Georgian Mathematical Journal Volume 9 (2002), Number 4, 709–721

THE WHITEHEAD CATEGORICAL GROUP OF DERIVATIONS

A. R. GARZÓN AND A. DEL RÍO

Dedicated to H. Inassaridze on his 70th birthday

Abstract. Given a categorical crossed module ℍ → G, where G is a group, we show that the category of derivations, Der(G, ℍ), from G into ℍ has a natural monoidal structure. We introduce the Whitehead categorical group of derivations as the Picard category of Der(G, ℍ) and then we characterize the invertible derivations, with respect to the tensor product, in this monoidal category.

2000 Mathematics Subject Classification: 18D10, 20J05.

Key words and phrases: Derivation, monoidal category, categorical group, crossed module.

1. Introduction

If G is a group and G → Aut(H) is a group homomorphism, that is, H is a G-group, there is no natural structure on the set Der(G, H) of derivations from G into H; however, in the course of investigations of the properties of relative homotopy groups, J. H. C. Whitehead [20] showed that if H is a Gcrossed module, then Der(G, H) has a natural monoid structure, and he further characterized the units in Der(G, H). The resulting group D(G, H) is called the Whitehead group of regular derivations. These results have then allowed the application of derivation groups to the variety of situations concerned with the notion of crossed module. Such an application, to holomorphs of groups, was given by A. S. T. Lue [12]; moreover, derivation groups appear, in the automorphism structures for crossed modules studied by K. Norrie [14] and R. Brown and N. D. Gilbert [2], and also in the development by N. D. Gilbert [9] of Whitehead’s ideas by using the equivalence between crossed modules and internal groups in the category of groupoids.

The object of this paper is to introduce the Whitehead “categorical group” of regular derivations in order to obtain analogous applications at the level of categorical groups.

Categorical groups are monoidal groupoids in which each object is invertible, up to isomorphism, with respect to the tensor product [1, 10, 16, 17] and they have been widely used in various fields such as ring theory, group cohomology and algebraic topology [3, 4, 6, 8, 11, 15, 17, 19].

In this setting of categorical groups, the notion of crossed module was introduced by L. Breen [1] and it has been used recently by P. Carrasco and J. Martínez [5] in order to study 2-fold extensions of a group G by a symmetric

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

G-categorical group 𝔸. Of course, this notion of crossed module recovers the classic one for groups when groups are seen as discrete categorical groups.

Derivations of 𝔾 into ℍ, when 𝔾 is a categorical group acting on a braided categorical group ℍ, have been studied in [7], where it is shown how the braiding in ℍ allows the definition of a categorical group structure in the category of derivations Der(𝔾, ℍ). However, if ℍ is not braided, there is no natural structure on the category Der(𝔾, ℍ).

Now, for any group G, we consider categorical crossed modules of the form ℍ → G[0], where G[0] denotes the discrete categorical group associated to G, which we call categorical G-crossed modules. Then we analyze the natural monoidal structure, in the category of derivations Der(G, ℍ), owing to the G-crossed module structure in the categorical group ℍ. For any monoidal category 𝒞, there is an associated categorical group 𝒫(𝒞) [19], called the Picard categorical group of 𝒞, consisting of invertible objects in 𝒞 and isomorphisms between them. Then we introduce the Whitehead categorical group of derivations D(G, ℍ) exactly as the Picard categorical group of the monoidal category Der(G, ℍ) and, finally, we characterize the objects of D(G, ℍ) in terms of the endomorphisms, of G and ℍ, associated to any derivation from G[0] into ℍ.

2. Preliminaries

Let us recall [13] that a monoidal category 𝔾 = (𝔾, ⊗, a, I, l, r) consists of a category 𝔾, a functor (tensor product) ⊗ : 𝔾 × 𝔾 → 𝔾, an object I of 𝔾, called the unit object, and natural isomorphisms called, respectively, the associativity, left-unit and right-unit constraints

a = a_{X,Y,Z} : (X ⊗ Y) ⊗ Z → X ⊗ (Y ⊗ Z),
l = l_{X} : I ⊗ X → X, r = r_{X} : X ⊗ I → X,

such that the usual coherence conditions are satisfied. In a monoidal category, an object X is said to be 2-regular, or invertible, if the functors Y ↦ X ⊗ Y and Y ↦ Y ⊗ X are equivalences. A categorical group 𝔾 is a monoidal small category where every arrow is invertible and every object is 2-regular. In this case, it is possible to define a contravariant functor (−)^{∗} : 𝔾 → 𝔾, X ↦ X^{∗}, f ↦ f^{∗}, and natural isomorphisms

γ_{X} : X ⊗ X^{∗} → I, ϑ_{X} : X^{∗} ⊗ X → I,

such that l_{X} · (γ_{X} ⊗ 1) = r_{X} · (1 ⊗ ϑ_{X}) · a_{X,X^{∗},X}, for all objects X ∈ 𝔾. The triple (X^{∗}, γ_{X}, ϑ_{X}) is termed an inverse for X.

Let us remark that, once an inverse for any object has been chosen, there is a natural isomorphism

υ_{X,Y} : (X ⊗ Y)^{∗} → Y^{∗} ⊗ X^{∗} ,

satisfying a_{Z^{∗},Y^{∗},X^{∗}} · (υ_{Y,Z} ⊗ 1) · υ_{X,Y ⊗Z} = (1 ⊗ υ_{X,Y}) · υ_{X⊗Y,Z} · a∗_{X,Y,Z}.

A functor T : 𝔾 → ℍ between categorical groups is termed a homomorphism if it is supplied with natural isomorphisms

μ = μ_{X,Y} : T (X ⊗ Y) → T (X) ⊗ T (Y),

compatible with a in the sense that:

(1 ⊗ μ_{Y,Z}) · μ_{X,Y ⊗Z} · T (a_{X,Y,Z}) = a_{T (X),T (Y ),T (Z)} · (μ_{X,Y} ⊗ 1) · μ_{X⊗Y,Z} . (1)

Below we will denote by 𝒞 𝒢 the category of categorical groups and homomorphisms between them.

Note that if (T, μ) is a homomorphism, then there exists a (unique) isomorphism [6], μ_{0} : T (I) → I, such that T (r_{X}) = r_{T (X)} · (1 ⊗ μ_{0}) · μ_{X,I} and T (l_{X}) = l_{T (X)} · (μ_{0} ⊗ 1) · μ_{I,X}. With respect to the inverses, there exist (unique) isomorphisms

λ_{X} : T (X^{∗}) → T (X)^{∗},

such that the equalities μ_{0} · T (γ_{X}) = γ_{T (X)} · (1 ⊗ λ_{X}) · μ_{X,X^{∗}} and μ_{0} · T (ϑ_{X}) = ϑ_{T (X)} · (λ_{X} ⊗ 1) · μ_{X^{∗},X} hold.

Given homomorphisms of categorical groups (T, μ), (T′μ′) : 𝔾 → ℍ, a morphism from (T, μ) to (T′, μ′) consists of a natural transformation ϵ : T → T′ such that, for any objects X, Y ∈ 𝔾, the following condition holds:

(ϵ_{X} ⊗ ϵ_{Y}) · μ_{X,Y} = μ′_{X,Y} · ϵ_{X⊗Y} .  (2)

If G is a group, a G-categorical group is defined [5] as a categorical group ℍ together with a homomorphism of categorical groups (a G-action)

(T, μ) : G[0] → ℰ q(ℍ)

from the discrete categorical group defined by G (denoted by G[0]) to the categorical group, ℰ q(ℍ), of equivalences of ℍ [1]. It is easy to see that giving a G-action on ℍ is equivalent to giving equivalences

x(−) : ℍ → ℍ, A ↦ ^{x}A , f ↦ ^{x}f

for each x ∈ G, together with natural isomorphisms

ψ = ψ_{x,A,A′} : ^{x}(A ⊗ A′) → ^{x}A ⊗ ^{x}A′,
ϕ = ϕ_{x,x′,A} : ^{xx′}A → ^{x}(^{x′}A),
ϕ_{0} = ϕ_{0,A} : ^{1}A → A,

satisfying suitable coherence conditions (see [6, 18]).

Moreover, for any x ∈ G, there exists a unique isomorphism

ψ_{0} = ψ_{0,x} : ^{x}I → I,

such that l_{^{x}A} · (ψ_{0,x} ⊗ 1) · ψ_{x,I,A} = ^{x}l_{A} and r_{^{x}A} · (1 ⊗ ψ_{0,x}) · ψ_{x,A,I} = ^{x}r_{A}, for any element x ∈ G and any object A ∈ ℍ .

A G-categorical group is termed strict when all the isomorphisms ψ_{x,A,A′}, ϕ_{x,x′,A}, ϕ_{0,A} and ψ_{0,x} are equalities.

Let ℍ and 𝔼 be G-categorical groups. A G-equivariant homomorphism (T, ν) : ℍ → 𝔼 consists of a categorical group homomorphism T = (T, μ) : ℍ → 𝔼 and a family of natural isomorphisms

ν = (ν_{x,A} : T (^{x}A) → ^{x}T (A))_{(x,A)∈G×Obj(ℍ)}

that are compatible with ψ, ϕ and ϕ_{0} in the appropriate sense (see [18]).

3. The Category of Derivations Der(G, ℍ)

Let G be a group and let ℍ be a G-categorical group. A derivation from G into ℍ is a pair (D, β) where D : G[0] → ℍ is a functor and

β = (β_{x,y} : D(xy) → D(x) ⊗ ^{x}D(y))_{x,y∈G}

is a family of isomorphisms such that, for any x, y, z ∈ G, the following coherence condition holds

(1 ⊗ ψ_{x,D(y),^{y}D(z)}) · (1 ⊗ ^{x}β_{y,z}) · β_{x,yz}
= (1 ⊗ (1 ⊗ ϕ_{x,y,D(z)})) · a_{D(x),^{x}D(y),^{xy}D(z)} · (β_{x,y} ⊗ 1) · β_{xy,z} .  (3)

If (D, β) is a derivation, there exists an isomorphism β̄_{0} : D(1) → I (where 1 is the neutral element of G) that is determined by the equalities

r_{D(x)} · (1 ⊗ ψ_{0,x}) · (1 ⊗ ^{x}β̄_{0}) · β_{x,1} = id_{D(x)} = l_{D(x)} · (1 ⊗ ϕ_{0,D(x)}) · (β̄_{0} ⊗ 1) · β_{1,x} .

Let D_{0} : G[0] → ℍ be the constant functor with value the unit object I ∈ ℍ. Then, if for any x ∈ G we consider the arrow in ℍ (β_{0})_{x} = (1 ⊗ ψ−1_{0,x}) · l−1_{I} : I → I ⊗ I → I ⊗ ^{x}I, the pair (D_{0}, β_{0}) is clearly a derivation, called the trivial derivation from G into ℍ.

Given two derivations (D, β), (D′, β′) : G[0] → ℍ, an arrow from (D, β) to (D′, β′) consists of a natural transformation ϵ : D → D′ such that, for any x, y, z ∈ G, the following condition holds:

(ϵ_{x} ⊗ ^{x}ϵ_{y}) · β_{x,y} = β′_{x,y} · ϵ_{xy} .  (4)

The vertical composition of natural transformations determines a composition for arrows between derivations, and therefore we can consider the category, Der(G, ℍ), of derivations from G into ℍ, which is actually a groupoid.

Note that, if H is a G-group, then H[0] is a G-categorical group and thus the category Der(G, H[0]) is exactly the discrete category associated to the set Der(G, H). Recall that, in this case, there is a bijection between the set Der(G, H) and the set of group homomorphisms f, from G to the semidirect product H ⋊ G, such that pr · f = id_{G}, where pr : H ⋊ G → G is the projection.

Now, if ℍ is a G-categorical group, we can consider the semidirect product ℍ⋊G[0] (see [6]), which is the categorical group where the underlying groupoid is ℍ × G[0] and with tensor product given by (A, x) ⊗ (B, y) = (A ⊗ ^{x}B, xy). Since 𝒞 𝒢 is a 2-category, where the 2-cells are the morphisms between homomorphisms, we can consider the category Hom_{𝒞 𝒢} (G[0], ℍ ⋊ G[0]) and then we have the following:

Proposition 3.1. For any group G and any G-categorical group ℍ there is an isomorphism of categories between Der(G, ℍ) and the full subcategory 𝒜 of Hom_{𝒞 𝒢} (G[0], ℍ ⋊ G[0]) whose objects are those homomorphisms T = (T, μ) : G[0] → ℍ ⋊ G[0] such that pT = id_{G[0]}, where p : ℍ ⋊ G[0] → G[0] is the projection.

Proof. We shall define functors

κ : Der(G, ℍ) ⟶ 𝒜 ; Σ : 𝒜 ⟶ Der(G, ℍ)

such that κ · Σ = id and Σ · κ = id.

For any object (D, β) ∈ Der(G, ℍ), we let κ((D, β)) = (T, μ), where T (x) = (D(x), x) ∈ ℍ ⋊ G[0], and μ_{x,y} : T (xy) → T (x) ⊗ T (y) is given by μ_{x,y} = (β_{x,y}, id_{xy}). For any arrow ϵ : (D, β) → (D′, β′), κ(ϵ) : (T, μ) → (T′, μ′) is the morphism determined by the natural transformation given by κ(ϵ)_{x} = (ϵ_{x}, id_{x}).

To define the functor Σ, let us consider the projection ∂ : ℍ ⋊ G[0] → ℍ, (A, x) ↦ A. Then, for any object (T, μ) ∈ 𝒜, Σ((T, μ)) = (D, β), where D(x) = ∂T (x), and β_{x,y} : D(xy) → D(x) ⊗ ^{x}D(y) is given by β_{x,y} = ∂(μ_{x,y}). For any morphism τ : (T, μ) → (T′, μ′), Σ(τ) : (D, β) → (D′, β′) is the arrow determined by the natural transformation whose component at X ∈ G is Σ(τ)_{x} = ∂(τ_{x}).

Now it is straightforward to check that κ and Σ are inverse to each other. □

4. The Monoidal Structure on Der(G, ℍ)

Recall that, if G is a group, a G-crossed module consists of a group homomorphism ρ : H → G together with an action (x, h) ↦ ^{x}h of G on H satisfying ρ(^{x}h) = xρ(h)x^{−1} and ^{ρ(h)}h′ = hh′h^{−1}, for all x ∈ G, h, h′ ∈ H.

This notion has the following generalization in the setting of categorical groups.

Definition 4.1 ([5]). Let G be a group. A categorical G-crossed module consists of a triad (ℍ, ρ, ξ), where ℍ is a G-categorical group, ρ : ℍ → G[0] is a G-equivariant homomorphism (necessarily strict), considering in G[0] the action given by conjugation (i.e. ρ(^{x}A) = xρ(A)x^{−1} for any x ∈ G and A ∈ ℍ), and

ξ = (ξ_{A,B} : ^{ρ(A)}B ⊗ A → A ⊗ B)_{(A,B)∈ℍ}

is a family of natural isomorphisms in ℍ, such that for all objects A, B, C ∈ ℍ the following diagrams are commutative:

\xymatrix{^{ρ(A)}(B ⊗ C) ⊗ A \ar[r]^{ξ_{A,B⊗C}} \ar[d]_{ψ⊗1} & A ⊗ B ⊗ C \\ ^{ρ(A)}B ⊗ ^{ρ(A)}C ⊗ A \ar[r]_{1⊗ξ_{A,C}} & ^{ρ(A)}B ⊗ A ⊗ C , \ar[u]_{ξ_{A,B} ⊗1}}  (5)

\xymatrix{^{ρ(A⊗B)}C ⊗ A ⊗ B \ar[r]^{ξ_{A⊗B,C}} \ar[d]_{ϕ⊗1} & A ⊗ B ⊗ C \\ ^{ρ(A)}(^{ρ(B)}C) ⊗ A ⊗ B \ar[r]_{ξ_{A,^{ρ(B)}C} ⊗1} & A ⊗ ^{ρ(B)}C ⊗ B , \ar[u]_{1⊗ξ_{B,C}}}  (6)

[figure: ^{x}(^{ρ(A)}B ⊗ A); ^{x}ξ_{A,B}; ψ; ^{x}(^{ρ(A)}B) ⊗ ^{x}A; ^{(xρ(A))}B ⊗ ^{x}A; ϕ⊗1; ^{x}(A ⊗ B); ψ; ^{(ρ(^{x}A)x)}B ⊗ ^{x}A .; ϕ⊗1; ^{x}A ⊗ ^{x}B; ^{ρ(^{x}A)}(^{x}B) ⊗ ^{x}A; ξ_{^{x}A,^{x}B}] (7)

Note that if H is a group and (H[0], ρ : H[0] → G[0], ξ) is a categorical G-crossed module, then ξ is necessarily an identity and therefore we obtain a G-equivariant group homomorphism ρ : H → G, which is a crossed module in the usual sense. In addition, Conduché’s 2-crossed modules are particular cases of categorical G-crossed modules (see [5]). Categorical G-modules (see [4]) are braided categorical groups [10] provided with a coherent left action of G[0]. If (𝔸, c) is a G-module then the trivial homomorphism 𝔸 → G[0] is clearly a categorical G-crossed module where ξ is given by the braiding c.

If H is a G-crossed module, then the set Der(G, H) of derivations from G into H has a natural monoid structure [20]. Now, if (ℍ, ρ, ξ) is a categorical G-crossed module, then ℍ is a G-categorical group and our aim below is to show that the category Der(G, ℍ) has a natural monoidal structure, that is inherited from the G-crossed module structure in ℍ. To do so, we first show the following:

Lemma 4.2. Let (ℍ, ρ, ξ) be a categorical G-crossed module. For any derivation (D, β) : G[0] → ℍ, there are endomorphisms of G[0] and of ℍ,

σ_{D} : G[0] → G[0] , θ_{D} = (θ_{D}, μ_{θ}) : ℍ ⟶ ℍ,

where σ_{D}(x) = ρ(D(x))x, x ∈ G, and θ_{D} is defined, for any object A ∈ ℍ, by θ_{D}(A) = D(ρ(A))⊗A, for any arrow f : A → B, by θ_{D}(f) = D(ρ(f))⊗f = 1⊗f and where (μ_{θ})_{A,B} : θ_{D}(A ⊗ B) → θ_{D}(A) ⊗ θ_{D}(B), A, B ∈ ℍ, is given by the following composition:

[figure: D(ρ(A ⊗ B)) ⊗ A ⊗ B; (μ_{θ})_{A,B}; D(ρ(A)) ⊗ A ⊗ D(ρ(B)) ⊗ B; D(ρ(A)ρ(B)) ⊗ A ⊗ B; β_{ρ(A),ρ(B)}⊗1; D(ρ(A)) ⊗ ^{ρ(A)}D(ρ(B)) ⊗ A ⊗ B .; 1⊗ξ_{A,D(ρ(B))}⊗1]

Moreover, if ϵ : (D, β) → (D′, β′) is an arrow in Der(G, ℍ), then σ_{D} = σ_{D′} and there is a morphism of categorical group homomorphisms τ : θ_{D} → θ_{D′} given, for any A ∈ ℍ, by τ_{A} = ϵ_{ρ(A)} ⊗ id_{A}.

Moreover, if D_{0} : G[0] → ℍ is the trivial derivation, then σ_{D_{0}} = id_{G} and there is a morphism θ_{D_{0}} → id_{ℍ} given, for any A ∈ ℍ, by l_{A} : I ⊗ A → A.

Proof. σ_{D} is a group homomorphism since, for any x, y ∈ G,

σ_{D}(xy) = ρ(D(xy)) xy = ρ[D(x) ⊗ ^{x}D(y)] xy
= ρ(D(x))ρ(^{x}D(y)) xy = ρ(D(x)) x ρ(D(y)) x^{−1}xy
= ρ(D(x)) x ρ(D(y)) y = σ_{D}(x)σ_{D}(y) .

As for θ_{D}, we obtain the coherence condition (1) for μ_{θ} as follows: (1 ⊗ (μ_{θ})_{B,C}) · (μ_{θ})_{A,B⊗C} · θ_{D}(a_{A,B,C}) =

=^{def} [(1 ⊗ (1 ⊗ ξ_{B,D(ρ(C))} ⊗ 1)) · (1 ⊗ (β_{ρ(B),ρ(C)} ⊗ 1))]·

[(1 ⊗ ξ_{A,D(ρ(B⊗C))} ⊗ 1) · (β_{ρ(A),ρ(B⊗C)} ⊗ 1)] · [1 ⊗ a_{A,B,C}]

=^{nat} (1 ⊗ (1 ⊗ ξ_{B,D(ρ(C))} ⊗ 1)) · (1 ⊗ ξ_{A,D(ρ(B))⊗^{ρ(B)}D(ρ(C))} ⊗ 1)·

((1 ⊗ ^{ρ(A)}β_{ρ(B),ρ(C)}) ⊗ 1 ⊗ 1) · (β_{ρ(A),ρ(B⊗C)} ⊗ 1) · [1 ⊗ a_{A,B,C}]

=^{(5)} (1 ⊗ (1 ⊗ ξ_{B,D(ρ(C))} ⊗ 1)) · (1 ⊗ ξ_{A,D(ρ(B))} ⊗ a_{θ_{D} (A),D(ρ(B))⊗^{ρ(B)}D(ρ(C)),B⊗C})·

(1 ⊗ 1 ⊗ ξ_{A,^{ρ(B)}D(ρ(C))} ⊗ 1) · (1 ⊗ ψ_{ρ(A),D(ρ(B)),^{ρ(B)}D(ρ(C))} ⊗ a−1_{A,B,C})·

((1 ⊗ ^{ρ(A)}β_{ρ(B),ρ(C)}) ⊗ 1 ⊗ 1) · (β_{ρ(A),ρ(B⊗C)} ⊗ 1) · [1 ⊗ a_{A,B,C}]

=^{(3)} (1 ⊗ (1 ⊗ ξ_{B,D(ρ(C))} ⊗ 1)) · (1 ⊗ ξ_{A,D(ρ(B))} ⊗ 1) · a_{θ_{D} (A),D(ρ(B))⊗^{ρ(B)}D(ρ(C)),B⊗C}·

(1 ⊗ 1 ⊗ ξ_{A,^{ρ(B)}D(ρ(C))} ⊗ 1) · (1 ⊗ (1 ⊗ ϕ_{ρ(A),ρ(B),D(ρ(C))}) ⊗ 1)·

(a_{D(ρ(A)),^{ρ(A)}D(ρ(B)),^{ρ(A⊗B)}D(ρ(C))} ⊗ 1) · ((β_{ρ(A),ρ(B)} ⊗ 1) ⊗ 1) · (β_{ρ(A⊗B),ρ(C)} ⊗ 1)

=^{(6)} (1 ⊗ (1 ⊗ ξ_{B,D(ρ(C))} ⊗ 1)) · (1 ⊗ ξ_{A,D(ρ(B))} ⊗ 1) · a_{θ_{D} (A),D(ρ(B))⊗^{ρ(B)}D(ρ(C)),B⊗C}·

(1 ⊗ (1 ⊗ ξ−1_{B,D(ρ(C))} ⊗ 1)) · (1 ⊗ (1 ⊗ ξ_{A⊗B,D(ρ(C))} ⊗ 1))·

(a_{D(ρ(A)),^{ρ(A)}D(ρ(B)),^{ρ(A⊗B)}D(ρ(C))} ⊗ 1) · ((β_{ρ(A),ρ(B)} ⊗ 1) ⊗ 1) · (β_{ρ(A⊗B),ρ(C)} ⊗ 1)

=^{nat} (1 ⊗ ξ_{A,D(ρ(B))} ⊗ a_{θ_{D} (A),θ_{D} (B),θ_{D} (C)}) · (a_{D(ρ(A)),^{ρ(A)}D(ρ(B)),A⊗B} ⊗ 1)·

(1 ⊗ (ξ_{A⊗B,D(ρ(C))} ⊗ 1)) · ((β_{ρ(A),ρ(B)} ⊗ 1) ⊗ 1) · (β_{ρ(A⊗B),ρ(C)} ⊗ 1)

=^{nat} a_{θ_{D} (A),θ_{D} (B),θ_{D} (C)} · [(1 ⊗ ξ_{A,D(ρ(B))} ⊗ 1) · ((β_{ρ(A),ρ(B)} ⊗ 1) ⊗ 1)]·

[(1 ⊗ ξ_{ρ(A⊗B),D(ρ(C))} ⊗ 1) · (β_{ρ(A⊗B),ρ(C)} ⊗ 1)]

=^{def} a_{θ_{D} (A),θ_{D} (B),θ_{D} (C)} · ((μ_{θ})_{A,B} ⊗ 1) · (μ_{θ})_{A⊗B,C} .

Finally, σ_{D} = σ_{D′} since, for any x ∈ G, ρ(ϵ_{x}) : ρ(D(x)) → ρ(D′(x)) is necessarily an identity, because G[0] is discrete, and therefore σ_{D}(x) = ρ(D(x))x = ρ(D′(x))x = σ_{D′}(x). On the other hand, to prove that τ : θ_{D} → θ_{D′}, given by τ_{A} = ϵ_{ρ(A)} ⊗ id_{A} : D(ρ(A)) ⊗ A → D′(ρ(A)) ⊗ A, is a morphism is routine. □

Now, using the endomorphism σ_{D} of G associated to any derivation D, we have:

Proposition 4.3. Let G be a group and (ℍ, ρ, ξ) a categorical G-crossed module. If (D_{1}, β_{1}) and (D_{2}, β_{2}) are derivations from G[0] into ℍ, then there is a derivation from G[0] into ℍ, (D_{1} ⊗ D_{2}, β_{1} ⊗ β_{2}), where D_{1} ⊗ D_{2} : G[0] → ℍ is given by (D_{1} ⊗ D_{2})(x) = D_{1}(σ_{D_{2}}(x)) ⊗ D_{2}(x), x ∈ G, and (β_{1} ⊗ β_{2}) = ((β_{1} ⊗ β_{2})_{x,y} : (D_{1} ⊗ D_{2})(xy) → (D_{1} ⊗ D_{2})(x) ⊗ ^{x}(D_{1} ⊗ D_{2})(y)) is the family of natural isomorphisms given by the following composition:

D_{1}(σ_{D_{2}}(x)σ_{D_{2}}(y)) ⊗ D_{2}(xy)
(β_{1})_{σD2 (x),σD2 (y)}⊗(β_{2})_{x,y}
D_{1}(σ_{D_{2}}(x)) ⊗^{ρ(D_{2}(x))x} D_{1}(σ_{D_{2}}(y)) ⊗ D_{2}(x) ⊗ ^{x}D_{2}(y)
1⊗ϕ⊗1
D_{1}(σ_{D_{2}}(x)) ⊗^{ρ(D_{2}(x))} (^{x}D_{1}(σ_{D_{2}}(y))) ⊗ D_{2}(x) ⊗ ^{x}D_{2}(y)
1⊗ξ_{D2(x),xD1(σD2 (y))}⊗1
D_{1}(σ_{D_{2}}(x)) ⊗ D_{2}(x) ⊗ ^{x}D_{1}(σ_{D_{2}}(y)) ⊗ ^{x}D_{2}(y)
1⊗ψ^{−1}
D_{1}(σ_{D_{2}}(x)) ⊗ D_{2}(x) ⊗ ^{x}(D_{1}(σ_{D_{2}}(y)) ⊗ D_{2}(y)).

Moreover, in such conditions, σ_{D_{1}⊗D_{2}} = σ_{D_{1}} · σ_{D_{2}}.

Proof. It is straightforward to check that β_{1} ⊗ β_{2} satisfies (3) so that (D_{1} ⊗ D_{2}, β_{1} ⊗ β_{2}) is a derivation from G[0] to ℍ.

As for the second statement we have:

σ_{D_{1}⊗D_{2}}(x) = ρ((D_{1} ⊗ D_{2})(x))x = ρ(D_{1}(σ_{D_{2}}(x)) ⊗ D_{2}(x)) x
= ρ(D_{1}(ρ(D_{2}(x)) x))ρ(D_{2}(x)) x = ρ(D_{1}(ρ(D_{2}(x)) x))σ_{D_{2}}(x)
= ρ(D_{1}(σ_{D_{2}}(x)))σ_{D_{2}}(x) = σ_{D_{1}}(σ_{D_{2}}(x)).
□

The derivation (D_{1} ⊗ D_{2}, β_{1} ⊗ β_{2}) built into the above proposition defines a functor

\xymatrix{Der(G, ℍ) × Der(G, ℍ) \ar[r]^{⊗} & Der(G, ℍ)}
(D_{1}, β_{1}) ⊗ (D_{2}, β_{2}) = (D_{1} ⊗ D_{2}, β_{1} ⊗ β_{2}),

that determines a monoidal structure in the category Der(G, ℍ). The isomorphism of associativity, ā : ((D_{1}, β_{1}) ⊗ (D_{2}, β_{2})) ⊗ (D_{3}, β_{3}) → (D_{1}, β_{1}) ⊗ ((D_{2}, β_{2}) ⊗ (D_{3}, β_{3})), is the arrow determined by the natural transformation ā : (D_{1} ⊗ D_{2}) ⊗ D_{3} → D_{1} ⊗ (D_{2} ⊗ D_{3}) given by ā_{x} = a_{D_{1}(σ_{D_{2}⊗D_{3}} (x)),D_{2}(σ_{D_{3}} (x)),D_{3}(x)},

x ∈ G; the unit object Ī, is the trivial derivation (D_{0}, β_{0}); the left unit constraint l̄ = l̄_{(D,β)} : (D_{0}, β_{0}) ⊗ (D, β) → (D, β), is the arrow determined by the natural transformation l̄: D_{0} ⊗ D → D given by l̄_{x} = l_{D(x)}; and the right unit constraint r̄ = r̄_{(D,β)} : (D, β) ⊗ (D_{0}, β_{0}) → (D, β), is the arrow determined by the natural transformation r̄: D ⊗ D_{0} → D given by r̄_{x} = r_{D(x)}.

Below we will see that another monoidal structure in Der(G, ℍ) can be defined alternatively, by using the endomorphism θ_{D} : ℍ → ℍ, given in Lemma 4.2, instead of σ_{D}. To do so, we first show the following:

Proposition 4.4. Let (ℍ, ρ, ξ) be a categorical G-crossed module and (D, β) a derivation from G[0] to ℍ. Then we have:

i) σ_{D} · ρ = ρ · θ_{D} : ℍ → G[0]. ii) Both θ_{D} · D and D · σ_{D} are derivations from G[0] into ℍ and there is an arrow in Der(G, ℍ), ϵ : θ_{D} · D → D · σ_{D}. iii) If ac : G[0] → Eq(ℍ) denotes the G-action on ℍ then, for any x ∈ G, there is a morphism of categorical group homomorphisms τ_{x} : θ_{D} · ac_{x} → ac_{σ_{D} (x)} · θ_{D}. iv) Given two derivations (D_{1}, β_{1}), (D_{2}, β_{2}) : G[0] → ℍ there is, for any x ∈ G, a natural isomorphism χ_{x} : D_{1}(σ_{D_{2}}(x)) ⊗ D_{2}(x) → θ_{D_{1}}(D_{2}(x)) ⊗ D_{1}(x).

Proof. i) (σ_{D} · ρ)(A) = ρ(D(ρ(A)))ρ(A) = ρ(D(ρ(A)) ⊗ A) = (ρ · θ_{D})(A). ii) To prove that θ_{D} · D and D · σ_{D} are derivations is straightforward. As for the arrow ϵ, it is given by the natural transformation whose component at x ∈ G, ϵ_{x} : (θ_{D} · D)(x) → (D · σ_{D})(x), is determined by the following diagram:

\xymatrix{θ_{D} (D(x)) = D(ρ(D(x))) ⊗ D(x) \ar@{-->}[r]^{ϵ_{x}} \ar[dd]_{1⊗r^{−1}} & D(σ_{D} (x)) = D(ρ(D(x))x) \ar[d]^{β_{ρ(D(x)),x}} \\  & D(ρ(D(x))) ⊗ ^{ρ(D(x))}D(x) \ar[d]^{1⊗r^{−1}} \\ D(ρ(D(x))) ⊗ D(x) ⊗ I \ar[d]_{1⊗γ^{−1}} & D(ρ(D(x))) ⊗ ^{ρ(D(x))}D(x) ⊗ I \ar[d]^{1⊗γ^{−1}} \\ D(ρ(D(x))) ⊗ D(x) ⊗ D(x) ⊗ D(x)^{∗} \ar[r]^{1⊗ξ_{D(x),D(x)}^{−1} ⊗1} & D(ρ(D(x))) ⊗ ^{ρ(D(x))}D(x) ⊗ D(x) ⊗ D(x)^{∗}.}

iii) For any object A ∈ ℍ, τ_{x,A} : θ_{D}(^{x}A) → ^{σ_{D} (x)}θ_{D}(A) is the arrow in ℍ given by the following diagram:

\xymatrix{θ_{D} (^{x}A) = D(ρ(^{x}A)) ⊗ ^{x}A = D(xρ(A)x^{−1}) ⊗ ^{x}A \ar@{-->}[r]^{τ_{x,A}} \ar[d]_{β_{xρ(A),x^{−1}} ⊗1} & ^{σ_{D} (x)}θ_{D} (A) = ^{(ρ(D(x)) x)}θ_{D} (A) \ar[d]^{ϕ} \\ D(xρ(A)) ⊗ ^{xρ(A)}D(x^{−1}) ⊗ ^{x}A \ar[d]_{1⊗ϕ⊗1} & ^{(ρ(D(x))}(^{x}θ_{D} (A)) \ar[d]^{(1⊗γ^{−1})·r^{−1}} \\ D(xρ(A)) ⊗ ^{x}(^{ρ(A)}D(x^{−1})) ⊗ ^{x}A \ar[d]_{1⊗ψ^{−1}} & ^{ρ(D(x))}(^{x}θ_{D} (A)) ⊗ D(x) ⊗ D(x)^{∗} \ar[d]^{ξ_{_{D(x),^{x}θ_{D}} (A)}⊗r^{−1}} \\ D(xρ(A)) ⊗ ^{x}(^{ρ(A)}D(x^{−1}) ⊗ A) \ar[d]_{β_{x,ρ(A)} ⊗^{x}ξ_{A,D(x^{−1})}} & D(x) ⊗ ^{x}(D(ρ(A)) ⊗ A) ⊗ D(x)^{∗} ⊗ I \ar[d]^{1⊗β̄_{0}^{−1}} \\ D(x) ⊗ ^{x}D(ρ(A)) ⊗ ^{x}(A ⊗ D(x^{−1})) \ar[d]_{1⊗ψ} & D(x) ⊗ ^{x}(D(ρ(A)) ⊗ A) ⊗ D(x)^{∗} ⊗ D(xx^{−1}) \ar[d]^{1⊗ψ⊗1⊗β_{x,x^{−1}}} \\ D(x) ⊗ ^{x}D(ρ(A)) ⊗ ^{x}A ⊗ ^{x}D(x^{−1}) \ar[r]_{(1⊗ϑ^{−1}⊗1)(r^{−1}⊗1)} & D(x) ⊗ ^{x}D(ρ(A)) ⊗ ^{x}A ⊗ D(x)^{∗} ⊗ D(x) ⊗ ^{x}D(x^{−1}).}

iv) The arrow χ_{x} : D_{1}(σ_{D_{2}}(x)) ⊗ D_{2}(x) → θ_{D_{1}}(D_{2}(x)) ⊗ D_{1}(x) is defined as the composite χ_{x} = (1 ⊗ ξ_{D_{2}(x),D_{1}(x)}) · ((β_{1})_{ρ(D_{2}(x)),x} ⊗ 1). □

Now, consider the functor

\xymatrix{Der(G, ℍ) × Der(G, ℍ) \ar[r]^{¯⊗} & Der(G, ℍ)}
(D_{1}, β_{1}) ¯⊗(D_{2}, β_{2}) = (D_{1} ¯⊗D_{2}, β_{1} ¯⊗β_{2}),

where D_{1} ¯⊗D_{2} : G[0] → ℍ is given by (D_{1} ¯⊗D_{2})(x) = θ_{D_{1}}(D_{2}(x)) ⊗ D_{1}(x) and

(β_{1} ¯⊗β_{2})_{x,y} : (D_{1} ¯⊗D_{2})(xy) → (D_{1} ¯⊗D_{2})(x) ⊗ ^{x}(D_{1} ¯⊗D_{2})(y)

is defined as the following composition:

θ_{D_{1}}(D_{2}(xy)) ⊗ D_{1}(xy)
θ_{D1} ((β_{2})_{x,y})⊗(β_{1})_{x,y}
θ_{D_{1}}(D_{2}(x) ⊗ ^{x}D_{2}(y)) ⊗ D_{1}(x) ⊗ ^{x}D_{1}(y)
μ_{θD1} ⊗1
θ_{D_{1}}(D_{2}(x)) ⊗ θ_{D_{1}}(^{x}D_{2}(y)) ⊗ D_{1}(x) ⊗ ^{x}D_{1}(y)
1⊗(τ_{1})_{x,D2(y)}⊗1
θ_{D_{1}}(D_{2}(x)) ⊗ ^{σ_{D_{1}} (x)}θ_{D_{1}}(D_{2}(y)) ⊗ D_{1}(x) ⊗ ^{x}D_{1}(y)
1⊗ϕ⊗1
θ_{D_{1}}(D_{2}(x)) ⊗ ^{ρ(D_{1}(x))}(^{x}θ_{D_{1}}(D_{2}(y))) ⊗ D_{1}(x) ⊗ ^{x}D_{1}(y)
1⊗ξ_{D1(x),xθD1 (D2(y))}⊗1
θ_{D_{1}}(D_{2}(x)) ⊗ D_{1}(x) ⊗ ^{x}θ_{D_{1}}(D_{2}(y)) ⊗ ^{x}D_{1}(y)
1⊗ψ^{−1}
θ_{D_{1}}(D_{2}(x)) ⊗ D_{1}(x) ⊗ ^{x}(θ_{D_{1}}(D_{2}(y)) ⊗ D_{1}(y)) .

To check that (D_{1} ¯⊗D_{2}, β_{1} ¯⊗β_{2}) is a derivation is straightforward. It is also easy to see that there is a morphism of categorical group homomorphisms

θ_{D_{1}} · θ_{D_{2}} → θ_{D_{1} ¯⊗D_{2}}  (8)

whose component at A ∈ ℍ is the arrow (μ_{θ_{D_{1}}} )_{D_{2}(ρ(A)),A}.

The functor ¯⊗ determines a monoidal structure in the category Der(G, ℍ) and, for any derivations (D_{1}, β_{1}) and (D_{2}, β_{2}), there is a natural isomorphism (D_{1} ⊗ D_{2})(x) → (D_{1} ¯⊗D_{2})(x) given by the isomorphism χ_{x} defined in Proposition 4.4 iv).

Then we summarize the above results in the following:

Theorem 4.5. If G is a group and (ℍ, ρ, ξ) is a categorical G-crossed module, the functors

⊗ : Der(G, ℍ) × Der(G, ℍ) ⟶ Der(G, ℍ)

and

¯⊗ : Der(G, ℍ) × Der(G, ℍ) ⟶ Der(G, ℍ)

determine, in the category Der(G, ℍ), two monoidal structures connected by the natural isomorphism χ_{x} defined in Proposition 4.4.

Let us recall now [19] that, for any monoidal category 𝒞, the Picard categorical group 𝒫(𝒞) of 𝒞 is the subcategory of 𝒞 given by invertible objects and isomorphisms between them. Clearly, 𝒫(𝒞) is a categorical group and any monoidal functor F : 𝒞 → 𝒟 restricts to a homomorphism of categorical groups 𝒫(F) : 𝒫(𝒞) → 𝒫(𝒟). In this way, there is a functor 𝒫 from the category of monoidal categories to 𝒞 𝒢 .

Definition 4.6. For any group G and any categorical G-crossed module (ℍ, ρ, ξ) we define the Whitehead categorical group of derivations D(G, ℍ) as the Picard categorical group, 𝒫(Der(G, ℍ)), of the monoidal category (Der(G, ℍ), ⊗).

Note that, when ℍ = H[0] is the discrete categorial group associated to a group H, the categorial group D(G, ℍ) is exactly the discrete one associated to the Whitehead group D(G, H) of the regular derivations from G into H. Also, note that if (𝔸, c) is a G-module then D(G, 𝔸) = Der(G, 𝔸), where the last is the categorical group of derivations studied in [7].

Finally, we characterize the objects of D(G, ℍ), that is, the invertible derivations, as follows:

Theorem 4.7. Let G be a group and (ℍ, ρ, ξ) a categorical G-crossed module. Then, the following statements are equivalent:

a) (D, β) ∈ D(G, ℍ),

b) σ_{D} ∈ Aut(G),

c) θ_{D} ∈ ℰ q(ℍ).

Proof. a) ⇒ b) If (D, β) ∈ D(G, ℍ), then there exists an inverse for (D, β), that is, a derivation (D, β)^{∗} = (D̄, β̄) ∈ D(G, ℍ) and isomorphisms (D, β)⊗(D̄, β̄) ≅ (D̄, β̄) ⊗ (D, β) ≅ (D_{0}, β_{0}). Thus, according to Lemma 4.2 and Proposition 4.3, σ_{D} · σ_{D̄} = σ_{D̄} · σ_{D} = σ_{D_{0}} = id_{G} and therefore σ_{D} is an automorphism.

a) ⇒ c) Using now (8) and Lemma 4.2, there exist morphisms of categorical group homomorphisms

θ_{D} · θ_{D̄} ⟶ θ_{D⊗D̄} ⟶ θ_{D_{0}} ⟶ id_{ℍ}

and

θ_{D̄} · θ_{D} ⟶ θ_{D̄⊗D} ⟶ θ_{D_{0}} ⟶ id_{ℍ}

and therefore θ_{D} is an equivalence of ℍ.

b) ⇒ a) Let (D, β) ∈ Der(G, ℍ) such that σ_{D} is an automorphism. An inverse (D, β)^{∗} = (D̄, β̄) for (D, β) is obtained as follows. Define D̄ : G[0] → ℍ by D̄(x) = (D(σ−1_{D}(x)))^{∗} and, by denoting x′ = σ−1_{D}(x), x ∈ G, let β̄_{x,y} : D̄(xy) → D̄(x) ⊗ ^{x}D̄(y) be the arrow in ℍ determined by the diagram:

\xymatrix{D̄(xy) = (D(x′y′))^{∗} \ar[d]_{(β_{x′,y′}^{−1} )^{∗}} \ar@{-->}[r]^{β̄_{x,y}} & D̄(x) ⊗ ^{x}D̄(y) = (D(x′))^{∗} ⊗ ^{x}(D(y′)^{∗}) \\ (D(x′) ⊗ ^{x′}D(y′))^{∗} \ar[d]_{(ξ_{D(x′),^{x′} D(y′)})^{∗}} & D(x′)^{∗} ⊗ (^{x}D(y′))^{∗} \ar[u]_{1⊗λ^{−1}} \\ (^{ρ(D(x′))}(^{x′}D(y′)) ⊗ D(x′))^{∗} \ar[r]_{(ϕ⊗1)^{∗}} & (^{σ_{D} (x′)}D(y′) ⊗ D(x′))^{∗}. \ar[u]_{υ}}

It is straightforward to see that (D̄, β̄) ∈ Der(G, ℍ). In addition, for any

x ∈ G, (D̄ ⊗ D)(x) = D̄(σ_{D}(x)) ⊗ D(x) = (D(σ−1_{D}(σ_{D}(x))))^{∗} ⊗ D(x) =

D(x)^{∗} ⊗ D(x) ≃ I = D_{0}(x) and therefore there is an arrow in Der(G, ℍ), (D̄, β̄) ⊗ (D, β) → (D_{0}, β_{0}). Thus, (D̄, β̄) is an inverse for (D, β) and so (D, β) ∈ D(G, ℍ).

c) ⇒ a) The proof is similar to the above one by defining D̄(x) = θ−1_{D}(D(x)^{∗}) where θ−1_{D} is a quasi-inverse of θ_{D}. □

Acknowledgments

This work was financially supported by DGI:BFM2001-2886.

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(Received 29.04.2002)

Authors’ address:

Departamento de Álgebra, Facultad de Ciencias Universidad de Granada, 18071 Granada Spain E-mail: agarzon@ugr.es