# Center construction for group-crossed tensor categories

Mizuki Oikawa The author is supported by JSPS KAKENHI Grant Number JP23KJ0540.
Graduate School of Mathematical Sciences
The University of Tokyo, Komaba, Tokyo 153-8914, Japan
E-mail<EMAIL_ADDRESS>
###### Abstract

We define the notion of the $(G,\Gamma)$-crossed center of a
$(G,\Gamma)$-crossed tensor category in the sense of Natale. We show that the
$(G,\Gamma)$-crossed center is a $(G\bowtie\Gamma,G\times\Gamma)$-braided
tensor category. This construction generalizes the graded center construction
for graded tensor categories and the equivariant center construction for
tensor categories with group actions.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Coherence theorems for crossed monoidal categories
  4. 4 Crossed center

##  1 Introduction

There are at least two generalizations of the Drinfeld center construction
that produce _$G$ -braided_ tensor categories in the sense of Turaev [Tur00].
One is the graded center construction for graded tensor categories
[GNN09][TV13], and another is the equivariant center construction for tensor
categories with group actions [BGM19][She19].

Then, it is natural to ask if there exists a center construction for group-
crossed tensor categories that depends on both a group action and a grading.
In this article, we define the notion of the _$(G,\Gamma)$ -crossed center_ of
a $(G,\Gamma)$-crossed pivotal tensor category in the sense of Natale [Nat16].
This construction reproduces the graded center construction when the group $G$
acting on the tensor category is trivial. Moreover, it reproduces the
equivariant center construction when the group $\Gamma$ that grades the tensor
category is trivial.

We show that the $(G,\Gamma)$-crossed center of a $(G,\Gamma)$-crossed pivotal
tensor category is a $(G\bowtie\Gamma,G\times\Gamma)$-braided tensor category,
where $G\bowtie\Gamma$ denotes the _Zappa–Szép product_ [Zap42] of $G$ and
$\Gamma$. Indeed, our construction produces a group-crossed tensor category in
the sense of Natale even when we start from one in the sense of Turaev.

This article is structured as follows. In Section 2, we review some
preliminaries on group-crossed and group-braided monoidal categories in the
sense of Natale. In Section 3, we show the coherence theorem for
$(G,\Gamma)$-crossed monoidal categories. In Section 4, we define the notion
of the _$(G,\Gamma)$ -crossed center_ of a $(G,\Gamma)$-crossed pivotal tensor
category.

##  2 Preliminaries

In this section, we recall the notion of a $(G,\Gamma)$-crossed monoidal
category in the sense of Natale [Nat16]. In short, it is a monoidal category
with a compatible group action and a grading in some sense. We also recall the
counterpart of a braiding on such a category.

Our terminologies on tensor category theory follow those in [EGNO15]. In
particular, a _multitensor category_ over a field $k$ is a locally finite
$k$-linear abelian rigid monoidal category with a bilinear monoidal product as
in [EGNO15, Definition 4.1.1]. A _tensor category_ is a multitensor category
such that the endomorphism space of the unit is one-dimensional. Here, an
_abelian monoidal category_ is a monoidal category that is an abelian category
with a biexact monoidal product. A _linear monoidal category_ is a monoidal
category that is a linear category (i.e. a category with linear hom spaces
whose composition of morphisms are bilinear) with a bilinear monoidal product.

First, we recall the notion of a _grading_ on an additive category. Let
$\\{\mathcal{A}_{i}\\}_{i\in I}$ be a family of additive categories. Then, its
_direct sum_ $\mathcal{C}=\bigoplus_{i\in I}\mathcal{A}_{i}$ is an additive
category with a family of additive functors
$\\{I_{i}:\mathcal{A}_{i}\to\mathcal{A}\\}_{i\in I}$ which has the
universality that for any additive category $\mathcal{B}$ and a family of
additive functors $\\{F_{i}:\mathcal{A}_{i}\to\mathcal{B}\\}_{i\in I}$ there
exists an additive functor $F:\mathcal{A}\to\mathcal{B}$ with a family of
natural isomorphisms $\sigma^{F}=\\{\sigma^{F,i}:FI_{i}\cong F_{i}\\}_{i\in
I}$ such that for any other such pair $(\tilde{F},\tilde{\sigma}^{F})$ and a
family of natural transformations $\\{\tau^{i}:F_{i}\to\tilde{F}_{i}\\}_{i\in
I}$ we have a unique natural transformation $\tau:F\to\tilde{F}$ with
$\tau^{i}\sigma^{F,i}=\sigma^{\tilde{F},i}(\tau\ast I_{i})$ for any $i\in I$,
where $\ast$ denotes the horizontal composition of natural transformations.
Such a category $\mathcal{A}$ always exists, see e.g. [EGNO15, Section 1.3].
The datum $(\\{\mathcal{A}_{i}\\}_{i\in I},\\{I_{i}\\}_{i\in I})$ is called an
_$I$ -grading_ on $\mathcal{A}$ and an additive category with an $I$-grading
is called an _$I$ -graded category_.

We can indeed take $\sigma^{F,i}$’s to be identities. In this case, we refer
to the functor $F$ as the _extension_ of the functors $\\{F_{i}\\}_{i\in I}$.
Also, we refer to the natural transformation $\tau$ obtained by universality
as the _extension_ of the natural transformations $\\{\tau^{i}\\}_{i\in I}$.
Thus, we often define functors on $\mathcal{C}$ and natural transformations
between them by defining them on $\mathcal{A}_{i}$’s.

More explicitly, the notion of an $I$-grading on an additive category is
equivalent to a family of additive subcategories $\\{\mathcal{A}_{i}\\}_{i\in
I}$ such that every object is a direct sum of objects of $\mathcal{A}_{i}$’s
and $\operatorname{Hom}_{\mathcal{A}}(\lambda,\mu)=\\{0\\}$ for
$\lambda\in\operatorname{Obj}(\mathcal{A}_{i})$ and
$\mu\in\operatorname{Obj}(\mathcal{A}_{j})$ with $i\neq j$. A decomposition of
an object into objects of $\mathcal{A}_{i}$’s is called a _homogeneous
decomposition_ , which is essentially unique by [Oik24, Lemma 2.10].

We write $\partial_{\mathcal{A}}\lambda=i$ or simply $\partial\lambda=i$ if
$\lambda\in\operatorname{Obj}(\mathcal{A}_{i})$. Since
$\operatorname{Obj}(\mathcal{A}_{i})\cap\operatorname{Obj}(\mathcal{A}_{j})$
consists of the zero objects of $\mathcal{A}$ for $i\neq j$, the assignment
$\partial$ is single-valued on nonzero objects. We put
$\operatorname{Homog}(\mathcal{A})\coloneqq\bigcup_{i\in
I}\operatorname{Obj}(\mathcal{A}_{i})$.

We are interested in the case where $\mathcal{A}$ is a monoidal category, $I$
is a group and they are compatible in an appropriate sense.

######  Definition 2.1.

Let $\Gamma$ be a group. A _$\Gamma$ -graded monoidal category_ is an additive
monoidal category $\mathcal{C}$ (i.e. a monoidal category that is an additive
category with a biadditive monoidal product) that is $\Gamma$-graded and
$\operatorname{Obj}(\mathcal{C}_{s})\otimes\operatorname{Obj}(\mathcal{C}_{t})\subset\operatorname{Obj}(\mathcal{C}_{st})$
for all $s,t\in\Gamma$.

A _$\Gamma$ -graded monoidal functor_ $F:\mathcal{C}\to\mathcal{D}$ between
$\Gamma$-graded monoidal category is an additive monoidal functor such that
$F(\operatorname{Obj}(\mathcal{C}_{s}))\subset\operatorname{Obj}(\mathcal{D}_{s})$
for all $s\in\Gamma$.

Next, we recall the notion of an _action_ of a group on a (not necessarily
monoidal) category.

######  Definition 2.2.

An _action_ of a group $G$ on a category $\mathcal{E}$ is a monoidal functor
from the monoidal category $\underline{G}$ of elements of $G$ with only
identity morphisms to the monoidal category of endofunctors on $\mathcal{E}$.

When $\mathcal{E}$ is additive (resp. linear), we require that an action of a
group values in the category of additive (resp. linear) endofunctors on
$\mathcal{E}$.

More explicitly, an action $\gamma$ of $G$ on $\mathcal{E}$ consists of an
assignment $g\mapsto\gamma(g)$ of endofunctors and natural isomorphisms
$\chi^{\gamma}_{g,h}:\gamma(g)\gamma(h)\cong\gamma(gh)$ for $g,h\in G$ and
$\iota^{\gamma}:\mathrm{id}_{\mathcal{E}}\cong\gamma(e)$ with certain
coherence relations, see e.g. [EGNO15, Section 2.4]. By the coherence theorem
for monoidal functors (see e.g. [Gur13, Subsection 2.3.3]), a natural
isomorphism obtained by vertical and horizontal compositions of
$\chi^{\gamma}_{g,h}$’s and $\iota^{\gamma}$ is canonical, see also [Gal17,
Proposition 4.1]. Therefore, we often suppress such a natural transformation
and its components.

For $\lambda\in\operatorname{Obj}(\mathcal{E})$ and $g\in G$, we let
${}^{g}\lambda$ denote $\gamma(g)(\lambda)$. We also use a similar notation
for morphisms.

In order to formulate compatibility between a group action and a grading, we
need the following notion, see e.g. [Kas95, Definition IX.1.1].

######  Definition 2.3.

A _matched pair of groups_ is a tuple
$(G,\Gamma)=(G,\Gamma,\rhd_{1},\rhd_{2})$ of groups $G$ and $\Gamma$, an
action $\rhd_{1}:G\times\Gamma\to\Gamma$ of $G$ on a set $\Gamma$ and an
action $\rhd_{2}:\Gamma\times G\to G$ of $\Gamma$ on a set $G$ such that
$g\rhd_{1}e=e$, $s\rhd_{2}e=e$ and

$\displaystyle g\rhd_{1}(st)$
$\displaystyle=((t\rhd_{2}g)\rhd_{1}s)(g\rhd_{1}t)$ $\displaystyle
s\rhd_{2}(gh)$ $\displaystyle=((h\rhd_{1}s)\rhd_{2}g)(s\rhd_{2}h)$

for all $g,h\in G$ and $s,t\in\Gamma$.

The notion of a matched pair of groups is equivalent to that of an _exact
factorization_ of a group. Namely, an _exact factorization_ of a group $H$ is
a pair $(G,\Gamma)$ of subgroups with $H=G\Gamma$ and $G\cap\Gamma=\\{e\\}$.
Given an exact factorization $(G,\Gamma)$ of a group $H$, we can obtain a
matched pair $(G,\Gamma,\rhd_{1},\rhd_{2})$ from the relation
$sg=(s\rhd_{2}g^{-1})^{-1}(g^{-1}\rhd_{1}s)$ for $g\in G$ and $s\in\Gamma$.
Conversely, given a matched pair $(G,\Gamma,\rhd_{1},\rhd_{2})$, this relation
defines the _Zappa–Szép product_ $H\coloneqq G\bowtie\Gamma$ of $G$ and
$\Gamma$ [Zap42] (see also [Kas95, Proposition IX.1.2]), which has an exact
factorization $(G,\Gamma)$.

######  Remark 2.4.

In the usual definition of a matched pair (e.g. [Kas95, Definition IX.1.1]),
$\rhd_{1}$ is a right action. Our definition is obtained by regarding it as a
left action by the inverse map $G^{\mathrm{op}}\cong G$.

We are interested in the following notion, which was introduced in [Nat16,
Definition 4.1].

######  Definition 2.5.

Let $(G,\Gamma)$ be a matched pair of groups. A _$(G,\Gamma)$ -crossed
monoidal category (in the sense of Natale)_ is a $\Gamma$-graded additive
monoidal category $\mathcal{C}=\bigoplus_{s\in\Gamma}\mathcal{C}_{s}$ with an
action $\gamma^{\mathcal{C}}$ of $G$ on a category $\mathcal{C}$, a family of
isomorphisms
$J^{\gamma^{\mathcal{C}}}=\\{J^{\gamma^{\mathcal{C}}(g)}_{\lambda,\mu}:{}^{\partial\mu\rhd_{2}g}\lambda{}^{g}\mu\cong{}^{g}(\lambda\mu)\\}_{\lambda\in\operatorname{Obj}(\mathcal{C}),\mu\in\operatorname{Homog}(\mathcal{C}),g\in
G}$ that is natural in $\lambda$ and $\mu$ and a family of isomorphisms
$\varphi^{\gamma^{\mathcal{C}}}=\\{\varphi^{\gamma^{\mathcal{C}}(g)}:\mathbf{1}_{\mathcal{C}}\cong{}^{g}\mathbf{1}_{\mathcal{C}}\\}_{g\in
G}$ with the following axioms:

  1. 1.

For all $s\in\Gamma$ and $g\in G$,
$\gamma^{\mathcal{C}}(g)(\operatorname{Obj}(\mathcal{C}_{s}))\subset\operatorname{Obj}(\mathcal{C}_{g\rhd_{1}s})$.

  2. 2.

For all $g\in G$, $\lambda\in\operatorname{Obj}(\mathcal{C})$ and
$\mu,\nu\in\operatorname{Homog}(\mathcal{C})$,

$\displaystyle(J^{\gamma^{\mathcal{C}}(\partial\nu\rhd_{2}g)}_{\lambda,\mu}\otimes\mathrm{id}_{{}^{g}\nu})J^{\gamma^{\mathcal{C}}(g)}_{\lambda\mu,\nu}$
$\displaystyle=(\mathrm{id}_{{}^{\partial(\mu\nu)\rhd_{2}g}\lambda}\otimes
J^{\gamma^{\mathcal{C}}(g)}_{\mu,\nu})J^{\gamma^{\mathcal{C}}(g)}_{\lambda,\mu\nu}$
$\displaystyle
J^{\gamma^{\mathcal{C}}(g)}_{\mathbf{1}_{\mathcal{C}},\mu}(\varphi^{\gamma^{\mathcal{C}}(\partial\mu\rhd_{2}g)}\otimes\mathrm{id}_{{}^{g}\mu})$
$\displaystyle=\mathrm{id}_{{}^{g}\mu}=J^{\gamma^{\mathcal{C}}(g)}_{\mu,\mathbf{1}_{\mathcal{C}}}(\mathrm{id}_{{}^{g}\mu}\otimes\varphi^{\gamma^{\mathcal{C}}(g)}).$

  3. 3.

For all $g,h\in G$, $\lambda\in\operatorname{Obj}(\mathcal{C})$ and
$\mu\in\operatorname{Homog}(\mathcal{C})$,

$\displaystyle(\chi^{\gamma^{\mathcal{C}}}_{g,h})_{\lambda\mu}{}^{g}J^{\gamma^{\mathcal{C}}(h)}_{\lambda,\mu}J^{\gamma^{\mathcal{C}}(g)}_{{}^{\partial\mu\rhd_{2}h}\lambda,{}^{h}\mu}$
$\displaystyle=J^{\gamma^{\mathcal{C}}(gh)}_{\lambda,\mu}((\chi^{\gamma^{\mathcal{C}}}_{(h\rhd_{1}\partial\mu)\rhd_{2}g,\partial\mu\rhd_{2}h})_{\lambda}\otimes(\chi^{\gamma^{\mathcal{C}}}_{g,h})_{\mu})$
$\displaystyle(\chi^{\gamma^{\mathcal{C}}}_{g,h})_{\mathbf{1}_{\mathcal{C}}}{}^{g}\varphi^{\gamma^{\mathcal{C}}(h)}\varphi^{\gamma^{\mathcal{C}}(g)}$
$\displaystyle=\varphi^{\gamma^{\mathcal{C}}(gh)}$
$\displaystyle\iota^{\gamma^{\mathcal{C}}}_{\lambda\mu}$
$\displaystyle=J^{\gamma^{\mathcal{C}}(e)}_{\lambda,\mu}(\iota^{\gamma^{\mathcal{C}}}_{\lambda}\otimes\iota^{\gamma^{\mathcal{C}}}_{\mu})$
$\displaystyle\iota^{\gamma^{\mathcal{C}}}_{\mathbf{1}_{\mathcal{C}}}$
$\displaystyle=\varphi^{\gamma^{\mathcal{C}}(e)}.$

For $\lambda,\mu\in\operatorname{Obj}(\mathcal{C})$ and $g\in G$, we let
${}^{g}\lambda\mu$ denote $\gamma^{\mathcal{C}}(g)(\lambda)\otimes\mu$. We
also use a similar notation for morphisms.

######  Remark 2.6.

One can see that axioms 2 and 3 give a generalization of
$\gamma^{\mathcal{C}}$ being an _action_ of $G$ on the monoidal category
$\mathcal{C}$ (i.e. an action of $G$ on the category $\mathcal{C}$ that values
in the category of monoidal endofunctors, see e.g. [EGNO15, Definition
2.7.1(ii)]). Indeed, when $\rhd_{2}$ is trivial, axiom 2 becomes the
monoidality of $\gamma^{\mathcal{C}}(g)$, and axiom 3 becomes the monoidality
of $\chi^{\gamma^{\mathcal{C}}}_{g,h}$ and $\iota^{\gamma}$.

In particular, if $\Gamma$ is trivial, then the notion of a
$(G,\Gamma)$-crossed monoidal category is equivalent to that of an additive
monoidal category with an action of $G$.

On the other hand, if $G$ is trivial, then by definition the notion of a
$(G,\Gamma)$-crossed monoidal category is equivalent to that of a
$\Gamma$-graded monoidal category.

We can also consider a generalization of a braiding, see [Nat16, Definition
7.1].

######  Definition 2.7.

Let $(G,\Gamma)=(G,\Gamma,\rhd_{1},\rhd_{2})$ be a matched pair of groups. A
_braiding_ on $(G,\Gamma)$ is the pair $(\phi,\psi)$ of group homomorphisms
$\phi,\psi:\Gamma\to G$ such that

$\displaystyle(\phi(s)\rhd_{1}t)s$ $\displaystyle=(\psi(t)\rhd_{1}s)t$
$\displaystyle(s\rhd_{2}g)\phi(s)$ $\displaystyle=\phi(g\rhd_{1}s)g$
$\displaystyle(s\rhd_{2}g)\psi(s)$ $\displaystyle=\psi(g\rhd_{1}s)g$
$\displaystyle s\rhd_{2}\phi(t)$ $\displaystyle=\phi(\psi(s)\rhd_{1}t)$
$\displaystyle s\rhd_{2}\psi(t)$ $\displaystyle=\psi(\phi(s)\rhd_{1}t)$

for all $g\in G$ and $s,t\in\Gamma$. A matched pair of groups with a braiding
is called a _braided matched pair_ of groups.

######  Definition 2.8.

Let $(G,\Gamma)\coloneqq(G,\Gamma,\phi,\psi)$ be a braided matched pair of
groups. A _$(G,\Gamma)$ -braided monoidal category (in the sense of Natale)_
is the pair $(\mathcal{C},b^{\mathcal{C}})$ of a $(G,\Gamma)$-crossed monoidal
category $\mathcal{C}$ and a natural isomorphism
$b^{\mathcal{C}}=\\{b^{\mathcal{C}}_{\lambda,\mu}:{}^{\phi(\partial\mu)}\lambda\mu\cong{}^{\psi(\partial\lambda)}\mu\lambda\\}_{\lambda,\mu\in\operatorname{Homog}(\mathcal{C})}$
such that

$\displaystyle{}^{g}b_{\lambda,\mu}^{\mathcal{C}}J^{\gamma^{\mathcal{C}}(g)}_{{}^{\phi(\partial\mu)}\lambda,\mu}((\chi^{\gamma^{\mathcal{C}}}_{\partial\mu\rhd_{2}g,\phi(\partial\mu)})^{-1}_{\lambda}(\chi^{\gamma^{\mathcal{C}}}_{\phi(g\rhd_{1}\partial\mu),g})_{\lambda}\otimes\mathrm{id}_{{}^{g}\mu})=J^{\gamma^{\mathcal{C}}(g)}_{{}^{\psi(\partial\lambda)}\mu,\lambda}((\chi^{\gamma^{\mathcal{C}}}_{\partial\lambda\rhd_{2}g,\psi(\partial\lambda)})_{\mu}^{-1}\otimes\mathrm{id}_{{}^{g}\lambda})b_{{}^{g}\lambda,{}^{g}\mu}^{\mathcal{C}}$
$\displaystyle
b^{\mathcal{C}}_{\lambda\mu,\nu}(J^{\gamma^{\mathcal{C}}(\phi(\partial\nu))}_{\lambda,\mu}\otimes\mathrm{id}_{\nu})=((\chi^{\gamma^{\mathcal{C}}}_{\psi(\partial\lambda),\psi(\partial\mu)})_{\nu}\otimes\mathrm{id}_{\lambda\mu})(b^{\mathcal{C}}_{\lambda,{}^{\psi(\partial\mu)}\nu}\otimes\mathrm{id}_{\mu})(\mathrm{id}_{{}^{\partial\mu\rhd_{2}\phi(\partial\nu)}\lambda}\otimes
b^{\mathcal{C}}_{\mu,\nu})$ $\displaystyle
b^{\mathcal{C}}_{\lambda,\mu\nu}((\chi^{\gamma^{\mathcal{C}}}_{\phi(\partial\mu),\phi(\partial\nu)})_{\lambda}\otimes\mathrm{id}_{\mu\nu})=(J^{\gamma^{\mathcal{C}}(\psi(\partial\lambda))}_{\mu,\nu}\otimes\mathrm{id}_{\lambda})(\mathrm{id}_{{}^{\psi(\phi(\partial\nu)\rhd_{1}\partial\lambda)}\mu}\otimes
b^{\mathcal{C}}_{\lambda,\nu})(b^{\mathcal{C}}_{{}^{\phi(\nu)}\lambda,\mu}\otimes\mathrm{id}_{\nu})$

for all $\lambda,\mu,\nu\in\operatorname{Homog}(\mathcal{C})$ and $g\in G$.

######  Remark 2.9.

  1. 1.

The components of a $(G,\Gamma)$-braiding $b^{\mathcal{C}}$ are grading
preserving by the first axiom for $(\phi,\psi)$ in Definition 2.7. The first
(resp. second, resp. third) axiom for $b^{\mathcal{C}}$ makes sense by the
second and third (resp. fourth, resp. fifth) axioms for $(\phi,\psi)$.

  2. 2.

Our definition of $(G,\Gamma)$-crossed and $(G,\Gamma)$-braided monoidal
categories coincides with that of [Nat16] if we take $\rhd_{1}$ to be a right
action, see Remark 2.4.

Indeed, the notion of a $(G,\Gamma)$-crossed monoidal category reproduces as a
special case that of a _$G$ -crossed monoidal category_, which was introduced
by Turaev [Tur00, Section 2.1], see [Nat16, Section 8.1].

######  Definition 2.10.

Let $G$ be a group. A _$G$ -crossed monoidal category (in the sense of
Turaev)_ is a $(G,G)$-crossed monoidal category, where $(G,G)$ denotes the
matched pair given by the adjoint action $\rhd_{1}$ and the trivial action
$\rhd_{2}$. A _$G$ -braided monoidal category (in the sense of Turaev)_ is a
$(G,G)$-braided monoidal category, where $(G,G)$ denotes the braided matched
pair given by the trivial homomorphism $\phi$ and the identity $\psi$.

Finally, we give some known examples.

######  Example 2.11.

For a matched pair $(G,\Gamma)$ of groups, the tensor category
$\mathrm{Vec}_{\Gamma}$ of finite dimensional $\Gamma$-graded vector spaces
can trivially be regarded as a $(G,\Gamma)$-crossed tensor category. Namely,
for $g\in G$ and a vector space $V$ with grading $s\in\Gamma$, define
$\gamma(g)(V)$ to be $V$ with grading $g\rhd_{1}s$. Then, $\gamma$ and
identity morphisms form a $(G,\Gamma)$-structure on $\mathrm{Vec}_{\Gamma}$.

Before introducing other examples, let us recall the notion of a _relative
center_ of a monoidal category [Maj91, Definition 3.2]. Let $\mathcal{C}$ be a
monoidal category and let $\mathcal{D}$ be its monoidal subcategory. Then, the
_relative center_ $\operatorname{Z}_{\mathcal{D}}(\mathcal{C})$ is defined as
follows. An object of $\operatorname{Z}_{\mathcal{D}}(\mathcal{C})$ is the
pair $(\lambda,h^{\lambda})$ of $\lambda\in\operatorname{Obj}(\mathcal{C})$
and a family
$h^{\lambda}=\\{h^{\lambda}_{\mu}:\lambda\mu\cong\mu\lambda\\}_{\mu\in\operatorname{Obj}(\mathcal{D})}$
of isomorphisms that is natural in $\mu$ such that
$h^{\lambda}_{\mu_{1}\mu_{2}}=(\mathrm{id}_{\mu_{1}}\otimes
h^{\lambda}_{\mu_{2}})(h^{\lambda}_{\mu_{1}}\otimes\mathrm{id}_{\mu_{2}})$ for
all $\mu_{1},\mu_{2}\in\operatorname{Obj}(\mathcal{D})$. The monoidal product
is given by
$h^{\lambda_{1}\lambda_{2}}_{\mu}\coloneqq(h^{\lambda_{1}}_{\mu}\otimes\mathrm{id}_{\lambda_{2}})(\mathrm{id}_{\lambda_{1}}\otimes
h^{\lambda_{2}}_{\mu})$ for
$(\lambda_{1},h^{\lambda_{1}}),(\lambda_{2},h^{\lambda_{2}})\in\operatorname{Obj}(\operatorname{Z}_{\mathcal{D}}(\mathcal{C}))$
and $\mu\in\operatorname{Obj}(\mathcal{D})$. The unit is given by
$h^{\mathbf{1}_{\mathcal{C}}}\coloneqq\mathrm{id}$.

We also recall that a _pivotal monoidal category_ is a rigid monoidal category
$\mathcal{C}$ with a monoidal isomorphism
$\delta^{\mathcal{C}}:\mathrm{id}\cong{}^{\vee\vee}$, where ∨ denotes the left
dual, see e.g. [EGNO15, Definition 4.7.7].

######  Example 2.12.

Let $\mathcal{C}$ be a $\Gamma$-graded tensor category. If $\mathcal{C}$ and
$\Gamma$ are finite [GNN09, Theorem 3.3] or $\mathcal{C}$ is (not necessarily
finite but) pivotal [TV13, Theorem 4.1], then the _$\Gamma$ -graded center_
$\operatorname{Z}_{\Gamma}(\mathcal{C})\coloneqq\operatorname{Z}_{\mathcal{C}_{e}}(\mathcal{C})$
of $\mathcal{C}$ is a $\Gamma$-braided tensor category in the sense of Turaev.

######  Example 2.13.

Let $\mathcal{C}$ be an additive monoidal category with an action
$\gamma^{\mathcal{C}}$ of a group $G$. Note that the assignment
$g\mapsto\gamma^{\mathcal{C}}(g^{-1})$ defines an action of $G^{\mathrm{op}}$
on $\mathcal{C}$. Then, the _$G$ -equivariant center_
$\operatorname{Z}^{G}(\mathcal{C})\coloneqq\operatorname{Z}_{\mathcal{C}}(\mathcal{C}\rtimes
G^{\mathrm{op}})$ of $\mathcal{C}$ is a $G$-braided monoidal category in the
sense of Turaev [BGM19, Theorem 6.2][She19, Proposition 4.4.6].

Here, we do not give the details of these two constructions because we give
their generalization in Section 4.

##  3 Coherence theorems for crossed monoidal categories

It is known that a group action on a monoidal category can be _strictified_ in
an appropriate sense [Gal17, Theorem 4.3] (see also [BGM19, Theorem 3.1]). In
this section, we generalize this fact in the setting of group-crossed monoidal
categories in the sense of Natale (Definition 2.5). As a consequence, we can
see that isomorphisms obtained from the crossed structure are indeed canonical
(Corollary 3.9), and therefore we may suppress them.

First, we define the appropriate notion of an _equivalence_ between group-
crossed monoidal categories. For a monoidal functor
$F:\mathcal{C}\to\mathcal{D}$ between monoidal categories, let $J^{F}$ and
$\varphi^{F}$ denote the monoidal structure of $F$. Namely, for a monoidal
functor $F$, we have the isomorphisms
$J^{F}_{\lambda,\mu}:F(\lambda)F(\mu)\cong F(\lambda\mu)$ for
$\lambda,\mu\in\operatorname{Obj}(\mathcal{C})$ and
$\varphi^{F}:\mathbf{1}_{\mathcal{D}}\cong F(\mathbf{1}_{\mathcal{C}})$, see
e.g. [EGNO15, Definition 2.4.5].

######  Definition 3.1.

Let $\mathcal{C}$ and $\mathcal{D}$ be $(G,\Gamma)$-crossed monoidal
categories for a matched pair $(G,\Gamma)$ of groups. A _$(G,\Gamma)$ -crossed
monoidal functor_ from $\mathcal{C}$ to $\mathcal{D}$ is the pair
$F=(F,\eta^{F})$ of a $\Gamma$-graded monoidal functor
$F:\mathcal{C}\to\mathcal{D}$ and a family of natural isomorphisms
$\eta^{F}=\\{\eta^{F}_{g}:F\gamma^{\mathcal{C}}(g)\cong\gamma^{\mathcal{D}}(g)F\\}_{g\in
G}$ such that

  1. 1.

$\eta^{F}_{gh}(\mathrm{id}_{F}\ast\chi^{\gamma^{\mathcal{C}}}_{g,h})=(\chi^{\gamma^{\mathcal{D}}}_{g,h}\ast\mathrm{id}_{F})(\mathrm{id}_{\gamma^{\mathcal{D}}(g)}\ast\eta^{F}_{h})(\eta^{F}_{g}\ast\mathrm{id}_{\gamma^{\mathcal{C}}(h)})$
for all $g,h\in G$.

  2. 2.

$(\eta^{F}_{g})_{\lambda\mu}F(J^{\gamma^{\mathcal{C}}(g)}_{\lambda,\mu})J^{F}_{{}^{\partial\mu\rhd_{2}g}\lambda,{}^{g}\mu}={}^{g}J^{F}_{\lambda,\mu}J^{\gamma^{\mathcal{D}}(g)}_{F(\lambda),F(\mu)}((\eta^{F}_{\partial\mu\rhd_{2}g})_{\lambda}\otimes(\eta^{F}_{g})_{\mu})$
for all
$\lambda\in\operatorname{Obj}(\mathcal{C}),\mu\in\operatorname{Homog}(\mathcal{C})$
and $g\in G$.

Let $F,F^{\prime}:\mathcal{C}\to\mathcal{D}$ be $(G,\Gamma)$-crossed monoidal
functors. A _$(G,\Gamma)$ -crossed natural transformation_ from $F$ to
$F^{\prime}$ is a monoidal natural transformation $\tau:F\to F^{\prime}$ such
that
$(\mathrm{id}_{\gamma^{\mathcal{D}}(g)}\ast\tau)\eta_{g}^{F}=\eta^{F^{\prime}}_{g}(\tau\ast\mathrm{id}_{\gamma^{\mathcal{C}}(g)})$
for all $g\in G$.

######  Remark 3.2.

When $G$ is trivial, the notion of a $(G,\Gamma)$-crossed functor is
equivalent to that of a $\Gamma$-graded functor (Definition 2.1) by taking
$\eta^{F}$ to be the identity transformation. When $\Gamma$ is trivial, the
notion of a $(G,\Gamma)$-crossed functor is equivalent to that of an additive
_monoidal $G$-functor_, see e.g. [Oik24, Definition 2.8]. When $\mathcal{C}$
and $\mathcal{D}$ are $G$-crossed monoidal categories in the sense of Turaev
(Definition 2.10), the notion of a $(G,\Gamma)$-crossed functor is equivalent
to that of a _$G$ -crossed monoidal functor_, see [Oik24, Definition 2.11].

######  Proposition 3.3.

Let $(G,\Gamma)$ be a matched pair of groups. Then, $(G,\Gamma)$-crossed
monoidal categories, $(G,\Gamma)$-crossed monoidal functors and
$(G,\Gamma)$-crossed monoidal natural transformations form a 2-category (i.e.
strict bicategory).

###### Proof.

It is easy to see that for $(G,\Gamma)$-crossed monoidal categories
$\mathcal{C}$ and $\mathcal{D}$, $(G,\Gamma)$-crossed monoidal functors from
$\mathcal{C}$ to $\mathcal{D}$ and $(G,\Gamma)$-crossed monoidal natural
transformations between them form a category, where the composition of
morphisms is given by the vertical composition.

Let $F_{1}:\mathcal{C}\to\mathcal{D}$, $F_{2}:\mathcal{D}\to\mathcal{E}$ be
$(G,\Gamma)$-crossed functors between $(G,\Gamma)$-crossed monoidal
categories. By a standard argument, $F_{2}F_{1}$ can be made into a
$\Gamma$-graded monoidal functor.

We make $F_{2}F_{1}$ into a $(G,\Gamma)$-crossed monoidal functor by putting
$\eta^{F_{2}F_{1}}_{g}\coloneqq(\eta^{F_{2}}_{g}\ast\mathrm{id}_{F_{1}})(\mathrm{id}_{F_{2}}\ast\eta_{g}^{F_{1}})$
for $g\in G$. Indeed, axiom 1 in Definition 3.1 for
$(F_{2}F_{1},\eta^{F_{2}F_{1}})$ follows by a standard argument. Namely, for
$g\in G$, let a crossing from $F\gamma^{\mathcal{C}}(g)$ to
$\gamma^{\mathcal{D}}(g)F$ graphically denote $\eta_{g}^{F}$ in the 2-category
of categories. Also, let a fork from
$\gamma^{\mathcal{C}}(g)\gamma^{\mathcal{D}}(h)$ to $\gamma^{\mathcal{C}}(gh)$
(resp. from $\gamma^{\mathcal{D}}(g)\gamma^{\mathcal{D}}(h)$ to
$\gamma^{\mathcal{D}}(gh)$) graphically denote
$\chi^{\gamma^{\mathcal{C}}}_{g,h}$ (resp.
$\chi^{\gamma^{\mathcal{D}}}_{g,h}$). Then, axiom 1 in Definition 3.1
graphically means that a fork can pass under a crossing. Then, axiom 1 for
$(F_{2}F_{1},\eta^{F_{2}F_{1}})$ follows since a fork can pass under two
crossings by a standard graphical calculation.

Axiom 2 follows since

$\displaystyle(\eta^{F_{2}F_{1}}_{g})_{\lambda\mu}F_{2}F_{1}(J^{\gamma^{\mathcal{C}}(g)}_{\lambda,\mu})J^{F_{2}F_{1}}_{{}^{\partial\mu\rhd_{2}g}\lambda,{}^{g}\mu}$
$\displaystyle=(\eta^{F_{2}}_{g})_{F_{1}(\lambda\mu)}F_{2}((\eta_{g}^{F_{1}})_{\lambda\mu}F_{1}(J^{\gamma^{\mathcal{C}}(g)}_{\lambda,\mu})J^{F_{1}}_{{}^{\partial\mu\rhd_{2}g}\lambda,{}^{g}\mu})J^{F_{2}}_{F_{1}({}^{\partial\mu\rhd_{2}g}\lambda),F_{1}({}^{g}\mu)}$
$\displaystyle=(\eta^{F_{2}}_{g})_{F_{1}(\lambda\mu)}F_{2}({}^{g}J^{F_{1}}_{\lambda,\mu}J^{\gamma^{\mathcal{D}}(g)}_{F_{1}(\lambda),F_{1}(\mu)}((\eta_{\partial\mu\rhd_{2}g}^{F_{1}})_{\lambda}\otimes(\eta_{g}^{F_{1}})_{\mu}))J^{F_{2}}_{F_{1}({}^{\partial\mu\rhd_{2}g}\lambda),F_{1}({}^{g}\mu)}$
$\displaystyle={}^{g}F_{2}(J^{F_{1}}_{\lambda,\mu})(\eta^{F_{2}}_{g})_{F_{1}(\lambda)F_{1}(\mu)}F_{2}(J^{\gamma^{\mathcal{D}}(g)}_{F_{1}(\lambda),F_{1}(\mu)})J^{F_{2}}_{{}^{\partial\mu\rhd_{2}g}F_{1}(\lambda),{}^{g}F_{1}(\mu)}(F_{2}((\eta_{\partial\mu\rhd_{2}g}^{F_{1}})_{\lambda})\otimes
F_{2}((\eta_{g}^{F_{1}})_{\mu}))$
$\displaystyle={}^{g}J^{F_{2}F_{1}}_{\lambda,\mu}J^{\gamma^{\mathcal{E}}(g)}_{F_{2}F_{1}(\lambda),F_{2}F_{1}(\mu)}((\eta^{F_{2}F_{1}}_{\partial\mu\rhd_{2}g})_{\lambda}\otimes(\eta^{F_{2}F_{1}}_{g})_{\mu})$

for $\lambda\in\operatorname{Obj}(\mathcal{C})$,
$\mu\in\operatorname{Homog}(\mathcal{C})$ and $g\in G$ by axiom 2 for $F_{1}$
and $F_{2}$ and the naturality of $\eta_{g}^{F_{2}}$ and $J^{F_{2}}$. Note
that we used that $F_{1}$ is $\Gamma$-graded at the final equality.

It is easy to see that the horizontal composition of $(G,\Gamma)$-crossed
monoidal transformation is again $(G,\Gamma)$-crossed monoidal, and therefore
the composition of $(G,\Gamma)$-crossed monoidal functors is a bifunctor.

We have $\eta^{(F_{3}F_{2})F_{1}}=\eta^{F_{3}(F_{2}F_{1})}$ for composable
$(G,\Gamma)$-crossed monoidal functors $F_{1},F_{2},F_{3}$ by definition.
Moreover, we have
$\eta^{\mathrm{id}_{\mathcal{D}}F}=\eta^{F\mathrm{id}_{\mathcal{C}}}=\eta^{F}$
for a $(G,\Gamma)$-crossed monoidal functor $F:\mathcal{C}\to\mathcal{D}$ by
definition. Thus, we obtain a 2-category. ∎

######  Definition 3.4.

A _$(G,\Gamma)$ -crossed monoidal equivalence_ between $(G,\Gamma)$-crossed
monoidal categories is an equivalence in the 2-category of
$(G,\Gamma)$-crossed monoidal categories (see e.g. [Lei04, Section 1.5]).
Equivalently, a $(G,\Gamma)$-crossed monoidal functor
$F:\mathcal{C}\to\mathcal{D}$ between $(G,\Gamma)$-crossed monoidal categories
is a _$(G,\Gamma)$ -crossed monoidal equivalence_ if there exists a
$(G,\Gamma)$-crossed monoidal functor $F^{-1}:\mathcal{D}\to\mathcal{C}$ with
$(G,\Gamma)$-crossed natural isomorphisms
$F^{-1}F\cong\mathrm{id}_{\mathcal{C}}$ and
$FF^{-1}\cong\mathrm{id}_{\mathcal{D}}$.

The following lemma is useful.

######  Lemma 3.5.

Let $F:\mathcal{C}\to\mathcal{D}$ be a $(G,\Gamma)$-crossed monoidal (resp.
tensor) functor between $(G,\Gamma)$-crossed monoidal (resp. multiring)
categories. Then, $F$ is a $(G,\Gamma)$-crossed monoidal equivalence if and
only if it is an equivalence as a functor.

###### Proof.

We show the “if” part since the “only if” part is trivial. Suppose $F$ is an
equivalence as a functor. Then, by a standard argument, an adjoint inverse
$F^{-1}$ of $F$ is additive (resp. linear exact faithful). Moreover, by a
standard argument (see e.g. [EGNO15, Remark 2.4.10]), we can make $F^{-1}$
into a monoidal functor and obtain a monoidal evaluation map
$\mathrm{ev}^{F}:F^{-1}F\cong\mathrm{id}_{\mathcal{C}}$ and a monoidal
coevaluation map $\mathrm{coev}^{F}:\mathrm{id}_{\mathcal{D}}\cong FF^{-1}$.

Since $\mathcal{C}$ and $\mathcal{D}$ are $\Gamma$-graded, for
$\lambda\in\operatorname{Obj}(\mathcal{D}_{s})$, we can consider the
homogeneous decomposition
$F^{-1}(\lambda)=\bigoplus_{t\in\Gamma}F^{-1}(\lambda)_{t}$ of
$F^{-1}(\lambda)$. Since $F$ is $\Gamma$-graded and
$FF^{-1}\cong\mathrm{id}_{\mathcal{D}}$ by assumption, $FF^{-1}(\lambda)_{t}$
is a zero object unless $t=s$. Thus, $F^{-1}FF^{-1}(\lambda)_{t}$ is a zero
object unless $t=s$, and therefore $F^{-1}$ is $\Gamma$-graded since
$F^{-1}\cong F^{-1}FF^{-1}$.

We make $F^{-1}$ into a $(G,\Gamma)$-crossed monoidal (resp. tensor) functor.
Put

$\displaystyle\eta^{F^{-1}}_{g}\coloneqq(\mathrm{ev}^{F}\ast\mathrm{id}_{\gamma^{\mathcal{C}}(g)F^{-1}})(\mathrm{id}_{F^{-1}}\ast(\eta^{F}_{g})^{-1}\ast\mathrm{id}_{F^{-1}})(\mathrm{id}_{F^{-1}\gamma^{\mathcal{D}}(g)}\ast\mathrm{coev}^{F})$

for $g\in G$. We check the axioms in Definition 3.1 for
$(F^{-1},\eta^{F^{-1}})$. Let us use the graphical notations in the proof of
Proposition 3.3. Then, $\eta^{F^{-1}}_{g}$ is graphically represented by a
rotated crossing, and therefore axiom 1 for $(F^{-1},\eta^{F^{-1}})$ follows
from axiom 1 and a conjugate equation for $F$. Axiom 2 follows since

$\displaystyle(\eta^{F^{-1}}_{g})_{\lambda\mu}F^{-1}(J^{\gamma^{\mathcal{D}}(g)}_{\lambda,\mu})J^{F^{-1}}_{{}^{\partial\mu\rhd_{2}g}\lambda,{}^{g}\mu}$
$\displaystyle=\mathrm{ev}^{F}_{{}^{g}F^{-1}(\lambda\mu)}F^{-1}((\eta^{F}_{g})^{-1}_{F^{-1}(\lambda\mu)}{}^{g}\mathrm{coev}^{F}_{\lambda\mu}J^{\gamma^{\mathcal{D}}(g)}_{\lambda,\mu})J^{F^{-1}}_{{}^{\partial\mu\rhd_{2}g}\lambda,{}^{g}\mu}$
$\displaystyle=\mathrm{ev}^{F}_{{}^{g}F^{-1}(\lambda\mu)}$ $\displaystyle\quad
F^{-1}((\eta^{F}_{g})^{-1}_{F^{-1}(\lambda\mu)}{}^{g}F(J^{F^{-1}}_{\lambda,\mu}){}^{g}J^{F}_{F^{-1}(\lambda),F^{-1}(\mu)}J^{\gamma^{\mathcal{D}}(g)}_{FF^{-1}(\lambda),FF^{-1}(\mu)}({}^{g}\mathrm{coev}^{F}_{\lambda}\otimes{}^{g}\mathrm{coev}^{F}_{\mu}))J^{F^{-1}}_{{}^{\partial\mu\rhd_{2}g}\lambda,{}^{g}\mu}$
$\displaystyle={}^{g}J^{F^{-1}}_{\lambda,\mu}\mathrm{ev}^{F}_{{}^{g}(F^{-1}(\lambda)F^{-1}(\mu))}$
$\displaystyle\quad
F^{-1}((\eta^{F}_{g})^{-1}_{F^{-1}(\lambda)F^{-1}(\mu)}{}^{g}J^{F}_{F^{-1}(\lambda),F^{-1}(\mu)}J^{\gamma^{\mathcal{D}}(g)}_{FF^{-1}(\lambda),FF^{-1}(\mu)}({}^{g}\mathrm{coev}^{F}_{\lambda}\otimes{}^{g}\mathrm{coev}^{F}_{\mu}))J^{F^{-1}}_{{}^{\partial\mu\rhd_{2}g}\lambda,{}^{g}\mu}$
$\displaystyle={}^{g}J^{F^{-1}}_{\lambda,\mu}\mathrm{ev}^{F}_{{}^{g}(F^{-1}(\lambda)F^{-1}(\mu))}F^{-1}F(J^{\gamma^{\mathcal{C}}(g)}_{F^{-1}(\lambda),F^{-1}(\mu)})F^{-1}(J^{F}_{{}^{\partial\mu\rhd_{2}g}\lambda,{}^{g}\mu})$
$\displaystyle\quad
F^{-1}((\eta^{F}_{\partial\mu\rhd_{2}g})^{-1}_{F^{-1}(\lambda)}{}^{g}\mathrm{coev}^{F}_{\lambda}\otimes(\eta^{F}_{g})^{-1}_{F^{-1}(\mu)}{}^{g}\mathrm{coev}^{F}_{\mu})J^{F^{-1}}_{{}^{\partial\mu\rhd_{2}g}\lambda,{}^{g}\mu}$
$\displaystyle={}^{g}J^{F^{-1}}_{\lambda,\mu}J^{\gamma^{\mathcal{C}}(g)}_{F^{-1}(\lambda),F^{-1}(\mu)}\mathrm{ev}^{F}_{{}^{g}F^{-1}(\lambda){}^{g}F^{-1}(\mu)}J^{F^{-1}F}_{{}^{\partial\mu\rhd_{2}g}\lambda,{}^{g}\mu}$
$\displaystyle\quad(F^{-1}((\eta^{F}_{\partial\mu\rhd_{2}g})^{-1}_{F^{-1}(\lambda)}{}^{g}\mathrm{coev}^{F}_{\lambda})\otimes
F^{-1}((\eta^{F}_{g})^{-1}_{F^{-1}(\mu)}{}^{g}\mathrm{coev}^{F}_{\mu}))$
$\displaystyle={}^{g}J^{F^{-1}}_{\lambda,\mu}J^{\gamma^{\mathcal{C}}(g)}_{F^{-1}(\lambda),F^{-1}(\mu)}((\eta^{F^{-1}}_{\partial\mu\rhd_{2}g})_{\lambda}\otimes(\eta^{F^{-1}}_{g})_{\mu})$

for $\lambda\in\operatorname{Obj}(\mathcal{D})$,
$\mu\in\operatorname{Homog}(\mathcal{D})$ and $g\in G$ by the naturality of
$J^{\gamma^{\mathcal{D}}(g)}$, $J^{F^{-1}}$, $\mathrm{ev}^{F}$ and
$\mathrm{coev}^{F}$, axiom 2 for $F$ and the monoidality of $\mathrm{ev}^{F}$
and $\mathrm{coev}^{F}$. Thus, $(F^{-1},\eta^{F^{-1}})$ is a
$(G,\Gamma)$-crossed monoidal (resp. tensor) functor.

Finally, recall that
$\eta^{F^{-1}F}_{g}=(\eta^{F^{-1}}_{g}\ast\mathrm{id}_{F})(\mathrm{id}_{F^{-1}}\ast\eta_{g}^{F})$
and
$\eta^{FF^{-1}}_{g}=(\eta^{F}_{g}\ast\mathrm{id}_{F^{-1}})(\mathrm{id}_{F}\ast\eta_{g}^{F^{-1}})$
for $g\in G$ by the proof of Proposition 3.3. Then, $\mathrm{ev}^{F}$ and
$\mathrm{coev}^{F}$ are $(G,\Gamma)$-crossed natural isomorphisms since

$\displaystyle(\mathrm{id}_{\gamma^{\mathcal{D}}(g)}\ast\mathrm{ev}^{F})(\eta^{F^{-1}}_{g}\ast\mathrm{id}_{F})(\mathrm{id}_{F^{-1}}\ast\eta_{g}^{F})$
$\displaystyle=\mathrm{ev}^{F}\ast\mathrm{id}_{\gamma^{\mathcal{C}}(g)}$
$\displaystyle(\eta^{F}_{g}\ast\mathrm{id}_{F^{-1}})(\mathrm{id}_{F}\ast\eta_{g}^{F^{-1}})(\mathrm{coev}^{F}\ast\mathrm{id}_{\gamma^{\mathcal{C}}(g)})$
$\displaystyle=\mathrm{id}_{\gamma^{\mathcal{D}}(g)}\ast\mathrm{coev}^{F}$

for $g\in G$ by conjugate equations for $F$. Thus, $(F^{-1},\eta^{F^{-1}})$ is
a $(G,\Gamma)$-crossed monoidal (resp. tensor) equivalence. ∎

######  Lemma 3.6.

Let $\mathcal{C}$ be a $(G,\Gamma)$-crossed monoidal category, and let
$\mathcal{D}$ be an additive monoidal category. Given an additive monoidal
equivalence $F:\mathcal{C}\to\mathcal{D}$, we can endow $\mathcal{D}$ a
$(G,\Gamma)$-crossed structure so that $F$ is a $(G,\Gamma)$-crossed monoidal
equivalence.

###### Proof.

First, we give a $\Gamma$-grading on $\mathcal{D}$. For $s\in\Gamma$, put
$\mathcal{D}_{s}\coloneqq\mathcal{C}_{s}$, where $\mathcal{C}_{s}$ is regarded
as an additive subcategory of $\mathcal{D}$ by using $F$. Then, since
$\mathcal{C}$ is $\Gamma$-graded and $F$ is essentially surjective, any object
of $\mathcal{D}$ is a direct sum of homogeneous objects. Since $F$ is moreover
an equivalence,
$\operatorname{Obj}(\mathcal{D}_{s})\cap\operatorname{Obj}(\mathcal{D}_{t})$
consists of zero objects if $s\neq t$. Thus,
$\mathcal{D}=\bigoplus_{s\in\Gamma}\mathcal{D}_{s}$. Note that $F$ is
$\Gamma$-graded by the definition of the $\Gamma$-grading on $\mathcal{D}$.

Next, we give a $G$-action on $\mathcal{D}$. Fix an adjoint inverse $F^{-1}$
and put $\gamma^{\mathcal{D}}(g)\coloneqq F\gamma^{\mathcal{C}}(g)F^{-1}$ for
$g\in G$. By a standard argument, $\gamma^{\mathcal{D}}$ defines a $G$-action
on the category $\mathcal{D}$ by putting
$\chi^{\gamma^{\mathcal{D}}}_{g,h}\coloneqq\mathrm{id}_{F\gamma^{\mathcal{C}}(g)}\ast\mathrm{ev}^{F}\ast\mathrm{id}_{\gamma^{\mathcal{C}}(h)F^{-1}}$
and $\iota^{\gamma^{\mathcal{D}}}\coloneqq\mathrm{coev}^{F}$. Moreover, axiom
1 in Definition 2.5 is satisfied since $F^{-1}$ is $\Gamma$-graded by the
proof of Lemma 3.5.

Then, we give a $(G,\Gamma)$-crossed structure. Put
$J^{\gamma^{\mathcal{D}}(g)}_{\lambda,\mu}\coloneqq
F(J^{\gamma^{\mathcal{C}}(g)}_{F^{-1}(\lambda),F^{-1}(\mu)})$ and
$\varphi^{\gamma^{\mathcal{D}}(g)}\coloneqq
F(\varphi^{\gamma^{\mathcal{C}}(g)})$ for
$\lambda\in\operatorname{Obj}(\mathcal{D})$,
$\mu\in\operatorname{Homog}(\mathcal{D})$ and $g\in G$. Note that
$J^{\gamma^{\mathcal{D}}(g)}_{\lambda,\mu}$ is a well-defined isomorphism
${}^{\partial\mu\rhd_{2}g}\lambda{}^{g}\mu\cong{}^{g}(\lambda\mu)$ since
$F^{-1}$ is $\Gamma$-graded. Then, axioms 2 and 3 in Definition 2.5 follow
from those for $\mathcal{C}$. Thus, $\mathcal{D}$ is a $(G,\Gamma)$-crossed
monoidal category.

Finally, we make $F$ into a $(G,\Gamma)$-crossed equivalence. By Lemma 3.5, it
suffices to make it into a $(G,\Gamma)$-crossed monoidal functor. We have
already seen that $F$ is $\Gamma$-graded. Put
$\eta^{F}_{g}\coloneqq\mathrm{id}_{F\gamma^{\mathcal{C}}(g)}\ast(\mathrm{ev}^{F})^{-1}$
for $g\in G$. It is easy to check that $(F,\eta^{F})$ satisfies axiom 1 in
Definition 3.1. Axiom 2 follow from the definition of
$J^{\gamma^{\mathcal{D}}}$ and $\varphi^{\gamma^{\mathcal{D}}}$ and the
monoidality of $\mathrm{ev}^{F}$. ∎

######  Definition 3.7.

A $(G,\Gamma)$-crossed monoidal category $\mathcal{C}$ is _strictly
$(G,\Gamma)$-crossed_ if the action $\gamma^{\mathcal{C}}$ is strict and
$J^{\gamma^{\mathcal{C}}}$ and $\varphi^{\gamma^{\mathcal{C}}}$ in Definition
2.5 consist of identity maps.

The following theorem should be called the coherence theorem for crossed
tensor categories.

######  Theorem 3.8.

Let $(G,\Gamma)$ be a matched pair of groups. Then, any $(G,\Gamma)$-crossed
monoidal category is $(G,\Gamma)$-crossed equivalent to a strictly
$(G,\Gamma)$-crossed strict monoidal category.

###### Proof.

Recall from [Gur13, Subsection 2.2.3] that for a monoidal category
$\mathcal{C}$, by letting $\operatorname{st}(\mathcal{C})$ be the strict
monoidal category whose objects are strings of objects of $\mathcal{C}$, we
obtain a monoidal equivalence
$\tilde{F}_{\mathcal{C}}:\mathcal{C}\simeq\operatorname{st}(\mathcal{C})$,
which is defined by regarding the objects of $\mathcal{C}$ as the strings of
length 1. By definition, when $\mathcal{C}$ is additive monoidal,
$\operatorname{st}(\mathcal{C})$ is again additive monoidal and
$\tilde{F}_{\mathcal{C}}$ is additive. Therefore,
$\operatorname{st}(\mathcal{C})$ can be made into a $(G,\Gamma)$-crossed
monoidal category so that $\tilde{F}_{\mathcal{C}}$ is a $(G,\Gamma)$-crossed
monoidal equivalence by Lemma 3.6. Thus, we may assume that $\mathcal{C}$ is a
strict monoidal category from the beginning.

For a $(G,\Gamma)$-crossed strict monoidal category $\mathcal{C}$, let us
define the category $\mathcal{C}(G)$ as in [Gal17, Section 4]. Namely, an
object of $\mathcal{C}(G)$ is a pair $L=(L,\xi^{L})$ of a family
$L=\\{L_{g}\\}_{g\in G}$ of objects of $\mathcal{C}$ and a family of
isomorphisms $\xi^{L}=\\{\xi_{g,h}^{L}:{}^{g}L_{h}\cong L_{gh}\\}_{g,h\in G}$
such that $\xi_{gh,k}^{L}=\xi_{g,hk}^{L}{}^{g}\xi_{h,k}^{L}$ and
$\xi_{e,h}^{L}=\mathrm{id}_{L_{h}}$ for all $g,h,k\in G$, where we suppressed
some canonical morphisms (see Section 2). A morphism $f:L\to L^{\prime}$ is a
family $f=\\{f_{g}:L_{g}\to L_{g}^{\prime}\\}_{g\in G}$ of morphisms of
$\mathcal{C}$ such that
$f_{gh}\xi^{L}_{g,h}=\xi^{L^{\prime}}_{g,h}{}^{g}f_{h}$ for all $g,h\in G$.

By a standard argument, $\mathcal{C}(G)$ is an additive category. As in
[Gal17, Section 4], we obtain a strict $G$-action $\gamma^{\mathcal{C}(G)}$ on
the additive category $\mathcal{C}(G)$ by putting

$\displaystyle\gamma^{\mathcal{C}(G)}(g)(L)\coloneqq(\\{L_{hg}\\}_{h},\\{\xi^{L}_{k,hg}\\}_{k,h})$

for $(L,\xi^{L})\in\operatorname{Obj}(\mathcal{C}(G))$ and $g\in G$. Moreover,
$\mathcal{C}(G)=\bigoplus_{s\in\Gamma}\mathcal{C}(G)_{s}$ by putting

$\displaystyle\operatorname{Obj}(\mathcal{C}(G)_{s})\coloneqq\\{L\in\operatorname{Obj}(\mathcal{C}(G))\mid
L_{e}\in\operatorname{Obj}(\mathcal{C}(G)_{s})\\}$

for $s\in\Gamma$ as in the proof of [Gal17, Corollary 5.3]. For any $g\in G$
and $s\in\Gamma$,
${}^{g}\operatorname{Obj}(\mathcal{C}(G)_{s})\subset\operatorname{Obj}(\mathcal{C}(G)_{g\rhd_{1}s})$
since for $L\in\operatorname{Obj}(\mathcal{C}(G)_{s})$ we have
$({}^{g}L)_{e}=L_{g}\cong{}^{g}L_{e}\in\operatorname{Obj}(\mathcal{C}_{g\rhd_{1}s})$.

We make $\mathcal{C}(G)$ into a monoidal category by putting

$\displaystyle LM\coloneqq(\\{L_{\partial
M\rhd_{2}h}M_{h}\\}_{h},\\{(\xi^{L}_{(h\rhd_{1}\partial M)\rhd_{2}g,\partial
M\rhd_{2}h}\otimes\xi^{M}_{g,h})J^{\gamma^{\mathcal{C}}(g)-1}_{L_{\partial
M\rhd_{2}h},M_{h}}\\}_{g,h})$

for $L\in\operatorname{Obj}(\mathcal{C}(G))$ and
$M\in\operatorname{Homog}(\mathcal{C}(G))$. Note that $\xi^{LM}_{g,h}$ is
indeed an isomorphism ${}^{g}(LM)_{h}\cong(LM)_{gh}$ by the matching equation
$((h\rhd_{1}\partial M)\rhd_{2}g)(\partial M\rhd_{2}h)=\partial
M\rhd_{2}(gh)$. By similar matching equations, we have

$\displaystyle\xi^{L}_{(k\rhd_{1}\partial M)\rhd_{2}(gh),\partial M\rhd_{2}k}$
$\displaystyle=\xi^{L}_{((hk\rhd_{1}\partial M)\rhd_{2}g)((k\rhd_{1}\partial
M)\rhd_{2}h),\partial M\rhd_{2}k}$ $\displaystyle=\xi^{L}_{(hk\rhd_{1}\partial
M)\rhd_{2}g,((k\rhd_{1}\partial M)\rhd_{2}h)(\partial
M\rhd_{2}k)}{}^{(hk\rhd_{1}\partial M)\rhd_{2}g}\xi^{L}_{(k\rhd_{1}\partial
M)\rhd_{2}h,\partial M\rhd_{2}k}$ $\displaystyle=\xi^{L}_{(hk\rhd_{1}\partial
M)\rhd_{2}g,\partial M\rhd_{2}hk}{}^{(hk\rhd_{1}\partial
M)\rhd_{2}g}\xi^{L}_{(k\rhd_{1}\partial M)\rhd_{2}h,\partial M\rhd_{2}k}$

and therefore

$\displaystyle\xi^{LM}_{gh,k}$ $\displaystyle=(\xi^{L}_{(hk\rhd_{1}\partial
M)\rhd_{2}g,\partial M\rhd_{2}hk}{}^{(hk\rhd_{1}\partial
M)\rhd_{2}g}\xi^{L}_{(k\rhd_{1}\partial M)\rhd_{2}h,\partial
M\rhd_{2}k}\otimes\xi^{M}_{g,hk}{}^{g}\xi^{M}_{h,k})J^{\gamma^{\mathcal{C}}(gh)-1}_{L_{\partial
M\rhd_{2}k},M_{k}}$ $\displaystyle=(\xi^{L}_{(hk\rhd_{1}\partial
M)\rhd_{2}g,\partial M\rhd_{2}hk}{}^{(hk\rhd_{1}\partial
M)\rhd_{2}g}\xi^{L}_{(k\rhd_{1}\partial M)\rhd_{2}h,\partial
M\rhd_{2}k}\otimes\xi^{M}_{g,hk}{}^{g}\xi^{M}_{h,k})$ $\displaystyle\qquad
J^{\gamma^{\mathcal{C}}(g)-1}_{{}^{(k\rhd_{1}\partial M)\rhd_{2}h}L_{\partial
M\rhd_{2}k},{}^{h}M_{k}}{}^{g}J^{\gamma^{\mathcal{C}}(h)-1}_{L_{\partial
M\rhd_{2}k},M_{k}}$ $\displaystyle=(\xi^{L}_{(hk\rhd_{1}\partial
M)\rhd_{2}g,\partial
M\rhd_{2}hk}\otimes\xi^{M}_{g,hk})J^{\gamma^{\mathcal{C}}(g)-1}_{L_{\partial
M\rhd_{2}hk},M_{hk}}{}^{g}((\xi^{L}_{(k\rhd_{1}\partial M)\rhd_{2}h,\partial
M\rhd_{2}k}\otimes\xi^{M}_{h,k})J^{\gamma^{\mathcal{C}}(h)-1}_{L_{\partial
M\rhd_{2}k},M_{k}})$ $\displaystyle=\xi^{LM}_{g,hk}{}^{g}\xi^{LM}_{h,k}$

for $g,h,k\in G$ by axiom 3 in Definition 2.5 and the naturality of
$J^{\gamma^{\mathcal{C}}(g)}$. We also have
$\xi^{LM}_{e,h}=(\xi^{L}_{e,\partial
M\rhd_{2}h}\otimes\xi^{M}_{e,h})=\mathrm{id}$ for $g\in G$ by axiom 3 in
Definition 2.5 and $(h\rhd_{1}\partial M)\rhd_{2}e=e$. Thus, $LM$ is well-
defined. Moreover, for morphisms $f:L\to L^{\prime}$ and $\tilde{f}:M\to
M^{\prime}$ of $\mathcal{C}(G)$, define a morphism $f\otimes\tilde{f}:LM\to
L^{\prime}M^{\prime}$ by putting $(f\otimes\tilde{f})_{g}\coloneqq f_{\partial
M\rhd_{2}g}\otimes\tilde{f}_{g}$ for $g\in G$. Then,

$\displaystyle(f\otimes\tilde{f})_{gh}\xi^{LM}_{g,h}$
$\displaystyle=(f_{\partial M\rhd_{2}gh}\xi^{L}_{(h\rhd_{1}\partial
M)\rhd_{2}g,\partial
M\rhd_{2}h}\otimes\tilde{f}_{gh}\xi^{M}_{g,h})J^{\gamma^{\mathcal{C}}(g)-1}_{L_{\partial
M\rhd_{2}h,M_{h}}}$ $\displaystyle=(\xi^{L^{\prime}}_{(h\rhd_{1}\partial
M^{\prime})\rhd_{2}g,\partial M^{\prime}\rhd_{2}h}{}^{(h\rhd_{1}\partial
M)\rhd_{2}g}f_{h}\otimes\xi^{M^{\prime}}_{g,h}{}^{g}\tilde{f}_{h})J^{\gamma^{\mathcal{C}}(g)-1}_{L_{\partial
M\rhd_{2}h,M_{h}}}$ $\displaystyle=(\xi^{L^{\prime}}_{(h\rhd_{1}\partial
M^{\prime})\rhd_{2}g,\partial
M^{\prime}\rhd_{2}h}\otimes\xi^{M^{\prime}}_{g,h})J^{\gamma^{\mathcal{C}}(g)-1}_{L^{\prime}_{\partial
M^{\prime}\rhd_{2}h,M^{\prime}_{h}}}{}^{g}(f\otimes\tilde{f})$

for $g,h\in G$ by the naturality of $J^{\gamma^{\mathcal{C}}(g)}$. Note that
we may assume $\partial M=\partial M^{\prime}$ since otherwise $\tilde{f}=0$.
Therefore, $f\otimes\tilde{f}$ is well-defined. Thus, the product $\otimes$
defines a biadditive bifunctor
$\mathcal{C}(G)\times\mathcal{C}(G)\to\mathcal{C}(G)$. Moreover, it makes
$\mathcal{C}(G)$ into a $\Gamma$-graded strict monoidal category with unit
$\mathbf{1}_{\mathcal{C}(G)}\coloneqq(\\{\mathbf{1}_{\mathcal{C}}\\}_{h},\\{\varphi^{\gamma^{\mathcal{C}}(g)-1}\\}_{g,h})$.
Indeed,

$\displaystyle((LM)N)_{h}=L_{\partial M\rhd_{2}(\partial
N\rhd_{2}h)}M_{\partial N\rhd_{2}h}N_{h}=L_{\partial(MN)\rhd_{2}h}M_{\partial
N\rhd_{2}h}N_{h}=(L(MN))_{h}$

for $L\in\operatorname{Obj}(\mathcal{C}(G))$,
$M,N\in\operatorname{Homog}(\mathcal{C}(G))$ and $h\in G$ since $\rhd_{2}$ is
an action and $\partial M\partial N=\partial M_{e}\partial
N_{e}=\partial(M_{e}N_{e})=\partial(M_{\partial
N\rhd_{2}e}N_{e})=\partial(MN)$. Moreover, since

$\displaystyle\xi^{(LM)N}_{g,h}$ $\displaystyle=(\xi^{LM}_{(h\rhd_{1}\partial
N)\rhd_{2}g,\partial
N\rhd_{2}h}\otimes\xi^{N}_{g,h})J^{\gamma^{\mathcal{C}}(g)-1}_{(LM)_{\partial
N\rhd_{2}h},N_{h}}$ $\displaystyle=(\xi^{L}_{((\partial
N\rhd_{2}h)\rhd_{1}\partial M)\rhd_{2}((h\rhd_{1}\partial
N)\rhd_{2}g),\partial M\partial N\rhd_{2}h}\otimes\xi^{M}_{(h\rhd_{1}\partial
N)\rhd_{2}g,\partial N\rhd_{2}h}\otimes\xi^{N}_{g,h})$
$\displaystyle\qquad(J^{\gamma^{\mathcal{C}}(g)-1}_{L_{\partial M\partial
N\rhd_{2}h},M_{\partial
N\rhd_{2}h}}\otimes\mathrm{id}_{{}^{g}N_{h}})J^{\gamma^{\mathcal{C}}(g)-1}_{L_{\partial
M\partial N\rhd_{2}h}M_{\partial N\rhd_{2}h},N_{h}}$
$\displaystyle=(\xi^{L}_{(h\rhd_{1}\partial M\partial N)\rhd_{2}g,\partial
M\partial N\rhd_{2}h}\otimes\xi^{M}_{(h\rhd_{1}\partial N)\rhd_{2}g,\partial
N\rhd_{2}h}\otimes\xi^{N}_{g,h})$
$\displaystyle\qquad(J^{\gamma^{\mathcal{C}}(g)-1}_{L_{\partial M\partial
N\rhd_{2}h},M_{\partial
N\rhd_{2}h}}\otimes\mathrm{id}_{{}^{g}N_{h}})J^{\gamma^{\mathcal{C}}(g)-1}_{L_{\partial
M\partial N\rhd_{2}h}M_{\partial N\rhd_{2}h},N_{h}}$
$\displaystyle\xi^{L(MN)}_{g,h}$ $\displaystyle=(\xi^{L}_{(h\rhd_{1}\partial
M\partial N)\rhd_{2}g,\partial M\partial
N\rhd_{2}h}\otimes\xi^{MN}_{g,h})J^{\gamma^{\mathcal{C}}(g)-1}_{L_{\partial
M\partial N\rhd_{2}h},(MN)_{h}}$ $\displaystyle=(\xi^{L}_{(h\rhd_{1}\partial
M\partial N)\rhd_{2}g,\partial M\partial
N\rhd_{2}h}\otimes\xi^{M}_{(h\rhd_{1}\partial N)\rhd_{2}g,\partial
N\rhd_{2}h}\otimes\xi^{N}_{g,h})$
$\displaystyle\qquad(\mathrm{id}_{{}^{(h\rhd_{1}\partial M\partial
N)\rhd_{2}g}L_{\partial M\partial N\rhd_{2}h}}\otimes
J^{\gamma^{\mathcal{C}}(g)-1}_{M_{\partial
N\rhd_{2}h},N_{h}})J^{\gamma^{\mathcal{C}}(g)-1}_{L_{\partial M\partial
N\rhd_{2}h},M_{\partial N\rhd_{2}h}N_{h}}$

for $g,h\in G$ by the matching relation $h\rhd_{1}\partial M\partial
N=((\partial N\rhd_{2}h)\rhd_{1}\partial M)(h\rhd_{1}\partial N)$, we obtain
$\xi^{(LM)N}=\xi^{L(MN)}$ by axiom 2 in Definition 2.5. The unit
$\mathbf{1}_{\mathcal{C}(G)}$ is indeed an object of $\mathcal{C}(G)$ by axiom
3 in Definition 2.5. We have
$(\mathbf{1}_{\mathcal{C}(G)}M)_{h}=M_{h}=(M\mathbf{1}_{\mathcal{C}(G)})_{h}$
since $\partial\mathbf{1}_{\mathcal{C}}=e$. Moreover,
$\xi^{\mathbf{1}_{\mathcal{C}(G)}M}=\xi^{M\mathbf{1}_{\mathcal{C}(G)}}=\xi^{M}$
follows from axiom 2 in Definition 2.5.

For $L\in\operatorname{Obj}(\mathcal{C}(G))$,
$M\in\operatorname{Homog}(\mathcal{C}(G))$ and $g,h\in G$, the matching
relation $\partial M\rhd_{2}hg=((g\rhd_{1}\partial M)\rhd_{2}h)(\partial
M\rhd_{2}g)$, we have

$\displaystyle({}^{g}(LM))_{h}$ $\displaystyle=(LM)_{hg}=L_{\partial
M\rhd_{2}hg}M_{hg}=L_{((g\rhd_{1}\partial M)\rhd_{2}h)(\partial
M\rhd_{2}g)}M_{hg}$ $\displaystyle=({}^{\partial
M\rhd_{2}g}L)_{(g\rhd_{1}\partial M)\rhd_{2}h}({}^{g}M)_{h}=({}^{\partial
M\rhd_{2}g}L)_{\partial{}^{g}M\rhd_{2}h}({}^{g}M)_{h}$
$\displaystyle=({}^{\partial M\rhd_{2}g}L{}^{g}M)_{h}.$

Moreover,

$\displaystyle\xi^{{}^{\partial M\rhd_{2}g}L{}^{g}M}_{k,h}$
$\displaystyle=(\xi^{{}^{\partial
M\rhd_{2}g}L}_{(h\rhd_{1}\partial{}^{g}M)\rhd_{2}k,\partial{}^{g}M\rhd_{2}h}\otimes\xi^{{}^{g}M}_{k,h})J^{\gamma^{\mathcal{C}}(k)-1}_{({}^{\partial
M\rhd_{2}g}L)_{\partial{}^{g}M\rhd_{2}h},({}^{g}M)_{h}}$
$\displaystyle=(\xi^{{}^{\partial M\rhd_{2}g}L}_{(hg\rhd_{1}\partial
M)\rhd_{2}k,(g\rhd_{1}\partial
M)\rhd_{2}h}\otimes\xi^{{}^{g}M}_{k,h})J^{\gamma^{\mathcal{C}}(k)-1}_{({}^{\partial
M\rhd_{2}g}L)_{(g\rhd_{1}\partial M)\rhd_{2}h},({}^{g}M)_{h}}$
$\displaystyle=(\xi^{L}_{(hg\rhd_{1}\partial M)\rhd_{2}k,\partial
M\rhd_{2}hg}\otimes\xi^{M}_{k,hg})J^{\gamma^{\mathcal{C}}(k)-1}_{L_{\partial
M\rhd_{2}hg},M_{hg}}$ $\displaystyle=\xi^{LM}_{k,hg}=\xi^{{}^{g}(LM)}_{k,h}$

for $g,h,k\in G$. Therefore, ${}^{g}(LM)={}^{\partial M\rhd_{2}g}L{}^{g}M$. We
also have ${}^{g}\mathbf{1}_{\mathcal{C}(G)}=\mathbf{1}_{\mathcal{C}(G)}$ by
definition. Thus, the $G$-action $\gamma^{\mathcal{C}(G)}$ and the
$\Gamma$-grading $\mathcal{C}(G)=\bigoplus_{s\in\Gamma}\mathcal{C}(G)_{s}$
make $\mathcal{C}(G)$ into a strictly $(G,\Gamma)$-crossed strict monoidal
category.

The functor
$I_{\mathcal{C}}:\mathcal{C}\to\mathcal{C}(G);\lambda\mapsto(\\{{}^{h}\lambda\\}_{h},\\{(\chi^{\gamma^{\mathcal{C}}}_{g,h})_{\lambda}\\}_{g,h})$
is an equivalence of categories as in the proof of [Gal17, Proposition 4.1].
It is also $\Gamma$-graded by definition. We can make it into a monoidal
functor by putting
$J^{I_{\mathcal{C}}}_{\lambda,\mu}\coloneqq\\{J^{\gamma^{\mathcal{C}}(h)}_{\lambda,\mu}\\}_{h}$
for $\lambda\in\operatorname{Obj}(\mathcal{C})$ and
$\mu\in\operatorname{Homog}(\mathcal{C})$ and
$\varphi^{I_{\mathcal{C}}}\coloneqq\\{\varphi^{\gamma^{\mathcal{C}}(h)}\\}_{h}$.
Indeed, they are morphisms of $\mathcal{C}(G)$ by axiom 3 in Definition 2.5
since
$\xi^{I_{\mathcal{C}}(\lambda)I_{\mathcal{C}}(\mu)}_{g,h}=J^{\gamma^{\mathcal{C}}(g)-1}_{{}^{\partial\mu\rhd_{2}h}\lambda,{}^{h}\mu}$
and
$\xi^{\mathbf{1}_{\mathcal{C}(G)}}_{g,h}=\varphi^{\gamma^{\mathcal{C}}(g)-1}$
by definition. They satisfy the coherence relations for a monoidal functor by
axiom 2 in Definition 2.5.

Moreover, we can make $I_{\mathcal{C}}$ into a $(G,\Gamma)$-crossed monoidal
functor and therefore prove the theorem by Lemma 3.5. Indeed, since

$\displaystyle I_{\mathcal{C}}({}^{g}\lambda)$
$\displaystyle=(\\{{}^{h}({}^{g}\lambda)\\}_{h},\\{(\chi^{\gamma^{\mathcal{C}}}_{k,h})_{{}^{g}\lambda}\\}_{k,h})$
$\displaystyle{}^{g}I_{\mathcal{C}}(\lambda)$
$\displaystyle=(\\{{}^{hg}\lambda\\}_{h},\\{(\chi^{\gamma^{\mathcal{C}}}_{k,hg})_{\lambda}\\}_{k,h})$

for $\lambda\in\operatorname{Obj}(\mathcal{C})$ and $g\in G$ by definition, we
can put
$(\eta^{I_{\mathcal{C}}}_{g})_{\lambda}\coloneqq\\{(\chi^{\gamma^{\mathcal{C}}}_{h,g})_{\lambda}\\}_{h}:I_{\mathcal{C}}({}^{g}\lambda)\cong{}^{g}I_{\mathcal{C}}(\lambda)$,
which is indeed a morphism of $\mathcal{C}(G)$ by coherence. Then, the first
two axioms in Definition 3.1 are satisfied by coherence. The remaining two
axioms follow by coherence from the definition of $J^{I_{\mathcal{C}}}$ and
$\varphi^{I_{\mathcal{C}}}$, respectively, and axiom 3 in Definition 2.5.
Thus, $(I_{\mathcal{C}},\eta^{I_{\mathcal{C}}})$ is a $(G,\Gamma)$-crossed
monoidal functor. ∎

######  Corollary 3.9.

Let $\mathcal{C}$ be a $(G,\Gamma)$-crossed monoidal category. Define a set
$W=\sqcup_{n\geq 0}W_{n}$ of words recursively by the following rules:
$\mathbf{1}\in W_{0}$, $-\in W_{1}$, $\otimes\in W_{2}$, $\gamma(g)\in W_{1}$
for $g\in G$ and $w^{\prime}((w_{i})_{i})\in W_{\sum_{i}n_{i}}$ for
$w^{\prime}\in W_{n}$ and a tuple $(w_{i})_{i=1}^{n}$ with $w_{i}\in
W_{n_{i}}$. For $w\in W_{n}$, let $w$ again denote the functor
$\mathcal{C}^{n}\to\mathcal{C}$ corresponding to $w$, where $\mathcal{C}^{0}$
denotes the category with only one object and its identity morphism. Define a
set $I$ of morphisms recursively by the following rules: the components of
$a^{\mathcal{C}},l^{\mathcal{C}},r^{\mathcal{C}},J^{\gamma^{\mathcal{C}}(g)}$,
$\chi^{\gamma^{\mathcal{C}}}_{g,h}$ and $\iota^{\gamma^{\mathcal{C}}}$ are in
$I$ for any $g,h\in G$, $\varphi^{\gamma^{\mathcal{C}}(g)}\in I$ for any $g\in
G$, $f^{-1}\in I$ for any $f\in I$, $f\circ f^{\prime}\in I$ for any
composable $f,f^{\prime}\in I$ and $w((f_{i})_{i})$ for any
$(f_{i})_{i=1}^{n}\subset I$ and $w\in W_{n}$. Then, for any $w,w^{\prime}\in
W_{n}$ and objects $\\{\lambda_{i}\\}_{i=1}^{n}$ of $\mathcal{C}$, a morphism
$w((\lambda_{i})_{i})\to w^{\prime}((\lambda_{i})_{i})$ in $I$ is unique if it
exists.

###### Proof.

By theorem 3.8, there exists a $(G,\Gamma)$-crossed monoidal equivalence $F$
from $\mathcal{C}$ to a strictly $(G,\Gamma)$-crossed strict monoidal category
$\tilde{\mathcal{C}}$. For $w\in W_{n}$ and
$\\{\lambda_{i}\\}_{i=1}^{n}\subset\operatorname{Obj}(\mathcal{C})$, we define
a natural isomorphism $C^{w}:Fw\cong\tilde{w}F^{n}$, where $\tilde{w}$ denotes
the functor $\tilde{\mathcal{C}}^{n}\to\tilde{\mathcal{C}}$ corresponding to
$w$, recursively by the following rules:
$C^{\mathbf{1}}\coloneqq\varphi^{F-1}$, $C^{\otimes}\coloneqq J^{F-1}$,
$C^{\gamma(g)}\coloneqq\eta^{F}_{g}$ for $g\in G$ and
$C^{v((w_{i})_{i})}\coloneqq(\mathrm{id}_{\tilde{v}}\ast(C^{w_{i}})_{i})(C^{v}\ast\mathrm{id}_{(w_{i})_{i}})$
for $\\{w_{i}\\}_{i=1}^{n}\subset W$ and $v\in W_{n}$.

We show that
$C^{w}_{(\lambda_{i})_{i}}=C^{w^{\prime}}_{(\lambda_{i})_{i}}F(f)$ for any
$f:w((\lambda_{i})_{i})\to w^{\prime}((\lambda_{i})_{i})$ in $I$, which
uniquely determines $f$. It suffices to prove this when $f$ is a generator of
$I$. The statement for $a^{\mathcal{C}},l^{\mathcal{C}},r^{\mathcal{C}}$
follows since $F$ is monoidal, which is standard. The statement for
$\chi^{\gamma^{\mathcal{C}}},\iota^{\gamma^{\mathcal{C}}},J^{\gamma^{\mathcal{C}}(g)}$
and $\varphi^{\gamma^{\mathcal{C}}(g)}$ follows from the axioms for
$F=(F,\eta^{F})$ in Definition 3.1. ∎

##  4 Crossed center

In this section, we give the group-crossed version of the Drinfeld center
construction (Theorem 4.8) by generalizing the graded center construction (see
Example 2.13) by Turaev and Virelizier [TV13].

######  Definition 4.1.

Let $\mathcal{C}$ be a $(G,\Gamma)$-crossed monoidal category. Then, the
_$(G,\Gamma)$ -crossed center_ $\operatorname{Z}^{G}_{\Gamma}(\mathcal{C})$ of
$\mathcal{C}$ is defined as follows. For $g\in G$, an object of
$\operatorname{Z}^{G}_{\Gamma}(\mathcal{C})_{g}$ is the pair
$(\lambda,h^{\lambda})$ of $\lambda\in\operatorname{Obj}(\mathcal{C})$ and a
natural isomorphism
$h^{\lambda}=\\{h^{\lambda}_{\nu}:\lambda\nu\cong{}^{g}\nu\lambda\\}_{\nu\in\operatorname{Obj}(\mathcal{C}_{e})}$
such that $h^{\lambda}_{\nu_{1}\nu_{2}}=(\mathrm{id}_{{}^{g}\nu_{1}}\otimes
h^{\lambda}_{\nu_{2}})(h^{\lambda}_{\nu_{1}}\otimes\mathrm{id}_{\nu_{2}})$ for
all $\nu_{1},\nu_{2}\in\operatorname{Obj}(\mathcal{C}_{e})$. A morphism
$f:(\lambda,h^{\lambda})\to(\mu,h^{\mu})$ of
$\operatorname{Z}^{G}_{\Gamma}(\mathcal{C})_{g}$ is a morphism
$f:\lambda\to\mu$ of $\mathcal{C}$ such that
$h^{\mu}_{\nu}(f\otimes\mathrm{id}_{\nu})=(\mathrm{id}_{{}^{g}\nu}\otimes
f)h^{\lambda}_{\nu}$ for all $\nu\in\operatorname{Obj}(\mathcal{C}_{e})$.
Then, $\operatorname{Z}^{G}_{\Gamma}(\mathcal{C})_{g}$ is an additive
category, where the composition of morphisms and identity morphisms are given
by those of $\mathcal{C}$. The category
$\operatorname{Z}^{G}_{\Gamma}(\mathcal{C})$ is defined to be $\bigoplus_{g\in
G}\operatorname{Z}^{G}_{\Gamma}(\mathcal{C})_{g}$.

######  Lemma 4.2.

Let $\mathcal{C}$ be a $(G,\Gamma)$-crossed monoidal category. Then, the
$(G,\Gamma)$-crossed center $\operatorname{Z}^{G}_{\Gamma}(\mathcal{C})$ of
$\mathcal{C}$ becomes a $G\times\Gamma$-graded monoidal category by putting

$\displaystyle\operatorname{Obj}(\operatorname{Z}^{G}_{\Gamma}(\mathcal{C})_{(g,s)})\coloneqq\\{(\lambda,h^{\lambda})\in\operatorname{Obj}(\operatorname{Z}^{G}_{\Gamma}(\mathcal{C})_{g})\mid\lambda\in\operatorname{Obj}(\mathcal{C}_{s})\\}$

for $(g,s)\in G\times\Gamma$,
$(\lambda,h^{\lambda})\otimes(\mu,h^{\mu})\coloneqq(\lambda\mu,(h^{\lambda}\otimes\mathrm{id})(\mathrm{id}\otimes
h^{\mu}))$ for
$(\lambda,h^{\lambda}),(\mu,h^{\mu})\in\operatorname{Homog}(\operatorname{Z}^{G}_{\Gamma}(\mathcal{C}))$
and
$\mathbf{1}_{\operatorname{Z}^{G}_{\Gamma}(\mathcal{C})}\coloneqq(\mathbf{1}_{\mathcal{C}},\mathrm{id})$.

###### Proof.

By coherence (Corollary 3.9), we can do the same graphical calculation as that
for the ordinary center of $\mathcal{C}$. ∎

######  Lemma 4.3.

Let $\mathcal{C}$ be a $(G,\Gamma)$-crossed monoidal category. If
$\lambda\in\operatorname{Homog}(\mathcal{C})$ has a left dual
$\lambda^{\vee}$, then, for any $g\in G$, ${}^{g}\lambda$ has a left dual
${}^{\partial\lambda\rhd_{2}g}(\lambda^{\vee})$ with evaluation and
coevaluation maps given by
$\mathrm{ev}_{{}^{g}\lambda}\coloneqq{}^{g}\mathrm{ev}_{\lambda}$ and
$\mathrm{coev}_{{}^{g}\lambda}\coloneqq{}^{\partial\lambda\rhd_{2}g}\mathrm{coev}_{\lambda}$,
where we suppressed canonical isomorphisms in Corollary 3.9. We also have a
similar statement for right duals.

###### Proof.

We only show the statement for left duals. Note that the target of
$\mathrm{coev}_{{}^{g}\lambda}$ is
${}^{\partial\lambda\rhd_{2}g}(\lambda\lambda^{\vee})={}^{\partial\lambda^{\vee}\rhd_{2}(\partial\lambda\rhd_{2}g)}\lambda{}^{\partial\lambda\rhd_{2}g}(\lambda^{\vee})={}^{g}\lambda{}^{\partial\lambda\rhd_{2}g}(\lambda^{\vee})$.
Then, we have

$\displaystyle(\mathrm{id}_{{}^{g}\lambda}\otimes{}^{g}\mathrm{ev}_{\lambda})({}^{\partial\lambda\rhd_{2}g}\mathrm{coev}_{\lambda}\otimes\mathrm{id}_{{}^{g}\lambda})$
$\displaystyle=({}^{e\rhd_{2}g}\mathrm{id}_{\lambda}\otimes{}^{g}\mathrm{ev}_{\lambda})({}^{\partial\lambda\rhd_{2}g}\mathrm{coev}_{\lambda}\otimes{}^{g}\mathrm{id}_{\lambda})$
$\displaystyle={}^{g}((\mathrm{id}_{\lambda}\otimes\mathrm{ev}_{\lambda})(\mathrm{coev}_{\lambda}\otimes\mathrm{id}_{\lambda}))=\mathrm{id}_{{}^{g}\lambda}$

and

$\displaystyle({}^{g}\mathrm{ev}_{\lambda}\otimes\mathrm{id}_{{}^{\partial\lambda\rhd_{2}g}(\lambda^{\vee})})(\mathrm{id}_{{}^{\partial\lambda\rhd_{2}g}(\lambda^{\vee})}{}^{\partial\lambda\rhd_{2}g}\mathrm{coev}_{\lambda})$
$\displaystyle=({}^{\partial\lambda^{\vee}\rhd_{2}(\partial\lambda\rhd_{2}g)}\mathrm{ev}_{\lambda}\otimes{}^{\partial\lambda\rhd_{2}g}\mathrm{id}_{\lambda^{\vee}})({}^{e\rhd_{2}(\partial\lambda\rhd_{2}g)}\mathrm{id}_{\lambda^{\vee}}{}^{\partial\lambda\rhd_{2}g}\mathrm{coev}_{\lambda})$
$\displaystyle={}^{\partial\lambda\rhd_{2}g}((\mathrm{ev}_{\lambda}\otimes\mathrm{id}_{\lambda^{\vee}})(\mathrm{id}_{\lambda^{\vee}}\otimes\mathrm{coev}_{\lambda}))=\mathrm{id}_{{}^{\partial\lambda\rhd_{2}g}\lambda^{\vee}}$

by coherence (Corollary 3.9). ∎

######  Lemma 4.4.

Let $\mathcal{C}$ be a $(G,\Gamma)$-crossed monoidal category. If
$\mathcal{C}$ is rigid, then so is
$\operatorname{Z}^{G}_{\Gamma}(\mathcal{C})$.

###### Proof.

We can rotate a crossing and do the usual graphical calculations by Lemma 4.3
and coherence (Corollary 3.9). ∎

Then, we give crossed structures.

######  Lemma 4.5.

The $(G,\Gamma)$-crossed center $\operatorname{Z}^{G}_{\Gamma}(\mathcal{C})$
of a $(G,\Gamma)$-crossed monoidal category $\mathcal{C}$ becomes a
$(G,G\times\Gamma)$-crossed monoidal category, where $(G,G\times\Gamma)$
denotes the matched pair $(G,G\times\Gamma,\rhd_{1}^{G},\rhd_{2}^{G})$ defined
by $g\rhd_{1}^{G}(h,t)\coloneqq((t\rhd_{2}g)hg^{-1},g\rhd_{1}t)$ and
$(h,t)\rhd_{2}g\coloneqq t\rhd_{2}g$, by putting
${}^{g}(\lambda,h^{\lambda})\coloneqq({}^{g}\lambda,h^{{}^{g}\lambda})$ and
$h^{{}^{g}\lambda}_{\nu}\coloneqq{}^{g}h^{\lambda}_{{}^{g^{-1}}\nu}$ for
$\nu\in\operatorname{Obj}(\mathcal{C}_{e})$.

###### Proof.

Immediate from coherence (Corollary 3.9). ∎

We define a _$(G,\Gamma)$ -crossed pivotal tensor category_ to be a
$(G,\Gamma)$-crossed tensor category $\mathcal{C}$ with
${}^{g}\delta^{\mathcal{C}}_{\lambda}=\delta^{\mathcal{C}}_{{}^{g}\lambda}$
for any $\lambda\in\operatorname{Obj}(\mathcal{C})$. Note that this is
meaningful thanks to Lemma 4.3.

The following lemma is the group-crossed version of [TV13, Section 4.4].
Recall that a pivotal $\Gamma$-graded tensor category $\mathcal{C}$ is _non-
singular_ if for any $s\in\Gamma$, $\mathcal{C}_{s}$ has at least one object
with nonzero dimension.

######  Lemma 4.6.

The $(G,\Gamma)$-crossed center $\operatorname{Z}^{(G,\Gamma)}(\mathcal{C})$
of a non-singular $(G,\Gamma)$-crossed pivotal tensor category $\mathcal{C}$
admits a $(\Gamma,G\times\Gamma)$-crossed structure, where
$(\Gamma,G\times\Gamma)$ denotes the matched pair
$(\Gamma,G\times\Gamma,\rhd_{1}^{\Gamma},\rhd_{2}^{\Gamma})$ defined by
$s\rhd_{1}^{\Gamma}(h,t)\coloneqq(s\rhd_{2}h,(h\rhd_{1}s)ts^{-1})$ and
$(h,t)\rhd_{2}^{\Gamma}s\coloneqq h\rhd_{1}s$.

###### Proof.

For
$\lambda=(\lambda,h^{\lambda})\in\operatorname{Z}^{(G,\Gamma)}(\mathcal{C})_{(h,t)}$
and $\mu\in\operatorname{Obj}(\mathcal{C}_{s})$ with dimension $d_{\mu}\neq
0$, put

$\displaystyle e_{\lambda}^{\mu}\coloneqq
d_{\mu}^{-1}(\mathrm{ev}_{{}^{s\rhd_{2}h}(\mu^{\vee})}\otimes\mathrm{id}_{{}^{h}\mu\lambda\mu^{\vee}})(\delta^{\mathcal{C}}_{{}^{h}\mu}\otimes
h^{\lambda}_{\mu^{\vee}\mu}\otimes\mathrm{id}_{\mu^{\vee}})(\mathrm{id}_{{}^{h}\mu\lambda\mu^{\vee}}\otimes\mathrm{coev}_{\mu}),$

which is well-defined since $({}^{h}\mu)^{\vee}={}^{s\rhd_{2}h}(\mu^{\vee})$
by Lemma 4.3. Then, $e^{\mu}_{\lambda}$ is an idempotent in
$\operatorname{End}_{\mathcal{C}}({}^{h}\mu\lambda\mu^{\vee})$ by the same
argument as in the proof of [TV13, Lemma 4.2]. Let
$(E_{\lambda}^{\mu},s^{\mu}_{\lambda},r^{\mu}_{\lambda})$ denote a retract of
$e^{\mu}_{\lambda}$ and put

$\displaystyle h^{E^{\mu}_{\lambda}}_{\nu}$ $\displaystyle\coloneqq
d_{\mu}^{-1}(\mathrm{id}_{{}^{s\rhd_{2}h}\nu}\otimes
r^{\mu}_{\lambda})(\mathrm{ev}_{{}^{s\rhd_{2}h}(\mu^{\vee})}\otimes\mathrm{id}_{{}^{s\rhd_{2}h}\nu{}^{h}\mu\lambda\mu^{\vee}})(\delta^{\mathcal{C}}_{{}^{h}\mu}\otimes
h^{\lambda}_{\mu^{\vee}\nu\mu}\otimes\mathrm{id}_{\mu^{\vee}})(\mathrm{id}_{{}^{h}\mu\lambda\mu^{\vee}\nu}\otimes\mathrm{coev}_{\mu})$
$\displaystyle\qquad(s^{\mu}_{\lambda}\otimes\mathrm{id}_{\nu})$

for $\nu\in\operatorname{Obj}(\mathcal{C}_{e})$. Then,
$h^{E^{\mu}_{\lambda}}_{\mathbf{1}_{\mathcal{C}}}=\mathrm{id}_{E^{\mu}_{\lambda}}$
by definition. Moreover, by the same argument as in the proof of [TV13, Lemma
4.3], we obtain
$h^{E^{\mu}_{\lambda}}_{\nu_{1}\nu_{2}}=(\mathrm{id}_{{}^{h}\nu_{1}}\otimes
h^{E^{\mu}_{\lambda}}_{\nu_{2}})(h^{E^{\mu}_{\lambda}}_{\nu_{1}}\otimes\mathrm{id}_{\nu_{2}})$
and therefore

$\displaystyle\gamma^{\Gamma}_{\mu}(\lambda)\coloneqq(E^{\mu}_{\lambda},h^{E^{\mu}_{\lambda}})\in\operatorname{Obj}(\operatorname{Z}^{(G,\Gamma)}(\mathcal{C})_{(s\rhd_{2}h,(h\rhd_{1}s)ts^{-1})}).$

By putting $\gamma^{\Gamma}_{\mu}(f)\coloneqq
r_{\lambda^{\prime}}^{\mu}fs_{\lambda}^{\mu}$ for
$f\in\operatorname{Hom}_{\operatorname{Z}^{(G,\Gamma)}(\mathcal{C})_{(h,t)}}(\lambda,\lambda^{\prime})$,
we can regard $\gamma^{\Gamma}_{\mu}$ as an endofunctor of $\mathcal{C}$ as in
the proof of [TV13, Lemma 4.3]. Then, the coherence theorem (Corollary 3.9)
makes graphical calculation possible, and the same argument as in the proof of
[TV13, Lemmata 4.3, 4.4 and 4.5] shows that for a family
$\\{\mu_{s}\\}_{s\in\Gamma}$ of homogeneous objects with nonzero dimensions,
the assignment $s\mapsto\gamma^{\Gamma}(s)\coloneqq\gamma^{\Gamma}_{\mu_{s}}$
defines a $(\Gamma,G\times\Gamma)$-crossed structure. ∎

By a tedious calculation, it can be checked that
$(G\bowtie\Gamma,G\times\Gamma,\tilde{\rhd}_{1},\tilde{\rhd}_{2})$, where
$(g,s)\tilde{\rhd}_{1}(h,t)\coloneqq g\rhd_{1}^{G}(s\rhd_{1}^{\Gamma}(h,t))$
and
$(h,t)\tilde{\rhd}_{2}(g,s)\coloneqq((s\rhd_{1}^{\Gamma}h)\rhd_{2}^{G}g,h\rhd_{2}^{\Gamma}s)$,
is a matched pair. It is easy to check that the homomorphisms
$\phi,\psi:G\times\Gamma\to G\bowtie\Gamma$ defined by putting
$\phi(h,t)\coloneqq(e,t)$ and $\psi(h,t)\coloneqq(h,t)$ for $(h,t)\in
G\times\Gamma$ give a braiding on the matched pair
$(G\bowtie\Gamma,G\times\Gamma)$.

We combine two crossed structures. The following lemma is the key.

######  Lemma 4.7.

Let $\mathcal{C}$ be a $G\times\Gamma$-graded monoidal category with
$(G,G\times\Gamma)$-crossed and $(\Gamma,G\times\Gamma)$-crossed structures.
Let $\gamma^{G}$ and $\gamma^{\Gamma}$ denote, respectively, the actions of
$G$ and $\Gamma$ on $\mathcal{C}$ given by the crossed structures. Suppose we
are given a family of natural isomorphisms

$\displaystyle\\{\sigma_{g,s}:\gamma^{\Gamma}(s)\gamma^{G}(g)\cong\gamma^{G}((s\rhd_{2}g^{-1})^{-1})\gamma^{\Gamma}(g^{-1}\rhd_{1}s)\\}_{g\in
G,s\in\Gamma}$

such that

$\displaystyle(\sigma_{g,s})_{\lambda\mu}({}^{s}J^{\gamma^{G}(g)}_{\lambda,\mu})J^{\gamma^{\Gamma}(s)}_{{}^{\partial\mu\rhd_{2}^{G}g}\lambda,{}^{g}\mu}$
$\displaystyle={}^{(s\rhd_{2}g^{-1})^{-1}}J^{\gamma^{\Gamma}(g^{-1}\rhd_{1}s)}_{\lambda,\mu}J^{\gamma^{G}((s\rhd_{2}g^{-1})^{-1})}_{{}^{\partial\mu\rhd_{2}^{\Gamma}(g^{-1}\rhd_{1}s)}\lambda,{}^{g^{-1}\rhd_{1}s}\mu}((\sigma_{\partial\mu\rhd_{2}^{G}g,(g\rhd_{1}^{G}\partial\mu)\rhd_{2}^{\Gamma}s})_{\lambda}\otimes(\sigma_{g,s})_{\mu})$
$\displaystyle(\sigma_{g,s})_{\mathbf{1}_{\mathcal{C}}}({}^{s}\varphi^{\gamma^{G}(g)})\varphi^{\gamma^{\Gamma}(s)}$
$\displaystyle={}^{(s\rhd_{2}g^{-1})^{-1}}\varphi^{\gamma^{\Gamma}(g^{-1}\rhd_{1}s)}\varphi^{\gamma^{G}((s\rhd_{2}g^{-1})^{-1})}$

and

$\displaystyle\sigma_{g,ss^{\prime}}(\chi^{\gamma^{\Gamma}}_{s,s^{\prime}}\ast\mathrm{id}_{\gamma^{G}(g)})$
$\displaystyle=(\mathrm{id}\ast\chi^{\gamma^{\Gamma}}_{(s^{\prime}\rhd_{2}g^{-1})^{-1}\rhd_{1}s,g\rhd_{1}s^{\prime}})(\sigma_{(s^{\prime}\rhd_{2}g^{-1})^{-1},s}\ast\mathrm{id}_{\gamma^{\Gamma}(g^{-1}\rhd_{1}s^{\prime})})(\mathrm{id}_{\gamma^{\Gamma}(s)}\ast\sigma^{g,s^{\prime}})$
$\displaystyle\sigma_{gg^{\prime},s}(\mathrm{id}_{\gamma^{\Gamma}(s)}\ast\chi^{\gamma^{G}}_{g,g^{\prime}})$
$\displaystyle=(\chi^{\gamma^{G}}_{(s\rhd_{2}g^{-1})^{-1},((g^{-1}\rhd_{1}s)\rhd_{2}g^{\prime-1})^{-1}}\ast\mathrm{id}_{(g^{\prime}(s\rhd_{2}g^{-1}))\rhd_{1}s})$
$\displaystyle\qquad(\mathrm{id}_{\gamma^{G}((s\rhd_{2}g^{-1})^{-1})}\ast\sigma_{g^{\prime},g^{-1}\rhd_{1}s})(\sigma_{g,s}\ast\mathrm{id}_{\gamma^{G}(g^{\prime})})$
$\displaystyle\sigma_{g,e}(\iota^{\gamma^{\Gamma}}\ast\mathrm{id}_{\gamma^{G}(g)})$
$\displaystyle=\mathrm{id}_{\gamma^{G}(g)}\ast\iota^{\gamma^{\Gamma}}$
$\displaystyle\sigma_{e,s}(\mathrm{id}_{\gamma^{\Gamma}(s)}\ast\iota^{\gamma^{G}})$
$\displaystyle=\iota^{\gamma^{G}}\ast\mathrm{id}_{\gamma^{\Gamma}(s)}$

for all $g,g^{\prime}\in G$ and $s,s^{\prime}\in\Gamma$. Then, $\mathcal{C}$
becomes a $(G\bowtie\Gamma,G\times\Gamma)$-crossed monoidal category.

###### Proof.

Put $\gamma(g,s)\coloneqq\gamma^{G}(g)\gamma^{\Gamma}(s)$ for $(g,s)\in
G\bowtie\Gamma$. Put

$\displaystyle\chi^{\gamma}_{(g,s),(g^{\prime},s^{\prime})}\coloneqq(\chi^{\gamma^{G}}_{g,(s\rhd_{2}g^{\prime-1})^{-1}}\ast\chi^{\gamma^{\Gamma}}_{g^{\prime-1}\rhd_{1}s,s^{\prime}})(\mathrm{id}_{\gamma^{G}(g)}\ast\sigma^{g^{\prime},s}\ast\mathrm{id}_{\gamma^{\Gamma}(s^{\prime})})$

for $(g,s),(g^{\prime},s^{\prime})\in G\bowtie\Gamma$. Put
$\iota^{\gamma}\coloneqq\iota^{\gamma^{G}}\ast\iota^{\gamma^{\Gamma}}$. Then,
$\gamma$ is a $G\bowtie\Gamma$-action on the category $\mathcal{C}$ as in the
proof that the direct product of two algebras is again an algebra. Namely,
$\chi^{\gamma^{G}}$, $\chi^{\gamma^{\Gamma}}$, $\iota^{\gamma^{G}}$ and
$\iota^{\gamma^{\Gamma}}$ correspond to the products and units of the
algebras. It corresponds to associativity and unit property that $\gamma^{G}$
and $\gamma^{\Gamma}$ are group actions. The natural isomorphism
$\sigma^{g,s}$ corresponds to the swap between algebras and the last four
axioms for $\sigma$ correspond to the commutativity between the swap and the
algebra structures. Then, the axioms for $\chi^{\gamma}$ and $\iota^{\gamma}$
follow, respectively, as in the proof of the associativity and unit property
of the direct product algebra.

For $\lambda\in\operatorname{Obj}(\mathcal{C})$,
$\mu\in\operatorname{Homog}(\mathcal{C})$ and $(g,s)\in G\bowtie\Gamma$, put

$\displaystyle J^{\gamma(g,s)}_{\lambda,\mu}$
$\displaystyle\coloneqq{}^{g}J^{\gamma^{\Gamma}(s)}_{\lambda,\mu}J^{\gamma^{G}(g)}_{{}^{\partial\mu\rhd_{2}^{\Gamma}s}\lambda,{}^{s}\mu}:{}^{h\tilde{\rhd}_{2}(g,s)}\lambda{}^{(g,s)}\mu\cong{}^{(g,s)}(\lambda\mu)$
$\displaystyle\varphi^{\gamma(g,s)}$
$\displaystyle\coloneqq{}^{g}\varphi^{\gamma^{\Gamma}(s)}\varphi^{\gamma^{G}(g)}:\mathbf{1}_{\mathcal{C}}\cong{}^{(g,s)}\mathbf{1}_{\mathcal{C}}.$

Then, axiom 2 in Definition 2.5 follows as in the proof that the composition
of monoidal functors is again a monoidal functor. We check axiom 3. For
$\lambda\in\operatorname{Obj}(\mathcal{C})$,
$\mu\in\operatorname{Homog}(\mathcal{C})$ and
$(g,s),(g^{\prime},s^{\prime})\in G\bowtie\Gamma$,

$\displaystyle(\chi^{\gamma}_{(g,s),(g^{\prime},s^{\prime})})_{\lambda\mu}{}^{(g,s)}J^{\gamma(g^{\prime},s^{\prime})}_{\lambda,\mu}J^{\gamma(g,s)}_{{}^{\partial\mu\tilde{\rhd}_{2}(g^{\prime},s^{\prime})}\lambda,{}^{(g^{\prime},s^{\prime})}\mu}$
$\displaystyle={}^{g(s\rhd_{2}g^{\prime-1})^{-1}}(\chi^{\gamma^{\Gamma}}_{g^{\prime-1}\rhd_{1}s,s^{\prime}})_{\lambda\mu}(\chi^{\gamma^{G}}_{g,(s\rhd_{2}g^{\prime-1})^{-1}})_{{}^{g^{\prime-1}\rhd_{1}s}({}^{s^{\prime}}(\lambda\mu))}{}^{g}(\sigma_{g^{\prime},s})_{{}^{s^{\prime}}(\lambda\mu)}$
$\displaystyle\qquad{}^{g}({}^{s}({}^{g^{\prime}}J^{\gamma^{\Gamma}(s^{\prime})}_{\lambda,\mu}J^{\gamma^{G}(g^{\prime})}_{{}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda,{}^{s^{\prime}}\mu})){}^{g}J^{\gamma^{\Gamma}(s)}_{{}^{\partial\mu\tilde{\rhd}_{2}(g^{\prime},s^{\prime})}\lambda,{}^{(g^{\prime},s^{\prime})}\mu}J^{\gamma^{G}(g)}_{{}^{((g^{\prime},s^{\prime})\tilde{\rhd}_{1}\partial\mu)\rhd_{2}^{\Gamma}s}({}^{\partial\mu\tilde{\rhd}_{2}(g^{\prime},s^{\prime})}\lambda),{}^{s}({}^{(g^{\prime},s^{\prime})}\mu)}$
$\displaystyle={}^{g(s\rhd_{2}g^{\prime-1})^{-1}}(\chi^{\gamma^{\Gamma}}_{g^{\prime-1}\rhd_{1}s,s^{\prime}})_{\lambda\mu}(\chi^{\gamma^{G}}_{g,(s\rhd_{2}g^{\prime-1})^{-1}})_{{}^{g^{\prime-1}\rhd_{1}s}({}^{s^{\prime}}(\lambda\mu))}{}^{g}({}^{(s\rhd_{2}g^{\prime-1})^{-1}}({}^{g^{\prime-1}\rhd_{1}s}J^{\gamma^{\Gamma}(s^{\prime})}_{\lambda,\mu}))$
$\displaystyle\qquad{}^{g}((\sigma_{g^{\prime},s})_{{}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda{}^{s^{\prime}}\mu}{}^{s}J^{\gamma^{G}(g^{\prime})}_{{}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda,{}^{s^{\prime}}\mu}J^{\gamma^{\Gamma}(s)}_{{}^{\partial\mu\tilde{\rhd}_{2}(g^{\prime},s^{\prime})}\lambda,{}^{(g^{\prime},s^{\prime})}\mu})J^{\gamma^{G}(g)}_{{}^{((g^{\prime},s^{\prime})\tilde{\rhd}_{1}\partial\mu)\rhd_{2}^{\Gamma}s}({}^{\partial\mu\tilde{\rhd}_{2}(g^{\prime},s^{\prime})}\lambda),{}^{s}({}^{(g^{\prime},s^{\prime})}\mu)}$
$\displaystyle={}^{g(s\rhd_{2}g^{\prime-1})^{-1}}((\chi^{\gamma^{\Gamma}}_{g^{\prime-1}\rhd_{1}s,s^{\prime}})_{\lambda\mu}{}^{g^{\prime-1}\rhd_{1}s}J^{\gamma^{\Gamma}(s^{\prime})}_{\lambda,\mu})(\chi^{\gamma^{G}}_{g,(s\rhd_{2}g^{\prime-1})^{-1}})_{{}^{g^{\prime-1}\rhd_{1}s}({}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda{}^{s^{\prime}}\mu)}$
$\displaystyle\qquad{}^{g}({}^{s\rhd_{2}g^{\prime-1}}J^{\gamma^{\Gamma}(g^{\prime-1}\rhd_{1}s)}_{{}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda,{}^{s^{\prime}}\mu}J^{\gamma^{G}((s\rhd_{2}g^{\prime-1})^{-1})}_{{}^{(s^{\prime}\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{\Gamma}(g^{\prime-1}\rhd_{1}s)}({}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda),{}^{g^{\prime-1}\rhd_{1}s}({}^{s^{\prime}}\mu)})$
$\displaystyle\qquad{}^{g}((\sigma_{(s^{\prime}\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{G}g^{\prime},((g^{\prime},s^{\prime})\tilde{\rhd}_{1}\partial\mu)\rhd_{2}^{\Gamma}s})_{{}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda}\otimes(\sigma_{g^{\prime},s})_{{}^{s^{\prime}}\mu})J^{\gamma^{G}(g)}_{{}^{((g^{\prime},s^{\prime})\tilde{\rhd}_{1}\partial\mu)\rhd_{2}^{\Gamma}s}({}^{\partial\mu\tilde{\rhd}_{2}(g^{\prime},s^{\prime})}\lambda),{}^{s}({}^{(g^{\prime},s^{\prime})}\mu)}$
$\displaystyle={}^{g(s\rhd_{2}g^{\prime-1})^{-1}}((\chi^{\gamma^{\Gamma}}_{g^{\prime-1}\rhd_{1}s,s^{\prime}})_{\lambda\mu}{}^{g^{\prime-1}\rhd_{1}s}J^{\gamma^{\Gamma}(s^{\prime})}_{\lambda,\mu}J^{\gamma^{\Gamma}(g^{\prime-1}\rhd_{1}s)}_{{}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda,{}^{s^{\prime}}\mu})$
$\displaystyle\qquad(\chi^{\gamma^{G}}_{g,(s\rhd_{2}g^{\prime-1})^{-1}})_{{}^{(s^{\prime}\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{\Gamma}(g^{\prime-1}\rhd_{1}s)}({}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda){}^{g^{\prime-1}\rhd_{1}s}({}^{s^{\prime}}\mu)}{}^{g}J^{\gamma^{G}((s\rhd_{2}g^{\prime-1})^{-1})}_{{}^{(s^{\prime}\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{\Gamma}(g^{\prime-1}\rhd_{1}s)}({}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda),{}^{g^{\prime-1}\rhd_{1}s}({}^{s^{\prime}}\mu)}$
$\displaystyle\qquad{}^{g}((\sigma_{(s^{\prime}\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{G}g^{\prime},((g^{\prime},s^{\prime})\tilde{\rhd}_{1}\partial\mu)\rhd_{2}^{\Gamma}s})_{{}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda}\otimes(\sigma_{g^{\prime},s})_{{}^{s^{\prime}}\mu})J^{\gamma^{G}(g)}_{{}^{((g^{\prime},s^{\prime})\tilde{\rhd}_{1}\partial\mu)\rhd_{2}^{\Gamma}s}({}^{\partial\mu\tilde{\rhd}_{2}(g^{\prime},s^{\prime})}\lambda),{}^{s}({}^{(g^{\prime},s^{\prime})}\mu)}$

by the naturality of $\sigma_{g^{\prime},s}$ and
$\chi^{\gamma^{G}}_{g,(s\rhd_{2}g^{\prime-1})^{-1}}$ and the first axiom for
$\sigma$. Then, we obtain

$\displaystyle(\chi^{\gamma}_{(g,s),(g^{\prime},s^{\prime})})_{\lambda\mu}{}^{(g,s)}J^{\gamma(g^{\prime},s^{\prime})}_{\lambda,\mu}J^{\gamma(g,s)}_{{}^{\partial\mu\tilde{\rhd}_{2}(g^{\prime},s^{\prime})}\lambda,{}^{(g^{\prime},s^{\prime})}\mu}$
$\displaystyle={}^{g(s\rhd_{2}g^{\prime-1})^{-1}}((\chi^{\gamma^{\Gamma}}_{g^{\prime-1}\rhd_{1}s,s^{\prime}})_{\lambda\mu}{}^{g^{\prime-1}\rhd_{1}s}J^{\gamma^{\Gamma}(s^{\prime})}_{\lambda,\mu}J^{\gamma^{\Gamma}(g^{\prime-1}\rhd_{1}s)}_{{}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda,{}^{s^{\prime}}\mu})$
$\displaystyle\qquad(\chi^{\gamma^{G}}_{g,(s\rhd_{2}g^{\prime-1})^{-1}})_{{}^{(s^{\prime}\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{\Gamma}(g^{\prime-1}\rhd_{1}s)}({}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda){}^{g^{\prime-1}\rhd_{1}s}({}^{s^{\prime}}\mu)}{}^{g}J^{\gamma^{G}((s\rhd_{2}g^{\prime-1})^{-1})}_{{}^{(s^{\prime}\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{\Gamma}(g^{\prime-1}\rhd_{1}s)}({}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda),{}^{g^{\prime-1}\rhd_{1}s}({}^{s^{\prime}}\mu)}$
$\displaystyle\qquad
J^{\gamma^{G}(g)}_{{}^{(((g^{\prime-1}\rhd_{1}s)s^{\prime})\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{G}(s\rhd_{2}g^{\prime-1})^{-1}}({}^{(s^{\prime}\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{\Gamma}(g^{\prime-1}\rhd_{1}s)}({}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda)),{}^{(s\rhd_{2}g^{\prime-1})^{-1}}({}^{g^{\prime-1}\rhd_{1}s}({}^{s^{\prime}}\mu))}$
$\displaystyle\qquad({}^{(s\rhd_{1}^{\Gamma}((g^{\prime},s^{\prime})\tilde{\rhd}_{1}\partial\mu))\rhd_{2}^{G}g}(\sigma_{(s^{\prime}\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{G}g^{\prime},((g^{\prime},s^{\prime})\tilde{\rhd}_{1}\partial\mu)\rhd_{2}^{\Gamma}s})_{{}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda}\otimes{}^{g}(\sigma_{g^{\prime},s})_{{}^{s^{\prime}}\mu})$
$\displaystyle={}^{g(s\rhd_{2}g^{\prime-1})^{-1}}(J^{\gamma^{\Gamma}((g^{\prime-1}\rhd_{1}s)s^{\prime})}_{\lambda,\mu}((\chi^{\gamma^{\Gamma}}_{(s^{\prime}\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{\Gamma}(g^{\prime-1}\rhd_{1}s),\partial\mu\rhd_{2}^{\Gamma}s^{\prime}})_{\lambda}\otimes(\chi^{\gamma^{\Gamma}}_{g^{\prime-1}\rhd_{1}s,s^{\prime}})_{\mu}))$
$\displaystyle\qquad
J^{\gamma^{G}(g(s\rhd_{2}g^{\prime-1})^{-1})}_{{}^{(s^{\prime}\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{\Gamma}(g^{\prime-1}\rhd_{1}s)}({}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda),{}^{g^{\prime-1}\rhd_{1}s}({}^{s^{\prime}}\mu)}$
$\displaystyle\qquad((\chi^{\gamma^{G}}_{(s\rhd_{1}^{\Gamma}((g^{\prime},s^{\prime})\tilde{\rhd}_{1}\partial\mu))\rhd_{2}^{G}g,(((g^{\prime-1}\rhd_{1}s)s^{\prime})\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{G}(s\rhd_{2}g^{\prime-1})^{-1}})_{{}^{(s^{\prime}\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{\Gamma}(g^{\prime-1}\rhd_{1}s)}({}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda)}$
$\displaystyle\qquad\otimes(\chi^{\gamma^{G}}_{g,(s\rhd_{2}g^{\prime-1})^{-1}})_{{}^{g^{\prime-1}\rhd_{1}s}({}^{s^{\prime}}\mu)})$
$\displaystyle\qquad({}^{(s\rhd_{1}^{\Gamma}((g^{\prime},s^{\prime})\tilde{\rhd}_{1}\partial\mu))\rhd_{2}^{G}g}(\sigma_{(s^{\prime}\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{G}g^{\prime},((g^{\prime},s^{\prime})\tilde{\rhd}_{1}\partial\mu)\rhd_{2}^{\Gamma}s})_{{}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda}\otimes{}^{g}(\sigma_{g^{\prime},s})_{{}^{s^{\prime}}\mu})$
$\displaystyle={}^{g(s\rhd_{2}g^{\prime-1})^{-1}}J^{\gamma^{\Gamma}((g^{\prime-1}\rhd_{1}s)s^{\prime})}_{\lambda,\mu}J^{\gamma^{G}(g(s\rhd_{2}g^{\prime-1})^{-1})}_{{}^{(s^{\prime}\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{\Gamma}(g^{\prime-1}\rhd_{1}s)}({}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda),{}^{g^{\prime-1}\rhd_{1}s}({}^{s^{\prime}}\mu)}$
$\displaystyle\qquad({}^{(((g^{\prime-1}\rhd_{1}s)s^{\prime})\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{G}(g(s\rhd_{2}g^{\prime-1})^{-1})}(\chi^{\gamma^{\Gamma}}_{(s^{\prime}\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{\Gamma}(g^{\prime-1}\rhd_{1}s),\partial\mu\rhd_{2}^{\Gamma}s^{\prime}})_{\lambda}\otimes{}^{g(s\rhd_{2}g^{\prime-1})^{-1}}(\chi^{\gamma^{\Gamma}}_{g^{\prime-1}\rhd_{1}s,s^{\prime}})_{\mu})$
$\displaystyle\qquad((\chi^{\gamma^{G}}_{(s\rhd_{1}^{\Gamma}((g^{\prime},s^{\prime})\tilde{\rhd}_{1}\partial\mu))\rhd_{2}^{G}g,(((g^{\prime-1}\rhd_{1}s)s^{\prime})\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{G}(s\rhd_{2}g^{\prime-1})^{-1}})_{{}^{(s^{\prime}\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{\Gamma}(g^{\prime-1}\rhd_{1}s)}({}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda)}$
$\displaystyle\qquad\otimes(\chi^{\gamma^{G}}_{g,(s\rhd_{2}g^{\prime-1})^{-1}})_{{}^{g^{\prime-1}\rhd_{1}s}({}^{s^{\prime}}\mu)})$
$\displaystyle\qquad({}^{(s\rhd_{1}^{\Gamma}((g^{\prime},s^{\prime})\tilde{\rhd}_{1}\partial\mu))\rhd_{2}^{G}g}(\sigma_{(s^{\prime}\rhd_{1}^{\Gamma}\partial\mu)\rhd_{2}^{G}g^{\prime},((g^{\prime},s^{\prime})\tilde{\rhd}_{1}\partial\mu)\rhd_{2}^{\Gamma}s})_{{}^{\partial\mu\rhd_{2}^{\Gamma}s^{\prime}}\lambda}\otimes{}^{g}(\sigma_{g^{\prime},s})_{{}^{s^{\prime}}\mu})$
$\displaystyle=J^{\gamma(g(s\rhd_{2}g^{\prime-1})^{-1},(g^{\prime-1}\rhd_{1}s)s^{\prime})}_{\lambda,\mu}((\chi^{\gamma}_{((g^{\prime},s^{\prime})\tilde{\rhd}_{1}\partial\mu)\tilde{\rhd}_{2}(g,s),\partial\mu\tilde{\rhd}_{2}(g^{\prime},s^{\prime})})_{\lambda}\otimes(\chi^{\gamma}_{(g,s),(g^{\prime},s^{\prime})})_{\mu})$

by the naturality of $J^{\gamma^{G}}$’s and the first condition of axiom 3 in
Definition 2.5 for $\gamma^{G}$ and $\gamma^{\Gamma}$. Thus, $\gamma$
satisfies the first condition of axiom 3. Next, the second condition of axiom
3 follows since

$\displaystyle(\chi^{\gamma}_{(g,s),(g^{\prime},s^{\prime})})_{\mathbf{1}_{\mathcal{C}}}{}^{(g,s)}\varphi^{\gamma(g^{\prime},s^{\prime})}\varphi^{\gamma(g,s)}$
$\displaystyle={}^{g(s\rhd_{2}g^{\prime-1})^{-1}}(\chi^{\gamma^{\Gamma}}_{g^{\prime-1}\rhd_{1}s,s^{\prime}})_{\mathbf{1}_{\mathcal{C}}}(\chi^{\gamma^{G}}_{g,(s\rhd_{2}g^{\prime-1})^{-1}})_{{}^{g^{\prime-1}\rhd_{1}s}({}^{s^{\prime}}\mathbf{1}_{\mathcal{C}})}$
$\displaystyle\qquad{}^{g}(\sigma_{g^{\prime},s})_{{}^{s^{\prime}}\mathbf{1}_{\mathcal{C}}}{}^{g}({}^{s}({}^{g^{\prime}}\varphi^{\gamma^{\Gamma}(s^{\prime})}\varphi^{\gamma^{G}(g^{\prime})})){}^{g}\varphi^{\gamma^{\Gamma}(s)}\varphi^{\gamma^{G}(g)}$
$\displaystyle={}^{g(s\rhd_{2}g^{\prime-1})^{-1}}(\chi^{\gamma^{\Gamma}}_{g^{\prime-1}\rhd_{1}s,s^{\prime}})_{\mathbf{1}_{\mathcal{C}}}(\chi^{\gamma^{G}}_{g,(s\rhd_{2}g^{\prime-1})^{-1}})_{{}^{g^{\prime-1}\rhd_{1}s}({}^{s^{\prime}}\mathbf{1}_{\mathcal{C}})}{}^{g}({}^{(s\rhd_{2}g^{\prime-1})^{-1}}({}^{g^{\prime-1}\rhd_{1}s}\varphi^{\gamma^{\Gamma}(s^{\prime})}))$
$\displaystyle\qquad{}^{g}((\sigma_{g^{\prime},s})_{\mathbf{1}_{\mathcal{C}}}{}^{s}\varphi^{\gamma^{G}(g^{\prime})}\varphi^{\gamma^{\Gamma}(s)})\varphi^{\gamma^{G}(g)}$
$\displaystyle={}^{g(s\rhd_{2}g^{\prime-1})^{-1}}((\chi^{\gamma^{\Gamma}}_{g^{\prime-1}\rhd_{1}s,s^{\prime}})_{\mathbf{1}_{\mathcal{C}}}{}^{g^{\prime-1}\rhd_{1}s}\varphi^{\gamma^{\Gamma}(s^{\prime})})(\chi^{\gamma^{G}}_{g,(s\rhd_{2}g^{\prime-1})^{-1}})_{{}^{g^{\prime-1}\rhd_{1}s}\mathbf{1}_{\mathcal{C}}}$
$\displaystyle\qquad{}^{g}({}^{(s\rhd_{2}g^{\prime-1})^{-1}}\varphi^{\gamma^{\Gamma}(g^{\prime-1}\rhd_{1}s)}\varphi^{\gamma^{G}((s\rhd_{2}g^{\prime-1})^{-1})})\varphi^{\gamma^{G}(g)}$
$\displaystyle={}^{g(s\rhd_{2}g^{\prime-1})^{-1}}((\chi^{\gamma^{\Gamma}}_{g^{\prime-1}\rhd_{1}s,s^{\prime}})_{\mathbf{1}_{\mathcal{C}}}{}^{g^{\prime-1}\rhd_{1}s}\varphi^{\gamma^{\Gamma}(s^{\prime})}\varphi^{\gamma^{\Gamma}(g^{\prime-1}\rhd_{1}s)})$
$\displaystyle\qquad(\chi^{\gamma^{G}}_{g,(s\rhd_{2}g^{\prime-1})^{-1}})_{\mathbf{1}_{\mathcal{C}}}{}^{g}\varphi^{\gamma^{G}((s\rhd_{2}g^{\prime-1})^{-1})}\varphi^{\gamma^{G}(g)}$
$\displaystyle={}^{g(s\rhd_{2}g^{\prime-1})^{-1}}\varphi^{\gamma^{\Gamma}((g^{\prime-1}\rhd_{1}s)s^{\prime})}\varphi^{\gamma^{G}(g(s\rhd_{2}g^{\prime-1})^{-1})}=\varphi^{\gamma(g(s\rhd_{2}g^{\prime-1})^{-1},(g^{\prime-1}\rhd_{1}s)s^{\prime})}$

by the naturality of $\sigma_{g^{\prime},s}$ and
$\chi^{\gamma^{G}}_{g,(s\rhd_{2}g^{\prime-1})^{-1}}$, the second axiom for
$\sigma$ and the second condition of axiom 3 for $\gamma^{G}$ and
$\gamma^{\Gamma}$. Then, the third condition of axiom 3 for $\gamma$ follows
since

$\displaystyle\iota^{\gamma}_{\lambda\mu}$
$\displaystyle=\gamma^{G}(e)(\iota^{\gamma^{\Gamma}}_{\lambda\mu})\iota^{\gamma^{G}}_{\lambda\mu}=\gamma^{G}(e)(J^{\gamma^{\Gamma}(e)}_{\lambda,\mu}(\iota^{\gamma^{\Gamma}}_{\lambda}\otimes\iota^{\gamma^{\Gamma}}_{\mu}))J^{\gamma^{G}(e)}_{\lambda,\mu}(\iota^{\gamma^{G}}_{\lambda}\otimes\iota^{\gamma^{G}}_{\mu})$
$\displaystyle=\gamma^{G}(e)J^{\gamma^{\Gamma}(e)}_{\lambda,\mu}J^{\gamma^{G}(e)}_{\gamma^{\Gamma}(e)(\lambda),\gamma^{\Gamma}(e)(\mu)}(\gamma^{G}(e)(\iota^{\gamma^{\Gamma}}_{\lambda})\otimes\gamma^{G}(e)(\iota^{\gamma^{\Gamma}}_{\mu}))=J^{\gamma(e)}_{\lambda,\mu}(\iota^{\gamma}_{\lambda}\otimes\iota^{\gamma}_{\mu})$

by the third condition of axiom 3 for $\gamma^{G}$ and $\gamma^{\Gamma}$ and
the naturality of $J^{\gamma^{G}(e)}$. Finally, the fourth condition of axiom
3 for $\gamma$ follows since
$\iota^{\gamma}_{\mathbf{1}_{\mathcal{C}}}=\gamma^{G}(e)(\iota^{\gamma^{\Gamma}}_{\mathbf{1}_{\mathcal{C}}})\iota^{\gamma^{G}}_{\mathbf{1}_{\mathcal{C}}}=\gamma^{G}(e)(\varphi^{\gamma^{\Gamma}(e)})\varphi^{\gamma^{G}(e)}=\varphi^{\gamma(e)}$
by the fourth condition in axiom 3 for $\gamma^{G}$ and $\gamma^{\Gamma}$. ∎

The following theorem is the main result of this article.

######  Theorem 4.8.

The $(G,\Gamma)$-crossed center $\operatorname{Z}^{G}_{\Gamma}(\mathcal{C})$
of a non-singular $(G,\Gamma)$-crossed pivotal tensor category $\mathcal{C}$
becomes a $(G\bowtie\Gamma,G\times\Gamma)$-braided pivotal tensor category.

###### Proof.

Let $\gamma^{G}$ (resp. $\gamma^{\Gamma}$) denote the actions of $G$ (resp.
$\Gamma$) with crossed structures given by Lemma 4.5 (resp. Lemma 4.6). Let
$\lambda\in\operatorname{Obj}(\operatorname{Z}^{G}_{\Gamma}(\mathcal{C})_{(h,t)})$
and let $\mu\in\operatorname{Obj}(\mathcal{C}_{g^{-1}\rhd_{1}s})$ for
$(g,s)\in G\bowtie\Gamma$. Since
${}^{(s\rhd_{2}g^{-1})^{-1}}(\mu^{\vee})=({}^{g}\mu)^{\vee}$,
${}^{(hg^{-1}\rhd_{1}s)t\rhd_{2}g}(({}^{h}\mu)^{\vee})=({}^{(t\rhd_{2}g)h}\mu)^{\vee}$,
${}^{(hg^{-1}\rhd_{1}s)t\rhd_{2}g}\mathrm{ev}_{({}^{h}\mu)^{\vee}}=\mathrm{ev}_{({}^{(t\rhd_{2}g)h}\mu)^{\vee}}$
and ${}^{(s\rhd_{2}g^{-1})^{-1}}\mathrm{coev}_{\mu}=\mathrm{coev}_{{}^{g}\mu}$
by Lemma 4.3, we obtain
${}^{(s\rhd_{2}g^{-1})^{-1}}e_{\lambda}^{\mu}=e_{{}^{g}\lambda}^{{}^{g}\mu}$
from the definition of $h^{{}^{g}\lambda}$, where $e^{\mu}_{\lambda}$ denotes
the idempotent defined in the proof of Lemma 4.6. Since $\partial{}^{g}\mu=s$,
we have an isomorphism
$(\sigma_{g,s})_{\lambda}:{}^{s}({}^{g}\lambda)\cong{}^{(s\rhd_{2}g^{-1})^{-1}}({}^{g^{-1}\rhd_{1}s}\lambda)$,
which is natural in $\lambda$, by the proof of Lemma 4.6. These
$\sigma_{g,s}$’s satisfy the conditions in Lemma 4.7 by the same argument as
in the proof that $\gamma^{\Gamma}$ does not depend on choices of retracts.
Thus, $\operatorname{Z}^{G}_{\Gamma}(\mathcal{C})$ becomes a
$(G,\Gamma)$-crossed pivotal multiring category.

Then, we endow $\operatorname{Z}^{G}_{\Gamma}(\mathcal{C})$ with a
$(G,\Gamma)$-braiding. Let us follow the graphical notations of [TV13],
although we draw an arrow from top to bottom. Then, for
$\lambda,\mu\in\operatorname{Homog}(\operatorname{Z}^{G}_{\Gamma}(\mathcal{C}))$,
we define the isomorphism $b_{\lambda,\mu}$ by Figure 1, where
$\\{\zeta_{s}\\}_{s\in\Gamma}$ is a family of homogeneous objects with nonzero
dimensions.

${}^{\partial_{\Gamma}\mu}\lambda$$\mu$${}^{\partial_{G}\lambda}({}^{e}\mu)$$\lambda$$\zeta_{e}^{\vee}$${}^{\partial_{G}\lambda}\zeta_{\partial_{\Gamma}\mu}$
Figure 1: The $(G\bowtie\Gamma,G\times\Gamma)$-braiding on
$\operatorname{Z}^{G}_{\Gamma}(\mathcal{C})$

Then, by the Yang–Baxter equation and the definition of $\gamma^{\Gamma}$,
$b_{\lambda,\mu}$ is a morphism of
$\operatorname{Z}^{G}_{\Gamma}(\mathcal{C})$. Moreover, it is easy to see that
the morphism of Figure 2 is the inverse of $b_{\lambda,\mu}$.

${}^{\partial_{\Gamma}\mu}\lambda$$\mu$${}^{\partial_{G}\lambda}({}^{e}\mu)$$\lambda$$\zeta_{e}^{\vee}$$d_{\zeta_{e}}^{-1}d_{\zeta_{\partial_{\Gamma}\mu}}^{-1}$${}^{\partial_{G}\lambda}\zeta_{\partial_{\Gamma}\mu}$
Figure 2: $b^{-1}_{\lambda,\mu}$

We show the second axiom. The left-hand side of the axiom is equal to the
morphism in Figure 3 for
$\lambda_{1},\lambda_{2},\mu\in\operatorname{Homog}(\operatorname{Z}_{\Gamma}^{G}(\mathcal{C}))$
by the definition of $J^{\gamma}$. On the other hand, by the naturality of
$h^{\lambda_{1}}$, the right-hand side of the axiom is equal to the
composition of
$(\chi^{\gamma}_{(\partial_{G}\lambda_{1},e),(\partial_{G}\lambda_{2},e)})_{\mu}{}^{\partial_{G}\lambda_{1}}\iota^{\Gamma}_{{}^{\partial_{G}\lambda_{2}}({}^{e}\mu)}$
and the morphism in Figure 3.

${}^{\partial_{\Gamma}\mu}\lambda_{2}$${}^{\partial\lambda_{2}\tilde{\rhd}_{2}\partial_{\Gamma}\mu}\lambda_{1}$$\mu$$\lambda_{1}$$\lambda_{2}$${}^{\partial_{G}\lambda_{1}\partial_{G}\lambda_{2}}({}^{e}\mu)$
Figure 3: The second axiom for $b_{\lambda,\mu}$

Then, since

$\displaystyle(\chi^{\gamma}_{(\partial_{G}\lambda_{1},e),(\partial_{G}\lambda_{2},e)})_{\mu}{}^{\partial_{G}\lambda_{1}}\iota^{\Gamma}_{{}^{\partial_{G}\lambda_{2}}({}^{e}\mu)}$
$\displaystyle={}^{\partial_{G}\lambda_{1}}({}^{\partial_{G}\lambda_{2}}(\chi^{\Gamma}_{e,e})_{\mu}(\sigma_{\partial_{G}\lambda_{2},e})_{{}^{e}\mu}\iota^{\Gamma}_{{}^{\partial_{G}\lambda_{2}}({}^{e}\mu)})$
$\displaystyle={}^{\partial_{G}\lambda_{1}\partial_{G}\lambda_{2}}((\chi^{\Gamma}_{e,e})_{\mu}\iota^{\Gamma}_{{}^{e}\mu})=\mathrm{id}$

by an axiom for $\sigma$ and a coherence relation for $\gamma^{\Gamma}$, the
second axiom follows.

Similarly, the first axiom follows from the axiom

$\displaystyle\sigma_{g,ss^{\prime}}(\chi^{\gamma^{\Gamma}}_{s,s^{\prime}}\ast\mathrm{id}_{\gamma^{G}(g)})=(\mathrm{id}\ast\chi^{\gamma^{\Gamma}}_{(s^{\prime}\rhd_{2}g^{-1})^{-1}\rhd_{1}s,g\rhd_{1}s^{\prime}})(\sigma_{(s^{\prime}\rhd_{2}g^{-1})^{-1},s}\ast\mathrm{id}_{\gamma^{\Gamma}(g^{-1}\rhd_{1}s^{\prime})})(\mathrm{id}_{\gamma^{\Gamma}(s)}\ast\sigma^{g,s^{\prime}})$

and the third axiom follows from the axiom
$\sigma_{e,s}(\mathrm{id}_{\gamma^{\Gamma}(s)}\ast\iota^{\gamma^{G}})=\iota^{\gamma^{G}}\ast\mathrm{id}_{\gamma^{\Gamma}(s)}$.
∎

## Acknowledgements

The author thanks his supervisor Yasuyuki Kawahigashi for his constant support
and advice. He thanks Yuki Arano and Kan Kitamura for useful comments. This
work was supported by JSPS KAKENHI Grant Number JP23KJ0540. He thanks their
financial support.

## Declarations

Funding: This work was supported by JSPS KAKENHI Grant Number JP23KJ0540.

Conflicts of interest/Competing interests: The author has no conflicts of
interest to declare that are relevant to the content of this article.

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