\text{pro}}: \text{Corr}(\text{Ind}(\text{dGeom}^{\text{qc}}))_{\text{rps,all}} \to \text{Pro}(\text{Cat}_{\infty}).$$

Combining [39, Theorem 4.4.6] and [12, Proposition 9.3.2.4] we conclude that we can upgrade these constructions to right-lax symmetric monoidal functors. ■

<figure><img src="image_38.png" /></figure>

*Acknowledgements.* First, we would like to thank Mattia Talpo for suggesting that we discuss about the use of derived algebraic geometry in the theory of cohomological Hall algebras. This was the starting point of our collaboration.

Part of this work was developed while the second-named author was visiting the Université de Strasbourg and was completed while the first-named author was visiting Kavli IPMU. We are grateful to both institutions for their hospitality and wonderful working conditions. We would like to thank Andrea Appel, Kevin Costello, Ben Davison, Adeel Khan, Olivier Schiffmann, Philippe Eyssidieux, Tony Pantev and Bertrand Toën for enlightening conversations. We especially thank Andrea Gagna and Ivan Di Liberti for discussions about the correct 2-categorical notion of the Ind-construction. Finally, we thank the anonymous referee for useful suggestions and comments.
The results of the present paper have been presented by the second-named author at the Work-
shop “Cohomological Hall algebras in Mathematics and Physics” (Perimeter Institute for Theoret-
ical Physics, Canada, February 2019). The second-named author is grateful to the organizers of this
event for the invitation to speak.

*Funding.* The work of the second-named author is partially supported by World Premier Inter-
national Research Center Initiative (WPI), MEXT, Japan, by JSPS KAKENHI Grant number
JP17H06598 and by JSPS KAKENHI Grant number JP18K13402.

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