2.1 Group Actions on Bicategories

In this subsection, we recall the notion of a group action on a monoidal category and that on a bicategory and give an appropriate notion of equivalence between bicategories with group actions, which turns out to be equivalent to the notion introduced in [6]. Moreover, we prove a coherence theorem for G-actions on bicategories, see Theorem 2.6, cf. [6, Theorem 3.1].

An action of a group G on a monoidal category \(\mathcal \) is a monoidal functor from the monoidal category \(\underline\) of the elements of G with only identity morphisms to the monoidal category of monoidal endofunctors on \(\mathcal \).

We also have to consider group actions on bicategories in this article (see Sects. 3.1 and 5.2). For basic notions of bicategories, see e.g. [33, Section 1.5]. Here we fix our notation: the composition of 1-cells of \(\mathcal \) is denoted by \(\otimes _\mathcal \) or \(\otimes \), for we regard bicategories as generalizations of monoidal categories, and often omitted. The associativity constraint is denoted by \(a^\mathcal \). The unit is denoted by \(\textbf^\mathcal \), and the left and right unit constraints are denoted, respectively, by \(l^\mathcal \) and \(r^\mathcal \). A pseudofunctor \(F: \mathcal \rightarrow \mathcal \) consists of a map F between 0-cells, functors \(F_: }_\mathcal (A,B) \rightarrow }_\mathcal (F(A),F(B))\) for \(A,B \in \,}}(\mathcal )\), natural isomorphisms \(J^F_: F_(-) \otimes F_(-) \cong F_(- \otimes -)\) for \(A,B,C \in \,}}(\mathcal )\) and \(\varphi ^F_A:\textbf^\mathcal _ \cong F_(\textbf^\mathcal _A)\) for \(A \in \,}}(\mathcal )\). A pseudonatural transformation \(\tau : F \rightarrow F'\) consists of 1-cells \(\tau ^0_C: F(C) \rightarrow F'(C)\) for \(C \in \,}}(\mathcal )\) and invertible 2-cells \(\tau _\lambda : \tau ^0_ F(\lambda ) \cong F'(\lambda ) \tau ^0_\) for 1-cells \(\lambda : C \rightarrow C'\). The vertical and horizontal compositions of pseudonatural transformations (see also [6, Section 1.1]) are denoted, respectively, by \(\circ \) and \(*\).

By stating the coherence theorem for pseudofunctors [26, Subsection 2.3.3] in the form of “all diagrams commute”, we can see that the 2-cells obtained by vertically and horizontally composing components of \(a^\mathcal \), \(l^\mathcal \) and \(r^\mathcal \) of bicategories \(\mathcal \) and \(J^F\)’s and \(\varphi ^F\)’s of pseudofunctors F are indeed canonical, which allows us to suppress these 2-cells. In particular, we may suppress the constraints of the tricategory of bicategories. We can also introduce some graphical representations in the tricategory of bicategories, which are used only in this subsection. A pseudonatural transformation is represented as an arrow from top to bottom, and the vertical and horizontal compositions of pseudonatural transformations are represented, respectively, by the vertical and horizontal concatenation of arrows. Note that we do not have to be careful of relative vertical positions when taking horizontal concatenations since the comparison constraints between pseudonatural transformations (see [6, Section 1.1]) are canonical by coherence.

An action of a group G on a bicategory \(\mathcal \) is a monoidal pseudofunctor (i.e. a trihomomorphism between monoidal bicategories, see [25, Chapters 2 and 3]) from \(\underline}\) to \(\,}}(\mathcal )\), where \(\underline}\) denotes G regarded as a monoidal bicategory and \(\,}}(\mathcal )\) denotes the monoidal bicategory of pseudofunctors from \(\mathcal \) to itself, see [6, Sections 1.1 and 2]. More explicitly, an action \(\gamma \) of G on \(\mathcal \) consists of pseudofunctors (biequivalences) \(\gamma (g)\) from \(\mathcal \) to itself, pseudonatural equivalences \(\chi ^_: \gamma (g) \gamma (h) \simeq \gamma (gh)\), a pseudonatural equivalence \(\iota ^\gamma : \textrm_\mathcal \simeq \gamma (e)\) and invertible modifications \(\omega ^_: \chi ^_ \circ (\chi ^_ *\textrm_) \cong \chi ^_ \circ (\textrm_ *\chi ^_)\), \(\kappa ^_: \chi ^_ \circ (\iota ^\gamma *\textrm_) \cong \textrm_\) and \(\zeta ^_: \chi ^_ \circ (\textrm_ *\iota ^\gamma ) \cong \textrm_\) for \(g,h,k \in G\). Graphically, let a fork with two inputs \(\gamma (g)\) and \(\gamma (h)\) and one output \(\gamma (g h)\) denote \(\chi ^\gamma _\), and let a small circle with one output \(\gamma (e)\) denote \(\iota ^\gamma \). Then, \(\omega ^\gamma \), \(\kappa ^\gamma \) and \(\zeta ^\gamma \) correspond, respectively, to associativity and left and right unit properties of an algebra in a weak sense, which satisfy the pentagon and triangle axioms [25, Definition 3.1].

For a 1-cell \(\lambda \) of \(\mathcal \) and \(g \in G\), the 1-cell \(\gamma (g) (\lambda )\) is often denoted by \(^g \lambda \). Moreover, \(^g \lambda \mu \) denotes \(^g (\lambda ) \otimes \mu \) for \(g \in G\) and 1-cells \(\lambda \) and \(\mu \), and \(^g f \otimes f'\) denotes \(^g (f) \otimes f'\) for \(g \in G\) and 2-cells f and \(f'\) in this article.

An appropriate notion of morphisms between (strict) 2-categories with unital (see [6, Definition 2.1]) group actions is given in [6, Definition 2.3]. Here, we define the notion of an equivalence in a general setting.

First, recall that for a biequivalence \(F: \mathcal \rightarrow \mathcal \), we can take a pseudofunctor \(F^: \mathcal \rightarrow \mathcal \) with pseudonatural equivalences \(\textrm^F: F^F \simeq \textrm_\mathcal \) and \(\textrm^F: \textrm_\mathcal \simeq FF^\) by fixing data that consist of 0-cells \(F^(D) \in \,}}(\mathcal )\), equivalence 1-cells \(\textrm^_D: D \rightarrow F(F^(D))\), left adjoint inverses \((\textrm^_D)^\vee \) of \(\textrm^_D\) (i.e. left duals of \(\textrm^_D\) in the bicategory \(\mathcal \) with invertible evaluation and coevaluation maps) for \(D \in \,}}(\mathcal )\) and left adjoint functors \(F_^\vee \) of \(F_\) for \(C,C' \in \,}}(\mathcal )\), see e.g. [33, Proposition 1.5.13]. The set of these data is also denoted by \(F^\) and referred to as an adjoint inverse of F. Another choice of an adjoint inverse only yields a pseudonaturally equivalent pseudofunctor \(F^\) by a standard duality argument. The pseudonatural equivalences \(\textrm^F\) and \(\textrm^F\) are graphically represented by arcs as in the case of duality in bicategories. Their adjoint inverses are represented by opposite arcs.

We also recall that we have a natural isomorphism \(J^_: F_^\vee (-) \otimes F_^\vee (-) \cong F_^\vee (- \otimes -)\) for \(A, B, C \in \,}}(\mathcal )\) and an isomorphism \(\varphi ^_A: \textbf^}_A \cong F^\vee _(\textbf^\mathcal _)\) for \(A \in \,}}(\mathcal )\) with the coherence conditions as that for pseudofunctors. Indeed, they are defined by putting \((J^_)_ F^\vee _((\widetilde}^_\lambda )^ \otimes (\widetilde}^F_\mu )^) \circ (\widetilde}^F_(\lambda ) F^\vee _(\mu )})^\) for \(\lambda : F(B) \rightarrow F(C)\) and \(\mu : F(A) \rightarrow F(B)\), and \(\varphi ^_A (\widetilde}^F_^\mathcal _A})^\), where \(\widetilde}^F_\nu \) and \(\widetilde}^F_\nu \), respectively, denote the components of the evaluation and coevaluation maps of \(F_\) for \(\nu : F(A) \rightarrow F(B)\). Then, as in the case of pseudofunctors, we may suppress \(J^\)’s and \(\varphi ^\)’s. By the definition of \(J^\)’s and \(\varphi ^\)’s, the naturality of \(\widetilde}^F\) and \(\widetilde}^F\) and conjugate equations for F, we can see that \(\widetilde}^F\) and \(\widetilde}^F\) are monoidal, i.e. \(\widetilde}^F_ = \widetilde}^F_ \otimes \widetilde}^F_\) for \(\lambda :F(B) \rightarrow F(C)\) and \(\mu :F(A) \rightarrow F(B)\), and \(\widetilde}^F__^\mathcal } = \textrm__^\mathcal }\) (and similar statements for \(\widetilde}^F\)), where \((J^_)_\) and \(\varphi ^_A\) are suppressed.

When a biequivalence \(F:\mathcal \rightarrow \mathcal \) is given, we can transport \(H \in \,}}(\mathcal )\) to \(\,}}(\mathcal )\) by fixing an adjoint inverse \(F^\) of F and putting \(\,}}(F)(H) F H \)\( F^ \in \,}}(\mathcal )\).

Lemma 2.1

Let \(F: \mathcal \rightarrow \mathcal \) be a biequivalence between bicategories. Then \(\,}}(F): \,}}(\mathcal ) \rightarrow \,}}(\mathcal )\) can be regarded as a monoidal pseudofunctor. Another choice of an adjoint inverse of F only yields a monoidally equivalent one (i.e. there exists a triequivalence with an identical 1-cell).

Proof

Let \(H, K, L, H', K' \in \,}}(\,}}(\mathcal ))\) and let \(\sigma \in }(H, H')\), \(\tau \in }(K, K')\) and \(\rho \in }(H', H'')\). Define functors \(\,}}(F)_\) to be

$$\begin \textrm_ *- *\textrm_}: }(H,H') \rightarrow }(\,}}(F)(H),\,}}(F)(H')). \end$$

Define \(J^\,}}(F)}_: \,}}(F)(\rho ) \circ \,}}(F)(\sigma ) \cong \,}}(F)(\rho \circ \sigma )\) and \(\varphi ^\,}}(F)}_H: \textrm_\,}}(F)(H)} \cong \,}}(F)(\textrm_H)\) to be canonical isomorphisms, which makes \(\,}}(F)\) into a pseudofunctor by coherence. Define pseudonatural equivalences \(\chi _^\,}}(F),0} \textrm_ *\textrm^F *\textrm_}: \,}}(F)(H) \,}}(F)(K) \simeq \,}}(F)(H K)\) and define invertible modifications \(\chi ^\,}}(F)}_: \chi _^\,}}(F),0} \circ (\,}}(F)(\sigma ) *\,}}(F)(\tau )) \cong \,}}(F)(\sigma *\tau ) \circ \chi _^\,}}(F),0}\) to be canonical isomorphisms, which give a pseudonatural equivalence \(\chi ^\,}}(F)} (\chi ^\,}}(F),0},\chi ^\,}}(F)}): \,}}(F)(-) \circ \,}}(F)(-) \simeq \,}}(F)(- \circ -)\) by coherence. Put \(\iota ^\,}}(F)} \textrm^F: \textrm_\mathcal \simeq \,}}(F)(\textrm_\mathcal ) = FF^\). Define modifications \(\omega ^\,}}(F)}_: \chi _^\,}}(F),0} \circ (\chi _^\,}}(F),0} *\textrm_\,}}(F)(L)}) \cong \chi _^\,}}(F),0} \circ (\textrm_\,}}(F)(H)} *\chi _^\,}}(F),0})\) to be canonical isomorphisms, which give an invertible modification \(\omega ^\,}}(F)}: \chi ^\,}}(F)} \circ (\chi ^\,}}(F)} *\textrm_\,}}(F)}) \cong \chi ^\,}}(F)} \circ (\textrm_\,}}(F)} *\chi ^\,}}(F)})\) by coherence. Then, Axiom (HTA1) in [25, Definition 3.1] holds by coherence.

Put \(f_D \textrm^_D\) for \(D \in \,}}(\mathcal )\). Let \(F^\vee (\lambda )\) denote \(F_^\vee (\lambda )\) for \(\lambda \in }_\mathcal (F(C),F(C'))\) and \(C,C' \in \,}}(\mathcal )\). Since \(\textrm^_C = F^\vee (f_^\vee )\) for \(C \in \,}}(\mathcal )\) by construction, we can define an invertible 2-cell

$$\begin \begin&\xi _C^F: ((\text _F *\text ^) \circ (\text ^ *\text _F))^0_C = F(\text ^_C) \text ^_ = FF^\vee (f_^\vee ) f_ \\ &\quad \rightarrow (\text _F)^0_C = }^\mathcal _ \end \end$$

to be \(\textrm_}((\widetilde}^F_})^ \otimes \textrm_})\). Since \(F^(\lambda ) = F^\vee (f_ \lambda f_D^\vee )\) for \(\lambda \in }_\mathcal (D,D')\) by construction, we can also define an invertible 2-cell

$$\begin&\tilde^F_D: ((\textrm^ *\textrm_}) \circ (\textrm_} *\textrm^))_D^0 = \textrm^_(D)} F^(\textrm^_D) \\&\quad = F^\vee (f^\vee _(D)} f_(D)} f_D f_D^\vee ) \\&\quad \quad \rightarrow (\textrm_})^0_D = \textbf^\mathcal _(D)} \end$$

to be \(F^\vee (\textrm_(D)}} \otimes \textrm_^)\) for \(D \in \,}}(\mathcal )\). We show that \(\xi ^F\) and \(\tilde^F\) are indeed modifications. For this, recall that \(\textrm^F_\nu = (\widetilde}^F_ \otimes \textrm_) (\textrm_ \lambda } \otimes \textrm^_)\) for \(\nu \in }_\mathcal (D,D')\) and \(\textrm^F_\rho = (\widetilde}^F_\rho \otimes \textrm_)}) F^\vee (\textrm_} \otimes \textrm_})\) for \(\rho \in }_\mathcal (C,C')\) by construction. Then, we can see by direct computations that \(\xi ^F\) and \(\tilde^F\) are modifications.

Then, put \(\kappa ^\,}}(F)}_H \xi ^F *\textrm_}\) and \(\zeta ^\,}}(F)}_H \textrm_ *\tilde^F\) for \(H \in \,}}(\,}}(\mathcal ))\), which define modifications \(\kappa ^\,}}(F)}: \chi ^\,}}(F)} \circ (\iota ^\,}}(F)} *\textrm_\,}}(F)}) \cong \textrm_\,}}(F)}\) and \(\zeta ^\,}}(F)}: \chi ^\,}}(F)} \circ (\textrm_\,}}(F)} *\iota ^\,}}(F)}) \cong \textrm_\,}}(F)}\) by coherence. To check that these modifications satisfy Axiom (HTA2) in [25, Definition 3.1], it is enough to show \(\textrm_^F} \otimes (\textrm_} *\xi ^F) = \textrm_^F} \otimes (\tilde^F *\textrm_F)\). For this, note that the component of the comparison constant \(\textrm^F \circ (\textrm_F} *\textrm^F) \cong \textrm^F \circ (\textrm^F *\textrm_F})\) for \(C \in \,}}(\mathcal )\) is given by \(F^\vee (\textrm_}\otimes (\widetilde}^F_})^ \otimes \textrm_F(C)})\). Then, we can find that it is enough to show \(\textrm_} \otimes \textrm_^\vee f_} = \textrm_^\vee f_} \otimes \textrm_}\), which follows since \(f_^\vee \) is an adjoint inverse. Thus, \(\,}}(F)\) is a monoidal pseudofunctor.

Finally, we show that another choice of an adjoint inverse \(F_2^\) yields an equivalent monoidal pseudofunctor \(\,}}_2 (F)\). Let \(\tilde_D\) for \(D \in \,}}(\mathcal )\) denote \(\textrm^_D\) for \(F_2^\), and let \(F^\vee _2\) denote \(F^\vee \) for \(F^_2\). Also, let \(\textrm_2^F\) and \(\textrm^F_2\), respectively, denote \(\textrm^F\) and \(\textrm^F\) for \(F_2^\). For \(H \in \,}}(\,}}(\mathcal ))\), put \(\eta _H^0 \textrm_ *\tau : \,}}(F)(H) \simeq \,}}_2(F)(H)\), where \(\tau : F^ \simeq F_2^\) denotes the pseudonatural equivalence obtained from \(\textrm^F_2 (\textrm^F)^\vee : FF^ \simeq FF_2^\) and \(^\vee \) denotes a left adjoint inverse in \(\,}}(\mathcal )\), by duality. For \(H, H' \in \,}}(\,}}(\mathcal ))\) and \(\sigma \in }(H, H')\), define \(\eta _\sigma : \eta ^0_ \circ \,}}(F)(\sigma ) \cong \,}}_2(F)(\sigma ) \circ \eta ^0_H\) to be the canonical isomorphism, which gives a pseudonatural equivalence \(\eta =(\eta ^0, \eta ): \,}}(F) \simeq \,}}_2(F)\) by coherence. Then, for \(C \in \,}}(\mathcal )\), define an invertible 2-cell

$$\begin \begin&\alpha _C: (\text ^F_2 \circ (\tau *\text _F))_C^0 \\ &= F^\vee _2(\tilde^\vee _) F^\vee (f_F(C)}^\vee f_F(C)} \tilde_ f^\vee _ f^\vee _F(C)} f_F(C)}) \\ &\quad \rightarrow \text ^_C = F^\vee (f_^\vee ) \end \end$$

to be

$$\begin&F^\vee ((\textrm__} \otimes \textrm_})(\textrm_^\vee _} \otimes \textrm_F(C)}} \otimes \textrm__ f^\vee _} \otimes \textrm_F(C)}}))\\&\quad (\beta _^\vee _} \otimes \textrm_}), \end$$

where \(\beta \) denotes the canonical isomorphism \(F^\vee _2 \cong F^\vee \). It is routine to check that \(\alpha \) is indeed a modification \(\textrm^F_2 \circ (\tau *\textrm_F) \cong \textrm^F\). For \(H,K \in \,}}(\,}}(\mathcal ))\), define \(\Pi _: \chi ^\,}}_2(F),0}_ \circ (\eta _H^0 *\eta _K^0) \cong \eta _^0 \circ \chi ^\,}}(F),0}_\) to be \(\textrm_ *\alpha *\textrm_K *\textrm_\), which gives a modification \(\Pi : \chi ^\,}}_2(F)} \circ (\eta *\eta ) \cong \eta \circ \chi ^\,}}(F)}\) by coherence. Then, \(\eta \) and \(\Pi \) satisfy the condition in [25, pp. 21–22] by coherence. Finally, define \(M: \eta __\mathcal }^0 \circ \iota ^\,}}(F)} \cong \iota ^\,}}_2(F)}\) to be the composition of the modifications in Fig. 1, where \(\tilde_2^F\) denotes \(\tilde^F\) for \(F_2^\). Note that e.g. \(\xi ^F\) indeed denotes \((\xi ^F *\textrm_}) \otimes \textrm_^F_2}\) etc. Then, \(\eta , \Pi \) and M satisfy the conditions in [25, pp. 23–24] by \(\textrm_^F} \otimes (\textrm_} *\xi ^F_2) = \textrm_^F} \otimes (\tilde^F_2 *\textrm_F)\). Thus, \((\eta ,\Pi ,M)\) is a monoidal pseudonatural equivalence \(\,}}(F) \simeq \,}}_2(F)\). \(\square \)

Fig. 1 Remark 2.2

The proof above shows that a biequivalence is indeed a 2-dualizable pseudofunctor in the sense of the tricategory version of [11, Definitions 5.1 and 6.1].

Definition 2.3

Let \((\mathcal ,\gamma ^\mathcal )\) and \((\mathcal ,\gamma ^\mathcal )\) be pairs of bicategories and actions of a group G. A G-biequivalence between \((\mathcal ,\gamma ^\mathcal )\) and \((\mathcal ,\gamma ^\mathcal )\) is the pair \(F = (F,\eta ^F)\) of a biequivalence \(F: \mathcal \rightarrow \mathcal \) and a monoidal equivalence \(\eta ^F: \,}}(F) \circ \gamma ^\mathcal \simeq \gamma ^\mathcal \).

The existence of a G-biequivalence between bicategories does not depend on a choice of \(F^\) by Lemma 2.1.

Lemma 2.4

G-biequivalence is indeed an equivalence relation.

Proof

Reflexivity follows since \(\,}}(\textrm_\mathcal ) \simeq \textrm_\,}}(\mathcal )}\) as monoidal pseudofunctors by coherence. We show transitivity. Let \(F: \mathcal \rightarrow \mathcal \) and \(H: \mathcal \rightarrow \mathcal \) be biequivalences. We fix \(F^\) and \(H^\), set \((HF)^ (E) F^ H^(E)\), \(\textrm^_E H(f_(E)}) h_E\) and \((\textrm^_E)^\vee h_E^\vee H(f_(E)}^\vee )\) for \(E \in \,}}(\mathcal )\), where \(f \textrm^\) and \(h \textrm^\), and set \((HF)^\vee F^\vee H^\vee \). Then, for \(\lambda \in }_}(E,E')\), we have an invertible 2-cell

$$\begin&F^\vee (\widetilde}^H_(E)}} \otimes \textrm_(\lambda )} \otimes \widetilde}^H_(E)}^\vee }): \ (HF)^(\lambda ) \\&\quad = F^\vee (H^\vee H(f_(E')}) H^(\lambda ) H^\vee H(f^\vee _(E)})) \\&\qquad \rightarrow F^ H^ (\lambda ) = F^\vee (f_(E')} H^(\lambda ) f^\vee _(E)}), \end$$

which defines a pseudonatural equivalence \(\tau : (HF)^ \simeq F^ H^\). Moreover, we define an invertible 2-cell

$$\begin&(\textrm^H \circ \,}}(H) (\textrm^F) \circ (\tau *\textrm_))_C^0 = F^\vee (f^\vee _ f_ H^\vee (h^\vee _) f^\vee _HF(C)}) \\&\quad \rightarrow \textrm^_C = F^\vee (H^\vee (h^\vee _) H^\vee H (f^\vee _HF(C)})) \end$$

for \(C \in \,}}(\mathcal )\) to be \(F^\vee ((\textrm_)} \otimes (\widetilde}^H_HF(C)}})^)(\textrm_} \otimes \)\( \textrm_) f^\vee _HF(C)}}))\), which indeed defines a modification \(\textrm^H \circ \,}}(H) (\textrm^F) \circ (\tau *\textrm_) \cong \textrm^\). Then, we obtain a monoidal equivalence \(\,}}(HF) \simeq \,}}(H) \,}}(F)\) as in the final part of the proof of Lemma 2.1, and therefore G-biequivalence is transitive. Finally, we show symmetry. Since \(\,}}(F^ F) \simeq \,}}(F^) \,}}(F)\) as monoidal pseudofunctors for a biequivalence F by the argument so far, it is enough to show that for biequivalences \(F, \tilde: \mathcal \rightarrow \mathcal \) with a pseudonatural equivalence \(\tau : F \simeq \tilde\), we have a monoidal equivalence \(\,}}(F) \simeq \,}}(\tilde)\). Define \(\tilde: F^ \simeq \tilde^\) to be \((\textrm^F *\textrm_^}) \circ (\textrm_} *\tau ^\vee *\textrm_^}) \circ (\textrm_} *\textrm^})\). Also, define an invertible modification \(\alpha : \textrm^} \circ (\tilde *\tau ) \cong \textrm^F\) by Fig. 2. Then, by putting \(\eta _H \tau *\textrm_H *\tilde\) and \(\Pi _ \textrm_ *\alpha *\textrm_}\) for \(H, K \in \,}}(\,}}(\mathcal ))\), we obtain a pseudonatural equivalence \(\eta \) and an invertible modification \(\Pi \) with the condition in [25, pp. 21–22] as in the proof of Lemma 2.1. Define a modification M by Fig. 3. We can see that \((\eta , \Pi , M)\) is a monoidal equivalence \(\,}}(F) \simeq \,}}(\tilde)\) by the duality of \(\tau \). \(\square \)

Fig. 2

A modification \(\alpha \)

Fig. 3

By definition [25, Section 3.3], for a G-biequivalence \(F: (\mathcal ,\gamma ^\mathcal ) \rightarrow (\mathcal ,\gamma ^\mathcal )\), the monoidal equivalence \(\eta ^F\) consists of pseudonatural equivalences \(\eta ^F_g: \,}}(F) (\gamma ^\mathcal (g)) \simeq \gamma ^\mathcal (g)\), invertible modifications \(\Pi ^_: \chi ^}_ \circ (\eta _g^F *\eta _h^F) \cong \eta ^F_ \circ \,}}(F)(\chi ^}_) \circ \chi ^\,}}(F),0}_(g),\gamma ^\mathcal (h)}\) for \(g,h \in G\) and an invertible modification \(M^F: \eta ^F_e \circ \,}}(F)(\iota ^}) \circ \iota ^\,}}(F)} \cong \iota ^}\). Let \(\eta ^F_g\) be graphically represented by a fork with three inputs F, \(\gamma ^\mathcal (g)\) and \(F^\) and one output \(\gamma ^\mathcal (g)\).

Next, we compare our definition with that in [6, Definition 2.3]. Suppose we are given a G-biequivalence \(F:(\mathcal , \gamma ^\mathcal ) \rightarrow (\mathcal , \gamma ^\mathcal )\). Then, we obtain \(\tilde^F_g: F \gamma ^\mathcal (g) \simeq \gamma ^\mathcal (g) F\) for \(g \in G\) by duality: namely, we put \(\tilde^F_g (\eta ^F_g *\textrm_F) \circ (\textrm_(g)} *(\textrm^)^)\). Moreover, we can define invertible modifications \(\tilde^F_: (\chi ^}_ *\textrm_F) \circ (\textrm_(g)} *\tilde_h^F) \circ (\tilde^F_g *\textrm_(h)}) \cong \tilde^F_ \circ (\textrm_F *\chi ^}_)\) for \(g,h \in G\) and \(\tilde^F:\tilde^F_e \circ (\textrm_F *\iota ^}) \cong \iota ^} *\textrm_F\) by Figs. 4 and 5. We can check that they satisfy the conditions in Figs. 6, 7 and 8, where a crossing from \(F \gamma ^\mathcal (g)\) to \(\gamma ^\mathcal (g) F\) denotes \(\tilde^F_g\) for \(g \in G\), with some standard computations, but note that we use \(\textrm_^F} \otimes (\textrm_} *\xi ^F) = \textrm_^F} \otimes (\tilde^F *\textrm_F)\) in the check for Fig. 8. Conversely, when a biequivalence F and a triple \((\tilde^F,\tilde^F,\tilde^F)\) with the conditions in Figs. 6, 7 and 8 are given, we can construct a triple \((\eta ^F, \Pi ^F, M^F)\) by putting \(\eta ^F_g (\textrm_(g)} *(\textrm^)^) \circ (\tilde^F_g *\textrm_})\) for \(g \in G\) and defining \(\Pi _^F\) for \(g, h \in G\) and \(M^F\) by Figs. 9 and 10. We can see that \((F, \eta ^F, \Pi ^F, M^F)\) is a G-biequivalence, using the following lemma to check the condition in [25, p. 24].

Fig. 4

Construction of \(\tilde^F\) from \(\Pi ^F\)

Fig. 5

Construction of \(\tilde^F\) from \(M^F\)

Fig. 6

The coherence for \(\omega \)’s

Fig. 7

The coherence for \(\kappa \)’s

Fig. 8

The coherence for \(\zeta \)’s

Fig. 9

Construction of \(\Pi ^F\) from \(\tilde^F\)

Fig. 10

Construction of \(M^F\) from \(\tilde^F\)

Lemma 2.5

For a biequivalence F, we have \((\xi ^F *\textrm_}) \otimes \textrm_^F} = (\textrm_F *\tilde^F) \otimes \textrm_^F}\).

Proof

We already proved \(\textrm_^F} \otimes (\textrm_} *\xi ^F) = \textrm_^F} \otimes (\tilde^F *\textrm_F)\) in the proof of Lemma 2.1. Then, from \((\textrm_^F} \otimes (\textrm_} *\xi ^F))(\textrm_^F *\textrm^F} \otimes (\tilde^F *\textrm_^F} *\textrm_F)) = (\textrm_^F} \otimes (\tilde^F *\textrm_F)) (\textrm_^F *\textrm^F} \otimes (\textrm_} *\textrm_^F} \xi ^F))\) we obtain \(\textrm_^F *\textrm^F} \otimes (\tilde^F *\textrm_^F} *\textrm_F) = \textrm_^F *\textrm^F} \otimes (\textrm_} *\textrm_^F} \xi ^F)\) and therefore \(\textrm_^F *\textrm^F} \otimes (\textrm_} *\xi ^F *\textrm_F}) \otimes (\textrm_ *\textrm_^F} *\textrm_) = \textrm_^F *\textrm^F} \otimes (\textrm_F} *\tilde^F *\textrm_F) \otimes (\textrm_ *\textrm_^F} *\textrm_)\). Thus, we obtain the statement since we may compose \(\xi ^ *\tilde^\) with the modifications in the statement. \(\square \)

Thus, we can equivalently define a G-equivalence to be a tuple (F, \(\tilde^F\), \(\tilde^F\), \(\tilde^F\)) with the conditions in Figs. 6, 7 and 8, which reduces to [6, Definition 2.3] when the action is unital.

Then, we state the coherence theorem for group actions on bicategories [6, Theorem 3.1] in the form of “all diagrams commute” for our convenience.

Theorem 2.6

Let \(\mathcal \) be a bicategory with an action \(\gamma ^\mathcal \) of a group G. Define a set \(W = \bigsqcup _ W_n\) of words recursively by the following rules: \(\textbf_C^\mathcal , (\chi ^}_)^0_C, (\iota ^})_C^0 \in W_0\) for any 0-cell C and \(g,h \in G\), \(- \in W_1\), \(\otimes \in W_\), \(\gamma ^\mathcal (g) \in W_1\) for \(g \in G\) and \(w'((w_i)_i) \in W_\) for \(w' \in W_n\) and a family \((w_i)_^n\) with \(w_i \in W_\). Let the functor \(\mathcal ^n \rightarrow \mathcal \), where \(\mathcal ^0\) denotes the category with only one object and its identity morphism, corresponding naturally to a word \(w \in W_n\) be denoted again by w. Define a set I of morphisms recursively by the following rules: the components of \(a^\mathcal ,l^\mathcal ,r^\mathcal , J^(g)}\) and \(\varphi ^(g)}\) are in I, \((\chi ^}_)_\lambda \) and \((\iota ^})_\lambda \) for any 1-cell \(\lambda \) and \(g,h \in G\) are in I, \(\omega ^}_, \kappa ^}_g, \zeta ^}_g \in I\) for \(g,h,k \in G\), \(f^ \in I\) for \(f \in I\), \(f \circ f' \in I\) for \(f,f' \in I\) if it is well-defined and \(w((f_i)_i) \in I\) for \(w \in W_n\) and \((f_i)_^n \subset I\). Then, for any \(w,w' \in W_n\) and 1-cells \((\lambda _i)_^n\), a 2-cell \(w((\lambda _i)_i) \rightarrow w'((\lambda _i)_i)\) in I is unique if it exists.

Proof

By the proof of [6, Theorem 3.1], there exists a G-biequivalence F from \(\mathcal \) to a 2-category \(\mathcal \) with a strict action \(\gamma ^\mathcal \) (see [6, Definition 2.2]) such that every equivalence 1-cell is indeed an isomorphism. We regard F as data \((F, \tilde^F, \tilde^F, \tilde^F)\) as above. We define 2-cells \(C^v_\) for \(v \in W_n\) and 1-cells \((\lambda _i)_^n\) recursively by the following rules: \(C^_C^\mathcal } (\varphi ^F_C)^: F(\textbf^\mathcal _C) \cong \textbf_^\mathcal \) for \(C \in \,}}(\mathcal )\), \(C^_ (J^F_)^: F(\lambda \mu ) \cong F(\lambda ) F(\mu )\) for 1-cells \(\lambda \) and \(\mu \) of \(\mathcal \), \(C^(g)}_\lambda \textrm\otimes (\tilde^F_g)_\lambda : F\gamma ^\mathcal (g)(\lambda ) \cong ((\tilde^F_g)_^0)^ \gamma ^\mathcal (g) F (\lambda ) (\tilde^F_g)_^0\) for \(\lambda \in }_\mathcal (C, C')\) and \(g \in G\), \(C^^})_C^0} \textrm\otimes (\tilde_^F)_C^: F((\chi _^})_C^0) \cong ((\tilde^_)^0_C)^ \gamma ^\mathcal (g)((\tilde^F_h)^0_C) (\tilde^F_g)^0_(h)(C)}\) for \(C \in \,}}(\mathcal )\) and \(g,h \in G\), \(C^})^0_C} \textrm \otimes \tilde^F_C: F((\iota ^})_C^0) \cong ((\tilde_e^F)_C^0)^\) for \(C \in \,}}(\mathcal )\) and \(C^_ v((C^_)_i) \circ C^v_\) for \(v \in W_n\), \((w_i)_^n \subset W\) and 1-cells \((\lambda _i)_^n\).

We show \(C^w_ = C^_ \circ F(f)\) for any 2-cell \(f:w((\lambda _i)_i) \rightarrow w'((\lambda _i)_i)\), which uniquely determines f. It is enough to prove this when f is a generator of I. The statement for \(a^\mathcal ,l^\mathcal ,r^\mathcal ,J^(g)}\) and \(\varphi ^(g)}\) follows since F is a pseudofunctor and \(\tilde^F_g\) is a pseudonatural transformation for any \(g \in G\), which is standard. The statement for \((\chi ^}_)_\) and \((\iota ^})_\lambda \) follows since \(\tilde^F_\) and \(\tilde^F\) are modifications. The statement for \(\omega ^}_, \kappa ^}_g\) and \(\zeta ^}_g\) follows from Figs. 6, 7 and 8. \(\square \)

Thanks to this theorem, we may hereafter suppress the 2-cells in I in this article.

Finally, we return to the case of monoidal categories.

Definition 2.7