Obstruction theory for extensions of categorical groups
Obstruction theory for extensions of categorical groups P. Carrasco and A. R. Garz´ on∗ ´ Departamento de Algebra, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain E-mail: [email protected] ; [email protected]
Abstract For any categorical group H, we introduce the categorical group Out(H) and then the well-known group exact sequence 1 → Z(H) → H → Aut(H) → Out(H) → 1 is raised to a categorical group level by using a suitable notion of exactness. Breen’s Schreier theory for extensions of categorical groups is codified in terms of homomorphism to Out(H) and then we develop a sort of Eilenberg-Mac Lane’s obstruction theory that solves the general problem of the classification of all categorical group extensions of a group G by a categorical group H, in terms of ordinary group cohomology. Key words: categorical group, extension, cohomology, abstract kernel, obstruction
MSC : 18D10; 18G50; 20J06
1
Introduction
There is a long history of studying obstruction theory for extensions in several contexts. p If 1 → H → E → G → 1 is a group extension, the epimorphism p induces a homomorphism ρ : G → Out(H)(= Aut(H)/Int(H)). Although not with this terminology, it was observed by Baer [1] that a pair (H, ρ : G → Out(H)) need not always be built from such an extension; the “obstruction” to forming an extension which would induce a given pair (H, ρ) was found by Eilenberg and Mac Lane [12]. They formulated the ∗
Financially supported by DGI of Spain and FEDER, Project: BFM2001-2886
1
general problem of group extensions as that of the construction and classification of all extensions which induce a given pair (H, ρ), which they termed a G-abstract kernel. The center, Z(H), of H is (via ρ) a G-module and the obstruction, to the existence of an extension inducing the G-abstract kernel (H, ρ), is given by an element of the third Eilenberg-Mac Lane cohomology group H 3 (G, Z(H)). Thus, an abstract kernel (H, ρ) is associated with an extension if, and only if, its obstruction vanishes. Moreover, in such a case, the set of equivalence classes of extensions that induce (H, ρ) is a principal and homogeneous space under a canonical action of the abelian group H 2 (G, Z(H)). Using this pattern, Dedecker [10] formalized an obstruction theory for extensions of a group by a crossed module of groups. This general obstruction theory for groups parallels the more classic one for groups, and it was translated to many other contexts, such as associative algebras [25], associative and commutative algebras [2] or Lie algebras [15]. Later, all these obstruction theories were unified by Duskin [11], who developed a low-dimensional obstruction theory in algebraic categories and, in turn, this theory was generalized to higher dimensions in [7]. More recently, graded extensions of categories [8] and graded extensions of monoidal categories [9] have been codified in terms of non-abelian cocycles and corresponding obstruction theories in both contexts have then been developed. In addition, an obstruction theory for split extensions has been given in [13] for categorical groups. Categorical groups are monoidal groupoids where every object is invertible, up to isomorphism, with respect to the tensor product [23, 16, 4] . The 2-category of categorical groups (sometimes equipped with braiding or symmetry) can be seen as the 2-dimensional analogue of the category of (abelian) groups. The analysis of the development of the results extending the classic results on group extensions reveals that the key lies in the consideration of (internal) groupoids instead of groups. In this sense, categorical groups provide an adequate context in which to approach problems of the same nature as the above-mentioned but from a more general point of view, obtaining thus a wider range of applications [26, 5, 6, 22, 27, 14]. After a first section, devoted to setting the notations and recalling several notions and basic results about categorical groups, we dedicate the next section to obtaining a categorical-group version of the well-known exact sequence 1 → Z(H) → H → Aut(H) → Out(H) → 1. To do so, we use the known categorical groups Eq(H) [4] and Z(H) [16] associated to any categorical group H, and we introduce the categorical group, Out(H), whose objects are still equivalences of H. Out(H) could be presented as a kind of 2
quotient of categorical groups and we will develop this topic in a forthcoming paper. Now, using the notion of 2-exactness given in [17], we conclude with the existence of a 2-exact sequence: j
p
i
Z(H) −→ H −→ Eq(H) −→ Out(H) that generalizes the above one for groups. In the last section we begin by recalling Breen’s Schreier theory for extensions of categorical groups. Note that the classification of such extensions is the algebraization of a topological problem consisting of the classification of those fibrations of spaces, with base space a K(G, 1)-space and whose fibre has the homotopy type of a categorical group. After pointing out that, for group extensions of G by H, the classic Schreier theory can be rewritten in terms of homomorphisms from the discrete categorical group G[0] to the categorical group Out(H[0]), we prove the analogous result for Breen’s theory. That is, for any group G and any categorical group H, there is a natural bijection (see Corollary 4.2) ∆ : [G[0], Out(H)] −→ Ext(G, H). Then, any extension H → E → G[0] induces a group homomorphism (that we call a G-abstract kernel of H), ρ : G → π0 (Out(H)), and our aim in this section is to develop an obstruction theory for extensions of categorical groups that induce a given homomorphism ρ. Since π0 (Z(H)) = π1 (Out(H)) is a G-module via ρ, we show that the obstruction is measured by an element of H 3 (G, π0 (Z(H))) and we conclude by formulating the corresponding obstruction theorems: “A G-abstract kernel ρ : G → π0 (Out(H)) is realizable if and only if its obstruction ∂(ρ) ∈ Hρ3 (G, π0 (Z(H))) vanishes.” “If a G-abstract kernel ρ : G → π0 (Out(H)) is realizable, then the set of equivalence classes of realizations of ρ, Extρ (G, H), is a principal homogeneous space under the abelian group Hρ2 (G, π0 (Z(H))). In particular, there is a (non-natural) bijection Extρ (G, H) ∼ = Hρ2 (G, π0 (Z(H))).”
2
Preliminaries
First, we recall [18, 23, 16, 20] that a monoidal category, G = (G, ⊗, a, I, l, r), consists of a category G, a functor ⊗ : G × G → G, an object I of G, called the unit object, and natural isomorphisms a = aX,Y,Z : (X ⊗ Y ) ⊗ Z → X ⊗ (Y ⊗ Z), 3
l = lX : I ⊗ X → X,
r = rX : X ⊗ I → X,
such that the usual coherence conditions are satisfied. In a monoidal category, an object X is said to be invertible if the functors Y 7→ X ⊗ Y and Y 7→ Y ⊗ X are equivalences. A categorical group G is a monoidal small category where every arrow and every object is invertible. In this case, it is possible to define a functor (−)∗ : G → G, X 7→ X ∗ , f 7→ f ∗ , and natural isomorphisms, mX : X ⊗ X ∗ → I and nX : X ∗ ⊗ X → I, such that lX · (mX ⊗ 1X ) = rX · (1X ⊗ nX ) · aX,X ∗ ,X , for all objects X ∈ G. The triple (X ∗ , mX , nX ) is termed an inverse for X. A categorical group G is said to be a braided categorical group if it is also equipped with a family of natural isomorphisms c = cX,Y : X ⊗ Y → Y ⊗ X (the braiding) that interacts with a, r and l in the sense of [16]. A homomorphism of categorical groups T = (T, µ) : G → H consists of a functor T : G → H and a family of natural isomorphisms µ = µX,Y : T (X ⊗Y ) −→ T (X)⊗T (Y ), satisfying the appropriate coherence conditions [16]. If (G, c) and (H, c) are braided categorical groups, compatibility with c is also required [16]. If T : G → H is a homomorphism, there exists an isomorphism [16, 6], µ0 : T (I) −→ I, such that, for any X ∈ G, the equalities T (rX ) = rX ·(1⊗µ0 )· µX,I and T (lX ) = lX · (µ0 ⊗ 1) · µI,X hold. With respect to the inverses there exist unique isomorphisms λX : T (X ∗ ) −→ T (X)∗ , such that the equalities µ0 · T (mX ) = mT (X) · (1 ⊗ λX ) · µX,X ∗ and µ0 · T (nX ) = nT (X) · (λX ⊗ 1) · µX ∗ ,X hold. We will denote by CG the category of categorical groups and by BCG the category of braided categorical groups. Recall that, if G ∈ CG, then the set of connected components of G, π0 (G), has a group structure (which is abelian if G ∈ BCG). Also, the abelian group π1 (G) = AutG (I) is associated to G and it is a π0 (G)-module . The action [26, 13] is given by the map π0 (G) × π1 (G) → π1 (G) ,
−1 ([X], u) 7→ γX δX (u) ,
(1)
where, for any X ∈ G, the maps (in fact group isomorphisms) γX , δX : π1 (G) → AutG (X) are defined, for each u ∈ π1 (G), as the unique arrows making the following
4
diagrams commutative: I ⊗X lX
u⊗idX
X
/I ⊗X
γX (u)
X ⊗I rX
lX
/X
idX ⊗u
X
/X ⊗I
δX (u)
rX
/X
that is, −1 −1 γX (u) = lX · (u ⊗ idX ) · lX , δX (u) = rX · (idX ⊗ u) · δX .
These isomorphisms [26, 13] are compatible with the functor ⊗ and for any two objects X, Y ∈ G and arrows f, g : X → Y there exists a unique element u ∈ π1 (G) such that the following equality is satisfied g = f · γX (u) = γY (u) · f .
(2)
Given homomorphisms of categorical groups (T, µ), (T 0 , µ0 ) : G → H, a morphism from (T, µ) to (T 0 , µ0 ) consists of a natural transformation : T → T 0 that interacts with µ and µ0 in the sense of [16]. An equivalence of a categorical group G is a homomorphism (T, µ) : G → G such that the endofunctor T : G → G is an equivalence of categories. These equivalences are the objects of the categorical group Eq(G) [4], whose arrows are the morphisms between them. The composition in Eq(G) is given by the usual vertical composition of natural transformations. The tensor product is given by the composition of homomorphisms and the horizontal composition of natural transformations. The associativity isomorphism is the identity, the unit object is I = idG , and an inverse for an object (T, µ) is obtained by taking a quasi-inverse T ∗ of T . This categorical group has a categorical subgroup Aut(G) whose objects, called automorphisms, are the equivalences (T, µ) that are strict (i.e., µX,Y and µ0 are identities) and where T is an isomorphism. The kernel of a homomorphism T = (T, µ) : G → H (see [17],[27]) consists of a universal triplet (K(T), j, ), where K(T) is a categorical group, j : K(T) → G is a homomorphism and : Tj → 0 is a morphism, and it is described explicitly in the following way: an object of K(T) is a pair (X, uX ) where X ∈ G and uX : T (X) → I is an arrow in H; an arrow f : (X, uX ) → (Y, uY ) is an arrow f : X → Y in G such that uX = uY · T (f ); the functor j sends f : (X, uX ) → (Y, uY ) to f : X → Y ; the tensor product is given by (X, uX ) ⊗ (Y, uY ) = (X ⊗ Y, uX · uY ), where uX · uY : T (X ⊗ Y ) → I is 5
µX,Y
uX ⊗uY
rI =lI
/ I ; the unit / T (X) ⊗ T (Y ) /I ⊗I the composite T (X ⊗ Y ) object is (I, µ0 ) and the associativity and left-unit and right-unit constraints are given by aX,Y,Z , lX and rX respectively; an inverse for any object (X, uX ) is given by (X ∗ , u∗X λX ); and the component at (X, uX ) of the morphism is given by uX . If G and H are braided categorical groups, then K(T) is also a braided categorical group where the braiding c = c(X,u ),(Y,u ) : X Y (X, uX ) ⊗ (Y, uY ) −→ (Y, uY ) ⊗ (X, uX ) is precisely the braiding in G cX,Y . Note that j : K(T) → G is a strict homomorphism of (braided) categorical groups. The categorical group K(T) just described is a standard homotopy kernel and so it is determined, up to isomorphism, by the following strict universal property: given a homomorphism F : K → G and a morphism τ : TF → 0, there exists a unique homomorphism F0 : K → K(T) such that jF0 = F and F0 = τ . Note that if (K(T), j, ) is the kernel of T = (T, µ) : G → H, the sequence j
T
K(T) −→ G −→ H induces (see [22]) an exact sequence of groups 1
/ π1 (G) π1 (T) / π1 (H) i iiii δiiiiii iiii tiiπi0i(j) / π0 (G) π0 (T) / π0 (H) π0 (K(T))
/ π1 (K(T))
π1 (j)
(3)
where, for any u ∈ π1 (H), δ(u) is the connected component of the object (I, u · µ0 ) ∈ K(T). If the functor T is essentially surjective, the above exact sequence is right exact.
3
The categorical group Out(H) and a 2-exact sequence
For any group H, it is well known that the kernel of the homomorphism i H → Aut(H), given by inner automorphisms, is Z(H), the center of H, and the cokernel is Out(H) = Aut(H)/Int(H), the group of outer automorphisms of H. Thus, there is an exact sequence of groups i
1 → Z(H) → H → Aut(H) → Out(H) → 1. 6
In this section we will obtain a categorical group version of this sequence. To do so, we start by recalling the homomorphism given by conjugation and the notion of center for categorical groups. We will also look at their relationship and some examples. If H is a categorical group, there is a homomorphism i = (i, µi ) : H → Eq(H) given by conjugation; that is, for any X ∈ H, iX : H → H is given by iX (Y ) = (X ⊗ Y ) ⊗ X ∗ , iX (f ) = (1 ⊗ f ) ⊗ 1. It is clear that iX is an equivalence with quasi-inverse iX ∗ . The natural isomorphisms iX (Y ⊗ Z) → iX (Y ) ⊗ iX (Z) are given by the composition of canonical morphisms. For any arrow f : X → Y , if : iX → iY is the morphism whose component at Z ∈ H is (if )Z = (f ⊗ 1) ⊗ f ∗ : (X ⊗ Z) ⊗ X ∗ → (Y ⊗ Z) ⊗ Y ∗ . Finally, for any X, Y ∈ H, (µi )X,Y : iX⊗Y → iX · iY is the morphism in Eq(H) also determined by the composition of canonical morphisms. The center, Z(H), of a categorical group H (see [16]) is the braided categorical group whose objects are pairs (X, v) where X ∈ H and v : X ⊗ − → − ⊗ X is a natural isomorphism such that the following two conditions hold:
lX · vI = rx ; aY,Z,X · vY ⊗Z · aX,Y,Z = (1 ⊗ vZ ) · aY,X,Z · (vY ⊗ 1). An arrow f : (X, v) → (X 0 , v 0 ) in Z(H) is an arrow f : X → X 0 in H such 0 · (f ⊗ 1) = (1 ⊗ f ) · v . The tensor product and the that, for all X ∈ H, vX X braiding are given by (X, v) ⊗ (X 0 , v 0 ) = (X ⊗ X 0 , (v ⊗ 1) · (1 ⊗ v 0 )) , c(X,v),(X 0 ,v0 ) = vX 0 . The kernel of the conjugation homomorphism i = (i, µi ) : H → Eq(H) is isomorphic to Z(H) where the isomorphism F : K(i) → Z(H) is given uX v by F (X, iX −→ I) = (X, X ⊗ − −→ − ⊗ X) where vY is the following composition: can / can / (X ⊗ Y ) ⊗ I (X ⊗ Y ) ⊗ (X ∗ ⊗ X) X ⊗ YN NN NN can vY N& (uX )Y ⊗1 ((X ⊗ Y ) ⊗ X ∗ ) ⊗ X . Y ⊗X o
7
The inverse is given by F −1 (X, v) = (X, uX ), where (uX )Y : (X ⊗ Y ) ⊗ X ∗ → Y is the following composition: vY ⊗1
/ (Y ⊗ X) ⊗ X ∗ can / Y ⊗ (X ⊗ X ∗ ) R R R R can R R R R (uX )Y can ) o
(X ⊗ Y ) ⊗ X ∗
Y ⊗I .
Y
Therefore, we have a sequence j
i
Z(H) −→ H −→ Eq(H) , that, according to (3), has associated an exact sequence of groups 1
/ π1 (H)
/ Der(π0 (H), π1 (H)) gg gggg gggg g g g gggg sgggg / π0 (H) / π0 (Eq(H)) π0 (Z(H))
/ π (H)π0 (H) 1
since it is easy to see that, on the one hand, π1 (Z(H)) = π1 (H)π0 (H) , the group of invariant elements in π( H), and, on the other hand, π1 (Eq(H)) = Der(π0 (H), π1 (H)), the group of derivations from π0 (H) into π1 (H). When H = H[0] is the strict discrete categorical group associated to a group H, the above sequence specializes to the exact sequence i
1 → Z(H) → H → Aut(H). δ
More generally, let L = (H → G) be a crossed module of groups and let H(L) be the strict categorical group associated to it (see [20]). It has as objects the elements of the group G and an arrow h : x → y is an element h ∈ H with x = δ(h)y. The composition is multiplication in H and the tensor product is given by h
h0
hy h0
(x −→ y) ⊗ (x0 −→ y 0 ) = (xx0 −→ yy 0 ) . Then we can consider Z(L) = Z(H(L)), which is a strict braided categorical group, explicitly described as follows: the objects are pairs (x, d) where x ∈ G and d : G → H is a derivation such that, for any y ∈ G, δd(y) = [x, y]; an arrow from (x, d) to (x0 , d0 ) is an element u ∈ H such that x = δ(u)x0 and, for any y ∈ G, d(y)(y u) = ud0 (y); the tensor product is given by (x, d) ⊗ (x0 , d0 ) = (xx0 , dd0 ) where (dd0 )(y) = x d0 (y)d(y). 8
The homomorphism i = (i, µi ) : H(L) → Eq(H(L)) factors through Aut(H(L)) and so Z(L) is the kernel of the homomorphism i = (i, µi ) : H(L) → Aut(H(L)). Through the equivalence between crossed modules of groups and strict categorical groups, Aut(H(L)) corresponds [4] to the crossed module Act(L), called the actor of L by Norrie [21]. There is a morphism of crossed modules L → Act(L) whose kernel, ZN (L), called the cenδres
/ StG (H) ∩ Z(G) ). ter of L by Norrie, is the crossed module ZN (L) = ( H G The associated categorical group H(ZN (L)) is the fiber in the unit object of the homomorphism i = (i, µi ) : H(L) → Aut(H(L)) and there is a homomorphism H(ZN (L)) → Z(L), x 7→ (x, d0 ), where d0 : G → H is such that d0 (x) = 1 for any x ∈ G. This homomorphism induces an injective group morphism π0 (H(ZN (L))) −→ π0 (Z(L)). On the other hand, the canonical homomorphism Z(L) → H(L), (x, d) 7→ x, induces the group morphism π0 (Z(L)) → π0 (H(L)) = Coker(δ) whose image is StCoker(δ) (Ker(δ)) ∩ Z(Coker(δ)) and whose kernel is the group of derivations module inner derivations Der(Coker(δ), Ker(δ))/IDer(Coker(δ), Ker(δ)). The group morphism π0 (H(ZN (L))) → StCoker(δ) (Ker(δ)) ∩ Z(Coker(δ)), x ¯ 7→ x ¯, is then the composite π0 (H(ZN (L))) π0 (Z(L)) StCoker(δ) (Ker(δ))∩Z(Coker(δ)). In the particular case of considering, for any G-module A, the crossed 0 module L = (A → G), then π0 (H(ZN (L))) = StG (A)∩Z(G) and π0 (Z(L)) ∼ = Der(G,A) π0 (H(ZN (L)))⊕ IDer(G,A) and sequence (3) specializes to the exact sequence
0
/A
/ Der(G, A) , ggg g g g gg gggg gggg g g g sg Der(G,A) /G / Aut(A, G, 0) (StG (A) ∩ Z(G)) ⊕ IDer(G,A) / AG
which is nothing more than the sequence obtained by gluing the exact sequences 0 → AG → A → IDer(A, G) → 0, 0 → IDer(G, A) → Der(G, A) → Der(G,A) IDer(G,A) → 0 and 0 → StG (A) ∩ Z(G) → G → Aut(A, G, 0) where the last morphism is given by x 7→ (a 7→ x a, y 7→ xyx−1 ). i If H is a group and L = (H → Aut(H)) is the crossed module defined by the homomorphism i given by conjugation, then Out(H) = Coker(i) = π0 (H(L)). In the same spirit, below we introduce the categorical group Out(H) associated to any categorical group H. The homomorphism i = (i, µi ) : H → Eq(H) given by conjugation defines a crossed module of categorical groups, a notion introduced by Breen in [4]. This crossed module has associated [22] a monoidal bicategory Bieq(H) 9
and then we define Out(H) = Cl(Bieq(H)) where Cl : Bicat → Cat is the classifying functor [3] which assigns to each bicategory A the category Cl(A) whose objects are those of A and whose arrows are 2-isomorphisms classes of 1-arrows of A. Let us note that if A is a monoidal bicategory, then Cl(A) is a monoidal category and the restriction of Cl to CG, seeing a categorical group as a monoidal bicategory with only one object, is just the functor π0 : CG → G. Also, note that if H is a group then Bieq(H[0]) is the monoidal bicategory, where all 2-arrows are identities, defined by the categorical group H(L) i associated to L = (H → Aut(H)). Thus, Out(H[0]) = Cl(H(L)) = H(L) and so π0 (Out(H[0])) = Out(H), whereas π1 (Out(H[0])) = Z(H). Next we explicitly describe the categorical group Out(H). The objects are the equivalences of H. Given equivalences (T, µ) to (T 0 , µ0 ), we consider 0 0 pairs (A, ϕA ) where A ∈ H and ϕA = ϕA X : T (X) ⊗ T (A) → T (A ⊗ X), X ∈ H, is a family of natural isomorphisms in H such that, for any X, Y ∈ H, the following diagram (where we have omitted the canonical morphisms of associativity) is commutative: T (X ⊗ Y ) ⊗ T 0 (A)
ϕA X⊗Y
mmm mmm m m mm v mm m
/
T 0 (A ⊗ X ⊗ Y )
RRR 0 RRRµA⊗X,Y RRR RRR RR) T 0 (A ⊗ X) ⊗ T 0 (Y ) ll5 lll l l l lll A lll ϕX ⊗1
µX,Y ⊗1
T (X) ⊗ T (Y ) ⊗ T 0 (A)
QQQ QQQ QQQ QQQ 1⊗ϕA Q( Y
/
T (X) ⊗ T 0 (A ⊗ Y )
T (X) ⊗ T 0 (A) ⊗ T 0 (Y ) .
1⊗µ0A,Y
(4)
(T 0 , µ0 )
(A, ϕA )
where Then an arrow from (T, µ) to is an equivalence class (A, ϕA ) = (B, ϕB ) if there exists an arrow in H, u : A → B, such that the following diagram is commutative: T (X) ⊗ T 0 (A) 1⊗T 0 (u)
ϕA X
T (X) ⊗ T 0 (B)
/ T 0 (A ⊗ X)
ϕB X
T 0 (u⊗1)
/ T 0 (B ⊗ X) .
The identity in the object (T, µ) is (I, ϕI ), where ϕIX is the following com-
10
posite of canonical arrows: can
T (X) ⊗ T (I)
ϕIX
/ T (X) ⊗ I
T (I ⊗ X) o
can
can
T (X),
and the composition of two arrows (A, ϕA ) : (T, µ) → (T 0 , µ0 ) and (B, ϕB ) : B⊗A (T 0 , µ0 ) → (T 00 , µ00 ) is (B ⊗ A, ϕB⊗A ), where ϕX is given by the following composition: T (X) ⊗ T 00 (B ⊗ A)
−1 1⊗(ϕB A)
ϕB⊗A X
/ T (X) ⊗ T 0 (A) ⊗ T 00 (B)
T 00 (B ⊗ A ⊗ X) o
ϕB A⊗X
ϕA X ⊗1
T 0 (A ⊗ X) ⊗ T 00 (B).
B⊗A satisfies the coherence condition (4) It is straightforward to see that ϕX and that the composition does not depend on the representatives. Then we have a category that is actually a groupoid, where the inverse of a morphism ∗ (A, ϕA ) is (A∗ , ϕA∗ ), with ϕA X given by the following composition:
/
can
T 0 (X) ⊗ T (A∗ )
T 0 (A∗ ) ⊗ T 0 (A ⊗ X) ⊗ T (A∗ )
−1 1⊗(ϕA ⊗1 X)
T 0 (A∗ ) ⊗ T (X) ⊗ T 0 (A) ⊗ T (A∗ )
can
I ⊗ T 0 (A∗ ) ⊗ T (X) ⊗ T 0 (A) ⊗ T (A∗ ) ⊗ I
∗
ϕA X
can
T 0 (A ⊗ A∗ ) ⊗ T 0 (A∗ ) ⊗ T (X) ⊗ T 0 (A) ⊗ T (A∗ ) ⊗ T 0 (A) ⊗ T 0 (A∗ )
−1 (ϕA ⊗1⊗ϕA A∗ ) A∗ ⊗1
T (A∗ ) ⊗ T 0 (A) ⊗ T 0 (A∗ ) ⊗ T (X) ⊗ T 0 (A) ⊗ T 0 (A ⊗ A∗ ) ⊗ T 0 (A∗ )
can
T (A∗ ) ⊗ T (X)
o
can
T (A∗ ) ⊗ I ⊗ T (X) ⊗ T 0 (A) ⊗ I ⊗ T 0 (A∗ ) .
Moreover, there is a functor ⊗ : Out(H) × Out(H) −→ Out(H) 11
that, on objects, is defined by the composition of equivalences and, for any arrows (A, ϕA ) : (T, µ) → (F, α) and (B, ϕB ) : (T 0 , µ0 ) → (F 0 , α0 ), (B, ϕB ) ⊗ (A, ϕA ) = (F ∗ (B) ⊗ A, ϕF ∗ (B)⊗A ), where F ∗ is a quasi-inverse of F ∗ (B)⊗A F and ϕX is given by the following composition: T 0 T (X) ⊗ F 0 F (F ∗ (B) ⊗ A)
can
/
T 0 T (X) ⊗ F 0 F F ∗ (B) ⊗ F 0 F (A)
can
T 0 T (X) ⊗ F 0 (B) ⊗ F 0 F (A)
ϕB T (X) ⊗1
F 0 (B ⊗ T (X)) ⊗ F 0 F (A)
F ∗ (B)⊗A
can
ϕX
F 0 (F F ∗ (B) ⊗ T (X)) ⊗ F 0 F (A)
can
F 0 (F F ∗ (B) ⊗ T (X) ⊗ F (A))
F 0 F (F ∗ (B) ⊗ A ⊗ X)
o
can
F 0 (1⊗ϕA X)
F 0 (F F ∗ (B) ⊗ F (A ⊗ X)) .
F ∗ (B)⊗A
It is straightforward to see that ϕX satisfies condition (4) and that the definition of ⊗ on arrows does not depend on the representatives. All these data define a categorical group Out(H) where the unit object is I = idH , the constraints of associativity and right and left unit are identities and an inverse for an object (T, µ) is obtained by taking a quasi-inverse of T. Note that there is an action [6] of Out(H) on the braided categorical group Z(H) given by the homomorphism Out(H) → Eq(Z(H)), T 7→ T . The elements of the group π0 (Out(H)) are the isomorphism classes of equivalences of H with product induced by the composition of equivalences, that is, [T][T0 ] = [TT0 ], for any equivalences T = (T, µ), T0 = (T 0 , µ0 ) : H → H. The elements of the abelian group π1 (Out(H)) are the arrows ∂ : idH → idH . Thus, an element ∂ ∈ π1 (Out(H)) is an equivalence class of a pair (A, ϕA ) where A ∈ H and, for any X ∈ H, ϕA X : X ⊗ A → A ⊗ X is a natural −1 · a A −1 isomorphism such that aX,Y,A · (ϕA ) A,X,Y = (1 ⊗ ϕY ) ) · aX,A,Y · X⊗Y −1 ⊗ 1) and l · ϕI = r . Moreover, (A, ϕA ) = (B, ϕB ) if there is an ((ϕA X X X) X
12
B arrow u : A → B such that (u ⊗ 1) · ϕA X = ϕX · (1 ⊗ u) and therefore
π1 (Out(H)) = π0 (Z(H)). For any categorical group H, there is a homomorphism p = (p, µp ) : Eq(H) −→ Out(H) that is the identity on objects and, for any arrow : (T, µ) → (T 0 , µ0 ) in Eq(H), p() is the class of the pair (I, ϕI ) where ϕIX : T (X) ⊗ T 0 (I) → T 0 (I ⊗ X) is given by the following composition T (X) ⊗ T 0 (I)
can
/ T (X) ⊗ I
can
/ T (X)
T 0 (I ⊗ X) o
X
T 0 (X).
can
For any equivalences T, T0 , (µp )T,T0 is given by the class of the pair (I, ϕI ), where ϕIX : T T 0 (X) ⊗ T T 0 (I) → T T 0 (I ⊗ X) is given by the composition can
T T 0 (X) ⊗ T T 0 (I)
/ T T 0 (X ⊗ I)
T T 0 (I ⊗ X) o
can
can
T T 0 (X).
Then, for any categorical group H, we have a sequence of homomorphisms j
p
i
Z(H) −→ H −→ Eq(H) −→ Out(H) and morphisms : ij → 0,
τ : pi → 0
where is determined by the fact that Z(H) is isomorphic to the kernel of i and τ is given by the natural transformation whose component at H ∈ H, τH : iH → idH , is given by the class of the pair (H, ϕH ), where ϕH X : ((H ⊗ X) ⊗ H ∗ ) ⊗ H → H ⊗ X is a composite of canonical morphisms. In [17], the following notion of exactness for homomorphisms of categorF
T
ical groups was introduced. Let G −→ H −→ K be two homomorphisms and : TF → 0 a morphism. From the universal property of the kernel of
13
T, there exists a homomorphism F0 making the following diagram commutative: K(T) F0
y
y
y
FF FF j FF FF " / H.
y
Abstract For any categorical group H, we introduce the categorical group Out(H) and then the well-known group exact sequence 1 → Z(H) → H → Aut(H) → Out(H) → 1 is raised to a categorical group level by using a suitable notion of exactness. Breen’s Schreier theory for extensions of categorical groups is codified in terms of homomorphism to Out(H) and then we develop a sort of Eilenberg-Mac Lane’s obstruction theory that solves the general problem of the classification of all categorical group extensions of a group G by a categorical group H, in terms of ordinary group cohomology. Key words: categorical group, extension, cohomology, abstract kernel, obstruction
MSC : 18D10; 18G50; 20J06
1
Introduction
There is a long history of studying obstruction theory for extensions in several contexts. p If 1 → H → E → G → 1 is a group extension, the epimorphism p induces a homomorphism ρ : G → Out(H)(= Aut(H)/Int(H)). Although not with this terminology, it was observed by Baer [1] that a pair (H, ρ : G → Out(H)) need not always be built from such an extension; the “obstruction” to forming an extension which would induce a given pair (H, ρ) was found by Eilenberg and Mac Lane [12]. They formulated the ∗
Financially supported by DGI of Spain and FEDER, Project: BFM2001-2886
1
general problem of group extensions as that of the construction and classification of all extensions which induce a given pair (H, ρ), which they termed a G-abstract kernel. The center, Z(H), of H is (via ρ) a G-module and the obstruction, to the existence of an extension inducing the G-abstract kernel (H, ρ), is given by an element of the third Eilenberg-Mac Lane cohomology group H 3 (G, Z(H)). Thus, an abstract kernel (H, ρ) is associated with an extension if, and only if, its obstruction vanishes. Moreover, in such a case, the set of equivalence classes of extensions that induce (H, ρ) is a principal and homogeneous space under a canonical action of the abelian group H 2 (G, Z(H)). Using this pattern, Dedecker [10] formalized an obstruction theory for extensions of a group by a crossed module of groups. This general obstruction theory for groups parallels the more classic one for groups, and it was translated to many other contexts, such as associative algebras [25], associative and commutative algebras [2] or Lie algebras [15]. Later, all these obstruction theories were unified by Duskin [11], who developed a low-dimensional obstruction theory in algebraic categories and, in turn, this theory was generalized to higher dimensions in [7]. More recently, graded extensions of categories [8] and graded extensions of monoidal categories [9] have been codified in terms of non-abelian cocycles and corresponding obstruction theories in both contexts have then been developed. In addition, an obstruction theory for split extensions has been given in [13] for categorical groups. Categorical groups are monoidal groupoids where every object is invertible, up to isomorphism, with respect to the tensor product [23, 16, 4] . The 2-category of categorical groups (sometimes equipped with braiding or symmetry) can be seen as the 2-dimensional analogue of the category of (abelian) groups. The analysis of the development of the results extending the classic results on group extensions reveals that the key lies in the consideration of (internal) groupoids instead of groups. In this sense, categorical groups provide an adequate context in which to approach problems of the same nature as the above-mentioned but from a more general point of view, obtaining thus a wider range of applications [26, 5, 6, 22, 27, 14]. After a first section, devoted to setting the notations and recalling several notions and basic results about categorical groups, we dedicate the next section to obtaining a categorical-group version of the well-known exact sequence 1 → Z(H) → H → Aut(H) → Out(H) → 1. To do so, we use the known categorical groups Eq(H) [4] and Z(H) [16] associated to any categorical group H, and we introduce the categorical group, Out(H), whose objects are still equivalences of H. Out(H) could be presented as a kind of 2
quotient of categorical groups and we will develop this topic in a forthcoming paper. Now, using the notion of 2-exactness given in [17], we conclude with the existence of a 2-exact sequence: j
p
i
Z(H) −→ H −→ Eq(H) −→ Out(H) that generalizes the above one for groups. In the last section we begin by recalling Breen’s Schreier theory for extensions of categorical groups. Note that the classification of such extensions is the algebraization of a topological problem consisting of the classification of those fibrations of spaces, with base space a K(G, 1)-space and whose fibre has the homotopy type of a categorical group. After pointing out that, for group extensions of G by H, the classic Schreier theory can be rewritten in terms of homomorphisms from the discrete categorical group G[0] to the categorical group Out(H[0]), we prove the analogous result for Breen’s theory. That is, for any group G and any categorical group H, there is a natural bijection (see Corollary 4.2) ∆ : [G[0], Out(H)] −→ Ext(G, H). Then, any extension H → E → G[0] induces a group homomorphism (that we call a G-abstract kernel of H), ρ : G → π0 (Out(H)), and our aim in this section is to develop an obstruction theory for extensions of categorical groups that induce a given homomorphism ρ. Since π0 (Z(H)) = π1 (Out(H)) is a G-module via ρ, we show that the obstruction is measured by an element of H 3 (G, π0 (Z(H))) and we conclude by formulating the corresponding obstruction theorems: “A G-abstract kernel ρ : G → π0 (Out(H)) is realizable if and only if its obstruction ∂(ρ) ∈ Hρ3 (G, π0 (Z(H))) vanishes.” “If a G-abstract kernel ρ : G → π0 (Out(H)) is realizable, then the set of equivalence classes of realizations of ρ, Extρ (G, H), is a principal homogeneous space under the abelian group Hρ2 (G, π0 (Z(H))). In particular, there is a (non-natural) bijection Extρ (G, H) ∼ = Hρ2 (G, π0 (Z(H))).”
2
Preliminaries
First, we recall [18, 23, 16, 20] that a monoidal category, G = (G, ⊗, a, I, l, r), consists of a category G, a functor ⊗ : G × G → G, an object I of G, called the unit object, and natural isomorphisms a = aX,Y,Z : (X ⊗ Y ) ⊗ Z → X ⊗ (Y ⊗ Z), 3
l = lX : I ⊗ X → X,
r = rX : X ⊗ I → X,
such that the usual coherence conditions are satisfied. In a monoidal category, an object X is said to be invertible if the functors Y 7→ X ⊗ Y and Y 7→ Y ⊗ X are equivalences. A categorical group G is a monoidal small category where every arrow and every object is invertible. In this case, it is possible to define a functor (−)∗ : G → G, X 7→ X ∗ , f 7→ f ∗ , and natural isomorphisms, mX : X ⊗ X ∗ → I and nX : X ∗ ⊗ X → I, such that lX · (mX ⊗ 1X ) = rX · (1X ⊗ nX ) · aX,X ∗ ,X , for all objects X ∈ G. The triple (X ∗ , mX , nX ) is termed an inverse for X. A categorical group G is said to be a braided categorical group if it is also equipped with a family of natural isomorphisms c = cX,Y : X ⊗ Y → Y ⊗ X (the braiding) that interacts with a, r and l in the sense of [16]. A homomorphism of categorical groups T = (T, µ) : G → H consists of a functor T : G → H and a family of natural isomorphisms µ = µX,Y : T (X ⊗Y ) −→ T (X)⊗T (Y ), satisfying the appropriate coherence conditions [16]. If (G, c) and (H, c) are braided categorical groups, compatibility with c is also required [16]. If T : G → H is a homomorphism, there exists an isomorphism [16, 6], µ0 : T (I) −→ I, such that, for any X ∈ G, the equalities T (rX ) = rX ·(1⊗µ0 )· µX,I and T (lX ) = lX · (µ0 ⊗ 1) · µI,X hold. With respect to the inverses there exist unique isomorphisms λX : T (X ∗ ) −→ T (X)∗ , such that the equalities µ0 · T (mX ) = mT (X) · (1 ⊗ λX ) · µX,X ∗ and µ0 · T (nX ) = nT (X) · (λX ⊗ 1) · µX ∗ ,X hold. We will denote by CG the category of categorical groups and by BCG the category of braided categorical groups. Recall that, if G ∈ CG, then the set of connected components of G, π0 (G), has a group structure (which is abelian if G ∈ BCG). Also, the abelian group π1 (G) = AutG (I) is associated to G and it is a π0 (G)-module . The action [26, 13] is given by the map π0 (G) × π1 (G) → π1 (G) ,
−1 ([X], u) 7→ γX δX (u) ,
(1)
where, for any X ∈ G, the maps (in fact group isomorphisms) γX , δX : π1 (G) → AutG (X) are defined, for each u ∈ π1 (G), as the unique arrows making the following
4
diagrams commutative: I ⊗X lX
u⊗idX
X
/I ⊗X
γX (u)
X ⊗I rX
lX
/X
idX ⊗u
X
/X ⊗I
δX (u)
rX
/X
that is, −1 −1 γX (u) = lX · (u ⊗ idX ) · lX , δX (u) = rX · (idX ⊗ u) · δX .
These isomorphisms [26, 13] are compatible with the functor ⊗ and for any two objects X, Y ∈ G and arrows f, g : X → Y there exists a unique element u ∈ π1 (G) such that the following equality is satisfied g = f · γX (u) = γY (u) · f .
(2)
Given homomorphisms of categorical groups (T, µ), (T 0 , µ0 ) : G → H, a morphism from (T, µ) to (T 0 , µ0 ) consists of a natural transformation : T → T 0 that interacts with µ and µ0 in the sense of [16]. An equivalence of a categorical group G is a homomorphism (T, µ) : G → G such that the endofunctor T : G → G is an equivalence of categories. These equivalences are the objects of the categorical group Eq(G) [4], whose arrows are the morphisms between them. The composition in Eq(G) is given by the usual vertical composition of natural transformations. The tensor product is given by the composition of homomorphisms and the horizontal composition of natural transformations. The associativity isomorphism is the identity, the unit object is I = idG , and an inverse for an object (T, µ) is obtained by taking a quasi-inverse T ∗ of T . This categorical group has a categorical subgroup Aut(G) whose objects, called automorphisms, are the equivalences (T, µ) that are strict (i.e., µX,Y and µ0 are identities) and where T is an isomorphism. The kernel of a homomorphism T = (T, µ) : G → H (see [17],[27]) consists of a universal triplet (K(T), j, ), where K(T) is a categorical group, j : K(T) → G is a homomorphism and : Tj → 0 is a morphism, and it is described explicitly in the following way: an object of K(T) is a pair (X, uX ) where X ∈ G and uX : T (X) → I is an arrow in H; an arrow f : (X, uX ) → (Y, uY ) is an arrow f : X → Y in G such that uX = uY · T (f ); the functor j sends f : (X, uX ) → (Y, uY ) to f : X → Y ; the tensor product is given by (X, uX ) ⊗ (Y, uY ) = (X ⊗ Y, uX · uY ), where uX · uY : T (X ⊗ Y ) → I is 5
µX,Y
uX ⊗uY
rI =lI
/ I ; the unit / T (X) ⊗ T (Y ) /I ⊗I the composite T (X ⊗ Y ) object is (I, µ0 ) and the associativity and left-unit and right-unit constraints are given by aX,Y,Z , lX and rX respectively; an inverse for any object (X, uX ) is given by (X ∗ , u∗X λX ); and the component at (X, uX ) of the morphism is given by uX . If G and H are braided categorical groups, then K(T) is also a braided categorical group where the braiding c = c(X,u ),(Y,u ) : X Y (X, uX ) ⊗ (Y, uY ) −→ (Y, uY ) ⊗ (X, uX ) is precisely the braiding in G cX,Y . Note that j : K(T) → G is a strict homomorphism of (braided) categorical groups. The categorical group K(T) just described is a standard homotopy kernel and so it is determined, up to isomorphism, by the following strict universal property: given a homomorphism F : K → G and a morphism τ : TF → 0, there exists a unique homomorphism F0 : K → K(T) such that jF0 = F and F0 = τ . Note that if (K(T), j, ) is the kernel of T = (T, µ) : G → H, the sequence j
T
K(T) −→ G −→ H induces (see [22]) an exact sequence of groups 1
/ π1 (G) π1 (T) / π1 (H) i iiii δiiiiii iiii tiiπi0i(j) / π0 (G) π0 (T) / π0 (H) π0 (K(T))
/ π1 (K(T))
π1 (j)
(3)
where, for any u ∈ π1 (H), δ(u) is the connected component of the object (I, u · µ0 ) ∈ K(T). If the functor T is essentially surjective, the above exact sequence is right exact.
3
The categorical group Out(H) and a 2-exact sequence
For any group H, it is well known that the kernel of the homomorphism i H → Aut(H), given by inner automorphisms, is Z(H), the center of H, and the cokernel is Out(H) = Aut(H)/Int(H), the group of outer automorphisms of H. Thus, there is an exact sequence of groups i
1 → Z(H) → H → Aut(H) → Out(H) → 1. 6
In this section we will obtain a categorical group version of this sequence. To do so, we start by recalling the homomorphism given by conjugation and the notion of center for categorical groups. We will also look at their relationship and some examples. If H is a categorical group, there is a homomorphism i = (i, µi ) : H → Eq(H) given by conjugation; that is, for any X ∈ H, iX : H → H is given by iX (Y ) = (X ⊗ Y ) ⊗ X ∗ , iX (f ) = (1 ⊗ f ) ⊗ 1. It is clear that iX is an equivalence with quasi-inverse iX ∗ . The natural isomorphisms iX (Y ⊗ Z) → iX (Y ) ⊗ iX (Z) are given by the composition of canonical morphisms. For any arrow f : X → Y , if : iX → iY is the morphism whose component at Z ∈ H is (if )Z = (f ⊗ 1) ⊗ f ∗ : (X ⊗ Z) ⊗ X ∗ → (Y ⊗ Z) ⊗ Y ∗ . Finally, for any X, Y ∈ H, (µi )X,Y : iX⊗Y → iX · iY is the morphism in Eq(H) also determined by the composition of canonical morphisms. The center, Z(H), of a categorical group H (see [16]) is the braided categorical group whose objects are pairs (X, v) where X ∈ H and v : X ⊗ − → − ⊗ X is a natural isomorphism such that the following two conditions hold:
lX · vI = rx ; aY,Z,X · vY ⊗Z · aX,Y,Z = (1 ⊗ vZ ) · aY,X,Z · (vY ⊗ 1). An arrow f : (X, v) → (X 0 , v 0 ) in Z(H) is an arrow f : X → X 0 in H such 0 · (f ⊗ 1) = (1 ⊗ f ) · v . The tensor product and the that, for all X ∈ H, vX X braiding are given by (X, v) ⊗ (X 0 , v 0 ) = (X ⊗ X 0 , (v ⊗ 1) · (1 ⊗ v 0 )) , c(X,v),(X 0 ,v0 ) = vX 0 . The kernel of the conjugation homomorphism i = (i, µi ) : H → Eq(H) is isomorphic to Z(H) where the isomorphism F : K(i) → Z(H) is given uX v by F (X, iX −→ I) = (X, X ⊗ − −→ − ⊗ X) where vY is the following composition: can / can / (X ⊗ Y ) ⊗ I (X ⊗ Y ) ⊗ (X ∗ ⊗ X) X ⊗ YN NN NN can vY N& (uX )Y ⊗1 ((X ⊗ Y ) ⊗ X ∗ ) ⊗ X . Y ⊗X o
7
The inverse is given by F −1 (X, v) = (X, uX ), where (uX )Y : (X ⊗ Y ) ⊗ X ∗ → Y is the following composition: vY ⊗1
/ (Y ⊗ X) ⊗ X ∗ can / Y ⊗ (X ⊗ X ∗ ) R R R R can R R R R (uX )Y can ) o
(X ⊗ Y ) ⊗ X ∗
Y ⊗I .
Y
Therefore, we have a sequence j
i
Z(H) −→ H −→ Eq(H) , that, according to (3), has associated an exact sequence of groups 1
/ π1 (H)
/ Der(π0 (H), π1 (H)) gg gggg gggg g g g gggg sgggg / π0 (H) / π0 (Eq(H)) π0 (Z(H))
/ π (H)π0 (H) 1
since it is easy to see that, on the one hand, π1 (Z(H)) = π1 (H)π0 (H) , the group of invariant elements in π( H), and, on the other hand, π1 (Eq(H)) = Der(π0 (H), π1 (H)), the group of derivations from π0 (H) into π1 (H). When H = H[0] is the strict discrete categorical group associated to a group H, the above sequence specializes to the exact sequence i
1 → Z(H) → H → Aut(H). δ
More generally, let L = (H → G) be a crossed module of groups and let H(L) be the strict categorical group associated to it (see [20]). It has as objects the elements of the group G and an arrow h : x → y is an element h ∈ H with x = δ(h)y. The composition is multiplication in H and the tensor product is given by h
h0
hy h0
(x −→ y) ⊗ (x0 −→ y 0 ) = (xx0 −→ yy 0 ) . Then we can consider Z(L) = Z(H(L)), which is a strict braided categorical group, explicitly described as follows: the objects are pairs (x, d) where x ∈ G and d : G → H is a derivation such that, for any y ∈ G, δd(y) = [x, y]; an arrow from (x, d) to (x0 , d0 ) is an element u ∈ H such that x = δ(u)x0 and, for any y ∈ G, d(y)(y u) = ud0 (y); the tensor product is given by (x, d) ⊗ (x0 , d0 ) = (xx0 , dd0 ) where (dd0 )(y) = x d0 (y)d(y). 8
The homomorphism i = (i, µi ) : H(L) → Eq(H(L)) factors through Aut(H(L)) and so Z(L) is the kernel of the homomorphism i = (i, µi ) : H(L) → Aut(H(L)). Through the equivalence between crossed modules of groups and strict categorical groups, Aut(H(L)) corresponds [4] to the crossed module Act(L), called the actor of L by Norrie [21]. There is a morphism of crossed modules L → Act(L) whose kernel, ZN (L), called the cenδres
/ StG (H) ∩ Z(G) ). ter of L by Norrie, is the crossed module ZN (L) = ( H G The associated categorical group H(ZN (L)) is the fiber in the unit object of the homomorphism i = (i, µi ) : H(L) → Aut(H(L)) and there is a homomorphism H(ZN (L)) → Z(L), x 7→ (x, d0 ), where d0 : G → H is such that d0 (x) = 1 for any x ∈ G. This homomorphism induces an injective group morphism π0 (H(ZN (L))) −→ π0 (Z(L)). On the other hand, the canonical homomorphism Z(L) → H(L), (x, d) 7→ x, induces the group morphism π0 (Z(L)) → π0 (H(L)) = Coker(δ) whose image is StCoker(δ) (Ker(δ)) ∩ Z(Coker(δ)) and whose kernel is the group of derivations module inner derivations Der(Coker(δ), Ker(δ))/IDer(Coker(δ), Ker(δ)). The group morphism π0 (H(ZN (L))) → StCoker(δ) (Ker(δ)) ∩ Z(Coker(δ)), x ¯ 7→ x ¯, is then the composite π0 (H(ZN (L))) π0 (Z(L)) StCoker(δ) (Ker(δ))∩Z(Coker(δ)). In the particular case of considering, for any G-module A, the crossed 0 module L = (A → G), then π0 (H(ZN (L))) = StG (A)∩Z(G) and π0 (Z(L)) ∼ = Der(G,A) π0 (H(ZN (L)))⊕ IDer(G,A) and sequence (3) specializes to the exact sequence
0
/A
/ Der(G, A) , ggg g g g gg gggg gggg g g g sg Der(G,A) /G / Aut(A, G, 0) (StG (A) ∩ Z(G)) ⊕ IDer(G,A) / AG
which is nothing more than the sequence obtained by gluing the exact sequences 0 → AG → A → IDer(A, G) → 0, 0 → IDer(G, A) → Der(G, A) → Der(G,A) IDer(G,A) → 0 and 0 → StG (A) ∩ Z(G) → G → Aut(A, G, 0) where the last morphism is given by x 7→ (a 7→ x a, y 7→ xyx−1 ). i If H is a group and L = (H → Aut(H)) is the crossed module defined by the homomorphism i given by conjugation, then Out(H) = Coker(i) = π0 (H(L)). In the same spirit, below we introduce the categorical group Out(H) associated to any categorical group H. The homomorphism i = (i, µi ) : H → Eq(H) given by conjugation defines a crossed module of categorical groups, a notion introduced by Breen in [4]. This crossed module has associated [22] a monoidal bicategory Bieq(H) 9
and then we define Out(H) = Cl(Bieq(H)) where Cl : Bicat → Cat is the classifying functor [3] which assigns to each bicategory A the category Cl(A) whose objects are those of A and whose arrows are 2-isomorphisms classes of 1-arrows of A. Let us note that if A is a monoidal bicategory, then Cl(A) is a monoidal category and the restriction of Cl to CG, seeing a categorical group as a monoidal bicategory with only one object, is just the functor π0 : CG → G. Also, note that if H is a group then Bieq(H[0]) is the monoidal bicategory, where all 2-arrows are identities, defined by the categorical group H(L) i associated to L = (H → Aut(H)). Thus, Out(H[0]) = Cl(H(L)) = H(L) and so π0 (Out(H[0])) = Out(H), whereas π1 (Out(H[0])) = Z(H). Next we explicitly describe the categorical group Out(H). The objects are the equivalences of H. Given equivalences (T, µ) to (T 0 , µ0 ), we consider 0 0 pairs (A, ϕA ) where A ∈ H and ϕA = ϕA X : T (X) ⊗ T (A) → T (A ⊗ X), X ∈ H, is a family of natural isomorphisms in H such that, for any X, Y ∈ H, the following diagram (where we have omitted the canonical morphisms of associativity) is commutative: T (X ⊗ Y ) ⊗ T 0 (A)
ϕA X⊗Y
mmm mmm m m mm v mm m
/
T 0 (A ⊗ X ⊗ Y )
RRR 0 RRRµA⊗X,Y RRR RRR RR) T 0 (A ⊗ X) ⊗ T 0 (Y ) ll5 lll l l l lll A lll ϕX ⊗1
µX,Y ⊗1
T (X) ⊗ T (Y ) ⊗ T 0 (A)
QQQ QQQ QQQ QQQ 1⊗ϕA Q( Y
/
T (X) ⊗ T 0 (A ⊗ Y )
T (X) ⊗ T 0 (A) ⊗ T 0 (Y ) .
1⊗µ0A,Y
(4)
(T 0 , µ0 )
(A, ϕA )
where Then an arrow from (T, µ) to is an equivalence class (A, ϕA ) = (B, ϕB ) if there exists an arrow in H, u : A → B, such that the following diagram is commutative: T (X) ⊗ T 0 (A) 1⊗T 0 (u)
ϕA X
T (X) ⊗ T 0 (B)
/ T 0 (A ⊗ X)
ϕB X
T 0 (u⊗1)
/ T 0 (B ⊗ X) .
The identity in the object (T, µ) is (I, ϕI ), where ϕIX is the following com-
10
posite of canonical arrows: can
T (X) ⊗ T (I)
ϕIX
/ T (X) ⊗ I
T (I ⊗ X) o
can
can
T (X),
and the composition of two arrows (A, ϕA ) : (T, µ) → (T 0 , µ0 ) and (B, ϕB ) : B⊗A (T 0 , µ0 ) → (T 00 , µ00 ) is (B ⊗ A, ϕB⊗A ), where ϕX is given by the following composition: T (X) ⊗ T 00 (B ⊗ A)
−1 1⊗(ϕB A)
ϕB⊗A X
/ T (X) ⊗ T 0 (A) ⊗ T 00 (B)
T 00 (B ⊗ A ⊗ X) o
ϕB A⊗X
ϕA X ⊗1
T 0 (A ⊗ X) ⊗ T 00 (B).
B⊗A satisfies the coherence condition (4) It is straightforward to see that ϕX and that the composition does not depend on the representatives. Then we have a category that is actually a groupoid, where the inverse of a morphism ∗ (A, ϕA ) is (A∗ , ϕA∗ ), with ϕA X given by the following composition:
/
can
T 0 (X) ⊗ T (A∗ )
T 0 (A∗ ) ⊗ T 0 (A ⊗ X) ⊗ T (A∗ )
−1 1⊗(ϕA ⊗1 X)
T 0 (A∗ ) ⊗ T (X) ⊗ T 0 (A) ⊗ T (A∗ )
can
I ⊗ T 0 (A∗ ) ⊗ T (X) ⊗ T 0 (A) ⊗ T (A∗ ) ⊗ I
∗
ϕA X
can
T 0 (A ⊗ A∗ ) ⊗ T 0 (A∗ ) ⊗ T (X) ⊗ T 0 (A) ⊗ T (A∗ ) ⊗ T 0 (A) ⊗ T 0 (A∗ )
−1 (ϕA ⊗1⊗ϕA A∗ ) A∗ ⊗1
T (A∗ ) ⊗ T 0 (A) ⊗ T 0 (A∗ ) ⊗ T (X) ⊗ T 0 (A) ⊗ T 0 (A ⊗ A∗ ) ⊗ T 0 (A∗ )
can
T (A∗ ) ⊗ T (X)
o
can
T (A∗ ) ⊗ I ⊗ T (X) ⊗ T 0 (A) ⊗ I ⊗ T 0 (A∗ ) .
Moreover, there is a functor ⊗ : Out(H) × Out(H) −→ Out(H) 11
that, on objects, is defined by the composition of equivalences and, for any arrows (A, ϕA ) : (T, µ) → (F, α) and (B, ϕB ) : (T 0 , µ0 ) → (F 0 , α0 ), (B, ϕB ) ⊗ (A, ϕA ) = (F ∗ (B) ⊗ A, ϕF ∗ (B)⊗A ), where F ∗ is a quasi-inverse of F ∗ (B)⊗A F and ϕX is given by the following composition: T 0 T (X) ⊗ F 0 F (F ∗ (B) ⊗ A)
can
/
T 0 T (X) ⊗ F 0 F F ∗ (B) ⊗ F 0 F (A)
can
T 0 T (X) ⊗ F 0 (B) ⊗ F 0 F (A)
ϕB T (X) ⊗1
F 0 (B ⊗ T (X)) ⊗ F 0 F (A)
F ∗ (B)⊗A
can
ϕX
F 0 (F F ∗ (B) ⊗ T (X)) ⊗ F 0 F (A)
can
F 0 (F F ∗ (B) ⊗ T (X) ⊗ F (A))
F 0 F (F ∗ (B) ⊗ A ⊗ X)
o
can
F 0 (1⊗ϕA X)
F 0 (F F ∗ (B) ⊗ F (A ⊗ X)) .
F ∗ (B)⊗A
It is straightforward to see that ϕX satisfies condition (4) and that the definition of ⊗ on arrows does not depend on the representatives. All these data define a categorical group Out(H) where the unit object is I = idH , the constraints of associativity and right and left unit are identities and an inverse for an object (T, µ) is obtained by taking a quasi-inverse of T. Note that there is an action [6] of Out(H) on the braided categorical group Z(H) given by the homomorphism Out(H) → Eq(Z(H)), T 7→ T . The elements of the group π0 (Out(H)) are the isomorphism classes of equivalences of H with product induced by the composition of equivalences, that is, [T][T0 ] = [TT0 ], for any equivalences T = (T, µ), T0 = (T 0 , µ0 ) : H → H. The elements of the abelian group π1 (Out(H)) are the arrows ∂ : idH → idH . Thus, an element ∂ ∈ π1 (Out(H)) is an equivalence class of a pair (A, ϕA ) where A ∈ H and, for any X ∈ H, ϕA X : X ⊗ A → A ⊗ X is a natural −1 · a A −1 isomorphism such that aX,Y,A · (ϕA ) A,X,Y = (1 ⊗ ϕY ) ) · aX,A,Y · X⊗Y −1 ⊗ 1) and l · ϕI = r . Moreover, (A, ϕA ) = (B, ϕB ) if there is an ((ϕA X X X) X
12
B arrow u : A → B such that (u ⊗ 1) · ϕA X = ϕX · (1 ⊗ u) and therefore
π1 (Out(H)) = π0 (Z(H)). For any categorical group H, there is a homomorphism p = (p, µp ) : Eq(H) −→ Out(H) that is the identity on objects and, for any arrow : (T, µ) → (T 0 , µ0 ) in Eq(H), p() is the class of the pair (I, ϕI ) where ϕIX : T (X) ⊗ T 0 (I) → T 0 (I ⊗ X) is given by the following composition T (X) ⊗ T 0 (I)
can
/ T (X) ⊗ I
can
/ T (X)
T 0 (I ⊗ X) o
X
T 0 (X).
can
For any equivalences T, T0 , (µp )T,T0 is given by the class of the pair (I, ϕI ), where ϕIX : T T 0 (X) ⊗ T T 0 (I) → T T 0 (I ⊗ X) is given by the composition can
T T 0 (X) ⊗ T T 0 (I)
/ T T 0 (X ⊗ I)
T T 0 (I ⊗ X) o
can
can
T T 0 (X).
Then, for any categorical group H, we have a sequence of homomorphisms j
p
i
Z(H) −→ H −→ Eq(H) −→ Out(H) and morphisms : ij → 0,
τ : pi → 0
where is determined by the fact that Z(H) is isomorphic to the kernel of i and τ is given by the natural transformation whose component at H ∈ H, τH : iH → idH , is given by the class of the pair (H, ϕH ), where ϕH X : ((H ⊗ X) ⊗ H ∗ ) ⊗ H → H ⊗ X is a composite of canonical morphisms. In [17], the following notion of exactness for homomorphisms of categorF
T
ical groups was introduced. Let G −→ H −→ K be two homomorphisms and : TF → 0 a morphism. From the universal property of the kernel of
13
T, there exists a homomorphism F0 making the following diagram commutative: K(T) F0
y
y
y
FF FF j FF FF " / H.
y
