Theory and Applications of Categories, Vol. 20, No. 16, 2008, pp. 543–604.

COHOMOLOGY THEORY IN 2-CATEGORIES

HIROYUKI NAKAOKA

Abstract. Recently, symmetric categorical groups are used for the study of the Brauer groups of symmetric monoidal categories. As a part of these efforts, some algebraic structures of the 2-category of symmetric categorical groups SCG are being investigated. In this paper, we consider a 2-categorical analogue of an abelian category, in such a way that it contains SCG as an example. As a main theorem, we construct a long cohomology 2-exact sequence from any extension of complexes in such a 2-category. Our axiomatic and self-dual definition will enable us to simplify the proofs, by analogy with abelian categories.

1. Introduction

In 1970s, B. Pareigis started his study on the Brauer groups of symmetric monoidal categories in [6]. Around 2000, the notion of symmetric categorical groups are introduced to this study by E. M. Vitale in [9] (see also [8]). By definition, a symmetric categorical group is a categorification of an abelian group, and in this sense the 2-category of symmetric categorical groups SCG can be regarded as a 2-dimensional analogue of the category Ab of abelian groups. As such, SCG and its variants (e.g. 2-category 𝔾-SMod of symmetric categorical groups with 𝔾-action where 𝔾 is a fixed categorical group) admit a 2-dimensional analogue of the homological algebra in Ab.

For example, E. M. Vitale constructed for any monoidal functor F : C → D between symmetric monoidal categories C and D, a 2-exact sequence of Picard and Brauer categorical groups

P(C) → P(D) → \overline{F} → B(C) → B(C).

By taking π_{0} and π_{1}, we can induce the well-known Picard-Brauer and Unit-Picard exact sequences of abelian groups respectively. In [7], A. del Río, J. Martínez-Moreno and E. M. Vitale defined a more subtle notion of the relative 2-exactness, and succeeded in constructing a cohomology long 2-exact sequence from any short relatively 2-exact sequence of complexes in SCG. In this paper, we consider a 2-categorical analogue of an abelian category, in such a way that it contains SCG as an example, so as to treat SCG and their variants in a more abstract, unified way.

The author is supported by JSPS. The author wishes to thank Professor Toshiyuki Katsura for his encouragement.

Received by the editors 2007-11-30 and, in revised form, 2008-11-16.

Transmitted by Ross Street. Published on 2008-11-25.

2000 Mathematics Subject Classification: 18D05.

Key words and phrases: symmetric categorical group, 2-category, cohomology, exact sequence.

\begin{matrix}c \\ ◯ \end{matrix} Hiroyuki Nakaoka, 2008. Permission to copy for private use granted.

In section 2, we review general definitions in a 2-category and properties of SCG, with simple comments. In section 3, we define the notion of a relatively exact 2-category as a generalization of SCG, also as a 2-dimensional analogue of an abelian category. We try to make the homological algebra in SCG ([7]) work well in this general 2-category. It will be worthy to note that our definition of a relatively exact 2-category is self-dual.

category  2-category
general theory abelian category relatively exact 2-category
example Ab SCG

In section 4, we show the existence of proper factorization systems in any relatively exact 2-category, which will make several diagram lemmas more easy to handle. In any abelian category, any morphism f can be written in the form f = e ∘ m (uniquely up to an isomorphism), where e is epimorphic and m is monomorphic. As a 2-dimensional analogue, we show that any 1-cell f in a relatively exact 2-category S admits the following two ways of factorization:

(1) i ∘ m ⟹ f where i is fully cofaithful and m is faithful.

(2) e ∘ j ⟹ f where e is cofaithful and j is fully faithful. (In the case of SCG, see [3].) In section 5, complexes in S and the relative 2-exactness are defined, generalizing those in SCG ([7]). Since we start from the self-dual definition, we can make good use of duality in the proofs. In section 6, as a main theorem, we construct a long cohomology 2-exact sequence from any short relatively 2-exact sequence (i.e. an extension) of complexes. Our proof is purely diagrammatic, and is an analogy of that for an abelian category. In section 5 and 6, several 2-dimensional diagram lemmas are shown. Most of them have 1-dimensional analogues in an abelian category, so we only have to be careful about the compatibility of 2-cells.

Since SCG is an example of a relatively exact 2-category, we expect some other 2categories constructed from SCG will be a relatively exact 2-category. For example, 𝔾-SMod, SCG × SCG and the 2-category of bifunctors from SCG are candidates. We will examine such examples in forthcoming papers.

2. Preliminaries

Definitions in a 2-category.

2.1. Notation. Throughout this paper, S denotes a 2-category (in the strict sense). We use the following notation. S^{0}, S^{1}, S^{2} : class of 0-cells, 1-cells, and 2-cells in S, respectively. S^{1}(A, B) : 1-cells from A to B, where A, B ∈ S^{0}. S^{2}(f, g) : 2-cells from f to g, where f, g ∈ S^{1}(A, B) for certain A, B ∈ S^{0}. S(A, B) : Hom-category between A and B (i.e. Ob(S(A, B)) = S^{1}(A, B), S(A, B)(f, g) = S^{2}(f, g)).

In diagrams, ⟶ represents a 1-cell, ⟹ represents a 2-cell, ∘ represents a horizontal composition, and · represents a vertical composition. We use capital letters A, B, … for 0-cells, small letters f, g, … for 1-cells, and Greek symbols α, β, … for 2-cells.

For example, one of the conditions in the definition of a 2-category can be written as follows (see for example [4]):

2.2. Remark. For any diagram in S

[figure: A; B; C; f_{1}; f_{2}; g_{1}; g_{2}; α; β] ,

we have

(f_{1} ∘ β) · (α ∘ g_{2}) = (α ∘ g_{1}) · (f_{2} ∘ β).  (1)

(Note: composition is always written diagrammatically.)

This equality is frequently used in later arguments.

Products, pullbacks, difference kernels and their duals are defined by the universality.

2.3. Definition. For any A_{1} and A_{2} ∈ S^{0}, their product (A_{1} × A_{2}, p_{1}, p_{2}) is defined as follows:

(a) A_{1} × A_{2} ∈ S^{0}, p_{i} ∈ S^{1}(A_{1} × A_{2}, A_{i}) (i = 1, 2). (b1) (existence of a factorization)

For any X ∈ S^{0} and q_{i} ∈ S^{1}(X, A_{i}) (i = 1, 2), there exist q ∈ S^{1}(X, A_{1} × A_{2}) and ξ_{i} ∈ S^{2}(q ∘ p_{i}, q_{i}) (i = 1, 2).

[figure: X; A_{1}; A_{1} × A_{2}; A_{2}; q_{1}; q; q_{2}; p_{1}; p_{2}; ξ_{1}; ξ_{2}]

(b2) (uniqueness of the factorization)

For any factorizations (q, ξ_{1}, ξ_{2}) and (q′, ξ′_{1}, ξ′_{2}) which satisfy (b1), there exists a unique 2-cell η ∈ S^{2}(q, q′) such that (η ∘ p_{i}) · ξ′_{i} = ξ_{i} (i = 1, 2).

[figure: q ∘ p_{i}; q′ ∘ p_{i}; q_{i}; η∘p_{i}; ξ_{i}; ξ′_{i}; ⟳]

The coproduct of A_{1}and A_{2} is defined dually.

2.4. Definition. For any A_{1}, A_{2}, B ∈ S^{0} and f_{i} ∈ S^{1}(A_{i}, B) (i = 1, 2), the pullback (A_{1} ×_{B} A_{2}, f′_{1}, f′_{2}, ξ) of f_{1} and f_{2} is defined as follows:

(a) A_{1} ×_{B} A_{2} ∈ S^{0}, f′_{1} ∈ S^{1}(A_{1} ×_{B} A_{2}, A_{2}), f′_{2} ∈ S^{1}(A_{1} ×_{B} A_{2}, A_{1}), ξ ∈ S^{2}(f′_{1} ∘ f_{2}, f′_{2} ∘ f_{1}).

[figure: A_{1} ×_{B} A_{2}; A_{2}; A_{1}; B; f′_{1}; f′_{2}; f_{1}; f_{2}; ξ]

(b1) (existence of a factorization)

For any X ∈ S^{0}, g_{1} ∈ S^{1}(X, A_{2}), g_{2} ∈ S^{1}(X, A_{1}) and η ∈ S^{2}(g_{1} ∘f_{2}, g_{2} ∘f_{1}), there exist g ∈ S^{1}(X, A_{1} × A_{2}), ξ_{i} ∈ S^{2}(g ∘ f′_{i}, g_{i}) (i = 1, 2) such that (ξ_{1} ∘ f_{2}) · η = (g ∘ ξ) · (ξ_{2} ∘ f_{1}).

[figure: A_{1} ×_{B} A_{2}; A_{2}; A_{1}; B; X; f′_{1}; f′_{2}; f_{1}; f_{2}; g; g_{1}; g_{2}; ξ; ξ_{1}; ξ_{2}; g ∘ f′_{1} ∘ f_{2}; g_{1} ∘ f_{2}; g ∘ f′_{2} ∘ f_{1}; g_{2} ∘ f_{1}; ξ_{1}∘f_{2}; g∘ξ; η; ξ_{2}∘f_{1}; ⟳]

(b2) (uniqueness of the factorization)

For any factorizations (g, ξ_{1}, ξ_{2}) and (g′, ξ′_{1}, ξ′_{2}) which satisfy (b1), there exists a unique 2-cell ζ ∈ S^{2}(g, g′) such that (ζ ∘ f′_{i}) · ξ′_{i} = ξ_{i} (i = 1, 2).

The pushout of f_{i} ∈ S^{1}(A, B_{i}) (i = 1, 2) is defined dually.

2.5. Definition. For any A, B ∈ S^{0} and f, g ∈ S^{1}(A, B), the difference kernel

(DK(f, g), d_{(f,g)}, φ_{(f,g)})

of f and g is defined as follows:

(a) DK(f, g) ∈ S^{0}, d_{(f,g)} ∈ S^{1}(DK(f, g), A), φ_{(f,g)} ∈ S^{2}(d_{(f,g)} ∘ f, d_{(f,g)} ∘ g).

[figure: DK(f, g); A; B; d_{(f,g)}; f; g] , [figure: DK(f, g); B; d_{(f,g)}∘f; d_{(f,g)}∘g; φ_{(f,g)}]

(b1) (existence of a factorization)

For any X ∈ S^{0}, d ∈ S^{1}(X, A), φ ∈ S^{2}(d∘f, d∘g), there exist \underline{d} ∈ S^{1}(X, DK(f, g)), \underline{φ} ∈ S^{2}(\underline{d} ∘ d_{(f,g)}, d) such that (\underline{d} ∘ φ_{(f,g)}) · (\underline{φ} ∘ g) = (\underline{φ} ∘ f) · φ. (b2) (uniqueness of the factorization)

For any factorizations (\underline{d}, \underline{φ}) and (\underline{d′}, \underline{φ′}) which satisfy (b1), there exists a unique 2-cell η ∈ S^{2}(\underline{d}, \underline{d′}) such that (η ∘ d_{(f,g)}) · \underline{φ′} = \underline{φ}.

The difference cokernel of f and g is defined dually.

The following definition is from [2].

2.6. Definition. Let f ∈ S^{1}(A, B).

(1) f is said to be faithful if f^{♭} := − ∘ f : S^{1}(C, A) → S^{1}(C, B) is faithful for any C ∈ S^{0}.

(2) f is said to be fully faithful if f^{♭} is fully faithful for any C ∈ S^{0}.

(3) f is said to be cofaithful if f^{♯} := f ∘ − : S^{1}(B, C) → S^{1}(A, C) is faithful for any C ∈ S^{0}.

(4) f is said to be fully cofaithful if f^{♯} is fully faithful for any C ∈ S^{0}.

Properties of SCG. By definition, a symmetric categorical group is a symmetric monoidal category (𝔾, ⊗, 0), in which each arrow is an isomorphism and each object has an inverse up to an equivalence with respect to the tensor ⊗. More precisely;

2.7. Definition. A symmetric categorical group (𝔾, ⊗, 0) consists of (a1) a category 𝔾 (a2) a tensor functor ⊗ : 𝔾 × 𝔾 → 𝔾 (a3) a unit object 0 ∈ Ob(𝔾) (a4) natural isomorphisms

α_{A,B,C} : A ⊗ (B ⊗ C) → (A ⊗ B) ⊗ C,
λ_{A} : 0 ⊗ A → A, ρ_{A} : A ⊗ 0 → A, γ_{A,B} : A ⊗ B → B ⊗ A

which satisfy certain compatibility conditions (cf. [5]), and the following two conditions are satisfied: (b1) For any A, B ∈ Ob(𝔾) and f ∈ 𝔾(A, B), there exists g ∈ 𝔾(B, A) such that f ∘ g = id_{A}, g ∘ f = id_{B}. (b2) For any A ∈ Ob(𝔾), there exist A^{∗} ∈ Ob(𝔾) and η_{A} ∈ 𝔾(0, A ⊗ A^{∗}).

In particular, there is a ‘zero categorical group’ 0, which consists of only one object 0 and one morphism id_{0}.

2.8. Definition. For symmetric categorical groups 𝔾 and ℍ, a monoidal functor F from 𝔾 to ℍ consists of (a1) a functor F : 𝔾 → ℍ (a2) natural isomorphisms

F_{A,B} : F (A ⊗ B) → F (A) ⊗ F (B) and F_{I} : F (0) → 0

which satisfy certain compatibilities with α, λ, ρ, γ. (cf. [5])

2.9. Remark. For any monoidal functors F : 𝔾 → ℍ and G : ℍ → 𝕂, their composition F ∘ G : 𝔾 → 𝕂 is defined by

(F ∘ G)_{A,B} := G(F_{A,B}) ∘ G_{F (A),F (B)}  (2)
(F ∘ G)_{I} := G(F_{I}) ∘ G_{I}.  (3)

In particular, there is a ‘zero monoidal functor’ 0_{𝔾,ℍ} : 𝔾 → ℍ for each 𝔾 and ℍ, which sends every object in 𝔾 to 0_{ℍ}, every arrow in 𝔾 to id_{0_{ℍ}}, and (0_{𝔾,ℍ})_{A,B} = λ_{0}^{−1} = ρ_{0}^{−1}, (0_{𝔾,ℍ})_{I} = id_{0}. It is easy to see that 0_{𝔾,ℍ} ∘ 0_{ℍ,𝕂} = 0_{𝔾,𝕂} (∀𝔾, ℍ, 𝕂).

2.10. Remark. Our notion of a monoidal functor is equal to that of a ‘γ-monoidal functor’ in [7].

2.11. Definition. For monoidal functors F, G : 𝔾 → ℍ, a natural transformation φ from F to G is said to be a monoidal transformation if it satisfies

φ_{A⊗B} ∘ G_{A,B} = F_{A,B} ∘ (φ_{A} ⊗ φ_{B})
F_{I} = φ_{0} ∘ G_{I}.  (4)

The following remark is from [9].

2.12. Remark. By condition (b2), it is shown that there exists a 2-cell ε_{A} ∈ 𝔾(A^{∗}⊗A, 0) for each object A, such that the following compositions are identities:

A ⟶_{λ_{A}^{−1}} 0 ⊗ A ⟶_{η_{A}⊗1} (A ⊗ A^{∗}) ⊗ A ⟶_{α^{−1}} A ⊗ (A^{∗} ⊗ A) ⟶_{1⊗ε_{A}} A ⊗ 0 ⟶_{ρ_{A}} A

A^{∗} ⟶_{ρ_{A^{∗}}^{−1}} A^{∗} ⊗ 0 ⟶_{1⊗η_{A}} A^{∗} ⊗ (A ⊗ A^{∗}) ⟶_{α} (A^{∗} ⊗ A) ⊗ A^{∗} ⟶_{ε_{A}⊗1} 0 ⊗ A^{∗} ⟶_{λ_{A^{∗}}} A^{∗}

For each monoidal functor F : 𝔾 → ℍ, there exists a natural morphism ι_{F,A} : F (A^{∗}) → F (A)^{∗}.

2.13. Definition. SCG is defined to be the 2-category whose 0-cells are symmetric categorical groups, 1-cells are monoidal functors, and 2-cells are monoidal transformations.

The following two propositions are satisfied in SCG (see for example [1]).

2.14. Proposition. For any symmetric categorical groups 𝔾 and ℍ, if we define a monoidal functor F ⊗_{𝔾,ℍ} G : 𝔾 → ℍ by

F ⊗_{𝔾,ℍ} G(A)  :=  F (A) ⊗_{ℍ} G(A)
(F ⊗_{𝔾,ℍ} G)_{A,B}  :=  (F (A ⊗ B) ⊗ G(A ⊗ B)
⟶^{F_{A,B}⊗G_{A,B}} F (A) ⊗ F (B) ⊗ G(A) ⊗ G(B)
⟶^{≃} F (A) ⊗ G(A) ⊗ F (B) ⊗ G(B))
(F ⊗_{𝔾,ℍ} G)_{I}  :=  (F (I) ⊗ G(I) ⟶^{F_{I}⊗G_{I}} I ⊗ I ⟶^{≃} I),

then (SCG(𝔾, ℍ), ⊗_{𝔾,ℍ}, 0_{𝔾,ℍ}) becomes again a symmetric categorical group with appropriately defined α, λ, ρ, γ, and

Hom = SCG(−, −) : SCG × SCG → SCG

becomes a 2-functor (cf. section 6 in [1]).

In SCG, by definition of the zero categorical group we have S^{1}(𝔾, 0) = {0_{𝔾,0}}, while S^{1}(0, 𝔾) may have more than one objects. In this point SCG might be said to have ‘non self-dual’ structure, but S^{1}(𝔾, 0) and S^{1}(0, 𝔾) have the following ‘self-dual’ property.

2.15. Remark. (1) For any symmetric categorical group 𝔾 and any monoidal functor F : 𝔾 → 0, there exists a unique 2-cell φ : F ⟹ 0_{𝔾,0}.

(2) For any symmetric categorical group 𝔾 and any monoidal functor F : 0 → 𝔾, there exists a unique 2-cell φ : F ⟹ 0_{0,𝔾}.

Proof. (1) follows from the fact that the zero categorical group has only one morphism id_{0}. (2) follows from condition (4) in Definition 2.11.

The usual compatibility arguments show the following Lemma.

2.16. Lemma. Let F : 𝔾 → ℍ be a monoidal functor. For any A, B ∈ Ob(𝔾),

[figure: Φ_{A,B} :; 𝔾(A, B); 𝔾(A ⊗ B^{∗}, 0); ∈; ∈; f; (f ⊗ 1_{B^{∗}}) ∘ η_{B}^{−1}]

and

[figure: Ψ_{A,B} :; 𝔾(A ⊗ B^{∗}, 0); 𝔾(A, B); ∈; ∈; g; ρ_{A}^{−1} ∘ (1_{A} ⊗ ε_{B}^{−1}) ∘ α_{A,B^{∗},B} ∘ (g ⊗ 1_{B}) ∘ λ_{B}]

are mutually inverse, and the following diagram is commutative;

[figure: 𝔾(A, B); 𝔾(A ⊗ B^{∗}, 0); ℍ(F (A), F (B)); ℍ(F (A) ⊗ F (B)^{∗}, 0); ℍ(F (A ⊗ B^{∗}), F (0)),; Φ_{A,B}; F; F; ⟳; Φ_{F (A),F (B)}; Θ_{A,B}^{F}]

where Θ_{A,B}^{F} is defined by

[figure: Θ_{A,B}^{F} :; ℍ(F (A ⊗ B^{∗}), F (0)); ℍ(F (A) ⊗ F (B)^{∗}, 0); ∈; ∈; h; (1_{F (A)} ⊗ (ι_{B}^{F})^{−1}) ∘ (F_{A,B^{∗}})^{−1} ∘ h ∘ F_{I}.]

3. Definition of a relatively exact 2-category

Locally SCG 2-category. We define a locally SCG 2-category not only as a 2-category whose Hom-categories are SCG, but with some more conditions, in order to let it be a 2-dimensional analogue of that of an additive category.

3.1. Definition. A locally small 2-category S is said to be locally SCG if the following conditions are satisfied: (A1) For every A, B ∈ S^{0}, there is a given functor ⊗_{A,B} : S(A, B) × S(A, B) → S(A, B), and a given object 0_{A,B} ∈ Ob(S(A, B)) = S^{1}(A, B) such that (S(A, B), ⊗_{A,B}, 0_{A,B}) becomes a symmetric categorical group, and the following naturality conditions are satisfied:

0_{A,B} ∘ 0_{B,C} = 0_{A,C} (∀A, B, C ∈ S^{0})

(A2) Hom = S(−, −) : S × S → SCG is a 2-functor which satisfies for any A, B, C ∈ S^{0},

(0_{A,B})_{I}^{♯} = id_{0_{A,C}} ∈ S^{2}(0_{A,C}, 0_{A,C})  (5)
(0_{A,B})_{I}^{♭} = id_{0_{C,B}} ∈ S^{2}(0_{C,B}, 0_{C,B}).  (6)

(A3) There is a 0-cell 0 ∈ S^{0} called a zero object, which satisfy the following conditions: (a3-1) S(0, 0) is the zero categorical group. (a3-2) For any A ∈ S^{0} and f ∈ S^{1}(0, A), there exists a unique 2-cell θ_{f} ∈ S^{2}(f, 0_{0,A}). (a3-3) For any A ∈ S^{0} and f ∈ S^{1}(A, 0), there exists a unique 2-cell τ_{f} ∈ S^{2}(f, 0_{A,0}). (A4) For any A, B ∈ S^{0}, their product and coproduct exist.

Let us explain about these conditions.

3.2. Remark. By condition (A1) of Definition 3.1, every 2-cell in a locally SCG 2category becomes invertible, as in the case of SCG (cf. [9]). This helps us to avoid being fussy about the directions of 2-cells in many propositions and lemmas, and we use the word ‘dual’ simply to reverse 1-cells.

3.3. Remark. By condition (A2) in Definition 3.1,

f^{♯} := f ∘ − : S(B, C) → S(A, C)
f^{♭} := − ∘ f : S(C, A) → S(C, B)

are monoidal functors (∀C ∈ S^{0}) for any f ∈ S^{1}(A, B), and the following naturality conditions are satisfied: (a2-1) For any f ∈ S^{1}(A, B), g ∈ S^{1}(B, C) and D ∈ S^{0}, we have (f ∘ g)^{♯} = g^{♯} ∘ f^{♯} as monoidal functors.

[figure: A; B; C; D; f; g; S(C, D); S(B, D); S(A, D); g^{♯}; (f∘g)^{♯}; f^{♯}; ⟳]

(a2-2) The dual of (a2-1) for −^{♭}. (a2-3) For any f ∈ S^{1}(A, B), g ∈ S^{1}(C, D), we have f^{♯} ∘ g^{♭} = g^{♭} ∘ f^{♯} as monoidal functors.

[figure: A; B; C; D; f; g; S(B, C); S(A, C); S(B, D); S(A, D); f^{♯}; g^{♭}; g^{♭}; f^{♯}; ⟳]

Since already (f ∘ g)^{♯} = g^{♯} ∘ f^{♯} as functors, (a2-1) means (f ∘ g)_{I}^{♯} = (g^{♯} ∘ f^{♯})_{I}, and by (3) in Remark 2.9, this is equivalent to

(f ∘ g)_{I}^{♯} = f^{♯}(g_{I}^{♯}) · f_{I}^{♯} = (f ∘ g_{I}^{♯}) · f_{I}^{♯}.

Similarly, we obtain

(f ∘ g)_{I}^{♭} = (f_{I}^{♭} ∘ g) · g_{I}^{♭},  (7)
(f_{I}^{♯} ∘ g) · g_{I}^{♭} = (f ∘ g_{I}^{♭}) · f_{I}^{♯}.  (8)

3.4. Remark. By condition (A2), for any f, g ∈ S^{1}(A, B) and any α ∈ S^{2}(f, g), α ∘ − : f^{♯} ⇒ g^{♯} becomes a monoidal transformation. So, the diagrams

[figure: f ∘ (k ⊗ h); (f ∘ k) ⊗ (f ∘ h); g ∘ (k ⊗ h); (g ∘ k) ⊗ (g ∘ h); α∘(k⊗h); f_{k,h}^{♯}; (α∘k)⊗(α∘h); g_{k,h}^{♯}; ⟳; and; f ∘ 0_{B,C}; g ∘ 0_{B,C}; 0_{A,C}; α∘0_{B,C}; f_{I}^{♯}; g_{I}^{♯}; ⟳]

are commutative for any C ∈ S^{0} and k, h ∈ S^{1}(B, C). Similar statement also holds for − ∘ α : f^{♭} ⇒ g^{♭}.

3.5. Corollary. In a locally SCG 2-category S, the following are satisfied: (1) For any diagram in S

[figure: C; A; B; h; 0_{C,A}; f; g; ε; α]

we have

h ∘ α = (ε ∘ f) · f_{I}^{♭} · g_{I}^{♭−1} · (ε^{−1} ∘ g).  (9)

(2) For any diagram in S

[figure: A; B; C; f; g; h; 0_{B,C}; α; ε] ,

we have

α ∘ h = (f ∘ ε) · f_{I}^{♯} · g_{I}^{♯−1} · (g ∘ ε^{−1}).  (10)

(3) For any diagram in S

[figure: A; B; C; f; 0_{A,B}; g; 0_{B,C}; α; β] ,

we have

(f ∘ β) · f_{I}^{♯} = (α ∘ g) · g_{I}^{♭}.  (11)

Proof. (1) (h ∘ α) =_{1} (ε ∘ f) · (0_{C,A} ∘ α) · (ε^{−1} ∘ g) = (ε ∘ f) · f_{I}^{♭} · g_{I}^{♭−1} · (ε^{−1} ∘ g). (2) is the

dual of (1). And (3) follows from (5), (6), (9), (10).

3.6. Remark. We don’t require a locally SCG 2-category to satisfy S^{1}(A, 0) = {0_{A,0}}, for the sake of duality (see the comments before Remark 2.15 ).

Relatively exact 2-category.

3.7. Definition. Let S be a locally SCG 2-category. S is said to be relatively exact if the following conditions are satisfied:

(B1) For any 1-cell f ∈ S^{1}(A, B), its kernel and cokernel exist.

(B2) For any 1-cell f ∈ S^{1}(A, B), f is faithful if and only if f = ker(cok(f)).

(B3) For any 1-cell f ∈ S^{1}(A, B), f is cofaithful if and only if f = cok(ker(g)). It is shown in [9] that SCG satisfies these conditions.

Let us explain about these conditions.

3.8. Definition. Let S be a locally SCG 2-category. For any f ∈ S^{1}(A, B), its kernel (Ker(f), ker(f), ε_{f}) is defined by universality as follows (we abbreviate ker(f) to k(f)) :

(a) Ker(f) ∈ S^{0}, k(f) ∈ S^{1}(Ker(f), A), ε_{f} ∈ S^{2}(k(f) ∘ f, 0). (b1) (existence of a factorization)

For any K ∈ S^{0}, k ∈ S^{1}(K, A) and ε ∈ S^{2}(k ∘ f, 0), there exist \underline{k} ∈ S^{1}(K, Ker(f)) and \underline{ε} ∈ S^{2}(\underline{k} ∘ k(f), k) such that (\underline{ε} ∘ f) · ε = (\underline{k} ∘ ε_{f}) · (\underline{k})_{I}^{♯}.

[figure: Ker(f); K; A; B; k(f); k; \underline{k}; f; 0; 0; ε; ε_{f}; \underline{ε}]

(b2) (uniqueness of the factorization)

For any factorizations (\underline{k}, \underline{ε}) and (\underline{k}′, \underline{ε}′) which satisfy (b1), there exists a unique 2-cell ξ ∈ S^{2}(\underline{k}, \underline{k}′) such that (ξ ∘ k(f)) · \underline{ε}′ = \underline{ε}.

3.9. Remark. (1) By its universality, the kernel of f is unique up to an equivalence. We write this equivalence class again Ker(f) = [Ker(f), k(f), ε_{f}].

(2) It is also easy to see that if f and f′ are equivalent, then

[Ker(f), k(f), ε_{f}] = [Ker(f′), k(f′), ε_{f′}].

For any f, its cokernel Cok(f) = [Cok(f), c(f), π_{f}] is defined dually, and the dual statements also hold for the cokernel.

3.10. Remark. Let S be a locally SCG 2-category, and let f be in S^{1}(A, B). For any pair (k, ε) with k ∈ S^{1}(0, A), ε ∈ S^{2}(k ∘ f, 0)

[figure: 0; A; B; k; f; 0; ε]

and for any pair (k′, ε′) with k′ ∈ S^{1}(0, A), ε′ ∈ S^{2}(k ∘ f, 0), there exists a unique 2-cell ξ ∈ S^{2}(k, k′) such that (ξ ∘ f) · ε′ = ε.

Proof. By condition (a3-2) of Definition 3.1, ε ∈ S^{2}(k ∘f, 0) must be equal to the unique 2-cell (θ_{k} ∘ f) · f_{I}^{♭}. Similarly we have ε′ = (θ_{k′} ∘ f) · f_{I}^{♭}, and, ξ should be the unique 2-cell θ_{k} · θ_{k′}^{−1} ∈ S^{2}(k, k′), which satisfies (ξ ∘ f) · ε′ = ε.

From this, it makes no ambiguity if we abbreviate Ker(f) = [0, 0_{0,A}, f_{I}^{♭}] to Ker(f) = 0, because [0, k, ε] = [0, k′, ε′] for any (k, ε) and (k′, ε′). Dually, we abbreviate Cok(f) = [0, 0_{A,0}, f_{I}^{♯}] to Cok(f) = 0.

By using condition (A3) of Definition 3.1, we can show the following easily:

3.11. Example. (1) For any A ∈ S^{0}, Ker(0_{A,0} : A → 0) = [A, id_{A}, id_{0}].

(2) For any A ∈ S^{0}, Cok(0_{0,A} : 0 → A) = [A, id_{A}, id_{0}].

3.12. Caution. (1) Ker(0_{0,A} : 0 → A) need not be equivalent to 0. Indeed, in the case of SCG, for any symmetric categorical group 𝔾, Ker(0_{0,𝔾} : 0 → 𝔾) is equivalent to an important invariant π_{1}(𝔾)[0].

(2) Cok(0_{A,0} : A → 0) need not be equivalent to 0 either. In the case of SCG, Cok(0_{𝔾,0} : 𝔾 → 0) is equivalent to π_{0}(𝔾)[1].

3.13. Remark. The precise meaning of condition (B2) in Definition 3.7 is that, for any 1-cell f ∈ S^{1}(A, B) and its cokernel [Cok(f), cok(f), π_{f}], f is faithful if and only if Ker(cok(f)) = [A, f, π_{f}]. Similarly for condition (B3).

Relative (co-)kernel and first properties of a relatively exact 2-category. Throughout this subsection, S is a relatively exact 2-category.

3.14. Definition. For any diagram in S

[figure: A; B; C; f; g; 0; φ] ,  (12)

its relative kernel (Ker(f, φ), ker(f, φ), ε_{(f,φ)}) is defined as follows (we abbreviate ker(f, φ) to k(f, φ)) :

(a) Ker(f, φ) ∈ S^{0}, k(f, φ) ∈ S^{1}(Ker(f, φ), A), ε_{(f,φ)} ∈ S^{2}(k(f, φ) ∘ f, 0). (b0) (compatibility of the 2-cells)

ε_{(f,φ)} is compatible with φ i.e. (k(f, φ) ∘ φ) · k(f, φ)_{I}^{♯} = (ε_{(f,φ)} ∘ g) · g_{I}^{♭}. (b1) (existence of a factorization)

For any K ∈ S^{0}, k ∈ S^{1}(K, A) and ε ∈ S^{2}(k ∘ f, 0) which are compatible with φ, i.e. (k ∘ φ) · k_{I}^{♯} = (ε ∘ g) · g_{I}^{♭}, there exist \underline{k} ∈ S^{1}(K, Ker(f, φ)) and \underline{ε} ∈ S^{2}(\underline{k} ∘ k(f, φ), k) such that (\underline{ε} ∘ f) · ε = (\underline{k} ∘ ε_{(f,φ)}) · (\underline{k})_{I}^{♯}.

[figure: Ker(f, φ); K; A; B; C; k(f,φ); k; \underline{k}; f; g; 0; 0; 0; φ; ε; ε_{(f,φ)}; \underline{ε}]

(b2) (uniqueness of the factorization)

For any factorizations (\underline{k}, \underline{ε}) and (\underline{k}′, \underline{ε}′) which satisfy (b1), there exists a unique 2-cell ξ ∈ S^{2}(\underline{k}, \underline{k}′) such that (ξ ∘ k(f, φ)) · \underline{ε}′ = \underline{ε}.

3.15. Remark. By its universality, the relative kernel of (f, φ) is unique up to an equivalence. We write this equivalence class [Ker(f, φ), k(f, φ), ε_{(f,φ)}].

3.16. Definition. Let S be a relatively exact 2-category. For any diagram (12) in S, its relative cokernel (Cok(g, φ), cok(g, φ), π_{(g,φ)}) is defined dually by universality. We abbreviate cok(g, φ) to c(g, φ), and write the equivalence class of the relative cokernel [Cok(g, φ), c(g, φ), π_{(g,φ)}].

3.17. Caution. In the rest of this paper, S denotes a relatively exact 2-category, unless otherwise specified. In the following propositions and lemmas, we often omit the statement and the proof of their duals. Each term should be replaced by its dual. For example, kernel and cokernel, faithfulness and cofaithfulness are mutually dual.

3.18. Remark. By using condition (A3) of Definition 3.1, we can show the following easily. (These are also corollaries of Proposition 3.20.)

(1) Ker(f, f_{I}^{♯}) = Ker(f) (and thus the ordinary kernel can be regarded as a relative kernel).

[figure: A; B; 0; f; 0; 0; f_{I}^{♯}]

(2) ker(f, φ) is faithful.

3.19. Lemma. Let f ∈ S^{1}(A, B) and take its kernel [Ker(f), k(f), ε_{f}]. If K ∈ S^{0}, k ∈ S^{1}(K, Ker(f)) and σ ∈ S^{2}(k ∘ k(f), 0)

[figure: K; Ker(f); A; B; k; k(f); f; 0; 0; ε_{f}; σ]

is compatible with ε_{f}, i.e. if σ satisfies

(σ ∘ f) · f_{I}^{♭} = (k ∘ ε_{f}) · k_{I}^{♯},  (13)

then there exists a unique 2-cell ζ ∈ S^{2}(k, 0) such that σ = (ζ ∘ k(f)) · k(f)_{I}^{♭}.

Proof. By (13), σ : k ∘ k(f) ⟹ 0 is a factorization compatible with ε_{f} and f_{I}^{♭}. On the other hand, by Remark 3.4, k(f)_{I}^{♭} : 0 ∘ k(f) ⇒ 0 is also a factorization compatible with ε_{f}, f_{I}^{♭}. So, by the universality of the kernel, there exists a unique 2-cell ζ ∈ S^{2}(k, 0) such that σ = (ζ ∘ k(f)) · k(f)_{I}^{♭}.

It is easy to see that the same statement also holds for relative (co-)kernels. In any relatively exact 2-category, the relative (co-)kernel always exist. More precisely, the following proposition holds.

3.20. Proposition. Consider diagram (12) in S. By the universality of Ker(g) = [Ker(g), ℓ, ε], f factors through ℓ uniquely up to an equivalence as \underline{φ} : \underline{f} ∘ ℓ ⟹ f, where \underline{f} ∈ S^{1}(A, Ker(g)) and \underline{φ} ∈ S^{2}(\underline{f} ∘ ℓ, f) :

(\underline{f} ∘ ε) · (\underline{f})_{I}^{♯} = (\underline{φ} ∘ g) · φ [figure: Ker(\underline{f}); Ker(g); A; B; C; ℓ; \underline{f}; k(\underline{f}); f; f; g; 0; 0; 0; φ; \underline{φ}; ε\underline{f}; ε]

Then we have Ker(f, φ) = [Ker(\underline{f}), k(\underline{f}), η], where η := (k(\underline{f}) ∘ \underline{φ}^{−1}) · (ε\underline{f} ∘ ℓ) · ℓ_{I}^{♭} ∈ S^{2}(k(\underline{f}) ∘ f, 0). We abbreviate this to Ker(f, φ) = Ker(\underline{f}).

Proof. For any K ∈ S^{0}, k ∈ S^{1}(K, A) and σ ∈ S^{2}(k ∘ f, 0) which are compatible with φ, i.e. (σ ∘ g) · g_{I}^{♭} = (k ∘ φ) · k_{I}^{♯}, if we put

ρ := (k ∘ \underline{φ}) · σ ∈ S^{2}(k ∘ \underline{f} ∘ ℓ, 0),

then ρ is compatible with ε. By Lemma 3.19, there exists a 2-cell ζ : k ∘ \underline{f} ⇒ 0 such that ρ = (ζ ∘ ℓ) · ℓ_{I}^{♭}. So, by the universality of Ker(\underline{f}), there exist \underline{k} ∈ S^{1}(K, Ker(\underline{f})) and \underline{σ} ∈ S^{2}(\underline{k} ∘ k(\underline{f}), k) such that (\underline{σ} ∘ \underline{f}) · ζ = (\underline{k} ∘ ε\underline{f}) · (\underline{k})_{I}^{♯}. Then, \underline{σ} is compatible with σ and η,

[figure: Ker(\underline{f}); K; A; B; k(\underline{f}); k; \underline{k}; f; 0; 0; σ; η; \underline{σ}]

and the existence of a factorization is shown. To show the uniqueness of the factorization, let (\underline{k}′, \underline{σ}′) be another factorization which is compatible with σ, η, i.e. (\underline{σ}′ ∘ f) · σ = (\underline{k}′∘η)·(\underline{k}′)_{I}^{♯}. Then, by using η = (k(\underline{f})∘\underline{ε}^{−1})·(ε\underline{f} ∘ℓ)·ℓ_{I}^{♭} and ζ ∘ℓ = ρ·ℓ_{I}^{♭−1} = (k∘\underline{ε})·σ·ℓ_{I}^{♭−1}, we can show ((\underline{σ}′ ∘ \underline{f}) · ζ) ∘ ℓ = ((\underline{k}′ ∘ ε\underline{f}) · (\underline{k}′)_{I}^{♯}) ∘ ℓ. Since ℓ is faithful, we obtain ((\underline{σ}′ ∘ \underline{f}) · ζ) = (\underline{k}′ ∘ ε\underline{f}) · (\underline{k}′)_{I}^{♯}. Thus, \underline{σ}′ is compatible with ζ and ε\underline{f}. By the universality of Ker(\underline{f}), there exists a 2-cell ξ ∈ S^{2}(\underline{k}, \underline{k}′) such that (ξ ∘ k(\underline{f})) · \underline{σ}′ = \underline{σ}. Uniqueness of such ξ ∈ S^{2}(\underline{k}, \underline{k}′) follows from the faithfulness of k(\underline{f}).

3.21. Proposition. Let f ∈ S^{1}(A, B), g ∈ S^{1}(B, C) and suppose g is fully faithful. Then, Ker(f ∘ g) = [Ker(f), k(f), (ε_{f} ∘ g) · g_{I}^{♭}]. We abbreviate this to Ker(f ∘ g) = Ker(f).

Proof. Since g is fully faithful, for any K ∈ S^{0}, k ∈ S^{1}(K, A) and σ ∈ S^{2}(k ∘ f ∘ g, 0), there exists ρ ∈ S^{2}(k ∘ f, 0) such that σ = (ρ ∘ g) · g_{I}^{♭}. And by the universality of Ker(f), there are \underline{k} ∈ S^{1}(K, Ker(f)) and \underline{σ} ∈ S^{2}(\underline{k} ∘ k(f), k) such that (\underline{σ} ∘ f) · ρ = (\underline{k} ∘ ε_{f}) · (\underline{k})_{I}^{♯}. Then, it can be easily seen that \underline{σ} is compatible with σ and (ε_{f} ∘ g) · g_{I}^{♭}:

(\underline{σ} ∘ f ∘ g) · σ = (\underline{k} ∘ ((ε_{f} ∘ g) · g_{I}^{♭})) · (\underline{k})_{I}^{♯} [figure: Ker(f); K; A; C; k(f); k; \underline{k}; f∘g; 0; 0; σ; \underline{σ}; (ε_{f} ∘g)·g_{I}^{♭}]

Thus we obtain a desired factorization. To show the uniqueness of the factorization, let (\underline{k}′, \underline{σ}′) be another factorization of k which satisfies

(\underline{σ}′ ∘ f ∘ g) · σ = (\underline{k}′ ∘ ((ε_{f} ∘ g) · g_{I}^{♭})) · (\underline{k}′)_{I}^{♯}

Then, we can show \underline{σ}′ is compatible with ρ and ε_{f}. By the universality of Ker(f), there exists a 2-cell ξ ∈ S^{2}(\underline{k}, \underline{k}′) such that (ξ ∘ k(f)) · \underline{σ}′ = \underline{σ}. Uniqueness of such ξ follows from the faithfulness of k(f).

By definition, f ∈ S^{1}(A, B) is faithful (resp. fully faithful) if and only if − ∘ f : S^{2}(g, h) → S^{2}(g ∘ f, h ∘ f) is injective (resp. bijective) for any K ∈ S^{0} and any g, h ∈ S^{1}(K, A). Concerning this, we have the following lemma.

3.22. Lemma. Let f ∈ S^{1}(A, B).

(1) f is faithful if and only if for any K ∈ S^{0} and k ∈ S^{1}(K, A),

− ∘ f : S^{2}(k, 0) → S^{2}(k ∘ f, 0 ∘ f) is injective.

(2) f is fully faithful if and only if for any K ∈ S^{0} and k ∈ S^{1}(K, A),

− ∘ f : S^{2}(k, 0) → S^{2}(k ∘ f, 0 ∘ f) is bijective.

Proof. By Lemma 2.16, we have the following commutative diagram for any g, h ∈ S^{1}(K, A):

[figure: S^{2}(g, h); S^{2}(g ⊗ h^{∗}, 0); S^{2}(g ∘ f, h ∘ f); S^{2}((g ∘ f) ⊗ (h ∘ f)^{∗}, 0); S^{2}((g ⊗ h^{∗}) ∘ f, 0 ∘ f); bij.; Φ_{g,h}; −∘f; −∘f; ⟳; bij.; Φ_{g∘f,h∘f}; bij.; Θ_{g,h}^{f^{♭}}; −∘f]

So we have

− ∘ f : S^{2}(g, h) → S^{2}(g ∘ f, h ∘ f) is injective (resp.bijective)
⇔ − ∘ f : S^{2}(g ⊗ h^{∗}, 0) → S^{2}((g ⊗ h^{∗}) ∘ f, 0 ∘ f) is injective (resp.bijective).

3.23. Corollary. For any f ∈ S^{1}(A, B), f is faithful if and only if the following condition is satisfied:

α ∘ f = id_{0∘f} ⇒ α = id_{0} (∀K ∈ S^{0}, ∀α ∈ S^{2}(0_{K,A}, 0_{K,A}))  (14)

Proof. If f is faithful, (14) is trivially satisfied, since we have id_{0∘f} = id_{0} ∘ f. To show the other implication, by Lemma 3.22, it suffices to show that − ∘ f : S^{2}(k, 0) → S^{2}(k ∘ f, 0 ∘ f) is injective. For any α_{1}, α_{2} ∈ S^{2}(k, 0) which satisfy α_{1} ∘f = α_{2} ∘ f, we have (α_{1}^{−1} · α_{2}) ∘ f = (α_{1} ∘ f)^{−1} · (α_{2} ∘ f) = id_{0∘f}. From the assumption we obtain α_{1}^{−1} · α_{2} = id_{0}, i.e. α_{1} = α_{2}.

The next corollary immediately follows from Lemma 3.22.

3.24. Corollary. For any f ∈ S^{1}(A, B), f is fully faithful if and only if for any K ∈ S^{0}, k ∈ S^{1}(K, A), and any σ ∈ S^{2}(k ∘ f, 0), there exists unique τ ∈ S^{2}(k, 0) such that σ = (τ ∘ f) · f_{I}^{♭}.

3.25. Corollary. For any f ∈ S^{1}(A, B), the following are equivalent:

(1) f is fully faithful.

(2) Ker(f) = 0.

Proof. \underline{(1)⇒(2)}

Since f is fully faithful, for any k ∈ S^{1}(K, A) and ε ∈ S^{2}(k ∘ f, 0), there exists a 2-cell \underline{ε} ∈ S^{2}(0_{K,A}, k) such that (\underline{ε} ∘ f) = (0 ∘ f_{I}^{♭}) · 0_{I}^{♯} · ε^{−1} = (0 ∘ f_{I}^{♭}) · ε^{−1}, and the existence of a factorization is shown. To show the uniqueness of the factorization, it suffices to show that for any other factorization (\underline{k}′, \underline{ε}′) with (\underline{ε}′ ∘ f) · ε = (\underline{k}′ ∘ f_{I}^{♭}) · (\underline{k}′)_{I}^{♯}, the unique 2-cell τ ∈ S^{2}(\underline{k}′, 0) (see condition (a3-2) in Definition 3.1) satisfies (τ ∘ 0) · \underline{ε} = \underline{ε}′. Since f is faithful, this is equivalent to (τ ∘ 0 ∘ f) · (\underline{ε} ∘ f) · ε = (\underline{ε}′ ∘ f) · ε, and this follows easily from τ ∘ 0 = (τ ∘ 0) · 0_{I}^{♯} = (\underline{k}′)_{I}^{♯} and (τ ∘ 0 ∘ f) · (0 ∘ f_{I}^{♭}) = (\underline{k}′ ∘ f_{I}^{♭}) · (τ ∘ 0). (see Corollary 3.5.) \underline{(2)⇒(1)} Since Ker(f) = [0, 0, f_{I}^{♭}], for any K ∈ S^{0}, k ∈ S^{1}(K, A) and any σ ∈ S^{2}(k ∘ f, 0), there exist \underline{k} ∈ S^{1}(K, 0) and \underline{σ} ∈ S^{2}(\underline{k} ∘ 0, k) such that (\underline{σ} ∘ f) · σ = (\underline{k} ∘ f_{I}^{♭}) · (\underline{k})_{I}^{♯}. Thus τ := \underline{σ}^{−1} · \underline{k}_{I}^{♯} satisfies σ = (τ ∘ f) · f_{I}^{♭}. If there exists another τ′ ∈ S^{2}(k, 0) satisfying σ = (τ′ ∘ f) · f_{I}^{♭}, then by the universality of the kernel, there exists υ ∈ S^{2}(\underline{k}, 0) such that (υ ∘ 0) · τ^{′−1} = \underline{τ}. Since υ ∘ 0 = \underline{k}_{I}^{♯} by (11), we obtain τ = τ′. Thus τ is uniquely determined.

3.26. Proposition. For any f ∈ S^{1}(A, B), the following are equivalent.

(1) f is an equivalence.

(2) f is cofaithful and fully faithful.

(3) f is faithful and fully cofaithful.

Proof. Since (1)⇔(3) is the dual of (1)⇔(2), we show only (1)⇔(2). (1)⇒(2) : trivial. (2)⇒(1) : Since f is cofaithful, we have f = cok(ker(f)), Cok(k(f)) = [B, f, ε_{f}]. On the other hand, since f is fully faithful, we have Ker(f) = [0, 0, f_{I}^{♭}], and so we have Cok(k(f)) = [A, id_{A}, id_{0}]. And by the uniqueness (up to an equivalence) of the cokernel, there is an equivalence from A to B, which is equivalent to f. Thus, f becomes an equivalence.

[figure: 0; A; A; B; _{id_{A}}; k(f)=0_{0,A}; f; ∃equiv.; 0; ε_{f}]

3.27. Lemma. Let f : A → B be a faithful 1-cell in S. Then, for any K ∈ S^{0} and k ∈ S^{1}(K, 0), we have S^{2}(k ∘ 0_{0,Ker(f)}, 0_{K,Ker(f)}) = {k_{I}^{♯}}.

[figure: K; 0; Ker(f); k; 0_{0,Ker(f)}; 0_{K,Ker(f)}; k_{I}^{♯}]

Proof. For any σ ∈ S^{2}(k ∘ 0_{0,Ker(f)}, 0_{K,Ker(f)}), we can show ((σ ∘ k(f)) · k(f)_{I}^{♭}) ∘ f = ((k ∘ k(f)_{I}^{♭}) · k_{I}^{♯}) ∘ f. By the faithfulness of f, we have (σ ∘ k(f)) · k(f)_{I}^{♭} = (k ∘ k(f)_{I}^{♭}) · k_{I}^{♯}. Thus, we have σ ∘ k(f) = k_{I}^{♯} ∘ k(f). By the faithfulness of k(f), we obtain σ = k_{I}^{♯}.

3.28. Corollary. f : A → B is faithful if and only if Ker(0_{0,A}, f_{I}^{♭}) = 0.

Proof. Since there is a factorization diagram with (0_{0,Ker(f)} ∘ ε_{f}) · (0_{0,Ker(f)})_{I}^{♯} = (k(f)_{I}^{♭} ∘ f) · f_{I}^{♭}

[figure: B; ,; A; Ker(f); 0; f; k(f); 0_{0,A}; 0_{0,Ker(f)}; 0; 0; ε_{f}; f_{I}^{♭}; k(f)_{I}^{♭}]

(see (a3-2) in Definition 3.1) we have Ker(0_{0,A}, f_{I}^{♭}) = Ker(0_{0,Ker(f)}) by Proposition 3.20. So, it suffices to show Ker(0_{0,Ker(f)}) = 0. For any K ∈ S^{0} and k ∈ S^{1}(K, 0), we have S^{2}(k ∘ 0_{0,Ker(f)}, 0_{K,Ker(f)}) = {k_{I}^{♯}} by the Lemma 3.27. So 0_{0,Ker(f)} becomes fully faithful, and thus Ker(0_{0,Ker(f)}) = 0.

Conversely, assume Ker(0_{0,A}, f_{I}^{♭}) = 0. For any K ∈ S^{0} and α ∈ S^{2}(0_{K,A}, 0_{K,A}) satisfying α ∘ f = id_{0∘f}, we show α = id_{0} (Corollary 3.23).

By α ∘ f = id_{0∘f}, α is compatible with f_{I}^{♭} :

[figure: B; A; 0; Ker(0_{0,A}, f_{I}^{♭}); = 0; K; f; 0_{0,A}; 0_{K,0}; ^{id^{0}}; 0; 0; 0; ^{id^{0}}; α; f_{I}^{♭}]

So there exist k ∈ S^{1}(K, 0) and ε ∈ S^{2}(k ∘ id_{0}, 0_{K,0}) satisfying

(ε ∘ 0_{0,A}) · α = (k ∘ id_{0}) · k_{I}^{♯}.

Since ε ∘ 0_{0,A} = k_{I}^{♯} by (5) and (10), we obtain α = id_{0}.

In any relatively exact 2-category S, the difference kernel of any pair of 1-cells g, h : A → B always exists. More precisely, we have the following proposition:

3.29. Proposition. For any g, h ∈ S^{1}(A, B), if we take the kernel Ker(g ⊗ h^{∗}) = [Ker(g ⊗ h^{∗}), k, ε] of g ⊗ h^{∗} and put ε̃:= Ψ_{k∘g,k∘h}(Θ_{g,h}^{k^{♯}}(ε · k_{I}^{♯−1})) ∈ S^{2}(k ∘ g, k ∘ h), then (Ker(g ⊗ h^{∗}), k, ε̃) is the difference kernel of g and h.

Proof. For any K ∈ S^{0} and ℓ ∈ S^{1}(K, A), there exists a natural isomorphism (Lemma 2.16)

[figure: S^{2}(ℓ ∘ (g ⊗ h^{∗}), 0); S^{2}(ℓ ∘ g, ℓ ∘ h); ∈; ∈; σ; σ̃ := Ψ_{ℓ∘g,ℓ∘h}(Θ_{g,h}^{ℓ^{♯}}(σ · ℓ_{I}^{♯})).]

So, to give a 2-cell σ ∈ S^{2}(ℓ ∘ (g ⊗ h^{∗}), 0) is equivalent to give a 2-cell σ̃ ∈ S^{2}(ℓ ∘ g, ℓ ∘ h). And, by using Remark 3.4 and Corollary 3.5, the usual compatibility argument shows the proposition.

In any relatively exact 2-category S, the pullback of any pair of morphisms f_{i} : A_{i} → B (i = 1, 2) always exists. More precisely, we have the following proposition:

3.30. Proposition. For any f_{i} ∈ S^{1}(A_{i}, B) (1 = 1, 2), if we take the product of A_{1} and A_{2} (A_{1} × A_{2}, p_{1}, p_{2}), and take the difference kernel (D, d, φ) of p_{1} ∘ f_{1} and p_{2} ∘ f_{2}

[figure: D; A_{1} × A_{2}; B; d; p_{1}∘f_{1}; p_{2}∘f_{2}] [figure: D; A_{1}; A_{2}; B; d∘p_{1}; d∘p_{2}; f_{1}; f_{2}; φ] ,

then, (D, d ∘ p_{1}, d ∘ p_{2}, φ) is the pullback of f_{1} and f_{2}.

Proof of condition (b1) (in Definition 2.4). For any X ∈ S^{0}, g_{i} ∈ S^{1}(X, A_{i}) (i = 1, 2) and η ∈ S^{2}(g_{1} ∘ f_{1}, g_{2} ∘ f_{2}), by the universality of A_{1} × A_{2}, there exist g ∈ S^{1}(X, A_{1} ×A_{2}) and ξ_{i} ∈ S^{2}(d∘p_{i}, g_{i}) (i = 1, 2). Applying the universality of the difference kernel to the 2-cell

ζ := (ξ_{1} ∘ f_{1}) · η · (ξ_{2}^{−1} ∘ f_{2}) ∈ S^{2}(g ∘ p_{1} ∘ f_{1}, g ∘ p_{2} ∘ f_{2}),  (15)

we see there exist \underline{g} ∈ S^{1}(X, D) and \underline{ζ} ∈ S^{2}(\underline{g} ∘ d, g)

[figure: D; A_{1} × A_{2}; B; X; d; p_{1}∘f_{1}; p_{2}∘f_{2}; g; \underline{g}; \underline{ζ}]  (16)

such that

(\underline{g} ∘ φ) · (\underline{ζ} ∘ p_{2} ∘ f_{2}) = (\underline{ζ} ∘ p_{1} ∘ f_{1}) · ζ.  (17)

By (15) and (17), we have (\underline{g} ∘ φ) · (((\underline{ζ} ∘ p_{2}) · ξ_{2}) ∘ f_{2}) = (((\underline{ζ} ∘ p_{1}) · ξ_{1}) ∘ f_{1}) · η, and thus condition (b1) is satisfied.

[figure: D; A_{1}; A_{2}; B; X; d∘p_{1}; d∘p_{2}; f_{2}; f_{1}; \underline{g}; g_{1}; g_{2}; φ; (\underline{ζ}∘p_{1})·ξ_{1}; (\underline{ζ}∘p_{2})·ξ_{2}]

proof of condition (b2)

If we take h ∈ S^{1}(X, D) and η_{i} ∈ S^{2}(h∘d∘p_{i}, g_{i}) (i = 1, 2) which satisfy (h∘φ)·(η_{2}∘f_{2}) = (η_{1} ∘f_{1})·η, then by the universality of A_{1} ×A_{2}, there exists a unique 2-cell κ ∈ S^{2}(h∘d, g) such that

(κ ∘ p_{i}) · ξ_{i} = η_{i} (i = 1, 2).  (18)

Then, κ becomes compatible with φ and ζ, i.e. (h ∘ φ) · (κ ∘ p_{2} ∘ f_{2}) = (κ ∘ p_{1} ∘ f_{1}) · ζ. So, comparing this with factorization (16), by the universality of the difference kernel, we see there exists a unique 2-cell χ ∈ S^{2}(h, \underline{g}) which satisfies

(χ ∘ d) · \underline{ζ} = κ  (19)

Then we have (χ ∘ d ∘ p_{i}) · (\underline{ζ} ∘ p_{i}) · ξ_{i} = (κ ∘ p_{i}) · ξ_{i} = η_{i} (i = 1, 2). Thus χ is the desired 2-cell in condition (b2), and the uniqueness of such a χ follows from the uniqueness of κ and χ which satisfy (18) and (19).

By the universality of the pullback, we have the following remark:

3.31. Remark. Let

[figure: A_{1} ×_{B} A_{2}; A_{2}; A_{1}; B; f′_{1}; f′_{2}; f_{2}; f_{1}; ξ]  (20)

be a pull-back diagram. Then, for any K ∈ S^{0}, g, h ∈ S^{1}(K, A_{1}×_{B}A_{2}) and α, β ∈ S^{2}(g, h), we have

α ∘ f′_{i} = β ∘ f′_{i} (i = 1, 2) ⟹ α = β.

Proof. To the diagram

[figure: K; A_{1}; A_{2}; B; g∘f′_{1}; g∘f′_{2}; f_{1}; f_{2}; g∘ξ] ,

the following diagram gives a factorization which satisfies condition (b1) in Definition 2.4.

[figure: A_{1} ×_{B} A_{2}; A_{2}; A_{1}; B; K; f′_{1}; f′_{2}; f_{1}; f_{2}; g; g∘f′_{1}; g∘f′_{2}; ξ; ⟳; ⟳]

Since each of id_{g} : g ⟹ g and α ∘ β^{−1} : g ⟹ g gives a 2-cell which satisfies condition (b2), we have α ∘ β^{−1} = id by the uniqueness. Thus α = β.

3.32. Proposition. (See also Proposition 5.12.) Let (20) be a pull-back diagram. We have

(1) f_{1}: faithful ⇒ f′_{1}: faithful.

(2) f_{1}: fully faithful ⇒ f′_{1}: fully faithful.

(3) f_{1}: cofaithful ⇒ f′_{1}: cofaithful.

Proof. \underline{proof of (1)} By Corollary 3.23, it suffices to show α∘f′_{1} = id_{0∘f′_{1}} ⇒ α = id_{0} for any K ∈ S^{0} and α ∈ S^{2}(0_{K,A_{1}×_{B}A_{2}}, 0_{K,A_{1}×_{B}A_{2}}). Since (0∘ξ)·(α∘f′_{2} ∘f_{1}) = (α∘f′_{1} ∘f_{2})·(0∘ξ) = id_{0∘f′_{1}∘f_{2}} · (0 ∘ ξ) = 0 ∘ ξ, we have α ∘ f′_{2} ∘ f_{1} = id_{0∘f′_{2}∘f_{1}} = id_{0∘f′_{2}} ∘ f_{1}. Since f_{1} is faithful, we obtain α ∘ f′_{2} = id_{0∘f′_{2}}. Thus, we have α ∘ f′_{i} = id_{0∘f′_{i}} = id_{0} ∘ f′_{i} (i = 1, 2). By Remark 3.31, this implies α = id_{0}. \underline{proof of (2)} By (1), f′_{1} is already faithful. By Corollary 3.23, it suffices to show that for any K ∈ S^{0}, k ∈ S^{1}(K, A_{1} ×_{B} A_{2}) and any σ ∈ S^{2}(k ∘ f′_{1}, 0), there exists a unique 2-cell κ ∈ S^{2}(k, 0) such that σ = (κ ∘ f′_{1}) · (f′_{1})_{I}^{♭}. Since f_{1} is fully faithful, for any K ∈ S^{0}, k ∈ S^{1}(K, A_{1} ×_{B} A_{2}) and any σ ∈ S^{2}(k ∘ f′_{1}, 0), there exists τ ∈ S^{2}(k ∘ f′_{2}, 0) such that (τ ∘ f_{1}) · (f_{1})_{I}^{♭} = (k ∘ ξ^{−1}) · (σ ∘ f_{2}) · (f_{2})_{I}^{♭}. Then, for the diagram

[figure: K; A_{1}; A_{2}; B; 0; 0; f_{2}; f_{1}; (f_{1})_{I}^{♭}·(f_{2})_{I}^{♭−1}] ,

each of the factorizations

[figure: A_{1} ×_{B} A_{2}; A_{2}; A_{1}; B; K; f′_{1}; f′_{2}; f_{1}; f_{2}; k; 0; 0; ξ; σ; τ; A_{1} ×_{B} A_{2}; A_{2}; A_{1}; B; K; f′_{1}; f′_{2}; f_{1}; f_{2}; 0; 0; 0; ξ; (f′_{1})^{♭}_{I}; (f′_{2})^{♭}_{I}]

satisfies condition (b1) in Definition 2.4. So there exists a 2-cell κ ∈ S^{2}(k, 0) such that σ = (κ ∘ f′_{1}) · (f′_{1})_{I}^{♭}. Uniqueness of such κ follows from the faithfulness of f′_{1}. \underline{proof of (3)} Let (A_{1} × A_{2}, p_{1}, p_{2}) be the product of A_{1} and A_{2}. For id_{A_{1}} ∈ S^{1}(A_{1}, A_{1}) and 0 ∈ S^{1}(A_{1}, A_{2}), by the universality of A_{1} × A_{2}, there exist i_{1} ∈ S^{1}(A_{1}, A_{1} × A_{2}), ξ_{1} ∈ S^{2}(i_{1} ∘ p_{1}, id_{A_{1}}) and ξ_{2} ∈ S^{2}(i_{2} ∘ p_{2}, 0). Similarly, there is a 1-cell i_{2} ∈ S^{1}(A_{2}, A_{1} × A_{2}) such that there are equivalences i_{2}∘p_{2} ≃ id_{A_{2}}, i_{2}∘p_{1} ≃ 0. If we put t := (p_{1}∘f_{1})⊗(p_{2}∘f_{2})^{∗}, then by Proposition 3.29 and 3.30, we have A_{1} ×_{B} A_{2} = Ker(t). So we may assume Ker(t) = [A_{1} ×_{B} A_{2}, d, ε_{t}] and f′_{1} = d ∘ p_{2}.

[figure: A_{1} ×_{B} A_{2}; A_{1} × A_{2}; B; d; t; 0; ε_{t}]

Since i_{1} ∘ t and f_{1} are equivalent;

i_{1} ∘ t ≃ (i_{1} ∘ p_{1} ∘ f_{1}) ⊗ (i_{1} ∘ p_{2} ∘ f_{2}^{∗}) ≃ (id_{A_{1}} ∘ f_{1}) ⊗ (0 ∘ f_{2}^{∗}) ≃ f_{1},

by the cofaithfulness of f_{1}, it follows that t is cofaithful. Thus, we have B = Cok(ker(t)), i.e. Cok(d) = [B, t, ε_{t}]. By (the dual of) Corollary 3.23, it suffices to show f′_{1} ∘ α = id_{f′_{1}∘0} ⇒ α = id_{0} for any C ∈ S^{0} and any α ∈ S^{2}(0_{A_{2},C}, 0_{A_{2},C}). For the 2-cell (d ∘ p_{2})_{I}^{♯} ∈ S^{2}(d ∘ p_{2} ∘ 0_{A_{2},C}, 0) (see the following diagram), by the universality of Cok(d), there exist u ∈ S^{1}(B, C) and γ ∈ S^{2}(t ∘ u, p_{2} ∘ 0) such that (d ∘ γ) · (d ∘ p_{2})_{I}^{♯} = (ε_{t} ∘ u) · u_{I}^{♭}. Thus, if we put γ′ := γ · (p_{2} ∘ α), we have

(d ∘ γ′) · (d ∘ p_{2})_{I}^{♯} = (d ∘ γ) · (d ∘ p_{2} ∘ α) · (d ∘ p_{2})_{I}^{♯}
= (d ∘ γ) · (f′_{1} ∘ α) · (d ∘ p_{2})_{I}^{♯} = (ε_{t} ∘ u) · u_{I}^{♭}.

So, γ and γ′ ∈ S^{2}(t ∘ u, p_{2} ∘ 0) give two factorization of p_{2} ∘ 0 compatible with ε_{t} and (d∘p_{2})_{I}^{♯}. By the universality of Cok(d) = [B, t, ε_{t}], there exists a unique 2-cell β ∈ S^{2}(u, u) such that

(t ∘ β) · γ = γ′.  (21)

[figure: A_{1} ×_{B} A_{2}; A_{1} × A_{2}; B; A_{2}; C; A_{1}; d; i_{1}; f_{1}; p_{2}; f′_{1}; t; u; u; 0; 0; α; γ; β; equivalence; ⟳]

Then we have (i_{1}∘t∘β)·(i_{1}∘γ) = i_{1}∘γ′ = (i_{1}∘γ)·(i_{1}∘p_{2}∘α) = (i_{1}∘γ)·(ξ_{2}∘0)·(0∘α)·(ξ_{2}^{−1}∘0) = (i_{1} ∘ γ), and thus, (i_{1} ∘ t) ∘ β = id_{i_{1}∘t∘u}. Since i_{1} ∘ t ≃ f_{1} is cofaithful, we obtain β = id_{u}. So, by (21), we have γ = γ′ = γ · (p_{2} ∘ α), and consequently p_{2} ∘ α = id_{p_{2}∘0}. Since p_{2} is cofaithful (because i_{2} ∘ p_{2} ≃ id_{A_{2}} is cofaithful), we obtain α = id_{0}.

3.33. Proposition. Consider diagram (12) in S. If we take Ker(f, φ) = [Ker(f, φ), ℓ, ε], then by the universality of Ker(f) = [Ker(f), k(f), ε_{f}], ℓ factors uniquely up to an equivalence as

[figure: Ker(f); Ker(f, φ); A; B; C; ,; k(f); ℓ; \underline{ℓ}; f; g; 0; 0; ε; ε_{f}; \underline{ε}]

where (\underline{ε} ∘ f) · ε = (\underline{ℓ} ∘ ε_{f}) · (\underline{ℓ})_{I}^{♯}. Then, \underline{ℓ} becomes fully faithful.

Proof. Since \underline{ℓ} ∘ k(f) is equivalent to a faithful 1-cell ℓ, so \underline{ℓ} becomes faithful. For any K ∈ S^{0}, k ∈ S^{1}(K, Ker(f, φ)) and σ ∈ S^{2}(k ∘ \underline{ℓ}, 0), if we put σ′ := (k ∘ \underline{ε}^{−1}) · (σ ∘ k(f)) · k(f)_{I}^{♭} ∈ S^{2}(k ∘ ℓ, 0), then σ′ becomes compatible with ε. So, by Lemma 3.19, there exists τ ∈ S^{2}(k, 0) such that σ′ = (τ ∘ ℓ) · ℓ_{I}^{♭}, i.e.

(k ∘ \underline{ε}^{−1}) · (σ ∘ k(f)) · (k(f))_{I}^{♭} = (τ ∘ ℓ) · ℓ_{I}^{♭}.  (22)

Now, since (k ∘ \underline{ε}) · (τ ∘ ℓ) · ℓ_{I}^{♭} = (τ ∘ \underline{ℓ} ∘ k(f)) · (\underline{ℓ} ∘ k(f))_{I}^{♭} by Corollary 3.5, (22) is equivalent to (σ ∘ k(f)) · (k(f))_{I}^{♭} = (τ ∘ \underline{ℓ} ∘ k(f)) · (\underline{ℓ}_{I}^{♭} ∘ k(f)) · (k(f))_{I}^{♭}.

Thus, we obtain σ ∘ k(f) = ((τ ∘ \underline{ℓ}) · \underline{ℓ}_{I}^{♭}) ∘ k(f). Since k(f) is faithful, it follows that σ = (τ ∘ \underline{ℓ}) · \underline{ℓ}_{I}^{♭}. Uniqueness of such τ follows from the faithfulness of \underline{ℓ}. Thus \underline{ℓ} becomes fully faithful by Corollary 3.24.

4. Existence of proper factorization systems

4.1. Definition. For any A, B ∈ S^{0} and f ∈ S^{1}(A, B), we define its image as Ker(cok(f)).

4.2. Remark. By the universality of the kernel, there exist i(f) ∈ S^{1}(A, Im(f)) and ι ∈ S^{2}(i(f) ∘ k(c(f)), f) such that (ι ∘ c(f)) · π_{f} = (i(f) ∘ ε_{c(f)}) · i(f)_{I}^{♯}. Coimage of f is defined dually, and we obtain a factorization through Coim(f).

4.3. Proposition. [1st factorization] For any f ∈ S^{1}(A, B), the factorization ι : i(f) ∘ k(c(f)) ⟹ f through Im(f)

[figure: A; B; Im(f); f; k(c(f)); i(f); ι]

satisfies the following properties: (A) i(f) is fully cofaithful and k(c(f)) is faithful. (B) For any factorization η : i ∘ m ⟹ f where m is faithful, following (b1) and (b2) hold: (b1) There exist t ∈ S^{1}(Im(f), C), ζ_{m} ∈ S^{2}(t ∘ m, k(c(f))), ζ_{i} ∈ S^{2}(i(f) ∘ t, i)

[figure: A; B; C; Im(f); t; k(c(f)); i(f); m; i; ζ_{m}; ζ_{i}]

such that (i(f) ∘ ζ_{m}) · ι = (ζ_{i} ∘ m) · η. (b2) If both (t, ζ_{m}, ζ_{i}) and (t′, ζ′_{m}, ζ′_{i}) satisfy (b1), then there is a unique 2-cell κ ∈ S^{2}(t, t′) such that (i(f) ∘ κ) · ζ′_{i} = ζ_{i} and (κ ∘ m) · ζ′_{m} = ζ_{m}.

Dually, we obtain the following proposition for the coimage factorization.

4.4. Proposition. [2nd factorization] For any f ∈ S^{1}(A, B), the factorization μ : c(k(f)) ∘ j(f) ⟹ f through Coim(f)

[figure: A; B; Coim(f); f; j(f); c(k(f)); μ]

satisfies the following properties: (A) j(f) is fully faithful and c(k(f)) is cofaithful. (B) For any factorization ν : e ∘ j ⟹ f where e is cofaithful, following (b1) and (b2) (the dual of the conditions in Proposition 4.3) hold:

(b1) There exists s ∈ S^{1}(C, Coim(f)), ζ_{e} ∈ S^{2}(e ∘ s, c(k(f))), and ζ_{j} ∈ S^{2}(s ∘ j(f), j)

[figure: A; B; Coim(f); C; s; j; e; j(f); c(k(f)); ζ_{j}; ζ_{e}]

such that (e ∘ ζ_{j}) · ν = (ζ_{e} ∘ j(f)) · μ. (b2) If both (s, ζ_{e}, ζ_{j}) and (s′, ζ′_{e}, ζ′_{j}) satisfy (b1), then there is a unique 2-cell λ ∈ S^{2}(t, t′) such that (λ ∘ j(f)) · ζ′_{j} = ζ_{j} and (e ∘ λ) · ζ′_{e} = ζ_{e}.

In the rest of this section, we show Proposition 4.3.

4.5. Lemma. For any f ∈ S^{1}(A, B), i(f) is cofaithful.

Proof. It suffices to show that for any C ∈ S^{0} and α ∈ S^{2}(0_{Im(f),C}, 0_{Im(f),C})

[figure: A; Im(f); C; i(f); 0; 0; α] ,

we have i(f) ∘ α = id_{i(f)∘0} ⟹ α = id_{0}. Take the pushout of k(c(f)) and 0_{Im(f),C}

[figure: Im(f); C ∐_{Im(f)} B; B; C; i_{C}; 0; i_{B}; k(c(f)); ξ]

and put

ξ_{1} := ξ · (ξ_{1} ∘ f_{2}) · η = (g ∘ ξ) · (ξ_{2} ∘ f_{1})(i_{C})_{I}^{♭} = (k(c(f)) ∘ i_{B} ⟹^{ξ} 0 ∘ i_{C} ⟹^{(i_{C})_{I}^{♭}} 0)
ξ_{2} := ξ · (α ∘ i_{C}) · (i_{C})_{I}^{♭} = (k(c(f)) ∘ i_{B} ⟹^{ξ} 0 ∘ i_{C} ⟹^{α∘i_{C}} 0 ∘ i_{C} ⟹^{(i_{C})_{I}^{♭}} 0).

Then, since i_{C} is faithful by (the dual of) Lemma 3.32, we have

α = id_{0} ⟺ α ∘ i_{C} = id_{0∘i_{C}} ⟺ ξ · (α ∘ i_{C}) · (i_{C})_{0}^{♭} = ξ · id_{0∘i_{C}} · (i_{C})_{I}^{♭} ⟺ ξ_{1} = ξ_{2}.

So, it suffices to show ξ_{1} = ξ_{2}. For each i = 1, 2, since Cok(k(c(f)) = [Cok(f), c(f), ε_{c(f)}], there exist e_{i} ∈ S^{1}(Cok(f), C ∐_{Im(f)}B) and ε_{i} ∈ S^{2}(c(f) ∘ e_{i}, i_{B}) such that

(k(c(f)) ∘ ε_{i}) · ξ_{i} = (ε_{c(f)} ∘ e_{i}) · (e_{i})_{I}^{♭}.  (23)

[figure: Im(f); C ∐_{Im(f)} B; B; Cok(f); 0; i_{B}; k(c(f); c(f); e_{i}; 0; ξ_{i}; ε_{i}; ε_{c(f)}]

Since by assumption i(f) ∘ α = id_{i(f)∘0}, we have

i(f) ∘ ξ_{2} = (i(f) ∘ ξ) · (i(f) ∘ α ∘ i_{C}) · (i(f) ∘ (i_{C})_{I}^{♭})
= (i(f) ∘ ξ) · (id_{i(f)∘0∘i_{C}}) · (i(f) ∘ (i_{C})_{I}^{♭}) = i(f) ∘ ξ_{1}.

So, if we put ϖ := (ι^{−1} ∘ i_{B}) · (i(f) ∘ ξ_{i}) · (i(f))_{I}^{♯} ∈ S^{2}(f ∘ i_{B}, 0), this doesn’t depend on i = 1, 2. We can show easily (f ∘ ε_{i}) · ϖ = (π_{f} ∘ e_{i}) · (e_{i})_{I}^{♭} (i = 1, 2). Thus (e_{1}, ε_{1}) and (e_{2}, ε_{2}) are two factorizations of i_{B} compatible with ϖ and π_{f}.

[figure: A; C ∐_{Imf} B; B; Cok(f); 0; i_{B}; f; c(f); e_{i}; 0; π_{f}; ϖ; ε_{i}]

By the universality of Cok(f), there exists a 2-cell β ∈ S^{2}(e_{1}, e_{2}) such that (c(f)∘β)·ε_{2} = ε_{1}, and thus we have ε_{1}^{−1} = ε_{2}^{−1} · (c(f) ∘ β^{−1}). So, by (23), we have

ξ_{1} = (k(c(f)) ∘ ε_{1}^{−1}) · (ε_{c(f)} ∘ e_{1}) · (e_{1})_{I}^{♭}
= (k(c(f)) ∘ ε_{2}^{−1}) · (k(c(f)) ∘ c(f) ∘ β^{−1}) · (ε_{c(f)} ∘ e_{1}) · (e_{1})_{I}^{♭}
=_{9} (k(c(f)) ∘ ε_{2}^{−1}) · (ε_{c(f)} ∘ e_{2}) · (e_{2})_{I}^{♭} = ξ_{2}.

4.6. Lemma. Let f ∈ S^{1}(A, B). Let ι : i(f) ∘ k(c(f)) ⟹ f be the factorization of f through Im(f) as before. If we are given a factorization η : i ∘ m ⟹ f of f where i ∈ S^{1}(A, C), m ∈ S^{1}(C, B) and m is faithful, then there exist t ∈ S^{1}(Im(f), C), ζ_{i} ∈ S^{2}(i(f) ∘ t, i) and ζ_{m} ∈ S^{2}(t ∘ m, k(c(f))) such that (ζ_{i} ∘ m) · η = (i(f) ∘ ζ_{m}) · ι.

Proof. By the universality of Cok(f), for π := (η^{−1} ∘ c(m)) · (i ∘ π_{m}) · i_{I}^{♯} ∈ S^{2}(f ∘ c(m), 0), there exist \overline{m} ∈ S^{1}(Cok(f), Cok(m)) and \overline{η} ∈ S^{2}(c(f) ∘ \overline{m}, c(m)) such that

(f ∘ \overline{η}) · π = (π_{f} ∘ \overline{m}) · (\overline{m})_{I}^{♭}.  (24)

[figure: A; Cok(m); B; Cok(f); 0; c(m); f; c(f); \overline{m}; 0; π_{f}; π; \overline{η}]

Since m is faithful by assumption, it follows Ker(c(m)) = [C, m, π_{m}]. By the universality of Ker(c(m)), for the 2-cell

ζ := (k(c(f)) ∘ \overline{η}^{−1}) · (ε_{c(f)} ∘ \overline{m}) · (\overline{m})_{I}^{♭} ∈ S^{2}(k(c(f)) ∘ c(m), 0),  (25)

there exist t ∈ S^{1}(Im(f), C) and ζ_{m} ∈ S^{2}(t ∘ m, k(c(f))) such that (ζ_{m} ∘ c(m)) · ζ = (t ∘ π_{m}) · t_{I}^{♯}.

If we put \overline{ζ} := (i(f) ∘ ζ_{m}) · ι, then the following claim holds:

4.7. Claim. Each of the two factorizations of f through Ker(c(m))

η : i ∘ m ⟹ f and \overline{ζ} : i(f) ∘ t ∘ m ⟹ f

is compatible with π_{m} and π.

[figure: A; C; B; Cok(m); f; m; c(m); 0; 0; π_{m}; π]

If the above claim is proven, then by the universality of Ker(c(m)) = [C, m, π_{m}], there exists a unique 2-cell ζ_{i} ∈ S^{2}(i(f)∘t, i) such that (ζ_{i} ∘m)·η = \overline{ζ}. Thus we obtain (t, ζ_{m}, ζ_{i}) which satisfies (ζ_{i} ∘ m) · η = \overline{ζ} = (i(f) ∘ ζ_{m}) · ι, and the lemma is proven. So, we show Claim 4.7.

(a) compatibility of η with π_{m}, π This follows immediately from the definition of π.

(b) compatibility of \overline{ζ} with π_{m}, π We have

i(f) ∘ ζ =_{25} (ι ∘ c(m)) · (f ∘ \overline{η}^{−1}) · (ι^{−1} ∘ c(f) ∘ \overline{m})
· (i(f) ∘ ε_{c(f)} ∘ \overline{m})) · (i(f) ∘ (\overline{m})_{I}^{♭})
=_{24} (ι ∘ c(m)) · π · i(f)_{I}^{♯−1}.

From this, we obtain (i(f) ∘ t ∘ π_{m}) · (i(f) ∘ t_{I}^{♯}) = (\overline{ζ} ∘ c(m)) · π · i(f)_{I}^{♯−1}. So we have

(\overline{ζ} ∘ c(m)) · π = (i(f) ∘ t ∘ π_{m}) · (i(f) ∘ t_{I}^{♯}) · i(f)_{I}^{♯−1} = (i(f) ∘ t ∘ π_{m}) · (i(f) ∘ t)_{I}^{♯}.

4.8. Lemma. Let A, B, C ∈ S^{0}, f, m, i ∈ S^{1}, η ∈ S^{2} be as in Lemma 4.6. If a triplet (t′, ζ′_{m}, ζ′_{i}) (where t′ ∈ S^{1}(Im(f), C), ζ′_{m} ∈ S^{2}(t′ ∘ m, k(c(f))), ζ′_{i} ∈ S^{2}(i(f) ∘ t′, i) satisfies

(i(f) ∘ ζ′_{m}) · ι = (ζ′_{i} ∘ m) · η,  (26)

then ζ′_{m} becomes compatible with ζ and π_{m} (in the notation of the proof of Lemma 4.6), i.e. we have (ζ′_{m} ∘ c(m)) · ζ = (t′ ∘ π_{m}) · (t′)_{I}^{♯}.

4.9. Remark. Since m is faithful, ζ′_{m} which satisfies (26) is uniquely determined by t′ and ζ′_{i} if it exists.

Proof of Lemma 4.8. Since we have

i(f) ∘ ((ζ′_{m} ∘ c(m) · ζ)
=_{26,1} (ζ′_{i} ∘ m ∘ c(m)) · (η ∘ c(m)) · (f ∘ \overline{η}^{−1}) · (ι^{−1} ∘ c(f) ∘ \overline{m})
· (i(f) ∘ ε_{c(f)} ∘ \overline{m}) · (i(f) ∘ (\overline{m})_{I}^{♭})
=_{24} ((i(f) ∘ t′ ∘ π_{m}) · (i(f) ∘ (t′)_{I}^{♯}),

we obtain (ζ′_{m} ∘ c(m)) · ζ = (t′ ∘ π_{m}) · (t′)_{I}^{♯} by the cofaithfulness of i(f).

4.10. Corollary. Let A, B, C, f, m, i, η as in Proposition 4.3. If both (t, ζ_{m}, ζ_{i}) and (t′, ζ′_{m}, ζ′_{i}) satisfy (b1), then there exists a unique 2-cell κ ∈ S^{2}(t, t′) such that (i(f)∘κ)·ζ′_{i} = ζ_{i} and (κ ∘ m) · ζ′_{m} = ζ_{m}.

Proof. By Lemma 4.8, there exists a 2-cell κ ∈ S^{2}(t, t′) such that (κ ∘ m) · ζ′_{m} = ζ_{m} by the universality of Ker(c(m)) = [C, m, π_{m}]. This κ also satisfies ζ_{i} = (i(f) ∘ κ) · ζ′_{i}, and unique by the cofaithfulness of i(f).

Considering the case of C = Im(f), we obtain the following corollary.

4.11. Corollary. For any t ∈ S^{1}(Im(f), Im(f)), ζ_{m} ∈ S^{2}(t ∘ k(c(f)), k(c(f))) and ζ_{i} ∈ S^{2}(i(f) ∘ t, i(f)) satisfying (ζ_{i} ∘ k(c(f))) · ι = (i(f) ∘ ζ_{m}) · ι, there exists a unique 2-cell κ ∈ S^{2}(t, id_{Im(f)}) such that i(f) ∘ κ = ζ_{i} and κ ∘ k(c(f)) = ζ_{m}.

Now, we can prove Proposition 4.3.

Proof of Proposition 4.3. Since all the other is already shown, it suffices to show the following:

4.12. Claim. For any C ∈ S^{0} and any g, h ∈ S^{1}(Im(f), C),

i(f) ∘ − : S^{2}(g, h) ⟶ S^{2}(i(f) ∘ g, i(f) ∘ h)

is surjective.

So, we show Claim 4.12. If we take the difference kernel of g and h;

d_{(g,h)} : DK(g, h) ⟶ Im(f), φ_{(g,h)} : d_{(g,h)} ∘ g ⟹ d_{(g,h)} ∘ h,

then by the universality of the difference kernel, for any β ∈ S^{2}(i(f) ∘ g, i(f) ∘ h) there exist i ∈ S^{1}(A, DK(g, h)) and λ ∈ S^{2}(i ∘ d_{(g,h)}, i(f))

[figure: DK(g, h); Im(f); C; A; d_{(g,h)}; g; h; i(f); i; λ]

such that (i ∘ φ_{(g,h)}) · (λ ∘ h) = (λ ∘ g) · β.

If we put m := d_{(g,h)} ∘ k(c(f)), then m becomes faithful since d_{(g,h)} and k(c(f)) are faithful. Applying Lemma 4.6 to the factorization η := (λ ∘ k(c(f))) · ι : i ∘ m ⟹ f, we obtain t ∈ S^{1}(Im(f), DK(g, h)), ζ_{m} ∈ S^{2}(t ∘ m, k(c(f))) and ζ_{i} ∈ S^{2}(i(f) ∘ t, i) such that (ζ_{i} ∘ m) · η = (i(f) ∘ ζ_{m}) · ι. Thus we have

(ζ_{i} ∘ d_{(g,h)} ∘ k(c(f))) · (λ ∘ k(c(f))) · ι = (i(f) ∘ ζ_{m}) · ι.

So, if we put \overline{ζ}_{i} := (ζ_{i} ∘ d_{(g,h)}) · λ ∈ S^{2}(i(f) ∘ t ∘ d_{(g,h)}, i(f)), then we have

(\overline{ζ}_{i} ∘ k(c(f))) · ι = (i(f) ∘ ζ_{m}) · ι.

By Corollary 4.11, there exists a 2-cell κ ∈ S^{2}(t∘d_{(g,h)}, id_{Im(f)}) such that κ∘k(c(f)) = ζ_{m} and i(f)∘κ = \overline{ζ}_{i}. If we put α := (κ^{−1} ∘g)·(t∘φ_{(g,h)})·(κ∘h) ∈ S^{2}(g, h), we can show that α satisfies i(f) ∘ α = β. Thus i(f) ∘ − : S^{2}(g, h) ⟶ S^{2}(i(f) ∘ g, i(f) ∘ h) is surjective.

4.13. Remark. In condition (B) of Proposition 4.3, if moreover i is fully cofaithful, then t becomes fully cofaithful since i and i(f) are fully cofaithful. On the other hand, t is faithful since k(c(f)) is faithful. So, in this case t becomes an equivalence by Proposition 3.26.

Together with Corollary 4.11, we can show easily the following corollary:

4.14. Corollary. For any f ∈ S^{1}(A, B), the following (b1) and (b2) hold: (b1) If in the factorizations

[figure: A; B; C; f; m; i; η; A; B; ,; C′; f; m′; i′; η′]

m, m′ are faithful and i, i′ are fully cofaithful, then there exist t ∈ S^{1}(C, C′), ζ_{m} ∈ S^{2}(t ∘ m′, m), and ζ_{i} ∈ S^{2}(i ∘ t, i′) such that (i ∘ ζ_{m}) · η = (ζ_{i} ∘ m′) · η′. (b2) If both (t, ζ_{m}, ζ_{i}) and (t′, ζ′_{m}, ζ′_{i}) satisfy (b1), then there is a unique 2-cell κ ∈ S^{2}(t, t′) such that (i ∘ κ) · ζ′_{i} = ζ_{i} and (κ ∘ m′) · ζ′_{m} = ζ_{m}.

4.15. Remark. Proposition 4.3 and Proposition 4.4 implies respectively the existence of (2,1)-proper factorization system and (1,2)-proper factorization system in any relatively exact 2-category, in the sense of [2].

In the notation of this section, condition (B2) and (B3) in Definition 3.7 can be written as follows:

4.16. Corollary. For any f ∈ S^{1}(A, B), we have;

(1) f is faithful iff i(f) : A ⟶ Im(f) is an equivalence.

(2) f is cofaithful iff j(f) : Coim(f) ⟶ B is an equivalence.

Proof. Since (1) is the dual of (2), we show only (2).

In the coimage factorization diagram

[figure: A; B; Coim(f); f; j(f); c(k(f)); μ_{f}] ,

since c(k(f)) is cofaithful and j(f) is fully faithful, we have f is cofaithful⟺ j(f) is cofaithful ⟺_{Prop. 3.26} j(f) is an equivalence.

5. Definition of relative 2-exactness

Diagram lemmas (1).

5.1. Definition. A complex A_{•} = (A_{n}, d_{n}^{A}, δ_{n}^{A}) is a diagram

[figure: ⋯ A_{n−2}; A_{n−1}; A_{n}; A_{n+1}; A_{n+2} ⋯; d_{n−2}^{A}; d_{n−1}^{A}; d_{n}^{A}; d_{n+1}^{A}; 0; 0; 0; 0; 0; δ_{n−1}^{A}; δ_{n}^{A}; δ_{n+1}^{A}]

where A_{n} ∈ S^{0}, d_{n}^{A} ∈ S^{1}(A_{n}, A_{n+1}), δ_{n}^{A} ∈ S^{2}(d_{n−1}^{A} ∘ d_{n}^{A}, 0), and satisfies the following compatibility condition for each n ∈ ℤ :

(d_{n−1}^{A} ∘ δ_{n+1}^{A}) · (d_{n−1}^{A})_{I}^{♯} = (δ_{n}^{A} ∘ d_{n+1}^{A}) ∘ (d_{n+1}^{A})_{I}^{♭}

5.2. Remark. We consider a bounded complex as a particular case of a complex, by adding zeroes.

[figure: ⋯; 0; 0; A_{0}; A_{1}; A_{2}; A_{3} ⋯; 0; 0; d_{0}^{A}; d_{1}^{A}; d_{2}^{A}; 0; 0; 0; 0; 0; (d_{0}^{A})_{I}^{♭}; δ_{1}^{A}; δ_{2}^{A}; id]

5.3. Definition. For any complexes A_{•} = (A_{n}, d_{n}^{A}, δ_{n}^{A}) and B_{•} = (B_{n}, d_{n}^{B}, δ_{n}^{B}), a complex morphism f_{•} = (f_{n}, λ_{n}) : A_{•} ⟶ B_{•} consists of f_{n} ∈ S^{1}(A_{n}, B_{n}) and λ_{n} ∈ S^{2}(d_{n}^{A} ∘ f_{n+1}, f_{n} ∘ d_{n}^{B}) for each n, satisfying

(δ_{n}^{A} ∘ f_{n+1}) · (f_{n+1})_{I}^{♭} = (d_{n−1}^{A} ∘ λ_{n}) · (λ_{n−1} ∘ d_{n}^{B}) · (f_{n−1} ∘ δ_{n}^{B}) · (f_{n−1})_{I}^{♯}.

[figure: ⋯ A_{n−2}; A_{n−1}; A_{n}; A_{n+1}; A_{n+2} ⋯; ⋯ B_{n−2}; B_{n−1}; B_{n}; B_{n+1}; B_{n+2} ⋯; d_{n−2}^{A}; d_{n−1}^{A}; d_{n}^{A}; d_{n+1}^{A}; d_{n−2}^{B}; d_{n−1}^{B}; d_{n}^{B}; d_{n+1}^{B}; f_{n−2}; f_{n−1}; f_{n}; f_{n+1}; f_{n+2}; λ_{n−2}; λ_{n−1}; λ_{n}; λ_{n+1}]

5.4. Proposition. Consider the following diagram in S.

[figure: A_{1}; A_{2}; B_{1}; B_{2}; f_{1}; a; f_{2}; b; λ]  (27)

If we take the cokernels of f_{1} and f_{2}, then there exist \overline{b} ∈ S^{1}(Cok(f_{1}), Cok(f_{2})) and \overline{λ} ∈ S^{2}(c(f_{1}) ∘ \overline{b}, b ∘ c(f_{2})) such that

(π_{f_{1}} ∘ \overline{b}) · (\overline{b})_{I}^{♭} = (f_{1} ∘ \overline{λ}) · (λ ∘ c(f_{2})) · (a ∘ π_{f_{2}}) · a_{I}^{♯}.

[figure: A_{1}; A_{2}; B_{1}; B_{2}; Cok(f_{1}); Cok(f_{2}); f_{1}; a; f_{2}; b; \overline{b}; c(f_{1}); c(f_{2}); 0; 0; λ; \overline{λ}; π_{f_{1}}; π_{f_{2}}]

If (\overline{b}′, \overline{λ}′) also satisfies this condition, there exists a unique 2-cell ξ ∈ S^{2}(\overline{b}, \overline{b}′) such that (c(f_{1}) ∘ ξ) · \overline{λ}′ = \overline{λ}.

Proof. This follows immediately if we apply the universality of Cok(f_{1}) to (λ ∘ c(f_{2})) · (a ∘ π_{f_{2}}) · a_{I}^{♯} ∈ S^{2}(f_{1} ∘ b ∘ c(f_{2}), 0).

5.5. Proposition. Consider the following diagrams in S,

[figure: A_{1}; A_{2}; A_{3}; B_{1}; B_{2}; B_{3}; f_{1}; a_{1}; f_{2}; b_{1}; f_{3}; a_{2}; b_{2}; a; b; λ_{1}; λ_{2}; α; β; A_{1}; A_{3}; B_{1}; B_{3}; f_{1}; a; f_{3}; b; λ]

which satisfy (f_{1} ∘ β) · λ = (λ_{1} ∘ b_{2}) · (a_{1} ∘ λ_{2}) · (α ∘ f_{3}). Applying Proposition 5.4, we obtain diagrams

[figure: B_{1}; B_{3}; Cok(f_{1}); Cok(f_{3}); c(f_{1}); b; c(f_{3}); \overline{b}; \overline{λ}; B_{1}; B_{2}; Cok(f_{1}); Cok(f_{2}); c(f_{1}); b_{1}; c(f_{2}); \overline{b}_{1}; \overline{λ}_{1}; B_{2}; B_{3}; Cok(f_{2}); Cok(f_{3}); c(f_{2}); b_{2}; c(f_{3}); \overline{b}_{2}; \overline{λ}_{2}]

with

(π_{f_{1}} ∘ \overline{b}) · (\overline{b})_{I}^{♭} = (f_{1} ∘ \overline{λ}) · (λ ∘ c(f_{3})) · (a ∘ π_{f_{3}}) · a_{I}^{♯}  (28)
(π_{f_{1}} ∘ \overline{b}_{1}) · (\overline{b}_{1})_{I}^{♭} = (f_{1} ∘ \overline{λ}_{1}) · (λ_{1} ∘ c(f_{2})) · (a_{1} ∘ π_{f_{2}}) · (a_{1})_{I}^{♯}
(π_{f_{2}} ∘ \overline{b}_{2}) · (\overline{b}_{2})_{I}^{♭} = (f_{2} ∘ \overline{λ}_{2}) · (λ_{2} ∘ c(f_{3})) · (a_{2} ∘ π_{f_{3}}) · (a_{2})_{I}^{♯}.

Then, there exists a unique 2-cell \overline{β} ∈ S^{2}(\overline{b}_{1} ∘ \overline{b}_{2}, \overline{b}) such that

(c(f_{1}) ∘ \overline{β}) · \overline{λ} = (\overline{λ}_{1} ∘ \overline{b}_{2}) · (b_{1} ∘ \overline{λ}_{2}) · (β ∘ c(f_{3})).

Proof. By (28), \overline{λ} is compatible with π_{f_{1}} and (λ ∘ c(f_{3})) · (a ∘ π_{f_{3}}) · a_{I}^{♯}.

[figure: A_{1}; B_{1}; Cok(f_{1}); Cok(f_{3}); f_{1}; c(f_{1}); b∘c(f_{3}); \overline{b}; 0; 0; π_{f_{1}}; \overline{λ}; (λ∘c(f_{3}))·(a∘π_{f_{3}})·a_{I}^{♯}]

On the other hand, \overline{λ}′ := (\overline{λ}_{1} ∘ \overline{b}_{2}) · (b_{1} ∘ \overline{λ}_{2}) · (β ∘ c(f_{3})) is also compatible with π_{f_{1}} and (λ ∘ c(f_{3})) · (a ∘ π_{f_{3}}) · a_{I}^{♯}. So, by the universality of the Cok(f_{1}), there exists a unique 2-cell \overline{β} ∈ S^{2}(\overline{b}_{1} ∘ \overline{b}_{2}, \overline{b}) such that (c(f_{1}) ∘ \overline{β}) · \overline{λ} = \overline{λ}′.

5.6. Corollary. Let (f_{n}, λ_{n}) : (A_{n}, d_{n}^{A}, δ_{n}^{A}) ⟶ (B_{n}, d_{n}^{B}, δ_{n}^{B}) be a complex morphism. Then, by taking the cokernels, we obtain a complex morphism (c(f_{n}), \overline{λ}_{n}) : (B_{n}, d_{n}^{B}, δ_{n}^{B}) ⟶ (Cok(f_{n}), \overline{d}_{n}^{B}, \overline{δ}_{n}^{B}) which satisfies

(d_{n}^{A} ∘ π_{f_{n+1}}) · (d_{n}^{A})_{I}^{♯} = (λ_{n} ∘ c(f_{n+1})) · (f_{n} ∘ \overline{λ}_{n}) · (π_{f_{n}} ∘ \overline{d}_{n}^{B}) · (\overline{d}_{n}^{B})_{I}^{♭}  (29)

for each n.

Proof. By Proposition 5.4, we obtain \overline{d}_{n}^{B} and \overline{λ}_{n} which satisfy (29). And by Proposition 5.5, for each n, there exists a unique 2-cell \overline{δ}_{n}^{B} ∈ S^{2}(\overline{d}_{n−1}^{B} ∘ \overline{d}_{n}^{B}, 0) such that ((δ_{n}^{B} ∘ c(f_{n+1})) · c(f_{n+1})_{I}^{♭} = (d_{n−1}^{B} ∘ \overline{λ}_{n}) · (\overline{λ}_{n−1} ∘ \overline{d}_{n}^{B}) · (c(f_{n−1}) ∘ \overline{δ}_{n}^{B}) · c(f_{n−1})_{I}^{♯}. By the uniqueness of \overline{β} in Proposition 5.5, it is easy to see that

(\overline{δ}_{n}^{B} ∘ \overline{d}_{n+1}^{B}) · (\overline{d}_{n+1}^{B})_{I}^{♭} = (\overline{d}_{n−1}^{B} ∘ \overline{δ}_{n+1}^{B}) · (\overline{d}_{n−1}^{B})_{I}^{♯}.

These are saying that (Cok(f_{n}), \overline{d}_{n}^{B}, \overline{δ}_{n}^{B}) is a complex and (c(f_{n}), \overline{λ}_{n}) is a complex morphism.

5.7. Proposition. Consider the following diagram in S.

[figure: A_{1}; A_{3}; B_{1}; B_{3}; f_{1}; a; f_{3}; b; λ]

By taking the cokernels of f_{1} and f_{2}, we obtain

[figure: A_{1}; A_{2}; B_{1}; B_{2}; Cok(f_{1}); Cok(f_{2}); ,; f_{1}; a; f_{2}; b; \overline{b}; c(f_{1}); c(f_{2}); λ; λ′]

and from this diagram, by taking the cokernels of a, b, \overline{b}, we obtain

[figure: A_{2}; B_{2}; Cok(a); Cok(b); Cok(f_{2}); Cok(\overline{b}); .; f_{2}; c(f_{2}); c(a); c(b); c(\overline{b}); \overline{c(f2)}; _{\overline{f}_{2}}; \overline{λ}; \overline{λ}′; 0; 0; π_{f2}; _{\overline{π}_{f2}}]

Then we have Cok(\overline{f}_{2}) = [Cok(\overline{b}), \overline{c(f_{2})}, \overline{π}_{f_{2}}]. We abbreviate this to Cok(\overline{f}_{2}) = Cok(\overline{b}).

Proof. Left to the reader.

5.8. Proposition. In the following diagram, assume f_{•} : A_{•} ⟶ B_{•} is a complex morphism.

[figure: A_{1}; B_{1}; A_{2}; B_{2}; A_{3}; B_{3}; d_{1}^{A}; f_{1}; d_{1}^{B}; f_{2}; f_{3}; d_{2}^{A}; d_{2}^{B}; 0; 0; λ_{1}; λ_{2}; δ_{2}^{A}; δ_{2}^{B}]  (30)

If we take the cokernels of d_{1}^{A} and d_{1}^{B},

[figure: A_{1}; B_{1}; A_{2}; B_{2}; Cok(d_{1}^{A}); Cok(d_{1}^{B}); d_{1}^{A}; f_{1}; d_{1}^{B}; f_{2}; \overline{f}_{2}; c(d_{1}^{A}); c(d_{1}^{B}); λ_{1}; \overline{λ}_{1}]

then by the universality of cokernel, we obtain \overline{d}_{2}^{A} ∈ S^{1}(Cok(d_{1}^{A}), A_{3}) and \overline{δ}_{2}^{A} ∈ S^{2}(c(d_{1}^{A}) ∘ \overline{d}_{2}^{A}, d_{2}^{A}) such that (d_{1}^{A}∘\overline{δ}_{2}^{A})·δ_{2}^{A} = (π_{d_{1}^{A}}∘\overline{d}_{2}^{A})·(\overline{d}_{2}^{A})_{I}^{♭}. Similarly, we obtain \overline{d}_{2}^{B} ∈ S^{1}(Cok(d_{1}^{B}), B_{3}),

\overline{δ}_{2}^{B} ∈ S^{2}(c(d_{1}^{B})∘\overline{d}_{2}^{B}, d_{2}^{B}) with (d_{1}^{B} ∘\overline{δ}_{2}^{B})·δ_{2}^{B} = (π_{d_{1}^{B}} ∘\overline{d}_{2}^{B})·(\overline{d}_{2}^{B})_{I}^{♭}. Then, there exists a unique

2-cell \overline{λ}_{2} ∈ S^{2}(\overline{d}_{2}^{A} ∘ f_{3}, \overline{f}_{2} ∘ \overline{d}_{2}^{B}) such that (c(d_{1}^{A}) ∘ \overline{λ}_{2}) · (\overline{λ}_{1} ∘ \overline{d}_{2}^{B}) · (f_{2} ∘ \overline{δ}_{2}^{B}) = (\overline{δ}_{2}^{A} ∘ f_{3}) · λ_{2}.

Proof. If we put δ := (d_{1}^{A} ∘ λ_{2}^{−1}) · (δ_{2}^{A} ∘ f_{3}) · (f_{3})_{I}^{♭}, then both the factorizations

(\overline{δ}_{2}^{A} ∘ f_{3}) · λ_{2} : c(d_{1}^{A}) ∘ (\overline{d}_{2}^{A} ∘ f_{3}) ⟹ f_{2} ∘ d_{2}^{B}
(\overline{λ}_{1} ∘ \overline{d}_{2}^{B}) · (f_{2} ∘ \overline{δ}_{2}^{B}) : c(d_{1}^{A}) ∘ (\overline{f}_{2} ∘ \overline{d}_{2}^{B}) ⟹ f_{2} ∘ d_{2}^{B}

are compatible with π_{d_{1}^{A}} and δ. So the proposition follows from the universality of Cok(d_{2}^{A}).

5.9. Proposition. In diagram (27), if we take the coimage factorization μ_{a} : c(k(a)) ∘ j(a) ⟹ a and μ_{b} : c(k(b)) ∘ j(b) ⟹ b, then there exist f ∈ S^{1}(Coim(a), Coim(b)), λ_{1} ∈ S^{2}(f_{1} ∘ c(k(b)), c(k(a)) ∘ f) and λ_{2} ∈ S^{2}(f ∘ j(b), j(a) ∘ f_{2}) such that (f_{1} ∘ μ_{b}) · λ = (λ_{1} ∘ j(b)) · (c(k(a)) ∘ λ_{2}) · (μ_{a} ∘ f_{2}).

[figure: A_{1}; Coim(a); A_{2}; B_{1}; Coim(b); B_{2}; f_{1}; f; f_{2}; c(k(a)); j(a); c(k(b)); j(b); a; b; λ_{1}; λ_{2}; μ_{a}; μ_{b}]  (31)

Moreover, for any other (f′, λ′_{1}, λ′_{2}) with this property, there exists a unique 2-cell ξ ∈ S^{2}(f, f′) such that λ_{1} · (c(k(a)) ∘ ξ) = λ′_{1} and (ξ ∘ j(b)) · λ′_{2} = λ_{2}.

Proof. Since the coimage factorization is unique up to an equivalence and is obtained by the factorization which fills in the following diagram, we may assume Ker(a) = [Ker(a), k(a), ε_{a}], Cok(k(a)) = [Coim(a), c(k(a)), π_{k(a)}], and (k(a) ∘ μ_{a}) · ε_{a} = (π_{k(a)} ∘ j(a)) · j(a)_{I}^{♭}.

[figure: Ker(a); A_{1}; Coim(a); A_{2}; k(a); c(k(a)); a; 0; 0; π_{k(a)}; ε_{a}; ^{∃}μ_{a}; ^{∃}j(a)]

Similarly, we may assume

Ker(b) = [Ker(b), k(b), ε_{b}],
Cok(k(b)) = [Coim(b), c(k(b)), π_{k(b)}]

and (k(b) ∘ μ_{b}) · ε_{b} = (π_{k(b)} ∘ j(b)) · j(b)_{I}^{♭}. By (the dual of) Proposition 5.4, there are \underline{f}_{1} ∈ S^{1}(Ker(a), Ker(b)) and \underline{λ} ∈ S^{2}(\underline{f}_{1} ∘ k(b), k(a) ∘ f_{1}) such that (\underline{λ} ∘ b) · (k(a) ∘ λ) · (ε_{a} ∘ f_{2}) · (f_{2})_{I}^{♭} = (\underline{f}_{1} ∘ ε_{b}) · (\underline{f}_{1})_{I}^{♯}. Applying Proposition 5.8, we can show the existence of (f, λ_{1}, λ_{2}). To show the uniqueness (up to an equivalence), let (f′, λ′_{1}, λ′_{2}) satisfy

(f_{1} ∘ μ_{b}) · λ = (λ′_{1} ∘ j(b)) · (c(k(a)) ∘ λ′_{2}) · (μ_{a} ∘ f_{2}).

From this, we can obtain

(\underline{f}_{1} ∘ π_{k(b)}) · (\underline{f}_{1})_{I}^{♯} = (\underline{λ} ∘ c(k(b))) · (k(a) ∘ λ′_{1}) · (π_{k(a)} ∘ f′) · f_{I}^{′♭}.

And the uniqueness follows from the uniqueness of 2-cells in Proposition 5.4 and Proposition 5.8.

5.10. Proposition. Let f_{•} : A_{•} ⟶ B_{•} be a complex morphism as in diagram (30). If we take the cokernels of f_{1}, f_{2}, f_{3} and relative cokernels of the complex A_{•} and B_{•} as in the following diagram, then we have Cok(\overline{f}_{3}) = Cok(\overline{d}_{2}^{B}, \overline{δ}_{2}^{B}).

[figure: A_{1}; A_{2}; A_{3}; Cok(d_{2}^{A}, δ_{2}^{A}); B_{1}; B_{2}; B_{3}; Cok(d_{2}^{B}, δ_{2}^{B}); Cok(f_{1}); Cok(f_{2}); Cok(f_{3}); d_{1}^{A}; d_{2}^{A}; c(d_{2}^{A},δ_{2}^{A}); f_{1}; f_{2}; f_{3}; _{\overline{f}_{3}}; d_{1}^{B}; d_{2}^{B}; c(d_{2}^{B},δ_{2}^{B}); c(f_{1}); c(f_{2}); c(f_{3}); _{\overline{d}{B \atop 1}}; _{\overline{d}{B \atop 2}}; 0; 0; λ_{1}; λ_{2}; ∃; δ_{2}^{A}; _{\overline{δ}{B \atop 2}}; _{\overline{λ}_{1}}; _{\overline{λ}_{2}}]

Proof. Immediately follows from Proposition 5.7, Proposition 5.8 and (the dual of) Proposition 3.20.

5.11. Proposition. In diagram (27), if a is fully cofaithful, then the following diagram obtained in Proposition 5.4 is a pushout diagram.

[figure: B_{1}; B_{2}; Cok(f_{1}); Cok(f_{2}); b; \overline{b}; c(f_{1}); c(f_{2}); \overline{λ}]

Proof. Left to the reader.

Concerning Proposition 3.32, we have the following proposition.

5.12. Proposition. Let

[figure: A_{1} ×_{B} A_{2}; A_{1}; A_{2}; B; f′_{1}; f′_{2}; f_{1}; f_{2}; ξ]

be a pullback diagram in S. If f_{1} is fully cofaithful, then f′_{1} is fully cofaithful.

Proof. Since f_{1} is cofaithful, in the notation of the proof of Proposition 3.32, Cok(i_{1}) = [A_{2}, p_{2}, ξ_{2}] and Cok(d) = [B, t, ε_{t}]. Applying Proposition 5.7 to the diagram

[figure: 0; A_{1} ×_{B} A_{2}; A_{1}; A_{1} × A_{2}; ,; 0; 0; d; i_{1}]

we obtain

Cok(f_{1}) = 0 ⟺ Cok(f′_{1}) = 0.

[figure: 0; A_{1} ×_{B} A_{2}; A_{1} ×_{B} A_{2}; A_{1}; A_{1} × A_{2}; A_{2}; A_{1}; B; 0; 0; 0; 0; 0; id; id; d; t; i_{1}; p_{2}; f_{1}; f′_{1}; ⟳]

5.13. Proposition. In diagram (27), assume a is cofaithful. By Proposition 5.9, we obtain a coimage factorization diagram as (31). If we take the cokernel of this diagram as

[figure: B_{1}; Coim(b); B_{2}; Cok(f_{1}); Cok(f); Cok(f_{2}); ,; c(f_{1}); c(f); c(f_{2}); c(k(b)); j(b); \overline{c(k(b))}; \overline{j(b)}; b; \overline{b}; \overline{λ_{1}}; \overline{λ_{2}}; μ_{b}; \overline{μ_{b}}]

then the factorization

[figure: Cok(f_{1}); Cok(f_{2}); Cok(f); \overline{b}; \overline{j(b)}; \overline{c(k(b))}; \overline{μ_{b}}]

becomes again a coimage factorization.

Proof. It suffices to show that \overline{c(k(b))} is cofaithful and \overline{j(b)} is fully faithful. Since c(k(b)) and c(f) are cofaithful, it follows that \overline{c(k(b))} is cofaithful. Since j(a) is an equivalence,

[figure: Coim(b); B_{2}; Cok(f); Cok(f_{2}); c(f); c(f_{2}); j(b); \overline{j(b)}; \overline{λ_{2}}]

is a pushout diagram by Proposition 5.11. By (the dual of) Proposition 5.12, \overline{j(b)} becomes fully faithful.

Definition of the relative 2-exactness.

5.14. Lemma. Consider the following diagram in S.

[figure: A; B; C; f; g; 0; φ]  (32)

If we factor it as

[figure: Ker(g); Cok(f); A; B; C; f; g; k(g); c(f); \underline{f}; \overline{g}; \underline{φ}; \overline{φ}; 0; φ]  (33)

with

(\underline{φ} ∘ g) · φ = (\underline{f} ∘ ε_{g}) · (\underline{f})_{I}^{♯}
(f ∘ \overline{φ}) · φ = (π_{f} ∘ \overline{g}) · (\overline{g})_{I}^{♭},

then Cok(\underline{f}) = 0 if and only if Ker(\overline{g}) = 0.

Proof. We show only Cok(\underline{f}) = 0 ⇒ Ker(\overline{g}) = 0, since the other implication can be shown dually. If Cok(\underline{f}) = 0, i.e. if \underline{f} is fully cofaithful, then we have

Cok(f) = Cok(\underline{f} ∘ k(g)) = Cok(k(g)) = Coim(g).

Thus the following diagram is a coimage factorization, and \overline{g} becomes fully faithful.

[figure: Cok(f); B; C; g; c(f); \overline{g}; \overline{φ}]

5.15. Definition. Diagram (32) is said to be 2-exact in B, if Cok(\underline{f}) = 0 (or equivalently Ker(\overline{g}) = 0 ).

5.16. Remark. In the notation of Lemma 5.14, the following are equivalent : (i) (32) is 2-exact in B. (ii) \underline{f} is fully cofaithful. (iii) \overline{g} is fully faithful. (iv) c(f) = cok(k(g)) (i.e. Cok(f) = Coim(g)).

(v) k(g) = ker(c(f)) (i.e. Ker(g) = Im(f)).

Proof. By the duality, we only show (i) ⇔ (iii) ⇔ (v).

(i) ⇔ (iii) follows from Corollary 3.25.

(iii) ⇒ (v) follows from Proposition 3.21.

(v) ⇒ (iii) follows from Proposition 4.3.

Let us fix the notation for relative (co-)kernels of a complex.

5.17. Definition. For any complex A_{•} = (A_{n}, d_{n}, δ_{n}) in S, we put

(1) [Z^{n}(A_{•}), z_{n}^{A}, ζ_{n}^{A}] := Ker(d_{n}, δ_{n+1}).

(2) [Q^{n}(A_{•}), q_{n}^{A}, ρ_{n}^{A}] := Cok(d_{n−1}, δ_{n−1}).

5.18. Remark. By the universality of Ker(d_{n}, δ_{n+1}) and Lemma 3.19, there exist k_{n} ∈ S^{1}(A_{n−1}, Z^{n}(A_{•})), ν_{n,1} ∈ S^{2}(k_{n} ∘ z_{n}, d_{n−1}) and ν_{n,2} ∈ S^{2}(d_{n−2} ∘ k_{n}, 0) such that

(ν_{n,1} ∘ d_{n}) · δ_{n} = (k_{n} ∘ ζ_{n}) · (k_{n})_{I}^{♯}
(d_{n−2} ∘ ν_{n,1}) · δ_{n−1} = (ν_{n,2} ∘ z_{n}) · (z_{n})_{I}^{♭}.

[figure: Z^{n}(A_{•}); A_{n−2}; A_{n−1}; A_{n}; A_{n+1}; A_{n+2}; d_{n−2}; d_{n−1}; k_{n}; d_{n}; d_{n+1}; z_{n}; ζ_{n}; ν_{n,2}; 0; 0; 0; 0; δ_{n}; δ_{n−1}; ν_{n,1}]

On the other hand, by the universality of Ker(d_{n}), we obtain a factorization diagram

[figure: Ker(d_{n}); A_{n−2}; A_{n−1}; A_{n}; A_{n+1}; A_{n+2}; d_{n−2}; d_{n−1}; _{\underline{d}_{n−1}}; d_{n}; d_{n+1}; k(d_{n}); ε_{dn}; _{\underline{δ}_{n−1}}; 0; 0; 0; 0; δ_{n}; δ_{n−1}; _{\underline{δ}_{n}}]

which satisfy

(\underline{δ}_{n} ∘ d_{n}) · δ_{n} = (\underline{d}_{n−1} ∘ ε_{d_{n}}) · (\underline{d}_{n−1})_{I}^{♯}
(d_{n−2} ∘ \underline{δ}_{n}) · δ_{n−1} = (\underline{δ}_{n−1} ∘ k(d_{n})) · (k(d_{n}))_{I}^{♭}

By Proposition 3.20, there exists a factorization of z_{n} through Ker(d_{n})

[figure: Z^{n}(A_{•}); Ker(d_{n}); A_{n}; A_{n+1}; d_{n}; k(d_{n}); \underline{z}^{n}; ε_{dn}; 0; 0; z_{n}; ζ_{n}; \underline{ζ}^{n}]

which satisfies

(\underline{ζ}_{n} ∘ d_{n}) · ζ_{n} = (\underline{z}_{n} ∘ ε_{d_{n}}) · (\underline{z}_{n})_{I}^{♯}.

Moreover \underline{z}_{n} is fully faithful by Proposition 3.33.

By the universality of Ker(d_{n}), we can show easily the following claim. 5.19. Claim. There exists a unique 2-cell ζ̂_{n} ∈ S(k_{n} ∘ z_{n}, \underline{d}_{n−1})

[figure: Z^{n}(A_{•}); Ker(d_{n}); A_{n−2}; A_{n−1}; A_{n}; A_{n+1}; d_{n−2}; k_{n}; d_{n}; z_{n}; _{\underline{d}_{n−1}}; k(d_{n}); \underline{z}^{n}; 0; 0; 0; 0; ζ_{n}; ν_{n,2}; ε_{dn}; _{\underline{δ}_{n−1}}; ζ̂_{n}; \underline{ζ}_{n}]

such that

(ζ̂_{n} ∘ k(d_{n})) · \underline{δ}_{n} = (k_{n} ∘ \underline{ζ}_{n}) · ν_{n,1}.

This ζ̂_{n} also satisfies

(d_{n−2} ∘ ζ̂_{n}) · \underline{δ}_{n−1} = (ν_{n,2} ∘ \underline{z}_{n}) · (\underline{z}_{n})_{I}^{♭}.

5.20. Remark. Dually, by the universality of the cokernels, we obtain the following two factorization diagrams, where \overline{q}_{n} is fully cofaithful.

[figure: Q^{n}(A_{•}); Cok(d_{n−1}); A_{n−2}; A_{n−1}; A_{n}; A_{n+1}; A_{n+2}; d_{n−2}; d_{n−1}; q_{n}; c(d_{n−1}); d_{n}; d_{n+1}; ℓ_{n}; _{\overline{d}_{n}}; μ_{n,2}; ρ_{n}; _{\overline{δ}_{n+1}}; π_{d_{n−1}}; 0; 0; 0; 0; μ_{n,1}; _{\overline{δ}_{n}}]

[figure: Q^{n}(A_{•}); Cok(d_{n−1}); A_{n−1}; A_{n}; A_{n+1}; A_{n+2}; q_{n}; c(d_{n−1}); d_{n+1}; \overline{q}_{n}; d_{n−1}; ℓ_{n}; _{\overline{d}_{n}}; μ_{n,2}; ρ_{n}; _{\overline{δ}_{n+1}}; π_{d_{n−1}}; 0; 0; 0; 0; \overline{ρ}_{n}; ρ̂_{n}]

We define relative 2-cohomology in the following two ways, which will be shown to be equivalent later.

5.21. Definition.

H_{1}^{n}(A_{•}) := Cok(k_{n}, ν_{n,2})
H_{2}^{n}(A_{•}) := Ker(ℓ_{n}, μ_{n,2})

5.22. Lemma. In the factorization diagram (33) in Lemma 5.14, if we take the cokernel of \underline{f} and the kernel of \overline{g}, then there exist w ∈ S^{1}(Cok(\underline{f}), Ker(\overline{g})) and ω ∈ S^{2}(c(\underline{f}) ∘ w ∘ k(\overline{g}), k(g) ∘ c(f)) such that

(\underline{f} ∘ ω) · (\underline{φ} ∘ c(f)) · π_{f} = (π\underline{f} ∘ w ∘ k(\overline{g})) · (w ∘ k(\overline{g}))_{I}^{♭}  (34)
(ω ∘ \overline{g}) · (k(g) ∘ \overline{φ}) · ε_{g} = (c(\underline{f}) ∘ w ∘ ε\overline{g}) · (c(\underline{f}) ∘ w)_{I}^{♯}.  (35)

[figure: Ker(g); Cok(f); A; B; C; Cok(\underline{f}); Ker(\overline{g}); w; f; g; k(g); c(f); k(\overline{g}); c(\underline{f}); \underline{f}; \overline{g}; 0; 0; 0; \underline{φ}; \overline{φ}; φ; ω; π\underline{f}; ε\overline{g}]

Moreover, for any other factorization (w′, ω′) with these properties, there exists a unique 2-cell κ ∈ S^{2}(w, w′) such that (c(\underline{f}) ∘ κ ∘ k(\overline{g})) · ω′ = ω.

Proof. Applying Proposition 5.4 to

[figure: A; Ker(g); A; B; ,; ^{id^{A}}; \underline{f}; k(g); f; \overline{φ}^{−1}]  (36)

we obtain w_{1} ∈ S^{1}(Cok(\underline{f}), Cok(f)) and ω_{1} ∈ S^{2}(c(\underline{f}) ∘ w_{1}, k(g) ∘ c(f)) which satisfy

(\underline{f} ∘ ω_{1}) · (\underline{φ} ∘ c(f)) · π_{f} = (π\underline{f} ∘ w_{1}) · (w_{1})_{I}^{♭}.  (37)

Then (ω_{1} ∘ \overline{g}) · (k(g) ∘ \overline{φ}) · ε_{g} ∈ S^{2}(c(\underline{f}) ∘ w_{1} ∘ \overline{g}, 0) becomes compatible with π\underline{f}.

[figure: A; Ker(g); Cok(\underline{f}); C; \underline{f}; c(\underline{f}); w_{1}∘\overline{g}; 0; 0; π\underline{f}; (ω_{1}∘\overline{g})·(k(g)∘\overline{φ})·ε_{g}]

By Lemma 3.19, there exists a 2-cell δ ∈ S(w_{1} ∘ \overline{g}, 0) such that

(c(\underline{f}) ∘ δ) · c(\underline{f})_{I}^{♯} = (ω_{1} ∘ \overline{g}) · (k(g) ∘ \overline{φ}) · ε_{g}.

So, if we take the cokernels of k(g) and w_{1}, then by Proposition 5.8, we obtain the following diagram:

[figure: Ker(g); B; Cok(\underline{f}); Cok(f); C; C; Coim(g); Cok(w_{1}); c(k(g)); c(w_{1}); j(g); ∃g^{†}; k(g); _{id_{C}}; w_{1}; ∃\overline{c}; c(\underline{f}); c(f); g; \overline{g}; ∃\overline{φ}_{2}; ∃\overline{φ}_{1}; ω_{1}; μ_{g}; ∃]

Applying Proposition 5.7 to (36), we obtain

[Cok(w_{1}), \overline{c}, (\overline{c})_{I}^{♭}] = Cok(0 ⟶^{0} Coim(g)).

Thus \overline{c} is an equivalence. Since j(g) is fully faithful, g^{†} becomes fully faithful. Thus the following diagram is 2-exact in Cok(f).

[figure: Cok(\underline{f}); Cok(f); C; w_{1}; \overline{g}; 0; δ]  (38)

So if we factor (38) by w ∈ S^{1}(Cok(\underline{f}), Ker(\overline{g})) and ω_{2} ∈ S^{2}(w∘k(\overline{g}), w_{1}) as in the diagram

[figure: Cok(\underline{f}); Cok(f); Ker(\overline{g}); C; w_{1}; w; k(\overline{g}); \overline{g}; 0; 0; δ; ω_{2}; ε\overline{g}]  (39)

which satisfies

(w ∘ ε\overline{g}) · w_{I}^{♯} = (ω_{2} ∘ \overline{g}) · δ,

then w becomes fully cofaithful by Lemma 5.14. If we put ω := (c(\underline{f}) ∘ ω_{2}) · ω_{1}, then (w, ω) satisfies conditions (34) and (35).

If (w′, ω′) satisfies

(\underline{f} ∘ ω′) · (\underline{φ} ∘ c(f)) · π_{f} = (π\underline{f} ∘ w′ ∘ k(\overline{g})) · (w′ ∘ k(\overline{g}))_{I}^{♭}  (40)
(ω′ ∘ \overline{g}) · (k(g) ∘ \overline{φ}) · ε_{g} = (c(\underline{f}) ∘ w′ ∘ ε\overline{g}) · (c(\underline{f}) ∘ w′)_{I}^{♯},

then, since both the factorization of k(g) ∘ c(f) through Cok(\underline{f})

ω′ : c(\underline{f}) ∘ w′ ∘ k(\overline{g}) ⟹ k(g) ∘ c(f)
ω_{1} : c(\underline{f}) ∘ w_{1} ⟹ k(g) ∘ c(f)

are compatible with π\underline{f} and (\underline{φ}∘c(f))·π_{f} by (37) and (40), there exists ω′_{2} ∈ S^{2}(w′∘k(\overline{g}), w_{1}) such that

(c(\underline{f}) ∘ ω′_{2}) · ω_{1} = ω′.

Then we can see ω′_{2} is compatible with ε\overline{g} and δ. So, comparing this with the factorization (39), by the universality of Ker(\overline{g}), we see there exists a unique 2-cell κ ∈ S^{2}(w, w′) such that (κ ∘ k(\overline{g})) · ω′_{2} = ω_{2}. Then κ satisfies (c(\underline{f}) ∘ κ ∘ k(\overline{g})) · ω′ = ω. Uniqueness of such κ follows from the fact that c(\underline{f}) is cofaithful and k(\overline{g}) is faithful.

5.23. Proposition. In Lemma 5.22, w is an equivalence.

Proof. We showed Lemma 5.22 by taking the cokernel first and the kernel second, but we obtain the same (w, ω) if we take the kernel first and the cokernel second, because of the symmetricity of the statement (and the uniqueness of (w, ω) up to an equivalence) of Lemma 5.22. As shown in the proof, since (38) is 2-exact in Cok(\underline{f}), w becomes fully cofaithful in the factorization (39). By the above remark, similarly w can be obtained also by the factorization

[figure: A; Ker(g); Cok(\underline{f}); Ker(\overline{g}); w; 0]

where the bottom row is 2-exact in Ker(g). So w becomes fully faithful. Thus, w is fully cofaithful and fully faithful, i.e. an equivalence.

5.24. Corollary. For any complex A_{•} = (A_{n}, d_{n}, δ_{n}), if we factor it as

[figure: Z^{n}(A_{•}); Q^{n}(A_{•}); A_{n−1}; A_{n}; A_{n+1}; A_{n−2}; A_{n+2}; H_{1}^{n}(A_{•}); H_{2}^{n}(A_{•}); d_{n−1}; d_{n}; z_{n}; q_{n}; d_{n−2}; d_{n+2}; k(ℓ_{n},μ_{n,2}); c(k_{n},ν_{n,2}); k_{n}; ℓ_{n}; ν_{n,1}; μ_{n,1}; 0; 0; 0; 0; 0; δ_{n}^{A}; ν_{n,2}; μ_{n,2}]

(in the notation of Definition 5.17, Remark 5.18 and Remark 5.20), then there exist w ∈ S^{1}(H_{1}^{n}(A_{•}), H_{2}^{n}(A_{•})) and ω ∈ S^{2}(c(k_{n}, ν_{n,2}) ∘ w ∘ k(ℓ_{n}, μ_{n,2}), z_{n} ∘ q_{n}) such that

(k_{n} ∘ ω) · (ν_{n,1} ∘ q_{n}) · ρ_{n} = (π_{(k_{n},ν_{n,2})} ∘ w ∘ k(ℓ_{n}, μ_{n,2})) · (w ∘ k(ℓ_{n}, μ_{n,2}))_{I}^{♭}
(ω ∘ ℓ_{n}) · (z_{n} ∘ μ_{n,1}) · ζ_{n} = (c(k_{n}, ν_{n,2}) ∘ w ∘ ε_{(ℓ_{n},μ_{n,2})}) · (c(k_{n}, ν_{n,2}) ∘ w)_{I}^{♯}.

[figure: Z^{n}(A_{•}); Q^{n}(A_{•}); A_{n−1}; A_{n}; A_{n+1}; A_{n−2}; A_{n+2}; H_{1}^{n}(A_{•}); H_{2}^{n}(A_{•}); w; d_{n−1}; d_{n}; z_{n}; q_{n}; d_{n−2}; d_{n+2}; k(ℓ_{n},μ_{n,2}); c(k_{n},ν_{n,2}); k_{n}; ℓ_{n}; 0; 0; 0; 0; 0; ν_{n,1}; μ_{n,1}; δ_{n}^{A}; ω; ν_{n,2}; μ_{n,2}]

For any other factorization (w′, ω′) with these conditions, there exists a unique 2-cell κ ∈ S^{2}(ω, ω′) such that (c(k_{n}, ν_{n,2}) ∘ κ ∘ k(ℓ_{n}, μ_{n,2})) · ω′ = ω. Moreover, this w becomes an equivalence.

Proof. For the factorization diagrams

[figure: Cok(d_{n−2}); A_{n−2}; A_{n−1}; A_{n}; A_{n+1}; d_{n−2}; c(d_{n−2}); d_{n−1}; d_{n}; _{\overline{d}_{n−1}}; 0; 0; 0; 0; _{δ_{n}}; π_{d_{n−2}}; δ_{n−1}; δ_{n}; _{δ_{n−1}}]

[figure: Ker(d_{n+1}); A_{n−1}; A_{n}; A_{n+1}; A_{n+2}; d_{n−1}; d_{n}; _{d_{n}}; d_{n+1}; k(d_{n+1}); 0; 0; 0; 0; ε_{d_{n+1}}; _{δ_{n}}; δ_{n+1}; δ_{n}; _{δ_{n+1}}]

which satisfy

(d_{n−2} ∘ \overline{δ}_{n−1}) · δ_{n−1} = (π_{d_{n−2}} ∘ \overline{d}_{n−1}) · (\overline{d}_{n−1})_{I}^{♭}
(\overline{δ}_{n−1} ∘ d_{n}) · δ_{n} = (c(d_{n−2}) ∘ \overline{δ}_{n}) · c(d_{n−2})_{I}^{♯}
(\underline{δ}_{n+1} ∘ d_{n+1}) · δ_{n+1} = (\underline{d}_{n} ∘ ε_{d_{n+1}}) · (\underline{d}_{n})_{I}^{♯}
(d_{n−1} ∘ \underline{δ}_{n+1}) · δ_{n} = (\underline{δ}_{n} ∘ k(d_{n+1})) · k(d_{n+1})_{I}^{♭},

there exists a unique 2-cell δ_{n}^{†} ∈ S^{2}(\overline{d}_{n−1} ∘ \underline{d}_{n}, 0) such that

(δ_{n}^{†} ∘ k(d_{n+1})) · k(d_{n+1})_{I}^{♭} = (\overline{d}_{n−1} ∘ \underline{δ}_{n+1}) · \overline{δ}_{n}
(c(d_{n−2}) ∘ δ_{n}^{†}) · c(d_{n−2})_{I}^{♯} = (\overline{δ}_{n−1} ∘ \underline{d}_{n}) · \underline{δ}_{n}.

By Proposition 3.20, applying Lemma 5.22 and Proposition 5.23 to the following diagram, we can obtain Corollary 5.24.

[figure: Z^{n}(A_{•}); Q^{n}(A_{•}); Cok(d_{n−2}); A_{n}; Ker(d_{n+1}); _{\overline{d}_{n−1}}; _{\underline{d}_{n}}; z_{n}; q_{n}; 0; δ_{n}^{†}]

Thus H_{1}^{n}(A_{•}) and H_{2}^{n}(A_{•}) are equivalent. We abbreviate this to H^{n}(A_{•}).

5.25. Definition. A complex A_{•} is said to be relatively 2-exact in A_{n} if H^{n}(A_{•}) is equivalent to zero.

5.26. Remark. If the complex is bounded, we consider the relative 2-exactness after adding zeroes as in Remark 5.2. For example, a bounded complex

[figure: A; B; C; f; g; 0; φ]

is relatively 2-exact in B if and only if

[figure: 0; A; B; C; 0; 0; f; g; 0; 0; 0; 0; f_{I}^{♭}; g_{I}^{♯}; φ]

is relatively 2-exact in B, and this is equivalent to the 2-exactness in B by Remark 3.18.

6. Long cohomology sequence in a relatively exact 2-category

Diagram lemmas (2).

6.1. Lemma. Let A_{•} be a complex in S, in which A_{5} = 0 and d_{4} = 0 :

[figure: A_{1}; A_{2}; A_{3}; A_{4}; 0; d_{1}^{A}; d_{2}^{A}; d_{3}^{A}; 0; 0; 0; 0; δ_{2}^{A}; δ_{3}^{A}; (d_{3}^{A})_{I}^{♯}]  (41)

Then, (41) is relatively 2-exact in A_{3} and A_{4} if and only if Cok(d_{2}, δ_{2}) = A_{4}, i.e. [Q^{3}(A_{•}), q_{3}, ρ_{3}] = [A_{4}, d_{3}, δ_{3}].

Proof. As in Remark 5.20, we have two factorization diagrams

[figure: Q^{3}(A_{•}); Cok(d_{2}); A_{2}; A_{3}; A_{4}; d_{2}; q_{3}; c(d_{2}); d_{3}; ℓ_{3}; _{\overline{d}_{3}}; ρ_{3}; π_{d_{2}}; 0; 0; μ_{3,1}; _{\overline{δ}_{3}}; Q^{3}(A_{•}); Cok(d_{2}); A_{2}; A_{3}; A_{4}; q_{3}; c(d_{2}); \overline{q}_{3}; d_{2}; ℓ_{3}; _{\overline{d}_{3}}; ρ_{3}; π_{d_{2}}; 0; 0; _{\overline{ρ}_{3}}; ρ̂_{3}]

where \overline{q}_{3} is fully cofaithful. We have

(41) is relatively 2-exact in A_{4}  ⇔ Cok(d_{3}, δ_{3}) = 0
⇔_{Prop. 3.20} Cok(\overline{d}_{3}) = 0 ⇔ Cok(\overline{q}_{3} ∘ ℓ_{3}) = 0
⇔_{Prop. 3.21} Cok(ℓ_{3}) = 0 ⇔ ℓ_{3} is fully cofaithful

and

(41) is relatively 2-exact in A_{3} ⇔ Ker(ℓ_{3}, (ℓ_{3})_{I}^{♯}) = 0
⇔_{Rem. 3.18} Ker(ℓ_{3}) = 0 ⇔ ℓ_{3} is fully faithful.

Thus, (41) is relatively 2-exact in A_{3} and A_{4} if and only if ℓ is fully cofaithful and fully faithful, i.e. ℓ is an equivalence.

By Remark 3.18, we have the following corollary:

6.2. Corollary. Let (A_{n}, d_{n}, δ_{n}) be a bounded complex in S, as follows:

[figure: A_{1}; A_{2}; A_{3}; 0; d_{1}^{A}; d_{2}^{A}; 0; 0; 0; δ_{2}^{A}; (d_{2}^{A})_{I}^{♯}]  (42)

Then, (42) is relatively 2-exact in A_{2} and A_{3} if and only if Cok(d_{1}^{A}) = [A_{3}, d_{2}^{A}, δ_{2}^{A}].

6.3. Lemma. Let A_{•} be a complex. As in Definition 5.17, Remark 5.18 and Remark 5.20, take a factorization diagram

[figure: Z^{n+1}(A_{•}); Q^{n}(A_{•}); A_{n−1}; A_{n}; A_{n+1}; A_{n+2}; d_{n−1}; k_{n+1}; q_{n}; d_{n}; d_{n+1}; z_{n+1}; ℓ_{n}; ζ_{n+1}; ν_{n+1,2}; μ_{n,2}; ρ_{n}; 0; 0; 0; 0; ν_{n+1,1}; μ_{n,1}]

which satisfies

(ν_{n+1,1} ∘ d_{n+1}) · δ_{n+1} = (k_{n+1} ∘ ζ_{n+1}) · (k_{n+1})_{I}^{♯}
(d_{n−1} ∘ ν_{n+1,1}) · δ_{n} = (ν_{n+1,2} ∘ z_{n+1}) · (z_{n+1})_{I}^{♭}
(d_{n−1} ∘ μ_{n,1}) · δ_{n} = (ρ_{n} ∘ ℓ_{n}) · (ℓ_{n})_{I}^{♭}
(μ_{n,1} ∘ d_{n+1}) · δ_{n+1} = (q_{n} ∘ μ_{n,2}) · (q_{n})_{I}^{♯}.

Then, there exist x_{n} ∈ S^{1}(Q^{n}(A_{•}), Z^{n+1}(A_{•})), ξ_{n} ∈ S^{2}(x_{n} ∘ z_{n+1}, ℓ_{n}) and η_{n} ∈ S^{2}(q_{n} ∘ x_{n}, k_{n+1}) such that

(ξ_{n} ∘ d_{n+1}) · μ_{n,2} = (x_{n} ∘ ζ_{n+1}) · (x_{n})_{I}^{♯}
(q_{n} ∘ ξ_{n}) · μ_{n,1} = (η_{n} ∘ z_{n+1}) · ν_{n+1,1}
(d_{n−1} ∘ η_{n}) · ν_{n+1,2} = (ρ_{n} ∘ x_{n}) · (x_{n})_{I}^{♭}.  (43)

Moreover, for any other (x′_{n}, ξ′_{n}, η′_{n}) with these properties, there exists a unique 2-cell κ ∈ S^{2}(x_{n}, x′_{n}) such that (κ ∘ z_{n+1}) · ξ′_{n} = ξ_{n} and (q_{n} ∘ κ) · η′_{n} = η_{n}.

Proof. By the cofaithfulness of q_{n}, we can show μ_{n,2} is compatible with δ_{n+2}. By the universality of the relative kernel Z^{n+1}(A_{•}), there exist x_{n} ∈ S^{1}(Q^{n}(A_{•}), Z^{n+1}(A_{•})) and ξ_{n} ∈ S^{2}(x_{n} ∘ z_{n+1}, ℓ_{n}) such that

(ξ_{n} ∘ d_{n+1}) · μ_{n,2} = (x_{n} ∘ ζ_{n+1}) · (x_{n})_{I}^{♯}.

Then, both the factorizations

ν_{n+1,1} : k_{n+1} ∘ z_{n+1} ⟹ d_{n}
(q_{n} ∘ ξ_{n}) · μ_{n,1} : q_{n} ∘ x_{n} ∘ z_{n+1} ⟹ d_{n}

are compatible with ζ_{n+1} and δ_{n+1}. Thus by the universality of relative kernel Z^{n+1}(A_{•}), there exists a unique 2-cell η_{n} ∈ S^{2}(q_{n}∘x_{n}, k_{n+1}) such that (q_{n}∘ξ_{n})·μ_{n,1} = (η_{n}∘z_{n+1})·ν_{n+1,1}. It can be easily seen that η_{n} also satisfies (43). Uniqueness (up to an equivalence) of (x_{n}, ξ_{n}, η_{n}) follows from the universality of the relative kernel Z^{n+1}(A_{•}) and the uniqueness of η_{n}.

6.4. Lemma. Consider the following complex diagram in S.

[figure: A; B; C; f; g; 0; φ]  (44)

If (44) is 2-exact in B and g is cofaithful, then we have Cok(f) = [C, g, φ].

Proof. If we factor (44) as

[figure: A; B; Cok(f); C; ,; f; c(f); g; \overline{g}; 0; 0; π_{f}; φ; \overline{φ}]

then, since (44) is 2-exact in B, \overline{g} becomes fully faithful. On the other hand, since g is cofaithful, \overline{g} is also cofaithful. Thus \overline{g} becomes an equivalence.

6.5. Lemma. Consider the following complex morphism in S.

[figure: A_{1}; A_{1}; A_{2}; B_{2}; A_{3}; B_{3}; 0; 0; d_{1}^{A}; id; d_{1}^{B}; f_{2}; f_{3}; d_{2}^{A}; d_{2}^{B}; 0; 0; 0; 0; λ; κ; δ_{2}^{A}; δ_{2}^{B}]

If the complexes are relatively 2-exact in A_{2}, A_{3} and B_{2}, B_{3} respectively, i.e. they satisfy Cok(d_{1}^{A}) = [A_{3}, d_{2}^{A}, δ_{2}^{A}] and Cok(d_{1}^{B}) = [B_{3}, d_{2}^{B}, δ_{2}^{B}] (see Corollary 6.2), then the following diagram obtained by taking the kernel of f_{2} becomes 2-exact in A_{3}.

[figure: Ker(f_{2}); A_{3}; B_{3}; k(f_{2})∘d_{2}^{A}; f_{3}; 0; (k(f_{2})∘κ)·(ε_{f_{2}}∘d_{2}^{B})·(d_{2}^{B})_{I}^{♭}]  (45)

Proof. By taking the kernels of id_{A_{1}} and f_{2} in the diagram

[figure: A_{1}; A_{2}; A_{1}; B_{2}; d_{1}^{A}; id; f_{2}; d_{1}^{B}; λ]

and taking the cokernels of 0_{0,A_{1}} and k(f_{2}), we obtain the following diagram by Proposition 5.8, where θ = k(f_{2})_{I}^{♭} · (d_{1}^{A})_{I}^{♭−1}:

[figure: 0; Ker(f_{2}); A_{1}; A_{2}; A_{1}; Coim(f_{2}); A_{1}; B_{2}; 0; 0; k(f_{2}); d_{1}^{A}; id; c(k(f_{2})); \overline{d}_{1}^{A}; id; j(f_{2}); d_{1}^{B}; f_{2}; θ; λ_{1}; λ_{2}; μ_{f_{2}}]  (46)

By taking the cokernels of 0_{0,Ker(f_{2})}, d_{1}^{A} and d_{1}^{B} in (46), we obtain the left of the following diagrams, while by Proposition 5.13 we obtain the right as a coimage factorization if we take the cokernels of d_{1}^{A}, \overline{d}_{1}^{A} and d_{1}^{B} in (46):

[figure: Ker(f_{2}); Ker(f_{2}); A_{2}; A_{3}; B_{2}; B_{3}; id; k(f_{2}); k(f_{2})∘d_{2}^{A}; d_{2}^{A}; f_{2}; f_{3}; d_{2}^{B}; 0; 0; ⟳; κ; ε_{f_{2}}; (k(f_{2})∘κ)·(ε_{f_{2}}∘d_{2}^{B})·(d_{2}^{B})_{I}^{♭}; A_{2}; A_{3}; Coim(f_{2}); Coim(f_{3}); B_{2}; B_{3}; d_{2}^{A}; c(k(f_{2})); c(k(f_{3})); \overline{d}_{2}^{A}; j(f_{2}); j(f_{3}); d_{2}^{B}; f_{2}; f_{3}; κ_{1}; κ_{2}; μ_{f_{2}}; μ_{f_{3}}]

On the other hand by Proposition 5.7, if we take the compatible 2-cell υ = (k(f_{2}) ∘ κ_{1}) · (π_{k(f_{2})} ∘ \overline{d}_{2}^{A}) · (\overline{d}_{2}^{A})_{I}^{♭} ∈ S^{2}(k(f_{2}) ∘ d_{2}^{A} ∘ c(k(f_{3})), 0),

[figure: Ker(f_{2}); Ker(f_{2}); A_{2}; A_{3}; Coim(f_{2}); Coim(f_{3}); id; k(f_{2}); k(f_{2})∘d_{2}^{A}; d_{2}^{A}; c(k(f_{2})); c(k(f_{3})); \overline{d}_{2}^{A}; 0; 0; ⟳; κ_{1}; π_{k(f2)}; υ]

then we have Cok(k(f_{2}) ∘ d_{2}^{A}) = [Coim(f_{3}), c(k(f_{3})), υ]. It can be easily shown that υ is compatible with μ_{f_{3}} and (k(f_{2}) ∘ κ) · (ε_{f_{2}} ∘ d_{2}^{B}) · (d_{2}^{B})_{I}^{♭}.

[figure: Ker(f_{2}); B_{3}; A_{3}; Coim(f_{3}); 0; f_{3}; k(f_{2})∘d_{2}^{A}; c(k(f_{3})); j(f_{3}); 0; υ; μ_{f_{3}}; (k(f_{2})∘κ)·(ε_{f_{2}}∘d_{2}^{B})·(d_{2}^{B})_{I}^{♭}]

Since Cok(k(f_{2}) ∘ d_{2}^{A}) = [Coim(f_{3}), c(k(f_{3})), υ] and j(f_{3}) is fully faithful by Proposition 4.4, this means (45) is 2-exact in A_{3}.

6.6. Lemma. Consider the following complex morphism in S.

[figure: A_{1}; B_{1}; A_{2}; B_{2}; A_{3}; B_{3}; 0; d_{1}^{A}; f_{1}; d_{1}^{B}; f_{2}; f_{3}; d_{2}^{A}; d_{2}^{B}; 0; 0; 0; λ_{1}; λ_{2}; δ_{2}^{A}; δ_{2}^{B}]  (47)

If the complexes are relatively 2-exact in A_{2} and B_{1}, B_{2} respectively, then the following diagram obtained by taking the kernels

[figure: Ker(f_{1}); Ker(f_{2}); Ker(f_{3}); _{\underline{d}_{A}_{1}}; _{\underline{d}_{A}_{2}}; 0; ^{\underline{δ}^{A}^{2}}]  (48)

is 2-exact in Ker(f_{2}).

Proof. If we decompose (47) into

[figure: A_{1}; B_{1}; Ker(d_{2}^{A}); Ker(d_{2}^{B}); 0; d_{1}^{A†}; f_{1}; d_{1}^{B†}; \underline{f}_{2}; 0; λ_{1}^{†}; and; Ker(d_{2}^{A}); Ker(d_{2}^{B}); A_{2}; B_{2}; A_{3}; B_{3}; ,; k(d_{2}^{A}); \underline{f}_{2}; k(d_{2}^{B}); f_{2}; f_{3}; d_{2}^{A}; d_{2}^{B}; \underline{λ}_{1}; λ_{2}]

then by (the dual of) Proposition 5.7, we have Ker(\underline{d}_{2}^{A}) = Ker(\underline{f}_{2}). Since d_{1}^{B†} is an equivalence by (the dual of) Corollary 6.2, the diagram obtained by taking the kernels of f_{1} and \underline{f}_{2}

[figure: Ker(f_{1}); A_{1}; Ker(\underline{f}_{2}); Ker(d_{2}^{A}); ^{\underline{d}{A† \atop 1}}; k(f_{1}); d_{1}^{A†}; k(\underline{f}_{2}); \underline{λ}_{1}^{†}]

becomes a pullback diagram by (the dual of) Proposition 5.11. Since d_{1}^{A†} is fully cofaithful, \underline{d}_{1}^{A†} becomes also fully cofaithful by Proposition 5.12. This means (48) is 2-exact in Ker(f_{2}).

6.7. Lemma. Consider the following complex morphism in S.

[figure: A_{1}; A_{1}; A_{2}; A_{2}; A_{3}; B_{3}; d_{1}^{A}; id; d_{1}^{B}; id; f_{3}; d_{2}^{A}; d_{2}^{B}; 0; 0; λ_{1}; λ_{2}; δ_{2}^{A}; δ_{2}^{B}]  (49)

If f_{3} is faithful and the bottom row is 2-exact in A_{2}, then the top row is also 2-exact in A_{2}.

Proof. By taking the cokernels of d_{1}^{A} and d_{1}^{B} in (49), we obtain (by Proposition 5.8)

[figure: Cok(d_{1}^{A}); A_{3}; Cok(d_{1}^{B}); B_{3}; .; ^{\overline{d}{A \atop 2}}; id; f_{3}; _{\overline{d}{B \atop 2}}; \overline{λ}_{2}]

Since \overline{d}_{2}^{B} is fully faithful, by taking the kernels in this diagram, we obtain the following diagram.

[figure: 0; Ker(f_{3}); Ker(\overline{d}_{2}^{A}); Cok(d_{1}^{A}); A_{3}; 0; Cok(d_{1}^{B}); B_{3}; 0; 0; k(f_{3}); k(\overline{d}_{2}^{A}); \overline{d}_{2}^{A}; 0; id; f_{3}; 0; _{\overline{d}{B \atop 2}}; ∃; ∃; _{\overline{λ}_{2}}]

In this diagram, we have

Ker(\overline{d}_{2}^{A}) = Ker(Ker(\overline{d}_{2}^{A}) ⟶ 0) =_{Prop. 5.7} Ker(0 ⟶ Ker(f_{3})) =_{Cor. 3.28} 0.

This means that the top row in (49) is 2-exact in A_{2}.

6.8. Corollary. Let

[figure: A_{1}; A_{2}; A_{3}; d_{1}^{A}; d_{2}^{A}; 0; δ_{2}^{A}; and; B_{1}; A_{2}; B_{3}; d_{1}^{B}; d_{2}^{B}; 0; δ_{2}^{B}]

be two complexes, and assume that there exist 1-cells f_{1}, f_{3} and 2-cells λ_{1}, λ_{2}, σ as in the following diagram

[figure: B_{1}; A_{1}; A_{2}; A_{3}; B_{3}; ,; f_{1}; d_{1}^{B}; d_{1}^{A}; d_{2}^{B}; d_{2}^{A}; f_{3}; 0; σ; λ_{1}; λ_{2}]

where f_{1} is cofaithful and f_{3} is faithful. Assume they satisfy

(d1) (λ_{1} ∘ d_{2}^{B}) · δ_{2}^{B} = (f_{1} ∘ σ) · (f_{1})_{I}^{♯}
(d2) (d_{1}^{A} ∘ λ_{2}) · σ = (δ_{2}^{A} ∘ f_{3}) · (f_{3})_{I}^{♭}.

Then, if the diagram

[figure: B_{1}; A_{2}; B_{3}; d_{1}^{B}; d_{2}^{B}; 0; δ_{2}^{B}]

is 2-exact in A_{2}, then the diagram

[figure: A_{1}; A_{2}; A_{3}; d_{1}^{A}; d_{2}^{A}; 0; δ_{2}^{A}]

is also 2-exact in A_{2}.

Proof. This follows if we apply Lemma 6.7 and its dual to the following diagrams:

[figure: B_{1}; A_{1}; A_{2}; A_{2}; B_{3}; B_{3}; ,; d_{1}^{B}; f_{1}; d_{1}^{A}; id; id; d_{2}^{B}; d_{2}^{B}; ⟳; 0; 0; λ_{1}^{−1}; δ_{2}^{B}; σ; A_{1}; A_{1}; A_{2}; A_{2}; A_{3}; B_{3}; d_{1}^{A}; id; d_{1}^{A}; id; f_{3}; d_{2}^{A}; d_{2}^{B}; ⟳; 0; 0; λ_{2}; δ_{2}^{A}; σ]

By Corollary 6.8, it can be shown that the 2-exactness plus compatibility implies the relative 2-exactness (see [7] in the case of SCG):

6.9. Corollary. Let A_{•} = (A_{n}, d_{n}, δ_{n}) be a complex in S. If

[figure: A_{n−1}; A_{n}; A_{n+1}; d_{n−1}^{A}; d_{n}^{A}; 0; δ_{n}^{A}]

is 2-exact in A_{n}, then A_{•} is relatively 2-exact in A_{n}.

Proof. This follows immediately if we apply Corollary 6.8 to the following diagram (see the proof of Corollary 5.24):

[figure: A_{n−1}; Cok(d_{n−2}); A_{n}; Ker(d_{n+1}); A_{n+1}; c(d_{n−2}); d_{n−1}; _{\overline{d}_{n−1}}; d_{n}; ^{\underline{d}^{n}}; k(d_{n+1}); 0; _{\overline{δ}_{n}}; ^{\overline{δ}^{n−1}}; _{\underline{δ}_{n+1}}]

Construction of the long cohomology sequence.

6.10. Definition. A complex in S

[figure: A; B; C; f; g; 0; φ]  (50)

is called an extension if it is relatively 2-exact in every 0-cell.

6.11. Remark. By Corollary 6.2 (and its dual), (50) is an extension if and only if Ker(g) = [A, f, φ] and Cok(f) = [C, g, φ].

6.12. Definition. Let (f_{•}, λ_{•}) : A_{•} ⟶ B_{•} and (g_{•}, κ_{•}) : B_{•} ⟶ C_{•} be complex morphisms and φ_{•} = {φ_{n} : f_{n} ∘ g_{n} ⟹ 0} be 2-cells. Then,

[figure: A_{•}; B_{•}; C_{•}; f_{•}; g_{•}; 0; φ_{•}]  (51)

is said to be an extension of complexes if it satisfies the following properties: (e1) For every n, the following complex is an extension:

[figure: A_{n}; B_{n}; C_{n}; f_{n}; g_{n}; 0; φ_{n}]

(e2) φ_{•} satisfies

(λ_{n} ∘ g_{n+1}) · (f_{n} ∘ κ_{n}) · (φ_{n} ∘ d_{n}^{C}) · (d_{n}^{C})_{I}^{♭} = (d_{n}^{A} ∘ φ_{n+1}) · (d_{n}^{A})_{I}^{♯}.

[figure: A_{n}; A_{n+1}; B_{n}; B_{n+1}; C_{n}; C_{n+1}; f_{n}; d_{n}^{A}; f_{n+1}; d_{n}^{B}; d_{n}^{C}; g_{n}; g_{n+1}; 0; 0; λ_{n}; κ_{n}; φ_{n}; φ_{n+1}]

Our main theorem is the following:

6.13. Theorem. For any extension of complexes in S

[figure: A_{•}; B_{•}; C_{•}; f_{•}; g_{•}; 0; φ_{•}] ,

we can construct a long 2-exact sequence:

[figure: ⋯; H^{n}(B_{•}); H^{n}(C_{•}); H^{n+1}(A_{•}); H^{n+1}(B_{•}); ⋯; 0; 0; 0; 0]

6.14. Caution. This sequence is not necessarily a complex. (See Remark 6.19.)

We prove this theorem in the rest of this section.

6.15. Lemma. In the notation of Lemma 6.3, we have

(1) Ker(x_{n}) = H^{n}(A_{•}),

(2) Cok(x_{n}) = H^{n+1}(A_{•}).

Proof. We only show (1), since (2) can be shown in the same way. In the notation of Lemma 6.3 and Remark 5.18, we can show that the factorization

(x_{n} ∘ \underline{ζ}_{n+1}) · ξ_{n} : (x_{n} ∘ \underline{z}_{n+1}) ∘ k(d_{n+1}) ⟹ ℓ_{n}

is compatible with ε_{d_{n+1}} and μ_{n,2}.

[figure: Q^{n}(A_{•}); Ker(d_{n+1}); A_{n+1}; A_{n+2}; d_{n+1}; k(d_{n+1}); x_{n}∘\underline{z}_{n+1}; ε_{d_{n+1}}; 0; 0; ℓ_{n}; μ_{n,2}; (x_{n}∘\underline{ζ}_{n+1})·ξ_{n}]

So, by Proposition 3.20, Proposition 3.21 and the fact that \underline{z}_{n+1} is fully faithful, we have H^{n}(A_{•}) = Ker(ℓ_{n}, μ_{n,2}) = Ker(x_{n} ∘ \underline{z}_{n+1}) = Ker(x_{n}).

6.16. Lemma. For any extension (51) of complexes in S, we can construct a complex morphism

[figure: Q^{n}(A_{•}); Z^{n+1}(A_{•}); Q^{n}(B_{•}); Z^{n+1}(B_{•}); Q^{n}(C_{•}); 0; Z^{n+1}(C_{•}); 0; Q^{n}(f_{•}); x_{n}^{A}; 0; 0; Z^{n+1}(f_{•}); x_{n}^{B}; x_{n}^{C}; Q^{n}(g_{•}); Z^{n+1}(g_{•}); 0; 0; λ̃_{n}; κ̃_{n}; Q^{n}(φ_{•}); Z^{n+1}(φ_{•})]

where the top line is a complex which is relatively 2-exact in Q^{n}(B_{•}), Q^{n}(C_{•}), and the bottom line is a complex which is relatively 2-exact in Z^{n+1}(A_{•}), Z^{n+1}(B_{•}).

Proof. If we take the relative cokernels Q^{n}(A_{•}), Q^{n}(B_{•}) and Q^{n}(C_{•}) of the complex diagram

[figure: A_{n−2}; A_{n−1}; A_{n}; A_{n+1}; B_{n−2}; B_{n−1}; B_{n}; B_{n+1}; C_{n−2}; C_{n−1}; C_{n}; C_{n+1}; ,; d_{n−2}^{A}; d_{n−1}^{A}; d_{n}^{A}; f_{n−2}; f_{n−1}; f_{n}; f_{n+1}; d_{n−2}^{B}; d_{n−1}^{B}; d_{n}^{B}; g_{n−2}; g_{n−1}; g_{n}; g_{n+1}; d_{n−2}^{C}; d_{n−1}^{C}; d_{n}^{C}; 0; 0; λ_{n−2}; λ_{n−1}; λ_{n}; δ_{n−1}^{A}; δ_{n−1}^{C}; κ_{n−2}; κ_{n−1}; κ_{n}]

then by (the dual of) Proposition 3.20, Proposition 5.4 and Proposition 5.5, we obtain a factorization diagram

[figure: A_{n−1}; A_{n}; Q^{n}(A_{•}); A_{n+1}; B_{n−1}; B_{n}; Q^{n}(B_{•}); B_{n+1}; C_{n−1}; C_{n}; Q^{n}(C_{•}); C_{n+1}; d_{n−1}^{A}; q_{n}^{A}; ℓ_{n}^{A}; f_{n−1}; f_{n}; Q^{n}(f_{•}); f_{n+1}; d_{n−1}^{B}; q_{n}^{B}; ℓ_{n}^{B}; g_{n−1}; g_{n}; Q^{n}(g_{•}); g_{n+1}; d_{n−1}^{C}; q_{n}^{C}; ℓ_{n}^{C}; λ_{n−1}; ^{\overline{λ}^{n−1,1}}; ^{\overline{λ}^{n−1,2}}; κ_{n−1}; _{\overline{κ}_{n−1,1}}; _{\overline{κ}_{n−1,2}}]

and a 2-cell Q^{n}(φ_{•}) ∈ S^{2}(Q^{n}(f_{•}) ∘ Q^{n}(g_{•}), 0), which satisfy compatibility conditions in Proposition 5.4 and Proposition 5.5. It is also easy to see by the universality of the relative cokernels that

(ℓ_{n}^{A} ∘ λ_{n+1}) · (\overline{λ}_{n−1,2} ∘ d_{n+1}^{B}) · (Q^{n}(f_{•}) ∘ μ_{n,2}^{B}) · (Q^{n}(f_{•}))_{I}^{♯} = (μ_{n,2}^{A} ∘ f_{n+2}) · (f_{n+2})_{I}^{♭}.

Now, since

[figure: A_{n}; B_{n}; C_{n}; 0; f_{n}; g_{n}; 0; 0; φ_{n}]

is relatively 2-exact in B_{n} and C_{n}, we have Cok(f_{n}) = [C_{n}, g_{n}, φ_{n}]. So, from Cok(f_{n}) = [C_{n}, g_{n}, φ_{n}] and Cok(f_{n−1}) = [C_{n−1}, g_{n−1}, φ_{n−1}], by Proposition 5.10 we obtain

Cok(Q^{n}(f_{•})) = [Q^{n}(C_{•}), Q^{n}(g_{•}), Q^{n}(φ_{•})],

i.e. the complex

[figure: Q^{n}(A_{•}); Q^{n}(B_{•}); Q^{n}(C_{•}); 0; Q^{n}(f_{•}); 0; Q^{n}(g_{•}); 0; Q^{n}(φ_{•})]

is relatively 2-exact in Q^{n}(B_{•}), Q^{n}(C_{•}). Dually, we obtain a factorization diagram

[figure: A_{n}; Z^{n+1}(A_{•}); A_{n+1}; A_{n+2}; B_{n}; Z^{n+1}(B_{•}); B_{n+1}; B_{n+2}; C_{n}; Z^{n+1}(C_{•}); C_{n+1}; C_{n+2}; k_{n+1}^{A}; z_{n+1}^{A}; d_{n+1}^{A}; f_{n}; Z^{n+1}(f_{•}); f_{n+1}; f_{n+2}; k_{n+1}^{B}; z_{n+1}^{B}; d_{n+1}^{B}; g_{n}; Z^{n+1}(g_{•}); g_{n+1}; g_{n+2}; k_{n+1}^{C}; z_{n+1}^{C}; d_{n+1}^{C}; ^{\underline{λ}^{n+1,2}}; ^{\underline{λ}^{n+1,1}}; λ_{n+1}; _{\underline{κ}_{n+1,2}}; _{\underline{κ}_{n+1,1}}; κ_{n+1}]

such that

(z_{n+1}^{A} ∘ λ_{n+1}) · (\underline{λ}_{n+1,1} ∘ d_{n+1}^{B}) · (Z^{n+1}(f_{•}) ∘ ζ_{n+1}^{B}) · Z^{n+1}(f_{•})_{I}^{♯}
= (ζ_{n+1}^{A} ∘ f_{n+2}) · (f_{n+2})_{I}^{♭}.

Then, it can be shown that each of the factorizations

Q^{n}(f_{•}) ∘ ξ_{n}^{B} : Q^{n}(f_{•}) ∘ x_{n}^{B} ∘ z_{n+1}^{B} ⟹ Q^{n}(f_{•}) ∘ ℓ_{n}^{B}
(x_{n}^{A} ∘ \underline{λ}_{n+1,1}^{−1}) · (ξ_{n}^{A} ∘ f_{n+1}) · \overline{λ}_{n−1,2} : x_{n}^{A} ∘ Z^{n+1}(f_{•}) ∘ z_{n+1}^{B} ⟹ Q^{n}(f_{•}) ∘ ℓ_{n}^{B}

are compatible with ζ_{n+1}^{B} and (Q^{n}(f_{•}) ∘ μ_{n,2}^{B}) · (Q^{n}(f_{•}))_{I}^{♯}.

[figure: Q^{n}(A_{•}); Z^{n+1}(B_{•}); B_{n+1}; B_{n+2}; d_{n+1}^{B}; z_{n+1}^{B}; 0; 0; Q^{n}(f_{•})∘x_{n}^{B}; Q^{n}(f_{•})∘ℓ_{n}^{B}; (Q^{n}(f_{•})∘μ_{n,2}^{B})·(Q^{n}(f_{•}))_{I}^{♯}; ζ_{n+1}^{B}; Q^{n}(f_{•})∘ξ_{n}^{B}; Q^{n}(A_{•}); Z^{n+1}(B_{•}); B_{n+1}; B_{n+2}; d_{n+1}^{B}; z_{n+1}^{B}; 0; 0; x_{n}^{A}∘Z^{n+1}(f_{•}); Q^{n}(f_{•})∘ℓ_{n}^{B}; ζ_{n+1}^{B}; (Q^{n}(f_{•})∘μ_{n,2}^{B})·(Q^{n}(f_{•}))_{I}^{♯}; (x_{n}^{A}∘\underline{λ}_{n+1,1}^{−1})·(ξ_{n}^{A}∘f_{n+1})·\overline{λ}_{n−1,2}]

So, by the universality of the relative kernel, there exists a unique 2-cell λ̃_{n} ∈ S^{2}(Q^{n}(f_{•}) ∘ x_{n}^{B}, x_{n}^{A} ∘ Z^{n+1}(f_{•})) such that

(λ̃_{n} ∘ z_{n+1}^{B}) · (x_{n}^{A} ∘ \underline{λ}_{n+1,1}^{−1}) · (ξ_{n}^{A} ∘ f_{n+1}) · \overline{λ}_{n−1,2} = Q^{n}(f_{•}) ∘ ξ_{n}^{B}.

This λ̃_{n} also satisfies (q_{n}^{A} ∘ λ̃_{n}) · (η_{n}^{A} ∘ Z^{n+1}(f_{•})) · \underline{λ}_{n+1,2} = (\overline{λ}_{n−1,1} ∘ x_{n}^{B}) · (f_{n} ∘ η_{n}^{B}) (see Remark 6.17). Similarly, we obtain a 2-cell κ̃_{n} ∈ S^{2}(Q^{n}(g_{•}) ∘ x_{n}^{C}, x_{n}^{B} ∘ Z^{n+1}(g_{•})) such that

(κ̃_{n} ∘ z_{n+1}^{C}) · (x_{n}^{B} ∘ \underline{κ}_{n+1,1}^{−1}) · (ξ_{n}^{B} ∘ g_{n+1}) · \overline{κ}_{n−1,2} = Q^{n}(g_{•}) ∘ ξ_{n}^{C}.

In the rest, we show the following:

(Q^{n}(f_{•}) ∘ κ̃_{n}) · (λ̃_{n} ∘ Z^{n+1}(g_{•})) · (x_{n}^{A} ∘ Z^{n+1}(φ_{•})) · (x_{n}^{A})_{I}^{♯} = (Q^{n}(φ_{•}) ∘ x_{n}^{C}) · (x_{n}^{C})_{I}^{♭}. (52)

We have the following equalities:

(Q^{n}(f_{•}) ∘ κ̃_{n} ∘ z_{n+1}^{C}) · (λ̃_{n} ∘ Z^{n+1}(g_{•}) ∘ z_{n+1}^{C})
= (Q^{n}(f_{•}) ∘ Q^{n}(g_{•}) ∘ ξ_{n}^{C}) · (Q^{n}(f_{•}) ∘ \overline{κ}_{n−1,2}^{−1}) · (\overline{λ}_{n−1,2}^{−1} ∘ g_{n+1})
· ((ξ_{n}^{A})^{−1} ∘ f_{n+1} ∘ g_{n+1}) · (x_{n}^{A} ∘ \underline{λ}_{n+1,1} ∘ g_{n+1}) · (x_{n}^{A} ∘ Z^{n+1}(f_{•}) ∘ \underline{κ}_{n+1,1}),
(Q^{n}(φ_{•}) ∘ x_{n}^{C} ∘ z_{n+1}^{C}) · (x_{n}^{C} ∘ z_{n+1}^{C})_{I}^{♭} · (x_{n}^{A} ∘ z_{n+1}^{A})_{I}^{♯−1} · (x_{n}^{A} ∘ z_{n+1}^{A} ∘ φ_{n+1}^{−1})
= (Q^{n}(f_{•}) ∘ Q^{n}(g_{•}) ∘ ξ_{n}^{C}) · (Q^{n}(f_{•}) ∘ \overline{κ}_{n−1,2}^{−1}) · (\overline{λ}_{n−1,2}^{−1} ∘ g_{n+1})
· ((ξ_{n}^{A})^{−1} ∘ f_{n+1} ∘ g_{n+1}),
(z_{n+1}^{C})_{I}^{♭−1} = (x_{n}^{A} ∘ z_{n+1}^{A})_{I}^{♯−1} · (x_{n}^{A} ∘ z_{n+1}^{A} ∘ φ_{n+1}^{−1}) · (x_{n}^{A} ∘ \underline{λ}_{n+1,1} ∘ g_{n+1})
· (x_{n}^{A} ∘ Z^{n+1}(f_{•}) ∘ \underline{κ}_{n+1,1}) · (x_{n}^{A} ∘ Z^{n+1}(φ_{•}) ∘ z_{n+1}^{C}) · ((x_{n}^{A})_{I}^{♯} ∘ z_{n+1}^{C}).

From these equalities and the faithfulness of z_{n+1}^{C}, we obtain (52).

6.17. Remark. It can be also shown that λ̃_{n} in the proof of Lemma 6.16 satisfies

(q_{n}^{A} ∘ λ̃_{n}) · (η_{n}^{A} ∘ Z^{n+1}(f_{•})) · \underline{λ}_{n+1,2} = (\overline{λ}_{n−1,1} ∘ x_{n}^{B}) · (f_{n} ∘ η_{n}^{B}).

By Lemma 6.15 and Lemma 6.16, Theorem 6.13 is reduced to the following Proposition:

6.18. Proposition. Consider the following diagram in S, where (A_{•}, d_{•}^{A}, δ_{•}^{A}) is a complex which is relatively 2-exact in A_{2} and A_{3}, and (B_{•}, d_{•}^{B}, δ_{•}^{B}) is a complex which is relatively 2-exact in B_{1} and B_{2}.

[figure: A_{1}; B_{1}; A_{2}; B_{2}; A_{3}; 0; B_{3}; 0; d_{1}^{A}; f_{1}; 0; 0; d_{1}^{B}; f_{2}; f_{3}; d_{2}^{A}; d_{2}^{B}; 0; 0; λ_{1}; λ_{2}; δ_{2}^{A}; δ_{2}^{B}]

Assume f_{•} : A_{•} ⟶ B_{•} is a complex morphism. Then there exist d ∈ S^{1}(Ker(f_{3}), Cok(f_{1})), α ∈ S^{2}(\underline{d}_{2}^{A} ∘ d, 0) and β ∈ S^{2}(d ∘ \overline{d}_{1}^{B}, 0) such that the sequence

[figure: Ker(f_{1}); Ker(f_{2}); Ker(f_{3}); Cok(f_{1}); Cok(f_{2}); Cok(f_{3}); _{\underline{d}_{A}_{1}}; _{\underline{d}_{A}_{2}}; d; ^{\overline{d}{B \atop 1}}; ^{\overline{d}{B \atop 2}}; 0; 0; 0; 0; ^{\underline{δ}^{A}^{2}}; α; β; _{\overline{δ}{B \atop 2}}]  (53)

is 2-exact in Ker(f_{2}), Ker(f_{3}), Cok(f_{1}), Cok(f_{2}).

6.19. Remark. This sequence does not necessarily become a complex. Indeed, for a relatively exact 2-category S, the following are shown to be equivalent by an easy diagrammatic argument:

(i) Any (53) obtained in Proposition 6.18 becomes a complex.

(ii) For any f ∈ S^{1}(A, B),

[figure: Ker(f); A; B; Cok(f); k(f); f; c(f); 0; 0; ε_{f}; π_{f}]  (54)

is a complex.

(Indeed, if (54) is a complex for each of f_{1}, f_{2} and f_{3}, then (53) becomes a complex.) Thus if S satisfies (ii), then the long cohomology sequence in Theorem 6.13 becomes

a complex. But this assumption is a bit too strong, since it is not satisfied by SCG. This is pointed out by the referee.

Proof of Proposition 6.18. Put Ker(d_{2}^{A} ∘ f_{3}) = [K, k, ζ]. If we take the kernel of the diagram

[figure: A_{1}; 0; A_{2}; B_{3}; A_{3}; B_{3}; 0; 0; d_{1}^{A}; 0; 0; d_{2}^{A}∘f_{3}; f_{3}; d_{2}^{A}; id; 0; 0; ⟳; 0; ξ_{0}; δ_{2}^{A}]  (55)

where ξ_{0} := (δ_{2}^{A} ∘ f_{3}) · (f_{3})_{I}^{♭}, then by Proposition 5.5 we obtain a diagram

[figure: A_{1}; A_{1}; K; A_{2}; Ker(f_{3}); A_{3}; 0; k_{1}; id; d_{1}^{A}; k; k(f_{3}); k_{2}; d_{2}^{A}; 0; 0; 0; ξ_{1}; ξ_{2}; α_{2}; δ_{2}^{A}]  (56)

which satisfies

(k_{2} ∘ ε_{f_{3}}) · (k_{2})_{I}^{♯} = (ξ_{2} ∘ f_{3}) · ζ
(ξ_{1} ∘ d_{2}^{A} ∘ f_{3}) · ξ_{0} = (k_{1} ∘ ζ) · (k_{1})_{I}^{♯}
(k_{1} ∘ ξ_{2}) · (ξ_{1} ∘ d_{2}^{A}) · δ_{2}^{A} = (α_{2} ∘ k(f_{3})) · k(f_{3})_{I}^{♭}.

By Lemma 6.6,

[figure: A_{1}; K; Ker(f_{3}); k_{1}; k_{2}; 0; α_{2}]

is 2-exact in K. On the other hand, by (the dual of) Proposition 5.11,

[figure: K; A_{2}; Ker(f_{3}); A_{3}; k_{2}; k; d_{2}^{A}; f_{3}; ξ_{2}]

is a pullback diagram, and k_{2} becomes cofaithful since d_{2}^{A} is cofaithful. Thus, we have Cok(k_{1}) = [Ker(f_{3}), k_{2}, α_{2}] by Lemma 6.4. Dually, if we put Cok(f_{1} ∘ d_{1}^{B}) = [Q, q, ρ], then we obtain the following diagram

[figure: 0; A_{1}; A_{1}; 0; 0; B_{1}; B_{2}; B_{3}; Cok(f_{1}); Q; B_{3}; 0; id; 0; f_{1}; f_{1}∘d_{1}^{B}; 0; 0; d_{1}^{B}; d_{2}^{B}; c(f_{1}); q; id; q_{1}; q_{2}; 0; ⟳; η_{0}; η_{1}; η_{2}; β_{2}]

which satisfies

η_{0} = (f_{1})_{I}^{♯−1} · (f_{1} ∘ δ_{2}^{B−1})
ρ = (f_{1} ∘ η_{1}) · (π_{f_{1}} ∘ q_{1}) ∘ (q_{1})_{I}^{♭}
id_{0} = η_{0} · (f_{1} ∘ d_{1}^{B} ∘ η_{2}) · (ρ ∘ q_{2}) · (q_{2})_{I}^{♭}
δ_{2}^{B} = (d_{1}^{B} ∘ η_{2}) · (η_{1} ∘ q_{2}) · (c(f_{1}) ∘ β_{2}) · c(f_{1})_{I}^{♯},

and we have Ker(q_{2}) = [Cok(f_{1}), q_{1}, β_{2}]. (The “un-duality” in appearance is simply because of the direction of the 2-cells.) Thus, we obtain complex morphisms:

[figure: A_{1}; K; Ker(f_{3}); A_{1}; A_{2}; A_{3}; B_{1}; B_{2}; B_{3}; Cok(f_{1}); Q; B_{3}; k_{1}; k_{2}; id; k; k(f_{3}); d_{1}^{A}; d_{2}^{A}; f_{1}; f_{2}; f_{3}; d_{1}^{B}; d_{2}^{B}; c(f_{1}); q; id; q_{1}; q_{2}; ξ_{1}; ξ_{2}; λ_{1}; λ_{2}; η_{1}; η_{2}]

If we put

c := k ∘ f_{2} ∘ q
α_{K} := (ξ_{1} ∘ f_{2} ∘ q) · (λ_{1} ∘ q) · ρ
β_{Q} := (k ∘ f_{2} ∘ η_{2}^{−1}) · (k ∘ λ_{2}^{−1}) · ζ,

then, it can be shown that the following diagram is a complex.

[figure: A_{1}; K; Q; B_{3}; k_{1}; c; q_{2}; 0; 0; α_{K}; β_{Q}]

Since Cok(k_{1}) = [Ker(f_{3}), k_{2}, α_{2}] as already shown, we have a factorization diagram

[figure: Ker(f_{3}); A_{1}; K; Q; B_{3}; k_{1}; c; k_{2}; q_{2}; \overline{c}; ^{\overline{β}^{Q}}; α_{2}; 0; 0; 0; 0; β_{Q}; α_{K}; ^{\overline{α}^{K}}]

which satisfies

(k_{1} ∘ \overline{α}_{K}) · α_{K} = (α_{2} ∘ \overline{c}) · (\overline{c})_{I}^{♭}
(\overline{α}_{K} ∘ q_{2}) · β_{Q} = (k_{2} ∘ \overline{β}_{Q}) · (k_{2})_{I}^{♯}.

Similarly, since Ker(q_{2}) = [Cok(f_{1}), q_{1}, β_{2}], we have a factorization diagram

[figure: Cok(f_{1}); A_{1}; K; Q; B_{3}; k_{1}; c; \underline{c}; q_{2}; q_{1}; β_{2}; _{\underline{α}_{K}}; 0; 0; 0; 0; β_{Q}; α_{K}; _{\underline{β}_{Q}}]

which satisfies

(\underline{β}_{Q} ∘ q_{2}) · β_{Q} = (\underline{c} ∘ β_{2}) · (\underline{c})_{I}^{♯}
(k_{1} ∘ \underline{β}_{Q}) · α_{K} = (\underline{α}_{K} ∘ q_{1}) · (q_{1})_{I}^{♭}.

Then, there exist d ∈ S^{1}(Ker(f_{3}), Cok(f_{1})), α^{†} ∈ S^{2}(k_{2} ∘ d, \underline{c}) and β^{†} ∈ S^{2}(d ∘ q_{1}, \overline{c}) such that

(k_{1} ∘ α^{†}) · \underline{α}_{K} = (α_{2} ∘ d) · d_{I}^{♭}
(β^{†} ∘ q_{2}) · \overline{β}_{Q} = (d ∘ β_{2}) · d_{I}^{♯}
(k_{2} ∘ β^{†}) · \overline{α}_{K} = (α^{†} ∘ q_{1}) · \underline{β}_{Q}

(note that Cok(k_{1}) = [Ker(f_{3}), k_{2}, α_{2}] and Ker(q_{2}) = [Cok(f_{1}), q_{1}, β_{2}] (cf. Lemma 6.3)):

[figure: Ker(f_{3}); Cok(f_{1}); A_{1}; K; Q; B_{3}; k_{2}; \underline{c}; q_{2}; d; k_{1}; \overline{c}; q_{1}; 0; 0; 0; 0; ^{\overline{β}^{Q}}; α_{2}; β_{2}; _{\underline{α}_{K}}; α^{†}; β^{†}]

Applying (the dual of) Proposition 5.8 to the diagram

[figure: Ker(f_{1}); Ker(f_{2}); Ker(f_{3}); A_{1}; A_{2}; A_{3}; 0; B_{3}; B_{3}; ,; ^{\underline{d}A 1}; ^{\underline{d}A 2}; k(f_{1}); k(f_{2}); k(f_{3}); d_{1}^{A}; d_{2}^{A}; 0; d_{2}^{A}∘f_{3}; f_{3}; 0; id; ⟳; ^{\underline{λ}^{1}}; ^{\underline{λ}^{2}}; ξ_{0}]

we see that there exist k′ ∈ S^{1}(Ker(f_{2}), K), ξ′_{1} ∈ S^{2}(\underline{d}_{1}^{A} ∘ k′, k(f_{1}) ∘ k_{1}), ξ′_{2} ∈ S^{2}(\underline{d}_{2}^{A}, k′ ∘ k_{2}) and ξ ∈ S^{2}(k′ ∘ k, k(f_{2})) such that

\underline{δ}_{2}^{A} = (\underline{d}_{1}^{A} ∘ ξ′_{2}) · (ξ′_{1} ∘ k_{2}) · (k(f_{1}) ∘ α_{2}) · k(f_{1})_{I}^{♯}
\underline{λ}_{2} = (ξ′_{2} ∘ k(f_{3})) · (k′ ∘ ξ_{2}) · (ξ ∘ d_{2}^{A})
(\underline{d}_{1}^{A} ∘ ξ) · \underline{λ}_{1} = (ξ′_{1} ∘ k) · (k(f_{1}) ∘ ξ_{1}).

Similarly, there exist q′ ∈ S^{1}(Q, Cok(f_{2})), η′_{1} ∈ S^{2}(q_{1} ∘ q′, \overline{d}_{1}^{B}), η′_{2} ∈ S^{2}(q_{2} ∘ c(f_{3}), q′ ∘ \overline{d}_{2}^{B}) and η ∈ S^{2}(q ∘ q′, c(f_{2})) such that

(β_{2} ∘ c(f_{3})) · c(f_{3})_{I}^{♭} = (q_{1} ∘ η′_{2}) · (η′_{1} ∘ \overline{d}_{2}^{B}) · \overline{δ}_{2}^{B}
(η_{1} ∘ q′) · (c(f_{1}) ∘ η′_{1}) = (d_{1}^{B} ∘ η) · \overline{λ}_{1}
(η_{2} ∘ c(f_{3})) · (q ∘ η′_{2}) · (η ∘ \overline{d}_{2}^{B}) = \overline{λ}_{2}.

If we put

α_{0} := (\underline{d}_{2}^{A} ∘ β^{†}) · (ξ′_{2} ∘ \overline{c}) · (k′ ∘ \overline{α}_{K}) · (ξ ∘ f_{2} ∘ q) · (ε_{f_{2}} ∘ q) · q_{I}^{♭},

then it can be shown that α_{0} : \underline{d}_{2}^{A} ∘ d ∘ q_{1} ⟹ 0 is compatible with β_{2}.

[figure: Ker(f_{2}); Cok(f_{1}); Q; B_{3}; \underline{d}_{2}^{A}∘d; q_{1}; q_{2}; 0; 0; α_{0}; β_{2}]

So by Lemma 3.19, there exists α ∈ S^{2}(\underline{d}_{2}^{A} ∘ d, 0) such that

(α ∘ q_{1}) · (q_{1})_{I}^{♭} = α_{0}.

Dually, if we put

β_{0} := (α^{†} ∘ \overline{d}_{1}^{B}) · (\underline{c} ∘ η_{1}^{′−1}) · (\underline{β}_{Q} ∘ q′) · (k ∘ f_{2} ∘ η) · (k ∘ π_{f_{2}}) · k_{I}^{♯},

then β_{0} : k_{2} ∘ d ∘ \overline{d}_{1}^{B} ⟹ 0 is compatible with α_{2}, and there exists β ∈ S^{2}(d ∘ \overline{d}_{1}^{B}, 0) such that

(k_{2} ∘ β) · (k_{2})_{I}^{♯} = β_{0}.

[figure: Ker(f_{1}); Ker(f_{2}); Ker(f_{3}); Cok(f_{1}); Cok(f_{2}); Cok(f_{3}); _{\underline{d}_{A}_{1}}; _{\underline{d}_{A}_{2}}; d; ^{\overline{d}{B \atop 1}}; ^{\overline{d}{B \atop 2}}; 0; 0; 0; 0; ^{\underline{δ}^{A}^{2}}; α; β; _{\overline{δ}{B \atop 2}}]

In the rest, we show that this is 2-exact in Ker(f_{2}), Ker(f_{3}), Cok(f_{1}), Cok(f_{2}). We show only the 2-exactness in Ker(f_{2}) and Ker(f_{3}), since the rest can be shown dually. The 2-exactness in Ker(f_{2}) follows immediately from Lemma 6.6. So, we show the 2-exactness in Ker(f_{3}). Since we have Cok(d_{1}^{A}) = [A_{3}, d_{2}^{A}, δ_{2}^{A}] and Cok(f_{1} ∘ d_{2}^{B}) = [Q, q, ρ], there exists a factorization (ℓ, ϖ_{1})

[figure: A_{1}; A_{1}; A_{2}; B_{2}; A_{3}; Q; 0; 0; d_{1}^{A}; id; f_{1}∘d_{1}^{B}; f_{2}; ℓ; d_{2}^{A}; q; 0; 0; 0; 0; λ_{1}; ϖ_{1}; δ_{2}^{A}; ρ]  (57)

such that

(d_{1}^{A} ∘ ϖ_{1}) · (λ_{1} ∘ q) · ρ = (δ_{2}^{A} ∘ ℓ) · ℓ_{I}^{♭}.

Applying Lemma 6.5 to diagram (57), we see that the following diagram becomes 2-exact in A_{3}:

[figure: Ker(f_{2}); A_{3}; Q; k(f_{2})∘d_{2}^{A}; ℓ; 0; (k(f_{2})∘ϖ_{1})·(ε_{f_{2}}∘q)·q_{I}^{♭}]  (58)

Then it can be shown that (ϖ_{1} ∘ q_{2}) · (f_{2} ∘ η_{2}^{−1}) : d_{2}^{A} ∘ ℓ ∘ q_{2} ⟹ f_{2} ∘ d_{2}^{B} is compatible with δ_{2}^{A} and (λ_{1} ∘ d_{2}^{B}) · (f_{1} ∘ d_{2}^{B}) · (f_{1})_{I}^{♯}. So, comparing the following two factorizations

[figure: A_{1}; B_{3}; A_{2}; A_{3}; f_{2}∘d_{2}^{B}; d_{1}^{A}; d_{2}^{A}; 0; ℓ∘q_{2}; 0; δ_{2}^{A}; (λ_{1}∘d_{2}^{B})·(f_{1}∘δ_{2}^{B})·(f_{1})_{I}^{♯}; (ϖ_{1}∘q_{2})·(f_{2}∘η_{2}^{−1}); A_{1}; B_{3}; ,; A_{2}; A_{3}; f_{2}∘d_{2}^{B}; d_{1}^{A}; d_{2}^{A}; 0; f_{3}; 0; δ_{2}^{A}; λ_{2}; (λ_{1}∘d_{2}^{B})·(f_{1}∘δ_{2}^{B})·(f_{1})_{I}^{♯}]

we see there exists a unique 2-cell ϖ_{2} ∈ S^{2}(ℓ ∘ q_{2}, f_{3}) such that

(d_{2}^{A} ∘ ϖ_{2}) · λ_{2} = (ϖ_{1} ∘ q_{2}) · (f_{2} ∘ η_{2}^{−1}).

Then it can be shown that each of the two factorizations

(1) (α^{†} ∘ q_{1}) · \underline{β}_{Q} : k_{2} ∘ d ∘ q_{1} ⟹ c
(2) (ξ_{2} ∘ ℓ) · (k ∘ ϖ_{1}) : k_{2} ∘ k(f_{3}) ∘ ℓ ⟹ k ∘ f_{2} ∘ q = c

is compatible with α_{2} and α_{K}.

[figure: A_{1}; Q; K; Ker(f_{3}); c; k_{1}; k_{2}; 0; d∘q_{1}; 0; α_{2}; α_{K}; (α^{†}∘q_{1})·\underline{β}_{Q}; A_{1}; Q; K; Ker(f_{3}); c; k_{1}; k_{2}; 0; k(f_{3})∘ℓ; 0; α_{2}; α_{K}; (ξ_{2}∘ℓ)·(k∘ϖ_{1})]

So there exists a unique 2-cell ϖ_{3} ∈ S^{2}(d ∘ q_{1}, k(f_{3}) ∘ ℓ) such that

(k_{2} ∘ ϖ_{3}) · (ξ_{2} ∘ ℓ) · (k ∘ ϖ_{1}) = (α^{†} ∘ q_{1}) · \underline{β}_{Q}

(recall that Cok(k_{1}) = [Ker(f_{3}), k_{2}, α_{2}]). Then we have (ϖ_{3} ∘ q_{2}) · (k(f_{3}) ∘ ϖ_{2}) · ε_{f_{3}} = (d ∘ β_{2}) · d_{I}^{♯}.

[figure: Ker(f_{3}); Cok(f_{1}); A_{3}; Q; B_{3}; B_{3}; d; k(f_{3}); q_{1}; ℓ; f_{3}; q_{2}; id; 0; 0; ϖ_{3}; ϖ_{2}; ε_{f_{3}}; β_{2}]  (59)

By taking kernels of d, ℓ and id_{B_{3}} in (59), we obtain the following diagram.

[figure: Ker(d); Ker(f_{3}); Cok(f_{1}); Ker(ℓ); A_{3}; Q; 0; B_{3}; B_{3}; k(d); d; \underline{k(f_{3})}; k(f_{3}); q_{1}; k(ℓ); ℓ; 0; f_{3}; q_{2}; 0; id; ^{\underline{ϖ}^{3}}; ϖ_{3}; _{\underline{ϖ}_{2}}; ϖ_{2}]

Since Ker(0 : Ker(ℓ) ⟶ 0) = Ker(d) by (the dual of) Proposition 5.7, so \underline{k(f_{3})} becomes an equivalence. On the other hand, the following is a complex morphism, where s := k(f_{2}) ∘ d_{2}^{A}.

[figure: Ker(f_{2}); Ker(f_{2}); Ker(f_{3}); A_{3}; Cok(f_{1}); Q; ^{\underline{d}^{A}^{2}}; id; s; k(f_{3}); q_{1}; d; ℓ; 0; 0; \underline{λ}_{2}; ϖ_{3}; α; (k(f_{2})∘ϖ_{1})·(ε_{f_{2}}∘q)·q_{I}^{♭}]  (60)

Thus by taking kernels of d and ℓ in diagram (60), we obtain the following factorization by (the dual of) Proposition 5.8.

[figure: Ker(f_{2}); Ker(f_{2}); Ker(d); Ker(ℓ); Ker(f_{3}); A_{3}; ∃(\underline{d}_{2}^{A})^{†}; id; ∃\underline{s}; \underline{k(f_{3})}; k(f_{3}); k(d); k(ℓ); ^{\underline{d}A 2}; s; ∃; \underline{ϖ}^{3}; ∃\underline{α}; ∃]

Since (58) is 2-exact in A_{3}, so \underline{s} becomes fully cofaithful. Since \underline{k(f_{3})} is an equivalence, this means (\underline{d}_{2}^{A})^{†} is fully cofaithful, and

[figure: Ker(f_{2}); Ker(f_{3}); Cok(f_{1}); _{\underline{d}_{A}_{2}}; d; 0; α]

becomes 2-exact in Ker(f_{3}).

References

[1] D. Bourn, E. M. Vitale, Extensions of symmetric cat-groups, Homol. Homotopy Appl. 4 (2002), 103-162.

[2] M. Dupont, E. M. Vitale, Proper factorization systems in 2-categories, J. Pure Applied Algebra 179 (2003), 65-86.

[3] S. Kasangian, E. M. Vitale, Factorization systems for symmetric cat-groups, Theory and Application of Categories 7 (2000), 47-70.

[4] S. Mac Lane, “Categories for the working mathematician”, Graduate Texts in Math., Vol. 5, Springer-Verlag, New York, 1998.

[5] B. Pareigis, Non-additive ring and module theory I, General theory of monoids, Publ. Math. Debrecen 24 (1977), 189-204.

[6] B. Pareigis, Non-additive ring and module theory IV, Brauer group of a symmetric monoidal category, in “Brauer groups”, Lecture Notes in Math., Vol. 549, pp. 112-133, Springer-Verlag, Berlin, 1976.

[7] A. del Río, J. Martínez-Moreno, E. M. Vitale, Chain complexes of symmetric categorical groups, J. Pure Applied Algebra 196 (2004), 279-312.

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

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

Graduate School of Mathematical Sciences, The University of Tokyo 3-8-1 Komaba, Meguro, Tokyo, 153-8914 Japan

Email: deutsche@ms.u-tokyo.ac.jp

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