Controlled objects as a symmetric monoidal functor

TL;DR

This paper constructs a functorial association between symmetric monoidal additive categories with group actions and coarse spaces, enabling a refined equivariant coarse algebraic K-homology.

Contribution

It introduces a new lax symmetric monoidal functor from $G$-bornological coarse spaces to additive categories, refining equivariant coarse algebraic K-homology.

Findings

Defines a functor $ extbf{V}_{ extbf{A}}^{G}$ for symmetric monoidal additive categories with $G$-action.

Establishes a lax symmetric monoidal structure on the equivariant coarse algebraic K-homology.

Provides a framework for functorial and monoidal refinement of equivariant coarse algebraic K-homology.

Abstract

The goal of this paper is to associate functorially to every symmetric monoidal additive category $\mathbf{A}$ with a strict $G$-action a lax symmetric monoidal functor $\mathbf{V}_{\mathbf{A}}^{G}:G\mathbf{BornCoarse}\to \mathbf{Add}_{\infty}$ from the symmetric monoidal category of $G$-bornological coarse spaces $G\mathbf{BornCoarse}$ to the symmetric monoidal $\infty$-category of additive categories $ \mathbf{Add}_{\infty}$. This allows to refine equivariant coarse algebraic $K$-homology to a lax symmetric monoidal functor.

Equations

$$E:G\mathbf{BornCoarse}\to \mathcal{C}$$

$$\mathrm{UK}{\mathcal{X}}_{\mathbf{A}}^G:G\mathbf{BornCoarse}\to {\mathcal{M}}_{loc}$$

$$\mathrm{UK}{\mathcal{X}}_{\mathbf{A}}^G:G\mathbf{BornCoarse}\stackrel{{\mathbf{V}}_{\mathbf{A}}^G}{\to }{\mathbf{Add}}_1\stackrel{\mathrm{UK}}{\to }{\mathcal{M}}_{loc},$$

$$\mathrm{UK}:{\mathbf{Add}}_1\stackrel{loc}{\to }{\mathbf{Add}}_{\infty }\stackrel{{\mathrm{UK}}_{\infty }}{\to }{\mathcal{M}}_{loc},$$

$${\mathbf{V}}_{\mathbf{A}}^{G,\otimes }:G\mathbf{BornCoarse}\to {\mathbf{Add}}_{\infty }^{\otimes }$$

$${\mathrm{UK}}_{\infty }^{\otimes }:{\mathbf{Add}}_{\infty }^{\otimes }\to {\mathcal{M}}_{loc}^{\otimes }.$$

$${\otimes }_{\mathbf{C}}:\mathbf{C}\times \mathbf{C}\to \mathbf{C},$$

$$F(C){\otimes }_{\mathbf{D}}F(C^{\prime })\to F(C{\otimes }_{\mathbf{C}}{C}^{\prime }),C,C^{\prime }\in \mathbf{C}$$

$${\mathtt{Hom}}_{\mathbf{A}{\otimes }_{\mathbf{Add}}\mathbf{B}}((A,B),(A^{\prime },B^{\prime })):={\mathtt{Hom}}_{\mathbf{A}}(A,A^{\prime }){\otimes }_{\mathbb{Z}}{\mathtt{Hom}}_{\mathbf{B}}(B,B^{\prime })$$

$${\mathbf{Cat}}_{\infty }:=\mathtt{N}({\mathbf{Cat}}_1)[W_{\mathbf{Cat}}^{-1}]$$

$${\mathtt{N}}_2(\mathbf{Cat}):=\mathcal{N}(\mathtt{N}(\mathbf{Cat})).$$

$${\mathtt{N}}_2(\mathbf{Cat})\simeq {\mathbf{Cat}}_{\infty }.$$

$${\mathbf{Cat}}_1^{\otimes }\to {\mathbf{Fin}}_{\ast },\text{ and }\mathtt{N}({\mathbf{Cat}}_1^{\otimes })\to \mathtt{N}({\mathbf{Fin}}_{\ast }),$$

$${\mathbf{Cat}}_{\infty }^{\otimes }:=\mathtt{N}({\mathbf{Cat}}_1^{\otimes })[W_{\mathbf{Cat}}^{\otimes ,-1}]\to \mathtt{N}({\mathbf{Fin}}_{\ast })$$

$${\mathbf{Add}}_{\infty }:=\mathtt{N}({\mathbf{Add}}_1)[W_{\mathbf{Add}}^{-1}]$$

$${\mathbf{Add}}_{\infty }\simeq {\mathtt{N}}_2(\mathbf{Add}).$$

$${\mathbf{Add}}_1^{\otimes }\to {\mathbf{Fin}}_{\ast },\mathtt{N}({\mathbf{Add}}_1^{\otimes })\to \mathtt{N}({\mathbf{Fin}}_{\ast }).$$

$${\mathbf{Add}}_{\infty }^{\otimes }:=\mathtt{N}({\mathbf{Add}}_1^{\otimes })[W_{\mathbf{Add}}^{\otimes ,-1}]\to \mathtt{N}({\mathbf{Fin}}_{\ast })$$

$$F_{\infty }:\mathtt{N}(\mathbf{C})\stackrel{\mathtt{N}(F)}{\to }\mathtt{N}({\mathbf{Cat}}_1)\to \mathtt{N}({\mathbf{Cat}}_1)[W_{\mathbf{Cat}}^{-1}]={\mathbf{Cat}}_{\infty }.$$

$$F^{\otimes }:\mathtt{N}({\mathbf{C}}^{\otimes })\to {\mathbf{Cat}}_{\infty }^{\otimes },(F^{\otimes }:\mathtt{N}({\mathbf{C}}^{\otimes })\to {\mathbf{Add}}_{\infty }^{\otimes })$$

$$F:\mathbf{C}\to {\mathbf{Cat}}_1$$

$${\pi }_F:\mathcal{F}\to \mathbf{C}.$$

$${\pi }_F((X,A){\otimes }_{\mathcal{F}}(X^{\prime },A^{\prime }))=X{\otimes }_{\mathbf{C}}{X}^{\prime },$$

$$(X,A){\otimes }_{\mathcal{F}}(X^{\prime },A^{\prime })=(X{\otimes }_{\mathbf{C}}{X}^{\prime },A{⊠}_{X,X^{\prime }}{A}^{\prime }).$$

$${⊠}_{X,X^{\prime }}:F(X)\times F(X^{\prime })\to F(X{\otimes }_{\mathbf{C}}{X}^{\prime })$$

$$f:X\to X^{\prime },g:Y\to Y^{\prime }$$

$$(f,{\mathtt{id}}_{F(f)(A)}):(X,A)\to (X^{\prime },F(f)(A)),(g,{\mathtt{id}}_{F(g)(B)}):(Y,B)\to (Y^{\prime },F(g)(B))$$

$$(f,{\mathtt{id}}_{F(f)(A)}){\otimes }_{\mathcal{F}}(g,{\mathtt{id}}_{F(g)(B)}):(X{\otimes }_{\mathbf{C}}Y,A{⊠}_{X,Y}B)\to (X^{\prime }{\otimes }_{\mathbf{C}}{Y}^{\prime },F(f)(A){⊠}_{X^{\prime },Y^{\prime }}F(g)(B))$$

$$F(f{\otimes }_{\mathbf{C}}g)(A{⊠}_{X,Y}B)\to F(f)(A){⊠}_{X^{\prime },Y^{\prime }}F(g)(B)$$

$$F(f{\otimes }_{\mathbf{C}}g)(A{⊠}_{X,Y}B)\to F(f)(A){⊠}_{X^{\prime },Y^{\prime }}F(g)(B).$$

Full text

1. 1 Introduction

2. 2 Symmetric monoidal functors to ${\mathbf{Cat}}$ and $\mathbf{Add}$

3. 2.1 From $2$- to $\infty$-categories

4. 2.2 Symmetric monoidal refinements of functors to ${\mathbf{Cat}}_{1}$ and $\mathbf{Add}_{1}$

5. 2.3 Proofs of Theorem 2.2 and Theorem 2.3.

6. 3 The symmetric monoidal functor of controlled objects

7. 3.1 Symmetric monoidal structures

8. 3.2 Bornological coarse spaces

9. 3.3 Controlled objects

10. 3.4 The symmetric monoidal refinement of ${\mathbf{V}}_{{\mathbf{A}}}^{G}$

11. 3.5 The symmetric monoidal $K$-theory functor for additive categories

1 Introduction

A ${\mathcal{C}}$-valued equivariant coarse homology theory is a functor

[TABLE]

satisfying a certain family of axioms [BEKW17, Def. 3.10]. Here, $G\mathbf{BornCoarse}$ is the category of $G$-bornological coarse spaces and ${\mathcal{C}}$ is a stable cocomplete $\infty$-category. We refer to [BEKW17, Sec. 2.1], or Section 3.2, for details. The category $G\mathbf{BornCoarse}$ has a symmetric monoidal structure $\otimes$, and if also ${\mathcal{C}}$ has a symmetric monoidal structure, then we can ask whether the functor $E$ can be refined to a lax symmetric monoidal functor. Such a refinement can simplify calculations or can be applied to obtain localization results, see [BC].

In the present paper, as an example for $E$, we consider the universal coarse algebraic $K$-homology

[TABLE]

associated to an additive category ${\mathbf{A}}$ with a strict action of the group $G$, where ${\mathcal{M}}_{loc}$ is the stable $\infty$-category on non-commutative motives, defined as the target of the universal localizing invariant ${\mathcal{U}}_{loc}$ of Blumberg-Gepner-Tabuada [BGT13]. The functor $\mathrm{UK}{\mathcal{X}}_{{\mathbf{A}}}^{G}$ has been introduced in [BC17] as the universal variant of the spectrum-valued coarse algebraic $K$-homology $K{\mathcal{X}}_{{\mathbf{A}}}^{G}$ constructed in [BEKW17, Ch. 8].

By [BGT14, Thm. 5.8], the $\infty$-category ${\mathcal{M}}_{loc}$ has a symmetric monoidal structure.

The main result of the present paper is the following theorem.

Theorem 1.1**.**

A symmetric monoidal structure on ${\mathbf{A}}$ induces a lax symmetric monoidal refinement of the functor $\mathrm{UK}{\mathcal{X}}_{{\mathbf{A}}}^{G}$.

The functor $\mathrm{UK}{\mathcal{X}}_{{\mathbf{A}}}^{G}$ is constructed as a composition of functors

[TABLE]

where $X\mapsto{\mathbf{V}}_{{\mathbf{A}}}^{G}(X)$ associates to a $G$-bornological coarse space $X$ its additive category of equivariant $X$-controlled ${\mathbf{A}}$-objects (see Definition 3.11), and ${\mathbf{C}}\mapsto\mathrm{UK}({\mathbf{C}})$ sends an additive category ${\mathbf{C}}$ to the motive ${\mathcal{U}}_{loc}({\mathbf{Ch}}^{b}({\mathbf{C}})[W^{-1}])$ of the associated stable $\infty$-category ${\mathbf{Ch}}^{b}({\mathbf{C}})[W^{-1}]$ of bounded chain complexes over ${\mathbf{C}}$, where $W$ is the set of homotopy equivalences. Since $\mathrm{UK}$ sends equivalences of additive categories to equivalences it has a factorization

[TABLE]

where $loc:\mathbf{Add}_{1}\to\mathbf{Add}_{\infty}:=\mathbf{Add}_{1}[W^{-1}_{\mathbf{Add}}]$ is the localization at the equivalences $W_{\mathbf{Add}}$ of additive categories, in the realm of $\infty$-categories.

Theorem 1.1 follows from the following two assertions:

Theorem 1.2** (Theorem 3.26).**

A symmetric monoidal structure on ${\mathbf{A}}$ induces a lax symmetric monoidal refinement

[TABLE]

of the functor $loc\circ{\mathbf{V}}_{{\mathbf{A}}}^{G}$.

Theorem 1.3** (Theorem 3.27).**

The functor $\mathrm{UK}_{\infty}$ admits a lax symmetric monoidal refinement

[TABLE]

The main difficulty in proving Theorem 1.2 is that the symmetric monoidal category of small additive categories is of a $2$-categorical nature. A pedestrian approach to the proof of this theorem would thus require to work with symmetric monoidal structures on $2$-categories and therefore tedious considerations of a large set of commuting diagrams. In this paper we prefer to use the language of symmetric monoidal $\infty$-categories. In Section 3.4, by using the Grothendieck construction, we encode the functor ${\mathbf{V}}_{{\mathbf{A}}}^{G}{\colon G\mathbf{BornCoarse}\to\mathbf{Add}_{1}}$ into a cocartesian fibration ${\mathcal{V}}_{{\mathbf{A}}}^{G}\to G\mathbf{BornCoarse}$ coming from an op-fibration of $1$-categories. We then encode a symmetric monoidal refinement of the functor ${\mathbf{V}}_{{\mathbf{A}}}^{G}$ into a symmetric monoidal structure on ${\mathcal{V}}_{{\mathbf{A}}}^{G}$ and a symmetric monoidal refinement of the functor to $G\mathbf{BornCoarse}$. This only requires $1$-categorical considerations. The machine of $\infty$-categories then produces, as explained in Section 2, the asserted symmetric monoidal refinement in Theorem 1.2.

The technical results Theorem 2.2 and Theorem 2.3 might be of independent interest in cases where one wants to construct symmetric monoidal refinements of functors from $1$-categories to ${\mathbf{Cat}}_{1}$ or $\mathbf{Add}_{1}$.

Theorem 1.3 is shown in Section 3.5 by combining various results in the literature on $dg$-categories.

Acknowledgements: We thank Denis-Charles Cisinksi and Thomas Nikolaus for helpful discussion. U.B. was supported by the SFB 1085 (Higher Invariants) and L.C. was supported by the GK 1692 (Curvature, Cycles, and Cohomology).

2 Symmetric monoidal functors to ${\mathbf{Cat}}$ and $\mathbf{Add}$

In this section, we construct lax symmetric monoidal refinements of functors from symmetric monoidal $1$-categories to the categories ${\mathbf{Cat}}_{1}$ (and $\mathbf{Add}_{1}$) of small (additive) categories.

2.1 From $2$- to $\infty$-categories

A symmetric monoidal structure on a $1$-category ${\mathbf{C}}$ consists of the tensor functor

[TABLE]

the tensor unit $1_{{\mathbf{C}}}$, and the associator, symmetry and unit-transformations, which must satisfy various compatibility relations. If ${\mathbf{C}}$ and ${\mathbf{D}}$ are symmetric monoidal $1$-categories, then we can consider lax symmetric monoidal functors from ${\mathbf{C}}$ to ${\mathbf{D}}$. Such a lax symmetric monoidal functor is given by a functor $F:{\mathbf{C}}\to{\mathbf{D}}$ together with a natural transformation

[TABLE]

that is compatible with the associators, symmetries and unit-transformations of ${\mathbf{C}}$ and ${\mathbf{D}}$ in a suitable way. We will list these structures and relations in Subsection 3.1 below.

The categories $\mathbf{Add}$ or ${\mathbf{Cat}}$ of small additive categories and small categories are naturally $2$-categories. Furthermore, the category ${\mathbf{Cat}}$ is symmetric monoidal with respect to the Cartesian symmetric monoidal structure $\times:=\otimes_{{\mathbf{Cat}}}$. The category $\mathbf{Add}$ has also a symmetric monoidal structure $\otimes_{\mathbf{Add}}$: if ${\mathbf{A}}$ and ${\mathbf{B}}$ are two additive categories, then the objects of the tensor product ${\mathbf{A}}\otimes_{\mathbf{Add}}{\mathbf{B}}$ are pairs $(A,B)$ of objects $A$ in ${\mathbf{A}}$ and $B$ in ${\mathbf{B}}$, and the morphisms are given by the tensor product

[TABLE]

of abelian groups.

In the case of a symmetric monoidal structure on a $2$-category, like ${\mathbf{Cat}}$ or $\mathbf{Add}$, we have the same compatibility relations between the structures (tensor functor, tensor unit, etc.) as in the $1$-categorical case, but they are satisfied up to $2$-morphisms only, which in turn must satisfy higher compatibility relations. A similar remark applies to the notion of a (lax) symmetric monoidal functor.

In the present paper we consider the $1$-categorical situation as explicitly manageable, and we will avoid to explicitly work with symmetric monoidal structures on $2$-categories.

Let ${\mathbf{C}}$ be a symmetric monoidal $1$-category. Our goal is to construct symmetric monoidal functors $F:{\mathbf{C}}\to{\mathbf{Cat}}$ or $F:{\mathbf{C}}\to\mathbf{Add}$ using $1$-categorical data only. Instead of working with the symmetric monoidal $2$-categories ${\mathbf{Cat}}$ or $\mathbf{Add}$ we will actually use the associated symmetric monoidal $\infty$-categories ${\mathbf{Cat}}_{\infty}$ or $\mathbf{Add}_{\infty}$.

We start with the ordinary category ${\mathbf{Cat}}_{1}$ of small categories. Let $W_{{\mathbf{Cat}}}$ be the equivalences in ${\mathbf{Cat}}_{1}$. The localization in large $\infty$-categories

[TABLE]

is the large $\infty$-category of categories. It models the $2$-category ${\mathbf{Cat}}$ in the following sense. The $2$-category ${\mathbf{Cat}}$ can be considered as a category enriched in categories. Applying the nerve functor ${\mathtt{N}}$ to the ${\mathtt{Hom}}$-categories in ${\mathbf{Cat}}$ we get a fibrant111i.e., the ${\mathtt{Hom}}$-complexes are Kan complexes simplicially enriched category ${\mathtt{N}}({\mathbf{Cat}})$. Applying the homotopy coherent nerve functor $\mathcal{N}$, we get an $\infty$-category

[TABLE]

Then, we have an equivalence of $\infty$-categories

[TABLE]

We refer to the appendix of [GHN17] for more details about ${\mathtt{N}}_{2}$.

The category ${\mathbf{Cat}}_{1}$ is a symmetric monoidal category and therefore gives rise to an op-fibration of $1$-categories [Lur14, Constr. 2.0.01], and to a symmetric monoidal $\infty$-category [Lur14, Def. 2.0.0.7 & Ex. 2.1.2.21]

[TABLE]

respectively. The equivalences $W_{{\mathbf{Cat}}}$ are preserved by the cartesian product. Hence we can form a symmetric monoidal localization [Hin16, Prop. 3.2.2]

[TABLE]

whose underlying $\infty$-category is equivalent to ${\mathbf{Cat}}_{\infty}$. Conseqently, the symmetric monoidal $\infty$-category ${\mathbf{Cat}}_{\infty}^{\otimes}\to{\mathtt{N}}(\mathbf{Fin}_{*})$ models the symmetric monoidal $2$-category ${\mathbf{Cat}}$. In this way we avoid to spell out the structures of a symmetric monoidal $2$-category explicitly.

A similar reasoning applies to $\mathbf{Add}$. We consider the large $1$-category $\mathbf{Add}_{1}$ of small additive categories and exact functors with the equivalences $W_{\mathbf{Add}}$. Then we define the large $\infty$-category

[TABLE]

and get an equivalence

[TABLE]

We can consider $\mathbf{Add}_{1}$ as a symmetric monoidal category giving rise to an op-fibration of $1$-categories and a symmetric monoidal $\infty$-category

[TABLE]

Since the equivalences $W_{\mathbf{Add}}$ are preserved by the tensor product $\otimes_{\mathbf{Add}}$, we get the symmetric monoidal localization

[TABLE]

whose underlying $\infty$-category is equivalent to $\mathbf{Add}_{\infty}$. Therefore $\mathbf{Add}_{\infty}^{\otimes}\to{\mathtt{N}}(\mathbf{Fin}_{*})$ models the symmetric monoidal $2$-category $\mathbf{Add}$.

Let ${\mathbf{C}}$ be an ordinary category. A functor $F:{\mathbf{C}}\to{\mathbf{Cat}}_{1}$ (or $F:{\mathbf{C}}\to\mathbf{Add}_{1}$) gives rise to a functor between $\infty$-categories $F_{\infty}:{\mathtt{N}}({\mathbf{C}})\to{\mathbf{Cat}}_{\infty}$ (or $F_{\infty}:{\mathtt{N}}({\mathbf{C}})\to\mathbf{Add}_{\infty}$) in the natural way, e.g. as the composition

[TABLE]

A symmetric monoidal $1$-category ${\mathbf{C}}$ gives rise to the symmetric monoidal $\infty$-category ${\mathtt{N}}({\mathbf{C}}^{\otimes})\to{\mathtt{N}}(\mathbf{Fin}_{*})$ whose underlying $\infty$-category is equivalent to ${\mathtt{N}}({\mathbf{C}})$. We now consider a functor $F:{\mathbf{C}}\to{\mathbf{Cat}}_{1}$ (or $F:{\mathbf{C}}\to\mathbf{Add}_{1}$). Recall that a map of $\infty$-operads [Lur14, Def. 2.1.2.7] can be thought of as a (lax) symmetric monoidal functor [Lur14, Def. 2.1.3.7] between the underlying categories.

Let $F$ and $F_{\infty}$ be as above.

Definition 2.1**.**

A lax symmetric monoidal refinement of $F$ is a morphism of $\infty$-operads

[TABLE]

that induces a functor equivalent to $F_{\infty}$ on the underlying $\infty$-categories.

Using this definition we avoid to spell out the details of the notion of a lax-symmetric functor from ${\mathbf{C}}$ to the $2$-category ${\mathbf{Cat}}$ or $\mathbf{Add}$.

2.2 Symmetric monoidal refinements of functors to ${\mathbf{Cat}}_{1}$ and $\mathbf{Add}_{1}$

In this subsection we state the technical results Theorem 2.2 and Theorem 2.3 which provide lax symmetric monoidal refinements of functors to ${\mathbf{Cat}}_{1}$ and $\mathbf{Add}_{1}$.

Let ${\mathbf{C}}$ be a $1$-category. A functor between $1$-categories

[TABLE]

can be interpreted, via the Grothendieck construction, as a cocartesian fibration

[TABLE]

An object of the $1$-category ${\mathcal{F}}$ is a pair $(X,A)$ with $X$ in ${\mathbf{C}}$ and $A$ in $F(X)$. A morphism $(X,A)\to(Y,B)$ is a pair $(f,\phi)$ of a morphism $f:X\to Y$ in ${\mathbf{C}}$ and a morphim $\phi:F(f)(A)\to B$ in $F(Y)$.

Assume that the categories ${\mathbf{C}}$ and ${\mathcal{F}}$ have symmetric monoidal structures such that

[TABLE]

i.e., $\pi_{F}$ preserves the tensor product strictly. Then, we can write

[TABLE]

For every two objects $X,X^{\prime}$ in ${\mathbf{C}}$ we obtain a bifunctor

[TABLE]

which is defined on morphisms in the canonical way. Let

[TABLE]

be morphisms in ${\mathbf{C}}$ and $A$ in $F(X)$ and $B$ in $F(Y)$. Then

[TABLE]

are morphisms in ${\mathcal{F}}$. Then the second component of their tensor product

[TABLE]

is a morphism

[TABLE]

in $F(X^{\prime}\otimes_{{\mathbf{C}}}Y^{\prime})$. This morphism will appear in the assumptions of the two theorems below.

We now consider the following data:

a symmetric monoidal $1$-category ${\mathbf{C}}$, 2. 2.

a functor $F:{\mathbf{C}}\to{\mathbf{Cat}}_{1}$, 3. 3.

a symmetric monoidal structure on the Grothendieck construction ${\mathcal{F}}$ of $F$.

Let $\pi_{F}:{\mathcal{F}}\to{\mathbf{C}}$ denote the associated projection.

Theorem 2.2**.**

Assume:

The functor $\pi_{F}$ strictly preserves the tensor product, the tensor unit as well as the associator, unit, and symmetry transformations. 2. 2.

For every two objects $(X,A)$ and $(Y,B)$ in ${\mathcal{F}}$ and morphisms $f:X\to X^{\prime}$ and $g:Y\to Y^{\prime}$ in ${\mathbf{C}}$ the morphism (2.3)

[TABLE]

is an isomorphism.

Then the data provide a lax symmetric monoidal refinement (Def. 2.1)

[TABLE]

of the functor $F$.

Note that Condition 1 in the theorem implies the Relation (2.1) so that the bifunctors $\boxtimes_{X,Y}$ appearing in Condition 2 are, in fact, defined.

The analoguous version for additive categories is the following.

Consider the following data:

a symmetric monoidal $1$-category ${\mathbf{C}}$, 2. 2.

a functor $F:{\mathbf{C}}\to\mathbf{Add}_{1}$, 3. 3.

a symmetric monoidal structure on the Grothendieck construction ${\mathcal{F}}$ of $F$.

Let $\pi_{F}:{\mathcal{F}}\to{\mathbf{C}}$ denote the associated projection.

Theorem 2.3**.**

Assume:

The functor $\pi_{F}$ strictly preserves the tensor product, the tensor unit as well as the associator, unit, and symmetry transformations. 2. 2.

The functors $\boxtimes_{X,X^{\prime}}$ are bi-additive for every $X,X^{\prime}$ in ${\mathbf{C}}$. 3. 3.

For every two objects $(X,A)$ and $(Y,B)$ in ${\mathcal{F}}$ and morphisms $f:X\to X^{\prime}$ and $g:Y\to Y^{\prime}$ in ${\mathbf{C}}$ the morphism (2.3)

[TABLE]

is an isomorphism.

Then the data provide a lax symmetric monoidal refinement

[TABLE]

of the functor $F$.

2.3 Proofs of Theorem 2.2 and Theorem 2.3.

We start with the proof of Theorem 2.2. Let

[TABLE]

denote the symmetric monoidal $\infty$-category corresponding to the symmetric monoidal category ${\mathbf{C}}$ [Lur14, Ex. 2.1.2.21]. Let

[TABLE]

be the cocartesian fibration corresponding to the symmetric monoidal category of small categories. Then the $\infty$-category

[TABLE]

corresponds to the $\infty$-category of lax symmetric monoidal functors

[TABLE]

see the text after [Lur14, Rem. 2.1.3.6]. We let

[TABLE]

denote the category of ${\mathtt{N}}({\mathbf{C}})$-monoids in ${\mathbf{Cat}}_{\infty}$ [Lur14, Def. 2.4.2.1]. By [Lur14, Prop. 2.4.2.5], we have an equivalence

[TABLE]

By [Lur14, Rem. 2.4.2.4], in order to provide an object of ${\mathbf{Mon}}_{{\mathtt{N}}({\mathbf{C}})}({\mathbf{Cat}}_{\infty})$, it suffices to present a cocartesian fibration

[TABLE]

which exhibits ${\mathcal{C}}^{\otimes}$ as a ${\mathtt{N}}({\mathbf{C}})$-monoidal category [Lur14, Rem. 2.1.2.13]. To this end we must show that the composition

[TABLE]

exhibits ${\mathcal{C}}^{\otimes}$ as an $\infty$-operad [Lur14, Prop. 2.1.2.12].

Let

[TABLE]

be a symmetric monoidal functor between $1$-categories as in the Theorem 2.2. We get an induced functor of symmetric monoidal categories

[TABLE]

and thus a morphism of $\infty$-operads

[TABLE]

Our task is then to show that ${\mathtt{N}}(\pi_{F}^{\otimes})$ exhibits ${\mathtt{N}}({\mathcal{F}}^{\otimes})$ as an ${\mathtt{N}}({\mathbf{C}})$-monoidal category, and we must only check that ${\mathtt{N}}(\pi_{F}^{\otimes})$ is a cocartesian fibration. It suffices to check that $\pi_{F}^{\otimes}$ is an op-fibration of $1$-categories.

By assumption, the underlying functor of $\pi_{F}$ (after forgetting the symmetric monoidal structures) arose from a Grothendieck construction for a functor

[TABLE]

Recall from [Lur14, Constr. 2.0.0.1] that the objects of ${\mathbf{C}}^{\otimes}$ in the fibre ${\mathbf{C}}\langle n\rangle$ of ${\mathbf{C}}^{\otimes}$ over $\langle n\rangle$ in $\mathbf{Fin}_{*}$ are $n$-tuples of objects of ${\mathbf{C}}$. Consider two objects

[TABLE]

in ${\mathbf{C}}^{\otimes}\langle n\rangle$ and ${\mathbf{C}}^{\otimes}\langle m\rangle$ and an object

[TABLE]

in ${\mathcal{F}}^{\otimes}\langle n\rangle$, where $A_{i}$ belongs to $F(X_{i})$. Let $\alpha\langle n\rangle\to\langle m\rangle$ be a morphism in $\mathbf{Fin}_{*}$ and

[TABLE]

be a morphism in ${\mathbf{C}}^{\otimes}$ over $\alpha$. Then $f$ is given by a collection of morphisms $f:=(f_{j})_{j\in\langle m\rangle}$ with

[TABLE]

We must provide a cocartesian lift of $f$. For $j$ in $\langle m\rangle$ we have a morphism

[TABLE]

in ${\mathcal{F}}$. One now checks in a straightforward (but tedious) manner that the collection $g:=(g_{j})_{j\in\langle m\rangle}$ is the cocartesian lift of $f$. The argument repeatedly uses the Condition 2. This finishes the proof of Theorem 2.2.

We now turn to the proof of Theorem 2.3. Consider the symmetric monoidal subcategory

[TABLE]

Indeed we can first consider the subcategory ${\mathbf{Cat}}_{\infty}(\coprod)$ of $\infty$-categories which admit finite coproducts and coproduct preserving functors. By [Lur14, Cor. 4.8.1.4] (applied to the collection $\mathcal{K}$ of finite sets) we get a symmetric monoidal subcategory

[TABLE]

In the next step we view $\mathbf{Add}_{\infty}$ as a full subcategory of ${\mathbf{Cat}}_{\infty}(\coprod)$ of pointed $1$-categories in which products and coproducts coincide. Using [Lur14, Cor. 2.2.1.1] one then shows that

[TABLE]

is again a suboperad. We now consider the diagram

[TABLE]

The lower horizontal map is a morphism of $\infty$-operads by Theorem 2.2. We first argue that the dotted lift exists. To this end we use [Lur14, Notation 4.8.1.2]. One must check that $F$ takes values in categories admitting finite coproducts (clear), and that the functors

[TABLE]

preserves sums in both variables separately, i.e., Assumption 2. Finally, for the dashed arrow we use that $F$ takes values in $\mathbf{Add}_{{1}}$.

3 The symmetric monoidal functor of controlled objects

3.1 Symmetric monoidal structures

In this subsection we write out, for later reference, the structures of a symmetric monoidal category and of a (lax) symmetric monoidal functor. Let ${\mathbf{C}}$ be a $1$-category:

Definition 3.1**.**

[Mac71, Sec. VII. 1. & 7.]** A symmetric monoidal structure on ${\mathbf{C}}$ is given by the following data:

a bifunctor $(-\otimes_{{\mathbf{C}}}-):{\mathbf{C}}\times{\mathbf{C}}\to{\mathbf{C}}$, 2. 2.

an object $1_{{\mathbf{C}}}$ (the tensor unit), 3. 3.

a natural isomorphism (the associativity constraint)

[TABLE] 4. 4.

a natural isomorphism $\eta^{{\mathbf{C}}}:1_{{\mathbf{C}}}\otimes_{{\mathbf{C}}}-\to{\mathtt{id}}_{{\mathbf{C}}}$ (the unit constraint), 5. 5.

a natural isomorphism (the symmetry) $\sigma^{{\mathbf{C}}}:(-\otimes_{{\mathbf{C}}}-)\circ T\to(-\otimes_{{\mathbf{C}}}-)$, where $T:{\mathbf{C}}\times{\mathbf{C}}\to{\mathbf{C}}\times{\mathbf{C}}$ is the flip functor.

This data have to satisfy the following relations:

the pentagon relation, 2. 2.

*the triangle relation, * 3. 3.

the inverse relation, 4. 4.

*the associativity coherence. *

A symmetric monoidal category is a category equipped with a symmetric monoidal structure.

We will use the name of the category as a superscript for the constraints, but if we evaluate e.g. the symmetry constraint $\sigma^{{\mathbf{C}}}$ at the objects $C,C^{\prime}$ of ${\mathbf{C}}$, then we write shortly $\sigma_{C,C^{\prime}}$ instead of $\sigma^{{\mathbf{C}}}_{C,C^{\prime}}$ since the type of objects in the subscript already determines the category in question.

Let ${\mathbf{C}}$ and ${\mathbf{D}}$ be symmetric monoidal categories, and let $F:{\mathbf{C}}\to{\mathbf{D}}$ be a functor.

Definition 3.2**.**

[Mac71, Sec. XI. 2.]** A symmetric monoidal structure on $F$ is given by the following data:

an isomorphism $\epsilon^{F}:1_{{\mathbf{D}}}\to F(1_{{\mathbf{C}}})$, 2. 2.

a natural isomorphism $\mu^{F}:(-\otimes_{{\mathbf{D}}}-)\circ(F\times F)\to F\circ(-\otimes_{{\mathbf{C}}}-)$.

This data have to satisfy the following relations:

associativity relation, 2. 2.

unitality relation, 3. 3.

symmetry relation.

Remark 3.3**.**

If we weaken the assumptions and we only require that $\epsilon^{F}$ and $\mu^{F}$ are natural transformations, then we get the definition of a lax symmetric monoidal functor.

3.2 Bornological coarse spaces

In this subsection we recall the definition of the symmetric monoidal category $G\mathbf{BornCoarse}$ of $G$-bornological coarse spaces [BE16, Sec. 2], [BEKW17, Sec. 2.1].

In the definitions below we will use the following notation:

For a set $Z$ we let $\mathcal{P}(Z)$ denote the power set of $Z$. 2. 2.

If a group $G$ acts on a set $X$, then it acts diagonally on $X\times X$ and therefore on $\mathcal{P}(X\times X)$. For $U$ in $\mathcal{P}(X\times X)$ we set

[TABLE] 3. 3.

For $U$ in $\mathcal{P}(X\times X)$ and $B$ in $\mathcal{P}(X)$ we define the $U$-thickening $U[B]$ by

[TABLE] 4. 4.

For $U$ in $\mathcal{P}(X\times X)$ we define its inverse by

[TABLE] 5. 5.

For $U,V$ in $\mathcal{P}(X\times X)$ we define their composition by

[TABLE]

Let $G$ be a group and let $X$ be a $G$-set.

Definition 3.4**.**

A $G$-coarse structure ${\mathcal{C}}$ on $X$ is a subset of $\mathcal{P}(X\times X)$ with the following properties:

${\mathcal{C}}$* is closed under composition, inversion, and forming finite unions or subsets. * 2. 2.

${\mathcal{C}}$* contains the diagonal ${\mathtt{diag}}(X)$ of $X$.* 3. 3.

For every $U$ in ${\mathcal{C}}$, the set $GU$ is also in ${\mathcal{C}}$.

The pair $(X,{\mathcal{C}})$ is called a $G$-coarse space, and the members of ${\mathcal{C}}$ are called (coarse) entourages of $X$.

Let $(X,{\mathcal{C}})$ and $(X^{\prime},{\mathcal{C}}^{\prime})$ be $G$-coarse spaces and let $f\colon X\to X^{\prime}$ be an equivariant map between the underlying sets.

Definition 3.5**.**

The map $f$ is controlled if for every $U$ in ${\mathcal{C}}$ we have $(f\times f)(U)\in{\mathcal{C}}^{\prime}$.

We obtain a category $G\mathbf{Coarse}$ of $G$-coarse spaces and controlled equivariant maps.

Let $G$ be a group and let $X$ be a $G$-set.

Definition 3.6**.**

A $G$-bornology ${\mathcal{B}}$ on $X$ is a subset of $\mathcal{P}(X)$ with the following properties:

${\mathcal{B}}$* is closed under forming finite unions and subsets.* 2. 2.

${\mathcal{B}}$* contains all finite subsets of $X$.* 3. 3.

${\mathcal{B}}$* is $G$-invariant.*

The pair $(X,{\mathcal{B}})$ is called a $G$-bornological space, and the members of ${\mathcal{B}}$ are called bounded subsets of $X$.

Let $(X,{{\mathcal{B}}})$ and $(X^{\prime},{{\mathcal{B}}^{\prime}})$ be $G$-bornological spaces and let $f\colon X\to X^{\prime}$ be an equivariant map between the underlying sets.

Definition 3.7**.**

The map $f$ is proper if for every $B^{\prime}$ in ${\mathcal{B}}^{\prime}$ we have $f^{-1}(B^{\prime})\in{\mathcal{B}}$.

We obtain a category $G\mathbf{Born}$ of $G$-bornological spaces and proper equivariant maps.

Let $X$ be a $G$-set equipped with a $G$-coarse structure ${\mathcal{C}}$ and a $G$-bornology ${\mathcal{B}}$.

Definition 3.8**.**

The coarse structure ${\mathcal{C}}$ and the bornology ${\mathcal{B}}$ are said to be compatible if for every $B$ in ${\mathcal{B}}$ and $U$ in ${\mathcal{C}}$ the $U$-thickening $U[B]$ (see (3.1)) lies in ${\mathcal{B}}$.

Definition 3.9**.**

A $G$-bornological coarse space is a triple $(X,{\mathcal{C}},{\mathcal{B}})$ consisting of a $G$-set $X$, a $G$-coarse structure ${\mathcal{C}}$ and a $G$-bornology ${\mathcal{B}}$ on $X$, such that ${\mathcal{C}}$ and ${\mathcal{B}}$ are compatible.

Usually we will denote a $G$-bornological coarse space by the symbol $X$ and write ${\mathcal{B}}(X)$ and ${\mathcal{C}}(X)$ for its bornology and coarse structures.

Definition 3.10**.**

A morphism $f:X\to X^{\prime}$ between $G$-bornological coarse spaces is an equivariant map of the underlying $G$-sets that is controlled and proper.

We obtain a category $G\mathbf{BornCoarse}$ of $G$-bornological coarse spaces and morphisms.

Next we describe the symmetric monoidal structure on $G\mathbf{BornCoarse}$ [BEKW17, Ex. 2.17]. We have a forgetful functor

[TABLE]

which associates to every $G$-bornological coarse space $X$ its underlying $G$-set. This functor is faithful. The category $G{\mathbf{Set}}$ is endowed with the cartesian symmetric monoidal structure. The symmetric monoidal structure on $G\mathbf{BornCoarse}$ will be defined in such a way that the functor $U$ preserves the unit and the tensor product strictly, i.e., the morphisms 1 and 2 in Definition 3.2 are identities. In other words, the associator, unit and symmetry constraints are imported from $G{\mathbf{Set}}$ and satisfy the relations required in Definition 3.1 automatically.

We start with the description of the bifunctor

[TABLE]

Let $X$ and $X^{\prime}$ be two $G$-bornological coarse spaces. Then their tensor product

[TABLE]

is the $G$-bornological coarse spaces defined as follows:

The underlying $G$-set of $X\otimes_{G\mathbf{BornCoarse}}X^{\prime}$ is the cartesian product of the underlying $G$-sets $X\times X^{\prime}$. 2. 2.

The $G$-bornology on $X\times X^{\prime}$ is generated by the subsets $B\times B^{\prime}$ for all $B$ in ${\mathcal{B}}(X)$ and $B^{\prime}$ in ${\mathcal{B}}(X^{\prime})$. 3. 3.

The $G$-coarse structure on $X\times X^{\prime}$ is generated by the entourages $U\times U^{\prime}$ for $U$ in ${\mathcal{C}}(X)$ and $U^{\prime}$ in ${\mathcal{C}}(X^{\prime})$.

Here a $G$-bornological (or coarse, respectivley) structure generated by a family of subsets (or entourages) is the minimal $G$-bornological (or $G$-coarse) structure containing these subsets (or entourages). Note that the underlying $G$-coarse space of the tensor product represents the cartesian product of the underlying $G$-coarse spaces of the factors in $G\mathbf{Coarse}$, but the tensor product is not the cartesian product in $G\mathbf{BornCoarse}$ in general.

From now on we will use the shorter notation $X\otimes X^{\prime}$ for the tensor product of $G$-bornological coarse spaces, i.e., we omit the subscript $G\mathbf{BornCoarse}$.

If $f:X\to Y$ and $f^{\prime}:X^{\prime}\to Y^{\prime}$ are morphisms of $G$-bornological coarse spaces, then their tensor product

[TABLE]

is induced by the equivariant map of underlying $G$-sets $(x,y)\mapsto(f(x),f(y))$. This finishes the description of the bifunctor 3.1.1

The tensor unit $1_{G\mathbf{BornCoarse}}$ (3.1.2) is given by the one-point space $*$.

As explained above, the associativity, unit and symmetry constraints are imported from $G{\mathbf{Set}}$. It is straightforward to check that they are implemented by morphisms of $G$-bornological coarse spaces.

This finishes the description of the symmetric monoidal structure $\otimes$ on the category $G\mathbf{BornCoarse}$.

3.3 Controlled objects

In this section, for every additive category ${\mathbf{A}}$ with a strict $G$-action, we describe the functor

[TABLE]

which sends a $G$-bornological coarse space $X$ to its additive category ${\mathbf{V}}_{{\mathbf{A}}}^{G}{(X)}$ of equivariant $X$-controlled ${\mathbf{A}}$-object [BEKW17, Sec. 8.2].

For a group $G$, let $BG$ be the category with one object $*$ and ${\mathtt{End}}_{BG}(*)\cong G$. Then ${\mathbf{Fun}}(BG,\mathbf{Add}_{1})$ is the category of additive categories with a strict $G$-action. Explicitly, an additive category with a strict $G$-action is an additive category ${\mathbf{A}}$ (the evaluation of the functor at the object $*$ in $BG$) together with an action of $G$ on ${\mathbf{A}}$ by exact functors, which is strictly associative. Our notation for the action of $g$ in $G$ on objects $A$ of ${\mathbf{A}}$ and morphisms $f$ is

[TABLE]

Let ${\mathbf{A}}$ be an additive category with a strict $G$-action and $X$ be a $G$-bornological coarse space. We consider the bornology ${\mathcal{B}}(X)$ of $X$ as a poset with a $G$-action $(g,B)\mapsto gB$, hence as a category with a strict $G$-action, i.e., an object of ${\mathbf{Fun}}(BG,{\mathbf{Cat}}_{1})$.

The category ${\mathbf{Fun}}({\mathcal{B}}(X),{\mathbf{A}})$ has an induced $G$-action which can explicitly be described as follows. If $M:{\mathcal{B}}\to{\mathbf{A}}$ is a functor and $g$ is an element of $G$, then $gM\colon{\mathcal{B}}(X)\to{\mathbf{A}}$ is the functor which sends a bounded set $B$ in ${\mathcal{B}}(X)$ to the object $gM(g^{-1}(B))$ of ${\mathbf{A}}$. If $\rho:M\to M^{\prime}$ is a natural transformation between two such functors, then we let $g\rho:gM\to gM^{\prime}$ denote the canonically induced natural transformation.

Definition 3.11**.**

[BEKW17, Def. 8.3]** An equivariant $X$-controlled ${\mathbf{A}}$-object is a pair $(M,\rho)$ consisting of a functor $M\colon{\mathcal{B}}(X)\to{\mathbf{A}}$ and a family $\rho=(\rho(g))_{g\in G}$ of natural isomorphisms $\rho(g)\colon M\to gM$ satisfying the following conditions:

$M(\emptyset)\cong 0$. 2. 2.

For all $B,B^{\prime}$ in ${\mathcal{B}}(X)$, the commutative square

[TABLE]

is a pushout square. 3. 3.

For all $B$ in ${\mathcal{B}}(X)$ there exists a finite subset $F$ of $B$ such that the inclusion $F\to B$ induces an isomorphism $M(F)\xrightarrow{\cong}M(B)$. 4. 4.

For all pairs of elements $g,g^{\prime}$ of $G$ we have the relation $\rho(gg^{\prime})=g\rho(g^{\prime})\circ\rho(g)$.

If $U$ is an invariant coarse entourage of $X$, i.e., an element of ${\mathcal{C}}(X)^{G}$, then we get a $G$-equivariant functor

[TABLE]

which sends a bounded subset $B$ of $X$ to its $U$-thickening $U[B]$, see (3.1). Indeed, the $U$-thickening $U[B]$ of a bounded subset $B$ is again bounded by the compatibility of the coarse structure ${\mathcal{C}}(X)$ and the bornology ${\mathcal{B}}(X)$ of $X$, and $U[-]$ preserves the inclusion relation. Since $U$ is $G$-invariant we have the equality $U[gB]=gU[B]$. It implies that $U[-]$ is $G$-equivariant. If $M:{\mathcal{B}}(X)\to{\mathbf{A}}$ is a functor, then we write $U[-]^{*}M:=M\circ U[-]$ for the pull-back of $M$ along $U[-]$.

Let $(M,\rho),(M^{\prime},\rho^{\prime})$ be two equivariant $X$-controlled ${\mathbf{A}}$-objects and $U$ be an invariant coarse entourage of $X$.

Definition 3.12**.**

An equivariant $U$-controlled morphism $\phi\colon(M,\rho)\to(M^{\prime},\rho^{\prime})$ is a natural transformation

[TABLE]

such that $\rho^{\prime}(g)\circ\phi=(g\phi)\circ\rho(g)$ for all elements $g$ of $G$.

We let $\mathrm{Mor}_{U}((M,\rho),(M^{\prime},\rho^{\prime}))$ denote the abelian group of equivariant $U$-controlled morphisms.

If $U^{\prime}$ is in ${\mathcal{C}}(X)^{G}$ such that $U\subseteq U^{\prime}$, then for every $B$ in ${\mathcal{B}}(X)$ we have $U[B]\subseteq U^{\prime}[B]$. These inclusions induce a transformation between functors $U[-]^{*}M^{\prime}\to U^{\prime}[-]^{*}M^{\prime}$ and therefore a map

[TABLE]

by postcomposition. Using these maps in the interpretation of the colimit we define the abelian group of equivariant controlled morphisms from $(M,\rho)$ to $(M^{\prime},\rho^{\prime})$ by

[TABLE]

We now consider a pair of morphisms in

[TABLE]

respectively, which are represented by

[TABLE]

We define the composition of the two morphisms to be represented by the morphism

[TABLE]

(see (3.2) for notation) where

[TABLE]

is defined in the canonical manner. We denote the resulting category of equivariant $X$-controlled ${\mathbf{A}}$-objects and equivariant controlled morphisms by ${\mathbf{V}}_{\mathbf{A}}^{G}(X)$. This category is additive [BEKW17, Lemma 8.7].

Let $f\colon X\to X^{\prime}$ be a morphism of $G$-bornological coarse spaces, and let $(M,\rho)$ be an equivariant $X$-controlled ${\mathbf{A}}$-object. Since $f$ is proper, it induces an equivariant functor $f^{-1}:{\mathcal{B}}(X^{\prime})\to{\mathcal{B}}(X)$, and we can define a functor $f_{*}M\colon{\mathcal{B}}(X^{\prime})\to{\mathbf{A}}$ by

[TABLE]

Furthermore, we define

[TABLE]

Let $U$ be in ${\mathcal{C}}(X)^{G}$ and let $\phi\colon(M,\rho)\to(M^{\prime},\rho^{\prime})$ be an equivariant $U$-controlled morphism. Then $V:=(f\times f)(U)$ belongs to ${\mathcal{C}}(X^{\prime})^{G}$ and we have $U[f^{-1}(B^{\prime})]\subseteq f^{-1}(V[B^{\prime}])$ for all bounded subsets $B^{\prime}$ of $X^{\prime}$. Therefore, we obtain an induced $V$-controlled morphism

[TABLE]

One checks that this construction defines an additive functor

[TABLE]

This completes the construction of the functor

[TABLE]

In the following we give a more explicit description of the objects and morphisms in ${\mathbf{V}}_{{\mathbf{A}}}^{G}(X)$ which will be used in the description of the symmetric monoidal structure on the Grothendieck construction associated to the functor ${\mathbf{V}}_{{\mathbf{A}}}^{G}$ in Section 3.4.

Convention 3.13**.**

We consider an additive category ${\mathbf{A}}$. If $(A_{i})_{i\in I}$ is a family of objects of ${\mathbf{A}}$ with at most finitely many non-zero members, then we use the symbol $\bigoplus_{i\in I}A_{i}$ in order to denote a choice of an object of ${\mathbf{A}}$ together with a family of morphisms $(A_{j}\to\bigoplus_{i\in I}A_{i})_{j\in I}$ representing the coproduct of the family.

Since in an additive category coproducts and products coincide, for every $j$ in $I$ we furthermore have a canonical projection

[TABLE]

such that the diagram

[TABLE]

commutes.

If $(A_{i^{\prime}}^{\prime})_{i^{\prime}\in I^{\prime}}$ is a second family of this type and $(\phi_{i,i^{\prime}}:A_{i^{\prime}}^{\prime}\to A_{i})_{i^{\prime}\in I^{\prime},i\in I}$ is a family of morphisms in ${\mathbf{A}}$, then we have a unique morphism $\oplus\phi_{i,i^{\prime}}$ such that the squares

[TABLE]

commute for every $i^{\prime}$ in $I^{\prime}$ and $i$ in $I$.∎

Let ${\mathbf{A}}$ be a small additive category with strict $G$-action. Let $X$ be a $G$-bornological coarse space (see Definition 3.9), and let $(M,\rho)$ be an equivariant $X$-controlled ${\mathbf{A}}$-object (see Definition 3.11). Let $B$ be in ${\mathcal{B}}(X)$ and $x$ be a point in $B$. The inclusion $\{x\}\to B$ induces a morphism $M(\{x\})\to M(B)$ in ${\mathbf{A}}$. The conditions 3.11.1 and 3.11.2 together imply that $M(\{x\})=0$ for all but finitely many points of $B$, and that the canonical morphism (induced by the universal property of the coproduct in ${\mathbf{A}}$)

[TABLE]

is an isomorphism.

Let now $U$ be in ${\mathcal{C}}(X)^{G}$, and let $\phi:(M,\rho)\to(M^{\prime},\rho^{\prime})$ be an equivariant $U$-controlled morphism. By Definition 3.12, the morphism $\phi$ is given by a natural transformation of functors $\phi:M\to U[-]^{*}M^{\prime}$ satisfying an equivariance condition. For every point $x$ in $X$ we get a morphism

[TABLE]

in ${\mathbf{A}}$. We let

[TABLE]

denote the composition of (3.6) with the projection onto the summand corresponding to $x^{\prime}$. In this way we get a family of morphisms $(\phi_{x^{\prime},x})_{x^{\prime},x\in X}$ in ${\mathbf{A}}$. In a similar manner, for $g$ in $G$, the transformation $\rho(g):M\to gM$ gives rise to a family of morphisms

[TABLE]

By construction the family $(\phi_{x^{\prime},x})_{x^{\prime},x\in X}$ satisfies the following conditions.

For all $x,x^{\prime}$ in $X$ the condition $\phi_{x^{\prime},x}\not=0$ implies that $(x^{\prime},x)\in U$. 2. 2.

We have $\rho^{\prime}(g)_{x^{\prime}}\circ\phi_{x^{\prime},x}=(g\phi)_{g^{-1}x^{\prime},g^{-1}x}\circ\rho(g)_{x}$ for all $x,x^{\prime}$ in $X$ and $g$ in $G$.

Lemma 3.14**.**

We have a bijection between equivariant $U$-controlled morphisms $\phi:(M,\rho)\to(M^{\prime},\rho^{\prime})$ and families $(\phi_{x^{\prime},x})_{x^{\prime},x\in X}$ of morphisms as in (3.7) satisfying the Conditions 1 and 2.

Proof.

Let $(M,\rho)$ and $(M^{\prime},\rho^{\prime})$ be in ${\mathbf{V}}_{{\mathbf{A}}}^{G}(X)$. We must show that a matrix $(\phi_{x^{\prime},x})_{x^{\prime},x\in X}$ of morphisms as in (3.7) which satisfies the Conditions 1 and 2 gives rise to an equivariant controlled morphism $\phi:(M,\rho)\to(M^{\prime},\rho^{\prime})$. Let $U$ be in ${\mathcal{C}}^{G}(X)$ such that Condition 1 holds true. We must construct an equivariant natural transformation $\phi:M\to U[-]^{*}M^{\prime}$.

We consider $B$ in ${\mathcal{B}}(X)$. Then $(M(\{x\})_{x\in B}$ and $(M^{\prime}(\{x^{\prime}\}))_{x^{\prime}\in U[B]}$ are families of objects in ${\mathbf{A}}$ with at most finitely many non-zero members. Using Convention 3.13, and in particular the notation from (3.4), we can define the morphism $\phi_{B}:M(B)\to M^{\prime}(U[B])$ such that

[TABLE]

commutes. It is now straightforward to check that the family $(\phi_{B})_{B\in{\mathcal{B}}(X)}$ assembles to a natural transformation $\phi:M\to U[-]^{*}M^{\prime}$ as required. By construction the morphism $\phi$ is $U$-controlled. Furthermore, the Condition 2 implies that $\phi$ satisfies the equivariance condition stated in Definition 3.12. ∎

Let ${f}:X_{0}\to X_{1}$ be a morphism of $G$-bornological coarse spaces and $(M_{i},\rho_{i})$ be objects of ${\mathbf{V}}_{{\mathbf{A}}}^{G}(X_{i})$ for $i=0,1$. Then a morphism

[TABLE]

induces a matrix

[TABLE]

To this end we observe that

[TABLE]

so that

[TABLE]

is the matrix representing $\phi$ according to Lemma 3.14. As a consequence of Lemma 3.14 we obtain:

Corollary 3.15**.**

A matrix (3.10) represents a morphism (3.9) iff the following conditions are satisfied:

There exists an entourage $U_{1}$ in ${\mathcal{C}}(X_{1})$ such that for every $x_{0}$ in $X_{0}$ and $x_{1}$ in $X_{1}$ the condition $\phi^{f}_{x_{1},x_{0}}\not=0$ implies that $(x_{1},f(x_{0}))\in U_{1}$. 2. 2.

For every $g$ in $G$ we have the equality

[TABLE]

3.4 The symmetric monoidal refinement of ${\mathbf{V}}_{{\mathbf{A}}}^{G}$

Let ${\mathbf{A}}$ be a small additive category with a strict $G$-action. Then we let

[TABLE]

denote the Grothendieck construction associated to the functor ${\mathbf{V}}_{{\mathbf{A}}}^{G}$ (see (3.3)) viewed as a functor from $G\mathbf{BornCoarse}$ to ${\mathbf{Cat}}_{1}$. The goal of this section is the construction of a symmetric monoidal structure (see Definition 3.1) on $\mathcal{V}_{{\mathbf{A}}}^{G}$ which satisfies the assumptions of Theorem 2.3.

Assumption 3.16**.**

We assume that ${\mathbf{A}}$ has a symmetric monoidal structure and that the strict action of $G$ on ${\mathbf{A}}$ has a refinement to an action by symmetric monoidal functors.

In order to introduce the notation for later arguments, we spell out the Assumption 3.16 explicitly. According to Definition 3.1 the category ${\mathbf{A}}$ comes with the following data:

a bifunctor $-\otimes_{{\mathbf{A}}}-$, 2. 2.

a tensor unit $1_{{\mathbf{A}}}$, 3. 3.

an associativity constraint $\alpha^{{\mathbf{A}}}$, 4. 4.

a unit constraint $\eta^{{\mathbf{A}}}$, 5. 5.

a symmetry constraint $\sigma^{{\mathbf{A}}}$.

This data satisfy the relations specified in Definition 3.1.

The strict action of $G$ on ${\mathbf{A}}$ by symmetric monoidal functors is implemented by the following data. For every $g$ in $G$ we have:

an additive functor $g:{\mathbf{A}}\to{\mathbf{A}}$, 2. 2.

an isomorphism $\epsilon^{g}:1_{{\mathbf{A}}}\to g1_{{\mathbf{A}}}$, 3. 3.

a natural isomorphism $\mu^{g}:(g-\otimes_{{\mathbf{A}}}g-)\to g(-\otimes_{{\mathbf{A}}}-)$,

satisfying the relations specified in Definition 3.2. We require that for all $g$ and $h$ in $G$ the following relation between the composition of symmetric monoidal functors and multiplication in $G$ holds true:

[TABLE]

The equality (as opposed to the additional data of a natural transformation) expresses the fact that the action of $G$ on ${\mathbf{A}}$ is strict.

We now describe the category $\mathcal{V}_{{\mathbf{A}}}^{G}$ explicitly.

The objects of $\mathcal{V}_{{\mathbf{A}}}^{G}$ are pairs $(X,(M,\rho))$ of objects $X$ in $G\mathbf{BornCoarse}$ and $(M,\rho)$ in ${\mathbf{V}}_{{\mathbf{A}}}^{G}(X)$. 2. 2.

A morphism $(f,\phi):(X,(M,\rho))\to(X^{\prime},(M^{\prime},\rho^{\prime}))$ consists of a morphism $f:X\to X^{\prime}$ in $G\mathbf{BornCoarse}$ and a morphism $\phi:f_{*}(M,\rho)\to(M^{\prime},\rho^{\prime})$ in ${\mathbf{V}}_{{\mathbf{A}}}^{G}(X^{\prime})$. 3. 3.

The composition of morphisms is given by

[TABLE]

The functor

[TABLE]

is the obvious functor which forgets the second component.

We now start with the description of the symmetric monoidal structure on $\mathcal{V}_{{\mathbf{A}}}^{G}$.

Let $*$ denote the one-point space. Then we can consider the equivariant $*$-controlled ${\mathbf{A}}$-object $1_{*}=(M^{unit},\rho^{unit})$ in ${\mathbf{V}}_{A}^{G}(*)$ defined as follows:

The functor $M^{unit}:{\mathcal{B}}(*)\to{\mathbf{A}}$ is uniquely determined by $M^{unit}(\{*\}):=1_{{\mathbf{A}}}$. 2. 2.

$\rho^{unit}(g):=\epsilon^{g}$ for all $g$ in $G$ (see 2.).

Definition 3.17**.**

The tensor unit of $\mathcal{V}_{{\mathbf{A}}}^{G}$ is defined to be the object $1_{\mathcal{V}_{{\mathbf{A}}}^{G}}:=(*,1_{*})$.

We now construct the bifunctor

[TABLE]

We start with its definition on objects. We consider two objects $(X,(M,\rho))$ and $(X^{\prime},(M^{\prime},\rho^{\prime}))$ in $\mathcal{V}_{{\mathbf{A}}}^{G}$. Then we define the functor

[TABLE]

as follows:

For every $B$ in ${\mathcal{B}}(X\otimes X^{\prime})$ we set (see Convention 3.13)

[TABLE]

Note that the sum has finitely many non-zero summands because of Definition 3.11 (3). 2. 2.

If $B^{\prime}$ is in ${\mathcal{B}}(X\otimes X^{\prime})$ such that $B^{\prime}\subseteq B$, then the morphism

[TABLE]

is given by the canonical map

[TABLE]

as described in Convention 3.13.

By using our Convention 3.13 and the universal property of the direct sum, one easily checks that this describes a functor satisfying the first three conditions of Definition 3.11.

We now define the family $\rho\boxtimes\rho^{\prime}$ as follows:

[TABLE]

using the notation (3.8). One checks using (3.11) that $(M\boxtimes M^{\prime},\rho\boxtimes\rho^{\prime})$ satisfies the remaining condition of Definition 3.11 and therefore belongs to ${\mathbf{V}}_{{\mathbf{A}}}^{G}(X\otimes X^{\prime})$.

Definition 3.18**.**

We define the bifunctor (3.12) on objects by

[TABLE]

Let $(f,\phi):(X_{0},(M_{0},\rho_{0}))\to(X_{1},(M_{1},\rho_{1}))$ be a morphism in $\mathcal{V}_{{\mathbf{A}}}^{G}$ (see 2). Then we define the morphism

[TABLE]

as follows.

We set $g:=f\otimes{\mathtt{id}}_{X^{\prime}}:X_{0}\otimes X^{\prime}\to X_{1}\otimes X^{\prime}$ using the tensor product in $G\mathbf{BornCoarse}$. 2. 2.

In order to describe the morphism

[TABLE]

we use Corollary 3.15. We must describe the matrix

[TABLE]

Now note that by definition

[TABLE]

so that we can set

[TABLE]

One easily checks that this matrix satisfies the conditions listed in Corollary 3.15 and therefore represents the desired morphism.

In a similar manner we define $(X,(M,\rho))\otimes(f^{\prime},\phi^{\prime})$ for a morphism $(f^{\prime},\phi^{\prime}):(X^{\prime}_{0},(M^{\prime}_{0},\rho^{\prime}_{0}))\to(X^{\prime}_{1},(M^{\prime}_{1},\rho_{1}))$.

Definition 3.19**.**

We define the bifunctor (3.12) on morphisms by the preceding description.

It is straightforward to check that (3.12) is a bifunctor, i.e., that its description on morphisms is compatible with composition.

Next we define the associativity constraint $\alpha^{\mathcal{V}_{{\mathbf{A}}}^{G}}$. We consider three objects $(X,(M,\rho))$, $(X^{\prime},(M^{\prime},\rho^{\prime}))$, and $(X^{\prime\prime},(M^{\prime\prime},\rho^{\prime\prime}))$. Then

[TABLE]

must be a morphism

[TABLE]

We set

[TABLE]

using the associativity constraint of $G\mathbf{BornCoarse}$. The second component $\phi$ is given via Corollary 3.15 by the matrix whose only non-trivial entries are

[TABLE]

using the associativity constraint of ${\mathbf{A}}$. The first condition of Corollary 3.15 is satisfied for the diagonal entourage of $X\times(X^{\prime}\times X^{\prime\prime})$, and for the second condition we use that $G$ acts on ${\mathbf{A}}$ by symmetric monoidal functors, in particular the first relation in Definition 3.2 for $\mu^{g}$ for all $g$ in $G$, see 3.

Definition 3.20**.**

We define the associativity constraint $\alpha^{\mathcal{V}_{{\mathbf{A}}}^{G}}$ by the description above.

It is straightforward but tedious to check that $\alpha^{\mathcal{V}_{{\mathbf{A}}}^{G}}$ is a natural transformation.

Following Definition 3.17 the unit constraint $\eta^{\mathcal{V}_{{\mathbf{A}}}^{G}}$ of $\mathcal{V}_{{\mathbf{A}}}^{G}$ is implemented by morphisms

[TABLE]

for all objects $(X,(M,\rho))$ of $\mathcal{V}_{{\mathbf{A}}}^{G}$. We set

[TABLE]

using the unit constraint of $G\mathbf{BornCoarse}$. Note that

[TABLE]

Hence, using Corollary 3.15, we can define morphism $\phi$ such that the non-trivial entries of its matrix are

[TABLE]

using the unit constraint of ${\mathbf{A}}$. It is easy to check that this matrix satisfies the first condition of Corollary 3.15 for the diagonal of $X$ and the second condition since the morphisms $\epsilon^{g}$ in 2 satisfy the relation of Definition 3.2.2 for all $g$ in $G$.

Definition 3.21**.**

We define the unit constraint $\eta^{\mathcal{V}_{{\mathbf{A}}}^{G}}$ by the description above.

It is straightforward to check that $\eta^{\mathcal{V}_{{\mathbf{A}}}^{G}}$ is a natural transformation.

Finally we define the symmetry constraint $\sigma^{\mathcal{V}_{{\mathbf{A}}}^{G}}$. We consider two objects $(X,(M,\rho))$ and $(X^{\prime},(M^{\prime},\rho^{\prime}))$ of $\mathcal{V}_{{\mathbf{A}}}^{G}$. Then we must define a morphism

[TABLE]

We set

[TABLE]

using the symmetry contraint for $G\mathbf{BornCoarse}$. The morphism $\phi$ is the given, using Corollary 3.15, by the matrix whose only non-trivial entries are

[TABLE]

using the symmetry constraint of ${\mathbf{A}}$. One easily checks that the first condition of Corollary 3.15 is satisfied for the diagonal entourage of $X^{\prime}\times X$. In order to verify the second condition we use that the transformations $\mu^{g}$ in 3 satisfy Definition 3.2.3 for every $g$ in $G$.

Definition 3.22**.**

We define the symmetry constraint $\sigma^{\mathcal{V}_{{\mathbf{A}}}^{G}}$ of $\mathcal{V}_{{\mathbf{A}}}^{G}$ by the description above.

It is straightforward to check that $\sigma^{\mathcal{V}_{{\mathbf{A}}}^{G}}$ is a natural transformation.

Proposition 3.23**.**

The functor $-\otimes_{\mathcal{V}_{{\mathbf{A}}}^{G}}-$ and the object $1_{\mathcal{V}_{{\mathbf{A}}}^{G}}$ together with the natural isomorphisms $\alpha^{\mathcal{V}_{{\mathbf{A}}}^{G}}$, $\eta^{\mathcal{V}_{{\mathbf{A}}}^{G}}$ and $\sigma^{\mathcal{V}_{{\mathbf{A}}}^{G}}$ define a symmetric monoidal structure on $\mathcal{V}_{{\mathbf{A}}}^{G}$.

The functor $\pi:\mathcal{V}_{{\mathbf{A}}}^{G}\to G\mathbf{BornCoarse}$ preserves the tensor product and the tensor unit as well as the associator, unit, and symmetry transformations.

Proof.

One verifies the relations listed in Definition 3.1 in a straightforward manner by inserting the definitions and using that the corresponding relations are satisfied for the symmetric monoidal structures on ${\mathbf{A}}$ and $G\mathbf{BornCoarse}$. ∎

Let $X$ and $X^{\prime}$ be $G$-bornological coarse spaces.

Proposition 3.24**.**

The functor

[TABLE]

obtained in (2.2) is additive in both variables.

Proof.

Let $(M_{i},\rho_{i})$ be in ${\mathbf{V}}_{{\mathbf{A}}}^{G}(X)$ for $i=0,1$ and $(M^{\prime},\rho)$ be in ${\mathbf{V}}_{{\mathbf{A}}}^{G}(X^{\prime})$. In view of the symmetry it suffices to show that the canonical morphism

[TABLE]

is an isomorphism. In view of Conditions 3.11.1 and 3.11.2 it suffices to show that

[TABLE]

is an isomorphism for every point $(x,x^{\prime})$ in $X\times X^{\prime}$. By inserting the definitions we see that this morphism is the same as

[TABLE]

But this last morphism is an isomorphism since the tensor product in ${\mathbf{A}}$ is additive in the first argument. ∎

Let $f\colon X\to X^{\prime}$ and $f^{\prime}\colon Y\to Y^{\prime}$ be two morphisms of $G$-bornolgical coarse spaces. Let $(M,\rho)$ be in ${\mathbf{V}}_{{\mathbf{A}}}^{G}(X)$ and $(N,\eta)$ be in ${\mathbf{V}}_{A}^{G}(Y)$.

Lemma 3.25**.**

The morphism

[TABLE]

in ${\mathbf{V}}_{{\mathbf{A}}}^{G}(X^{\prime}\otimes Y^{\prime})$ (see (2.3)) is an isomorphism.

Proof.

In view of Conditions 3.11.1 and 3.11.2 it suffices to show that

[TABLE]

is an isomorphism for every point $(x^{\prime},y^{\prime})$ in $X^{\prime}\times Y^{\prime}$. Inserting the definitions this morphism is given by

[TABLE]

which for every $(x,y)$ in $(f\times f^{\prime})^{-1}(\{x^{\prime},y^{\prime}\})$ is the morphism

[TABLE]

induced by the inclusions of the respective summands of the tensor factors. Since the tensor product in ${\mathbf{A}}$ preserves sums in both arguments we conclude that (3.13) is an isomorphism. ∎

In view of Theorem 2.3 the Propositions 3.23 and 3.24 and Lemma 3.25 now imply:

Theorem 3.26**.**

If ${\mathbf{A}}$ is a symmetric monoidal additive category with a strict action of $G$ by symmetric monoidal functors, then the functor $loc\circ{\mathbf{V}}_{{\mathbf{A}}}^{G}:G\mathbf{BornCoarse}\to\mathbf{Add}_{\infty}$ admits a refinement to a lax symmetric monoidal functor

[TABLE]

3.5 The symmetric monoidal $K$-theory functor for additive categories

In [BC17] a universal $K$-theory functor

[TABLE]

was considered, where ${\mathcal{M}}_{loc}$ is the category of non-commutative motives of Blumberg-Gepner-Tabuada [BGT13]. This functor was defined as the upper horizontal composition in the diagram

[TABLE]

where ${\mathcal{U}}_{loc}$ is the universal localizing invariant, and ${\mathbf{Ch}}^{b}[-]_{\infty}$ sends an additive category ${\mathbf{A}}$ to the stable $\infty$-category of bounded chain complexes over ${\mathbf{A}}$ with homotopy equivalences inverted. Since the functor $\mathrm{UK}$ preserves equivalences of additive categories we have the indicated factorization ${\mathrm{UK}_{\infty}}$.

Theorem 3.27**.**

The functor $\mathrm{UK}_{\infty}$ admits a symmetric monoidal refinement

[TABLE]

Proof.

The proof of this theorem will be finished at the end of the present section. As a first step we observe that it suffices to construct a symmetric monoidal refinement $\mathrm{UK}^{\otimes}$ of $\mathrm{UK}$. Then we obtain the symmetric monoidal refinement $\mathrm{UK}^{\otimes}_{\infty}$ of $\mathrm{UK}_{\infty}$ from the universal property of the symmetric monoidal localization $loc^{\otimes}:{\mathtt{N}}(\mathbf{Add}_{1}^{\otimes})\to\mathbf{Add}_{\infty}^{\otimes}$ using [Hin16].

By [BGT14] the universal localizing invariant ${\mathcal{U}}_{loc}$ refines to a symmetric monoidal functor

[TABLE]

It therefore remains to produce a symmetric monoidal functor

[TABLE]

refining ${\mathbf{Ch}}^{b}[-]_{\infty}$. We use the symbol $St$ in order to indicate that this functor is related with stabilization.

We are going to use the following notation. The category $\mathbf{dgCat}_{1}$ is the $1$-category of small dg-categories. The set $W_{Morita}$ is the set of Morita equivalences, i.e., functors between $dg$-categories ${\mathcal{C}}\to{\mathcal{D}}$ which induce an equivalence of derived categories [Kel06, Sec. 4.6], [Coh, Def. 2.29].

The category $\mathbf{dgCat}_{1}$ contains the full subcategory $\mathbf{dgCat}_{1,flat}$ of locally flat dg-categories, i.e., dg-categories ${\mathbf{C}}$ with the property that for every two objects $C,C^{\prime}$ in ${\mathbf{C}}$ the complex ${\mathtt{Hom}}_{{\mathbf{C}}}(C,C^{\prime})$ consists of flat ${\mathbb{Z}}$-modules. It furthermore contains the full subcategory of pre-triangulated $dg$-categories [Kel06, Sec. 4.5], [BK91].

Furthermore, ${\mathbf{Cat}}^{ex}_{\infty,H{\mathbb{Z}}}$ is the category of $H{\mathbb{Z}}$-linear stable idempotent complete $\infty$-categories and $H{\mathbb{Z}}$-linear exact functors, and ${\mathcal{F}}$ forgets the $H{\mathbb{Z}}$-linear structure. For the equivalence marked by $DK$ (for Dold-Kan) we refer to [Coh].

Proposition 3.28**.**

We have the bold part of the following commuting diagram:

[TABLE]

Proof.

For every $dg$-category the canonical inclusion ${\mathcal{C}}\to{\mathbf{Ch}}^{b}({\mathcal{C}})$ represents the pretriangulated hull [Kel06, Sec. 4.5], [BK91]. In particular, the functor ${\mathbf{Ch}}^{b}$ has values in pretriangulated $dg$-categories. 2. 2.

The two triangles in the corresponding square commute since for every $dg$-category ${\mathcal{C}}$ the canonical inclusion induces a Morita equivalence ${\mathcal{C}}\to{\mathbf{Ch}}^{b}({\mathcal{C}})$. To this end we use that the inclusion of ${\mathcal{C}}$ into its triangulated hull is a Morita equivalence [Kel06, Sec. 4.6]. 3. 3.

The $dg$-nerve ${\mathtt{N}}^{dg}:{\mathtt{N}}(\mathbf{dgCat}_{1}^{pre})\to{\mathbf{Cat}}^{ex}_{\infty}$ preserves Morita equivalences and therefore descends to ${\mathtt{N}}^{dg}_{\infty}$ as indicated. 4. 4.

We have an equivalence ${\mathcal{F}}\circ DK\circ\iota\simeq{\mathtt{N}}_{\infty}^{dg}$, [Fao17, Prop. 3.3.2], see also [Tab10], [Toe07]. In order to provide more details we consider the functor $Z^{0}:\mathbf{dgCat}_{1}^{pre}\to{\mathbf{Cat}}_{\infty}$ which associates to a $dg$-category its underlying category (with ${\mathtt{Hom}}_{Z^{0}({\mathcal{C}})}(A,B)=Z^{0}({\mathtt{Hom}}_{{\mathcal{C}}}(A,B))$) considered as an $\infty$-category. We furthermore let $W_{{\mathcal{C}}}$ be the morphisms in $Z^{0}({\mathcal{C}})$ which become isomorphisms in the homotopy category $H^{0}({\mathcal{C}})$ (with ${\mathtt{Hom}}_{H^{0}({\mathcal{C}})}(A,B)=H^{0}({\mathtt{Hom}}_{{\mathcal{C}}}(A,B))$). Then both functors

[TABLE]

present the localization $Z^{0}({\mathcal{C}})\to Z^{0}({\mathcal{C}})[W_{{\mathcal{C}}}^{-1}]$.

∎

The horizontal composition given by the middle row in (3.14) defines a functor $St$.

Lemma 3.29**.**

The functor $St$ is equivalent to the functor ${\mathbf{Ch}}^{b}(-)_{\infty}$ constructed in [BC17, Prop. 2.11].

Proof.

By [BC17, Rem. 2.9] we have the first equivalence of functors in the chain

[TABLE]

from $\mathbf{Add}_{1}$ to ${\mathbf{Cat}}^{ex}_{\infty}$. This implies the Lemma in view of the commutativity of (3.14). ∎

Proposition 3.30**.**

The functor $St$ has a symmetric monoidal refinement $St^{\otimes}$.

Proof.

All $1$-categories in the lower two lines of the diagram (3.14) have symmetric monoidal structures and the functors connecting them have canonical symmetric monoidal refinements. The same is true for the $\infty$-categories and the remaining functors except for ${\mathtt{N}}(\mathbf{dgCat}_{1})[W^{-1}_{Morita}]$ and the corresponding functors. The problem is that the tensor product of dg-categories is not compatible with Morita equivalences and therefore does not descend to the localization directly. For this reason one considers the subcategory of locally flat dg-categories and uses the equivalence $!!$ in order to transfer the symmetric monoidal structures. So in order to construct the symmetric monoidal refinement of $St$ we must bypass this node of the diagram. To this end we use a symmetric monoidal flat resolution functor $\mathbf{Q}$ as indicated. The left triangle in (3.14) is filled by a natural transformation (not an isomorphism), but the square

[TABLE]

does commute. We then get the following commuting diagram of symmetric monoidal functors

[TABLE]

defining the symmetric monoidal refinement $St^{\otimes}$ of $St$.

It remains to argue that a symmetric monoidal flat resolution functor $\mathbf{Q}$ exists. We start with the following well-known fact.

Lemma 3.31**.**

There exists a functor $Q$ fitting into the commuting diagram

[TABLE]

such that the filler is a quasi-isomorphism. 2. 2.

The functor $Q$ has a lax symmetric monoidal structure.

Proof.

The natural idea works. The functor $Q$ sends $A$ in ${\mathbf{Ab}}$ to

[TABLE]

in ${\mathbf{Ch}}$, where $F_{0}(A):={\mathbb{Z}}[A]$ is the free abelian group generated by the underlying set of $A$, $F_{1}(A)$ is the kernel of the canonical homomorphism $F_{0}(A)\to A$, and $d_{A}$ is the inclusion. ∎

We define the flat resolution functor for additive categories by

[TABLE]

where the dotted arrow is the natural functor induced from the lax symmetric monoidal functor $Q$ which provides a functor from ${\mathbf{Ab}}$-enriched categories to ${\mathbf{Ch}}$-enriched categories with flat ${\mathtt{Hom}}$-complexes. Furthermore, the symmetric monoidal structure on $Q$ induces naturally a symmetric monoidal structure on $\mathbf{Q}$. ∎

This finishes the proof of Theorem 3.27. ∎