Comm. in Alg. 19 (1991), 379–417.

DERIVED CATEGORIES AND UNIVERSAL PROBLEMS

Bernhard Keller

Mathematik, G 28.2 ETH-Zentrum 8092 Zurich, Switzerland

Introduction

In this paper we search for a universal property of the (bounded positive) derived category of an exact category. We thereby hope to obtain a better understanding of the category of S-functors [10] starting from the derived category. Such S-functors play an essential rôle in the study of hearts of tstructures [2] [1], in J. Rickard’s ’Morita theory for derived categories‘ [13] and in D. Happel’s description of the derived category of a finite-dimensional algebra [7].

We briefly outline the contents of the paper. Let A be an exact category and DA the bounded positive derived category (cf. section 1). We start with what we consider the most natural approach, namely the question whether the canonical ∂-functor A → DA is universal among the ∂-functors from A to suspended categories S. This, however, is not the case. We analyse the situation in section 1. Our conclusion is that the concepts of suspended category and S-functor alone do not provide rich enough a framework for an adequate treatment of the question. As a supplement, we propose ’towers‘ of suspended (resp. exact) categories: A tower T consists of a sequence

T_{0}, T_{1}, …, T_{n}, …, n ∈ N

of categories of the respective type and of a multitude of compatible functors joining them. A tower A^{∧} with base A_{0}^{∧} = A is associated with the exact category A. The derived categories D(A_{n}^{∧}) form a tower of suspended categories DA^{∧} and the tower of ∂-functors A^{∧} → DA^{∧} is universal (Theorem 2.6). Passing from the tower DA^{∧} to its base DA_{0}^{∧} = DA we obtain new information on the problem of extending a ∂-functor A → S to an S-functor DA → S. For example, if Λ and Γ are derived equivalent rings [13], then the S-equivalence H_{b}P_{Γ} → H_{b}P_{Λ} constructed in [loc. cit.] is unique among the S-functors which map Γ to the tilting complex T and occur as the base of a tower of S-functors. These ’basic‘ S-functors apparently form a very large class. For example, all left derived functors are basic and all left S-adjoint functors of basic S-functors are basic (2.8). In fact, we do not know any example of a non-basic S-functor. Neither do we know an example of a suspended category which does not occur as the base of a tower of suspended categories.

The essential ingredient of the proof of the universal property of DA^{∧} is the description of the derived category as a localisation of a category of presheaves. We refer to section 3 for details.

The description of DA by presheaves also leads to a characterization of the ’construction D’, i.e. the ’hyperfunctor‘ ( = 2-functor) which assigns DB to B for each exact category B (section 4). In fact, D is universal among the constructions D′ which map J : IB → MB to an S-equivalence and

0 → B →^{I} IB →^{P} B → 0

to an exact sequence of suspended categories, for each B. Here MB is the category of morphisms of B, J is the inclusion of the subcategory IB of inflations of B, I maps B to 0 → B and P maps i : A → B to A.

1. Extending ∂-functors

By an additive, exact [8] [12] or suspended category we shall always mean a svelte (=equivalent to a small) category of the respective type. We shall also assume that idempotents split in the exact categories we consider. Following [4] we use the words ’inflation‘, ’deflation‘ and ’conflation‘ instead of ’admissible monomorphism‘, ’admissible epimorphism‘ and ’admissible short exact sequence‘ [12], respectively. If B is an exact category with enough injectives, we denote by \underline{B} the residue class category of B modulo the ideal of morphisms factoring through an injective. The exact structure on B yields a structure of suspended category on \underline{B} [10].

Let A be an exact category and D_{b}A the bounded derived category of A : It is the localisation [15] of the homotopy category H_{b}A of bounded chain complexes

… → K_{n+1} →^{d} K_{n} →^{d} K_{n−1} → …, dd = 0, K_{n} = 0 ∀ n >> 0 and ∀ n << 0,

at the subcategory of acyclic complexes, i.e. complexes A admitting conflations

Z_{n+1} ⟶^{j_{n}} A_{n} ⟶^{q_{n}} Z_{n−1}

such that d_{n} = j_{n−1}q_{n}, ∀ n (we always suppress the zeroes at the ends of a conflation). The positive derived category DA is the full subcategory of D_{b}A consisting of the positive complexes K, i.e. K_{n} = 0, ∀ n < 0. Note that our use of the notation DA does not agree with the usual conventions. We shall see in 5.1 that DA identifies with a localisation of the positive homotopy category H_{0]}^{b} A, the full subcategory of HA consisting of the positive complexes. Clearly DA is a suspended category in the sense of [10]. We shall work with DA rather than with D_{b}A since this is technically simpler and at the same time leads to slightly more general results. In fact, it is usually easy to pass from DA to D_{b}A using the fact that the latter category identifies with the smallest triangulated category containing DA as a suspended subcategory (cf. [10, 2.1]).

By sending A ∈ A to the complex K with K_{0} = A and K_{n} = 0 for all n ≠ 0, we obtain an additive functor can from A to DA. Moreover, for each conflation

ε : A →^{i} B →^{d} C

of A, there is a unique [2, 1.1.10] connecting morphism ∂ε : C → SA such that the sequence

A →^{i} B →^{d} C →^{∂ε} SA

is a triangle of DA (we omit can from the notation whenever the context makes it clear that we are speaking of complexes rather than of objects of A). Thus, (can , ∂) is a ∂-functor in the sense of the following definition: If S is a suspended category, a ∂-functor from A to S consists of an additive functor D : A → S and a natural transformation δ which assigns a morphism δε : DC → SDA to each conflation ε in such a way that

DA →^{Di} DB →^{Dd} DC →^{δε} SDA

is a triangle of S. A morphism of ∂-functors (D, δ) → (D′, δ′) is given by a morphism of functors μ : D → D′ such that the square

DC →^{δε} SDA
μC ↓  ↓ SμA
D′C →^{δ′ε} SD′A

commutes for each conflation ε. It is clear how to compose two such morphisms. We see that the ∂-functors from A to S form a category Δ (A, S). Each Sfunctor (F, φ) : DA → S yields a ∂-functor (F can , δ) whose second component assigns δε = (φcan A)(F ∂ε) to the conflation ε. Also, if ν is a morphism of S-functors from DA to S, can ν is a morphism of the associated ∂-functors. Clearly these assignments define a functor

can^{∗} : Susp (DA, S) → Δ (A, S),

where the left hand side is the category of S-functors from DA to S. It is reasonable to ask the

Question: Is can^{∗} an equivalence ?

An affirmative answer would entitle us to call DA the solution of a universal problem. The answer, however, is no. The following simple example shows that even if each conflation of A splits, there are ∂-functors A → S which do not ’extend‘ to S-functors DA → S.

Example. Let B be an exact category,

A →^{i} B →^{d} C

a non-split conflation of B and ∂ : C → SA the connecting morphism in DB. Let A be the full additive subcategory of DB consisting of the objects ∐_{i∈I} X_{i}, where I is finite and each X_{i} belongs to {A, B, C, SA}. We equip A with the split conflations. Suppose there is an S-functor F from DA = H_{0]}^{b} A to DB whose restriction to A is isomorphic to the inclusion A ⊂ DB. Then it is not hard to see that

F ( … → 0 → A →^{i} B →^{d} C) = 0.

Since the morphism ∂ : C → SA of A factors through

… → 0 → A →^{i} B →^{d} C

in DA, this implies F ∂ = 0, a contradiction.

For a general suspended category S, we know nothing about the image of can^{∗}. The problem only becomes tractable when we make more specific assumptions about S: If S is the stable category of an exact category with enough injectives then a ∂-functor D is isomorphic to F can for some F : DA → S if it satisfies

Condition 1: For each n > 0 and all A, B ∈ A we have S (S^{n}DA, DB) = 0.

This is an easy consequence of [10, 3.2]. We shall also reprove it in this paper. The condition is not necessary. For example, if A has enough injectives, one can define (cf. [14] or example 2.6) an S-functor F : DA → \underline{A} which extends the canonical projection A → \underline{A}. However, the latter does not satisfy condition 1 unless S : \underline{A} → \underline{A} is zero, i.e. the injective dimension of A is 1.

Now let us consider two S-functors F, F′ : DA → S and a morphism μ from D = F can to D′ = F′can . Even if S is the stable category of an exact category with enough injectives and D and D′ satisfy condition 1, we do not know how to extend μ to a morphism of S-functors ν : F → F′. We illustrate the state of our ignorance by posing the

Problem: If B is an additive category and F : H_{0]}^{b} B → H_{0]}^{b} B is an S-functor whose restriction to B is isomorphic to can : B → H_{0]}^{b} B, is it true that F is isomorphic to the identity of H_{0]}^{b} B ?

We remark that it would be enough to produce any ν : F → 1 such that can ν is an isomorphism since such a ν would automatically be invertible, as an easy induction argument shows. We should also mention the following positive result: If D ≅ D′ and D satisfies the above condition, then there is an isomorphism F K →^{∼} F′K for each K ∈ DA. It is not hard to prove this for general S by induction on the maximal n with K_{n} ≠ 0. While this is certainly useful in the applications, it does not solve the problem we are considering since there seems to be no way to ensure the functoriality of the isomorphisms

F K →^{∼} F′K.

Let us return to the problem of extending μ : D → D′ to ν : F → F′. Suppose that S is the stable category of an exact category with enough injectives, that D and D′ satisfy condition 1 and that F and F′ were obtained from D and D′ by the method of [10, 3.2]. Then one can prove that ν exists if D and D′ satisfy

Condition 2: For each n > 0 and all A, B ∈ A we have S(S^{n}DA, D′B) = 0.

Let us summarize our findings in vague terms: Extending a ∂-functor is possible if S is of a special type and condition 1 is satisfied; extending a morphism between the restrictions of two S-functors is possible if S is of a special type, the S-functors are of a special type and their restrictions satisfy conditions 1 and 2.

Thus suspended categories and S-functors ’of a special type‘ behave better than general suspended categories and S-functors. For example, the above problem becomes trivial if F is assumed to be ’of a special type‘. In the following section, we shall introduce the class of basic S-functors (resp. suspended categories), which contains all those of ’special type‘. The class of basic Sfunctors is closed under left adjoints and contains all left derived functors. The good properties of S-functors ’of a special type‘ are shared by all basic S-functors. They derive from the good properties of towers of S-functors.

2. Towers

2.1 For each n ∈ N let P_{n} be the partially ordered set

{0 < 1}^{n} = {x = (x_{1}, …, x_{n}) : x_{i} ∈ {0, 1}}

and let P_{0} = {∗}. Let P be the category whose objects are the P_{n}, n ∈ N and whose morphisms are all possible compositions of the maps

p_{ε}^{j} : P_{n} → P_{n+1} , (x_{1}, …, x_{n}) ↦ (x_{1}, …, x_{j−1}, ε, x_{j}, …, x_{n}) ,
q^{j} : P_{n+1} → P_{n} , (x_{1}, …, x_{n+1}) ↦ (x_{1}, …, x_{j−1}, x_{j+1}, …, x_{n+1}) ,

where n ∈ N and 1 ≤ j ≤ n + 1. More explicitly, an order preserving map a : P_{m} → P_{n} is a morphism of P iff for all 1 ≤ i < j ≤ n we have

π_{i} a = π_{u} and π_{j} a = π_{v} ⟹ u < v ,

where π_{i} x = x_{i} for x ∈ P_{n}. The category P is the ’cubical category‘ implicit in the definition of cubical homology (see e.g. [3]). As for the simplicial category one proves that any relation between its generators is a consequence of

\begin{matrix}p_{ε}^{j} p_{η}^{k} = p_{η}^{k+1} p_{ε}^{j} & j ≤ k \\ q^{j} q^{k} = q^{k} q^{j+1} & j ≥ k \end{matrix} q^{j} p_{ε}^{k} = \{ \begin{matrix}p_{ε}^{k−1} q^{j} & j < k \\ 1 & j = k \\ p_{ε}^{k} q^{j+1} & j > k. \end{matrix}

We define an order relation on the morphisms from P_{l} to P_{m} by b ≤ c ⇔ b(x) ≤ c(x) , ∀ x. Note that b ≤ c implies ab ≤ ac and be ≤ ce for all morphisms a : P_{m} → P_{n} and e : P_{k} → P_{l}.

Remarks. a) The ’juxtaposition functor‘

⊔ : P × P → P , (P_{n}, P_{m}) ↦ P_{n+m}

makes P into a strictly monoidal category with neutral object P_{0} (cf. [11]). If (C, ⊔, e) is another strictly monoidal category and if F : {P_{0}, P_{1}} → C is a functor defined on the full subcategory {P_{0}, P_{1}} and sending P_{0} to e, there is a unique extension G : P → C of F which commutes with ⊔.

b) Taking the inequalities b ≤ c as 2-morphisms b → c we can view P as a 2-category [5, V.1]. Its 2-morphisms are generated by the

φ^{j} : p_{0}^{j} q^{j} → 1 , ψ^{j} : 1 → p_{1}^{j} q^{j}

subject to the relations

q^{j} φ^{j} = 1 , φ^{j} p_{0}^{j} = 1 , q^{j}ψ^{j} = 1 , ψ^{j} p_{1}^{j} = 1.

The functor ⊔ is indeed a 2-functor and P is a strictly monoidal 2-category. The analogue of a) holds with respect to the full 2-subcategory whose nonidentical 1-morphisms are

\begin{aligned}& P_{0} \\ p_{0}^{1} ↓ & q^{1} ↑ & ↓ p_{1}^{1} \\ & P_{1} \end{aligned} .

For example, if Add is the category of additive categories, there is a unique 2functor from P to Fun (Add, Add) (cf. A.1) which carries ⊔ to the composition of 2-functors and sends P_{0} to the identity, P_{1} to M (cf. example 2.2 a), p_{0}^{1}, q^{1}, p_{1}^{1} to P_{0}^{1}, Q_{0}^{1}, P_{1}^{1} and φ^{1}, ψ^{1} to the adjunction morphisms.

2.2 A tower T of additive categories is given by the following data

• a sequence of additive categories T_{n}, n ∈ N, • an additive functor T a : T_{n} → T_{m} for each morphism a : P_{m} → P_{n},

• a morphism of functors T (c, b) : T c → T b for each pair of morphisms b ≤ c from P_{l} to P_{m}.

These data are subject to the following conditions

• T 1 = 1 and T bT a = T ab,

• T (b, b) = 1 and T (c, b) T (d, c) = T (d, b),

• T (c, b)T a = T (ac, ab) and T eT (c, b) = T (ce, be),

whenever a is a morphism P_{m} → P_{n}, b ≤ c ≤ d are morphisms P_{l} → P_{m} and e is a morphism P_{k} → P_{l}. Note that the first condition implies that T gives rise to a functor from P^{op} to the category of additive categories and that the second condition implies that T gives rise to a functor from P (P_{l}, P_{m})^{op} to the category of additive functors from T_{m} to T_{l}. Here (as always) we regard an ordered set S as a category with objects s ∈ S and a morphism s → t for each pair s ≤ t. This definition can be summed up by saying that T is a 2-functor from P^{OP} to Add (cf. remark 2.1 b) and A.1), where P^{OP} denotes the 2-category P with reversed 1- and 2-morphisms.

In a completely analogous fashion, one defines towers of exact categories and towers of suspended categories.

For any tower T , we denote T p_{ε}^{j} by P_{ε}^{j} and T q^{j} by Q_{0}^{j}.

Examples. a) Let B be an additive category and B_{m}^{∧} the category of contravariant functors from P_{m} to B. Thus B_{m}^{∧} consists of the m-dimensional commutative hypercubes in B. If X ∈ B_{m}^{∧} and b : P_{l} → P_{m} is a morphism, we define (B^{∧}bX)(x) = Xb(x). If c is another morphism P_{l} → P_{m} and b ≤ c then

B^{∧}(c, b) : B^{∧}c → B^{∧}b

is furnished by the morphisms

Xc(x) → Xb(x) , x ∈ P_{l}.

Clearly B^{∧} is a tower of additive categories. Note that B_{0}^{∧} is isomorphic to B and that B_{n}^{∧} is isomorphic to

M^{n}B = MM … MB

where MB is the category of morphisms of B. More precisely, we have an isomorphism

M^{j} : B_{n+1}^{∧} → MB_{n}^{∧} , X ↦ (P_{1}^{j}X → P_{0}^{j}X)

for each 1 ≤ j ≤ n + 1. For later reference, we record the functors

Q_{−1}^{j} : B_{n}^{∧} → P_{n+1}^{∧} and Q_{1}^{j} : P_{n}^{∧} → P_{n+1}^{∧} ,

which we define by their compositions with M^{j}:

M^{j}Q_{−1}^{j}X = (0 → X) , M^{j}Q_{1}^{j}X = (X → 0).

Observe that we have a chain of adjoint functors

Q_{−1}^{j} ⊣ P_{0}^{j} ⊣ Q_{0}^{j} ⊣ P_{1}^{j} ⊣ Q_{1}^{j}.

b) Let A be an exact category. We convert A^{∧} into a tower of exact categories by endowing A_{n}^{∧} with the pairs whose evaluation at each x ∈ P_{n} is a conflation of A. Now we inductively define exact subcategories I^{n}A of A^{∧}: I^{0}A = A_{0}^{∧} = A and I^{n}A consists of the X ∈ A_{n}^{∧} such that the morphism

P_{1}^{1}X → P_{0}^{1}X

is an inflation of I^{n−1}A, n > 0. Using the snake lemma it is easy to verify that this is equivalent to requiring that

P_{1}^{j}X → P_{0}^{j}X

be an inflation for each j. It is easy to see that A^{∧}aX lies in I^{m}A if X lies in I^{n}A and a : P_{m} → P_{n} is a morphism. Thus the I^{n}A yield an exact ’subtower‘ I^{∗}A of A^{∧}.

c) If A is an exact category, then clearly the DA_{n}^{∧} form a tower DA^{∧} of suspended categories. If moreover A has enough injectives, then I^{n}A has enough injectives (namely the objects with injective components) and the \underline{I^{n}A} form a tower \underline{I^{∗}A} of suspended categories: This is due to the fact that the passage from exact to stable categories is compatible with composition of functors and with morphisms of functors.

2.3 Now let S and T be towers of additive categories. A tower of additive functors F : S → T consists of • a sequence of additive functors F_{n} : S_{n} → T_{n}, n ∈ N,

• an isomorphism F a : T aF_{n} →^{∼} F_{m}Sa for each a : P_{m} → P_{n} .

[figure: T_{n}; S_{n}; T_{m}; S_{m}; F_{n}; F_{m}; Sa; T a; F a]

We require that the F a be compatible with compositions in the sense that F ab = (F bSa)(T bF a) and that they be compatible with the morphisms S(c, b) and T (c, b) in the sense that F_{l} S(c, b) · F c = F b T (c, b) · F_{m}. Note that F does not give rise to a morphism between the functors P_{m} ↦ S_{m} and P_{m} ↦ T_{m} from P to the category of additive categories unless F a = 1 for all morphisms a. We may view F as a 1-morphism of Fun (P^{OP} , Add). It is clear how to define towers of exact functors and towers of S-functors.

We compose two towers of functors F : S → T and G : R → S by setting (F G)_{n} = F_{n}G_{n} and F Ga = (F_{m}Ga)(F aG_{n}).

Examples. a) Each additive functor F : B → C yields a tower of additive functors F^{∧} from B^{∧} to C^{∧}: F_{n}^{∧} is the induced functor on the category of n-dimensional hypercubes and F^{∧}a is the identity for each morphism a.

b) For each x ∈ P_{n} let a_{x} : P_{0} → P_{n} be the morphism with a_{x}(∗) = x. If T is a tower of additive categories, we define Φ : T → T_{0}^{∧} by

Φ_{n} : T_{n} → (T_{0}^{∧})_{n} , X ↦ (x ↦ (T a_{x})X)

and by Φa = 1 for each morphism a.

c) Let S be a tower of suspended categories and let S| be the tower of additive categories obtained from S by forgetting the suspended structure. The suspension functors S_{n} : S_{n} → S_{n} yield a tower of additive functors S : S| → S| where (Sa)^{−1} is the commutation isomorphism (Sa)S_{n} → S_{n}(Sa) for each morphism a of P.

2.4 Let E be a tower of exact categories and T a tower of suspended categories. A tower of ∂-functors D : E → T consists of

• a sequence of ∂-functors D_{n} : E_{n} → T_{n}, n ∈ N

• an isomorphism of ∂-functors F a : T aD_{n} → D_{m}Ea for each a : P_{m} → P_{n}.

As in 2.3 we require Dab = (DbEa)(T bDa) and Db·T (c, b)D_{m} = D_{n}E(c, b)·Dc. The composition of a tower of ∂-functors D : E → T with a tower of S-functors F : T → T is defined in analogy with 2.3.

Examples. In the situation of 2.2, Example b), we have an obvious tower of ∂-functors D : A^{∧} → DA^{∧} with Da = 1 for each morphism a. If A has enough injectives we also have a ∂-functor D : I^{∗}A → \underline{I^{∗}A} and again Da = 1 for each morphism a. In this case, there is also a less obvious tower D : A^{∧} → \underline{I^{∗}A} : D_{0} is the canonical ∂-functor A → \underline{A}, the underlying additive functor of D_{1} : MA → \underline{I^{1}A} sends

f : X_{1} → X_{0} to [ \begin{matrix}f \\ i \end{matrix} ] : X_{1} → X_{0} ⊕ I,

where i : X_{1} → I is an inflation into an injective. The connecting morphism of D_{1} is obtained by applying this construction to the category EA of conflations of A: It provides us with an additive functor

EMA →^{∼} MEA → \underline{I^{1}EA} ,

which we compose with the canonical functor from \underline{I^{1}EA} →^{∼} \underline{EI^{1}A} to the category of triangles of \underline{I^{1}A}. The construction of the higher D_{n} is similar. One can give a more rigorous treatment of this tower using 7.1 and A.2.

2.5 If F and G are towers of functors from S to T , a morphism μ : F → G is given by a sequence μ_{n} : F_{n} → G_{n} of morphisms of functors such that (μ_{m}T a)F a = Ga (T aμ_{n}) for each morphism a : P_{m} → P_{n}. Thus μ is a 2morphism of Fun (P^{OP} , Add). It is clear how to compose such morphisms. The towers of additive functors from S to T and their morphisms are easily seen to form a category Hom_{add}(S, T ). Morphisms of towers of S-functors and of ∂-functors are defined similarly. If S and T are towers of suspended categories, we denote the corresponding category of towers of S-functors by Hom_{S}(S, T ). If E is a tower of exact categories, we denote the category of towers of ∂-functors by Hom_{∂}(E, T ).

Examples. a) If μ : F → G is a morphism of additive functors from B to C, there is an obvious morphism μ^{∧} : F^{∧} → G^{∧} of towers.

b) We continue 2.3, Example b). Let A be an additive category and F : S → A^{∧} a tower of additive functors. There is an isomorphism φF : F → F_{0}^{∧}Φ, which is obtained as follows: For each X ∈ S_{n} and each x ∈ P_{n} we have (F_{n}X)(x) = (A_{0}^{∧}a_{x})(F_{n}X) by definition; we set

(φF)_{n}X(x) = (F a_{x})X : (A_{0}^{∧}a_{x})(F_{n}X) →^{∼} F_{0}Sa_{x}X.

It is straightforward to verify that φ is a well-defined morphism. Moreover if μ : F → F′ is a morphism of towers, we have (μ^{∧}Φ)(φF) = (φF′)μ. This shows that G ↦ G^{∧}Φ is a quasiinverse for the functor

Hom_{add}(S, A^{∧}) → Add (S_{0}, A) , F ↦ F_{0} ,

where Add (S_{0}, A) is the category of additive functors S_{0} → A. Observe that by the construction of φ, this functor induces an isomorphism of the full subcategory of the F with F a = 1, ∀ a onto Add (S_{0}, A).

2.6 Let T be an epivalent tower of suspended categories, i.e. for each n and for each 1 ≤ j ≤ n + 1 the functor

M^{j} : T_{n+1} → MT_{n} , X ↦ (P_{1}^{j}X → P_{0}^{j}X)

is an epivalence (i.e. M^{j} is full and essentially surjective and a morphism f is invertible iff M^{j}f is). If the T_{n} are triangulated categories, this means that T_{n+1} is a recollement of two copies of T_{n} (P_{0}^{j} = i^{!} and P_{1}^{j} = j^{∗} in the notations of [2, 1.4.3]) as we shall see in 6.1. Now let A be an exact category.

Theorem. The canonical functor Hom_{S}(DA^{∧}, T ) → Hom_{∂}(A^{∧}, T ) is an equivalence.

We shall prove this in section 9.

Examples. With the notations of 2.2, Example c), DA^{∧} and \underline{I^{∗}A} are epivalent towers, as we shall see in 6.1. The towers A^{∧} → DA^{∧} and A^{∧} → \underline{I^{∗}A} of 2.4 yield the identical tower DA^{∧} → DA^{∧} and a tower DA^{∧} → \underline{I^{∗}A}, respectively. The base DA → \underline{A} of the latter extends the canonical projection A → \underline{A}.

2.7 A suspended category is basic if it occurs as the base of an epivalent tower of suspended categories; an S-functor between two basic suspended categories is basic if it occurs as the base of a tower of S-functors; a morphism between two basic S-functors ... . Finally, if A is an exact category and T a tower of suspended categories, a ∂-functor from A to T_{0} is basic if it occurs as the base of a tower of ∂-functors from A^{∧} to T and similarly for morphisms. We point out that ’basic‘ always refers to a fixed choice of the respective towers.

Theorem 2.6 is to be conceived of as a means for studying basic S-functors, ∂-functors ... . Thus it is crucial to know how large the classes of these basic entities are. Empirically, we have found that they are quite large (see 2.8). Indeed we do not know of an example of a non-basic entity. On the other hand, we are not able to prove that, for example, each suspended category is basic, or that a tower over a given base is ’unique‘ if it exists.

The following theorem shows that certain ∂-functors are necessarily basic and that all morphisms between certain pairs of ∂-functors are basic. We use the above notations.

Theorem.

a) If D : A → T_{0} is a ∂-functor with T_{0}(S^{n}DA, DB) = 0 for all n > 0, A, B ∈ A, there is a tower of ∂-functors D^{+} : A^{∧} → T with D_{0}^{+} →^{∼} D. The tower D^{+} is unique up to unique isomorphism.

b) If D, D′ are towers of ∂-functors A^{∧} → T with T_{0}(S^{n}D_{0}A, D′_{0}B) = 0 for all n > 0, A, B ∈ A, the map

Hom (D, D′) → Hom (D_{0}, D′_{0}) , μ ↦ μ_{0}

is bijective.

Combined with 2.6 this immediately yields the Corollary.

a) If D : A → T_{0} is a ∂-functor with T_{0}(S^{n}DA, DB) = 0 for all n > 0, A, B ∈ A, there is a basic S-functor F : DA → T_{0} extending D. It is unique up to a unique basic isomorphism.

b) If F and F′ are two basic S-functors DA → T_{0} with T_{0}(S^{n}F A, F′A) = 0 for all n > 0, A, B ∈ A and μ is a morphism between the restrictions of F and F′ to A, there is a unique basic morphism of S-functors ν : F → F′ extending μ.

2.8 The following theorem and its corollary account for the all-pervasive-ness of basic S-functors. Suppose that S and T are basic suspended categories.

Theorem. If L : T → S is left S-adjoint to a basic S-functor, it is basic.

Note that the dual statement holds for basic co-suspended categories and that both hold for basic triangulated categories (we entrust the reader with the definition of these concepts). In view of the desription of the morphisms of P, the theorem follows from A.2 and 6.5.

Now assume A is an exact category with enough projectives. The localisation functor H_{+}A → D_{+}A is of course basic and has the projective resolution functor (dual to [9, 4.1]) as a left adjoint. By the theorem, projective resolution is a basic S-functor. Hence we have the

Corollary. If F : A → B is an additive functor, the left derived functor \underline{L}F : D_{+}A → D_{+}B is a basic S-functor.

3. Presheaves and the derived category

Let A be an exact category and N_{1} the set of natural numbers n ≥ 1. We endow N_{1} with the topology of the cofinite sets. A rough presheaf F on N_{1} with values in A^{op} consists of objects F(U) ∈ A^{op} for each open subset U of N_{1} and of morphisms ρ_{UV} : F(U) → F(V) of A^{op} for each inclusion U ⊂ V of open sets of N_{1}. We require that

• ρ_{UU} = 1 for each open U ⊂ N_{1},

• ρ_{UW} = ρ_{V W} ρ_{UV} whenever U ⊂ V ⊂ W are open in N_{1} and • there is an open M ∈ N_{1} such that F(U) vanishes if U does not contain

M.

In particular we have F(∅) = 0. We endow the category RA of rough presheaves with the exact structure consisting of the pairs

F′ →^{i} F →^{d} F′′

such that

F′(U) →^{iU} F(U) →^{dU} F′′(U)

is a conflation of A^{op} for each open U ⊂ N_{1}. By definition the class Σ consists of those morphisms s : F → F′′ which fit into a conflation

F′ →^{i} F →^{s} F′′ ,

where F′ admits a j ∈ N_{1} such that the restriction

F′(U) → F′(U ∩ U_{j})

is invertible for all open U, where U_{j} = N_{1} − j.

The open sets U_{j}, j ∈ N_{1} clearly form an open covering U of N_{1}. If F is a rough presheaf, the associated Čech-complex

CF = C(U, F)

has the components C^{0}F = F(N_{1}) and

C^{p}F = ∏_{i_{1}<...<i_{p}} F(U_{i_{1}} ∩ … ∩ U_{i_{p}}) , p > 0

and the differential given by the matrix of the

(−1)^{k}ρ : F(U_{i_{1}} ∩ … U_{i_{k−1}} ∩ U_{i_{k+1}} ∩ … ∩ U_{i_{p+1}}) → F(U_{i_{1}} ∩ … ∩ U_{i_{p+1}}).

Observe that there are only finitely many non-zero terms in the definition of C^{p}F and that CF is a positive differential complex over A.

Theorem. The functor C induces an equivalence (RA)[Σ^{−1}] → DA.

This is an ’abstract‘ localisation [5, I, 1.1]. We shall give the proof in 5.4.

4. The universal property of the construction D

In this section, we view the assignment A ↦ DA as a 2-functor (cf. A.1) from the 2-category of exact categories Ex to the 2-category of suspended categories Susp. This simply means that D is defined on exact categories, exact functors and morphisms of exact functors and is compatible with the various composition functors. The canonical ∂-functors A → DA, A ∈ Ex combine into a 2-∂-functor can in the sense of the following definition: If F : Ex → Susp is a 2-functor, a 2-∂-functor D to F consists of

• a ∂-functor DA : A → FA for each exact category A

• an isomorphism of ∂-functors DF : (FF)(DA) → (DB)F for each exact functor F : A → B.

These data are required to be compatible with compositions of exact functors and with morphisms of exact functors in the already familiar fashion:

D(GF) = (DG)F · (FG)(DF) and (DB)μ · DF = DG · FμDA

whenever we have exact functors

A →^{F} B →^{G} C

or a morphism μ : F → G of exact functors from A to B. A morphism α : D → D′ of 2-∂-functors assigns a morphism of ∂-functors αA : DA → D′A to each exact A. The αA are required to satisfy

D′F · FF αA = αBF · DF

for each exact functor F : A → B. The 2-∂-functors to a given F form a category Hom_{∂}(F). Similarly, the 1-morphisms D → F of Fun (Ex, Susp) form a category Hom_{S} (D, F). The canonical 2-∂-functor induces a functor

can^{∗} : Hom_{S}(D, F) → Hom_{∂}(F).

We shall give a sufficient condition for can^{∗} to be an equivalence. For each A ∈ Ex let MA be the category of morphisms of A, JA : IA → MA the inclusion of the subcategory of MA consisting of the inflations of A and

0 → A ⟶^{QA} IA ⟶^{P A} A → 0

the exact sequence of exact categories with (QA)B = (0 → B) and (P A)(A → B) = A. We shall see that DJA is an equivalence (10.1) and that Im DQA = Ker DP A (Example 6.1 c).

Theorem. The functor can^{∗} is an equivalence if FJA is an equivalence and Im FQA = Ker FP A for each exact A.

We shall prove this in section 10.

5. Proof of Theorem 3

5.1 Proposition.

a) DA is isomorphic to the localisation of H_{0]}^{b} A (1.1) at the class of morphisms \overline{s} which fit into a triangle

X →^{\overline{s}} Y → A → SX

of H_{0]}^{b} A with acyclic A.

b) DA is ismorphic to the localisation of C_{0]}^{b} A (the category of bounded positive complexes) at the class of morphisms d which fit into a sequence

A →^{i} X →^{d} X′

of C_{0]}^{b} A such that (i_{n}, d_{n}) is a conflation of A for all n ∈ N and A is of the form

… → 0 → A_{j} →^{∼} A_{j−1} → 0 → …

for some j ≥ 1.

Proof. a) As in the classical case [15, I, §2, no. 1] one sees that the class of morphisms \overline{s} admits a calculus of left fractions [5, I, 2.3]. Now suppose X is a positive complex and X′ appears in a triangle

B → X →^{\overline{t}} X′ → SB

of H_{b}A with an acyclic B. Realising X′ as a mapping cone we see that \overline{t} factors as \overline{r}\overline{s}, where \overline{s} is as in the assertion and \overline{r} : X′_{≥0} → X′ induces an isomorphism

H_{b}A (K, X′_{≥0}) →^{∼} H_{b}A (K, X′)

for each positive complex K. Using left fractions to compute the morphism groups D_{b}A (K, X) we infer

DA (K, X) →^{∼} D_{b}A (K, X).

b) Let us first assume that each conflation of A splits. We have to show that H_{0]}^{b} A identifies with the localisation of C_{0]}^{b} A at the class Θ of morphisms d. Obviously the functor (C_{0]}^{b} A)[Θ^{−1}] → H_{0]}^{b} A is full and bijective on objects. Since the d ∈ Θ admit right inverses in C_{0]}^{b} A, the localisation functor C_{0]}^{b} A → (C_{0]}^{b} A)[Θ^{−1}] is full as well and it only remains to be shown that two morphisms of complexes f, g : X → Y have the same image in (C_{0]}^{b} A)[Θ^{−1}] if they are homotopic. Let f − g = ki_{X}, where i_{X} : X → IX has the components

[ \begin{matrix}1 \\ d_{n} \end{matrix} ] : X_{n} → X_{n} ⊕ X_{n+1} = (IX)_{n} and d^{IX} = [ \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ] .

Since [1_{X} 0] : X ⊕ IX → X is in Θ, the morphisms [1_{X} i]^{t} and [1_{X} 0]^{t} : X → X ⊕ IX have the same image in (C_{0]}^{b} A)[Θ^{−1}]. Hence the same is true of f = [g k][1_{X} i]^{t} and g = [g k][1_{X} 0]^{t}.

If A is a general exact category, we conclude from the above special case that (C_{0]}^{b} A)[Θ^{−1}] identifies with the localisation of H_{0]}^{b} A at the image of Θ. We have to show that the saturation of the image contains each morphism \overline{s} as in a). Since the mapping cone over s is acyclic, there is a commutative diagram

[figure: …; X_{n+1}; Z_{n+1}; Y_{n+1}; X_{n}; Z_{n}; Y_{n}; X_{n−1}; Z_{n−1}; Y_{n−1}; …; X_{1}; Z_{1}; Y_{1}; X_{0}; Z_{0}; Y_{0}; d_{n+1}; d_{n}; d_{1}; j′_{n}; q′_{n}; j′_{1}; q′_{1}; j′′_{n}; q′′_{n}; j′′_{1}; q′′_{1}; d_{n+1}; d_{n}; d_{1}]

where s_{n} = j′′_{n} q′_{n+1}, j′_{n} q′′_{n+1} = 0 and

Z_{n} →^{j_{n}} X_{n−1} ⊕ Y_{n} →^{q_{n}} Z_{n−1} , j_{n} = [−j′_{n} j′′_{n}]^{t} , q_{n} = [q′_{n} q′′_{n}] ,

is a conflation ∀ n > 0. We see that s = t^{0} · t^{1} · t^{2} …, where

(t_{0}^{0}, t_{1}^{0}, t_{2}^{0}, …) = (j′′_{0}, 1_{Y_{1}}, 1_{Y_{2}}, …)
(t_{0}^{1}, t_{1}^{1}, t_{2}^{1}, …) = (q′_{1}, j′′_{1}, 1_{Y_{2}}, …)
⋮
(t_{0}^{n}, t_{1}^{n}, t_{2}^{n}, …) = (1_{X_{0}}, 1_{X_{1}}, …, 1_{X_{n−2}}, q′_{n}, j′′_{n}, 1_{Y_{n+1}}, …)
⋮

We may therefore assume that there is an n ≥ 1 such that s_{i} = 1_{X_{i}} for i > n and i < n − 1 and that the sequence ([d_{n} − s_{n}]^{t}, [s_{n−1} d_{n}]) is a conflation. The sequence of complexes

… 0 → X_{n}  ⟶^{1}  X_{n}  → 0 …
↓  u_{n} ↓  ↓ u_{n−1}  ↓
… X_{n+1} → X_{n} ⊕ Y_{n} ⟶^{d_{n}⊕1} X_{n−1} ⊕ Y_{n} → X_{n−2} …
↓  v_{n} ↓  ↓ v_{n−1}  ↓
… Y_{n+1} →  Y_{n}  ⟶  Y_{n−1}  → Y_{n−2} …

where u_{n} = [1 − s_{n}]^{t}, v_{n} = [s_{n} 1], u_{n−1} = [d_{n} − s_{n}]^{t}, v_{n−1} = [s_{n−1} d_{n}], shows that such an \overline{s} becomes invertible in (C_{0]}^{b} A)[Θ^{−1}].

5.2 Lemma. The composition

RA →^{C} C_{0]}^{b} A → DA

makes the s ∈ Σ invertible.

Proof. The image of the conflation

F′ →^{i} F →^{s} F′′

of section 3 fits into a triangle of DA. It is therefore enough to show that CF′ is acyclic. Indeed, CF′ is split acyclic, as it is not hard to verify.

5.3 Let K ∈ C_{0]}^{b} A. Using the notations of section 3 we define a rough presheaf RK on N_{1}:

• (RK)(U) = 0 unless U = U_{1} ∩ … ∩ U_{n} for some n ∈ N in which case (RK)(U) = K_{n},

• ρ_{UV} = 0 unless U = U_{1} ∩ … ∩ U_{n}, V = U ∩ U_{n+1} for some n ∈ N in which case ρ_{UV} = d_{n+1}.

Lemma. The composition

C_{0]}^{b} A →^{R} RA → (RA)[Σ^{−1}]

induces a functor DA → (RA)[Σ^{−1}].

Proof. If

A →^{i} X →^{d} X′

is a sequence as in 5.1 b), the pair (Ri, Rd) is a conflation of RA and

(RA)(U) → (RA)(U ∩ U_{j})

is clearly invertible for each open U ⊂ N_{1}. Hence Rd ∈ Σ.

5.4 We prove theorem 3. According to 5.2 and 5.3, C and R induce functors

(RA)[Σ^{−1}] \overrightarrow{←} DA ,

which we also denote by C and R. It is clear that CR is isomorphic to the identity. We shall produce an isomorphism between RC and the identity of (RA)[Σ^{−1}]. More precisely, we shall first construct an isomorphism

1 →^{Ψ} G

in (RA)[Σ^{−1}], where G is a functor RA → RA such that Gs becomes invertible in (RA)[Σ^{−1}] for all s ∈ Σ and that (GF)(U) ≠ 0 only if U is of the form U_{1} ∩ … ∩ U_{n} for some n ∈ N. It is then clear that we have an isomorphism RCG →^{∼} G of functors RA → RA, hence an isomorphism

RC ⟶^{RCΨ} RCG →^{∼} G ←^{Ψ} 1

of functors (RA)[Σ^{−1}] → (RA)[Σ^{−1}]. We first have to introduce some notation: If M ⊂ N_{1} is a finite subset and V ⊂ N_{1} its complement then, since N_{1} = M ∐ V, a rough presheaf F on N_{1} is given by the presheaf W ↦ F_{W} on the discrete space M whose value at W ⊂ M is the rough presheaf

U ↦ F_{W} (U) = F(W ∪ U) , U ⊂ V

on V. Thus if M = {m, n}, we may describe F by the commutative square

F_{{m,n}} →^{a} F_{{m}}
b ↓  ↓ c
F_{{n}} →^{d} F_{∅} ,

where a, …, d are the restriction morphisms. We now define a presheaf G = F′_{n,m}F by the square

\begin{aligned}G_{\{m,n\}} & → & G_{\{m\}} \\ ↓ & & ↓ \\ G_{\{n\}} & → & G_{∅} \end{aligned} := \begin{matrix}F_{\{m,n\}} & ⟶^{a} & F_{\{m\}} \\ [b − a]^{t} ↓ & & ↓ [c − 1]^{t} \\ F_{\{n\}} ⊕ F_{\{m\}} & ⟶^{d⊕1} & F_{∅} ⊕ F_{\{m\}} \end{matrix}

and a presheaf H = F_{n,m}F by the square

\begin{aligned}H_{\{m,n\}} & → & H_{\{m\}} \\ ↓ & & ↓ \\ H_{\{n\}} & → & H_{∅} \end{aligned} := \begin{matrix}F_{\{m,n\}} & ⟶ & 0 \\ [b − a]^{t} ↓ & & ↓ \\ F_{\{n\}} ⊕ F_{\{m\}} & ⟶^{[d c]} & F_{∅} \end{matrix} .

We have a morphism α : F′_{n,m}F → F whose components are the obvious projections and a morphism β : F′_{n,m}F → F_{n,m}F whose components are obvious except for

[1 c] : F_{∅} ⊕ F_{{m}} → F_{∅}

in the lower right corner. Clearly α and β are deflations,

Ker α(U) → Ker α(U ∩ U_{m})

is invertible ∀ U and

Ker β(U) → Ker β(U ∩ U_{n})

is invertible ∀ U. Hence α, β lie in Σ. This implies that F_{n,m}s is invertible in RA[Σ^{−1}] for all s ∈ Σ and that we have an isomorphism

1 → F_{n,m}

of functors RA[Σ^{−1}] → RA[Σ^{−1}]. Observe that if F has width

w = min {p : C^{q}F = 0 ∀ q > p}

then F_{n,m}F has width w and that F_{n,m}F →^{∼} F in RA if n > w. Moreover if (F_{n,m}F)(U) ≠ 0, then U ∩ {n, m} ≠ {m} and we have F(U) ≠ 0 or F(U ∪ {m} − {n}) ≠ 0. Using this it is not hard to verify that μ(F_{n,m}F) ≥ μ(F) if (n, m) ≤ μ(F) and that μ(F_{n,m}F) > (n, m) if (n, m) = μ(F), where μ(F) ≤ ∞ is minimal (with respect to the lexicographic ordering) among the pairs (k, l), k > l such that there is an open U ⊂ N_{1} with U ∩ {k, l} = l and F(U) ≠ 0. Now we define functors G_{n} : RA → RA , n ∈ N_{1} by

G_{1} = 1 , G_{2} = F_{2,1} , …, G_{n} = F_{n,n−1} F_{n,n−2} … F_{n,1} G_{n−1}.

It is clear that μ(G_{n}F) ≥ (n, n − 1) and that G_{n}F →^{∼} G_{n+1}F in RA if n is greater than the width of F. The functor G defined by

GF = lim_{⟶} G_{n}F , n > w

has the required properties.

6. Epivalence and Recollement

6.1 Let S_{0} and S_{1} be suspended categories and Q_{0} : S_{0} → S_{1} a fully faithful S-functor admitting a left S-adjoint P_{0} and a right S-adjoint P_{1}. The functor

M : S_{1} → MS_{0} , X ↦ (MX : P_{1}X → P_{0}X)

from S_{1} to the category of morphisms of S_{0} is defined by requiring that Q_{0}MX equal the composition of the adjunction morphisms

Q_{0}P_{1}X → X → Q_{0}P_{0}X.

We consider the conditions

(E) The functor M is an epivalence (I, 5.2).

(R) There are chains of S-adjoint functors

P_{−1} ⊣ Q_{−1} ⊣ P_{0} and Q_{0} ⊣ P_{1} ⊣ Q_{1} ,

Q_{−1} and Q_{1} are fully faithful and Ker P_{1} = Im Q_{−1}.

If the suspension functors of S_{0} and S_{1} are equivalences, condition (R) is equivalent to the recollement setup of [2, 1.4.3] with P_{0} = i^{!} and P_{1} = j^{∗}, as it is not hard to verify.

Examples. a) If T is an epivalent tower the S-functor Q_{0}^{j} : T_{n} → T_{n+1} satisfies (E).

b) If A is an exact category with enough injectives, then IA, the full subcategory of MA consisting of the inflations of A, is exact with enough injectives (I, 5.1). The functor

Q_{0} : \underline{A} → \underline{IA} , A ↦ (A →^{1} A)

has the left adjoint

P_{0} : \underline{IA} → \underline{A} , (A_{1} →^{i} A_{0}) ↦ A_{0}

and the right adjoint

P_{1} : \underline{IA} → \underline{A} , (A_{1} →^{i} A_{0}) ↦ A_{1}.

The functor M is simply given by

\underline{IA} → M\underline{A} , (A_{1} →^{i} A_{0}) ↦ (A_{1} ⟶^{\overline{i}} A_{0}).

Condition (E) holds (compare [9, 5.2]). We infer that the tower T = \underline{I^{∗}A} is epivalent.

c) Let A be an exact category. The functor

A → MA , A ↦ (A →^{1} A)

has two successive left adjoints given by

(A_{1} →^{f} A_{0}) ↦ A_{0} , A ↦ (0 → A)

and two successive right adjoints given by

(A_{1} →^{f} A_{0}) ↦ A_{1} , A ↦ (A → 0).

These functors induce a chain of S-adjoint functors

Q_{−1} ⊣ P_{0} ⊣ Q_{0} ⊣ P_{1} ⊣ Q_{1}

between DA and DMA, Q_{0} being induced by A ↦ (A →^{1} A). It is clear that 1 →^{∼} P_{0}Q_{−1}, 1 →^{∼} P_{1}Q_{0} and P_{1}Q_{1} →^{∼} 1, which means that Q_{−1}, Q_{0} and Q_{1} are fully faithful. Moreover for each complex K, there is a triangle

Q_{−1}P_{0}K → K → Q_{1}P_{1}K → SQ_{−1}P_{0}K ,

which shows that Ker P_{1} = Im Q_{−1} since the suspension functor of DMA is fully faithful. In order to construct the missing left adjoint P_{−1}, we use that DMA ←^{∼} DIA (10.1). P_{−1} is ’induced‘ by

Cok : IA → A , (A_{1} →^{i} A_{0}) ↦ Cok i.

We have thus shown that Q_{0} : DA → DMA satisfies (R). By the following lemma Q_{0} also satisfies (E). We infer that T = DA^{∧} is epivalent.

Lemma.

a) Suppose (E) holds. Then (R) holds and the functors Q_{−1}, Q_{1} and Q_{0} induce S-equivalences from S_{0} to the full subcategories of S_{1} consisting of the X with P_{1}X = 0, P_{0}X = 0 and with invertible MX, respectively. Moreover for each X ∈ S_{1} there is an exact sequence

S_{0} (SP_{1}X, P_{0}Y) → S_{1} (X, Y) →^{M} MS_{0} (MX, MY) → 0 ,

which is functorial in Y ∈ S_{1}.

b) Suppose (R) holds. For each X ∈ S_{1} there is a unique morphism ζX such that the S-sequence

Q_{0}P_{1}X → X → Q_{−1}P_{−1}X →^{ζX} SQ_{0}P_{1}X

is a triangle. There is a triangle

P_{1}X →^{MX} P_{0}X → P_{−1}X → SP_{1}X

functorial in X ∈ S_{1}. There is a canonical isomorphism η : P_{−1}Q_{1} →^{∼} S. If the suspension functor of S_{1} is fully faithful, then (E) holds.

Proof. a) Construction of Q_{−1}: Let Y ∈ S_{0}. We choose X ∈ S_{1} such that there is an isomorphism

0 → Y
↓  ↓ f_{0}
P_{1}X → P_{0}X.

Since M is an epivalence, f_{0} yields a surjection

S_{1} (X, U) → S_{0} (Y, P_{0}U) , g ↦ (P_{0}g)f_{0}.

Suppose that g is mapped to 0. We form a triangle

X →^{g} U →^{h} V → SX

in S_{1}. Since P_{0}g = 0, P_{0}h admits a retraction. Since P_{0}X = 0, P_{1}h is invertible. Hence Mh admits a retraction. Since M is an epivalence, this implies that h admits a retraction, so g = 0. Using 6.7 we complete Q_{−1} to a left S-adjoint. By the construction, we have P_{1}Q_{−1} = 0 and 1 →^{∼} P_{0}Q_{−1}. If P_{1}U = 0, then the construction shows that the image of U → Q_{−1}P_{0}U under M is invertible. Since M detects isomorphisms, U →^{∼} Q_{−1}P_{0}U, so Ker P_{1} = Im Q_{−1}. The construction of Q_{1} is similar. It shows that Q_{1} is an equivalence of S_{0} onto the full subcategory of S_{1} consisting of the X with P_{0}X = 0. – Since Q_{0} is fully faithful, MQ_{0}X is invertible for all Y ∈ S_{0}. Conversely, if MX is invertible, then the image of Q_{0}P_{1}X → X under M is invertible.

Construction of P_{−1}: Let X ∈ S_{1}. We form a triangle

Q_{0}P_{1}X → X → U → SQ_{0}P_{1}X

over the adjunction morphism. The associated long exact Hom-sequence shows that Hom (X, Q_{−1}Y) ←^{∼} Hom (U, Q_{−1}Y) for each Y ∈ S_{0} since P_{1}Q_{−1} = 0. Applying P_{1} to the triangle we see that P_{1}U = 0 hence U ∈ Im Q_{−1}. Since Q_{−1} is fully faithful, we can conclude that P_{−1} exists as an additive functor. Using 6.7 we turn it into a left S-adjoint. Now let X ∈ S_{1}, a = −Q_{−1}MX and let b : Q_{−1}P_{1}X → Q_{0}P_{1}X be the morphism such that P_{0}b is the composition

P_{0}Q_{−1}P_{1}X ←^{∼} P_{1}X →^{∼} P_{0}Q_{0}P_{1}X.

We form a triangle

Q_{−1}P_{1}X ⟶^{[a b]^{t}} Q_{−1}P_{0}X ⊕ Q_{0}P_{1}X →^{c} U → SQ_{−1}P_{1}X.

Let

e : Q_{−1}P_{0}X ⊕ Q_{0}P_{1}X → X

have the adjunction morphisms as components. Then e[a b]^{t} = 0, so e = fc for some f. Since the images of the triangle under P_{0} and P_{1} are split exact sequences, Mf is invertible. So f is invertible and we have a triangle

Q_{−1}P_{1}X → Q_{−1}P_{0}X ⊕ Q_{0}P_{1}X → X → SQ_{−1}P_{1}X.

Applying S_{1} (?, Y) to this triangle we get the sequence of the assertion.

b) As in [2, 1.4.3] one sees that the ’morphism of degree 1‘ is unique if it exists. To derive the first triangle, we form a triangle

Q_{0}P_{1}X → X → U → SQ_{0}P_{1}X

over the adjunction morphism. Applying P_{1} to the triangle we see that P_{1}U = 0 hence U ∈ Im Q_{−1}. Since the triangle also shows that

Hom (X, Q_{−1}Y) ←^{∼} Hom (U, Q_{−1}Y) , ∀ Y ∈ S_{0}

this implies that U is canonically isomorphic to Q_{−1}P_{−1}X. We obtain the second triangle by applying P_{0} to the first. If we apply the second triangle to X = Q_{1}Y we get an isomorphism P_{−1}Q_{1}Y → SP_{1}Q_{1}Y since P_{0}Q_{1}Y = 0 by adjunction. The required isomorphism is the composition

P_{−1}Q_{1}Y → SP_{1}Q_{1}Y → SY.

Now suppose the suspension functor of S_{1} is fully faithful. Let X ∈ S_{1}. We form a triangle

X → Q_{1}P_{1}X → U → SX

in S_{1}. Applying P_{1} we see that P_{1}U = 0, hence U ∈ Im Q_{−1}. Moreover

Hom (Q_{−1}Y, U) →^{∼} Hom (Q_{−1}Y, SX) , ∀ Y ∈ S_{0}.

So U is canonically isomorphic to Q_{−1}P_{0}SX and we have a triangle

Q_{−1}P_{0}X → X → Q_{1}P_{1}X → SQ_{−1}P_{0}X.

Now it is obvious that a morphism f : X → X′ is invertible iff P_{0}f and P_{1}f are invertible, i.e. M detects isomorphisms. In fact, this was all we needed besides condition (R) to derive the exact sequence of a). Hence M is an epivalence.

6.2 Suppose Q_{0} : S → T satisfies condition (E) and X, Y ∈ S_{1}.

Lemma.

a) The map

S_{1} (X, Y) → MS_{0} (MX, MY)

is bijective if S_{0} (SP_{1}X, P_{0}Y) = 0. We have

S_{1} (S^{k}X, Y) = 0 , ∀ k > 0

if S_{0} (S^{k}P_{1}X, P_{1}Y) = S_{0} (S^{k}P_{0}X, P_{0}Y) = S_{0} (S^{k}P_{1}X, P_{0}Y) = 0 for each k > 0.

b) If

X →^{f} Y →^{g} Z

are morphisms of S_{1} such that gf = 0, that there are triangles

P_{1}X →^{P_{1}f} P_{1}Y →^{P_{1}g} P_{1}Z → SP_{1}X , P_{0}X →^{P_{0}f} P_{0}Y →^{P_{0}g} P_{0}Z → SP_{0}X

in S_{0} and that S_{0} (SP_{1}X, P_{1}Z) = S_{0} (SP_{0}X, P_{0}Z) = 0, then there is a triangle

X →^{f} Y →^{g} Z → SX

in S_{1}.

Proof. a) is an immediate consequence of 6.1 a). b) Let

X →^{f} Y →^{h} U →^{k} SX

be a triangle in S_{1}. Its ’images‘ under P_{1} and P_{0} are isomorphic to the given triangles. Hence there are isomorphisms i_{1} : P_{1}U → P_{1}Z and i_{0} : P_{0}U → P_{0}Z with i_{1} P_{1}h = P_{1}g and i_{0} P_{0}h = P_{0}g, respectively. Since S_{0} (SP_{1}X, P_{1}Z) = 0 = S_{0} (SP_{0}X, P_{0}Z), i_{0} and i_{1} are uniquely determined by these equalities. Now gf = 0, so g = jh for some j : U → Z. We have P_{0}j = i_{0} and P_{1}j = i_{1}. Since M is an epivalence, j is invertible. Hence (f, g, kj^{−1}) is a triangle.

6.3 We prepare for the proof of the redundancy of the connecting morphisms (section 7). Let

S_{0} →^{F_{0}} S′_{0}
Q_{0} ↓  ↓ Q_{0}
S_{1} →^{F_{1}} S′_{1}

be a diagram of suspended categories and S-functors and let

μ : F_{1}Q_{0} →^{∼} Q_{0}F_{0}

be an isomorphism of S-functors. We assume that both functors Q_{0} satisfy condition (R) of 6.1, that the morphisms

P_{0}F_{1} → F_{0}P_{0} , P_{1}F_{1} ← F_{0}P_{1}

associated (A.4) with μ and μ^{−1} are invertible and that the morphisms

F_{1}Q_{−1} ← Q_{−1}F_{0} , F_{1}Q_{1} → Q_{1}F_{0}

associated with the inverses of the above morphisms are also invertible. We now infer from the first triangle of 6.1 b) that the associated morphism

P_{−1}F_{1} → F_{0}P_{−1}

is also invertible.

Remark. If the functors Q_{0} satisfy condition (E) and the morphisms

P_{0}F_{1} → F_{0}P_{0} , P_{1}F_{1} ← F_{0}P_{1}

are invertible, the morphisms

F_{1}Q_{−1} ← Q_{−1}F_{0} , F_{1}Q_{1} → Q_{1}F_{0}

are automatically invertible (apply M to these morphisms and use 6.1 a).

Let φ_{0} : SF_{0} → F_{0}S be the commutation isomorphism. A straightforward verification shows that the diagram

\begin{matrix}F_{0}S & & ⟶^{φ_{0}} & & SF_{0} \\ F_{0}η ↓ & & & & ↓ ηF_{0} \\ F_{0}P_{−1}Q_{1} & →^{∼} & P_{−1}F_{1}Q_{1} & →^{∼} & P_{−1}Q_{1}F_{0} \end{matrix} (∗)

is commutative. This means that φ_{0} is uniquely determined by the η and by the underlying additive functors of the S-functors at hand. We make this more precise. Suppose that we are given additive functors

F_{0} : S_{0} → S′_{0} and F_{1} : S_{1} → S′_{1}

and an isomorphism of functors

μ : Q_{0}F_{0} →^{∼} F_{1}Q_{0}.

We assume that

a) the associated morphisms P_{0}F_{1} → F_{0}P_{0}, P_{1}F_{0} → F_{0}P_{1} are invertible,

b) the morphisms F_{1}Q_{−1} → Q_{−1}F_{0}, F_{1}Q_{1} → Q_{1}F_{0} associated to the inverses of the morphisms of a) are invertible and

c) the associated morphism P_{−1}F_{1} → F_{0}P_{−1} is invertible.

Observe that b) follows from a) if the functors Q_{0} satisfy condition (E) (apply M to the morphisms of b). Moreover, assuming a) and b), a sufficient condition for c) is that for each triangle

X →^{u} Y →^{v} Z →^{w} SX

of S_{1} there is some triangle

F_{1}X →^{F_{1}u} F_{1}Y →^{F_{1}v} F_{1}Z →^{w′} SF_{1}X

of S′_{1}. This follows from the unicity of the first triangle of 6.1 b).

Under the hypotheses a), b) and c), (F_{0}, φ_{0}) is an S-functor if φ_{0} is defined by diagram (∗) and Q_{0} satisfies condition (E). The proof is a straightforward verification based on 6.1 b). We shall also need to know that (F_{0}, φ_{0}) ’functorially depends‘ on the triple consisting of F_{0}, F_{1}, and μ. That is to say that if we are given another triple

G_{0} : S_{0} → S′_{0} , G_{1} : S_{1} → S′_{1} , ν : Q_{0}G_{0} →^{∼} G_{1}Q_{0}

satisfying the analogues of a), b) and c), and if

α_{0} : F_{0} → G_{0} and α_{1} : F_{1} → G_{1}

are morphisms of functors such that the diagram

Q_{0}F_{0} ⟶^{μ} F_{1}Q_{0}
Q_{0}α_{0} ↓  ↓ α_{1}Q_{0}
Q_{0}G_{0} ⟶^{ν} G_{1}Q_{0}

commutes, then α_{0} gives rise to a morphism of S-functors (F_{0}, φ_{0}) → (G_{0}, γ_{0}). Again we omit the straightforward proof. Finally we have to consider compositions of functors. Suppose that we are given additive functors

S_{0} →^{F_{0}} S′_{0} →^{G_{0}} S′′_{0} , S_{1} →^{F_{1}} S′_{1} →^{G_{1}} S′′_{1}

and commutation isomorphisms

μ : Q_{0}F_{0} → F_{1}Q_{0} , ν : Q_{0}G_{0} → G_{1}Q_{0}

yielding invertible associated morphisms. Then the triple consisting of G_{0}F_{0}, G_{1}F_{1} and (G_{1}μ)(νF_{0}) yields an S-functor (G_{0}F_{0}, θ). It is not hard to verify that this S-functor is the composition of the S-functors (G_{0}, γ_{0}) and (F_{0}, φ_{0}) constructed from F_{0}, F_{1}, μ and G_{0}, G_{1}, ν, respectively.

6.4 Let A be an exact category. We have the functors

Q_{−1} : A → MA , A ↦ (0 → A) , P_{0} : MA → A , (A_{1} → A_{0}) ↦ A_{0} ,
Q_{0} : A → MA , A ↦ (A →^{1} A) , P_{1} : MA → A , (A_{1} → A_{0}) ↦ A_{1} ,

Q_{1} : A → MA , A ↦ (A → 0).

Moreover if X = (A →^{i} B) is an inflation, we have a well defined P_{−1}X = Cok i in A. Now let Q_{0} : S_{0} → S_{1} be an S-functor satisfying condition (R), D_{0} : A → S_{0} and D_{1} : MA → S_{1} ∂-functors and λ : Q_{0}D_{0} →^{∼} D_{1}Q_{0} an isomorphism of ∂-functors. As in 6.3, we require that the associated morphisms

P_{0}D_{1} → D_{0}P_{1} , P_{1}D_{1} ← D_{0}P_{1} , D_{1}Q_{−1} ← Q_{−1}D_{0} , D_{1}Q_{1} → Q_{1}D_{0}

are invertible. If Q_{0} : S_{0} → S_{1} even satisfies condition (E), then we only have to require this for the first pair of morphisms. It follows for the second pair by 6.1 a). If

ε : A →^{i} B →^{d} C

is a conflation of A, we have a conflation

ε^{+} : \begin{aligned}A_{1} & →^{1} & A_{1} & → & 0 \\ ∥ & & ↓ i & & ↓ \\ A_{1} & →^{i} & A_{0} & → & Cok i \end{aligned}

in MA. Its ’codification‘ is

Q_{0}P_{1}X → X → Q_{−1}P_{−1}X ,

where X = (A →^{i} B). The image of the morphism Q_{0}P_{1}X → X under D_{1} identifies with Q_{0}P_{1}D_{1}X → D_{1}X by assumption. So the ’image‘ of the conflation is isomorphic to the first triangle of 6.1 b) applied to D_{1}X. We obtain an isomorphism

Q_{−1}P_{−1}D_{1}X →^{∼} D_{1}Q_{−1}P_{−1}X ,

hence an isomorphism P_{−1}D_{1}X → D_{0}P_{−1}X, which is easily seen to be associated with the above D_{1}Q_{−1} ←^{∼} Q_{−1}D_{0}. Thus, as for S-functors, the associated morphism

P_{−1}D_{1} → D_{0}P_{−1}

is invertible whenever it is defined. Moreover we have commutative diagrams

Q_{−1}P_{−1}D_{1}X ⟶^{ζD_{1}X} SQ_{0}P_{1}D_{1}X
↓  ↑
D_{1}Q_{−1}P_{−1}X ⟶^{δ_{1}ε^{+}} SD_{1}Q_{0}P_{1}X

and

\begin{matrix}P_{0}D_{1}Q_{−1}P_{−1}X & ⟶^{P_{0}δ_{1}ε^{+}} & P_{0}SD_{1}Q_{0}P_{1}X & →^{∼} & SP_{0}D_{1}Q_{0}P_{1}X \\ ↓ & & & & ↓ \\ D_{0}C & & ⟶^{δ_{0}ε} & & SD_{0}A \end{matrix} (∗)

which show how to compute δ_{0}ε from D_{1}X. Now suppose that D_{0} : A → S_{0} and D_{1} : MA → S_{1} are additive functors and λ : D_{1}Q_{0} → Q_{0}D_{0} is an isomorphism of functors. We assume that

a) the associated morphisms P_{0}D_{1} → D_{0}P_{0}, P_{1}D_{1} → D_{0}P_{1} are invertible,

b) the morphisms D_{1}Q_{−1} → Q_{−1}D_{0}, D_{1}Q_{1} → Q_{1}D_{0} are invertible and

c) for each inflation X = (A →^{i} B) the morphism P_{−1}D_{1}X → D_{0}P_{−1}X is invertible.

Observe that b) follows from a) if the functor Q_{0} satisfies condition (E). Moreover, assuming a) and b), a sufficient condition for c) is that for each conflation

X →^{j} Y →^{q} Z

of MA there is some triangle

D_{1}X →^{D_{1}j} D_{1}Y →^{D_{1}q} D_{1}Z → SD_{1}X.

Under the hypotheses a), b) and c), one can verify that the diagrams (∗) define a ’connecting morphism‘ δ_{0} such that (D_{0}, δ_{0}) is a ∂-functor. Moreover this construction transforms ’morphisms of triples D_{0}, D_{1}, λ’ to morphisms of ∂-functors and is compatible with ’composition with exact functors from the right’ and with ’composition with S-functors from the left‘. All of these statements are easy to make precise (compare 6.3) and to prove. We omit the details.

6.5 We prepare for the proof of the theorem on left adjoints (2.8). We use the notations and hypotheses of the beginning of 6.3. In addition, we assume that both functors Q_{0} satisfy condition (E).

Lemma. If F_{0} : S_{0} → S′_{0} admits a left S-adjoint G_{0}, then F_{1} admits a left S-adjoint G_{1}. The morphisms

G_{1}Q_{0} → Q_{0}G_{0} , G_{0}P_{0} → P_{0}G_{1} , G_{0}P_{1} → P_{1}G_{1}

associated with

Q_{0}F_{0} → F_{1}Q_{0} , P_{0}F_{1} → F_{0}P_{0} , P_{1}F_{1} ← F_{0}P_{1}

are invertible.

Proof. Let X ∈ S′_{1}. The functor S′_{1} (X, F_{1} ?) is represented by S^{k}Q_{0}G_{0}Y for X = S^{k}Q_{0}Y and by S^{k}Q_{−1}G_{0}Y for X = S^{k}Q_{−1}Y. Since for any X ∈ S′_{1} we have a triangle (cf. proof of 6.1 a)

Q_{−1}P_{1}X → Q_{−1}P_{0}X ⊕ Q_{0}P_{1}X → X → SQ_{−1}P_{1}X ,

we conclude from 6.7 that the functor is representable for arbitrary X and that F_{1} has a left S-adjoint. Using that F_{1} ’commutes‘ with the right adjoints of Q_{0}, P_{0} and P_{1} it is easy to see that the associated morphisms are invertible.

6.6 We prove the results about S-adjoints that we have been using in this section. Let S and T be suspended categories. We study right S-adjoints. Let (L, λ) : S → T be an S-functor, R a right adjoint of the underlying additive functor of L and Φ : LR → 1_{T}, Ψ : 1_{S} → RL compatible adjunction morphisms.

Lemma. If the composition \overline{ρ} = (RSΦ)(RλR)(ΨSR) is invertible and ρ = \overline{ρ}^{−1}, then (R, ρ) is a right S-adjoint to (L, λ).

We omit the proof since it is quite similar to that of lemma 6.7

6.7 We study left S-adjoints. Let S, T be suspended categories and (R, ρ) : S → T an S-functor. We say that the left adjoint L is defined in X ∈ T if the functor T (X, R?) is representable, i.e. if there is an object LX ∈ S and a morphism ΨX : X → RLX which induces a bijection

(ΨX)^{∗}R(LX, ?) : S(LX, ?) →^{∼} T (X, R?) , f ↦ (Rf)(ΨX).

If L is defined in X and in SX, we have a canonical morphism λX : LSX → SLX defined by

(Rλ)(ΨSX) = (ρ^{−1}L)(SΨX).

Let D be the full subcategory of T consisting of the objects X satisfying

• L is defined in S^{n}X, ∀ n ∈ N and

• λS^{n}X is invertible, ∀ n.

By definition we have SD ⊂ D. The following lemma shows that D is a suspended subcategory of T and that (L, λ) : D → S is an S-functor. If RS ⊂ D, then (R, ρ) and (L, λ) yield a pair of S-adjoint functors between S and D.

Lemma. (cf. [10, 1.5]) If X and Y lie in D and

X →^{u} Y →^{v} Z →^{w} SX

is a triangle of T , then Z lies in D and

LX →^{Lu} LY →^{Lv} LZ ⟶^{(λX)(Lw)} SLX

is a triangle of S.

Proof. 1st step: L is defined in S^{n}Z, ∀ n ∈ N. We form a triangle

LX →^{Lu} LY →^{v′} Z′ →^{w′} SLX

in S. By SP3 [10, 1.1] there is a morphism of triangles

X →^{u} Y →^{v} Z  ⟶^{w} SX
ΨX ↓  ΨY ↓  f ↓  ↓ SΨX
RLX →^{RLu} RLY →^{Rv′} RZ′ ⟶^{(ρL)(Rw′)} SRLX.

It yields a morphism of exact sequences

T (X, R?) ← T (Y, R?) ← T (Z, R?) ← T (SX, R?) ← T (SY, R?)
↑ α_{1} ↑ α_{2} ↑ α_{3} ↑ α_{4} ↑ α_{5}
S(LX, ?) ← S(LY, ?) ← S(Z′, ?) ← S(SLX, ?) ← S(SLY, ?),

where α_{1} and α_{2} are given by Ψ^{∗}R(L, ?), α_{4} and α_{5} by

(SΨ)^{∗}(ρ^{−1}L)^{∗}R(SL, ?)

and α_{3} by f^{∗}R(Z′, ?). By assumption, the evaluations of Ψ^{∗}R(L, ?) and

(SΨ)^{∗}(ρ^{−1}L)^{∗}R(SL, ?) = (ΨS)^{∗}(Rλ)^{∗}R(SL, ?) = (ΨS)^{∗}R(LS, ?)λ^{∗}

at X resp. Y are invertible. By the 5-lemma, α_{3} is invertible, so L is defined in X. Since (Sv, Sw, −Su) is a triangle as well, we can use the same argument to conclude that L is defined in SX ….

2nd step: (Lu, Lv, (λX)(Lw)) is a triangle. In the above notations, there is an isomorphism g : LZ → Z′ such that (Rg)(ΨZ) = f. We claim that

LX →^{Lu} LY →^{Lv} LZ ⟶^{(λX)(Lw)} SLX
∥ ∥ ↓ g ∥
LX →^{Lu} LY →^{v′} Z′  ⟶^{w′} SLX

is a morphism of S-sequences, which implies that the first row is a triangle of T . Indeed, g(Lv) = v′ follows from

(Rg)(RLv)(ΨY) = (Rg)(ΨZ)v = fv = (Rv′)(ΨY)

and w′g = (λX)(Lw) from

(Rw′)(Rg)(ΨZ) = (Rw′)f = (ρLX)^{−1}(SΨX)w
= (RλX)(ΨSX)w = (RλX)(RLw)(ΨZ).

3rd step: Z lies in D. Since λZ occurs in the morphism of triangles

\begin{matrix}LSX & →^{LSu} & LSY & →^{LSv} & LSZ & ⟶^{−(λSX)(LSw)} & SLSX \\ ↓ λX & & ↓ λY & & ↓ λZ & & ↓ SλX \\ SLX & →^{SLu} & SLY & →^{SLv} & SLZ & ⟶^{−(SλX)(SLw)} & SSLX \end{matrix} ,

λZ is invertible. The same argument shows that λS^{n}Z is invertible for each n ∈ N. The assertion now follows from the 1st step.

7. Redundancy of the connecting morphisms

7.1 Let A be an exact category and S, T epivalent towers of suspended categories. We denote the underlying towers of additive categories of A^{∧}, S and T by A^{∧}|, S| and T |, respectively.

Lemma.

a) The forgetful functor

Hom_{S} (S, T ) → Hom_{add} (S|, T |)

is an isomorphism onto the full subcategory consisting of the towers F such that for each triangle

X →^{u} Y →^{v} Z →^{w} SX

of S_{n} there is some triangle

F_{n}X →^{F_{n}u} F_{n}Y →^{F_{n}v} F_{n}Z →^{w′} SF_{n}X

of T_{n}.

b) The forgetful functor

Hom_{∂} (A^{∧}, S) → Hom_{add} (A^{∧}|, S|)

is an isomorphism onto the full subcategory consisting of the towers D such that for each conflation

A →^{i} B →^{d} C

of A_{n}^{∧} there is some triangle

D_{n}A →^{D_{n}i} D_{n}B →^{D_{n}d} D_{n}C → SD_{n}A

of S_{n}.

Proof. In order to produce an inverse of the forgetful functor, we consider the functor Θ : P → P which associates P_{n+1} to P_{n} and maps a morphism a : P_{m} → P_{n} to

Θa : P_{m+1} → P_{n+1} , (x_{1}, …, x_{m+1}) ↦ (x_{1}, a(x_{2}, …, x_{m+1})).

We have a natural transformation φ : Θ → 1_{P} whose value at P_{n} is

q_{0}^{1} : P_{n+1} → P_{n} , (x_{1}, …, x_{n+1}) ↦ (x_{2}, …, x_{n+1}).

Now Θ is really a 2-functor and Θ is a morphism of 2-functors. So we obtain a morphism of towers

Sφ : S → SΘ

whose components are the

Q_{0}^{1} : S_{n} → S_{n+1}.

Of course we have S|Θ=SΘ|. Let F : S| → T | be a tower as in the assertion. We have a diagram

S| →^{F} T |
Sφ ↓  ↓ T φ
SΘ| →^{F Θ} T Θ|

of towers of additive categories and an isomorphism

F φ : (T φ)F → F (ΘSφ).

We apply 6.3 to complete each F_{n} to an S-functor F̃_{n} and to combine the F̃_{n} into a tower of S-functors F̃. The proof of b) is completely analogous. We omit the details.

7.2 We use the notations and hypotheses of Theorem 4. We denote the compositions of D and F with the forgetful 2-functor U : Susp → Add by D| and F|.

Lemma.

a) The forgetful functor

Hom_{S} (D, F) → Hom_{add} (D|, F|)

is an isomorphism.

b) The forgetful functor

Hom_{∂} (F) → Hom_{add} (U, F|)

is an isomorphism.

Proof. a) We produce an inverse of the forgetful functor. For each exact A we consider the functor

Q : A → MA.

If F : D| → F| is given, we have a diagram

DA →^{F A} FA
DQ ↓  ↓ FQ
DMA →^{F MA} FMA

and an isomorphism

F Q : (FMA)(DQ) →^{∼} (FQ)(F A).

By example 6.1 c), DQ satisfies condition (E) and similarly we see that FQ satisfies condition (R). It is clear that conditions 6.3 a) and b) are satisfied since the functors P_{0}, P_{1}, Q_{1} and Q_{−1} are all induced by exact functors between A and MA (compare 6.4). Since by assumption F and D carry the inclusion IA → MA to an equivalence, the functor P_{−1} is ’induced‘ by the cokernel functor IA → A (cf. 6.4) and therefore condition 6.3 c) is also satisfied.

So we can complete F A to an S-functor F̃A and combine the F̃A into a 1morphism F̃ : D → F. The proof of b) is completely analogous. We omit the details.

8. Proof of Theorem 2.7

8.1 We prove b). It is not hard to see (use [2, 1.1.9]) that under the hypotheses of b), morphisms of ∂-functors from D_{0} to D′_{0} bijectively correspond to morphisms between the underlying additive functors. Therefore, according to 7.1 b), we only have to show that the functor

Hom_{add} (A^{∧}|, T |) → Add (A|, T_{0}|) , D| ↦ D|_{0} ,

where the | denotes underlying additive categories resp. functors, induces a bijection

Hom (D|, D′|) → Hom (D|_{0}, D′|_{0}).

From now on we omit the | . We factor the above functor as

Hom_{add} (A^{∧}, T ) → Hom_{add} (A^{∧}, T_{0}^{∧}) → Hom (A, T_{0}) ,

where the first functor is induced by Φ : T → T_{0}^{∧} (Example 2.3 b). The second functor is an equivalence by Example 2.3 b). So it remains to be shown that Φ induces a bijection

Hom (D, D′) → Hom (ΦD, ΦD′).

By A.5 it is enough to check this locally, i.e. we have to show that Φ_{n} : T_{n} → (T_{0}^{∧})_{n} induces a bijection

Hom (D_{n}T a, D′_{n}T a) → Hom (Φ_{n}D_{n}T a, Φ_{n}D′_{n}T a)

for each morphism a : P_{m} → P_{n}. This certainly holds if for all X, Y ∈ A_{n}^{∧}, the map

Hom (D_{n}X, D′_{n}Y) → Hom (Φ_{n}D_{n}X, Φ_{n}D′_{n}Y)

is bijective. This follows from lemma 6.2 a) by induction.

8.2 We prove a). The functor D yields D^{∧} : A^{∧} → T_{0}^{∧}. Let V_{n} ⊂ (T_{0}^{∧})_{n} be the image of D_{n}^{∧} and U_{n} ⊂ T_{n} the preimage of V_{n} under Φ_{n} : T_{n} → (T_{0}^{∧})_{n}. Obviously we have a diagram of towers of additive categories

A^{∧} → V ← U → T .

We shall show that U → V is an equivalence and that the quasiinverse provides the required D^{+} upon composition with A^{∧} → V and U → T . By remark A.2 we only have to show that U_{n} → V_{n} is an equivalence for each n. This is clear for n = 0. Suppose it has been shown up to n − 1. Moreover suppose that we have shown that Hom (S^{k}X, Y) = 0, ∀ k > 0, ∀ X, Y ∈ U_{n−1}. We have a diagram

MA_{n−1}^{∧} ⟶ MV_{n−1} ←^{∼} MU_{n−1} → MT_{n−1}
∼↑ ↑∼ ↑ ↑ M^{1}
A_{n}^{∧} ⟶^{(D_{0}^{∧})_{n}} V_{n} ← U_{n} → T_{n}

which commutes up to isomorphism. By lemma 6.2 a), M^{1} induces an equivalence U_{n} → MU_{n−1} and Hom (S^{k}X, Y) = 0, ∀ k > 0, ∀ X, Y ∈ U_{n}. This implies the assertion. The tower of additive functors D^{+} thus constructed obviously satisfies D_{0}^{+} →^{∼} D. It follows by induction from lemma 6.2 b) that the image of a conflation under D_{n}^{+} can be embedded into a triangle. Hence D^{+} yields a tower of ∂-functors by 7.1 b).

9. Proof of Theorem 2.6

9.1 We establish the connection between towers and presheaves. Let A be an exact category. We have a full embedding from A_{m}^{∧} to RA which with X ∈ A_{m}^{∧} associates the rough presheaf F such that

• F(U) = 0 if U does not contain U_{1} ∩ … ∩ U_{m},

• F(U) = X(x_{1}, …, x_{m}) if U contains U_{1} ∩ … ∩ U_{m} and x_{i} = 1 iff U is contained in U_{i}.

We define R_{m}A ⊂ RA to be the image of A_{m}^{∧}. It identifies with the category of presheaves on the discrete set {1, …, m}. By ’transport of structure‘ we combine the R_{m}A into a tower R_{∗}A. Note that the functor

Q_{−1}^{m+1} : R_{m}A → R_{m+1}A

of example 2.2 a) coincides with the canonical embedding. Thus RA is the limit of the direct system

R_{0}A ⟶^{Q_{−1}^{1}} R_{1}A → … → R_{m}A ⟶^{Q_{−1}^{n+1}} R_{m+1}A → ….

Now let n ∈ N. Applying the above to R_{n}A instead of A we find that the direct limit of

R_{0}R_{n}A → R_{1}R_{n}A → … → R_{m}R_{n}A → R_{m+1}R_{n}A → …

is RR_{n}A. We shall identify R_{m}R_{n}A with R_{m+n}A in the canonical fashion: A presheaf F on

{1, …, m + n} →^{∼} {1, …, m} ∐{1, …, n}

is given by the presheaf W ↦ F_{W} on {1, …, m} whose value at W is the presheaf

U ↦ F_{W} (U) = F (W ∪ U).

In the following paragraph we shall suppress A in the symbols RA, R_{m}A, R_{∗}A, R_{m}R_{n}A and RR_{n}A.

9.2 We want to prove theorem 2.6 by constructing a quasiinverse of

Hom_{S} (DR_{∗}, T ) → Hom_{∂} (R_{∗}, T ).

Let D be a tower of ∂-functors R_{∗} → T . We first describe the components F_{n} of the image F of D under the quasiinverse. By example 6.1 a), the functors

Q_{−1}^{n+1} : T_{n} → T_{n+1}

admit left adjoints P_{−1}^{n+1}. For n < l we put

L_{n,l} = P_{−1}^{n+1} … P_{−1}^{l−1}P_{−1}^{l} : T_{l} → T_{n}.

We obtain a diagram

R_{0}R_{n} ⟶^{Q_{−1}^{n+1}} R_{1}R_{n} → … R_{m}R_{n} ⟶ …
D_{n} ↓  ↓ L_{n,n+1}D_{n+1} ↓ L_{n,n+m}D_{n+m}
T_{n} ⟶^{1} T_{n} ⟶ … T_{n} ⟶ …,

which commutes up to isomorphism by 6.4. By A.4 and 9.1, we obtain a functor E_{n} : RR_{n} → T_{n} which ’extends‘ the L_{n,n+m}D_{n+m}. Clearly the image of a conflation under E_{n} embeds into a triangle. So in order to show that the s ∈ Σ are made invertible by E_{n}, it is enough to show that E_{n}F′ vanishes if F′ is a presheaf as in section 3. This is equivalent to showing that L_{n,l}D_{l}Q_{0}^{j} = 0 for each l > n and each 1 ≤ j ≤ l. But we have D_{n}Q_{0}^{j} →^{∼} Q_{0}^{j}D_{n} since D is a

tower, P_{−1}^{k}Q_{0}^{j} →^{∼} Q_{0}^{j}P_{−1}^{k} for k > j (apply 6.3 to the square Q_{0}^{j}Q_{0}^{k} →^{∼} Q_{0}^{k}Q_{0}^{j}) and

P_{−1}^{j}Q_{0}^{j} = 0. We conclude that E_{n} induces an F_{n} : DR_{n} → T_{n}. It is clear by the construction that the image of a triangle of DR_{n} under F_{n} can be embedded into a triangle of T_{n}. So once we have shown that the F_{n} combine into a tower F of additive functors it will follow from 7.1 a) that this tower gives rise to a unique tower of S-functors.

9.2 We keep the notations and hypotheses of the preceding paragraph. In order to construct F as a tower and to make it clear that F depends on D in a functorial way, we have to add one layer of abstraction.

Let Θ : P → P be the functor which carries P_{n} to P_{n+1} and maps a morphism a : P_{m} → P_{n} to

Θa : P_{m+1} → P_{n+1} , (x_{1}, …, x_{m+1}) ↦ (a(x_{1}, …, x_{m}), x_{m+1}).

We have a natural transformation τ : 1_{P} → Θ taking the values

τP_{n} = p_{0}^{n+1} : P_{n} → P_{n+1}, (x_{1}, …, x_{n}) ↦ (x_{1}, …, x_{n}, 0).

Of course, Θ is really a 2-functor and τ a morphism of 2-functors. For each m we obtain a tower of exact functors

R_{∗}τΘ^{m} : R_{∗}Θ^{m} ← R_{∗}Θ^{m+1}.

By example 2.2 a) and A.2, this tower admits a left adjoint, which we denote by R_{∗}σΘ^{m}, by abuse of notation. Its components are the

Q_{−1}^{n+m+1} : R_{n+m} → R_{n+m+1} .

We form the direct system R_{N} :

R_{∗} ⟶^{R_{∗}σ} R_{∗}Θ → … → R_{∗}Θ^{m} ⟶^{R_{∗}σΘ^{m}} …,

which we view as an object of Fun (N, Fun (P^{OP} , Ex)), where P^{OP} denotes the 2-category P with reversed 1- and 2-morphisms. By 9.1 and A.4 its direct limit is isomorphic to RR_{∗}, the tower with components (RR_{∗})_{n} = RR_{n}. Using the notations of A.4 we have two ’universal arrows‘

R_{N} → ΔRR_{∗} → ΔDR_{∗}

in Fun (N, Fun (P^{OP} , Ex)). We denote their composition by Can. Let us return to T . By remark 6.3 and A.2, the tower

T τΘ^{m} : T Θ^{m} ← T Θ^{m+1}

admits a left adjoint, which we denote by T σΘ^{m}. We consider σ^{+} : ΔT → T_{N} given by

T ⟶^{1} T ⟶ … T ⟶ …
∥ ↓ ↓
T ⟶^{T σ} T Θ ⟶^{T σΘ} … T Θ^{m} ⟶^{T σΘ^{m}} ….

Here the vertical arrows are defined so as to make the squares commutative. By A.2 and 6.3, σ^{+} admits a left adjoint ρ^{+} : T_{N} → ΔT

T ⟶^{T σ} T Θ ⟶^{T σΘ} … T Θ^{m} ⟶^{T σΘ^{m}} …
∥ ↓ ρ_{1}^{+} ↓ ρ_{m}^{+}
T ⟶^{1} T ⟶ … T ⟶ ….

Of course, the n-th component of ρ_{m}^{+} is simply

(ρ_{m}^{+})_{n} = L_{n,n+m} : T_{n+m} → T_{n}.

We compose ρ^{+} : T_{N} → ΔT with D_{N} : R_{N} → T_{N}. By A.4 the composition D_{N}ρ^{+} gives rise to a tower of additive functors F : DR_{∗} → T which makes the square

R_{N} ⟶^{Can} ΔDR_{∗}
D_{N} ↓  ↓ ΔF
T_{N} ⟶^{ρ^{+}} ΔT

commutative up to isomorphism. By construction, F depends on D in a functorial manner and F can →^{∼} D canonically. There only remains to be constructed a functorial isomorphism F →^{∼} G for the case where D = Gcan for some tower of S-functors G : DR_{∗} → T . It is enough to produce an isomorphism between (ΔF) Can and (ΔG) Can, i.e. between ρ^{+}D_{N} and (ΔG)Can. We illustrate the situation by the diagram

R_{N} ⟶^{canN} (DR_{∗})_{N} ⟶^{GN} T_{N}
∥ ↓ ρ^{+} ↓ ρ^{+}
R_{N} ⟶^{Can} ΔDR_{∗} ⟶^{ΔG} ΔT .

It is clear from A.3 that D_{N} = G_{N}can_{N} if D = Gcan . Hence ρ^{+}D_{N} = ρ^{+}G_{N}can_{N}. We also have τ^{+}G_{N} →^{∼} (ΔG)τ^{+}, canonically, where τ^{+} denotes the obvious 1-morphisms T_{N} → ΔT and (DR_{∗})_{N} → ΔDR_{∗}. By ’twofold association‘ (A.3 and 6.3), we obtain an isomorphism ρ^{+}G_{N} →^{∼} (ΔG)ρ^{+}, which by A.3 is functorial in G. So it only remains to be shown that ρ^{+}can_{N} →^{∼} Can. We fix a choice of ρ^{+} : The

ρ_{m}^{+} : DR_{∗}Θ^{m} → DR_{∗}

are determined by the

(ρ_{m}^{+})_{n} : DR_{m+n} → DR_{n}

which are to be induced by the canonical isomorphisms

RR_{m+n} → RR_{n}

provided by the partitions

{1, …, m + n} ∐ N_{1} →^{∼} {1, …, n} ∐ N_{1} .

It is then clear that ρ^{+}can_{N} = Can but we still have to show that (ρ_{m}^{+})_{n} is really left adjoint to the composition

Q_{−1}^{m+n} … Q_{−1}^{n} : DR_{n} → DR_{m+n} .

It is obviously enough to consider the case m = 1. After replacing A by R_{n}A we may also assume that n = 0. The assertion then means that

Z : RR_{1} → RA

induces a left adjoint of the functor

Q_{−1}^{1} : DA → DR_{1}A →^{∼} DMA .

Now Z induces the mapping cone functor so that the assertion follows from 10.1 b).

10. Proof of Theorem 4

10.1 Lemma.

a) The inclusion J : IA → MA induces an S-equivalence DIA →^{∼} DMA.

b) The mapping cone functor C : DMA → DA is left adjoint to the embedding DA → DMA induced by A ↦ (0 → A).

Proof. a) Let I_{cs}A and M_{cs}A be the categories IA and MA endowed with the componentwise split conflations. For each (f : A_{1} → A_{0}) ∈ M_{cs}A we have a conflation

\begin{matrix}0 & ⟶ & A_{1} & ⟶ & A_{1} \\ ↓ & & ↓ i & & ↓ f \\ A_{1} & ⟶^{[1 −f]^{t}} & A_{1} ⊕ A_{0} & ⟶^{[f 1]} & A_{0} \end{matrix} , i = [1 0]^{t}

in M_{cs}A. We see that each object of M_{cs}A (hence of I_{cs}A) admits a projective resolution of lenghth one by objects contained in U, the full subcategory of I_{cs}A consisting of the split conflations. Therefore

DI_{cs}A ←^{∼} H_{0]}^{b} U →^{∼} DM_{cs}A.

It follows from 5.1 a) that DIA (resp. DMA) identifies with the localisation of DI_{cs}A (resp. DM_{cs}A) at the class of morphisms s which fit into a triangle

X →^{s} Y → A → SX

with an IA-acyclic (resp. MA-acyclic) A. The assertion follows because the preimages in H_{0]}^{b} U of the respective classes coincide.

b) It is enough to show the assertion for the restriction C|DIA. Now Q : DA → DIA obviously has the functor Cok : DIA → DA as a left adjoint. Since the mapping cone over the canonical morphism CX → Cok X, X ∈ DIA is acyclic, C and Cok are isomorphic as functors DIA → DA.

10.2 We construct a quasiinverse of

can^{∗} : Hom_{S} (D, F) → Hom_{∂} (F).

We first remark that the functor

Q_{0} : FA → FMA

induced by A ↦ (A → 0) satisfies condition (R) of 6.1. To see this, we can use the same argument as in example 6.1 c). Now let D ∈ Hom_{∂} (F) and let U : Ex → Add be the forgetful 2-functor. We consider F as a 2-functor Ex → Add and D as a morphism of 2-functors U → F. We define the 1morphism D_{N} : U_{N} → F_{N} to be

U ⟶^{UQ} UM ⟶^{UMQ} … UM^{n} ⟶^{UM^{n}Q} …
D ↓ ↓ ↓ DM^{n}
F ⟶^{FQ} FM ⟶^{FMQ} … FM^{n} ⟶^{FM^{n}Q} …,

where Q is the 1-morphism whose value at A is the exact embedding

A → MA , A ↦ (0 → A)

and the squares are commutative up to canonical isomorphism. From 6.1 it is clear that the 1-morphism σ^{+} : ΔF → F_{N} given by

F ⟶^{1} F ⟶^{1} … F ⟶^{F} …
↓ ↓ ↓
F ⟶^{FQ} FM ⟶^{FMQ} … FM^{n} ⟶^{FM^{n}Q} …

(the vertical arrows are defined so as to make the squares commutative) admits a left adjoint ρ^{+} : F_{N} → ΔF. The composition ρ^{+}D_{N} ’factors‘ through the ’universal arrow‘ (A.5)

U_{N} → ΔR ,

where R assigns the additive category of rough presheaves RA to A ∈ Ex and the functors

M^{n}A → RA

are defined in analogy with 9.1. So we have E : R → F such that

U_{N} → ΔR ⟶^{ΔE} ΔF

is isomorphic to ρ^{+}D_{N}. As in 9.2, one sees that

EA : RA → FA

makes all the s ∈ Σ invertible. This clearly implies that E ’factors‘ through

R → D

giving rise to F : D → F. From 7.2 we see that F corresponds to a 1-morphism of Fun(Ex, Susp). Now one can imitate the end of the proof in 9.3. We omit the details.

Appendix : 2-Functor-Categories

A.1 Let C and D be 2-categories [5, V. 1]. We set out to define a sub-2-category of the 2-functor-category of [6, I, 2.4] from C to D.

An object of the 2-category Fun (C, D) is a 2-functor X : C → D, i.e. a map X : C_{0} → D_{0} together with functors

X(x, y) : Hom_{C} (x, y) → Hom_{D} (Xx, Xy) , f ↦ Xf ,

which are compatible with units and compositions. A 1-morphism F : X → Y assigns a 1-morphism F x : Xx → Xy to each object x ∈ C and a an invertible 2-morphism F f : Y fF x → F yXf to each 1-morphism f : x → y

[figure: Y x; Xx; Y y; Xy; F x; F y; Xf; Y f; F f]

such that F 1_{x} = 1_{F x}, ∀ x ∈ C_{0}, F fg = (F fXg)(Y fF g) for each pair of composable 1-morphisms f, g of C and (F yXμ)(F f) = (F g)(Y μF x) for each 2-morphism μ : f → g of C. The composition of two 1-morphisms F, G is defined by F Gx = F xGx for each x ∈ C_{0} and F Gf = (F yGf)(F fGx) for each 1-morphism f : x → y. A 2-morphism Θ : F → G assigns a 2-morphism Θx : F x → Gx to each object x of C such that the equation Gf(Y fΘx) = (ΘyXf)F f holds for each 1-morphism f : x → y. The two compositions of 2-morphisms are (ΘΦ)x = ΘxΦx and (Θ ∗ Φ)x = Θx ∗ Φx.

Example. Let C be the 2-category with a single object x having only identical 2-morphisms and such that Hom_{C} (x, x) is a free monoid on s : x → x. Then Fun (C, Cat) is isomorphic to the 2-category whose objects are the pairs (X , S) of categories X with an endofunctor S : X → X, whose morphisms are the ’Sfunctors‘ and whose 2-morphisms are the ’morphisms of S-functors‘ (compare [10, 1.4]).

A.2 We keep the assumptions and notations of 6.1. An adjoint pair in a 2-category consists of 1-morphisms l : x → y, r : y → x and 2-morphisms φ : lr → 1_{y}, ψ : 1_{x} → rl such that (rφ)(ψr) = 1_{r} and (φl)(lψ) = 1_{l}. If φ and ψ are invertible, l and r are quasiinverse equivalences.

Now let R : X → Y be a 1-morphism of Fun (C, D) and suppose that for each x ∈ C_{0} we are given an adjoint pair Rx, Lx, Φx, Ψx such that

Mf = (ΦyXfLx)(LyRfLx)(LyY fΨx) : LyY f → XfLx

is invertible for each 1-morphism f : x → y of C.

Proposition. (compare [10, 1.6]) The assignments

x ↦ Lx , f ↦ Lf = (Mf)^{−1}

define a 1-morphism L of Fun (C, D), the assignments

x ↦ Φx , x ↦ Ψx

define 2-morphisms Φ, Ψ and R, L, Φ, Ψ is an adjoint pair in Fun (C, D).

Remark. In particular, R has a quasiinverse iff each Rx has a quasiinverse.

Proof. Substituting into the definitions we obtain statements which immediately follow from A.3. We omit the details.

A.3 Let l, r, φ, ψ and l′, r′, φ′, ψ′ be adjoint pairs in a 2-category C which appear in a diagram

x ⟶^{f} x′
l ↓↑ r  l′ ↓↑ r′
y ⟶^{g} y′.

We call two 2-morphisms α : fr → r′g and β : l′f → gl associated if the following equivalent conditions hold (compare [10, 1.6]) :

i) α = (r′gφ)(r′βr)(ψ′fr) ii) (r′β)(ψ′f) = (αl)(fψ)
iii) β = (φ′gl)(l′αl)(l′fψ) iv) (gφ)(βr) = (φ′g)(l′α).

We can interpret this as follows: i) and iii) define inverse bijections between the 1-morphisms from l to l′ and from r to r′, where we consider l, l′ as 2functors from {0 < 1} (having only identical 2-morphisms) to C and r, r′ as 2-functors from {0 < 1} to C^{oP}, the 2-category C ’with reversed 2-morphisms‘. In fact, these bijections are part of an isomorphism of categories

Hom (l, l′) →^{≅} Hom (r, r′)^{op}.

We make this more precise: Let f_{1} : x → x′ and g_{1} : y → y′ be another pair of morphisms and μ : f → f_{1}, ν : g → g_{1} 2-morphisms. If α : fr → r′g, β : l′f → gl and α_{1} : f_{1}r → r′g_{1}, β_{1} : l′f_{1} → g_{1}l are associated pairs, then

(r′ν)α = α_{1}(μr) ⟺ (νl)β = β_{1}(l′μ) ,

i.e. μ, ν define a 2-morphism (f, g, α) → (f_{1}, g_{1}, α_{1}) iff they define a 2morphism (f, g, β) → (f_{1}, g_{1}, β_{1}).

In 9.2 and A.2 we also need that the isomorphisms

Hom (l, l′) →^{≅} Hom (r, r′)^{op}

are compatible with compositions, i.e. if a diagram

x ⟶^{f} x′  ⟶^{h} x′′
l ↓↑ r l′ ↓↑ r′ l′′ ↓↑ r′′
y ⟶^{g} y′  ⟶^{i} y′′

is given, where l′′, r′′, φ′′, ψ′′ is another adjoint pair, and if α : fr → r′g, β : l′f → gl and γ : hr′ → r′′i, δ : l′′h → il′ are associated pairs, then (γg)(hα) : (hf)r → r′′(ig) and (iβ)(δf) : l′′(hf) → (ig)l are associated.

A.4 We consider the partially ordered set N as a 2-category having only identical 2-morphisms. Let E be another 2-category and let Δ : E → Fun (N, E) be the obvious ’diagonal‘ 2-functor. The limit of a 2-functor X : N → E consists of an object lim_{⟶} X ∈ E and a 1-morphism f : X → Δlim_{⟶} X inducing an equivalence of categories

Hom_{E} (lim_{⟶} X, y) → Hom_{Fun (N,E)}(X, Δy) , g ↦ f · Δg

for each object y of E.

Example. a) Let A be an exact category. In E = Add we consider the sequence of embeddings (9.1)

R_{0} ⟶^{Q_{−1}^{1}} R_{1} → … → R_{n} ⟶^{Q_{−1}^{n+1}} R_{n+1} → ….

For an exact category B, a 1-morphism to ΔB is given by a family of functors

G_{n} : R_{n} → B

and of isomorphisms

γ_{n} : G_{n} →^{∼} G_{n+1}Q_{−1}^{n+1}.

Using a well-known technique we now exhibit a category LA and a 1-morphism to ΔLA which even induces an isomorphism

Hom (LA, B) →^{≅} Hom(R_{0} → R_{1} → …, ΔB).

The objects of LA are the pairs (X, n) of natural numbers n and of objects X ∈ R_{n}. The morphisms from (X, n) to (Y, m) bijectively correspond to the elements of RA (X, Y) (we identify X, Y with their images in RA). The functor can_{n} : R_{n} → LA associates the pair (X, n) with X ∈ R_{n}. The isomorphism can_{n} →^{∼} can_{n+1}Q_{−1}^{n+1} is produced by the identities of RA.

Obviously LA is equivalent to RA. Hence RA is also a limit of the sequence of the R_{n}. We conclude by theorem 3 that for each additive category T_{0} the canonical functor

Hom (DA, T_{0}) → Hom (R_{0} → R_{1} → …, ΔT_{0})

is an equivalence onto the full subcategory of the ’compatible families‘ (G_{n}, γ_{n}) such that G_{n} makes the s ∈ Σ lying in R_{n} invertible.

Now we consider the case where E = Fun (C, D) for two 2-categories C, D. A 2-functor X : N → Fun (C, D) yields a 2-functor X_{c} : N → D defined by

n ↦ X_{c}(n) = (Xn)(c)

for each c ∈ C. Let us suppose that for each c there is a strict limit, i.e. a limit lim_{⟶} X_{c} furnishing isomorphisms

Hom (lim_{⟶} X_{c}, y) →^{≅} Hom (X_{c}, Δy) , ∀ y ∈ D.

In this case, the assignment c ↦ lim_{⟶} X_{c} can be completed to a 2-functor L : C → D in a natural way, and the 1-morphisms X_{c} → Δlim_{⟶} X_{c} can be completed to a 1-morphism X → ΔL. A tedious exercise shows that this 1-morphism induces an isomorphism

Hom_{Fun (C,D)}(L, Y) → Hom_{Fun (N,Fun (C,D))}(X, ΔY)

for each Y ∈ Fun (C, D).

Example. b) We consider the 2-functor R_{∗} : P^{OP} → Add and the sequence

R_{∗} ⟶^{R_{∗}σ} R_{∗}Θ → … → R_{∗}Θ^{m} ⟶^{R_{∗}σΘ^{m}} ….

of 9.2. By definition, its evaluation at P_{n} is the sequence

R_{n} ⟶^{Q_{−1}^{n+1}} R_{n+1} → … → R_{n+m} ⟶^{Q_{−1}^{n+m+1}} R_{n+m+1} → ….

As in example a), we see that this sequence has a strict limit L_{n}. The L_{n} combine into a 2-functor L : P^{OP} → Add. The canonical morphisms R_{∗}Θ^{m} → RR_{∗} induce a 1-morphism L → RR_{∗}. Since its components are equivalences, it is an equivalence itself by A.2. Hence RR_{∗} is a limit of the seqence of the R_{∗}Θ^{m}. In particular, it follows that, for each tower of additive categories T , the functor

Hom (DR_{∗}, T ) → Hom(R_{∗} → R_{∗}Θ → …, ΔT )

is an equivalence onto the full subcategory of the ’compatible families‘ G_{m} : R_{∗}Θ^{m} → T such that (G_{m})_{n} : R_{n+m} → T makes all the s ∈ Σ (with respect to RR_{n}) lying in R_{n+m} invertible.

A.5 Let C, D be 2-categories, X, Y, Z ∈ Fun (C, D), F, G : X → Y and H : Y → Z 2-functors. Suppose that, for each 1-morphism f : x → y of C, Hy induces a bijection

Hom_{D} (Xx, Y y)(F yXf, GyXf) →^{∼} Hom_{D} (Xx, Zy)(HF yXf, HGyXf).

Lemma. H induces a bijection of the classes of 2-morphisms

Hom_{Fun (C,D)}(X, Y)(F, G) →^{∼} Hom_{Fun (C,D)}(X, Z)(HF, HG).

We omit the straightforward proof.

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