Homology, Homotopy and Applications, vol.4(1), 2002, pp.103–162

EXTENSIONS OF SYMMETRIC CAT-GROUPS

D. BOURN and E.M. VITALE

(communicated by Antonio Cegarra)

Abstract This paper is an attempt to study extensions of symmetric categorical groups from a structural point of view. Using in a systematic way bilimits in the 2-category of symmetric categorical groups, we develop a theory which closely follows the classical theory of abelian group extensions. The basic results are established for any proper class of extensions, and a cohomological classification is obtained for those extensions whose epi part has a categorical section.

Introduction

Extensions of categorical groups have been extensively studied in [2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 20, 25], and remarkable applications to the classification of homotopy types and to equivariant group cohomology have been found. In most of the works on this subject, the chosen class of “epimorphisms” between categorical groups is that of Grothendieck dense fibrations. This class does not have good 2categorical properties, basically because it is not stable under natural isomorphisms. We have therefore tried to replace the class of Grothendieck dense fibrations with a class having a better 2-categorical behaviour.

Since an epimorphism of abelian groups is a morphism with a set-theoretical section, our first idea was been to consider, as epimorphisms between symmetric categorical groups, those symmetric monoidal functors having a categorical section. This class is stable not only under natural isomorphisms, but also under bi-pullbacks. These facts allowed us to use in a systematic way bi-limits and bi-colimits, so that to develop the basic algebra for extensions in a way which closely follows the classical theory of abelian group extensions.

While doing this, we realized soon that, at least as far as the basic algebra is concerned, only few stability properties are needed. We decided then to rewrite the theory in terms of “proper classes” of extensions, having as examples extensions with a categorical section, extensions with a graph-theoretical section, and also the more general class of extensions in which the epi part is simply a functor essentially surjective on objects. This last class seems to be the most general one which supports the constructions of basic algebra. It has been independently considered by A. Rousseau in his Ph. D. Thesis [20], where extensions of (not necessarily symmetric)

Received April 5, 2002, revised October 24, 2002; published on November 20, 2002. 2000 Mathematics Subject Classification: 18D10, 18G99, 20L05. Key words and phrases: symmetric categorical groups, extensions, bilimits, homological algebra. \begin{matrix}c \\ ◯ \end{matrix} 2002, D. Bourn and E.M. Vitale. Permission to copy for private use granted.

categorical groups are classified following Breen-Grothendieck approach in terms of monoidal fibrations of bitorsors.

The present paper is organized as follows: After recalling basic facts on categorical groups (Section 1), in Section 2 we give the definition of extension of symmetric categorical groups and we show that the 2category EXT(ℂ, 𝔸) of extensions of 𝔸 by ℂ is a 2-groupoid. In Section 3 we compare extensions of symmetric categorical groups with extensions of abelian groups. Section 4 is devoted to trivial and split extensions. In Section 5 we discuss bi-pullbacks and bi-pushouts of symmetric categorical groups. They are used in Sections 6 and 7 to compare EXT and Hom and to show that EXT measures whether a morphism can be extended or lifted. Sections 8 and 9 contain definition, basic facts and some examples of proper classes of extensions. In Section 10, we establish the fundamental 2-exact Hom-Ext sequences obtained by an extension. Section 11 is devoted to projective (and injective) objects. In Section 12 we show that EXT measures the non-exactness of Hom. In Section 13 we define the 2-dimensional analogue of Baer sum, making Ext(ℂ, 𝔸) into a symmetric categorical group. The last two sections are devoted to a cohomological classifications of F-extensions (i.e. those extensions of symmetric categorical groups whose epi part has a categorical section). We define a convenient notion of cobord and symmetric cocycle and we obtain a categorical equivalence between the symmetric categorical group of F-extensions and the symmetric categorical group of cocycles modulo cobords.

All along the paper, we restrict our attention to symmetric categorical groups and monoidal functors compatible with the symmetry. Several results contain a part stated in terms of bi-limits only (or proved using only bi-limits) and a dual part involving bi-colimits. It is a general fact that the part involving only bi-limits holds also for not necessarily symmetric categorical groups and arbitrary monoidal functors.

To end this introduction, let us point out two major open problems.

Clearly, working with symmetric categorical groups instead of braided categorical groups, we loss some relevant examples. The reason why we restrict our attention to the symmetric case is that the cokernel of a morphism (as well as basic properties on 2-exact sequences based on the duality kernel-cokernel, see [16]) is a main ingredient in our analysis, and its description is known, up to now, only for morphisms between symmetric categorical groups. Moreover, several definitions and constructions we use seem much more delicate if we have only a braiding instead of a symmetry (they are so delicate that we suspect that the right context to study the non-symmetric theory could be that of bigroupoids, instead of categorical groups).

Another problem concerns projective objects (in the sense of Definition 11.1) in the 2-category of symmetric categorical groups. The notion of projectivity is crucial in the classical theory, but, unfortunately, we do not know if the 2-category of symmetric categorical groups has enough projective objects. (It would be interesting to solve this problem in order to appreciate the strong specialization done in Sections 14 and 15, where we consider only F-extensions.)

Finally, we would like to thank the members of the Granada school in category theory for several stimulating and useful discussions the second author had with them when he was visiting Granada in summer 2001. We also thank the referee for his help in comparing F-extensions with the classification of extensions established in [20].

1. The 2-category SCG

Let us fix some notations. In any category, the composition is written dia-

grammatically, that is \xymatrix{X \ar[r]^{f} & Y \ar[r]^{g} & Z} is written f · g. The identity arrow is 1 = 1_{X}: X → X (but the identity natural transformation on a functor F is called F). If ℂ is a monoidal category, we write I = I_{ℂ} for the unit object, ⊗ for the tensor product, a = a_{X,Y,Z}: (X ⊗ Y) ⊗ Z → X ⊗ (Y ⊗ Z) for the associativity isomorphism, l = l_{X}: I ⊗ X → X and r = r_{Y} : Y ⊗ I → Y for the unit isomorphisms. (Ab)using the coherence theorem for monoidal categories, we often assume the associativity isomorphism to be the identity. If ℂ is symmetric, we write γ = γ_{X,Y} : X ⊗ Y → Y ⊗ X for the symmetry. If F: ℂ → 𝔻 is a monoidal functor, we denote its monoidal structure by F_{X,Y} : F (X) ⊗ F (Y) → F (X ⊗ Y) and F_{I}: I → F (I). For any category ℂ, we write π_{0}ℂ for the (possibly large) set of isomorphism classes of objects. If ℂ is monoidal, π_{1}ℂ is the commutative (see [18]) monoid ℂ(I, I). A categorical group (cat-group, for short) is a monoidal groupoid in which each object is regular, i.e. it is invertible, up to isomorphism, with respect to the tensor product. If 𝔾 is a cat-group, we fix, for each object X, a dual object X^{∗}, with unit and counit η_{X}: I → X ⊗ X^{∗}, ϵ_{X}: X^{∗} ⊗ X → I. If f: X → Y is in ℂ, f^{∗}: Y^{∗} → X^{∗} is given by

Y^{∗} ≃ Y^{∗} ⊗ \xymatrix{I \ar[r]^{1⊗η_{X}} & Y^{∗}} ⊗ (X ⊗ \xymatrix{X^{∗}) \ar[r]^{1⊗(f⊗1)} & Y^{∗}} ⊗ (Y ⊗ X^{∗}) ≃

≃ (Y^{∗} ⊗ Y) ⊗ \xymatrix{X^{∗} \ar[r]^{ϵ_{Y} ⊗1} & I} ⊗ X^{∗} ≃ X^{∗} .

Basic facts on monoidal categories and cat-groups can be found in [11, 12, 13, 15, 16, 19, 21, 23, 24, 26]. The 2-category SCG has symmetric cat-groups as objects, monoidal functors compatible with the symmetry as 1-cells, and monoidal natural transformations as 2cells (observe that they are natural isomorphisms, because cat-groups are groupoids). Note that π_{0} and π_{1} extend to two 2-functors from SCG to the discrete 2-category Ab of abelian groups. Moreover, a 1-cell F: 𝔸 → 𝔹 in SCG is an equivalence if and only if π_{0}(F) and π_{1}(F) are isomorphisms in Ab (see [16]).

Let us recall now, from [16], the universal property and a construction for kernels and cokernels in SCG. (As a matter of convention, each time that we consider some kind of limit or colimit in SCG, it is to be understood in the sense of bi-limit or bi-colimit, see [22].) Given a 1-cell Σ: 𝔹 → ℂ, its kernel is given by a triple (KerΣ, e_{Σ}, ϵ_{Σ})

[figure: KerΣ; 0; e_{Σ}; ℂ; 𝔹; Σ; ⇑^{ϵ_{Σ}}]

(where 0: KerΣ → ℂ is defined by 0(f) = 1_{I} for each arrow f in KerΣ) such that, for any other triple (𝔻, F, φ)

[figure: 𝔻; 0; F; ℂ; 𝔹; Σ; ⇑^{φ}]

there are F′: 𝔻 → KerΣ and φ′: F′ · e_{Σ} ⇒ F such that the following diagram commutes

\xymatrix{F′ · e_{Σ} · Σ \ar@{=>}[r]^{F′·ϵ_{Σ}} \ar@{=>}[d]_{φ′·Σ} & F′ · 0 \ar@{=>}[d] \\ F · Σ \ar@{=>}[r]_{φ} & 0}

Moreover, if F′′: 𝔻 → KerΣ and φ′′: F′ · e_{Σ} ⇒ F make commutative the analogous diagram, then there is a unique ψ: F′ ⇒ F′′ making commutative the following diagram

\xymatrix{F′ · e_{Σ} \ar@{=>}[rr]^{ψ·e_{Σ}} \ar@{=>}[dr]_{φ′} &  & F′′ · e_{Σ} \ar@{=>}[dl]^{φ′′} \\  & F}

As any bilimit, the kernel is determined, up to equivalence, by its universal property. It can be described as the comma category having as objects pairs of the form (X ∈ 𝔹, ϵ_{X}: Σ(X) → I). The functor e_{Σ} forgets ϵ_{X} and ϵ_{Σ}(X, ϵ_{X}) is ϵ_{X}. In fact, this description could be called the standard kernel, because it satisfies another universal property: if F and φ are as before, there is a unique F′ such that F′ · e_{Σ} = F and F′ · ϵ_{Σ} = φ. We always use the first universal property, but the second one is sometimes useful to avoid φ′ and then to simplify notations. The cokernel is defined by the dual universal property. In the following diagram, we fix the notations for the cokernel of a 1-cell Γ: 𝔸 → 𝔹

[figure: 𝔸; 0; Γ; CokerΓ; 𝔹; P_{Γ}; ⇑^{π_{Γ}}]

Objects of CokerΓ are those of 𝔹. An arrow X → Y in CokerΓ is an equivalence class of pairs of the form (f, A), with A an object of 𝔸 and f: X → Y ⊗ Γ(A) an arrow in 𝔹. Two pairs (f, A) and (f′, A′) are equivalent if there is g: A → A′ in 𝔸 such that f · (1_{Y} ⊗ Γg) = f′. Once again, we get in this way a standard cokernel.

2. The 2-category of extensions

We want now to define extensions of symmetric cat-groups. For this, consider the following diagram in SCG :

[figure: 𝔸; 0; Γ; ℂ; 𝔹; Σ; ⇑^{φ}]

and the corresponding factorizations through the kernel of Σ and the cokernel of Γ

\xymatrix{ & CokerΓ \ar[dr]^{Σ_{0}} \\ 𝔸 \ar[r]^{Γ} \ar[dr]_{Γ_{0}} & 𝔹 \ar[u]^{P_{Γ}} \ar[r]_{Σ} & ℂ \\  & KerΣ \ar[u]_{e_{Σ}}}

Recall, from [16], that the triple (Γ, φ, Σ) is 2-exact when Γ_{0} is full and essentially surjective on objects or, equivalently, when Σ_{0} is full and faithful.

Proposition 2.1. The following conditions are equivalent :

1) The triple (Γ, φ, Σ) is 2-exact, Γ is faithful and Σ is essentially surjective;

2) Γ_{0} is an equivalence and Σ is essentially surjective;

3) Γ is faithful and Σ_{0} is an equivalence.

Proof. Obvious, because P_{Γ} is essentially surjective and e_{Σ} is faithful.

We are ready to give the definition of extension (compare with Definition 3.2.1 in [20]).

Definition 2.2. Let 𝔸, ℂ be in SCG ;

1) An extension of 𝔸 by ℂ is a diagram (Γ, φ, Σ) in SCG

[figure: 𝔸; 0; Γ; ℂ; 𝔹; Σ; ⇑^{φ}]

which satisfies the equivalent conditions of Proposition 2.1. We write also (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ for such an extension;

2) If (Γ, φ, Σ) and (Γ′, φ′, Σ′) are two extensions of 𝔸 by ℂ, a 1-cell

(α, β, γ): (Γ, φ, Σ) → (Γ′, φ′, Σ′)

is given by a 1-cell β and two 2-cells α, γ in SCG as in the following diagram

[figure: 𝔹; β; Σ; ⇓^{α}; 𝔸; Γ; Γ′; ℂ; 𝔹′; Σ′; ⇑^{γ}]

such that the following diagram commutes

\xymatrix{Γ · β · Σ′ \ar@{=>}[r]^{Γ·γ} \ar@{=>}[d]_{α·Σ′} & Γ · Σ \ar@{=>}[d]^{φ} \\ Γ′ · Σ′ \ar@{=>}[r]_{φ′} & 0 ;}

3) If (α, β, γ), (α′, β′, γ′): (Γ, φ, Σ) → (Γ′, φ′, Σ′) are 1-cells, a 2-cell

b: (α, β, γ) ⇒ (α′, β′, γ′)

is a 2-cell b: β ⇒ β′ in SCG such that the following diagrams commute

[figure: Γ · β; Γ·b; α; Γ · β′; α′; β · Σ′; b·Σ′; γ; β′ · Σ′; γ′; Γ′; Σ .]

Proposition 2.3. With the obvious compositions and identities, the data of Definition 2.2 define a 2-category EXT(ℂ, 𝔸).

We write Ext(ℂ, 𝔸) for the classifying category of EXT(ℂ, 𝔸), i.e. Ext(ℂ, 𝔸) has the same objects as EXT(ℂ, 𝔸) and, as arrows, 2-isomorphism classes of 1-cells in EXT(ℂ, 𝔸).

Example 2.4. Let F: 𝔸 → 𝔹 be a morphism in SCG. The following diagrams are extensions

[figure: KerF; 0; e_{F}; Coker(e_{F}); 𝔸; P_{e_{F}}; ⇑^{πe_{F}}; Ker(P_{F}); 0; e_{P_{F}}; CokerF; 𝔹; P_{F}; ⇑^{ϵ_{P_{F}}}]

Our first result on extensions will be the 2-dimensional analogue of the Short Five Lemma, that is the fact that EXT(ℂ, 𝔸) is a 2-groupoid. This has been independently proved also in [20]. We sketch the proof for the reader’s convenience and because we deduce this fact from a slightly more general argument. We need a notation: if A is an abelian group, we write A[1] for the symmetric cat-group with only one object and such that π_{1}(A[1]) = A; we write A[0] for the discrete symmetric cat-group such that π_{0}(A[0]) = A. Both (−)[0] and (−)[1] extend to morphisms. (Of course, A[0] and A[1] are strict cat-groups. Compare with [1], where classical homological algebra is developed in terms of higher strict n-cat-groups.)

Lemma 2.5. Let (f, g): A → B → C be an exact sequence of abelian groups.

1) If f is a monomorphism, then (f[0], =, g[0]): A[0] → B[0] → C[0] is a 2-exact sequences of symmetric cat-groups;

2) If g is an epimorphism, than (f[1], =, g[1]): A[1] → B[1] → C[1] is a 2-exact sequences of symmetric cat-groups.

Proposition 2.6. Consider a morphism in SCG together with its kernel and its cokernel,

\xymatrix{KerF \ar[r]^{e_{F}} & 𝔾 \ar[r]^{F} & ℍ \ar[r]^{P_{F}} & CokerF} .

Then the following diagram (where 1 is the symmetric cat-group with only one arrow, and λ and μ are defined in the proof) is a 2-exact sequence of symmetric cat-groups

[figure: 1; π_{1}(KerF)[0]; π_{1}(e_{F})[0]; π_{1}(𝔾)[0]; π_{1}(F)[0]; π_{1}(ℍ)[0]; λ; KerF; e_{F}; 𝔾; F; ℍ; P_{F}; CokerF; μ; π_{0}(𝔾)[1]; π_{0}(F)[1]; π_{0}(ℍ)[1]; π_{0}(P_{F})[1]; π_{0}(CokerF)[1]; 1]

Proof. Since π_{1} preserves kernels, π_{0} preserves cokernels and both send 2-exact sequences into exact sequences [27], 2-exactness in π_{1}(KerF)[0], π_{1}(𝔾)[0], π_{0}(ℍ)[1] and π_{0}(CokerF)[1] follows from the previous lemma. 2-exactness in 𝔾 and ℍ is obvious. It remains to check 2-exactness in π_{1}(ℍ)[0], KerF, CokerF and π_{0}(𝔾)[1]. This is straightforward, once λ and μ defined. From [16], recall that π_{0}(Ker) and π_{1}(CokerF) are isomorphic (in fact, they are equal if we use the description of Ker and Coker given in Section 1). The functor λ corresponds to π_{1}(P_{F}): π_{1}(ℍ) → π_{1}(CokerF) = π_{0}(KerF); explicitly, λ: π_{1}(ℍ)[0] → KerF sends h: I → I to (I, F_{I}^{−1} · h: F (I) → I). The functor μ corresponds to π_{0}(e_{F}): π_{1}(CokerF) = π_{0}(KerF) → π_{0}(𝔾); explicitly, μ: CokerF → π_{0}(𝔾)[1] sends [f, G]: H_{1} → H_{2} to [G]: ∗ → ∗.

Corollary 2.7. Let F: 𝔾 → ℍ be a morphism in SCG. The following is an exact sequence of abelian groups

[figure: 0; π_{1}(KerF); π_{1}(e_{F}); π_{1}(𝔾); π_{1}(F); π_{1}(ℍ); π_{1}(P_{F}); π_{1}(CokerF); 0; π_{0}(CokerF); π_{0}(ℍ); π_{0}(P_{F}); π_{0}(𝔾); π_{0}(F); π_{0}(KerF); π_{0}(e_{F})]

Proposition 2.8. Let 𝔸 and ℂ be in SCG.

1) The 2-category EXT(ℂ, 𝔸) is a 2-groupoid, i.e. each 2-cell is an isomorphism and each 1-cell is an equivalence;

2) The category Ext(ℂ, 𝔸) is a groupoid.

Proof. 1) We use the notations of Definition 2.2. To prove that 2-cells are isomorphisms, one simply checks that, if b: (α, β, γ) ⇒ (α′, β′, γ′) is a 2-cell in EXT(ℂ, 𝔸), then b^{−1} : β′ ⇒ β is a 2-cell (α′, β′, γ′) ⇒ (α, β, γ) in EXT(ℂ, 𝔸). Consider now a 1cell (α, β, γ): (Γ, φ, Σ) → (Γ′, φ′, Σ′) in EXT(ℂ, 𝔸). Consider also the factorizations \overline{β}: KerΣ → KerΣ′ and \underline{β}: CokerΓ → CokerΓ′ induced, respectively, by (β, γ) and (α, β). A straightforward direct computation shows that the following four squares commute

\xymatrix{π_{0}(KerΣ) \ar[d]_{π_{0}(\overline{β})} & π_{0}(𝔸) \ar[l]_{π_{0}(Γ_{0})} \ar[d]_{1} & π_{1}(CokerΓ) \ar[l]_{π_{1}(μ)} \ar[d]_{π_{1}(\underline{β})} \ar[r]^{π_{1}(Σ_{0})} & π_{1}(ℂ) \ar[d]^{1} \ar[r]^{π_{0}(λ)} & π_{0}(KerΣ) \ar[d]^{π_{0}(\overline{β})} \\ π_{0}(KerΣ′) & π_{0}(𝔸) \ar[l]^{π_{0}(Γ′_{0})} & π_{1}(CokerΓ′) \ar[l]^{π_{1}(μ′)} \ar[r]_{π_{1}(Σ′_{0})} & π_{1}(ℂ) \ar[r]_{π_{0}(λ′)} & π_{0}(KerΣ′)}

The commutativity of the first and the third square means that π_{0}(\overline{β}) and π_{1}(\underline{β}) are isomorphisms. Now we can particularize Corollary 2.7 taking, as F, the functors Σ and Σ′. We obtain the following commutative diagram with exact rows

\xymatrix{π_{1}(ℂ) \ar[r]^{π_{0}(λ)} \ar[d]_{1} & π_{0}(KerΣ) \ar[r]^{π_{0}(e_{Σ})} \ar[d]_{π_{0}(\overline{β})} & π_{0}(𝔹) \ar[r]^{π_{0}(Σ)} \ar[d]^{π_{0}(β)} & π_{0}(ℂ) \ar[d]^{1} \ar[r] & π_{0}(CokerΣ) = 0 \ar[d]^{1} \\ π_{1}(ℂ) \ar[r]_{π_{0}(λ′)} & π_{0}(KerΣ′) \ar[r]_{π_{0}(e_{Σ′})} & π_{0}(𝔹′) \ar[r]_{π_{0}(Σ′)} & π_{0}(ℂ) \ar[r] & π_{0}(CokerΣ′) = 0}

(the zeros are due to the fact that Σ and Σ′ are essentially surjective). By the Five Lemma, π_{0}(β) is an isomorphism. In an analogous way, we can particularize Corollary 2.7 taking, as F, the functors Γ and Γ′. We have the following commutative diagram with exact rows

\xymatrix{0 = π_{1}(KerΓ) \ar[d]_{1} \ar[r] & π_{1}(𝔸) \ar[d]_{1} \ar[r]^{π_{1}(Γ)} & π_{1}(𝔹) \ar[d]_{π_{1}(β)} \ar[r]^{π_{1}(P_{Γ})} & π_{1}(CokerΓ) \ar[d]^{π_{1}(\underline{β})} \ar[r]^{π_{1}(μ)} & π_{0}(𝔸) \ar[d]^{1} \\ 0 = π_{1}(KerΓ′) \ar[r] & π_{1}(𝔸) \ar[r]_{π_{1}(Γ′)} & π_{1}(𝔹′) \ar[r]_{π_{1}(P_{Γ′})} & π_{1}(CokerΓ′) \ar[r]_{π_{1}(μ′)} & π_{0}(𝔸)}

(where the zeros are due to the fact that Γ and Γ′ are faithful). The Five Lemma implies now that π_{1}(β) is an isomorphism. This implies that β: 𝔹 → 𝔹′ is an equivalence in SCG. Consider the adjoint equivalence in SCG

β^{−1} : 𝔹′ → 𝔹 ; η: 1_{𝔹} ⇒ β · β^{−1} ; ϵ: β^{−1} · β ⇒ 1_{𝔹′}

and put

\xymatrix{x^{−1} : Γ \ar@{=>}[r]^{Γ·η} & Γ · β · β^{−1} \ar@{=>}[r]^{α·β^{−1}} & Γ′ · β^{−1} \\ y^{−1} : Σ′ \ar@{=>}[r]^{ϵ^{−1}·Σ′} & β^{−1} · β · Σ′ \ar@{=>}[r]^{β^{−1}·γ} & β^{−1} · Σ}

It is now straightforward, using triangular identities for (η, ϵ), to check that (x, β^{−1}, y): (Γ′, φ′, Σ′) → (Γ, φ, Σ) is a 1-cell and η, ϵ are 2-cells in EXT(ℂ, 𝔸).

2) Obvious from 1).

3. Abelian group extensions

The comparison between abelian group extensions and extensions of symmetric cat-groups is the object of this short section. First, observe that we can complete Lemma 2.5 in the following way.

Lemma 3.1. Let (f, g): A → B → C be an extension of abelian groups. Then (f[0], =, g[0]): A[0] → B[0] → C[0] and (f[1], =, g[1]): A[1] → B[1] → C[1] are extensions of symmetric cat-groups.

Proof. If 0 → A → B → C → 0 is exact, then 1 → A[0] → B[0] → C[0] → 1 is 2-exact. In particular, the 2-exactness in A[0] means that f[0] is full and faithful, and the 2-exactness in C[0] means that g[0] is full and essentially surjective. The same argument works for A[1] → B[1] → C[1] .

The converse is not true, in the sense that if (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ is an extension of symmetric cat-groups, then in general neither (π_{0}(Γ), π_{0}(Σ)): π_{0}(𝔸) → π_{0}(𝔹) → π_{0}(ℂ) nor (π_{1}(Γ), π_{1}(Σ)): π_{1}(𝔸) → π_{1}(𝔹) → π_{1}(ℂ) are extensions of abelian groups (respectively because π_{0}(Γ) in general is not injective and π_{1}(Σ) in general is not surjective.

Lemma 3.2. Let (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ be an extension of symmetric cat-groups.

1) If π_{1}(𝔸) = 0 = π_{1}(ℂ) then π_{1}(𝔹) = 0 and (π_{0}(Γ), π_{0}(Σ)): π_{0}(𝔸) → π_{0}(𝔹) → π_{0}(ℂ) is an extension of abelian groups;

2) If π_{0}(𝔸) = 0 = π_{0}(ℂ) then π_{0}(𝔹) = 0 and (π_{1}(Γ), π_{1}(Σ)): π_{1}(𝔸) → π_{1}(𝔹) → π_{1}(ℂ) is an extension of abelian groups.

Proof. 1) If we apply π_{1}, we get an exact sequence 0 → π_{1}(𝔹) → 0, so that π_{1}(𝔹) = 0. Moreover, since π_{1}(ℂ) = 0, then e_{Σ} is full (see [16]). But then also Γ is full, so that π_{0}(Γ) is injective.

2) If we apply π_{0}, we get an exact sequence 0 → π_{0}(𝔹) → 0, so that π_{0}(𝔹) = 0. Moreover, since π_{0}(𝔸) = 0, then P_{Γ} is full (see [16]). But then also Σ is full, so that π_{1}(Σ) is surjective.

Note that, in the situation of Lemma 3.2, if π_{1}(𝔹) = 0, then π_{1}(𝔸) = 0 (because π_{1} preserves kernels), but in general π_{1}(ℂ) ≠ 0 (because the cokernel of a morphism between discrete cat-groups in general is not dicrete). Dually, if π_{0}(𝔹) = 0, then π_{0}(ℂ) = 0 (because π_{0} preserves cokernels), but in general π_{0}(𝔸) ≠ 0.

We can summarize the previous discussion in the following proposition.

Proposition 3.3. Let A and C be two abelian groups. The (discrete) 2-groupoids EXT(C[0], A[0]), EXT(C[1], A[1]) and EXT(C, A) (the classical groupoid of extensions of A by C) are equivalent.

4. Trivial extensions

Consider two symmetric cat-groups 𝔸 and ℂ; the product category 𝔸 × ℂ with the obvious projections and injections

\xymatrix{𝔸 & 𝔸 × ℂ \ar[l]_{p_{𝔸}} \ar[r]^{p_{ℂ}} & ℂ & 𝔸 \ar[r]^{i_{𝔸}} & 𝔸 × ℂ & ℂ \ar[l]_{i_{ℂ}}}

satisfies the universal properties of the product and of the coproduct in SCG.

Example 4.1. If 𝔸, ℂ are in SCG, then

T (𝔸, ℂ): \xymatrix{𝔸 \ar[r]^{i_{𝔸}} & 𝔸 × ℂ \ar[r]^{p_{ℂ}} & ℂ}

(with the identity 2-cell i_{𝔸} · p_{ℂ} = 0) is an extension of 𝔸 by ℂ. We call it the trivial extension of 𝔸 by ℂ.

Definition 4.2. An extension (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ is a split extension of 𝔸 by ℂ (or, in short, splits) if it is equivalent, in EXT(ℂ, 𝔸), to the trivial extension T (𝔸, ℂ).

Lemma 4.3.

1) Let

[figure: ℂ; 1_{ℂ}; S; ℂ; 𝔹; Σ; ⇑^{σ}]

be in SCG; then (e_{Σ}, ϵ_{Σ}, Σ): KerΣ → 𝔹 → ℂ is a split extension of KerΣ by ℂ ;

2) Let

[figure: 𝔸; 1_{𝔸}; Γ; 𝔸; 𝔹; R; ⇑^{ρ}]

be in SCG; then (Γ, π_{Γ}, P_{Γ}): 𝔸 → 𝔹 → CokerΓ is a split extension of 𝔸 by CokerΓ.

Proof. We give a detailed proof of part 1) (part 2) is similar) because it is our first example of proof using the universal property of a (bi)limit. We need a 1-cell in EXT(ℂ, KerΣ)

[figure: KerΣ × ℂ; β; p_{ℂ}; ⇓^{α}; KerΣ; i_{KerΣ}; e_{Σ}; ℂ; 𝔹; Σ; ⇑^{γ}]

By the universal property of the coproduct KerΣ × ℂ, we get a 1-cell β and two 2-cells α, δ in SCG

[figure: KerΣ × ℂ; β; ⇓^{α}; KerΣ; i_{KerΣ}; e_{Σ}; ℂ; i_{ℂ}; S; 𝔹; ⇓^{δ}]

Now we have two 2-cells in SCG

\xymatrix{γ_{1} : i_{KerΣ} · β · Σ \ar@{=>}[r]^{α·Σ} & e_{Σ} · Σ \ar@{=>}[r]^{ϵ_{Σ}} & 0 \ar@{=>}[r] & i_{KerΣ} · p_{ℂ} \\ γ_{2} : i_{ℂ} · β · Σ \ar@{=>}[r]^{δ·Σ} & S · Σ \ar@{=>}[r]^{σ} & 1_{ℂ} \ar@{=>}[r] & i_{ℂ} · p_{ℂ}}

By the universal property of the coproduct KerΣ × ℂ, there exists a unique 2-cell γ: β · Σ ⇒ p_{ℂ} in SCG such that i_{KerΣ} · γ = γ_{1} and i_{ℂ} · γ = γ_{2}. It remains to check that (α, β, γ): T (KerΣ, ℂ) → (e_{Σ}, ϵ_{Σ}, Σ) is a 1-cell in EXT(ℂ, KerΣ). This means to check the commutativity of

\xymatrix{i_{KerΣ} · β · Σ \ar@{=>}[r]^{i_{KerΣ}·γ} \ar@{=>}[d]_{α·Σ} & i_{KerΣ} · p_{ℂ} \ar@{=>}[d] \\ e_{Σ} · Σ \ar@{=>}[r]_{ϵ_{Σ}} & 0}

which, by definition of γ_{1}, amounts to i_{KerΣ} · γ = γ_{1}.

Corollary 4.4. Let

[figure: 𝔸; 0; Γ; ℂ; 𝔹; Σ; ⇑^{φ}]

be in EXT(ℂ, 𝔸). The following conditions are equivalent :

1) The extension (Γ, φ, Σ) splits ;

2) There exist S: ℂ → 𝔹 and σ: S · Σ ⇒ 1_{ℂ} in SCG ;

3) There exist R: 𝔹 → 𝔸 and ρ: Γ · R ⇒ 1_{𝔸} in SCG.

Proof. 1) ⇒ 2) : Given (α, β, γ): T (𝔸, ℂ) → (Γ, φ, Σ) in EXT(ℂ, 𝔸), put S = i_{ℂ} · β

and σ: S · Σ = i_{ℂ} · β · \xymatrix{Σ \ar@{=>}[r]^{i_{ℂ}·γ} & i_{ℂ} · p_{ℂ} \ar@{=>}[r] & 1_{ℂ} .}

2) ⇒ 1) : By Lemma 4.3, using that the equivalence Γ_{0} : 𝔸 → KerΣ induces a biequivalence EXT(ℂ, 𝔸) ≃ EXT(ℂ, KerΣ) in which (Γ, φ, Σ) corresponds to (e_{Σ}, ϵ_{Σ}, Σ).

1) ⇒ 3) ⇒ 1) : Similar.

In view of some applications to 2-exact sequences, we need a more precise formulation of Corollary 4.4. For this, fix an extension (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ of 𝔸 by ℂ and consider the category Split(Σ) :

- objects are pairs (S: ℂ → 𝔹, σ: S · Σ ⇒ 1_{ℂ}) in SCG ;

- an arrow λ: (S, σ) ⇒ (S′, σ′) is a 2-cell λ: S ⇒ S′ in SCG such that the following diagram commutes

[figure: S · Σ; λ·Σ; σ; S′ · Σ; σ′; 1_{ℂ}]

Consider also the trivial extension T (𝔸, ℂ).

Lemma 4.5. Composition with i_{ℂ}: ℂ → 𝔸 × ℂ induces an equivalence

i_{ℂ} · −: EXT(ℂ, 𝔸)(T (𝔸, ℂ), (Γ, φ, Σ)) → Split(Σ)

Proof. Let us describe explicitly the functor i_{ℂ} · −:

- Given a 1-cell (α, β, γ): T (𝔸, ℂ) → (Γ, φ, Σ) in EXT(ℂ, 𝔸), we obtain an ob-

ject ( \xymatrix{ℂ \ar[r]^{i_{ℂ}} & 𝔸 × ℂ \ar[r]^{β} & 𝔹} , i_{ℂ} · γ: i_{ℂ} · β · Σ ⇒ i_{ℂ} · p_{ℂ} = 1_{ℂ}) in Split(Σ) ;

- Given a 2-cell b: (α, β, γ) ⇒ (α′, β′, γ′): T (𝔸, ℂ) → (Γ, φ, Σ) in EXT(ℂ, 𝔸), then the second condition on b in Definition 2.2 means that i_{ℂ}·b: (i_{ℂ}·β, i_{ℂ}·γ) ⇒ (i_{ℂ} · β′, i_{ℂ} · γ′) is an arrow in Split(Σ).

Now we check that i_{ℂ} · − is an equivalence : Faithfulness : let \overline{b}: (α, β, γ) ⇒ (α′, β′, γ′) be another 2-cell in EXT(ℂ, 𝔸). Since α′ is a natural isomorphism, by the first condition on b and \overline{b} in Definition 2.2 we have that i_{𝔸} · b = i_{𝔸} · \overline{b}. If, moreover, we assume that i_{ℂ} · b = i_{ℂ} · \overline{b}, then b = \overline{b}. Fullness : let λ: (i_{ℂ} · β, i_{ℂ} · γ) ⇒ (i_{ℂ} · β′, i_{ℂ} · γ′) be an arrow in Split(Σ) and consider

\xymatrix{i_{𝔸} · β \ar@{=>}[r]^{α} & Γ \ar@{=>}[r]^{(α′)^{−1}} & i_{𝔸} · β′} . By the universal property of the coproduct 𝔸 × ℂ, we get a unique 2-cell b: β ⇒ β′ in SCG such that

\xymatrix{i_{𝔸} · β \ar@{=>}[rr]^{i_{𝔸}·b} \ar@{=>}[dr]_{α} &  & i_{𝔸} · β′ · Σ \\  & ℂ \ar@{=>}[ur]_{(α′)^{−1}}}

commutes and i_{ℂ} · b = λ. It remains to check the commutativity of

[figure: β · Σ′; b·Σ′; γ; β′ · Σ′; γ′; Σ]

One can do this precomposing with i_{𝔸} and i_{ℂ} and using, respectively, that (α, β, γ) and (α′, β′, γ′) are 1-cells in EXT(ℂ, 𝔸) and the condition on λ to be an arrow in Split(Σ). Essential surjectivity : this is the part already proved in Lemma 4.3 (write everywhere 𝔸, φ and Γ insted of KerΣ, ϵ_{Σ} and e_{Σ}). It remains only to check that δ: (i_{ℂ} · β, i_{ℂ} · γ) ⇒ (S, σ) is an arrow in Split(Σ), but this is exactly the second condition on γ in the proof of Lemma 4.3.

For later use, let us write explicitly a consequence of Lemma 4.5. Let (x, L, y): (Γ_{1}, φ_{1}, Σ_{1}) → (Γ_{2}, φ_{2}, Σ_{2}) be a 1-cell in EXT(ℂ, 𝔸)

[figure: 𝔹_{1}; L; Σ_{1}; ⇓^{x}; 𝔸; Γ_{1}; Γ_{2}; ℂ; 𝔹_{2}; Σ_{2}; ⇑^{y}]

For each pair of 1-cells

(α_{1}, β_{1}, γ_{1}): T (𝔸, ℂ) → (Γ_{1}, φ_{1}, Σ_{1}) (α_{2}, β_{2}, γ_{2}): T (𝔸, ℂ) → (Γ_{2}, φ_{2}, Σ_{2})

in EXT(ℂ, 𝔸), there is a bijection between 2-cells b: (α_{1}, β_{1}, γ_{1}) · (x, L, y) ⇒ (α_{2}, β_{2}, γ_{2}) in EXT(ℂ, 𝔸) and arrows

λ: ( \xymatrix{ℂ \ar[r]^{S_{1}} & 𝔹_{1} \ar[r]^{L} & 𝔹_{2}} , S_{1} · L · \xymatrix{Σ_{2} \ar@{=>}[r]^{S_{1}·y} & S_{1} · Σ_{1} \ar@{=>}[r]^{σ_{1}} & 1_{ℂ})} ⇒

\xymatrix{(ℂ \ar[r]^{S_{2}} & 𝔹_{2}, & S_{2} · Σ_{2} \ar@{=>}[r]^{σ_{2}} & 1_{ℂ}),}

in Split(Σ_{2}), where (S_{1}, σ_{1}) and (S_{2}, σ_{2}) correspond to the extensions (Γ_{1}, φ_{1}, Σ_{1}) and (Γ_{2}, φ_{2}, Σ_{2}) via the equivalences

i_{ℂ} · −: EXT(ℂ, 𝔸)(T (𝔸, ℂ), (Γ_{1}, φ_{1}, Σ_{1})) → Split(Σ_{1})
i_{ℂ} · −: EXT(ℂ, 𝔸)(T (𝔸, ℂ), (Γ_{2}, φ_{2}, Σ_{2})) → Split(Σ_{2}) .

This bijection sends b: β_{1} · L ⇒ β_{2} into λ: S_{1} · L = i_{ℂ} · β_{1} · \xymatrix{L \ar@{=>}[r]^{i_{ℂ}·b} & i_{ℂ}} · β_{2} = S_{2} .

Clearly, given an extension (Γ, φ, Σ), instead of Σ one could consider Γ and obtain a dual lemma perfectly analogous to Lemma 4.5. We leave this to the reader.

5. Pullbacks and pushouts in SCG

To compare EXT and Hom, as well as to define the 2-dimensional analogue of Baer sum, we need pullbacks and pushouts in SCG (in the sense of bilimits, of course).

Let us start with pushouts. First, we recall the universal property: given two 1-cells F: 𝔸 → 𝔹 and G: 𝔸 → ℂ in SCG, their pushout is a diagram in SCG of the form

[figure: 𝔸; G; F; ⇒^{i_{F,G}}; ℂ; i_{G}; 𝔹; i_{F}; F ∪ G]

such that, for any other diagram in SCG

[figure: 𝔸; G; F; ⇒^{φ}; ℂ; K; 𝔹; H; 𝔻 ,]

there exist a 1-cell φ^{H,K}: F ∪ G → 𝔻 and two 2-cells φ^{H}: i_{F} · φ^{H,K} ⇒ H , φ^{K}: i_{G} · φ^{H,K} ⇒ K in SCG such that the following diagram commutes

\xymatrix{F · i_{F} · φ^{H,K} \ar@{=>}[r]^{F ·φ^{H}} \ar@{=>}[d]_{i_{F,G}·φ^{H,K}} & F · H \ar@{=>}[d]^{φ} \\ G · i_{G} · φ^{H,K} \ar@{=>}[r]_{G·φ^{K}} & G · K;}

moreover, if \overline{φ}^{H,K}: F ∪ G → 𝔻 , \overline{φ}^{H}: i_{F} · \overline{φ}^{H,K} ⇒ H , \overline{φ}^{K}: i_{G} · \overline{φ}^{H,K} ⇒ K satisfy the same condition, then there is a unique 2-cell ψ: \overline{φ}^{H,K} ⇒ φ^{H,K} in SCG such that the following diagrams commute

\xymatrix{i_{F} · \overline{φ}^{H,K} \ar@{=>}[rr]^{i_{F} ·ψ} \ar@{=>}[dr]_{_{\overline{φ}_{H}}} &  & i_{F} · φ^{H,K} \ar@{=>}[dl]^{φ^{H}} & i_{G} · \overline{φ}^{H,K} \ar@{=>}[rr]^{i_{G}·ψ} \ar@{=>}[dr]_{_{\overline{φ}_{K}}} &  & i_{G} · φ^{H,K} \ar@{=>}[dl]^{φ^{K}} \\  & H &  &  & K .}

Passing through the description of cokernels in SCG given in [16] and the obvious description of coproducts in SCG, we get an explicit description for pushouts :

- Objects of F ∪ G are those of 𝔹 × ℂ ;

- A pre-morphism is a triple (f, A, g): (B_{1}, C_{1}) → (B_{2}, C_{2}), with A an object of 𝔸, f: B_{1} → F A ⊗ B_{2} in 𝔹 and g: C_{1} ⊗ GA → C_{2} in ℂ ;

- A morphism is an equivalence class of pre-morphisms : (f, A, g), (\overline{f}, \overline{A}, \overline{g}): (B_{1}, C_{1}) → (B_{2}, C_{2}) are equivalent if there exists α: A → \overline{A} in 𝔸 such that f ·(F α⊗1) = \overline{f} and (1⊗Gα)·\overline{g} = g ; we write [f, A, g] for the equivalence class of a pre-morphism (f, A, g) ;

- The composition of

\xymatrix{(B_{1}, C_{1}) \ar[r]^{[f,A,g]} & (B_{2}, C_{2}) \ar[r]^{[f′,A′,g′]} & (B_{3}, C_{3})}

is, up to associativity, [f · (1 ⊗ f′) · (F_{A,A′} ⊗ 1), A ⊗ A′, (1 ⊗ G_{A,A′}^{−1}) · (g ⊗ 1) · g] ;

- The tensor product of

\xymatrix{(B_{1}, C_{1}) \ar[r]^{[f,A,g]} & (B_{2}, C_{2}) & with & (B′_{1}, C′_{1}) \ar[r]^{[f′,A′,g′]} & (B′_{2}, C′_{2})}

is [(f ⊗ f′) · (1 ⊗ γ ⊗ 1) · (1 ⊗ 1 ⊗ F_{A,A′}), A ⊗ A′, (1 ⊗ 1 ⊗ G_{A,A′}^{−1}) · (1 ⊗ γ ⊗ 1) · (g ⊗ g′)]: (B_{1} ⊗ B′_{1}, C_{1} ⊗ C′_{1}) → (B_{2} ⊗ B′_{2}, C_{2} ⊗ C′_{2}) ;

- i_{F} : 𝔹 → F ∪ G sends f: B_{1} → B_{2} into [f · l_{B_{2}}^{−1} · (F_{I} ⊗ 1), I, (1 ⊗ G_{I}^{−1}) · l_{I}]: (B_{1}, I) → (B_{2}, I) ;

- i_{G}: ℂ → F ∪ G sends g: C_{1} → C_{2} into [r_{I}^{−1} · (1 ⊗ F_{I}), I, (1 ⊗ G_{I}^{−1}) · r_{C_{1}} · g]: (I, C_{1}) → (I, C_{2}) ;

- i_{F,G}: F · i_{F} ⇒ G · i_{G} is defined, for each A in 𝔸, by i_{F,G}(A) = [r_{F A}^{−1}, A, l_{GA}]: i_{F} (F A) = (F A, I) → (I, GA) = i_{G}(GA) ;

- φ^{H,K}: F ∪ G → 𝔻 sends [f, A, g]: (B_{1}, C_{1}) → (B_{2}, C_{2}) into

H(B_{1}) ⊗ K(C_{1}) ≃ K(C_{1}) ⊗ H(B_{1})
1⊗Hf
K(C_{1}) ⊗ H(F (A) ⊗ B_{2}) ≃ K(C_{1}) ⊗ H(F (A)) ⊗ H(B_{2})
1⊗φ_{A}⊗1
K(C_{1}) ⊗ K(G(A)) ⊗ H(B_{2}) ≃ K(C_{1} ⊗ G(A)) ⊗ H(B_{2})
Kg⊗1
K(C_{2}) ⊗ H(B_{2}) ≃ H(B_{2}) ⊗ K(C_{2})

- φ^{H}: i_{F} · φ^{H,K} ⇒ H is defined, for each B in 𝔹, by

φ^{H}(B): φ^{H,K}(i_{F} (B)) = \xymatrix{H(B) ⊗ K(I) \ar[r]^{1⊗K_{I}^{−1}} & H(B) ⊗ I} ≃ H(B) ;

- φ^{K}: i_{G} · φ^{H,K} ⇒ K is defined, for each C in ℂ, by

φ^{K}(C): φ^{H,K}(i_{G}(C)) = \xymatrix{H(I) ⊗ K(C) \ar[r]^{H_{I}^{−1}⊗1} & I ⊗ K(C)} ≃ K(C) ;

- ψ: \overline{φ}^{H,K} ⇒ φ^{H,K} is defined, for (B, C) in F ∪ G, by ψ(B, C): \overline{φ}^{H,K}(B, C) ≃

\overline{φ}^{H,K}(i_{F} (B)) ⊗ \xymatrix{\overline{φ}^{H,K}(i_{G}(C)) \ar[r]^{\overline{φ}^{H}(B)⊗\overline{φ}^{K}(C)} & H(B) ⊗ K(C)} = φ^{H,K}(B, C).

All what we need about pushouts is the next proposition.

Proposition 5.1. Consider the following pushout diagram in SCG

[figure: 𝔸; G; F; ⇒^{i_{F,G}}; ℂ; i_{G}; 𝔹; i_{F}; F ∪ G]

1) If G is faithful, then i_{F} is faithful;

2) If G is full, then i_{F} is full;

3) If G is essentially surjective, then i_{F} is essentially surjective.

Proof. Everything can be checked directly using the previous explicit description. For example, let us prove point 1. Consider two arrows f, h: B_{1} → B_{2} in 𝔹 and assume that i_{F} (f) = i_{F} (h) in F ∪ G. This means that there exists α: I → I in 𝔸 making commutative the following diagrams

\xymatrix{I ⊗ GI ≃ GI \ar[r]^{G_{I}^{−1}} \ar[d]_{1⊗G(α)} & I \ar[d]^{1} & B_{1} \ar[r]^{f} \ar[d]_{1} & B_{2} ≃ I ⊗ B_{2} \ar[r]^{F_{I}⊗1} & F I ⊗ B_{2} \ar[d]^{F (α)⊗1} \\ I ⊗ GI ≃ GI \ar[r]_{G_{I}^{−1}} & I & B_{1} \ar[r]_{h} & B2 ≃ I ⊗ B_{2} \ar[r]_{F_{I}⊗1} & F I ⊗ B_{2}}

¿From the first equation, we have G(α) = 1_{GI}. If G is faithful, this implies that α = 1_{I}. But then, from the second equation, we get f = h.

Pullbacks in SCG are easy, in fact they are computed as in the 2-category of groupoids. We fix the notations for future references and we leave to the reader to write the universal property and the explicit description.

[figure: F ∩ G; p_{G}; p_{F}; ⇒^{p_{F,G}}; ℂ; G; 𝔹; F; 𝔸]

For pullbacks, the dual of Proposition 5.1 holds.

Proposition 5.2. With the previous notations.

1) If G is essentially surjective, then p_{F} is essentially surjective;

2) If G is full, then p_{F} is full;

3) If G is faithful, then p_{F} is faithful.

6. EXT and Hom

This section is devoted to the construction of new extensions from a given one, using pullbacks and pushouts.

Let ℂ, 𝔻 be two symmetric cat-groups; the hom-category Hom(𝔻, ℂ) has an obvious structure of symmetric cat-group. This plainly extends to a 2-functor

Hom: SCG × SCG → SCG

which reverses the direction of 1-cells in the first variable. Fix now a third symmetric cat-group 𝔸 and an extension E = (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ of 𝔸 by ℂ.

We need a 2-functor

− · E: Hom(𝔻, ℂ) → EXT(𝔻, 𝔸),

where we consider Hom(𝔻, ℂ) as a 2-category with no non-trivial 2-cells.

For this, consider a 1-cell G: 𝔻 → ℂ in SCG and the pullback of Σ and G, together with the comparison, as in the following diagram

[figure: 𝔸; φ_{Γ,0}; 0; Γ; ⇒^{φ_{0}}; ⇓^{φ_{Γ}}; Σ ∩ G; ⇒^{p_{Σ,G}}; p_{Σ}; p_{G}; 𝔻; G; 𝔹; Σ; ℂ]

where φ_{Γ,0}, φ_{Γ} and φ_{0} make commutative the following diagram

\xymatrix{φ_{Γ,0} · p_{Σ} · Σ \ar@{=>}[rr]^{φ_{Γ,0}·p_{Σ,G}} \ar@{=>}[d]_{φ_{Γ}·Σ} &  & φ_{Γ,0} · p_{G} · G \ar@{=>}[d]^{φ_{0}·G} \\ Γ · Σ \ar@{=>}[r]_{φ} & 0 \ar@{=>}[r] & 0 · G .}

Lemma 6.1. The diagram in SCG

[figure: 𝔸; 0; φ_{Γ,0}; 𝔻; Σ ∩ G; p_{G}; ⇑^{φ_{0}}]

is an extension of 𝔸 by 𝔻 ; we denote it by G · E.

Proof. From Proposition 5.2, we know that p_{G} is essentially surjective. The fact that

[figure: 𝔸; 0; φ_{Γ,0}; 𝔻; Σ ∩ G; p_{G}; ⇑^{φ_{0}}]

is a kernel of p_{G} can be checked using the universal property of the pullback Σ ∩ G and the universal property of

[figure: 𝔸; 0; Γ; ℂ; 𝔹; Σ; ⇑^{φ}]

as kernel of Σ. The detailed proof is long but essentially straightforward, and we omit it.

Consider now another 1-cell G′: 𝔻 → ℂ in SCG and the extension G′·E obtained by the pullback

[figure: 𝔸; φ′_{Γ,0}; 0; Γ; ⇒^{φ′_{0}}; ⇓^{φ′_{Γ}}; Σ ∩ G′; ⇒^{p_{Σ,G′}}; p′_{Σ}; p_{G′}; 𝔻; G′; 𝔹; Σ; ℂ]

Let μ: G ⇒ G′ be a 2-cell in SCG ; by the universal property of Σ ∩ G′, we obtain a 1-cell β: Σ ∩ G → Σ ∩ G′ and two 2-cells s: β · p′_{Σ} ⇒ p_{Σ}, γ: β · p_{G′} ⇒ p_{G} in SCG such that the following diagram commutes

\xymatrix{β · p′_{Σ} · Σ \ar@{=>}[rr]^{β·p_{Σ,G′}} \ar@{=>}[d]_{s·Σ} &  & β · p_{G′} · G′ \ar@{=>}[d]^{γ·G′} \\ p_{Σ} · Σ \ar@{=>}[r]_{p_{Σ,G}} & p_{G} · G \ar@{=>}[r]_{p_{G}·μ} & p_{G} · G′ .}

We are looking for a 1-cell in EXT(𝔻, 𝔸) induced by μ :

[figure: Σ ∩ G; β; p_{G}; ⇓^{α}; 𝔸; φ_{Γ,0}; φ′_{Γ,0}; 𝔻; Σ ∩ G′ .; p_{G′}; ⇑^{γ}]

As far as α is concerned, we have

\xymatrix{α_{𝔻}: φ_{Γ,0} · β · p_{G′} \ar@{=>}[r]^{φ_{Γ,0}·γ} & φ_{Γ,0} · p_{G} \ar@{=>}[r]^{φ_{0}} & 0 \ar@{=>}[r]^{(φ′_{0})^{−1}} & φ′_{Γ,0} · p_{G′} \\ α_{𝔹}: φ_{Γ,0} · β · p′_{Σ} \ar@{=>}[r]^{φ_{Γ,0}·s} & φ_{Γ,0} · p_{Σ} \ar@{=>}[r]^{φ_{Γ}} & 0 \ar@{=>}[r]^{(φ′_{Γ})^{−1}} & φ′_{Γ,0} · p′_{Σ}.}

Since

\xymatrix{φ_{Γ,0} · β · p′_{Σ} · Σ \ar@{=>}[d]_{α_{𝔹}·Σ} \ar@{=>}[r]^{φ_{Γ,0}·β·p_{Σ,G′}} & φ_{Γ,0} · β · p_{G′} · G′ \ar@{=>}[d]^{α_{𝔻}·G′} \\ φ′_{Γ,0} · p′_{Σ} · Σ \ar@{=>}[r]_{φ′_{Γ,0}·p_{Σ,G′}} & φ′_{Γ,0} · p_{G′} · G′}

commutes, there is a unique 2-cell α: φ_{Γ,0} · β ⇒ φ′_{Γ,0} such that α · p_{G′} = α_{𝔻} and α · p′_{Σ} = α_{𝔹}.

Lemma 6.2. With the previous notations,

(α, β, γ): (φ_{Γ,0}, φ_{0}, p_{G}) → (φ′_{Γ,0}, φ′_{0}, p′_{G})

is a 1-cell in EXT(𝔻, 𝔸); we denote it by μ · E.

Proof. The condition to be a 1-cell in EXT(𝔻, 𝔸) is precisely the equation α · p_{G′} = α_{𝔻}.

We pass now to the dual construction, involving pushouts. For this, consider an extension E = (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ in SCG and fix a 1-cell F: 𝔸 → 𝔻 in SCG. Consider also the pushout of F and Γ and the factorization as in the following diagram

[figure: 𝔸; Γ; F; ⇒^{i_{F,Γ}}; 𝔹; i_{Γ}; Σ; 𝔻; i_{F}; 0; F ∪ Γ; φ^{0,Σ}; ⇒^{φ^{Σ}}; ⇓^{φ^{0}}; ℂ]

with φ^{0,Σ}, φ^{0} and φ^{Σ} making commutative the diagram

\xymatrix{F · i_{F} · φ^{0,Σ} \ar@{=>}[rr]^{i_{F,Γ}·φ^{0,Σ}} \ar@{=>}[d]_{F ·φ^{0}} &  & Γ · i_{Γ} · φ^{0,Σ} \ar@{=>}[d]^{Γ·φ^{Σ}} \\ F · 0 \ar@{=>}[r] & 0 & Γ · Σ . \ar@{=>}[l]^{φ}}

Lemma 6.3. The diagram in SCG

[figure: 𝔻; 0; i_{F}; ℂ; F ∪ Σ; φ^{0,Σ}; ⇑^{φ^{0}}]

is an extension of 𝔻 by ℂ ; we denote it by E · F.

Proof. From Proposition 5.1, we know that i_{F} is faithful. Moreover, one can check directly the universal property of the cokernel

[figure: 𝔻; 0; i_{F}; ℂ; F ∪ Σ; φ^{0,Σ}; ⇑^{φ^{0}}]

using the universal property of the pushout F ∪ Γ and that of the cokernel

[figure: 𝔸; 0; Γ; ℂ; 𝔹; Σ; ⇑^{φ}]

The construction of the extension E · F of Lemma 6.3 fits into a 2-functor

− · F: EXT(ℂ, 𝔸) → EXT(ℂ, 𝔻)

Let us describe it in detail. Consider another extension E′ = (Γ′, φ′, Σ′): 𝔸 → 𝔹′ → ℂ with the corresponding factorization

[figure: 𝔸; Γ′; F; ⇒^{i_{F,Γ′}}; 𝔹′; i_{Γ′}; Σ′; 𝔻; i′_{F}; 0; F ∪ Γ′; φ^{0,Σ′}; ⇒^{φ^{Σ′}}; ⇓^{φ^{′0}}; ℂ]

and let (α, β, γ): (Γ, φ, Σ) → (Γ′, φ′, Σ′) be a 1-cell in EXT(ℂ, 𝔸). By the universal property of the pushout F ∪ Γ, we get β_{F} : F ∪ Γ → F ∪ Γ′, α_{F} : i_{F} · β_{F} ⇒ i′_{F} , δ_{F} : i_{Γ} · β_{F} ⇒ β · i_{Γ′} making commutative the following diagram

\xymatrix{Γ · i_{Γ} · β_{F} \ar@{=>}[rr]^{Γ·δ_{F}} &  & Γ · β · i_{Γ′} \ar@{=>}[d]^{α·i_{Γ′}} \\ F · i_{F} · β_{F} \ar@{=>}[u]^{i_{F,Γ}·β_{F}} \ar@{=>}[r]_{F ·α_{F}} & F · i′_{F} \ar@{=>}[r]_{i_{F,Γ′}} & Γ′ · i_{Γ′} .}

Consider now the 2-cells

\xymatrix{γ_{𝔹}: i_{Γ} · β_{F} · φ^{0,Σ′} \ar@{=>}[r]^{δ_{F} ·φ^{0,Σ′}} & β · i_{Γ′} · φ^{0,Σ′} \ar@{=>}[r]^{β·φ^{Σ′}} & β · Σ′ \ar@{=>}[r]^{γ} & Σ \ar@{=>}[d]^{(φ^{Σ})^{−1}} \\  &  &  & i_{Γ} · φ^{0,Σ} \\ γ_{𝔻}: i_{F} · β_{F} · φ^{0,Σ′} \ar@{=>}[r]^{α_{F} ·φ^{0,Σ′}} & i′_{F} · φ^{0,Σ′} \ar@{=>}[r]^{φ^{′0}} & 0 \ar@{=>}[r]^{(φ^{0})^{−1}} & i_{F} · φ^{0,Σ} .}

Since they are compatible with i_{F,Γ}, there exists a unique γ_{F} : β_{F} · φ^{0,Σ′} ⇒ φ^{0,Σ} such that i_{F} · γ_{F} = γ_{𝔻} and i_{Γ} · γ_{F} = γ_{𝔹}. In particular,

(α_{F} , β_{F} , γ_{F}): (i_{F} , φ^{0}, φ^{0,Σ}) → (i′_{F} , φ^{′0}, φ^{0,Σ′})

is a 1-cell in EXT(ℂ, 𝔻). Starting from another 1-cell (α′, β′, γ′): (Γ, φ, Σ) → (Γ′, φ′, Σ′) in EXT(ℂ, 𝔸), repeat the previous argument. Consider also a 2-cell b: (α, β, γ) ⇒ (α′, β′, γ′). Because of the first condition in the definition of 2-cell, the universal property of the pushout F ∪Γ gives a unique 2-cell b_{F} : β_{F} ⇒ β′_{F} making commutative the following diagram

[figure: i_{F} · β_{F}; i_{F} ·b_{F}; α_{F}; i_{F} · β′_{F}; α′_{F}; i_{Γ} · β_{F}; i_{Γ}·b_{F}; δ_{F}; i_{Γ} · β′_{F}; δ′_{F}; i′_{F}; β · i_{Γ′}; b·i_{Γ′}; β′ · i_{Γ′} .]

To have a 2-cell b_{F} : (α, β_{F} , γ_{F}) ⇒ (α′_{F} , β′_{F} , γ′_{F}) in EXT(ℂ, 𝔻), it remains to check the commutativity of

\xymatrix{β_{F} · φ^{0,Σ′} \ar@{=>}[rr]^{b_{F} ·φ^{0,Σ′}} \ar@{=>}[dr]_{γ_{F}} &  & β′_{F} · φ^{0,Σ′} \ar@{=>}[dl]^{γ′_{F}} \\  & φ^{0,Σ} & ;}

this can be done, via the definition of γ_{F} , using the second condition on the 2-cell b and the two equations satisfied by b_{F} .

To end this section, let us summarize the situation just discussed, as well as the dual situation.

Proposition 6.4. Consider two 1-cells F: 𝔸 → 𝔻, G: 𝔻 → ℂ in SCG and an extension E = (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ. The pullback construction gives two 2functors

\xymatrix{− · E: Hom(𝔻, ℂ) \ar[r] & EXT(𝔻, 𝔸) \\ G · −: EXT(ℂ, 𝔸) \ar[r] & EXT(𝔻, 𝔸) .}

The pushout construction gives two 2-functors

\xymatrix{E · −: Hom(𝔸, 𝔻) \ar[r] & EXT(ℂ, 𝔻) \\ − · F: EXT(ℂ, 𝔸) \ar[r] & EXT(ℂ, 𝔻) .}

7. First application

As a first, simple application, let us show that EXT measures if a 1-cell can be lifted (or extended).

Proposition 7.1. Consider a 1-cell G: 𝔻 → ℂ in SCG and an extension E = (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ of 𝔸 by ℂ. There exist \overline{G}: 𝔻 → 𝔹 and ψ: \overline{G} · Σ ⇒ G in SCG if and only if the extension G · E in EXT(𝔻, 𝔸) splits.

Proof. If G·E splits, there is a section S: 𝔻 → Σ∩G, σ: S ·p_{G} ⇒ 1_{𝔻} of p_{G} in SCG. Then one can put \overline{G} = S · p_{Σ} and

\xymatrix{ψ: S · p_{Σ} · Σ \ar@{=>}[r]^{S·p_{Σ,G}} & S · p_{G} · G \ar@{=>}[r]^{σ·G} & 1_{𝔻} · G = G.}

Conversely, consider

[figure: 𝔻; 1_{𝔻}; \overline{G}; ⇒^{ψ}; 𝔻; G; 𝔹; Σ; ℂ]

By the universal property of the pullback Σ ∩ G, we get

[figure: 𝔻; 1_{𝔻}; S; 𝔻; Σ ∩ G; p_{G}; ⇑^{σ}]

in SCG

We state the dual proposition, whose proof is left to the reader.

Proposition 7.2. Consider a 1-cell F: 𝔸 → 𝔻 in SCG and an extension E = (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ of 𝔸 by ℂ. There exist \overline{F}: 𝔹 → 𝔻 and ψ: Γ · \overline{F} ⇒ F in SCG if and only if the extension E · F in EXT(ℂ, 𝔻) splits.

8. Proper classes of extensions

As explained in the Introduction, we define now the notion of proper class of extensions, in order to make results of Sections 10, 11 and 13 available for some special classes of extensions.

The next definition extends Definition 2.2. It is useful to define proper classes, and also to define Baer sum (see Section 13).

Definition 8.1. Consider two extensions E = (Γ, φ, Σ) and E′ = (Γ′, φ′, Σ′) in SCG

[figure: 𝔸; 0; Γ; ℂ; 𝔹; Σ; ⇑^{φ}; 𝔸′; 0; Γ′; ℂ′; 𝔹′ .; Σ′; ⇑^{φ′}]

1) A 1-cell (F, α, β, γ, G): E → E′ is pictured in the following diagram

[figure: 𝔸; Γ; F; ⇓^{α}; 𝔹; β; Σ; ⇓^{γ}; ℂ; G; 𝔸′; Γ′; 𝔹′; Σ′; ℂ′]

with F, α, β, γ, G in SCG and such that the following diagram commutes

\xymatrix{Γ · Σ · G \ar@{=>}[r]^{Γ·γ} \ar@{=>}[d]_{φ·G} & Γ · β · Σ′ \ar@{=>}[r]^{α·Σ′} & F · Γ′ · Σ′ \ar@{=>}[d]^{F ·φ′} \\ 0 · G \ar@{=>}[r] & 0 & G · 0 \ar@{=>}[l]}

2) If (F, α, β, γ, G), (F′, α′, β′, γ′, G′): E → E′ are two 1-cells, a 2-cell

(x, y, z): (F, α, β, γ, G) ⇒ (F′, α′, β′, γ′, G′)

is a triple x: F ⇒ F′ , y: β ⇒ β′ , z: G ⇒ G′ of 2-cells in SCG making commutative the following diagrams

\xymatrix{Γ · β \ar@{=>}[r]^{α} \ar@{=>}[d]_{Γ·y} & F · Γ′ \ar@{=>}[d]^{x·Γ′} & Σ · G \ar@{=>}[r]^{γ} \ar@{=>}[d]_{Σ·z} & β · Σ′ \ar@{=>}[d]^{y·Σ′} \\ Γ · β′ \ar@{=>}[r]_{α′} & F′ · Γ′ & Σ · G′ \ar@{=>}[r]_{γ′} & β′ · Σ′ .}

Observe that 2-cells are invertible, and that a 1-cell (F, α, β, γ, G) is invertible as soon as F and G are equivalences. The proof is similar to that of Proposition 2.8.

Definition 8.2. A proper class P of extensions is the assignement, for every pair of symmetric cat-groups 𝔸 and ℂ, of a full (at the level of 1-cells and 2-cells) sub-2-category P(ℂ, 𝔸) of EXT(ℂ, 𝔸) in such a way that:

1) The trivial extension T (𝔸, ℂ) is in P(ℂ, 𝔸); 2) If E ∈ P(ℂ, 𝔸), E′ ∈ EXT(ℂ′, 𝔸′) and (F, α, β, γ, G): E → E′ is an equiva-

lence (in the sense of Definition 8.1), then E′ ∈ P(ℂ′, 𝔸′);

3) The 2-functors of Proposition 6.4 restrict to P

\xymatrix{Hom(𝔻, ℂ) × EXT(ℂ, 𝔸) \ar[r]^{−·−}_{(pullback)} & EXT(𝔻, 𝔸) \\ Hom(𝔻, ℂ) × P(ℂ, 𝔸) \ar@{.>}[r] \ar[u] & P(𝔻, 𝔸) \ar[u]}

\xymatrix{EXT(ℂ, 𝔸) × Hom(𝔸, 𝔻) \ar[r]^{−·−}_{(pushout)} & EXT(ℂ, 𝔻) \\ P(ℂ, 𝔸) × Hom(𝔸, 𝔻) \ar@{.>}[r] \ar[u] & P(ℂ, 𝔻) \ar[u]}

To prepare the study of injective and projective symmetric cat-groups, let us observe that a proper class P of extensions gives rise to two classes of morphisms in SCG.

Definition 8.3. Let P be a proper class of extensions.

1) A morphism Γ: 𝔸 → 𝔹 in SCG is a P-mono if there exist Σ: 𝔹 → ℂ and φ: Γ · Σ ⇒ 0 in SCG such that (Γ, φ, Σ) ∈ P(ℂ, 𝔸). Equivalently, Γ is a P-mono if (Γ, π_{Γ}, P_{Γ}) ∈ P(CokerΓ, 𝔸).

2) A morphism Σ: 𝔹 → ℂ in SCG is a P-epi if there exist Γ: 𝔸 → 𝔹 and φ: Γ · Σ ⇒ 0 in SCG such that (Γ, φ, Σ) ∈ P(ℂ, 𝔸). Equivalently, Σ is a P-epi if (e_{Σ}, ϵ_{Σ}, Σ) ∈ P(ℂ, KerΣ).

The classes P-mono and P-epi inherit some good 2-categorical property from those of the proper class P.

Proposition 8.4. Let P be a proper class of extensions.

0) For any 𝔸 ∈ SCG, the canonic morphism 1 → 𝔸 is a P-mono; 0’) For any 𝔸 ∈ SCG, the canonic morphism 𝔸 → 1 is a P-epi;

1) Equivalences ⊆ P-mono ⊆ faithful functors; 1’) Equivalences ⊆ P-epi ⊆ essentially surjective functors;

2) P-monos are stable under composition with equivalences and under natural isomorphisms;

2’) P-epis are stable under composition with equivalences and under natural iso-

morphisms;

3) P-monos are stable under pushouts; 3’) P-epis are stable under pullbacks;

4) Given F: 𝔸 → 𝔼, H: 𝔼 → 𝔹 in SCG, if F · H is a P-mono then F is a P-mono;

4’) Given L: 𝔹 → 𝔼, G: 𝔼 → ℂ in SCG, if L · G is a P-epi then G is a P-epi.

The non trivial points are 4 and 4’. Their proof is based on the next lemma

Lemma 8.5.

1) Consider the following diagram in SCG, where \overline{H} and \overline{h} are induced by the universal property of CokerF,

[figure: ⇓^{h}; 𝔼; P_{F}; H; ⇓^{\overline{h}}; CokerF; \overline{H}; 𝔸; F; G; 𝔹; P_{G}; CokerG]

If G is faithful, then the square is a pullback.

2) Consider the following diagram in SCG, where \overline{L} and \overline{l} are induced by the universal property of KerG,

[figure: KerF; e_{F}; \overline{L}; ⇓^{\overline{l}}; 𝔹; F; L; ⇓^{l}; ℂ; KerG; e_{G}; 𝔼; G]

If F is essentially surjective, then the square is a pushout.

Proof. Let us prove part 2, the proof of part 1 is dual. Observe that, since F is essentially surjective, (e_{F} , ϵ_{F} , F): KerF → 𝔹 → ℂ and (e_{G}, ϵ_{G}, G): KerG → 𝔼 → ℂ are extensions. Consider now the factorization through the pushout (Section 5) and the pushout extension (Section 6) as in the following diagrams

[figure: KerF; e_{F}; \overline{L}; ⇒^{i}; 𝔹; i_{2}; L; KerG; i_{1}; e_{G}; \overline{L} ∪ e_{F}; K; ⇒^{b}; ⇓^{a}; 𝔼; KerF; e_{F}; \overline{L}; ⇒^{i}; 𝔹; i_{2}; F; KerG; i_{1}; 0; \overline{L} ∪ e_{F}; T; ⇒^{d}; ⇓^{c}; ℂ]

Using the universal property of the pushout \overline{L} ∪ e_{F} , one can complete the following diagram with a 2-cell e: K · G ⇒ T and check that it is a 1-cell between extensions

[figure: \overline{L} ∪ e_{F}; K; T; ⇓^{a}; KerG; i_{1}; e_{G}; ℂ; 𝔼; G; ⇑^{e}]

By Proposition 2.8, the proof is complete.

The proof of 8.4.4 and 8.4.4’ is now obvious. Consider the diagram of 8.5.2 with F = L · G and l = L · G. Since F is a P-epi, the extension (e_{F} , ϵ_{F} , F) is in P(ℂ, KerF). But (e_{G}, ϵ_{G}, G) is the pushout extension of (e_{F} , ϵ_{F} , F) along \overline{L}, so that it is in P(ℂ, KerG), that is G is a P-epi. The proof of 8.4.4 is similar.

We have constructed the classes P-mono and P-epi starting from a proper class of extensions. It is possible to work in the opposite direction.

Proposition 8.6. Let M and E be two classes of morphisms in SCG. Assume that M satisfies conditions 0, 1, 2, 3 and 4 of Proposition 8.4 and that E satisfies conditions 0’, 1’, 2’, 3’ and 4’ of Proposition 8.4. Define P(ℂ, 𝔸) as the full sub-2-category of EXT(ℂ, 𝔸) of those extensions (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ such that Γ ∈ M and Σ ∈ E. Then P is a proper class of extensions.

Proof. Let us check, for example, that if E = (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ is in P(ℂ, 𝔸), then for any G: 𝔻 → ℂ the pullback extension G · E is in P(𝔻, 𝔸). Consider the diagram defining G · E (Section 6)

[figure: 𝔸; φ_{Γ,0}; 0; Γ; ⇒^{φ_{0}}; ⇓^{φ_{Γ}}; Σ ∩ G; ⇒^{p_{Σ,G}}; p_{Σ}; p_{G}; 𝔻; G; 𝔹; Σ; ℂ]

By condition 3’, p_{G} is in E; by condition 4 and the fact that M is closed under natural isomorphisms, we have that φ_{Γ,0} is in E.

Finally, let us observe that the class E alone suffices to construct a proper class of extensions.

Proposition 8.7. Let E be a class of morphisms in SCG satisfying conditions 0’, 1’, 2’, 3’ and 4’ of Proposition 8.4. Define the class M as the class of those faithful morphisms Γ: 𝔸 → 𝔹 in SCG whose cokernel is in E (equivalently, which are the kernel of some morphism Σ: 𝔹 → ℂ in E). Then M satisfies conditions 0, 1, 2, 3 and 4 of Proposition 8.4.

Proof. Let us check condition 3. Consider an extension (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ with Γ in E. Consider also the pushout extension along any F: 𝔸 → 𝔻 in SCG (Section 6)

[figure: 𝔸; Γ; F; ⇒^{i_{F,Γ}}; 𝔹; i_{Γ}; Σ; 𝔻; i_{F}; 0; F ∪ Γ; φ^{0,Σ}; ⇒^{φ^{Σ}}; ⇓^{φ^{0}}; ℂ]

Since Γ ∈ M, then Σ ∈ E. By condition 4’, φ^{0,Σ} ∈ E and then its kernel i_{Σ} is in M. As far as condition 4 is concerned, it follows from part 1 of Lemma 8.5 and the assumption that E is stable under pullbacks.

Dually, the class E can be constructed by the class M, so that the class M alone suffices to get a proper class of extensions.

9. Examples of proper classes

Example 9.1. Clearly, EXT is a proper class of extensions. In this case, P-epis are the essentially surjective functors and P-monos are the faithful functors.

Example 9.2. Define an extension (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ to be a G-extension if Σ: 𝔹 → ℂ is a Grothendieck fibration. This class is not proper (for example because Grothendieck fibrations are not stable under natural isomorphisms). Moreover, the smallest proper class containing G is EXT. Indeed, if Σ: 𝔹 → ℂ is any functor between groupoids, the standard (bi)pullback of Σ along the identity on ℂ is a Grothendieck fibration equivalent to Σ.

Example 9.3. We arrive now to the example which motivates the abstract approach of Section 8.

Definition 9.4. An extension (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ is an F-extension if there exist a functor S: ℂ → 𝔹 and a natural transformation σ: S · Σ ⇒ 1_{ℂ}.

Let us insist on the fact F-extensions are not trivial, because we do not require S and σ to be monoidal. The fact that F is a proper class is quite obvious. We only point out that the stability under pullbacks is due to the fact (already observed) that pullbacks in SCG are computed as in the 2-category of groupoids, so that a functorial section for Σ: 𝔹 → ℂ gives rise to a functorial section for any pullback of Σ. The last section of this paper is devoted to a cohomological classification of F-extensions.

We have defined the class F using F-epis. We want now to characterize F-monos.

Proposition 9.5. Let (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ be an extension. The following conditions are equivalent:

1) (Γ, φ, Σ) is an F-extension;

2) There exists a functor R: 𝔹 → 𝔸 and a coherent (w.r.t. the monoidal structure of Γ) natural transformation

ν = (ν_{B,A}: R(B ⊗ Γ(A)) → R(B) ⊗ A)_{(B,A)∈𝔹×𝔸}

To prove Proposition 9.5, we need the following lemma, which says that the section (S, σ) can be normalized, and that the pair R, ν induces a normalized retraction for Γ.

Lemma 9.6.

1) Let Σ: 𝔹 → ℂ, S: ℂ → 𝔹, σ: S · Σ ⇒ 1_{ℂ} be as in Definition 9.4; there is a functor \overline{S}: ℂ → 𝔹, a natural transformation \overline{σ}: \overline{S} · Σ ⇒ 1_{ℂ} and a morphism \overline{S}_{I}: I → \overline{S}(I) such that the following diagram commutes

\xymatrix{I \ar[r]^{Σ_{I}} \ar[d]_{1} & Σ(I) \ar[d]^{Σ(\overline{S}_{I})} \\ I & Σ(\overline{S}(I)) \ar[l]^{_{\overline{σ}_{I}}}}

2) Let Γ: 𝔸 → 𝔹, R: 𝔹 → 𝔸 and ν be as in Proposition 9.5; there is a functor \overline{R}: 𝔹 → 𝔸, a natural transformation \overline{ρ}: Γ · \overline{R} ⇒ 1_{𝔸} and a morphism \overline{R}_{I}: I → \overline{R}(I) such that the following diagram commutes

\xymatrix{I \ar[r]^{^{\overline{R}^{I}}} \ar[d]_{1} & \overline{R}(I) \ar[d]^{\overline{R}(Γ_{I})} \\ I & \overline{R}(Γ(I)) \ar[l]^{_{\overline{ρ}_{I}}}}

Proof. 1) Define

\overline{S}: f: X → Y ↦ S(f) ⊗ 1: S(X) ⊗ S(I)^{∗} → S(Y) ⊗ S(I)^{∗}
\overline{S}_{I} = η_{S(I)} : I → S(I) ⊗ S(I)^{∗} = \overline{S}(I)

\overline{σ}_{X}: Σ(\overline{S}(X)) ≃ Σ(S(X)) ⊗ \xymatrix{Σ(S(I))^{∗} \ar[r]^{σ_{X}⊗(σ_{I}^{−1})^{∗}} & X ⊗ I^{∗}} ≃ X ⊗ I ≃ X

2) Define \overline{R}(X) = R(X) ⊗ R(I)^{∗} and \overline{R}_{I} = η_{R(I)} : I → R(I) ⊗ R(I)^{∗} = \overline{R}(I). We get a natural transformation \overline{ν}_{B,A} = ν_{B,A} · (1 ⊗ γ): \overline{R}(B ⊗ Γ(A)) = R(B ⊗ Γ(A)) ⊗ R(I)^{∗} → R(B) ⊗ A ⊗ R(I)^{∗} → R(B) ⊗ R(I)^{∗} ⊗ A = \overline{R}(B) ⊗ A. Finally, we have ρ_{A} = \overline{ν}_{I,A} · \overline{R}_{I}^{−1} : \overline{R}(Γ(A)) ≃ \overline{R}(I ⊗ Γ(A)) → \overline{R}(I) ⊗ A → I ⊗ A ≃ A.

Now we can come back to the proof of Proposition 9.5.

Proof. 1) ⇒ 2) : Since Γ_{0} : 𝔸 → KerΣ is an equivalence, to prove condition 2) we look for a functor R: 𝔹 → KerΣ. Let B be an object of 𝔹, we define

R(B) = (B ⊗ S(Σ(B))^{∗}, R_{B}),

where R_{B} is

Σ(B ⊗ S(Σ(B))^{∗}) ≃ Σ(B) ⊗ \xymatrix{Σ(S(Σ(B)))^{∗} \ar[r]^{1⊗(σ_{Σ(B)}^{−1})^{∗}} & Σ(B) ⊗ Σ(B)^{∗} \ar[r]^{η_{Σ(B)}^{−1}} & I;}

if f: B → C is an arrow in 𝔹, we define R(f) = f ⊗ S(Σ(f^{−1}))^{∗}. Now, to obtain a natural transformation ν, let us consider an object (N, n: Σ(N) → I) in KerΣ.

Up to the monoidal structure of Σ, the component of ν at the point (B, (N, n)) is given by the following composition

B ⊗ N ⊗ S(Σ(B) ⊗ Σ(N))^{∗}
1⊗1⊗S(1⊗n^{−1})^{∗}
B ⊗ N ⊗ S(Σ(B) ⊗ I)^{∗}

B ⊗ N ⊗ S(Σ(B))^{∗}
1⊗γ
B ⊗ S(Σ(B))^{∗} ⊗ N

2) ⇒ 1) : Now we look for a functor S: CokerΓ → 𝔹 (the natural transformation σ: S · P_{Σ} ⇒ 1_{CokerΓ} will be the obvious one). Let X be an object in CokerΓ, we put S(X) = X ⊗ Γ(R(X))^{∗}. Consider now an arrow [f, A]: X → Y in CokerΓ, with A in 𝔸 and f: X → Y ⊗ Γ(A) in 𝔹. We define S[f, A] by

X ⊗ Γ(R(X))^{∗}
f⊗Γ(R(f^{−1}))^{∗}
Y ⊗ Γ(A) ⊗ Γ(R(Y ⊗ Γ(A)))^{∗}
1⊗1⊗Γ(ν_{Y,A}^{−1})^{∗}
Y Γ(A) ⊗ Γ(R(Y) ⊗ A)^{∗} ≃ Y ⊗ Γ(R(Y))^{∗}

It is easy to check that S is well defined; its functoriality follows from the coherence of ν.

The lack of symmetry in Proposition 9.5 can be corrected. Indeed, since Σ is monoidal, the natural transformation σ can be equivalently replaced by a natural transformation Σ(B ⊗ S(C)) → Σ(B) ⊗ C.

The difference between extensions and F-extensions is stressed by the following fact.

Proposition 9.7. Let (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ be an F-extension in SCG.

1) \xymatrix{π_{0}𝔸 \ar[r]^{π_{0}Γ} & π_{0}𝔹 \ar[r]^{π_{0}Σ} & π_{0}ℂ} is an extension of abelian groups ;

2) \xymatrix{π_{1}𝔸 \ar[r]^{π_{1}Γ} & π_{1}𝔹 \ar[r]^{π_{1}Σ} & π_{1}ℂ} is a split extension of abelian groups.

Proof. Since (Γ, φ, Σ) is an extension, in particular it is 2-exact. This implies the exactness of the two sequences of abelian groups (see Proposition 3.1 in [27]). Moreover, π_{1}Γ is injective, because Γ is faithful, and π_{0}Σ is surjective, because Σ is essentially surjective. By part 2 in Lemma 9.6, π_{0}Γ · π_{0}\overline{R} = 1_{π_{0}𝔸}, so that π_{0}Γ is injective and the π_{0}-sequence is an extension of abelian groups. As far as the π_{1}-sequence is concerned, we assume that the section (S, σ) is normalized (Lemma 9.6). In this way, we can still define π_{1}S: π_{1}ℂ → π_{1}𝔹 by

\xymatrix{I \ar[r]^{x} & I} ↦ \xymatrix{I \ar[r]^{S_{I}} & S(I) \ar[r]^{S(x)} & S(I) \ar[r]^{S_{I}^{−1}} & I}

and it is a morphism of groups because S is a functor. Finally, using once again the condition of Lemma 9.6, one checks that π_{1}S · π_{1}Σ = 1_{π_{1}ℂ}, so that the π_{1}- sequence is a split extension of abelian groups.

Example 9.8. If, in Definition 9.4, we ask that the section (S, σ) is only at a graph-theoretical level, we get another proper class strictly contained between F and EXT.

Example 9.9. An extension (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ of symmetric cat-groups is a π_{1}-extension if (π_{1}Γ, π_{1}Σ): π_{1}𝔸 → π_{1}𝔹 → π_{1}ℂ is an extension of abelian groups. Equivalently, (Γ, φ, Σ) is a π_{1}-extension when π_{1}Σ is surjective. Since π_{1} preserves pullbacks, π_{1}-extensions are a proper class.

Example 9.10. An extension (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ of symmetric cat-groups is a π_{0}-extension if (π_{0}Γ, π_{0}Σ): π_{0}𝔸 → π_{0}𝔹 → π_{0}ℂ is an extension of abelian groups. Equivalently, (Γ, φ, Σ) is a π_{0}-extension when π_{0}Γ is injective. Since π_{0} preserves pushouts, π_{0}-extensions are a proper class.

Example 9.11. From Propositions 5.1 and 5.2, we know that morphisms in SCG which are full functors are stable under pullbacks and pushouts. Therefore, one can wonder if, adding the conditions that Γ and Σ are full, one gets a proper class of extensions. This is not the case, because the trivial extension 𝔸 → 𝔸 × ℂ → ℂ is not of this kind, apart when 𝔸 is equivalent to a one-object cat-group and ℂ is equivalent to a discrete cat-group. There are no other possibilities. In fact, we have the following lemma.

Lemma 9.12. Let (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ be an extension.

1) Γ is full if and only if π_{1}(ℂ) = 0;

2) Σ is full if and only π_{0}(𝔸) = 0.

Proof. We prove part 1, part 2 is dual. If: It follows from 1.3.8 in [16], because ℂ is equivalent to CokerΓ. Only if: It follows from 1.3.2 in [16], because Γ is equivalent to e_{Σ}.

Full extensions, that is extensions with Γ and Σ full, provide a quick classification of symmetric cat-groups in the following way. Let A and B be two abelian groups and call Ep(A, B) (Ep stays for “Epinglage”, see [21]) the following 2-category:

- an object is a triple (𝔹, a, c) with 𝔹 in SCG and a: π_{1}(𝔹) → A, c: π_{0}(𝔹) → C two isomorphisms of groups;

- a 1-cell F: (𝔹, a, c) → (𝔹′, a′, c′) is a morphism F: 𝔹 → 𝔹′ in SCG such that the following diagrams commute

\xymatrix{π_{1}(𝔹) \ar[rr]^{π_{1}(F)} \ar[dr]_{a} &  & π_{1}(𝔹′) \ar[dl]^{a′} & π_{0}(𝔹) \ar[rr]^{π_{0}(F)} \ar[dr]_{c} &  & π_{0}(𝔹′) \ar[dl]^{c′} \\  & A &  &  & C}

- a 2-cell λ: F ⇒ F′ is simply a 2-cell in SCG.

Proposition 9.13. Let A and C be two abelian groups. The 2-categories Ep(A, C) and EXT(C[0], A[1]) are biequivalent.

Proof. Given a symmetric cat-group 𝔹, there is a canonical extension π_{1}(𝔹)[1] → 𝔹 → π_{0}(𝔹)[0]. Conversely, if (Γ, φ, Σ): A[1] → 𝔹 → C[0] is an extension, by Corollary 2.7 we have an exact sequence

\xymatrix{0 \ar[r] & A \ar[r]^{π_{1}(Γ)} & π_{1}(𝔹) \ar[r] & 0 \ar[r] & 0 \ar[r] & π_{0}(𝔹) \ar[r]^{π_{0}(Σ)} & C \ar[r] & 0}

so that π_{1}(Γ) and π_{0}(Σ) are isomorphisms. Finally, to check that these two constructions give a biequivalence, observe that in a 1-cell

[figure: 𝔹; β; Σ; ⇓^{α}; A[1]; Γ; Γ′; C[0]; 𝔹′; Σ′; ⇑^{γ}]

α is reduced to α_{I} which, by monoidality, is determined by Γ, β and Γ′; moreover, since C[0] is discrete, γ simply amount to say that π_{0}(β) · π_{0}(Σ′) = π_{0}(Σ).

This classification does not seem very deep. Nevertheless, observe that a full extension (Γ, φ, Σ): A[1] → 𝔹 → C[0] always has a functorial section for Σ, so that full extensions are a special case of F-extensions. Therefore, we can apply to full extensions the cohomological classification of Section 15. Since the choice of a functorial section of Σ amount to the choice of a set-theoretical section of π_{0}(Σ) together with a clivage of 𝔹, Theorem 15.2 and the previous proposition give the classification of symmetric cat-groups obtained by Sinh in [21].

10. The fundamental 2-exact sequences

In this section we use the functors induced by the 2-functors of Proposition 6.4 on the classifying groupoids. We use the same notations :

\xymatrix{− · E: Hom(𝔻, ℂ) \ar[r] & Ext(𝔻, 𝔸) \\ G · −: Ext(ℂ, 𝔸) \ar[r] & Ext(𝔻, 𝔸) \\ E · −: Hom(𝔸, 𝔻) \ar[r] & Ext(ℂ, 𝔻) \\ − · F: Ext(ℂ, 𝔸) \ar[r] & Ext(ℂ, 𝔻) .}

All these groupoids are pointed groupoids, and the various functors are functors of pointed groupoids (in fact, they are all in SCG, but the structure of symmetric cat-group of Ext(ℂ, 𝔸) will be discussed later) : the point in Hom(𝔻, ℂ) is the zero functor 0: 𝔻 → ℂ, the point in Ext(ℂ, 𝔸) is the trivial extension T (𝔸, ℂ). We say that a diagram in the 2-category Grpd* of pointed groupoids

[figure: 𝔸; 0; Γ; ℂ; 𝔹; Σ; ⇑^{φ}]

is a 2-exact sequence if the comparison functor Γ_{0} : 𝔸 → KerΣ is full and essentially surjective.

The next lemma is a step towards Proposition 10.2. It is an obvious exercise on the left exactness of representable 2-functors.

Lemma 10.1. Consider a symmetric cat-group 𝔻, a 1-cell Σ: 𝔹 → ℂ in SCG and its kernel

[figure: KerΣ; 0; e_{Σ}; ℂ; 𝔹.; Σ; ⇑^{ϵ_{Σ}}]

In the next diagram, the comparison functor K is an equivalence

\xymatrix{Hom(𝔻, KerΣ) \ar[r]^{−·e_{Σ}} \ar[dr]_{K} & Hom(𝔻, 𝔹) \ar[r]^{−·Σ} & Hom(𝔻, ℂ) \\  & Ker(− · Σ) . \ar[u]_{e_{(−·Σ)}}}

Proposition 10.2. Let E = (Γ, φ, Σ)

[figure: 𝔸; 0; Γ; ℂ; 𝔹; Σ; ⇑^{φ}]

be an extension in SCG and fix a symmetric cat-group 𝔻. The sequence in Grpd*

\xymatrix{Hom(𝔻, 𝔸) \ar[r]^{−·Γ} & Hom(𝔻, 𝔹) \ar[r]^{−·Σ} & Hom(𝔻, ℂ) \ar@{-->}[dll]_{−·E} \\ Ext(𝔻, 𝔸) \ar[r]_{−·Γ} & Ext(𝔻, 𝔹) \ar[r]_{−·Σ} & Ext(𝔻, ℂ)}

(equipped with the pointed natural transformations specified in the proof) is 2-exact at each point. If P is a proper class of extensions and if E ∈ P(ℂ, 𝔸), the same result holds replacing EXT by P.

Proof. The 2-exactness of

[figure: Hom(𝔻, 𝔸); 0; −·Γ; Hom(𝔻, ℂ); Hom(𝔻, 𝔹); −·Σ; ⇑^{−·φ}]

is attested by Lemma 10.1. We give now a detailed proof of the 2-exactness in Ext(𝔻, 𝔸). The argument for the 2-exactness in Hom(𝔻, ℂ) and Ext(𝔻, 𝔹) is similar and we give only the description of the pointed natural transformations involved. We use Lemma 4.5 and its consequence discussed at the end of Section 4, as wall as the dual argument.

2-exactness in Ext(𝔻, 𝔸) : let G: 𝔻 → ℂ be in Hom(𝔻, ℂ) and consider the pullback along Σ and the pushout along Γ, i.e. the extension (G · E) · Γ, as in the following diagrams

[figure: ℙ; p_{G}; p_{Σ}; ⇒^{p_{Σ,G}}; 𝔻; G; 𝔸; φ_{Γ,0}; Γ; 𝔹; ⇓^{φ_{Γ}}; Σ; ℂ; 𝔸; φ_{Γ,0}; Γ; ⇒^{i_{Γ,φ}}; ℙ; p_{G}; i_{φ}; ⇑^{ψ_{G}}; 𝔻; 𝔹; i_{Γ}; ℚ; ψ]

with φ_{0} : φ_{Γ,0} · p_{G} ⇒ 0 and ψ_{0} : i_{Γ} · ψ ⇒ 0. Using φ_{Γ}, we get a retraction in SCG in the following way

[figure: 𝔸; φ_{Γ,0}; Γ; ⇒^{i_{Γ,φ}}; ℙ; i_{φ}; p_{Σ}; 𝔹; i_{Γ}; 1_{𝔹}; ℚ; R; ⇒^{π}; ⇓^{ρ}; 𝔹]

This gives the component at G of a pointed natural transformation ξ: (− · E) · (− · Γ) ⇒ 0. Consider now the factorization through the kernel of (−·Γ) induced by the previous natural transformation

\xymatrix{Hom(𝔻, ℂ) \ar[r]^{−·E} \ar[dr]_{Λ} & Ext(𝔻, 𝔸) \ar[r]^{−·Γ} & Ext(𝔻, 𝔹) \\  & Ker(− · Γ) \ar[u]}

1) Λ is essentially surjective on objects : consider an extension (L, μ, M): 𝔸 → 𝔼 → 𝔻, its pushout along Γ

[figure: 𝔸; L; Γ; ⇒^{i_{Γ,L}}; 𝔼; i_{L}; 𝔹; j_{L}; 𝕃]

and the factorization

[figure: 𝕃; Σ_{L}; ⇓^{jμ}; 𝔹; j_{L}; 0; 𝔼; i_{L}; M; 𝔻; ⇓^{iμ}]

making commutative the following diagram

\xymatrix{Γ · j_{L} · Σ_{L} \ar@{=>}[r]^{Γ·j_{μ}} \ar@{=>}[d]_{i_{Γ,L}·Σ_{L}} & Γ · 0 \ar@{=>}[r] & 0 \\ L · i_{L} · Σ_{L} \ar@{=>}[rr]_{L·i_{μ}} &  & L · M . \ar@{=>}[u]_{μ}}

Assume that there is a retraction in SCG

[figure: 𝔹; 1_{𝔹}; j_{L}; 𝔹; 𝕃; T; ⇑^{t}; .]

Using this retraction, we can construct a 2-cell in SCG

\xymatrix{τ: L · i_{L} · T · Σ \ar@{=>}[r]^{i_{Γ,L}^{−1}·T ·Σ} & Γ · j_{L} · T · Σ \ar@{=>}[r]^{Γ·t·Σ} & Γ · Σ \ar@{=>}[r]^{φ} & 0 .}

Since (M, μ) is a cokernel of L, we get G: 𝔻 → ℂ in Hom(𝔻, ℂ) and a 2-cell τ′: M · G ⇒ i_{L} · T · Σ in SCG such that the following diagram commutes

\xymatrix{L · M · G \ar@{=>}[r]^{μ·G} \ar@{=>}[d]_{L·τ′} & 0 · G \ar@{=>}[d] \\ L · i_{L} · T · Σ \ar@{=>}[r]_{τ} & 0 .}

Consider now the extension G · E as at the beginning of the proof. We look for a 1-cell (α, β, γ): (L, μ, M) → G · E in EXT(𝔻, 𝔸). The universal property of the pullback ℙ gives a factorization

[figure: 𝔼; i_{L}·T; β; M; 𝔹; ℙ; p_{Σ}; p_{G}; ⇐^{δ}; ⇒^{γ}; 𝔻]

such that

\xymatrix{β · p_{Σ} · Σ \ar@{=>}[r]^{δ·Σ} \ar@{=>}[d]_{β·p_{Σ,G}} & i_{L} · T · Σ \\ β · p_{G} · G \ar@{=>}[r]_{γ·G} & M · G \ar@{=>}[u]_{τ′}}

commutes. We can construct two 2-cells in SCG

\xymatrix{α_{G}: L · β · p_{G} \ar@{=>}[r]^{L·γ} & L · M \ar@{=>}[r]^{μ} & 0 \ar@{=>}[r]^{φ_{0}^{−1}} & φ_{Γ,0} · p_{G} \\ α_{Σ} : L · β · p_{Σ} \ar@{=>}[r]^{L·δ} & L · i_{L} · T \ar@{=>}[r]^{i_{Γ,L}^{−1}·T} & Γ · j_{L} · T \ar@{=>}[r]^{Γ·t} & Γ \ar@{=>}[r]^{φ_{Γ}^{−1}} & φ_{Γ,0} · p_{Σ} .}

Since α_{G} and α_{Σ} are compatible with p_{Σ,G}, the universal property of the pullback ℙ gives a unique 2-cell α: L · β ⇒ φ_{Γ,0} in SCG such that α · p_{G} = α_{G} and α · p_{Σ} = α_{Σ}. In particular, the equation α · p_{G} = α_{G} express the condition on (α, β, γ) to be a 1-cell (L, μ, M) → G · E in EXT(𝔻, 𝔸). It remains to prove that (α, β, γ) is a morphism in the kernel of (− · Γ). For this, observe that the universal property of the pushout 𝕃 gives a factorization q: 𝕃 → ℚ, m: i_{L} · q ⇒ β · i_{φ}, n: j_{L} · q ⇒ i_{Γ} such that

\xymatrix{Γ · j_{L} · q \ar@{=>}[rr]^{i_{Γ,L}·q} \ar@{=>}[d]_{Γ·n} &  & L · i_{L} · q \ar@{=>}[d]^{L·m} \\ Γ · i_{Γ} \ar@{=>}[r]_{i_{Γ,φ}} & φ_{Γ,0} · i_{φ} & L · β · i_{φ} \ar@{=>}[l]^{α·i_{φ}}}

commutes. Now we have two retractions of j_{L} : (T, t) and the one given by

[figure: 𝔹; 1_{𝔹}; i_{Γ}; j_{L}; 𝔹; ⇑^{ρ}; 𝕃; q; ⇒^{n}; ℚ .; R]

We have also a 2-cell in SCG

\xymatrix{i_{L} · q · R \ar@{=>}[r]^{m·R} & β · i_{φ} · R \ar@{=>}[r]^{β·π} & β · p_{Σ} \ar@{=>}[r]^{δ} & i_{L} · T}

and, since the following diagram commutes

\xymatrix{Γ · j_{L} · q · R \ar@{=>}[r]^{i_{Γ,L}·q·R} \ar@{=>}[d]_{Γ·n·R} & L · i_{L} · R \ar@{=>}[r]^{L·m·R} & L · β · i_{φ} · R \ar@{=>}[d]^{L·β·π} \\ Γ · i_{Γ} · R \ar@{=>}[d]_{Γ·ρ} &  & L · β · p_{Σ} \ar@{=>}[d]^{L·δ} \\ Γ \ar@{=>}[r]_{Γ·t^{−1}} & Γ · j_{L} · T \ar@{=>}[r]_{i_{Γ,L}·T} & L · i_{L} · T ,}

the universal property of the pushout 𝕃 gives a unique 2-cell in SCG, λ: T ⇒ q · R, such that

\xymatrix{j_{L} · T \ar@{=>}[r]^{j_{L}·λ} \ar@{=>}[d]_{t} & j_{L} · q · R \ar@{=>}[d]^{n·R} & i_{L} · T \ar@{=>}[r]^{i_{L}·λ} & i_{L} · q · R \ar@{=>}[d]^{m·R} \\ 1_{𝔹} & i_{Γ} · R \ar@{=>}[l]^{ρ} & β · p_{Σ} \ar@{=>}[u]^{δ} & β · i_{φ} · R \ar@{=>}[l]^{β·π}}

commute. In particular, the first equation exactly means that λ commutes with the retraction.

2) Λ is full : consider G, \overline{G} in Hom(𝔻, ℂ), the extensions G · E and \overline{G} · E and a 1-cell (α, β, γ): G · E → \overline{G} · E in EXT(𝔻, 𝔸). Consider also the retractions in SCG

[figure: 𝔹; 1_{𝔹}; i_{Γ}; 𝔹; ℚ; R; ⇑^{ρ}; 𝔹; 1_{𝔹}; _{\overline{i}_{Γ}}; 𝔹; \overline{ℚ}; \overline{R}; ⇑^{\overline{ρ}}; .]

The universal property of the pushout ℚ gives a factorization q, a, b as in the following diagram

[figure: ℚ; q; ⇒^{b}; ℙ; β; i_{φ}; 𝔹; i_{Γ}; _{\overline{i}_{Γ}}; \overline{ℚ}; ⇓^{a}; \overline{ℙ} .; _{\overline{i}_{φ}}]

In this way we have a second retraction for i_{Γ} :

[figure: 𝔹; i_{Γ}; 1_{𝔹}; ⇑^{a}; 𝔹; ℚ; _{\overline{i}_{Γ}}; q; \overline{ℚ} .; \overline{R}; ⇐^{\overline{ρ}}]

Assume that there is a 2-cell λ: R ⇒ q·\overline{R} in SCG making commutative the following diagram

\xymatrix{i_{Γ} · R \ar@{=>}[r]^{i_{Γ}·λ} \ar@{=>}[d]_{ρ} & i_{Γ} · q · \overline{R} \ar@{=>}[d]^{a·\overline{R}} \\ 1_{𝔹} & \overline{i}_{Γ} · \overline{R} . \ar@{=>}[l]^{\overline{ρ}}}

We can build up the following 2-cell

[figure: τ: p_{G} · Σ; π^{−1}·Σ; i_{φ} · R · Σ; i_{φ}·λ·Σ; i_{φ} · q · \overline{R} · Σ; b·\overline{R}·Σ; β · \overline{i}_{φ} · \overline{R} · Σ; β·\overline{π}·Σ; β · \overline{p}_{Σ} · Σ; β·p_{Σ,\overline{G}}; β · p\overline{G} · \overline{G}; γ·\overline{G}; p_{G} · \overline{G} .]

Since the diagram

\xymatrix{φ_{Γ,0} · p_{Σ} · Σ \ar@{=>}[rr]^{φ_{Γ,0}·τ} \ar@{=>}[d]_{φ_{Γ}·Σ} &  & φ_{Γ,0} · p_{G} · \overline{G} \ar@{=>}[d]^{φ_{0}·\overline{G}} \\ Γ · Σ \ar@{=>}[r]_{φ} & 0 & 0 · \overline{G} \ar@{=>}[l]}

commutes, the universal property of (p_{G}, φ_{0}) (it is the cokernel of φ_{Γ,0}) gives a unique 2-cell μ: G ⇒ \overline{G} in SCG making commutative the following diagram

\xymatrix{p_{G} · G \ar@{=>}[rr]^{p_{G}·μ} &  & p_{G} · \overline{G} \\  & p_{Σ} · Σ \ar@{=>}[ul]^{p_{Σ,G}} \ar@{=>}[ur]_{τ}}

It remains to prove that (α, β, γ) and μ · E coincide as arrows in Ext(𝔻, 𝔸). The functor part of μ · E is given by the following factorization through the pullback \overline{ℙ}

[figure: ℙ; p_{Σ}; M; p_{G}; 𝔹; \overline{ℙ}; _{\overline{p}_{Σ}}; p\overline{G}; ⇐^{n}; ⇒^{m}; 𝔻 .]

We need a 2-cell x: M ⇒ β in EXT(𝔻, 𝔸) and, for this, we can consider

[figure: ℙ; p_{Σ}; β; p_{G}; 𝔹; \overline{ℙ}; _{\overline{p}_{Σ}}; p\overline{G}; ⇐^{y}; ⇒^{γ}; 𝔻]

where y is given by the following composition

\xymatrix{β · \overline{p}_{Σ} \ar@{=>}[r]^{β·\overline{π}^{−1}} & β · \overline{i}_{φ} · \overline{R} \ar@{=>}[r]^{b^{−1}·\overline{R}} & i_{φ} · q · \overline{R} \ar@{=>}[r]^{i_{φ}·λ^{−1}} & i_{φ} · R \ar@{=>}[r]^{π} & p_{Σ}}

Since the following diagram commutes

\xymatrix{β · \overline{p}_{Σ} · Σ \ar@{=>}[rr]^{β·p_{Σ,\overline{G}}} \ar@{=>}[d]_{y·Σ} &  & β · p\overline{G} · \overline{G} \ar@{=>}[d]^{γ·\overline{G}} \\ p_{Σ} · Σ \ar@{=>}[r]_{p_{Σ,G}} & p_{G} · G \ar@{=>}[r]_{p_{G}·μ} & p_{G} · \overline{G} ,}

the universal property of the pullback \overline{ℙ} gives a unique 2-cell x: M ⇒ β in SCG such that

[figure: M · \overline{p}_{Σ}; x·\overline{p}_{Σ}; n; β · \overline{p}_{Σ}; y; p_{Σ}; M · p\overline{G}; x·p\overline{G}; m; β · p\overline{G}; γ; p_{G}]

commute. The verification that x satisfies also the first condition of 2-cell in EXT(𝔻, 𝔸) needs a full description of μ · E (see Lemma 6.2) and we omit it.

2-exactness in Hom(𝔻, ℂ) : let F: 𝔻 → 𝔹 be in Hom(𝔻, 𝔹) and consider the following pullback

[figure: ℙ; p_{2}; p_{1}; ⇒^{p}; 𝔻; F ·Σ; 𝔹; Σ; ℂ .]

We get a section of p_{2} in SCG in the following way

[figure: 𝔻; S_{F}; 1_{𝔻}; F; ⇒^{σ_{F}}; ⇓^{p_{F}}; ℙ; ⇒^{p}; p_{1}; p_{2}; 𝔻; F ·Σ; 𝔹; Σ; ℂ .]

Via Lemma 4.5, the section (S_{F} , σ_{F}) gives the component at F of a pointed natural transformation ζ: (− · Σ) · (− · E) ⇒ 0.

2-exactness in Ext(𝔻, 𝔹) : let L = (F, ψ, G): 𝔸 → 𝔼 → 𝔻 be an extension of 𝔸 by 𝔻. Consider the following pushouts and factorizations

[figure: 𝔸; F; Γ; ⇒^{i_{Γ,F}}; 𝔼; i_{F}; G; 𝔹; i_{Γ}; 0; ℚ; i_{ψ}; ⇒^{ψ_{Γ}}; ⇓^{ψ_{0}}; 𝔻; 𝔹; i_{Γ}; Σ; ⇒^{i_{Σ,ℚ}}; ℚ; i_{ℚ}; i_{ψ}; ℂ; i_{Σ}; 0; ℝ; j_{ψ}; ⇒^{ψ_{Σ}}; ⇓^{Σ_{0}}; 𝔻]

which gives (L · Γ) · Σ. We get a factorization

[figure: ℚ; σ; 𝔹; i_{Γ}; Σ; ℂ; ⇓^{s}; ⇓^{f}; 𝔼; i_{F}; 0]

and, finally, a factorization

[figure: ℝ; R; ℂ; i_{Σ}; 1_{ℂ}; ℂ; ⇓^{ρ}; ⇓^{r}; ℚ; i_{ℚ}; σ]

making commutative the following diagram

\xymatrix{Σ · i_{Σ} · R \ar@{=>}[r]^{Σ·ρ} \ar@{=>}[d]_{i_{Γ,ℚ}·R} & Σ \\ i_{Γ} · i_{ℚ} · R \ar@{=>}[r]_{i_{Γ}·r} & i_{Γ} · σ . \ar@{=>}[u]_{s}}

In particular, via the dual of Lemma 4.5, the retraction (R, ρ) of i_{Σ} gives the component at the point L of a pointed natural transformation χ: (− · Γ) · (− · Σ) ⇒ 0.

Let us write explicitly the duals of Lemma 10.1 and Proposition 10.2.

Lemma 10.3. Consider a symmetric cat-group 𝔻, a 1-cell Γ: 𝔸 → 𝔹 in SCG and its cokernel

[figure: 𝔸; 0; Γ; CokerΓ; 𝔹.; P_{Γ}; ⇑^{π_{Γ}}]

In the next diagram, the comparison functor K is an equivalence

\xymatrix{Hom(CokerΓ, 𝔻) \ar[r]^{P_{Γ}·−} \ar[dr]_{K} & Hom(𝔹, 𝔻) \ar[r]^{Γ·−} & Hom(𝔸, 𝔻) \\  & Ker(Γ · −) . \ar[u]_{P_{(Γ·−)}}}

Proposition 10.4. Let E = (Γ, φ, Σ)

[figure: 𝔸; 0; Γ; ℂ; 𝔹; Σ; ⇑^{φ}]

be an extension in SCG and fix a symmetric cat-group 𝔻. The sequence in Grpd*

\xymatrix{Hom(ℂ, 𝔻) \ar[r]^{Σ·−} & Hom(𝔹, 𝔻) \ar[r]^{Γ·−} & Hom(𝔸, 𝔻) \ar@{-->}[dll]_{E·−} \\ Ext(ℂ, 𝔻) \ar[r]_{Σ·−} & Ext(𝔹, 𝔻) \ar[r]_{Γ·−} & Ext(𝔸, 𝔻)}

(equipped with four specific pointed natural transformations) is 2-exact at each point. If P is a proper class of extensions and if E ∈ P(ℂ, 𝔸), the same result holds replacing EXT by P.

11. Projective objects

In this section we consider projective objects in SCG. Definitions and properties for injective objects are dual and are left to the reader. We fix a proper class P of extensions. The notion of P-projective object corresponding to our definition of extension is the following one.

Definition 11.1. Let 𝔸 be a symmetric cat-group ; 𝔸 is P-projective if, for any P-epi F: 𝔹 → 𝔸, there exists a section in SCG (i.e. a 1-cell S: 𝔸 → 𝔹 and a 2-cell σ: S · F ⇒ 1_{𝔸} in SCG).

Using pullbacks in SCG (see Section 5), the previous definition can be restated in the following way : a symmetric cat-group 𝔸 is P-projective if, for each 1-cell G: 𝔸 → ℂ in SCG and for each P-epi F: 𝔹 → ℂ, there exist G′: 𝔸 → 𝔹 and g: G′ · F ⇒ G in SCG.

Corollary 11.2. Let 𝔸 be in SCG ; the following conditions are equivalent : 1) 𝔸 is P-projective ;

2) For each 𝔹 in SCG, any P-extension of 𝔹 by 𝔸 splits ;

3) For each 𝔹 in SCG, the classifying groupoid clP(𝔸, 𝔹) of P(𝔸, 𝔹) is connected.

In the next corollary we use that the sequence of Proposition 10.4 (as well as that of Proposition 10.2) is a 2-exact sequence of symmetric cat-groups, and not only of pointed groupoids (the cat-group structure of clP is discussed in Section 13). By Proposition 6.2 in [16], we can then translate its 2-exactness saying that the factorizations through the appropriate cokernels are full and faithful functors.

Corollary 11.3. Consider an extension E = (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ of 𝔸 by ℂ. Assume that 𝔹 is a P-projective object in SCG and that E ∈ P(ℂ, 𝔸). Then, for any symmetric cat-group 𝔻, clP(ℂ, 𝔻) is equivalent in SCG to the cokernel of

Γ · −: Hom(𝔹, 𝔻) → Hom(𝔸, 𝔻).

Proof. By Proposition 10.4, we have a 2-exact sequence

\xymatrix{Hom(𝔸, 𝔻) \ar[r]^{E·−} & clP(ℂ, 𝔻) \ar[r]^{Σ·−} & clP(𝔹, 𝔻)}

and then its image by π_{0} : SCG → Ab is an exact sequence of abelian groups. Moreover, by Corollary 11.2, π_{0}(clP(𝔹, 𝔻)) = 0, so that π_{0}(E · −) is surjective. Equivalently, E·− is essentially surjective on objects. Consider now the factorization ϵ of E · − through the cokernel of Γ · −

\xymatrix{Hom(𝔹, 𝔻) \ar[r]^{Γ·−} & Hom(𝔸, 𝔻) \ar[r]^{E·−} \ar[d] & clP(ℂ, 𝔻) \\  & Coker(Γ · −) \ar[ur]_{ϵ}}

By 2-exactness in Hom(𝔸, 𝔻) of the sequence in Proposition 10.4, ϵ is full and faithful. It is also essentially surjective, because E · − is essentially surjective.

12. Exactness of Hom

In this section we show that EXT measures the non exactness of representable 2-functors. We consider the case of covariant representable 2-functors and we leave to the reader the controvariant case.

Lemma 12.1. Let

[figure: 𝔾; 0; F; 𝕂; ℍ; G; ⇑^{λ}]

be a 2-exact sequence in SCG. The functor F is essentially surjective if and only if π_{0}(G) = 0.

Proof. Only if : Since π_{0}(F) is surjective, Cokerπ_{0}(F) = 0. But Cokerπ_{0}(F) ≃ π_{0}(CokerF), and π_{0}(G) factors through π_{0}(CokerF). If : Since (π_{0}(F), π_{0}(G)): π_{0}(𝔾) → π_{0}(ℍ) → π_{0}(𝕂) is exact and π_{0}(G) = 0, then π_{0}(F) is surjective, that is F is essentially surjective.

Proposition 12.2. Let E = (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ be an extension. Consider a symmetric cat-group 𝔻 and the morphism − · E: Hom(𝔻, ℂ) → Ext(𝔻, 𝔸). The sequence (−·Γ, −·φ, −·Σ): Hom(𝔻, 𝔸) → Hom(𝔻, 𝔹) → Hom(𝔻, ℂ) is an extension if and only if π_{0}(− · E) = 0.

Proof. By Lemma 10.1, we know that − · Γ is equivalent to the kernel of − · Σ. Hence, the result follows applying the previous lemma to the sequence

\xymatrix{Hom(𝔻, 𝔹) \ar[r]^{−·Σ} & Hom(𝔻, ℂ) \ar[r]^{−·E} & Ext(𝔻, 𝔸)}

which is 2-exact by Proposition 10.2.

Corollary 12.3. Let 𝔻 be a symmetric cat-group. The following conditions are equivalent :

1) 𝔻 is projective (i.e. P-projective for P=EXT) ;

2) For any essentially surjective morphism Σ: 𝔹 → ℂ, the functor

− · Σ: Hom(𝔻, 𝔹) → Hom(𝔻, ℂ)

is essentially surjective ;

3) For any extension (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ, the sequence

(− · Γ, − · φ, − · Σ): Hom(𝔻, 𝔸) → Hom(𝔻, 𝔹) → Hom(𝔻, ℂ)

is an extension.

Proof. The non-trivial implication is 1) ⇒ 2), which follows from Proposition 12.2 and Corollary 11.2.

We consider now F-extensions (Example 9.3). In this case, Proposition 12.2 has a slightly different (and a little bit more difficult to prove) formulation.

Proposition 12.4. Consider an F-extension E = (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ in SCG and a symmetric cat-group 𝔻. The diagram (𝔻, E):

[figure: Hom(𝔻, 𝔸); 0; −·Γ; Hom(𝔻, ℂ); Hom(𝔻, 𝔹); −·Σ; ⇑^{−·φ}]

is an F-extension if and only if the 2-functor

− · E: Hom(𝔻, ℂ) → F(𝔻, 𝔸)

is isomorphic to the 2-functor which sends each 1-cell G: 𝔻 → ℂ into the trivial extension T (𝔸, 𝔻) and each 2-cell μ: G ⇒ G′ into the identity of T (𝔸, 𝔻).

Proof. By Lemma 10.1, we have to deal only with a functorial section for − · Σ. We use the notations of Lemma 6.1 and Lemma 6.2. (⇐) Assume that we have a 2-natural transformation

\xymatrix{Hom(𝔻, ℂ) \ar[r]^{0} \ar[r]^{⇓^{θ}}_{−·E} & F(𝔻, 𝔸) .}

This means that, for each G: 𝔻 → ℂ in Hom(𝔻, ℂ), we have a 1-cell (α_{G}, θ_{G}, γ_{G}): T (𝔸, 𝔻) → G · E in F(𝔻, 𝔸)

[figure: 𝔸 × 𝔻; θ_{G}; p_{𝔻}; ⇓^{α_{G}}; 𝔸; i_{𝔸}; φ_{Γ,0}; 𝔻; Σ ∩ G; p_{G}; ⇑^{γ_{G}}]

and, for each μ: G ⇒ G′ in Hom(𝔻, ℂ), we have a 2-cell θ_{μ}: (α_{G}, θ_{G}, γ_{G}) · (μ · E) ⇒ (α_{G′}, θ_{G′}, γ_{G′}). Now we can define T: Hom(𝔻, ℂ) → Hom(𝔻, 𝔹) by

[figure: Σ ∩ G; β; p_{Σ}; ⇓^{θμ}; T[𝔻; G; ⇓^{μ}; G′; ℂ] = 𝔻; i_{𝔻}; 𝔸 × 𝔻; θ_{G}; θ_{G′}; 𝔹; Σ ∩ G′; p_{Σ′}; ⇓^{s^{−1}}]

and it is functorial by 2-naturality of θ. We need a natural transformation

[figure: Hom(𝔻, ℂ); 1; T; Hom(𝔻, ℂ); Hom(𝔻, 𝔹) .; −·Σ; ⇑^{τ}]

For each G ∈ Hom(𝔻, ℂ), we define τ(G): T (G) · Σ ⇒ G by the following diagram

[figure: 𝔻; i_{𝔻}; 1_{𝔻}; 𝔸 × 𝔻; θ_{G}; p_{𝔻}; =; Σ ∩ G; p_{G}; p_{Σ}; ⇓^{γ_{G}}; ⇓^{p_{Σ,G}}; 𝔹; Σ; ℂ; 𝔻; G]

The naturality of τ comes from the compatibility of β with p_{Σ,G} and p_{Σ,G′} and from the fact that θ_{μ} is a 2-cell in F(𝔻, 𝔸).

(⇒) Assume that we have a functorial section

[figure: Hom(𝔻, ℂ); 1; T; Hom(𝔻, ℂ); Hom(𝔻, 𝔹) .; −·Σ; ⇑^{τ}]

and consider, for G in Hom(𝔻, ℂ), the 2-cell τ(G): T (G) · Σ ⇒ G in SCG. By the universal property of the pullback Σ ∩ G, we obtain g, g_{1}, g_{2} in SCG as in the following diagram

[figure: 𝔻; g; 1_{𝔻}; T (G); ⇒^{g_{1}}; ⇓^{g_{2}}; Σ ∩ G; ⇒^{p_{Σ,G}}; p_{Σ}; p_{G}; 𝔻; G; 𝔹; Σ; ℂ]

making commutative the following diagram

\xymatrix{g · p_{Σ} · Σ \ar@{=>}[r]^{g·p_{Σ,G}} \ar@{=>}[d]_{g_{2}·Σ} & g · p_{G} · G \ar@{=>}[d]^{g_{1}·G} \\ T (G) · Σ \ar@{=>}[r]_{τ(G)} & G .}

In particular, (g, g_{1}) is an object of Split(p_{G}). By Lemma 4.5, (g, g_{1}) corresponds to a 1-cell (α_{G}, θ_{G}, γ_{G}): T (𝔸, 𝔻) → G · E in F(𝔻, 𝔸). Now, let μ: G ⇒ G′ be in Hom(𝔻, ℂ). We need a 2-cell in F(𝔻, 𝔸) as in the following diagram

[figure: T (𝔸, 𝔻); (α_{G},θ_{G},γ_{G}); (α_{G′} ,θ_{G′} ,γ_{G′}); G · E; μ·E; ⇒^{θμ}; G′ · E .]

By Lemma 4.5, we can equivalently look for an arrow

λ_{μ}: (g · β, \xymatrix{g · β · p_{G′} \ar@{=>}[r]^{g·γ} & g · p_{G} \ar@{=>}[r]^{g_{1}} & 1_{𝔻}} ) ⇒ (g′, \xymatrix{g′ · p_{G′} \ar@{=>}[r]^{g′_{1}} & 1_{𝔻})}

in Split(p_{G′}), where g′, g′_{1} and g′_{2} are as in the following diagram

[figure: 𝔻; g′; 1_{𝔻}; T (G′); ⇒^{g′_{1}}; ⇓^{g′_{2}}; Σ ∩ G′; ⇒^{p_{Σ,G′}}; p′_{Σ}; p_{G′}; 𝔻; G′; 𝔹; Σ; ℂ .]

Consider the 2-cells

\xymatrix{λ_{𝔻}: g · β · p_{G′} \ar@{=>}[r]^{g·γ} & g · p_{G} \ar@{=>}[r]^{g_{1}} & 1_{𝔻} \ar@{=>}[r]^{(g′_{1})^{−1}} & g′ · p_{G′} \\ λ_{𝔹}: g · β · p′_{Σ} \ar@{=>}[r]^{g·s} & g · p_{Σ} \ar@{=>}[r]^{g_{2}} & T (G) \ar@{=>}[r]^{T (μ)} & T (G′) \ar@{=>}[r]^{(g′_{2})^{−1}} & g′ · p′_{Σ} .}

Using the three equations satisfied by β, γ, s, g, g_{1}, g_{2}, g′, g′_{1}, g′_{2} and the naturality of τ, one can check that the following diagram commutes

\xymatrix{g · β · p′_{Σ} · Σ \ar@{=>}[r]^{g·β·p_{Σ,G′}} \ar@{=>}[d]_{λ_{𝔹}·Σ} & g · β · p_{G′} · G′ \ar@{=>}[d]^{λ_{𝔻}·G′} \\ g′ · p′_{Σ} · Σ \ar@{=>}[r]_{g′·p_{Σ,G′}} & g′ · p_{G′} · G′ .}

The universal property of the pullback Σ∩G′ gives then a unique 2-cell λ_{μ}: g·β ⇒ g′ such that λ_{μ} · p_{G′} = λ_{𝔻} and λ_{μ} · p′_{Σ} = λ_{𝔹}. The first of these equations precisely means that λ_{μ} is an arrow in Split(p_{G′}).

13. Baer sum

In this section, we introduce Baer sum for extensions of symmetric cat-groups. The construction closely follows the 1-dimensional analogue and we don’t give the proof. If A, C are two abelian groups, the group of extensions of A by C is in fact the group of connected components of the category of extensions of A by C, which is a symmetric cat-group. In the same way, the structure of symmetric cat-group on Ext(ℂ, 𝔸), for 𝔸 and ℂ two symmetric cat-groups, could be extended to a monoidal structure on the 2-groupoid EXT(ℂ, 𝔸). We limit our attention to the groupoid Ext(ℂ, 𝔸) only for sake of simplicity.

Lemma 13.1. With the notations of Definition 8.1.

1) Consider an extension E = (Γ, φ, Σ), a 1-cell G: 𝔻 → ℂ in SCG and the following 1-cell G · E → E (see Lemma 6.1)

[figure: 𝔸; φ_{Γ,0}; 1_{𝔸}; ⇓^{φ_{Γ}}; Σ ∩ G; p_{Σ}; p_{G}; ⇓^{p{−1 \atop Σ,G}}; 𝔻; G; 𝔸; Γ; 𝔹; Σ; ℂ .]

For any 1-cell (F, α, β, γ, G): E′ → E

[figure: 𝔸′; Γ′; F; ⇓^{α}; 𝔹′; β; Σ′; ⇓^{γ}; 𝔻; G; 𝔸; Γ; 𝔹; Σ; ℂ]

there is a 1-cell (F′, α′, β′, γ′, G′): E′ → G · E and a 2-cell

(x, y, z): (F′, α′, β′, γ′, G′) · (1_{𝔸}, φ_{Γ}, p_{Σ}, p_{Σ,G}^{−1}, G) ⇒ (F, α, β, γ, G).

Moreover, for any other 1-cell (F′′, α′′, β′′, γ′′, G′′): E′ → G · E with a 2-cell

(u, v, w): (F′′, α′′, β′′, γ′′, G′′) · (1_{𝔸}, φ_{Γ}, p_{Σ}, p_{Σ,G}^{−1}, G) ⇒ (F, α, β, γ, G),

there is a unique 2-cell (a, b, c): (F′, α′, β′, γ′, G′) ⇒ (F′′, α′′, β′′, γ′′, G′′) making commutative the following diagram

\xymatrix{(F′, α′, β′, γ′, G′) · (1_{𝔸}, φ_{Γ}, p_{Σ}, p_{Σ,G}^{−1}, G) \ar@{=>}[dr]^{(x,y,z)} \ar@{=>}[dd]_{(a,b,c)·(1_{𝔸},p_{Σ},G)} \\  & (F, α, β, γ, G) \\ (F′′, α′′, β′′, γ′′, G′′) · (1_{𝔸}, φ_{Γ}, p_{Σ}, p_{Σ,G}^{−1}, G) . \ar@{=>}[ur]_{(u,v,w)}}

2) An analogue universal property holds if one considers an extension E = (Γ, φ, Σ), a 1-cell F: 𝔸 → 𝔻 in SCG and the 1-cell E → E · F as in Lemma 6.3

[figure: 𝔸; Γ; F; ⇓^{i{−1 \atop F,Γ}}; 𝔹; i_{Γ}; Σ; ⇓^{(φ^{Σ})^{−1}}; ℂ; 1_{ℂ}; 𝔻; i_{F}; F ∪ Γ; φ^{0,Σ}; ℂ .]

Corollary 13.2. Consider an extension E = (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ and four 1-cells in SCG

\xymatrix{𝔸 \ar[r]^{F} & 𝔻 \ar[r]^{H} & 𝔽 & 𝔾 \ar[r]^{K} & 𝔼 \ar[r]^{G} & ℂ .}

The following pairs of functors are naturally isomorphic in a canonical way :

1) \xymatrix{Ext(ℂ, 𝔸) \ar[r]^{−·F} & Ext(ℂ, 𝔻) \ar[r]^{−·H} & Ext(ℂ, 𝔽)}

and \xymatrix{Ext(ℂ, 𝔸) \ar[r]^{−·(F ·H)} & Ext(ℂ, 𝔽) ;}

2) \xymatrix{Ext(ℂ, 𝔸) \ar[r]^{G·−} & Ext(𝔼, 𝔸) \ar[r]^{K·−} & Ext(𝔾, 𝔸)}

and \xymatrix{Ext(ℂ, 𝔸) \ar[r]^{(K·G)·−} & Ext(𝔾, 𝔸) ;}

3) \xymatrix{Ext(ℂ, 𝔸) \ar[r]^{−·F} & Ext(ℂ, 𝔻) \ar[r]^{G·−} & Ext(𝔼, 𝔻)}

and \xymatrix{Ext(ℂ, 𝔸) \ar[r]^{G·−} & Ext(𝔼, 𝔸) \ar[r]^{−·F} & Ext(𝔼, 𝔻) ;}

4) \xymatrix{Hom(𝔸, 𝔻) \ar[r]^{E·−} & Ext(ℂ, 𝔻) \ar[r]^{G·−} & Ext(𝔼, 𝔻)}

and \xymatrix{Hom(𝔸, 𝔻) \ar[r]^{(G·E)·−} & Ext(𝔼, 𝔻) ;}

5) \xymatrix{Hom(𝔼, ℂ) \ar[r]^{−·E} & Ext(𝔼, 𝔸) \ar[r]^{−·F} & Ext(𝔼, 𝔻)}

and \xymatrix{Hom(𝔼, ℂ) \ar[r]^{−·(E·F)} & Ext(𝔼, 𝔻) .}

We can now describe the monoidal structure of the groupoid Ext(ℂ, 𝔸). Given two extensions

[figure: 𝔸; 0; Γ_{1}; ℂ; 𝔹_{1}; Σ_{1}; ⇑^{φ_{1}}; 𝔸; 0; Γ_{2}; ℂ; 𝔹_{2}; Σ_{2}; ⇑^{φ_{2}}; ,]

we get a new extension considering

[figure: 𝔸 × 𝔸; 0; Γ_{1}×Γ_{2}; ℂ × ℂ; 𝔹_{1} × 𝔹_{2}; Σ_{1}×Σ_{2}; ⇑^{φ_{1}×φ_{2}}; .]

This construction extends to a 2-functor

⊕: EXT(ℂ, 𝔸) × EXT(ℂ, 𝔸) → EXT(ℂ × ℂ, 𝔸 × 𝔸) .

Consider now the diagonal and codiagonal 1-cells in SCG

Δ: ℂ → ℂ × ℂ ∇: ℂ × ℂ → ℂ .

Using part 3 of Corollary 13.2, we can define the tensor product in Ext(ℂ, 𝔸) in the following way

⊗: \xymatrix{Ext(ℂ, 𝔸) × Ext(ℂ, 𝔸) \ar[r]^{⊕} & Ext(ℂ × ℂ, 𝔸 × 𝔸) \ar[r]^{Δ·−·∇} & Ext(ℂ, 𝔸) .}

The unit object is given by the trivial extension T (𝔸, ℂ) and the inverse of an extension E is E · (1_{𝔸}^{∗}), where 1_{𝔸}^{∗}: 𝔸 → 𝔸 is the 1-cell in SCG defined by

[figure: X; f; Y; X^{∗}; (f^{−1})^{∗}; Y^{∗} .] It is a (very) long but essentially straightforward work to complete this definition with associativity, commutativity and unit isomorphisms, and to check the axioms for a symmetric cat-group. For example, the associativity isomorphism is given by the canonical natural isomorphisms in the following diagram (where we write (ℂ, 𝔸) instead of Ext(ℂ, 𝔸)) :

[figure: (ℂ, 𝔸) × (ℂ, 𝔸) × (ℂ, 𝔸); 1×⊕; ⊕×1; ≃; (ℂ, 𝔸) × (ℂ × ℂ, 𝔸 × 𝔸); 1×(Δ·−·∇); ⊕; (ℂ × ℂ, 𝔸 × 𝔸) × (ℂ, 𝔸); (Δ·−·∇)×1; ⊕; (ℂ × ℂ × ℂ, 𝔸 × 𝔸 × 𝔸); ≃; ≃; ≃; (1×Δ)·−·(1×∇); (Δ×1)·−·(∇×1); (ℂ, 𝔸) × (ℂ, 𝔸); ⊕; (ℂ, 𝔸) × (ℂ, 𝔸); ⊕; (ℂ × ℂ, 𝔸 × 𝔸); Δ·−·∇; (ℂ × ℂ, 𝔸 × 𝔸); Δ·−·∇; (ℂ, 𝔸)]

Remark 13.3. Observe that the functors listed at the beginning of Section 10 are now 1-cells in SCG. In particular, the sequences of Propositions 10.2 and 10.4 are 2-exact sequences in SCG.

Remark 13.4. The results of this section hold if we replace EXT by a proper class P of extensions, provided that the 2-functor ⊕: EXT(ℂ, 𝔸) × EXT(ℂ, 𝔸) → EXT(ℂ × ℂ, 𝔸 × 𝔸) restricts to P. This is the case for all the examples given in Section 9.

14. Cobords and cocycles

In [20], extensions of (non necessarily symmetric) cat-groups are classified. A catgroup 𝔸 is a bicategory with only one object, and we can consider the bicategory Bieq(𝔸) of biequivalences from 𝔸 to 𝔸. Clearly, Bieq(𝔸) is a monoidal bicategory under composition. Given another cat-group ℂ, we can look at it as a monoidal bicategory with only identity 2-arrows. The main result in [20] is a biequivalence EXT(ℂ, 𝔸) ≃ Bimon(ℂ, Bieq(𝔸)), where an object in Bimon(ℂ, Bieq(𝔸)) is a morphism of bicategories ℂ → Bieq(𝔸) which takes into account the monoidal structures in a sense made precise in [20]. (The referee pointed out to us that F-extensions (Definition 9.4) correspond to those objects of Bimon(ℂ, Bieq(𝔸)) such that the underlying homomorphism of bicategories is equivalent to the null homomorphism.)

In this section and in the next one, we look for a cohomological classification of extensions more on the line of Schreier theory. Having in mind factor sets, the class of extensions which seems appropriate for such a theory is the class of F-extensions. Once again, for the reasons explained in the introduction, we restrict our attention to symmetric cat-groups.

The first step towards a cohomological classification of F-extensions is to find an appropriate notion of cocycle. For this, consider 𝔸 and ℂ in SCG and let us look for a monoidal structure on 𝔸 × ℂ such that

[figure: 𝔸; 0; i_{𝔸}; ℂ; 𝔸 × ℂ; p_{ℂ}; =]

is an extension (necessarily, an F-extension). Let us call for a moment ∗: 𝔸 × ℂ × 𝔸 × ℂ → 𝔸 × ℂ the new tensor product in 𝔸 × ℂ. If we assume that

(A, X) ∗ (B, Y) = ((A, I) ∗ (B, I)) ⊗ ((I, X) ∗ (I, Y))

(where ⊗ is the point-wise tensor product in 𝔸×ℂ), we get that (A, X)∗(B, Y) must be equal to (A ⊗ B ⊗ F (X, Y), X ⊗ Y), for F: ℂ × ℂ → 𝔸 any functor. Moreover, the associativity, unit and symmetry isomorphisms for (𝔸 × ℂ, ∗) give rise to four natural isomorphisms (compare with 5.2.1.11, 5.2.1.14 and 5.2.1.15 in [20])

- c_{F} (X, Y, Z): F (X, Y) ⊗ F (X ⊗ Y, Z) → F (Y, Z) ⊗ F (X, Y ⊗ Z) ;

- r_{F} (X): F (X, I) → I;

- l_{F} (Y): F (I, Y) → I;

- s_{F} (X, Y): F (X, Y) → F (Y, X) .

Finally, the four coherence axioms for the isomorphisms in (𝔸 × ℂ, ∗) are equivalent to the following four axioms on c_{F} , r_{F} , l_{F} and s_{F} , expressed in terms of commutative diagrams (we omit the associativity isomorphism in 𝔸) : (C1) The two following compositions coincide

\xymatrix{F (X, Y) ⊗ F (X ⊗ Y, Z) ⊗ F ((X ⊗ Y) ⊗ Z, W) \ar[d]_{1⊗c_{F} (X⊗Y,Z,W)} \\ F (X, Y) ⊗ F (Z, W) ⊗ F (X ⊗ Y, Z ⊗ W) \ar[d]_{γ⊗1} \\ F (Z, W) ⊗ F (X, Y) ⊗ F (X ⊗ Y, Z ⊗ W) \ar[d]_{1⊗c_{F} (X,Y,Z⊗W)} \\ F (Z, W) ⊗ F (Y, Z ⊗ W) ⊗ F (X, Y ⊗ (Z ⊗ W))}

F (X, Y) ⊗ F (X ⊗ Y, Z) ⊗ F ((X ⊗ Y) ⊗ Z, W)
c_{F} (X,Y,Z)⊗F (a,1)
F (Y, Z) ⊗ F (X, Y ⊗ Z) ⊗ F (X ⊗ (Y ⊗ Z), W)
1⊗c_{F} (X,Y ⊗Z,W)
F (Y, Z) ⊗ F (Y ⊗ Z, W) ⊗ F (X, (Y ⊗ Z) ⊗ W)
c_{F} (Y,Z,W)⊗F (1,a)
F (Z, W) ⊗ F (Y, Z ⊗ W) ⊗ F (X, Y ⊗ (Z ⊗ W))

(C2)

\xymatrix{F (X, I) ⊗ F (X ⊗ I, Z) \ar[rr]^{c_{F} (X,I,Z)} \ar[dr]_{r_{F} (X)⊗F (r_{X},1)} &  & F (I, Z) ⊗ F (X, I ⊗ Z) \ar[dl]^{l_{F} (Z)⊗F (1,l_{Z})} \\  & I ⊗ F (X, Z) ;}

(C3)

\xymatrix{F (X, Y) ⊗ F (X ⊗ Y, Z) \ar[d]_{s_{F} (X,Y)⊗F (γ,1)} \ar[r]^{c_{F} (X,Y,Z)} & F (Y, Z) ⊗ F (X, Y ⊗ Z) \ar[d]^{1⊗s_{F} (X,Y ⊗Z)} \\ F (Y, X) ⊗ F (Y ⊗ X, Z) \ar[d]_{c_{F} (Y,X,Z)} & F (Y, Z) ⊗ F (Y ⊗ Z, X) \ar[d]^{c_{F} (Y,Z,X)} \\ F (X, Z) ⊗ F (Y, X ⊗ Z) \ar[r]_{s_{F} (X,Z)⊗F (1,γ)} & F (Z, X) ⊗ F (Y, Z ⊗ X) ;}

(C4)

\xymatrix{F (X, Y) \ar[rr]^{1} \ar[dr]_{s_{F} (X,Y)} &  & F (X, Y) \\  & F (Y, X) . \ar[ur]_{s_{F} (Y,X)}}

Definition 14.1. Let 𝔸, ℂ be two symmetric cat-groups.

1) A symmetric 2-cocycle of ℂ with coefficients in 𝔸 (a cocycle, for short) is given by F = (F, c_{F} , r_{F} , l_{F} , s_{F}) where F: ℂ × ℂ → 𝔸 is a functor and - c_{F} (X, Y, Z): F (X, Y) ⊗ F (X ⊗ Y, Z) → F (Y, Z) ⊗ F (X, Y ⊗ Z) ,

- r_{F} (X): F (X, I) → I ,

- l_{F} (Y): F (I, Y) → I ,

- s_{F} (X, Y): F (X, Y) → F (Y, X) are natural isomorphisms satisfying the axioms (C1),(C2), (C3) and (C4).

2) Given two cocycles F and F′, a morphism of cocycles is a natural transformation α: F ⇒ F′ making commutative the following diagrams (MC1)

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

(MC2)

\xymatrix{F (X, I) \ar[rr]^{α_{X,I}} \ar[dr]_{r_{F} (X)} &  & F′(X, I) \ar[dl]^{r_{F′} (X)} & F (I, Y) \ar[rr]^{α_{I,Y}} \ar[dr]_{l_{F} (Y)} &  & F′(I, Y) \ar[dl]^{l_{F′} (Y)} \\  & I &  &  & I ;}

(MC3)

\xymatrix{F (X, Y) \ar[r]^{s_{F} (X,Y)} \ar[d]_{α_{X,Y}} & F (Y, X) \ar[d]^{α_{Y,X}} \\ F′(X, Y) \ar[r]_{s′_{F} (X,Y)} & F′(Y, X) .}

3) A normalized 2-cobord of ℂ with coefficients in 𝔸 (a cobord, for short) is given by G = (G, G_{I}), where G: ℂ → 𝔸 is a functor and G_{I}: I → GI is an arrow in 𝔸.

4) Given two cobords G and G′, a morphism of cobords is a natural transformation α: G ⇒ G′ making commutative the following diagram

\xymatrix{GI \ar[rr]^{α_{I}} &  & G′I \\  & I . \ar[ul]^{G_{I}} \ar[ur]_{G′_{I}}}

Proposition 14.2. Let 𝔸, ℂ be in SCG.

1) The data of the previous definition give a category Z^{2}(ℂ, 𝔸) of cocycles and a category B^{2}(ℂ, 𝔸) of cobords ;

2) Z^{2}(ℂ, 𝔸) and B^{2}(ℂ, 𝔸) have a structure of symmetric cat-groups induced pointwise by that of 𝔸.

If G ∈ B^{2}(ℂ, 𝔸), we can construct a functor δG: ℂ × ℂ → 𝔸 in the following way :

- (δG)(X, Y) = G(X ⊗ Y) ⊗ G(Y)^{∗} ⊗ G(X)^{∗}

- if f: X → X′ and g: Y → Y′ are in ℂ, then (δG)(f, g) = G(f ⊗g)⊗G(g^{−1})^{∗} ⊗

G(f^{−1})^{∗} . Moreover (using G_{I}, the symmetry of ℂ and 𝔸 and, for c_{δG}, unit and counit of G(X ⊗Y) ⊣ G(X ⊗Y)^{∗} and G(Y ⊗Z) ⊣ G(Y ⊗Z)^{∗}) one can construct four natural isomorphisms c_{δG}, r_{δG}, l_{δG}, s_{δG} so that δG becomes a cocycle. More precisely, we have the following lemmas.

Lemma 14.3. The previous construction gives rise to a 1-cell in SCG

δ: B^{2}(ℂ, 𝔸) → Z^{2}(ℂ, 𝔸) .

Lemma 14.4. The kernel of δ is equivalent, in SCG, to Hom(ℂ, 𝔸).

Proof. We only explain the construction of the equivalence. First, observe that there is a forgetful functor U: Hom(ℂ, 𝔸) → B^{2}(ℂ, 𝔸) which sends G = (G, G_{I}, G_{X,Y}) into G = (G, G_{I}). Clearly, U is a 1-cell in SCG. Moreover, G_{X,Y} precisely gives a 2-cell in SCG

[figure: Hom(ℂ, 𝔸); 0; U; Z^{2}(ℂ, 𝔸); B^{2}(ℂ, 𝔸); δ; ⇑]

so that U factors through the kernel of δ. This factorization is in fact an equivalence.

To end this section, let us point out some formal consequences of the axioms defining a cocycle. They are useful to complete some proof in the next section.

Lemma 14.5. Let F = (F, c_{F} , r_{F} , l_{F} , s_{F}) be a cocycle. The following equations hold: (C5) l_{F} (I) = r_{F} (I) ; (C6)

\xymatrix{F (I, Y) ⊗ F (I ⊗ Y, Z) \ar[rr]^{c_{F} (I,Y,Z)} \ar[d]_{l_{F} (Y)⊗F (l_{Y} ,1)} &  & F (Y, Z) ⊗ F (I, Y ⊗ Z) \ar[d]^{1⊗l_{F} (Y ⊗Z)} \\ I ⊗ F (Y, Z) \ar[r]_{l_{F (Y,Z)}} & F (Y, Z) & F (Y, Z) ⊗ I ; \ar[l]^{r_{F (Y,Z)}}}

(C7)

\xymatrix{F (X, Y) ⊗ F (X ⊗ Y, I) \ar[rr]^{c_{F} (X,Y,I)} \ar[d]_{1⊗r_{F} (X⊗Y)} &  & F (Y, I) ⊗ F (X, Y ⊗ I) \ar[d]^{r_{F} (Y)⊗F (1,r_{Y} )} \\ F (X, Y) ⊗ I \ar[r]_{r_{F (X,Y )}} & F (X, Y) & I ⊗ F (X, Y) ; \ar[l]^{l_{F (X,Y )}}}

(C8) s_{F} (X, I) · l_{F} (X) = r_{F} (X) , s_{F} (I, Y) · r_{F} (Y) = l_{F} (Y) ;

(C9) s_{F} (I, I) = 1_{F (I,I)} .

Proof. Similar to that for monoidal categories (see [17] or [15]).

15. Classification of F-extensions

¿From the discussion at the beginning of Section 14, we know that each cocycle F ∈ Z^{2}(ℂ, 𝔸) induces a symmetric monoidal structure on the groupoid 𝔸×ℂ, where the tensor product

⊗_{F} : (𝔸 × ℂ) × (𝔸 × ℂ) → 𝔸 × ℂ

is defined by

(A, X) ⊗_{F} (B, Y) = (A ⊗ B ⊗ F (X, Y), X ⊗ Y) ,

the identity object is (I, I) and the structural isomorphisms are induced by the natural isomorphisms c_{F} , r_{F} , l_{F} and s_{F} . We denote by 𝔸 ×_{F} ℂ the category 𝔸 × ℂ equipped with this structure. It is in fact a symmetric cat-group, a dual being given by (A, X)^{∗} = (A^{∗} ⊗ F (X, X^{∗})^{∗}, X^{∗}).

Proposition 15.1. Let 𝔸 and ℂ be in SCG.

1) The previous construction extends to a functor

E: Z^{2}(ℂ, 𝔸) → F(ℂ, 𝔸) ;

2) The functor E is essentially surjective on objects ;

3) The functor E is a 1-cell in SCG.

Proof. 1) It is a routine verification. Let us only write explicitly the action of E. If F ∈ Z^{2}(ℂ, 𝔸), then E(F) is

[figure: 𝔸; 0; i_{𝔸}; ℂ; 𝔸 ×_{F} ℂ; p_{ℂ}; =; .]

If λ: F ⇒ F′ is a morphism in Z^{2}(ℂ, 𝔸), then E(λ) is

[figure: 𝔸 ×_{F} ℂ; β_{λ}; p_{ℂ}; ⇓^{α_{λ}}; 𝔸; i_{𝔸}; i_{𝔸}; ℂ; 𝔸 ×_{F′} ℂ; p_{ℂ}; ⇑^{γ_{λ}}]

where α_{λ} and γ_{λ} are identity natural transformations and β_{λ} is the identity as a functor, but its monoidal structure is determined by λ in the following way

(A, X) ⊗_{F} (B, Y) = (A ⊗ B ⊗ F (X, Y), X ⊗ Y)
(1⊗1⊗λ_{X,Y} ,1⊗1)
(A, X) ⊗_{F′} (B, Y) = (A ⊗ B ⊗ F′(X, Y), X ⊗ Y) .

2) Let E = (Γ, φ, Σ): 𝔸 → 𝔹 → ℂ be an F-extension with a section

[figure: ℂ; 1_{ℂ}; S; ℂ; 𝔹; Σ; ⇑^{σ}]

pointed by S_{I}: I → S(I). Since Γ_{0} : 𝔸 → KerΣ is an equivalence, we look for a cocycle F_{E}: ℂ × ℂ → KerΣ. If X and Y are in ℂ, we define F_{E}(X, Y) = (S(X) ⊗ S(Y) ⊗ S(X ⊗ Y)^{∗}, σ_{X,Y}), where

σ_{X,Y} : Σ(S(X) ⊗ S(Y) ⊗ S(X ⊗ Y)^{∗}) ≃ \xymatrix{Σ(S(X)) ⊗ Σ(S(Y)) ⊗ Σ(S(X ⊗ Y))^{∗} \ar[d]^{σ_{X}⊗σ_{Y} ⊗(σ_{X⊗Y}^{−1} )^{∗}} \\ X ⊗ Y ⊗ (X ⊗ Y)^{∗} \ar[d]^{η_{X⊗Y}^{−1}} \\ I ,}

and F_{E}(f, g) = S(f) ⊗ S(g) ⊗ S(f^{−1} ⊗ g^{−1})^{∗} for f and g two arrows in ℂ. Now we describe the structure of cocycle of F_{E}:

- r_{F_{E}}(X): F_{R}(X, I) → (I, Σ_{I}^{−1} : Σ(I) → I) is given by

\xymatrix{S(X) ⊗ S(I) ⊗ S(X ⊗ I)^{∗} \ar[rr]^{1⊗S_{I}^{−1}⊗S(r_{X}^{−1})^{∗}} &  & S(X) ⊗ I ⊗ S(X)^{∗} \ar[d]^{≃} \\  & I & S(X) ⊗ S(X)^{∗} ; \ar[l]^{η_{S(X)}^{−1}}}

- l_{F_{E}}(Y) is similar ;

- s_{F_{E}}(X, Y): F_{E}(X, Y) → F_{E}(Y, X) is given by

γ_{S(X),S(Y)}⊗S(γ_{X,Y}^{−1} )^{∗}: S(X)⊗S(Y)⊗S(X⊗Y)^{∗} → S(Y)⊗S(X)⊗S(Y ⊗X)^{∗} ;

- c_{F_{E}}(X, Y, Z): F_{E}(X, Y) ⊗ F_{E}(X ⊗ Y, Z) → F_{E}(Y, Z) ⊗ F_{E}(X, Y ⊗ Z) is given by

S(X) ⊗ S(Y) ⊗ S(X ⊗ Y)^{∗} ⊗ S(X ⊗ Y) ⊗ S(Z) ⊗ S(X ⊗ Y ⊗ Z)^{∗}
1⊗1⊗ϵ_{S(X⊗Y )}⊗1⊗1
S(X) ⊗ S(Y) ⊗ S(Z) ⊗ S(X ⊗ Y ⊗ Z)^{∗}
1⊗1⊗1⊗ϵ_{S(Y ⊗Z)}^{−1}⊗1
S(X) ⊗ S(Y) ⊗ S(Z) ⊗ S(Y ⊗ Z)^{∗} ⊗ S(Y ⊗ Z) ⊗ S(X ⊗ Y ⊗ Z)^{∗}
γ_{S(X),S(Y )⊗S(Z)⊗S(Y ⊗Z)∗} ⊗1⊗1
S(Y) ⊗ S(Z) ⊗ S(Y ⊗ Z)^{∗} ⊗ S(X) ⊗ S(Y ⊗ Z) ⊗ S(X ⊗ Y ⊗ Z)^{∗} .

It remains to find a 1-cell in F(ℂ, KerΣ) as in the following diagram

[figure: 𝔹; β; Σ; ⇓^{α}; KerΣ; e_{Σ}; i_{KerΣ}; ℂ; KerΣ ×_{F_{E}} ℂ; p_{ℂ}; ⇑^{γ}; .]

As far as β is concerned, if B is in 𝔹, put

β(B) = (B ⊗ S(Σ(B))^{∗}, (1_{Σ(B)} ⊗ (σ_{Σ(B)}^{−1})^{∗}) · η_{Σ(B)}^{−1}, Σ(B))

and β(g) = (g ⊗ S(Σ(g^{−1}))^{∗}, Σ(g)) if g is an arrow in 𝔹. The monoidal structure of β is given by that of Σ on the second component, and, up to symmetry, by the counits ϵ_{S(Σ(A))} and ϵ_{S(Σ(B))} on the first component (A, B being two objects in 𝔹). Finally, the 2-cell γ is the identity natural transformation, and the 2-cell α is given, for an object (B, b: Σ(B) → I) in KerΣ, by (1 ⊗ S(Σ(b^{−1}))^{∗}, b).

3) Consider now F, F′ ∈ Z^{2}(ℂ, 𝔸), the direct sum E(F)⊕E(F′) and the F-extension built up using the pullback along the diagonal Δ: ℂ → ℂ × ℂ as in the following diagram

[figure: ℙ; p_{Δ}; p; ⇒; ℂ; Δ; 𝔸 × 𝔸; i; i_{𝔸}×i_{𝔸}; 𝔸 ×_{F} ℂ × 𝔸 ×_{F′} ℂ; ⇓; p_{ℂ}×p_{ℂ}; ℂ × ℂ .]

An object in ℙ is given by (A, X, B, Y, C, x, y), with A, B in 𝔸, X, Y, C in ℂ and x: X → C, y: Y → C; the arrows are the obvious ones. The functor i: 𝔸 × 𝔸 → ℙ sends (A, B) into (A, I, B, I, I, 1, 1). Consider also F ⊗ F′ in Z^{2}(ℂ, 𝔸) and the Fextension

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

The monoidal structure of E results from showing that the following diagram is a pushout in SCG :

[figure: 𝔸 × 𝔸; i; ∇; =; ℙ; Λ; 𝔸; i_{𝔸}; 𝔸 ×_{F ⊗F′} ℂ ,]

where Λ sends (A, X, B, Y, C, x, y) into (A ⊗ B, C). If (A′, X′, B′, Y′, C′, x′, y′) is a second object in ℙ, the monoidal structure of Λ is given, on the second component, by F (x, x′) and F′(y, y′). To show that the previous diagram is a pushout in SCG, consider any diagram of the form

[figure: 𝔸 × 𝔸; i; ∇; ⇒^{τ}; ℙ; H; 𝔸; K; 𝔼 .]

We need a factorization in SCG

[figure: 𝔸 ×_{F ⊗F′} ℂ; Φ; 𝔸; i_{𝔸}; K; 𝔼; ⇓^{k}; ⇓^{h}; ℙ; H; Λ]

and, for that, we put

- Φ(A, X) = K(A)⊗_{𝔼}H(I, X, I, X, X, 1, 1), with the monoidal structure induced by τ ;

- k is induced by H_{I}: I → H(I) :

- h is defined, using l_{F} , l_{F′} and the monoidal structure of H, in the following way : Φ(Λ(A, X, B, Y, C, x, y)) =

\xymatrix{= K(A ⊗ B) ⊗_{𝔼} H(C) \ar[r]^{τ_{A,B}^{−1}⊗1} & H(A, I, B, I, I, 1, 1) ⊗_{𝔼} H(C) ≃}

≃ H((A, I, B, I, I, 1, 1) ⊗_{ℙ} (I, C, I, C, C, 1, 1)) ≃ ≃ H(A ⊗ F (I, C), C, B ⊗ F′(I, C), C, C, 1, 1) ≃

\xymatrix{≃ H(A, C, B, C, C, 1, 1) \ar[r]^{H(1,x^{−1},1,y^{−1},1)} & H(A, X, B, Y, C, x, y) .}

Finally, given another factorization in SCG

[figure: 𝔸 ×_{F ⊗F′} ℂ; Ψ; 𝔸; i_{𝔸}; K; 𝔼; ⇓^{m}; ⇓^{l}; ℙ ,; H; Λ]

since i_{𝔸}(A)⊗_{F ⊗F′} Λ(I, C, I, C, C, 1, 1) = (A⊗I ⊗F (I, C)⊗F′(I, C), I ⊗C) ≃ (A, C), we have

Ψ(A, C) ≃ Ψ(i_{𝔸}(A)) ⊗_{𝔼} Ψ(Λ(I, C, I, C, C, 1, 1))
m_{A}⊗l_{(I,C,I,C,C,1,1)}
Φ(A, C) = K(A) ⊗_{𝔼} H(I, C, I, C, C, 1, 1) .

To get a cohomological classification of F-extensions, we need a 2-cell in SCG

[figure: B^{2}(ℂ, 𝔸); 0; δ; F(ℂ, 𝔸); Z^{2}(ℂ, 𝔸); E; ⇑^{ϕ}; .]

For this, let us fix a cobord G ∈ B^{2}(ℂ, 𝔸) and let us describe the following 1-cell in F(ℂ, 𝔸)

[figure: 𝔸 ×_{δG} ℂ; ϕ_{G}; p_{ℂ}; ⇓^{α_{G}}; 𝔸; i_{𝔸}; i_{𝔸}; ℂ; 𝔸 × ℂ; p_{ℂ}; ⇑^{γ_{G}}; .]

We put ϕ_{G}(A, X) = (A⊗G(X)^{∗}, X) for (A, X) an object of 𝔸×_{δG}ℂ, and ϕ_{G}(f, g) = (f ⊗ G(g^{−1})^{∗}, g) for (f, g) an arrow in 𝔸 ×_{δG} ℂ. Its monoidal structure is induced by the unit η_{G(X⊗Y)}. As far as the 2-cells are concerned, γ_{G} is the identity natural transformation, and α_{G} is induced by G_{I}: I → G(I). If λ: G ⇒ G′ is a morphism in B^{2}(ℂ, 𝔸), the naturality of ϕ is attested by the following 2-cell in F(ℂ, 𝔸) :

[figure: 𝔸 ×_{δG} ℂ; β_{δλ}; ϕ_{G}; 𝔸 ×_{δG′} ℂ; ϕ_{G′}; 𝔸 × ℂ; ⇓^{ϕ_{λ}}]

where, for (A, X) an object in 𝔸 ×_{δG} ℂ, we have :

ϕ_{λ}(A, X) = (1 ⊗ λ_{X}^{∗}, 1)^{−1} : (A ⊗ G′(X)^{∗}, X) → (A ⊗ G(X)^{∗}, X) .

Consider now the cokernel of δ :

\xymatrix{B^{2}(ℂ, 𝔸) \ar[r]^{δ} & Z^{2}(ℂ, 𝔸) \ar[r]^{P} & Cokerδ .}

Using its universal property and the 2-cell ϕ: δ · E ⇒ 0, we get a 1-cell

E′: Cokerδ → F(ℂ, 𝔸)

which extends E: Z^{2}(ℂ, 𝔸) → F(ℂ, 𝔸).

Theorem 15.2. The 1-cell in SCG

E′: Cokerδ → F(ℂ, 𝔸)

is an equivalence.

Proof. Because of point 2 of Proposition 15.1, it is enough to prove that E′ is full and faithful. By Proposition 6.2 in [16], this is equivalent to prove that the factorization

δ′: B^{2}(ℂ, 𝔸) → KerE

of δ through the kernel of E (factorization induced by the 2-cell ϕ: δ · E ⇒ 0) is full and essentially surjective on objects. Essentially surjective : Consider an object in KerE, that is F ∈ Z^{2}(ℂ, 𝔸) together with a section in SCG

[figure: ℂ; 1_{ℂ}; S; ℂ; 𝔸 ×_{F} ℂ; p_{ℂ}; ⇑^{σ}]

We get a pointed functor

G = S · p_{A} : \xymatrix{ℂ \ar[r] & 𝔸 ×_{F} ℂ \ar[r] & 𝔸}

and we need an isomorphism F ≃ δG. For this, observe that S(X) ≃ (G(X), X) (because of σ), so that the monoidal structure of S gives the following chain of natural isomorphisms : (G(X ⊗ Y), X ⊗ Y) ≃ S(X ⊗ Y) ≃ S(X) ⊗_{F} S(Y) ≃ (G(X), X) ⊗_{F} (G(Y), Y) = (G(X) ⊗ G(Y) ⊗ F (X, Y), X ⊗ Y). Its first component gives a natural isomorphism F (X, Y) ≃ G(X ⊗Y)⊗G(Y)^{∗}⊗G(X)^{∗}, as requested. It remains to prove that this isomorphism is an arrow in KerE. But this is attested by the following chain of natural isomorphisms, where β: 𝔸×ℂ → 𝔸×_{F} ℂ is induced by i_{𝔸}: 𝔸 → 𝔸 ×_{F} ℂ and S: ℂ → 𝔸 ×_{F} ℂ : ϕ_{G}(β(A, X)) ≃ ϕ_{G}((A, I) ⊗_{F} (G(X), X)) = ϕ_{G}(A ⊗ G(X) ⊗ F (I, X), I ⊗ X) ≃ ϕ_{G}(A ⊗ G(X), X) = (A ⊗ G(X) ⊗ G(X)^{∗}, X) ≃ (A, X). Full : Consider now G_{1}, G_{2} ∈ B^{2}(ℂ, 𝔸) and let λ: δG_{1} ⇒ δG_{2} be in Z^{2}(ℂ, 𝔸). Assume also that there is a 2-cell s: ϕ_{G_{1}} ⇒ β_{λ} ·ϕ_{G_{2}} in EXT(ℂ, 𝔸). In particular, for each X in ℂ, there is a natural isomorphism s_{(I,X)} : (G_{1}(X)^{∗}, X) → (G_{2}(X)^{∗}, X). Its first component induces a natural isomorphism \overline{s}_{X}: G_{1}(X) → G_{2}(X) and, using that s is monoidal, one can check that δ(\overline{s}) = λ.

References

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This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at

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D. Bourn bourn@lmpa.univ-littoral.fr

Laboratoire de Mathématiques, Université du Littoral 220 av. de l’Université, BP 5526, 59379 Dunkerque, France.

E.M. Vitale vitale@math.ucl.ac.be

Département de Mathématiques, Université catholique de Louvain Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium.