---
abstract: 'A graded tensor category over a group $G$ will be called a crossed product tensor category if every homogeneous component has at least one multiplicatively invertible object.  Our main result is a description of the crossed product tensor categories, graded monoidal functors, monoidal natural transformations, and braidings in terms of coherent outer $G$-actions over tensor categories.' 
address: 'Departamento de MatemáticasPontificia Universidad Javeriana Bogotá, Colombia'
author:
- 'C'' esar Galindo'
title: Crossed product tensor categories
---

Introduction
============

A $G$-graded ring $A= \bigoplus_{\sigma\in G}A_\sigma$ is called a $G$-crossed product if each $A_\sigma$ has an invertible element.  Some important classes of rings like skew group-rings and twisted group-rings are special cases of crossed product rings.  One of the basic examples is the group algebra of a group $F$, it is graded by a quotient group of $F$, see [@CR Subsection 11C].  In this case the theory of representation of $F$ can be analyzed using Clifford theory, see [@CR Subsection 11A].  In analogy with graded rings, a $G$-graded tensor category (see Subsection \[defincion graded an crossed\]) ${\mathcal{C}}=\bigoplus_{\sigma\in G}{\mathcal{C}}_\sigma$ will be called $G$-crossed product tensor category, if there is an invertible object in each homogeneous component of ${\mathcal{C}}$. The graded tensor categories appear naturally in classification problems of fusion categories and finite tensor categories [@ENO], [@finite-categories].  One of the most interesting examples of $G$-crossed product tensor categories is the semi-direct product tensor category associated to an action of a group over a tensor category, see [@tambara].  Semi-direct tensor product category has been used in order to solve an important open problem in semisimple Hopf algebras theory [@nik].  In [@G] was proposed a Clifford theory categorification for crossed product tensor categories, in order to describe simple module categories in terms of subgroups and induced module categories.  We stress that in [@G] crossed product tensor categories were called strongly graded tensor categories.  Recently, $G$-graded fusion categories (that include crossed product fusion categories) were classified by Etingof, Nikshych, and Ostrik in [@ENO3], using invertible bimodule categories over fusion categories.  The crossed product rings are commonly described using crossed systems [@MoGR].  Crossed systems can be interpreted in terms of monoidal functors in the following way: if $A$ is a ring, let we denote by $\overline{Out(A)}$ the monoidal category (in fact, it is a categorical-group) of outer automorphisms, were the objects are automorphisms of $A$, and the arrows between automorphisms $\sigma$ and $\tau$ are invertible elements $a\in A$, such $a\sigma(x)= \tau(x)a$ for all $x\in A$.  Given a group $G$, the data that define $G$-crossed systems are the same as the data that define monoidal functors from $\underline{G}$ to $\overline{Out(A)}$, where $\underline{G}$ is the discrete monoidal category associated with $G$. We develop crossed product system theory for crossed product tensor categories using higher category theory.  If ${\mathcal{C}}$ is a tensor category, the bicategory $\text{Bieq}({\mathcal{C}})$ is a monoidal bicategory (in fact, it is a weak 3-group).  So, a $G$-crossed system or coherent outer $G$-action over ${\mathcal{C}}$ must be a trihomomorphism from $\underline{\underline{G}}$ (the discrete 3-category associated to $G$, see *Remark* \[3-group discreto\]) to $\text{Bieq}({\mathcal{C}})$. The main goal of this paper is to describe the 2-category of $G$-crossed product tensor categories in terms of coherent outer $G$-actions over a tensor category (Theorem \[main theorem\]), and describe the braidings of $G$-graded tensor categories.  The organization of the paper is as follows: in Section \[preliminares\] we recall the main definitions of bicategory theory, as well as the definitions of categorical-groups, graded tensor categories, and the monoidal structure over $\text{Bieq}({\mathcal{C}})$. In Section \[section action y correspondecia\] we define incoherent and coherent outer $G$-actions over a tensor category ${\mathcal{C}}$, and we show an explicit bijective correspondence between equivalence classes of $G$-crossed product tensor categories and coherent outer $G$-actions.  In Section \[section classif\] we show a biequivalence between the 2-category of crossed product tensor categories and the 2-category of coherent outer $G$-actions.  Finally, in Section \[seccion 6\] we give some necessary and sufficient conditions for the existence of a braiding over a crossed product tensor category. 

preliminaries {#preliminares}
=============

General conventions
-------------------

Throughout this article we work over a field $k$. By a tensor category $({\mathcal{C}},\otimes,\alpha, I )$ we understand a $k$-linear abelian category ${\mathcal{C}}$, endowed with a $k$-bilinear exact bifunctor $\otimes:{\mathcal{C}}\times {\mathcal{C}}\to {\mathcal{C}}$, an object $I\in {\mathcal{C}}$, and an associativity constraint $\alpha_{V,W,Z}: (V\otimes W)\otimes Z\to V\otimes (W\otimes Z)$, such that Mac Lane’s pentagon axiom holds [@BaKi], $V\otimes I= I\otimes V = V$, $\alpha_{V, I,W}= \text{id}_{V\otimes
W}$ for all $V, W\in {\mathcal{C}}$ and $\dim_k {\mbox{\rm End\,}}_{\mathcal{C}}(I)=1$. We shall considerer only monoidal categories which constraint of unit are identities.  So, without loss of generality, we shall suppose that for every monoidal functor $(F,\psi):{\mathcal{C}}\to {\mathcal{D}}$, we have $F(I_{\mathcal{C}})=I_{\mathcal{D}}$ and $\psi_{V,I}=\psi_{I,V}={\mathrm{id}}_V$, since each monoidal functor is monoidally equivalent to one with these properties. 

Bicategories {#seccion bicategorias}
------------

In this section we review some definitions on bicategory theory that we shall need later.  We refer the reader to [@Be] for a detailed exposition on the subject. 

A bicategory $\mathcal{B}$ consists of the following data

-   a set ${\text{Obj}}(\mathcal{B})$ (with elements $A,B,\ldots $ called 0-cells ),

-   for each pair $A, B \in {\text{Obj}}(\mathcal{B})$, a category $\mathcal{B}(A,B)$ (with objects $V, W, \ldots$ called 1-cells and morphisms $f, g, \ldots$ called 2-cells),

-   for each $A, B, C\in {\text{Obj}}(\mathcal{B})$, a bifunctor $$\overline{\circ}^{ABC}:\mathcal{B}(A,B)\times \mathcal{B}(B,C)\to \mathcal{B}(A,C),$$

-   for each 0-cell $A\in {\text{Obj}}(\mathcal{B})$, a 1-cell $I_A\in \mathcal{B}(A,A)$,

-   for each $A, B,C, D \in {\text{Obj}}(\mathcal{B})$, natural isomorphisms (constraint of associativity) $$\begin{aligned}
        \alpha^{A,B,C,D}:  - \overline{\circ}^{ABD} ( - \overline{\circ}^{BCD} - ) &\to ( - \overline{\circ}^{ABC} - ) \overline{\circ}^{ACD} -:\\
        \mathcal{B}(A,B)\times \mathcal{B}(B,C)\times \mathcal{B}(C,D)&\to \mathcal{B}(A,D). 
        \end{aligned}$$

Subject to the following axioms

-   coherence of the associativity: if $(S,T,U,V)$ is an object in $\mathcal{B}(A,B)\times
    \mathcal{B}(B,C)\times \mathcal{B}(C,D)\times \mathcal{B}(D,E)$, the next diagram commutes $$\begin{diagram}
    \node{}\node{S\overline{\circ}(T\overline{\circ}(U\overline{\circ} V))} \arrow{sw,l}{\alpha_{S,T,U\overline{\circ} V}}\arrow{se,l}{{\mathrm{id}}\overline{\circ} \alpha_{T,U,V}}\node{}\\
    \node{(S\overline{\circ} T)\overline{\circ} (U\overline{\circ} V)}\arrow{s,r}{\alpha_{S\overline{\circ} T,U,V}} \node{}\node{S\overline{\circ}((T\overline{\circ} U)\overline{\circ} V)} \arrow{s,r}{\alpha_{S,T\overline{\circ} U,V}}\\
    \node{((S\overline{\circ} T)\overline{\circ} U)\overline{\circ} V} \node{} \node{(S\overline{\circ} (T\overline{\circ} U))\overline{\circ} V} \arrow[2]{w,t}{\alpha_{S,T,U}\overline{\circ} {\mathrm{id}}}
    \end{diagram}$$

-   coherence of the unity $$\alpha_{S,I_B,T}={\mathrm{id}}_{S\overline{\circ} T}.$$

If $\alpha$ is the identity, we have $(S\overline{\circ} T)\overline{\circ} U = S\overline{\circ} (T\overline{\circ} U)$ and similarly for morphisms, in this case we shall say that $\mathcal{B}$ is a 2-category.  A monoidal category $({\mathcal{C}}, \otimes, \alpha, I)$ is the same as a bicategory with only one 0-cell, and in this case $\overline{\circ}=\otimes$.

Let $\mathcal{B}=(\overline{\circ}, I,\alpha)$ and $\mathcal{B}'=(\overline{\circ}, I',\alpha')$ be bicategories.  A pseudofunctor $\Phi=(F,\phi)$ from $\mathcal{B}$ to $\mathcal{B}'$ consists of the following data

-   a function $F: {\text{Obj}}(\mathcal{B})\to {\text{Obj}}(\mathcal{B}')$, $A\mapsto F(A)$,

-   for each pair $A,B\in {\text{Obj}}(\mathcal{B})$, functors $$F_{AB}:\mathcal{B}(A,B)\to \mathcal{B}'(F(A),F(B)),\   \ S\mapsto F(S), \  f\mapsto F(f),$$

-   for each triple $A, B, C  \in {\text{Obj}}(\mathcal{B})$, a natural isomorphism $$\psi^{ABC}:  F_{AC}(- \overline{\circ}^{ABC} -)\to F_{AB}(-)\overline{\circ}^{F(A)F(B)F(C)} F_{BC}(-),$$

Subject to the following axioms

-   $F_{AA}(I_A)= I'_{F(A)}$,

-   if $(S,T,U)$ is an object in $\mathcal{B}(A,B)\times \mathcal{B}(B,C)\times \mathcal{B}(C,D)$, the following diagram commutes (where the indexes have been omitted) $$\begin{diagram}
    \node{F(S\overline{\circ} (T\overline{\circ} U))}\arrow{s,l}{\psi}\arrow[3]{e,t}{F(\alpha)}\node[2]{}\node{F((S\overline{\circ} T)\overline{\circ} U )}\arrow{s,r}{ \psi}\\
    \node{F(S)\overline{\circ} F(T\overline{\circ} U)}\arrow{s,l}{{\mathrm{id}}\overline{\circ}\psi}\node[2]{}\node{F(S\overline{\circ} T)\overline{\circ} F(U)}\arrow{s,r}{\psi\overline{\circ} {\mathrm{id}}}\\
    \node{F(S)\overline{\circ} (F(T)\overline{\circ} F(U))}\arrow[3]{e,t}{\alpha'}\node[2]{}\node{(F(S)\overline{\circ} F(T))\overline{\circ} F(U)}\\
    \end{diagram}$$

-   if $S$ is an object in $\mathcal{B}(A,B)$, then $\psi_{S,I_B} ={\mathrm{id}}_{F(S)}$ and $\psi_{I_A,S}= F(S)$, for each pair of 0-cells $A, B \in {\text{Obj}}(\mathcal{B})$.

1.   The notion of pseudofunctor can be in some manner dualized by reversing the direction of the 2-cells $F_{A,B}$, this notion will be called op-pseudofunctor. 

2.   A pseudofunctor between monoidal categories is just a monoidal functor and an op-pseudofunctor is an op-monoidal functor. 

Let $F,G:\mathcal{B}_0\to \mathcal{B}_1$ be pseudofunctors between bicategories $\mathcal{B}_0$ and $\mathcal{B}_1$. A pseudonatural transformation $\sigma: F\to G$, consists of the following data

-   for each $A\in {\text{Obj}}(\mathcal{B}_0)$, 1-cells $\sigma_A\in \mathcal{B}_1(F(A),G(A))$,

-   for each pair $A, B\in {\text{Obj}}(\mathcal{B}_0)$, and each 1-cell $V\in \mathcal{B}_0(A,B)$ a natural isomorphism $$\sigma_V:F^{AB}(V)\overline{\circ}^{F(A)F(B)G(B)} \sigma_B \to \sigma_A \overline{\circ}^{F(A)G(A)G(B)} G^{AB}(V),$$

such that $\sigma_{I_A}={\mathrm{id}}_{I_A}$ for all $A\in {\text{Obj}}(\mathcal{B}_0)$ and for all $S\in \mathcal{B}_0(A,B)$, $T\in
\mathcal{B}_0(B,C)$, the following diagram $$\begin{diagram}
\node{ F(S T)\sigma}\arrow{s,l}{ \psi^F\overline{\circ} {\mathrm{id}}_\sigma}\arrow[2]{e,t}{\sigma_{ST}}\node[2]{\sigma G(S T) }\arrow{s,r}{{\mathrm{id}}_\sigma\overline{\circ} \psi^{G}}\\
\node{ F(S)F(T)\sigma}\arrow{e,t}{{\mathrm{id}}_{F(S)}\overline{\circ}\sigma_T} \node{F(S) \sigma G(T)}\arrow{e,t}{ \sigma_S\overline{\circ} {\mathrm{id}}_{G(T)}}\node{\sigma G(S) G(T)}
\end{diagram}$$ commutes (where associativity constraint, indexes, and the symbols $\overline{\circ}$ between objects have been omitted as a space-saving measure). 

Again, the notion of pseudonatural transformation can be dualized by reversing the order of the natural isomorphisms $\sigma_V$, this notion will be called op-pseudonatural transformation. 

Pseudonatural transformations may be composed in the obvious way.  If $\sigma:F\to G$, and $\tau: G\to H$ are pseudonatural transformations, then we define a new pseudonatural transformation $\sigma\overline{\circ}\tau:F\to H$ by $(\sigma\overline{\circ}\tau)_A=\sigma_A\overline{\circ}\tau_A$, and $(\sigma\overline{\circ}\tau)_V$ is defined by the commutativity of the diagram:

$$\begin{diagram}
\node{F(V)\overline{\circ}(\sigma_A\overline{\circ}\tau_A)} \arrow{s,l}{\alpha_{F(V),\sigma_A,\tau_A}} \arrow[3]{e,t}{(\sigma\overline{\circ}\tau)_V}\node[2]{} \node{(\sigma_B\overline{\circ}\tau_B)\overline{\circ} H(V)}\\
\node{(F(V)\overline{\circ}\sigma_A)\overline{\circ}\tau_A} \arrow{s,l}{\sigma_V\overline{\circ}{\mathrm{id}}_{\tau_A}}\node[2]{} \node{\sigma_B\overline{\circ}(\tau_B\overline{\circ} H(V))}\arrow{n,r}{\alpha_{\sigma_B,\tau_B,H(V)}}\\
\node{(\sigma_B\overline{\circ} G(V))\overline{\circ}\tau_A}\arrow[3]{e,t}{\alpha^{-1}_{\sigma_B,G(V),\tau_A}} \node[2]{}\node{\sigma_B\overline{\circ}(G(V)\overline{\circ}\tau_A)} \arrow{n,r}{{\mathrm{id}}_{\sigma_B}\overline{\circ}\tau_V}\\
\end{diagram}$$ where the index have been omitted. 

A modification between two pseudonatural transformations $\Gamma:\sigma\to \widetilde{\sigma} $, consists of 2-cells $\Gamma_A: \sigma_A \to \widetilde{\sigma}_A$ in $\mathcal{B}_1(F(A),G(A))$, such that for all 1-cell $V\in \mathcal{B}_0(A,B)$ the diagram $$\begin{diagram}
\node{F^{AB}(V)\overline{\circ} \sigma_B}\arrow{s,l}{ \text{id}\overline{\circ} \Gamma_B}\arrow{e,t}{\sigma_V}\node{\sigma_A \overline{\circ} G^{AB}(V)} \arrow{s,r}{ \Gamma_A\overline{\circ}{\mathrm{id}}}\\
\node{ F^{AB}(V)\overline{\circ} \widetilde{\sigma}_B}\arrow{e,t}{\widetilde{\sigma}_V}\node{\widetilde{\sigma}_A\overline{\circ} G^{AB}(V)}
\end{diagram}$$commutes. 

The Monoidal bicategory Bieq(${\mathcal{C}}$) of a tensor category {#Bieq(C)}
------------------------------------------------------------------

Given a pair of bicategories ${\mathcal{B}}$ and ${\mathcal{B}}'$, we can define the “functor bicategory" $[{\mathcal{B}},{\mathcal{B}}']$, whose 0-cells are pseudofunctors ${\mathcal{B}}\to {\mathcal{B}}'$, whose 1-cells are pseudonatural equivalence, and whose 2-cells are invertible modifications.  The bicategory $[{\mathcal{B}},{\mathcal{B}}']$ is not usually a 2-category, because composition of 1-cells in $[{\mathcal{B}},{\mathcal{B}}']$ involves composition of 1-cells in ${\mathcal{B}}'$, but in the case that ${\mathcal{B}}'$ is a 2-category, $[{\mathcal{B}},{\mathcal{B}}']$ is a 2-category.  When ${\mathcal{B}}={\mathcal{B}}'$, the bicategory $[{\mathcal{B}},{\mathcal{B}}]$ will be denoted by $\text{Bieq}({\mathcal{B}})$, and it has a monoidal structure in the sense of [@GPS].  Now, we shall describe the monoidal bicategory $\text{Bieq}({\mathcal{C}})$ associated to a tensor category $({\mathcal{C}},\otimes,I)$. The tensor product $\overline{\otimes}$ of monoidal endofunctors is defined by the composition of monoidal functors.  If $(\theta',\theta'_{(-)}): K\to K'$, $(\theta,\theta_{(-)}):H\to H'$ are pseudonatural transformations, the tensor product $(\theta',\theta'_{(-)})\overline{\otimes} (\theta,\theta_{(-)}): KH\to K'H'$ is defined as $(\theta',\theta'_{(-)})\overline{\otimes} (\theta,\theta_{(-)}):=(K(\theta)\otimes \theta',\theta'\overline{\otimes}\theta)$, where $\theta'\overline{\otimes}\theta$ is given by the commutativity of the following diagram $$\begin{diagram}
\node{ KH(V)\otimes K(\theta)\otimes \theta'} \arrow[2]{e,t}{(\theta\overline{\otimes}\theta')_V}\node[2]{K(\theta)\otimes \theta' \otimes K'H'(V)}\\ \node{K(H(V)\otimes \theta)\otimes \theta'}\arrow{n,l}{\psi^K_{H(V),\theta}\otimes{\mathrm{id}}_{\theta'}}  \arrow{e,t}{K(\theta_V)\otimes{\mathrm{id}}_{\theta'}}\node{K(\theta\otimes H'(V))\otimes \theta' } \arrow{e,t}{\psi^K_{\theta,H'(V)}\otimes{\mathrm{id}}_{\theta'}} \node{K(\theta)\otimes K(H'(V))\otimes \theta'} \arrow{n,r}{{\mathrm{id}}_{K(\theta)}\otimes \theta'_{H'(V)}}
\end{diagram}$$The tensor product of modifications $g$ and $f$ is defined as $g\overline{\otimes} f := K(f)\otimes g$. If $\chi_1=(\theta,\theta_{(-)}):F\to F'$ and $\chi_2=(\theta',\theta'_{(-)}): H\to H'$ are pseudonatural transformations, where $F,F',H,H':{\mathcal{C}}\to {\mathcal{C}}$ are monoidal functors, then the comparison constraint $$\begin{diagram}
\node{}\node{F'\circ H}\arrow{se,l}{{\mathrm{id}}_{F'}\overline{\otimes}\chi_2}\node{}\\
\node{F\circ H}\arrow{ne,l}{\chi_1\overline{\otimes} {\mathrm{id}}_H}\arrow{se,r}{{\mathrm{id}}_F\overline{\otimes}\chi_2}\node{\Downarrow c_{\chi_1,\chi_2}}\node{F'\circ H'}\\
\node{}\node{F\circ H'}\arrow{ne,r}{\chi_1\overline{\otimes}{\mathrm{id}}_H'}
\end{diagram}$$is given by $$c_{\chi_1,\chi_2}:=\theta_{\theta'}^{-1}: \theta\otimes F'(\theta')\to F(\theta')\otimes \theta.$$ The constraint of associativity $a_{f,g,h}:(f\otimes g)\otimes h\to f\otimes (g\otimes h)$ of the tensor product of pseudonatural transformations $f:K\to K'$, $g:H\to H'$, $h:G\to G'$ is given by the modification $$\psi^K_{H(h),g}\otimes {\mathrm{id}}_f:KH(h)\otimes K(g)\otimes f\to K(H(h)\otimes g)\otimes f,$$ and it is easy to see that $a$ satisfies the pentagonal identity. 

The data (TD6), (TD7), and (TD8) of [@GPS], in the monoidal bicategory $\text{Bieq}({\mathcal{C}})$ are trivial, since we only considerer monoidal functor $(F,\psi):{\mathcal{C}}\to {\mathcal{C}}$ such that $F(I)=I$ and $\psi_{V,I}=\psi_{I,V}={\mathrm{id}}_V$, for all $V\in {\mathcal{C}}$. The category $\text{Bieq}({\mathcal{C}})({\mathrm{id}}_{\mathcal{C}},{\mathrm{id}}_{\mathcal{C}})$ is exactly the center of ${\mathcal{C}}$, *i.e.*, the braided monoidal category $\mathcal{Z}({\mathcal{C}})$, see [@Kassel pag.  330]. 

Categorical-groups
------------------

A categorical-group $\mathcal{G}$ is a monoidal category where every object, and every arrow is invertible, *i.e.* for all $X\in {\text{Obj}}(\mathcal{G})$ there is $X^*\in {\text{Obj}}(\mathcal{G})$, such that $X\otimes X^*\cong X^*\otimes X\cong I$. We refer the reader to [@Baez] for a detailed exposition on the subject. 

A trivial example of a categorical-group is the discrete categorical-group $\underline{G}$, associated to a group $G$. The objects of $\underline{G}$ are the elements of $G$, the arrows are only the identities, and the tensor product is the multiplication of $G$. A more interesting examples is the following. 

Let $G$ be a group, $A$ a $G$-module, and $\omega\in Z^3(G,A)$ a normalized 3-cocycle.  We shall define the category ${\mathcal{C}}(G,A,\omega)$ by:

1.   ${\text{Obj}}({\mathcal{C}}(G,A, \omega))=G$,

2.   ${\text{Hom}}_{{\mathcal{C}}(G,A, \omega)}(g,h)=\left\{
                         \begin{array}{ll}
                           A, & \hbox{if $g=h$} \\
                           \emptyset, & \hbox{if $g\neq h$.}
                         \end{array}
                       \right.$

We define a monoidal structure in ${\mathcal{C}}(G,A,\omega)$ as follows:

Let $g\in {\mbox{\rm End\,}}(a)$ and $h\in {\mbox{\rm End\,}}(b)$, $a, b \in A$, $g, h \in
G$. Thus, $a\otimes b= a+ \  ^gb$ and $g\otimes h = gh$. We define the associator as $\Phi_{g,h,k}= \omega(g,h,k)$.

The 3-cocycle condition is equivalent to the pentagon axiom, and the condition of normality implies that $e$ is the unit object for this category. 

Complete invariants of a categorical-group $\mathcal{G}$ with respect to monoidal equivalences are $$\pi_0(\mathcal{G}), \pi_1(\mathcal{G}), \phi(\mathcal{G})\in H^3(\pi_0(\mathcal{G}),\pi_1(\mathcal{G})),$$ where $\pi_0(\mathcal{G})$ is the group of isomorphism classes of objects, $\pi_1(\mathcal{G})$ is the abelian group of automorphisms of the unit object.  The group $\pi_1(\mathcal{G})$ is a $\pi_0(\mathcal{G})$-module in the natural way, and $\phi(\mathcal{G})$ is a third cohomology class given by the associator, see [@Baez Subsection 8.3] for details on how to obtain $\phi(\mathcal{G})$. If $\mathcal{G}$ is a categorical group by [@Baez Theorem 43] there is an equivalence of monoidal categories between $\mathcal{G}$ and ${\mathcal{C}}(\pi_0(\mathcal{G}),\pi_1(\mathcal{G}),\phi)$, where $\phi$ is a 3-cocycle in the class $\phi(\mathcal{G})$. Also, it is easy to see that there is a bijective correspondence between monoidal functors $$F : {\mathcal{C}}(G,A,\omega) \to {\mathcal{C}}(G',A',\omega')$$ and triples $(\pi_0(F),\pi_1(F),\theta(F))$ that consist of:

-   a group morphism $\pi_0(F) : G \to G'$,

-   a $G$-module morphism $\pi_1(F) \colon A \to A'$,

-   a normalized 2-cochain $k(F) \colon G^2 \to A'$, such that $dk(F) = \pi_1(F) \omega - \omega' \pi_0(F)^3$.

For monoidal functors $F,F' : {\mathcal{C}}(G,A,\omega) \to {\mathcal{C}}(G',A',\omega')$, there is bijective correspondence between monoidal natural isomorphisms $\theta \colon F \to F'$ and normalized 1-cochains $p(\theta) \colon G \to A'$, where $dp(\theta) = k(F) - k(F')$.

The next result follows from the last discussion or from [@Baez Theorem 43]. 

\[prop cat-groups\] Let $\mathcal{G}$ be a categorical group and let $f:G\to \pi_0(\mathcal{G})$ be a morphism of groups.  Then there is a monoidal functor $F:\underline{G}\to \mathcal{G}$, such that $f=\pi_0(F)$ if and only if the cohomology class of $\phi f^3$ is zero, where $\phi$ is a 3-cocycle in the class $\phi(\mathcal{G})$.

If $\phi f^3$ is zero, the classes of equivalence of monoidal functors $F:\underline{G}\to \mathcal{G}$ such that $\pi_0(F)=f$ are in one to one correspondence with $H^2(G,\pi_1(\mathcal(\mathcal{G})))$.

Crossed product tensor categories {#defincion graded an crossed}
---------------------------------

Let $G$ be a group and let ${\mathcal{C}}$ be a tensor category.  We shall say that ${\mathcal{C}}$ is $G$-graded, if there is a decomposition $${\mathcal{C}}=\oplus_{x\in G}{\mathcal{C}}_x$$ into a direct sum of full abelian subcategories, such that for all $\sigma, x\in G$, the bifunctor $\otimes$ maps ${\mathcal{C}}_\sigma\times {\mathcal{C}}_x$ to ${\mathcal{C}}_{\sigma
x}$, see [@ENO]. 

Let ${\mathcal{C}}$ be a tensor category graded over a group $G$. We shall say that ${\mathcal{C}}=\bigoplus_{\sigma \in G}{\mathcal{C}}_\sigma$ is a crossed product tensor category over $G$, if every ${\mathcal{C}}_\sigma$ has a multiplicatively invertible object. 

Given a group $G$, we define the 2-category of $G$-crossed product tensor categories.  The 0-cells are crossed product tensor categories over $G$, 1-cells are graded op-monoidal functors, *i.e.*, op-monoidal functors $F: {\mathcal{C}}\to {\mathcal{D}}$ such that $F$ maps ${\mathcal{C}}_\sigma$ to ${\mathcal{D}}_\sigma$ for all $\sigma \in G$, and 2-cells are monoidal natural transformations between the graded op-monoidal functors.  The composition of 1-cells and 2-cells is the obvious. 

The existence of some extra properties of a crossed product tensor category ${\mathcal{C}}$, can be deduced from the tensor subcategory ${\mathcal{C}}_e$. For example ${\mathcal{C}}$ is semisimple or rigid if and only if ${\mathcal{C}}_e$ is semisimple or rigid.  However, if ${\mathcal{C}}_e$ is a braided tensor category, not necessary ${\mathcal{C}}$ is braided, see Section \[seccion 6\].  A crossed product tensor category ${\mathcal{C}}$ is a fusion category [@ENO] or finite tensor category [@finite-categories], if and only if $G$ is finite, and ${\mathcal{C}}_e$ is a fusion category or a finite tensor category, respectively. 

Outer $G$-actions over tensor categories {#section action y correspondecia}
========================================

Incoherent outer $G$-actions
----------------------------

Let ${\mathcal{C}}$ be a tensor category.  We define the categorical-group $\underline{2Out_\otimes({\mathcal{C}})}$, where objects are monoidal autoequivalences of ${\mathcal{C}}$, and arrows are equivalence classes of invertible pseudonatural isomorphisms up to invertible modifications.  The composition of arrows in $\underline{2Out_\otimes({\mathcal{C}})}$ is the equivalence class of pseudonatural isomorphisms composition, and the tensor product is the composition of monoidal functors and pseudonatural transformations. 

Let $G$ be a group and let ${\mathcal{C}}$ be a monoidal category.  An incoherent outer $G$-action over ${\mathcal{C}}$, is an *op-monoidal functor* $*:\underline{G}\to \underline{2Out_\otimes({\mathcal{C}})}$. Two incoherent outer $G$-actions are equivalent if the associated monoidal functors are monoidally equivalent. 

We shall analyze the incoherent outer $G$-action using the Subsection \[categorical-groups\].  Complete invariants for the categorical group $\underline{2Out_\otimes({\mathcal{C}})}$ are $\pi_0(\underline{2Out_\otimes({\mathcal{C}})})$ the equivalences classes of monoidal functor under invertible modification, $\pi_1(\underline{2Out_\otimes({\mathcal{C}})})=Inv(\mathcal{Z}({\mathcal{C}}))$ the abelian group of isomorphisms classes of invertible objects of the center of ${\mathcal{C}}$, and a third cohomology class $\phi(\underline{2Out_\otimes({\mathcal{C}})}) \in H^3(\pi_0(\underline{2Out_\otimes({\mathcal{C}})}),Inv(\mathcal{Z}({\mathcal{C}})))$.

Every incoherent outer $G$-action over a tensor category induces a group morphism $f : G \to \pi_0(\underline{2Out_\otimes({\mathcal{C}})})$. We shall say that a group morphism $f : G \to \pi_0(\underline{2Out_\otimes({\mathcal{C}})})$ is realizable if there is some incoherent outer $G$-action such that the induced group morphism coincides with $f$.

Let $G$ be a group and let $f:G\to \pi_0(\underline{Out_\otimes({\mathcal{C}})})$ be a group morphism.  Then there is an incoherent outer $G$-action over ${\mathcal{C}}$ that realize the morphism $f$ if and only if the cohomology class of $\phi f^3$ is zero, where $\phi$ is some 3-cocycle in the class of $\phi(\underline{2Out_\otimes({\mathcal{C}})})$.

If $\phi f^3$ is zero, the classes of equivalence of monoidal functors $F:\underline{G}\to \underline{2Out_\otimes({\mathcal{C}})}$ such that $\pi_0(F)=f$ are in one to one correspondence with $H^2(G, Inv(\mathcal{Z}({\mathcal{C}})))$.

See Proposition \[prop cat-groups\]. 

Coherent outer $G$-actions
--------------------------

Let ${\mathcal{C}}$ be a monoidal category and let $F: G\to \underline{2Out_\otimes({\mathcal{C}})}$ be an incoherent outer $G$-action.  We define a crossed system associated to $F$ as the following data

-   monoidal functors $(\sigma_*,\psi^{\sigma_*}): {\mathcal{C}}\to {\mathcal{C}}$ for all $\sigma \in G$,

-   pseudonatural isomorphisms $(U_{\sigma,\tau},\chi_{\sigma,\tau}): \sigma_*\circ \tau_*\to (\sigma\tau)_*$ for all $\sigma,\tau  \in G$,

-   invertible modifications $\omega_{\sigma,\tau,\rho}: \chi_{\sigma,\tau\rho}\overline{\circ} ({\mathrm{id}}_{\sigma_*} \overline {\otimes}\chi_{\tau,\rho}) \to \chi_{\sigma\tau,\rho}\overline{\circ} (\chi_{\sigma,\tau}\overline{\otimes}{\mathrm{id}}_{\rho_*})$ for all $\sigma,\tau,\rho \in G$.

that realize the incoherent outer $G$-action $F$ (recall that the symbol $\overline{\circ}$ is the composition of 1-cells in the bicategory Bieq$({\mathcal{C}})$). 

1.   By abuse of notation, we write $\chi_{\sigma,\tau}$ instead of pseudonatural transformation $(U_{\sigma,\tau},\chi_{\sigma,\tau})$, when no confusion can arise. 

2.   By the definition of $\underline{2Out_\otimes({\mathcal{C}})}$, there are several crossed systems that realize an incoherent $G$-action. 

3.   For every crossed system, without loss of generality, we can and shall assume that

    -   $e_*= {\mathrm{id}}_{\mathcal{C}}$  the monoidal identity functor,

    -   $\chi_{e,\sigma}=\chi_{\sigma,e}=(I,{\mathrm{id}}_{\sigma_*})$ the identity pseudonatural isomorphism,

    -   $ \omega_{\sigma,e,\tau}={\mathrm{id}}_{\chi_{\sigma,\tau}}$ the identity modification,

    for all $\sigma,\tau \in G$.

Let ${\mathcal{C}}$ be a tensor category and let $F: \underline{G}\to \underline{2Out_\otimes({\mathcal{C}})}$ be an incoherent outer $G$-action.  A coherent outer $G$-action associated to $F$, is a crossed system $(\{ \widetilde{\sigma}\}_{\sigma\in G},\chi,\omega)$ associated to $F$, such that for the pseudonatural isomorphisms $\chi_{\sigma,\tau}$ and the invertible modifications $\omega_{\sigma,\tau,\rho}$, the diagram

$$\label{coherencia}
\begin{diagram}
\node{\sigma_*(\tau_*(U_{\rho,\mu})U_{\tau,\rho\mu})U_{\sigma,\tau\rho\mu}} \arrow{s,r}{\psi^{\sigma_*}_{\tau_*(U_{\rho,\mu}),U_{\tau,\rho\mu}}{\mathrm{id}}_{U_{\sigma,\tau\rho\mu}}} \arrow{e,t}{\sigma_*(\omega_{\tau,\rho,\mu}){\mathrm{id}}_{U_{\sigma,\tau\rho\mu}}} \node{\sigma_*(U_{\tau,\rho}U_{\tau\rho,\mu})U_{\sigma,\tau\rho\mu}} \arrow{e,t}{\psi^{\sigma_*}_{U_{\tau,\rho},U_{\tau\rho,\mu}}{\mathrm{id}}_{U_{\sigma,\tau\rho\mu}}} \node{\sigma_*(U_{\sigma,\tau})\sigma_*(U_{\tau\rho,\mu})U_{\sigma,\tau\rho\mu}} \arrow{s,r}{{\mathrm{id}}_{\sigma_*(U_{\sigma,\tau})}\omega_{\sigma,\tau\rho,\mu}}\\
\node{\sigma_*(\tau_*(U_{\sigma,\mu}))\sigma_*(U_{\tau,\rho\mu})U_{\sigma,\tau\rho\mu}} \arrow{s,r}{{\mathrm{id}}_{\sigma_*(\tau_*(U_{\sigma,\tau}))}\omega_{\sigma,\tau,\rho\mu}} \node{}\node{\sigma_*(U_{\tau,\rho})U_{\sigma,\tau\rho}U_{\sigma\tau\rho,\mu}} \arrow{s,r}{\omega_{\sigma,\tau,\rho}{\mathrm{id}}_{U_{\sigma\tau\rho,\mu}}}\\
\node{\sigma_*(\tau_*(U_{\rho,\mu}))U_{\sigma,\tau}U_{\sigma\tau,\rho\mu}} \arrow{e,t}{\chi_{\sigma,\tau}(U_{\rho,\mu}){\mathrm{id}}_{U_{\sigma\tau,\rho\mu}}} \node{U_{\sigma,\tau}(\sigma\tau)_*(U_{\rho,\mu})U_{\sigma\tau,\rho\mu}} \arrow{e,t}{{\mathrm{id}}_{U_{\sigma,\tau}}\omega_{\sigma\tau,\rho,\mu}} \node{U_{\sigma,\tau}U_{\sigma\tau,\rho}U_{\sigma\tau\rho,\mu}}
\end{diagram}$$

commutes for all $\sigma,\tau,\rho,\mu \in G$ (where tensor symbols among objects and arrows have been omitted as a space-saving measure). 

\[3-group discreto\] For every group $G$, we can associate a discrete 3-category $\underline{\underline{G}}$, where objects are elements of $G$, and

$$\underline{\underline{G}}(g,h)=\left\{
                       \begin{array}{ll}
                         \{*\}, & \hbox{if $g=h$} \\
                         \emptyset, & \hbox{if $g\neq h$.}
                       \end{array}
                     \right.$$ $$\begin{gathered}
\overline{\otimes}: \underline{\underline{G}}\times \underline{\underline{G}}\to \underline{\underline{G}},\\
\underline{\otimes}_{g,h}:\underline{\underline{G}}(\sigma,\sigma)\times \underline{\underline{G}}(\tau,\tau) \to \underline{\underline{G}}(\sigma\tau,\sigma\tau).\end{gathered}$$ The definition of a coherence outer $G$-action over ${\mathcal{C}}$, is equivalent to the definition of a trihomorphism from $\underline{\underline{G}}$ to $\text{Bieq}({\mathcal{C}})$ (see [@GPS] for the definition of trihomomorphism). 

Given a crossed system associated to an incoherent outer $G$-action, we can define a monoidal bicategory.  In order to describe the monoidal bicategory in a simple way we can suppose, without loss of generality that ${\mathcal{C}}$ is skeletal, so for every pair $\sigma,\tau \in G$ there is only one pseudonatural transformation $\chi_{\sigma,\tau}^{-1}:(\sigma\tau)_*\to \sigma_*\tau_*$, such that $\chi_{\sigma,\tau}^{-1} \circ \chi_{\sigma,\tau }= {\mathrm{id}}_{\sigma_*\tau_*}$, and $\chi_{\sigma,\tau} \circ\chi_{\sigma,\tau}^{-1} = {\mathrm{id}}_{(\sigma\tau)_*}$ for all $\sigma, \tau\in G$. Let $<G>\subseteq \text{Bieq}({\mathcal{C}})$ be the full sub-bicategory where objects are $\{\sigma_*\}_{\sigma\in G}$. We define a homomorphism of bicategories $\otimes_G: <G> \times <G> \to <G>$ by $\sigma_*\otimes_G \tau_* = (\sigma\tau)_*$, and the commutativity of the diagram $$\begin{diagram}
\node{\sigma_*\tau_*} \arrow{s,l}{\chi_{\sigma,\tau}} \arrow{e,t}{f\overline{\otimes} g}\node{\sigma_*\tau_*}\arrow{s,r}{\chi_{\sigma,\tau}}\\
\node{(\sigma\tau)_*}\arrow{e,t}{f\otimes_G g}\node{(\sigma\tau)_*}
\end{diagram}$$ where $f\in \text{Bieq}(\sigma_*,\sigma_*), g\in \text{Bieq}({\mathcal{C}})(\tau_*,\tau_*)$.

We define a pseudonatural equivalence in the bicategory $[<G>\times <G>\times <G>, <G>]$ by the commutativity of the diagram $$\begin{diagram}
\node{(\sigma\tau\rho)_*}\arrow[2]{e,t}{a_{\sigma,\tau,\rho}}\node{} \node{(\sigma\tau\rho)_*}\\
\node{\sigma_*\circ(\tau\rho)_*}\arrow{n,l}{\chi_{\sigma,\tau\rho}} \node{}\node{(\sigma\tau)_*\rho}\arrow{n,r}{\chi_{\sigma\tau,\rho}}\\
\node{}  \node{\sigma_*\tau_*\rho_*} \arrow{nw,r}{{\mathrm{id}}_\sigma\overline{\otimes}\chi_{\tau,\rho}} \arrow{ne,r}{\chi_{\sigma,\tau}\overline{\otimes}{\mathrm{id}}_\rho}
\end{diagram}$$ The diagram define a modification $$\pi: a_{\sigma\tau,\rho,\mu}\circ a_{\sigma,\tau,\rho\mu} \to a_{\sigma,\tau,\rho}\otimes_G {\mathrm{id}}_\mu \circ a_{\sigma,\tau\rho,\mu}\circ {\mathrm{id}}_{\sigma}\otimes_G a_{\tau,\rho,\mu},$$ in the bicategory $[<G>\times <G>\times <G>\times <G>, <G>]$.

Since ${\mathcal{C}}$ is skeletal for every invertible object $U\in {\mathcal{C}}$, we can identify ${\text{Aut}}_{\mathcal{C}}(U)$ with ${\text{Aut}}(I)=k^*$, so the modification $\omega$ is identified by a map $$\omega: G\times G\times G\to k^*.$$ The modification $\pi$ defines a map $$\pi:G\times G\times G\times G\to k^*,$$ given by $$\pi(\sigma,\tau,\rho,\mu)= \delta(\omega)(\sigma,\tau,\rho,\mu)\psi^{\sigma_*}_{\tau_*(U_{\rho,\mu}),U_{\tau,\rho\mu}}\chi_{\sigma,\tau}(U_{\rho,\mu}) (\psi^{\sigma_*}_{U_{\tau,\rho},U_{\tau\rho,\mu}})^{-1}.$$ It is straightforward to see that $\pi$ is a 4-cocycle, see [@ENO3 Subsection 8.4].  It is also possible to see the 4-cocycle condition directly for the nonabelian 4-cocycle condition [@GPS (TA1)] in the monoidal bicategory $<G>$. It is clear that if the chosen modification is changed, the 4-cocycle $\pi$ only change for a 4-coboundary, so an incoherent outer $G$-action defines a fourth cohomology class. 

Let ${\mathcal{C}}$ be a tensor category.  An incoherent outer $G$-action over ${\mathcal{C}}$ is coherent if and only if the associated fourth cohomology class is trivial. 

If an outer $G$-action is coherent, the diagram commutes, so the map $\pi$ is trivial.  Conversely, if there is a 3-coboundary $\lambda: G\times G\times G\to G$, such that $\delta(\lambda)= \pi$, then the modification defined by the map $\lambda^{-1}\omega$ defines a coherent outer $G$-action. 

The crossed product tensor category associated to a coherent outer $G$-action {#from action to crossed}
-----------------------------------------------------------------------------

If a group $G$ acts over a monoidal category ${\mathcal{C}}$, we shall define a $G$-crossed product tensor category associated to this action, denoted as ${\mathcal{C}}\rtimes G$. We set ${\mathcal{C}}\rtimes G=\bigoplus_{\sigma}{\mathcal{C}}_\sigma$ as an abelian category, where ${\mathcal{C}}_\sigma ={\mathcal{C}}$. We shall denote by $[V,\sigma]$ the object $V\in {\mathcal{C}}_{\sigma}$, and a morphism from $\bigoplus_{\sigma\in G}[V_\sigma,\sigma]$ to $\bigoplus_{\sigma\in G}[W_\sigma,\sigma]$ is expressed as $\bigoplus_{\sigma\in G}[f_\sigma,\sigma]$ with $f_\sigma: V_\sigma\to W_\sigma$ a morphism in ${\mathcal{C}}$. The tensor product $\cdot:{\mathcal{C}}\rtimes G\times {\mathcal{C}}\rtimes G\to {\mathcal{C}}\rtimes G$ is defined by $$\begin{aligned}
    [V,\sigma]\cdot [W,\tau] &:= [V\otimes\sigma_*(W)\otimes U_{\sigma,\tau},\sigma\tau] \  \  \text{ for objects, and}\\
    [f,\sigma]\cdot [g,\tau] &:= [f\otimes \sigma_*(g)\otimes {\mathrm{id}}_{U_{\sigma,\tau}},\sigma\tau]\  \  \text{ for morphisms.}\end{aligned}$$

It is easy to see that the unit object is $(I,e)$. The associativity is given by

$$\begin{diagram}
\node{[V,\sigma]\cdot([W,\tau]\cdot[Z,\rho])} \arrow{e,t,=}{}\arrow{s,l}{\alpha_{[V,\sigma],[W,\tau],[Z,\rho]}} \node{[V\otimes\sigma_*(W\otimes\tau_*(Z)\otimes U_{\tau,\rho})\otimes U_{\sigma,\tau\rho},\sigma\tau\rho]} \arrow{s,r}{[\alpha^{{\mathcal{C}}\rtimes G}(V,\sigma,W,\tau,Z,\rho),\sigma\tau\rho]}\\
\node{([V,\sigma]\cdot[W,\tau])\cdot[Z,\rho]}\arrow{e,=}{} \node{[V\otimes\sigma_*(W)\otimes U_{\sigma,\tau}\otimes(\sigma\tau)_*(Z)\otimes U_{\sigma\tau,\rho},\sigma\tau\rho]}
\end{diagram}$$

where $\alpha^{{\mathcal{C}}\rtimes G}(V,\sigma,W,\tau,Z,\rho)$ is the composition $$\begin{diagram}
\node{V\otimes\sigma_*(W\otimes\tau_*(Z)\otimes U_{\tau,\rho})\otimes U_{\sigma,\tau\rho}}\arrow{s,l} {{\mathrm{id}}_V\otimes\psi^{\sigma_*}_{W,\tau_*(Z)\otimes U_{\tau,\sigma}}\otimes {\mathrm{id}}_{U_{\sigma,\tau\rho}}}\\
\node{V\otimes\sigma_*(W)\otimes\sigma_*(\tau_*(Z)\otimes U_{\tau,\rho})\otimes U_{\sigma,\tau\rho}}  \arrow{s,l}{{\mathrm{id}}_{V\otimes\sigma_*(W)}\otimes \psi^{\sigma_*}_{\tau_*(Z),U_{\tau,\rho}}\otimes {\mathrm{id}}_{U_{\sigma,\tau\rho}}}\\
\node{V\otimes\sigma_*(W)\otimes\sigma_*(\tau_*(Z))\otimes \sigma_*(U_{\tau,\rho})\otimes U_{\sigma,\tau\rho}} \arrow{s,l}{{\mathrm{id}}_{{V\otimes\sigma_*(W)\otimes\sigma_*(\tau_*(Z))}}\otimes \omega_{\sigma,\tau,\rho}}\\
\node{V\otimes\sigma_*(W)\otimes\sigma_*(\tau_*(Z))\otimes U_{\sigma,\tau}\otimes U_{\sigma\tau,\rho}} \arrow{s,l}{{\mathrm{id}}_{V\otimes \sigma_*(W)}\otimes\chi_{\sigma,\tau}\otimes{\mathrm{id}}_{U_{\sigma\tau,\rho}}}\\
\node{V\otimes\sigma_*(W)\otimes U_{\sigma,\tau}\otimes (\sigma\tau)_*(Z)\otimes U_{\sigma\tau,\rho}}
\end{diagram}$$ The associativity constraint have been omitted as a space-saving measure.  As we shall see, the coherence condition over an outer $G$-action, is exactly the pentagonal identity for ${\mathcal{C}}\rtimes G$.

### Pentagonal identity for ${\mathcal{C}}\rtimes G$

For a category ${\mathcal{D}}$ with a bifunctor $\otimes:{\mathcal{D}}\times{\mathcal{D}}\to {\mathcal{D}}$ and natural isomorphisms $\alpha_{A,B,C}: A\otimes(B\otimes C)\to (A\otimes B)\otimes C$, we shall denote by $P(A,B,C,D)$ the following pentagonal diagram $$\begin{diagram}
\node{}\node{A\otimes(B\otimes(C\otimes D))} \arrow{sw,l}{\alpha_{A,B,C\otimes D}}\arrow{se,l}{{\mathrm{id}}\otimes \alpha_{B,C,D}}\node{}\\
\node{(A\otimes B)\otimes (C\otimes D)}\arrow{s,l}{\alpha_{A\otimes B,C,D}} \node{}\node{A\otimes((B\otimes C)\otimes D)} \arrow{s,r}{\alpha_{A,B\otimes C,D}}\\
\node{((A\otimes B)\otimes C)\otimes D} \node{} \node{(A\otimes (B\otimes C))\otimes D} \arrow[2]{w,t}{\alpha_{A,B,C}\otimes {\mathrm{id}}}
\end{diagram}$$

\[obs notacion\] ***From now on, we shall denote $[V]:= [V,e]$ and $[\sigma]:= [I,\sigma]$, for all $V\in {\mathcal{C}}, \sigma \in G$. Analogously, $[f]:= [f,e]: [V]\to [W]$ for all arrow $f:V\to W$ in ${\mathcal{C}}$. Note that $[V]\cdot[\sigma]= [V,\sigma]$ and $[\sigma]\cdot[V]=[\sigma_*(V),\sigma]$ for all $V\in {\mathcal{C}}, \sigma \in G$.***

In order to prove the coherence of ${\mathcal{C}}\rtimes G$, is sufficient to see the pentagonal identity for the $[V], [\sigma]$, $V\in {\mathcal{C}}, \sigma \in G$, since every object in ${\mathcal{C}}\rtimes G$ is a direct sum of tensor products of $[V], [\sigma]$. First, see the next equality

$$\label{igualdad asociatividad}
\alpha_{[V,\sigma],[W,\tau],[Z,\rho]}={\mathrm{id}}_{[V]}\cdot\alpha_{[I,\sigma],[W,\tau],[Z,\rho]},$$

so $\alpha_{[V,e],[W,\tau],[Z,\rho]}={\mathrm{id}}$. The eight pentagonal identities of the form $P([V],\ldots)$ follows from .  The pentagon $P([\sigma],[V],[W],[\tau])$ commutes because $$\alpha_{[\sigma],[W],[Z,\rho]}=\alpha_{[\sigma],[W],[Z]}\cdot{\mathrm{id}}_{[\rho]}.$$ The pentagons $P([\sigma],[V],[\tau],[W]), P([\sigma],[\tau],[\rho])$, $P([\sigma],[\tau],[V],[\rho])$ commute by the definition of $\alpha$. The following table explains the commutativity of the other pentagons

                Pentagons                                    Pentagonal identity equivalence
  -------------------------------------- -----------------------------------------------------------------------
        $P([\sigma],[V],[W],[Z])$                  $(\sigma_*,\psi^{\sigma_*})$ is a monoidal functor
       $P([\sigma],[\tau],[V],[W])$       $(\chi_{\sigma,\tau},U_{\sigma,\tau})$ is a pseudonatural equivalence
     $P([\sigma],[\tau],[\rho],[V])$                  $\omega_{\sigma,\tau,\rho}$ is a modification
   $P([\sigma],[\tau],[\rho],[\gamma])$                      commutativity of the diagram . 

The coherent outer $G$-action associated to a $G$-crossed product tensor category
---------------------------------------------------------------------------------

Let ${\mathcal{C}}$ be a $G$-crossed product tensor category.  In order to show more clearly the associated coherent outer $G$-action, we shall make some reductions.  Let we choose a family $\{N_\sigma\}_{\sigma\in G}$ of homogeneous invertible objects, where $N_e=I$. The family $\{N_\sigma\}_{\sigma\in G}$ defines the equivalences of categories $$\begin{aligned}
    N_\sigma\otimes (-): {\mathcal{C}}_e &\to {\mathcal{C}}_\sigma\\
     V &\mapsto V\otimes N_\sigma\\
     f&\mapsto f\otimes {\mathrm{id}}_{N_\sigma}.\end{aligned}$$ Using these equivalences, we have an equivalence of categories $${\mathcal{C}}=\bigoplus_{\sigma\in G}{\mathcal{C}}_\sigma\to \bigoplus_{\sigma\in G}{\mathcal{C}}_e^\sigma,$$ where ${\mathcal{C}}_e^\sigma={\mathcal{C}}_e$, for all $\sigma\in G$. Now, we can transport the monoidal structure of ${\mathcal{C}}$ to $\bigoplus_{\sigma\in G}{\mathcal{C}}_e$. Then, without loss of generality we can suppose that the graded tensor category ${\mathcal{C}}=\bigoplus_{\sigma\in G}{\mathcal{C}}_\sigma$ has the following properties:

-   ${\mathcal{C}}_\sigma={\mathcal{C}}_e$ for all $\sigma \in G$ (so we can and will use the same notations of the Remark \[obs notacion\]),

-   the objects $[\sigma]\in {\mathcal{C}}_\sigma$ are invertible for all $\sigma\in G$,

-   $[V]\cdot[W,\sigma]=[V\otimes W,\sigma]$, for all $V, W \in {\mathcal{C}}_e$, $\sigma\in G$.

For each pair $\sigma, \tau \in G$, we have that $[\sigma]\cdot [\tau] \in {\mathcal{C}}_{\sigma\tau}$, so there is an unique invertible object $U_{\sigma\tau}\in {\mathcal{C}}_e$, such that $[\sigma]\cdot [\tau]=[U_{\sigma,\tau},\sigma\tau]$. Analogously, the objects $[\sigma]$ define functors $\sigma_*:{\mathcal{C}}_e\to {\mathcal{C}}_e, V\mapsto \sigma_*(V)$ by the rule $[\sigma]\cdot [V]=[\sigma_*(V),\sigma]$ for all $V\in {\mathcal{C}}_e$, and ${\mathrm{id}}_{[\sigma]}\cdot [f] =[\sigma_*(f),\sigma]$, for all arrow $f$ in ${\mathcal{C}}_e$.

\[lema producto\] If the category $({\mathcal{C}}_e,\otimes, I)$ is skeletal, then $$[V,\sigma]\cdot[W,\tau]=[V\otimes\sigma_*(W)\otimes U_{\sigma,\tau},\sigma\tau]$$ for all $V, W\in {\mathcal{C}}_e, \   \sigma, \tau \in G$.

Since ${\mathcal{C}}_e$ is skeletal, the category ${\mathcal{C}}=\bigoplus_{\sigma}{\mathcal{C}}_\sigma$ is skeletal.  Then we do not need to parenthesize tensor products for objects in ${\mathcal{C}}$. Also, recall that $[\sigma]\cdot[V]=[\sigma_*(V),\sigma]=[\sigma_*(V)]\cdot[\sigma]$, for all $V\in {\mathcal{C}}_e$, $\sigma \in G$.

Hence, $$\begin{aligned}
    [V,\sigma]\cdot[W,\tau] &= [V]\cdot[\sigma]\cdot[W]\cdot[\tau]\\
    &=[V]\cdot[\sigma_*(W)]\cdot[\sigma]\cdot[\tau] \\
    &=[W\otimes \sigma_*(W)]\cdot[U_{\sigma,\tau},\sigma\tau]\\
    &=[W\otimes \sigma_*(W)]\cdot[U_{\sigma,\tau}]\cdot[\sigma\tau]\\
    &=[W\otimes \sigma_*(W)\otimes U_{\sigma,\tau}]\cdot[\sigma\tau]\\
    &=[W\otimes \sigma_*(W)\otimes U_{\sigma,\tau},\sigma\tau]\\\end{aligned}$$for all $V, W \in {\mathcal{C}}_e$, $\sigma \in G$.

Under this reduction and using the Lemma \[lema producto\] we can describe the coherent outer $G$-action as the reciprocal construction of the Subsection \[from action to crossed\].  Suppose that ${\mathcal{C}}_e$ is skeletal, then the data that define the coherent outer $G$-action associated to ${\mathcal{C}}$ are the following:

-   monoidal equivalences: $(\sigma_*,\psi^{\sigma_*}): {\mathcal{C}}_e\to {\mathcal{C}}_e$, where $$[\psi^{\sigma_*}_{W,Z},\sigma]:=\alpha_{[\sigma],[W],[Z]}:[\sigma_*(W\otimes Z),\sigma]\to [\sigma_*(W)\otimes \sigma_*(Z),\sigma],$$

-   pseudonatural transformations: $ (U_{\sigma,\tau},\chi_{\sigma,\tau}):\sigma_*\circ \tau_*\to (\sigma\tau)_*$, where $$[\chi_{\sigma,\tau}(Z),\sigma\tau]:=\alpha_{[\sigma],[\tau],[Z]}: [\sigma_*(\tau_*(Z))\otimes U_{\sigma,\tau},\sigma\tau] \to [U_{\sigma,\tau}\otimes(\sigma\tau)_*(Z),\sigma\tau],$$

-   modifications $\omega_{\sigma,\tau\rho}:  \chi_{\sigma,\tau\rho}\overline{\circ} ({\mathrm{id}}_{\sigma_*} \overline {\otimes}\chi_{\tau,\rho}) \to \chi_{\sigma\tau,\rho}\overline{\circ} (\chi_{\sigma,\tau}\overline{\otimes}{\mathrm{id}}_{\rho_*})$, where $$[\omega_{\sigma,\tau,\rho},\sigma\tau\rho]:= \alpha_{[\sigma],[\tau],[\rho]}:[\sigma_*(U_{\tau,\rho})\otimes U_{\sigma,\tau\rho},\sigma\tau\rho] \to [U_{\sigma,\tau}\otimes U_{\sigma\tau,\rho},\sigma\tau\rho].$$

Classification of crossed product tensor category {#section classif}
=================================================

In this section, we shall see that the 2-category of crossed product tensor category over a fixed group $G$, is equivalent to the 2-category of all coherent outer $G$-actions.  The 0-cells of the 2-category of coherent outer $G$-actions are coherent outer $G$-action over a tensor category.  Let $(\{ \widetilde{\sigma}\}_{\sigma\in G},\chi,\omega)$ and $(\{\widehat{\sigma}\}_{\sigma \in G},\chi',\omega')$ be coherent outer $G$-actions over tensor categories ${\mathcal{C}}$ and ${\mathcal{D}}$, respectively.  An arrow from $(\{ \widetilde{\sigma}\}_{\sigma\in G},\chi,\omega)$ to $(\{\widehat{\sigma}\}_{\sigma \in G},\chi',\omega')$, is a triple $(H,\theta,\Pi)$, where $(H,\psi^H):{\mathcal{C}}\to {\mathcal{D}}$ is an op-monoidal functor, $(\theta^\sigma,\theta_\sigma): \widehat{\sigma}\circ H\to H\circ \widetilde{\sigma}$ is a pseudonatural equivalence for each $\sigma\in G$, and $\Pi$ is a modification

$$\begin{diagram}
\node{}\node{\widehat{\sigma} H\widetilde{\tau}}\arrow{se,t}{\theta_\sigma\overline{\otimes}id_{\widetilde{\tau}} } \node{}\\
\node{\widehat{\sigma}\widehat{\tau}H}\arrow{ne,t}{{\mathrm{id}}_{\widehat{\sigma}}\overline{\otimes}\theta_\tau} \arrow{s,t}{\chi_{\sigma,\tau}'\overline{\otimes}{\mathrm{id}}_H}\node{\Uparrow\Pi_{\sigma,\tau}} \node{H \widetilde{\sigma}\widetilde{\tau}}\arrow{s,r}{{\mathrm{id}}_H\overline{\otimes} \chi_{\sigma,\tau}}\\
\node{\widehat{\sigma\tau}H}\arrow[2]{e,t}{\theta_{\sigma\tau}}\node{} \node{H\widetilde{\sigma\tau}}
\end{diagram}$$ such that: $(\theta^e, \theta_e)=(I,{\mathrm{id}})$, $\Pi_{\sigma,e}=\Pi_{e,\sigma}={\mathrm{id}}_{\theta_\sigma}$ for all $\sigma \in G$, and the diagram

$$\begin{diagram}
\node{\theta_\sigma\widehat{\sigma}(\theta_\tau\widehat{\tau}(\theta_\rho)U_{\tau,\rho}')U_{\sigma,\tau\rho}'} \arrow{s,l}{{\mathrm{id}}_{\theta_\sigma}\otimes\widehat{\sigma}(\Pi_{\tau,\rho})\otimes{\mathrm{id}}_{U_{\sigma,\tau\rho}'}} \arrow{e,t}{\alpha^{{\mathcal{D}}\rtimes G}(\theta_\sigma,\sigma,\theta_\tau,\tau,\theta_\rho,\rho)} \node{\theta_\sigma\widehat{\sigma}(\theta_\tau)U_{\sigma,\tau}'\widehat{\sigma\tau}(\theta_\rho)U_{\sigma\tau,\rho}'} \arrow{s,r}{\Pi_{\sigma,\tau}\otimes{\mathrm{id}}_{\widehat{\sigma\tau}(\theta_\rho)U_{\sigma\tau,\rho}'}}\\
\node{\theta_\sigma\widehat{\sigma}(H(U_{\tau,\rho})\theta_{\tau\rho})U_{\sigma,\tau\rho}'} \arrow{s,l}{{\mathrm{id}}_{\theta_\sigma}\otimes\psi^{\widehat{\sigma}}_{H(U_{\tau,\rho}),\theta_{\tau\rho}}\otimes{\mathrm{id}}_{U_{\sigma,\tau\rho}'}} \node{H(U_{\sigma,\tau})\theta_{\sigma\tau}\widehat{\sigma\tau}(\theta_\rho)U_{\sigma\tau,\rho}'} \arrow{s,r}{{\mathrm{id}}_{H(U_{\sigma,\tau})}\otimes\Pi_{\sigma\tau,\rho}}\\
\node{\theta_\sigma\widehat{\sigma}(H(U_{\tau,\rho}))\widehat{\sigma}(\theta_{\tau\rho})U_{\sigma,\tau\rho}'} \arrow{s,l}{\theta^\sigma_{U_{\tau,\rho}}\otimes{\mathrm{id}}_{\widehat{\sigma}(\theta_{\tau\rho})U_{\sigma,\tau\rho}'}} \node{H(U_{\sigma,\tau})H(U_{\sigma\tau,\rho})\theta_{\sigma\tau\rho}} \arrow{s,r}{\psi^H_{U_{\sigma,\tau},U_{\sigma\tau,\rho}}\otimes{\mathrm{id}}_{\theta_{\sigma\tau\rho}}}\\
\node{H(\widetilde{\sigma}(U_{\tau,\rho}))\theta_\sigma\widehat{\sigma}(\theta_{\tau\rho})U_{\sigma,\tau\rho}'} \arrow{s,l}{{\mathrm{id}}_{H(\widetilde{\sigma}(U_{\tau,\rho}))}\otimes\Pi_{\sigma,\tau\rho}} \node{H(U_{\sigma,\tau}U_{\sigma\tau,\rho})\theta_{\sigma\tau\rho}}\\
\node{H(\widetilde{\sigma}(U_{\tau,\rho}))\otimes H(U_{\sigma,\tau\rho})\theta_{\sigma\tau\rho}} \arrow{e,t}{\psi^H_{\widetilde{\sigma}(U_{\tau,\rho}),U_{\sigma,\tau\rho}}} \node{H(\sigma_*(U_{\tau,\rho})U_{\sigma,\tau\rho})\theta_{\sigma\tau\rho}} \arrow{n,r}{H(\omega_{\sigma,\tau,\rho})}
\end{diagram}$$commutes for all $\sigma,\tau, \rho \in G$ (where tensor symbols among objects have been omitted as a space-saving measure).  A 2-cell from $(H,\theta,\Pi)$ to $(\widetilde{H},\widetilde{\theta},\widetilde{\Pi})$ consist of the data $\{m_\sigma, m\}_{\sigma \in G}$, where $m: H\to \widetilde{H}$ is a monoidal natural transformation and $m_\sigma:\theta_\sigma\to \widetilde{\theta}_\sigma$ are morphisms in ${\mathcal{C}}$. The previous data are subject to the following axioms: $m_e={\mathrm{id}}_I$ and the diagrams $$\begin{diagram}
\node{\theta_\sigma\widehat{\sigma}(\theta_\tau)U_{\sigma,\tau}'}\arrow{s,l}{\Pi_{\sigma,\tau}} \arrow{e,t}{m_\sigma\otimes \widehat{\sigma}(m_\tau)\otimes{\mathrm{id}}_{U'_{\sigma,\tau}}} \node{\widetilde{\theta}_\sigma\widehat{\sigma}(\widetilde{\theta}_\tau)U_{\sigma,\tau}'} \arrow{s,r}{\widetilde{\Pi}_{\sigma,\tau}} \\
\node{H(U_{\sigma,\tau})\theta_{\sigma\tau}}\arrow{e,t}{m_{U_{\sigma,\tau}}\otimes m_{\sigma\tau}} \node{\widetilde{H}(U_{\sigma,\tau})\widetilde{\theta}_{\sigma\tau}}
\end{diagram}
\begin{diagram}
\node{\theta_\sigma\widehat{\sigma}(H(V))} \arrow{s,l}{\theta^\sigma_V}\arrow{e,t}{m_\sigma\otimes\widehat{\sigma}(m_V)} \node{\widetilde{\theta}_\sigma\widehat{\sigma}(\widetilde{H}(V))} \arrow{s,r}{\widetilde{\theta}^\sigma_V}\\
\node{H(\widetilde{\sigma}(V))\theta_\sigma} \arrow{e,t}{m_{\widetilde{\sigma}(V)}\otimes m_\sigma} \node{\widetilde{H}(\widetilde{\sigma}(V))\widetilde{\theta}_\sigma}
\end{diagram}$$commute for all $\sigma,\tau \in G, V\in {\mathcal{C}}$ (where tensor symbols among objects have been omitted). 

\[main theorem\] There is a biequivalence between the 2-category of coherent outer $G$-actions and the 2-category of $G$-crossed product tensor categories. 

The bijective correspondence between $G$-crossed product tensor categories and coherent outer $G$-action was described in the Section \[section action y correspondecia\]. 

If $T=(H,\theta,\Pi)$ is a 1-cell between coherent outer $G$-action $(\{ \widetilde{\sigma}\}_{\sigma\in G}\chi,\omega)$ and $(\{\widehat{\sigma}\}_{\sigma \in G},\chi',\omega')$ over ${\mathcal{C}}$ and ${\mathcal{D}}$ respectively, then we define an op-monoidal functor $(T,\psi^T): {\mathcal{C}}\rtimes G \to {\mathcal{D}}\rtimes G$ as

-   $T([V,\sigma])= [H(V)\otimes \theta_\sigma,\sigma], T([f,\sigma])=[H(f)\otimes {\mathrm{id}}_{\theta_\sigma},\sigma]$ for all $V\in {\mathcal{C}}, \sigma\in G$,

-   $\psi^T:T([V,\sigma])\cdot T([W,\sigma])\to T([V,\sigma]\cdot[W,\tau]),$

where $$\begin{diagram}
\node{T([V,\sigma])\cdot T([W,\sigma])} \arrow[4]{s,l}{\psi^T_{[V,\sigma],[W,\tau]}} \arrow{e,t,=}{}
\node{H(V)\theta_\sigma\widehat{\sigma}(H(W)\theta_\tau)U'_{\sigma,\tau}} \arrow{s,r}{{\mathrm{id}}_{H(V)\theta_\sigma}\otimes\psi^{\widehat{\sigma}}_{H(W),\theta_\tau}\otimes {\mathrm{id}}_{U'_{\sigma,\tau}}}\\
\node{} \node{H(V)\theta_\sigma\widehat{\sigma}(H(W))\widehat{\sigma}(\theta_\tau)U'_{\sigma,\tau}} \arrow{s,r}{{\mathrm{id}}_{H(V)}\otimes\theta^{\sigma}_W\otimes{\mathrm{id}}_{\widehat{\sigma}(\theta_\tau)U'_{\sigma,\tau}}}\\
\node{} \node{H(V)H(\widetilde{\sigma}(W))\theta_\sigma\widehat{\sigma}(\theta_\tau)U'_{\sigma,\tau}}\arrow{s,r}{\psi^H_{V,\widetilde{\sigma}(W)}\otimes \Pi_{\sigma,\tau}}\\
\node{}\node{H(V\widetilde{\sigma}(W))H(U_{\sigma,\tau})\theta_{\sigma\tau}} \arrow{s,r}{\psi^H_{V\widetilde{\sigma}(W),U_{\sigma,\tau}}\otimes {\mathrm{id}}_{\theta_{\sigma\tau}}}\\
\node{T([V,\sigma]\cdot[W,\tau])}\arrow{e,t,=}{} \node{H(V\widetilde{\sigma}(W)U_{\sigma,\tau})\theta_{\sigma\tau}}
\end{diagram}$$ (where tensor symbols among objects of ${\mathcal{C}}$ have been omitted as a space-saving measure).  Conversely, given a graded op-monoidal functor $(T,\psi^T):{\mathcal{C}}\rtimes G\to {\mathcal{D}}\rtimes G$, we define a 1-cell $(H,\theta,\Pi)$ as $[H(V),e]=T([V,e])$, $[\theta_\sigma,\sigma]=T([I,\sigma])$, $[\theta^\sigma_V,\sigma]=\psi^T_{[I,\sigma],[V,e]}$, $[\Pi_{\sigma,\tau},\sigma\tau]=\psi^T_{[I,\sigma],[I,\tau]}$. Given a 2-cell $\{m_\sigma, m\}_{\sigma \in G}$ between 1-cells $T=(H,\theta,\Pi)$ and $T'=(H',\theta',\Pi')$, we define a monoidal natural isomorphism $m:T\to T'$ between the associated op-monoidal functors by $m_{[V,\sigma]}=[m_V\otimes m_\sigma,\sigma]$. Conversely, given a monoidal natural isomorphism $m:T\to T'$, we define a 2-cell by $[m_V,e]= m_{[V,e]}$, $[m_\sigma,\sigma]=m_{[I,\sigma]}$. Finally, in order to see that the 2-categories are biequivalent, note that every crossed product tensor category is equivalent to one of the form ${\mathcal{C}}_e\rtimes G$. So, every functor between ${\mathcal{C}}\rtimes G$ and ${\mathcal{D}}\rtimes G$ is monoidally equivalent to one induced by a 1-cell of the coherent outer $G$-action 2-category, and every monoidal natural transformation is equal to one induced by a 2-cell. 

Braided crossed product tensor categories {#seccion 6}
=========================================

Recall that a braiding for a monoidal category $({\mathcal{C}},\otimes ,I, \alpha)$ is a natural isomorphism $c: \otimes\to \otimes \tau$, where $\tau: {\mathcal{C}}\times {\mathcal{C}}\to {\mathcal{C}}\times {\mathcal{C}}$ is the flip, and the hexagons

$$\label{H1}
\begin{diagram}
\node{}\node{(U\otimes V)\otimes W}\arrow{e,t}{c_{U\otimes V, W}}\node{W\otimes (U\otimes V)}\arrow{se,l}{a_{W,U,V}} \\
\node{U\otimes (V\otimes W)} \arrow{se,r}{{\mathrm{id}}_{U}\otimes c_{V,W}}\arrow{ne,l}{a_{U,V,W}}\node{}\node{}\node{(W\otimes U)\otimes V}\\
\node{}\node{U\otimes (W\otimes V)}\arrow{e,t}{a_{U,W,V}}\node{(U\otimes W)\otimes V}\arrow{ne,r}{c_{U,W}\otimes {\mathrm{id}}_V}
\end{diagram}$$

$$\label{H2}
\begin{diagram}
\node{}\node{U\otimes (V\otimes W)}\arrow{e,t}{c_{U,V\otimes W}}\node{(V\otimes W)\otimes U}\arrow{se,l}{a_{V,W,U}^{-1}} \\
\node{(U\otimes V)\otimes W} \arrow{se,r}{ c_{U,V}\otimes {\mathrm{id}}_W}\arrow{ne,l}{a_{U,V,W}^{-1}}\node{}\node{}\node{V\otimes (W\otimes U)}\\
\node{}\node{(V\otimes U)\otimes W}\arrow{e,t}{a_{V,U,W}^{-1}}\node{V \otimes (U\otimes W)}\arrow{ne,r}{ {\mathrm{id}}_V \otimes c_{U,W}}
\end{diagram}$$

commute for all $U, V,W\in {\mathcal{C}}$. If a $G$-crossed product tensor category admits a braiding, the group $G$ must be abelian.  So, from now on we shall suppose that $G$ is abelian.  Let ${\mathcal{C}}$ be a tensor category with a coherent outer $G$-action, such that the tensor category ${\mathcal{C}}\rtimes G$ admits a braiding $c$. The braiding $c_{[V],[\sigma]}:[V,\sigma]\to [\sigma_*(V),\sigma]$ defines natural isomorphisms $c_{V,\sigma}:V\to \sigma_*(V)$. The commutativity of the hexagon is equivalent to $c_{V,\sigma}$ is a monoidal natural isomorphism from ${\mathrm{id}}_{\mathcal{C}}$ to $\sigma_*$. For that reason, if ${\mathcal{C}}\rtimes G$ has a braiding, we can suppose that $\sigma_*={\mathrm{id}}_{\mathcal{C}}$ for all $\sigma\in G$.

A coherent outer $G$-action shall be called central if $\sigma_*={\mathrm{id}}_{\mathcal{C}}$ for all $\sigma \in G$.

For a central coherent outer $G$-action, the pseudonatural transformations $(\chi_{\sigma,\tau},U_{\sigma,\tau})$ are just elements in $\mathcal{Z}({\mathcal{C}})$ (the center of ${\mathcal{C}}$), and the modifications $\omega$ are morphisms in $\mathcal{Z}({\mathcal{C}})$.

Let $({\mathcal{C}},c)$ be a braided tensor category, and let $G$ be an abelian group.  A braiding for a central coherent $G$-action over ${\mathcal{C}}$ is a triple $(\theta^\sigma, \overline{\theta}^\sigma, t_{\sigma,\tau})_{\sigma,\tau\in G}$, where

-   $\theta^\sigma, \overline{\theta}^\sigma:{\mathrm{id}}_{\mathcal{C}}\to {\mathrm{id}}_{\mathcal{C}}$ are monoidal natural isomorphisms,

-   $t_{\sigma,\tau}: \chi_{\sigma,\tau}\to \chi_{\tau,\sigma}$ are isomorphisms in $\mathcal{Z}({\mathcal{C}})$ for all $\sigma,\tau \in G$,

such that $\theta^e =\overline{\theta}^e= {\mathrm{id}}$, $\theta_I^\sigma ={\mathrm{id}}_I$, $t_{\sigma,e}=t_{e,\sigma}={\mathrm{id}}_I$, and the diagrams

$$\label{B1}
\begin{diagram}
\node{Z\otimes U_{\sigma,\tau}}\arrow{se,r}{((\overline{\theta}_Z^{\sigma\tau})^{-1}\overline{\theta}_Z^{\sigma}\overline{\theta}_Z^{\tau})\otimes {\mathrm{id}}_{U_{\sigma,\tau}}} \arrow{e,t}{\chi_{\sigma,\tau}(Z)}\node{U_{\sigma,\tau}\otimes Z} \arrow{s,r}{c_{U_{\sigma,\tau},Z}} \\
\node{}\node{Z\otimes U_{\sigma,\tau}}
\end{diagram}$$

$$\label{B2}
\begin{diagram}
\node{Z\otimes U_{\sigma,\tau}}\arrow{se,r}{((\theta_Z^{\sigma\tau})^{-1}\theta_Z^{\sigma}\theta_Z^{\tau})\otimes {\mathrm{id}}_{U_{\sigma,\tau}}} \arrow{e,t}{c_{U_{\sigma,\tau},Z}}\node{U_{\sigma,\tau}\otimes Z} \arrow{s,r}{\chi_{\sigma,\tau}(Z)} \\
\node{}\node{Z\otimes U_{\sigma,\tau}}
\end{diagram}$$

$$\label{B3}
\begin{diagram}
\node{}\node{U_{\sigma,\tau}U_{\sigma\tau,\rho}}\arrow[2]{e,t}{\theta_{U_{\sigma,\tau}}^\rho\otimes t_{\sigma\tau,\rho}} \node{}\node{U_{\sigma,\tau}U_{\rho,\sigma\tau}} \arrow{se,l}{w_{\rho,\sigma,\tau}}\\
\node{U_{\tau,\rho}U_{\sigma,\tau\rho}}\arrow{ne,l}{\omega_{\sigma,\tau,\rho}} \arrow{se,r}{t_{\tau,\rho}\otimes {\mathrm{id}}_{U_{\sigma,\rho\tau}}} \node{}\node{}\node{}\node{U_{\rho,\sigma}U_{\sigma\rho,\tau}}\\
\node{} \node{U_{\rho,\tau}U_{\sigma,\rho\tau}} \arrow[2]{e,t}{\omega_{\sigma,\rho,\tau}}\node{} \node{U_{\sigma,\rho}U_{\sigma\rho,\tau}} \arrow{ne,r}{t_{\sigma,\tau}\otimes {\mathrm{id}}_{U_{\sigma\rho,\tau}}}
\end{diagram}$$

$$\label{B4}
\begin{diagram}
\node{}\node{U_{\tau,\rho}U_{\sigma,\tau\rho}}\arrow[2]{e,t}{\overline{\theta}^\sigma_{U_{\tau,\rho}}\otimes t_{\sigma,\tau\rho}}
\node{} \node{U_{\tau,\rho}U_{\tau\rho,\sigma}} \arrow{se,l}{\omega_{\tau,\rho,\sigma}^{-1}}\\
\node{U_{\sigma,\tau}U_{\sigma\tau,\rho}}\arrow{ne,l}{\omega^{-1}_{\sigma,\tau,\rho}} \arrow{se,r}{t_{\sigma,\tau}\otimes{\mathrm{id}}_{U_{\sigma\tau,\rho}}} \node{}\node{}\node{}\node{U_{\rho,\sigma}U_{\tau,\rho\sigma}}\\
\node{}\node{U_{\tau,\sigma}U_{\tau\sigma,\rho}}\arrow[2]{e,t}{\omega_{\tau,\sigma,\rho}^{-1}}\node{}\node{U_{\sigma,\rho}U_{\tau,\sigma\rho}} \arrow{ne,r}{t_{\sigma,\rho}\otimes {\mathrm{id}}_{U_{\tau,\sigma\rho}}}
\end{diagram}$$

commute for all $\sigma,\tau, \rho \in G, Z\in {\mathcal{C}}$ (where tensor symbols among objects have been omitted as a space-saving measure). 

Let $({\mathcal{C}},c)$ be a braided tensor category with a coherent central outer $G$-action.  Then, there is a bijective correspondence between braidings over ${\mathcal{C}}\rtimes G$ and braidings over the central coherent outer $G$-action of ${\mathcal{C}}$.

Let $(\theta,\overline{\theta},t)$ be a braiding for a central coherent outer $G$-action $(\chi,\omega)$. Then, we define a braiding over ${\mathcal{C}}\rtimes G$ by $$\begin{aligned}
c_{[V,\sigma],[W,\tau]} &= (c_{V,W}\circ (\theta^\tau_V\otimes\overline{\theta}^\sigma_W))\otimes t_{\sigma,\tau}\\
&= ((\overline{\theta}^\sigma_W\otimes \theta^\tau_V)\circ c_{V,W} )\otimes t_{\sigma,\tau}.\end{aligned}$$ Conversely, given a braiding $c$ over ${\mathcal{C}}\rtimes G$, we define a braiding for the coherent central outer $G$-action by $$\begin{aligned}
    [\theta_V^\sigma,\sigma] := c_{[V],[\sigma]}, \  \  \  [\overline{\theta}_V^\sigma,\sigma]:= c_{[\sigma],[V]},\  \  \text{and}\ \  \ [t_{\sigma,\tau},\sigma\tau]:= c_{[\sigma],[\tau]}.\end{aligned}$$

Let we denote by $H(U,V,W)$ and $H'(U,V,W)$ the hexagons and , respectively.  Let $\theta^\sigma,\overline{\theta}^\sigma :{\mathrm{id}}_{\mathcal{C}}\to {\mathrm{id}}_{\mathcal{C}}$ be natural isomorphisms for each $\sigma\in G$, and let $t_{\sigma,\tau}:U_{\sigma,\tau}\to U_{\tau,\sigma}$ be isomorphisms in ${\mathcal{C}}$. If we set the following definitions of natural isomorphisms $$\begin{aligned}
      c_{[V],[\sigma]}:=[\theta_V^\sigma,\sigma], \  \  \  c_{[\sigma],[V]}:=[\overline{\theta}_V^\sigma,\sigma],\  \  \ c_{[\sigma],[\tau]}:=[t_{\sigma,\tau},\sigma\tau],\end{aligned}$$ it is easy to see that the commutativity of $H([V],[W],\sigma)$ and $H'([\sigma], [V],[W])$ is equivalent to $\theta^\sigma$ and $\overline{\theta}^\sigma$ be monoidal natural isomorphisms, respectively.  The commutativity of $H'([\sigma],[\tau],[Z])$ and $H([\sigma],[V],[\tau])$ is equivalent to $t_{\sigma,\tau}$ be a morphism in $\mathcal{Z}({\mathcal{C}})$. The commutativity of $H([\sigma],[\tau],[Z])$ and $H([Z],[\sigma],[\tau])$ is equivalent to the commutativity of and , respectively.  The commutativity of $H([\sigma],[\tau],[\rho])$ and $H'([\sigma],[\tau],[\rho])$ is equivalent to the commutativity of , , respectively. 

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