Theory and Applications of Categories, Vol. 30, No. 56, 2015, pp. 1999–2047.

ON THE 3-REPRESENTATIONS OF GROUPS AND THE 2-CATEGORICAL TRACES

WEI WANG

Abstract. To 2-categorify the theory of group representations, we introduce the notions of the 3-representation of a group in a strict 3-category and the strict 2-categorical action of a group on a strict 2-category. We also 2-categorify the concept of the trace by introducing the 2-categorical trace of a 1-endomorphism in a strict 3-category. For a 3-representation ρ of a group G and an element f of G, the 2-categorical trace 𝕋r_{2}ρ_{f} is a category. Moreover, the centralizer of f in G acts categorically on this 2-categorical trace. We construct the induced strict 2-categorical action of a finite group, and show that the 2-categorical trace 𝕋r_{2} takes an induced strict 2-categorical action into an induced categorical action of the initia groupoid. As a corollary, we get the 3-character formula of the induced strict 2-categorical action.

Contents

1 Introduction  1999
2 The 3-representations of groups  2002
3 The 2-categorical traces of 3-representations  2016
4 The induced strict 2-categorical action on the induced 2-category  2026
5 The 3-character of the induced strict 2-categorical action  2029
6 The categorical action of the centralizer of f on 𝕋r_{2}ρ_{f}  2034

1. Introduction

The notion of a group acting on a category goes back to Grothendieck’s Tohoku paper [15]. Recently Ganter, Kapranov [13] and Bartlett [5] categorified the concept of the trace of a linear transformation by introducing the notion of the category trace. This is a set associated to any endofunctor on a small category, and is a vector space in the linear case. Moreover, a functor commuting with the endofunctor defines a linear transformation on this vector space, whose ordinary trace defines a joint trace. This allowed these authors to define 2-characters. When a group acts on a k-linear category, the joint trace of a commuting pair of group elements is the 2-character of the categorical

Supported by National Nature Science Foundation in China (No. 11171298; No. 11571305) Received by the editors 2015-02-14 and, in revised form, 2015-12-23. Transmitted by Larry Breen. Published on 2015-12-31. 2010 Mathematics Subject Classification: 18D05; 18D99; 20J99; 20C99. Key words and phrases: the 3-representation of a group in a 3-category; the 2-categorical trace; the

3-cocycle condition; the induced strict 2-categorical actions; the 3-character; 2-categorification.. \begin{matrix}c \\ ◯ \end{matrix} Wei Wang, 2015. Permission to copy for private use granted.

action. This is an analogue of the character of the representation of a group on a vector space and is a 2-class function. In general, an n-class function is a function defined on ntuples of commuting elements of a group and invariant under simultaneous conjugation. Such functions already appear in equivariant Morava E-theory [16]. The theory of 2representations was developed further in [6] [10] [11] [12] [14] [23] [26] etc..

During the past two decades an active direction of research has been the categorification of some algebraic, geometric or analytic concepts. For example, 2-vector spaces, 2-bundles (gerbes), 2-connections and 2-curvatures. All involve 2-categorical constructions and have various applications, such as a geometric definition of elliptic cohomology [1], 2-gauge theory [3] [4] and the 2-dimensional Langlands correspondence [17] [22]. It is believed that higher categorification is necessary for many geometric and physical applications. 3-categorical constructions already appear in the theory of 2-gerbes (3-bundles) [7]

[8] and in 3-gauge theory [21] [24] [27], which involves more general Gray-categories. The purpose of this paper is to 2-categorify the theory of group representations and characters by introducing the notions of the 3-representation of a group in a 3-category, the strict 2-categorical action of a group on a 2-category and the 2-categorical trace. The problem of investigating representations of groups in higher categories has already been mentioned in [13].

A geometric motivation for considering higher representations of groups is as follows. Suppose that G is a Lie group and that H is a Lie subgroup. Let V be a finite dimensional representation of H. We can construct a homogeneous vector bundle G ×_{H} V over the homogeneous space G/H as G × V modulo the equivalent relation

(g, v) ∼ (gh, h^{−1}.v) for g ∈ G, h ∈ H, v ∈ V.

The space of sections of this bundle is exactly the space Ind_{H}^{G}V of the induced representation. When V is a 2- or 3-representation of H, a similar construction will give us a homogeneous 2- or 3-bundle over the homogeneous space G/H. This will provide us good examples of higher bundles in higher differential geometry and higher gauge theory. But for a higher representation π of the Lie group H, the functors π(h) usually depend on h ∈ H “discontinuously”. Thus it is not easy to describe the space of “sections” of the resulting higher bundles. However, when G and H are finite, G/H is discrete, and so we have a clear picture. This is why we only consider 3-representations of a finite group in this paper.

For simplicity, we only consider strict 2- and 3-categories. A 3-representation of a group G in a 3-category is given by a 1-isomorphism for each element of G, a 2-isomorphism for each pair of elements of G, and a 3-isomorphism for each triple of elements of G. These 3isomorphisms must satisfy the 3-cocycle condition. This condition has a simple geometric interpretation: the composition of 3-isomorphisms corresponding to 5 tetrahedrons in the boundary of a 4-simplex is equal to the identity 3-arrow. Given a 2-category V, a strict 2-categorical action of G on V is given by an endofunctor of V for each element of G, a pseudonatural transformation between functors for each pair of elements of G, and a modification for each triple of elements of G. Details are given in Section 2.3-2.4.

Recall that given a 2-representation ϱ of a finite group G in a 2-category V and an element f of G, we have a 1-isomorphism

ϱ_{f} : x → x,

where x is an object of V that G acts on. In [5] [13], the authors introduced the notion of the categorical trace 𝕋rϱ_{f}. This is the set of 2-arrows in V, whose 1-source is the unit arrow 1_{x} and whose 1-target is ϱ_{f}. The centralizer of f in G acts on this set naturally. In our case, given a 3-representation ρ of G in a 3-category C and an element f of G, we have a 1-isomorphism ρ_{f} : x → x in C. The 2-categorical trace 𝕋r_{2}ρ_{f} is a category. Its objects are 2-arrows with 1-source the unit arrow 1_{x} and 1-target ρ_{f}, and its morphisms are 3-isomorphisms between such 2-isomorphisms:

[figure: x; x; 1_{x}; ρ_{f}; χ] , [figure: x; x; 1_{x}; ρ_{f}; χ; χ′] .

Moreover, the centralizer of f in G, denoted by C_{G}(f), acts categorically on the 2categorical trace 𝕋r_{2}ρ_{f} in the following sense. We can define an invertible functor ψ_{g} acting on 𝕋r_{2}ρ_{f} for each g ∈ C_{G}(f), and for any h, g ∈ C_{G}(f), define a natural isomorphism

Γ_{h,g} : ψ_{h} ∘ ψ_{g} ⟶ ψ_{hg}

between such functors on the category 𝕋r_{2}ρ_{f}. This construction is given in Section 3. To prove the action to be categorical, we have to show the associativity in the definition of categorical action, i.e.,

Γ_{k,hg}#(ψ_{k} ∘ Γ_{h,g}) = Γ_{kh,g}#(Γ_{k,h} ∘ ψ_{g}) : ψ_{k} ∘ ψ_{h} ∘ ψ_{g} ⟶ ψ_{khg},  (1)

for any k, h, g ∈ C_{G}(f), where # is the composition of natural transformations between functors on the category 𝕋r_{2}ρ_{f}. This is the most difficult and technical part of this paper. By applying the 3-cocycle identity (15) repeatedly, we prove in Section 6 that

{ψ_{g}, Γ_{h,g}}_{g,h∈C_{G}(f)}

is a categorical action of the centralizer C_{G}(f) on the category 𝕋r_{2}ρ_{f}.

An easy and interesting example of 3-representations is the 1-dimensional one, which is given by a 3-cocycle on a finite group G. A 3-cocycle is a function c : G × G × G ⟶ k^{∗} such that

c(g_{3}, g_{2}, g_{1})c(g_{4}, g_{3}g_{2}, g_{1})c(g_{4}, g_{3}, g_{2}) = c(g_{4}, g_{3}, g_{2}g_{1})c(g_{4}g_{3}, g_{2}, g_{1})  (2)

for any g_{4}, …, g_{1} ∈ G. Here k is a field of characteristic 0. Such a 3-cocycle gives us a strict action of G on a 2-category with only one object, one 1-arrow and the set of 2arrows isomorphic to k^{∗}. For an element f of G, its 2-categorical trace 𝕋r_{2}ρ_{f} is a category with only one object and the set of 1-arrows isomorphic to k^{∗}. For any h and g in the centralizer C_{G}(f), we can construct an element Γ_{h,g} (48) from the 3-cocycle c in (2) such that Γ_{∗,∗} is a 2-cocycle on the centralizer. This can be proved quite easily and elementarily by using the condition (2) for 3-cocycles repeatedly in Section 6.1. This corresponds step by step to the proof of the general case carried out in Section 6.4. It can be viewed as a simple model of the proof of (1). The difficulty in the general case is that we have to handle diagrams, while in the 1-dimensional case we only need to handle element of the field k.

Suppose that C is a k-linear 3-category. Then 𝕋r_{2}ρ_{f} is also a k-linear category. If k, g and f are pairwise commutative, then ψ_{k} and ψ_{g} are k-linear endofunctors acting on 𝕋r_{2}ρ_{f}. We define the 3-character of a 3-representation ρ to be

χ_{ρ}(f, g, k) := the joint trace of functors ψ_{k} and ψ_{g} on 𝕋r_{2}ρ_{f}.

It is the trace of the linear transformation induced by the functor ψ_{k} on the k-vector space 𝕋rψ_{g}.

Suppose that a subgroup H of a finite group G acts strictly 2-categorically on a 2category V. In Section 4, we define the induced 2-category Ind_{H}^{G}(V) and strict 2-categorical action of G on it. In Section 5, we calculate the 2-categorical trace of the induced strict 2-categorical action as

𝕋r_{2}(Ind_{H}^{G}ρ) = Ind_{Λ(H)}^{Λ(G)}𝕋r_{2}(ρ),  (3)

where Λ(H) and Λ(G) are initia groupoids associated to groups H and G, respectively. As a corollary, we derive the 3-character of the induced strict 2-categorical action, which coincides with the formula in [16] for n-characters when n = 3. These results are the generalization of induced categorical action and the 2-character formula in [13].

It would be interesting to investigate the m-representation of a group in an m-category, the m-cocycle condition and (m − 1)-categorical trace for a positive integer m > 3.

I would like to thank the anonymous referee for his/her many inspiring and valuable suggestions.

2. The 3-representations of groups

2.1. Strict 2-categories. A 2-category is a category enriched over the category of all small categories. In particular, a strict 2-category C consists of collections C_{0} of objects, C_{1} of arrows and C_{2} of 2-arrows, together with

• functions s_{n}, t_{n} : C_{i} → C_{n} for all 0 ≤ n < i ≤ 2, called n-source and n-target,

• functions #_{n} : C_{n+1} × C_{n+1} → C_{n+1} for all n = 0, 1, called vertical composition,

• a function #_{0} : C_{2} × C_{2} → C_{2}, called the horizontal composition,

• a function 1_{∗} : C_{i} → C_{i+1} for i = 0, 1, called the identity.

For a 1-arrow [figure: x; y; A], its 0-source and 0-target are x and y, respectively. For a 2-arrow [figure: x; y; φ; A; B] in C_{2}, its 1-source and 1-target are [figure: x; y; A] and [figure: x; y; B],

respectively, while its 0-source and 0-target are x and y, respectively.

Two 1-arrows A and A′ are called 0-composable if the 0-target of A coincides with the 0-source of A′. In this case, their vertical composition is A#_{0}A′ : [figure: x; y; z; A; A′]. Two 2-arrows ϕ and ψ are called 1-composable if the 1-target of ϕ coincide with the 1-source of ψ. In this case, their vertical composition ϕ#_{1}ψ is

[figure: x; y; B; A; C; ϕ; ψ],

where A = s_{1}(ϕ), B = t_{1}(ϕ) = s_{1}(ψ), C = t_{1}(ψ), x = s_{0}(ϕ) = s_{0}(ψ), y = t_{0}(ϕ) = t_{0}(ψ). In general, two arrows are composable if the target matching condition is satisfied.

Two 2-arrows ϕ and ψ are called horizontally composable (0-composable) if the 0-target of ϕ coincides with the 0-source of ψ. In this case, their horizontal composition ϕ#_{0}ψ is

[figure: x; y; z; A; C; ϕ; B; D; ψ].

In particular, when ϕ = 1_{A} we call 1_{A}#_{0}ψ whiskering from left by 1-arrow A, and denote it by

A#_{0}ψ : [figure: y; x; z; ψ; B; D; A],

Similarly, we define whiskering from right by a 1-arrow.

The identities satisfy

1_{x}#_{0}A = A = A#_{0}1_{y}, for any 1 − arrow A : x ⟶ y;
1_{A}#_{1}ϕ = ϕ = ϕ#_{1}1_{B}, for any 2 − arrow ϕ : A ⟹ B.

(4)

The composition #_{p} satisfies the associativity

(ϕ#_{p}ψ)#_{p}ω = ϕ#_{p}(ψ#_{p}ω),  (5)

if the corresponding arrows are p-composable, for p = 0 or 1.

The horizontal composition satisfies the interchange law:

(A#_{0}ψ)#_{1}(ϕ#_{0}D) = ϕ#_{0}ψ = (ϕ#_{0}B)#_{1}(C#_{0}ψ).  (6)

Namely,

[figure: x; y; z; A; B; D; ψ; x; y; z; A; C; ϕ; D; x; y; z; A; C; ϕ; B; x; y; z; C; B; D; ψ]

the vertical composition of left two 2-arrows coincides with the vertical composition of right two 2-arrows. They are both equal to the horizontal composition ϕ#_{0}ψ. The interchange law allows us to change the order of compositions of 2-arrows, up to whiskerings. This is essentially the paste theorem for 2-categories (cf. §2.13 in [18]).

The interchange law (6) is a special case of the following more general compatibility condition for different compositions. If (β, β′), (γ, γ′) ∈ C_{k} × C_{k} are p-composable and (β, γ), (β′, γ′) ∈ C_{k} × C_{k} are q-composable, p, q = 0, 1, then we have

(β#_{p}β′)#_{q}(γ#_{p}γ′) = (β#_{q}γ)#_{p}(β′#_{q}γ′).  (7)

The the left-hand side of the interchange law (6) is exactly the compatibility condition (7) with p = 0, q = 1, β = 1_{A}, β′ = ψ, γ = ϕ, γ′ = 1_{D}, by using the property (4) of identities.

(4) (5) and (7) are the main axioms that a strict 2-category satisfies.

A 1-arrow A : x → y is called invertible or a 1-isomorphism, if there exists another 1-arrow B : y → x such that 1_{x} = A#_{0}B and B#_{0}A = 1_{y}. A strict 2-category in which every 1-arrow is invertible is called a strict 2-groupoid. A 2-arrow φ : A ⇒ B is called invertible or a 2-isomorphism if there exists another 2-arrow ψ : B ⇒ A such that ψ#_{1}φ = 1_{B} and φ#_{1}ψ = 1_{A}. ψ is uniquely determined and called the inverse of φ.

Let S and T be two strict 2-categories. A (strict) 2-functor F : S → T is an assignment of a 2-arrow

[figure: F (X); F (Y ); F (φ); F (f); F (g)]

to each 2-arrow [figure: X; Y; φ; f; g] such that F preserves compositions #_{p} and identities. More

explicitly, we have

• F (φ#_{1}ψ) = F (φ)#_{1}F (ψ) and F (1_{f}) = 1_{F (f)} for all composable 2-arrows φ and ψ and any 0- or 1-arrow f;

• F (g)#_{0}F (f) = F (g#_{0}f) for all composable 1-arrows g and f, and F (φ)#_{0}F (ψ) = F (φ#_{0}ψ) for all horizontally composable 2-arrows φ and ψ.

Let F_{1} and F_{2} be two 2-functors from S to T . A pseudonatural transformation ρ : F_{1} → F_{2} is an assignment of a 1-arrow ρ(X) in T to each object X in S and a 2-isomorphism ρ(f)

[figure: F_{1}(X); F_{1}(Y ); F_{2}(X); F_{2}(Y ); F_{1}(f); F_{2}(f); ρ(X); ρ(Y ); ρ(f)]  (8)

in T to each 1-arrow f : X → Y in S such that they satisfy two axioms • The composition of 1-arrows in S:

[figure: F_{1}(X); F_{1}(Y ); F_{2}(X); F_{2}(Y ); F_{1}(f); F_{2}(f); ρ(X); ρ(Y ); ρ(f); F_{1}(Z); F_{2}(Z); F_{1}(g); F_{2}(g); ρ(Z); ρ(g)]  [figure: F_{1}(X); F_{1}(Z); F_{2}(X); F_{2}(Z); F_{1}(f#_{0}g); F_{2}(f#_{0}g); ρ(X); ρ(Z); ρ(f#_{0}g)] ;  (9)

• The compatibility with 2-arrows:

[figure: F_{1}(X); F_{1}(Y ); F_{2}(X); F_{2}(Y ); F_{1}(f); F_{2}(f); ρ(X); ρ(Y ); ρ(f); F_{2}(g); F_{2}(φ); F_{1}(X); F_{1}(Y ); F_{2}(X); F_{2}(Y ); F_{1}(g); F_{2}(g); ρ(X); ρ(Y ); ρ(g); F_{1}(f); F_{1}(φ)]  (10)

for any 2-arrow φ : f ⇒ g.

Let F_{1}, F_{2} : S → T be two strict 2-functors and let ρ_{1}, ρ_{2} : F_{1} → F_{2} be pseudonatural transformations. A modification Φ : ρ_{1} [figure] ρ_{2} is an assignment of a 2-arrow

[figure: F_{1}(X); F_{2}(X); Φ(X); ρ_{1}(X); ρ_{2}(X)]

in T to any object X in S, which satisfies

[figure: F_{1}(X); F_{1}(Y ); F_{2}(X); F_{2}(Y ); F_{1}(f); F_{2}(f); ρ_{1}(X); ρ_{1}(Y ); ρ_{1}(f); ρ_{2}(X); Φ(X)]  [figure: F_{1}(X); F_{1}(Y ); F_{2}(X); F_{2}(Y ); F_{1}(f); F_{2}(f); ρ_{2}(X); ρ_{2}(Y ); ρ_{2}(f); ρ_{1}(Y ); Φ(Y )] .  (11)

2.2. Strict 3-categories. A 3-category is a category enriched over the category of all small strict 2-categories. In particular, a strict 3-category C consists of collections C_{0} of objects, C_{1} of 1-arrows, C_{2} of 2-arrows, and C_{3} of 3-arrows, together with

• functions s_{n}, t_{n} : C_{i} → C_{n} for all 0 ≤ n < i ≤ 3, called n-source and n-target,

• functions #_{n} : C_{n+1} × C_{n+1} → C_{n+1} for all n = 0, 1, 2, called vertical composition,

• a function #_{p} : C_{i} × C_{i} → C_{i}, p + 2 ≤ i, called the horizontal composition,

• a function 1_{∗} : C_{i} → C_{i+1} for i = 0, 1, called identity.

For a 3-arrow φ : [figure: x; y; f; f′; γ; γ′], its 2-source and 2-target are γ and γ′ respectively.

The 3-arrows φ and φ′ : [figure: x; y; f; f′; γ′; γ′′] are 2-composable, and their composition φ#_{2}φ′ is

[figure: x; y; f; f′; γ; γ′′].

In a strict 3-category, 0-, 1- and 2-arrows behave as in a 2-category. We call two 3arrows φ and ψ horizontally p-composable if the p-target of φ coincides with the p-source of ψ, p = 0, 1, and denote their horizontal composition as φ#_{p}ψ.

For a 2-arrow δ, 3-arrows 1_{δ} and φ are horizontally 1-composable if the 1-target of δ coincides with the 1-source of φ. In this case,

δ#_{1}φ := 1_{δ}#_{1}φ [figure: x; y; f; g; δ; f′; γ; γ′],

is called whiskering from above by a 2-arrow δ. It is similar to define whiskering from below:

[figure: x; y; f; g′; η; f′; γ; γ′]

There is also whiskering from left ( or right) by a 1-arrow A#_{0}φ := 1_{1_{A}}#_{0}φ (or φ#_{0}B):

[figure: x; z; w; y; f; A; B; f′; γ; γ′]

The properties of identities, the associativity and the compatibility condition for different compositions, similar to (4) (5) and (7) for a strict 2-category, also hold in a strict 3-category. See page 8 of [19] for an explicit definition of a strict m-category.

A strict 3-functor (or a functor) is a map preserving compositions and identities.

2.3. Remark. In a strict 3-category, the interchange law (6) for the horizontal composition of 2-arrows is also satisfied. But in general, a 3-category does not satisfy the interchange law. Gray-categories are the greatest possible semi-strictification of 3-categories, and appear naturally in 3-gauge theory [27]. The 3-representation in a Gray-category is more natural, but is much more complicated. So we restrict to the 3-representation in strict 3-categories in this paper.

In a strict 3-category C, a 1-arrow B : x → y is called a 1-isomorphism if there exists 1-arrow C : y → x such that there exist 2-isomorphisms u : 1_{y} ⟹ C#_{k}B and v : 1_{x} ⟹ B#_{k}C. We call C a quasi-inverse to B, and vise versa. However, when k = 2 or 3, we call a k-arrow a k-isomorphism if it is strictly invertible.

2.4. The 3-representations of a group in a strict 3-category . Let C be a strict 3-category and let G be a group. G can be viewed as a strict 3-category with only one object •, G as the set of 1-arrows g : • → •, the set of 2-arrows consisting of the identities of 1-arrows, and the set of 3-arrows consisting of the identities of 2-arrows. A 3-representation of a group G in C is a weak functor ρ from G to C in the following sense. We have

(1) an object x of C;

(2) for each g ∈ G, a 1-isomorphism ρ_{g} : x → x;

(3) for each h, g ∈ G, a 2-isomorphism ϕ_{h,g} : ρ_{h}ρ_{g} ⟹ ρ_{hg} (here and in the following we write ρ_{h}#_{0}ρ_{g} as ρ_{h}ρ_{g} for simplicity), corresponding to the 2-cell

[figure: •; •; •; ρ_{hg}; ρ_{h}; ρ_{g}; ϕ_{h,g}];

(4) for each g_{3}, g_{2}, g_{1} ∈ G, a 3-isomorphism, called the associator,

Φ_{g_{3},g_{2},g_{1}} : (ρ_{g_{3}}#_{0}ϕ_{g_{2},g_{1}})#_{1}ϕ_{g_{3},g_{2}g_{1}}  [figure](ϕ_{g_{3},g_{2}}#_{0}ρ_{g_{1}})#_{1}ϕ_{g_{3}g_{2},g_{1}},  (12)

corresponding to the 3-cell

[figure: •; •; •; •; ρg_{3}; ρg_{3}g_{2}; ρg_{2}; ρg_{3}g_{2}g_{1}; ρg_{2}g_{1}; ρg_{1}].

It can be viewed as exchanging the diagonals of the quadrilateral:

[figure: •; •; •; •; ρg_{3}; ρg_{2}; ρg_{3}g_{2}g_{1}; ρg_{2}g_{1}; ρg_{1}; ϕg_{2},g_{1}; ϕg_{3},g_{2}g_{1}]  [figure] [figure: •; •; •; •; ρg_{3}; ρg_{3}g_{2}; ρg_{2}; ρg_{3}g_{2}g_{1}; ρg_{1}; ϕg_{3}g_{2},g_{1}; ϕg_{3},g_{2}],  (13)

which can also be drawn in the following form:

[figure: x; x; x; x; ϕg_{3},g_{2}g_{1}; ϕg_{2},g_{1}; ρg_{1}; ρg_{2}; ρg_{3}; ρg_{3}g_{2}g_{1}; ρg_{2}g_{1}]  [figure] [figure: x; x; x; x; ϕg_{3}g_{2},g_{1}; ϕg_{3},g_{2}; ρg_{1}; ρg_{2}; ρg_{3}; ρg_{3}g_{2}g_{1}; ρg_{3}g_{2}]; (14)

(5) a 2-isomorphism ϕ_{1} : ρ_{1} ⟹ 1_{x}; such that the following conditions are satisfied:

• ϕ_{1,g} = ϕ_{1}#_{0}ρ_{g}, ϕ_{g,1} = ρ_{g}#_{0}ϕ_{1}.

• the 3-cocycle condition that for any g_{4}, …, g_{1} ∈ G, we have

\begin{matrix}\{[ρ_{g_{4}}#_{0}Φ_{g_{3},g_{2},g_{1}}]#_{1}ϕ_{g_{4},g_{3}g_{2}g_{1}}\} #_{2} \{[ρ_{g_{4}}#_{0}ϕ_{g_{3},g_{2}}#_{0}ρ_{g_{1}}]#_{1}Φ_{g_{4},g_{3}g_{2},g_{1}}\} \\ #_{2} \{[Φ_{g_{4},g_{3},g_{2}}#_{0}ρ_{g_{1}}]#_{1}ϕ_{g_{4}g_{3}g_{2},g_{1}}\} \\ = \{[(ρ_{g_{4}}ρ_{g_{3}})#_{0}ϕ_{g_{2},g_{1}}]#_{1}Φ_{g_{4},g_{3},g_{2}g_{1}}\} #_{2} \{[ϕ_{g_{4},g_{3}}#_{0}(ρ_{g_{2}}ρ_{g_{1}})]#_{1}Φ_{g_{4}g_{3},g_{2},g_{1}}\} . \end{matrix}  (15)

[figure: •; •; •; •; •; ρg_{4}; ρg_{3}; ρg_{2}; ρg_{4}g_{3}g_{2}g_{1}; ρg_{2}g_{1}; ρg_{1}].

Equivalently, the composition of the 3-isomorphisms represented by 5 tetrahedrons above in the boundary of a 4-simplex is the identity. This comes from the fact that the boundary of the corresponding 4-simplex in the 3-category G is the identity 3-arrow.

2.5. Remark. (1) For simplicity, we assume in this paper that ρ_{1} = 1_{x} and that ϕ_{1} is the identity.

(2) The 3-cocycle {Φ_{g_{3},g_{2},g_{1}}} defines an element of the 3-dimensional non-abelian cohomology. A first attempt at an explicit description of the 3-dimensional non-abelian cohomology of a group goes back to Dedecker [9]. See section 4 of [7] for 3-dimensional non-abelian Čech cocycles, which can be used to construct a 2-gerbe.

2.6. The 3-cocycle condition . We will give a clear geometric description of the 3-cocycle condition (15) in terms of 5 tetrahedrons in the boundary of a 4-simplex above. This is equivalent to triviality of the 3-holonomy. See section 5 C of [27] for the 3-holonomy in the lattice 3-gauge theory (the cubical case), where 3-gauge theory from the point of view of Gray-categories is investigated.

In the left-hand side of the 3-cocycle condition (15), the first 3-isomorphism is

A_{1} = [ρ_{g_{4}}#_{0}Φ_{g_{3},g_{2},g_{1}}]#_{1}ϕ_{g_{4},g_{3}g_{2}g_{1}}.  (16)

Here Φ_{g_{3},g_{2},g_{1}} is a 3-isomorphism whiskered from left by the 1-isomorphism ρ_{g_{4}}, and ρ_{g_{4}}#_{0}Φ_{g_{3},g_{2},g_{1}} is whiskered from below by the 2-isomorphism ϕ_{g_{4},g_{3}g_{2}g_{1}}. A_{1} corresponds to the 3-cell

[figure: •; •; •; •; •; ρ_{b}; ρa; ρg_{4}; ρg_{3}; ρg_{4}g_{3}g_{2}g_{1}; ρg_{2}; ρg_{2}g_{1}; ρ_{g_{1}}; The 3−arrow A_{1}; ϕg_{4},g_{3}g_{2}g_{1}]

whose 2-source and 2-target are the 2-isomorphisms

\begin{matrix}s_{2}(A_{1}) = [(ρ_{g_{4}}ρ_{g_{3}})#_{0}ϕ_{g_{2},g_{1}}]#_{1}[ρ_{g_{4}}#_{0}ϕ_{g_{3},g_{2}g_{1}}]#_{1}ϕ_{g_{4},g_{3}g_{2}g_{1}} : ρ_{g_{4}}ρ_{g_{3}}ρ_{g_{2}}ρ_{g_{1}} ⟶ ρ_{g_{4}g_{3}g_{2}g_{1}}, \\ t_{2}(A_{1}) = [ρ_{g_{4}}#_{0}ϕ_{g_{3},g_{2}}#_{0}ρ_{g_{1}}]#_{1}[ρ_{g_{4}}#_{0}ϕ_{g_{3}g_{2},g_{1}}]#_{1}ϕ_{g_{4},g_{3}g_{2}g_{1}} : ρ_{g_{4}}ρ_{g_{3}}ρ_{g_{2}}ρ_{g_{1}} ⟶ ρ_{g_{4}g_{3}g_{2}g_{1}}, \end{matrix}

(17)

corresponding to 2-cells

[figure: •; •; •; •; •; ρg_{4}; ρg_{3}; ρg_{4}g_{3}g_{2}g_{1}; ρg_{2}; ρg_{2}g_{1}; ρg_{1}; The 2−arrow s_{2}(A_{1}); •; •; •; •; •; ρ_{b}; ρa; ρg_{4}; ρg_{3}; ρg_{2}; ρg_{4}g_{3}g_{2}g_{1}; ρg_{1}; The 2−arrow t_{2}(A_{1})]

respectively, where ρ_{a} := ρ_{g_{3}g_{2}}, ρ_{b} := ρ_{g_{3}g_{2}g_{1}}. It is fundamental in this paper to write down the p-arrow corresponding to p-cells as whiskered vertical compositions. For example, s_{2}(A_{1}) in (17) is the composition of the following three whiskered 2-isomorphisms.

[figure: •; •; •; •; •; ρg_{4}; ρg_{3}; ρg_{2}; ρg_{2}g_{1}; ρg_{1}; ϕg_{2},g_{1}]  [figure: •; •; •; •; ρg_{4}; ρg_{3}; ρg_{3}g_{2}g_{1}; ρg_{2}g_{1}; ϕg_{3},g_{2}g_{1}]  [figure: •; •; •; ρg_{4}; ρg_{4}g_{3}g_{2}g_{1}; ρg_{3}g_{2}g_{1}; ϕg_{4},g_{3}g_{2}g_{1}].

The second 3-isomorphism in the left-hand side of the 3-cocycle condition (15) is A_{2} = [ρ_{g_{4}}#_{0}ϕ_{g_{3},g_{2}}#_{0}ρ_{g_{1}}]#_{1}Φ_{g_{4},g_{3}g_{2},g_{1}}, corresponding to the 3-cell

[figure: •; •; •; •; •; ρg_{4}; ρg_{3}; ρ_{a}; ρ_{b}; ρg_{2}; ρg_{1}; The 3−arrow A_{2}]

(here ρ_{a} := ρ_{g_{4}g_{3}g_{2}}, ρ_{b} := ρ_{g_{3}g_{2}}) with 2-source s_{2}(A_{2}) = t_{2}(A_{1}) in (17) and 2-target

t_{2}(A_{2}) = [ρ_{g_{4}}#_{0}ϕ_{g_{3},g_{2}}#_{0}ρ_{g_{1}}]#_{1}[ϕ_{g_{4},g_{3}g_{2}}#_{0}ρ_{g_{1}}]#_{1}ϕ_{g_{4}g_{3}g_{2},g_{1}}  (18)

corresponding to 2-cells

[figure: •; •; •; •; •; ρg_{4}; ρg_{3}; ρa; ρ_{b}; ρg_{2}; ρg_{4}g_{3}g_{2}g_{1}; ρg_{1}; The 2−arrow t_{2}(A_{2})].

And the third 3-isomorphism in the left-hand side of the 3-cocycle condition (15) is

A_{3} = [Φ_{g_{4},g_{3},g_{2}}#_{0}ρ_{g_{1}}]#_{1}ϕ_{g_{4}g_{3}g_{2},g_{1}},

corresponding to the 3-cell

[figure: •; •; •; •; •; ρg_{4}; ρg_{3}; ρa; ρg_{2}; ρg_{1}; ϕ_{b}; The 3−arrow A_{3}]

(here ρ_{a} := ρ_{g_{4}g_{3}g_{2}}, ϕ_{b} := ϕ_{g_{4}g_{3}g_{2},g_{1}}) with 2-source s_{2}(A_{3}) = t_{2}(A_{2}) in (18) and 2-target

t_{2}(A_{3}) = [ϕ_{g_{4},g_{3}}#_{0}(ρ_{g_{2}}ρ_{g_{1}})]#_{1}[ϕ_{g_{4}g_{3},g_{2}}#_{0}ρ_{g_{1}}]#_{1}ϕ_{g_{4}g_{3}g_{2},g_{1}},  (19)

corresponding to the 2-cells

[figure: •; •; •; •; •; ρg_{4}; ρ_{b}; ρg_{3}; ρa; ρg_{2}; ρg_{4}g_{3}g_{2}g_{1}; ρg_{1}; The 2−arrow t_{2}(A_{3})],  (20)

where ρ_{a} := ρ_{g_{4}g_{3}g_{2}}, ρ_{b} := ρ_{g_{4}g_{3}}. Then the composition A_{1}#_{2}A_{2}#_{2}A_{3} of 3-isomorphisms is the left-hand side of the 3-cocycle condition (15), whose 2-source is s_{2}(A_{1}) in (17) and 2-target is t_{2}(A_{3}) in (19).

On the right-hand side of the 3-cocycle condition (15), the first 3-isomorphism is

A′_{1} = [(ρ_{g_{4}}ρ_{g_{3}})#_{0}ϕ_{g_{2},g_{1}}]#_{1}Φ_{g_{4},g_{3},g_{2}g_{1}},

corresponding to the 3-cell

[figure: •; •; •; •; •; ρg_{4}; ρg_{3}; ρg_{2}; ρg_{2}g_{1}; ρg_{1}; The 3−arrow A′_{1}]

with 2-source s_{2}(A_{1}) in (17) and 2-target

t_{2}(A′_{1}) = [(ρ_{g_{4}}ρ_{g_{3}})#_{0}ϕ_{g_{2},g_{1}}]#_{1}[ϕ_{g_{4},g_{3}}#_{0}ρ_{g_{2}g_{1}}]#_{1}ϕ_{g_{4}g_{3},g_{2}g_{1}},  (21)

corresponding to the left 2-cells in the following diagram:

[figure: •; •; •; •; •; (2); (1); (3); ρg_{4}; ρg_{4}g_{3}; ρg_{3}; ρg_{2}; ρg_{2}g_{1}; ρg_{1}; The 2−arrow t_{2}(A′_{1})] [figure: •; •; •; •; •; (1); (2); (3); ρg_{4}; ρg_{4}g_{3}; ρg_{3}; ρg_{2}; ρg_{1}; The 2−arrow s_{2}(A′_{2})].

By the interchange law (6) for horizontal compositions, we can interchange 2-isomorphism (1) and (2) identically in the left 2-cells above to get the 2-isomorphism

s_{2}(A′_{2}) = [ϕ_{g_{4},g_{3}}#_{0}(ρ_{g_{2}}ρ_{g_{1}})]#_{1}[ρ_{g_{4}g_{3}}#_{0}ϕ_{g_{2},g_{1}}]#_{1}ϕ_{g_{4}g_{3},g_{2}g_{1}},  (22)

corresponding to the right 2-cells above. The last 3-isomorphism is

A′_{2} = [ϕ_{g_{4},g_{3}}#_{0}(ρ_{g_{2}}ρ_{g_{1}})]#_{1}Φ_{g_{4}g_{3},g_{2},g_{1}}

[figure: •; •; •; •; •; ρg_{4}; ρg_{4}g_{3}; ρg_{3}; ρg_{2}; ρg_{1}; ϕg_{4},g_{3}; The 3−arrow A′_{2}]

whose 2-target is exactly the 2-isomorphism t_{2}(A_{3}) in (19)-(20).

It is not easy to draw several 3-cells corresponding to the composition of 3-arrows in a 3-category C. For this reason, let us consider the associated 2-category C^{+} such that

(C^{+})_{i} := C_{i+1},

and i-source and i-target are s_{i+1} and t_{i+1}, i = 0, 1, 2, respectively. Functions ˜#_{p} : C_{k}^{+} × C_{k}^{+} ⟶ C_{k}^{+} are described by arrows #_{p+1} : C_{k+1} × C_{k+1} ⟶ C_{k+1}, and identities ˜1 : C_{k−1}^{+} → C_{k}^{+} are defined in a similar manner. C^{+} is a strict 2-category since Hom_{C}(x, y) is a strict 2-category for any objects x, y of C, by the fact that a strict 3-category is a category enriched over the category of all small strict 2-categories. We also define C^{++} to be the category with

(C^{++})_{i} := C_{i+2}

and the i-source and i-target are now s_{i+2} and t_{i+2}, i = 0, 1, respectively. The function ˜#_{0} : C_{1}^{++} × C_{1}^{++} ⟶ C_{1}^{++} becomes #_{2} : C_{3} × C_{3} ⟶ C_{3}. C^{++} is a category by the same reason.

In the corresponding strict 2-category C^{+}, 3-isomorphism A_{1} in (16) is represented by the following 2-isomorphism:

[figure: ρ_{g_{4}}ρ_{g_{3}}ρ_{g_{2}}ρ_{g_{1}}; ρ_{g_{4}}ρ_{g_{3}g_{2}}ρ_{g_{1}}; ρ_{g_{4}}ρ_{g_{3}}ρ_{g_{2}g_{1}}; ρ_{g_{4}}ρ_{g_{3}g_{2}g_{1}}; ρ_{g_{4}g_{3}g_{2}g_{1}}; ρ_{g_{4}}#_{0}ϕ_{g_{3},g_{2}}#_{0}ρ_{g_{1}}; ρ_{g_{4}}#_{0}ϕ_{g_{3},g_{2}g_{1}}; (ρ_{g_{4}}ρ_{g_{3}})#_{0}ϕ_{g_{2},g_{1}}; ρ_{g_{4}}#_{0}ϕ_{g_{3}g_{2},g_{1}}; ϕ_{g_{4},g_{3}g_{2}g_{1}}; A_{1}]  (23)

Here the upper and lower boundaries in (23) (as 1-arrows in C^{+}) represent the source s_{2}(A_{1}) and target t_{2}(A_{1}) in (17) (as 2-isomorphisms in C) respectively. To draw the picture neatly, we omit the whiskering parts. Then the 3-cocycle condition (15) can be expressed simply as an identity of 2-isomorphisms in C^{+} as follows:

[figure: •; •; •; •; •; •; •; ϕg_{4},g_{3}; ϕg_{3},g_{2}; ϕg_{2},g_{1}; ϕg_{4}g_{3},g_{2}; ϕa; ϕg_{4},g_{3}g_{2}; ϕg_{4}g_{3}g_{2},g_{1}; ϕg_{3},g_{2}g_{1}; ϕg_{4},g_{3}g_{2}g_{1}; A_{1}; A_{2}; A_{3}; •; •; •; •; •; •; •; ϕg_{4},g_{3}; ϕg_{2},g_{1}; ϕg_{2},g_{1}; ϕg_{4}g_{3},g_{2}; ϕ_{b}; ϕg_{4},g_{3}; ϕg_{4}g_{3}g_{2},g_{1}; ϕg_{3},g_{2}g_{1}; ϕg_{4},g_{3}g_{2}g_{1}; A′_{2}; A′_{1}]

(24)

where ϕ_{a} := ϕ_{g_{3}g_{2},g_{1}}, ϕ_{b} := ϕ_{g_{4}g_{3},g_{2}g_{1}}. Here •’s above represent 1-isomorphisms in C. The 2-isomorphisms in (17), (18), (19), (21) and (22) are represented by 1-isomorphisms in (24). Now the 3-cocycle condition (24) can be viewed as the commutativity of the 2-isomorphisms in the boundary of the following cube in C^{+}:

[figure: •; •; •; •; •; •; •; •; ϕg_{4},g_{3}; ϕg_{3},g_{2}; ϕg_{2},g_{1}; ϕg_{4}g_{3},g_{2}; ϕa; ϕg_{4}g_{3}g_{2},g_{1}; ϕg_{3},g_{2}g_{1}; ϕg_{4},g_{3}g_{2}g_{1}; A_{1}; A_{2}; A_{3}; ϕg_{2},g_{1}; ϕ_{b}; A′_{2}; A′_{1}]  (25)

2.7. Remark. (1) In the upper boundaries of diagrams in (24), the number of group elements in the second subscripts of ϕ_{∗,∗}’s is increasing: g_{1}, g_{2}g_{1} g_{3}g_{2}g_{1}, while in the lower boundaries it is the number of group elements in the first subscripts of ϕ_{∗,∗}’s which are increasing: g_{4}, g_{4}g_{3} g_{4}g_{3}g_{2}.

(2) (24) or (25) is similar to the pentagon condition of bicategories, but here we actually have more complicated whiskering (cf. (23)).

Given a strict 2-category V, there exists an associated 3-category V^{∗} for which V_{0}^{∗} consists of one object V, V_{1}^{∗} consists of all functors from V to V, V_{2}^{∗} consists of all pseudonatural transformations and V_{3}^{∗} consists of all modifications. This is a 3-category. Because only 3-representations of a group in a strict 3-category are developed, we have to consider a strict 3-subcategory W of V^{∗} for a strict 2-categories V. We call a 3-representation of G in such a strict 3-subcategory W a strict 2-categorical action of G on V. In particular, we have an endofunctor ρ_{g} : V → V for each g ∈ G, a pseudonatural transformation ϕ_{h,g} : ρ_{h}#_{0}ρ_{g} ⟹ ρ_{hg} for each h, g ∈ G, and a modification Φ_{g_{3},g_{2},g_{1}} (the associator in (12)) for each g_{3}, g_{2}, g_{1} ∈ G. Here ρ_{h}#_{0}ρ_{g} is the composition of functors:

ρ_{h}#_{0}ρ_{g}(w) := ρ_{h}(ρ_{g}(w))

for w ∈ V. By the definition of 3-representations, the endofunctor ρ_{g}, the pseudonatural transformation ϕ_{h,g} and the modification Φ_{g_{3},g_{2},g_{1}} must all be invertible in W ⊂ V^{∗}.

For example, for the 2-category V used in the 1-dimensional 3-representation in Subsection 3.8, its V^{∗} is a strict 3-category. For the general action of G on a 2-category V, we need to develop 3-representation of a group in a Gray-category, since the semistrictification of a 3-category is a Gray-category.

When a 2-category V is viewed as a 3-category with only identity 3-arrow, a 3representation of G in V is a 2-representation if the the associator 3-isomorphism in (12) is the identity, so that the 3-cocycle condition (15) holds trivially. This coincides with the definition of the 2-representation in the strict sense in section 2.2 of [13]. And for a category V, a 2-representation of G in the 2-category V^{∗} is a categorical action of G on V.

3. The 2-categorical traces of 3-representations

3.1. The 2-categorical trace of a 1-endomorphism. Let C be a 3-category, x ∈ C and A : x → x be a 1-endomorphism. Then A is an object of the 2-category Hom_{C}(x, x). The 2-categorical trace of A is defined as

𝕋r_{2}(A) = Hom_{C}(1_{x}, A),

which is a category. This is a subcategory of C^{++}.

Let A : x → x be a 1-endomorphism for x ∈ C_{0}, and let the 1-arrow C : y → x be a quasi-inverse to a 1-arrow B : x → y. Then for any 2-arrow χ : 1_{x} ⟹ A in 𝕋r_{2}(A)_{0}, the composition

1_{y} ⟶^{u} C#_{0}B C#_{0}1_{x}#_{0}B ⟶^{C#_{0}χ#_{0}B} C#_{0}A#_{0}B

defines a functor

\begin{matrix}Ψ(C, B, u) : 𝕋r_{2}(A)_{0} & ⟶ 𝕋r_{2}(C#_{0}A#_{0}B)_{0}, \\ (χ : 1_{x} ⟹ A) & ⟼ u#_{1}[C#_{0}χ#_{0}B], \end{matrix}

corresponding to the diagram

[figure: x; x; y; y; C; B; 1_{x}; A; χ; u; 1_{y}];

and for any 3-arrow γ : χ [figure]χ′ in 𝕋r_{2}(A)_{1}, we have

\begin{matrix}𝕋r_{2}(A)_{1} & ⟶ 𝕋r_{2}(C#_{0}A#_{0}B)_{1}, \\ γ & ⟼ u#_{1}[C#_{0}γ#_{0}B], \end{matrix} [figure: x; x; y; y; C; B; 1_{x}; A; u; 1_{y}; χ; χ′].

3.2. Proposition. Ψ(C, B, u) : 𝕋r_{2}(A) ⟶ 𝕋r_{2}(C#_{0}A#_{0}B) is a functor.

Proof. For 2-arrows χ, χ′, χ̃ : 1_{x} ⟹ A and 3-arrows γ : χ [figure]χ′, γ̃ : χ′ [figure] χ̃, we have the composition γ#_{2}γ̃ : χ [figure]χ̃. Then by using repeatedly the compatibility condition (7) for compositions, we find

Ψ(C, B, u)(γ)#_{2}Ψ(C, B, u)(γ̃) = {u#_{1}[C#_{0}γ#_{0}B]}#_{2}{u#_{1}[C#_{0}γ̃#_{0}B]}
= u#_{1}[C#_{0}(γ#_{2}γ̃)#_{0}B]
= Ψ(C, B, u)(γ#_{2}γ̃).

Thus Ψ(C, B, u) is a functor.

3.3. The 2-categorical trace 𝕋r_{2}ρ_{f}. Let ρ be a 3-representation of G in a 3-category C. Fix an object x in C that G acts on. For f ∈ G, let ρ_{f} : x → x be a 1-isomorphism in C. Recall that 𝕋r_{2}ρ_{f} is a category whose objects are 2-arrows with source 1_{x} and target ρ_{f} and the morphisms are 3-arrows between them. In the sequel, we will use the notation

g^{∗} := g^{−1}

for simplicity. For any g commuting with f and a 2-arrow χ : 1_{x} [figure]ρ_{f} in (𝕋r_{2}ρ_{f})_{0}, we define a 2-arrow ψ_{g}(χ) : 1_{x} [figure]ρ_{f} by

ψ_{g}(χ) := u_{g}#_{1} [ρ_{g}#_{0}χ#_{0}ρ_{g^{∗}}] #_{1} [ϕ_{g,f}#_{0}ρ_{g^{∗}}] #_{1}ϕ_{gf,g^{∗}}.  (26)

This is given by the composition of 2-arrows in the following diagram

[figure: x; x; x; x; ϕ_{gf,g^{∗}}; ρ_{g}; ρ_{g^{∗}}; 1_{x}; ρ_{f}; 1_{x}; χ; u_{g}; ρ_{gf}; ρ_{gfg^{∗}}=ρ_{f}; ϕ_{g,f}],  (27)

where u_{g} = ϕ_{g,g^{∗}}^{−1} : 1_{x} ⟹ ρ_{g}ρ_{g^{∗}}. For a 3-arrow Θ : χ [figure] χ′, we define ψ_{g}(Θ) as a 3-arrow whiskered by corresponding 2-isomorphisms in (27). In other words,

ψ_{g}(Θ) = u_{g}#_{1} [ρ_{g}#_{0}Θ#_{0}ρ_{g^{∗}}] #_{1} [(ϕ_{g,f}#_{0}ρ_{g^{∗}})#_{1}ϕ_{gf,g^{∗}}] : ψ_{g}(χ) [figure] ψ_{g}(χ′)  (28)

is a 3-arrow corresponding to the diagram

[figure: x; x; x; x; ϕ_{gf,g^{∗}}; ρ_{g}; ρ_{g^{∗}}; 1_{x}; ρ_{f}; 1_{x}; u_{g}; ρ_{gf}; ρ_{gfg^{∗}}=ρ_{f}; ϕ_{g,f}; χ; χ′]

in the 3-category C. Then ψ_{g} defines an endofunctor ψ_{g} on 𝕋r_{2}ρ_{f} by the proof of Proposition 3.2. Namely, we have

ψ_{g}(Θ ˜#_{0}Θ′) = ψ_{g}(Θ) ˜#_{0}ψ_{g}(Θ′)

for any 3-arrow Θ′ : χ′ [figure]χ′′, where ˜#_{0} is the composition in the category C^{++} ( ˜#_{0} = #_{2}).

In Section 3.4, we will construction a natural isomorphism Γ_{h,g} : ψ_{h} ∘ ψ_{g} ⟶ ψ_{hg} for given g, h ∈ C_{G}(f). It gives us natural isomorphisms Γ_{g^{∗},g} : ψ_{g^{∗}} ∘ ψ_{g} ⟶ ψ_{1} and Γ_{g,g^{∗}} : ψ_{g} ∘ ψ_{g^{∗}} ⟶ ψ_{1}. Thus ψ_{g} for each g ∈ C_{G}(f) is an equivalence of the category 𝕋r_{2}ρ_{f}.

3.4. The adjoint 2-isomorphisms. For a 2-isomorphism [figure: y; x; χ_{1}; χ_{2}; ϕ] in a 2-category

V, we define the adjoint 2-isomorphism ϕ^{†} to be [figure: x; y; χ_{1}^{−1}; χ_{2}^{−1}; ϕ^{†}] by the composition of arrows

[figure: x; y; y; x; χ_{2}; χ_{1}; ϕ^{−1}; χ_{1}^{−1}; χ_{2}^{−1}] .  (29)

This is a 2-isomorphism with inverted 1-source and 1-target. This operation will be used later. See also section 2 of [20] for the definition of similar adjoint 2-arrows, but ϕ^{−1} in (29) is replaced there by ϕ.

3.5. Proposition. (1) For any pair of 2-isomorphisms [figure: y; x; χ_{1}; χ_{2}; ϕ] and [figure: y; x; χ_{2}; χ_{3}; ψ], we

have (ϕ#_{1}ψ)^{†} = ϕ^{†}#_{1}ψ^{†}.

(2) For any 1-isomorphism χ_{0} : z ⟶ x, we have (χ_{0}#_{0}ϕ)^{†} = ϕ^{†}#_{0}χ_{0}^{−1}; and for 1-isomorphism χ̃_{0} : y ⟶ z, we have (ϕ#_{0}χ̃_{0})^{†} = χ̃_{0}^{−1}#_{0}ϕ^{†}.

(3) For a 2-isomorphism [figure: z; y; χ̃_{1}; χ̃_{2}; ϕ̃], we have (ϕ#_{0}ϕ̃)^{†} = ϕ̃^{†}#_{0}ϕ^{†}, i.e., [figure: y; z; x; χ_{1}^{−1}; χ_{2}^{−1}; ϕ̃^{†}; χ_{1}^{−1}; χ_{2}^{−1}; ϕ^{†}].

Proof. (1) ϕ^{†}#_{1}ψ^{†} = (ϕ#_{1}ψ)^{†} follows from

[figure: y; x; x; y; y; x; χ_{2}; χ_{1}; ϕ^{−1}; χ_{1}^{−1}; χ_{2}^{−1}; χ_{3}; χ_{2}; ψ^{−1}; χ_{3}^{−1}; x; y; y; x; χ_{2}; χ_{1}; ϕ^{−1}; χ_{1}^{−1}; χ_{3}^{−1}; χ_{3}; ψ^{−1}]

by [figure: x; y; y; x; χ_{3}; χ_{2}; ψ^{−1}; χ_{2}^{−1}; χ_{2}] [figure: y; x; χ_{3}; χ_{2}; ψ^{−1}] and the interchange law (6) for hor-

izontal compositions.

(2) follows from the fact that (χ_{0}#_{0}ϕ)^{†} is

[figure: y; x; z; x; y; z; x; χ_{0}^{−1}; χ_{1}^{−1}; χ_{0}; χ_{2}; χ_{1}; ϕ^{−1}; χ_{2}^{−1}; χ_{0}^{−1}],

since χ_{0}^{−1}#_{0}χ_{0} is equal to the identity 1_{x}.

(3) Note that ϕ#_{0}ϕ̃ = (χ_{1}#_{0}ϕ̃)#_{1}(ϕ#_{0}χ̃_{2}) by using the interchange law (6) . We see that

(ϕ#_{0}ϕ̃)^{†} = (χ_{1}#_{0}ϕ̃)^{†} #_{1} (ϕ#_{0}χ̃_{2})^{†} = (ϕ̃^{†}#_{0}χ_{1}^{−1}) #_{1} (χ̃_{2}^{−1}#_{0}ϕ^{†}) = ϕ̃^{†}#_{0}ϕ^{†}

by using (1), (2) and the interchange law (6) again.

3.6. The categorical action of the centralizer of f on 𝕋r_{2}ρ_{f}. To construct a categorical action of the centralizer C_{G}(f) of f on the category 𝕋r_{2}ρ_{f}, let us write down the composition law for the functors ψ_{h} and ψ_{g},

ψ_{h} ∘ ψ_{g} : 𝕋r_{2}ρ_{f} ⟶ 𝕋r_{2}ρ_{f},

where h, g ∈ C_{G}(f). For a fixed χ ∈ (𝕋r_{2}ρ_{f})_{0} and Θ ∈ (𝕋r_{2}ρ_{f})_{1}, by using the definition (26)-(28) of ψ_{∗} twice, we see that ψ_{h} ∘ ψ_{g}(χ) = ψ_{h}(ψ_{g}(χ)) is the composition of 2-arrows in C in the following diagram:

[figure: x; x; x; x; x; x; ρ_{g}; ρ_{g^{∗}}; ρ_{h}; ρ_{h^{∗}}; 1_{x}; ρ_{f}; χ; u_{h}; u_{g}; 1_{x}; 1_{x}; ρ_{gf}; ρ_{f}; ρ_{hf}; ρ_{f}; ϕ_{g,f}; ϕ_{h,f}; ϕ_{hf,h^{∗}}; ϕ_{gf,g^{∗}}],  (30)

and ψ_{h} ∘ ψ_{g}(Θ) = ψ_{h}(ψ_{g}(Θ)) is a 3-arrow in C defined similarly. Recall that we assume ρ_{g1} = ρ_{g}1_{x} and ρ_{h1} = ρ_{h}1_{x}. The upper half part of (30) is the same as the lower half with f replaced by 1_{x} and 2-isomorphisms inverted:

[figure: x; x; x; x; x; x; ρg; ρ_{g^{∗}}; ρ_{h}; ρ_{h^{∗}}; 1_{x}; ϕ{−1 \atop h,1}; ϕ{−1 \atop g1,g^{∗}}; ϕ{−1 \atop g,1}; 1_{x}; 1_{x}; ρ_{g1}; ρ_{h1}; ϕ{−1 \atop h1,h^{∗}}],  (31)

namely, we have u_{h} = ϕ_{h1,h^{∗}}^{−1}#_{1}[ϕ_{h,1}^{−1}#_{0}ρ_{h^{∗}}] and similar identity for u_{g}. Note that ϕ_{h,1} and ϕ_{g,1} are identities by our assumptions in Remark 2.2 (1).

Now let us write down the natural isomorphism

Γ_{h,g} : ψ_{h} ∘ ψ_{g} ⟶ ψ_{hg}

between functors on the category 𝕋r_{2}ρ_{f}. The lower half of diagram (30) is

[figure: x; x; x; x; x; x; ρ_{g}; ρ_{g^{∗}}; ρ_{h}; ρ_{h^{∗}}; ρ_{f}; ρ_{gf}; ρ_{gg^{∗}}; ρ_{hg^{∗}}; ρ_{hh^{∗}}]  (32)

Here and in the following, for simplicity, we will use the notation

ρ\underline{g_{1}g_{2}} := ρ_{g_{1}...g_{2}},

i.e., we omit the group elements between g_{1} and g_{2} in the sequence h, g, f, g^{∗}, h^{∗} in diagram (32).

Recall that the associator 3-isomorphism Φ_{g_{3},g_{2},g_{1}} in (12)-(13) can be drawn in the form (14). By definition, the 3-isomorphism

Λ̂_{1} = γ_{1}#_{1}[Φ_{h,gf,g^{∗}}#_{0}ρ_{h^{∗}}]#_{1}γ_{2},  (33)

is the associator Φ_{h,gf,g^{∗}}#_{0}ρ_{h^{∗}} whiskered by two 2-isomorphisms

\begin{matrix}γ_{1} = [ρ_{h}#_{0}ϕ_{g,f}#_{0}(ρ_{g^{∗}}ρ_{h^{∗}})] : & [figure: x; x; x; x; x; x; ρ_{g}; ρ_{h^{∗}}; ρ_{f}; ρ_{gf}; ρ_{g^{∗}}; ρ_{h}] , \\ γ_{2} = ϕ_{\underline{hg^{∗}},h^{∗}} : & [figure: x; x; x; ρ_{h^{∗}}; ρ\underline{hg^{∗}}; ρ\underline{hh^{∗}}], \end{matrix} (34)

from above and below, respectively. This replaces the diagonal ρ\underline{gg^{∗}} of the dotted quadrilateral in diagram (32) by the wavy diagonal ρ\underline{hf} of the same quadrilateral in the following diagram:

[figure: x; x; x; x; x; x; ρ_{g}; ρ_{g^{∗}}; ρ_{h}; ρ_{h^{∗}}; ρ_{f}; ρ_{gf}; ρ\underline{hf}; ρ\underline{hg^{∗}}; ρ\underline{hh^{∗}}].  (35)

Λ̂_{1} in (33) is the following 3-isomorphism

[figure: x; x; γ_{2}; γ_{1}; χ; χ′]  (36)

where χ is the 2-arrow corresponding to the dotted quadrilateral in diagram (32), χ′ is the 2-arrow corresponding to the same quadrilateral in diagram (35) with the diagonal changed, and 2-arrows γ_{1} and γ_{2} are given by (34).

The 3-isomorphism

Λ̂_{2} = [Φ_{h,g,f}#_{0}(ρ_{g^{∗}}ρ_{h^{∗}})]#_{1}\{[ϕ_{\underline{hf},g^{∗}}#_{0}ρ_{h^{∗}}]#_{1}ϕ_{\underline{hg^{∗}},h^{∗}}\},  (37)

as a whiskered associator (14), then changes the diagonal ρ_{gf} of the dotted-wavy quadrilateral in diagram (35) to the wavy diagonal ρ_{hg} of the same quadrilateral in the following diagram:

[figure: x; x; x; x; x; x; ρ_{g}; ρ_{g^{∗}}; ρ_{h}; ρ_{h^{∗}}; ρ_{f}; ρ_{hg}; ρ\underline{hf}; ρ\underline{hg^{∗}}; ρ\underline{hh^{∗}}].  (38)

Similarly, the 3-isomorphism

Λ̂_{3} = \{[ϕ_{h,g}#_{0}(ρ_{f}ρ_{g^{∗}}ρ_{h^{∗}})]#_{1}[ϕ_{hg,f}#_{0}(ρ_{g^{∗}}ρ_{h^{∗}})]\}#_{1}Φ_{\underline{hf},g^{∗},h^{∗}}^{−1},  (39)

which is the whiskered associator Φ_{\underline{hf},g^{∗},h^{∗}}^{−1}, changes the diagonal ρ_{hg^{∗}} of the dotted quadrilateral in diagram (38) to the wavy diagonal ρ_{g^{∗}h^{∗}} of the same quadrilateral in the following diagram:

[figure: x; x; x; x; x; x; ρ_{g}; ρ_{g^{∗}}; ρ_{h}; ρ_{h^{∗}}; ρ_{f}; ρ_{hg}; ρ\underline{hf}; ρ_{g^{∗}h^{∗}}; ρ\underline{hh^{∗}}].  (40)

Recall that the upper half of diagram (30) is the same as the lower half with f replaced by 1 and 2-isomorphisms inverted. So by the corresponding 3-isomorphisms, denoted by Λ̂′_{1}, Λ̂′_{2}, Λ̂′_{3}, the upper half of (31) is changed to

[figure: x; x; x; x; x; x; ρ_{g}; ρ_{g^{∗}}; ρ_{h}; ρ_{h^{∗}}; ρ_{1}; ρ_{hg}; ρ_{gh}; ρ_{g^{∗}h^{∗}}; ρ_{1}].  (41)

Note that

[figure: x; x; x; ρ_{g}; ρ_{h}; ρ_{hg}; ρ_{hg}; ϕ_{h,g}^{−1}; ϕ_{h,g}; x; x; ρ_{hg}]  (42)

and the part involving ρ_{g^{∗}}ρ_{h^{∗}} is also cancelled. As a result, the composition of (41) and (40), together with 2-arrow χ : 1_{x} [figure] ρ_{f}, gives us the diagram (27) with g replaced by gh. This is exactly ψ_{gh}(χ). Therefore, the composition of suitable whiskered 3-isomorphisms Λ̂′_{1}, Λ̂′_{2}, Λ̂′_{3}, Λ̂_{1}, Λ̂_{2} and Λ̂_{3} gives a natural isomorphism Γ_{h,g} : ψ_{h} ∘ ψ_{g} ⟶ ψ_{hg} such that for χ ∈ (𝕋r_{2}ρ_{f})_{0}

Γ_{h,g}(χ) : ψ_{h}(ψ_{g}(χ)) [figure] ψ_{hg}(χ)

is a 3-isomorphism in C.

It is not easy to draw 3-arrows Λ̂_{j}’s in the 3-category C. But in the 2-category C^{+}, the first 3-arrow Λ̂_{1} in (33) can be drawn as the 2-isomorphism corresponding to the following diagram:

[figure: ρ_{h}ρgρ_{f} ρ_{g^{∗}} ρ_{h^{∗}}; ρ_{h}ρ_{gf} ρ_{g^{∗}} ρ_{h^{∗}}; ρ_{h}ρ\underline{gg^{∗}} ρ_{h^{∗}}; ρ\underline{hg^{∗}} ρ_{h^{∗}}; ρ\underline{hh^{∗}}; ρ\underline{hf} ρ_{g^{∗}} ρ_{h^{∗}}; ρ_{h}#_{0}ϕ_{g,f} #_{0}(ρ_{g^{∗}} ρ_{h^{∗}}); ρ_{h}#_{0}ϕ_{gf,g^{∗}} #_{0}ρ_{h^{∗}}; ϕ_{h,\underline{gg^{∗}}#_{0}ρ_{h^{∗}}}; ϕ_{\underline{hg^{∗}},h^{∗}}; ϕ_{h,gf} #_{0}(ρ_{g^{∗}} ρ_{h^{∗}}); ϕ_{\underline{hf},g^{∗}} #_{0}ρ_{h^{∗}}; Λ̂_{1}] .

Here the upper path

[figure: ρ_{h}ρ_{g}ρ_{f}ρ_{g^{∗}}ρ_{h^{∗}}; ρ_{h}ρ_{gf}ρ_{g^{∗}}ρ_{h^{∗}}; · · · · · ·; ρ_{h}#_{0}ϕ_{g,f} #_{0}(ρ_{g^{∗}}ρ_{h^{∗}}); ρ_{h}#_{0}ϕ_{gf,g^{∗}}#_{0}ρ_{h^{∗}}]

corresponds to the 2-isomorphisms in C in (32) (the lower half of ψ_{h}(ψ_{g}(χ))), while the lower paths corresponds to the 2-isomorphisms in C in (35) (the lower half of ψ_{hg}(χ))). And the 2-isomorphism Λ̂_{1} corresponds to the 3-isomorphism in C in (33). Since 𝕋r_{2}ρ_{f} is a subcategory of C^{++}, diagrams in the 2-category C^{+} are sufficient for our purpose. In the sequel, to simplify diagrams,

ρ_{h} ⋯ ρ\underline{g_{1}g_{2}} ⋯ ρ_{h^{∗}} is simply written as ρ\underline{g_{1}g_{2}},

as an object in the 2-category C^{+}. For simplicity, we also omit the whiskering part of 1-isomorphisms ϕ_{∗,∗}’s in diagrams. The 3-isomorphisms Λ̂_{1} : (32) [figure] (35), Λ̂_{2} :

(35) [figure] (38) and Λ̂_{3} : (38) [figure] (40) in the 3-category C correspond to 2-isomorphisms in the 2-category C^{+} in the following diagram:

𝒟_{f} :  [figure: ρ_{f}; ρ_{gf}; ρ\underline{gg^{∗}}; ρ\underline{hg^{∗}}; ρ\underline{hh^{∗}}; ρ\underline{hf}; ρ_{hg}; ρ\underline{hf}ρ_{g^{∗}h^{∗}}; ϕ_{g,f}; ϕ_{gf,g^{∗}}; ϕ_{h,\underline{gg^{∗}}}; ϕ_{\underline{hg^{∗}},h^{∗}}; ϕ_{h,gf}; ϕ_{\underline{hf},g^{∗}}; ϕ_{h,g}; ϕ_{hg,f}; ϕ_{g^{∗},h^{∗}}; ϕ_{\underline{hf},g^{∗}h^{∗}}; Λ̂_{1}; Λ̂_{2}; Λ̂_{3}] ,  (43)

respectively. Just as for the upper half of diagram (30), diagram (31) is changed to diagram (41). In C^{+}, this is the composition of 2-isomorphisms given by the following diagram

[figure: ρ_{1}; ρ_{g1}; ρ\underline{gg^{∗}}; ρ\underline{hg^{∗}}; ρ\underline{hh^{∗}}; ρ\underline{h1}; ρ_{hg}; ρ\underline{h1}ρ_{g^{∗}h^{∗}}; ϕ_{g,1}^{−1}; ϕ_{g1,g^{∗}}^{−1}; ϕ_{h,\underline{gg^{∗}}}^{−1}; ϕ_{\underline{hg^{∗}},h^{∗}}^{−1}; ϕ{−1 \atop h,g1}; ϕ{−1 \atop \underline{h1},g^{∗}}; ϕ_{h,g}^{−1}; ϕ_{hg,1}^{−1}; ϕ_{g^{∗},h^{∗}}^{−1}; ϕ_{\underline{h1},g^{∗}h^{∗}}^{−1}; ˆΛ̂_{1}^{†}; ˆΛ̂_{2}^{†}; ˆΛ̂_{3}^{†}]  : 𝒟_{1}^{†}  (44)

where ˆ Λ̂_{j} is the 2-isomorphism previously denoted by Λ̂_{j} (with f replaced by 1), and ˆ Λ̂_{j}^{†}

(previously denoted by Λ̂′_{j}) is the 2-isomorphism adjoint to ˆ Λ̂_{j}, defined in §3.4 . Recall that the adjoint 2-isomorphism is the inverse one with 1-source and 1-target inverted. We apply the adjoint operation to diagram (43) to get diagram (44), the mirror-symmetric diagram of (43), by using Proposition 3.5. Given χ : 1_{x} [figure] ρ_{f}, we connect the diagrams (44) and (43) to get Γ_{h,g}(χ) as a 2-isomorphism in C^{+}:

𝒟_{1}^{†} [figure: χ] 𝒟_{f}.  (45)

For objects χ, χ′ ∈ (𝕋r_{2}ρ_{f})_{0} and a morphism Θ : χ → χ′ in (𝕋r_{2}ρ_{f})_{1} (i.e., a 3-arrow in C), Γ_{h,g}(Θ) is also a 3-arrow. We connect diagrams (43) and (44) to get Γ_{h,g}(Θ) as the following diagram in the 2-category C^{+}:

[figure: ρ_{1}; ρ_{g1}; · · · · · ·; ρ\underline{h1}; _{· · · · · ·}; ρ_{hg}; ϕ_{g,1}^{−1}; ϕ_{h,g1}^{−1}; ϕ_{h,g}^{−1}; ϕ_{hg,1}^{−1}; ˆΛ̂_{2}^{†}; ρ_{f}; ρ_{gf}; · · · · · ·; ρ\underline{hf}; _{· · · · · ·}; ρ_{hg}; ϕ_{g,f}; ϕ_{h,gf}; ϕ_{h,g}; ϕ_{hg,f}; Λ̂_{2}; χ; χ′; Θ] .  (46)

Note that ψ_{h}∘ψ_{g}(χ) in (30) is the upper boundary of diagram (46) and ψ_{hg}(χ′) is the lower boundary of diagram (46). Γ_{h,g}(χ) is the diagram (46) with the 2-arrow Θ : χ ⟹ χ′ deleted, but 1-arrow χ : ρ_{1} ⟶ ρ_{f} remains, whereas Γ_{h,g}(χ′) is the diagram (46) with the 2-arrow Θ : χ ⟹ χ′ deleted, but 1-arrow χ′ : ρ_{1} ⟶ ρ_{f} remains. Applying the interchange law (6) to the diagram (46), we see that Γ_{h,g} is a natural isomorphism in the category 𝕋r_{2}ρ_{f} ⊂ C^{++}, i.e. the diagram

[figure: ψ_{h} ∘ ψ_{g}(χ); ψ_{h} ∘ ψ_{g}(χ′); ψ_{hg}(χ); ψ_{hg}(χ′); ψ_{h}∘ψ_{g}(Θ); ψ_{hg}(Θ); Γ_{h,g}(χ); Γ_{h,g}(χ′)]

is commutative, where the 2-arrows ψ_{h} ∘ ψ_{g}(Θ) and ψ_{hg}(Θ) in C^{+} are Θ whiskered by 1-isomorphisms corresponding to the upper and lower boundaries of diagram (46), respectively.

3.7. Theorem. {ψ_{g}, Γ_{h,g}}_{g,h∈C_{G}(f)} is a categorical action of the centralizer C_{G}(f) on the category 𝕋r_{2}ρ_{f}.

This theorem will be proved in Section 6 by checking the associative law (1) for Γ_{∗,∗}, which is an identity of natural transformations between functors on 𝕋r_{2}ρ_{f}. Note that in (1), s_{0}(Γ_{k,hg}) = ψ_{k} ∘ ψ_{hg} = t_{0}(ψ_{k} ∘ Γ_{h,g}). So the composition of natural transformations used in (1) is in the usual order, not in the natural order which we assumed in Remark 2.1 (1).

3.8. 1-dimensional 3-representations. We fix a field k of characteristic 0 containing all roots of unity. Let A be a 2-category with only one object, one 1-arrow and 2-arrows A_{2} ≅ k^{∗}. Fix a 3-cocycle c satisfying the condition (2). Let ϱ^{c} be the strict 2-categorical action of G on A as follows: ϱ_{g}^{c} is the identity functor for each g ∈ G;

ϕ_{h,g} : 1_{A} = ϱ_{h}^{c}ϱ_{g}^{c} ⟹ ϱ_{hg}^{c} = 1_{A}

is also the identity pseudonatural isomorphism for any h, g ∈ G; and

Φ_{g_{3},g_{2},g_{1}} : id = (ϱ_{g_{3}}^{c}#_{0}ϕ_{g_{2},g_{1}})#_{1}ϕ_{g_{3},g_{2}g_{1}}  [figure](ϕ_{g_{3},g_{2}}#_{0}ϱ_{g_{1}}^{c})#_{1}ϕ_{g_{3}g_{2},g_{1}} = id,  (47)

is a modification determined by the element c(g_{3}, g_{2}, g_{1}) ∈ k^{∗} for any g_{3}, g_{2}, g_{1} ∈ G. Then the 3-cocycle condition (24) for Φ is reduced to the equation (2). The cohomology classes of 3-cocycles are classified by H^{3}(G, k^{∗}).

For f ∈ G, it is easy to see that 𝕋r_{2}ϱ_{f}^{c} is a category with a single object given by the identity pseudonatural isomorphism χ_{0} : 1_{A} → ϱ_{f}^{c} = 1_{A}, and morphisms (𝕋r_{2}ρ_{f})_{1} ≅ k^{∗} (an element of k^{∗} provides a modification). For g ∈ C_{G}(f), ψ_{g} : 𝕋r_{2}ϱ_{f}^{c} → 𝕋r_{2}ϱ_{f}^{c} is the identity functor by the definitions (26)-(28). And

Γ_{h,g} : χ_{0} = ψ_{h} ∘ ψ_{g}(χ_{0}) ⟶ ψ_{hg}(χ_{0}) = χ_{0}

is a natural isomorphism given by the element (also denoted by Γ_{h,g} by abuse of notations)

Γ_{h,g} = \frac{c(h, gf, g^{∗})c(h, g, f)c(\underline{hf}, g^{∗}, h^{∗})^{−1}}{c(h, g, g^{∗})c(h, g, 1)c(hg, g^{∗}, h^{∗})^{−1}} .  (48)

This element is obtained by replacing Φ_{g_{3},g_{2},g_{1}} by the element c(g_{3}, g_{2}, g_{1}) and all other isomorphisms by 1 in Λ̂_{j}’s in (33) (37) (39), and using the adjoint operation (29), .

3.9. Proposition. Γ given by (48) is a 2-cocycle on the centralizer C_{G}(f).

This proposition will be proved in Section 6.1.

3.10. Remark. There exists a transgression map that maps a 3-cocycle c on a finite group G to a 2-cocycle on the inertia groupoid of G [26]. It is given by

C_{h,g} := \frac{c(h, g, f)c(hgfg^{−1}h^{−1}, h, g)}{c(h, gfg^{−1}, g)}

for given f ∈ G (cf. Remark 3.17 in [14]). Note that for h, g ∈ C_{G}(f) we have C_{h,g} := c(h, g, f)c(f, h, g)/c(h, f, g). So our 2-cocycle Γ_{h,g} in (48) is different from the transgressed one. On the other hand, our 2-cocycle is only defined for elements which commute with a given element f, not on the entire inertia groupoid of G.

Let ϱ be a categorical action of a finite group G on a k-linear category W. For a commuting pair of elements g and f in G, the 2-character χ_{ϱ}(f, g) of a categorical action ϱ is the joint trace of functors ϱ_{f} and ϱ_{g}, i.e., the trace of the linear transformation induced by the functor ϱ_{g} on the categorical trace 𝕋rϱ_{f} (a k-vector space, which we assume to be finite dimensional).

Now let ρ be a strict 2-categorical action of a finite group G on a k-linear 2-category V. Then 𝕋r_{2}ρ_{f} is a k-linear category and ψ defines a categorical action of the centralizer of f in G on it by Theorem 3.7. If k, g, f ∈ G are pairwise commutative, we define the 3-character of the 2-categorical action ρ to be

χ_{ρ}(f, g, k) := χ_{ψ}(g, k),  (49)

the joint trace of functors ψ_{g} and ψ_{k} acting on the k-linear category 𝕋r_{2}ρ_{f}, i.e., the trace of the linear transformation induced by the functor ψ_{k} on the k-vector space 𝕋rψ_{g}, which we assume to be finite dimensional.

By using the 2-character formula for 1-dimensional 2-representation in proposition 5.1 of [13], the 3-character of the 3-representation ϱ^{c} for pairwise commutative k, g, f ∈ G is given by

χ_{ϱ^{c}}(f, g, k) = \frac{Γ_{k,g}Γ_{kg,k^{−1}}}{Γ_{k,1}Γ_{k,k^{−1}}} ,

where the expressions Γ_{∗,∗}’s are defined by (48). It can also be derived from diagram (27).

4. The induced strict 2-categorical action on the induced 2-category

4.1. The induced 2-category. Let H ⊂ G be a subgroup of a finite group G and let ρ : H → V^{∗} be a strict 2-categorical action of H on a strict 2-category V (cf. definitions at the end of Section 2.4). Ind_{H}^{G}(V) is a strict 2-category where • objects are maps ϑ : G ⟶ V_{0} together with a 1-isomorphism

u_{g,h} : ϑ(gh) ⟶ ρ_{h^{∗}}ϑ(g)

for each g ∈ G, h ∈ H, satisfying the condition:

(1) u_{g,1} : ϑ(g) ⟶ ρ_{1}ϑ(g) coincides with ϕ_{1}^{−1}[ϑ(g)];

(2) for each g ∈ G, h_{1}, h_{2} ∈ H, we have a 2-isomorphism:

[figure: ρ_{(h_{1}h_{2})^{∗}}ϑ(g); ρ_{h_{2}^{∗}}ρ_{h_{1}^{∗}}ϑ(g); ϑ(gh_{1}h_{2}); ρ_{h_{2}^{∗}}ϑ(gh_{1}); u_{g,h_{1}h_{2}}; ϕ_{h^{∗}_{2},h^{∗}_{1}}^{−1}ϑ(g); ρ_{h^{∗}_{2}}u_{g,h_{1}}; u_{gh_{1},h_{2}}]

• 1-arrows F : (ϑ, u) → (ϑ′, u′) between objects;

• 2-arrows γ : F → F̃.

For k ∈ G, the action (ind_{H}^{G}ρ)_{k} on the 2-category Ind_{H}^{G}(V) is given by

[(ind_{H}^{G}ρ)_{k}ϑ] (g) = ϑ(k^{−1}g), [(ind_{H}^{G}ρ)_{k}u]_{g,h} = u_{k^{−1}g,h},

for an object (ϑ, u) in Ind_{H}^{G}(V). And (ind_{H}^{G}ρ)_{k}(F) for a 1-arrows F : (ϑ, u) → (ϑ′, u′) and (ind_{H}^{G}ρ)_{k}(γ) for a 2-arrow γ : F → F̃ can be defined similarly. In general, each commutative diagram in the definition of the induced category in section 7.1 of [13] is replaced by a 2-isomorphism.

We will not write down the definition of the induced 2-category Ind_{H}^{G}(V) explicitly. It is a little bit complicated. Since we only work on finite groups, we can simply identify Ind_{H}^{G}(V) with V^{m} as a 2-category, where m is the index of H in G. For a strict 2-category V, V^{m} is also a strict 2-category with

objects V_{0}^{m} := {(x_{1}, …, x_{m}); x_{j} ∈ V_{0}},
p − arrows V_{p}^{m} := {(γ_{1}, …, γ_{m}) : (x_{1}, …, x_{m}) → (y_{1}, …, y_{m}); V_{p} ∋ γ_{j} : x_{j} → y_{j}},

p = 1, 2. The compositions are defined as

(…, γ_{j}, …)#_{p}(…, γ′_{j}, …) := (…, γ_{j}#_{p}γ′_{j}, …),  (50)

if γ_{j} and γ′_{j} are p-composable. The axioms for functions #_{p} and identities of V^{m} are obviously satisfied. The identification Ind_{H}^{G}(V) ≅ V^{m} can be obtained by choosing a system of representatives

ℛ = {r_{1}, …, r_{m}}

of left cosets of H in G, and associating to each map ϑ : G → V_{0} an object (ϑ(r_{1}), …, ϑ(r_{m})) in V_{0}^{m}.

Let a_{jk} : V → V be functors such that the m × m matrix F = (a_{jk}) has only one nonvanishing entry in each row or column. Then F defines a strict functor from V^{m} to V^{m} by

F (…, δ_{j}, …) = (…, ∑_{k} a_{jk}(δ_{k}), …) ,

where we write ∑_{k} a_{jk}(δ_{k}) formally for δ_{k} ∈ V_{p}, p = 0, 1, 2, since there exists only one term in this sum. But when the 2-category is k-linear, such sums exist. If F̃ = (ã_{jk}) : V^{m} → V^{m} is another such functor, then we have

(F #_{0}F̃)_{jk} := ∑_{l} a_{jl}ã_{lk}.

Moreover, a pseudonatural transformation ϕ : F → F̃ is given by an m × m matrix ϕ = (ϕ_{jk}) with ϕ_{jk} : a_{jk} → ã_{jk} a pseudonatural transformation between functors on V.

Let ϕ̃ = (ϕ̃_{jk}) : F̃ → ˜ F̃ be another pseudonatural transformation. Then their composition is ϕ#_{1}ϕ̃:= (ϕ_{jk}#_{1}ϕ̃_{jk}).

4.2. The induced strict 2-categorical action . Suppose that ρ is a strict 2categorical action of H on the 2-category V. For f ∈ G, we define (ind_{H}^{G}ρ)_{f} to be a functor from V^{m} to V^{m} as follows. It is an m × m matrix whose entries are functors from V to V, i.e., the (j, i)-entry is

[(ind_{H}^{G}ρ)_{f}]_{ji} = \{ \begin{matrix}ρ_{h},  if fr_{i} = r_{j}h, for h ∈ H, \\ 0,  otherwise. \end{matrix}  (51)

This corresponds to the fact that for a map ϑ : G → V_{0}, we have [(ind_{H}^{G}ρ)_{f}(ϑ)](r_{j}) = ϑ(f^{−1}r_{j}) and ϑ(f^{−1}r_{j}) = ϑ(r_{i}h^{−1}) → ρ_{h}ϑ(r_{i}). It is clear that only one entry in each row or column of the m × m matrix (ind_{H}^{G}ρ)_{f} is nonvanishing. Then,

(ind_{H}^{G}ρ)_{f}(…, δ_{j}, …) = (…, ∑ ((ind_{H}^{G}ρ)_{f})_{ji} (δ_{i}), …) ,  (52)

where δ_{j} ∈ V_{p}, for p = 0, 1, 2.

For simplicity, from now on the induced object will be denoted by the hatted one, e.g. ind_{H}^{G}ρ is denoted by ρ̂. The composition functor ρ̂_{g_{2}} and ρ̂_{g_{1}} is defined as

(ρ̂_{g_{2}} ρ̂_{g_{1}})_{ki} = \{ \begin{matrix}ρ_{h_{2}}ρ_{h_{1}},  if g_{1}r_{i} = r_{j}h_{1}, g_{2}r_{j} = r_{k}h_{2}, for some h_{1}, h_{2} ∈ H, \\ 0,  otherwise. \end{matrix}

(53)

Thus ρ̂_{g_{2}} ρ̂_{g_{1}} can be viewed as the product of two m × m matrices of functors. On the other hand,

(ρ̂_{g_{2}g_{1}})_{ki} = ρ_{h_{2}h_{1}}  (54)

since (g_{2}g_{1})r_{i} = r_{k}(h_{2}h_{1}) by (53). We define the pseudonatural transformation (as a 2-isomorphism in (V^{m})^{∗})

ϕ̂_{g_{2},g_{1}} : ρ̂_{g_{2}} ρ̂_{g_{1}} [figure] ρ̂_{g_{2}g_{1}},

as the m × m matrix whose (k, i)-entry is the 2-isomorphism

(ϕ̂_{g_{2},g_{1}})_{ki} = ϕ_{h_{2},h_{1}} : ρ_{h_{2}}ρ_{h_{1}} ⟶ ρ_{h_{2}h_{1}},  (55)

and all other entries vanish. For g_{1}, g_{2}, g_{3} ∈ G, the 3-isomorphism in (V^{m})^{∗}

Φ̂_{g_{3},g_{2},g_{1}} : [ρ̂_{g_{3}}#_{0}ϕ̂_{g_{2},g_{1}}]#_{1}ϕ̂_{g_{3},g_{2}g_{1}} [figure] [ϕ̂_{g_{3},g_{2}}#_{0}ρ̂_{g_{1}}]#_{1}ϕ̂_{g_{3}g_{2},g_{1}}

is a modification. Write

g_{3}r_{k} = r_{l}h_{3}

for some h_{3} ∈ H. Then we have

[ρ̂_{g_{3}}#_{0}ϕ̂_{g_{2},g_{1}}]_{li} = ρ_{h_{3}}#_{0}ϕ_{h_{2},h_{1}} and [ϕ̂_{g_{3},g_{2}}#_{0}ρ̂_{g_{1}}]_{li} = ϕ_{h_{3},h_{2}}#_{0}ρ_{h_{1}},  (56)

etc.. We define Φ̂_{g_{3},g_{2},g_{1}} as an m × m matrix whose (l, i)-entry is the modification (as a 3-isomorphism in V^{∗})

(Φ̂_{g_{3},g_{2},g_{1}})_{li} = Φ_{h_{3},h_{2},h_{1}} : [ρ_{h_{3}}#_{0}ϕ_{h_{2},h_{1}}]#_{1}ϕ_{h_{3},h_{2}h_{1}} [figure] [ϕ_{h_{3},h_{2}}#_{0}ρ_{h_{1}}]#_{1}ϕ_{h_{3}h_{2},h_{1}},

and all other entries vanish.

For g_{4} ∈ G, write

g_{4}r_{l} = r_{t}h_{4}

for some h_{4} ∈ H. The (t, i)-entry of the m × m matrix [ρ̂_{g_{4}}#_{0}Φ̂_{g_{3},g_{2},g_{1}}]#_{1}ϕ̂_{g_{4},g_{3}g_{2}g_{1}} is the modification

[ρ_{h_{4}}#_{0}Φ_{h_{3},h_{2},h_{1}}]#_{1}ϕ_{h_{4},h_{3}h_{2}h_{1}}

of V, and similarly we obtain other terms in the 3-cocycle condition (15) for Φ̂. So the 3-cocycle condition (15) for Φ̂ is reduced to the 3-cocycle condition for Φ. Note that functors or pseudonatural transformation or modification we consider are matrices, of which entries are in a strict 3-subcategory W of V^{∗}. It follows from the strictness of W that ρ̂ is a strict 2-categorical action of G on V^{m} ≈ Ind_{H}^{G}(V).

5. The 3-character of the induced strict 2-categorical action

5.1. The 2-categorical trace of the induced strict 2-categorical action. As above ρ is a strict 2-categorical action of H on the 2-category V. Let R be a system of representatives of G/H. We have the decomposition

R = R′ ∪ R′′,

where R′ := {r ∈ R; r^{−1}fr ∈ H}, R′′ := {r ∈ R; r^{−1}fr ∉ H}. For a fixed element f of G, the decomposition

[f]_{G} ∩ H = [h_{1}]_{H} ∪ ⋯ [h_{n}]_{H}

induces a decomposition

R′ = ⋃_{i=1}^{n} R_{i} with R_{i} = {r ∈ R; r^{−1}fr ∈ [h_{i}]_{H}} .

For fixed i, we pick r_{i} ∈ R_{i} and write h_{i} = r_{i}^{−1}fr_{i}. For r ∈ R_{i}, we have r^{−1}fr = h^{−1}h_{i}h for some h ∈ H. From now on, by replacing r by rh^{−1} in the representatives of R_{i} ⊂ G/H, we can assume

r^{−1}fr = h_{i} for all r ∈ R_{i}.  (57)

Denote

m_{i} := |R_{i}|,  m′ := |R′| = ∑_{i=1}^{n} m_{i},  m′′ := |R′′|,  m := m′ + m′′.

It follows from the definition (51)-(52) of ρ̂_{f} that

ρ̂_{f} = ( \begin{matrix}A_{00} & A_{01} & A_{02} & ⋯ & A_{0n} \\ A_{10} & A_{11} & 0 & ⋯ & 0 \\ A_{20} & 0 & A_{22} & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ A_{n0} & 0 & 0 & ⋯ & A_{nn} \end{matrix} ) ,  A_{ii} = ( \begin{matrix}ρ_{h_{i}} \\ & . & . & . \\ & & & & ρ_{h_{i}} \end{matrix} )_{m_{i}×m_{i}} ,  (58)

where i = 1, …, n, and A_{00} is a off-diagonal m′′ × m′′ matrix. So an object of 𝕋r_{2}ρ̂_{f} is a pseudonatural transformation χ : 1_{V^{m}} → ρ̂_{f} of the form

χ = ( \begin{matrix}0_{m′′×m′′} \\ & . & . & . \\ & & & & D_{i} \\ & & & & & . & . & . \end{matrix} ) ,  D_{i} = ( \begin{matrix}χ_{m_{1}+···+m_{i−1}+1} \\ & . & . & . \\ & & & & χ_{m_{1}+···+m_{i}} \end{matrix} ) , (59)

where χ_{m_{1}+···+m_{i−1}+α} : 1_{V} → ρ_{h_{i}} is an object of 𝕋r_{2}ρ_{h_{i}}, α = 1, …, m_{i}. Also morphisms in 𝕋r_{2}ρ̂_{f} are diagonal. So we have

𝕋r_{2}ρ̂_{f} = ⨁_{i=1}^{n}(𝕋r_{2}ρ_{h_{i}})^{m_{i}}.

5.2. Lemma. ([13], Lemma 7.7) Left multiplication with r_{i}^{−1} maps R_{i} into a system of representatives of C_{G}(h_{i})/C_{H}(h_{i}).

For g ∈ C_{G}(f) and r ∈ R_{i}, we write

gr = r̃h,  (60)

for some r̃ ∈ R and h ∈ H. Also, r is uniquely determined by r̃ for fixed g. Then

r̃^{−1}fr̃ = hr^{−1}g^{−1}fgrh^{−1} = hh_{i}h^{−1}

by (57). Hence r̃ ∈ R_{i} and so r̃^{−1}fr̃ = h_{i} by assumption (57). It follows that h ∈ C_{H}(h_{i}). Then

gr = r̃h  gives (ρ̂_{g})_{r̃r} = ρ_{h},
fr = rh_{i}  gives (ρ̂_{f})_{rr} = ρ_{h_{i}},
g^{−1}r̃ = rh^{−1}  gives (ρ̂_{g^{−1}})_{rr̃} = ρ_{h^{∗}},

and all other entries vanish. Thus

(ρ̂_{g}ρ̂_{f} ρ̂_{g^{∗}})_{r̃r̃} = ρ_{h}ρ_{h_{i}}ρ_{h^{∗}}

and all other entries in the last (m′ × m′)-block vanish (see (58)).

We denote by ψ̃ the categorical action of the centralizer C_{G}(f) of f on the category 𝕋r_{2}ρ̂_{f}. By definition (26), ψ̃_{g} for g ∈ C_{G}(f) is an invertible functor as follows. For a pseudonatural transformation diag(…, χ_{r}, …) = χ : 1_{V^{m}} → ρ̂_{f} in (59), ψ̃_{g}(χ) is a pseudonatural transformation given by

diag(1_{V}, …, 1_{V}) \underrightarrow{ϕ̂_{g,g^{∗}}^{−1}} ρ̂_{g}ρ̂_{g^{∗}} ⟶^{ρ̂_{g}#_{0}χ#_{0}ρ̂_{g^{∗}}} ρ̂_{g}ρ̂_{f} ρ̂_{g^{∗}} ⟶^{ϕ̂_{g,f} #_{0}ρ̂_{g^{∗}}} ρ̂_{gf} ρ̂_{g^{∗}} \underrightarrow{ϕ̂_{gf,g^{∗}}} ρ̂_{gfg^{∗}} = ρ̂_{f},

where the first m′′ diagonal terms of ψ̃_{g}(χ) must vanish, and other diagonal terms are

\begin{aligned}\binom{}{ϕ_{g,g^{∗}}^{−1}}_{r̃r̃} & = ϕ_{h,h^{∗}}^{−1} : 1_{V} → ρ_{h}ρ_{h^{∗}}, \\ (ρ̂_{g}#_{0}χ#_{0}ρ̂_{g^{∗}})_{r̃r̃} & = (ρ̂_{g})_{r̃r}#_{0}χ_{rr}#_{0}(ρ̂_{g^{∗}})_{rr̃} = ρ_{h}#_{0}χ_{r}#_{0}ρ_{h^{∗}} : ρ_{h}ρ_{h^{∗}} → ρ_{h}ρ_{h_{i}}ρ_{h^{∗}}, \\ (ϕ̂_{g,f}#_{0}ρ̂_{g^{∗}})_{r̃r̃} & = ϕ_{h,h_{i}}#_{0}ρ_{h^{∗}} : ρ_{h}ρ_{h_{i}}ρ_{h^{∗}} → ρ_{hh_{i}}ρ_{h^{∗}} \\ \binom{}{ϕ_{gf,g^{∗}}}_{r̃r̃} & = ϕ_{hh_{i},h^{∗}} : ρ_{hh_{i}}ρ_{h^{∗}} → ρ_{hh_{i}h^{∗}} = ρ_{h_{i}}. \end{aligned}

All other entries vanish by definitions (54)-(55). Therefore, ψ̃_{g}(χ) is a diagonal m × m matrix of pseudonatural transformations, whose (r̃, r̃)-entry for r̃ ∈ R′ is

(ψ̃_{g}(χ))_{r̃r̃} = ϕ_{h,h^{∗}}^{−1}#_{1}[ρ_{h}#_{0}χ_{r}#_{0}ρ_{h^{∗}}]#_{1}[ϕ_{h,h_{i}}#_{0}ρ_{h^{∗}}]#_{1}ϕ_{hh_{i},h^{∗}} : 1_{V} → ρ_{h_{i}},  (61)

and vanishes for all r̃ ∈ R′′.

Now denote by ψ^{(i)} the categorical action of the centralizer C_{H}(h_{i}) on the category 𝕋r_{2}ρ_{h_{i}}, which is constructed from the strict 2-categorical action ρ of H on V. Recall that by definition (26), we have a functor ψ_{h}^{(i)} for each h ∈ C_{H}(h_{i}). For h ∈ C_{H}(h_{i}) and a pseudonatural transformation ω : 1_{V} → ρ_{h_{i}} , the pseudonatural transformation ψ_{h}^{(i)}(ω) is again by definition (26) the composition of the following pseudonatural transformations between functors:

1_{V} ⟶^{ϕ_{h,h^{∗}}^{−1}} ρ_{h}ρ_{h^{∗}} ⟶^{ρ_{h}#_{0}ω#_{0}ρ_{h^{∗}}} ρ_{h}ρ_{h_{i}}ρ_{h^{∗}} ⟶^{ϕ_{h,h_{i}}#_{0}ρ_{h^{∗}}} ρ_{hh_{i}}ρ_{h^{∗}} ⟶^{ϕ_{hh_{i},h^{∗}}} ρ_{hh_{i}h^{∗}} = ρ_{h_{i}}.

Then we see that (61) can be written as

(ψ̃_{g} (χ))_{r̃r̃} = ψ_{h}^{(i)}(χ_{r}) : 1_{V} → ρ_{h_{i}},  (62)

with r, r̃ ∈ R_{i} and h determined by (60). Namely, the resulting r̃-th diagonal term is the image of the r-th diagonal term under the action of the functor ψ_{h}^{(i)}.

Note that we have the identification

Ind_{C_{H}(h_{i})}^{C_{G}(h_{i})}𝕋r_{2}ρ_{h_{i}} ≅ (𝕋r_{2}ρ_{h_{i}})^{m_{i}},  (63)

since |C_{G}(h_{i})/C_{H}(h_{i})| = m_{i} by Lemma 5.2, and that (60) is equivalent to

(r_{i}^{−1}gr_{i})(r_{i}^{−1}r) = (r_{i}^{−1}r̃)h,  h ∈ C_{H}(h_{i}).  (64)

The coset C_{G}(h_{i})/C_{H}(h_{i}) are represented by r_{i}^{−1}r for r ∈ R_{i} by Lemma 5.2 again, and an element of C_{G}(h_{i}) can always be written as r_{i}^{−1}gr_{i} for some g ∈ C_{G}(f). As above we denote by ψ̂^{(i)} the induced action of the centralizer C_{G}(h_{i}) of h_{i} on the category Ind_{C_{H}(h_{i})}^{C_{G}(h_{i})}𝕋r_{2}ρ_{h_{i}}. Recall the definition (51)-(52) of the induced action. So the action of r_{i}^{−1}gr_{i} ∈ C_{G}(h_{i}) on the induced category (63) is given by the functor ψ̂^{(i)}_{r_{i}^{−1}gr_{i}} on (𝕋r_{2}ρ_{h_{i}})^{m_{i}} with

(ψ̂^{(i)}_{r_{i}^{−1}gr_{i}}(χ))_{r_{i}^{−1}r̃,r_{i}^{−1}r} = ψ_{h}^{(i)} (χ_{r_{i}^{−1}r}) : 1_{V} → ρ_{h_{i}}.  (65)

for χ ∈ (𝕋r_{2}ρ_{h_{i}})^{m_{i}}, where h is given by (64), and all other entries vanish. Here we use the expressions r_{i}^{−1}r as indices of the components of (𝕋r_{2}ρ_{h_{i}})^{m_{i}}. Comparing (62) with (65), we find that the action of g ∈ C_{G}(f) on (𝕋r_{2}ρ_{h_{i}})^{m_{i}} coincides with the induced action of r_{i}^{−1}gr_{i} ∈ C_{G}(h_{i}) on it, and so the action of the centralizer C_{G}(f) on 𝕋r_{2}ρ̂_{f} decomposes into actions on

⨁_{i}(𝕋r_{2}ρ_{h_{i}})^{m_{i}} = ⨁_{i} Ind_{C_{H}(h_{i})}^{C_{G}(h_{i})}𝕋r_{2}ρ_{h_{i}}.  (66)

Recall that the initia groupoid Λ(G) of a group G has as objects, the elements of G, and for two such elements u and v, there is one morphism in Λ(G) from u to v for every g ∈ G such that v = gug^{−1}. Note that the initia groupoid Λ(G) is equivalent to the groupoid with the set of objects consisting of the conjugacy classes [g_{i}] and the set of morphisms consisting of g : [g_{i}] → [g_{i}] for g ∈ C_{G}(g_{i}). Therefore the above result can be summarized as follows.

5.3. Theorem. Let V be a k-linear 2-category. The 2-categorical trace 𝕋r_{2} takes induced strict 2-categorical action into the induced categorical action of the associated initia groupoids, i.e. (3) holds.

5.4. Remark. Even for the categorical action, Section 4 above and the present subsection provide some details not written down explicitly in section 7.2 of [13].

5.5. The 3-character formula. Recall the 2-character formula for an induced categorical action.

5.6. Theorem. ([18], Corollary 7.6) Let ϱ be a categorical action of a subgroup H of a finite group G on a k-linear category W. Suppose that 𝕋rϱ_{h} is finite dimensional for each h ∈ H. Then the 2-character of the induced categorical action of G is given by

χ_{ind}(f, g) = \frac{1}{|H|} ∑_{\begin{matrix}s∈G \\ s^{−1}(f,g)s∈H×H \end{matrix}} χ_{ϱ}(s^{−1}fs, s^{−1}gs)  (67)

for g ∈ C_{G}(f).

We now state:

5.7. Theorem. Let H be a subgroup of a finite group G and let ρ be a strict 2-categorical action of H on the 2-category V. Let ψ be the categorical actions of the centralizers on the 2-categorical trace. Suppose that 𝕋rψ_{h} is finite dimensional for each h ∈ H. Then the 3-character of the induced strict 2-categorical action of G is given by

χ_{ind}(f, g, k) = \frac{1}{|H|} ∑_{\begin{matrix}s∈G \\ s^{−1}(f,g,k)s∈H×H×H \end{matrix}} χ_{ρ}(s^{−1}fs, s^{−1}gs, s^{−1}ks)  (68)

for f, g and k pairwise commutative.

Proof. By the decomposition (66) of the action of C_{G}(f) on 𝕋r_{2}ρ̂_{f} and (62)-(65), we have

χ_{ind}(f, g, k) = ∑_{i=1}^{n} χψ̂^{(i)}(r_{i}^{−1}gr_{i}, r_{i}^{−1}kr_{i}).

Now apply Theorem 5.6 to the categorical action ψ̂^{(i)} (65) of C_{G}(h_{i}), which is induced from the categorical action ψ^{(i)} of C_{H}(h_{i}) on 𝕋r_{2}ρ_{h_{i}}, to get

χ_{ind}(f, g, k) = ∑_{i=1}^{n} \frac{1}{|C_{H}(h_{i})|} ∑_{\begin{matrix}t∈C_{G}(h_{i}) \\ t^{−1}r_{i}^{−1}(g,k)r_{i}t∈C_{H}(h_{i})×C_{H}(h_{i}) \end{matrix}} χ_{ψ^{(i)}}(t^{−1}r_{i}^{−1}gr_{i}t, t^{−1}r_{i}^{−1}kr_{i}t).

Recall that ψ^{(i)} is the categorical action of C_{H}(h_{i}) on 𝕋r_{2}ρ_{h_{i}} constructed from the strict 2-categorical action ρ of H. So we have

χ_{ψ^{(i)}}(t^{−1}r_{i}^{−1}gr_{i}t, t^{−1}r_{i}^{−1}kr_{i}t) = χ_{ρ}(h_{i}, t^{−1}r_{i}^{−1}gr_{i}t, t^{−1}r_{i}^{−1}kr_{i}t)

by the definition of the 3-character (49) for the strict 2-categorical action ρ of group H. Moreover, the decomposition of the action of C_{G}(f) on 𝕋r_{2}ρ̂_{f} in Section 5.1 is independent of the choice of h_{i} ∈ [h_{i}]_{H}, conjugacy class of h_{i} in H. Therefore,

χ_{ind}(f, g, k) = ∑_{h∈H} \frac{1}{|[h]_{H}|} \frac{1}{|C_{H}(h)|} ∑_{\begin{matrix}s^{−1}fs=h,s∈G, \\ s^{−1}(g,k)s∈C_{H}(h)×C_{H}(h) \end{matrix}} χ_{ρ}(h, s^{−1}gs, s^{−1}ks).

Here we have used the fact that h_{i} = s^{−1}fs = s^{−1}r_{i}h_{i}r_{i}^{−1}s if and only if r_{i}^{−1}s ∈ C_{G}(h_{i}). Note that for s ∈ G, we have s^{−1}gs (resp. s^{−1}ks) ∈ H if and only if s^{−1}gs (resp. s^{−1}ks) ∈ C_{H}(h) since g and k commute with f = shs^{−1}. The 3-character formula (68) follows.

6. The categorical action of the centralizer of f on 𝕋r_{2}ρ_{f}

6.1. A model: the 1-dimensional case. Let us prove by using the condition (2) for 3-cocycles repeatedly that the expression Γ given in (48) is a 2-cocycle on the centralizer C_{G}(f). This proof corresponds step by step to that of the general case carried out in Section 6.4.

Proof of Proposition 3.9. By the definition of Γ_{∗,∗} in (48), we see that

Γ_{h,g}Γ_{k,hg} = \frac{Π_{f}}{Π_{1}} ,

where

\begin{matrix}Π_{f} := c(h, gf, g^{∗})c(h, g, f)c(hgf, g^{∗}, h^{∗})^{−1} · c(k, \underline{hf}, g^{∗}h^{∗})c(k, hg, f)c(\underline{kf}, g^{∗}h^{∗}, k^{∗})^{−1}, \\ (69) \end{matrix}

and Π_{1} is just Π_{f} with f replaced by 1. Similarly, we have

Γ_{k,h}Γ_{kh,g} = \frac{Π′_{f}}{Π′_{1}} ,

where

Π′_{f} = c(k, hf, h^{∗})c(k, h, f)c(khf, h^{∗}, k^{∗})^{−1} · c(kh, gf, g^{∗})c(kh, g, f)c(\underline{kf}, g^{∗}, (kh)^{∗})^{−1}, (70)

and Π′_{1} is just Π′_{f} with f replaced by 1.

Apply the 3-cocycle condition (2) to the product of the two boldface terms in (70) with g_{4} = k, g_{3} = h, g_{2} = gf, g_{1} = g^{∗} to get

Π′_{f} = c(h, gf, g^{∗})c(k, hgf, g^{∗})c(k, h, gf)

·c(k, hf, h^{∗})c(khf, h^{∗}, k^{∗})^{−1}c(kh, g, f)c(\underline{kf}, g^{∗}, (kh)^{∗})^{−1}. (71)

Here the second line above is the right-hand side of (70) with the two boldface terms deleted. Note that \underline{kf}g^{∗} = khf. Apply the 3-cocycle condition (2) to the product of the two boldface terms in (71) with g_{4} = \underline{kf}, g_{3} = g^{∗}, g_{2} = h^{∗}, g_{1} = k^{∗} to get

Π′_{f} = c(g^{∗}, h^{∗}, k^{∗})^{−1}c(\underline{kf}, g^{∗}h^{∗}, k^{∗})^{−1}c(\underline{kf}, g^{∗}, h^{∗})^{−1}

·c(h, gf, g^{∗})c(k, \underline{hf}, g^{∗})c(k, h, gf) · c(k, hf, h^{∗}) · c(kh, g, f), (72)

Here the second line above is the right-hand side of (71) with the two boldface terms deleted. Apply the 3-cocycle condition (2) to the product of the three boldface terms in (72) with g_{4} = k, g_{3} = \underline{hf}, g_{2} = g^{∗},g_{1} = h^{∗} to get

Π′_{f} = c(k, \underline{hf}, g^{∗}h^{∗})c(\underline{hf}, g^{∗}, h^{∗})^{−1}

·c(g^{∗}, h^{∗}, k^{∗})^{−1}c(\underline{kf}, g^{∗}h^{∗}, k^{∗})^{−1}c(h, gf, g^{∗})c(k, h, gf)c(kh, g, f), (73)

by k\underline{hf} = \underline{kf} and \underline{hf}g^{∗} = hf. Here the second line above is the right-hand side of (72) with the three boldface terms deleted. Apply the 3-cocycle condition (2) to the product of the two boldface terms in (73) with g_{4} = k, g_{3} = h, g_{2} = g, g_{1} = f to get

Π′_{f} = c(h, g, f) · c(k, hg, f)c(k, h, g)

·c(k, \underline{hf}, g^{∗}h^{∗})c(\underline{hf}, g^{∗}, h^{∗})^{−1}c(g^{∗}, h^{∗}, k^{∗})^{−1}c(\underline{kf}, g^{∗}h^{∗}, k^{∗})^{−1}c(h, gf, g^{∗}). (74)

For f = 1 in (74), we see that Π′_{1} also has the product c(k, h, g)c(g^{∗}, h^{∗}, k^{∗})^{−1} of the two boldface terms, which is independent of f. They are cancelled in Π′_{f}/Π′_{1}. So we get

\frac{Π′_{f}}{Π′_{1}} = \frac{Π_{f}}{Π_{1}}

by comparing (74) and (69). Proposition 3.9 is proved.

6.2. The natural isomorphism Γ_{k,hg}#(ψ_{k} ∘ Γ_{h,g}) . Let us write down the natural isomorphism Γ_{k,hg}#(ψ_{k} ∘ Γ_{h,g}). For a fixed χ ∈ (𝕋r_{2}ρ_{f})_{0}, by using the definition of compositions in (30) twice, we see that ψ_{k} ∘ ψ_{h} ∘ ψ_{g}(χ) is the composition of the following 2-arrows:

[figure: x; x; x; x; x; x; x; x; ρ_{g}; ρ_{g^{∗}}; ρ_{h}; ρ_{h^{∗}}; ρ_{k}; ρ_{k^{∗}}; 1_{x}; ρ_{f}; χ; u_{g}; u_{h}; u_{k}; 1_{x}; 1_{x}; 1_{x}; ρ_{gf}; ρ\underline{gg^{∗}}; ρ\underline{hg^{∗}}; ρ\underline{hh^{∗}}; ρ\underline{kh^{∗}}; ρ_{kk^{∗}}].  (75)

Let us calculate the 3-isomorphism

[Γ_{k,hg}#(ψ_{k} ∘ Γ_{h,g})](χ) : ψ_{k} ∘ ψ_{h} ∘ ψ_{g}(χ) [figure] ψ_{khg}(χ)  (76)

for a fixed 2-arrow χ ∈ 𝕋r_{2}ρ_{f} ⊂ C^{++}. We consider the lower half part of (75) first. The 3-isomorphism

Λ_{1} = ♢#_{1}[ρ_{k}#_{0}Φ_{h,gf,g^{∗}}#_{0}(ρ_{h^{∗}}ρ_{k^{∗}})]#_{1}♢,  (77)

the associator Φ_{h,gf,g^{∗}} (14) whiskered by 2-isomorphisms ♢ which we do not write down explicitly, changes the diagonal ρ\underline{gg^{∗}} of the dotted quadrilateral in (75) to the wavy diagonal ρ\underline{hf} of the same quadrilateral in the following diagram:

[figure: x; x; x; x; x; x; x; x; ρ_{g}; ρ_{g^{∗}}; ρ_{h}; ρ_{h^{∗}}; ρ_{k}; ρ_{k^{∗}}; ρ_{f}; ρ_{gf}; ρ\underline{hf}; ρ\underline{hg^{∗}}; ρ\underline{hh^{∗}}; ρ\underline{kh^{∗}}; ρ\underline{kk^{∗}}].  (78)

This is a 3-arrow as (36). The 2-arrows outside the quadrilateral are fixed as the whiskering parts. The 3-isomorphism

Λ_{2} = ♢#_{1}[ρ_{k}#_{0}Φ_{h,g,f}#_{0}(ρ_{g^{∗}}ρ_{h^{∗}}ρ_{k^{∗}})]#_{1}♢,  (79)

as a whiskered associator (14), changes the diagonal ρ_{gf} of the above dotted-wavy quadrilateral to the wavy diagonal ρ_{hg} of the same quadrilateral in the following diagram:

[figure: x; x; x; x; x; x; x; x; ρ_{g}; ρ_{g^{∗}}; ρ_{h}; ρ_{h^{∗}}; ρ_{k}; ρ_{k^{∗}}; ρ_{f}; ρ_{hg}; ρ\underline{hf}; ρ\underline{hg^{∗}}; ρ\underline{hh^{∗}}; ρ\underline{kh^{∗}}; ρ_{kk^{∗}}].  (80)

The 3-isomorphism

Λ_{3} = ♢#_{1}[ρ_{k}#_{0}Φ_{hf,g^{∗},h^{∗}}^{−1}#_{0}ρ_{k^{∗}}]#_{1}♢,  (81)

as a whiskered associator (14), changes the diagonal ρ\underline{hg^{∗}} of the above dotted quadrilateral to the wavy diagonal ρ_{g^{∗}h^{∗}} of the same quadrilateral in the following diagram:

[figure: x; x; x; x; x; x; x; x; ρ_{g}; ρ_{g^{∗}}; ρ_{h}; ρ_{h^{∗}}; ρ_{k}; ρ_{k^{∗}}; ρ_{f}; ρ_{hg}; ρ\underline{hf}; ρ_{g^{∗}h^{∗}}; ρ\underline{hh^{∗}}; ρ\underline{kh^{∗}}; ρ\underline{kk^{∗}}].  (82)

Note that the diagrams (78), (80) and (82) are exactly the diagrams (35), (38) and (40) by adding from below to each of these the arrows:

[figure: x; x; x; x; ρ\underline{hh^{∗}}; ρ_{k}; ρ_{k^{∗}}; ρ\underline{kh^{∗}}; ρ\underline{kk^{∗}}],

By definition, the composition Λ_{1}#_{2}Λ_{2}#_{2}Λ_{3} is the 3-isomorphism

[ψ_{k} ∘ Γ_{h,g}](χ) : ψ_{k} ∘ ψ_{h} ∘ ψ_{g}(χ) [figure]ψ_{k} ∘ ψ_{hg}(χ)

corresponding to the lower half of (75).

The 3-isomorphism

Λ_{4} = ♢#_{1}[Φ_{k,hf,g^{∗}h^{∗}}#_{0}ρ_{k^{∗}}]#_{1}♢,

as a whiskered associator (14), changes the diagonal ρ_{hh^{∗}} of the dotted-wavy quadrilateral in (82) to the wavy diagonal ρ_{kf} of the same quadrilateral in the following diagram:

[figure: x; x; x; x; x; x; x; x; ρ_{g}; ρ_{g^{∗}}; ρ_{h}; ρ_{h^{∗}}; ρ_{k}; ρ_{k^{∗}}; ρ_{f}; ρ_{hg}; ρ_{hf}; ρ_{g^{∗}h^{∗}}; ρ_{kf}; ρ_{kh^{∗}}; ρ_{kk^{∗}}].  (83)

The 3-isomorphism

Λ_{5} = ♢#_{1} [Φ_{k,hg,f}#_{0} (ρ_{g^{∗}}ρ_{h^{∗}}ρ_{k^{∗}})] #_{1}♢,

as a whiskered associator (14), changes the diagonal ρ\underline{hf} of the above dotted-wavy quadrilateral to the wavy diagonal ρ\underline{kg} of the same quadrilateral in the following diagram:

[figure: x; x; x; x; x; x; x; x; ρ_{g}; ρ_{g^{∗}}; ρ_{h}; ρ_{h^{∗}}; ρ_{k}; ρ_{k^{∗}}; ρ_{f}; ρ_{hg}; ρ\underline{kg}; ρ_{g^{∗}h^{∗}}; ρ\underline{kf}; ρ\underline{kh^{∗}}; ρ\underline{kk^{∗}}].  (84)

The 3-isomorphism

Λ_{6} = ♢#_{1}Φ_{\underline{kf},g^{∗}h^{∗},k^{∗}}^{−1},

as a whiskered associator (14), changes the diagonal ρ\underline{kh^{∗}} of the above dotted quadrilateral to the wavy diagonal ρ\underline{g^{∗}k^{∗}} of the same quadrilateral in the following diagram:

[figure: x; x; x; x; x; x; x; x; ρ_{g}; ρ_{g^{∗}}; ρ_{h}; ρ_{h^{∗}}; ρ_{k}; ρ_{k^{∗}}; ρ_{f}; ρ_{hg}; ρ\underline{kg}; ρ_{g^{∗}h^{∗}}; ρ\underline{kf}; ρ\underline{g^{∗}k^{∗}}; ρ\underline{kk^{∗}}].  (85)

The composition Λ_{4}#_{2}Λ_{5}#_{2}Λ_{6} is the 3-isomorphism

Γ_{k,hg}(χ) : ψ_{k} ∘ ψ_{hg}(χ) [figure] ψ_{khg}(χ)

corresponding to the lower half of (75).

In the 2-category C^{+}, the composition Λ_{1}#_{2} ⋯ #_{2}Λ_{6} of 3-isomorphisms corresponds to the following diagram 𝒟_{f}^{l} := where the symbol = in this diagram follows from the interchange law (6) for a horizontal composition: the commutativity of ϕ_{k,\underline{hf}} and ϕ_{g^{∗},h^{∗}}. Note that the part involving Λ_{1}, Λ_{2}, Λ_{3} is just the diagram (43). Let 𝒟_{1}^{l} be the corresponding diagram in C^{+} with f replaced by 1, by using adjoint operations as in (44). Then as in (45), the 2-isomorphism in C^{+} corresponding to the morphism [Γ_{k,hg}#(ψ_{k} ∘ Γ_{h,g})](χ) in 𝕋r_{2}ρ_{f} is

[figure: ρ_{f}; ρ_{gf}; ρ\underline{gg^{∗}}; ρ\underline{hg^{∗}}; ρ\underline{hh^{∗}}; ρ\underline{kh^{∗}}; ρ\underline{kk^{∗}}; ρ\underline{hf}; ρ_{hg}; ρ\underline{hf}ρ_{g^{∗}h^{∗}}; ρ\underline{kg}; ρ\underline{kf}ρ_{g^{∗}h^{∗}}; ρ\underline{kf}; ρ\underline{kf}ρ_{g^{∗}h^{∗}k^{∗}}; ϕ_{g,f}; ϕ_{gf,g^{∗}}; ϕ_{h,\underline{gg^{∗}}}; ϕ_{\underline{hg^{∗}},h^{∗}}; ϕ_{k,\underline{hh^{∗}}}; ϕ_{\underline{kh^{∗}},k^{∗}}; ϕ_{h,gf}; ϕ_{\underline{hf},g^{∗}}; ϕ_{h,g}; ϕ_{hg,f}; ϕ_{g^{∗},h^{∗}}; ϕ_{\underline{hf},g^{∗}h^{∗}}; ϕ_{k,\underline{hf}}; ϕ_{\underline{kf},g^{∗}h^{∗}}; ϕ_{k,\underline{hf}}; ϕ_{g^{∗},h^{∗}}; ϕ_{g^{∗}h^{∗},k^{∗}}; ϕ_{\underline{kf},\underline{g^{∗}k^{∗}}}; ϕ_{\underline{kg},f}; ϕ_{k,hg}; Λ_{1}; Λ_{2}; Λ_{3}; Λ_{4}; Λ_{5}; Λ_{6}]

𝒟_{1}^{l} [figure: χ] 𝒟_{f}^{l}.  (87)

6.3. The natural isomorphism Γ_{kh,g}#(Γ_{k,h} ∘ ψ_{g}) . To calculate Γ_{k,h} ∘ ψ_{g}, we fix the part

[figure: x; x; x; x; ρ_{g}; ρ_{g^{∗}}; ρ_{f}; ρ_{gf}; ρ\underline{gg^{∗}}]

in the lower half of (75), which corresponds to ψ_{g}. The 3-isomorphism

Λ̃_{1} = ♢#_{1}Φ_{k,\underline{hg^{∗}},h^{∗}}#_{1}♢,  (88)

as a whiskered associator (14), changes the 1-isomorphism ρ\underline{hh^{∗}} in the lower part of (75) to the wavy diagonal ρ\underline{kg^{∗}} of the same quadrilateral in the following diagram:

[figure: x; x; x; x; x; x; x; x; ρ_{g}; ρ_{g^{∗}}; ρ_{h}; ρ_{h^{∗}}; ρ_{k}; ρ_{k^{∗}}; ρ_{f}; ρ_{gf}; ρ\underline{gg^{∗}}; ρ\underline{hg^{∗}}; ρ\underline{kg^{∗}}; ρ\underline{kh^{∗}}; ρ\underline{kk^{∗}}].

The 3-isomorphism

Λ̃_{2} = ♢#_{1}[Φ_{k,h,\underline{gg^{∗}}}#_{0}(ρ_{h^{∗}}ρ_{k^{∗}})]#_{1}♢,  (89)

as a whiskered associator (14), changes the diagonal ρ\underline{hg^{∗}} of the above dotted-wavy quadrilateral to the wavy diagonal ρ_{kh} of the same quadrilateral in the following diagram:

[figure: x; x; x; x; x; x; x; x; ρ_{g}; ρ_{g^{∗}}; ρ_{h}; ρ_{h^{∗}}; ρ_{k}; ρ_{k^{∗}}; ρ_{f}; ρ_{gf}; ρ_{gg^{∗}}; ρ_{kh}; ρ_{kg^{∗}}; ρ_{kh^{∗}}; ρ_{kk^{∗}}].

The 3-isomorphism

Λ̃_{3} = ♢#_{1}Φ_{\underline{kg^{∗}},h^{∗},k^{∗}}^{−1},  (90)

as a whiskered associator (14), changes the diagonal ρ\underline{kh^{∗}} of the above dotted quadrilateral to the wavy diagonal ρ_{h^{∗}k^{∗}} of the same quadrilateral in the following diagram:

[figure: x; x; x; x; x; x; x; x; ρ_{g}; ρ_{g^{∗}}; ρ_{h}; ρ_{h^{∗}}; ρ_{k}; ρ_{k^{∗}}; ρ_{f}; ρ_{gf}; ρ\underline{gg^{∗}}; ρ_{kh}; ρ\underline{kg^{∗}}; ρ_{h^{∗}k^{∗}}; ρ\underline{kk^{∗}}].  (91)

The composition Λ̃_{1}#_{2}Λ̃_{2}#_{2}Λ̃_{3} is the 3-isomorphism

Γ_{k,h} ∘ ψ_{g} : ψ_{k} ∘ ψ_{h} ∘ ψ_{g}(χ) [figure] ψ_{kh} ∘ ψ_{g}(χ),

corresponding to the lower half of (75).

The 3-isomorphism

Λ̃_{4} = ♢#_{1}[Φ_{kh,gf,g^{∗}}#_{0}(ρ_{h^{∗}}ρ_{k^{∗}})]#_{1}♢,  (92)

as a whiskered associator (14), changes the diagonal ρ\underline{gg^{∗}} of the dotted quadrilateral in (91) to the wavy diagonal ρ\underline{kf} of the same quadrilateral in the following diagram:

[figure: x; x; x; x; x; x; x; x; ρ_{g}; ρ_{g^{∗}}; ρ_{h}; ρ_{h^{∗}}; ρ_{k}; ρ_{k^{∗}}; ρ_{f}; ρ_{gf}; ρ\underline{kf}; ρ_{kh}; ρ\underline{kg^{∗}}; ρ_{h^{∗}k^{∗}}; ρ\underline{kk^{∗}}].

The 3-isomorphism

Λ̃_{5} = ♢#_{1}[Φ_{kh,g,f}#_{0}(ρ_{g^{∗}}ρ_{h^{∗}}ρ_{k^{∗}})]#_{1}♢,  (93)

as a whiskered associator (14), changes the diagonal ρ_{gf} of the above dotted quadrilateral to the wavy diagonal ρ\underline{kg} of the same quadrilateral in the following diagram:

[figure: x; x; x; x; x; x; x; x; ρ_{g}; ρ_{g^{∗}}; ρ_{h}; ρ_{h^{∗}}; ρ_{k}; ρ_{k^{∗}}; ρ_{f}; ρ\underline{kg}; ρ\underline{kf}; ρ_{kh}; ρ\underline{kg^{∗}}; ρ_{h^{∗}k^{∗}}; ρ\underline{kk^{∗}}].

At last, the 3-isomorphism

Λ̃_{6} = ♢#_{1}Φ_{\underline{kf},g^{∗},h^{∗}k^{∗}}^{−1},  (94)

as a whiskered associator (14), changes the diagonal ρ\underline{kg^{∗}} of the above dotted quadrilateral to the wavy diagonal ρ\underline{g^{∗}k^{∗}} of the same quadrilateral in the following diagram:

[figure: x; x; x; x; x; x; x; x; ρ_{g}; ρ_{g^{∗}}; ρ_{h}; ρ_{h^{∗}}; ρ_{k}; ρ_{k^{∗}}; ρ_{f}; ρ\underline{kg}; ρ\underline{kf}; ρ_{kh}; ρ\underline{g^{∗}k^{∗}}; ρ_{h^{∗}k^{∗}}; ρ\underline{kk^{∗}}].

The composition Λ̃_{4}#_{2}Λ̃_{5}#_{2}Λ̃_{6} is the 3-isomorphism Γ_{kh,g}(χ) : ψ_{kh} ∘ ψ_{g}(χ) [figure] ψ_{khg}(χ) corresponding to the lower half of (75).

The composition of Λ̃_{1}#_{2} ⋯ #_{2}Λ̃_{6} in the 2-category C^{+} is the following diagram 𝒟_{f}^{r} :=

[figure: ρ_{f}; ρ_{gf}; ρ\underline{gg^{∗}}; ρ\underline{hg^{∗}}; ρ\underline{hh^{∗}}; ρ\underline{kh^{∗}}; ρ\underline{kk^{∗}}; ρ\underline{kg^{∗}}; ρ_{kh}ρ\underline{gg^{∗}}; ρ\underline{kg^{∗}}ρ_{h^{∗}k^{∗}}; ρ_{kh}ρ_{gf}; ρ\underline{kf}; ρ_{kh}; ρ\underline{kg}; ρ\underline{kf}ρ_{h^{∗}k^{∗}}; ρ\underline{kf}ρ\underline{g^{∗}k^{∗}}; ϕ_{g,f}; ϕ_{gf,g^{∗}}; ϕ_{h,\underline{gg^{∗}}}; ϕ_{\underline{hg^{∗}},h^{∗}}; ϕ_{k,\underline{hh^{∗}}}; ϕ_{\underline{kh^{∗}},k^{∗}}; ϕ_{k,\underline{hg^{∗}}}; ϕ_{\underline{kg^{∗}},h^{∗}}; ϕ_{k,h}; ϕ_{kh,\underline{gg^{∗}}}; ϕ_{h^{∗},k^{∗}}; ϕ_{\underline{kg^{∗}},h^{∗}k^{∗}}; ϕ_{k,h}; ϕ_{gf,g^{∗}}; ϕ_{kh,gf}; ϕ_{\underline{kf},g^{∗}}; ϕ_{k,h}; ϕ_{g,f}; ϕ_{kh,g}; ϕ_{\underline{kg},f}; ϕ_{h^{∗},k^{∗}}; ϕ_{\underline{kf},g^{∗}}; ϕ_{g^{∗},h^{∗}k^{∗}}; ϕ_{\underline{kf},\underline{g^{∗}k^{∗}}}; Λ̃_{1}; Λ̃_{2}; Λ̃_{3}; Λ̃_{4}; Λ̃_{5}; Λ̃_{6}] .

(95)

Let 𝒟_{1}^{r} be the corresponding diagram in C^{+} with f replaced by 1, by using adjoint operations as in (44). Then the 2-isomorphism in C^{+} corresponding to the morphism [Γ_{kh,g}#(Γ_{k,h} ∘ ψ_{g})](χ) in 𝕋r_{2}ρ_{f} is

𝒟_{1}^{r} [figure: χ] 𝒟_{f}^{r}.  (96)

6.4. The proof of the associativity. Let us show the identity (1), i.e., that diagrams 𝒟_{1}^{l} ⟶^{χ} 𝒟_{f}^{l} in (87) and 𝒟_{1}^{r} ⟶^{χ} 𝒟_{f}^{r} in (96) are identical in the 2-category C^{+}, by using the 3-cocycle identity (24) repeatedly. This proof corresponds to that of the 1-dimensional case in Section 6.1 step by step.

Apply the 3-cocycle identity (24) to the dotted diagram in (95) with g_{4} = k, g_{3} =

2042  WEI WANG

h, g_{2} = gf, g_{1} = g^{∗} to get wavy isomorphisms in the following diagram

[figure: ρ_{f}; ρ_{gf}; ρ\underline{gg^{∗}}; ρ\underline{hg^{∗}}; ρ\underline{hh^{∗}}; ρ\underline{kh^{∗}}; ρ\underline{kk^{∗}}; ρ\underline{kg^{∗}}; ρ\underline{hf}; ρ\underline{kg^{∗}}ρ_{h^{∗}k^{∗}}; ρ_{kh}ρ_{gf}; ρ\underline{kf}; ρ_{kh}; ρ\underline{kg}; ρ\underline{kf}ρ_{h^{∗}k^{∗}}; ρ\underline{kf}ρ\underline{g^{∗}k^{∗}}; ϕ_{g,f}; ϕ_{gf,g^{∗}}; ϕ_{h,\underline{gg^{∗}}}; ϕ_{\underline{hg^{∗}},h^{∗}}; ϕ_{k,\underline{hh^{∗}}}; ϕ_{\underline{kh^{∗}},k^{∗}}; ϕ_{k,\underline{hg^{∗}}}; ϕ_{\underline{kg^{∗}},h^{∗}}; ϕ_{h,gf}; ϕ_{\underline{hf},g^{∗}}; ϕ_{h^{∗},k^{∗}}; ϕ_{\underline{kg^{∗}},h^{∗}k^{∗}}; ϕ_{k,h}; ϕ_{k,\underline{hf}}; ϕ_{kh,gf}; ϕ_{\underline{kf},g^{∗}}; ϕ_{k,h}; ϕ_{g,f}; ϕ_{kh,g}; ϕ_{\underline{kg},f}; ϕ_{h^{∗},k^{∗}}; ϕ_{\underline{kf},g^{∗}}; ϕ_{g^{∗},h^{∗}k^{∗}}; ϕ_{\underline{kf},\underline{g^{∗}k^{∗}}}; Λ̃_{3}; Λ̃_{6}]

(97)

Note that Λ̃_{3} in (90) and Λ̃_{6} in (94) are the inverse of associators. Apply the 3-cocycle identity, the inverse version of (24) (the lower and upper boundaries are exchanged), to the above dotted diagram with g_{4} = \underline{kf}, g_{3} = g^{∗}, g_{2} = h^{∗}, g_{1} = k^{∗} to get wavy isomorphisms in the following:

[figure: ρ_{f}; ρ_{gf}; ρ\underline{gg^{∗}}; ρ\underline{hg^{∗}}; ρ\underline{hh^{∗}}; ρ\underline{kh^{∗}}; ρ\underline{kk^{∗}}; ρ\underline{kg^{∗}}; ρ\underline{hf}; ρ\underline{kf}ρ_{g^{∗}h^{∗}}; ρ_{kh}ρ_{gf}; ρ\underline{kf}; ρ_{kh}; ρ\underline{kg}; ρ\underline{kf}ρ_{h^{∗}k^{∗}}; ρ\underline{kf}ρ\underline{g^{∗}k^{∗}}; ϕ_{g,f}; ϕ_{gf,g^{∗}}; ϕ_{h,\underline{gg^{∗}}}; ϕ_{\underline{hg^{∗}},h^{∗}}; ϕ_{k,\underline{hh^{∗}}}; ϕ_{\underline{kh^{∗}},k^{∗}}; ϕ_{k,\underline{hg^{∗}}}; ϕ_{\underline{kg^{∗}},h^{∗}}; ϕ_{h,gf}; ϕ_{\underline{hf},g^{∗}}; ϕ_{g^{∗},h^{∗}}; ϕ_{\underline{kf},g^{∗}h^{∗}}; ϕ_{k,h}; ϕ_{k,\underline{hf}}; ϕ_{kh,gf}; ϕ_{\underline{kf},g^{∗}}; ϕ_{k,h}; ϕ_{g,f}; ϕ_{kh,g}; ϕ_{\underline{kg},f}; ϕ_{h^{∗},k^{∗}}; ϕ_{g^{∗}h^{∗},k^{∗}}; ϕ_{g^{∗},h^{∗}k^{∗}}; ϕ_{\underline{kf},\underline{g^{∗}k^{∗}}}; Λ̃_{1}; Λ̂]

(98)

where Λ̂ is the inverse of a whiskered associator. Note that the commutative cube in (25) implies the following identity.

[figure: •; •; •; •; •; •; •; A{′−1 \atop 2}; ϕ_{b}; ϕ_{g_{4},g_{3}g_{2}}; ϕa; ϕ_{g_{3}g_{2},g_{1}}; ϕ_{g_{4}g_{3},g_{2}g_{1}}; ϕ_{g_{4},g_{3}g_{2}g_{1}}; ϕ_{g_{2},g_{1}}; A_{2}; ϕ_{g_{3},g_{2}}; ϕ_{g_{4},g_{3}}; A_{3}; •; •; •; •; •; •; •; ϕ_{g_{3},g_{2}}; ϕ_{g_{3}g_{2},g_{1}}; ϕg_{2},g_{1}; ϕ_{g_{4},g_{3}}; ϕg_{2},g_{1}; ϕ_{c}; ϕ_{g_{4},g_{3}}; A_{1}^{−1}; A′_{1}; ϕ_{g_{4}g_{3},g_{2}g_{1}}; ϕ_{g_{4},g_{3}g_{2}g_{1}}]

where ϕ_{a} := ϕ_{g_{4}g_{3}g_{2},g_{1}}, ϕ_{b} := ϕ_{g_{4}g_{3},g_{2}}, ϕ_{c} := ϕ_{g_{3},g_{2}g_{1}}. The left-hand side is the back, bottom and right (this 2-isomorphism is inverted) faces of the cube in (25), while the right-hand side is the left (this 2-isomorphism is inverted), top and front faces of the cube. Apply this identity to the dotted-wavy diagram in (98) with g_{4} = k, g_{3} = \underline{hf}, g_{2} = g^{∗},g_{1} = h^{∗} to get wavy isomorphisms in the following:

[figure: ρ_{f}; ρ_{gf}; ρ\underline{gg^{∗}}; ρ\underline{hg^{∗}}; ρ\underline{hh^{∗}}; ρ\underline{kh^{∗}}; ρ\underline{kk^{∗}}; ρ\underline{hf}ρ_{g^{∗}h^{∗}}; ρ\underline{hf}; ρ\underline{kf}ρ_{g^{∗}h^{∗}}; ρ_{kh}ρ_{gf}; ρ\underline{kf}; ρ_{kh}; ρ\underline{kg}; ρ\underline{kf}ρ_{h^{∗}k^{∗}}; ρ\underline{kf}ρ\underline{g^{∗}k^{∗}}; ϕ_{g,f}; ϕ_{gf,g^{∗}}; ϕ_{h,\underline{gg^{∗}}}; ϕ_{\underline{hg^{∗}},h^{∗}}; ϕ_{k,\underline{hh^{∗}}}; ϕ_{\underline{kh^{∗}},k^{∗}}; ϕ_{g^{∗},h^{∗}}; ϕ_{\underline{hf},g^{∗}h^{∗}}; ϕ_{h,gf}; ϕ_{\underline{hf},g^{∗}}; ϕ_{g^{∗},h^{∗}}; ϕ_{\underline{kf},g^{∗}h^{∗}}; ϕ_{k,h}; ϕ_{k,\underline{hf}}; ϕ_{kh,gf}; ϕ_{k,\underline{hf}}; ϕ_{k,h}; ϕ_{g,f}; ϕ_{kh,g}; ϕ_{\underline{kg},f}; ϕ_{h^{∗},k^{∗}}; ϕ_{g^{∗}h^{∗},k^{∗}}; ϕ_{g^{∗},h^{∗}k^{∗}}; ϕ_{\underline{kf},\underline{g^{∗}k^{∗}}}; Λ̃_{5}]

(99)

Apply the 3-cocycle identity (24) to the above dotted diagram with g_{4} = k, g_{3} = h, g_{2} = g, g_{1} = f to get wavy isomorphisms in the following diagram 𝒟̃_{f}^{r} :=

[figure: ρ_{f}; ρ_{gf}; ρ\underline{gg^{∗}}; ρ\underline{hg^{∗}}; ρ\underline{hh^{∗}}; ρ\underline{kh^{∗}}; ρ\underline{kk^{∗}}; ρ\underline{hf}ρ_{g^{∗}h^{∗}}; ρ\underline{hf}; ρ\underline{kf}ρ_{g^{∗}h^{∗}}; ρ_{hg}; ρ\underline{kf}; ρ_{kh}; ρ\underline{kg}; ρ\underline{kf}ρ_{h^{∗}k^{∗}}; ρ\underline{kf}ρ\underline{g^{∗}k^{∗}}; ϕ_{g,f}; ϕ_{gf,g^{∗}}; ϕ_{h,\underline{gg^{∗}}}; ϕ_{\underline{hg^{∗}},h^{∗}}; ϕ_{k,\underline{hh^{∗}}}; ϕ_{\underline{kh^{∗}},k^{∗}}; ϕ_{g^{∗},h^{∗}}; ϕ_{\underline{hf},g^{∗}h^{∗}}; ϕ_{h,gf}; ϕ_{\underline{hf},g^{∗}}; ϕ_{g^{∗},h^{∗}}; ϕ_{\underline{hf},g^{∗}h^{∗}}; ϕ_{h,g}; ϕ_{k,\underline{hf}}; ϕ_{hg,f}; ϕ_{k,\underline{hf}}; ϕ_{k,h}; ϕ_{k,hg}; ϕ_{kh,g}; ϕ_{\underline{kg},f}; ϕ_{h^{∗},k^{∗}}; ϕ_{g^{∗}h^{∗},k^{∗}}; ϕ_{g^{∗},h^{∗}k^{∗}}; ϕ_{\underline{kf},\underline{g^{∗}k^{∗}}}; Ξ_{2}; Ξ_{1}]

(100)

With f replaced by 1, by using adjoint operations as in (44), the diagram 𝒟_{1}^{r} corresponding to the upper half is identically changed to the following diagram 𝒟̃_{1}^{r} :=

[figure: ρ_{1}; ρ_{g1}; ρ\underline{gg^{∗}}; ρ\underline{hg^{∗}}; ρ\underline{hh^{∗}}; ρ\underline{kh^{∗}}; ρ\underline{kk^{∗}}; ρ\underline{h1}ρ_{g^{∗}h^{∗}}; ρ\underline{h1}; ρ\underline{k1}ρ_{g^{∗}h^{∗}}; ρ_{hg}; ρ\underline{k1}; ρ_{kh}; ρ\underline{kg}; ρ\underline{k1}ρ_{h^{∗}k^{∗}}; ρ\underline{k1}ρ\underline{g^{∗}k^{∗}}; ϕ{−1 \atop g,1}; ϕ{−1 \atop g1,g^{∗}}; ϕ{−1 \atop h,\underline{gg^{∗}}}; ϕ{−1 \atop \underline{hg^{∗}},h^{∗}}; ϕ{−1 \atop k,\underline{hh^{∗}}}; ϕ{−1 \atop \underline{kh^{∗}},k^{∗}}; ϕ{−1 \atop g^{∗},h^{∗}}; ϕ{−1 \atop \underline{h1},g^{∗}h^{∗}}; ϕ{−1 \atop h,g1}; ϕ{−1 \atop \underline{h1},g^{∗}}; ϕ{−1 \atop g^{∗},h^{∗}}; ϕ{−1 \atop \underline{h1},g^{∗}h^{∗}}; ϕ{−1 \atop h,g}; ϕ{−1 \atop k,\underline{h1}}; ϕ{−1 \atop hg,1}; ϕ{−1 \atop k,\underline{h1}}; ϕ{−1 \atop k,h}; ϕ{−1 \atop k,hg}; ϕ{−1 \atop kh,g}; ϕ{−1 \atop \underline{kg},1}; ϕ{−1 \atop h^{∗},k^{∗}}; ϕ{−1 \atop g^{∗}h^{∗},k^{∗}}; ϕ{−1 \atop g^{∗},h^{∗}k^{∗}}; ϕ{−1 \atop \underline{k1},\underline{g^{∗}k^{∗}}}; Ξ_{2}^{†}; Ξ_{1}^{†}],

(101)

where Ξ_{j}^{†} is the adjoint of Ξ_{j}, j = 1, 2. Then the whole diagram 𝒟_{1}^{r} ⟶^{χ} 𝒟_{f}^{r} in (96) is identically changed to

𝒟̃_{1}^{r} ⟶^{χ} 𝒟̃_{f}^{r} ,

namely,

[figure: ρ_{f}; · · · · · ·; ρ_{kh}; ρ\underline{kg}; ρ_{hg}; _{· · · · · ·}; ϕ_{h,g}; ϕ_{k,h}; ϕ_{k,hg}; ϕ_{kh,g}; ρ_{1}; · · · · · ·; ρ_{kh}; ρ\underline{kg}; ρ_{hg}; _{· · · · · ·}; ϕ_{h,g}^{−1}; ϕ_{k,h}^{−1}; ϕ_{k,hg}^{−1}; ϕ_{kh,g}^{−1}; χ; Ξ_{1}; Ξ_{1}^{†}]  (102)

Note that (100) is exactly 𝒟_{f}^{l} in (86) with two extra 2-isomorphisms Ξ_{1} and Ξ_{2}. But by definition, the 2-isomorphisms Ξ_{1} and Ξ_{1}^{†} are the associators (13) corresponding to the 3-isomorphisms in C, which change

[figure: x; x; x; x; x; ρ_{g}; ρ_{h}; ρ_{k}; ρ_{f}; ρ\underline{kg}; ρ_{hg}; ρ\underline{kg}; ρ_{hg}; 1_{x}; χ; ϕ{−1 \atop k,hg}; ϕ_{k,hg}; ϕ{−1 \atop h,g}; ϕ_{h,g}] to [figure: x; x; x; x; x; ρ_{g}; ρ_{h}; ρ_{k}; ρ_{f}; ρ\underline{kg}; ρ_{kh}; ρ\underline{kg}; ρ_{kh}; 1_{x}; χ; ϕ_{kh,g}; ϕ_{k,h}],

and we have

[figure: x; x; x; x; ρ_{g}; ρ_{h}; ρ_{k}; ρ\underline{kg}; ρ_{hg}; ρ_{kg}; ρ_{hg}] [figure: x; x; ρ_{kg}] ,

by cancellation (42). So Ξ_{1} and Ξ_{1}^{†} are cancelled. More precisely, as a 3-isomorphism, Ξ_{1}^{†}#_{0}χ#_{0}Ξ_{1} is

(Ξ_{1}^{†}#_{0}χ)#_{1}(Ξ_{1}#_{0}χ) = (Ξ_{1}^{†}#_{1}Ξ_{1})#_{0}χ = 1_{ρ_{kg}}#_{0}χ,

and we have

(ϕ_{k,hg}^{−1}#_{0}ϕ_{h,g}^{−1})#_{0}(ϕ_{h,g}#_{0}ϕ_{k,hg})#_{0}χ = (ϕ_{k,hg}^{−1}#_{0}ϕ_{h,g}^{−1})#_{0}χ#_{0}(ϕ_{h,g}#_{0}ϕ_{k,hg})

in the 2-category C^{+}, up to whiskering, by the interchange law. Namely, 𝒟̃_{1}^{r} ⟶^{χ} 𝒟̃_{f}^{r} in (102) is identical to

[figure: ρ\underline{kg}; ρ\underline{kg}; ρ_{f}; · · · · · ·; _{· · · · · ·}; ρ_{hg}; _{· · · · · ·}; ϕ_{h,g}; ϕ_{k,hg}; ρ_{1}; · · · · · ·; _{· · · · · ·}; ρ_{hg}; _{· · · · · ·}; ϕ_{h,g}^{−1}; ϕ_{k,hg}^{−1}; χ]

Similarly, the 2-isomorphisms Ξ_{2} (100) and Ξ_{2}^{†} in (101) are also cancelled. The resulting diagram is exactly the diagram 𝒟_{1}^{l} ⟶^{χ} 𝒟_{f}^{l} in (87). This completes the proof of Theorem 3.7.

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Department of Mathematics, Zhejiang University, Zhejiang 310027, P. R. China

Email: wwang@zju.edu.cn
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