The Deligne Tensor Productsec:1Deligne

Let us fix $\mathds{k}$ a field. We begin by recalling a few definitions from
DSPS13.

  * •

A $\mathds{k}$-linear category is called finite if it is equivalent to the
category of finite dimensional modules over a finite $\mathds{k}$-algebra.

  * •

A finite category is semisimple if every object splits as a finite direct sum
of simple objects.

  * •

A tensor category is a rigid monoidal linear category.

We are now ready to give the definition of the (1-)Deligne tensor product.
Given $\mathcal{C}$ and $\mathcal{D}$ two finite categories, their Deligne
tensor product, if it exists, is the bilinear bifunctor
$\boxtimes:\mathcal{C}\times\mathcal{D}\rightarrow\mathcal{C}\boxtimes\mathcal{D}$,
right-exact in both variables, satisfying the following 2-universal property:

  1. 1.

For any finite category $\mathcal{E}$ and every bilinear bifunctor right-exact
in both variables $F:\mathcal{C}\times\mathcal{D}\rightarrow\mathcal{E}$,
there exists a right-exact functor
$F^{\prime}:\mathcal{C}\boxtimes\mathcal{D}\rightarrow\mathcal{E}$ and a
natural isomorphism $u:F^{\prime}\circ\boxtimes\Rightarrow F$.

  2. 2.

Moreover, for any right-exact functors
$G,H:\mathcal{C}\boxtimes\mathcal{D}\rightarrow\mathcal{E}$ and natural
transformation $t:G\circ\boxtimes\Rightarrow H\circ\boxtimes$, there exists a
unique natural transformation $t^{\prime}:G\Rightarrow H$ such that
$t^{\prime}\circ\boxtimes=t$.

We can reformulate definition • as follows: Given $\mathcal{C}$ and
$\mathcal{D}$ two finite categories, their Deligne tensor product, denoted by

$\boxtimes:\mathcal{C}\times\mathcal{D}\rightarrow\mathcal{C}\boxtimes\mathcal{D},$

is the bilinear bifunctor of finite categories right-exact in both variables
such that precomposition with $\boxtimes$ induces a equivalence of categories

$Hom(\mathcal{C}\boxtimes\mathcal{D},\mathcal{E})\simeq
Hom_{bil}(\mathcal{C}\times\mathcal{D},\mathcal{E}),$

from the category of right exact functors
$\mathcal{C}\boxtimes\mathcal{D}\rightarrow\mathcal{E}$ to the category of
bilinear bifunctor $\mathcal{C}\times\mathcal{D}\rightarrow\mathcal{E}$ right-
exact in both variables. This reformulation is well-known (for instance, see
[DSPS14]), whence we have the following result. The Deligne tensor product of
two finite categories exists, is a finite category, and is unique up to a
unique natural isomorphism. In order to motivate our construction of the
2-Deligne tensor product of two compact semisimple 2-categories, we will now
explain how the Deligne tensor product of two finite semisimple categories
over a perfect field can be constructed using a completed tensor product. As
the latter construction is quite general, let us now consider $\mathcal{A}$
and $\mathcal{B}$ two $\mathds{k}$-linear categories. We begin by defining the
category $\mathcal{A}\otimes\mathcal{B}$ whose set of objects is
$Ob(\mathcal{A})\times Ob(\mathcal{B})$, and for which the space of morphisms
from $(A_{1},B_{1})$ to $(A_{2},B_{2})$ is

$Hom_{\mathcal{A}\otimes\mathcal{B}}((A_{1},B_{1}),(A_{2},B_{2})):=Hom_{\mathcal{A}}(A_{1},A_{2})\otimes_{\mathds{k}}Hom_{\mathcal{B}}(B_{1},B_{2}).$

The universal property of the tensor product ensures that this is a category.
Further, it has the following 2-universal property. Given $\mathcal{A}$, and
$\mathcal{B}$ be two $\mathds{k}$-linear categories. There is a 2-universal
bilinear functor
$\beta:\mathcal{A}\times\mathcal{B}\rightarrow\mathcal{A}\otimes\mathcal{B}$,
i.e. for every $\mathds{k}$-linear category $\mathcal{E}$, precomposition with
$\beta$ induces an equivalence of categories

$Hom(\mathcal{A}\otimes\mathcal{B},\mathcal{E})\simeq
Hom_{bil}(\mathcal{A}\times\mathcal{B},\mathcal{E})$

from the category of linear functors
$\mathcal{A}\otimes\mathcal{B}\rightarrow\mathcal{E}$ to the category of
bilinear functors $\mathcal{A}\times\mathcal{B}\rightarrow\mathcal{E}$. The
universal property of the tensor product proves that the canonical bilinear
functor
$\beta:\mathcal{A}\times\mathcal{B}\rightarrow\mathcal{A}\otimes\mathcal{B}$
is 2-universal. The second ingredient in the construction of the completed
tensor product of two $\mathds{k}$-linear categories is the notion of Cauchy
completion, i.e. completion under direct sums and splittings of idempotents.
By lemma A.1.8 and proposition A.1.9 of [D1], Cauchy completion has the
following 2-universal property. Let $\mathcal{A}$ be a $\mathds{k}$-linear
category. Then, there exists a 2-universal linear functor
$\kappa:\mathcal{A}\rightarrow Cau(\mathcal{A})$ to a Cauchy complete
category, i.e for every Cauchy complete $\mathds{k}$-linear category
$\mathcal{E}$, precomposition by $\kappa$ induces an equivalence of categories

$Hom(Cau(\mathcal{A}),\mathcal{E})\simeq Hom(\mathcal{A},\mathcal{E})$

between the categories of linear functors. Putting lemma • and • together, we
find that for every $\mathds{k}$-linear categories $\mathcal{A}$ and
$\mathcal{B}$, the bilinear bifunctor $\mathcal{A}\times\mathcal{B}\rightarrow
Cau(\mathcal{A}\otimes\mathcal{B})$ is 2-universal amongst bilinear bifunctors
from $\mathcal{A}\times\mathcal{B}$ to a Cauchy complete $\mathds{k}$-linear
category. If we now take as input two finite semisimple categories
$\mathcal{C}$ and $\mathcal{D}$, more can be said. Let $\mathcal{C}$ and
$\mathcal{D}$ be two finite semisimple $\mathds{k}$-linear categories, then
the completed tensor product $Cau(\mathcal{C}\otimes\mathcal{D})$ is a finite
semisimple category. In particular, it agrees with the Deligne tensor product
$\mathcal{C}\boxtimes\mathcal{D}$. Let $\mathcal{E}$ be an arbitrary finite
category, and $F:\mathcal{C}\times\mathcal{D}\rightarrow\mathcal{E}$ be a
bilinear bifunctor. (It is necessarily right-exact in both variables as
$\mathcal{C}$ and $\mathcal{D}$ are semisimple). Thanks to lemmas • and •, $F$
can be factored up to natural isomorphism through $\beta$ and $\kappa$:

$\tikzcd[sep=small]\mathcal{C}\times\mathcal{D}\arrow[rd,"\beta"^{\prime}]\arrow[rrrr,"F"]&\mathcal{E}.\\\
\mathcal{C}\otimes\mathcal{D}\arrow[rd,"\kappa"^{\prime}]\arrow[rrru,dotted]\\\
Cau(\mathcal{C}\otimes\mathcal{D})\arrow[rruu,dotted,"F^{\prime}"^{\prime}]$

The functor $F^{\prime}$ is 2-universal by construction. Hence, if we can show
that $Cau(\mathcal{C}\otimes\mathcal{D})$ is a finite category, we will get
that the composite

$\mathcal{C}\times\mathcal{D}\rightarrow\mathcal{C}\otimes\mathcal{D}\rightarrow
Cau(\mathcal{C}\otimes\mathcal{D})$

is the Deligne tensor product of $\mathcal{C}$ and $\mathcal{D}$. In fact, we
will now show that $Cau(\mathcal{C}\otimes\mathcal{D})$ is finite semisimple.
Pick objects $C$ in $\mathcal{C}$ and $D$ in $\mathcal{D}$ such that the
corresponding fully faithful inclusions
$\kappa_{C}:\mathrm{B}End_{\mathcal{C}}(C)\hookrightarrow\mathcal{C}$ and
$\kappa_{D}:\mathrm{B}End_{\mathcal{D}}(D)\hookrightarrow\mathcal{D}$ are
Cauchy completions. This is possible because $\mathcal{C}$ and $\mathcal{D}$
are finite semisimple categories. Firstly, observe that

$\mathrm{B}End_{\mathcal{C}}\otimes\mathrm{B}End_{\mathcal{D}}\cong\mathrm{B}(End_{\mathcal{C}}(C)\otimes
End_{\mathcal{D}}(D))$

by construction. Secondly, note that the image of
$\kappa_{C}\otimes\kappa_{D}$ generates $\mathcal{C}\otimes\mathcal{D}$ under
direct sums and idempotent splittings. Using lemma A.2.4 of [D1], this implies
that

$Cau(\mathrm{B}(End_{\mathcal{C}}(C)\otimes End_{\mathcal{D}}(D)))\simeq
Cau(\mathcal{C}\otimes\mathcal{D}).$

As $\mathds{k}$ is perfect, corollary 18 of [ERZ] implies that
$End_{\mathcal{C}}(C)\otimes End_{\mathcal{D}}(D)$ is a finite semisimple
algebra. Thus, the left hand-side is the category of finite dimensional
modules over a finite semisimple algebra. In particular, we have that
$Cau(\mathcal{C}\otimes\mathcal{D})$ is a finite semisimple category. This
proves the claim, and concludes the proof. We now recall two properties of the
Deligne tensor product that will have analogues for the 2-Deligne tensor
product. The first result gives us a partial description of the $Hom$-spaces
in a Deligne tensor product (for instance, see [EGNO]). Given $\mathcal{C}$
and $\mathcal{D}$ two finite categories, for any $C_{1},C_{2}$ in
$\mathcal{C}$ and $D_{1},D_{2}$ in $\mathcal{D}$, we have:

$Hom_{\mathcal{C}\boxtimes\mathcal{D}}(C_{1}\boxtimes D_{1},C_{2}\boxtimes
D_{2})\cong Hom_{\mathcal{C}}(C_{1},C_{2})\otimes
Hom_{\mathcal{C}}(D_{1},D_{2}).$

The second tells us how the Deligne tensor product behaves if we input
monoidal finite tensor categories, and then finite tensor categories. Let
$\mathcal{C}$ and $\mathcal{D}$ be finite monoidal categories. Then, the
finite category $\mathcal{C}\boxtimes\mathcal{D}$, and the 2-universal functor
$\boxtimes:\mathcal{C}\times\mathcal{D}\rightarrow\mathcal{C}\boxtimes\mathcal{D}$
admit canonical monoidal structures. Further, $\boxtimes$ is 2-universal as a
monoidal bilinear bifunctor right-exact in both variables. It is known that
the Deligne tensor product of two finite monoidal categories is monoidal, and
that the functor $\boxtimes$ is indeed monoidal (see [Del] or [DSPS13]). Using
the 2-universal property of the 2-Deligne tensor product, the remaining part
of the statement follows. Let us also recall the following result, which is
proposition 5.17 of [Del]. Assume $\mathds{k}$ is perfect. Let $\mathcal{C}$
and $\mathcal{D}$ be finite tensor categories. Then, the canonical monoidal
structure on $\mathcal{C}\boxtimes\mathcal{D}$ is rigid, i.e.
$\mathcal{C}\boxtimes\mathcal{D}$ is a finite tensor category. As is argued in
subsection 2.2.3 of [DSPS13], the hypothesis that $\mathds{k}$ is perfect
cannot be removed in corollary •. This explains why we will focus our
attention on perfect fields in what follows.

##  1 A Review of Compact Semisimple 2-Categories

As we have seen above, one important feature of finite semisimple categories
is that they are Cauchy complete, i.e. they admit direct sums and idempotents
split. It is therefore desirable that any categorification of this notion
should be suitably Cauchy complete. Defining directs sums for objects in a
2-category, by which we mean a weak 2-category or bicategory, is not
difficult. However, categorifying the notion of idempotents and their
splittings requires is not nearly as straightforward. One possible way of
achieving this is using separable monads and their splittings as explained in
[DR]. However, we prefer the notions of 2-condensation monads and
2-condensations introduced in [GJF] (see subsection 1.1 of [D1] for spelled
out versions of these two definitions), as they enjoy better categorical
properties. Note that this preference is essentially a matter of taste in the
context of semisimple 2-categories as can be seen from theorem 3.3.3 of [GJF].
Now, let $\mathds{k}$ be a field. As it will be extensively used in the next
section, we begin by spelling out the 3-universal property of the Cauchy
completion of $\mathds{k}$-linear 2-categories in the spirit of definition
1.2.1 of [D1]. Let $\mathfrak{A}$ be a $\mathds{k}$-linear 2-category which is
locally Cauchy complete, i.e. its $Hom$-categories are Cauchy complete. A
Cauchy completion is a 3-functor $\kappa:\mathfrak{A}\rightarrow
Cau(\mathfrak{A})$ to a locally Cauchy complete and Cauchy complete 2-category
$Cau(\mathfrak{A})$ satisfying the following 3-universal property:

  1. 1.

For any locally Cauchy complete and Cauchy complete $\mathds{k}$-linear
2-category $\mathfrak{E}$ and every linear 2-functor
$F:\mathfrak{A}\rightarrow\mathfrak{E}$, there exists a linear 2-functor
$F^{\prime}:Cau(\mathfrak{A})\rightarrow\mathfrak{E}$ and a 2-natural
equivalence $u:F^{\prime}\circ\kappa\Rightarrow F$.

  2. 2.

For every linear 2-functors $G,H:Cau(\mathfrak{A})\rightarrow\mathfrak{E}$ and
2-natural transformation $t:G\circ\kappa\Rightarrow H\circ\kappa$, there
exists a 2-natural transformation $t^{\prime}:G\Rightarrow H$ and an
invertible modification $\lambda:t^{\prime}\circ\kappa\Rrightarrow t$.

  3. 3.

Furthermore, for any 2-natural transformations $r,s:G\Rightarrow H$ and
modification $\lambda:r\circ\kappa\Rightarrow s\circ\kappa$, there exists a
unique modification $\lambda^{\prime}:r\Rrightarrow s$ such that
$\lambda^{\prime}\circ\kappa=\lambda$.

We move one to recall the definition of a semisimple 2-category over
$\mathds{k}$ from [D5] (see [DR] for the original definition). A semisimple
2-category is a $\mathds{k}$-linear 2-category, which has right and left
adjoints for 1-morphisms, is Cauchy complete, and whose $Hom$-categories are
semisimple abelian, i.e. every object splits as a finite direct sum of simple
objects. An object $C$ of a semisimple 2-category $\mathfrak{C}$ is called
simple if $Id_{C}$ is a simple object of finite semisimple tensor category
$End_{\mathfrak{C}}(C)$. Over an algebraically closed field of characteristic
zero, the following definition was introduced in [DR] to categorify the notion
of a finite semisimple 1-category. It was generalized in [D5] to arbitrary
fields. A semisimple 2-category is called finite if it is locally finite, i.e.
its $Hom$-categories are finite, and it has only finitely many equivalence
classes of simple objects. However, as shown in theorem 2.1.2 of [D5], finite
semisimple 2-categories do not exist over fields that are not separably
closed. As we wish to work over as general a field as possible, we need to
recall a generalization of the notion of finite semisimple 2-category. Namely,
following [D5], we say that two simple objects of $\mathfrak{C}$ are connected
if there exists a non-zero 1-morphism between them. As observed in [DR], being
connected defines an equivalence relation on the set of simple objects of
$\mathfrak{C}$, and we write $\pi_{0}(\mathfrak{C})$ for the quotient. We are
now ready to recall the definition of a compact semisimple 2-category from
[D5]. A semisimple 2-category $\mathfrak{C}$ is called compact if it is
locally finite, and $\pi_{0}(\mathfrak{C})$ is finite. Given a finite
semisimple tensor category $\mathcal{C}$, theorem 1.3.4 of [D5] proves that
$Cau(\mathrm{B}\mathcal{C})\simeq\mathbf{Mod}(\mathcal{C})$ is a compact
semisimple 2-category, which we can equivalently think of as the 2-category of
separable right $\mathcal{C}$-module categories (see definition 2.5.1 of
[DSPS13]). Moreover, theorem 1.3.6 of [D5] characterises compact semisimple
2-categories completely, and provides further justification as to why the last
definition is the appropriate categorification of the notion of a finite
semisimple 1-category over an arbitrary field. We give the statement below for
the reader’s convenience. Let $\mathfrak{C}$ be a compact semisimple
2-category. Then, there exists an object $C$ of $\mathfrak{C}$ such that the
inclusion $\mathrm{B}End_{\mathfrak{C}}(C)\hookrightarrow\mathfrak{C}$ is a
Cauchy completion. In particular, every compact semisimple 2-category
$\mathfrak{C}$ is equivalent to $\mathbf{Mod}(\mathcal{C})$ from some finite
semisimple tensor category $\mathcal{C}$. Let us also mention that corollary
2.2.3 of [D5] shows that, over an algebraically closed field or a real closed
field, every compact semisimple 2-category is finite. Finally, we will also
use the following variant of the notion of compact semisimple 2-category
introduced in [D5]. A compact semisimple 2-category $\mathfrak{C}$ is locally
separable if for every object $C$ of $\mathfrak{C}$, $End_{\mathfrak{C}}(C)$
is a separable tensor category (see definition 2.5.8 of [DSPS13]).

##  2 The 2-Deligne Tensor Product

Currently, there is not adequate definition of finite 2-category available. In
particular, we cannot mimick definition • in order to obtain a definition of
the 2-Deligne tensor product. However, we can adapt the second construction
given in section LABEL:sec:1Deligne. Namely, we begin by defining a completed
tensor product of $\mathds{k}$-linear 2-categories over an arbitrary field
$\mathds{k}$. Then, we show that the completed tensor product of two compact
semisimple 2-categories is compact semisimple 2-category, this proves that the
2-Deligne tensor product of two compact semisimple 2-categories exists. Given
$\mathfrak{A}$ and $\mathfrak{B}$ two $\mathds{k}$-linear 2-categories. If it
exists, we write
$\widehat{\otimes}:\mathfrak{A}\times\mathfrak{B}\rightarrow\mathfrak{A}\widehat{\otimes}\mathfrak{B}$
for the bilinear 2-functor to a locally Cauchy complete and Cauchy complete
$\mathds{k}$-linear 2-category satisfying the following 3-universal property:

  1. 1.

For any Cauchy complete and locally Cauchy complete $\mathds{k}$-linear
2-category $\mathfrak{E}$, and every bilinear 2-functor
$F:\mathfrak{A}\times\mathfrak{B}\rightarrow\mathfrak{E}$, there exists a
linear 2-functor
$F^{\prime}:\mathfrak{A}\widehat{\otimes}\mathfrak{B}\rightarrow\mathfrak{E}$
and a 2-natural equivalence $u:F^{\prime}\circ\widehat{\otimes}\Rightarrow F$.

  2. 2.

For every 2-functors
$G,H:\mathfrak{A}\widehat{\otimes}\mathfrak{B}\rightarrow\mathfrak{E}$, and
2-natural transformation $t:G\circ\widehat{\otimes}\Rightarrow
H\circ\widehat{\otimes}$, there exists a 2-natural equivalence
$t^{\prime}:G\Rightarrow H$ together with an invertible modification
$\gamma:t^{\prime}\circ\widehat{\otimes}\Rrightarrow t$.

  3. 3.

Furthermore, for any 2-natural transformation $r,s:G\Rightarrow H$ and
modification $\delta:r\circ\widehat{\otimes}\Rrightarrow
s\circ\widehat{\otimes}$, there exists a unique invertible modification
$\delta:r\Rrightarrow s$ such that
$\delta^{\prime}\circ\widehat{\otimes}=\delta$.

Definition 2 can be reformulated more concisely as saying that precomposition
with $\widehat{\otimes}$ induces an equivalence of 2-categories

$Hom(\mathfrak{A}\widehat{\otimes}\mathfrak{B},\mathfrak{E})\simeq
Hom_{bil}(\mathfrak{A}\times\mathfrak{B},\mathfrak{E}),$

from the 2-category of linear 2-functors
$\mathfrak{A}\widehat{\otimes}\mathfrak{B}\rightarrow\mathfrak{E}$ to the
2-category of bilinear 2-functors
$\mathfrak{A}\times\mathfrak{B}\rightarrow\mathfrak{E}$. We now wish to show
that the 3-universal bilinear 2-functor
$\widehat{\otimes}:\mathfrak{A}\times\mathfrak{B}\rightarrow\mathfrak{A}\widehat{\otimes}\mathfrak{B}$
of definition 2 exists. We begin with the following lemma. Given
$\mathfrak{A}$, and $\mathfrak{B}$ two $\mathds{k}$-linear 2-categories. There
is a 3-universal bilinear 2-functor
$\beta:\mathfrak{A}\times\mathfrak{B}\rightarrow\mathfrak{A}\widetilde{\otimes}\mathfrak{B}$,
i.e. for every $\mathds{k}$-linear and locally Cauchy complete 2-category
$\mathfrak{E}$, precomposition with $\beta$ induces an equivalence of
2-categories

$Hom(\mathfrak{A}\widetilde{\otimes}\mathfrak{B},\mathfrak{E})\simeq
Hom_{bil}(\mathfrak{A}\times\mathfrak{B},\mathfrak{E})$

from the 2-category of linear 2-functors
$\mathfrak{A}\widetilde{\otimes}\mathfrak{B}\rightarrow\mathfrak{E}$ to the
2-category of bilinear 2-functors
$\mathfrak{A}\times\mathfrak{B}\rightarrow\mathfrak{E}$. Let
$\mathfrak{A}\otimes\mathfrak{B}$ be the $\mathds{k}$-linear 2-category with
objects given by the pairs $(A,B)$ with $A$ an object of $\mathfrak{A}$, and
$B$ an object of $\mathfrak{B}$, and with $Hom$-categories from
$(A_{1},B_{1})$ to $(A_{2},B_{2})$ given by

$Hom_{\mathfrak{A}\otimes\mathfrak{B}}((A_{1},B_{1}),(A_{2},B_{2})):=Hom_{\mathfrak{A}}(A_{1},A_{2})\otimes
Hom_{\mathfrak{B}}(B_{1},B_{2}).$

The 2-universal property of $\otimes$ explained in lemma • shows that this
does define a 2-category, and that precomposition with the canonical bilinear
2-functor
$\mathfrak{A}\times\mathfrak{B}\rightarrow\mathfrak{A}\otimes\mathfrak{B}$
induces an equivalence of linear 2-categories

$Hom(\mathfrak{A}\otimes\mathfrak{B},\mathfrak{F})\simeq
Hom_{bil}(\mathfrak{A}\times\mathfrak{B},\mathfrak{F})$

for every linear 2-category $\mathfrak{F}$. Now, let
$\mathfrak{A}\widetilde{\otimes}\mathfrak{B}$ be the local Cauchy completion
of $\mathfrak{A}\otimes\mathfrak{B}$, i.e. the 2-category with objects given
by the pairs $(A,B)$, and with $Hom$-categories given by the Cauchy
completions of
$Hom_{\mathfrak{A}\otimes\mathfrak{B}}((A_{1},B_{1}),(A_{2},B_{2}))$. Using
the 2-universal property of the Cauchy completion of $\mathds{k}$-linear
categories of [D1] recalled in lemma • above, we see that this does define a
2-category, and that precomposition with the canonical inclusion
$\mathfrak{A}\otimes\mathfrak{B}\rightarrow\mathfrak{A}\widetilde{\otimes}\mathfrak{B}$
induces an equivalence of 2-categories

$Hom(\mathfrak{A}\widetilde{\otimes}\mathfrak{B},\mathfrak{E})\simeq
Hom_{bil}(\mathfrak{A}\otimes\mathfrak{B},\mathfrak{E})$

for every locally Cauchy complete linear 2-category $\mathfrak{E}$. For any
$\mathds{k}$-linear 2-categories $\mathfrak{A}$ and $\mathfrak{B}$, the
3-universal bilinear 2-functor
$\widehat{\otimes}:\mathfrak{A}\times\mathfrak{B}\rightarrow\mathfrak{A}\widehat{\otimes}\mathfrak{B}$
of definition 2 exists. Using the notations of lemma 2, we set
$\mathfrak{A}\widehat{\otimes}\mathfrak{B}$ to be the Cauchy completion of the
2-category $\mathfrak{A}\widetilde{\otimes}\mathfrak{B}$, and we let
$\widehat{\otimes}:\mathfrak{A}\times\mathfrak{B}\rightarrow\mathfrak{A}\widehat{\otimes}\mathfrak{B}$
be equal to the composite

$\mathfrak{A}\times\mathfrak{B}\xrightarrow{\beta}\mathfrak{A}\widetilde{\otimes}\mathfrak{B}\xrightarrow{\kappa}Cau(\mathfrak{A}\widetilde{\otimes}\mathfrak{B})=\mathfrak{A}\widehat{\otimes}\mathfrak{B}.$

Now, for any locally Cauchy complete and Cauchy complete $\mathds{k}$-linear
2-category $\mathfrak{E}$, precomposition by $\widehat{\otimes}$ induces an
equivalence of 2-categories

$Hom(\mathfrak{A}\widehat{\otimes}\mathfrak{B},\mathfrak{E})\simeq
Hom(\mathfrak{A}\widetilde{\otimes}\mathfrak{B},\mathfrak{E})\simeq
Hom_{bil}(\mathfrak{A}\times\mathfrak{B},\mathfrak{E}).$

This proves the result. Thanks to its definition using a 3-universal property,
the bilinear 2-functor
$\widehat{\otimes}:\mathfrak{A}\times\mathfrak{B}\rightarrow\mathfrak{A}\widehat{\otimes}\mathfrak{B}$
is unique in the sense that the 3-category formed by such 2-functors is a
contractible 3-groupoid. From now on, we take $\mathds{k}$ to be a perfect
field. By analogy with the decategorified situation recalled in section
LABEL:sec:1Deligne, we make the following definition. Given two compact
semisimple 2-categories $\mathfrak{C}$ and $\mathfrak{D}$ over a perfect field
$\mathds{k}$, their 2-Deligne tensor product, if it exists, is the 3-universal
bilinear 2-functor
$\boxdot:\mathfrak{C}\times\mathfrak{D}\rightarrow\mathfrak{C}\boxdot\mathfrak{D}$
to a compact semisimple 2-category. Observe that, if we can prove that the
completed tensor product $\mathfrak{C}\widehat{\otimes}\mathfrak{D}$ is a
compact semisimple 2-category, we immediately get that the 2-Deligne tensor
product of $\mathfrak{C}$ and $\mathfrak{D}$ exists. Over a perfect field, the
2-Deligne tensor product
$\boxdot:\mathfrak{C}\times\mathfrak{D}\rightarrow\mathfrak{C}\boxdot\mathfrak{D}$
of two compact semisimple 2-categories $\mathfrak{C}$ and $\mathfrak{D}$
exists. It is enough to prove that
$\mathfrak{C}\widehat{\otimes}\mathfrak{D}=Cau(\mathfrak{C}\widetilde{\otimes}\mathfrak{D})$
is a compact semisimple 2-category, as then,
$\widehat{\otimes}:\mathfrak{C}\times\mathfrak{D}\rightarrow\mathfrak{C}\widehat{\otimes}\mathfrak{D}$
has the desired 3-universal property. To this end, let $C$ be a generator of
$\mathfrak{C}$, and $D$ be a generator of $\mathfrak{D}$, meaning that the
canonical inclusions
$\kappa_{C}:\mathrm{B}End_{\mathfrak{C}}(C)\rightarrow\mathfrak{C}$ and
$\kappa_{D}:\mathrm{B}End_{\mathfrak{D}}(D)\rightarrow\mathfrak{D}$ are Cauchy
completions. (These exist by theorem 1.) Note that the linear 2-functor

$\kappa_{C}\widetilde{\otimes}\kappa_{D}:\mathrm{B}End_{\mathfrak{C}}(C)\widetilde{\otimes}\mathrm{B}End_{\mathfrak{D}}(D)\rightarrow\mathfrak{C}\widetilde{\otimes}\mathfrak{D}$

is fully faithful by construction. We claim that every object of
$\mathfrak{C}\widetilde{\otimes}\mathfrak{D}$ is the splitting of a
2-condensation monad supported on the image $C\widetilde{\otimes}D$ of
$\kappa_{C}\widetilde{\otimes}\kappa_{D}$. To see this, observe that as $C$ is
a generator for $\mathfrak{C}$, every simple object of $\mathfrak{C}$ is the
splitting of a 2-condensation monad supported on $C$. Now, if
$(C,C_{1},f_{1},g_{1},\phi_{1},\gamma_{1})$ and
$(C,C_{2},f_{2},g_{2},\phi_{2},\gamma_{2})$ are two 2-condensations in
$\mathfrak{C}$ (in the notation of subsection 1.1 of [D1]), note that

$\Big{(}C,C_{1}\boxplus C_{1},\pmatrix{f}_{1}\\\
f_{2},\pmatrix{g}_{1}&g_{2},\pmatrix{\phi}_{1}&0\\\
0\phi_{2},\pmatrix{\gamma}_{1}&0\\\ 0\gamma_{2}\Big{)}$

is a 2-condensation. In particular, $C_{1}\boxplus C_{2}$ is the splitting of
a 2-condensation monad supported on $C$. As every object of $\mathfrak{C}$ is
a finite direct sum of simple objects by proposition 1.1.7 of [D5], we find
that every object $C^{\prime}$ of $\mathfrak{C}$ is the splitting of a
condensation monad supported on $C$. Likewise, every object $D^{\prime}$ of
$\mathfrak{D}$ is the splitting of a 2-condensation monad supported on $D$.
Thus, we find that $(C^{\prime},D^{\prime})$ is the splitting of a
2-condensation monad supported on $(C,D)$. Now, using that splittings of
2-condensation monads are preserved by all 2-functors, we get that
$C^{\prime}\widetilde{\otimes}D^{\prime}$ is the splitting of a 2-condensation
monad supported on $C\widetilde{\otimes}D$ as desired. Using lemma A.2.4 of
[D1], we see that $Cau(\kappa_{C}\widetilde{\otimes}\kappa_{D})$ is an
equivalence of 2-categories. Moreover, by construction, we have

$BEnd_{\mathfrak{C}}(C)\widetilde{\otimes}BEnd_{\mathfrak{D}}(D)\simeq
B(End_{\mathfrak{C}}(C)\boxtimes End_{\mathfrak{D}}(D)).$

But $End_{\mathfrak{C}}(C)\boxtimes End_{\mathfrak{D}}(D)$ is a finite
semisimple tensor category by corollary •. This completes the proof. Let
$\mathcal{C}$ and $\mathcal{D}$ be two finite semisimple tensor categories.
Then, we have

$\mathbf{Mod}(\mathcal{C})\boxdot\mathbf{Mod}(\mathcal{D})\simeq\mathbf{Mod}(\mathcal{C}\boxtimes\mathcal{D}).$

Let $\mathfrak{C}$ and $\mathfrak{D}$ be two locally separable compact
semisimple 2-categories. Then, $\mathfrak{C}\boxdot\mathfrak{D}$ is locally
separable. By theorem 1.4.7 of [D5], there exists separable tensor categories
$\mathcal{C}$ and $\mathcal{D}$ such that
$\mathfrak{C}\simeq\mathbf{Mod}(\mathcal{C})$ and
$\mathfrak{D}\simeq\mathbf{Mod}(\mathcal{D})$. By corollary 2, we find that
$\mathfrak{C}\boxdot\mathfrak{D}\simeq\mathbf{Mod}(\mathcal{C}\boxdot\mathcal{D})$.
The result now follows from corollary 2.5.11 of [DSPS13]. Observe that under
the 3-equivalence of theorem 3.1.4 of [D5], the 1-Deligne tensor product
$\mathcal{C}\boxtimes\mathcal{D}$ of two finite semisimple tensor categories
is sent to the 2-Deligne tensor product of their compact semisimple categories
of separable modules
$\mathbf{Mod}(\mathcal{C})\boxdot\mathbf{Mod}(\mathcal{D})$. Thus, taking for
granted that the 3-category $\mathbf{TC}^{rsep}$ of finite semisimple tensor
categories and right separable bimodules admits a symmetric monoidal structure
given by the Deligne tensor product, then $\mathbf{CSS2C}$ the 3-category of
compact semisimple 2-categories admits a symmetric monoidal structure given by
the 2-Deligne tensor product. In particular, as every separable tensor
category is a fully dualizable object $\mathbf{TC}^{rsep}$ by results of
[DSPS13], every locally separable compact semisimple 2-category is a fully
dualizable object $\mathbf{CSS2C}$.

##  3 Properties of the 2-Deligne Tensor Product

Now that we have shown that the 2-Deligne tensor product of two compact
semisimple 2-categories is a compact semisimple 2-category, we can examine its
properties. Throughout, we work over a perfect field $\mathds{k}$ unless
otherwise specified. Let $\mathfrak{C}$ and $\mathfrak{D}$ be two compact
semisimple 2-categories. For any objects $C_{1},C_{2}$ in $\mathfrak{C}$ and
$D_{1},D_{2}$ in $\mathfrak{D}$, there is an equivalence of categories:

$Hom_{\mathfrak{C}\boxdot\mathfrak{D}}(C_{1}\boxdot D_{1},C_{2}\boxdot
D_{2})\simeq Hom_{\mathfrak{C}}(C_{1},C_{2})\boxtimes
Hom_{\mathfrak{C}}(D_{1},D_{2}).$

Moreover, this equivalence commutes with composition up to natural
isomorphism. By theorem 1.3.6 of [D5] recalled in theorem 1 above, it is
enough to prove that given two finite semisimple tensor categories
$\mathcal{C}$ and $\mathcal{D}$, and separable algebras
$P_{1},P_{2}\in\mathcal{C}$ and $Q_{1},Q_{2}\in\mathcal{D}$, then

$Bimod_{\mathcal{C}\boxtimes\mathcal{D}}(P_{1}\boxtimes Q_{1},P_{2}\boxtimes
Q_{2})\simeq Bimod_{\mathcal{C}}(P_{1},P_{2})\boxtimes
Bimod_{\mathcal{D}}(Q_{1},Q_{2}).$

Observe that there is a natural functor $F$, as in the diagram below, given by
taking the Deligne tensor product of two bimodules. As $F$ is bilinear, it can
be factored through the Deligne tensor product up to natural isomorphism, i.e.
there exist a functor $F^{\prime}$ making the diagram

$\tikzcd{Bimod_{\mathcal{C}}(P_{1},P_{2})\times
Bimod_{\mathcal{D}}(Q_{1},Q_{2})}\arrow[r,"F"]\arrow[d,"\boxtimes"^{\prime}]&{Bimod_{\mathcal{C}\boxtimes\mathcal{D}}(P_{1}\boxtimes
Q_{1},P_{2}\boxtimes Q_{2})}\\\ {Bimod_{\mathcal{C}}(P_{1},P_{2})\boxtimes
Bimod_{\mathcal{D}}(Q_{1},Q_{2}).}\arrow[ru,dotted,"F^{\prime}"^{\prime}]$

commute up to natural isomorphism. Let us prove that the functor $F^{\prime}$
is fully faithful. Let $M$ be an object of $Bimod_{\mathcal{C}}(P_{1},P_{2})$,
$N$ an object of $Bimod_{\mathcal{D}}(Q_{1},Q_{2})$, and $C\boxtimes D$ an
object of $\mathcal{C}\boxtimes\mathcal{D}$. There are natural isomorphisms:
Hom_P_1Q_1 , P_2Q_2&((P_1Q_1)⊗(CD) ⊗(P_2Q_2), MN)
≅Hom_CD(CD, MN)
≅Hom_C(C, M)⊗Hom_D(D, N)
≅Hom_P_1 , P_2(P_1⊗C⊗P_2, M)⊗Hom_Q_1 , Q_2(Q_1⊗D⊗Q_2, N). As every object of
$Bimod_{\mathcal{C}}(P_{1},P_{2})\boxtimes Bimod_{\mathcal{D}}(Q_{1},Q_{2})$
is the splitting of an idempotent on an object of the form $(P_{1}\otimes
C\otimes P_{2})\boxtimes(Q_{1}\otimes D\otimes Q_{2})$ for some objects $C$ in
$\mathcal{C}$ and $D$ in $\mathcal{D}$, we get that the functor $F^{\prime}$
is fully faithful. Now, observe that the objects $(P_{1}\boxtimes
Q_{1})\otimes(C\boxtimes D)\otimes(P_{2}\boxtimes Q_{2})$ with $C\boxtimes D$
varying form a set of projective generators for the finite semisimple category
$Bimod_{\mathcal{C}\boxtimes\mathcal{D}}(P_{1}\boxtimes Q_{1},P_{2}\boxtimes
Q_{2})$. Given that we have shown above that $F^{\prime}$ is fully faithful,
this implies that $F^{\prime}$ is essentially surjective. Finally, let us
comment on the naturality of this equivalence. Given $P_{1},P_{2},P_{3}$
separable algebras in $\mathcal{C}$, and $Q_{1},Q_{2},Q_{3}$ separable
algebras in $\mathcal{D}$, for every bimodules $A_{1}$ in
$Bimod(P_{1},P_{2})$, $A_{2}$ in $Bimod(P_{2},P_{3})$, $B_{1}$ in
$Bimod(Q_{1},Q_{2})$, and $B_{2}$ in $Bimod(Q_{2},Q_{3})$, using the
2-universal property of the Deligne tensor product, it is enough to show that
there is a natural isomorphism

$(A_{1}\otimes_{P_{2}}A_{2})\boxtimes(B_{1}\otimes_{Q_{2}}B_{2})\cong(A_{1}\boxtimes
B_{1})\otimes_{P_{2}\boxtimes Q_{2}}(A_{2}\boxtimes B_{2}).$

This follows by comparing the universal properties. Let us fix a compact
semisimple 2-category $\mathfrak{D}$. Then, for every compact semisimple
2-categories $\mathfrak{C}$, and $\mathfrak{E}$, there is a natural
equivalence of 2-categories

$Hom(\mathfrak{C}\boxdot\mathfrak{D},\mathfrak{E})\simeq
Hom(\mathfrak{C},Hom(\mathfrak{D},\mathfrak{E})).$

We have the following natural equivalences of 2-categories:

$Hom(\mathfrak{C}\boxdot\mathfrak{D},\mathfrak{E})\simeq
Hom_{bil}(\mathfrak{C}\times\mathfrak{D},\mathfrak{E})\simeq
Hom(\mathfrak{C},Hom(\mathfrak{D},\mathfrak{E})),$

where the first equivalence is given by theorem 2, and the second holds by
inspection. One may also ask how the connected components of a compact
semisimple 2-category interact with the 2-Deligne tensor product. Over an
algebraically closed field, this is the same thing as asking what are the
connected components of the 2-Deligne tensor product of two finite semisimple
2-categories. In this case, we simply get the product of the connected
components of the finite semisimple 2-categories. Let $\mathds{k}$ be an
algebraically closed field. Let $\mathfrak{C}$, $\mathfrak{D}$ be two finite
semisimple 2-categories. Then, we have

$\pi_{0}(\mathfrak{C}\boxdot\mathfrak{D})\cong\pi_{0}(\mathfrak{C})\times\pi_{0}(\mathfrak{D}).$

Observe that any finite semisimple 2-category can be decomposed into a direct
sum of connected finite semisimple 2-categories (i.e. $\pi_{0}$ is a
singleton). Therefore, it is enough to prove that if $\mathfrak{C}$ and
$\mathfrak{D}$ are connected, then so $\mathfrak{C}\boxdot\mathfrak{D}$. But,
a finite semisimple 2-category is connected if and only if it is equivalent to
the 2-category of separable module categories over a fusion category. Thus,
there exists fusion categories $\mathcal{C}$ and $\mathcal{D}$ such that
$\mathfrak{C}\simeq\mathbf{Mod}(\mathcal{C})$ and
$\mathfrak{D}\simeq\mathbf{Mod}(\mathcal{D})$. By corollary 2, we have

$\mathfrak{C}\boxdot\mathfrak{D}\simeq\mathbf{Mod}(\mathcal{C}\boxtimes\mathcal{D}).$

The result now follows from lemma •, which implies that
$\mathcal{C}\boxtimes\mathcal{D}$ is a fusion category. Let us work over
$\mathbb{R}$. Then, we have Mod(Vect_C)&Mod(Vect_C)≃Mod(Vect_CVect_C)
≃Mod(Vect_CVect_C)≃Mod(Vect_C)Mod(Vect_C). In particular, lemma 3 fails over
non-algebraically closed fields. Moreover, note that
$\mathbf{Vect}_{\mathbb{C}}\boxdot\mathbf{Vect}_{\mathbb{C}}$ is the direct
sum of the two distinct simple objects of
$\mathbf{Mod}(\mathbf{Vect}_{\mathbb{C}})\boxplus\mathbf{Mod}(\mathbf{Vect}_{\mathbb{C}})$.
Thus, the 2-functor $\boxdot$ may send the product of two simple objects to an
object that is not simple. (Note however that this can not happen over an
algebraically closed field as, in this case, the Deligne tensor product of two
fusion categories is a fusion category by lemma •.) It is useful to know when
the functor $\boxdot$ hits every equivalence class of simple objects of the
target. Over an algebraically closed field, we give a criterion ensuring this
property. Firstly, recall from subsection 4.5 [EGNO] that the Frobenus-Perron
dimension of a finite tensor category is an algebraic integer that only
depends on the Grothendieck ring. Secondly, observe that given $\mathfrak{C}$
a connected finite semisimple 2-category, we have that
$Hom_{\mathfrak{C}}(C,D)$ is non-zero for every pair of simple objects $C,D$.
Thus, the endomorphism fusion categories of any two simple objects are Morita
equivalent, whence they have the same Frobenius-Perron dimension by corollary
7.16.7 of [EGNO]. This means that the following definition is sensible. The
Frobenius-Perron dimension of a connected finite semisimple 2-category over an
algebraically closed field is the Frobenius-Perron dimension of the
endomorphism fusion category of any of its simple objects. Let $\mathfrak{C}$
and $\mathfrak{D}$ be connected finite semisimple 2-categories over an
algebraically closed field. If $\mathfrak{C}$ and $\mathfrak{D}$ have coprime
Frobenius-Perron dimension, then, for any simple object $A$ of
$\mathfrak{C}\boxdot\mathfrak{D}$, there exists simple objects objects $C$ in
$\mathfrak{C}$ and $D$ in $\mathfrak{D}$ unique up to equivalence such that
$C\boxdot D\simeq A$. The theorem follows readily from the following claim:
Given $\mathcal{C}$ and $\mathcal{D}$ two fusion categories of coprime
Frobenius-Perron dimension. Let $\mathcal{P}$ be an indecomposable right
$\mathcal{C}\boxtimes\mathcal{D}$-module category. Then, there exists
indecomposable right module categories $\mathcal{M}$ over $\mathcal{C}$ and
$\mathcal{N}$ over $\mathcal{D}$ unique up to equivalence such that
$\mathcal{P}\simeq\mathcal{M}\boxtimes\mathcal{N}$. Furthermore, if
$\mathcal{P}$ is separable, so are $\mathcal{M}$ and $\mathcal{N}$. We prove
the first part of this statement using a variant of the proof of proposition
8.55 of [ENO], which deals with the characteristic zero case. Given
$\mathcal{P}$ an indecomposable right $\mathcal{C}\boxtimes\mathcal{D}$-module
category, we can restrict $\mathcal{P}$ to $\mathcal{C}$, and write
$\mathcal{P}|_{\mathcal{C}}=\oplus_{i}\mathcal{M}_{i}$ for some indecomposable
right $\mathcal{C}$-module categories $\mathcal{M}_{i}$. Then, the action of
$\mathcal{D}$ can be viewed as a tensor functor
$F:\mathcal{D}\rightarrow\oplus_{i,j}Fun_{\mathcal{C}}(\mathcal{M}_{i},\mathcal{M}_{j})$.
We write $Im(F)$ for the image of $F$, i.e. the multifusion category generated
by the summands of the objects in $F(\mathcal{D})$. Now, if $Im(F)$ was
decomposable, a decomposition of $Im(F)$ would induce a decomposition of
$\mathcal{P}$ as a right $\mathcal{C}\boxtimes\mathcal{D}$-module category. By
hypothesis, this is not the case, thus $Im(F)$ is indecomposable. In
particular, if we write $Im(F)=\oplus_{i,j}Im(F)_{ij}$ with $Im(F)_{ij}$ a
sub-category of $Fun_{\mathcal{C}}(\mathcal{M}_{i},\mathcal{M}_{j})$, then we
find that $Im(F)_{ij}$ is non-zero for every $i$ and $j$. Now, as
$F:\mathcal{D}\rightarrow Im(F)$ is surjective (see definition 1.8.3 of
[EGNO]), it follows from theorem 6.2.1 of [EGNO] that the Frobenius-Perron
dimension of $Im(F)$ divides that of $\mathcal{D}$. But, for every $i$, the
fusion subcategory $Im(F)_{ii}$ of $Im(F)$ is also a fusion subcategory of
$Fun_{\mathcal{C}}(\mathcal{M}_{i},\mathcal{M}_{i})$. Thus, the Frobenius-
Perron dimension of $Im(F)_{ii}$ divides the Frobenius-Perron dimensions of
$\mathcal{C}$ and $\mathcal{D}$. (This follows from corollary 7.16.7 and
theorem 7.17.4 of [EGNO].) As these algebraic integers are coprime, the
Frobenius-Perron dimension of $Im(F)_{ii}$ must be equal to $1$. Whence,
$Im(F)_{ii}\simeq\mathbf{Vect}$, and so $Im(F)_{ij}\simeq\mathbf{Vect}$ for
every $i$ and $j$, as we are working over an algebraically closed field. This
implies that the $\mathcal{M}_{i}$ must all be equivalent to a single right
$\mathcal{C}$-module category $\mathcal{M}$. The functor
$F:\mathcal{D}\rightarrow\oplus_{i,j}\mathbf{Vect}=End(\boxplus_{i}\mathbf{Vect})$
then defines a finite semisimple right $\mathcal{D}$-module category
$\mathcal{N}$, and we have $\mathcal{P}\simeq\mathcal{M}\boxtimes\mathcal{N}$.
Now, it is clear from the construction that $\mathcal{M}$ and $\mathcal{N}$
are unique. Let us now assume that $\mathcal{P}$ is separable. We wish to show
that $\mathcal{M}$ and $\mathcal{N}$ are separable. Let us begin by proving
that $\mathcal{M}$ is separable. In order to see this, observe that there is
an equivalence of finite tensor categories

$\mathcal{Z}_{\mathcal{D}}(End_{\mathcal{C}}(\mathcal{P}))\simeq
End_{\mathcal{C}\boxtimes\mathcal{D}}(\mathcal{P})$

between the relative Drinfel’d center of $End_{\mathcal{C}}(\mathcal{P})$ over
$F:\mathcal{D}\rightarrow End_{\mathcal{C}}(\mathcal{P})$, and the finite
tensor category $End_{\mathcal{C}\boxtimes\mathcal{D}}(\mathcal{P})$. But,
$End_{\mathcal{C}\boxtimes\mathcal{D}}(\mathcal{P})$ is a fusion category as
$\mathcal{P}$ is an indecomposable separable right
$\mathcal{C}\boxtimes\mathcal{D}$-module category. Now, the forgetful functor

$\mathcal{Z}_{\mathcal{D}}(End_{\mathcal{C}}(\mathcal{P}))\rightarrow
End_{\mathcal{C}}(\mathcal{P})$

is surjective. Thence, we find that $End_{\mathcal{C}}(\mathcal{P})$ is finite
semisimple by theorem 6.1.16 of [EGNO]. Furthermore, for any $i$, there is a
surjective tensor functor $End_{\mathcal{C}}(\mathcal{P})\rightarrow
End_{\mathcal{C}}(\mathcal{M}_{i})$, so we find that
$\mathcal{M}_{i}\simeq\mathcal{M}$ is separable. Finally, in order to prove
that $\mathcal{N}$ is separable, one can proceed similarly. Namely, by
uniqueness of $\mathcal{M}$ and $\mathcal{N}$, $\mathcal{P}|_{\mathcal{D}}$
decomposes into a direct sum of right $\mathcal{D}$-module categories which
are all equivalent to $\mathcal{N}$. The above argument then shows that
$\mathcal{N}$ is indeed separable. Example 3 shows that theorem 3 fails over
fields that are not algebraically closed. Let $\mathds{k}$ be an algebraically
closed field of characteristic different from $2$ and $3$. Let us consider the
finite semisimple 2-category of modules over the separable fusion category
$\mathcal{C}=\mathbf{Vect}_{\mathds{Z}/6\mathds{Z}}$. Thanks to our assumption
on the characteristic of $\mathds{k}$, all these module categories are
separable. Now, by example 7.4.10 of [EGNO] (see also [Nat]), we know that the
equivalence classes of simple objects of $\mathbf{Mod}(\mathcal{C})$ are given
by pair consisting of a subgroup of $\mathds{Z}/6\mathds{Z}$ and a cohomology
class on this subgroup. Denoting by $triv$ the trivial cohomology class, we
get the following list:

$(\langle 0\rangle,triv)$ | $(\langle 3\rangle,triv)$ | $(\langle 2\rangle,triv)$ | $(\langle 1\rangle,triv).$
---|---|---|---

Now, observe that we have an equivalence

$\mathcal{C}\simeq\mathbf{Vect}_{\mathds{Z}/2\mathds{Z}}\boxtimes\mathbf{Vect}_{\mathbb{Z}/3\mathbb{Z}}$

of fusion categories. Thus, combining proposition 3, theorem 3, we find that
the equivalence classes of simple objects of $\mathbf{Mod}(\mathcal{C})$ are
described by pair consisting of an equivalence class of simple objects in
$\mathbf{Mod}(\mathbf{Vect}_{\mathbb{Z}/2\mathbb{Z}})$ and in
$\mathbf{Mod}(\mathbf{Vect}_{\mathbb{Z}/3\mathbb{Z}})$. We derive the
following description in the notation of example 2.1.12 of [D1]:

$(\mathbf{Vect}_{\mathbb{Z}/2\mathbb{Z}},\mathbf{Vect}_{\mathbb{Z}/3\mathbb{Z}})\
\ \ (\mathbf{Vect}_{\mathbb{Z}/3\mathbb{Z}},\mathbf{Vect})\ \ \
(\mathbf{Vect},\mathbf{Vect}_{\mathbb{Z}/2\mathbb{Z}})\ \ \
(\mathbf{Vect},\mathbf{Vect}).$

The corresponding irreducible right $\mathcal{C}$-module categories are given
by taking the 1-Deligne tensor product of the above pairs. In this case, these
can be made very explicit, and we obtain the following list of right
$\mathcal{C}$-module categories:

$\mathbf{Vect}_{\mathds{Z}/6\mathds{Z}}$ | $\mathbf{Vect}_{\mathds{Z}/3\mathds{Z}}$ | $\mathbf{Vect}_{\mathds{Z}/2\mathds{Z}}$ | $\mathbf{Vect}.$
---|---|---|---

The module structures arise from the obvious tensor functors. Let $\mathds{k}$
be an algebraically closed field of characteristic not equal to $2$. We now
consider the finite semisimple 2-category of right module categories over
$\mathcal{D}=\mathbf{Vect}_{\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}}$.
Let us write $a=(1,0)$, $b=(0,1)$, and $c=(1,1)$ the non-zero elements of
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$. The equivalence classes
of simple objects of $\mathbf{Mod}(\mathcal{D})$ are given as follows:

$(\langle 0\rangle,triv)$ | $(\langle a\rangle,triv)$ | $(\langle b\rangle,triv)$
---|---|---
$(\langle c\rangle,triv)$ | $(\langle a,b\rangle,triv)$ | $(\langle a,b\rangle,\nu).$

We have used $\nu$ to denote the non-trivial class in
$H^{2}(\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z},\mathds{k}^{*})$.
Observe that we have an equivalence

$\mathcal{D}\simeq\mathbf{Vect}_{\mathbb{Z}/2\mathbb{Z}}\boxtimes\mathbf{Vect}_{\mathbb{Z}/2\mathbb{Z}}$

of fusion categories. By construction, the essential image of the canonical
2-functor

$\boxdot:\mathbf{Mod}(\mathbf{Vect}_{\mathds{Z}/2\mathbb{Z}})\times\mathbf{Mod}(\mathbf{Vect}_{\mathds{Z}/2\mathbb{Z}})\rightarrow\mathbf{Mod}(\mathcal{D})$

contains exactly the following equivalence classes of simple objects:

$(\langle 0\rangle,triv)$ | $(\langle a\rangle,triv)$ | $(\langle b\rangle,triv)$ | $(\langle a,b\rangle,triv).$
---|---|---|---

In particular, the image of $\boxdot$ does not contain every equivalence class
of simple objects.

##  4 The 2-Deligne Tensor Product of Compact Semisimple Tensor 2-Categories

Let $\mathds{k}$ be an arbitrary field. We prove that the completed tensor
product $\widehat{\otimes}$ of two monoidal linear 2-categories inherits a
monoidal structure. But, before doing so, let us point out the following
lemma. Let $\mathfrak{A}_{1}$, …, $\mathfrak{A}_{n}$ be linear 2-categories.
There is a 3-universal multilinear 2-functor, which we denote by

$\widehat{\otimes}:\mathfrak{A}_{1}\times...\times\mathfrak{A}_{n}\rightarrow\mathfrak{A}_{1}\widehat{\otimes}...\widehat{\otimes}\mathfrak{A}_{n}.$

In particular, all the different ways of parenthesizing
$\mathfrak{A}_{1}\widehat{\otimes}...\widehat{\otimes}\mathfrak{A}_{n}$ are
canonically equivalent. One can define the functor $\widehat{\otimes}$
inductively by

$\xymatrix{\mathfrak{A}_{1}\times...\times\mathfrak{A}_{n}\ar[rr]^{-}{\widehat{\otimes}\times
id\times...\times
id}&\mathfrak{A}_{1}\widehat{\otimes}\mathfrak{A}_{2}\times...\times\mathfrak{A}_{n}\ar[r]...\ar[r]^{-}{\widehat{\otimes}}\mathfrak{A}_{1}\widehat{\otimes}...\widehat{\otimes}\mathfrak{A}_{n}.}$

The 3-universal property is derived by induction. Let $\mathfrak{A}$,
$\mathfrak{B}$ be monoidal linear 2-categories. Then, the compact semisimple
2-category $\mathfrak{A}\widehat{\otimes}\mathfrak{B}$ admits a canonical
monoidal structure such that the bilinear 2-functor
$\widehat{\otimes}:\mathfrak{A}\times\mathfrak{B}\rightarrow\mathfrak{A}\widehat{\otimes}\mathfrak{B}$
is monoidal. Further, the bilinear monoidal 2-functor $\widehat{\otimes}$ is
the 3-universal bilinear monoidal 2-functor from
$\mathfrak{A}\times\mathfrak{B}$ to a locally Cauchy complete and Cauchy
complete monoidal linear 2-category. In the proof below, we will construct
many adjoint 2-natural equivalences. In order not complicate notations
unnecessarily, we shall only construct one of the two adjoints; The others can
be constructed similarly using the 3-universal property of the 2-Deligne
tensor product supplied by theorem 2. Concisely, the idea behind the proof is
to observe that the Cartesian product $\mathfrak{A}\times\mathfrak{B}$ is
monoidal, and use the 3-universal property of $\widehat{\otimes}$ to transfer
this monoidal structure onto $\mathfrak{A}\widehat{\otimes}\mathfrak{B}$. In
more details, in the notation of [SP], write
$(\mathfrak{A},\Box_{\mathfrak{A}},I_{\mathfrak{A}},\alpha_{\mathfrak{A}},l_{\mathfrak{A}},r_{\mathfrak{A}})$,
respectively
$(\mathfrak{B},\Box_{\mathfrak{B}},I_{\mathfrak{B}},\alpha_{\mathfrak{B}},l_{\mathfrak{B}},r_{\mathfrak{B}})$
for the monoidal structure on $\mathfrak{A}$, respectively $\mathfrak{B}$. Let
us consider the following solid arrow diagram:

<EMAIL_ADDRESS>\mathfrak{A}\times\mathfrak{B}\ar[rr]_{-}{\widehat{\otimes}}\mathfrak{A}\widehat{\otimes}\mathfrak{B},}$

where $\Box_{\mathfrak{A}\times\mathfrak{B}}$ is the composite of the
canonical permutation 2-functor

$\mathfrak{A}\times\mathfrak{B}\times\mathfrak{A}\times\mathfrak{B}\rightarrow\mathfrak{A}\times\mathfrak{A}\times\mathfrak{B}\times\mathfrak{B},$

and the 2-functor $\Box_{\mathfrak{A}}\times\Box_{\mathfrak{B}}$. Using
proposition 4, the dotted 2-functor $\Box$ exists, and there is an 2-natural
equivalence $\chi$ witnessing the commutativity of the diagram.

$\tikzcd[columnsep=1ex]&(\mathfrak{A}\times\mathfrak{B})^{\times
3}\arrow[ld,"\Box_{\mathfrak{A}\times\mathfrak{B}}\times
Id"^{\prime}]\arrow[rdd,nearstart,"Id\times\Box_{\mathfrak{A}\times\mathfrak{B}}",crossingover]\arrow[rrr,"\widehat{\otimes}^{3}"](\mathfrak{A}\widehat{\otimes}\mathfrak{B})^{\times
3}\arrow[ld,"\Box_{\mathfrak{A}\widehat{\otimes}\mathfrak{B}}\times
Id"^{\prime},nearend]\arrow[rdd,"Id\times\Box_{\mathfrak{A}\widehat{\otimes}\mathfrak{B}}"]\\\
(\mathfrak{A}\times\mathfrak{B})^{\times
2}\arrow[rdd,"\Box_{\mathfrak{A}\times\mathfrak{B}}"^{\prime}]\arrow[rrr,nearend,"\widehat{\otimes}^{2}"^{\prime},crossingover](\mathfrak{A}\widehat{\otimes}\mathfrak{B})^{\times
2}\\\ (\mathfrak{A}\times\mathfrak{B})^{\times
2}\arrow[ld,"\Box_{\mathfrak{A}\times\mathfrak{B}}"^{\prime}]\arrow[rrr,nearstart,"\widehat{\otimes}^{2}"^{\prime}](\mathfrak{A}\widehat{\otimes}\mathfrak{B})^{\times
2}\arrow[ld,"\Box_{\mathfrak{A}\widehat{\otimes}\mathfrak{B}}"]\\\
\mathfrak{A}\times\mathfrak{B}\arrow[rrr,"\widehat{\otimes}"^{\prime}]\mathfrak{A}\widehat{\otimes}\mathfrak{B}\arrow[from=luu,nearstart,"\Box_{\mathfrak{A}\widehat{\otimes}\mathfrak{B}}",crossingover].$
Figure : Construction of the associator (Part 1)

We claim that the 2-functor
$\Box_{\mathfrak{A}\widehat{\otimes}\mathfrak{B}}:=\Box\circ\widehat{\otimes}$
provides $\mathfrak{A}\widehat{\otimes}\mathfrak{B}$ with a monoidal
structure. Let us show how the associator is constructed. We begin by
considering the diagram depicted in figure 4. The left face is filled using
the associator of $\mathfrak{A}\times\mathfrak{B}$, and the front, back, top,
and bottom faces are filled using $\chi$. Now, we want to use the 3-universal
property of the completed tensor product derived in proposition 2 to fill in
the right face. However, in order for this operation to be licit, we need to
partially fill the right face using the diagram depicted in figure 4.

$\adjustbox{maxwidth=}{\tikzcd[columnsep=0]&(\mathfrak{A}\widehat{\otimes}\mathfrak{B})^{\times
3}\arrow[ld,"\widehat{\otimes}\times
Id"]\arrow[rd,"Id\times\widehat{\otimes}"^{\prime}]\arrow[lldd,"(\Box\circ\widehat{\otimes})\times
Id",bendright=49]\arrow[rrdd,"Id\times(\Box\circ\widehat{\otimes})"^{\prime},bendleft=49]\\\
(\mathfrak{A}\widehat{\otimes}\mathfrak{B})^{\widehat{\otimes}2}\times(\mathfrak{A}\widehat{\otimes}\mathfrak{B})\arrow[rd,"\widehat{\otimes}"]\arrow[ld,"\Box\times
Id"](\mathfrak{A}\widehat{\otimes}\mathfrak{B})\times(\mathfrak{A}\widehat{\otimes}\mathfrak{B})^{\widehat{\otimes}2}\arrow[ld,"\widehat{\otimes}"^{\prime}]\arrow[rd,"Id\times\Box"^{\prime}]\\\
(\mathfrak{A}\widehat{\otimes}\mathfrak{B})^{\times
2}\arrow[rd,"\widehat{\otimes}"]\arrow[rrdd,"\Box\circ\widehat{\otimes}",bendright=49](\mathfrak{A}\widehat{\otimes}\mathfrak{B})^{\widehat{\otimes}3}\arrow[ld,dotted,"\Box\widehat{\otimes}Id"^{\prime}]\arrow[rd,dotted,"Id\widehat{\otimes}\Box"](\mathfrak{A}\widehat{\otimes}\mathfrak{B})^{\times
2}\arrow[ld,"\widehat{\otimes}"^{\prime}]\arrow[lldd,"\Box\circ\widehat{\otimes}"^{\prime},bendleft=49]\\\
(\mathfrak{A}\widehat{\otimes}\mathfrak{B})^{\widehat{\otimes}2}\arrow[rd,"\Box"](\mathfrak{A}\widehat{\otimes}\mathfrak{B})^{\widehat{\otimes}2}\arrow[ld,"\Box"^{\prime}]\\\
\mathfrak{A}\widehat{\otimes}\mathfrak{B}.}$ Figure : Construction of the
associator (Part 2)

There is a canonical 2-natural equivalence filling in the top square of figure
4 by lemma 4. The existence of the dotted arrows, as well as that of 2-natural
transformations witnessing the commutativity of the lateral diamonds follow
using the 3-universal property of the completed tensor product (see definition
2). Observe that the bottom square of the diagram depicted in figure 4 is
empty. Gluing together the right face of the diagram represented in figure 4
and the exterior face of that given in figure 4, we obtain a diagram on which
we can use the 3-universal property of lemma 4. Doing so, we find a 2-natural
equivalence $\alpha_{\mathfrak{A}\widehat{\otimes}\mathfrak{B}}$ that fills
the empty square in figure 4, and the the commutativity of the whole polyhedra
is witnessed by an invertible modification $\omega$. The modification $\omega$
shows that $\widehat{\otimes}$ and $\chi$ are compatible with the associators.
Similarly, one constructs the pentagonator
$\pi_{\mathfrak{A}\widehat{\otimes}\mathfrak{B}}$ of
$\mathfrak{A}\widehat{\otimes}\mathfrak{B}$ out of the pentagonator of
$\mathfrak{A}\times\mathfrak{B}$, simultaneously proving that
$\widehat{\otimes}$, $\chi$, and $\omega$ are compatible with the
pentagonators. One can then proceed to show that it satisfies the required
axioms. Let us briefly comment on the monoidal unit
$I_{\mathfrak{A}\widehat{\otimes}\mathfrak{B}}$ of
$\mathfrak{A}\widehat{\otimes}\mathfrak{B}$ and on the unitors. In order to
defines $I_{\mathfrak{A}\widehat{\otimes}\mathfrak{B}}$, it is convenient to
think of objects of a compact semisimple 2-category as linear 2-functors from
$\mathrm{B}^{2}\mathds{k}$, the linear 2-category with one object and one
1-morphism. Consider the following solid arrow diagram:

$\tikzcd\mathrm{B}^{2}\mathds{k}\times\mathrm{B}^{2}\mathds{k}\arrow[d,"I_{\mathfrak{A}}\times
I_{\mathfrak{B}}"^{\prime}]\arrow[r,"\otimes"]&\mathrm{B}^{2}\mathds{k}\arrow[d,dotted,"I_{\mathfrak{A}\widehat{\otimes}\mathfrak{B}}"]\\\
\mathfrak{A}\times\mathfrak{B}\arrow[r,"\widehat{\otimes}"^{\prime}]\mathfrak{A}\widehat{\otimes}\mathfrak{B}.$

Using the 3-universal property of $\otimes$ given in lemma •, the diagram can
be filled by a 2-natural equivalence $\iota$, which we can also think of as an
equivalence $I_{\mathfrak{A}\widehat{\otimes}\mathfrak{B}}\rightarrow
I_{\mathfrak{A}}\widehat{\otimes}I_{\mathfrak{B}}$. Using this together with
arguments analogous to the ones presented above, we can define the unitors,
and prove that they satisfy the relevant axioms. This concludes the proof of
the first part of the statement. Now, we shall prove that the monoidal
2-functor $\widehat{\otimes}$ we have constructed above is 3-universal. Let
$(\mathfrak{E},\Box_{\mathfrak{E}},I_{\mathfrak{E}},\alpha_{\mathfrak{E}},l_{\mathfrak{E}},r_{\mathfrak{E}})$
be a monoidal $\mathds{k}$-linear 2-category that is locally Cauchy complete,
and Cauchy complete. Let

$(F,\chi_{F},\iota_{F},\omega_{F},\gamma_{F},\delta_{F}):\mathfrak{A}\times\mathfrak{B}\rightarrow\mathfrak{E}$

be a monoidal bilinear 2-functor in the notation of [SP]. Immediately, we see
that there is a 2-natural equivalence
$\phi:F^{\prime}\circ\widehat{\otimes}\Rightarrow F$ for some linear 2-functor
$F^{\prime}:\mathfrak{A}\widehat{\otimes}\mathfrak{B}\rightarrow\mathfrak{E}$.
We want to show that $F^{\prime}$ and $\phi$ are monoidal. To this end,
consider the diagram depicted in figure 4.

$\tikzcd&\mathfrak{A}\times\mathfrak{B}\times\mathfrak{A}\times\mathfrak{B}\arrow[r,"F\times
F"]\arrow[ld,"\widehat{\otimes}\times\widehat{\otimes}"^{\prime}]\arrow[dd,"\Box_{\mathfrak{A}\times\mathfrak{B}}"]\mathfrak{E}\times\mathfrak{E}\arrow[dd,"\Box_{\mathfrak{E}}"]\\\
(\mathfrak{A}\widehat{\otimes}\mathfrak{B})\times(\mathfrak{A}\widehat{\otimes}\mathfrak{B})\arrow[rru,pos=0.1,crossingover,"F^{\prime}\times
F^{\prime}"^{\prime}]\arrow[d,"\widehat{\otimes}"^{\prime}]\\\
\mathfrak{A}\widehat{\otimes}\mathfrak{B}\widehat{\otimes}\mathfrak{A}\widehat{\otimes}\mathfrak{B}\arrow[d,"\Box"^{\prime}]\arrow[rr,dotted,bendleft=20]\mathfrak{A}\times\mathfrak{B}\arrow[r,nearstart,"F"]\arrow[ld,"\widehat{\otimes}"^{\prime}]\mathfrak{E}\\\
\mathfrak{A}\widehat{\otimes}\mathfrak{B}.\arrow[rru,"F^{\prime}"^{\prime}]$
Figure : Monoidality of $F^{\prime}$

The left face can be filled using $\chi$, the bottom face using $\phi$, the
top face using $\phi\times\phi$, and the back face using $\chi_{F}$. Using the
3-universality of $\boxdot$ obtained in theorem 2, we can factor the composite
$\Box_{\mathfrak{E}}\circ(F^{\prime}\times F^{\prime})$ through $\boxdot$ up
to a 2-natural equivalence. This fills in the upper half of the front face of
the diagram represented in figure 4, and we can then use lemma 4 to complete
the bottom half, and get an invertible modification $\Pi$ witnessing the
commutativity of the resulting prism. In particular, we denote the 2-natural
equivalence corresponding to the composite of the front face of the diagram
given in figure 4 by $\chi^{\prime}_{F}$. One now checks that
$\chi^{\prime}_{F}$ endow the 2-functor $F^{\prime}$ with a monoidal
structure, and that $\phi$ together with $\Pi$ define a monoidal 2-natural
equivalence $\phi:F\Rightarrow F^{\prime}$. The remainder of the proof uses
the same techniques, but involves big diagrams, which we omit. We are mainly
interested in tensor 2-categories, i.e. rigid monoidal linear 2-categories
(see [D5]). After having proven theorem 4, it is therefore natural to ask
whether the completed tensor product of two tensor 2-categories is again a
tensor 2-category. It turns out that the answer is positive. If $\mathfrak{A}$
and $\mathfrak{B}$ be tensor 2-categories, then
$\mathfrak{A}\widehat{\otimes}\mathfrak{B}$ with the monoidal structure of
theorem 4 is a tensor 2-category. We prove that the objects of
$\mathfrak{A}\widehat{\otimes}\mathfrak{B}$ have right duals. The existence of
left duals can be treated similarly. As the 2-functor
$\mathfrak{A}\times\mathfrak{B}\rightarrow\mathfrak{A}\widehat{\otimes}\mathfrak{B}$
is monoidal, every object of the form $A\widehat{\otimes}B$ for some $A$ in
$\mathfrak{A}$, and $B$ in $\mathfrak{B}$ has a right dual given explicitly by
$A^{\sharp}\widehat{\otimes}B^{\sharp}$. Let $E$ be any object of
$\mathfrak{A}\widehat{\otimes}\mathfrak{B}$. We claim that there exists a
2-condensation monad supported on an object that is a direct sum of objects in
the image of $\widehat{\otimes}$, whose splitting is $E$. Namely, by
construction, we have $\widehat{\otimes}=\kappa\circ\beta$, and $\beta$ is
esentially surjective. The claim now follows from the definition of the Cauchy
completion as the composition of the direct sum completion and the Karoubi
envelope (see appendix A.2 of [D1]). Thence it is enough to prove that the
splitting of a 2-condensation monad supported on an object that has a right
dual has a right dual. This is precisely the content of lemma 4 below. The
following lemma should be compared with theorem 4.1.1 of [GJF]. Because our
context is different, we provide a proof in the appendix. Let $\mathfrak{C}$
be a monoidal 2-category (not necessarily linear) that is locally idempotent
complete and in which every 2-condensation monad splits. Let $C$ be an object
of $\mathfrak{C}$ that has a right dual, and $(C,e,\mu,\delta)$ a
2-condensation monad. Then, the splitting of $(C,e,\mu,\delta)$ has a right
dual. See the appendix. We now assume that our base field $\mathds{k}$ is
perfect. We have recalled in propositions • and • that the 1-Deligne tensor
product of two finite tensor categories is itself a finite tensor category. We
now prove a categorified version of this statement for finite semisimple
tensor categories. Given $\mathfrak{C}$ and $\mathfrak{D}$ two compact
semisimple tensor 2-categories over a perfect field. Their 2-Deligne tensor
product $\mathfrak{C}\boxdot\mathfrak{D}$ is a compact semisimple tensor
2-category. Further, the 2-functor
$\boxdot:\mathfrak{C}\times\mathfrak{D}\rightarrow\mathfrak{C}\boxdot\mathfrak{D}$
is monoidal. This follows by combining theorems 2 and 4 together with
proposition 4. Over an algebraically closed field or a real closed, it is
shown in [D5] that every compact semisimple 2-category is finite. In this
case, we use the term multifusion 2-category in lieu of compact semisimple
tensor 2-category. Further, a fusion 2-category is a multifusion 2-category
whose monoidal unit is a simple object. Let $\mathds{k}$ be an algebraically
closed field. If $\mathfrak{C}$ and $\mathfrak{D}$ are two fusion
2-categories, then $\mathfrak{C}\boxdot\mathfrak{D}$ is a fusion 2-category.
This follows from proposition 3. We end by examining some examples. Let us
work over $\mathbb{R}$. Then, $\mathbf{Mod}(\mathbf{Vect}_{\mathbb{C}})$ is a
fusion 2-category with a single equivalence class of simple object. But, we
found in example 3 that

$\mathbf{Mod}(\mathbf{Vect}_{\mathbb{C}})\boxdot\mathbf{Mod}(\mathbf{Vect}_{\mathbb{C}})\simeq\mathbf{Mod}(\mathbf{Vect}_{\mathbb{C}})\boxplus\mathbf{Mod}(\mathbf{Vect}_{\mathbb{C}}).$

In particular, the monoidal unit of this multifusion 2-category is the direct
sum of the two non-equivalent simple objects of the right hand-side. Thus,
$\mathbf{Mod}(\mathbf{Vect}_{\mathbb{C}})\boxdot\mathbf{Mod}(\mathbf{Vect}_{\mathbb{C}})$
is not a fusion 2-category. We use the notation of example 3. In particular,
we work over an algebraically closed field of characteristic distinct from $2$
and $3$. Let us endow $\mathcal{C}$ with a braiding via

$\mathcal{C}\simeq\mathbf{Vect}_{\mathbb{Z}/2\mathbb{Z}}^{\beta}\boxtimes\mathbf{Vect}_{\mathbb{Z}/3\mathbb{Z}}^{triv},$

where $triv$ is the trivial symmetric braiding, and $\beta$ denotes the super
braiding. The induced monoidal product on $\mathbf{Mod}(\mathcal{C})$ is given
by the relative Deligne tensor product $\boxtimes_{\mathcal{C}}$. Using
theorem 4 and theorem 3, we see that the table depicting the action of
$\boxtimes_{\mathcal{C}}$ on the equivalence classes of simple objects of
$\mathbf{Mod}(\mathcal{C})$ is the product of the tables for
$\mathbf{Mod}(\mathbf{Vect}_{\mathbb{Z}/2\mathbb{Z}}^{\beta})$ and
$\mathbf{Mod}(\mathbf{Vect}_{\mathbb{Z}/3\mathbb{Z}}^{triv})$ derived in
example 3.3.7 of [D5] (see also example 2.5.3 of [D2]). The resulting table is
given below, where we write $a:=3\in\mathbb{Z}/6\mathbb{Z}$, and
$b:=2\in\mathbb{Z}/6\mathbb{Z}$. Further, we have removed all the trivial
cohomology classes to make the table more legible.

$\boxtimes_{\mathcal{C}}$ | $\langle 0\rangle$ | $\langle a\rangle$ | $\langle b\rangle$ | $\langle a+b\rangle$
---|---|---|---|---
$\langle 0\rangle$ | $\langle 0\rangle$ | $\langle a\rangle$ | $\langle b\rangle$ | $\langle a+b\rangle$
$\langle a\rangle$ | $\langle a\rangle$ | $\langle 0\rangle$ | $\langle a+b\rangle$ | $\langle b\rangle$
$\langle b\rangle$ | $\langle b\rangle$ | $\langle a+b\rangle$ | $3\langle b\rangle$ | $3\langle a+b\rangle$
$\langle a+b\rangle$ | $\langle a+b\rangle$ | $\langle b\rangle$ | $3\langle a+b\rangle$ | $3\langle b\rangle$

We follow the notations of examples 3. In particular, we work over an
algebraically closed field of characteristic not equal to $2$. The equivalence
of fusion categories

$\mathcal{D}\simeq\mathbf{Vect}_{\mathbb{Z}/2\mathbb{Z}}^{\beta}\boxtimes\mathbf{Vect}_{\mathbb{Z}/2\mathbb{Z}}^{\beta}$

induces a braiding on $\mathcal{D}$. The induced monoidal structure on
$\mathbf{Mod}(\mathcal{D})$ is given by $\boxtimes_{\mathcal{D}}$. Using
theorem 4 we can partially compute the action of $\boxtimes_{\mathcal{D}}$ on
the equivalence classes of simple objects of $\mathbf{Mod}(\mathcal{D})$.
Namely, we can compute the monoidal products (depicted in the table below)
involving the objects $\langle 0\rangle$, $\langle a\rangle$, $\langle
b\rangle$, and $\langle a,b\rangle$. We have omitted the trivial cohomology
classes to declutter the table, and denoted the simple object $(\langle
a,b\rangle,\nu)$ succinctly by $\nu$.

$\boxtimes_{\mathcal{D}}$ | $\langle 0\rangle$ | $\langle a\rangle$ | $\langle b\rangle$ | $\langle c\rangle$ | $\langle a,b\rangle$ | $\nu$
---|---|---|---|---|---|---
$\langle 0\rangle$ | $\langle 0\rangle$ | $\langle a\rangle$ | $\langle b\rangle$ | $\langle c\rangle$ | $\langle a,b\rangle$ | $\nu$
$\langle a\rangle$ | $\langle a\rangle$ | $\langle 0\rangle$ | $\langle a,b\rangle$ | $\nu$ | $\langle b\rangle$ | $\langle c\rangle$
$\langle b\rangle$ | $\langle b\rangle$ | $\langle a,b\rangle$ | $\langle 0\rangle$ | $\nu$ | $\langle a\rangle$ | $\langle c\rangle$
$\langle c\rangle$ | $\langle c\rangle$ | $\nu$ | $\nu$ | $2\langle c\rangle$ | $\langle c\rangle$ | $2\nu$
$\langle a,b\rangle$ | $\langle a,b\rangle$ | $\langle b\rangle$ | $\langle a\rangle$ | $\langle c\rangle$ | $\langle 0\rangle$ | $\nu$
$\nu$ | $\nu$ | $\langle c\rangle$ | $\langle c\rangle$ | $2\nu$ | $\nu$ | $2\langle c\rangle$

Provided our base field has characteristic zero, the remaining entries can be
filled using proposition 2.5.1 of [D2]. Over an algebraically closed field of
characteristic greater than 2, they can be computed using a similar method. As
the fusion categories $\mathcal{C}$ and $\mathcal{D}$ of examples 4 and 4 are
symmetric, so are $\mathbf{Mod}(\mathcal{C})$ and $\mathbf{Mod}(\mathcal{D})$.
This explains why their respective multiplication tables are symmetric along
the diagonal.


# Wavefunction collapse induced by gravity in a relativistic Schródinger-
Newton model

Luis A. Poveda
Departemento de Física Centro Federal de Educação Tecnológica de Minas Gerais
Amazonas $5253$, $30421-169$ Belo Horizonte, MG, Brasil
<EMAIL_ADDRESS>
Luis Grave de Peralta
Department of Physics and Astronomy
Texas Tech University
Lubbock, TX 79409
<EMAIL_ADDRESS>
Arquimedes Ruiz-Columbié
Wind Energy Program
Texas Tech University
Lubbock, TX $79409$
<EMAIL_ADDRESS>
###### Abstract

A relativistic version of the Schrödinger-Newton equation is analyzed within
the recently proposed Grave de Peralta approach [L. Grave de Peralta, Results
Phys. 18 (2020) 103318], which include relativistic effects by a
parametrization of the non-relativistic hamiltonian, so as to impose that the
average kinetic energy of the system coincide with its relativistic kinetic
energy. The reliability of this method is tested for the particle in a box. By
applying this method to the Schrödinger-Newton equation we shows that the
characteristic length of the model [L. Diósi, Phys. Lett. 105A (1984) 199]
goes to zero for a mass of the order of the Planck mass, suggesting a collapse
of the wavefuncton, induced by gravity.

##  1 Introduction

The Schrödinger-Newton equation (SNE) [1, 2, 3, 4, 5, 6] is a model which
describe the time evolution of a Schrödinger quantum field coupled to a
Newtonian gravitational field, aimed to elucidate the role of gravity on
quantum state reduction [7, 8, 9]. For a single-point particle the equation is
written as

$i\hbar\frac{\partial\Psi(\mathbf{x},t)}{\partial
t}=\left[-\frac{\hbar^{2}}{2m}\nabla^{2}-m\Phi(\mathbf{x},t)\right]\Psi(\mathbf{x},t),$
(1)

where $\Phi(\mathbf{x},t)$ is the gravitational potential determined by the
mass distribution $m|\Psi(\mathbf{x},t)|^{2}$.

Eqs. (1) involves three parameters: $\hbar$, the action quantum constant, $G$,
the gravitational constant, and $m$ a free parameter that couples both fields.
Therefore, the solutions of the SNE are length scaled by

$l_{D}=\frac{\hbar^{2}}{Gm^{3}},$ (2)

here called the Diósi length, as it was initially proposed by Diósi [1], as a
measure of the quantum uncertainty of the stationary soliton-like solutions of
the SNE. As observed by Diósi [1], for a distance of the order of $l_{D}$ the
gravitational force balances the dispersion of the quantum field $\Psi$ by the
action of the $p^{2}/2m$ operator, leading to a stationary state. The
$l_{D}\sim m^{-3}$ dependence point out to the suppression of the macroscopic
quantum behavior due to gravitational collapse for a enough massive particle.

Note that when $m$ is equal to the Planck mass ($m_{P}=\sqrt{\hbar c/G}$),
$l_{D}$ match the Planck distance ($l_{P}=\sqrt{\hbar G/c^{3}}$), and the
reduced Compton wavelength ($\lambdabar_{C}=\hbar/mc$) of the particle.
Considering that $\lambdabar_{C}$ is the cut-off for a relativistic quantum
field description and that $l_{P}$ is a fundamental distance, below which
gravity is no longer a classical field, the SNE should break down when the
mass approach the Planck mass.

As suggested elsewhere [6], the Eqs. (1) is the Newtonian limit of the semi-
classical Einstein equations [10, 11]. Hence, a relativistic treatment require
to deal with the Einstein field equations, couple to a Dirac or Klein-Gordon
field. Rather, in the present work we use the Grave de Peralta approach [12,
13, 14, 15, 16] which include relativistic effects to the non-relativistic
hamiltonian by introducing a parameter which value is fixed by the condition
that the average kinetic energy of the system should coincide with its
relativistic kinetic energy.

The paper is organized as follow. In section 2, the Grave de Peral approach to
relativity is briefly summarized; then, the method is apply to well known
quantum problem of a particle an infinite well in section 3; after that, a
relativistic version of the Schrödinger-Newton eqaution is given and the
relativistic Diósi length is derived in section 4; finally, some conclusions
are given in section 5.

##  2 The Grave de Peralta equation

Consider the hamiltonian

$\hat{H}=\hat{K}^{\mathrm{(GP)}}+\hat{V},$ (3)

where

$\hat{K}^{\mathrm{(GP)}}=-\frac{\hbar^{2}}{\left(1+\gamma\right)m}\nabla^{2},$
(4)

is the Grave de Peralta (GP) kinetic energy operator [12, 13, 14, 15, 16]
obtained by first quantization ($p\to\hat{p}\equiv\hbar\nabla$) of the
relativistic kinetic energy [17]

$\frac{p^{2}}{\left(1+\gamma\right)m}$ (5)

where $\gamma=\left({1+v^{2}/c^{2}}\right)^{-1/2}$ is the Lorentz’s factor.

Note that

$\hat{K}^{\mathrm{(GP)}}=\frac{2}{\left(1+\gamma\right)}\hat{K}^{\mathrm{(S)}},$
(6)

where $\hat{K}^{\mathrm{(S)}}$ is the well known Schrödinger ($\mathrm{S}$)
kinetic energy operator, which result from the quantization of the non-
relativistic version of Eq. (5).

In Eqs. (4) and (6), $\gamma$ is not an operator, instead is a parameter which
value is chosen by the condition

$\langle\hat{K}^{\mathrm{(GP)}}\rangle=(\gamma-1)mc^{2}$ (7)

where $\langle~{}\cdot~{}\rangle$ is the average in an appropriated quantum
state.

Substituting Eq. (6) in (7) and solving for the parameter $\gamma$ we obtain,

$\gamma=\sqrt{1+\frac{2\langle\hat{K}^{\mathrm{(S)}}\rangle}{mc^{2}}}$ (8)

and note that in general, $\gamma$ has a coordinate dependence.

The Grave de Peralta equation (GPE) is the Schrödinger-like equation,

$i\hbar\frac{\partial\Psi(\mathbf{x},t)}{\partial
t}=\left[-\frac{\hbar^{2}}{(1+\gamma)m}\nabla^{2}+V(\mathbf{x})\right]\Psi(\mathbf{x},t)$
(9)

with $\gamma$ given by Eq. (8). The corresponding stationary solutions
fulfills the eigenvalue equation

$\left[-\frac{\hbar^{2}}{(1+\gamma)m}\nabla^{2}+V(\mathbf{x})\right]\psi(\mathbf{x})=E\psi(\mathbf{x}),$
(10)

In the next section we show that the GdPE include relativistic corrections in
a reliable way, for the elementary system of a particle in a one-dimensional
box of size $L$[13].

##  3 Relativistic particle in a box

In this case, $V(x)=0$ for $0\leq x\leq L$, and the non-relativistic energy
eigenvalues are given by [18],

$E_{n}^{\mathrm{(S)}}=\frac{\hbar^{2}n^{2}\pi^{2}}{2mL^{2}}$ (11)

Considering that
$\langle\hat{H}^{\mathrm{(S)}}\rangle=\langle\hat{K}^{\mathrm{(S)}}\rangle$
inside the box, and taking the average in a Schrödinger eigenstate, Eq. (8)
adopt the form,

$\gamma=\sqrt{1+\frac{\hbar^{2}n^{2}\pi^{2}}{m^{2}c^{2}L^{2}}}$ (12)

Figure 1: Comparison of the present results (solid dots, solid lines) from Eq.
(13) and the reported in ref. [19] (empty dots, dashed lines), for different
sizes of the well. In square dots the corresponding non-relativistic energies.
The thin line indicates the energy threshold $2mc^{2}$.

Finally the energy of the $n$-th level, within the relativistic Grave de
Peralta approach will be [13]

$E_{n}^{\mathrm{(GP)}}=\frac{\hbar^{2}n^{2}\pi^{2}}{\left[1+\sqrt{1+\frac{\hbar^{2}n^{2}\pi^{2}}{m^{2}c^{2}L^{2}}}\right]mL^{2}}$
(13)

First, is easy to see that by expanding Eq. (13) in power of $(p/mc)^{2}$,
with $p=\hbar n\pi/L$, up to the second order we obtain,

$E_{n}=\frac{\hbar^{2}n^{2}\pi^{2}}{2mL^{2}}-\frac{\hbar^{4}n^{4}\pi^{4}}{8m^{3}c^{2}L^{4}}+\mathcal{O}(L^{-6})$
(14)

where the first order coincides with the non-relativistic energy, Eq. (11),
and the second order gives the mass-velocity relativistic correction to the
kinetic energy.

A comparison of the energies computed with Eq. (13) and those reported in
reference [19], is given in Figure 1, for different sizes of the well. Apart
from disagreements for lower $n$, the present model reliably predict, with
very good agreement, most of the reported data and describe the tendency when
the energy increase and the size of the well reduce.

Figure 2: A plot of Eq. (15). See text for details.

In general por a particle confined in a region of size $l$, the relativistic
parameter $\gamma$ of Eq. (8) can be evaluated as,

$\gamma=\sqrt{1+\left(\frac{\lambdabar_{C}}{l}\right)^{2}}$ (15)

where $\lambdabar_{C}=\hbar/mc$ is the reduced Compton wavelength of the
particle. Figure 2 show the behavior of $\gamma$ as a function of the particle
seize in units of $\lambdabar_{C}$. From the figure, the parameter appreciable
deviates from $1$ when the localization region of the particle approach the
Compton wavelength. Moreover, when the particle extent to a spatial region
several orders larger than its Compton wavelength, the relativistic effects
becomes negligible. It is worth noting that for a particle with a mass close
to the Planck mass, the relativistic effects start to become relevant when the
particle is localized in a region of the order of the Planck length, which
coincide with $\lambdabar_{C}$ for $m=m_{P}$.

##  4 The relativistic Diósi length

Now consider the eigenvalue equation,

$\left[\frac{\hbar^{2}}{\left(1+\gamma\right)m}\nabla^{2}-{Gm^{2}}\int\frac{~{}|\psi(\mathbf{x^{\prime}})|^{2}}{|\mathbf{x}-\mathbf{x^{\prime}}|}d\mathbf{x^{\prime}}\right]\psi=E\psi,$
(16)

which is the Schrödinger-Newton model with the Grave de Peralta modification
to include relativistic effects.

Using Eq. (15) we can qualitatively evaluate the energy of a stationary
solution of Eq. (16), for a particle localized in a region of size $l$, in the
form,

$E\approx\frac{\hbar^{2}}{\left[1+\sqrt{1+\left(\frac{\lambdabar}{l}\right)^{2}}\right]ml^{2}}-\frac{Gm^{2}}{l}.$
(17)

This equation is the relativistic version of those previously obtained by
Diósi [1], namely

$E\approx\frac{\hbar^{2}}{2ml^{2}}-\frac{Gm^{2}}{l}.$ (18)

By minimizing this expression with respect to $l$, Diósi arrived to the
distance $l_{D}$ given by Eq. (2), which represent the characteristic length
of a stationary solution of the SNE. Indeed, this length give a measure of the
size of the quantum field when its expansion, due to the dispersive $p^{2}/2m$
energy term, is stopped by the gravitational self-force.

Following Diósi, it is straightforward to shows that the value of $l$ for
which Eq. (17) has a minimum is,

$l_{D}^{(r)}=l_{D}\sqrt{1-\left(\frac{\lambdabar_{C}}{l_{D}}\right)^{2}},$
(19)

where the supra-index $r$ in $l_{D}^{(r)}$ stand for "relativistic".

Figure 3: Length scales in units of the Planck length as a function of the
mass in units of the Planck mass. See text for details.

In the same way, $l_{D}^{(r)}$ represent the length scale for the solutions of
Eq. (16), and clearly indicate that this characteristic length goes to zero
when the Diósi length approach the Compton wavelength of the particle. This
happens for a mass of the particle equal to the Planck mass, when it is
verified that $l_{D}=\lambdabar_{C}=l_{P}$. Figure 3 shows the behavior of the
different length scales in units of the Planck scale, where it is evident that
close to the Planck mass the relativistic corrected Diósi length start to
depart from the non-relativistic one and sharply drop to zero, then becoming
undefined for larger vales. This sort of collapse behavior observed for the
characteristic length, may be given a glimpse of a truly collapse behavior of
a quantum field induced by gravity, in a full relativistic approach.

The Eq. (19) may be written in the following appealing form

$l_{D}^{(r)}=l_{D}\sqrt{1-\left(\frac{m_{0}}{m_{P}}\right)^{4}}$ (20)

##  5 Conclusions

The Grave de Peralta proposal to account for relativistic effects by the Eqs.
(8), (9) and (10), shows a good agreement with the values obtained by
integration of the Dirac equation for a particle in a box. Although some
appreciable disagreements are observed above the $2m_{0}c^{2}$ threshold, our
values reproduce with good agreement most of the reported data. The major
advantage of this method rely on the possibility of include relativistic
corrections by solving just an equation beyond a Schrödinger-like equation.

After extend the Schrödinger-Newton model to the relativistic domain using the
Grave de Peralta approach, we obtained that length scale of the model is
bounded from above at a mass of the particle of the order of the Planck mass.
This indicates that the coupling between a relativistic quantum field and a
classical gravitational field may lead to a collapse of the wavefunction
induced by gravity.

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