2.1 Ambidexterity

Throughout the paper, by an $(\infty , 2)$-category $\mathcal {C}$ we will mean a complete 2-fold Segal space. We refer to [Reference Johnson-Freyd and ScheimbauerJS17, § 6] for a definition of a symmetric monoidal $(\infty , 2)$-category, so that $h_2\mathcal {C}$ becomes a symmetric monoidal 2-category.

Definition 2.1 Suppose $\mathcal {E}_1$ and $\mathcal {E}_2$ are $(\infty , 2)$-categories. An adjunction

\[ {F\colon \mathcal{E}_1}\rightleftarrows{\mathcal{E}_2\colon G} \]

is an adjunction in the homotopy 2-category of $(\infty , 2)$-categories. It is called ambidextrous if the unit $\eta \colon \mathrm {id} \to GF$ and the counit $\epsilon \colon FG\to \mathrm {id}$ have right adjoints $\eta ^\mathrm {R}\colon GF\to \mathrm {id}$ and $\epsilon ^\mathrm {R}\colon \mathrm {id}\to FG$.

Remark 2.4 Recall that if $\eta \colon \mathrm {id}\to GF$ exhibits $G$ as right adjoint to $F$, then we can find a counit $\epsilon \colon FG\to \mathrm {id}$ and invertible modifications

called triangulators, satisfying the swallowtail axioms

in the homotopy 2-categories $h_2\mathcal {E}_1$ and $h_2\mathcal {E}_2$. See, for example, [Reference GurskiGur12, Remark 2.2].

Definition 2.5 Suppose $\mathcal {E}_1,\mathcal {E}_2$ are symmetric monoidal $(\infty , 2)$-categories. A symmetric monoidal ambidextrous adjunction

\[ {F\colon \mathcal{E}_1}\rightleftarrows{\mathcal{E}_2\colon G} \]

is an ambidextrous adjunction where $F$ is symmetric monoidal, satisfying the projection formula: for any objects $\mathcal {M}_1\in \mathcal {E}_1$ and $\mathcal {M}_2\in \mathcal {E}_2$, the composite

\[ \mathcal{M}_1\otimes G\mathcal{M}_2\xrightarrow{\eta} GF(\mathcal{M}_1\otimes G\mathcal{M}_2)\cong G(F\mathcal{M}_1\otimes FG\mathcal{M}_2)\xrightarrow{\mathrm{id}\otimes\epsilon} G(F\mathcal{M}_1\otimes \mathcal{M}_2) \]

is an equivalence.

Since $G$ is right adjoint to $F$, it has a natural lax monoidal structure and since it is also left adjoint to $F$, it has a natural oplax monoidal structure. Let us now work out compatibilities between the two.

The lax monoidal structure on $G$ is given by the composite

\[ G\mathcal{M}_1\otimes G\mathcal{M}_2\xrightarrow{\sim}G(FG\mathcal{M}_1\otimes \mathcal{M}_2)\xrightarrow{\epsilon\otimes\mathrm{id}} G(\mathcal{M}_1\otimes\mathcal{M}_2), \]

which we denote by $\alpha$. Similarly, the oplax monoidal structure on $G$ is given by the composite

\[ G(\mathcal{M}_1\otimes \mathcal{M}_2) \xrightarrow{\epsilon^\mathrm{R}\otimes\mathrm{id}} G(FG\mathcal{M}_1\otimes \mathcal{M}_2)\xrightarrow{\sim} G\mathcal{M}_1\otimes G\mathcal{M}_2, \]

which we denote by $\alpha ^\mathrm {R}$.

The lax compatibility with the units is expressed by the morphisms

\[ 1_{\mathcal{E}_1}\xrightarrow{\eta} GF(1_{\mathcal{E}_1})\cong G(1_{\mathcal{E}_2}),\qquad G(1_{\mathcal{E}_2})\cong GF(1_{\mathcal{E}_1})\xrightarrow{\eta^\mathrm{R}} 1_{\mathcal{E}_1}. \]

For a triple of objects $\mathcal {M}_1,\mathcal {M}_2,\mathcal {M}_3\in \mathcal {E}_2$ we have a 2-isomorphism

which gives rise to a modification

(2.7)\begin{equation} (\mathrm{id}\otimes\alpha)\circ(\alpha^\mathrm{R}\otimes\mathrm{id})\cong \alpha^\mathrm{R}\circ\alpha. \end{equation}

For a pair of objects $\mathcal {M}_1,\mathcal {M}_2\in \mathcal {E}_2$ we have a 2-isomorphism

which gives rise to a modification

(2.8)\begin{equation} \alpha\circ (\mathrm{id}\otimes\epsilon^\mathrm{R})\cong \epsilon^\mathrm{R}\circ (\epsilon\otimes\mathrm{id}). \end{equation}

Similarly, we have a 2-isomorphism

which gives rise to a modification

(2.9)\begin{equation} (\mathrm{id}\otimes\epsilon)\circ \alpha^\mathrm{R}\cong (\epsilon^\mathrm{R}\otimes\mathrm{id})\circ \epsilon. \end{equation}

Proposition 2.11 Let $\mathcal {E}_1,\mathcal {E}_2$ be symmetric monoidal $(\infty ,2)$-categories and $F\colon \mathcal {E}_1\rightleftarrows \mathcal {E}_2\colon G$ a symmetric monoidal ambidextrous adjunction. Then $G\colon \mathcal {E}_2\rightarrow \mathcal {E}_1$ preserves dualizable objects. Moreover, given a dualizable object $\mathcal {M}\in \mathcal {E}_2$ with the dual $\mathcal {M}^\vee$, the dual of $G(\mathcal {M})$ is given by $G(\mathcal {M}^\vee )$ with the evaluation map given by

\[ G(\mathcal{M})\otimes G(\mathcal{M}^\vee)\xrightarrow{\alpha} G(\mathcal{M}\otimes \mathcal{M}^\vee)\xrightarrow{\mathrm{ev}} G(1_{\mathcal{E}_2})\xrightarrow{\eta^\mathrm{R}} 1_{\mathcal{E}_1} \]

and the coevaluation map given by

\[ 1_{\mathcal{E}_1}\xrightarrow{\eta} G(1_{\mathcal{E}_2})\xrightarrow{\mathrm{coev}} G(\mathcal{M}\otimes\mathcal{M}^\vee)\xrightarrow{\alpha^\mathrm{R}} G(\mathcal{M})\otimes G(\mathcal{M}^\vee). \]

Proof. We construct triangulators using the diagrams

and

using (2.7). Here the corner 2-isomorphisms are constructed as

If $\mathcal {E}$ is a symmetric monoidal $(\infty ,2)$-category, we denote by $\mathcal {E}^{\mathrm {dual}}\subset \iota _1\mathcal {E}$ the non-full subcategory whose objects are the 1-dualizable objects and whose morphisms are the right-adjointable morphisms.

Proposition 2.12 If $F\colon \mathcal {E}_1\rightleftarrows \mathcal {E}_2\colon G$ is a symmetric monoidal ambidextrous adjunction, it restricts to an adjunction

\[ F\colon \mathcal{E}_1^{\mathrm{dual}}\rightleftarrows \mathcal{E}_2^{\mathrm{dual}}\colon G. \]

In the future we will also need a certain ‘coherent’ version of duality.

Definition 2.13 Suppose $\mathcal {E}$ is a symmetric monoidal $\infty$-category. A coherent dual pair is given by the following data:

• • objects $\mathcal {M},\mathcal {M}^\vee \in \mathcal {E}$;

• • 1-morphisms $\mathrm {coev}\colon 1\rightarrow \mathcal {M}\otimes \mathcal {M}^\vee$ and $\mathrm {ev}\colon \mathcal {M}^\vee \otimes \mathcal {M}\rightarrow 1$;

• • invertible 2-morphisms

  

These are required to satisfy the swallowtail axioms

which are understood as equalities in the homotopy 2-category $h_2\mathcal {E}$.

By [Reference Pstra̧gowskiPst14, Theorem 2.14], every dualizable object is part of a coherent dual pair.

2.2 Rigidity

In this section we define the notion of a rigid symmetric monoidal functor, generalizing the discussion of [Reference GaitsgoryGai15, § D].

Let $\mathrm {Mod}_k=\mathrm {Mod}_k(\mathrm {Sp})\in \mathcal {P}\mathrm {r}^{\mathrm {St}}$ be the $\infty$-category of $k$-modules. The $\infty$-category

\[ \mathcal{P}\mathrm{r}^{\mathrm{St}}_k=\mathrm{Mod}_{\mathrm{Mod}_k}(\mathcal{P}\mathrm{r}^{\mathrm{St}}) \]

has an induced symmetric monoidal structure. Let us also introduce the symmetric monoidal $(\infty , 2)$-category

\[ \mathbf{Pr}^{\mathrm{St}}_k = \mathrm{Mod}_{\mathrm{Mod}_k}(\mathbf{Pr}^{\mathrm{St}}). \]

Let $\mathcal {C}$ be a $k$-linear symmetric monoidal $\infty$-category. Then $\mathcal {C}$ is an object in

\[ \mathrm{CAlg}(\mathcal{P}\mathrm{r}^{\mathrm{St}}_k) = \mathrm{CAlg}(\iota_1\mathbf{Pr}^{\mathrm{St}}_k). \]

We denote by $\mathrm {Mod}_\mathcal {C} = \mathrm {Mod}_\mathcal {C}(\mathcal {P}\mathrm {r}^{\mathrm {St}}_k)$ its $\infty$-category of modules. As explained in [Reference Hoyois, Scherotzke and SibillaHSS17, § 4.4], there exists a functor

\[ \mathrm{CAlg}(\iota_1\mathbf{Pr}^{\mathrm{St}}_k) \to \mathrm{Cat}^\otimes_{(\infty,2)} \quad \mathcal{C} \mapsto \mathrm{Mod}_\mathcal{C}(\mathbf{Pr}^{\mathrm{St}}_k) \]

sending $\mathcal {C}$ to the monoidal $(\infty ,2)$-category of $\mathcal {C}$-modules in $\mathbf {Pr}^{\mathrm {St}}_k$. Further, as shown in [Reference Hoyois, Scherotzke and SibillaHSS17, § 4.4], this construction has the property that

\[ \iota_1 \mathrm{Mod}_\mathcal{C}(\mathbf{Pr}^{\mathrm{St}}_k) \simeq \mathrm{Mod}_\mathcal{C}. \]

Thus $\mathrm {Mod}_\mathcal {C}(\mathbf {Pr}^{\mathrm {St}}_k)$ gives a natural $(\infty , 2)$-categorical enhancement of $\mathrm {Mod}_\mathcal {C}$.

Given a symmetric monoidal functor $f\colon \mathcal {D}\rightarrow \mathcal {C}$, we get an induced adjunction

\[ {f^*\colon \mathrm{Mod}_\mathcal{D}(\mathbf{Pr}^{\mathrm{St}}_k)}\rightleftarrows{\mathrm{Mod}_\mathcal{C}(\mathbf{Pr}^{\mathrm{St}}_k)\colon f_*} \]

where the functor $f^*$ sends a $\mathcal {D}$-module category $\mathcal {M}$ to $\mathcal {C}\otimes _\mathcal {D} \mathcal {M}$ and $f_*$ is the forgetful functor. The counit of the adjunction

\[ \epsilon_\mathcal{M}\colon f^*f_*\mathcal{M}\cong \mathcal{C}\otimes_\mathcal{D}\mathcal{M}\longrightarrow \mathcal{M} \]

is given by the action map, and the unit

\[ \eta_\mathcal{N}\colon \mathcal{N}\longrightarrow f_*f^* \mathcal{N}\cong \mathcal{N}\otimes_\mathcal{D} \mathcal{C} \]

is given by $n\mapsto n\boxtimes 1_\mathcal {C}$.

Let

\[ \Delta\colon \mathcal{C}\otimes_\mathcal{D}\mathcal{C}\longrightarrow \mathcal{C} \]

be the tensor product functor. It can naturally be enhanced to a morphism in $\mathrm {Mod}_{\mathcal {C}\otimes _\mathcal {D}\mathcal {C}}$. Since the underlying functor preserves colimits, it has a possibly discontinuous right adjoint. Moreover, the right adjoint a priori is only lax compatible with the action of $\mathcal {C}\otimes _\mathcal {D}\mathcal {C}$.

Given a rigid symmetric monoidal functor $f\colon \mathcal {D}\rightarrow \mathcal {C}$, we denote the corresponding right adjoints by

\[ f^\mathrm{R}\colon \mathcal{C}\longrightarrow \mathcal{D},\qquad \Delta^\mathrm{R}\colon \mathcal{C}\longrightarrow \mathcal{C}\otimes_\mathcal{D}\mathcal{C}. \]

Proposition 2.17 The functors

\[ \mathrm{ev}\colon \mathcal{C}\otimes_{\mathcal{D}}\mathcal{C}\stackrel{\Delta}\longrightarrow \mathcal{C}\stackrel{f^\mathrm{R}}\longrightarrow \mathcal{D} \]

and

\[ \mathrm{coev}\colon \mathcal{D}\stackrel{f}\longrightarrow \mathcal{C}\stackrel{\Delta^\mathrm{R}}\longrightarrow \mathcal{C}\otimes_\mathcal{D}\mathcal{C} \]

exhibit a self-duality $\mathcal {C}\simeq \mathcal {C}^\vee$ in $\mathrm {Mod}_{\mathcal {D}}$.

Proof. We have to check that the composite

\begin{align*} \mathcal{C}&\xrightarrow{\mathrm{id}\otimes f} \mathcal{C}\otimes_{\mathcal{D}} \mathcal{C} \\ &\xrightarrow{\mathrm{id}\otimes\Delta^\mathrm{R}} \mathcal{C}\otimes_{\mathcal{D}}\mathcal{C}\otimes_{\mathcal{D}} \mathcal{C} \\ &\xrightarrow{\Delta\otimes\mathrm{id}} \mathcal{C}\otimes_{\mathcal{D}}\mathcal{C} \\ &\xrightarrow{f^\mathrm{R}\otimes\mathrm{id}} \mathcal{C} \end{align*}

is naturally isomorphic to the identity. Indeed, by the first axiom of rigidity we know that $\Delta ^\mathrm {R}$ lies in $\mathrm {Mod}_{\mathcal {C}\otimes _\mathcal {D} \mathcal {C}}$. Using the canonical algebra map $\mathcal {C} \to \mathcal {C}\otimes _\mathcal {D} \mathcal {C}$ given by $x\mapsto x\boxtimes 1$, we see that $\Delta ^\mathrm {R}$ also lies in $\mathrm {Mod}_\mathcal {C}$. Hence we have a commutative diagram of $\infty$-categories

and the claim follows since the composite

\[ \mathcal{C}\xrightarrow{\mathrm{id}\otimes f} \mathcal{C}\otimes_{\mathcal{D}}\mathcal{C} \xrightarrow{\Delta} \mathcal{C} \]

is naturally isomorphic to the identity.

Given an object $x\in \mathcal {C}$, the functor $\mathcal {D}\rightarrow \mathcal {C}$ given by $d\mapsto f(d)\otimes x$ preserves colimits, so it admits a right adjoint $\underline {\mathrm {Hom}}(x, -)\colon \mathcal {C}\to \mathcal {D}$, which is a lax $\scr D$-module functor.

Proof. Suppose $x\in \mathcal {C}$ is dualizable. Then we have a sequence of equivalences

\begin{align*} \mathrm{Map}_{\mathcal{C}}(f(d)\otimes x, y)&\simeq \mathrm{Map}_{\mathcal{C}}(f(d), x^\vee\otimes y) \\ &\simeq \mathrm{Map}_{\mathcal{D}}(d, f^\mathrm{R}(x^\vee\otimes y)) \end{align*}

and hence $\underline {\mathrm {Hom}}(x, y)\simeq f^\mathrm {R}(x^\vee \otimes y)$. Therefore, $\underline {\mathrm {Hom}}(x, -)$ preserves colimits since $f^\mathrm {R}$ does, and it is $\mathcal {D}$-linear since $f^\mathrm {R}$ is.

Conversely, suppose $\underline {\mathrm {Hom}}(x, -)$ preserves colimits and is $\mathcal {D}$-linear. Consider the functor

\[ \underline{\mathrm{Hom}}_2(x, -)\colon \mathcal{C}\otimes_{\mathcal{D}}\mathcal{C}\rightarrow \mathcal{C} \]

obtained from $\underline {\mathrm {Hom}}(x,-)$ by extending scalars from $\scr D$ to $\scr C$.

We define the duality data as follows. Let

\[ x^\vee = \underline{\mathrm{Hom}}_2(x, \Delta^\mathrm{R}(1)). \]

By construction we have an evaluation morphism $x^\vee \boxtimes x\rightarrow \Delta ^\mathrm {R}(1)$.

We define the coevaluation to be the composite

\begin{align*} 1 &\longrightarrow \underline{\mathrm{Hom}}_2(x, 1\boxtimes x) \\ &\longrightarrow \underline{\mathrm{Hom}}_2(x, \Delta^\mathrm{R}(x)) \\ &\stackrel{\sim}\longleftarrow \underline{\mathrm{Hom}}_2(x, x\otimes_1 \Delta^\mathrm{R}(1)) \\ &\simeq x\otimes \underline{\mathrm{Hom}}_2(x, \Delta^\mathrm{R}(1)) \\ &= x\otimes x^\vee.\end{align*}

Here we use the unit $1\boxtimes x\rightarrow \Delta ^\mathrm {R}\circ \Delta (1\boxtimes x)\simeq \Delta ^\mathrm {R}(x)$ in the second line, the first axiom of rigidity in the third line, and the $\mathcal {C}$-linearity of $\underline {\mathrm {Hom}}_2(x,-)$ in the fourth line.

The evaluation is defined to be the composite

\begin{align*} x^\vee\otimes x &= \Delta(x^\vee\boxtimes x) \\ &\longrightarrow \Delta\circ \Delta^\mathrm{R}(1) \\ &\longrightarrow 1.\end{align*}

The duality axioms follow from the naturality of the unit and counit morphisms.

Conversely, one has the following statement.

Let us state several important properties of rigid symmetric monoidal functors which we will need.

Proposition 2.21 Suppose $f\colon \mathcal {D}\rightarrow \mathcal {C}$ is rigid. Then the adjunction

\[ {f^*\colon \mathrm{Mod}_\mathcal{D}(\mathbf{Pr}^{\mathrm{St}}_k)}\rightleftarrows{\mathrm{Mod}_\mathcal{C}(\mathbf{Pr}^{\mathrm{St}}_k)\colon f_*} \]

is a symmetric monoidal ambidextrous adjunction.

Proof. Recall that the first axiom of rigidity states that $\Delta \colon \mathcal {C}\otimes _\mathcal {D}\mathcal {C}\rightarrow \mathcal {C}$ has a right adjoint $\Delta ^\mathrm {R}$ in $\mathrm {Mod}_{\mathcal {C}\otimes _\mathcal {D} \mathcal {C}}(\mathbf {Pr}^{\mathrm {St}}_k)$. In particular, it has a right adjoint in $\mathrm {Mod}_\mathcal {C}(\mathbf {Pr}^{\mathrm {St}}_k)$, where $\mathcal {C}\rightarrow \mathcal {C}\otimes _\mathcal {D}\mathcal {C}$ is given by $c\mapsto 1\otimes c$. We can identify $\epsilon \cong \Delta \otimes _\mathcal {C}\mathrm {id}_{(-)}$, therefore it has a right adjoint given by $\Delta ^\mathrm {R}\otimes _\mathcal {C}\mathrm {id}_{(-)}$ which is obviously a strict natural transformation.

Similarly, the second axiom of rigidity states that $f\colon \mathcal {D}\rightarrow \mathcal {C}$ has a right adjoint $f^\mathrm {R}$ in $\mathrm {Mod}_\mathcal {D}(\mathbf {Pr}^{\mathrm {St}}_k)$. But we can identify $\eta \cong f\otimes _\mathcal {D}\mathrm {id}_{(-)}$ and hence it has a right adjoint given by $f^\mathrm {R}\otimes _\mathcal {D}\mathrm {id}_{(-)}$.

Finally, given $\mathcal {M}_1\in \mathrm {Mod}_\mathcal {D}$ and $\mathcal {M}_2\in \mathrm {Mod}_\mathcal {C}$, the composite

\[ \mathcal{M}_1\otimes_\mathcal{D} f_*\mathcal{M}_2\xrightarrow{\eta} f_*f^*(\mathcal{M}_1\otimes_\mathcal{D} f_*\mathcal{M}_2)\cong f_*(f^*\mathcal{M}_1\otimes_\mathcal{C} f^*f_*\mathcal{M}_2)\xrightarrow{\mathrm{id}\otimes\epsilon} f_*(f^*\mathcal{M}_1\otimes_\mathcal{C} \mathcal{M}_2) \]

is an equivalence if and only if it is so in $\mathcal {P}\mathrm {r}^{\mathrm {St}}_k$. But its image in $\mathcal {P}\mathrm {r}^{\mathrm {St}}_k$ is

\[ \mathcal{M}_1\otimes \mathcal{M}_2\xrightarrow{1\otimes\mathrm{id}\otimes\mathrm{id}} \mathcal{C}\otimes \mathcal{M}_1\otimes \mathcal{M}_2\cong \mathcal{M}_1\otimes (\mathcal{C}\otimes \mathcal{M}_2)\xrightarrow{\mathrm{id}\otimes\epsilon}\mathcal{M}_1\otimes\mathcal{M}_2 \]

which is equivalent to the identity by the unit axiom.

Let us now show that rigid functors are stable under compositions and pushouts.

Proof. Since $\mathcal {D}\rightarrow \mathcal {C}$ is right adjointable as a $\mathcal {D}$-module, it is also right adjointable as an $\mathcal {E}$-module. Therefore, the composite $\mathcal {E}\rightarrow \mathcal {D}\rightarrow \mathcal {C}$ is right adjointable in $\mathrm {Mod}_\mathcal {E}(\mathbf {Pr}^{\mathrm {St}}_k)$.

The tensor product $\mathcal {C}\otimes _\mathcal {E}\mathcal {C}\rightarrow \mathcal {C}$ can be written as a composite

\[ \mathcal{C}\otimes_\mathcal{E}\mathcal{C}\rightarrow \mathcal{C}\otimes_\mathcal{D}\mathcal{C}\rightarrow \mathcal{C}. \]

The first functor can be identified with

\[ \mathcal{C}\otimes_\mathcal{E}\mathcal{C}\rightarrow \mathcal{D}\otimes_{\mathcal{D}\otimes_\mathcal{E}\mathcal{D}}(\mathcal{C}\otimes_{\mathcal{E}} \mathcal{C}). \]

Since $\mathcal {D}\otimes _\mathcal {E} \mathcal {D}\rightarrow \mathcal {D}$ is right adjointable as an $\mathcal {D}\otimes _\mathcal {E}\mathcal {D}$-module, we therefore see that $\mathcal {C}\otimes _\mathcal {E}\mathcal {C}\rightarrow \mathcal {C}\otimes _\mathcal {D}\mathcal {C}$ is right adjointable as a $\mathcal {C}\otimes _\mathcal {E}\mathcal {C}$-module.

The second functor $\mathcal {C}\otimes _\mathcal {D} \mathcal {C}\rightarrow \mathcal {C}$ is right adjointable as a $\mathcal {C}\otimes _\mathcal {D}\mathcal {C}$-module and hence as a $\mathcal {C}\otimes _\mathcal {E}\mathcal {C}$-module.

Proposition 2.23 Suppose $f\colon \mathcal {D}\rightarrow \mathcal {C}$ is rigid and $\mathcal {D}\rightarrow \mathcal {E}$ is an arbitrary symmetric monoidal functor. Then

\[ \mathcal{E}\longrightarrow \mathcal{E}\otimes_\mathcal{D} \mathcal{C} \]

is rigid.

Proof. The morphism $\mathcal {D}\rightarrow \mathcal {C}$ is right adjointable in $\mathrm {Mod}_\mathcal {D}$. The functor

\[ \mathcal{E}\otimes_\mathcal{D}(-)\colon \mathrm{Mod}_\mathcal{D}\rightarrow \mathrm{Mod}_\mathcal{E} \]

sends it to $\mathcal {E}\rightarrow \mathcal {E}\otimes _\mathcal {D}\mathcal {C}$ which is therefore also right adjointable.

Let $\mathcal {P} = \mathcal {E}\otimes _\mathcal {D} \mathcal {C}$. We can identify

\[ \mathcal{P}\otimes_\mathcal{E}\mathcal{P}\cong \mathcal{E}\otimes_{\mathcal{E}\otimes\mathcal{E}}(\mathcal{P}\otimes\mathcal{P})\cong \mathcal{E}\otimes_\mathcal{D} \mathcal{D}\otimes_{\mathcal{D}\otimes \mathcal{D}} (\mathcal{C}\otimes\mathcal{C}). \]

Therefore, we can upgrade $\mathcal {E}\otimes _{\mathcal {D}}(-)$ to a functor

\[ \mathrm{Mod}_{\mathcal{C}\otimes_\mathcal{D}\mathcal{C}}\rightarrow \mathrm{Mod}_{\mathcal{P}\otimes_\mathcal{E} \mathcal{P}}. \]

This sends the tensor functor $\mathcal {C}\otimes _\mathcal {D}\mathcal {C}\rightarrow \mathcal {C}$ to the tensor functor $\mathcal {P}\otimes _\mathcal {E}\mathcal {P}\rightarrow \mathcal {P}$ which is therefore right adjointable.

2.3 Loop spaces

Since the $\infty$-category $\mathrm {CAlg}(\mathcal {P}\mathrm {r}^{\mathrm {St}}_k)$ has all small colimits, it is naturally tensored over spaces.

Definition 2.24 Let $\mathcal {C}$ be a $k$-linear symmetric monoidal $\infty$-category. Its loop space is defined to be the $k$-linear symmetric monoidal $\infty$-category

\[ \mathcal{L}\mathcal{C} = S^1\otimes \mathcal{C}\cong \mathcal{C}\otimes_{\mathcal{C}\otimes \mathcal{C}} \mathcal{C}. \]

The inclusion of the basepoint $\mathrm {pt}\rightarrow S^1$ gives rise to a symmetric monoidal functor

\[ p_\mathcal{C}\colon \mathcal{C}\longrightarrow \mathcal{L}\mathcal{C}. \]

Given a functor of symmetric monoidal $\infty$-categories $f\colon \mathcal {D}\rightarrow \mathcal {C}$, we get an induced symmetric monoidal functor

\[ \mathcal{L} f\colon \mathcal{L}\mathcal{D}\longrightarrow \mathcal{L}\mathcal{C}. \]

Note that we can identify it with the composite

(2.26)\begin{equation} \mathcal{L}\mathcal{D}\rightarrow \mathcal{L}\mathcal{D}\otimes_\mathcal{D}\mathcal{C}\cong \mathcal{C}\otimes_{\mathcal{C}\otimes\mathcal{C}}(\mathcal{C}\otimes_\mathcal{D}\mathcal{C})\rightarrow \mathcal{C}\otimes_{\mathcal{C}\otimes\mathcal{C}}\mathcal{C}\cong \mathcal{L}\mathcal{C}. \end{equation}

We will also denote by

\[ p\colon \mathcal{L}\mathcal{D}\otimes_\mathcal{D}\mathcal{C}\rightarrow \mathcal{L}\mathcal{C} \]

the functor induced by $\mathcal {L} f$ and $p_\mathcal {C}$.

Proof. Suppose $\mathcal {D}\rightarrow \mathcal {C}$ is rigid. Therefore, $\mathcal {L}\mathcal {D}\rightarrow \mathcal {L}\mathcal {D}\otimes _\mathcal {D}\mathcal {C}$ is rigid by Proposition 2.23. Similarly, since $\mathcal {C}\otimes _\mathcal {D}\mathcal {C}\rightarrow \mathcal {C}$ is right adjointable in $\mathrm {Mod}_{\mathcal {C}\otimes _\mathcal {D}\mathcal {C}}(\mathbf {Pr}^{\mathrm {St}}_k)$,

\[ \mathcal{L}\mathcal{D}\otimes_\mathcal{D}\mathcal{C}\cong \mathcal{C}\otimes_{\mathcal{C}\otimes\mathcal{C}}(\mathcal{C}\otimes_\mathcal{D}\mathcal{C})\rightarrow \mathcal{C}\otimes_{\mathcal{C}\otimes\mathcal{C}} \mathcal{C} \]

is right adjointable in $\mathrm {Mod}_{\mathcal {L}\mathcal {D}\otimes _\mathcal {D}\mathcal {C}}(\mathbf {Pr}^{\mathrm {St}}_k)$. Therefore, the composite (2.26) is right adjointable in $\mathrm {Mod}_{\mathcal {L}\mathcal {D}}(\mathbf {Pr}^{\mathrm {St}}_k)$.

Now suppose in addition that $\mathcal {C}\otimes _\mathcal {D}\mathcal {C}\rightarrow \mathcal {C}$ is rigid. Then $\mathcal {C}\otimes _{\mathcal {C}\otimes \mathcal {C}}(\mathcal {C}\otimes _\mathcal {D}\mathcal {C})\rightarrow \mathcal {C}\otimes _{\mathcal {C}\otimes \mathcal {C}} \mathcal {C}$ is rigid by Proposition 2.23. Therefore, by Proposition 2.22 the functor $\mathcal {L}\mathcal {D}\rightarrow \mathcal {L}\mathcal {C}$ is rigid as well.

Let $\lambda \colon \mathrm {id}\rightarrow (\mathcal {L} f)_*(\mathcal {L} f)^*$ be the unit of the adjunction $ {(\mathcal {L} f)^*\colon \mathrm {Mod}_{\mathcal {L}\mathcal {D}}}\rightleftarrows{\mathrm {Mod}_{\mathcal {L}\mathcal {C}}\colon (\mathcal {L} f)_*}\!\!\!.$

2.4 Smooth and proper modules

In this section we introduce further finiteness conditions on functors and modules relevant to the uncategorified GRR theorem. The notions of smooth, proper and saturated category go back to works of Bondal, Kapranov and others in the setting of classical triangulated categories; see, for instance, [Reference Bondal and KapranovBK90]. We refer the reader to [Reference LurieLur17, § 4.6.4] for a discussion of closely related concepts in the $\infty$-categorical setting.

Proof. Since $\mathcal {D}\rightarrow \mathcal {C}$ and $\mathcal {C}\otimes _\mathcal {D}\mathcal {C}\rightarrow \mathcal {C}$ are rigid, $\mathcal {L} f$ is rigid by Proposition 2.27.

Decompose $\mathcal {L} f$ using (2.26) as

\[ \mathcal{L}\mathcal{D}\rightarrow \mathcal{L}\mathcal{D}\otimes_\mathcal{D}\mathcal{C}\cong \mathcal{C}\otimes_{\mathcal{C}\otimes\mathcal{C}}(\mathcal{C}\otimes_\mathcal{D}\mathcal{C})\rightarrow \mathcal{C}\otimes_{\mathcal{C}\otimes\mathcal{C}} \mathcal{C}. \]

Since $\mathcal {D}\rightarrow \mathcal {C}$ is twice right adjointable in $\mathrm {Mod}_{\mathcal {D}}(\mathbf {Pr}^{\mathrm {St}}_k)$ by properness of $f$, $\mathcal {L}\mathcal {D}\rightarrow \mathcal {L}\mathcal {D}\otimes _\mathcal {D}\mathcal {C}$ is twice right adjointable in $\mathrm {Mod}_{\mathcal {L}\mathcal {D}}(\mathbf {Pr}^{\mathrm {St}}_k)$.

Similarly, since $\mathcal {C}\otimes _\mathcal {D}\mathcal {C}\rightarrow \mathcal {C}$ is twice right adjointable in $\mathrm {Mod}_{\mathcal {C}\otimes _\mathcal {D}\mathcal {C}}(\mathbf {Pr}^{\mathrm {St}}_k)$ by smoothness of $f$, $\mathcal {C}\otimes _{\mathcal {C}\otimes \mathcal {C}}(\mathcal {C}\otimes _\mathcal {D}\mathcal {C})\rightarrow \mathcal {C}\otimes _{\mathcal {C}\otimes \mathcal {C}}\mathcal {C}$ is also twice right adjointable in $\mathrm {Mod}_{\mathcal {L}\mathcal {D}\otimes _\mathcal {D}\mathcal {C}}(\mathbf {Pr}^{\mathrm {St}}_k)$.

2.5 Geometric setting

Recall that a derived prestack is a functor from the $\infty$-category of connective $\mathbb {E}_\infty$ algebras over $k$ to the $\infty$-category of spaces. Our main source of rigid symmetric monoidal functors is given by considering passable morphisms of derived prestacks.

If $f\colon X\rightarrow Y$ is a morphism of prestacks, then the pullback $f^*\colon \mathrm {QCoh}(Y)\rightarrow \mathrm {QCoh}(X)$ is a symmetric monoidal functor.

Proof. The proof is similar to the proof of [Reference GaitsgoryGai15, Proposition 5.1.7].

First of all, the second axiom of passability for $f\colon X\rightarrow Y$ is exactly the second axiom of rigidity for $f^*\colon \mathrm {QCoh}(Y)\rightarrow \mathrm {QCoh}(X)$.

Since $\mathrm {QCoh}(X)$ is dualizable as a $\mathrm {QCoh}(Y)$-module category, the functor $\mathrm {QCoh}(X)\otimes _{\mathrm {QCoh}(Y)}(-)$ preserves limits. Therefore, the functor

\begin{align*} \mathrm{QCoh}(X)\otimes_{\mathrm{QCoh}(Y)}\mathrm{QCoh}(X) &= \Big(\lim_{S\rightarrow X} \mathrm{QCoh}(S) \Big)\otimes_{\mathrm{QCoh}(Y)} \mathrm{QCoh}(X) \\ &\rightarrow \lim_{S\rightarrow X} \big(\mathrm{QCoh}(S)\otimes_{\mathrm{QCoh}(Y)} \mathrm{QCoh}(X)\big) \end{align*}

is an equivalence where the limit is over affine derived schemes $S$ with a morphism to $X$. Since the diagonal of $Y$ is quasi-affine, the morphism $S\rightarrow X\rightarrow Y$ is quasi-affine as well. But then by [Reference GaitsgoryGai15, Proposition B.1.3], the functor $\mathrm {QCoh}(S)\otimes _{\mathrm {QCoh}(Y)} \mathrm {QCoh}(X)\rightarrow \mathrm {QCoh}(S\times _Y X)$ is an equivalence. Since

\[ \mathrm{QCoh}(X\times_Y X)\simeq \lim_{S\rightarrow X} \mathrm{QCoh}(S\times_Y X), \]

this proves that the natural functor $\mathrm {QCoh}(X)\otimes _{\mathrm {QCoh}(Y)} \mathrm {QCoh}(X)\rightarrow \mathrm {QCoh}(X\times _Y X)$ is an equivalence.

Since the diagonal $X\rightarrow X\times _Y X$ is quasi-compact and representable, the functor

\[ \Delta_*\colon \mathrm{QCoh}(X)\rightarrow \mathrm{QCoh}(X)\otimes_{\mathrm{QCoh}(Y)} \mathrm{QCoh}(X)\simeq \mathrm{QCoh}(X\times_Y X) \]

is continuous and satisfies the projection formula. This immediately implies the first axiom of rigidity of $f^*\colon \mathrm {QCoh}(Y)\rightarrow \mathrm {QCoh}(X)$.

3.1 Traces

In this section we recall the definition of the trace functor given in [Reference Hoyois, Scherotzke and SibillaHSS17, § 2.2] and state some of its properties.

Let $\mathcal {C}$ be a symmetric monoidal $(\infty ,n)$-category. In [Reference Hoyois, Scherotzke and SibillaHSS17, § 2.3] we define a symmetric monoidal $(\infty , n-1)$-category $\mathrm {Aut}(\mathcal {C}),$ which carries a canonical $S^1$-action. The objects and 1-morphisms of $\mathrm {Aut}(\mathcal {C})$ can be described as follows.

1. (i) An object of $\mathrm {Aut}(\mathcal {C})$ is a pair $(A, a),$ where $A$ is a dualizable object of $\mathcal {C},$ and $a$ is an automorphism of $A.$

2. (ii) A $1$-morphism $(A, a) \to (B,b)$ in $\mathrm {Aut}(\mathcal {C})$ is a commutative diagram

   
   where $\phi : A \to B$ is right dualizable, and $\alpha$ is an invertible 2-cell.

We also define a trace functor

\[ \mathrm{Tr}\colon \mathrm{Aut}(\mathcal{C}) \longrightarrow \Omega\mathcal{C} \]

which is symmetric monoidal and natural in $\mathcal {C}$ [Reference Hoyois, Scherotzke and SibillaHSS17, Definitions 2.9 and 2.11].

Proposition 3.4 [Reference Hoyois, Scherotzke and SibillaHSS17, Theorem 2.14]

The symmetric monoidal trace functor

\[ \mathrm{Tr}\colon \mathrm{Aut}(\mathcal{C}) \longrightarrow \Omega\mathcal{C} \]

is $S^1$-invariant with respect to the canonical $S^1$-action on $\mathrm {Aut}(\mathcal {C})$ and the trivial $S^1$-action on $\Omega \mathcal {C}.$

Proposition 3.5 Let $\mathcal {C}$ be a symmetric monoidal $(\infty , n)$-category. The composite

\[ \mathrm{Aut}(\Omega\mathcal{C})\cong \Omega\mathrm{Aut}(\mathcal{C})\xrightarrow{\Omega\mathrm{Tr}}\Omega^2\mathcal{C} \]

is equivalent to the trace functor $\mathrm {Tr}\colon \mathrm {Aut}(\Omega \mathcal {C})\rightarrow \Omega (\Omega \mathcal {C})$ as an $S^1$-equivariant symmetric monoidal functor.

Proof. Consider the diagram

where the lower vertical maps are evaluation at the trace $\mathrm {Tr}(u)\in \Omega \mathrm {Fr}^\mathrm {rig}(S^1)$ of the universal automorphism $u$. By construction, the vertical composites are the respective trace functors, whose $S^1$-equivariance is induced by an $S^1$-invariant refinement of $\mathrm {Tr}(u)$ (see the proof of [Reference Hoyois, Scherotzke and SibillaHSS17, Theorem 2.14]). This is therefore a commutative diagram of $S^1$-equivariant symmetric monoidal functors, which proves the claim.

3.2 Chern character

Let $\mathcal {C}$ be a $k$-linear symmetric monoidal $\infty$-category. The identity functor $S^1 \otimes \mathcal {C} \simeq \mathcal {L} \mathcal {C} \to \mathcal {L} \mathcal {C}$ induces by adjunction a symmetric monoidal functor $\mathcal {C}\to \mathrm {Fun}(S^1, \mathcal {L}\mathcal {C})$ and a functor $S^1\to \mathrm {Fun}^\otimes (\mathcal {C}, \mathcal {L}\mathcal {C})$. Choosing once and for all a basepoint $p\colon \mathrm {pt}\to S^1$, the latter is equivalent to the following data:

• • a symmetric monoidal functor $p_\mathcal {C}\colon \mathcal {C} \to \mathcal {L} \mathcal {C}$, which is induced by the inclusion of the basepoint;

• • a natural equivalence of symmetric monoidal functors $\mathrm {mon}\colon p_\mathcal {C} \xrightarrow {\simeq } p_\mathcal {C}.$

Definition 3.7 The Chern character of $\mathcal {C}$ is the composite

\[ \mathrm{ch}\colon \mathcal{C}^\mathrm{dual} \longrightarrow \mathrm{Fun}(S^1, (\mathcal{L} \mathcal{C})^\mathrm{dual}) \simeq \mathrm{Aut}(\mathcal{L} \mathcal{C}) \stackrel{\mathrm{Tr}} \longrightarrow \Omega \mathcal{L} \mathcal{C}. \]

By Proposition 3.4, $\mathrm {ch}$ is $S^1$-equivariant and hence factors through the fixed points of the $S^1$-action on $\Omega \mathcal {L} \mathcal {C}$:

\[ \mathrm{ch}\colon \mathcal{C}^\mathrm{dual} \longrightarrow (\Omega\mathcal{L}\mathcal{C})^{S^1}. \]

We will now relate the monodromy automorphism to certain cocartesian diagrams. Let $p\colon \mathrm {pt}\rightarrow S^1$ be the basepoint. We have $\mathrm {Aut}(p) = \mathbb {Z}$. Consider morphisms of spaces

(3.8)

The two composites $\mathrm {pt}\coprod \mathrm {pt}\rightarrow \mathrm {pt}\rightarrow S^1$ are equal, so the space of $2$-cells completing this diagram to a square is given by $\mathrm {Aut}(p)\times \mathrm {Aut}(p)\cong \mathbb {Z}\times \mathbb {Z}$. For every pair of integers $(n, m)$, we thus obtain a commutative square of the form above. Given $\mathcal {C}\in \mathrm {CAlg}(\mathcal {P}\mathrm {r}^{\mathrm {St}}_k)$, the above square of spaces gives rise to a square

in $\mathrm {CAlg}(\mathcal {P}\mathrm {r}^{\mathrm {St}}_k)$.

Proposition 3.9 Let $\mathcal {C}\in \mathrm {CAlg}(\mathcal {P}\mathrm {r}^{\mathrm {St}}_k)$ be a $k$-linear symmetric monoidal $\infty$-category. The $(1, 0)$ square

is cocartesian.

Proof. It is enough to prove the claim in the $\infty$-category $\mathcal {S}$ of spaces.

We have a homotopy pushout diagram of groupoids

which is a cocartesian square in $\mathcal {S}$.

We can complete it to a diagram

where the isomorphism at the bottom sends the morphism $a$ to the non-trivial automorphism of the point and the morphism $b$ to the identity. The total diagram is exactly a diagram (3.8) of type $(1,0)$, which proves the claim.

Remark 3.10 Note that, whiskering the $(1, 0)$ square

along $\mathcal {C}\rightarrow \mathcal {C}\otimes \mathcal {C}$ given by $x\mapsto x\boxtimes 1$, we obtain the monodromy automorphism of $p_\mathcal {C}\colon \mathcal {C}\rightarrow \mathcal {L}\mathcal {C}$. Whiskering the same $(1, 0)$ square along $x\mapsto 1\boxtimes x$, we obtain the identity.

3.3 Categorified Chern character

Let $\mathcal {C}$ be a $k$-linear symmetric monoidal $\infty$-category, and let

(3.12)

be the monodromy automorphism. Applying the functor $\mathrm {Mod}(-)$ to (

) gives a natural equivalence

(3.13)

Remark 3.15 It follows from Remark 3.10 that the categorified monodromy automorphism can also be obtained by whiskering the $(1, 0)$ square

(3.16)

along $\pi _1^* : \mathrm {Mod}_\mathcal {C}\rightarrow \mathrm {Mod}_ {\mathcal {C}\otimes \mathcal {C}}.$ In particular, evaluating (

3.16

) on $\mathcal {M} \boxtimes \mathcal {N}\in \mathrm {Mod}_{\mathcal {C} \otimes \mathcal {C}}$ induces the automorphism

\[ \mathrm{Mon}_{\mathcal{M}} \otimes \mathrm{id}: p^*\mathcal{M} \otimes_{\mathcal{L} \mathcal{C}} p^* \mathcal{N} \to p^* \mathcal{M} \otimes_{\mathcal{L} \mathcal{C}} p^* \mathcal{N} . \]

Restricting to dualizable objects, we get an equivalence

and this determines a map

\[ B\mathbb{Z} \simeq S^1 \to \mathrm{Fun}^\otimes(\mathrm{Mod}_{\mathcal{C}}^\mathrm{dual},\mathrm{Mod}_{\mathcal{L}\mathcal{C}}^\mathrm{dual}). \]

By adjunction we obtain a symmetric monoidal functor

\[ \mathrm{Mod}^{\mathrm{dual}}_{\mathcal{C}} \to \mathrm{Fun}(S^1, \mathrm{Mod}^{\mathrm{dual}}_{\mathcal{L} \mathcal{C}}). \]

Definition 3.17 The categorified Chern character is the composite

\[ \mathrm{Ch}: \mathrm{Mod}^{\mathrm{dual}}_{\mathcal{C}} \longrightarrow \mathrm{Fun}(S^1, \mathrm{Mod}^{\mathrm{dual}}_{\mathcal{L} \mathcal{C}}) \cong \mathrm{Aut}(\mathrm{Mod}^ \mathrm{dual} _{\mathcal{L} \mathcal{C}}) \xrightarrow{\mathrm{Tr}} \mathcal{L} \mathcal{C}. \]

By Proposition 3.4, $\mathrm {Ch}$ is $S^1$-equivariant and hence factors through the fixed points of the $S^1$-action on $\mathcal {L} \mathcal {C}$, that is, the $\infty$-category of $S^1$-equivariant objects of $\mathcal {L}\mathcal {C}$:

\[ \mathrm{Ch}: \mathrm{Mod}^{\mathrm{dual}}_{\mathcal{C}} \longrightarrow (\mathcal{L}\mathcal{C})^{S^1}. \]

3.4 Decategorifying the Chern character

Note that the categorified Chern character being symmetric monoidal induces a map of spaces

\[ \mathrm{Ch}: \Omega\mathrm{Mod}_\mathcal{C}^{\mathrm{dual}}\longrightarrow \Omega(\mathcal{L}\mathcal{C})^{S^1}\simeq (\Omega\mathcal{L}\mathcal{C})^{S^1}. \]

We will show that this map coincides with the uncategorified Chern character.

Lemma 3.18 The composite

\[ \mathrm{CAlg}(\mathcal{P}\mathrm{r}^{\mathrm{St}}_k)\xrightarrow{\mathrm{Mod}}\mathrm{CAlg}(\mathrm{Cat}_{(\infty, 2)})\xrightarrow{\Omega} \mathrm{CAlg}(\mathrm{Cat}_{(\infty, 1)}) \]

is equivalent to the forgetful functor.

Theorem 3.19 Let $\mathcal {C}$ be a $k$-linear symmetric monoidal $\infty$-category. The composite

\[ \mathcal{C}^\mathrm{dual} \cong \Omega\mathrm{Mod}_\mathcal{C}^{\mathrm{dual}}\xrightarrow{\mathrm{Ch}} \Omega\mathcal{L}\mathcal{C} \]

is equivalent to the Chern character

\[ \mathrm{ch}\colon \mathcal{C}^\mathrm{dual} \longrightarrow \Omega\mathcal{L}\mathcal{C} \]

as $S^1$-equivariant $\mathbb {E}_\infty$ maps.

Proof. The composite

\[ S^1\xrightarrow{\mathrm{mon}}\mathrm{Fun}^{\otimes}(\mathcal{C}, \mathcal{L}\mathcal{C})\xrightarrow{\mathrm{Mod}}\mathrm{Fun}^{\otimes}(\mathrm{Mod}_\mathcal{C}, \mathrm{Mod}_{\mathcal{L}\mathcal{C}})\xrightarrow{\Omega} \mathrm{Fun}^{\otimes}(\mathcal{C}, \mathcal{L}\mathcal{C}) \]

is equivalent by Lemma 3.18 to $\mathrm {mon}\colon S^1\rightarrow \mathrm {Fun}^{\otimes }(\mathcal {C}, \mathcal {L}\mathcal {C})$. Therefore, by adjunction we get a commutative diagram

of $S^1$-equivariant symmetric monoidal $\infty$-categories.

Consider the diagram

of $S^1$-equivariant $\mathbb {E}_\infty$ spaces. The bottom square commutes by Proposition 3.5. The vertical composite on the left coincides with the Chern character $\mathrm {ch}\colon \mathcal {C}^\mathrm {dual}\rightarrow \Omega \mathcal {L}\mathcal {C}$ and the claim follows from the commutativity of the diagram.

4.1 Statement

We have an obvious functoriality of the Chern character with respect to symmetric monoidal functors.

Proposition 4.1 Let $f\colon \mathcal {D}\rightarrow \mathcal {C}$ be a symmetric monoidal functor. Then there is a commutative diagram of $\infty$-categories

(4.2)

Moreover, if $f$ is rigid, by Propositions 2.12 and 2.27 we can pass to right adjoints of the vertical functors in (4.2); the resulting diagram a priori only commutes up to a natural transformation.

Theorem 4.3 Let $f\colon \mathcal {D}\rightarrow \mathcal {C}$ be a rigid symmetric monoidal functor. Then, passing to right adjoints of the vertical functors in (4.2), we obtain a diagram

(4.4)

which commutes up to an invertible $2$-morphism.

The rest of the section is devoted to the proof of this theorem. Let $\mathcal {M}$ be a dualizable $\mathcal {C}$-module category. Without loss of generality (see [Reference Pstra̧gowskiPst14, Theorem 2.14]) we may assume that the duality data for $\mathcal {M}$ is coherent in the sense of Definition 2.13.

The natural transformation in (4.4) is obtained as the composite

\[ \mathrm{Ch}(f_*\mathcal{M})\longrightarrow (\mathcal{L} f)^\mathrm{R} (\mathcal{L} f) \mathrm{Ch}(f_*\mathcal{M})\cong (\mathcal{L} f)^\mathrm{R} \mathrm{Ch}(f^* f_*\mathcal{M}) \longrightarrow (\mathcal{L} f)^\mathrm{R} \mathrm{Ch}(\mathcal{M}). \]

In turn, this is obtained as the composite 2-morphism in the diagram

(4.5)

in $\mathrm {Mod}_{\mathcal {L}\mathcal {D}}(\mathbf {Pr}^{\mathrm {St}}_k)$, where the columns are given by individual Chern characters. We will prove that the composite 2-morphism in (

) is a 2-isomorphism. All equalities of 2-morphisms in this section are to be understood in the homotopy 2-category.

As a first step, we will analyze the subdiagrams in (4.5) containing $(\epsilon ^\mathrm {R})^\vee$. We have a 2-isomorphism $(\epsilon ^\mathrm {R}_\mathcal {M})^\vee \cong \epsilon _{\mathcal {M}^\vee }$ constructed via the following diagram:

(4.6)

where the bottom-left square has the 2-isomorphism given by (2.8).

4.2 Analyzing the (co)evaluation

We will first apply the isomorphism $(\epsilon ^\mathrm {R}_\mathcal {M})^\vee \cong \epsilon _{\mathcal {M}^\vee }$ to the top part of diagram (4.5).

Lemma 4.7 Under the identification $(\epsilon ^\mathrm {R}_\mathcal {M})^\vee \cong \epsilon _{\mathcal {M}^\vee }$ given by (4.6) the diagram

becomes equal to

in $h_2\mathrm {Mod}_\mathcal {C}(\mathbf {Pr}^{\mathrm {St}}_k)$, where the bottom rectangle is given by (2.9).

Proof. The original diagram can be expanded to

Using the fact that the 2-isomorphisms (2.8) and (2.9) are modifications, we get

Using the fact that the triangulator $\tau _1$ is a modification of natural transformations $\epsilon \eta \cong \mathrm {id}$, we obtain

Canceling out the two triangulators $\tau _1$ appearing in the modifications

\[ (\mathrm{id}\otimes\eta^\mathrm{R})\circ\alpha^\mathrm{R}\cong\mathrm{id},\qquad (\mathrm{id}\otimes\eta^\mathrm{R})\circ (\mathrm{id}\otimes\epsilon^\mathrm{R})\cong\mathrm{id}, \]

we obtain

Using naturality of $\epsilon ^\mathrm {R}$, we get

Using the fact that (2.9) is a modification, we get

Applying naturality of $\epsilon$, we get

Using the swallowtail axiom for the coherent duality data for $\mathcal {M}$ we get the result.

From Lemma 4.7 we obtain that the diagram

is equal to that obtained by applying $(\mathcal {L} f)_*p^*_\mathcal {C}$ to

Using the obvious equivalence $\epsilon \otimes \epsilon \cong \epsilon \circ \alpha$ and Lemma 2.10(2), it is equal to

Now we will analyze the bottom part of diagram (4.5) in a similar way.

Lemma 4.8 Under the identification $(\epsilon ^\mathrm {R}_\mathcal {M})^\vee \cong \epsilon _{\mathcal {M}^\vee }$ given by (4.6) the diagram

becomes equal to

where the top rectangle is given by (2.8).

From Lemma 4.8 we obtain that the diagram

is equal to that obtained by applying $(\mathcal {L} f)_*p^*_\mathcal {C}$ to

Using the equivalence $\epsilon \otimes \epsilon \cong \epsilon \circ \alpha$ and Lemma 2.10 (1), it is equivalent to

4.3 Reduction to $\mathcal {M}=\mathcal {C}$

Observe that the diagram

is equivalent to

Therefore, applying previous simplifications and removing invertible 2-morphisms from (4.5), we get

We have a diagram

which we have to prove commutes up to an invertible 2-morphism. It will be enough to prove that

commutes up to an invertible 2-morphism for any pair of $\mathcal {C}$-modules $\mathcal {M},\mathcal {N}$.

Note that all functors are $\mathcal {P}\mathrm {r}^{\mathrm {St}}_k$-linear and commute with geometric realizations, so by [Reference LurieLur17, Theorem 4.8.4.1] it is enough to prove the assertion for $\mathcal {M}=\mathcal {N}=\mathcal {C}$.

Substituting $\mathcal {M}=\mathcal {N}=\mathcal {C}$ and interchanging the first two columns, we obtain a diagram

in $\mathbf {Pr}^{\mathrm {St}}_k$. By construction this is the mate of the 2-isomorphism in the rectangle

4.4 Reduction to rigidity

We will now simplify the above rectangle to show that it is right adjointable.

Let $\mathcal {E}$ be an $\infty$-category with finite colimits and consider a diagram

(4.9)

The morphism $A_{12}\rightarrow B_{12}$ gives rise to a natural transformation $A_{12}\amalg _{A_2}(-)\rightarrow B_{12}\amalg _{A_2}(-)$ of functors $\mathcal {E}_{A_2/}\rightarrow \mathcal {E}$. The universal property of the pushout gives a map $A_2\amalg _{A_0} B_0\rightarrow B_2$. Therefore, we get morphisms

\[ A_{12}\amalg_{A_2}(A_2\amalg_{A_0} B_0)\rightarrow B_{12}\amalg_{A_2}(A_2\amalg_{A_0} B_0)\rightarrow B_{12}\amalg_{A_2}B_2\rightarrow B_{12}. \]

Replacing $A_2$ by $A_1$ and $B_2$ by $B_1$, we similarly get morphisms

\[ A_{12}\amalg_{A_1}(A_1\amalg_{A_0} B_0)\rightarrow B_{12}\amalg_{A_1}(A_1\amalg_{A_0} B_0)\rightarrow B_{12}\amalg_{A_1}B_1\rightarrow B_{12}. \]

Using the commutativity data of diagram (4.9), we obtain

(4.10)

We can exchange the first two columns to obtain another diagram:

(4.11)

which we will draw as

(4.12)

which gives a square in $\mathcal {E}$, that is, a functor $\Delta ^1\times \Delta ^1\rightarrow \mathcal {E}$.

Lemma 4.13 Suppose that the top and bottom squares in (4.9) are cocartesian. Then the square (4.12) is equal to the square

obtained using naturality of the transformation $A_1\amalg _{A_0}(-)\rightarrow B_1\amalg _{B_0}(-)$ of functors $\mathcal {E}_{B_0/}\rightarrow \mathcal {E}$ with respect to the morphism $B_0\amalg _{A_0} A_2\rightarrow B_2$.

Now consider the case $\mathcal {E}=\mathrm {CAlg}(\mathcal {P}\mathrm {r}^{\mathrm {St}}_k)$. Using the functor $f\colon \mathcal {D}\rightarrow \mathcal {C}$ of $k$-linear symmetric monoidal $\infty$-categories, we obtain a cube

where the top and bottom squares are of type $(1, 0)$ (see § 3.2 for what this means). In this case the isomorphisms

\[ A_{12}\amalg_{A_2}(A_2\amalg_{A_0} B_0)\xrightarrow{\sim} A_{12}\amalg_{A_1}(A_1\amalg_{A_0} B_0) \]

and

\[ A_{12}\amalg_{A_2}(A_2\amalg_{A_0} B_0)\xrightarrow{\sim} A_{12}\amalg_{A_1}(A_1\amalg_{A_0} B_0) \]

become by § 3.3 the categorified monodromy maps

\[ \mathrm{Mon}_{f_*\mathcal{C}}\otimes\mathrm{id}\colon \mathcal{L}\mathcal{D}\otimes_\mathcal{D}(\mathcal{C}\otimes_\mathcal{D}\mathcal{C})\xrightarrow{\sim} \mathcal{L}\mathcal{D}\otimes_\mathcal{D}(\mathcal{C}\otimes_\mathcal{D}\mathcal{C}) \]

and

\[ \mathrm{Mon}_{f^*f_*\mathcal{C}}\otimes\mathrm{id}\colon \mathcal{L}\mathcal{C}\otimes_\mathcal{D}(\mathcal{C}\otimes_\mathcal{D}\mathcal{C})\xrightarrow{\sim} \mathcal{L}\mathcal{C}\otimes_\mathcal{D}(\mathcal{C}\otimes_\mathcal{D}\mathcal{C}). \]

The diagram (4.10) in this case becomes

Similarly, the diagram (4.11) in this case becomes

which we may draw as a rectangle (4.12):

which we have to show is right adjointable.

By Proposition 3.9 the bottom and top squares in the cube are cocartesian, so Lemma 4.13 applies and the above rectangle is equivalent to the square

expressing naturality of the transformation $\mathcal {D}\otimes _{\mathcal {D}\otimes \mathcal {D}}(-)\rightarrow \mathcal {C}\otimes _{\mathcal {C}\otimes \mathcal {C}}(-)$ with respect to the morphism of $\mathcal {C}\otimes \mathcal {C}$-modules $\Delta \colon \mathcal {C}\otimes _\mathcal {D}\mathcal {C}\rightarrow \mathcal {C}$. Since $f$ is rigid, $\Delta$ admits a right adjoint in $\mathrm {Mod}_{\mathcal {C}\otimes _\mathcal {D}\mathcal {C}}(\mathbf {Pr}^{\mathrm {St}}_k)$ and hence in $\mathrm {Mod}_{\mathcal {C}\otimes \mathcal {C}}(\mathbf {Pr}^{\mathrm {St}}_k)$, so the square is right adjointable.

5.1 The Ben-Zvi–Nadler Chern character

Ben-Zvi and Nadler give a construction of a Chern character based on the functoriality properties of traces in symmetric monoidal $(\infty ,2)$-categories. Let us recall their definition.

Suppose $\mathcal {C}$ is a $k$-linear rigid symmetric monoidal $\infty$-category. By Proposition 2.17, $\mathcal {C}$ is dualizable in $\mathrm {Mod}_{\mathrm {Mod}_k}$ and we have an equivalence

\[ \mathcal{C}^\mathrm{dual}\cong \mathrm{Hom}_{\mathrm{Mod}_{\mathrm{Mod}_k}^{\mathrm{dual}}}(\mathrm{Mod}_k, \mathcal{C}) \]

given by sending a dualizable object $x\in \mathcal {C}$ to the functor $k\mapsto x$.

Let $\dim \colon \mathrm {Mod}_{\mathrm {Mod}_k}^\mathrm {dual} \to \mathrm {Mod}_k$ be the composite

\[ \mathrm{Mod}_{\mathrm{Mod}_k}^\mathrm{dual} \to \mathrm{Aut}(\mathrm{Mod}_{\mathrm{Mod}_k}) \xrightarrow{\mathrm{Tr}} \Omega\mathrm{Mod}_{\mathrm{Mod}_k} \cong \mathrm{Mod}_k, \]

where the first functor sends $\mathcal {M}$ to $\mathrm {id}_\mathcal {M}$. Then we have $\dim (\mathrm {Mod}_k)\cong k$ and $\dim (\mathcal {C})\cong \Omega \mathcal {L}\mathcal {C}$. Therefore, we also get a map

\[ \dim\colon \mathrm{Hom}_{\mathrm{Mod}_{\mathrm{Mod}_k}^{\mathrm{dual}}}(\mathrm{Mod}_k, \mathcal{C})\rightarrow \Omega\mathcal{L}\mathcal{C}. \]

The Chern character defined in [Reference Ben-Zvi and NadlerBN13a] is given by the composite

\[ \mathcal{C}^\mathrm{dual}\rightarrow \mathrm{Hom}_{\mathrm{Mod}_{\mathrm{Mod}_k}^{\mathrm{dual}}}(\mathrm{Mod}_k, \mathcal{C})\xrightarrow{\dim}\Omega\mathcal{L}\mathcal{C}. \]

Theorem 5.1 Suppose $\mathcal {C}$ is a $k$-linear rigid symmetric monoidal $\infty$-category. Then the uncategorified Chern character $\mathrm {ch}\colon \mathcal {C}^\mathrm {dual}\rightarrow \Omega \mathcal {L}\mathcal {C}$ is equivalent to the composite

\[ \mathcal{C}^\mathrm{dual}\rightarrow \mathrm{Hom}_{\mathrm{Mod}_{\mathrm{Mod}_k}^{\mathrm{dual}}}(\mathrm{Mod}_k, \mathcal{C})\xrightarrow{\dim}\Omega\mathcal{L}\mathcal{C}. \]

Proof. The categorified GRR Theorem 4.3 applied to $\mathrm {Mod}_k\rightarrow \mathcal {C}$ gives a commutative square

Evaluating it on the endomorphisms of $\mathcal {C}\in \mathrm {Mod}_\mathcal {C}^{\mathrm {dual}}$, we get the top square in the diagram

Here the morphism $\Omega \mathcal {L}\mathcal {C}\rightarrow \mathrm {Hom}(\Omega \mathcal {L}\mathcal {C}, \Omega \mathcal {L}\mathcal {C})$ is adjoint to the multiplication map on $\Omega \mathcal {L}\mathcal {C}$, $\mathrm {Hom}(\Omega \mathcal {L}\mathcal {C}, \Omega \mathcal {L}\mathcal {C})\rightarrow \Omega \mathcal {L}\mathcal {C}$ is given by the evaluation on the identity element, and

\[ \mathrm{Hom}_{\mathrm{Mod}_{\mathrm{Mod}_k}^{\mathrm{dual}}}(\mathcal{C}, \mathcal{C})\rightarrow \mathrm{Hom}_{\mathrm{Mod}_{\mathrm{Mod}_k}^{\mathrm{dual}}}(\mathrm{Mod}_k, \mathcal{C}) \]

is given by precomposition with the unit $\mathrm {Mod}_k\rightarrow \mathcal {C}$. The bottom square commutes by the functoriality of dimensions.

The composite

\[ \mathcal{C}^\mathrm{dual}\rightarrow \Omega\mathrm{Mod}_\mathcal{C}^\mathrm{dual}\rightarrow \Omega\mathcal{L}\mathcal{C} \]

is the uncategorified Chern character by Theorem 3.19, so the claim follows from the commutativity of the diagram.

5.2 From the categorified GRR to the classical GRR

The classical Grothendieck–Riemann–Roch theorem states the functoriality of the uncategorified Chern character with respect to the pushforward functor $f_*\colon \mathrm {Perf}(X)\rightarrow \mathrm {Perf}(Y)$ for a suitable morphism of schemes $f\colon X \to Y$. In this section we will prove its generalization with values in a sheaf of categories.

Let $f\colon \mathcal {D}\rightarrow \mathcal {C}$ be a rigid symmetric monoidal functor. Let $\mathcal {T}$ be a dualizable $\mathcal {C}$-module category, $\mathcal {T}'$ a dualizable $\mathcal {D}$-module category and $g\colon f_*\mathcal {T}\rightarrow \mathcal {T}'$ a right adjointable morphism in $\mathrm {Mod}_\mathcal {D}$, that is, a morphism in $\mathrm {Mod}_\mathcal {D}^{\mathrm {dual}}$.

Then we have Chern characters

\[ \mathrm{Ch}\colon \mathrm{Hom}_{\mathrm{Mod}_\mathcal{C}^{\mathrm{dual}}}(\mathcal{C}, \mathcal{T})\rightarrow \mathrm{Hom}_{(\mathcal{L}\mathcal{C})^{S^1}}(\mathbf{1}_{\mathcal{L}\mathcal{C}}, \mathrm{Ch}(\mathcal{T})) \]

and

\[ \mathrm{Ch}\colon \mathrm{Hom}_{\mathrm{Mod}_\mathcal{D}^{\mathrm{dual}}}(\mathcal{D}, \mathcal{T}')\rightarrow \mathrm{Hom}_{(\mathcal{L}\mathcal{D})^{S^1}}(\mathbf{1}_{\mathcal{L}\mathcal{D}}, \mathrm{Ch}(\mathcal{T}')). \]

Moreover, we can define pushforward maps as follows. The map

\[ \mathrm{Hom}_{\mathrm{Mod}_\mathcal{C}^{\mathrm{dual}}}(\mathcal{C}, \mathcal{T})\rightarrow \mathrm{Hom}_{\mathrm{Mod}_\mathcal{D}^{\mathrm{dual}}}(\mathcal{D}, \mathcal{T}') \]

sends a morphism $x\colon \mathcal {C}\rightarrow \mathcal {T}$ to the composite

\[ \mathcal{D}\rightarrow f_*\mathcal{C}\xrightarrow{x}f_*\mathcal{T}\xrightarrow{g} \mathcal{T}'. \]

Similarly, the map

\[ \mathrm{Hom}_{(\mathcal{L}\mathcal{C})^{S^1}}(\mathbf{1}_{\mathcal{L}\mathcal{C}}, \mathrm{Ch}(\mathcal{T}))\rightarrow \mathrm{Hom}_{(\mathcal{L}\mathcal{D})^{S^1}}(\mathbf{1}_{\mathcal{L}\mathcal{D}}, \mathrm{Ch}(\mathcal{T}')) \]

is given by sending a morphism $\mathbf {1}_{\mathcal {L}\mathcal {C}}\rightarrow \mathrm {Ch}(\mathcal {T})$ to the composite

\[ \mathbf{1}_{\mathcal{L}\mathcal{D}}\rightarrow (\mathcal{L} f)^\mathrm{R} \mathbf{1}_{\mathcal{L}\mathcal{C}}\rightarrow (\mathcal{L} f)^\mathrm{R} \mathrm{Ch}(\mathcal{T})\rightarrow \mathrm{Ch}(\mathcal{T}'), \]

where the last morphism is $\mathrm {Ch}(f_*\mathcal {T}\rightarrow \mathcal {T}')\colon (\mathcal {L} f)^\mathrm {R}\mathrm {Ch}(\mathcal {T})\rightarrow \mathrm {Ch}(\mathcal {T}')$.

Theorem 5.2 We have a commutative diagram of spaces

Proof. We have a commutative diagram of spaces

where the top square commutes by the categorified GRR theorem, Theorem 4.3, applied to $f\colon \mathcal {D}\rightarrow \mathcal {C}$ and the rest of the squares commute by functoriality of the Chern character $\mathrm {Ch}$.

Corollary 5.3 Suppose $f\colon \mathcal {D}\rightarrow \mathcal {C}$ is a rigid symmetric monoidal functor which is, moreover, proper in the sense of Definition 2.29. Then we have a commutative diagram of spaces

Remark 5.4 Let $X\rightarrow Y$ be a morphism of perfect stacks which is representable, proper, of finite Tor-amplitude, and locally almost of finite presentation. Then by Example 2.36 the pullback functor $\mathrm {QCoh}(Y)\rightarrow \mathrm {QCoh}(X)$ is rigid and proper. Therefore, the corollary in this case produces a commutative diagram

Moreover, if we assume that $X\rightarrow Y$ is a smooth morphism of smooth schemes over a field $k$ of characteristic zero, then by the results of Markarian [Reference MarkarianMar09] the HKR isomorphisms $\mathcal {O}(\mathcal {L} X)\cong \Omega ^{-\bullet }(X)$ intertwine the integration map $\int _f\colon \mathcal {O}(\mathcal {L} X)\rightarrow \mathcal {O}(\mathcal {L} Y)$ and the integration map $\int _f\colon \Omega ^{-\bullet }(X)\rightarrow \Omega ^{-\bullet }(Y)$ on differential forms twisted by the relative Todd class $\mathrm {Td}_{X/Y}$. Therefore, we obtain a commutative diagram

5.3 The Grothendieck–Riemann–Roch Theorem for the secondary Chern character

In this section we prove a GRR theorem for the secondary Chern character. If $\mathcal {C}$ is a $k$-linear symmetric monoidal $\infty$-category, we denote by $\mathrm {Mod}^\mathrm {sat}_\mathcal {C}$ the full subcategory of $\mathrm {Mod}^\mathrm {dual}_\mathcal {C}$ spanned by the saturated $\mathcal {C}$-modules (Definition 2.30).

Definition 5.5 Let $\mathcal {C}$ be a $k$-linear symmetric monoidal $\infty$-category. We define the secondary Chern character to be the composite

\[ \mathrm{ch}^{(2)}\colon \iota_0 \mathrm{Mod}^\mathrm{sat}_{\mathcal{C}} \to ((\mathcal{L} \mathcal{C})^{\mathrm{dual}})^{S^1} \to \Omega(\mathcal{L}^2 \mathcal{C}) ^{(S^1\times S^1)} \]

where the first map is the categorified Chern character for $\mathcal {C}$ and the second map is the classical Chern character for $\mathcal {L}\mathcal {C}$.

Theorem 5.6 (Secondary GRR) Let $f \colon \mathcal {D}\to \mathcal {C}$ be a smooth and proper functor of symmetric monoidal $\infty$-categories. Then the square

commutes.

Proof. Since $f$ is smooth and proper, the pushforward $f_*\colon \mathrm {Mod}_\mathcal {C} \to \mathrm {Mod}_\mathcal {D}$ preserves saturated $\infty$-categories (Lemma 2.32). Restricting Theorem 4.3 to saturated $\infty$-categories, we therefore obtain a commutative square

By Lemma 2.33, $\mathcal {L} f \colon \mathcal {L}\mathcal {D} \to \mathcal {L}\mathcal {C}$ is proper. Hence, composing the above commutative diagram with the classical GRR as in Corollary 5.3 yields the statement.

Let us assume that $\mathcal {C}$ is compactly generated and rigid in the sense of Definition 2.16. We denote the subcategory of compact objects by $\mathcal {C}^\omega$. By Corollary 2.19, we have that $\mathcal {C}^\mathrm {dual}=\iota _0\mathcal {C}^\omega$. We will write $\mathrm {mod}_{\mathcal {C}^\omega }$ for the $\infty$-category of small stable idempotent complete $\mathcal {C}^\omega$-linear $\infty$-categories. The $\mathrm {ind}$-completion functor identifies $\mathrm {mod}_{\mathcal {C}^\omega }$ with a full subcategory of $\mathrm {Mod}_{\mathcal {C}}^\mathrm {dual}$. In [Reference Hoyois, Scherotzke and SibillaHSS17], we considered the $\infty$-category $\mathrm {mod}^{\mathrm {sat}}_{\mathcal {C}^{\omega }}$ of small saturated $\mathcal {C}^\omega$-linear $\infty$-categories, which is the intersection $\mathrm {mod}_{\mathcal {C}^\omega }\cap \mathrm {Mod}_\mathcal {C}^\mathrm {sat}$. As proved in [Reference Hoyois, Scherotzke and SibillaHSS17, Theorem 6.20], $\mathrm {ch}^{(2)}$ descends to a morphism of $\mathbb {E}_\infty$ ring spectra

\[ \mathrm{ch}^{(2)}\colon \mathbb{K}^{(2)}(\mathcal{C}^\omega) \to \Omega_\mathrm{Sp}(\mathcal{L}^2\mathcal{C})^{(S^1\times S^1)}, \]

where $\Omega _\mathrm {Sp}$ is the spectrum of endomorphisms of the unit object. This is the diagonal composition in the diagram

where a dotted arrow means a map to the infinite loop space of the target; see [Reference Hoyois, Scherotzke and SibillaHSS17, Diagram (6.19)].

Theorem 5.7 (Motivic GRR) Let us assume that $\mathcal {C}$ and $\mathcal {D}$ are compactly generated and rigid, and let $f \colon \mathcal {D}\to \mathcal {C}$ be a rigid symmetric monoidal functor. Then the square

commutes. If $f$ is smooth and proper, the square

commutes.

Proof. Note that the functor $f_*\colon \mathrm {Mod}^\mathrm {dual}_{\mathcal {C}} \to \mathrm {Mod}^\mathrm {dual}_{\mathcal {D}}$ preserves compactly generated $\infty$-categories. The functor $\mathrm {mod}_{\mathcal {C}^\omega } \to \mathbb {M}\mathrm {ot}(\mathcal {C}^\omega )$ is by definition the universal functor to a presentable stable $\infty$-category that preserves zero objects, exact sequences, and filtered colimits. The functors

\[ \mathrm{mod}_{\mathcal{C}^{\omega}} \xrightarrow{f_*} \mathrm{mod}_{\mathcal{D}^{\omega}}\xrightarrow{\mathrm{Ind}} \mathrm{Mod}^\mathrm{dual}_{\mathcal{D}} \xrightarrow{\mathrm{Ch}} (\mathcal{L}\mathcal{D})^{S^1} \]

and

\[ \mathrm{mod}_{\mathcal{C}^{\omega}} \xrightarrow{\mathrm{Ind}} \mathrm{Mod}^\mathrm{dual}_{\mathcal{C}} \xrightarrow{\mathrm{Ch}} (\mathcal{L}\mathcal{C})^{S^1} \]

satisfy these conditions and therefore the categorified GRR factors through the commutative square as in the statement. The commutativity of the second square is proved in the same way, using that $\mathrm {mod}^\mathrm {sat}_{\mathcal {C}^\omega }\to \mathrm {Mot}^\mathrm {sat}(\mathcal {C}^\omega )$ is the universal functor to a stable idempotent complete $\infty$-category that preserves zero objects and exact sequences.

Corollary 5.8 Assume that $\mathcal {C}$ and $\mathcal {D}$ are compactly generated and rigid, and let $f \colon \mathcal {D}\to \mathcal {C}$ be a proper symmetric monoidal functor. Then we have a commutative square of spectra

Theorem 5.9 (Secondary motivic GRR) Let $\mathcal {C}$ and $\mathcal {D}$ be compactly generated rigid categories and let $f \colon \mathcal {D}\to \mathcal {C}$ be a smooth and proper symmetric monoidal functor. Then the square

commutes.

Proof. Applying non-connective $K$-theory to the second square in Theorem 5.7, we get a commutative square of spectra

Now $\mathcal {L} f\colon \mathcal {L} \mathcal {D} \to \mathcal {L} \mathcal {C}$ is also a rigid symmetric functor which is proper (Lemma 2.33). Hence, Corollary 5.8 applied to $\mathcal {L} f\colon \mathcal {L} \mathcal {D} \to \mathcal {L} \mathcal {C}$ yields the commutative square

Combining the two squares yields the statement.

5.4 Secondary Chern character and the motivic Chern class

In this section we establish a comparison between the secondary Chern character and Brasselet, Schürmann and Yokura's motivic Chern class [Reference Brasselet, Schürmann and YokuraBSY10]. The motivic Chern class is an enhancement of MacPherson's total Chern class of singular varieties [Reference MacPhersonMac74] and, as explained in [Reference SchürmannSch09], it specializes to other well-known invariants of singular varieties.

Throughout the section, $k$ is a field of characteristic zero. A variety is an integral separated scheme of finite type over $\mathrm {Spec}(k)$. For $X$ a variety, we write $\mathrm {Mot}(X)$ for the presentable stable $\infty$-category $\mathbb {M}\mathrm {ot}(\mathrm {Perf}(X))$ of localizing $\mathrm {Perf}(X)$-motives [Reference Hoyois, Scherotzke and SibillaHSS17, Definition 5.14].

Definition 5.10 We denote by $\mathrm {Mot}_{\mathrm {BM}}(X)$ the smallest stable idempotent complete full subcategory of $\mathrm {Mot}(X)$ such that for every proper map $f\colon Y \to X$ from a smooth variety the pushforward factors as

We call $\mathrm {Mot}_{\mathrm {BM}}(X)$ the $\infty$-category of Borel–Moore non-commutative motives over $X$.

Lemma 5.12 The restriction of $\mathrm {Ch}$ to $\mathrm {Mot}_{\mathrm {BM}}(X)$ factors as

Proof. Let $f\colon Y \to X$ be a proper map from a smooth variety $Y$. As $f\colon Y \to X$ is proper, $\mathcal {L} f$ is proper and there is a well-defined pushforward $\mathcal {L} f_*\colon \mathrm {Coh}(\mathcal {L} Y) \to \mathrm {Coh}(\mathcal {L} X).$ Further, since $Y$ is smooth, $\mathcal {L} Y$ is eventually coconnective (see Lemma 6.9), and therefore there is an inclusion $\mathrm {Perf}(\mathcal {L} Y) \subset \mathrm {Coh}(\mathcal {L} Y).$ The motivic GRR theorem (Theorem 5.7) yields a commutative square

(5.13)

By the previous discussion, the upper composition $\mathcal {L} f_* \circ \mathrm {Ch}$ lands in $\mathrm {Coh}(\mathcal {L} X)$:

\[ \mathrm{Mot}^{\mathrm{sat}}(Y) \xrightarrow{ \mathcal{L} f_* \circ \mathrm{Ch} } \mathrm{Coh}(\mathcal{L} X) \subset \mathrm{QCoh}(\mathcal{L} X). \]

Then, since (

5.13

) is commutative, the lower composition $\mathrm {Ch} \circ f_*$ also corestricts to $\mathrm {Coh}(\mathcal {L} X).$

Now, by definition $\mathrm {Mot}_{\mathrm {BM}}(X)$ is generated under fibers, cofibers, and retracts by the images of the pushforward functors

\[ f_*\colon \mathrm{Mot}^{\mathrm{sat}}(Y) \to \mathrm{Mot}(X) \]

as $Y\to X$ ranges over all proper maps with smooth domain. We conclude that $\mathrm {Ch}$ restricted to $\mathrm {Mot}_{\mathrm {BM}}(X)$ lands in $\mathrm {Coh}(\mathcal {L} X),$ which is what we wanted to prove.

Definition 5.14 We denote by $K^{(2)}_{\mathrm {BM}}(X)$ the algebraic K-theory of $\mathrm {Mot}_{\mathrm {BM}}(X),$

\[ K^{(2)}_{\mathrm{BM}}(X) := K(\mathrm{Mot}_{\mathrm{BM}}(X)). \]

Let $i_X\colon X \to \mathcal {L} X$ be the embedding of the trivial loops. By [Reference BarwickBar15, Proposition 9.2] the pushforward in $G$-theory, $i_{X*}\colon G(X) \longrightarrow G(\mathcal {L} X),$ is an equivalence.

Definition 5.15 We denote by $\mathrm {ch}^{(2)}_{\mathrm {BM}}$ the map

\[ \mathrm{ch}^{(2)}_{\mathrm{BM}}\colon K^{(2)}_{\mathrm{BM}}(X) = K(\mathrm{Mot}_{\mathrm{BM}}(X)) \xrightarrow{K(\mathrm{Ch})} K(\mathrm{Coh}(\mathcal{L} X)) \simeq G(X). \]

We call $\mathrm {ch}^{(2)}_{\mathrm {BM}}$ the BM secondary Chern character.

The categorified GRR theorem implies a GRR statement for the BM secondary Chern character.

Proposition 5.16 Let $f\colon Y \to X$ be a proper map of algebraic varieties. Then there is a commutative square

In [Reference BittnerBit04] Bittner obtains a presentation of the Grothendieck group of varieties over $X,$ $K_0(\mathrm {Var}_X)$, which we recall next. She proves that $K_0(\mathrm {Var}_X)$ is isomorphic to the free abelian group on isomorphism classes of proper maps $[Y \longrightarrow X],$ such that $Y$ is smooth and equidimensional, subject to the following two relations.

1. (i) $[\varnothing \longrightarrow X] = 0$.

2. (ii) For every diagram

   
   where $j$ is a closed embedding of smooth equidimensional algebraic varieties, $\mathrm {Bl}_{Z}(Y)$ is the blow-up of $Y$ along $Z,$ and $E$ is the exceptional divisor,
   \[ [\mathrm{Bl}_ZY \longrightarrow X] - [E \longrightarrow X] = [Y \longrightarrow X] - [Z \longrightarrow X] \quad \text{in } K_0(\mathrm{Var}_X). \]

If $\mathcal {C}$ is an $\infty$-category over $X$ having the property that its motive lies in $\mathrm {Mot}_{\mathrm {BM}}(X),$ we denote by $[\mathcal {C}]$ its class in $K_{\mathrm {BM}}^{(2)}(X)$.

Proposition 5.17 There is a homomorphism of groups $\mu \colon K_0(\mathrm {Var}_X) \to K^{(2)}_{\mathrm {BM}, 0}(X)$ given by the assignment

\[ \quad [Y \stackrel{f} \longrightarrow X] \in K_0(\mathrm{Var}_X) \mapsto f_*[\mathrm{Perf}(Y)] \in K^{(2)}_{\mathrm{BM}, 0}(X). \]

Proof. The proof is the same as that given in [Reference Bondal, Larsen and LuntsBLL04] for the case $X= \mathrm {Spec}(k).$ The key ingredient is Orlov's formula for the category of perfect complexes of blow-ups; see [Reference Bondal, Larsen and LuntsBLL04, Proposition 7.5]. The only thing to prove is that the assignment

\[ \quad [Y \stackrel{f} \longrightarrow X] \in K_0(\mathrm{Var}_X) \mapsto f_*[\mathrm{Perf}(Y)] \in K^{(2)}_{\mathrm{BM}, 0}(X). \]

is compatible with relations $(1)$ and $(2)$ from Bittner's presentation of $K_0(\mathrm {Var}_X)$.

Relation $(1)$ reduces to the fact that $\mathrm {Perf}(\varnothing )$ is the $0$ category. Now let $Z \stackrel {j} \to Y$ be as in relation $(2)$, and let $s$ be the codimension of $Z$ in $Y$. Orlov's formula, proved in [Reference OrlovOrl93], gives a $\mathrm {Perf}(Y)$-linear semi-orthogonal decomposition of $\mathrm {Perf}(\mathrm {Bl}_{Z}(Y))$ with $s$ factors: one copy of $\mathrm {Perf}(Y)$ and $s-1$ copies of $\mathrm {Perf}(Z)$. The exceptional divisor $E \subset \mathrm {Bl}_{Z}(Y)$ is a projective bundle over $E$ of rank $s-1$. Thus, by [Reference OrlovOrl93], $\mathrm {Perf}(E)$ has a $\mathrm {Perf}(Z)$-linear semi-orthogonal decomposition with $s$ factors all equivalent to $\mathrm {Perf}(Z)$.

We obtain the following identities in $K^{(2)}_{\mathrm {BM}, 0}(X)$:

\[ (f q)_*[\mathrm{Perf}(\mathrm{Bl}_{Z}(Y))] = f_*[\mathrm{Perf}(Y)] + (s-1)(f j)_*[\mathrm{Perf}(Z)], \quad (f q i)_*[\mathrm{Perf}(E)] = (i f)_*[\mathrm{Perf}(Z)]. \]

This immediately implies that

\[ f_*[\mathrm{Perf}(\mathrm{Bl}_{Z}(Y))] - (f q i)_*[\mathrm{Perf}(E)] = f_*[\mathrm{Perf}(Y)] - (fj)_*[\mathrm{Perf}(Z)], \]

which can be rewritten as the identity

\[ \mu(\mathrm{Bl}_{Z}(Y)) - \mu(E) = \mu(Y) - \mu(Z).\]

This shows that relation $(2)$ is satisfied, and concludes the proof.

5.4.1 The motivic Chern class

Let $X$ be a variety. The motivic Chern class was defined in [Reference Brasselet, Schürmann and YokuraBSY10]. It is the morphism

\[ mC_*\colon K_0(\mathrm{Var}_X) \longrightarrow G_0(X) \otimes \mathbb{Z}[y] \]

which is uniquely determined by the following two properties.

1. (i) If $X$ is smooth, $mC_*([X \xrightarrow {1_X} X]) = \sum [\Omega ^ i_X] \cdot y^ i \in G_0(X) \otimes \mathbb {Z}[y]$.

2. (ii) If $Y \to X$ is a proper map and $Y$ is a smooth algebraic variety there is a commutative diagram

   (5.18)

   

Theorem 5.19 Let $X$ be a variety. Then there is a commutative diagram

(5.20)

where the vertical map on the right is the quotient map

\[ G_0(X) \otimes \mathbb{Z}[y] \rightarrow G_0(X) \otimes \mathbb{Z}[y]/(y+1)=G_0(X). \]

Proof. By Proposition 5.16 the BM secondary Chern character satisfies a GRR theorem for pushforwards along proper maps. Then, in view of the defining properties (1) and (2) of the motivic Chern class, to prove the claim it is sufficient to verify the following two compatibilities. The first is that, if $X$ is smooth, diagram (5.20) commutes when evaluated on $[X \xrightarrow {1_X} X].$ This holds, since

\[ \mathrm{ch}_{\mathrm{BM}}^{(2)} \circ \mu([X \xrightarrow{1_X} X]) = \mathrm{ch}_{\mathrm{BM}}^{(2)}([\mathrm{Perf}(X)]) = [i_X^*\mathcal{O}_{\mathcal{L} X}] = \sum (-1)^ i \Omega^ i_X = s \circ mC_*([X \xrightarrow{1_X} X]). \]

Finally, we need to check that if $f\colon Y \rightarrow X$ is a proper map, the square

commutes. This is clear, and this concludes the proof.

6.1 Ind-coherent sheaves on loop spaces and crystals

In this section we review definitions and results from Preygel's article [Reference PreygelPre15].

Let $\mathcal {C}$ be a $k$-linear $\infty$-category equipped with an $S^1$-action. The invariant category $\mathcal {C}^{S^1}$ is linear over $C^*(BS^1, k) \simeq k[[u]]$, with $u$ in (homological) degree $-2$, and we set

\[ \mathcal{C}^{\mathrm{Tate}}:=\mathcal{C}^{S^1} \otimes_{k[[u]]} k((u)). \]

If $\mathcal {C}$ is large the Tate construction is often not quite the right concept. Under the assumption that $\mathcal {C}$ is a stable $\infty$-category with a coherent t-structure [Reference PreygelPre15, Definition 4.2.7], Preygel introduces the tTate construction $\mathcal {C}^{\mathrm {tTate}}$ as a better-behaved alternative. It is defined by

\[ \mathcal{C}^\mathrm{tTate} := \mathrm{Ind}(\mathrm{Coh}(\mathcal{C})^{S^1})\otimes_{k[[u]]}k((u)), \]

where $\mathrm {Coh}(\mathcal {C})\subset \mathcal {C}$ is the full subcategory of bounded almost perfect objectsFootnote ¹ and $(-)\otimes _{k[[u]]}k((u))$ is extension of scalars for presentable linear $\infty$-categories.

The notation $\mathrm {Coh}(\mathcal {C})$ is motivated by geometric applications. Indeed, let $X$ be a derived scheme with an $S^1$-action. If $\mathcal {C} = \mathrm {Ind}(\mathrm {Coh}(X))$ then $\mathrm {Coh}(\mathcal {C})$ coincides with the stable category of coherent complexes on $X$, $\mathrm {Coh}(X)$. Furthermore, by [Reference PreygelPre15, Remark 4.5.6] we have that

\[ \mathrm{IndCoh}(X)^\mathrm{tTate} \simeq \mathrm{Ind}(\mathrm{Coh}(X)^{S^1}) \otimes_{k[[u]]} k((u)) \simeq \mathrm{Ind}(\mathrm{Coh}(X)^\mathrm{Tate}). \]

Let $X$ be a derived scheme and $X_\mathrm {dR}$ the associated de Rham prestack. Recall that a crystal on $X$ is by definition a quasi-coherent sheaf on $X_\mathrm {dR}$, and that the functor

\[ \Upsilon \colon \mathrm{QCoh}(X_\mathrm{dR}) \to \mathrm{IndCoh}(X_\mathrm{dR}) \]

is an equivalence [Reference Gaitsgory and RozenblyumGR14, Proposition 2.4.4]. The inclusion of the constant loops $X\to \mathcal {L} X$ is a nil-isomorphism, and hence it induces an equivalence of de Rham prestacks. We therefore have an $S^1$-equivariant map

\[ \pi_X\colon \mathcal{L} X \rightarrow (\mathcal{L} X)_{\mathrm{dR}} \simeq X_{\mathrm{dR}}, \]

where the $S^1$-action is given by loop rotation on $\mathcal {L} X$ and is trivial on $X_{\mathrm {dR}}$. The morphism $\pi _X$ is an inf-schematic nil-isomorphism and hence induces an adjunction

\[ \pi_{X,*}: \mathrm{IndCoh}(\mathcal{L} X) \rightleftarrows \mathrm{IndCoh}(X_{\mathrm{dR}}): \pi_X^!, \]

where the right adjoint is symmetric monoidal.

Theorem 6.1 [Reference PreygelPre15, Theorem 1.3.5]

For every derived scheme $X$, the morphism $\pi _X$ induces inverse equivalences of symmetric monoidal $\infty$-categories

\[ (\pi_{X,*})^{\mathrm{tTate}} : \mathrm{IndCoh}(\mathcal{L} X)^{\mathrm{tTate}} \stackrel{\simeq} \longleftrightarrow \mathrm{IndCoh}(X_{\mathrm{dR}})^{\mathrm{tTate}} : (\pi^!_X)^{\mathrm{tTate}}. \]

Definition 6.2 If $\mathcal {C}$ is a presentable stable $k$-linear $\infty$-category, we denote by $\mathcal {C}_{\mathbb {Z}/2}$ its $\mathbb {Z}/2$-folding,

\[ \mathcal{C}_{\mathbb{Z}/2} := \mathcal{C} \otimes_k k((u)), \]

where $u$ is in degree $-2$.

If $S^1$ acts trivially on a stable $\infty$-category $\mathcal {C}$ with coherent t-structure, we have [Reference PreygelPre15, Lemma 4.5.4]

\[ \mathcal{C}^\mathrm{tTate} \simeq \mathrm{Ind}(\mathrm{Coh}(\mathcal{C}))_{\mathbb{Z}/2}. \]

In particular, Theorem 6.1 gives equivalences of symmetric monoidal $\infty$-categories

(6.3)\begin{equation} \mathrm{IndCoh}(\mathcal{L} X)^\mathrm{tTate} \simeq \mathrm{IndCoh}(X_\mathrm{dR})_{\mathbb{Z}/2} \simeq \mathrm{QCoh}(X_\mathrm{dR})_{\mathbb{Z}/2}. \end{equation}

We will not distinguish notationally between an object of $\mathcal {C}$ and its image in the $\mathbb {Z}/2$-folding $\mathcal {C}_{\mathbb {Z}/2}$, as it will always be clear from the context which is meant.

We now discuss the functoriality of the construction $\mathcal {C}\mapsto \mathcal {C}^\mathrm {tTate}$, following [Reference PreygelPre15, § 4.6]. An exact functor $F\colon \mathcal {C}\to \mathcal {D}$ between stable $\infty$-categories with coherent t-structures is called coherent if it is left t-exact up to a shift and $F|\mathcal {C}_{<0}$ preserves filtered colimits. For such a functor there is an induced commutative diagram

where $\mathcal {C}_{<\infty }=\bigcup _n \mathcal {C}_{\leq n}\subset \mathcal {C}$ is the subcategory of homologically bounded-above objects. Note that the functors $f^!\colon \mathrm {IndCoh}(X)\to \mathrm {IndCoh}(Y)$ and $f_*\colon \mathrm {IndCoh}(Y)\to \mathrm {IndCoh}(X)$ are coherent for any morphism of derived schemes $f\colon Y\to X$. Similarly, if $\mathcal {C}$ has a symmetric monoidal structure whose unit is bounded above and such that $x\otimes (-)$ is coherent for every $x\in \mathcal {C}_{<0}$, there is an induced symmetric monoidal structure on $\mathrm {Ind}(\mathrm {Coh}(\mathcal {C}))$ that restricts to the original one on $\mathcal {C}_{<\infty }$.

If $\mathcal {C}$ has an $S^1$-action, the diagram of functors

is thus natural in $\mathcal {C}$ with respect to coherent functors. Replacing the functor $\mathrm {Ind}(\mathrm {Coh}(\mathcal {C})^{S^1}) \to \mathcal {C}^{S^1}$ by its right adjoint, we obtain a canonical functor

\[ \Theta \colon \mathcal{C}^{S^1} \to \mathcal{C}^\mathrm{tTate}, \]

which is left-lax natural in $\mathcal {C}$, and whose restriction to $\mathcal {C}^{S^1}_{<\infty }$ is strictly natural by [Reference PreygelPre15, Lemma 4.6.1 and Remark 4.6.3]. If $\mathcal {C}$ is symmetric monoidal as before, the above diagram is one of symmetric monoidal functors. It follows that $\Theta$ is right-lax symmetric monoidal, and that its restriction to $\mathcal {C}^{S^1}_{<\infty }$ is symmetric monoidal; in particular, $\Theta$ is unital.

Lemma 6.4 Suppose that $\mathcal {C}=\mathrm {Ind}(\mathrm {Coh}(\mathcal {C}))$ and that $S^1$ acts trivially on $\mathcal {C}$. Then the functor $\Theta \colon \mathcal {C}^{S^1} \to \mathcal {C}^\mathrm {tTate}\simeq \mathcal {C}_{\mathbb {Z}/2}$ sends an $S^1$-equivariant object $E$ to its Tate construction

\[ E^{tS^1}=E^{S^1}\otimes_{k[[u]]}k((u)). \]

Proof. There is a commutative square of left-adjoint functors

where the horizontal functors equip an object with trivial $S^1$-action. Under the equivalence $\mathrm {Ind}(\mathrm {Coh}(\mathcal {C})^{S^1})\simeq \mathcal {C}\otimes _k k[[u]]$ of [Reference PreygelPre15, Lemma 4.5.4], the lower horizontal functor is given by extension of scalars. Passing to right adjoints, we deduce that the functor

\[ \mathcal{C}^{S^1}\to \mathrm{Ind}(\mathrm{Coh}(\mathcal{C})^{S^1})\simeq \mathcal{C}\otimes_k k[[u]] \]

sends an object E to its $S^1$-fixed points $E^{S^1}$ with their $k[[u]]$-module structure.

6.2 The categorified de Rham Chern character

For $X$ smooth over $k$, the categorified de Rham Chern character will be a functor

\[ \mathrm{Ch}^\mathrm{dR}\colon \mathrm{Mod}^\mathrm{dual}_{\mathrm{QCoh}(X)} \longrightarrow \mathcal{D}_X\mathrm{{-}mod}_{\mathbb{Z}/2} \]

associating to every dualizable sheaf of $\infty$-categories on $X$ a $2$-periodic $\mathcal {D}_X$-module. More generally, for $X$ not necessarily smooth, we will define $\mathrm {Ch}^\mathrm {dR}$ as a functor valued in the $\infty$-category $\mathrm {QCoh}(X_\mathrm {dR})_{\mathbb {Z}/2}$ of $2$-periodic crystals. The relationship between crystals and D-modules will be reviewed in § 6.3.

Definition 6.5 Let $X$ be a derived scheme. The categorified de Rham Chern character is the functor

\[ \mathrm{Mod}^{\mathrm{dual}}_{\mathrm{QCoh}(X)} \xrightarrow{\mathrm{Ch}} \mathrm{QCoh}(\mathcal{L} X)^{S^1} \xrightarrow{\Upsilon}\mathrm{IndCoh}(\mathcal{L} X)^{S^1} \xrightarrow{\Theta} \mathrm{IndCoh}(\mathcal{L} X)^{\mathrm{tTate}} \simeq \mathrm{QCoh}(X_{\mathrm{dR}})_{\mathbb{Z}/2}, \]

where the last equivalence is (6.3). We denote the de Rham Chern character by $\mathrm {Ch}^{\mathrm {dR}}$.

Note that $\mathrm {Ch}^\mathrm {dR}$ is a unital right-lax symmetric monoidal functor, being a composition of such functors. Moreover, the restriction of $\mathrm {Ch}^\mathrm {dR}$ to fully dualizable $\mathrm {QCoh}(X)$-modules is strictly symmetric monoidal, since $\Upsilon \circ \mathrm {Ch}$ takes such modules to $\mathrm {Coh}(\mathcal {L} X)^{S^1}$.

When $X$ is a smooth scheme, $\mathrm {Ch}^\mathrm {dR}$ is an enhancement of periodic cyclic homology:

Lemma 6.6 Let $X$ be a smooth scheme and let $\pi \colon X\to X_\mathrm {dR}$ be the canonical map. Then the composite functor

\[ \mathrm{Mod}^{\mathrm{dual}}_{\mathrm{QCoh}(X)} \xrightarrow{\mathrm{Ch}^\mathrm{dR}} \mathrm{QCoh}(X_\mathrm{dR})_{\mathbb{Z}/2} \xrightarrow{\pi^*} \mathrm{QCoh}(X)_{\mathbb{Z}/2} \]

sends $\mathcal {M}$ to its relative periodic cyclic homology $\mathrm {HP}(\mathcal {M}/X)=\mathrm {HH}(\mathcal {M}/X)^{tS^1}$.

Proof. Let $e\colon X\to \mathcal {L} X$ be the inclusion of the constant loops. Since $e$ is proper, the functor $e^!$ admits a coherent left adjoint $e_*$, so that it commutes with $\Theta$. Since $X$ is smooth, the functor $\Upsilon \colon \mathrm {QCoh}(X)\to \mathrm {IndCoh}(X)$ is an equivalence. Using these facts, one can identify $\pi ^*\circ \mathrm {Ch}^\mathrm {dR}$ with the composition

\[ \mathrm{Mod}^\mathrm{dual}_{\mathrm{QCoh}(X)} \xrightarrow{\mathrm{Ch}} \mathrm{QCoh}(\mathcal{L} X)^{S^1} \xrightarrow{e^*} \mathrm{QCoh}(X)^{S^1} \xrightarrow{\Theta} \mathrm{QCoh}(X)_{\mathbb{Z}/2}. \]

By definition, $\mathrm {Ch}(\mathcal {M})$ is the trace of the monodromy automorphism of $p^*\mathcal {M}$, where $p\colon \mathcal {L} X\to X$. Since $p\circ e=\mathrm {id}_X$, we have that $e^*(\mathrm {Ch}(\mathcal {M}))$ is the trace of the identity on $\mathcal {M}$, that is, the Hochschild homology $\mathrm {HH}(\mathcal {M}/X)$ with its canonical $S^1$-action.

By Lemma 6.4, the last functor sends an $S^1$-equivariant object $E$ to its Tate fixed points $E^{tS^1}$, which completes the proof.