Applied Categorical Structures (2005) 13: 131–140 DOI: 10.1007/s10485-005-4383-1

© Springer 2005

Equivariant Extensions of Categorical Groups
A. R. GARZÓN and A. DEL RÍO Dpto. Álgebra, Fac. Ciencias, Univ. Granada, 18071, Granada, Spain. e-mail: [email protected] (Received: 4 March 2004; accepted: 20 March 2005) Abstract. If  is a group, then the category of -graded categorical groups is equivalent to the category of categorical groups supplied with a coherent left-action from . In this paper we use this equivalence and the homotopy classification of graded categorical groups and their homomorphisms to develop a theory of extensions of categorical groups when a fixed group of operators is acting. For this kind of extensions we show a suitable Schreier’s theory and a precise theorem of classification, including obstruction theory, which generalizes both known results when the group of operators is trivial (categorical group extensions theory) or when the involved categorical groups are discrete (equivariant group extensions theory). Mathematics Subject Classifications (2000): 18D10, 18B40, 20J05, 20J06. Key words: monoidal groupoid, graded categorical group, equivariant cohomology, groups with operators, extensions, obstruction.

1. Introduction Categorical groups are monoidal groupoids where every object is invertible, up to isomorphism, with respect to the tensor product. In [1] L. Breen developed a theory of extensions of categorical groups generalizing the classical Schreier’s theory [12] on the classification of group extensions with non-abelian kernel and, more recently, an obstruction theory for these categorical group extensions has been also showed in [2]. The cohomology sets defined by Breen to codify the set of equivalence classes of categorical group extensions provided then, as a particular instance, a suitable definition of a 3-cohomology of a group G with coefficients in a non-abelian group H . The geometric interpretation of such a third cohomology when H is abelian (i.e., the usual Eilenberg–Mac Lane cohomology group) was already known after Sinh’s thesis [13] since, when G = π0 (G) is the group of connected components of a categorical group G and H = π1 (G) is the group of automorphisms of the unit object, an element of that cohomology group, called the Postnikov invariant of G, allowed the classification, up to equivalence, of G. If  is a group, -graded monoidal categories and, in particular, -graded categorical groups were originally introduced by Frölich and Wall in [7] where they presented a suitable abstract setting to study Brauer groups in equivariant situa
Partially supported by MTM2004-01060.

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tions, but another interesting examples of graded categorical groups also appear in algebraic topology and ring theory (see [4, 5]). In [15], K. H. Ulbrich proved that the category of -graded categorical groups is equivalent to the one of categorical groups supplied with a coherent left action from  (-categorical groups in our terminology) and, in [4], A. Cegarra et al. have proved a precise classification theorem of these graded categorical groups and their homomorphisms, where the corresponding Postnikov invariant is given by a three-dimensional equivariant cohomology class (see [3] for the general definition and properties about the cohomology of groups with operators). These results were applied then to give an appropriate treatment of the equivariant group extensions with a non-abelian kernel that parallels the well-known for group extensions by Schreier [12] and Eilenberg and Mac Lane [6] which appear now as particular instances when  is trivial. The main objective of this paper is to prove a classification theorem (Theorem 3.3), including obstruction theory, of the set of equivalence classes of equivariant categorical group extensions (see Definition 3.1). This theory generalizes both the theory of extensions of categorical groups and the one of extensions of groups with operators and our result bases heavily, on the one hand, on the classification given in [4] of the set of homotopy classes of graded monoidal funtors and, on the other hand, in the Schreier’s theory developed in Theorem 3.2. Factor sets for equivariant extensions of the -group G by the -categorical group H are graded monoidal functors with discrete domain and whose codomain is the -graded categorical group Out (H) that is introduced and studied in Section 2. We remark that this factor set theory provides, as in the non-equivariant case Breen’s Schreier theory did, a suitable definition of an equivariant 3-cohomology of a -group G with coefficients in a -equivariant G-group H just by considering H the categorical group associated to the crossed module H → Aut(H ). 2. Preliminaries Hereafter  is a fixed group and  CGgr will denote the 2-category of -graded categorical groups (see [7, 4] for details). If G is a -group (i.e., a group G enriched with a left -action by automorphisms), then the discrete -graded categorical group dis G has the elements of G as objects and their morphisms σ : x → y are the elements σ ∈  with σ x = y. Composition is multiplication in  and the grading gr : dis G →  is the obvious map gr(σ ) = σ . The graded tensor σ σ σ product is given by (x → y) ⊗ (x  → y  ) = (xx  → yy  ), and the graded unit, σ σ I :  → dis G, by I (∗ → ∗) = (1 → 1); the associativity and unit isomorphisms are identities. If G is a -graded categorical group, then the subcategory Ker G consisting of all morphisms of grade 1 is a categorical group whose homotopy groups are the so-called homotopy groups of G, that is, πi G = πi Ker G, i = 0, 1. Actually, because the -graded enrichment, π0 G is a -group (the -action is defined by σ [X] = [Y ] whenever there exists a morphism X → Y in G of grade σ ) and

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π1 G is a -equivariant π0 G-module (the -action is σ u = I (σ )uI (σ )−1 ) (see [4, Proposition 1.2] for details). Thus, for example, if G is any -group, then π0 dis G = G, as a -group, while π1 dis G = 0. A categorical group H endowed with a coherent left-action from the (discrete) categorical group dis  associated to the group  is termed a -categorical group (for more details see [14] where the categorical group H is assumed to be symmetric, or [8], where G-categorical groups, for general categorical group of operators G, are considered). -categorical groups are the objects of a 2-category denoted by  CG where the 1-cells are the equivariant monoidal functors and the two-cells are the equivariant monoidal natural transformations. If G is a -graded categorical group then Ker G is a -categorical group and, conversely, any -categorical group H determines a -graded categorical group Hgr whose objects are the same as those of H and whose arrows of grade σ are the arrows f : σ X → Y in H. These two constructions determine actually [15, Theorem 1.2] an equivalence of categories 

CG   CGgr ,

H → Hgr

that points out the interest of graded categorical groups in the study of problems of equivariant nature. Note that if H is a -group then dis H is a -categorical group and (dis H )gr = dis H . If H is a -categorical group, we introduce below a graded version, denoted by Out (H), of the categorical group Out(H) defined in [2] associated to any categorical group H. This graded categorical group provides the key to codify the equivariant extensions we study in next section. The objects of Out (H) are the equivalences of the categorical group H. A premorphism of grade σ ∈ , from T to T  , is a triple (A, ϕ A , σ ) where A ∈ H and ϕ A : σ T → iA T  is a monoidal natural transformation (where σ T and iA are the equivalences of H given respectively, −1 for every object X ∈ H, by (σ T )(X) = σ T (σ X) and iA (X) = A ⊗ X ⊗ A∗ ).  A morphism of grade σ from T to T is then an equivalence class of premorphisms [A, ϕ A , σ ] where [A, ϕ A , σ ] = [B, ϕ B , σ ] if there exists a morphism in H, u : A → B, such that, for all X ∈ H, the following diagram is conmutative: (σ T )(X)

A ϕX

B ϕX

A ⊗ T  (X) ⊗ A∗ u⊗1⊗(u∗ )−1

B ⊗ T  (X) ⊗ B ∗ .

The identity on an object T is the class [I, ϕ I , 1] where, for every X ∈ H, ϕXI can is the following composition of canonical isomorphisms (1 T )(X) = 1 T (1 X) −→ can T (X) −→ I ⊗ T (X) ⊗ I . The composition is given by T

[A,ϕ A ,σ ]

T

[B,ϕ B ,τ ]

T  = T

[τ A⊗B,ϕ

τ A⊗B

,τ σ ]

T 

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´ AND A. DEL R´IO A. R. GARZON τ

where, for every X ∈ H, ϕXA⊗B is the dotted arrow in the following diagram: (τ σ T )(X)

τ

A ⊗ B ⊗ T  (X) ⊗ (τ A ⊗ B)∗

can

can −1

τ

τ

((σ T )(τ X))

A ⊗ B ⊗ T  (X) ⊗ B ∗ ⊗ τ A∗

τ ϕA τ −1 X

τ

B ⊗1 1⊗ϕX

can

−1

(A ⊗ T  (τ X) ⊗ A∗ )

τ

A ⊗ (τ T  )(X) ⊗ τ A∗ .

τ

It is straightforward to see that ϕ A⊗B is a monoidal natural transformation and that the composition does not depend on the representatives. Moreover, in Out (H) there is a graded monoidal structure determined by -graded functors Out (H) × Out (H) −→ Out (H)

and

I :  −→ Out (H)

defined as follows. If [A, ϕ A , σ ] : T → F and [B, ϕ B , σ ] : T  → F  then 

[B, ϕ B , σ ] ⊗ [A, ϕ A , σ ] = [B ⊗ F  (A), ϕ B⊗F (A) , σ ] : T  T → F  F where, for every X ∈ H, 

ϕXB⊗F (A) : (σ (T  T ))(X) −→ B ⊗ F  (A) ⊗ (F  F )(X) ⊗ (B ⊗ F  (A))∗ is the following composition: B⊗F  (A)

(σ (T  T ))(X)

ϕX

B ⊗ F  (A) ⊗ (F  F )(X) ⊗ (B ⊗ F  (A))∗

can

(σ T  )((σ T )(X))

can

B ⊗ F  (A) ⊗ B ∗ ⊗ B ⊗ F  (F (X)) ⊗ B ∗ ⊗ B ⊗ F  (A∗ ) ⊗ B ∗

A) (σ T  )(ϕX

(σ T  )(A ⊗ F (X) ⊗ A∗ )

B ⊗ϕ B B ϕA F (X) ⊗ϕA∗

can

(σ T  )(A) ⊗ (σ T  )(F (X)) ⊗ (σ T  )(A∗ ). σ

[I,ϕ I ,σ ]

As for I :  −→ Out (H), it is given by I (∗ −→ ∗) = idH −−−→ idH where, for each X ∈ H, ϕXI : (σ idH )(X) −→ I ⊗ idH (X) ⊗ I is a composite of canonical isomorphisms. In this way Out (H) is a -graded categorical group and there exists an isomorphism of categorical groups (in fact of -categorical groups) between Ker Out (H) and Out(H). Thus Out (H) is a groupoid because Ker Out (H) ∼ = Out(H) is a groupoid. In a similar way it is possible to introduce a graded version Z  (H) of the centre Z(H) of a categorical group (see [9, Example 2.3]) and it is easy to observe that π1 (Out (H)) = π1 (Ker Out (H)) = π1 (Out(H)) = π0 (Z(H)) = π0 (Ker Z  (H)) = π0 (Z  (H)) as -modules.

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In the particular case that H is a -group and we consider the -categorical group dis H , we have that Out (dis H ) is the -graded categorical group whose objects are the elements of the -group Aut(H ); a morphism of grade σ from f to g is a pair (a, σ ) where a ∈ H is such that, for every x ∈ H , (σ f )(x) = ag(x)a −1 (that is, σ f = Ca · g where Ca is conjugation by a). Composition is given by (a,σ )

(τ ab,τ σ )

(b,τ )

(a,σ )

(b,σ )

f −→ g −→ h = f −−−→ h, the graded tensor by (f −→ f  ) ⊗ (g −→ g  ) (af  (b),σ )

(1,σ )

σ

= f g −−−−→ f  g  and the graded unit by I (∗ −→ ∗) = idH −→ idH . Let us remark that Out (dis H ) is just the -graded categorical group Hol H defined in [4] and π0 (Out (dis H )) is the -group Out(H ) whereas π1 (Out (dis H )) is the -equivariant Out(H )-module Z(H ). 3. Equivariant Extensions of Categorical Groups: Obstruction Theory In this section we develop a theory of extensions of -categorical groups which extends both categorical group extensions theory [1, 11, 2] and equivariant group extensions theory [4]. This general theory provides, as an example, a theory of equivariant extensions of a -group G by a -crossed module of groups L and, in particular, when L is the crossed module H → Aut(H ) given by inner automorphisms, one obtains a definition of a three-dimensional cohomology of a -group G with coefficients in a non-abelian -group H . DEFINITION 3.1. Let  be a group and suppose that G is a -group and H is a -categorical group. A -equivariant extension of G by H is a sequence of -categorical groups and equivariant monoidal functors j

p

E : H −→ E −→ dis G such that p is surjective and j induces a monoidal equivalence between H and K(p) = p−1 (1). j

p

j

p

Given two extensions E : H −→ E −→ dis G and E : H −→ E −→ dis G, a morphism between them is a pair (h, ν) : E → E H

j

ν⇓

H

j

E

p

dis G

h

E

p

dis G

where h : E → E is an equivariant monoidal functor such that p h = p and ν : hj ⇒ j  is an equivariant monoidal transformation. We denote by Ext (G, H)

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the set of equivalence classes of -equivariant extensions of G by H. Note that if  = 1, the trivial group, then the set Ext1 (G, H) is the set of categorical group extensions Ext(G, H) studied in [1, 11, 2] whereas if H = dis H for some -group H , then Ext (G, dis H ) is just the set Ext (G, H ) of equivalence classes of equivariant group extensions of G by H studied in [4]. In the following theorem we introduce an equivariant factor set theory for equivariant extensions of categorical groups. An equivariant factor set for an equivariant extension must be an appropriate set of data to rebuild, by a suitable crossed product construction, the equivariant extension up to equivalence. Below we will see that these factor sets are graded monoidal functors from the discrete -graded categorical group dis G to the -graded categorical group Out (H) introduced in Preliminaries. We should remark that Schreier’s theories for the classification of the sets of extensions Ext(G, H) and Ext (G, H ) were respectively established in [2, Corollary 4.1] and [4, Theorem 4.2] and now, since Out1 (H) is the categorical group Out(H) (see [2]) and Out (dis H ) is just the -graded categorical group Hol H (see [4]), both theories appear as particular cases of the following theorem. THEOREM 3.2 (Schreier theory for equivariant extensions of categorical groups). Let  be a group and suppose that G is a -group and H is a -categorical group. There exists a natural bijection (1) [dis G, Out (H)] ∼ = Ext (G, H), between the set of homotopy classes of -graded monoidal functors from dis G to Out (H) and the set of equivalence classes of -equivariant extensions of G by H. Proof. We can restrict our attention, without loss of generality (see [4]), to -graded monoidal functors (T , µ) : dis G → Out (H) such that T (1) = idH and µ0 = ididH . It is not very difficult to observe that the system of data describing such a graded monoidal functor consists of mappings: α Obj(E q(H)), (i) G T (α), with T (1) = idH ; Obj(H) and G × G Obj(H) (ii) G ×  Aα,σ (α, σ ) with Aα,1 = I Mor(E q(H)) (iii) G × 

Aα,β (α, β) with Aα,1 = I = A1,β ; Mor(E q(H)) and G × G

(α, σ ) ϕ Aα,σ ϕ Aα,σ : σ (T (α)) → iAα,σ · T (σ α), (iv) G ×  ×  (α, τ, σ )

(α, β) ϕ Aα,β ϕ Aα,β : 1 (T (α)T (β)) → iAα,β · T (αβ);

Mor(H)

G×G×

Mor(H)

uα,τ,σ ,

(α, β, σ )

uα,β,σ

and

G×G×G

Mor(H)

(α, β, γ )

uα,β,γ

with uα,τ,σ : Aα,τ σ −→ τ Aα,σ ⊗ Aσ α,τ , uα,β,σ : σ Aα,β ⊗ Aαβ,σ −→ Aα,σ ⊗ (σ α) (Aβ,σ ) ⊗ Aσ α,σ β and uα,β,γ : Aα,β ⊗ Aαβ,γ −→ α Aβ,γ ⊗ Aα,βγ satisfying those coherence conditions easily deduced from those for (T , µ).

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EQUIVARIANT EXTENSIONS

Every graded monoidal functor (T , µ) : dis G → Out (H) gives rise to a -equivariant extension of G by H j

p

E(T ,µ) : H −→ H ×(T ,µ) G −→ dis G that we call a crossed product equivariant extension of G by H. The categorical group H ×(T ,µ) G has, as objects, all pairs (A, α) with A ∈ H and α ∈ G and in H ×(T ,µ) G there is a morphism f : (A, α) → (B, β) if α = β and f : A → B is a f

f

morphism in H. The functors j and p are given by j (A → B) = (A, 1) → (B, 1) f

and p((A, α) → (B, α)) = idα . The tensor product ⊗ : (H ×(T ,µ) G) × (H ×(T ,µ) G) −→ H ×(T ,µ) G is defined, on objects, by (A, α) ⊗ (B, β) = (A ⊗ T (α)(B) ⊗ Aα,β , αβ), and, on arrows, by 

f g (A, α) → (A , α) ⊗ (B, β) → (F  , β) = (A ⊗ T (α)(B) ⊗ Aα,β , αβ)

f ⊗T (α)(g)⊗1

(A ⊗ T (α)(B  ) ⊗ Aα,β , αβ).

The natural isomorphisms of associativity are deduced using the data ϕ Aα,β and uα,β,γ . An inverse for every object (A, α) is the object (A, α)∗ = (T (α)∗ (A∗ ⊗ A∗α,α−1 ), α −1 ) where A∗ is an inverse of A and T (α)∗ is a quasi-inverse of T (α). H ×(T ,µ) G is actually a -categorical group with -action given by σ


f (A, α) → (B, α) = (σ A ⊗ Aα,σ , σ α)

σ f ⊗1

(σ B ⊗ Aα,σ , σ α).

The natural isomorphisms of the action φτ,σ,(A,α) : τ σ (A, α) −→ τ (σ (A, α)) and ψσ,(A,α),(B,β) : σ ((A, α) ⊗ (B, β)) −→ σ (A, α) ⊗ σ (B, β) are given using the data ϕ Aα,σ , uα,τ,σ and uα,β,σ . Moreover, it is straightforward to see that the system of data describing a homotopy (i.e., a graded monoidal natural equivalence) between two -graded monoidal functors from dis G to Out (H) yields to equivalent crossed product -equivariant extensions and viceverse. j p Finally, we prove that any equivariant extension of G by H, E : H −→ E −→ dis G, is equivalent to a crossed product extension of G by H associated with a certain -graded monoidal functor (T , µ) : dis G → Out (H). There is no loss of generality in assuming that H = p−1 (I ) and j is the inclusion. For every α ∈ G we choose a representative Xα ∈ E, with X1 = I , such that p(Xα ) = α. Thus there is a map G → Obj(E q(H)), α → T (α), with T (α)(A) =

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iXα = Xα ⊗ A ⊗ Xα∗ . Besides, since p(σ Xα ) = σ p(Xα ) = σ α = p(Xσ α ), Aα,σ = σ Xα ⊗ Xσ∗α ∈ H and, using that p(Xα ⊗ Xβ ) = p(Xα ) ⊗ p(Xβ ) = αβ = p(Xαβ ), ∗ ∈ H. Then, we have well-defined mappings we have that Aα,β = Xα ⊗ Xβ ⊗ Xαβ G×

Obj(H),

G×G

Obj(H)

(α, σ )

Aα,σ

(α, β)

Aα,β

with Aα,1 = I

with Aα,1 = I = A1,β .

The map α → iXα is not, in general, an equivariant monoidal functor from dis G to E q  (H), however the existence of canonical morphisms Aα,σ ⊗ Xσ α → σ Xα

and

Aα,β ⊗ Xα,β → Xα ⊗ Xβ

(2)

determines morphisms in E q(H), ϕ Aα,σ : σ (T (α)) → iAα,σ · T (σ α) and

ϕ Aα,β : T (α)T (β) → iAα,β · T (αβ)

and therefore there are mappings G×

Mor(E q(H)),

G×G

Mor(E q(H))

(α, σ )

ϕ Aα,σ

(α, β)

ϕ Aα,β .

If X ∈ E is such that p(X) = α, then p(X⊗Xα∗ ) = p(X)⊗p(Xα )−1 = αα −1 = 1 so that X ⊗ Xα∗ = A ∈ H and therefore there exists a canonical morphism X → A ⊗ Xα . This observation together with the morphisms (2) assure the existence, for any X, Y ∈ E and σ ∈ , of canonical morphisms X ⊗ Y → A ⊗ Xα ⊗ B ⊗ Xβ → A ⊗ T (α)(B) ⊗ Aα,β ⊗ Xαβ and σ

X → σ (A ⊗ Xα ) → σ A ⊗ σ Xα → σ A ⊗ Aα,σ ⊗ Xσ α

and therefore the structure of -categorical group in E can be described in terms of the ones of dis G and H together with the equivalences T (α), the objects Aα,σ and Aα,β and the morphisms between equivalences ϕ Aα,σ and ϕ Aα,β . It is straightforward to deduce, from the associativity isomorphism (X ⊗ Y ) ⊗ Z −→ X ⊗ (Y ⊗ Z) in E, the existence of morphisms uα,β,γ : Aα,β ⊗ Aαβ,γ −→ α Aβ,γ ⊗ Aα,βγ which, due to the pentagon axiom, satisfy the required coherence condition. Analogously, using the isomorphisms of the coherent -action in E, φτ,σ,X : τ σ X → τ (σ X) and ψσ,X,Y : σ (X ⊗ Y ) → σ X ⊗ σ Y , one deduces the existence of morphisms uα,τ,σ : Aα,τ σ → τ Aα,σ ⊗ Aσ α,τ and uα,β,σ : σ Aα,β ⊗ σ Aαβ,σ −→ Aα,σ ⊗ ( α) (Aβ,σ ) ⊗ Aσ α,σ β satisfying the required conditions. All in all we have a -graded monoidal functor (T , µ) = (T (α), Aα,σ , Aα,β , ϕ Aα,σ , ϕ Aα,β , uα,τ,σ , uα,β,σ , uα,τ,γ ) : dis G → Out (H)

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EQUIVARIANT EXTENSIONS

whose associated crossed product -equivariant extension E(T ,µ) is equivalent to 2 the given one by the morphism E(T ,µ) → E given by h(A, α) = A ⊗ Xα . j

p

Let us remark that if E : H −→ E −→ dis G is a -equivariant extension of G by H, the assignation showed in the above proof, α → iXα with p(Xα ) = α, induces a homomorphism of -groups ρE : G → π0 (Out (H)). For each -group G and each -graded categorical group G, if [dis G, G] denotes the set of homotopy classes of graded monoidal functors from dis G to G, then there is a canonical map π0 : [dis G, G] −→ Hom (G, π0 G)

[T ] → π0 T ,

(3)

where Hom (G, π0 G) is the set of equivariant homomorphisms from the -group G to the -group π0 G. A -group homomorphism ρ : G → π0 G is said to be realizable whenever it is in the image of the above map π0 , that is, if ρ = π0 T for some graded monoidal functor T : dis G → G. The map (3) produces a partitioning  [dis G, G; ρ], [dis G, G] = ρ

where, for each ρ ∈ Hom (G, π0 G), [dis G, G; ρ] = π0−1 (ρ) is the set of homotopy classes of realizations of ρ. When a -equivariant homomorphism ρ : G → π0 G is specified, it is possible that there is no graded monoidal functor that realizes ρ, and this leads to a problem of obstructions which (even in a more general form) is solved in [4, Theorem 3.2] by means of a 3-dimensional equivariant group cohomology class (the obstruction cohomology class) Obs(ρ) ∈ H3 (G, π1 G) (note that π1 G is, via ρ, a -equivariant G-module and so the equivariant cohomology groups Hn (G, π1 G) [3] are defined). As a consequence of that theorem (see [5, Theorem 2.2]), a -group homomorphism ρ : G → π0 G is realizable, that is, [dis G, G; ρ] = ∅, if, and only if, its obstruction Obs(ρ) vanishes. Moreover, if Obs(ρ) = 0, then there is a bijection [dis G, G; ρ] ∼ = H2 (G, π1 G). Recalling that π0 (Out (H)) is a -group and π1 (Out (H)) = π0 (Z  (H)) is a equivariant π0 (Out (H))-module and defining (with the notion of abstract kernel of S. Mac Lane in mind) an equivariant G-abstract kernel of a -categorical group H as an equivariant homomorphism of -groups ρ : G → π0 (Out (H)), bijection (1) allows now to state the following classification theorem about equivariant categorical group extensions. THEOREM 3.3. Let  be a group and suppose that G is a -group and H is a -categorical group. (i) There is a canonical partition  Ext (G, (H, ρ)), Ext (G, H) = ρ

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´ AND A. DEL R´IO A. R. GARZON

where, for each equivariant homomorphism ρ : G → π0 (Out (H)), Ext (G, (H, ρ)) is the set of classes of equivariant extensions E of G by H which realize ρ, that is, with ρE = ρ. (ii) Each equivariant G-abstract kernel ρ : G → π0 (Out (H)) determines a 3-dimensional cohomology class Obs(ρ) ∈ H3 (G, π0 (Z  (H))) of G with coefficients in the G-module (via ρ) π0 (Z  (H)). This invariant is called the obstruction of the equivariant G-abstract kernel ρ. (iii) An equivariant G-abstract kernel ρ is realizable, that is, Ext (G, (H, ρ)) = ∅, if and only if its obstruction Obs(ρ) vanishes. (iv) If the obstruction of an equivariant G-abstract kernel ρ vanishes, then there is a bijection Ext (G, (H, ρ)) ∼ = H2 (G, π0 (Z  (H))).

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