Theory and Applications of Categories, Vol. 30, No. 27, 2015, pp. 933–984.

ON THE SECOND COHOMOLOGY CATEGORICAL GROUP AND A HOCHSCHILD-SERRE 2-EXACT SEQUENCE

A.R. GARZON AND E.M. VITALE

Résumé. We introduce the second cohomology categorical group of a categorical group 𝔾 with coefficients in a symmetric 𝔾-categorical group and we show that it classifies extensions of 𝔾 with symmetric kernel and a functorial section. Moreover, from an essentially surjective homomorphism of categorical groups we get 2-exact sequences à la Hochschild-Serre connecting the categorical groups of derivations and the first and the second cohomology categorical groups.

Contents

1 Introduction 933 2 The group case revisited 936 3 A reminder on categorical groups 941 4 The second cohomology categorical group 947 5 The Hochschild-Serre sequences for categorical groups 951 6 The middle point of the Hochschild-Serre 2-exact sequences 955 7 Categorical crossed modules 962 8 Extensions with symmetric kernel 969 9 Classification of extensions with symmetric kernel 974

1. Introduction

Let E → G be a surjective homomorphism of groups with kernel N, and let A be a G-module. In [21], Hochschild and Serre obtained a 5-term exact sequence involving the group of derivations and the second cohomology group of E and G with coefficients in A

\xymatrix{0 → Der(G, A) \ar[r] & Der(E, A) \ar[r] & Hom_{G}(N_{ab}, A) \ar[r] & H^{2}(G, A) \ar[r] & H^{2}(E, A)}

where the middle point Hom_{G}(N_{ab}, A) is the group of G-module homomorphisms from the abelianization of N to A. The Hochschild-Serre sequence can also be modified to get

Financial support from MCI of Spain (Project: MTM2011-22554) and Consejería de Innovación de J. de Andalucía (FQM-168) is gratefully acknowledged

Received by the editors 2014-04-15 and, in revised form, 2015-06-27. Transmitted by Ieke Moerdijk. Published on 2015-07-07. 2010 Mathematics Subject Classification: 18D10, 18B40, 18D35, 18G50, 20J06. Key words and phrases: (symmetric) categorical group, 2-exactness, cohomology categorical group,

extension.

\begin{matrix}c \\ ◯ \end{matrix} A.R. Garzon and E.M. Vitale, 2015. Permission to copy for private use granted.

a 5-term exact sequence involving the first and the second cohomology group

\xymatrix{0 → H^{1}(G, A) \ar[r] & H^{1}(E, A) \ar[r] & Hom_{G}(N_{ab}, A) \ar[r] & H^{2}(G, A) \ar[r] & H^{2}(E, A)}

(These sequences are indeed part of long exact sequences, see [27, 2, 20, 22, 26, 14].) In the present paper, our first aim is to give a 2-dimensional version of these exact sequences, replacing groups with categorical groups.

In order to construct cohomology categorical groups, a first step has been done by Ulbrich in [32] (see also [31, 13]) : if G is a group, [G]_{0} is the discrete categorical group associated with it and 𝔸 is a symmetric [G]_{0}-categorical group (a [G]_{0}-module in the terminology of [32]), Ulbrich defined cohomology groups H^{n}(G, 𝔸) by considering the cocomplex of symmetric categorical groups ℂ(G, 𝔸) that, in dimension n, consists of functors from G^{n} to 𝔸, where G^{n} = G × ... × G is seen as a discrete category, and ∂ is obtained by taking alternating sums (i.e., tensor) of the homomorphisms d_{i}: ℂ^{n}(G, 𝔸) → ℂ^{n+1}(G, 𝔸), i = 0, ..., n + 1 defined, for all x_{1}, ..., x_{n} ∈ G and F ∈ ℂ^{n}(G, 𝔸), by

- d_{0}(F)(x_{1}, ..., x_{n+1}) = ^{x_{1}}F (x_{2}, ..., x_{n+1}),

- d_{i}(F)(x_{1}, ..., x_{n+1}) = F (x_{1}, ..., x_{i}x_{i+1}, ..., x_{n+1}), 1 ≤ i ≤ n, and

- d_{n+1}(F)(x_{1}, ..., x_{n+1}) = F (x_{1}, ..., x_{n}). This process can be generalized replacing the group G by any categorical group 𝔾 and using a corresponding cocomplex of symmetric categorical groups ℂ(𝔾, 𝔸). Moreover, instead of cohomology groups, using relative kernels and relative cokernels we can construct from ℂ(𝔾, 𝔸) all the cohomology categorical groups ℍ^{n}(𝔾, 𝔸) as explained in [15]. What we do in this paper is to give an explicit description in terms of cobords and factor sets of the first and the second cohomology categorical groups (and of the categorical group of derivations), that is, those categorical groups entering in the Hochschild-Serre sequences.

Since we dispose of an explicit description of the second cohomology categorical group, it is natural to look at what kind of extensions this categorical group classifies. This leads us to revisit categorical crossed modules (in the sense of [12]) and finally to a classification of extensions with symmetric kernel and functorial section which generalizes the classification obtained in [5] for extensions of symmetric categorical groups.

More in detail, the layout of this paper is as follows : in Section 2 we revisit the case of groups and we give a proof of the Hochschild-Serre exact sequence slightly different from the one available in the literature (see for instance [20]) and more convenient to prepare our 2-dimensional version. In particular, we separate the construction of the sequence, which is a special case of the Kernel-Cokernel Lemma and could be obtained even if the group homomorphism E → G is not surjective (see [34]), from the interpretation of the middle point of the sequence, which definitely depends on the fact that the homomorphism is surjective. Section 3 is a reminder on categorical groups. We fix notation and we recall convenient versions for categorical groups of actions, derivations, kernel and cokernel, and exactness. In Section 4 we discuss cobords and factor sets of a categorical group 𝔾 with coefficients in a symmetric 𝔾-categorical group 𝔸, and define the cohomology categorical group ℍ^{2}(𝔾, 𝔸, φ). Specializing in different ways 𝔾 and 𝔸 and taking the group π_{0}(ℍ^{2}(𝔾, 𝔸, φ)) of connected components, we recover Eilenberg-Mac Lane and Ulbrich cohomology groups. Section 5 is devoted to the construction of two Hochschild-Serre exact sequences of categorical groups from an essentially surjective homomorphism of categorical groups 𝔼 → 𝔾 and a symmetric 𝔾-categorical group 𝔸. In Section 6 we concentrate on an interpretation of the middle point of the Hochschild-Serre exact sequences similar to the one established in the group case. In order to introduce extensions with a symmetric kernel, in Section 7 we discuss categorical crossed modules, a notion introduced in [12] (see also [9]). The reason why we need categorical crossed modules deserves an explication : in the case of groups, an extension can be presented as a surjective homomorphism together with its kernel or, equivalently, as a normal subgroup together with the corresponding quotient group. Even for categorical groups the two approaches are equivalent ; this is due to some general results on categorical crossed modules established in [12] and summarized in the present paper as Proposition 7.9. Having this fact in mind, in Section 8 we define an extension as a normal sub-categorical group together with the corresponding quotient categorical group. The advantage with this definition is that, when the normal sub-categorical group is symmetric, it is easy to construct an action of the quotient on the kernel, much more easy than to do the same if one defines an extension as an essentially surjective homomorphism with a symmetric kernel. The price to pay is that (contrarily to what happens with groups) to be a normal sub-categorical group is a structure, and not a property, and the only reasonable way to handle such a structure is to see it as a special case of the general structure of categorical crossed module. Finally, Section 9 is devoted to the cohomological classification of extensions with symmetric kernel. We give only the constructive part of the proof and we omit the long diagrammatic arguments needed to check that the various constructions fulfill the requested coherence conditions.

1.1. Remark. The cohomology categorical group ℍ^{2}(𝔾, 𝔸, φ) introduced in Section 4 is one of two possible variants. In order to understand this point, let us look at the simpler case where the action φ is trivial, and let us start with the classical case of groups. For a group G and an abelian group A, to construct the cohomology groups H^{n}(G, A) one can start considering the chain cocomplex of maps (in any dimension) from the simplicial set K(G, 1) (the Grothendieck’s nerve of G seen as a category with only one object) to A, with differentials given by alternating sums of the induced cofaces. The same process can be carried out for a categorical group 𝔾 and a symmetric categorical group 𝔸. What changes in the case of categorical groups is that at the beginning of the story two different choices are possible, and these different choices lead to different cohomology categorical groups. Indeed, one can consider as starting point either Ner(𝔾), the simplicial set of a categorical group 𝔾 (see [8]), or the pseudo-simplicial category that 𝔾 defines by the familiar bar construction, where face and degeneracy functors are defined in a standard way and where simplicial identities hold up to isomorphism (see [7, 10] and the references therein). When the action φ is trivial, our cohomology categorical group ℍ^{2}(𝔾, 𝔸, φ) is the one coming from this pseudo-simplicial nerve. The Ner(𝔾) version is related to singular

936  A.R. GARZON AND E.M. VITALE

extensions of categorical groups ; it deserves a separate study which will not be developed here. Acknowledgments : We thank the referee for pointing out a mistake in the statement of Corollary 6.5. We also thank him for some valuable suggestions involving the homotopy classification of categorical groups established in [30], suggestions which we hope to develop in further works.

2. The group case revisited

Let G be a group and A = (A, φ) an abelian group together with a G-module structure φ: G × A → A. We write φ(x, a) = ^{x}a. The starting point to construct the Hochschild-Serre sequence and the H^{1}-H^{2} sequence is described by the following commutative diagram of abelian groups and homomorphisms

\xymatrix{A \ar[r]^{d_{G}} & Der(G, A) \ar[r]^{i_{G}} \ar[d]_{π_{G}} & C^{1}(G, A) \ar[d]_{_{\underline{π}_{G}}} \ar[r]^{δ_{G}} & Z^{2}(G, A) \ar[r]^{q_{G}} \ar[dr]_{_{\underline{q}_{G}}} & H^{2}(G, A) \ar[d]^{s_{G}} \\  & H^{1}(G, A) \ar[r]_{_{\underline{i}_{G}}} & \underline{C}^{1}(G, A) \ar@{-->}[ur]_{_{\underline{δ}_{G}}} &  & \underline{H}^{2}(G, A)}

where - C^{1}(G, A) is the group of 1-cochains, i.e. maps g: G → A such that g(1) = 0;

- Z^{2}(G, A) is the group of 2-cocycles, i.e. maps f: G × G → A such that

f(x, 1) = 0 = f(1, y) , ^{x}f(y, z) + f(x, yz) = f(x, y) + f(xy, z)

- δ_{G}: C^{1}(G, A) → Z^{2}(G, A) is defined by δ_{G}(g)(x, y) = ^{x}g(y) − g(xy) + g(x);

- Der(G, A) and H^{2}(G, A) are, respectively, the kernel and the cokernel of δ_{G} (the element of Der(G, A) are called derivations) ;

- d_{G}: A → Der(G, A) is defined by the inner derivation d_{G}(a)(x) = ^{x}a − a;

- H^{1}(G, A) is the cokernel of d_{G}, and \underline{C}^{1}(G, A) is the cokernel of i_{G} · d_{G};

- \underline{i}_{G} and \underline{δ}_{G} are induced by the universal properties of H^{1}(G, A) and \underline{C}^{1}(G, A);

- \underline{H}^{2}(G, A) is the cokernel of \underline{δ}_{G};

- s_{G} is induced by the universal property of H^{2}(G, A). 2.1. Lemma. In the previous diagram

1. \underline{i}_{G}: H^{1}(G, A) → \underline{C}^{1}(G, A) is the kernel of \underline{δ}_{G}: \underline{C}^{1}(G, A) → Z^{2}(G, A);

2. s_{G}: H^{2}(G, A) → \underline{H}^{2}(G, A) is an isomorphism.

Proof. Apply the Kernel-Cokernel Lemma to the diagram

\xymatrix{A \ar[r]^{i_{G}·d_{G}} \ar[d]_{!} & C^{1}(G, A) \ar[d]_{δ_{G}} \ar[r]^{^{\underline{π}^{G}}} & \underline{C}^{1}(G, A) \ar[d]^{_{\underline{δ}_{G}}} \\ 0 \ar[r]_{!} & Z^{2}(G, A) \ar[r]_{id} & Z^{2}(G, A)}

In the resulting exact sequence

\xymatrix{A \ar[r]^{d_{G}} & Der(G, A) \ar[r] & Ker(\underline{δ}_{G}) \ar[r] & 0 \ar[r] & H^{2}(G, A) \ar[r]^{s_{G}} & \underline{H}^{2}(G, A) \ar[r] & 0}

the exactness in Der(G, A) and in Ker(\underline{δ}_{G}) gives that H^{1}(G, A) ≃ Ker(\underline{δ}_{G}), and the exactness in H^{2}(G, A) and in \underline{H}^{2}(G, A) gives that s_{G} is an isomorphism.

Consider now an epimorphism of groups p: E → G and the induced E-module structure E × A → G × A → A on A. Composition with p: E → G induces also a monomorphism p_{1} : C^{1}(G, A) → C^{1}(E, A), and composition with p × p: E × E → G × G induces a monomorphism p_{2} : Z^{2}(G, A) → Z^{2}(E, A). We have all the ingredients to construct the following commutative diagram of abelian groups and homomorphisms

\xymatrix{A \ar[d]_{d_{G}} \ar[rr]^{id} &  & A \ar[d]_{d_{E}} \\ Der(G, A) \ar[dd]_{i_{G}} \ar[dr]^{π_{G}} \ar[rr]^{a} &  & Der(E, A) \ar[dd]_{i_{E}} \ar[dr]^{π_{E}} \ar[rr]^{b} &  & Ker(δ_{p}) \ar[dd]_{i_{p}} \ar[dr]^{π_{p}} \\  & H^{1}(G, A) \ar[dd]_{_{\underline{i}_{G}}} \ar[rr]^{\underline{a}} &  & H^{1}(E, A) \ar[dd]_{_{\underline{i}_{E}}} \ar[rr]^{\underline{b}} &  & Ker(\underline{δ}_{p}) \ar[dd]^{_{\underline{i}_{p}}} \\ C^{1}(G, A) \ar[dd]_{δ_{G}} \ar@{-->}[dr]^{_{\underline{π}_{G}}} \ar[rr]^{p_{1}} &  & C^{1}(E, A) \ar[dd]_{δ_{E}} \ar@{-->}[dr]^{_{\underline{π}_{E}}} \ar[rr]^{q_{1}} &  & C^{1}(p, A) \ar[dd]_{δ_{p}} \ar@{-->}[dr]^{_{\underline{π}_{p}}} \\  & \underline{C}^{1}(G, A) \ar[dl]_{_{\underline{δ}_{G}}} \ar[rr]^{_{\underline{p}_{1}}} &  & \underline{C}^{1}(E, A) \ar[dl]_{_{\underline{δ}_{E}}} \ar[rr]^{_{\underline{q}_{1}}} &  & \underline{C}^{1}(p, A) \ar[dl]^{_{\underline{δ}_{p}}} \\ Z^{2}(G, A) \ar[dd]_{q_{G}} \ar[dr]^{_{\underline{q}_{G}}} \ar[rr]^{p_{2}} &  & Z^{2}(E, A) \ar[dd]_{q_{E}} \ar[dr]^{_{\underline{q}_{E}}} \ar[rr]^{q_{2}} &  & Z^{2}(p, A) \ar[dd]_{q_{p}} \ar[dr]^{_{\underline{q}_{p}}} \\  & \underline{H}^{2}(G, A) \ar[rr]^{\underline{c}} &  & \underline{H}^{2}(E, A) \ar[rr]^{\underline{d}} &  & Coker(\underline{δ}_{p}) \\ H^{2}(G, A) \ar[rr]_{c} \ar[ur]^{s_{G}} &  & H^{2}(E, A) \ar[rr]_{d} \ar[ur]^{s_{E}} &  & Coker(δ_{p}) \ar[ur]_{s_{p}}}

where - \underline{p}_{1} : \underline{C}^{1}(G, A) → \underline{C}^{1}(E, A) is induced by the universal property of \underline{C}^{1}(G, A);

- C^{1}(p, A) is the cokernel of p_{1}, \underline{C}^{1}(p, A) is the cokernel of \underline{p}_{1}, and Z^{2}(p, A) is the cokernel of p_{2};

- δ_{p} and \underline{δ}_{p} are induced by the universal properties of C^{1}(p, A) and \underline{C}^{1}(p, A), π_{p} and \underline{π}_{p} are induced by the universal properties of Ker(\underline{δ}_{p}) and C^{1}(p, A), and s_{p} is induced by the universal property of Coker(δ_{p}).

2.2. Lemma. In the previous diagram 1. \underline{p}_{1} : \underline{C}^{1}(G, A) → \underline{C}^{1}(E, A) is a monomorphism ;

2. \underline{π}_{p}: C^{1}(p, A) → \underline{C}^{1}(p, A) is an isomorphism and, therefore,

3. π_{p} and s_{p} are isomorphisms.

Proof. Consider

\xymatrix{A \ar[dd]_{id} \ar[dr]_{q^{G}} \ar[rr]^{i_{G}·d_{G}} &  & C^{1}(G, A) \ar[dd]^{p_{1}} \ar[r]^{^{\underline{π}^{G}}} & \underline{C}^{1}(G, A) \ar[dd]^{_{\underline{p}_{1}}} \\  & Ker(\underline{π}_{G}) \ar@{-->}[ur] \ar[dd]^{h} \\ A \ar[dr]_{q^{E}} \ar[rr]^{i_{E}·d_{E}} &  & C^{1}(E, A) \ar[r]_{_{\underline{π}_{E}}} & \underline{C}^{1}(E, A) \\  & Ker(\underline{π}_{E}) \ar@{-->}[ur]}

Observe that h is an epimorphism because q^{E} is an epimorphism. Now apply the Kernel-Cokernel Lemma to the diagram

\xymatrix{Ker(\underline{π}_{G}) \ar[d]_{h} \ar[r] & C^{1}(G, A) \ar[r]^{^{\underline{π}^{G}}} \ar[d]_{p_{1}} & \underline{C}^{1}(G, A) \ar[d]^{_{\underline{p}_{1}}} \\ Ker(\underline{π}_{E}) \ar[r] & C^{1}(E, A) \ar[r]_{_{\underline{π}_{E}}} & \underline{C}^{1}(E, A)}

The resulting exact sequence

\xymatrix{0 \ar[r] & Ker(h) \ar[r] & 0 \ar[r] & Ker(\underline{p}_{1}) \ar[r] & 0 \ar[r] & C^{1}(p, A) \ar[r]^{^{\underline{π}^{p}}} & \underline{C}^{1}(p, A) \ar[r] & 0}

gives that Ker(\underline{p}_{1}) = 0, so that \underline{p}_{1} is a monomorphism, and that \underline{π}_{p} is an isomorphism.

2.3. Corollary. Consider an epimorphism of groups p: E → G and a G-module A. There are two exact sequences of abelian groups connected by a morphism of complexes

\xymatrix{0 → Der(G, A) \ar[d]_{π_{G}} \ar[r]^{a} & Der(E, A) \ar[d]_{π_{E}} \ar[r]^{b} & Ker(δ_{p}) \ar[d]_{π_{p}} \ar[r]^{Δ} & H^{2}(G, A) \ar[d]^{s_{G}} \ar[r]^{c} & H^{2}(E, A) \ar[d]^{s_{E}} \ar[r]^{d} & Coker(δ_{p}) → 0 \ar[d]^{s_{p}} \\ 0 → H^{1}(G, A) \ar[r]_{\underline{a}} & H^{1}(E, A) \ar[r]_{\underline{b}} & Ker(\underline{δ}_{p}) \ar[r]_{\underline{Δ}} & \underline{H}^{2}(G, A) \ar[r]_{\underline{c}} & \underline{H}^{2}(E, A) \ar[r]_{\underline{d}} & Coker(\underline{δ}_{p}) → 0}

Moreover, π_{G} and π_{E} are epimorphisms, and π_{p}, s_{G}, s_{E} and s_{p} are isomorphisms.

Proof. Apply the Kernel-Cokernel Lemma to the diagrams

\xymatrix{C^{1}(G, A) \ar[d]_{δ_{G}} \ar[r]^{p_{1}} & C^{1}(E, A) \ar[d]^{δ_{E}} \ar[r]^{q_{1}} & C^{1}(p, A) \ar[d]^{δ_{p}} & \underline{C}^{1}(G, A) \ar[d]_{_{\underline{δ}_{G}}} \ar[r]^{^{\underline{p}^{1}}} & \underline{C}^{1}(E, A) \ar[d]^{_{\underline{δ}_{E}}} \ar[r]^{^{\underline{q}^{1}}} & \underline{C}^{1}(p, A) \ar[d]^{_{\underline{δ}_{p}}} \\ Z^{2}(G, A) \ar[r]_{p_{2}} & Z^{2}(E, A) \ar[r]_{q_{2}} & Z^{2}(p, A) & Z^{2}(G, A) \ar[r]_{p_{2}} & Z^{2}(E, A) \ar[r]_{q_{2}} & Z^{2}(p, A)}

and use Lemma 2.1 and Lemma 2.2.

We look now for a different description of the group Ker(δ_{p}). Let N be the kernel of p: E → G and N_{ab} = N/[N, N] its abelianization. Since p is surjective, N_{ab} has a Gmodule structure induced by conjugation in E:

x\overline{n} = \overline{ene^{−1}} for n ∈ N, x ∈ G and p(e) = x

In the next result, we denote by Hom_{G}(N_{ab}, A) the abelian group of G-module homomorphisms N_{ab} → A, and by \underline{Hom}(N, A) the abelian group of equivariant homomorphisms, that is, group homomorphisms h: N → A such that h(ene^{−1}) = ^{p(e)}h(n) for all e ∈ E, n ∈ N.

2.4. Proposition. Let p: E → G be a group epimorphism with kernel N. The abelian groups Hom_{G}(N_{ab}, A), \underline{Hom}(N, A) and Ker(δ_{p}) are isomorphic.

Proof. 1) The isomorphism \underline{Hom}(N, A) ≃ Hom_{G}(N_{ab}, A) is just the restriction of the natural isomorphism Grp(N, A) ≃ Ab(N_{ab}, A) given by the universal property of N_{ab}.

2) Consider now the kernel of δ_{p}: C^{1}(p, A) → Z^{2}(p, A). Explicitly

Ker(δ_{p}) = {[g: E → A] | ∃f: G × G → A : δ_{E}(g) = p_{2}(f)}

with [g] = [g′] if there exists u: G → A such that g = p_{1}(u) + g′, and with δ_{E}(g) = p_{2}(f) meaning that ^{p(e_{1})}g(e_{2}) − g(e_{1}e_{2}) + g(e_{1}) = f(p(e_{1}), p(e_{2})) for all e_{1}, e_{2} ∈ E. We put

Σ: Ker(δ_{p}) → \underline{Hom}(N, A) [g] ↦ (g · i: N → E → A)

where i: N → G is the inclusion.

- Σ is well-defined : if [g] = [g′], then condition g = p_{1}(u) + g′ gives for all n ∈ N

g(n) = u(p(n)) + g′(n) = u(1) + g′(n) = g′(n)

- Σ[g]: N → A is a group homomorphism : condition δ_{E}(g) = p_{2}(f) gives for all n ∈ N and e ∈ E

g(ne) = ^{p(n)}g(e) + g(n) − f(p(n), p(e)) = ^{1}g(e) + g(n) − f(1, p(e)) = g(e) + g(n) (1)

- Σ[g] ∈ \underline{Hom}(N, A): using condition δ_{E}(g) = p_{2}(f) once again we have for all e ∈ E

p(e)g(e^{−1}) = g(ee^{−1}) − g(e) + f(p(e), p(e^{−1})) = −g(e) + f(p(e), p(e^{−1}))  (2)

Now if n ∈ N and e ∈ E, from (2) and δ_{E}(g) = p_{2}(f) it follows that

p(e)g(ne^{−1}) = g(ene^{−1}) − g(e) + f(p(e), p(ne^{−1})) = g(ene^{−1}) + ^{p(e)}g(e^{−1})

and therefore using (1) we get

g(ene^{−1}) = ^{p(e)}g(ne^{−1}) − ^{p(e)}g(e^{−1}) = ^{p(e)}(g(n) + g(e^{−1})) − ^{p(e)}g(e^{−1}) =

= ^{p(e)}g(n) + ^{p(e)}g(e^{−1}) − ^{p(e)}g(e^{−1}) = ^{p(e)}g(n)

- Σ is surjective : let h: N → A be in \underline{Hom}(N, A) and choose a set-theoretical section s: G → E of p such that s(1) = 1. We can construct g_{s} ∈ C^{1}(E, A) and f_{s} ∈ Z^{2}(G, A) as follows

g_{s}: E → A: g_{s}(e) = h(es(p(e))^{−1}) , f_{s}: G × G → A: f_{s}(x, y) = h(s(xy)s(y)^{−1}s(x)^{−1})

To check that δ_{E}(g_{s}) = p_{2}(f_{s}) is long but easy, so let us check that Σ[g_{s}] = h: for all n ∈ N we have g_{s}(n) = h(ns(p(n))^{−1}) = h(ns(1)^{−1}) = h(n).

- Σ is injective : let [g] ∈ Ker(δ_{p}) such that the restriction of g to N is 0. Fix once again a set-theoretical section s of p such that s(1) = 1 and put u_{s} = g · s: G → A. We have to show that g = u_{s} · p, so that [g] = 0. For this, observe that from δ_{E}(g) = p_{2}(f) we deduce

0 = g(1) = g(s(x)s(x)^{−1}) = g(s(x)) + ^{x}g(s(x)^{−1}) − f(x, x^{−1})  (3)

for all x ∈ G. Finally, since es(p(e))^{−1} ∈ N for all e ∈ E, using condition δ_{E}(g) = p_{2}(f) one more time and putting x = p(e) in (3) we have

0 = g(es(p(e))^{−1}) = g(e) + ^{p(e)}g(s(p(e))^{−1}) − f(p(e), p(e)^{−1}) =
= g(e) − g(s(p(e))) = g(e) − u_{s}(p(e))

Putting together Corollary 2.3 and Proposition 2.4, we get the exact sequences originally due to Hochschild and Serre [21].

2.5. Corollary. Consider an epimorphism of groups p: E → G with kernel N and a G-module A. There are exact sequences of abelian groups

0 → Der(G, A) → Der(E, A) → Hom_{G}(N_{ab}, A) → H^{2}(G, A) → H^{2}(E, A)

0 → H^{1}(G, A) → H^{1}(E, A) → Hom_{G}(N_{ab}, A) → H^{2}(G, A) → H^{2}(E, A)

3. A reminder on categorical groups

In this section we collect some basic facts on categorical groups and on the corresponding notions of action, derivation, kernel, cokernel, and exactness. Starting from the works of Deligne [16], Frohlich and Wall [17] and Sinh [30], categorical groups have been studied extensively in the literature (they already appear in the work of Whitehead [35] in their strict form of crossed modules of groups). We refer to [4] and [28] for an introduction to monoidal categories, and to [1] and [33] for basic facts on categorical groups. Actions of a categorical group on a (symmetric) categorical group have been introduced in [6] and [11]. The corresponding notion of derivation has been introduced in [18] and [19]. Convenient notions of kernel, cokernel and exactness for homomorphisms of (symmetric) categorical groups have been studied in [25] and [33].

3.1. . Categorical groups. A categorical group (sometimes called gr-category or 2-group in the literature) is a monoidal groupoid 𝔾 = (𝔾, ⊗, I, a, l, r) such that every object X is invertible, that is, the functor

X ⊗ (−): 𝔾 → 𝔾 , Y ↦ X ⊗ Y

is an equivalence. It is then possible to choose, for each X ∈ 𝔾, an object X^{∗} ∈ 𝔾 (called an inverse of X) and arrows η_{X}: I → X ⊗ X^{∗} and ε_{X}: X^{∗} ⊗ X → I such that the usual triangular identities are satisfied. The choice of a system of inverses (X^{∗}, η_{X}, ϵ_{X}), X ∈ 𝔾, induces a monoidal equivalence

(−)^{∗}: 𝔾 → 𝔾 f: X → Y ↦ f^{∗}: X^{∗} → Y^{∗}

where f^{∗} is defined as follows

\xymatrix{X^{∗} ≃ X^{∗} ⊗ I \ar[r]^{id⊗η_{Y}} & X^{∗} ⊗ Y ⊗ Y^{∗} \ar[r]^{id⊗f^{−1}⊗id} & X^{∗} ⊗ X ⊗ Y^{∗} \ar[r]^{ε_{X}⊗id} & I ⊗ Y^{∗} ≃ Y^{∗}}

The following simple lemma is quite useful, it will be tacitly used in several calculations all along the paper.

3.2. Lemma. Let f: X → Y be an arrow in a categorical group. The following diagrams commute

\xymatrix{X^{∗} ⊗ X \ar[rr]^{f^{∗}⊗f} \ar[dr]_{ε_{X}} &  & Y^{∗} ⊗ Y \ar[dl]^{ε_{Y}} & X ⊗ X^{∗} \ar[rr]^{f⊗f^{∗}} &  & Y ⊗ Y^{∗} \\  & I &  &  & I \ar[ul]^{η_{X}} \ar[ur]_{η_{Y}}}

In fact, the commutativity of these diagrams characterizes f^{∗}: X^{∗} → Y^{∗} once f is given.

(Note that, if in the definition of f^{∗} we use f instead of f^{−1}, then the monoidal equivalence (−)^{∗}: 𝔾 → 𝔾 is contravariant and the previous lemma holds in any monoidal category with duals.) A categorical group 𝔾 is said to be symmetric if it is symmetric as a monoidal category, the symmetry being usually denoted by c_{X,Y} : X ⊗ Y → Y ⊗ X. We will denote by CG (respectively SCG) the 2-category whose objects are categorical groups (respectively, symmetric categorical groups). The 1-arrows, called homomorphisms, are monoidal functors T = (T, T_{2}): 𝔾 → ℍ (respectively, symmetric monoidal functors), and the 2arrows, called morphisms, are monoidal natural transformations. Note that a canonical arrow T_{0} : I → T I can be constructed from the natural and coherent family of arrows T_{2}^{X,Y} : T X ⊗ T Y → T (X ⊗ Y). Note also that in CG and SCG, the 2-arrows are invertible.

If F: 𝔾 → ℍ is any functor between categorical groups, we denote by F^{∗}: 𝔾 → ℍ the composite functor

\xymatrix{𝔾 \ar[r]^{F} & ℍ \ar[r]^{(−)^{∗}} & ℍ}

that is, F^{∗}(X) = (F X)^{∗} and F^{∗}(f) = (F f)^{∗}. In the same way, from a natural transformation α: F ⇒ G we get a natural transformation α^{∗}: F^{∗} ⇒ G^{∗} defined by α_{X}^{∗} = (α_{X})^{∗}. When F is monoidal, we can assume that F^{∗}(X) = F (X^{∗}) and F^{∗}(f) = F (f^{∗}). Finally, if α: F ⇒ G is monoidal, then α_{X}^{∗} = α_{X^{∗}} (to check this last fact, use Lemma 3.2).

If 𝔾 ∈ CG, then the set π_{0}(𝔾) of its connected components is a group (abelian if 𝔾 is symmetric) with operation induced by tensor product. We denote by π_{1}(𝔾) the abelian group Aut_{𝔾}(I) of automorphisms of the object I (to check that π_{1}(𝔾) is abelian, use the Eckmann-Hilton argument : in π_{1}(𝔾) composition coincide, up to the canonical isomorphism I ≃ I ⊗ I, with tensor product). Finally, for a group G we denote by [G]_{0} the discrete categorical group on G; if G is abelian, we denote by [G]_{1} the symmetric categorical group with one object and such that π_{1}([G]_{1}) = G.

3.3. Remark. Throughout the paper we will use several variants of the following simple fact. If 𝔾 and ℍ are categorical groups with ℍ symmetric, then CG(𝔾, ℍ) is a symmetric categorical group with structure induced pointwise from that of ℍ. Moreover,

π_{1}(CG(𝔾, ℍ)) ≃ Grp(π_{0}(𝔾), π_{1}(ℍ))

Indeed, let 0: 𝔾 → ℍ be the constant homomorphism sending each arrow on the identity arrow on I, and let α: 0 ⇒ 0 be a morphism. By naturality, we have that α_{X} = α_{Y} : I → I if [X] = [Y] in π_{0}(𝔾), so that α induces a map π_{0}(𝔾) → π_{1}(ℍ). By monoidality, we have that α_{X⊗Y} = α_{X} ⊗ α_{Y} , so that such a map π_{0}(𝔾) → π_{1}(ℍ) is a group homomorphism.

3.4. Notation. When no confusion arises, in order to simplify notation we will

- omit the associativity isomorphism a_{X,Y,Z}: (X ⊗ Y) ⊗ Z → X ⊗ (Y ⊗ Z) and the right and left unit isomorphisms r_{X}: X → X ⊗ I and l_{X}: X → I ⊗ X

- omit indexes, writing for example c: X ⊗ Y → Y ⊗ X instead of c_{X,Y}

- denote canonical arrows as “can” or even as unlabelled arrows ; for example, we write can: X → Y ⊗ X ⊗ Y^{∗} or X → Y ⊗ X ⊗ Y^{∗} or just X ≃ Y ⊗ X ⊗ Y^{∗} instead of

\xymatrix{X \ar[r]^{r_{X}} & X ⊗ I \ar[r]^{id⊗η_{Y}} & X ⊗ Y ⊗ Y^{∗} \ar[r]^{c_{X,Y} ⊗id} & Y ⊗ X ⊗ Y^{∗}}

or of

\xymatrix{X \ar[r]^{l_{X}} & I ⊗ X \ar[r]^{η_{Y} ⊗id} & Y ⊗ Y^{∗} ⊗ X \ar[r]^{id⊗c_{Y^{∗},X}} & Y ⊗ X ⊗ Y^{∗}}

3.5. . Actions. If 𝔾 and 𝔸 are categorical groups, an action of 𝔾 on 𝔸 is a homomorphism of categorical groups φ: 𝔾 → 𝔼q(𝔸), where 𝔼q(𝔸) is the categorical group of monoidal autoequivalences of 𝔸. When such a 𝔾-action is given, we will say that 𝔸 = (𝔸, φ) is a 𝔾-categorical group. To give a 𝔾-categorical group structure on 𝔸 is equivalent to giving a functor

φ: 𝔾 × 𝔸 → 𝔸 , \xymatrix{(X, A) \ar[r]^{(f,u)} & (Y, B)} ↦ \xymatrix{φ(X)(A) = ^{X}A \ar[r]^{φ(f)(u)=^{f}u} & φ(Y)(B) = ^{Y}B}

(when f = id_{X} or u = id_{A}, we write respectively ^{X}u and ^{f}A instead of ^{f}u) together with natural families of arrows

φ_{2}^{X,A,B}: ^{X}A ⊗ ^{X}B → ^{X}(A ⊗ B) , φ_{1}^{X,Y,A}: ^{X}(^{Y}A) → ^{X⊗Y}A , φ_{0}^{A}: A → ^{I}A

satisfying the following coherence conditions (see Definition 2.1 in [11])^{1} : (act1) φ_{2}^{X,A,B⊗C} · (id ⊗ φ_{2}^{X,B,C}) · a = ^{X}a · φ_{2}^{X,A⊗B,C} · (φ_{2}^{X,A,B} ⊗ id)

(act2) φ_{1}^{X,Y ⊗Z,A} · ^{X}φ_{1}^{Y,Z,A} = ^{a}A · φ_{1}^{X⊗Y,Z,A} · φ_{1}^{X,Y,^{Z}A} (act3) φ_{1}^{I,Y,A} · φ_{0}^{^{Y}A} = ^{l}A , φ_{1}^{X,I,A} · ^{X}φ_{0}^{A} = ^{r}A

(act4) φ_{1}^{X,Y,A⊗B} · ^{X}φ_{2}^{Y,A,B} · φ_{2}^{X,^{Y}A,^{Y}B} = φ_{2}^{X⊗Y,A,B} · (φ_{1}^{X,Y,A} ⊗ φ_{1}^{X,Y,B}) (act5) φ_{2}^{I,A,B} · (φ_{0}^{A} ⊗ φ_{0}^{B}) = φ_{0}^{A⊗B} One can construct a unique natural family of arrows φ_{•}^{X}: I → ^{X}I such that

1. All along the paper, we will consider several coherence conditions. When they are already available in the literature, we give a precise reference and express the condition as an equation. This is the case in Subsections 3.5 and 3.11 (categorical actions and derivations) and in Subsections 7.2 and 7.3 (categorical precrossed and crossed modules). Otherwise, we express the condition in the more readable form of a commutative diagram.

φ_{2}^{X,I,B} ·(φ_{•}^{X} ⊗id)·l = ^{X}l , φ_{2}^{X,A,I} ·(id⊗φ_{•}^{X})·r = ^{X}r , φ_{1}^{X,Y,I} · ^{X}φ_{•}^{Y} ·φ_{•}^{X} = φ_{•}^{X⊗Y} , φ_{0}^{I} = φ_{•}^{I}

We say that an action φ: 𝔾 → 𝔼q(𝔸) is symmetric (or that 𝔸 is a symmetric 𝔾-categorical group) if 𝔸 is symmetric and φ factorizes through φ: 𝔾 → 𝔼q_{s}(𝔸), where 𝔼q_{s}(𝔸) is the categorical group of symmetric monoidal autoequivalences of 𝔸. To express an action φ: 𝔾 → 𝔼q_{s}(𝔸) as a functor φ: 𝔾 × 𝔸 → 𝔸, we need one more condition (see Definition 2.1 in [11]) : (act6) φ_{2}^{X,B,A} · c = ^{X}c · φ_{2}^{X,A,B}

3.6. Remark. In Section 6 we will use the fact that any action φ: 𝔾 → 𝔼q(𝔸) induces two group actions

π_{0}(φ): π_{0}(𝔾) × π_{0}(𝔸) → π_{0}(𝔸) and \underline{φ}: π_{0}(𝔾) × π_{1}(𝔸) → π_{1}(𝔸)

The first one is defined in the obvious way because π_{0}(𝔾 × 𝔸) = π_{0}(𝔾) × π_{0}(𝔸). The second one sends a pair ([X], a) ∈ π_{0}(𝔾) × π_{1}(𝔸) on the composite

\xymatrix{I \ar[r]^{φ_{•}^{X}} & ^{X}I \ar[r]^{^{X}a} & ^{X}I \ar[r]^{(φ_{•}^{X})^{−1}} & I}

and is well-defined thanks to the naturality of φ_{•} and the functoriality of φ.

3.7. . Kernel. Given a homomorphism T: 𝔾 → ℍ in CG, its kernel is the following diagram in CG

[figure: 𝔾; T; ⇓k(T); 𝕂er(T); K(T); 0; ℍ]

where - an object of 𝕂er(T) is a pair (X ∈ 𝔾, x: T X → I);

- an arrow f: (X, x) → (Y, y) in 𝕂er(T) is an arrow f: X → Y in 𝔾 such that y · T (f) = x;

- the faithful (but in general not full) homomorphism K(T): 𝕂er(T) → 𝔾 is defined by K(T)(f: (X, x) → (Y, y)) = (f: X → Y);

- the component at (X, x) of the morphism k(T) is given by x: T (K(T)(X, x)) =

T X → I = 0(X, x). The kernel of T is a bilimit in the sense of [3] and also a standard homotopy kernel. Bilimit means that for any other diagram in CG of the form

[figure: 𝔾; T; ⇓ρ; 𝔸; R; 0; ℍ]

there exists a homomorphism R′: 𝔸 → 𝕂er(T) and a morphism ρ′: K(T) · R′ ⇒ R such that ρ · (T ∘ ρ′) = k(T) ∘ R′ (where ∘ is the horizontal or Godement composition of morphisms) ; moreover, if R′′: 𝔸 → 𝕂er(T) and ρ′′: K(T) · R′′ ⇒ R are such that ρ · (T ∘ ρ′′) = k(T) ∘ R′′, then there exists a unique morphism r: R′ ⇒ R′′ such that ρ′′ · (K(T) ∘ r) = ρ′. Standard homotopy kernel means that, in the same situation, there exists a unique homomorphism R′: 𝔸 → 𝕂er(T) such that K(T) ∘ R′ = R and k(T) · R′ = ρ. Finally, if T: 𝔾 → ℍ and R: 𝔸 → 𝔾 are in SCG, then 𝕂er(T), K(T) and R′ also are in SCG. Note that, because of the double universal property of the kernel, we do not pay too much attention to the fact that a diagram in CG or in SCG involving kernels (or cokernels) commutes strictly or just up to a 2-arrow.

3.8. . Cokernel. Let now T: 𝔾 → ℍ be a homomorphism in SCG. Its cokernel is the following diagram in SCG

[figure: ℍ; C(T); ⇓c(T); 𝔾; T; 0; ℂoker(T)]

and it satisfies two universal properties dual to those of the kernel. It can be described as follows :

- the objects of ℂoker(T) are those of ℍ;

- a prearrow from A to B is a pair (X ∈ 𝔾, f: A → T X ⊗ B);

- an arrow [X, f]: \xymatrix{A \ar[r]|{∘} & B} is an equivalence class of prearrows, where two prearrows (X, f), (X′, f′) fromA to B are equivalent if there exists an arrow x: X → X′ in 𝔾 such that (T (x) ⊗ id) · f = f′;

- the tensor product of two arrows [X, f]: \xymatrix{A \ar[r]|{∘} & B} and [Y, g]: \xymatrix{C \ar[r]|{∘} & D} is given by the class of the prearrow with object part X ⊗ Y and arrow part

\xymatrix{A ⊗ C \ar[r]^{f⊗g} & T X ⊗ B ⊗ T Y ⊗ D \ar[r]^{id⊗c⊗id} & T X ⊗ T Y ⊗ B ⊗ D ≃ T (X ⊗ Y) ⊗ B ⊗ D}

- the essentially surjective on objects homomorphism C(T): ℍ → ℂoker(T) sends an arrow f: A → B to the class of the prearrow

(I, \xymatrix{A \ar[r]^{f} & B \ar[r] & T I ⊗ B} )

C(T) in general is not a full functor ;

- the component at X ∈ 𝔾 of the morphism c(T) is given by the class of the prearrow (X, T X → I ⊗ T X).

3.9. Exercise. We leave to the reader the following exercise, which is easy but meaningful to grasp the difference between groups and categorical groups when kernels and cokernels are involved. Let 𝔸 be a categorical group and consider the canonical homomorphisms 𝔸 → 0 and 0 → 𝔸. We have

𝕂er(𝔸 → 0) = 𝔸 , 𝕂er(0 → 𝔸) = [π_{1}(𝔸)]_{0}

If 𝔸 is symmetric, we also have

ℂoker(𝔸 → 0) = [π_{0}(𝔸)]_{1} , ℂoker(0 → 𝔸) = 𝔸

3.10. . Exactness. A diagram

[figure: 𝔹; G; ⇓ρ; 𝔸; F; 0; ℂ]

in CG is 2-exact if the comparison homomorphism F′: 𝔸 → 𝕂er(G) is full and essentially surjective on objects. When the above diagram is in SCG, its 2-exactness is equivalent to ask that the comparison homomorphism G′: ℂoker(F) → ℂ is full and faithful. Obvious examples of 2-exact complexes are

\xymatrix{𝕂er(T) \ar[r]^{K(T)} & 𝔾 \ar[r]^{T} & ℍ & and & 𝔾 \ar[r]^{T} & ℍ \ar[r]^{C(T)} & ℂoker(T)}

equipped respectively with k(T): T · K(T) ⇒ 0 and c(T): C(T) · T ⇒ 0. More interesting, if

\xymatrix{𝔸 \ar[r]^{F} & 𝔹 \ar[r]^{G} & ℂ} with ρ: G · F ⇒ 0

is 2-exact, then

π_{0}( \xymatrix{𝔸 \ar[r]^{F} & 𝔹 \ar[r]^{G} & ℂ} ) and π_{1}( \xymatrix{𝔸 \ar[r]^{F} & 𝔹 \ar[r]^{G} & ℂ} )

are exact sequences of groups (the converse implication is not true).

3.11. . Derivations. If 𝔾 is a categorical group and (𝔸, φ) is a 𝔾-categorical group, a derivation from 𝔾 to 𝔸 is a functor D: 𝔾 → 𝔸 together with a natural family of arrows

β_{X,Y} : DX ⊗ ^{X}DY → D(X ⊗ Y)

satisfying the following coherence condition (see Definition 3.1 in [19]) : (der1) D(a) · β_{X⊗Y,Z} · (β_{X,Y} ⊗ id) · a^{−1} · (id ⊗ (id ⊗ φ_{1}^{X,Y,DZ})) =

= β_{X,Y ⊗Z} · (id ⊗ ^{X}β_{Y,Z}) · (id ⊗ φ_{2}^{X,DY,^{Y}DZ})

One can construct a unique arrow β_{0} : I → DI such that

β_{X,I} · (id ⊗ ^{X}β_{0}) · (id ⊗ φ_{•}^{X}) · r = D(r) , β_{I,X} · (id ⊗ φ_{0}^{DX}) · (β_{0} ⊗ id) · l = Dl

Given two derivations (D, β), (D′, β′): 𝔾 → 𝔸, a morphism of derivations ϵ: (D, β) ⇒ (D′, β′) is a natural transformation ϵ: D ⇒ D′ compatible with β and β′: (see Definition 3.1 in [19]) (der2) ϵ_{X⊗Y} · β_{X,Y} = β′_{X,Y} · (ϵ_{X} ⊗ ^{X}ϵ_{Y}) From the previous conditions, it follows that ϵ_{I} · β_{0} = β′_{0}. When (𝔸, φ) is a symmetric 𝔾-categorical group, the category of derivations 𝔻er(𝔾, 𝔸, φ) is a symmetric categorical group, with structure induced by that of 𝔸. Moreover, in this case there exists a homomorphism d_{𝔾}: 𝔸 → 𝔻er(𝔾, 𝔸, φ) defined, on an object A, by the inner derivation d_{𝔾}(A)(X) = ^{X}A ⊗ A^{∗}. For such a functor, the structure of derivation d_{𝔾}(A)(X) ⊗ ^{X}d_{𝔾}(A)(Y) → d_{𝔾}(A)(X ⊗ Y) is provided by

\xymatrix{^{X}A ⊗ A^{∗} ⊗ ^{X}(^{Y}A ⊗ A^{∗}) \ar[r]^{id⊗(φ_{2}^{X,^{Y}A,A^{∗}})^{−1}} & (1) \ar[r]^{φ_{1}^{X,Y,A}⊗φ_{2}^{X,A,A^{∗}}} & (2) \ar[r]^{id⊗φ_{•}^{X}} & ^{X⊗Y}A ⊗ A^{∗}}

where

(1) is ^{X}A ⊗ A^{∗} ⊗ ^{X}(^{Y}A) ⊗ ^{X}A^{∗} ≃ ^{X}(^{Y}A) ⊗ A^{∗} ⊗ ^{X}A ⊗ ^{X}A^{∗}

(2) is ^{X⊗Y}A ⊗ A^{∗} ⊗ ^{X}(A ⊗ A^{∗}) ≃ ^{X⊗Y}A ⊗ A^{∗} ⊗ ^{X}I For an arrow f: A → B in 𝔸, the natural transformation d_{𝔾}(f): d_{𝔾}(A) ⇒ d_{𝔾}(B) has component at X given by ^{X}f ⊗ f^{∗}: ^{X}A ⊗ A^{∗} → ^{X}B ⊗ B^{∗}. We denote the kernel and the cokernel of d_{𝔾} by

\xymatrix{ℍ^{0}(𝔾, 𝔸, φ) \ar[r]^{K(d_{𝔾})} & 𝔸 \ar[r]^{d_{𝔾}} & 𝔻er(𝔾, 𝔸, φ) \ar[r]^{C(d_{𝔾})} & ℍ^{1}(𝔾, 𝔸, φ)}

4. The second cohomology categorical group

In this section we introduce a possible second cohomology categorical group ℍ^{2}(𝔾, 𝔸, φ). To avoid confusion with the second cohomology categorical group H^{2}(𝔾, 𝔸) studied in [?], we adopt the terminology here of “cobord” and “factor set” to describe ℍ^{2}(𝔾, 𝔸, φ), and we leave the terminology “cochain” and “cocycle” for H^{2}(𝔾, 𝔸). We fix a symmetric 𝔾-categorical group 𝔸 = (𝔸, φ: 𝔾 → 𝔼q_{s}(𝔸)) as in Subsection 3.5. We start by describing the symmetric categorical group ℂ^{1}(𝔾, 𝔸) of cobords and the symmetric categorical group ℤ^{2}(𝔾, 𝔸, φ) of factor sets. In both cases, the structure of symmetric categorical group is inherited from that of 𝔸.

4.1. Definition. A cobord of 𝔾 with coefficients in 𝔸 is a pair G = (G, G_{0}) with G: 𝔾 → 𝔸 a functor and G_{0} : I → GI an arrow. Given two cobords G, G′: 𝔾 → 𝔸, a morphism of cobords is a natural transformation β: G ⇒ G′ such that the following diagram commutes

(cob1) \xymatrix{GI \ar[rr]^{β_{I}} &  & G′I \\  & I \ar[ul]^{G_{0}} \ar[ur]_{G′_{0}}}

We denote by ℂ^{1}(𝔾, 𝔸) the symmetric categorical group of cobords. 4.2. Definition. A factor set of 𝔾 with coefficients in 𝔸 is a 4-tuple F = (F, a_{F} , r_{F} , l_{F}) where F: 𝔾 × 𝔾 → 𝔸 is a functor and

a_{F}^{X,Y,Z}: F (X, Y) ⊗ F (X ⊗ Y, Z) → ^{X}F (Y, Z) ⊗ F (X, Y ⊗ Z)

r_{F}^{X}: I → F (X, I) l_{F}^{Y} : I → F (I, Y)

are natural families of arrows such that the following diagrams commute

(fs1) \xymatrix{I ⊗ F (X, Z) \ar[d]_{r_{F}^{X}⊗F (r_{X},id)} \ar[r]^{φ_{•}^{X}⊗id} & ^{X}I ⊗ F (X, Z) \ar[d]^{^{X}l_{F}^{Z}⊗F (id,l_{Z})} \\ F (X, I) ⊗ F (X ⊗ I, Z) \ar[r]_{a_{F}^{X,I,Z}} & ^{X}F (I, Z) ⊗ F (X, I ⊗ Z)}

(fs2)

\xymatrix{(1) \ar[d]_{a_{F}^{X,Y,Z}⊗F (a,id)} \ar[r]^{id⊗a_{F}^{X⊗Y,Z,W}} & (6) \ar[r]^{c⊗id} & (7) \ar[r]^{id⊗a_{F}^{X,Y,Z⊗W}} & (8) \ar[d]^{(φ_{1}^{X,Y,F (Z,W)})^{−1}⊗id} \\ (2) \ar[d]_{id⊗a_{F}^{X,Y ⊗Z,W}} &  &  & (9) \ar[d]^{φ_{2}^{X,^{Y}F (Z,W),F (Y,Z⊗W)}} \\ (3) \ar[r]_{φ_{2}^{X,F (Y,Z),F (Y ⊗Z,W)}⊗id} & (4) \ar[rr]_{^{X}a_{F}^{Y,Z,W} ⊗F (id,a)} &  & (5)}

where

(1) = F (X, Y) ⊗ F (X ⊗ Y, Z) ⊗ F ((X ⊗ Y) ⊗ Z, W)

(2) = ^{X}F (Y, Z) ⊗ F (X, Y ⊗ Z) ⊗ F (X ⊗ (Y ⊗ Z), W)

(3) = ^{X}F (Y, Z) ⊗ ^{X}F (Y ⊗ Z, W) ⊗ F (X, (Y ⊗ Z) ⊗ W)

(4) = ^{X}(F (Y, Z) ⊗ F (Y ⊗ Z, W)) ⊗ F (X, (Y ⊗ Z) ⊗ W)

(5) = ^{X}(^{Y}F (Z, W) ⊗ F (Y, Z ⊗ W)) ⊗ F (X, Y ⊗ (Z ⊗ W))

(6) = F (X, Y) ⊗ ^{X⊗Y}F (Z, W) ⊗ F (X ⊗ Y, Z ⊗ W)

(7) = ^{X⊗Y}F (Z, W) ⊗ F (X, Y) ⊗ F (X ⊗ Y, Z ⊗ W)

(8) = ^{X⊗Y}F (Z, W) ⊗ ^{X}F (Y, Z ⊗ W) ⊗ F (X, Y ⊗ (Z ⊗ W))

(9) = ^{X}(^{Y}F (Z, W)) ⊗ ^{X}F (Y, Z ⊗ W) ⊗ F (X, Y ⊗ (Z ⊗ W))

ON THE SECOND COHOMOLOGY CATEGORICAL GROUP  949

Given two factor sets F, F′: 𝔾 × 𝔾 → 𝔸, a morphism of factor sets is a natural transformation α: F ⇒ F′ such that the following diagrams commute

(fs3) \xymatrix{F (X, Y) ⊗ F (X ⊗ Y, Z) \ar[r]^{a_{F}^{X,Y,Z}} \ar[d]_{α_{X,Y} ⊗α_{X⊗Y,Z}} & ^{X}F (Y, Z) ⊗ F (X, Y ⊗ Z) \ar[d]^{^{X}α_{Y,Z}⊗α_{X,Y ⊗Z}} \\ F′(X, Y) ⊗ F′(X ⊗ Y, Z) \ar[r]_{a_{F′}^{X,Y,Z}} & ^{X}F′(Y, Z) ⊗ F′(X, Y ⊗ Z)}

(fs4) \xymatrix{F (X, I) \ar[dd]_{α_{X,I}} &  & F (I, Y) \ar[dd]^{α_{I,Y}} \\  & I \ar[ul]_{r_{F}^{X}} \ar[ur]^{l_{F}^{Y}} \ar[dl]^{r_{F′}^{X}} \ar[dr]_{l_{F′}^{Y}} \\ F′(X, I) &  & F′(I, Y)}

We denote by ℤ^{2}(𝔾, 𝔸, φ) the symmetric categorical group of factor sets. 4.3. Remark. The definition of factor set comes from the following fact. Given a functor F: 𝔾 × 𝔾 → 𝔸 and three natural families a_{F}^{X,Y,Z}, r_{F}^{X}, l_{F}^{Y} , we can define a tensor product

⊗_{F} : 𝔸 × 𝔾 × 𝔸 × 𝔾 → 𝔸 × 𝔾 , (A, X) ⊗_{F} (B, Y) = (A ⊗ ^{X}B ⊗ F (X, Y), X ⊗ Y)

a unit objet (I, I), an inverse (A, X)^{∗} = (^{X^{∗}}A^{∗} ⊗ ^{X^{∗}}F^{∗}(X, X^{∗}), X^{∗}) and three natural families

- (A, X) → (A, X) ⊗_{F} (I, I) with components

\xymatrix{A \ar[r] & A ⊗ I ⊗ I \ar[r]^{id⊗φ_{•}^{X}⊗r_{F}^{X}} & A ⊗ ^{X}I ⊗ F (X, I)} , \xymatrix{X \ar[r] & X ⊗ I}

- (B, Y) → (I, I) ⊗_{F} (B, Y) with components

\xymatrix{B \ar[r] & I ⊗ B ⊗ I \ar[r]^{id⊗φ_{0}^{B}⊗l_{F}^{Y}} & I ⊗ ^{I}B ⊗ F (I, Y)} , \xymatrix{Y \ar[r] & I ⊗ Y}

- ((A, X) ⊗_{F} (B, Y)) ⊗_{F} (C, Z) → (A, X) ⊗_{F} ((B, Y) ⊗_{F} (C, Z)) with first component

\xymatrix{(1) \ar[r]^{id⊗c⊗id} & (2) \ar[r]^{id⊗(φ_{1}^{X,Y,C})^{−1}⊗a_{F}^{X,Y,Z}} & (3) \ar[r]^{id⊗φ_{2}^{X,B,^{Y}C,F (Y,Z)}⊗id} & (4)}

where

(1) = A ⊗ ^{X}B ⊗ F (X, Y) ⊗ ^{X⊗Y}C ⊗ F (X ⊗ Y, Z)

(2) = A ⊗ ^{X}B ⊗ ^{X⊗Y}C ⊗ F (X, Y) ⊗ F (X ⊗ Y, Z)

(3) = A ⊗ ^{X}B ⊗ ^{X}(^{Y}C) ⊗ ^{X}F (Y, Z) ⊗ F (X, Y ⊗ Z)

(4) = A ⊗ ^{X}(B ⊗ ^{Y}C ⊗ F (Y, Z)) ⊗ F (X, Y ⊗ Z)

and second component (X ⊗ Y) ⊗ Z → X ⊗ (Y ⊗ Z).

Conditions (fs1) and (fs2) in Definition 4.2 are precisely what is needed to make such families coherent in the usual sense of monoidal categories, so that (𝔸 × 𝔾, ⊗_{F} , (I, I), …) is a monoidal category (in fact, a categorical group). In Section 9 we will denote such a categorical group by 𝔸 ×_{F} 𝔾. Observe that if the factor set F is the 0-functor, then 𝔸 ×_{F} 𝔾 is the semi-direct product 𝔸 ⋊_{φ} 𝔾, see [19, 24].

4.4. Lemma. There exists a homomorphism δ = δ_{𝔾}: ℂ^{1}(𝔾, 𝔸) → ℤ^{2}(𝔾, 𝔸, φ) defined, on an object G, by

δ(G)(X, Y) = ^{X}GY ⊗ G^{∗}(X ⊗ Y) ⊗ GX

r_{δG}^{X}: \xymatrix{I \ar[r]^{φ_{•}^{X}} & ^{X}I \ar[r]^{^{X}G_{0}} & ^{X}GI \ar[r] & ^{X}GI ⊗ G^{∗}(X ⊗ I) ⊗ GX}

l_{δG}^{Y}: \xymatrix{I \ar[r] & GY ⊗ G^{∗}(I ⊗ Y) ⊗ I \ar[r]^{φ_{0}^{GY} ⊗id⊗G_{0}} & ^{I}GY ⊗ G^{∗}(I ⊗ Y) ⊗ GI}

a_{δG}^{X,Y,Z}: \xymatrix{(1) \ar[r] & (2) \ar[r]^{(φ_{1}^{X,Y,GZ})^{−1}⊗can} & (3) \ar[r]^{φ_{2}^{X,^{Y}GZ,G^{∗}(Y ⊗Z)}⊗id} & (4)}

where

(1) = ^{X}GY ⊗ G^{∗}(X ⊗ Y) ⊗ GX ⊗ ^{X⊗Y}GZ ⊗ G^{∗}(X ⊗ Y ⊗ Z) ⊗ G(X ⊗ Y)

(2) = ^{X⊗Y}GZ ⊗ ^{X}GY ⊗ G^{∗}(X ⊗ Y ⊗ Z) ⊗ GX

(3) = ^{X}(^{Y}GZ) ⊗ ^{X}G^{∗}(Y ⊗ Z) ⊗ ^{X}GY ⊗ ^{X}G(Y ⊗ Z) ⊗ G^{∗}(X ⊗ Y ⊗ Z) ⊗ GX

(4) = ^{X}(^{Y}GZ ⊗ G^{∗}(Y ⊗ Z) ⊗ GY) ⊗ ^{X}G(Y ⊗ Z) ⊗ G^{∗}(X ⊗ Y ⊗ Z) ⊗ GX and, on a morphism β: G ⇒ G′, by

δ(β)_{X,Y} : \xymatrix{^{X}GY ⊗ G^{∗}(X ⊗ Y) ⊗ GX \ar[r]^{^{X}β_{Y} ⊗β_{X⊗Y}^{∗} ⊗β_{X}} & ^{X}G′Y ⊗ G^{′∗}(X ⊗ Y) ⊗ G′X}

Moreover, the kernel of δ_{𝔾} is equivalent to 𝔻er(𝔾, 𝔸, φ).

The fact that the families r_{δG}, l_{δG} and a_{δG} constructed in Lemma 4.4 are canonical depends on the fact that 𝔸 is symmetric, a braiding on 𝔸 is not sufficient here.

4.5. Definition. The second cohomology categorical group ℍ^{2}(𝔾, 𝔸, φ) of 𝔾 with coefficients in 𝔸 is the cokernel of δ_{𝔾}. Therefore, we have

\xymatrix{𝔻er(𝔾, 𝔸, φ) \ar[r]^{K(δ)} & ℂ^{1}(𝔾, 𝔸) \ar[r]^{δ} & ℤ^{2}(𝔾, 𝔸, φ) \ar[r]^{C(δ)} & ℍ^{2}(𝔾, 𝔸, φ)}

4.6. Example.

1. If 𝔾 is the discrete categorical group [G]_{0} associated with a group G, then the set of objects of ℤ^{2}([G]_{0}, 𝔸) is identified with the set of Ulbrich’s 3-cocycles of G with coefficients in 𝔸, see [32]. In particular (see Example 3 in [11]), if 𝔸 is the discrete symmetric [G]_{0}-categorical group [A]_{0} associated with a G-module A, then ℤ^{2}([G]_{0}, [A]_{0}) is the discrete symmetric categorical group [Z^{2}(G, A)]_{0} associated with the abelian group of Eilenberg-Mac Lane 2-cocycles of G with coefficients in A.

2. If 𝔾 = [G]_{0} and, moreover, 𝔸 = [A]_{1} is the symmetric [G]_{0}-categorical group with only one object associated with a G-module A, then ℤ^{2}([G]_{0}, [A]_{1}) is the discrete symmetric categorical group associated with the abelian group of Eilenberg-Mac Lane 3-cocycles of G with coefficients in the G-module A.

This shows that the cohomology groups introduced by Eilenberg-Mac Lane [27] and Ulbrich [32] can be obtained as special instances of the symmetric categorical group ℍ^{2}(𝔾, 𝔸, φ) via the functor π_{0}.

5. The Hochschild-Serre sequences for categorical groups

In order to prepare the Hochschild-Serre 2-exact sequences, we generalise Lemma 2.1 and Lemma 2.2 to symmetric categorical groups.

5.1. Lemma. Consider a diagram in SCG

\xymatrix{𝔸 \ar[r]^{D} & 𝕂er(δ) \ar[r]^{K(δ)} \ar[d]_{C(D)} & ℂ \ar[d]_{C(K(δ)·D)} \ar[r]^{δ} & ℤ \ar[r]^{C(δ)} \ar[dr]_{C(\underline{δ})} & ℂoker(δ) \ar[d]^{S} \\  & ℂoker(D) \ar[r]_{\underline{K}(δ)} & ℂoker(K(δ) · D) \ar[ur]_{\underline{δ}} &  & ℂoker(\underline{δ})}

where \underline{K}(δ), \underline{δ} and S are induced by the universal properties of ℂoker(D), ℂoker(K(δ) · D) and ℂoker(δ). Then

1. \underline{K}(δ): ℂoker(D) → ℂoker(K(δ) · D) is the kernel of \underline{δ}: ℂoker(K(δ) · D) → ℤ;

2. S: ℂoker(δ) → ℂoker(\underline{δ}) is full and essentially surjective.

Proof. Consider the diagram

\xymatrix{𝔸 \ar[d] \ar[r]^{K(δ)·D} & ℂ \ar[d]_{δ} \ar[r]^{C(K(δ)·D)} & Coker(K(δ) · D) \ar[d]^{\underline{δ}} \\ 0 \ar[r] & ℤ \ar[r]_{Id} & ℤ}

Apply the Kernel-Cokernel Lemma for symmetric categorical groups (Proposition 6.3 in [15]) and use Exercise 3.9. We get a 2-exact sequence

\xymatrix{𝔸 \ar[r]^{D} & 𝕂er(δ) \ar[r]^{T} & 𝕂er(\underline{δ}) \ar[r] & [π_{0}(𝔸)]_{1} \ar[r]^{R} & ℂoker(δ) \ar[r]^{S} & ℂoker(\underline{δ})}

1. Consider the factorization through the cokernel

\xymatrix{𝔸 \ar[r]^{D} & 𝕂er(δ) \ar[d]_{C(D)} \ar[r]^{T} & 𝕂er(\underline{δ}) \\  & ℂoker(D) \ar@{-->}[ur]_{T′}}

By 2-exactness in 𝕂er(δ), T′ is full and faithful. Moreover, the sequence

\xymatrix{π_{0}(𝕂er(δ)) \ar[r]^{π_{0}(T)} & π_{0}(𝕂er(\underline{δ})) \ar[r] & π_{0}([π_{0}(𝔸)]_{1}) = 0}

is exact, so that π_{0}(T) is surjective. This means that T is essentially surjective, and then T′ also is essentially surjective.

2. Consider the factorization through the kernel

\xymatrix{[π_{0}(𝔸)]_{1} \ar[dr]_{R′} \ar[r]^{R} & ℂoker(δ) \ar[r]^{S} & ℂoker(\underline{δ}) \\  & 𝕂er(S) \ar[u]_{K(S)}}

By 2-exactness in ℂoker(δ), R′ is (full and) essentially surjective. Therefore,

π_{0}(R′): π_{0}([π_{0}(𝔸)]_{1}) = 0 → π_{0}(𝕂er(S))

is surjective, and then π_{0}(𝕂er(S)) = 0. This means that S is full (see Proposition 2.1 in [33]). Finally, the fact that S is essentially surjective is obvious because S · C(δ) = C(\underline{δ}) and C(\underline{δ}) is essentially surjective.

5.2. Lemma. Consider a diagram in SCG

\xymatrix{𝔸 \ar[d]_{D} \ar[r]^{F} & 𝔸′ \ar[d]^{D′} \\ ℂ \ar[r]_{G} \ar[d]^{≃}_{C(D)} & ℂ′ \ar[d]^{C(D′)} \ar[r]^{C(G)} & ℂoker(G) \ar[d]^{R} \\ ℂoker(D) \ar[r]_{\underline{G}} & ℂoker(D′) \ar[r]_{C(\underline{G})} & ℂoker(\underline{G})}

with F full and essentially surjective, and G faithful. Then 1. the comparison \underline{G}: ℂoker(D) → ℂoker(D′) is faithful ;

2. the comparison R: ℂoker(G) → ℂoker(\underline{G}) is an equivalence.

Proof. Using the factorization \underline{D} of D through the kernel of C(D), and the factorization \underline{D}′ of D′ through the kernel of C(D′), we split the upper part of the previous diagram as

[figure: 𝔸; F; \underline{D}; D; 𝔸′; _{\underline{D}_{′}}; D′; Ker(C(D)); \underline{F}; K(C(D)); ≃; 𝕂er(C(D′)); K(C(D)); ℂ; G; ℂ′]

Following Proposition 2.1 in [25], we have that \underline{D} and \underline{D}′ are full and essentially surjective. Therefore, ℂoker(D) is also a cokernel of K(C(D)), ℂoker(D′) is also a cokernel of K(C(D′)), and \underline{F} is full and essentially surjective. Apply now the Kernel-Cokernel Lemma to the diagram

\xymatrix{𝕂er(C(D)) \ar[r]^{K(C(D))} \ar[d]_{\underline{F}} & ℂ \ar[r]^{C(D)} \ar[d]^{G} & ℂoker(D) \ar[d]^{\underline{G}} \\ 𝕂er(C(D′)) \ar[r]_{K(C(D′))} & ℂ′ \ar[r]_{C(D′)} & ℂoker(D′)}

We get a 2-exact sequence

\xymatrix{𝕂er(\underline{F}) \ar[r] & 𝕂er(G) \ar[r] & 𝕂er(\underline{G}) \ar[r] & 0 \ar[r] & ℂoker(G) \ar[r]^{R} & ℂoker(\underline{G})}

where ℂoker(\underline{F}) is equivalent to 0 because \underline{F} is full and essentially surjective (see Proposition 2.2 in [33]).

1. From the previous 2-exact sequence, we get an exact sequence of abelian groups

0 = π_{1}(𝕂er(G)) → π_{1}(𝕂er(\underline{G})) → 0

where 0 = π_{1}(𝕂er(G)) because G is faithful (see Proposition 2.1 in [33]). Therefore, π_{1}(𝕂er(\underline{G})) = 0, and then \underline{G} is faithful.

2. The 2-exactness in ℂoker(G) of the previous sequence of symmetric categorical groups gives that R is full and faithful (use Exercise 3.9). Finally, R is also essentially surjective because R · C(G) = C(\underline{G}) · C(D′).

5.3. . Hochschild-Serre 2-exact sequences. In the rest of this section, we fix an essentially surjective homomorphism of categorical groups P: 𝔼 → 𝔾 and a symmetric 𝔾-categorical group 𝔸 = (𝔸, φ: 𝔾 → 𝔼q_{s}(𝔸)). Clearly, P induces a symmetric action φ · P of 𝔼 on 𝔸 (by abuse of notation, we will write φ instead of φ · P). Moreover, P induces by composition two faithful homomorphisms

P_{ℂ}: ℂ^{1}(𝔾, 𝔸) → ℂ^{1}(𝔼, 𝔸) and P_{ℤ}: ℤ^{2}(𝔾, 𝔸, φ) → ℤ^{2}(𝔼, 𝔸, φ)

We can now construct the following diagram in SCG (each part is either commutative or commutative up to a 2-arrow) :

[figure: 𝔸; d_{𝔾}; Id; 𝔸; d_{𝔼}; 𝔻er(𝔾, 𝔸, φ); K(δ_{𝔾}); C(d_{𝔾}); 𝔻er(𝔼, 𝔸, φ); K(δ_{𝔼}); C(d_{𝔼}); 𝕂er(δ_{P}); K(δ_{P}); π_{P}; ℍ^{1}(𝔾, 𝔸, φ); \underline{K}(δ_{𝔾}); ℍ^{1}(𝔼, 𝔸, φ); \underline{K}(δ_{𝔼}); 𝕂er(\underline{δ}_{P}); K(\underline{δ}_{P}); ℂ^{1}(𝔾, 𝔸); δ_{𝔾}; C(K(δ_{𝔾})·d_{𝔾}); P_{ℂ}; ℂ^{1}(𝔼, 𝔸); δ_{𝔼}; C(K(δ_{𝔼})·d_{𝔼}); C(P_{ℂ}); ℂ^{1}(P, 𝔸); δ_{P}; _{\underline{π}_{P}}; \underline{ℂ}^{1}(𝔾, 𝔸); _{\underline{δ}_{𝔾}}; _{\underline{P}_{ℂ}}; \underline{ℂ}^{1}(𝔼, 𝔸); _{\underline{δ}_{𝔼}}; C(\underline{P}_{ℂ}); \underline{ℂ}^{1}(P, 𝔸); _{\underline{δ}_{P}}; ℤ^{2}(𝔾, 𝔸, φ); C(δ_{𝔾}); C(\underline{δ}_{𝔾}); P_{ℤ}; ℤ^{2}(𝔼, 𝔸, φ); C(δ_{𝔼}); C(\underline{δ}_{𝔼}); C(P_{ℤ}); ℤ^{2}(P, 𝔸, φ); C(δ_{P}); C(\underline{δ}_{P}); \underline{ℍ}^{2}(𝔾, 𝔸, φ); \underline{ℍ}^{2}(𝔼, 𝔸, φ); ℂoker(\underline{δ}_{P}); ℍ^{2}(𝔾, 𝔸, φ); s_{𝔾}; ℍ^{2}(𝔼, 𝔸, φ); s_{𝔼}; ℂoker(δ_{P}); s_{P}]

where - \underline{ℂ}^{1}(𝔾, 𝔸) is the cokernel of K(δ_{𝔾}) · d_{𝔾} and \underline{ℍ}^{2}(𝔾, 𝔸, φ) is the cokernel of \underline{δ}_{𝔾};

- ℂ^{1}(P, 𝔸) is the cokernel of P_{ℂ}, \underline{ℂ}^{1}(P, 𝔸) is the cokernel of \underline{P}_{ℂ}, and ℤ^{2}(P, 𝔸, φ) is the cokernel of P_{ℤ}.

5.4. Proposition. Let P: 𝔼 → 𝔾 be an essentially surjective homomorphism of categorical groups and 𝔸 a symmetric 𝔾-categorical group. There exist 2-exact sequences of symmetric categorical groups

\xymatrix{𝔻er(𝔾, 𝔸, φ) \ar[d]_{C(d_{𝔾})} \ar[r] & 𝔻er(𝔼, 𝔸, φ) \ar[d]_{C(d_{𝔼})} \ar[r] & 𝕂er(δ_{P}) \ar[d]_{π_{P}} \ar[r] & ℍ^{2}(𝔾, 𝔸, φ) \ar[d]^{s_{𝔾}} \ar[r] & ℍ^{2}(𝔼, 𝔸, φ) \ar[d]^{s_{𝔼}} \ar[r] & ℂoker(δ_{P}) \ar[d]^{s_{P}} \\ ℍ^{1}(𝔾, 𝔸, φ) \ar[r] & ℍ^{1}(𝔼, 𝔸, φ) \ar[r] & 𝕂er(\underline{δ}_{P}) \ar[r] & \underline{ℍ}^{2}(𝔾, 𝔸, φ) \ar[r] & \underline{ℍ}^{2}(𝔼, 𝔸, φ) \ar[r] & ℂoker(\underline{δ}_{P})}

Moreover, for the connecting homomorphisms, we have :

- C(d_{𝔾}) and C(d_{𝔼}) are essentially surjective ;

- s_{𝔾} and s_{𝔼} are full and essentially surjective ;

- π_{P} and s_{P} are equivalences.

Proof. Apply the Kernel-Cokernel Lemma to the diagrams

\xymatrix{ℂ^{1}(𝔾, 𝔸) \ar[d]_{δ_{𝔾}} \ar[r]^{P_{ℂ}} & ℂ^{1}(𝔼, 𝔸) \ar[d]^{δ_{𝔼}} & \underline{ℂ}^{1}(𝔾, 𝔸) \ar[d]_{_{\underline{δ}_{𝔾}}} \ar[r]^{^{\underline{P}^{ℂ}}} & \underline{ℂ}^{1}(𝔼, 𝔸) \ar[d]^{_{\underline{δ}_{𝔼}}} \\ ℤ^{2}(𝔾, 𝔸, φ) \ar[r]_{P_{ℤ}} & ℤ^{2}(𝔼, 𝔸, φ) & ℤ^{2}(𝔾, 𝔸, φ) \ar[r]_{P_{ℤ}} & ℤ^{2}(𝔼, 𝔸, φ)}

in order to construct the 2-exact sequences. Then apply Lemma 5.1 to the diagram

\xymatrix{𝔸 \ar[r]^{d_{𝔾}} & 𝔻er(𝔾, 𝔸, φ) \ar[r]^{K(δ_{𝔾})} \ar[d]_{C(d_{𝔾})} & ℂ^{1}(𝔾, 𝔸) \ar[d]_{C(K(δ_{𝔾})·d_{𝔾})} \ar[r]^{δ_{𝔾}} & ℤ^{2}(𝔾, 𝔸, φ) \ar[r]^{C(δ_{𝔾})} \ar@{-->}[dr]_{C(\underline{δ}_{𝔾})} & ℍ^{2}(𝔾, 𝔸, φ) \ar[d]^{s_{𝔾}} \\  & ℍ^{1}(𝔾, 𝔸, φ) \ar[r]_{\underline{K}(δ_{𝔾})} & \underline{ℂ}^{1}(𝔾, 𝔸) \ar[ur]_{_{\underline{δ}_{𝔾}}} &  & \underline{ℍ}^{2}(𝔾, 𝔸, φ)}

(and to the similar diagram with 𝔾 replaced by 𝔼) and Lemma 5.2 to the diagram

\xymatrix{𝔸 \ar[d]_{K(δ_{𝔾})·d_{𝔾}} \ar[r]^{Id} & 𝔸 \ar[d]^{K(δ_{𝔼})·d_{𝔼}} \\ ℂ^{1}(𝔾, 𝔸) \ar[d]_{C(K(δ_{𝔾})·d_{𝔾})} \ar[r]^{P_{ℂ}} & ℂ^{1}(𝔼, 𝔸) \ar[d]^{C(K(δ_{𝔼})·d_{𝔼})} \ar[r]^{C(P_{ℂ})} & ℂ^{1}(P, 𝔸) \ar[d]^{_{\underline{π}_{P}}} \\ \underline{ℂ}^{1}(𝔾, 𝔸) \ar[r]_{_{\underline{P}_{ℂ}}} & \underline{ℂ}^{1}(𝔼, 𝔸) \ar[r]_{C(\underline{P}_{ℂ})} & \underline{ℂ}^{1}(P, 𝔸)}

The fact that \underline{π}_{P} is an equivalence immediately implies that π_{P} and s_{P} also are equivalences.

6. The middle point of the Hochschild-Serre 2-exact sequences

In this section we look for a description of 𝕂er(δ_{P}) similar to the one obtained with Proposition 2.4 in the case of groups. We fix the following data

\xymatrix{ℕ \ar[r]^{K(P)} & 𝔼 \ar[r]^{P} & 𝔾 \ar[r]^{φ} & 𝔼q_{s}(𝔸)}

where P is a homomorphism of categorical groups, ℕ is the kernel of P, and (𝔸, φ) is a symmetric 𝔾-categorical group. From Example 2.6.v in [12], recall that there is a canonical action •: 𝔼 × ℕ → ℕ with constraints •_{0}, •_{1}, •_{2}. Explicitly, E • (N, n: P N → I) is given by

(E ⊗ N ⊗ E^{∗}, \xymatrix{P (E ⊗ N ⊗ E^{∗}) \ar[r]^{P_{2}^{−1}} & P E ⊗ P N ⊗ P E^{∗} \ar[r]^{id⊗n⊗id} & P E ⊗ I ⊗ P E^{∗} ≃ I)}

6.1. . The equivariant hom. We denote by \underline{ℍom}(ℕ, 𝔸) the symmetric categorical group having equivariant homomorphisms as objects. This means that an object is a homomorphism (H, H_{2}): ℕ → 𝔸 in CG together with a natural family of arrows

H_{φ}^{E,(N,n)} : H(E • (N, n)) → ^{P E}H(N, n)

satisfying the following coherence conditions^{2} : (act7’) H_{φ}^{I,(N,n)} · H(•_{0}^{(N,n)}) = ^{P_{0}}H(N, n) · φ_{0}^{H(N,n)}

(act8’) ^{P E}H_{2}^{(N,n),(M,m)} · φ_{2}^{P E,H(N,n),H(M,m)} · (H_{φ}^{E,(N,n)} ⊗ H_{φ}^{E,(M,m)}) = = H_{φ}^{E,(N,n)⊗(M,m)} · H(•_{2}^{E,(N,n),(M,m)}) · H_{2}^{E•(N,n),E•(M,m)}

(act9’) ^{P_{2}^{E,F}}H(N, n) · φ_{1}^{P E,P F,H(N,n)} · ^{P E}H_{φ}^{F,(N,n)} · H_{φ}^{E,F •(N,n)} = H_{φ}^{E⊗F,(N,n)} · H(•_{1}^{E,F,(N,n)})

A morphism

α: (H, H_{2}, H_{φ}) ⇒ (K, K_{2}, K_{φ}): ℕ → 𝔸

in \underline{ℍom}(ℕ, 𝔸) is a morphism α: H ⇒ K in CG such that

(act10’) ^{P E}α_{(N,n)} · H_{φ}^{E,(N,n)} = K_{φ}^{E,(N,n)} · α_{E•(N,n)}

In order to prove that \underline{ℍom}(ℕ, 𝔸) and 𝕂er(δ_{P}) are equivalent symmetric categorical groups (Proposition 6.4), we need two lemmas.

6.2. Lemma. Consider the group actions

\xymatrix{π_{0}(𝔼) × π_{0}(ℕ) \ar[r]^{π_{0}(•)} & π_{0}(ℕ) & and & π_{0}(𝔼) × π_{1}(𝔸) \ar[r]^{π_{0}(P)×id} & π_{0}(𝔾) × π_{1}(𝔸) \ar[r]^{\underline{φ}} & π_{1}(𝔸)}

(see Remark 3.6 for the action \underline{φ}), and the abelian group Hom_{π_{0}(𝔼)}(π_{0}(ℕ), π_{1}(𝔸)) of equivariant group homomorphisms. Then

π_{1}(\underline{ℍom}(ℕ, 𝔸)) ≃ Hom_{π_{0}(𝔼)}(π_{0}(ℕ), π_{1}(𝔸))

Proof. As in Remark 3.3, a morphism α: 0 ⇒ 0 in \underline{ℍom}(ℕ, 𝔸) induces a group homomorphism

π_{0}(ℕ) → π_{1}(𝔸) , [N, n] ↦ (α_{(N,n)} : I → I)

Moreover, by condition (act10’), such a homomorphism is equivariant.

2. We write these conditions just as equations, because they are special cases of conditions (act7) – (act10) defining homomorphisms and morphisms of actions, see Subsection 7.1.

6.3. Lemma. Consider the following diagram in CG

[figure: 𝔼; P; 0; ⇓k(P); 𝔸; ℕ; K(P); 0; 𝔾; U; ⇑β]

If P is essentially surjective and if β and k(P) are compatible, that is, for every (N, n) ∈ ℕ

\xymatrix{UP N \ar[rr]^{U(n)} \ar[dr]_{β_{N}} &  & UI \\  & I \ar[ur]_{U_{0}}}

commutes, then there exists a unique morphism \overline{β}: U ⇒ 0 in CG such that \overline{β} ∘ P = β.

Proof. Using part 2 of Proposition 7.9, this Lemma can be seen as a special case of Lemma 7.5. For the reader convenience, we sketch here the direct argument. The uniqueness of \overline{β} immediately follows from the fact that P is essentially surjective. To construct \overline{β}, consider an object X ∈ 𝔾, fix E ∈ 𝔼 and x: X → P E and put

\overline{β}_{X}: \xymatrix{UX \ar[r]^{U(x)} & UP E \ar[r]^{β_{E}} & I}

The naturality of \overline{β} amounts to the equation β_{F} ·U(f) = β_{E} for every arrow f: P E → P F in 𝔾. To check this equation, consider the following object in ℕ

(F ⊗ E^{∗}, n_{f}: \xymatrix{P (F ⊗ E^{∗}) ≃ P F ⊗ P E^{∗} \ar[r]^{f⊗id} & P E ⊗ P E^{∗} ≃ I} )

The compatibility between β and k(P) applied to the object (F ⊗ E^{∗}, n_{f}) gives the commutativity of the exterior of the following diagram

[figure: U(P E ⊗ P E^{∗}); U(f⊗id); UI; U(η_{P E}); (∗); I; U_{0}; can; id; UP E ⊗ UP E^{∗}; β_{E}⊗β_{E^{∗}}; U_{2}^{P E,P E^{∗}}; U(f)⊗id; I ⊗ I; id; UP F ⊗ UP E^{∗}; U_{2}^{P F,P E^{∗}}; β_{F} ⊗β_{E^{∗}}; I ⊗ I; U(P F ⊗ P E^{∗}); U(P_{2}^{F,E^{∗}}); UP (F ⊗ E^{∗}); β_{F ⊗E^{∗}}; I; can]

Since the four unlabelled regions commute by naturality and monoidality of β, the region labelled (∗) also commutes. Tensoring (∗) with UP E we get β_{F} · U(f) = β_{E}, as requested. The rest of the proof is now easy.

Consider now ℕ = 𝕂er(P) → 𝔼 → 𝔾 → 𝔼q_{s}(𝔸) as at the beginning of this section, and 𝕂er(δ_{P}) → ℂ^{1}(P, 𝔸) → ℤ^{2}(P, 𝔸, φ) as in Subsection 5.3. We have the following result.

6.4. Proposition. There is a homomorphism of symmetric categorical groups

Σ: 𝕂er(δ_{P}) → \underline{ℍom}(ℕ, 𝔸)

Moreover :

1. If P is essentially surjective, then Σ is faithful.

2. If P has a functorial section, then Σ is an equivalence.

Proof. Construction of the functor Σ. Recall that an object in 𝕂er(δ_{P}) is given by G: 𝔼 → 𝔸 in ℂ^{1}(𝔼, 𝔸) together with an equivalence class [L, α] with L: 𝔾 × 𝔾 → 𝔸 in ℤ^{2}(𝔾, 𝔸, φ) and α: δ_{𝔼}(G) ⇒ P_{ℤ}(L) in ℤ^{2}(𝔼, 𝔸, φ). We define Σ(G, [L, α]) to be the restriction of G to the kernel of P

Σ(G, [L, α]) = \xymatrix{ℕ \ar[r]^{K(P)} & 𝔼 \ar[r]^{G} & 𝔸}

The monoidal structure G(K(P)(M, m)) ⊗ G(K(P)(N, n)) → G(K(P)((M, m) ⊗ (N, n))) is given by

\xymatrix{GM ⊗ GN \ar[r]^{c} & GN ⊗ GM \ar[r]^{φ_{0}^{GN} ⊗id} & ^{I}GN ⊗ GM \ar[r]^{^{m^{−1}}GN⊗id} & ^{P M}GN ⊗ GM \ar[d]^{_{\overline{α}_{M,N}}} \\ G(M ⊗ N) \ar[r]^{r_{L}^{I}=l_{L}^{I}} & L(I, I) ⊗ G(M ⊗ N) &  & L(P M, P N) ⊗ G(M ⊗ N) \ar[ll]_{L(m,n)⊗id}}

where \overline{α}_{M,N} is obtained in a canonical way from

α_{M,N}: ^{P M}GN ⊗ G^{∗}(M ⊗ N) ⊗ GM → L(P M, P N)

The equivariant structure G(K(P)(E • (N, n))) → ^{P E}G(K(P)(N, n)) is obtained in a similar way, using φ_{2}, α_{N,E}, α_{E,N⊗E^{∗}} and α_{E,E^{∗}}. The functor Σ is well-defined on objects. Indeed, [L, α] = [L′, α′] means that there exists λ: L ⇒ L′ in ℤ^{2}(𝔾, 𝔸, φ) such that P_{ℤ}(λ) · α = α′. This implies the commutativity of the following diagram, which expresses the fact that the monoidal structure of Σ(G, [L, α]) does not depend on the choice of L and α (and similarly for the equivariant structure)

\xymatrix{L(P M, P N) ⊗ G(M ⊗ N) \ar[r]^{L(m,n)⊗id} \ar[dd]^{λ_{P M,P N} ⊗id} & L(I, I) ⊗ G(M ⊗ N) \ar[dd]_{λ_{I,I}⊗id} \\ ^{P M}GN ⊗ GM \ar[u]^{_{\overline{α}_{M,N}}} \ar[d]_{_{\overline{α′}_{M,N}}} & G(M ⊗ N) \ar[u]_{r_{L}^{I}⊗id} \ar[d]^{r_{L′}^{I}⊗id} \\ L′(P M, P N) ⊗ G(M ⊗ N) \ar[r]_{L′(m,n)⊗id} & L′(I, I) ⊗ G(M ⊗ N)}

Recall now that an arrow [U, β]: (G, [L, α]) ⇒ (G′, [L′, α′]) in 𝕂er(δ_{P}) is a class of pairs with U: 𝔾 → 𝔸 in ℂ^{1}(𝔾, 𝔸) and β: G ⇒ P_{ℂ}(U) ⊗ G′ in ℂ^{1}(𝔼, 𝔸) such that there exists λ: L ⇒ δ_{𝔾}(U)⊗L′ in ℤ^{2}(𝔾, 𝔸, φ) such that P_{ℤ}(λ)·α = (δ_{𝔼}(P_{ℂ}(U))⊗α′)·δ_{𝔼}(β). We define

Σ[U, β]: Σ(G, [L, α]) = G · K(P) ⇒ G′ · K(P) = Σ(G′, [L′, α′])

Σ[U, β]_{(N,n)} : \xymatrix{GN \ar[r]^{β_{N}} & UP N ⊗ G′N \ar[r]^{U(n)⊗id} & UI ⊗ G′N ≃ G′N}

The functor Σ is well-defined on arrows. Indeed, [U, β] = [U′, β′] means that there exists γ: U ⇒ U′ in ℂ^{1}(𝔾, 𝔸) such that (P_{ℂ}(γ) ⊗ G′) · β = β′. This implies the commutativity of the following diagram, which expresses the fact that Σ[U, β] does not depend on the choice of U and β

\xymatrix{UP N ⊗ G′N \ar[r]^{U(n)⊗id} \ar[dd]^{γ_{N} ⊗id} & UI ⊗ G′N \ar[dd]_{γ_{I}⊗id} \\ GN \ar[u]^{β_{N}} \ar[d]_{β′_{N}} & I ⊗ G′N \ar[u]_{U_{0}⊗id} \ar[d]^{U′_{0}⊗id} \\ U′P N ⊗ G′N \ar[r]_{U′(n)⊗id} & U′I ⊗ G′N}

Since the symmetric monoidal structure of 𝕂er(δ_{P}) and of \underline{ℍom}(ℕ, 𝔸) are both induced by that of 𝔸, the functor Σ is a homomorphism in SCG.

1. Σ is faithful. This is equivalent to prove that π_{1}(Σ): π_{1}(𝕂er(δ_{P})) → π_{1}(\underline{ℍom}(ℕ, 𝔸)) is injective (Proposition 1.1 in [33]). Using Lemma 6.2, we get the following explicit description of π_{1}(Σ): for an element [U: 𝔾 → 𝔸, β: 0 ⇒ P_{ℂ}(U)] in π_{1}(𝕂er(δ_{P})),

π_{1}(Σ)[U, β]: π_{0}(ℕ) → π_{0}(𝔸) , [N, n] ↦ \xymatrix{I \ar[r]^{β_{N}} & UP N \ar[r]^{U(n)} & UI \ar[r]^{U_{0}^{−1}} & I}

Assume now that [U, β] is in the kernel of π_{1}(Σ). This means that, for any object (N, n) in the kernel of P, one has that

\xymatrix{I \ar[r]^{β_{N}} \ar[dr]_{U_{0}} & UP N \ar[d]^{U(n)} \\  & UI}

commutes. This is precisely the compatibility condition requested in Lemma 6.3. Since P is essentially surjective, from Lemma 6.3. we get a (unique) γ: 0 ⇒ U in ℂ^{1}(𝔾, 𝔸) such that γ ∘ P = β. Therefore, [U, β] = 0 in π_{1}(𝕂er(δ_{P})) and π_{1}(Σ) is injective. Observe that we can apply Lemma 6.3 even if U in general is not monoidal. Indeed, since [U, β] is in the kernel of π_{1}(Σ), there exists λ: 0 ⇒ δ_{𝔾}(U) in ℤ^{2}(𝔾, 𝔸, φ) such that P_{ℤ}(λ) = δ_{𝔼}(β). Explicitly, this gives

λ_{X,Y} : I → ^{X}UY ⊗ U^{∗}(X ⊗ Y) ⊗ UX

so that U is a derivation and β is a morphism of derivations, and derivations correspond to homomorphisms into the semi-direct product (see [19] or [24]).

2. Assume now that P has a functorial section, that is, there exists a functor S: 𝔾 → 𝔼 and a natural transformation s: Id ⇒ P · S. With no lost of generality, we can assume that there exists an arrow S_{0} : I → SI such that P (S_{0}) · P_{0} = s_{I}. The functor Σ is essentially surjective. Let

(H, H_{2}): ℕ → 𝔸 , H_{φ}^{E,(N,n)} : H(E • (N, n)) → ^{P E}H(N, n)

be an object in \underline{ℍom}(ℕ, 𝔸). We look for an object (G_{S}, [L_{S}, α_{S}]) in 𝕂er(δ_{P}) such that Σ(G_{S}, [L_{S}, α_{S}]) ≃ (H, H_{2}, H_{φ}). From an object E ∈ 𝔼, we construct an arrow

e_{s}: \xymatrix{P (E ⊗ (SP E)^{∗}) ≃ P E ⊗ (P SP E)^{∗} \ar[r]^{id⊗s_{P E}^{∗}} & P E ⊗ (P E)^{∗} ≃ I}

and we define

G_{S}: 𝔼 → 𝔸 , G_{S}(E) = H(E ⊗ (SP E)^{∗}, e_{s})

The functor G_{S} is defined in a similar way on arrows. Using S_{0} we get that G_{S} is in ℂ^{1}(𝔼, 𝔸). From objects X, Y ∈ 𝔾, we construct now an arrow

(x, y)_{s}: P (S(X ⊗ Y) ⊗ (SY)^{∗} ⊗ (SX)^{∗}) → I

Up to the constraint P_{2}, (x, y)_{s} is given by

\xymatrix{P S(X ⊗ Y) ⊗ (P SY)^{∗} ⊗ (P SX)^{∗} \ar[r]^{s_{X⊗Y} ⊗s_{Y}^{∗} ⊗s_{X}^{∗}} & X ⊗ Y ⊗ Y^{∗} ⊗ X^{∗} ≃ I}

We define

L_{S}: 𝔾 × 𝔾 → 𝔸 , L_{S}(X, Y) = H(S(X ⊗ Y) ⊗ (SY)^{∗} ⊗ (SX)^{∗}, (x, y)_{s})

L_{S} is defined in a similar way on arrows. To have a factor set, we still need the constraints a_{L}, r_{L}, l_{L} as in Definition 4.2. We will go back to this point in few lines, let us now construct α_{S}: δ_{𝔼}(G_{S}) ⇒ P_{ℤ}(L_{S}). The idea is to go back to the proof of Proposition 2.4, check that δ_{E}(g_{s}) = p_{2}(f_{s}) and then follow step-by-step such a proof. The output is the following natural family of arrows (without writing the arrow-part of the objects of ℕ)

\xymatrix{(1) \ar[r]^{H_{φ}^{−1}⊗can⊗id} & (2) \ar[r]^{c⊗id} & (3) \ar[r]^{H_{2}} & (4) \ar[r]^{can} & (5)}

where

(1) = ^{P X}H(Y ⊗ (SP Y)^{∗}) ⊗ H(X ⊗ Y ⊗ (SP (X ⊗ Y))^{∗})^{∗} ⊗ H(X ⊗ (SP X)^{∗})

(2) = H(X ⊗ Y ⊗ (SP Y)^{∗} ⊗ X^{∗}) ⊗ H(SP (X ⊗ Y) ⊗ Y^{∗} ⊗ X^{∗}) ⊗ H(X ⊗ (SP X)^{∗})

(3) = H(SP (X ⊗ Y) ⊗ Y^{∗} ⊗ X^{∗}) ⊗ H(X ⊗ Y ⊗ (SP Y)^{∗} ⊗ X^{∗}) ⊗ H(X ⊗ (SP X)^{∗})

(4) = H(SP (X ⊗ Y) ⊗ Y^{∗} ⊗ X^{∗} ⊗ X ⊗ Y ⊗ (SP Y)^{∗} ⊗ X^{∗} ⊗ X ⊗ (SP X)^{∗})

(5) = H(SP (X ⊗ Y) ⊗ (SP Y)^{∗} ⊗ (SP X)^{∗}) In the same way, to construct the constraints a_{L}, r_{L}, l_{L} one can make explicit the fact that the map f_{s}: G × G → A considered in the proof of Proposition 2.4 is a 2-cocycle, and then follow step-by-step such a proof. We omit this part, as well as the straightforward construction of an isomorphism Σ(G_{S}, [L_{S}, α_{S}]) ≃ (H, H_{2}, H_{φ}). The functor Σ is full. This is equivalent to prove that π_{0}(𝕂er(Σ)) = 0 (Proposition 2.1 in [33]). An object in 𝕂er(Σ) is an object (G, [L, α]) in 𝕂er(δ_{P}) together with a morphism μ: 0 ⇒ Σ(G, [L, α]) in \underline{ℍom}(ℕ, 𝔸), that is, μ: 0 ⇒ G·K(P) is a monoidal and equivariant natural transformation. To prove that [G, [L, α]] = 0, we need U_{S}: 𝔾 → 𝔸 in ℂ^{1}(𝔾, 𝔸), β_{S}: G ⇒ P_{ℂ}(U_{S}) in ℂ^{1}(𝔼, 𝔸) and λ_{S}: L ⇒ δ_{𝔾}(U_{S}) in ℤ^{2}(𝔾, 𝔸, φ). We define

U_{S}: \xymatrix{𝔾 \ar[r]^{S} & 𝔼 \ar[r]^{G} & 𝔸}

We construct β_{S} in two steps. Consider first the following natural family of arrows depending on the object X ∈ 𝔾

\xymatrix{I \ar[r]^{G_{0}} & GI \ar[r]^{G(η_{SX})} & G(SX ⊗ (SX)^{∗}) \ar[r]^{\overline{α}} & GSX ⊗ ^{P SX}G(S^{∗}X) ⊗ L^{∗}(P SX, (P SX)^{∗}) \ar[d]_{id⊗^{s_{X}}G(S^{∗}X)⊗L^{∗}(s_{X},s_{X}^{∗})} \\  &  &  & GSX ⊗ ^{X}G(S^{∗}X) ⊗ L^{∗}(X, X^{∗})}

with \overline{α} is obtained in a canonical way from α. Tensoring with G^{∗}SX, we get

b_{S}(X): G^{∗}SX → ^{X}G(S^{∗}X) ⊗ L^{∗}(X, X^{∗})

Fix now an object E ∈ 𝔼 and consider the object (E ⊗ S^{∗}P E, e_{s}: P (E ⊗ S^{∗}P E) → I) of ℕ already used to prove that Σ is essentially surjective. We can construct the following natural family of arrows depending on E ∈ 𝔼

\xymatrix{I \ar[rr]^{μ_{(E⊗S^{∗}P E,es)}} &  & G(E ⊗ S^{∗}P E) \ar[r]^{α̃} & GE ⊗ ^{P E}G(S^{∗}P E) ⊗ L^{∗}(P E, (P SP E)^{∗}) \ar[d]_{id⊗L^{∗}(id,s_{P E}^{∗})} \\  & GE ⊗ G^{∗}SP E &  & GE ⊗ ^{P E}G(S^{∗}P E) ⊗ L^{∗}(P E, (P E)^{∗}) \ar[ll]_{id⊗b_{S}(P E)^{−1}}}

with α̃ obtained once again from α in a canonical way. Tensoring with GSP E we get

β_{S}(E): U_{S}P E = GSP E → GE

as required. It remains to construct

λ_{S}(X, Y): L(X, Y) → ^{X}U_{S}X ⊗ U_{S}^{∗}(X ⊗ Y) ⊗ U_{S}X

The argument is similar to the one used to prove that Σ is faithful, and we only sketch it. Using β_{S} and α, we construct

τ_{E,F} : \xymatrix{L(P E, P F) \ar[r]^{α_{E,F}^{−1}} & ^{P E}GF ⊗ G^{∗}(E ⊗ F) ⊗ GE \ar[d]^{^{P E}β_{S}(F)⊗β_{S}^{∗}(E⊗F)⊗β_{S}(E)} \\ & ^{P E}U_{S}P F ⊗ U_{S}^{∗}P (E ⊗ F) ⊗ U_{S}P E}

Now we have to prove that τ is compatible with k(P). This means that for every (N, n), (M, m) ∈ ℕ, the following diagram must be commutative

\xymatrix{L(P N, P M) \ar[d]_{L(n,m)} \ar[rr]^{τ_{N,M}} &  & ^{P N}U_{S}P M ⊗ U_{S}^{∗}(P N ⊗ P M) ⊗ U_{S}P N \ar[d]^{^{n}U_{S}(m)⊗U_{S}^{∗}(n⊗m)⊗U_{S}(n)} \\ L(I, I) & I \ar[l]^{can} \ar[r]_{can} & ^{I}U_{S}I ⊗ U_{S}^{∗}(I ⊗ I) ⊗ U_{S}I}

If this is the case, then there exists a unique λ_{S}: L ⇒ δ_{𝔾}(U) such that λ_{S}∘(P ×P) = μ, and we have done. Through a (quite intricate) diagram chasing, the commutativity of the above diagram reduces to the fact that if E = K(P)(N, n), then the arrow e_{s}: P (E⊗S^{∗}P E) → I is equal to

\xymatrix{P (N ⊗ S^{∗}P N) \ar[r]^{P (id⊗S^{∗}(n))} & P (N ⊗ I) ≃ P N \ar[r]^{n} & I}

and this is easy to prove using the naturality of s and Lemma 3.2 applied to the arrow n: P N → I.

Putting together Proposition 5.4 and Proposition 6.4, we get the following result.

6.5. Corollary. Let P: 𝔼 → 𝔾 be a homomorphism of categorical groups and 𝔸 a symmetric 𝔾-categorical group, and assume that P has a functorial section. There exist 2-exact sequences of symmetric categorical groups

\xymatrix{𝔻er(𝔾, 𝔸, φ) \ar[r] & 𝔻er(𝔼, 𝔸, φ) \ar[r] & \underline{ℍom}(ℕ, 𝔸) \ar[r] & ℍ^{2}(𝔾, 𝔸, φ) \ar[r] & ℍ^{2}(𝔼, 𝔸, φ) \ar[r] & ℂoker(δ_{P}) \\ ℍ^{1}(𝔾, 𝔸, φ) \ar[r] & ℍ^{1}(𝔼, 𝔸, φ) \ar[r] & \underline{ℍom}(ℕ, 𝔸) \ar[r] & \underline{ℍ}^{2}(𝔾, 𝔸, φ) \ar[r] & \underline{ℍ}^{2}(𝔼, 𝔸, φ) \ar[r] & ℂoker(\underline{δ}_{P})}

7. Categorical crossed modules

Categorical precrossed modules and categorical crossed modules have been introduced in [12], and used in [9] as algebraic models for connected homotopy 3-types. In this section we complete the definitions of [12] so to organize categorical (pre)crossed modules in a 2-category. We also add some results on the quotient categorical group associated with a categorical crossed module.

7.1. . The 2-category of actions. Consider two actions

(φ: 𝔾 × 𝔸 → 𝔸, φ_{0}, φ_{1}, φ_{2}) and (φ′: 𝔾′ × 𝔸′ → 𝔸′, φ′_{0}, φ′_{1}, φ′_{2})

in the sense of Subsection 3.5. A homomorphism of actions is a triple

(R, F, λ): (𝔾, 𝔸, φ) → (𝔾′, 𝔸′, φ′)

with R: 𝔾 → 𝔾′ and F: 𝔸 → 𝔸′ two homomorphisms in CG (with F in SCG if the actions φ and φ′ are symmetric), and

λ^{X,A}: F (^{X}A) → ^{RX}F A

a natural family of arrows satisfying the following coherence conditions :

(act7) \xymatrix{F A \ar[r]^{F (φ_{0}^{A})} \ar[d]_{_{φ′{F A \atop 0}}} & F (^{I}A) \ar[d]^{λ^{I,A}} \\ ^{I}F A \ar[r]_{^{R_{0}}F A} & ^{RI}F A}

(act8) \xymatrix{F (^{X}A) ⊗ F (^{X}B) \ar[d]_{λ^{X,A}⊗λ^{X,B}} \ar[r]^{F_{2}^{^{X}A,^{X}B}} & F (^{X}A ⊗ ^{X}B) \ar[r]^{F (φ_{2}^{X,A,B})} & F (^{X}(A ⊗ B)) \ar[d]^{λ^{X,A⊗B}} \\ ^{RX}F A ⊗ ^{RX}F B \ar[r]_{_{φ′{RX,F A,F B \atop 2}}} & ^{RX}(F A ⊗ F B) \ar[r]_{^{RX}F_{2}^{A,B}} & ^{RX}F (A ⊗ B)}

(act9) \xymatrix{^{RX}F (^{Y}A) \ar[d]_{^{RX}λ^{Y,A}} & F (^{X}(^{Y}A)) \ar[l]_{λ^{X,^{Y}A}} \ar[r]^{F (φ_{1}^{X,Y,A})} & F (^{X⊗Y}A) \ar[d]^{λ^{X⊗Y,A}} \\ ^{RX}(^{RY}F A) \ar[r]_{_{φ′{RX,RY,F A \atop 1}}} & ^{RX⊗RY}F A \ar[r]_{^{R{X,Y \atop 2}} F A} & ^{R(X⊗Y)}F A}

It follows that λ^{X,I} · F (φ_{•}^{X}) · F_{0} = ^{RX}F_{0} · φ′_{•}^{RX}. Given two homomorphisms of actions

(R, F, λ), (R′, F′, λ′): (𝔾, 𝔸, φ) → (𝔾′, 𝔸′, φ′)

a morphism of actions is a pair of morphisms

β: R ⇒ R′ α: F ⇒ F′

in CG such that the following diagram commutes

(act10) \xymatrix{F · φ \ar@{=>}[r]^{λ} \ar@{=>}[d]_{α∘φ} & φ′ · (R × F) \ar@{=>}[d]^{φ′∘(β×α)} \\ F′ · φ \ar@{=>}[r]_{λ′} & φ′ · (R′ × F′)}

Actions with their homomorphisms and morphisms form a 2-category denoted ACT (or SACT if we restrict to symmetric actions). Observe that, if (R, F, λ): (𝔾, 𝔸, φ) → (𝔾′, 𝔸′, φ′) is a homomorphism of actions with 𝔸 = 𝔸′ and F = Id_{𝔸}, then λ becomes a morphism in CG:

[figure: 𝔾′; φ′; ⇑λ; 𝔾; R; φ; 𝔼q(𝔸); or; 𝔾′; φ′; ⇑λ; 𝔾; R; φ; 𝔼q_{s}(𝔸)]

7.2. . The 2-category of categorical precrossed modules. A categorical precrossed module is a 5-tuple

(𝔾, 𝔸, φ: 𝔾 → 𝔼q(𝔸), T: 𝔸 → 𝔾, ν)

with (𝔾, 𝔸, φ) an action, T: 𝔸 → 𝔾 a homomorphism in CG, and

ν_{X,A}: T (^{X}A) ⊗ X → X ⊗ T A

a natural family of arrows satisfying the following coherence conditions (see Definition 2.2 in [12]) :

(pcm1) ν_{X⊗Y,A} · (T (φ_{1}^{X,Y,A}) ⊗ id) = (id ⊗ ν_{Y,A} ⊗ id) · (ν_{X,^{Y}A} ⊗ id)

(pcm2) (id ⊗ T_{2}^{A,B}) · (ν_{X,A} ⊗ id) · (id ⊗ ν_{X,B}) = ν_{X,A⊗B} · (T (φ_{2}^{X,A,B}) ⊗ id) · (T_{2}^{^{X}A,^{X}B} ⊗ id) (pcm3) ν_{I,A} · T (φ_{0}^{A} ⊗ id) · T (l_{A}) = l_{T A}

It follows that ν_{X,I} · T (φ_{•}^{X} ⊗ id) · (T_{0} ⊗ id) · l_{X} = (id ⊗ T_{0}) · r_{X}. Given two categorical precrossed modules

(𝔾, 𝔸, φ: 𝔾 → 𝔼q(𝔸), T: 𝔸 → 𝔾, ν) and (𝔾′, 𝔸′, φ′: 𝔾′ → 𝔼q(𝔸′), T′: 𝔸′ → 𝔾′, ν′)

a homomorphism of categorical precrossed modules is a 4-tuple

(R, F, λ, τ): (𝔾, 𝔸, φ, T, ν) → (𝔾′, 𝔸′, φ′, T′, ν′)

with (R, F, λ): (𝔾, 𝔸, φ) → (𝔾′, 𝔸′, φ′) a homomorphism in ACT , and τ: R · T ⇒ T′ · F a morphism in CG such that the following diagram commutes

(pcm4) \xymatrix{R(T (^{X}A) ⊗ X) \ar[rr]^{R(ν_{X,A})} &  & R(X ⊗ T A) \ar[d]^{(R_{2}^{X,T A})^{−1}} \\ RT (^{X}A) ⊗ RX \ar[u]^{R_{2}^{T (^{X}A),X}} \ar[d]_{τ_{^{X}A}⊗id} &  & RX ⊗ RT A \ar[d]^{id⊗τ_{A}} \\ T′F (^{X}A) ⊗ RX \ar@{-->}[dr]_{T′(λ^{X,A})⊗id} &  & RX ⊗ T′(F A) \ar[dl]^{ν′_{RX,F A}} \\  & T′(^{RX}F A) ⊗ RX}

Given two homomorphisms of categorical precrossed modules

(R, F, λ, τ), (R′, F′, λ′, τ′): (𝔾, 𝔸, φ, T, ν) → (𝔾′, 𝔸′, φ′, T′, ν′)

a morphism of categorical precrossed modules is a morphism

(β, α): (R, F, λ) ⇒ (R′, F′, λ′)

in ACT such that the following diagram commutes

(pcm5) \xymatrix{R · T \ar@{=>}[r]^{τ} \ar@{=>}[d]_{β∘T} & T′ · F \ar@{=>}[d]^{T′∘α} \\ R′ · T \ar@{=>}[r]_{τ′} & T′ · F′}

Categorical precrossed modules with their homomorphisms and morphisms form a 2category denoted CPCM (or SCPCM if we restrict to symmetric actions).

7.3. . The 2-category of categorical crossed modules. A categorical crossed module is a 6-tuple

(𝔾, 𝔸, φ: 𝔾 → 𝔼q(𝔸), T: 𝔸 → 𝔾, ν, χ)

with (𝔾, 𝔸, φ, T, ν) a categorical precrossed module, and

χ_{A,B}: ^{T A}B ⊗ A → A ⊗ B

a natural family of arrows satisfying the following coherence conditions (see Definition 2.4 in [12]) :

(cm1) (id ⊗ χ_{B,C}) · (χ_{A,^{T B}C} ⊗ id) · ((φ_{1}^{T A,T B,C})^{−1} ⊗ id) · (^{T_{2}^{A,B}} C ⊗ id) = χ_{A⊗B,C}

(cm2) (χ_{A,B} ⊗ id) · (id ⊗ χ_{A,C}) = χ_{A,B⊗C} · (φ_{2}^{T A,B,C} ⊗ id)

(cm3) ^{X}χ_{A,B} · φ_{2}^{X,^{T A}B,A} · (φ_{1}^{X,T A,B} ⊗ id)^{−1} · (^{ν_{X,A}}B ⊗ id) · (φ_{1}^{T (^{X}A),X,B} ⊗ id) = φ_{2}^{X,A,B} · χ_{^{X}A,^{X}B}

(cm4) T_{2}^{A,B} · ν_{T A,B} = T (χ_{A,B}) · T_{2}^{^{T A}B,A} (cm5) χ_{I,B} · r_{^{T I}B} · ^{T_{0}}B · φ_{0}^{B} = l_{B} It follows that χ_{A,I} · (φ_{•}^{T A} ⊗ id) · l_{A} = r_{A}. Given two categorical crossed modules

(𝔾, 𝔸, φ: 𝔾 → 𝔼q(𝔸), T: 𝔸 → 𝔾, ν, χ) and (𝔾′, 𝔸′, φ′: 𝔾′ → 𝔼q(𝔸′), T′: 𝔸′ → 𝔾′, ν′, χ′)

a homomorphism of categorical crossed modules is a homomorphism

(R, F, λ, τ): (𝔾, 𝔸, φ, T, ν) → (𝔾′, 𝔸′, φ′, T′, ν′)

in CPCM such that the following diagram commutes

(cm6) \xymatrix{F (^{T A}B ⊗ A) \ar[r]^{F (χ_{A,B})} & F (A ⊗ B) \\ F (^{T A}B) ⊗ F A \ar[u]^{F_{2}^{^{T A}B,A}} & F A ⊗ F B \ar[u]_{F_{2}^{A,B}} \\ ^{RT A}F B ⊗ F A \ar[u]^{λ^{T A,B}⊗id} \ar[r]_{^{τ_{A}}F B⊗id} & ^{T′EA}F B ⊗ F A \ar[u]_{χ′_{F A,F B}}}

Finally, morphisms of categorical crossed modules are the same as morphisms of categorical precrossed modules. Categorical crossed modules with their homomorphisms and morphisms form a 2-category denoted CCM (or SCCM if we restrict to symmetric actions).

7.4. . The quotient categorical group. Let us recall from [12] that, given a categorical crossed module (φ: 𝔾 → 𝔼q(𝔸), T: 𝔸 → 𝔾, ν, χ), we can construct the quotient categorical group

[figure: 𝔾; C(T); ⇓c(T); 𝔸; T; 0; 𝔾/T]

Its construction is the same as for the cokernel of a homomorphism in SCG, see Subsection 3.8, the only difference is the tensor product of arrows : given two arrows [A, f]: \xymatrix{X \ar[r]|{∘} & Y} and [B, g]: \xymatrix{H \ar[r]|{∘} & K} , their tensor product is given by the class of the prearrow with object part A ⊗ ^{Y}B and arrow part

\xymatrix{X ⊗ H \ar[r]^{f⊗g} & T A ⊗ Y ⊗ T B ⊗ K \ar[r]^{id⊗ν_{Y,B}^{−1}⊗id} & T A ⊗ T (^{Y}B) ⊗ Y ⊗ K ≃ T (A ⊗ ^{Y}B) ⊗ Y ⊗ K}

Here is the universal property (in the bilimit style) of the quotient categorical group : given a diagram in CG

[figure: 𝔾; G; ⇓δ; 𝔸; T; 0; ℍ]

if the following diagram commutes

(qcg1) \xymatrix{GX ⊗ GT A \ar[d]_{id⊗δ_{A}} \ar[r]^{G_{2}^{X,T A}} & G(X ⊗ T A) & G(T (^{X}A) ⊗ X) \ar[l]_{G(ν_{X,A})} \\ GX ⊗ I \ar[r]_{can} & I ⊗ GX & GT (^{X}A) ⊗ GX \ar[l]^{δ_{^{X}A}⊗id} \ar[u]_{G_{2}^{T (^{X}A),X}}}

then there exists

[figure: 𝔾/T; G′; ⇓δ′; 𝔾; C(T); G; ℍ]

in CG such that the following diagram commutes

(qcg2) \xymatrix{G′ · C(T) · T \ar@{=>}[r]^{δ′∘T} \ar@{=>}[d]_{G′∘c(T)} & G · T \ar@{=>}[d]^{δ} \\ G′ · 0 & 0 \ar@{=>}[l]^{can}}

Moreover, the pair (G′, δ′) is essentially unique. Conversely, given a pair (G′, δ′) as above, condition (qcg2) defines a morphism δ: G·T ⇒ 0 in CG satisfying condition (qcg1).

As a consequence of its universal property, the quotient categorical group satisfies the following cancellation property.

7.5. Lemma. Let (φ: 𝔾 → 𝔼q(𝔸), T: 𝔸 → 𝔾, ν, χ) be in CCM, and consider a diagram in CG

[figure: 𝔾; C(T); C(T); 𝔾/T; H; 𝔾/T; K; ⇓β; ℍ]

such that the following diagram commutes

(qcg3) [figure: H · C(T) · P; H∘c(T); β∘T; H · 0; can; K · C(T) · T; K∘c(T); K · 0]

There exists a unique morphism α: H ⇒ K such that α ∘ C(T) = β.

Here is how homomorphisms of categorical crossed modules extend to the quotient categorical groups.

7.6. Lemma. Consider a homomorphism in CCM

(R, F, λ, τ): (φ: 𝔾 → 𝔼q(𝔸), T: 𝔸 → 𝔾, ν, χ) → (φ′: 𝔾 → 𝔼q(𝔸′), T′: 𝔸′ → 𝔾′, ν′, χ′)

There exists a diagram in CG

[figure: 𝔾; C(T); R; 𝔾/T; R̃; 𝔾′; C(T′); ⇓τ̃; 𝔾′/T′]

such that the following diagram commutes

(qcg4) [figure: R̃ · C(T) · T; R̃∘c(T); τ̃∘T; C(T′) · R · T; C(T′)∘τ; C(T′) · T′ · F; c(T′)∘F; R̃ · 0; can; 0 · F]

Proof. Define

δ: \xymatrix{C(T′) · R · T \ar@{=>}[r]^{C(T′)∘τ} & C(T′) · T′ · F \ar@{=>}[r]^{c(T′)∘F} & 0 · F \ar@{=>}[r]^{can} & 0}

Check condition (qcg1) on δ using condition (pcm4) on τ, and apply the universal property of 𝔾/T. Condition (qcg4) is now a special case of condition (qcg2).

7.7. Remark. To complete Lemma 7.6, consider a morphism in CCM

(β, α): (R, F, λ, τ) ⇒ (R′, F′, λ′, τ′): (𝔾, 𝔸, φ, T, ν, χ) → (𝔾′, 𝔸′, φ′, T′, ν′, χ′)

There exists a unique morphism β̃: R̃ ⇒ R̃′ such that the following diagram commutes

(qcg5) \xymatrix{R̃ · C(T) \ar@{=>}[r]^{τ̃} \ar@{=>}[d]_{β̃∘C(T)} & C(T′) · R \ar@{=>}[d]^{C(T′)∘β} \\ R̃′ \ar@{=>}[r]_{τ̃′} & C(T′) · R′}

We adopt the following terminology, introduced in [12].

7.8. Definition. Let 𝔾 be a categorical group. A normal sub-categorical group of 𝔾 is a categorical crossed module (𝔾, 𝔸, φ: 𝔾 → 𝔼q(𝔸), T: 𝔸 → 𝔾, ν, χ) with T faithful.

From Example 2.6.v in [12], we know that the kernel of a homomorphism F: 𝔾 → ℍ in CG has a canonical structure of normal sub-categorical group. Moreover, condition (qcg1) is verified if we take δ = k(F). Here is the relation between kernels and quotients (Propositions 3.6 and 3.8 in [12]).

7.9. Proposition.

1. Let (φ: 𝔾 → 𝔼q(𝔸), T: 𝔸 → 𝔾, ν, χ) be in CCM and consider the factorization through the kernel as in the following diagram

\xymatrix{𝔸 \ar[r]^{T} \ar[dr]_{T′} & 𝔾 \ar[r]^{C(T)} & 𝔾/T \\  & 𝕂er(C(T)) \ar[u]_{K(C(T))}}

(a) T′ is a full and essentially surjective homomorphism of 𝔾-categorical crossed modules ;

(b) if T is faithful, then 𝔸 and Ker(C(T)) are equivalent normal sub-categorical

groups of 𝔾.

2. Let F: 𝔾 → ℍ be in CG and consider the factorization through the quotient as in the following diagram

\xymatrix{𝕂er(F) \ar[r]^{K(F)} & 𝔾 \ar[d]_{C(K(F))} \ar[r]^{F} & ℍ \\  & 𝔾/K(F) \ar[ur]_{F′}}

(a) F′ is full and faithful ;

(b) in particular, F is a quotient (i.e., F′ is an equivalence) iff F is essentially surjective.

To end this section, we point out a simple but relevant situation (see Remark 8.5) where the universal property of the quotient categorical group holds.

7.10. Lemma. Let (φ: 𝔾 → 𝔼q_{s}(𝔸), T: 𝔸 → 𝔾, ν, χ) be in SCCM. The morphism in CG

[figure: 𝔾; φ; ⇓δ; 𝔸; T; 0; 𝔼q_{s}(𝔸)]

with δ_{A,B}: ^{T A}B → B defined by

\xymatrix{^{T A}B ≃ ^{T A}B ⊗ A ⊗ A^{∗} \ar[r]^{χ_{A,B}⊗id} & A ⊗ B ⊗ A^{∗} \ar[r]^{c_{A,B}⊗id} & B ⊗ A ⊗ A^{∗} ≃ B}

satisfies condition (qcg1).

Proof. This follows from conditions (act6) in Subsection 3.5 and (cm3) in Subsection 7.3.

8. Extensions with symmetric kernel

8.1. Definition. Let 𝔸 and 𝔾 be categorical groups, with 𝔸 symmetric. An extension of 𝔾 by 𝔸 is a 6-tuple

(𝔹, ψ: 𝔹 → 𝔼q_{s}(𝔸), T: 𝔸 → 𝔹, ν, χ, S)

where (𝔹, 𝔸, ψ, T, ν, χ) is an object in SCCM with T faithful (that is, 𝔸 is a symmetric normal sub-categorical group of 𝔹), and S: 𝔹/T → 𝔾 is an equivalence in CG. When ψ, ν and χ are understood, we denote an extension of 𝔾 by 𝔸 by

\xymatrix{𝔸 \ar[r]^{T} & 𝔹 \ar[r]^{C(T)} & 𝔹/T \ar[r]^{S} & 𝔾}

A homomorphism of extensions is a 4-tuple

(R, λ, τ, m): (𝔹, 𝔸, ψ, T, ν, χ, S) → (𝔹′, 𝔸, ψ′, T′, ν′, χ′, S′)

where m: S ⇒ S′ · R̃ is a morphism in CG (with R̃ induced by λ and τ as in Lemma 7.6), and (R, Id_{𝔸}, λ, τ): (𝔹, 𝔸, ψ, T, ν, χ) → (𝔹′, 𝔸, ψ′, T′, ν′, χ′,) is a homomorphism in SCCM

[figure: 𝔹; R; ψ; C(T); 𝔹/T; R̃; S; 𝔸; T; T′; ⇓τ; 𝔼q_{s}(𝔸); ⇓λ; 𝔾; ⇓m; 𝔹′; ψ′; C(T′); 𝔹′/T′; S′]

A morphism of extensions

β: (R, λ, τ, m) ⇒ (R′, λ′, τ′, m′): (𝔹, 𝔸, ψ, T, ν, χ, S) → (𝔹′, 𝔸, ψ′, T′, ν′, χ′, S′)

is a morphism (β, id): (R, Id_{𝔸}, λ, τ) ⇒ (R′, Id_{𝔸}, λ′, τ′) in SCCM such that the following diagram commutes

(ext1) \xymatrix{S′ · R̃ \ar@{=>}[rr]^{S′∘β̃} &  & S′ · R̃′ \\  & S \ar@{=>}[ul]^{m} \ar@{=>}[ur]_{m′}}

where β̃ is obtained by β as in Remark 7.7. Extensions of 𝔾 by 𝔸 with their homomorphisms and morphisms form a 2-category denoted 𝔼xt(𝔾, 𝔸).

8.2. Remark. Because of Proposition 7.9, in any extension of 𝔾 by 𝔸 the symmetric categorical group 𝔸 is equivalent, as a normal sub-categorical group, to the kernel of S · C(T): 𝔹 → 𝔹/T → 𝔾.

8.3. Remark. Using Lemma 7.5, in the definition of homomorphism of extensions we can replace the morphism m: S ⇒ S′ · R̃ by a morphism

[figure: 𝔹; C(T); R; 𝔹/T; S; ⇓μ; 𝔾; 𝔹′; C(T′); 𝔹′/T′; S′]

such that the following diagram commutes

(ext2) [figure: S · C(T) · T; μ∘T; S∘c(T); S′ · C(T′) · R · T; S′·C(T′)∘τ; S′ · C(T′) · T′; S′∘c(T); S · 0; can; S′ · 0]

In this case, condition (ext1) for a morphism of extensions is to be written as

(ext1’) \xymatrix{S′ · C(T′) · R \ar@{=>}[rr]^{S′·C(T′)∘β} &  & S′ · C(T′) · R′ \\  & S · C(T) \ar@{=>}[ul]^{μ} \ar@{=>}[ur]_{μ′}}

8.4. Remark. In fact, the 2-category 𝔼xt(𝔾, 𝔸) is a 2-groupoid, that is, any morphism is invertible and any homomorphism is an equivalence. The proof of this fact follows the same lines of the proof of Proposition 2.8 in [5], just replace the cokernel of a homomorphism in SCG used in [5] by the quotient categorical group (see also Proposition 10.3 in [29]).

8.5. Remark. Consider an extension of 𝔾 by 𝔸

\xymatrix{𝔸 \ar[r]^{T} & 𝔹 \ar[r]^{C(T)} & 𝔹/T \ar[r]^{S} & 𝔾}

Since 𝔸 is symmetric, we get a symmetric action of 𝔾 on 𝔸 as follows

\xymatrix{𝔾 \ar[r]^{S^{−1}} & 𝔹/T \ar[r]^{ψ̃} & 𝔼q_{s}(𝔸)}

where (ψ̃, δ̃) is the factorization through the quotient

[figure: 𝔸; T; 0; 𝔹; C(T); ψ; 𝔹/T; ψ̃; ⇓δ; 𝔼q_{s}(𝔸); ⇓δ̃]

the morphism δ: ψ · T ⇒ 0 is as in Lemma 7.10, and S^{−1} is the essentially unique adjoint quasi-inverse of S. We recall ψ̃ · S^{−1} : 𝔾 → 𝔼q_{s}(𝔸) as the action of 𝔾 on 𝔸 induced by a given extension of 𝔾 by 𝔸.

The next proposition allows us to use the action induced by an extension as a parameter to classify the extensions with symmetric kernel.

8.6. Proposition. Consider a homomorphism of extensions as in Definition 8.1

[figure: 𝔹; R; ψ; C(T); 𝔹/T; R̃; S; 𝔸; T; T′; ⇓τ; 𝔼q_{s}(𝔸); ⇓λ; 𝔾; ⇓m; 𝔹′; ψ′; C(T′); 𝔹′/T′; S′]

The actions of 𝔾 on 𝔸 induced by the two extensions are equivalent.

This statement means that there exists an equivalence of the form

(Id, Id, ≃): (𝔾, 𝔸, ψ̃ · S^{−1}) → (𝔾, 𝔸, ψ̃′ · S^{′−1})

in the 2-category SACT of symmetric actions described in Subsection 7.1.

To help reading the proof, let us resume the whole situation in the following diagram

[figure: 𝔼q_{s}(𝔸); 𝔹; C(T); R; ψ; ⇐δ; ⇐δ̃; 𝔹/T; R̃; S; ψ̃; 𝔸; 0; 0; T; T′; ⇓τ; ⇓τ̃; ⇓m; 𝔾; 𝔹′; C(T′); ψ′; ⇐δ′; ⇐δ̃′; 𝔹′/T′; S′; ψ̃′; 𝔼q_{s}(𝔸)]

Proof. The proof essentially consists in finding a morphism λ̃ to fill-in the following diagram in CG

[figure: 𝔾; ⇒m; 𝔹/T; R̃; ψ̃; S; 𝔹′/T′; ψ̃′; S′; 𝔼q_{s}(𝔸); ⇒λ̃]

We start constructing the following morphism

\xymatrix{ψ̃′ · R̃ · C(T) \ar@{=>}[r]^{ψ̃′∘τ̃} & ψ̃′ · C(T′) · R \ar@{=>}[r]^{δ̃′∘R} & ψ′ · R & ψ \ar@{=>}[l]_{λ} & ψ̃ · C(T) \ar@{=>}[l]_{δ̃}}

In order to use the cancellation property of the quotient (Lemma 7.5), we have to check condition (qcg3). Using condition (qcg2) on δ̃ and on δ̃′ and condition (qcg4) on τ̃, condition (qcg3) reduces to the commutativity of

\xymatrix{ψ′ · T′ \ar@{=>}[d]_{δ′} & ψ′ · R · T \ar@{=>}[l]_{ψ′∘τ} \ar@{=>}[d]^{λ∘T} \\ 0 & ψ · T \ar@{=>}[l]^{δ}}

Finally, using the description of δ and δ′ given in Lemma 7.10, the commutativity of the previous diagram amounts to condition (cm6).

In the next definition we specialize extensions (Definition 8.1) to get φ-extensions for φ a fixed symmetric action.

8.7. Definition. Let φ: 𝔾 → 𝔼q_{s}(𝔸) be a symmetric action. A φ-extension of 𝔾 by 𝔸 is a 7-tuple

(𝔹, ψ: 𝔹 → 𝔼q_{s}(𝔸), T: 𝔸 → 𝔹, ν, χ, S, s)

where

\xymatrix{𝔸 \ar[r]^{T} & 𝔹 \ar[r]^{C(T)} & 𝔹/T \ar[r]^{S} & 𝔾}

is an extension of 𝔾 by 𝔸, and

[figure: 𝔾; φ; ⇓s; 𝔹/T; S; ψ̃; 𝔼q_{s}(𝔸)]

is a morphism in CG (ψ̃ is as in Remark 8.5). A homomorphism of φ-extensions

(R, λ, τ, m): (𝔹, 𝔸, ψ, T, ν, χ, S, s) → (𝔹′, 𝔸, ψ′, T′, ν′, χ′, S′, s′)

is a homomorphism of extensions (R, λ, τ, m) such that the following diagram commutes

(φext1) \xymatrix{φ · S \ar@{=>}[r]^{s} \ar@{=>}[d]_{φ∘m} & ψ̃ \ar@{=>}[d]^{λ̃} \\ φ · S′ · R̃ \ar@{=>}[r]_{s′∘R̃} & ψ̃′ · R̃}

A morphism of φ-extensions

β: (R, λ, τ, m) ⇒ (R′, λ′, τ′, m′): (𝔹, 𝔸, ψ, T, ν, χ, S, s) → (𝔹′, 𝔸, ψ′, T′, ν′, χ′, S′, s′)

is just a morphism of extensions. φ-Extensions of 𝔾 by 𝔸 with their homomorphisms and morphisms form a 2-groupoid denoted 𝕆pext(φ, 𝔾, 𝔸).

8.8. Remark. Using Lemma 7.5 once again, we can reformulate Definition 8.7 as follows : a φ-extension of 𝔾 by 𝔸 is a 7-tuple

(𝔹, ψ: 𝔹 → 𝔼q_{s}(𝔸), T: 𝔸 → 𝔹, ν, χ, S, σ)

where

\xymatrix{𝔸 \ar[r]^{T} & 𝔹 \ar[r]^{C(T)} & 𝔹/T \ar[r]^{S} & 𝔾}

is an extension of 𝔾 by 𝔸, and

[figure: 𝔼q_{s}(𝔸); ⇐σ; 𝔾; φ; 𝔹; ψ; C(T); 𝔹/T; S]

is a morphism in CG such that the following diagram commutes

(φext2) \xymatrix{φ · S · C(T) · T \ar@{=>}[r]^{σ∘T} \ar@{=>}[d]_{φ·S∘c(T)} & ψ · T \ar@{=>}[d]^{δ} \\ φ · S · 0 & 0 \ar@{=>}[l]^{can}}

A homomorphism of φ-extensions

(R, λ, τ, μ): (𝔹, 𝔸, ψ, T, ν, χ, S, σ) → (𝔹′, 𝔸, ψ′, T′, ν′, χ′, S′, σ′)

is a homomorphism of extensions (R, λ, τ, μ) as in Remark 8.3, such that the following diagram commutes

(φext1’) \xymatrix{φ · S · C(T) \ar@{=>}[r]^{φ∘μ} \ar@{=>}[d]_{σ} & φ · S′ · C(T′) · R \ar@{=>}[d]^{σ′∘R} \\ ψ \ar@{=>}[r]_{λ} & ψ′ · R}

Finally, a morphism of φ-extensions

β: (R, λ, τ, μ) ⇒ (R′, λ′, τ′, μ′): (𝔹, 𝔸, ψ, T, ν, χ, S, σ) → (𝔹′, 𝔸, ψ′, T′, ν′, χ′, S′, σ′)

is just a morphism of extensions as in Remark 8.3.

9. Classification of extensions with symmetric kernel

In Subsection 3.8 we have described the cokernel ℂoker(T) of a morphism T: 𝔾 → ℍ in SCG as a categorical group. In fact, ℂoker(T) is obtained from a monoidal bicategory (that, by abuse of notation, we still denote ℂoker(T)) by taking 2-isomorphism classes of 1-arrows as arrows. Explicitly, the objects of ℂoker(T) are those of ℍ, a 1-arrow from A to B is a pair (X ∈ 𝔾, f: A → T X ⊗ B), and a 2-arrow from (X, f: A → T X ⊗ B) to (X′, f′: A → T X′ ⊗ B) is an arrow x: X → X′ in 𝔾 such that (T (x) ⊗ id) · f = f′. (The fact that ℂoker(T) is a monoidal bicategory is mentioned in the introduction of [33], and the whole proof has been done in [23]. The same argument can be developed for the quotient categorical group associated with a categorical crossed module.) In order to compare ℍ^{2}(𝔾, 𝔸, φ) with 𝕆pext(φ, 𝔾, 𝔸), in the next proposition we look at ℍ^{2}(𝔾, 𝔸, φ) as a monoidal bicategory.

9.1. Proposition. Let 𝔾 be a categorical group and φ: 𝔾 → 𝔼q_{s}(𝔸) a symmetric action. There exists a homomorphism of bicategories

\overline{E}: ℍ^{2}(𝔾, 𝔸, φ) → 𝕆pext(φ, 𝔾, 𝔸)

which is locally an equivalence.

Proof. We split the proof in four steps. Step 1. We construct a 2-functor

E: ℤ^{2}(𝔾, 𝔸, φ) → 𝕆pext(φ, 𝔾, 𝔸)

where ℤ^{2}(𝔾, 𝔸, φ) is seen as a 2-category with only identity 2-arrows. Let F: 𝔾 × 𝔾 → 𝔸 be a factor set and consider the categorical group 𝔸 ×_{F} 𝔾 described in Remark 4.3, together with the canonical injection and the canonical projection

\xymatrix{𝔸 \ar[r]^{i_{𝔸}} & 𝔸 ×_{F} 𝔾 \ar[r]^{p_{𝔾}} & 𝔾}

Since i_{𝔸}: 𝔸 → 𝔸 ×_{F} 𝔾 is equivalent to the kernel of p_{𝔾}, we get a canonical structure of normal sub-categorical group on i_{𝔸}: 𝔸 → 𝔸 ×_{F} 𝔾. In particular, the symmetric action ψ: 𝔸 ×_{F} 𝔾 → 𝔼q_{s}(𝔸) is given by

(A,X)B = p_{𝔸}((A, X) ⊗_{F} (B, I) ⊗_{F} (A, X)^{∗})

Moreover, since p_{𝔾} is surjective on objects, by Proposition 7.9 we get a monoidal equivalence S: (𝔸 ×_{F} 𝔾)/i_{𝔸} → 𝔾 such that S · C(i_{𝔸}) = p_{𝔾}. In this way, we get an extension E(F) of 𝔾 by 𝔸

\xymatrix{𝔸 \ar[r]^{i_{𝔸}} & 𝔸 ×_{F} 𝔾 \ar[r]^{C(i_{𝔸})} & (𝔸 ×_{F} 𝔾)/i_{𝔸} \ar[r]^{S} & 𝔾}

which in fact is a φ-extension because there is an obvious morphism σ: φ · p_{𝔾} ⇒ ψ in CG obtained using φ_{2}, φ_{1} and the symmetry of 𝔸. Let now α: F ⇒ F′: 𝔾 × 𝔾 → 𝔸 be a morphism of factor sets. We get a homomorphism of extensions E(α): E(F) ⇒ E(F′) as follows

[figure: 𝔸 ×_{F} 𝔾; Id; p_{𝔾}; 𝔸 ×_{F} 𝔾; Id; ψ; 𝔸; i_{𝔸}; i_{𝔸}; 𝔾; ⇓λ; 𝔼q_{s}(𝔸); 𝔸 ×_{F′} 𝔾; p_{𝔾}; 𝔸 ×_{F′} 𝔾; ψ′]

where λ is given by λ_{(A,X),B} = id ⊗ ^{X}(id ⊗ ^{X^{∗}}α_{X,X^{∗}}^{∗}) ⊗ α_{X,X^{∗}} and Id: 𝔸 ×_{F} 𝔾 → 𝔸 ×_{F′} 𝔾 is the identity functor with monoidal structure given by

\xymatrix{(A, X) ⊗_{F} (B, Y) \ar[r]^{(id⊗α_{X,Y} ,id)} & (A, X) ⊗_{F′} (B, Y)}

In order to check that E(α) is a homomorphism in 𝕆pext(φ, 𝔾, 𝔸), one has to apply Lemma 3.2 to the arrow α_{X,X^{∗}}: F (X, X^{∗}) → F′(X, X^{∗}). Step 2. We construct now a 2-natural transformation

[figure: ℤ^{2}(𝔾, 𝔸, φ); E; ⇓Φ; ℂ^{1}(𝔾, 𝔸); 0; δ; 𝕆pext(φ, 𝔾, 𝔸)]

Let G: 𝔾 → 𝔸 be a cobord. We get a homomorphism of extensions as follows

[figure: 𝔸 ×_{δG} 𝔾; Φ_{G}; p_{𝔾}; 𝔸 ×_{δG} 𝔾; Φ_{G}; ψ; 𝔸; i_{𝔸}; i_{𝔸}; ⇓τ; 𝔾; ⇓λ; 𝔼q_{s}(𝔸); 𝔸 ×_{φ} 𝔾; p_{𝔾}; 𝔸 ×_{φ} 𝔾; ψ′]

where 𝔸 ×_{φ} 𝔾 is constructed by taking F = 0 in Remark 4.3, and the symmetric action ψ′: 𝔸 ×_{φ} 𝔾 → 𝔼q_{s}(𝔸) is constructed as the action ψ in Step 1, but taking once again F = 0; explicitly :

\xymatrix{^{(A,X)}B = A ⊗ ^{X}B ⊗ A^{∗} \ar[r]^{can} & ^{X}B}

The functor Φ_{G} is defined on arrows by

(f, h): (A, X) → (B, Y) ↦ (f ⊗ G(h), h): (A ⊗ GX, X) → (B ⊗ GY, Y)

and its monoidal structure is obtained using φ_{2}, φ_{1} and the symmetry of 𝔸. Finally, the morphism τ: Φ_{G} · i_{𝔸} ⇒ i_{𝔸} is defined by

\xymatrix{Φ_{G}(i_{𝔸}(A)) \ar[r]^{(id⊗G_{0}^{−1},id)} & (A ⊗ I, I) ≃ i_{𝔸}(A)}

and the morphism λ: ψ ⇒ ψ′ · Φ_{G} is the opposite of the morphism σ used in Step 1. Step 3. Now we extend the 2-functor E: ℤ^{2}(𝔾, 𝔸, φ) → 𝕆pext(φ, 𝔾, 𝔸), constructed in Step 1, to a homomorphism of bicategories

\overline{E}: ℍ^{2}(𝔾, 𝔸, φ) → 𝕆pext(φ, 𝔾, 𝔸)

(we limit ourselves to define \overline{E} on objects, 1-arrows and 2-arrows ; to check that \overline{E} is indeed a homomorphism of bicategories is long but essentially straightforward). On objects, \overline{E} is defined as E. Consider now a 1-arrow in ℍ^{2}(𝔾, 𝔸, φ), that is, a pair (G, α) with G in ℂ^{1}(𝔾, 𝔸) and α: F ⊗δG ⇒ F′ in ℤ^{2}(𝔾, 𝔸, φ); the homomorphism \overline{E}(G, α): E(F) ⇒ E(F′) factors through E(F ⊗ δG) and is described in the following diagram, where Φ_{G} and τ are as in Step 2, and Id is the identity functor with monoidal structure determined by α as in Step 1.

[figure: ⇑τ; 𝔸 ×_{F} 𝔾; p_{𝔾}; 𝔸; i_{𝔸}; i_{𝔸}; i_{𝔸}; 𝔸 ×_{F ⊗δG} 𝔾; p_{𝔾}; Φ_{G}; Id; 𝔾; 𝔸 ×_{F′} 𝔾; p_{𝔾}]

Consider now a 2-arrow β: (G, α) ⇒ (G′, α′) in ℍ^{2}(𝔾, 𝔸, φ), that is, a morphism β: G ⇒ G′ in ℂ^{1}(𝔾, 𝔸) such that

\xymatrix{F ⊗ δG \ar@{=>}[rr]^{F ⊗δ(β)} \ar@{=>}[dr]_{α} &  & F ⊗ δG′ \ar@{=>}[dl]^{α′} \\  & F′}

commutes ; the morphism \overline{E}(β) is described in the diagram

[figure: 𝔸 ×_{F} 𝔾; \overline{β}⇒; 𝔸 ×_{F ⊗δG} 𝔾; Φ_{G}; Id; Id; 𝔸 ×_{F ⊗δG′} 𝔾; Φ_{G′}; 𝔸 ×_{F′} 𝔾; Id]

where \overline{β}_{A,X} = (id⊗β_{X}, id): (A⊗GX, X) → (A⊗G′X, X), and the monoidal structure on the identity functor Id is determined by F ⊗ δ(β) as in Step 1. Finally, the fact that the identity natural transformation in the bottom triangle is monoidal is precisely condition α′ · (F ⊗ δ(β)) = α above, and the fact that the natural transformation \overline{β} is monoidal follows from Lemma 3.2 applied to the arrow β_{X⊗Y} : G(X ⊗ Y) → G′(X ⊗ Y). Step 4. Finally, we prove that the homomorphism \overline{E}: ℍ^{2}(𝔾, 𝔸, φ) → 𝕆pext(φ, 𝔾, 𝔸), constructed in Step 3, is locally an equivalence. \overline{E} is locally faithful : if β, β′: G ⇒ G′ are such that \overline{E}(β) = \overline{E}(β′), then for all A ∈ 𝔸 and X ∈ 𝔾 we have

id ⊗ β_{X} = id ⊗ β′_{X}: A ⊗ GX → A ⊗ G′X

and taking A = I we get β_{X} = β′_{X}. \overline{E} is locally full : let (G, α), (G′, α′): F → F′ be 1-arrows in ℍ^{2}(𝔾, 𝔸, φ) and consider a 2arrow \overline{β}: \overline{E}(G, α) ⇒ \overline{E}(G′, α′) in 𝕆pext(φ, 𝔾, 𝔸). Explicitly, \overline{β} is a natural transformation

\overline{β}_{A,X}: Φ_{G}(A, X) = (A ⊗ GX, X) → (A ⊗ G′X, X) = Φ_{G′}(A, X)

and so it has two components

\overline{β}_{1}(A, X): A ⊗ GX → A ⊗ G′X , \overline{β}_{2}(A, X): X → X

We define β_{X}: GX → G′X as follows :

\xymatrix{GX ≃ I ⊗ GX \ar[r]^{\overline{β}_{1}(I,X)} & I ⊗ G′X ≃ G′X}

In this way, condition (pcm5) on \overline{β} gives condition (cob1) on β, that is, β: G ⇒ G′ is a morphism in ℂ^{1}(𝔾, 𝔸), and the monoidal character of \overline{β} gives condition α′ ·(F ⊗δ(β)) = α on β, so that β: (G, α) ⇒ (G′, α′) is a 2-arrow in ℍ^{2}(𝔾, 𝔸, φ). Moreover, condition (act10) on \overline{β} gives that

\overline{β}_{1}(A, X) = id ⊗ β_{X}: A ⊗ GX → A ⊗ G′X

and condition (ext1’) on \overline{β} gives that

\overline{β}_{2}(A, X) = id: X → X

so that \overline{E}(β) = \overline{β}. \overline{E} is locally essentially surjective : consider two cocycles F, F′: 𝔾 × 𝔾 → 𝔸 and a homomorphism (R, τ, μ): \overline{E}(F′) → \overline{E}(F) in 𝕆pext(φ, 𝔾, 𝔸)

[figure: 𝔸 ×_{F} 𝔾; p_{𝔾}; 𝔸; i_{𝔸}; i_{𝔸}; ⇓τ; 𝔾; ⇑μ; 𝔸 ×_{F′} 𝔾; R; p_{𝔾}]

We are going to construct G in ℂ^{1}(𝔾, 𝔸), α: F ⊗δG ⇒ F′ in ℤ^{2}(𝔾, 𝔸, φ) and β: Φ_{G} ⇒ R in 𝕆pext(φ, 𝔾, 𝔸). Let us write

R(A, X) = (R^{1}(A, X), R^{2}(A, X))

for the two components of R, and

τ_{A} = (τ^{1}(A), τ^{2}(A)): (A, I) → (R^{1}(A, I), R^{2}(A, I))

for the two components of τ, and observe that

μ_{A,I} = τ^{2}(A): I → R^{2}(A, I)

because of condition (ext2). Now we put

G = R^{1}(I, −): 𝔾 → 𝔸

As far as the natural transformation β is concerned, it is of the form

β_{A,X} = (β^{1}(A, X), β^{2}(A, X)): Φ_{G}(A, X) = (A ⊗ GX, X) → (R^{1}(A, X), R^{2}(A, X))

and we put

β^{2}(A, X) = μ_{A,X}: X → R^{2}(A, X)

To construct the first component β^{1}(A, X), observe that for any cocycle F one has (A, I) ⊗_{F} (I, X) ≃ (A, X), so that the monoidal structure of R provides a natural family of arrows R_{2}^{A,X} with two components of the form

(ρ^{1}(A, X), ρ^{2}(A, X)): (R^{1}(A, I)⊗R^{1}(I, X), R^{2}(A, I)⊗R^{2}(I, X)) → (R^{1}(A, X), R^{2}(A, X))

and we put

β^{1}(A, X): \xymatrix{A ⊗ R^{1}(I, X) \ar[r]^{τ^{1}(A)⊗id} & R^{1}(A, I) ⊗ R^{1}(I, X) \ar[r]^{ρ^{1}(A,X)} & R^{1}(A, X)}

To construct the natural transformation α, consider the following arrow, where we use μ_{I,X}, μ_{I,Y} and the definition of ⊗_{F} in the first step, the monoidal structure of R in the second step, and φ_{•}^{X} and the definition of ⊗_{F′} in the third step

(R^{1}(I, X) ⊗ ^{X}R^{1}(I, Y) ⊗ F (X, Y), R^{2}(I, X) ⊗ R^{2}(I, Y))

(R^{1}(I, X), R^{2}(I, X)) ⊗_{F} (R^{1}(I, Y), R^{2}(I, Y))

R((I, X) ⊗_{F} (I, Y))

(R^{1}(F′(X, Y), X ⊗ Y), R^{2}(F′(X, Y), X ⊗ Y))

Composing the first component of such arrow with

β^{1}(F′(X, Y), X ⊗ Y)^{−1} : R^{1}(F′(X, Y), X ⊗ Y) → F′(X, Y) ⊗ R^{1}(I, X ⊗ Y)

we get an arrow

GX ⊗ ^{X}GY ⊗ F (X, Y) → F′(X, Y) ⊗ G(X ⊗ Y)

and then, using symmetry and inverses in 𝔸, the needed arrow

α_{X,Y} : F (X, Y) ⊗ ^{X}GY ⊗ G^{∗}(X ⊗ Y) ⊗ GX → F′(X, Y)

This concludes the proof of Proposition 9.1.

9.2. Remark. For any cocycle F: 𝔾 × 𝔾 → 𝔸, the φ-extension

E(F): \xymatrix{𝔸 \ar[r]^{i_{𝔸}} & 𝔸 ×_{F} 𝔾 \ar[r]^{p_{𝔾}} & 𝔾}

described in Step 1 of the proof of Proposition 9.1, has a functorial section given by the canonical injection i_{𝔾}: 𝔾 → 𝔸 ×_{F} 𝔾 (note that i_{𝔾} is not monoidal, unless F = 0). We will say that an extension

\xymatrix{𝔸 \ar[r]^{T} & 𝔹 \ar[r]^{C(T)} & 𝔹/T \ar[r]^{S} & 𝔾}

has a functorial section if there exists a (non necessarily monoidal) functor U: 𝔾 → 𝔹 and a natural transformation

[figure: 𝔹; S·C(T); ⇓u; 𝔾; Id; U; 𝔾]

and we will denote by 𝕆pext_{F S}(φ, 𝔾, 𝔸) the full and 2-full sub-2-category of 𝕆pext(φ, 𝔾, 𝔸) of those φ-extensions which have a functorial section.

9.3. Theorem. Let 𝔾 be a categorical group and φ: 𝔾 → 𝔼q_{s}(𝔸) a symmetric action. The corestriction

\overline{E}: ℍ^{2}(𝔾, 𝔸, φ) → 𝕆pext_{F S}(φ, 𝔾, 𝔸)

of the homomorphism of bicategories \overline{E}: ℍ^{2}(𝔾, 𝔸, φ) → 𝕆pext(φ, 𝔾, 𝔸) of Proposition 9.1, is a biequivalence.

Proof. We have to prove that \overline{E} is essentially surjective on objects, that is, surjective up to equivalence. Let

[figure: 𝔼q_{s}(𝔸); ⇐σ; 𝔾; φ; 𝔸; T; 𝔹; ψ; C(T); 𝔹/T; S]

be a φ-extension with functorial section. Up to the equivalences 𝔾 ≃ 𝔹/T and 𝔸 ≃ 𝕂er(C(T)), we can describe a cocycle F: 𝔹/T × 𝔹/T → 𝕂er(C(T)) using a section

[figure: 𝔹; C(T); ⇓u; 𝔹/T; U; Id; 𝔹/T]

The functor F is defined by

F (X, Y) = (UX ⊗ UY ⊗ U^{∗}(X ⊗ Y), u_{X} ⊗ u_{Y} ⊗ u_{X⊗Y}^{∗})

Let us look for example to its associator

a_{F}^{X,Y,Z}: F (X, Y) ⊗ F (X ⊗ Y, Z) → ^{X}F (Y, Z) ⊗ F (X, Y ⊗ Z)

It is given by the following arrow, where the first step is canonical (use the explicit definition of F) and the second step comes from the fact that the equivalence 𝔸 ≃ 𝕂er(C(T)) is equivariant (first part of Proposition 7.9)

F (X, Y) ⊗ F (X ⊗ Y, Z)

UX ⊗ F (Y, Z) ⊗ U^{∗}X ⊗ F (X, Y ⊗ Z)

XF (Y, Z) ⊗ F (X, Y ⊗ Z)
(^{σ(X)}F (Y,Z))^{−1}⊗id
XF (Y, Z) ⊗ F (X, Y ⊗ Z)

Finally, the needed arrow in 𝕆pext_{F S}(φ, 𝔾, 𝔸) is given by

[figure: 𝔸; T; ≃; ≃ can; 𝔹; C(T); R; 𝔹/T; 𝕂er(C(T)); i_{𝕂er(C(T))}; 𝕂er(C(T)) ×_{F} 𝔹/T; p_{𝔹/T}]

with

RX = ((X ⊗ U^{∗}X, \xymatrix{X ⊗ U^{∗}X \ar[r]^{id⊗u_{X}^{∗}} & X ⊗ X^{∗} ≃ I} ), X)

The previous theorem allows us to define the Baer sum of extensions with symmetric kernel and functorial section. More precisely, if we denote by Opext_{F S}(φ, 𝔾, 𝔸) the groupoid obtained from 𝕆pext_{F S}(φ, 𝔾, 𝔸) taking as arrows 2-equivalence classes of homomorphisms, we get the following corollary.

9.4. Corollary. Let 𝔾 be a categorical group and φ: 𝔾 → 𝔼q_{s}(𝔸) a symmetric action. The biequivalence

\overline{E}: ℍ^{2}(𝔾, 𝔸, φ) → 𝕆pext_{F S}(φ, 𝔾, 𝔸)

induces a structure of symmetric categorical group on the groupoid Opext_{F S}(φ, 𝔾, 𝔸).

References

[1] J.C. Baez and D.A. Lauda, Higher dimensional algebra v : 2-groups. Theory and Applications of Categories 12 (2004) 423–491.

[2] M. Barr and J. Beck, Homology and standard constructions. In Lecture Notes in Mathematics 80, Springer-Verlag 1969.

[3] J. Bénabou, Some remarks on 2-categorical algebra. Bulletin de la Société Mathématique de Belgique 41 (1989) 127–194.

[4] F. Borceux, Handbook of categorical algebra II : categories and structures. Encyclopedia of Mathematics and its Applications 50, Cambridge University Press 1994.

[5] D. Bourn and E.M. Vitale, Extensions of symmetric cat-groups. Homology, Homotopy and Applications 4 (2002) 103–162.

[6] L. Breen, Théorie de Schreier supérieure. Annales Scientifiques de l’Ecole Normale Supérieure 25 (1992) 465–514.

[7] M. Bullejos and A.M. Cegarra, Classifying spaces for monoidal categories through geometric nerves. Canadian Mathematical Bulletin 47 (2004) 321–331.

[8] P. Carrasco and A.M. Cegarra, (Braided) tensor structures on homotopy groupoids and nerves of (braided) categorical groups. Communications in Algebra 24 (1996) 3995– 4058.

[9] P. Carrasco, A.M. Cegarra and A.R. Garzon, The classifying space of a categorical crossed module. Mathematische Nachrichten 283 (2010) 544–567.

[10] P. Carrasco, A.M. Cegarra and A.R. Garzon, Nerves and the classifying spaces for bicategories. Algebraic and Geometric Topology 10 (2010) 219–274.

[11] P. Carrasco, A.R. Garzon and J.G. Miranda, Schreier theory for singular extensions of categorical groups. Communications in Algebra 28 (2000) 2585–2613.

[12] P. Carrasco, A.R. Garzon and E.M. Vitale, On categorical crossed modules. Theory and Applications of Categories 16 (2006) 585–618.

[13] P. Carrasco and J. Martínez-Moreno, Simplicial cohomology with coefficients in symmetric categorical groups. Applied Categorical Structures 12 (2004) 257–285.

[14] A.M. Cegarra and E. Aznar, An exact sequence in the first variable for torsor cohomology : the 2-dimensional theory of obstructions. Journal of Pure and Applied Algebra 39 (1986) 197–250.

[15] A. del Rio, J. Martínez-Moreno and E.M. Vitale, Chain complexes of symmetric categorical groups. Journal of Pure and Applied Algebra 196 (2005) 279–312.

[16] P. Deligne, La formule de dualité globale. In Lecture Notes in Mathematics 305, Springer-Verlag 1973.

[17] A. Frohlich and C.T.C. Wall, Graded monoidal categories. Compositio Mathematica 28 (1974) 229–285.

[18] A.R. Garzon and A. del Rio, Low-dimensional cohomology for categorical groups. Cahiers de Topologie et Géométrie Différentielle Catégorique 44 (2003) 247–280.

[19] A.R. Garzon, H. Inassaridze and A. del Rio, Derivations of categorical groups. Theory and Applications of Categories 13 (2004) 86–105.

[20] P.J. Hilton and U. Stammbach, A course in homological algebra. Graduate Texts in Mathematics 4, Springer-Verlag 1971.

[21] G. Hochschild and J.-P. Serre, Cohomology of group extensions. Transactions of the American Mathematical Society 74 (1953) 110–134.

[22] J. Huebschmann, Extensions de groupes et paires croisés. Comptes Rendus de l’Académie des Sciences de Paris 285 (1977) 993–995.

[23] N. Johnson and A. Osorno, Modeling stable one-types. Theory and Applications of Categories 26 (2012) 520–537.

[24] S. Kasangian, G. Metere and E.M. Vitale, Split extensions, semidirect product, and holomorph of categorical groups. Homology, Homotopy and Applications 8 (2006) 145– 167.

[25] S. Kasangian and E.M. Vitale, Factorization systems for symmetric categorical groups. Theory and Applications of Categories 7 (2000) 47–70.

[26] J.L. Loday, Cohomologie et groupe de Steinberg relatifs. Journal of Algebra 54 (1978) 178–202.

[27] S. Mac Lane, Homology. Die Grundlehren der Mathematischen Wissenschaften 114, Springer-Verlag 1963.

[28] S. Mac Lane, Categories for the working mathematician. Graduate Texts in Mathematics 5, Springer-Verlag 1998 (2nd edition).

[29] A. Rousseau, Bicatégories monoidales et extensions de gr-catégories. Homology, Homotopy and Applications 5 (2003) 437–547.

[30] H.X. Sinh, Gr-catégories. Ph.D. Thesis, Université Paris VII 1975.

[31] M. Takeuchi and K.H. Ulbrich, Complexes of categories with abelian group structure. Journal of Pure and Applied Algebra 27 (1983) 61–73.

[32] K.H. Ulbrich, Group cohomology for Picard categories. Journal of Algebra 91 (1984) 464–498.

[33] E.M. Vitale, A Picard-Brauer exact sequence of categorical groups. Journal of Pure and Applied Algebra 175 (2002) 383–408.

[34] E.M. Vitale, The Snail Lemma. Preprint (2014).

[35] J.H.C. Whitehead, Combinatorial homotopy II. Bulletin of the American Mathematical Society 55 (1949) 453–496.

Departamento de Álgebra, Facultad de Ciencias Universidad de Granada 18071 Granada, Spain Institut de recherche en mathématique et physique Université catholique de Louvain B 1348 Louvain-la-Neuve, Belgium

Email: agarzon@ugr.es, enrico.vitale@uclouvain.be
This article may be accessed at http://www.tac.mta.ca/tac/

THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods. Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication. Full text of the journal is freely available in .dvi, Postscript and PDF from the journal’s server at http://www.tac.mta.ca/tac/ and by ftp. It is archived electronically and in printed paper format.

Subscription information Individual subscribers receive abstracts of articles by e-mail as they are published. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. For institutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh, rrosebrugh@mta.ca.

Information for authors The typesetting language of the journal is TEX, and L^{A}TEX2e is strongly encouraged. Articles should be submitted by e-mail directly to a Transmitting Editor. Please obtain detailed information on submission format and style files at http://www.tac.mta.ca/tac/.

Managing editor. Robert Rosebrugh, Mount Allison University : rrosebrugh@mta.ca

TEXnical editor. Michael Barr, McGill University : barr@math.mcgill.ca

Assistant TEX editor. Gavin Seal, Ecole Polytechnique Fédérale de Lausanne :

gavin seal@fastmail.fm

Transmitting editors.

Clemens Berger, Université de Nice-Sophia Antipolis : cberger@math.unice.fr Richard Blute, Université d’ Ottawa : rblute@uottawa.ca Lawrence Breen, Université de Paris 13 : breen@math.univ-paris13.fr Ronald Brown, University of North Wales : ronnie.profbrown(at)btinternet.com

Valeria de Paiva : valeria.depaiva@gmail.com

Ezra Getzler, Northwestern University : getzler(at)northwestern(dot)edu Kathryn Hess, Ecole Polytechnique Fédérale de Lausanne : kathryn.hess@epfl.ch Martin Hyland, University of Cambridge : M.Hyland@dpmms.cam.ac.uk Anders Kock, University of Aarhus : kock@imf.au.dk Stephen Lack, Macquarie University : steve.lack@mq.edu.au F. William Lawvere, State University of New York at Buffalo : wlawvere@buffalo.edu Tom Leinster, University of Edinburgh : Tom.Leinster@ed.ac.uk Ieke Moerdijk, Radboud University Nijmegen : i.moerdijk@math.ru.nl Susan Niefield, Union College : niefiels@union.edu Robert Paré, Dalhousie University : pare@mathstat.dal.ca Jiri Rosicky, Masaryk University : rosicky@math.muni.cz Giuseppe Rosolini, Università di Genova : rosolini@disi.unige.it Alex Simpson, University of Edinburgh : Alex.Simpson@ed.ac.uk James Stasheff, University of North Carolina : jds@math.upenn.edu Ross Street, Macquarie University : street@math.mq.edu.au Walter Tholen, York University : tholen@mathstat.yorku.ca Myles Tierney, Rutgers University : tierney@math.rutgers.edu R. J. Wood, Dalhousie University : rjwood@mathstat.dal.ca