Theory and Applications of Categories, Vol. 7, No. 5, 2000, pp. 47–70.

FACTORIZATION SYSTEMS FOR SYMMETRIC CAT-GROUPS

S. KASANGIAN AND E.M. VITALE

Transmitted by Walter Tholen

ABSTRACT. This paper is a first step in the study of symmetric cat-groups as the 2-dimensional analogue of abelian groups. We show that a morphism of symmetric catgroups can be factorized as an essentially surjective functor followed by a full and faithful one, as well as a full and essentially surjective functor followed by a faithful one. Both these factorizations give rise to a factorization system, in a suitable 2-categorical sense, in the 2-category of symmetric cat-groups. An application to exact sequences is given.

Introduction

A cat-group is a monoidal groupoid in which each object is invertible, up to isomorphisms, with respect to the tensor product [7, 10, 16, 24]. Cat-groups are a useful tool for ring theory, group cohomology and algebraic topology (for example, small and strict cat-groups correspond to crossed modules) [2, 5, 6, 13, 15, 21, 22, 25]. Symmetric cat-groups, together with symmetric monoidal functors and monoidal natural transformations, constitute a 2category which can be seen as a 2-dimensional analogue of the category of abelian groups. Several algebraic problems can be considered in this setting, and some of them, especially the study of exact sequences and of extensions, lead in a natural way to the search for convenient classes of surjections and injections between cat-groups. The aim of this note is to discuss two different factorization systems for symmetric cat-groups. The problem of factorizing a (monoidal) functor has been discussed, from different points of view, also in [1, 8, 18, 19, 20].

After recalling some preliminary facts on cat-groups (section 1), in section 2 we show how to factorize a morphism of symmetric cat-groups. The idea is quite simple : in any abelian category, a morphism can be factorized through the kernel of its cokernel or through the cokernel of its kernel, and these two factorizations are essentially the same. But this is a gift of the one-dimensional (abelian) world ! Since we have the appropriate notion of kernel and cokernel for symmetric cat-groups, we can do the same with a morphism between symmetric cat-groups, and we obtain two different factorizations. In section 3 we develop a little bit of the theory of factorization systems in a 2-category and finally, in section 4, we prove that both factorizations constructed in section 2 give rise to factorization systems. In the last section we use our results to discuss the notion of 2-exact sequence of cat-groups. This notion has been introduced in [24] in order to study the Brauer and Picard cat-groups of a commutative ring.

Received by the editors 2000 January 10 and, in revised form, 2000 March 24.

Published on 2000 April 3.

2000 Mathematics Subject Classification: 18A32, 18D05, 18D10, 18G99, 20L05.

Key words and phrases: Cat-groups, factorization systems in a 2-category.

\begin{matrix}c \\ ◯ \end{matrix} S. Kasangian and E.M. Vitale, 2000. Permission to copy for private use granted.

The reason why the two factorizations obtained in section 2 are different is a quite surprising fact, explained in detail in section 5 : in the 2-category of symmetric cat-groups, we can give a suitable definition of mono, epi, kernel and cokernel. As expected in the 2-dimensional analogue of the category of abelian groups, each epi is a cokernel and each mono is a kernel, but in general cokernels fail to be epis and kernels fail to be monos.

Acknowledgment: We would like to thank W. Tholen and the referee for some useful comments and suggestions. We especially thank the referee for having suggested to us a link with homotopical algebra. This link is probably deeper and more interesting than the few lines at the end of section 1 show.

1. Categorical groups

We start recalling some basic facts about cat-groups.

1.1. Definition. A (symmetric) cat-group 𝔾 is a (symmetric) monoidal groupoid 𝔾 = (𝔾, ⊗, I, …) such that for each object A there exists an object A^{∗} and an arrow η_{A}: I → A ⊗ A^{∗}.

The asymmetry in the previous definition is only apparent, in fact we have the following

1.2. Proposition. Let 𝔾 be a cat-group ; for each object A it is possible to find a morphism ϵ_{A}: A^{∗} ⊗ A → I such that

D = (A^{∗} ⊣ A, η_{A}, ϵ_{A})

is a duality in 𝔾. The choice, for each A, of such a duality induces a monoidal equivalence

( )^{∗}: 𝔾^{op} → 𝔾 .

One of the main tools of this paper is the cokernel of a morphism of cat-groups. Since we have an explicit description of it only in the case of symmetric cat-groups, from now on we limit our attention to symmetric cat-groups. We denote by SCG the 2-category having symmetric cat-groups as objects, monoidal functors preserving the symmetry as arrows, and monoidal natural transformations as 2-cells. Observe that in this 2-category, 2-cells are invertible. We use the same name, SCG, for the category obtained from the previous 2-category forgetting 2-cells. We write H(SCG) for the category with the same objects as SCG, but with 2-isomorphism classes of morphisms as arrows. There is an obvious functor

H: SCG → H(SCG) .

If 𝔾 is a symmetric cat-group, π_{0}(𝔾) is the (possibly large) set of isomorphism classes of objects of 𝔾, and π_{1}(𝔾) = 𝔾(I, I) is the set of endomorphisms of the unit object. Both π_{0}(𝔾) and π_{1}(𝔾) are abelian groups and give rise to two functors

π_{0} : SCG → Ab , π_{1} : SCG → Ab

(where Ab is the category of abelian groups) which factor through H.

If G is an abelian group, \underline{D}(G) is the discrete symmetric cat-group with the elements of G as objects, and G! is the symmetric cat-group with a single object and with the elements of G as arrows. In this way we obtain two functors

\underline{D}: Ab → SCG , !: Ab → SCG

which are full and faithful, as well as their composite with H. Moreover, the following adjunctions hold :

! · H ⊣ π_{1} , π_{0} ⊣ \underline{D} · H ⊣ π_{0} .

More precisely, for each symmetric cat-group 𝔾 we have

- a full and faithful morphism π_{1}(𝔾)! → 𝔾 which is an equivalence iff π_{0}(𝔾) = 0 ;

- a full and essentially surjective morphism 𝔾 → \underline{D}(π_{0}(𝔾)) which is an equivalence iff π_{1}(𝔾) = 0 .

Let F: 𝔾 → ℍ be a morphism of symmetric cat-groups. From [24] we recall an explicit description of its kernel and its cokernel.

\underline{Kernel:} the kernel of F is given by a symmetric cat-group KerF, a morphism e_{F} : KerF → 𝔾, and a 2-cell λ_{F} : e_{F} · F ⇒ 0 (where 0: KerF → ℍ is the zero-morphism, i.e. the functor which sends each arrow of KerF to the identity of the unit object of ℍ) :

- the objects of KerF are pairs (X, λ_{X}) where X is an object of 𝔾 and λ_{X}: F (X) → I is an arrow of ℍ;

- an arrow f: (X, λ_{X}) → (Y, λ_{Y}) in KerF is an arrow f: X → Y in 𝔾 such that

\xymatrix{F (X) \ar[rr]^{F (f)} \ar[dr]_{λ_{X}} &  & F (Y) \ar[dl]^{λ_{Y}} \\  & I}

commutes;

- the functor e_{F} forgets the arrow λ_{X} of (X, λ_{X});

- the component at (X, λ_{X}) of λ_{F} is given by λ_{X}.

\underline{Cokernel:} the cokernel of F is given by a symmetric cat-group CokerF, a morphism P_{F} : ℍ → CokerF, and a 2-cell π_{F} : F · P_{F} ⇒ 0 :

- the objects of CokerF are those of ℍ; - an arrow [f, N]: \xymatrix{X \ar[r]|{•} & Y} in CokerF is a class of equivalence of pair (f, N) with N an object of 𝔾 and f: X → Y ⊗F (N) an arrow of ℍ; two pairs (f, N) and (g, M) are equivalent if there exists an arrow α: N → M in 𝔾 such that

\xymatrix{ & X \ar[dl]_{f} \ar[dr]^{g} \\ Y ⊗ F (N) \ar[rr]_{1⊗F (α)} &  & Y ⊗ F (M)}

commutes;

- the functor P_{F} sends an arrow X → Y of ℍ to [X → Y ≃ Y ⊗ I ≃ Y ⊗

F (I), I]: \xymatrix{X \ar[r]|{•} & Y} ;

- if N is an object of 𝔾, the component at N of π_{F} is [F (N) → I ⊗ F (N), N]:

\xymatrix{F (N) \ar[r]|{•} & I} .

The kernel and the cokernel are special instances of bilimits (see [17]) and are determined, up to monoidal equivalences, by their universal property, which is discussed in detail in [24]. In section 2 we use the explicit description of the kernel and of the cokernel, but in section 4 we essentially use their universal property. For this we recall here the universal property of the kernel (that of the cokernel is dual). For any morphism G: 𝕂 → 𝔾 and for any 2-cell φ: G · F ⇒ 0, there exists a morphism G′: 𝕂 → KerF and a 2-cell φ′: G′ · e_{F} ⇒ G such that the following diagram commutes

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

If moreover (G′′, φ′′) satisfies the same condition as (G′, φ′), then there exists a unique 2-cell ψ: G′′ ⇒ G′ such that

\xymatrix{ & G \\ G′′ · e_{F} \ar@{=>}[ur]^{φ′′} \ar@{=>}[rr]_{ψ·e_{F}} &  & G′ · e_{F} \ar@{=>}[ul]_{φ′}}

commutes.

We list here some facts we will use later.

1.3. Proposition. Let F: 𝔾 → ℍ be a morphism of symmetric cat-groups;

1) π_{0}(KerF) and π_{1}(CokerF) are isomorphic groups;

2) the functor e_{F} is faithful; it is full iff π_{1}(ℍ) = 0; 3) the functor P_{F} is essentially surjective; it is full iff π_{0}(𝔾) = 0;

4) the factorization of π_{1}(e_{F}) through the kernel of π_{1}(F) is an isomorphism;

5) the factorization of π_{0}(P_{F}) through the cokernel of π_{0}(F) is an isomorphism;

6) F is faithful iff π_{1}(F) is injective iff π_{1}(KerF) = 0;

7) F is essentially surjective iff π_{0}(F) is surjective iff π_{0}(CokerF) = 0;

8) F is full iff π_{0}(F) is injective and π_{1}(F) is surjective iff π_{0}(KerF) = 0 iff π_{1}(CokerF) = 0;

9) F is an equivalence iff π_{0}(F) and π_{1}(F) are isomorphisms iff KerF and CokerF are equivalent to the cat-group with a single arrow.

To end this section, let us establish a link with homotopical algebra. The cat-group KerF just described satisfies also the following strict universal property (and a dual one is satisfied by CokerF) : given a morphism G: 𝕂 → 𝔾 and a 2-cell φ: G · F ⇒ 0, there exists a unique morphism G′: 𝕂 → KerF such that G′ · e_{F} = G. In other words, KerF is also a standard homotopy kernel, or homotopy fibre, and CokerF is also a standard homotopy cokernel, or mapping cone (and they are determined up to isomorphisms by these universal properties), while the more general universal properties we are considering can be viewed as providing a “general” homotopy kernel or cokernel, determined up to equivalences, consistently with Mather’s original definition for topological spaces [14]. M. Grandis has developed in [9] an axiomatic for homotopical algebra based on homotopy kernels and cokernels. Following his approach, the discrete cat-groups

\underline{D}(π_{1}(𝔾)) = Ker(1 → 𝔾)

(1 is the cat-group with a single arrow) is the object of loops Ω(𝔾), so that π_{1}(𝔾) = π_{0}(Ω(𝔾)). Moreover, the cat group

π_{0}(𝔾)! = Coker(𝔾 → 1)

is the suspension Σ(𝔾), so that π_{0}(𝔾) = π_{1}(Σ(𝔾)). Finally, the morphisms π_{1}(𝔾)! → 𝔾 and 𝔾 → \underline{D}(π_{0}(𝔾)) are precisely the counit and the unit of the adjunction Σ ⊣ Ω; this adjunction is a straightforward consequence of the strict universal properties defining these functors.

2. Factorizations of a morphism

In this section we construct two factorizations of a morphism of symmetric cat-groups. Anticipating what will be explained in section 5, we can think of the first factorization as a (epi, regular mono) factorization and of the second one as a (regular epi, mono) factorization.

\underline{First factorization:} let F: 𝔾 → ℍ be a morphism of symmetric cat-groups. Consider its cokernel P_{F} : ℍ → CokerF and the kernel of P_{F} , which we call image, j_{F} : ImF → ℍ. The universal property of ImF gives us a morphism F̂: 𝔾 → ImF and a 2-cell F̂·j_{F} ⇒ F. Moreover, it is possible to choose (in a unique way) the morphism F̂ so that the 2-cell is the identity. In other words, the following is a commutative diagram in SCG

\xymatrix{𝔾 \ar[rr]^{F} \ar[dr]_{F̂} &  & ℍ \\  & ImF \ar@{-->}[ur]_{j_{F}}}

2.1. Proposition. With the above notations, 1) the functor j_{F} is faithful;

2) the functor F̂ is full and essentially surjective. Proof. 1) j_{F} is a kernel, so it is faithful by point 2 of Proposition 1.3.

2) Explicitly, an object in ImF is a class of triples (X, f, N) with X an object of ℍ, N an object of 𝔾 and f: X → F (N); we identify (X, f, N) with (X, f′, N′) if there exists α: N → N′ such that

\xymatrix{ & X \ar[dl]_{f} \ar[dr]^{f′} \\ F (N) \ar[rr]_{F (α)} &  & F (N′)}

commutes. An arrow λ: \xymatrix{[X, f, N] \ar[r]|{•} & [Y, g, M]} is an arrow λ: X → Y such that there exists an arrow l: N → M making commutative the diagram

\xymatrix{X \ar[r]^{λ} \ar[d]_{f} & Y \ar[d]^{g} \\ F (N) \ar[r]_{F (l)} & F (M)}

The functor F̂: 𝔾 → ImF sends μ: A → B into

F (μ): \xymatrix{[F (A), 1_{F (A)}, A] \ar[r]|{•} & [F (B), 1_{F (B)}, B]} .

• F̂ is essentially surjective: let [X, f, N] be an object of ImF, then

f: \xymatrix{[X, f, N] \ar[r]|{•} & F̂(N)}

is an arrow in ImF because

\xymatrix{X \ar[r]^{f} \ar[d]_{f} & F (N) \ar[d]^{1_{F (N)}} \\ F (N) \ar[r]_{F (1_{N})} & F (N)}

commutes.

• F̂ is full: let A, B be in 𝔾 and consider an arrow λ: \xymatrix{F̂(A) \ar[r]|{•} & F̂(B)} in ImF. This means that λ: F (A) → F (B) is an arrow in ℍ and there exists l: A → B in 𝔾 such that

\xymatrix{F (A) \ar[r]^{λ} \ar[d]_{1_{F (A)}} & F (B) \ar[d]^{1_{F (B)}} \\ F (A) \ar[r]_{F (l)} & F (B)}

commutes. And then λ = F̂(l) in ImF.

2.2. Corollary. Let F: 𝔾 → ℍ be a morphism of symmetric cat-groups; the following conditions are equivalent:

1) F is a kernel, that is there exists a morphism G: ℍ → 𝕂 such that F ·G is isomorphic to the zero-morphism and the induced morphism 𝔾 → KerG is an equivalence;

2) F is the kernel of its cokernel, that is F̂ is an equivalence;

3) F is faithful.

Proof. If F is faithful, then F̂ is faithful and then it is an equivalence. The implications 2) ⇒ 1) ⇒ 3) are obvious.

\underline{Second factorization:} let F: 𝔾 → ℍ be a morphism of symmetric cat-groups. Consider its kernel e_{F} : KerF → 𝔾 and the cokernel of e_{F} , which we call coimage, F′: 𝔾 → CoimF. As for the first factorization, the universal property of CoimF gives us a morphism i_{F} : CoimF → ℍ making commutative the following diagram

\xymatrix{𝔾 \ar[rr]^{F} \ar[dr]_{F′} &  & ℍ \\  & CoimF \ar[ur]_{i_{F}}}

2.3. Proposition. With the above notations,

1) the functor F′ is essentially surjective;

2) the functor i_{F} is full and faithful.

Proof. 1) F′ is a cokernel, so it is essentially surjective by point 3) of Proposition 1.3.

2) Objects of CoimF are those of 𝔾. An arrow [f, N, λ_{N}]: \xymatrix{X \ar[r]|{•} & Y} in CoimF is a class of triple (f, N, λ_{N}) with N in 𝔾, λ_{N}: F (N) → I and f: X → Y ⊗ N; we identify (f, N, λ_{N}) with (g, M, λ_{M}) if there exists α: N → M such that

\xymatrix{ & X \ar[dl]_{f} \ar[dr]^{g} &  & and & F (N) \ar[rr]^{F (α)} \ar[dr]_{λ_{N}} &  & F (M) \ar[dl]^{λ_{M}} \\ Y ⊗ N \ar[rr]_{1⊗α} &  & Y ⊗ M &  &  & I}

commute. The functor i_{F} sends [f, N, λ_{N}]: \xymatrix{X \ar[r]|{•} & Y} into

\xymatrix{F (X) \ar[r]^{F (f)} & F (Y ⊗ N)} ≃ \xymatrix{F (Y) ⊗ F (N) \ar[r]^{1⊗λ_{N}} & F (Y) ⊗ I} ≃ F (Y).

• i_{F} is faithful : consider two parallel arrows in CoimF,

[f, N, λ_{N}], [g, M, λ_{M}]: \xymatrix{X \ar[r]|{•} & Y}

and suppose i_{F} [f, N, λ_{N}] = i_{F} [g, M, λ_{M}]. Using f: X → Y ⊗N and g: X → Y ⊗M, we obtain β = f^{−1} · g: Y ⊗ N → Y ⊗ M and

α: N ≃ \xymatrix{I ⊗ N \ar[r]^{ϵ_{Y}^{−1}⊗1} & Y^{∗} ⊗ Y ⊗ N \ar[r]^{β} & Y^{∗} ⊗ Y ⊗ M \ar[r]^{ϵ_{Y} ⊗1} & I ⊗ M} ≃ M

It remains to check the two conditions on α :

- the first one, that is the equation f · (1 ⊗ α) = g, becomes β = 1 ⊗ α, which follows from a diagram chase ;

- as far as the second one is concerned, consider the following diagram

\xymatrix{ & F (Y ⊗ N) \ar[dd]^{F (1⊗α)} \ar[r] & F (Y) ⊗ F (N) \ar[dd]^{1⊗F (α)} \ar[r]^{1⊗λ_{N}} & F (Y) ⊗ I \ar[dd]^{1} \ar[dr] \\ F (X) \ar@{-->}[ur]^{F (f)} \ar[dr]_{F (g)} &  &  &  & F (Y) \\  & F (Y ⊗ M) \ar[r] & F (Y) ⊗ F (M) \ar[r]_{1⊗λ_{M}} & F (Y) ⊗ I \ar[ur]}

The left-hand triangle commutes because f · (1 ⊗ α) = g, the first square commutes because F is monoidal, and the hypothesis i_{F} [f, N, λ_{N}] = i_{F} [g, M, λ_{M}] precisely means that the outer diagram is commutative. Since all the arrows are isomorphisms, it follows that the second square is commutative. Finally, this commutativity implies the second condition on α, that is F (α) · λ_{M} = λ_{N}, because F (Y) is invertible, so that tensoring with F (Y) is an autoequivalence of ℍ.

• i_{F} is full : recall that we have a natural transformation

π_{e_{F}} : e_{F} · F′ ⇒ 0

and that F′ is the identity on objects. Consider two objects X, Y in CoimF and an arrow h: i_{F} (X) = F (X) → F (Y) = i_{F} (Y) in ℍ. We can build up an object (Y^{∗} ⊗ X, λ) of KerF with λ defined by

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

Using the component at (Y^{∗} ⊗ X, λ) of the transformation π_{e_{F}} , we obtain the following arrow in CoimF

\xymatrix{k: X \ar[r]|{•} & I ⊗ X \ar[r]|{•} & Y ⊗ Y^{∗} ⊗ X \ar[r]|{•}^{1⊗π_{e_{F}} (Y^{∗}⊗X,λ)} & Y ⊗ I \ar[r]|{•} & Y.}

Finally, using the triangular identities of the duality (Y^{∗} ⊣ Y, η_{Y} , ϵ_{Y}), one checks that i_{F} (k) = h.

2.4. Corollary. Let F: 𝔾 → ℍ be a morphism of symmetric cat-groups; the following conditions are equivalent:

1) F is a cokernel;

2) F is the cokernel of its kernel;

3) F is essentially surjective.

Proof. If F is essentially surjective, then i_{F} is an equivalence.

3. Factorization systems in a 2-category

The notion of factorization system in a category is a well-established one. To give its natural 2-categorical version, and to prove some elementary facts, we closely follow the review of factorization systems given in [3] (see also [12]).

In this section, C is a 2-category with invertible 2-cells. Given two arrows f: A → B and g: C → D, we say that f has the fill-in property with respect to g, in symbols f ↓ g, if for each pair of arrows u: A → C, v: B → D and for each 2-cell φ: f · v ⇒ u · g

[figure: A; f; u; B; v; φ; C; g; D]

there exists an arrow w: B → C and two 2-cells α: f · w ⇒ u , β: w · g ⇒ v such that the following diagram commutes

\xymatrix{f · w · g \ar@{=>}[rr]^{f·β} \ar@{=>}[dr]_{α·g} &  & f · v \ar@{=>}[dl]^{φ} \\  & u · g}

(we say that (α, w, β) is a fill-in for (u, φ, v)); moreover, if (α′, w′, β′) is another fill-in for (u, φ, v), then there exists a unique ψ: w ⇒ w′ such that

[figure: f · w; f·ψ; α; f · w′; α′; u; and; w · g; ψ·g; β; w′ · g; β′; v]

Theory and Applications of Categories, Vol. 7, No. 5 56

commute. (See [4] for an interpretation of this conditions in terms of bilimits.) If H is a class of arrows of C, we use the following notations :

H^{↓} = {g s.t. h ↓ g for each h in H}

H^{↑} = {f s.t. f ↓ h for each h in H}

Clearly, if H_{1} ⊂ H_{2} then H_{1}^{↓} ⊃ H_{2}^{↓} and H_{1}^{↑} ⊃ H_{2}^{↑} 3.1. Definition. A factorization system (E, M) in C is given by two classes E and M of arrows in C such that :

1) E and M contain equivalences and are closed under composition with equivalences ;

2) E and M are stable under 2-cells (this means that if e is in E and there is a 2-cell e′ ⇒ e, then also e′ is in E, and the same holds for M) ;

3) for each arrow f of C, there exist e ∈ E , m ∈ M and a 2-cell φ

[figure: I; m; φ; A; e; f; B]

(we say that (e, φ, m) is a (E, M)-factorization for f) ;

4) for each e ∈ E and for each m ∈ M, one has e ↓ m. 3.2. Proposition. Let (E, M) be a factorization system in C ;

1) if f is in E and in M, then f is an equivalence ;

2) E = M^{↑} and M = E^{↓} ;

3) E and M are closed under composition ;

4) E is stable under bi-pushout and M is stable under bi-pullback ;

5) if (E′, M′) is another factorization system in C, then E ⊂ E′ iff M ⊃ M′. Proof. The proof is a quite long but essentially straightforward 2-categorical version of that of the corresponding 1-categorical properties. To give the flavor of the proof, we check that M^{↑} is closed under composition. Consider the following diagram

\xymatrix{A \ar[r]^{f} \ar[d]_{u} & B \ar[r]^{g} & C \ar[d]^{v} \ar@{=>}[dll]_{φ} \\ D \ar[rr]_{m} &  & E}

with f, g ∈ M^{↑} and m ∈ M. Since f ↓ m, there exists a fill-in (α, w, β) for (u, φ, g · v). Since g ↓ m, there exists a fill-in (ϵ, z, μ) for (w, β^{−1}, v). In this way we obtain a fill-in

( \xymatrix{f · g · z \ar@{=>}[r]^{f·ϵ} & f · w \ar@{=>}[r]^{α} & u} , z, μ)

for (u, φ, v). Now suppose that (\overline{η}, \overline{z}, \overline{μ}) is another fill-in for (u, φ, v). This implies that (\overline{η}, g · \overline{z}, g · \overline{μ}) is a fill-in for (u, φ, g · v), so that there exists a unique ψ: w ⇒ g · \overline{z} such that the following diagrams commute

\xymatrix{f · w \ar@{=>}[rr]^{f·ψ} \ar@{=>}[dr]_{α} &  & f · g · \overline{z} \ar@{=>}[dl]^{\overline{η}} & w · m \ar@{=>}[rr]^{ψ·m} \ar@{=>}[dr]_{β} &  & g · \overline{z} · m \ar@{=>}[dl]^{g·\overline{μ}} \\  & u &  &  & g · v}

This implies that (ψ^{−1}, \overline{z}, \overline{μ}) is a fill-in for (w, β^{−1}, v), so that there exists a unique σ: z ⇒ \overline{z} such that the following diagrams commute

[figure: g · z; g·σ; ϵ; g · \overline{z}; ψ^{−1}; w; z · m; σ·m; μ; \overline{z} · m; \overline{μ}; v]

and then commutes also

\xymatrix{f · g · z \ar@{=>}[r]^{f·g·σ} \ar@{=>}[d]_{f·ϵ} & f · g · \overline{z} \ar@{=>}[d]^{\overline{η}} \\ f · w \ar@{=>}[r]_{α} & u}

It remains to check that if σ′: z ⇒ \overline{z} is another 2-cell making commutative

[figure: f · g · z; f·g·σ′; f·ϵ; f · g · \overline{z}; \overline{η}; f · w; α; u; z · m; σ′·m; μ; \overline{z} · m; \overline{μ}; v]

then σ = σ′. For this, it suffices to prove the commutativity of

[figure: g · z; g·σ′; ϵ; g · \overline{z}; ψ^{−1}; w]

that is to prove that \xymatrix{w \ar@{=>}[r]^{ϵ^{−1}} & g · z \ar@{=>}[r]^{g·σ′} & g · \overline{z}} satisfies the conditions which characterize ψ. The first one coincides with the first assumption on σ′, the second one follows from the fact that (ϵ, z, μ) is a fill-in for (w, β^{−1}, v).

We leave it as an exercise for the reader to formulate the correct notion of functoriality and uniqueness for the (E, M)-factorization. To end this section, observe that in general a factorization system in C does not induce a factorization system (in the usual sense) neither in the underlying category of C, nor in the homotopy category of C.

4. Factorization systems in SCG

In this section we show that the factorizations of a morphism between symmetric catgroups constructed in section 2 satisfy the axioms introduced in section 3.

\underline{First factorization:} we need two simple lemmas.

4.1. Lemma. Consider the following morphisms in SCG

\xymatrix{𝕃 \ar[r]^{G} & 𝕂 \ar[r]^{K} \ar[r]_{H} & ℍ}

and assume that G is full and essentially surjective ; for each 2-cell μ: G · K ⇒ G · H there exists a unique 2-cell \overline{μ}: K ⇒ H such that G · \overline{μ} = μ.

Proof. Let X be an object of 𝕂 and choose an object A_{X} in 𝕃 and a morphism x: X → G(A_{X}) ; we can put

\xymatrix{\overline{μ}_{X}: K(X) \ar[r]^{K(x)} & K(G(A_{X})) \ar[r]^{μ_{A_{X}}} & H(G(A_{X})) \ar[r]^{H(x^{−1})} & H(X)}

This definition does not depend on the choice of A_{X} and x because G is full: if x′: X → G(A′_{X}) is another choice, there exists a z: A_{X} → A′_{X} such that G(z) = x^{−1} · x′; now the naturality of μ implies that K(x) · μ_{A_{X}} · H(x^{−1}) = K(x′) · μ_{A′_{X}} · H(x′^{−1}). In particular, if X = G(A), we can choose A_{X} = A and x = 1_{A}, so that \overline{μ}_{G(A)} = μ_{A}, that is G·\overline{μ} = μ. Now let f: X → Y be a morphism in 𝕂 and choose x: X → G(A_{X}) , y: Y → G(A_{Y}). Once again, the fullness of G gives us a morphism φ: A_{X} → A_{Y} such that G(φ) = x^{−1} · f · y. The following diagram commutes in each part, so that \overline{μ} is natural

\xymatrix{K(X) \ar[r]^{K(x)} \ar[d]_{K(f)} & K(G(A_{X})) \ar[r]^{μ_{A_{X}}} \ar[d]_{K(G(φ))} & H(G(A_{X})) \ar[r]^{K(x^{−1})} \ar[d]_{H(G(φ))} & H(X) \ar[d]^{H(f)} \\ K(Y) \ar[r]_{K(y)} & K(G(A_{Y})) \ar[r]_{μ_{A_{Y}}} & H(G(A_{Y})) \ar[r]_{H(y^{−1})} & H(Y)}

As far as the uniqueness of \overline{μ} is concerned, observe that its naturality implies the commutativity of

\xymatrix{K(X) \ar[r]^{^{\overline{μ}^{X}}} \ar[d]_{K(x)} & H(X) \ar[d]^{H(x)} \\ K(G(A_{X})) \ar[r]_{_{\overline{μ}_{G(A_{X})}}} & H(G(A_{X}))}

so that the condition G · \overline{μ} = μ forces our definition of \overline{μ}. Finally, it is easy to check that if μ is monoidal, then also \overline{μ} is monoidal.

4.2. Lemma. Consider three morphisms H, K and F in SCG and the kernel of F, as in the diagram

[figure: 𝔾; λ_{F}; F; 𝕃; K; H; KerF; e_{F}; 0; ℍ]

let ρ: K · e_{F} ⇒ H · e_{F} be a 2-cell in SCG such that

\xymatrix{K · e_{F} · F \ar@{=>}[r]^{ρ·F} \ar@{=>}[d]_{K·λ_{F}} & H · e_{F} · F \ar@{=>}[d]^{H·λ_{F}} \\ K · 0 \ar@{=>}[r] & H · 0}

commutes. There exists a unique 2-cell \overline{ρ}: K ⇒ H such that \overline{ρ} · e_{F} = ρ.

Proof. It suffices to check the particular case where H = 0, and this case follows easily from the universal property of the kernel.

4.3. Proposition. Consider the following classes of morphisms in SCG :

E_{1} = full and essentially surjective morphisms ;

M_{1} = faithful morphisms.

(E_{1}, M_{1}) is a factorization system in the sense of Definition 3.1.

Proof. Clearly, E_{1} and M_{1} contain equivalences, are closed under composition with equivalences and are stable under 2-cells. The existence of an (E_{1}, M_{1})-factorization has been proved in Proposition 2.1, so that it remains to prove the fill-in condition. For this, consider G: 𝕃 → 𝕂 in E_{1} and let

[figure: 𝔾; F; λ_{F}; KerF; e_{F}; 0; ℍ]

be the kernel of a morphism F. We start showing that G ↓ e_{F} . Consider two morphisms U: 𝕃 → KerF , V : 𝕂 → 𝔾 and a 2-cell φ: G · V ⇒ U · e_{F} . We obtain a new 2-cell

\xymatrix{μ: G · V · F \ar@{=>}[r]^{φ·F} & U · e_{F} · F \ar@{=>}[r]^{U·λ_{F}} & U · 0 \ar@{=>}[r] & G · 0}

and, by Lemma 4.1, there exists a unique 2-cell \overline{μ}: V · F ⇒ 0 such that G · \overline{μ} = μ. Now the universal property of the kernel gives us a morphism W: 𝕂 → KerF and a 2-cell β: W · e_{F} ⇒ V such that

\xymatrix{W · e_{F} · F \ar@{=>}[r]^{β·e_{F}} \ar@{=>}[d]_{W ·λ_{F}} & V · F \ar@{=>}[d]^{\overline{μ}} \\ W · 0 \ar@{=>}[r] & 0}

commutes. We obtain a 2-cell

\xymatrix{ρ: G · W · e_{F} \ar@{=>}[r]^{G·β} & G · V \ar@{=>}[r]^{φ} & U · e_{F}}

making commutative the following diagram

\xymatrix{G · W · e_{F} · F \ar@{=>}[r]^{ρ·F} \ar@{=>}[d]_{G·W ·λ_{F}} & U · e_{F} · F \ar@{=>}[d]^{U·λ_{F}} \\ G · W · 0 \ar@{=>}[r] & U · 0}

By Lemma 4.2, there exists a unique 2-cell \overline{ρ}: G · W ⇒ U such that \overline{ρ} · e_{F} = ρ. This means that (\overline{ρ}, W, β) is a fill-in for (U, φ, V). Suppose now that (\overline{α}, \overline{W} , \overline{β}) is another fill-in for (U, φ, V). The condition to be a fill-in implies the commutativity of

\xymatrix{G · \overline{W} · e_{F} · F \ar@{=>}[r]^{G·\overline{β}·F} \ar@{=>}[d]_{G·\overline{W} ·λ_{F}} & G · V · F \ar@{=>}[d]^{G·\overline{μ}} \\ G · \overline{W} · 0 \ar@{=>}[r] & G · 0}

By Lemma 4.1, this implies the commutativity of

\xymatrix{\overline{W} · e_{F} · F \ar@{=>}[r]^{\overline{β}·F} \ar@{=>}[d]_{\overline{W} ·λ_{F}} & V · F \ar@{=>}[d]^{\overline{μ}} \\ \overline{W} · 0 \ar@{=>}[r] & 0}

so that, by the universal property of the kernel, there exists a unique ψ: W ⇒ \overline{W} such that

[figure: W · e_{F}; ψ·e_{F}; β; \overline{W} · e_{F}; \overline{β}; U]

commutes. It remains only to check the commutativity of

and to do this we use once again the universal property of the kernel. Consider the 2-cell

\xymatrix{U · e_{F} · F \ar@{=>}[r]^{U·λ_{F}} & U · 0 \ar@{=>}[r] & 0} ;

[figure: G · W; G·ψ; \overline{ρ}; G · \overline{W}; \overline{α}; U]

it factors through the kernel of F in two ways : the obvious one, that is (U, U · e_{F}), but also (G · W, \overline{ρ} · e_{F}). By the universal property, there exists a unique σ: G · W ⇒ U such that

\xymatrix{G · W · e_{F} \ar@{=>}[rr]^{σ·e_{F}} \ar@{=>}[dr]_{\overline{ρ}·e_{F}} &  & U · e_{F} \ar@{=>}[dl]^{U·e_{F}} \\  & U · e_{F}}

commutes. Clearly, we can take \overline{ρ} as σ, but also

\xymatrix{G · W \ar@{=>}[r]^{G·ψ} & G · W \ar@{=>}[r]^{\overline{α}} & U}

can be taken as σ. By uniqueness of σ, we have finished. Now we turn to the general case. Let G be in E_{1} and F in M_{1} ; we have to prove that G ↓ F. For this we consider the factorization of F

\xymatrix{𝔾 \ar[rr]^{F} \ar[dr]_{F̂} &  & ℍ \\  & ImF \ar@{-->}[ur]_{j_{F}}}

By Proposition 2.1, F̂ is an equivalence. By a general argument (see [11]) we can choose a quasi-inverse \overline{F}: ImF → 𝔾 and two natural transformations ϵ: F̂·\overline{F} ⇒ Id , η: Id ⇒ \overline{F} ·F̂ in such a way that (\overline{F} , F̂, η, ϵ) is an adjoint equivalence in SCG. By the first part of the proof, we have G ↓ j_{F}; it is now straightforward, even if quite long, to prove G ↓ F.

\underline{Second factorization:} as for the first factorization, we need two preliminary lemmas.

4.4. Lemma. Consider the following morphisms in SCG

\xymatrix{ℍ \ar[r]^{K} \ar[r]_{H} & 𝕂 \ar[r]^{G} & 𝕃}

and assume that G is full and faithful ; for each 2-cell μ: K · G ⇒ H · G there exists a unique 2-cell \overline{μ}: K ⇒ H such that \overline{μ} · G = μ.

Proof. For each object A in ℍ, there exists a unique \overline{μ}_{A}: K(A) → H(A) such that G(\overline{μ}_{A}) = μ_{A}. The naturality of \overline{μ} as well as its uniqueness follows from the faithfulness of G. It is easy to check that if μ is monoidal, then also \overline{μ} is monoidal.

4.5. Lemma. Consider three morphisms H, K and F in SCG and the cokernel of F, as in the diagram

[figure: ℍ; π_{F}; P_{F}; 𝔾; F; 0; CokerF; K; H; 𝕃]

Theory and Applications of Categories, Vol. 7, No. 5 62

let ρ: P_{F} · K ⇒ P_{F} · H be a 2-cell in SCG such that

\xymatrix{F · P_{F} · K \ar@{=>}[r]^{F ·ρ} \ar@{=>}[d]_{π_{F} ·K} & F · P_{F} · H \ar@{=>}[d]^{π_{F} ·H} \\ 0 · K \ar@{=>}[r] & 0 · H}

commutes. There exists a unique 2-cell \overline{ρ}: K ⇒ H such that P_{F} · \overline{ρ} = ρ. 4.6. Proposition. Consider the following classes of morphisms in SCG :

E_{2} = essentially surjective morphisms ; M_{2} = full and faithful morphisms.

(E_{2}, M_{2}) is a factorization system in the sense of Definition 3.1. Proof. The proof follows the same lines as the proof of Proposition 4.3. Once again, it is easier to start showing that, if G is in M_{2} and F is any morphism, then P_{F} ↓ G. The general case follows from this using that, by Proposition 2.3, the morphism i_{F} in the second factorization of F

\xymatrix{𝔾 \ar[rr]^{F} \ar[dr]_{F′} &  & ℍ \\  & CoimF \ar[ur]_{i_{F}}}

is an equivalence if F is essentially surjective.

5. More on morphisms in SCG

In this section we want to invert Lemma 4.1 and Lemma 4.4. 5.1. Lemma. Let F: 𝔾 → ℍ be a morphism in SCG; the following conditions are equivalent :

1) F is faithful ;

2) for each symmetric cat-group 𝕂, the functor induced by F

− · F: SCG(𝕂, 𝔾) → SCG(𝕂, ℍ)

is faithful ;

3) given two morphisms K, H: 𝕃 → 𝔾 and a 2-cell ρ: K · F ⇒ H · F such that

\xymatrix{K · F · P_{F} \ar@{=>}[r]^{ρ·P_{F}} \ar@{=>}[d]_{K·π_{F}} & H · F · P_{F} \ar@{=>}[d]^{H·π_{F}} \\ K · 0 \ar@{=>}[r] & H · 0}

commutes, then there exists a unique 2-cell \overline{ρ}: K ⇒ H such that \overline{ρ} · F = ρ.

Proof. 1) ⇒ 2) : This follows from Lemma 4.2 because, by Corollary 2.2, F is the kernel of P_{F} .

3) ⇒ 2) : If ρ is of the form α · F for a 2-cell α: H ⇒ H, then the condition on ρ holds and then β · F = ρ = α · F implies β = \overline{ρ} = α.

2) ⇒ 1) : Consider the abelian group ℤ and take 𝕂 = ℤ!, so that − · F becomes π_{1}(F). Now − · F faithful means π_{1}(F) injective, so that F is faithful by point 6) in Proposition 1.3.

5.2. Proposition. Let F: 𝔾 → ℍ be a morphism in SCG; the following conditions are equivalent :

1) F is full and faithful ;

2) for each symmetric cat-group 𝕂, the functor induced by F

− · F: SCG(𝕂, 𝔾) → SCG(𝕂, ℍ)

is full and faithful ;

Proof. 1) ⇒ 2) : This is exactly Lemma 4.4.

2) ⇒ 1) : By Lemma 5.1, we already know that F is faithful. Now take 𝕂 = \underline{D}(ℤ), and the fullness of − · F means that F is full.

5.3. Proposition. Let G: 𝕃 → 𝕂 be a morphism in SCG; the following conditions are equivalent :

1) G is full and essentially surjective ;

2) for each symmetric cat-group ℍ, the functor induced by G

G · −: SCG(𝕂, ℍ) → SCG(𝕃, ℍ)

is full and faithful ;

Proof. 1) ⇒ 2) : This is exactly Lemma 4.1.

2) ⇒ 1) : Consider the class E′_{1} of the morphisms which satisfy condition 2). Since E_{1} ⊂ E′_{1} (because 1) ⇒ 2)), each morphism in SCG has a (E′_{1}, M_{1})-factorization. Moreover, clearly E′_{1} contains equivalences, it is closed under composition with equivalences and it is stable under 2-cells. To prove that E_{1} = E′_{1}, it remains to show that, for each G in E′_{1} and for each F in M_{1}, one has G ↓ F. But this is exactly what we have done in the proof of Proposition 4.3.

Let us close this section with a point of terminology. In view of Proposition 5.2, it is reasonable to call “monomorphism” a full and faithful morphism (and, dually, “epimorphism” a full and essentially surjective morphism). This is because in any 2-category C with invertible 2-cells, the following conditions on an arrow f: A → B are equivalent :

i) for each object C of C, the functor

− · f: C(C, A) → C(C, B)

is full and faithful ;

ii) the diagram

[figure: A; 1_{A}; 1_{A}; A; f; f; A; f; B]

is a bi-pullback ;

iii) there exist two morphisms f_{0}, f_{1} : P → A and a 2-cell φ: f_{0} ⇒ f_{1} such that the diagram

[figure: P; f_{0}; f_{1}; A; f; φ·f; A; f; B]

is a bi-pullback.

In the introduction, we say that SCG can be seen as the 2-dimensional analogue of the category of abelian groups. The results stated in this paper follow this idea, and in fact cokernels coincide with strong epis (= morphisms orthogonal to monos) and kernels coincide with strong monos (= morphisms orthogonal to epis), each morphism can be factorized as an epi followed by a kernel or as a cokernel followed by a mono, each mono is a kernel and each epi is a cokernel. The surprise is that in this 2-dimensional world a kernel fails to be a mono and a cokernel fails to be an epi (the failure being measured respectively by π_{1} and π_{0}).

6. Exact sequences

In order to study some classical exact sequences of abelian groups associated with a morphism of commutative unital rings, in [24] the notion of 2-exact sequence of symmetric cat-groups has been introduced.

For a sequence of morphisms of abelian groups, the notion of exactness can be stated in several equivalent ways, and this is a relevant fact to make exact sequences easy to use. We want to show here that the same can be done in the framework of symmetric cat-groups, but using in the appropriate way both the factorizations we have.

We need some preliminary constructions. Consider two morphisms in SCG together with their factorizations as in the following diagram

[figure: KerG; 0; e_{G}; CoimG; G′; g; 𝔸; F; F̂; 𝔹; G; P_{F}; λ_{P_{F}}; P_{e_{G}}; π_{e_{G}}; ℂ; ImF; e_{P_{F}}; f; 0; CokerF]

with the following commutative diagrams

\xymatrix{F̂ · e_{P_{F}} · P_{F} \ar@{=>}[r]^{f·P_{F}} \ar@{=>}[d]_{F̂·λ_{P_{F}}} & F · P_{F} \ar@{=>}[d]^{π_{F}} & e_{G} · P_{e_{G}} · G′ \ar@{=>}[r]^{e_{G}·g} \ar@{=>}[d]_{π_{e_{G}}·G′} & e_{G} · G \ar@{=>}[d]^{λ_{G}} \\ F̂ · 0 \ar@{=>}[r] & 0 & 0 · G′ \ar@{=>}[r] & 0}

Now fix a 2-cell φ: F · G ⇒ 0 and consider the factorizations given by the universal properties of KerG and CokerF

[figure: 𝔸; 0; F; CokerF; G̃; φ̃; 𝔹; π_{F}; G; P_{F}; ℂ; 𝔸; F; \overline{F}; 𝔹; G; λ_{G}; KerG; \overline{φ}; 0; e_{G}; ℂ]

with the following commutative diagram

\xymatrix{\overline{F} · e_{G} · G \ar@{=>}[r]^{\overline{φ}·G} \ar@{=>}[d]_{\overline{F} ·λ_{G}} & F · G \ar@{=>}[d]_{φ} & F · P_{F} · G̃ \ar@{=>}[l]_{F ·φ̃} \ar@{=>}[d]^{π_{F} ·G̃} \\ \overline{F} · 0 \ar@{=>}[r] & 0 & 0 · G̃ \ar@{=>}[l]}

Consider the 2-cell

\xymatrix{β: F̂ · e_{P_{F}} · G \ar@{=>}[r]^{f·G} & F · G \ar@{=>}[r]^{φ} & 0 \ar@{=>}[r] & F̂ · 0} ;

since F̂ is full and essentially surjective, by Lemma 4.1 there exists a unique α: e_{P_{F}} ·G ⇒ 0 such that F̂ · α = β. The universal property of KerG gives us a factorization

[figure: KerG; e_{G}; μ; ImF; P; e_{P_{F}}; 𝔹]

such that the following diagram commutes

\xymatrix{P · e_{G} · G \ar@{=>}[r]^{μ·G} \ar@{=>}[d]_{P ·λ_{G}} & e_{P_{F}} · G \ar@{=>}[d]^{α} \\ P · 0 \ar@{=>}[r] & 0}

Since the diagram

\xymatrix{F̂ · P · e_{G} · G \ar@{=>}[r]^{F̂·μ·G} \ar@{=>}[d]_{F̂·P ·λ_{G}} & F̂ · e_{P_{F}} · G \ar@{=>}[r]^{f·G} & F · G \ar@{=>}[d]^{φ} \\ F̂ · P · 0 \ar@{=>}[rr] &  & 0}

commutes, the universal property of KerG gives a unique 2-cell ν: \overline{F} ⇒ F̂ · P such that the following diagram commutes

\xymatrix{\overline{F} · e_{G} \ar@{=>}[d]_{\overline{φ}} \ar@{=>}[r]^{ν·e_{G}} & F̂ · P · e_{G} \ar@{=>}[d]^{F̂·μ} \\ F & F̂ · e_{P_{F}} \ar@{=>}[l]^{f}}

Analogously, start with the 2-cell

\xymatrix{δ: F · P_{e_{G}} · G′ \ar@{=>}[r]^{F ·g} & F · G \ar@{=>}[r]^{φ} & 0 \ar@{=>}[r] & 0 · G′} ;

since G′ is full and faithful, by Lemma 4.4 there exists a unique γ: F · P_{e_{G}} ⇒ 0 such that γ · G′ = δ. The universal property of CokerF gives a factorization

[figure: CokerF; Q; θ; 𝔹; P_{F}; P_{e_{G}}; CoimG]

with the commutative diagram

\xymatrix{F · P_{F} · Q \ar@{=>}[d]_{π_{F} ·Q} \ar@{=>}[r]^{F ·θ} & F · P_{e_{G}} \ar@{=>}[d]^{γ} \\ 0 · Q \ar@{=>}[r] & 0}

Furthermore, the universal property of CokerF gives a unique η: G̃ ⇒ Q · G′ with the commutative diagram

\xymatrix{P_{F} · G̃ \ar@{=>}[d]_{φ̃} \ar@{=>}[r]^{P_{F} ·η} & P_{F} · Q · G′ \ar@{=>}[d]^{θ·G′} \\ G & P_{e_{G}} · G′ \ar@{=>}[l]^{g}}

Finally, consider the 2-cell

\xymatrix{x: F̂ · e_{P_{F}} · P_{e_{G}} \ar@{=>}[r]^{f·P_{e_{G}}} & F · P_{e_{G}} \ar@{=>}[r]^{γ} & 0 \ar@{=>}[r] & F̂ · 0} ;

since F̂ is full and essentially surjective, there exists a unique u: e_{P_{F}} · P_{e_{G}} ⇒ 0 such that F̂ · u = x. In the same way, consider

\xymatrix{y: e_{P_{F}} · P_{e_{G}} · G′ \ar@{=>}[r]^{e_{P_{F}} ·g} & e_{P_{F}} · G \ar@{=>}[r]^{α} & 0 \ar@{=>}[r] & 0 · G′} ;

since G′ is full and faithful, there exists a unique v: e_{P_{F}} · P_{e_{G}} ⇒ 0 such that v · G′ = y.

6.1. Lemma. With the previous notations, u = v.

Proof. Since F̂ is full and essentially surjective and G′ is full and faithful, by Lemmas 4.1 and 4.4 it is enough to check that F̂ · u · G′ = F̂ · v · G′. This easily follows from the previous commutative diagrams.

6.2. Proposition. With the previous notations, the following conditions are equivalent :

1) \overline{F}: 𝔸 → KerG is full and essentially surjective ;

2) P: ImF → KerG is an equivalence ;

3) Q: CokerF → CoimG is an equivalence ;

4) Ĝ: CokerF → ℂ is full and faithful.

Proof. 1) ⇒ 2) : Using μ: P · e_{G} ⇒ e_{P_{F}} , we can deduce the faithfulness of P from that of e_{P_{F}} . Using ν: \overline{F} ⇒ F̂ · P, we can deduce the essential surjectivity of P from that of \overline{F} , and the fullness of P from the fact that \overline{F} is full and F̂ is essentially surjective.

2) ⇒ 1) : Since F̂ is full and essentially surjective, if P is an equivalence, then the composite F̂ · P is full and essentially surjective. But then also \overline{F} is full and essentially surjective because of ν: \overline{F} ⇒ F̂ · P. In a similar way, one can prove the equivalence of conditions 3) and 4).

3) ⇒ 4) : Consider the following diagram in SCG

[figure: KerP_{F}; e_{P_{F}}; P; 𝔹; _{Id_{𝔹}}; P_{F}; CokerF; Q; KerG; e_{G}; 𝔹; P_{e_{G}}; G; Coker(e_{G}); KerP_{e_{G}}; e_{Pe_{G}}; ℂ]

since e_{G} is faithful, by Corollary 2.2 we know that the comparison KerG → KerP_{e_{G}} is an equivalence. Now if Q is an equivalence, also P is an equivalence.

2) ⇒ 3) : Similar to 3) ⇒ 2), but using Corollary 2.4.

6.3. Definition. A sequence in SCG

[figure: 𝔹; φ; G; 𝔸; F; 0; ℂ]

is 2-exact if it satisfies one of the equivalent conditions of the previous proposition.

Observe that the first condition in Proposition 6.2 is the definition of 2-exact sequence used in [24].

6.4. Proposition. (with the previous notations) Consider the following sequences in SCG

[figure: 𝔹; φ; G; 𝔸; F; 0; ℂ; 𝔹; u=v; P_{e_{G}}; ImF; e_{P_{F}}; 0; CoimG]

[figure: 𝔹; α; G; ImF; e_{P_{F}}; 0; ℂ; 𝔹; γ; P_{e_{G}}; 𝔸; F; 0; CoimG]

The following conditions are equivalent :

1) the sequence (F, φ, G) is 2-exact ;

2) the sequence (e_{P_{F}} , α, G) is 2-exact ;

3) the sequence (F, γ, P_{e_{G}}) is 2-exact ;

4) the sequence (e_{P_{F}} , u = v, P_{e_{G}}) is 2-exact.

Proof. 1) ⟺ 2) : Same argument as in 1) ⟺ 2) of Proposition 6.2, because P is always faithful.

1) ⟺ 3) : Same argument as in 3) ⟺ 4) of Proposition 6.2.

1) ⟺ 4) : Come back to the diagram in the proof of Proposition 6.2 and consider the factorization H: KerP_{F} → KerP_{e_{G}} of e_{P_{F}} through e_{P_{e_{G}}}. We have: (F, φ, G) is 2exact iff P is full and essentially surjective iff H is full and essentially surjective (because KerG → KerP_{e_{G}} is an equivalence) iff (e_{P_{F}} , u, P_{e_{G}}) is 2-exact.

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[20] R. Street, R.F.C. Walters, The comprehensive factorization of a functor, Bull. Am. Math. Soc. 79 (1973) 936-941.

[21] M. Takeuchi, On Villamayor and Zelinsky’s long exact sequence, Memoirs Am. Math. Soc. 249 (1981).

[22] O.E. Villamayor, D. Zelinsky, Brauer groups and Amitsur cohomology for general commutative ring extensions, J. Pure Applied Algebra 10 (1977) 19-55.

[23] E.M. Vitale, The Brauer and Brauer-Taylor groups of a symmetric monoidal category, Cahiers Topologie Géométrie Différentielle Catégoriques 37 (1996) 91-122.

[24] E.M. Vitale, A Picard-Brauer exact sequence of categorical groups, submitted.

[25] E.M. Vitale, On the categorical structure of H^{2}, submitted.

Dipartimento di Matematica, Università di Milano, Via Cesare Saldini 50, 20133 Milano, Italy and Département de Mathématique, Université catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium

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