Theory and Applications of Categories, Vol. 16, No. 20, 2006, pp. 529–557.

CATEGORICAL REPRESENTATIONS OF CATEGORICAL GROUPS

JOHN W. BARRETT AND MARCO MACKAAY

Abstract. A representation theory for (strict) categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these bicategories contain representations which are indecomposable but not irreducible. A simple example is computed in explicit detail.

1. Introduction

In three-dimensional topology there is a very successful interaction between category theory, topology, algebra and mathematical physics which is reasonably well understood. Namely, monoidal categories play a central role in the construction of invariants of threemanifolds (and knots, links and graphs in three-manifolds), which can be understood using quantum groups and, from a physics perspective, the Chern-Simons functional integral. The monoidal categories determined by the quantum groups are all generalisations of the idea that the representations of a group form a monoidal category.

The corresponding situation for four-manifold topology is less coherently understood and one has the feeling that the current state of knowledge is very far from complete. The complexity of the algebra increases dramatically in increasing dimension (though it might eventually stabilise). Formalisms exist for the application of categorical algebra to fourdimensional topology, for example using Hopf categories [CF], categorical groups [Y-HT] or monoidal 2-categories [CS, BL, M-S]. Since braided monoidal categories are a special type of monoidal 2-category (ones with only one object), then there are examples of the latter construction given by the representation theory of quasi-triangular Hopf algebras. This leads to the construction of the four-manifolds invariants by Crane, Yetter, Broda and Roberts which give information on the homotopy type of the four-manifold [CKY, R, R-EX]. At present it seems that categorical invariants which delve further into the smooth or combinatorial structure of four-manifolds will require different types of examples of monoidal 2-categories.

In this paper we determine a new set of examples of monoidal 2-categories. We show that the categorical representations of a categorical group form a monoidal 2-category, by direct analogy with the way in which the representations of a group form a monoidal category. The categorical definitions are given in section 2 and 3. In section 4, an abstract definition of categorical representations, and their morphisms, is given and then unpacked for the case of strict representations. An extended example is calculated in section 4 with explicit matrices, illustrating many of the complexities of more general examples. In section 5 we give a fairly complete characterisation of the one-dimensional categorical representations, outlining a number of examples. Finally in section 6 we make a number of remarks about the structure of the N-dimensional categorical representations, in particular the phenomenon of representations which are indecomposable but not irreducible. These remarks generalise some of the features of the example in section 4.

Received by the editors 2004-09-21 and, in revised form, 2006-08-24.

Transmitted by Ross Street. Published on 2006-09-04.

2000 Mathematics Subject Classification: 18D05.

Key words and phrases: categorical group, categorical representations, monoidal 2-categories.

\begin{matrix}c \\ ◯ \end{matrix} John W. Barrett and Marco Mackaay, 2006. Permission to copy for private use granted.

One particular example of a monoidal category leads to a state-sum model for quantum gravity in three-dimensional space-time [TV, B-O]. The motivation for this work grew out of enquiring whether there is a corresponding model in the more realistic fourdimensional space-time. The first attempts at doing this [BC, DFKR] used a braided monoidal category and suffer from several problems, one of which is that there is no ‘data’ on the edges of a triangulation of the manifold, which is where one might expect to find the combinatorial version of the metric tensor [REG]. Thus we arrived at the idea of constructing the monoidal 2-category of representations for the example of the categorical Lie group determined by the Lorentz group and its action on the translation group of Minkowski space, generalising the construction of [M-FG]. An early draft of this paper is the reference cited by Crane and Yetter [CY-2G, CY-MC, Y-MC, CSH] who developed the particular example, and the machinery of measurable categories to handle the Lie aspect, much further.

2. Categorical groups

By the term ‘categorical group’ we mean what some authors call a ‘strict categorical group’.

2.1. Definition. A categorical group is by definition a group-object in the category of groupoids.

This means that a categorical group is a groupoid G, with a set of objects G_{0} ⊂ G, together with functors which implement the group product, ∘: G ×G → G, and the inverse −1 : G → G, together with an identity object 1 ∈ G_{0}. These satisfy the usual group laws:

a ∘ (b ∘ c) = (a ∘ b) ∘ c
a ∘ 1 = 1 ∘ a = a
a ∘ a^{−1} = a^{−1} ∘ a = 1

for all a, b, c in G. In particular, G is a strict monoidal category.

2.2. Definition. A functorial homomorphism between two categorical groups is a strict monoidal functor.

Categorical groups are equivalent to crossed modules of groups. This equivalence, and the basic properties of categorical groups, are explained in [BS]. Here we give a brief outline.

In a categorical group G with hom-sets G(X, Y), the categorical composition f · g and the group product ∘ are related by the interchange law

(f ∘ g) · (h ∘ k) = (f · h) ∘ (g · k).

For all categories in this paper the diagrammatic order of composition is used. This means that f · g is defined when the target of f is equal to the source of g.

In fact the composition is determined by the product. If f ∈ G(X, Y) and g ∈ G(Y, Z), then

f · g = f ∘ 1_{Y^{−1}} ∘ g = g ∘ 1_{Y^{−1}} ∘ f.

In particular, G(1, 1) is an abelian group, the composition and product coinciding.

2.3. Definition. A crossed module is a homomorphism of groups

∂: E → G

together with an action ▷ of G on E by automorphisms, such that

∂(X▷e) = X(∂e)X^{−1}
(∂e)▷e′ = ee′e^{−1}.

We call E the principal group and G the base group.

There is a natural notion of a mapping between crossed modules.

2.4. Definition. A homomorphism of crossed modules

(E, G, ∂, ▷) → (E′, G′, ∂′, ▷′)

is given by two vertical homomorphisms

E ⟶^{∂} G
F_{p}↓  ↓^{F_{b}}
E′ ⟶^{∂′} G′

which commute and satisfy F_{p}(X▷e) = F_{b}(X)▷′F_{p}(e). We call the latter condition the action condition.

The equivalence with categorical groups is as follows.

2.5. Theorem. (Verdier) The category of categorical groups and functorial homomorphisms and the category of crossed modules of groups and homomorphisms between them are equivalent.

The proof is sketched. Given a categorical group, a crossed module is defined by taking the base group G(G) to be the objects,

G(G) = (G_{0}, ∘),

and the principal group E(G), to be the subset of morphisms which are morphisms from the object 1 (to any object),

E(G) = ⋃_{X} G(1, X),

again with the product operation ab = a ∘ b. The homomorphism E → G of the crossed module is e ∈ G(1, X) ↦ X and the action is Y ▷e = 1_{Y} ∘ e ∘ 1_{Y^{−1}}.

Conversely, given a crossed module (E, G, ∂, ▷), a categorical group is constructed in a canonical way by taking G_{0} = G, G(X, Y) = ∂^{−1}(Y X^{−1}). If f ∈ G(X, Y) and g ∈ G(Z, T), then the tensor product is defined as

f ∘ g = f(X▷g) ∈ G(XZ, Y T)

and the composition, for Y = Z,

f · g = gf ∈ G(X, T).

It is worth noting that the definition of a crossed module and of a categorical group makes sense when the groups and groupoids are Lie groups and the equivalence between Lie categorical groups and Lie crossed modules holds in the same way.

2.6. Examples.

1. Let K be a group, then we define \overline{K}, the closure of K, to be the groupoid with one object • and hom-space \overline{K}(•, •) = K. If K is abelian, then the group operation also defines a monoidal structure, so that \overline{K} becomes a categorical group. The corresponding crossed module has principal group E(\overline{K}) = K and trivial base group.

2. A categorical group is connected if there is a morphism between any pair of objects. The corresponding crossed module ∂: E → G is surjective. This implies that E is a central extension of G, and the action of G on E is determined by conjugation in E.

3. A categorical group is skeletal if there are no morphisms between distinct objects (generalising (1)). In this case the crossed module ∂: E → G has ∂ = 1 and is determined entirely by the action of G on the abelian group E. An example is the wreath product, where G is the symmetric group S_{n} and E = (ℂ^{∗})^{n}, with the action of S_{n} being the permutation of the factors in E.

4. A categorical group is discrete if there is at most one morphism between any pair of objects. The crossed module ∂: E → G is injective and E is a normal subgroup of G. Again the action of G on E is determined by conjugation.

5. The transformation categorical group of a category C is defined by

G_{0} = {functorial isomorphisms on C} and G = {natural isomorphisms}.

Here we consider, as usual, G_{0} to be a subset of G by identifying a functorial isomorphism with the identity natural isomorphism of it. For example, if C = \overline{K}, then the crossed module corresponding to the transformation categorical group is K → AutK, where ∂ maps a group element to the corresponding inner automorphism.

3. Bicategories

In this section we recall the definitions of 2-dimensional category theory [G, KS]. First we define 2-categories, sometimes called strict 2-categories, and then indicate the changes required to give the weaker notion of bicategories. Finally we discuss monoidal structures on bicategories and the example of 2-Vect.

3.1. Definition. A 2-category C is given by:

1. A set of objects C_{0}.

2. A small category C(X, Y), for each pair X, Y ∈ C_{0}. The set of objects in C(X, Y) we denote by C_{1}(X, Y). The elements of C_{1}(X, Y) are called 1-morphisms. For each pair f, g ∈ C_{1}(X, Y), we denote the set of morphisms in C(X, Y) from f to g by C_{2}(f, g). The elements of C_{2}(f, g) are called 2-morphisms. The composition in C(X, Y) is called the vertical composition and denoted by a small dot, e.g. μ · ν (or sometimes by simple concatenation without dot).

3. A functor

∘: C(X, Y) × C(Y, Z) → C(X, Z),

for each triple X, Y, Z ∈ C_{0}. Together these are called the horizontal composition. The horizontal composition is required to be associative and unital. The last condition means that there is a 1-morphism 1_{X} ∈ C_{1}(X, X), for each X ∈ C_{0}, which is a right and left unit for horizontal composition.

3.2. Example. Let G be a categorical group. Then we define the closure of G, which we denote by \overline{G}, to be the 2-category with one object denoted •, such that \overline{G}(•, •) = G. The horizontal composition is defined by the monoidal structure in G. Note that there is a slightly confusing mixture of subscripts now, because \overline{G}_{0} = \{•\}, whereas \overline{G}_{1} = \overline{G}_{1}(•, •) = G_{0}. Unfortunately this renumbering seems unavoidable in this subject and we hope that the context will always avoid confusion in this paper.

Next we recall the definition of 2-functors, pseudonatural transformations and modifications.

3.3. Definition. Given two 2-categories C and D, a 2-functor between them, F: C → D, consists of:

1. A function F_{0} : C_{0} → D_{0}.

2. A functor F_{1}(X, Y): C(X, Y) → D(F (X), F (Y)), for each pair X, Y ∈ C_{0}. These functors are required to be compatible with the horizontal composition and the unit 1-morphisms. By abuse of notation we sometimes denote both F_{0} and F_{1}(X, Y) simply by F.

3.4. Definition. Given two 2-functors between two 2-categories F, G: C → D, a pseudonatural transformation between F and G, denoted h: F ⇒ G, consists of:

1. A 1-morphism h(X) ∈ D_{1}(F (X), G(X)), for each X ∈ C_{0}.

2. For each pair X, Y ∈ C_{0}, a natural isomorphism h̃: F_{1}(X, Y) ∘ h(Y) → h(X) ∘ G_{1}(X, Y), where we consider F_{1}(X, Y) ∘ h(Y) and h(X) ∘ G_{1}(X, Y) both as functors from C(X, Y) to D(F (X), G(Y)). We require two coherence conditions to be satisfied:

(a) h̃(f ∘ g) = (1_{F (f)} ∘ h̃(g)) · (h̃(f) ∘ 1_{G(g)}).

(b) h̃(1_{X}) = 1_{h(X)}.

This notion of pseudonatural transformation is almost the same as the notion of quasinatural transformation in [G], which differs in that the 2-cells are not required to be isomorphisms and have the arrows reversed.

3.5. Definition. Let F, G, H: C → D be three 2-functors and let h: F ⇒ G and k: G ⇒ H be two pseudonatural transformations. The horizontal composite of h and k, denoted h ∘ k, is defined by

1. For each X ∈ C_{0}, (h ∘ k)(X) = h(X) ∘ k(X).

2. For each pair X, Y ∈ C_{0}, { \atop h ∘ k} = (h̃ ∘ 1_{k(Y)}) · (1_{h(X)} ∘ k̃).

3.6. Definition. Let F, G: C → D be two 2-functors and h, k: F ⇒ G be two pseudonatural transformations. A modification ϕ: h ⇛ k is given by a 2-morphism ϕ(X) ∈ D_{2}(h(X), k(X)), for each X ∈ C_{0}. These 2-morphisms are required to satisfy

h̃(f) · (ϕ(X) ∘ 1_{G(f)}) = (1_{F (f)} ∘ ϕ(Y)) · k̃(f),

for any X, Y ∈ C_{0} and f ∈ C_{1}(X, Y).

3.7. Definition. The horizontal and vertical compositions of modifications are directly induced by the corresponding compositions in D.

3.8. Weakenings. The notion of a 2-category can be weakened to the notion of a bicategory, in which the associativity of horizontal composition is not given by an equation between 1-morphisms, but by 2-isomorphisms which are the components of a natural isomorphism called the 1-associator,

α_{f,g,h}: f ∘ (g ∘ h) ⇒ (f ∘ g) ∘ h,

which satisfies a certain coherence law. Similarly, the unital nature of the horizontal composition can be weakened by introducing natural isomorphisms [BEN]. However, in all the examples in this paper the unital isomorphisms are all identities. We call this sort of bicategory a strictly unital bicategory

The notions of 2-functor, pseudonatural transformation and modification have suitable generalisations to the case of bicategories. One new phenomenon which occurs is that, for the 2-functors, the horizontal composition is no longer preserved exactly, but only up to a family of natural isomorphisms, defined as follows. If F is a 2-functor and f and g composable 1-mophisms in its domain, then the 2-isomorphisms are

F̃_{fg}: F (f) ∘ F (g) → F (f ∘ g).  (1)

In general a 2-functor between bicategories involves weakening the condition F (1_{f}) = 1_{F (f)} to an isomorphism. This notion of 2-functor is called a homomorphism in [BEN, STR]. It is also often called a pseudofunctor [KS].

3.9. Definition. A strictly unitary homomorphism is a 2-functor between bicategories for which the isomorphisms F (1_{f}) ≅ 1_{F (f)} are all identity 2-morphisms. If additionally, for all f, g, F̃_{fg} is an identity 2-morphism we call F a strict homomorphism

The strictly unitary condition is also called normal or normalized in some references. The pseudonatural transformations have a straightforward generalisation to bicategories.

Given two bicategories C and D, then the following result gives a construction of a new bicategory [STR, GPS].

3.10. Theorem. The 2-functors from C to D, together with their pseudonatural transformations and their modifications form a bicategory bicat(C, D).

3.11. Monoidal structures. The notion of a monoidal structure on a 2-category is straightforward to define.

3.12. Definition. A monoidal 2-category is a 2-category C together with a 2-functor ⊠: C × C → C the monoidal product, which is associative and unital. The latter means that there is a unit object I which is a left and right unit for the monoidal product.

By C × C we mean the cartesian product 2-category. The requirement that ⊠ be a 2functor means that the interchange law is satisfied as an identity. This is too strict for the purposes of this paper, as we need to use bicategories rather than 2-categories. This gives the notion of a monoidal bicategory, for which the definition is the same as the definition for a monoidal 2-category, but with the appropriate notion of 2-functor for bicategories and pseudonatural transformations which give the associative and unital conditions. This means the 2-functor ⊠: C ×C → C carries natural 2-isomorphisms ˜⊠ called the tensorator. The monoidal structures in the examples of interest to us are not strictly associative either. There are natural 1-isomorphisms called the 0-associator. We remark that in our examples the 0-associator is strictly natural and satisfies the pentagon equation on the nose, so all the other coherers in the general definition of a monoidal bicategory besides the ones mentioned above are always trivial in this paper. This entire structure, together with the axioms it obeys, is a special case of the definition of a tricategory given by [GPS], in which the tricategory has only one object.

3.13. 2-Vector spaces. In this section we recall the definition, due to Kapranov and Voevodsky [KV], of the monoidal bicategory of 2-vector spaces in the completely coordinatized version.

3.14. Definition. We define the monoidal bicategory 2-Vect as follows:

1. 2-Vect_{0} = ℕ, the set of natural numbers including zero.

2. For any N, M ∈ 2-Vect_{0}, we define the category 2-Vect(N, M) as follows:

i) The set of objects of 2-Vect_{1}(N, M) consists of all N × M matrices with coefficients in ℕ.

ii) For any a, b ∈ 2-Vect_{1}(N, M), the set 2-Vect_{2}(a, b) consists of all N × M matrices whose coefficients are complex matrices such that the i, j-coefficient has dimension a_{j}^{i} × b_{j}^{i}. For any a ∈ 2-Vect_{1}(N, M), we define 1_{a} to be the N × M matrix such that (1_{a})_{j}^{i} is the identity matrix of dimension a_{j}^{i} The vertical composition of two composable 2-morphisms is defined by componentwise matrix multiplication

(μ · ν)_{j}^{i} = μ_{j}^{i}ν_{j}^{i}.

3. The unit 1-morphism on an object N, denoted 1_{N}, is given by the N × N identity matrix. The (horizontal) composition of two 1-morphisms is defined by matrix multiplication.

The horizontal composition of two composable 2-morphisms is defined by

(μ ∘ ν)_{j}^{i} = ⊕_{k}μ_{k}^{i} ⊗ ν_{j}^{k}.

4. The horizontal composition is not strictly associative, and so there are associativity isomorphisms which carry out the corresponding permutations of bases. In detail, if f, g, h are composable 1-morphisms, then the associativity 2-morphism α_{f,g,h} has components the permutation matrices [α_{f,g,h}]_{j}^{i} :

⨁_{k} [(⨁_{l} (C^{f_{l}^{i}} ⊗ C^{g_{k}^{l}})) ⊗ C^{h_{j}^{k}}] → ⨁_{l} [C^{f_{l}^{i}} ⊗ (⨁_{k} (C^{g_{k}^{l}} ⊗ C^{h_{j}^{k}}))]

which map any vector of the form

⨁_{k} [(⨁_{l} (x_{l}^{i} ⊗ y_{k}^{l})) ⊗ z_{j}^{k}]

to the corresponding element

⨁_{l} [x_{l}^{i} ⊗ (⨁_{k} (y_{k}^{l} ⊗ z_{j}^{k}))] .

5. The monoidal product of two objects is defined by

N ⊠ M = NM.

For two 1-morphisms we define

(a ⊠ b)_{kl}^{ij} = a_{k}^{i}b_{l}^{j}.

Finally, for two 2-morphisms we define

(μ ⊠ ν)_{kl}^{ij} = μ_{k}^{i} ⊗ ν_{l}^{j}.

The unit object is 1.

This monoidal product has a tensorator ˜⊠ and an 0-associator which are again given by the corresponding permutations of basis elements.

6. The remaining coherers for a monoidal bicategory are trivial.

So far we have defined the structure of the monoidal 2-category. There are also linear structures in 2-Vect, which we define below. Although we have not defined linear structures in general, we hope that the definitions below are clear.

3.15. Definition. There are three levels of linear structure in 2-Vect, which we call the monoidal sum, the direct sum and the sum respectively.

1. The monoidal sum defines a monoidal structure on 2-Vect. For two objects it is

defined as

N ⊞ M = N + M,

for two 1-morphisms as

a ⊞ b = a ⊕ b

and for two 2-morphisms as

μ ⊞ ν = μ ⊕ ν.

The zero object is 0. Unlike ˜⊠, which is non-trivial, ˜⊞ is just the identity.

2. The direct sum defines a monoidal structure on each 2-Vect(N, M). The direct sum of two 1-morphisms a, b in 2-Vect(N, M) is defined by

(a ⊕ b)_{j}^{i} = a_{j}^{i} + b_{j}^{i}.

The direct sum of two 2-morphisms in 2-Vect(N, M) is defined by

(μ ⊕ ν)_{j}^{i} = μ_{j}^{i} ⊕ ν_{j}^{i}.

The zero 1-morphism, denoted (0), is given by the N × M-dimensional zero matrix.

3. The sum defines the linear structure in each 2-Vect(N, M). For two 2-morphisms

we define

(μ + ν)_{j}^{i} = μ_{j}^{i} + ν_{j}^{i}.

The zero 2-morphism, denoted ((0)), is given by the matrix all of whose entries are zero matrices of the right size.

3.16. Definition. Let N ∈ ℕ. We define the general linear categorical group GL(N) ⊂ 2-Vect(N, N) to be the categorical group consisting of all invertible 1- and 2-morphisms in 2-Vect(N, N).

This definition makes sense because the horizontal composition of invertible 1- and 2-morphisms in 2-Vect is in fact strictly associative, as proved in the following lemma.

3.17. Lemma. The monoidal category GL(N) is a categorical group, and the associated crossed module of groups is given by

(ℂ^{∗})^{N} ⟶^{1} S_{N},

where S_{N} is the symmetric group on N letters and 1 the trivial group homomorphism which maps everything to 1 ∈ S_{N}. The action of S_{N} on (ℂ^{∗})^{N} is given by the permutations of the coordinates.

Proof. The only invertible 1-morphisms in 2-Vect(N, N) are the permutation matrices. The 1-associator for a triple of permutation matrices is trivial, because it is a set of permutation matrices on 0 or 1 dimensional vector spaces. Hence the horizontal composition in GL(N) is strictly associative and it forms a categorical group.

Between two permutation matrices there can only be an invertible 2-morphism if they are equal. The reason is that, if the corresponding entries in two permutation matrices are different, then one of them has to be zero.

This shows that 2-Vect_{2}(1_{N}, 1_{N}) has its only non-trivial entries on the diagonal and these are 1-dimensional invertible complex matrices. Hence 2-Vect_{2}(1_{N}, 1_{N}) is isomorphic to (ℂ^{∗})^{N}. Theorem 2.5 shows that the action of a permutation matrix P ∈ 2-Vect_{1}(N, N) on a diagonal matrix a ∈ 2-Vect_{2}(1_{N}, 1_{N}) is given by

P aP^{−1}.

Here we mean the ordinary matrix multiplication, which is what 1_{P} ⊗ a ⊗ 1_{P^{−1}} amounts to in this case. Thus the element in S_{N}, which corresponds to P, acts on the element in (ℂ^{∗})^{N}, which corresponds to a, by permutation of its coordinates.

4. Categorical representations

In this section we give the abstract definitions of categorical representations, the functors between them, the natural transformations between such functors and the monoidal product. These definitions are analogues of Neuchl’s [N] definitions for Hopf categories. We also work out a concrete example.

Let G be a group and \overline{G} its closure (defined in Example 1). One can check that a representation of G is precisely a functor \overline{G} → Vect and an intertwiner precisely a natural transformation between two such functors. This observation motivates the following definition for categorical groups.

The starting point is the selection of a monoidal bicategory as the category in which the categorical group is represented. In this paper we discuss only 2-Vect, although other monoidal bicategories could be used instead (see the remarks in the final section).

Let G be an arbitrary categorical group and let \overline{G} be its closure, defined in Ex. 3.2. We first give a conceptual definition.

4.1. Definition. a) A categorical representation of G is a strictly unitary homomorphism (R, R̃): \overline{G} → 2-Vect. We call the non-negative integer R(•) ∈ 2-Vect_{0} the dimension of the categorical representation.

b) A 1-intertwiner is a pseudonatural transformation between two categorical representations.

c) A 2-intertwiner is a modification between two 1-intertwiners with the same source and target.

Note that Neuchl [N] uses the terms G-functors and G-transformations instead of 1- and 2-intertwiners.

Theorem 3.10 shows that the categorical representations of G, together with the 1- and 2-intertwiners, form a bicategory bicat(\overline{G}, 2-Vect). To give this the structure of a monoidal bicategory, we first promote \overline{G} and 2-Vect to tricategories and use some general results about those. We consider \overline{G} as a strict tricategory, denoted \underline{\overline{G}} by adding only identity 3-morphisms to the existing strict bicategory. For 2-Vect we take the closure \overline{2-Vect} as the tricategory. By a general result of Gordon, Power and Street [GPS] about tricategories we know that tricat(\underline{\overline{G}}, \overline{2-Vect}) forms a tricategory. The objects of this tricategory are trihomomorphisms (functors between tricategories). The constant trihomomorphism is the one that sends every 3-morphism to 1_{1_{1•}}.

4.2. Lemma. \overline{bicat(\overline{G}, 2-Vect)} is the subtricategory of tricat(\underline{\overline{G}}, \overline{2-Vect}) determined by the 1-, 2- and 3-morphisms on the unique constant trihomomorphism.

Proof. Just check the diagrams in [GPS].

Unfortunately Gordon, Power and Street [GPS] do not give explicit definitions of the composition rules for the various morphisms in tricategories. Therefore we spell out the tensor product in bicat(\overline{G}, 2-Vect) below, which corresponds to the horizontal composition in tricat(\underline{\overline{G}}, \overline{2-Vect}). The first definition can also be found in [ROU].

4.3. Lemma. Let (R, R̃) and (T, T̃) be two categorical representations. Then the monoidal product (R ⊠ T, { \atop R ⊠ T}) of (R, R̃) and (T, T̃) is given by

R ⊠ T (X) = R(X) ⊠ T (X)

and the following diagram

R(X)R(Y) ⊠ T (X)T (Y) ⟶^{R̃_{X,Y} ⊠T̃_{X,Y}} R(XY) ⊠ T (XY) = R ⊠ T (XY)
↓˜⊠_{(R(X),T (X)),(R(Y ),T (Y ))}  ↑_{R⊠T}^{}_{X,Y}
(R(X) ⊠ T (X))(R(Y) ⊠ T (Y))  (R ⊠ T (X))(R ⊠ T (Y))

4.4. Lemma. Let (h, h̃): R_{1} → R_{2} and (k, k̃): T_{1} → T_{2} be 1-intertwiners. The monoidal product (h ⊠ k, { \atop h ⊠ k}) is given by

h ⊠ k(X) = h(X) ⊠ k(X)

using on the right the monoidal product in 2-Vect, and by the following diagram

R_{1} ⊠ T_{1}(f) ∘ h ⊠ k(Y)  ⟶^{{ \atop h⊠k}}  (h ⊠ k(X)) ∘ (R_{2} ⊠ T_{2}(f))
∥  ∥
(R_{1}(f) ⊠ T_{1}(f)) ∘ (h(Y) ⊠ k(Y))  (h(X) ⊠ k(X)) ∘ (R_{2}(f) ⊠ T_{2}(f))
˜⊠↓  ˜⊠↓
(R_{1}(f) ∘ h(Y)) ⊠ (T_{1}(f) ∘ k(Y)) ⟶^{h̃⊠k̃} (h(X) ∘ R_{2}(f)) ⊠ (k(X) ∘ T_{2}(f))

4.5. Lemma. Let α and β be two 2-intertwiners. The monoidal product is given by

α ⊠ β,

using the monoidal product in 2-Vect.

There is also a natural way of defining the monoidal sum of two categorical representations.

4.6. Definition. Let (R, R̃) and (T, T̃) be two categorical representations. Their monoidal sum is defined by

R ⊞ T (X) = R(X) ⊞ T (X),

and

R̃ ⊞ T̃(X, Y) = R̃(X, Y) ⊞ T̃(X, Y).

Note that the latter makes sense, because

(R(X) ∘ R(Y)) ⊞ (T (X) ∘ T (Y)) = (R(X) ⊞ T (X)) ∘ (R(Y) ⊞ T (Y))

holds on the nose, i.e. the 2-isomorphism between both sides of the equation is the identity, as already noticed in definition 3.15.

Since ˜⊞ is trivial, the definition of their monoidal sum of 1- and 2-intertwiners is much simpler than that of their monoidal product.

4.7. Definition. Let (h, h̃): R_{1} → R_{2} and (k, k̃): T_{1} → T_{2} be 1-intertwiners. We define their monoidal sum, (h ⊞ k, { \atop h ⊞ k}), to be

h ⊞ k(X) = h(X) ⊞ k(X)

and

h̃ ⊞ k̃,

using on the right the monoidal sum in 2-Vect.

4.8. Definition. Let α and β be two 2-intertwiners. We define their monoidal sum as

α ⊞ β,

using the monoidal product in 2-Vect.

We do not give a precise definition of the direct sum of 1- and 2-intertwiners and the sum of 2-intertwiners, because we do not need them.

4.9. Strict categorical representations. These definitions can be unpacked by applying the general definitions of bicategories to this particular situation and expressing the result in terms of categorical groups. The categorical representations can be formulated in terms of crossed modules. In this way a strictly unitary homomorphism leads to a weakened notion of morphism between crossed modules, involving a group 2-cocycle on the object group G corresponding to R̃. The extra conditions on this cocycle appear to be very complicated. Therefore we restrict our attention to a subclass of categorical representations in the rest of this paper, which we call strict categorical representations.

4.10. Definition. A strict categorical representation is a strict homomorphism R: \overline{G} → 2-Vect. This means that R̃_{X,Y} are all identity 2-morphisms.

Restricting to strict categorical representations still gives a monoidal bicategory, as proved by the following lemma.

4.11. Lemma. The monoidal product and the monoidal sum of two strict categorical representations yield strict categorical representations.

Proof. The first claim follows from the fact that the top and the left-hand side of the diagram in lemma 4.3 are trivial. Note that ˜⊠_{(R(X),T (X)),(R(Y),T (Y))} is trivial, because all matrices in the subscript are permutation matrices.

The second claim is obvious as well, because the monoidal sum of two identity 2morphisms is an identity 2-morphism.

Let us now unpack the definition of a strict categorical representation and 1- and 2-intertwiners.

• A strict categorical representation of G amounts to a choice of a non-negative integer N and a strict homomorphism R: \overline{G} → \overline{GL(N)}, or equivalently a functorial homomomorphism between categorical groups R: G → GL(N). It can also be described as a homomorphism between the corresponding crossed modules,

\begin{matrix}E(G) & ⟶^{∂} & G(G) \\ ^{R_{p}}↓ & & ↓^{R_{b}} \\ (ℂ^{∗})^{N} & ⟶^{1} & S_{N} \end{matrix}  (2)

according to theorem 2.5 and lemma 3.17.

• Let R: G → GL(N) and T: G → GL(M) be two strict categorical representations of G. A 1-intertwiner between them consists of a 1-morphism h_{•} ∈ 2-Vect_{1}(N, M) together with a 2-isomorphism h̃(X) ∈ 2-Vect_{2}(R(X) ∘ h_{•}, h_{•} ∘ T (X)), for each X ∈ G_{0}.

The 1-morphism h_{•} can be thought of as a vector bundle (with fibres of varying dimension) over the finite set N × M, the cartesian product of the N-element set with the M-element set. The data above determine an action of the group G_{0} on this vector bundle in the following way.

There is a right action of X ∈ G_{0} on the set N, i ↦ iR_{b}(X), given by the permutation matrix R_{b}(X) (acting on the right) and similarly a right action of G_{0} on the set M given by T_{b}.

4.12. Lemma. The collection of linear maps

h̃(X)_{j}^{i}: h_{•}_{j}^{iR_{b}(X)} → h_{•}_{jT_{b}(X)^{−1}}^{i}

determines a left action of G_{0} on the vector bundle h_{•}.

Proof. The two coherence conditions in definition 3.4 are

h̃(X ∘ Y) = (1_{R(X)} ∘ h̃(Y)) · (h̃(X) ∘ 1_{T (Y)})

h̃(1) = 1_{h_{•}}.

Note that the 1-associators in these expressions are trivial because R and T take values in the permutation matrices. Taking the (i, j)-th component of h̃ gives

h̃(X ∘ Y)_{j}^{i} = h̃(Y)_{j}^{iR_{b}(X)} · h̃(X)_{jT_{b}(Y)^{−1}}^{i},

which is the condition for a left action. This covers the left action of G_{0} on N × M given by (iR_{b}(X), j) ↦ (i, jT_{b}(X)^{−1}).

The remaining condition satisfied by the 1-intertwiner is the naturality condition. This is that the diagram

\begin{matrix}R(X) ∘ h_{•} & ⟶^{h̃(X)} & h_{•} ∘ T (X) \\ ^{R(f)∘1_{h•}} ↓ & & ↓^{1_{h•}∘T (f)} \\ R(Y) ∘ h_{•} & ⟶^{h̃(Y)} & h_{•} ∘ T (Y) \end{matrix}  (3)

commutes.

Since h̃(X) is invertible, this implies the equation on 1-morphisms

R(X) ∘ h_{•} = h_{•} ∘ T (X).  (4)

Also restricting naturality to the crossed module data (X = 1), gives the equation

(R(e) ∘ 1_{h_{•}}) · h̃(∂e) = 1_{h_{•}} ∘ T (e)  (5)

for e ∈ E(G).

It is possible to show that this last condition is actually sufficient to recover all of (3).

4.13. Lemma. For intertwiners of strict categorical representations, the naturality condition (3) follows from condition (5) for all e: 1 → Y and the coherence condition 2(a) of definition 3.4.

Proof. The naturality condition follows from the following computation. Suppose f: X → X ∘ Y is an arbitrary morphism in G_{1}. We can write f = 1_{X} ∘ e, where e: 1 → Y. Then

(R(f) ∘ 1_{h_{•}}) · h̃(X ∘ Y)
= (1_{R(X)} ∘ R(e) ∘ 1_{h_{•}}) · ((1_{R(X)} ∘ h̃(Y)) · (h̃(X) ∘ 1_{T (Y)})

(using 2(a) of definition 3.4)

= (1_{R(X)} ∘ ((R(e) ∘ 1_{h_{•}}) · h̃(Y))) · (h̃(X) ∘ 1_{T (Y)})
= (1_{R(X)} ∘ 1_{h_{•}} ∘ T (e)) · (h̃(X) ∘ 1_{T (Y)})

(using (5))

= h̃(X) ∘ T (e)
= (h̃(X) ∘ 1_{1}) · (1_{h_{•}} ∘ 1_{T (X)} ∘ T (e))
= h̃(X) · (1_{h_{•}} ∘ T (f)) .

In the computation, all 1-associators which occur are the identity.

• Let h = (h_{•}, h̃), k = (k_{•}, k̃): R → T be 1-intertwiners. A 2-intertwiner between them consists of a single 2-morphism ϕ ∈ 2-Vect_{2}(h_{•}, k_{•}).

The condition that this satisfies is that for each X ∈ G_{0} the following diagram commutes:

\begin{matrix}R(X) ∘ h_{•} & ⟶^{h̃(X)} & h_{•} ∘ T (X) \\ ^{1_{R(X)}∘ϕ}↓ & & ↓^{ϕ∘1_{T (X)}} \\ R(X) ∘ k_{•} & ⟶^{k̃(X)} & k_{•} ∘ T (X) \end{matrix}  (6)

4.14. Example.

4.15. Convention. From now on a categorical representation will always mean a strict categorical representation.

In this section we work out the categorical representations of a concrete example of a finite skeletal categorical group. Recall that the crossed modules corresponding to skeletal categorical groups are determined by a group G and an abelian group E on which G acts by automorphisms. The simplest example is G = C_{2}, the cyclic group with two elements ±1, and E = C_{3} = {1, x, x^{−1}}, with the non-trivial action of C_{2} on C_{3}, −1▷x = x^{−1}. We call this categorical group G(2, 3). Another way of looking at this example is by defining the total space of all the morphisms

G(2, 3) =

\{(\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) , (\begin{matrix}x & 0 \\ 0 & x^{−1} \end{matrix}) , (\begin{matrix}x^{−1} & 0 \\ 0 & x \end{matrix}) , (\begin{matrix}0 & x \\ x^{−1} & 0 \end{matrix}) , (\begin{matrix}0 & x^{−1} \\ x & 0 \end{matrix})\} .

The source and target of each morphism is determined by the place of the non-zero coefficients of the corresponding matrix, e.g. the sources and targets of the first two matrices are equal to 1 and −1 respectively. In this matrix notation the monoidal product, corresponding to the horizontal composition in \overline{G(2, 3)} and denoted by ∘, is defined by matrix multiplication (giving the dihedral group D_{3}) and the composition, corresponding to the vertical composition in \overline{G(2, 3)} and denoted by simple concatenation, by coefficientwise multiplication.

The constructions will be carried out in this example. The main features of the general case are apparent in this example; some comments on the generalisations are given at the end of the section.

We classify all 1- and 2-dimensional categorical representations of G(2, 3).

1. V(1). This is the identity representation defined by R_{b}(±1) = 1 ∈ S_{1} and R_{p}(x) = 1 ∈ ℂ^{∗}. It is the only 1-dimensional categorical representation, due to the following argument. Obviously R_{b}(±1) = 1 ∈ S_{1} has to hold. By (2) we see that R(x) = ξ(x), where ξ is a complex group character on C_{3}. By the action condition in definition 2.4 we see that ξ(x^{−1}) = ξ(x), so ξ has to be the trivial character and R_{p}(x) = 1.

2. V(2). This is the trivial 2-dimensional categorical representation defined by R_{b}(±1) = 1 ∈ S_{2} and R_{p}(x) = (1, 1) ∈ (ℂ^{∗})^{2}. As in the 1-dimensional case, the action condition forces R_{p} to be trivial if R_{b} is trivial. Below we show that V(2) is isomorphic to V(1) ⊞ V(1).

3. V(2)_{ξ}. These are the non-trivial 2-dimensional categorical representations, where R_{b}(±1) = ±1 ∈ S_{2} and R_{p} is determined by one complex group character, ξ, on C_{3}.

By (2) we see that R_{p}(x) = (ξ(x), ψ(x)), where ξ and ψ are both complex group characters on C_{3}. The action condition now becomes (ξ(x)^{−1}, ψ(x)^{−1}) = (ψ(x), ξ(x)), so we have ξ = ψ^{−1}. There are no further restriction on ξ. A nice way of picturing the strict homomorphisms R of these representations is by using the matrix definition of G(2, 3):

R ((\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix})) = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) , R ((\begin{matrix}x & 0 \\ 0 & x^{−1} \end{matrix})) = (\begin{matrix}ξ(x) & 0 \\ 0 & ξ(x)^{−1} \end{matrix}) .

The image of the other endomorphisms is obtained via horizontal composition.

Next we study all the 1-intertwiners between these categorical representations. In Lemma 4.12 we showed that these can be seen as vector bundles with a left action of G_{0}. In particular, equation (4) holds. The naturality condition simplifies a bit further in this example. Let (h_{•}, h̃) be a 1-intertwiner between two categorical representations R and T. By lemma 4.13 the naturality condition reduces to the equation

R(e) ∘ 1_{h_{•}} = 1_{h_{•}} ∘ T (e),  (7)

for any e ∈ E(G(2, 3)). Note that this shows that the action on the vector bundle does not have to satisfy any additional conditions.

First we study the 1-intertwiners between categorical representations of the same dimension.

1. C(V(1), V(1)): A 1-intertwiner, in this case, is given by h_{•} = (n), i.e. a 1-dimensional matrix with a non-negative integer coefficient, and an n-dimensional representation of C_{2} denoted by h̃. The naturality condition for 1-endomorphisms does not impose any restrictions in this case, as one can easily check.

2. C(V(2), V(2)): A 1-intertwiner is given by

h_{•} = (\begin{matrix}n_{1} & n_{2} \\ n_{3} & n_{4} \end{matrix}) ,

with n_{i} ∈ ℕ for i = 1, 2, 3, 4, and h̃. The naturality condition (7) does not impose any restrictions in this case, because R is trivial. Just as in the one-dimensional case we see that h̃ defines a representation of C_{2} on ℂ^{n_{i}}, for all i = 1, 2, 3, 4.

3. C(V(2)_{ξ}, V(2)_{ψ}): In this case a 1-intertwiner is given by

h_{•} = (\begin{matrix}n_{1} & n_{2} \\ n_{3} & n_{4} \end{matrix})

and h̃. The naturality condition imposes restrictions on h. We need consider only

equations (4) and (7). First, let X = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}). This gives

(\begin{matrix}1_{n_{2}} & 1_{n_{1}} \\ 1_{n_{4}} & 1_{n_{3}} \end{matrix}) = (\begin{matrix}1_{n_{1}} & 1_{n_{2}} \\ 1_{n_{3}} & 1_{n_{4}} \end{matrix}) ∘ (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix})
= (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) ∘ (\begin{matrix}1_{n_{1}} & 1_{n_{2}} \\ 1_{n_{3}} & 1_{n_{4}} \end{matrix})  (8)
= (\begin{matrix}1_{n_{3}} & 1_{n_{4}} \\ 1_{n_{1}} & 1_{n_{2}} \end{matrix}) .

This shows that n_{1} = n_{4} = n and n_{2} = n_{3} = m.

Now consider e = (\begin{matrix}x & 0 \\ 0 & x^{−1} \end{matrix}) in equation (7). In the same way, this gives

(\begin{matrix}ξ(x)1_{n} & ξ(x)1_{m} \\ ξ(x)^{−1}1_{m} & ξ(x)^{−1}1_{n} \end{matrix}) = (\begin{matrix}ψ(x)1_{n} & ψ(x)^{−1}1_{m} \\ ψ(x)1_{m} & ψ(x)^{−1}1_{n} \end{matrix}) .

Therefore there are three possible cases: a) ψ = ξ ≢ 1, b) ψ = ξ^{−1} ≢ 1 and c) ψ = ξ ≡ 1.

a) Since the character group of C_{3} is isomorphic to C_{3} we see that ξ ≠ ξ^{−1}. Equation (8) holds if and only if n ∈ ℕ is arbitrary, and m = 0. Thus we have

h_{•} = (\begin{matrix}n & 0 \\ 0 & n \end{matrix}) .

The condition that h̃ is an action on the vector bundle defined by h implies that we have

h̃(1) = (\begin{matrix}1_{n} & 0 \\ 0 & 1_{n} \end{matrix}) and h̃(−1) = (\begin{matrix}0 & A \\ A^{−1} & 0 \end{matrix}) ,

where A ∈ GL(n, ℂ) is arbitrary.

b) Equation (8) holds if and only if n = 0, and m ∈ ℕ is arbitrary. Thus we have

h_{•} = (\begin{matrix}0 & m \\ m & 0 \end{matrix}) .

Again we can determine h̃ explicitly:

h̃(1) = (\begin{matrix}0 & 1_{m} \\ 1_{m} & 0 \end{matrix}) and h̃(−1) = (\begin{matrix}A & 0 \\ 0 & A^{−1} \end{matrix}) ,

where A ∈ GL(m, ℂ) is arbitrary. Taking m = 1 shows that V(2)_{ξ} and V(2)_{ξ^{−1}} are isomorphic.

c) Equation (8) holds with no restriction on n and m. Thus we have

h_{•} = (\begin{matrix}n & m \\ m & n \end{matrix}) .

Obviously we can decompose this matrix as

(\begin{matrix}n & m \\ m & n \end{matrix}) = (\begin{matrix}n & 0 \\ 0 & n \end{matrix}) ⊕ (\begin{matrix}0 & m \\ m & 0 \end{matrix}) .

As in the previous two cases we see that

h̃(1) = (\begin{matrix}1_{n} & 1_{m} \\ 1_{m} & 1_{n} \end{matrix}) and h̃(−1) = (\begin{matrix}A & B \\ B^{−1} & A^{−1} \end{matrix}) ,

with arbitrary A ∈ GL(n, ℂ) and B ∈ GL(m, ℂ).

Finally let us describe the 1-intertwiners between categorical representations of different type.

1. Let (h_{•}, h̃): V(1) → V(2) be a 1-intertwiner. Then h_{•} has the form h_{•} = (n m), where n, m ∈ ℕ. The naturality condition does not impose any restrictions because both categorical representations are trivial. As before, h̃ simply defines two representations of C_{2} of dimensions n and m respectively.

Analogously we see that any 1-intertwiner (h_{•}, h̃): V(2) → V(1) is of the form

h_{•} = (\begin{matrix}n \\ m \end{matrix}) ,

with h̃ defining two representations of C_{2} again. Taking n = 1, m = 0 and n = 0, m = 1 respectively, and h̃ the trivial representation in both cases, shows that V(2) is isomorphic to V(1) ⊞ V(1).

2. Let (h_{•}, h̃): V(1) → V(2)_{ξ} be a 1-intertwiner. Again h_{•} has the form h_{•} = (n m). The naturality condition now imposes the following restriction:

(1_{n} 1_{m}) ∘ (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) = (1_{m} 1_{n}) = (1_{n} 1_{m}) .

This holds if and only if n = m. Taking e = (\begin{matrix}x & 0 \\ 0 & x^{−1} \end{matrix}), we see that ξ has to be

equal to 1. Just as before h̃ defines an action on a vector bundle over 2 with fibre ℂ^{n}.

Likewise non-zero 1-intertwiners (h_{•}, h̃): V(2)_{ξ} → V(1) can be seen to exist if and only if ξ is trivial and the intertwiners are the transposes of the previous ones.

3. Let (h_{•}, h̃): V(2) → V(2)_{ξ} be a 1-intertwiner. We already know that V(2) ≅ V(1) ⊞ V(1). Therefore this case reduces to the direct sum of the previous case. Again by transposition we get the classification of all 1-intertwiners between V(2)_{ξ} and V(2).

The 2-intertwiners are very easy to describe. Given two 1-intertwiners between two categorical representations, we know that we can interpret them as homogeneous vector bundles by the above results. A 2-intertwiner between them can then be interpreted as a bundle map between these homogeneous vector bundles which commutes with the actions of C_{2}. This interpretation follows immediately from the above and diagram (6).

Let us now have a look at the monoidal product of the above categorical representations.

1. Clearly we have V(1) ⊠ V ≅ V, for any categorical representation V. Because we also know that V(2) ≅ V(1) ⊞ V(1), we see that V(2) ⊠ V ≅ V ⊞ V.

2. We now study V(2)_{ξ} ⊠ V(2)_{ψ}. The easiest way to understand this tensor product is by looking at

(\begin{matrix}ξ(x) & 0 \\ 0 & ξ(x)^{−1} \end{matrix}) ⊠ (\begin{matrix}ψ(x) & 0 \\ 0 & ψ(x)^{−1} \end{matrix}) =

(\begin{matrix}ξ(x)ψ(x) & 0 & 0 & 0 \\ 0 & ξ(x)ψ(x)^{−1} & 0 & 0 \\ 0 & 0 & ξ(x)^{−1}ψ(x) & 0 \\ 0 & 0 & 0 & ξ(x)^{−1}ψ(x)^{−1} \end{matrix})

and

(\begin{matrix}0 & ξ(x) \\ ξ(x)^{−1} & 0 \end{matrix}) ⊠ (\begin{matrix}0 & ψ(x) \\ ψ(x)^{−1} & 0 \end{matrix}) =

(\begin{matrix}0 & 0 & 0 & ξ(x)ψ(x) \\ 0 & 0 & ξ(x)ψ(x)^{−1} & 0 \\ 0 & ξ(x)^{−1}ψ(x) & 0 & 0 \\ ξ(x)^{−1}ψ(x)^{−1} & 0 & 0 & 0 \end{matrix}) .

It is now easy to check that

h_{•} = (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix})

and

h̃ = (\begin{matrix}(1) & 0 & 0 & 0 \\ 0 & 0 & 0 & (1) \\ 0 & (1) & 0 & 0 \\ 0 & 0 & (1) & 0 \end{matrix})

define an invertible 1-intertwiner

V(2)_{ξ} ⊠ V(2)_{ψ} → V(2)_{ξψ} ⊞ V(2)_{ξψ^{−1}}.

Note that ξψ or ξψ^{−1} is trivial, for any choice of ξ and ψ.

5. Categorical characters

In this section we study the 1-dimensional (strict) categorical representations of an arbitrary categorical group G.

5.1. Theorem. (a) A one-dimensional categorical representation R of G is completely determined by a group character ξ_{R} on E = E(G) which is invariant under the action of G = G(G).

(b) A 1-intertwiner (h_{•}, h̃) between two one-dimensional categorical representations R and T is either zero or given by a representation h̃: G → GL(h_{•}) such that

h̃(X) = ξ_{R}(e)^{−1}ξ_{T} (e),  (9)

for X = ∂e and any e ∈ E. The right-hand side of (9) should be read as a scalar matrix of the right size for the equation to make sense. In particular there exists no non-zero 1-intertwiner if the restrictions of ξ_{R} and ξ_{T} to ker ∂ ⊂ E are different. If R = T then h̃ has to be trivial on ∂(E). The composition of two 1-intertwiners corresponds to the tensor product of the two respective representations of G.

(c) Suppose we have two 1-intertwiners between the same pair of categorical representations, then a 2-intertwiner between them is given by an ordinary intertwiner between the corresponding representations. The horizontal composition of two 2-intertwiners corresponds to the tensor product of the respective ordinary intertwiners, whereas the vertical composition of two 2-intertwiners corresponds to the ordinary (matrix) product of the two respective intertwiners.

Proof. (a) The character is ξ_{R}(e) = R(e) restricted to e ∈ E. The first claim follows immediately from diagram (2).

(b) Now consider a 1-intertwiner (h_{•}, h̃) between R and T. In this particular case h_{•} ∈ Vect_{1}(1, 1) ≅ ℕ, so we can identify h with a natural number. Just as in the previous section we see that h̃ defines a representation of G on ℂ^{h_{•}}. Condition (9) expresses the naturality condition for h and can be read off from diagram (5).

The composite of two 1-intertwiners is given by the tensor product of the respective representations of G, say h̃_{1} and h̃_{2}, because, in the case of one-dimensional categorical representations, we have (see definition 3.5)

((h̃_{1}) ∘ 1) · (1 ∘ (h̃_{2})) = (h̃_{1} ⊗ h̃_{2}),

where the 1’s denote the identity matrices of the right size.

(c) This follows directly from condition (6).

We now apply our results to the examples in Sect. 2.

5.2. Example. Let G be a categorical group and (E, G, ∂, ▷) the corresponding crossed module of groups. A one-dimensional categorical representation is determined by a Ginvariant group character on E, say ξ, and is denoted by V_{ξ}. We determine the Homcategories between an arbitrary pair V_{ξ} and V_{ψ}. Let (h_{•}, h̃) be an arbitrary 1-intertwiner between V_{ξ} and V_{ψ}.

1. Suppose G is connected. Then theorem 5.1 says that h̃ is a scalar representation of G, which is completely determined by ψξ^{−1}. Therefore, for any two characters ξ and ψ which coincide on the kernel of ∂, there is exactly one 1-intertwiner between V_{ξ} and V_{ψ}. The 2-intertwiners are just the ordinary intertwiners between these scalar representations of G.

2. Suppose G is skeletal. Let Rep(G) be the monoidal category of representations of G. Then Hom(V_{ξ}, V_{ψ}) = Rep(G), if ξ = ψ, and zero otherwise.

3. Suppose G is discrete, so that E is a normal subgroup of G. Note that in this case ξ (and ϕ) have to be characters which are constant on each conjugacy class of E. Theorem 5.1 says that the restriction of h̃ to E is the scalar representation of G determined by ψξ^{−1}. A concrete example of some interest is the case where E = {±1} ⊂ SU(2) = G. We have two characters on E, namely the trivial one, say ξ, and the inclusion {±1} ⊂ ℂ^{∗}, say ψ. Now an easy exercise reveals that we have

Hom(ξ, ξ) = Hom(ψ, ψ) = Rep(SU(2))even = Rep(SO(3))

and

Hom(ξ, ψ) = Hom(ψ, ξ) = Rep(SU(2))odd,

where the last expression denotes the subcategory of Rep(SU(2)) generated by the odd spins only.

6. Concluding remarks

In this section we give a rather incomplete sketch of some features of the categorical representations of general categorical groups and also make some remarks about possible generalizations of our constructions.

As already proved, a (strict) categorical representation corresponds precisely to a homomorphism of crossed modules

\begin{matrix}E(G) & ⟶^{∂} & G(G) \\ ^{R_{p}}↓ & & ↓^{R_{b}} \\ (ℂ^{∗})^{N} & ⟶^{1} & S_{N}. \end{matrix}  (10)

This leads to the following concrete description of a categorical representation.

6.1. Lemma. An N-dimensional categorical representation of G consists of a group homomorphism R_{b}: G(G) = G_{0} → S_{N} whose kernel contains the image of ∂, together with N group characters ξ_{1}, …, ξ_{n} on E = E(G) satisfying

ξ_{i}(X ▷ e) = ξ_{R_{b}(X)i}(e),  (11)

for any X ∈ G(G), e ∈ E and i = 1, …, N. Here R_{b}(X) denotes the left action on the set of N elements.

Proof. Both R_{b} and R_{p} in (10) are group homomorphisms. Clearly R_{p} is a group homomorphism if and only if it defines N group characters. Also (10) is commutative if and only if the kernel of R_{b} contains the image of ∂. Finally, (11) is equivalent to the action condition on (R_{b}, R_{p}).

6.2. Remark. Note that (11) implies that each ξ_{i} has to be invariant under the action of ker(R_{b}), i.e.

ξ_{i}(X ▷ e) = ξ_{i}(e),

for any X ∈ ker(R_{b}) and e ∈ E.

6.3. Definition. Let R be a categorical representation.

a) R is called decomposable if there are two non-zero categorical representations S and T such that R ≅ S ⊞ T holds. R is called indecomposable if it is not decomposable.

b) R is called reducible if there exist a categorical representation S of dimension less than R and two 1-intertwiners h: S → R and h′: R → S such that h ∘ h′: S → S is isomorphic to the identity 1-intertwiner on S. R is called irreducible if it is not reducible.

Note that any irreducible categorical representation is indecomposable as well, but the converse is false as the following example shows.

6.4. Example. In the example G(2, 3) of the previous section, we saw that V(2) ≅ V(1) ⊞ V(1) is decomposable and V(2)_{ξ} is irreducible for ξ ≠ 1. For ξ = 1, we see that V(2)_{1} is indecomposable, but reducible. The lemma below shows that all categorical representations of G(2, 3) of dimension greater than two are decomposable.

6.5. Remark. The appearance of indecomposable reducible categorical representations is of course due to the fact that the 1-morphisms in 2-Vect are matrices with only nonnegative integer entries. For example, to decompose the non-trivial 2-dimensional representation R_{b}: C_{2} → S_{2} as a representation of groups, one uses an intertwiner with negative and fractional entries. Consequently the monoidal 2-category of categorical representations of G(2, 3) is not semi-simple (see [M-S] for the precise definition of semi-simplicity). Therefore it is not clear if it can be used for the construction of topological state-sums, because semi-simplicity is an essential ingredient in the proof of topological invariance of these state-sums. Note that one cannot simply ignore the indecomposable reducible categorical representations, because, as we showed in the previous section, one of the summands in the decomposition of V(2)_{ξ} ⊠ V(2)_{ψ} is equal to V(2)_{1}, for any choice of ξ and ψ. It seems that this problem also exists in Crane and Yetter’s generalization of categorical representations [CY-MC].

There are two elementary results about indecomposable categorical representations that we decided to include in these remarks because they are very easy to prove.

6.6. Lemma. An N-dimensional categorical representation, R, is indecomposable if and only if the action of R_{b}(G(G)) on {1, …, N} is transitive.

Proof. Suppose we have R ≅ S ⊞ T and dim S = K and dim T = N − K. Clearly we can write R_{b} ≅ S_{b} ⊕ T_{b}, where ⊕ means the composite of S_{b} × T_{b} and the canonical map between the symmetric groups S_{K} × S_{N−K} → S_{K+(N−K)} = S_{N}, and ≅ means equal up to conjugation by a fixed permutation. Thus we see that the action of R_{b}(G(G)) on {1, …, N} is not transitive.

Conversely, suppose that {1, …, N} can be written as the union of two non-empty subsets A and B which are both invariant under the action of R_{b}(G(G)). By reordering we may assume that A = {1, …, K} and B = {K + 1, …, N}. The restrictions of R_{b}(G(G)) to A and B respectively yield a decomposition R_{b} = S_{b} ⊕ T_{b}. Take S_{p} = (ξ_{1}, …, ξ_{K}) and T_{p} = (ξ_{K+1}, …, ξ_{N}). Then condition (11) shows that (S_{b}, S_{p}) and (T_{b}, T_{p}) are both categorical representations and R = S ⊕T. Note that ≅ had become the identity here because of our assumption that A = {1, …, K} and B = {K + 1, …, N}, which corresponds to the choice of a fixed permutation.

6.7. Lemma. If R is indecomposable, then it is completely determined by R_{b} and just one character on E(G) which is invariant under the action of ker(R_{b}).

Proof. Suppose R is indecomposable. Let i ≠ j ∈ {1, …, N} be arbitrary. By assumption there exists an X ∈ G(G) such that R_{b}(X)i = j. By (11) we have ξ_{i}(X ▷ f) = ξ_{j}(f). Thus, if you fix ξ_{i}, then ξ_{j} is uniquely determined. Since i, j were arbitrary, this shows that one character determines uniquely all the others. We already remarked that any such character has to be invariant under ker(R_{b}).

6.8. Remark. Given a subgroup H ⊂ G = G(G) of index N, the action of G on G/H by left (or right) multiplication is transitive. Given an ordering on the elements of G/H we thus obtain a group homomorphism R_{b}: G → S_{N} such that R_{b}(G) acts transitively on {1, …, N}. Another choice of ordering leads to a conjugate group homomorphism. For any X ∈ G, the construction above applied to XHX^{−1} yields the homomorphism XR_{b}X^{−1}. It is easy to check that the converse is also true: given R_{b} satisfying the above condition, the kernel of R_{b} has index N in G and any group homomorphism G → S_{N}, determined by an ordering on G/ ker(R_{b}), is conjugate to R_{b}. This sets up a bijective correspondence between conjugacy classes of group homomorphisms G → S_{N} and conjugacy classes of subgroups of G of index N.

There is quite some literature on the theory of permutation representations of finite groups. In this theory the building blocks are the transitive permutation representations. The key observation about them, from the point of view of representation theory, is that Mackey’s theory of induced representations carries over to the context of transitive permutation representations without problems [LLC, LLBC]. This allows for a complete description of the decomposition of the tensor product of two transitive permutation representations into a direct sum of transitive permutation representations, for example. Clearly categorical representations, as defined in this paper, are a generalization of permutation representations, the indecomposable ones being the generalizations of the transitive permutation representations, and one could probably generalize some of the techniques used for the latter to study categorical representations.

Although we have not worked out all the details about general categorical representations, we can say some more about the categorical representations of skeletal categorical groups G, of which G(2, 3) is a very simple example. Despite its simplicity the case of G(2, 3) reveals the general features: an indecomposable categorical representation V corresponds to an orbit, G(G)ξ, in the set of characters of E(G), and a homomorphism G(G) → S_{N} corresponding to a subgroup 1 ⊆ H ⊆ G(G)_{ξ}, where G(G)_{ξ} is the stabilizer of ξ in G(G). V is irreducible if and only H = G(G)_{ξ}. 1-Intertwiners between two categorical representations can be interpreted as homogeneous vector bundles on the cartesian product of the two orbits in the spaces of characters of E(G). 2-Intertwiners can be interpreted as maps between homogeneous bundles. The decomposition of the monoidal product of two indecomposable categorical representions V_{1} and V_{2} can be obtained by looking at the decomposition into orbits of the cartesian product of the two orbits, corresponding to V_{1} and V_{2} respectively, and can, in general, contain indecomposable reducible categorical representations.

A short remark on the fact that we have only worked out examples of strict categorical representations is in place. If G is skeletal, then nothing terribly interesting happens when considering general categorical representations. Each N-dimensional indecomposable categorical representations is determined by a homomorphism of crossed modules, just as in the strict case, together with an additional group 2-cocycle on G(G) with values in (ℂ^{∗})^{N}. Because we have only considered strict units, the 2-cocycles have to be normalized, but that is the only restriction. The 1-intertwiners between two indecomposable categorical representations V_{1} and V_{2} become projective homogeneous vector bundles on the cartesian product of the orbits corresponding to V_{1} and V_{2}, with a projective action of G(G) which fails to be an ordinary action by the quotient of the 2-cocycles of V_{1} and V_{2}. The 2-intertwiners are just maps between these vector bundles which intertwine the projective actions. However, for general G the categorical representations are not so easy to describe. We tried to weaken the notion of homomorphism between crossed modules by introducing a 2-cocycle, but we found that this 2-cocycle has to satisfy an additional independent equation. Because the interpretation of this equation is not clear, we decided to work out the strict categorical representations only.

Finally we want to comment on possible generalizations of our framework. One can consider generalizations of the notion of categorical representation and generalizations of the notion of categorical group. As an example of the former one could consider monoidal bicategories other than 2-Vect. For example, one could consider Crane and Yetter’s monoidal bicategory of measurable categories [CY-MC]. As they remark in their introduction, this allows for more interesting categorical representations of categorical Lie groups because the base group can be represented in more general topological symmetry groups than S_{N}. However, as we remarked above, indecomposable reducible categorical representations seem to appear in this setting as well, which puts its applicability for state-sums at risk. Somehow the discreteness of the non-negative integer entries in the 1-morphisms in 2-Vect has not been solved completely in this new setting.

If one only considers 1-dimensional categorical representations, then, of course, no indecomposable reducible categorical representations appear, e.g. the last example of the previous section yields a semi-simple monoidal 2-category and could be used for the construction of topological state-sums of 3- and 4-dimensional manifolds. We have not worked out what these invariants are, but possibly they are connected to Yetter’s [Y-EX] and Roberts [R-EX] refined invariants. One idea for a generalization would be to replace Vect by a more interesting braided monoidal category, such as the ones appearing in the representation theory of quantum groups.

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School of Mathematical Sciences University Park Nottingham NG7 2RD, UK

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