On exact functors for Heller triangulated categories

Matthias Künzer January 13, 2013

Abstract

We show certain standard constructions of the theory of Verdier triangulated categories to be valid in the Heller triangulated framework as well; viz. Karoubi hull, exactness of adjoints, localisation.

Contents

0 Introduction  2
0.1 Extending from Verdier to Heller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2
0.2 Desirables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3
0.3 Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3
1 Exact functors  3
2 Idempotents and cones  6
2.1 A general remark on residue classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  6
2.2 A Heller triangulation on the Karoubi hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  6
2.3 Functoriality of the Karoubi hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  9
2.4 Closed Heller triangulated categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Heller triangulated subcategories  14
4 Functors are exact if and only if they are compatible with n-triangles  15
5 Adjoints  19
5.1 Adjoints and shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2 An adjoint of an exact functor is exact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.3 A functor shiftcompatibly adjoint to a strictly exact functor is strictly exact . . . . . . . . . . . . 23
6 Localisation  23
A Some general assertions  30
A.1 Remarks on coretractions and retractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
A.2 Two lemmata on subexact functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
A.3 Karoubi hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
A.4 Multiplicative systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

MSC2010: 18E30.

0 Introduction

0.1 Extending from Verdier to Heller

The following facts are part of the classical theory that Verdier triangulated categories.

• Verdier triangulated categories are stable under formation of the Karoubi hull [1].

• The Karoubi hull construction is functorial within Verdier triangulated categories and exact functors [1].

• Verdier triangulated categories are stable under localisation at a thick subcategory [13].

• Such a localisation has a universal property within Verdier triangulated categories and exact functors [13].

• An adjoint functor of an exact functor is exact [11, App. 2, Prop. 11], [5, 1.6].

We extend these assertions somewhat to fit into the Heller triangulated setting.

• Heller triangulated categories are stable under formation of the Karoubi hull; cf. Proposition 12.

• The Karoubi hull construction is functorial within Heller triangulated categories and exact functors; cf. Proposition 13.

• Closed Heller triangulated categories are stable under localisation at a thick subcategory; cf. Proposition 36. (Concerning closedness, see remark below.)

• Such a localisation has a universal property within closed Heller triangulated categories and exact functors; cf. Proposition 38.

• An adjoint functor of an exact functor is exact; cf. Proposition 28.

In a general Heller triangulated category, it is unknown whether there exists a cone on a given morphism. This however is true if all idempotents split [8, Lem. 3.1]. It is technically convenient to extend this assertion in the following manner. Define a Heller triangulated category to be closed if this property holds; cf. [9, Def. 13], Definition 14, Remark 15, Lemma 20. Prove that certain constructions yield closed Heller triangulated categories or preserve closedness; cf. [9, Cor. 21], Proposition 36.

An exact functor between Heller triangulated categories (C, T, ϑ) and (C′, T′, ϑ′) is a pair (F, a) consisting of a subexact functor F and an isotransformation a : T F \overline{}F T′ such that ϑ, ϑ′ and a are compatible; cf. [5, Def. 1.4], Definition 1. Exactness of such a pair can also be characterised via n-triangles; cf. Proposition 25. The deeper reason behind that fact is that closed Heller triangulated categories can, alternatively, be defined via sets of n-triangles for n ⩾ 0 with suitable properties with respect to quasicyclic and folding operations, as S. Thomas informed me.

The proof of the exactness of an adjoint of an exact functor does not have to make recourse to n-triangles. Neither does the construction of the Heller triangulation on the Karoubi hull. This shows the convenience of the definition of a triangulation via a tuple ϑ = (ϑ_{n})_{n⩾0} of isomorphisms between certain shift functors, and to view the n-triangles as accessory, if useful; which is no longer the point of view taken in [7].

0.2 Desirables

Still missing is a precise formulation in which sense the dual of a Heller triangulated category is again a Heller triangulated category, and also in which sense the constructions above are compatible with duality. Moreover, we do not treat exactness of derived functors, except implicitly, in those cases where a derived functor can be written as a composite of an adjoint of a localisation functor, an exact functor and another localisation functor. Still missing, in the Heller triangulated context, is furthermore the exactness of the lift of the inclusion of the heart to a functor on the bounded derived category [2, Prop. 3.1.10], or more generally, the functor Z appearing in the construction of [5, Ex. 2.3] ; cf. [5, Th. 3.2].

0.3 Notations and conventions

We use the notations and conventions from [8]. In particular, we write composition of morphisms and functors in the natural order; viz. morphisms as ^{\overline{}^{f}} ^{\overline{}^{g}} = ^{\overline{}^{fg}} = ^{\overline{}^{f·g}} and functors as \overline{}^{F} ^{\overline{}^{G}} = ^{\overline{}^{F G}} = ^{\overline{}^{F ⋆ G}} . Similarly for transformations.

Epic and epimorphic are synonymous, and so are monic and monomorphic.

1 Exact functors

Let (C, T, ϑ), (C′, T′, ϑ′) and (C′′, T′′, ϑ′′) be Heller triangulated categories; cf. [8, Def. 1.5.(ii)].

In [8, Def. 1.5.(ii)], we required a strictly exact functor C \overline{}C′ to satisfy F T′ = T F. The adjoint functor of a strictly exact functor does not always seem to be strictly exact. Following Keller and Vossieck, we shall prove below that if we call a functor exact, if it satisfies the weakened condition F T′ ≃ T F instead (and an accordingly modified compatibility with the Heller triangulations), then an adjoint of an exact functor is exact; cf. [5, 1.4]. Nonetheless, generally speaking, usually one deals with strictly exact functors. Hence we shall also state an extra condition of shiftcompatibility on the adjunction that ensures a shiftcompatibly adjoint functor of a strictly exact functor to be strictly exact.

Given n ⩾ 0 and a transformation G ^{\overline{}^{a}} G′ between subexact additive functors C [figure: G; G′] C′, we

denote by G^{+}(Δ̄_{n}^{#}) ^{\overline{}^{a^{+}(Δ̄_{n}^{#})}} G^{′+}(Δ̄_{n}^{#}) the transformation given by

(X(a^{+}(Δ̄_{n}^{#})))_{β/α} := X_{β/α}a : X_{β/α}G \overline{}X_{β/α}G′

for X ∈ Ob C(Δ̄_{n}^{#}), and for β/α ∈ Δ̄_{n}^{#} , i.e. for α, β ∈ Δ̄_{n} with β^{−1} ⩽ α ⩽ β ⩽ α^{+1}. Moreover,

we denote by \underline{G^{+}(Δ̄_{n}^{#})} ^{\overline{}^{\underline{a^{+}(Δ̄_{n}^{#})}}} \underline{G^{′+}(Δ̄_{n}^{#})} the induced transformation between the induced functors on the stable categories.

Sometimes, we abbreviate (\underline{G} ^{\overline{}^{\underline{a}}} \underline{G′}) := (\underline{G^{+}(Δ̄_{n}^{#})} ^{\overline{}^{\underline{a^{+}(Δ̄_{n}^{#})}}} \underline{G^{′+}(Δ̄_{n}^{#})}).

Definition 1

A pair (F, a), consisting of an additive functor C ^{\overline{}^{F}} C′ and a transformation T F ^{\overline{}^{a}} F T′, is called an exact pair, or an exact functor, if the following conditions hold.

(1) a is an isotransformation.
(2) F is subexact, i.e. its induced functor Ĉ ^{\overline{}^{F̂}} Ĉ′ on the Freyd categories is exact.

(3) We have

(ϑ_{n} ⋆ \underline{F^{+}(Δ̄_{n}^{#})}) · \underline{a^{+}(Δ̄_{n}^{#})} = \underline{F^{+}(Δ̄_{n}^{#})} ⋆ ϑ′_{n}

for all n ⩾ 0.

In particular, provided T F = F T′, then (F, 1) is exact if and only if F is strictly exact; cf. [8, Def. 1.5.(iii)]. In this case, we sometimes identify F and (F, 1).

Calling a pair (F, a) an exact functor instead of an exact pair is an abuse of notation.

We shall not discuss whether condition (1) is redundant; we need it for the construction of Ỹ in §4, but that may be due to the order of our arguments. Condition (3) asserts that the following cylindrical diagram commutes for all n ⩾ 0.

[figure: \underline{C^{+}(Δ̄_{n}^{#})}; \underline{F^{+}(Δ̄_{n}^{#})}; [−]^{+1}; [−^{+1}]; \underline{C^{′+}(Δ̄_{n}^{#})}; [−]^{+1}; [−^{+1}]; 1; \underline{a^{+}(Δ̄_{n}^{#})}; \underline{C^{+}(Δ̄_{n}^{#})}; \underline{F^{+}(Δ̄_{n}^{#})}; \underline{C^{′+}(Δ̄_{n}^{#})}; ϑ_{n}; ϑ′_{n}]

I.e., using the abbreviation just introduced, we require Xϑ_{n}\underline{F} · X\underline{a} = X\underline{F} ϑ′_{n} to hold in \underline{C^{′+}(Δ̄_{n}^{#})} for all X ∈ Ob \underline{C^{+}(Δ̄_{n}^{#})} = Ob C^{+}(Δ̄_{n}^{#}).

Definition 2 Suppose given exact functors (F, a) from C to C′, and (F′, a′) from C′ to C′′. The composite of (F, a) and (F′, a′) is defined to be

(F, a) ⋆ (F′, a′) = (F, a)(F′, a′) := (F F′, (aF′)(F a′)) = (F ⋆ F′, (a ⋆ F′) · (F ⋆ a′)) .

Composition is associative.

Remark 3 If (F, a) and (F′, a′) are exact, then so is their composite (F, a)(F′, a′).

Proof. To be able to distinguish more easily, we shall make use, from the second to the last but first step, of the notation a ⋆ F = aF, F ⋆ F′ = F F′ etc. Given n ⩾ 0, we obtain

(ϑ_{n} ⋆ \underline{(F ⋆ F′)^{+}(Δ̄_{n}^{#})}) · \underline{((a ⋆ F′) · (F ⋆ a′))^{+}(Δ̄_{n}^{#})}
= (ϑ_{n} ⋆ \underline{F^{+}(Δ̄_{n}^{#})} ⋆ \underline{F^{′+}(Δ̄_{n}^{#})}) · (\underline{a^{+}(Δ̄_{n}^{#})} ⋆ \underline{F^{′+}(Δ̄_{n}^{#})}) · (\underline{F^{+}(Δ̄_{n}^{#})} ⋆ \underline{a^{′+}(Δ̄_{n}^{#})})
= (\underline{F^{+}(Δ̄_{n}^{#})} ⋆ ϑ′_{n} ⋆ \underline{F^{′+}(Δ̄_{n}^{#})}) · (\underline{F^{+}(Δ̄_{n}^{#})} ⋆ \underline{a^{′+}(Δ̄_{n}^{#})})
= \underline{F^{+}(Δ̄_{n}^{#})} ⋆ \underline{F^{′+}(Δ̄_{n}^{#})} ⋆ ϑ′′_{n}
= \underline{(F ⋆ F′)^{+}(Δ̄_{n}^{#})} ⋆ ϑ′′_{n} .

Definition 4 Suppose given exact functors (F, a) and (G, b) from (C, T, ϑ) to (C′, T′, ϑ′).

A transformation F ^{\overline{}^{s}} G such that (T ⋆ s) · b = a · (s ⋆ T′) holds, is called periodic.

The periodicity condition requires that

\xymatrix{X^{+1}F \ar[r]^{X^{+1}s} \ar[d]_{Xa}^{≀} & X^{+1}G \ar[d]^{Xb}_{≀} \\ (XF)^{+1} \ar[r]^{(Xs)^{+1}} & (XG)^{+1}}

commute for all X ∈ Ob C.

Remark 5 Suppose given exact functors (F, a), (G, b) and (H, c) from C to C′, and periodic

transformations F [figure: s; s′] G ^{\overline{}^{t}} H.

(1) The composite F ^{\overline{}^{s·t}} H is periodic.

(2) The identity F ^{\overline{}^{1}} F is periodic.

(3) If s is a periodic isotransformation from (F, a) to (G, b), then s^{−} is a periodic isotransformation from (G, b) to (F, a).

(4) The difference F ^{\overline{}^{s−s′}} G of two periodic transformations is periodic.

(5) The direct sum (F, a) ⊕ (G, b) := (F ⊕ G, a ⊕ b) = (F ⊕ G, (\begin{matrix}a & 0 \\ 0 & b \end{matrix})) is exact, with periodic inclusions from and periodic projections to (F, a) and (G, b).

Definition 6 Write C, C′ _{ex} for the category of the exact functors and periodic transformations from C to C′; cf. Definitions 1, 4, Remark 5.

Write C, C′ _{st ex} for the full subcategory of C, C′ _{ex} of the strictly exact functors and periodic transformations from C to C′.

2 Idempotents and cones

Let (C, T, ϑ) be a Heller triangulated category; cf. [8, Def. 1.5].

2.1 A general remark on residue classes

Concerning Frobenius categories, cf. e.g. [8, Sec. A.2.3].

Remark 7 Given a full and faithful exact functor G : F \overline{}F ′ of Frobenius categories that sends all bijective objects to bijective objects. Then the induced functor \underline{G} : \underline{F} \overline{}\underline{F} ′ on the classical stable categories is full and faithful.

Proof. By construction, it is full. We claim that it is faithful. Suppose given X \overline{}Y in F such that

(XG \overline{}Y G) = (XG \overline{}B′ \overline{}Y G)

in C′ for some bijective object B′ of C′. Choose X [figure] B in C with B bijective in C. Since G preserves pure monomorphy, XG \overline{}B′ factors over XG [figure] BG, whence XG \overline{}Y G factors over XG [figure] BG, whence X \overline{}Y factors over X [figure] B.

Suppose given weakly abelian categories A and A′. Suppose given a subexact functor A ^{\overline{}^{F}} A′. Suppose given n ⩾ 0. We obtain an induced functor

\underline{F^{+}(Δ̄_{n}^{#})} : \underline{A^{+}(Δ̄_{n}^{#})} \overline{}\underline{A^{′+}(Δ̄_{n}^{#})}

on the respective stable categories of n-pretriangles. Cf. [8, §1.2.1.3, §A.6.3].

Remark 8 If F is full and faithful, so is \underline{F^{+}(Δ̄_{n}^{#})} .

In particular, if F is the embedding of a full subcategory, we may and will also consider \underline{F^{+}(Δ̄_{n}^{#})} to be the embedding of a full subcategory.

Proof. By [8, Prop. 5.5], both A^{+}(Δ̄_{n}^{#}) and A^{′+}(Δ̄_{n}^{#}) are Frobenius categories; and the full and faithful functor F^{+}(Δ̄_{n}^{#}) : A^{+}(Δ̄_{n}^{#}) \overline{}A^{′+}(Δ̄_{n}^{#}) induced by F preserves bijective objects, viz. split objects, and pure short exact sequences, viz. pointwise split short exact sequences. So by Remark 7, the assertion follows.

2.2 A Heller triangulation on the Karoubi hull

Let Ĉ denote the Freyd category of C; cf. e.g. [8, A.6.3]. We consider the full and faithful functor C \overline{}Ĉ as an embedding of a full subcategory. Let C̃ denote the full subcategory of bijectives in the abelian Frobenius category Ĉ. So we have full subcategories

C ⊆ C̃ ⊆ Ĉ

Since the image of C in Ĉ is a big enough subcategory of bijectives, the embedding C [figure] C̃ is a Karoubi hull of C; cf. [6, III.II]. Cf. also Remark 43, Lemma 44 – which we will not use and argue directly instead.

We shall give a Heller triangulation on this Karoubi hull C̃ of C. The Verdier triangulated version of this construction is due to Balmer and Schlichting; cf. [1, Th. 1.12].

As a full subcategory of bijective objects in abelian Frobenius category, the category C̃ is weakly abelian.

The shift T on C induces a shift T̂ on Ĉ, which restricts to a shift T̃ on C̃.

Remark 9 Suppose given n ⩾ 0 and X ∈ Ob \underline{C̃^{+}(Δ̄_{n}^{#})} . There exists Z ∈ Ob \underline{C^{+}(Δ̄_{n}^{#})} such that X is isomorphic to a direct summand of Z in \underline{C̃^{+}(Δ̄_{n}^{#})} . In other words, there exists Z ∈ Ob \underline{C^{+}(Δ̄_{n}^{#})} and a split monomorphism X ^{\overline{}^{i}} Z in \underline{C̃^{+}(Δ̄_{n}^{#})} .

Proof. By [8, Prop. 2.6], it suffices to prove that given X ∈ Ob \underline{C̃(Δ̇_{n})}, there exists Z ∈ Ob \underline{C(Δ̇_{n})} such that X is isomorphic, in \underline{C̃(Δ̇_{n})}, to a direct summand of Z.

It suffices to prove the existence of a split monomorphism X \overline{}Z in C̃(Δ̇_{n}) with Z ∈ Ob C(Δ̇_{n}).

For i ∈ [1, n], let Y_{i} ∈ Ob C̃ be such that X_{i} ⊕ Y_{i} is isomorphic to an object in C. Let Y ∈ Ob C̃(Δ̇_{n}) have entry Y_{i} at position i for 1 ⩽ i ⩽ n and the morphism from position i to position j be zero for 1 ⩽ i < j ⩽ n. The diagram X ⊕ Y has X as a summand and is isomorphic to an object in C(Δ̇_{n}).

Remark 10 Given n ⩾ 0, a diagram X ∈ Ob \underline{C̃^{+}(Δ̄_{n}^{#})} , a split monomorphism X \overline{}Z with Z ∈ Ob \underline{C^{+}(Δ̄_{n}^{#})} and a morphism X ^{\overline{}^{x}} X′, then there exists a commutative quadrangle

[figure: X; •; x; X′; •; Z; Z′]

in \underline{C̃^{+}(Δ̄_{n}^{#})} with Z′ ∈ Ob \underline{C^{+}(Δ̄_{n}^{#})} .

Moreover, if X \overline{}X′ is a split monomorphism, we may choose Z \overline{}Z′ to be a split monomorphism.

Proof. We form

[figure: X; •; _{(_{1 0})}; x; X′; •; •; _{(_{1 0})}; X ⊕ Y; _{(\begin{matrix}x & 0 \\ 0 & 1 \end{matrix})}; X′ ⊕ Y ,]

where X ⊕ Y ≃ Z. By Remark 9, there is a split monomorphism from X′ ⊕ Y to an object Z′ of Ob \underline{C^{+}(Δ̄_{n}^{#})} .

Moreover, if X ^{\overline{}^{x}} X′ is split monic, so is the composite (X ⊕ Y ^{\overline{}^{(\begin{matrix}x & 0 \\ 0 & 1 \end{matrix})}} X′ ⊕ Y \overline{}Z′).

Construction 11 Given n ⩾ 0, we define [−]^{+1} ^{\overline{}^{ϑ̃_{n}}} [−^{+1}] on \underline{C̃^{+}(Δ̄_{n}^{#})} as follows.

Given X ∈ Ob \underline{C̃^{+}(Δ̄_{n}^{#})} , choose a split monomorphism X ^{\overline{}^{i}} Z with Z ∈ Ob \underline{C^{+}(Δ̄_{n}^{#})} , existent by Remark 9, and choose a retraction p to i. Define

([X]^{+1} ^{\overline{}^{Xϑ̃_{n}}} [X^{+1}]) := ([X]^{+1} ^{\overline{}^{[i]^{+1}}} [Z]^{+1} ^{\overline{}^{Zϑ_{n}}} [Z^{+1}] ^{\overline{}^{[p^{+1}]}} [X^{+1}]) .

To prove that Xϑ̃_{n} is welldefined, we shall first show that it is independent of the choice of the retraction p. Given d : Z \overline{}X with id = 0, we have to show that [i]^{+1}Zϑ_{n}[d^{+1}] = 0. Since [i^{+1}] is monic, it suffices to show that [i]^{+1}Zϑ_{n}[d^{+1}][i^{+1}] = 0. In fact,

[i]^{+1}Zϑ_{n}[d^{+1}][i^{+1}] = [i]^{+1}Zϑ_{n}[(di)^{+1}] = [i]^{+1}[di]^{+1}Zϑ_{n} = [id]^{+1}[i^{+1}]Zϑ_{n} = 0 ,

since di is in \underline{C^{+}(Δ̄_{n}^{#})} .

Now assume given another split monomorphism X \overline{}Z′ with Z′ ∈ Ob \underline{C^{+}(Δ̄_{n}^{#})} . By Remark 10, we may assume that this split monomorphism factors into two split monomorphisms X ^{\overline{}^{i}} Z ^{\overline{}^{i′}} Z′. Let ip = 1 and i′p′ = 1. Then (ii′)(p′p) = 1, and we may conclude

[ii′]^{+1}(Z′ϑ_{n})[(p′p)^{+1}] = [i]^{+1}[i′]^{+1}(Z′ϑ_{n}[p^{′+1}])[p^{+1}] = [i]^{+1}[i′]^{+1}([p′]^{+1}Zϑ_{n})[p^{+1}] = [i]^{+1}Zϑ_{n}[p^{+1}] ,

since p′ is in \underline{C^{+}(Δ̄_{n}^{#})} .

To show that ϑ_{n} is a transformation, we suppose given a morphism X ^{\overline{}^{f}} X′ in \underline{C̃^{+}(Δ̄_{n}^{#})} and have to show that Xϑ̃_{n}[f^{+1}] =^{!} [f]^{+1}X′ϑ̃_{n}. By Remarks 9 and 10, we find a commutative quadrangle

[figure: X; •; i; f; X′; •; i′; Z; g; Z′]

in \underline{C̃^{+}(Δ̄_{n}^{#})} with Z, Z′ ∈ Ob \underline{C̃^{+}(Δ̄_{n}^{#})} . Choose p and p′ such that ip = 1 and i′p′ = 1. It suffices to show that Xϑ̃_{n}[f^{+1}][i^{′+1}] =^{!} [f]^{+1}X′ϑ̃_{n}[i^{′+1}] by monomorphy of [i^{′+1}]. In fact,

Xϑ̃_{n}[f^{+1}][i^{′+1}]  = Xϑ̃_{n}[i^{+1}][g^{+1}]  = [i]^{+1}Zϑ_{n}[p^{+1}][i^{+1}][g^{+1}]
= [i]^{+1}Zϑ_{n}[(pig)^{+1}]  = [i]^{+1}[pig]^{+1}Z′ϑ_{n} = [ig]^{+1}Z′ϑ_{n}
= [fi′]^{+1}Z′ϑ_{n}  = [fi′]^{+1}[p′i′]^{+1}Z′ϑ_{n} = [fi′]^{+1}Z′ϑ_{n}[(p′i′)^{+1}]
= [f]^{+1}[i′]^{+1}Z′ϑ_{n}[p^{′+1}][i^{′+1}] = [f]^{+1}X′ϑ̃_{n}[i^{′+1}] .

Note that Zϑ̃_{n} = Zϑ_{n} for Z ∈ Ob \underline{C^{+}(Δ̄_{n}^{#})} .

End of construction.

Proposition 12

(1) The tuple ϑ̃ := (ϑ̃_{n})_{n⩾0} is the unique Heller triangulation on (C̃, T̃) such that the full and faithful inclusion functor C [figure] C̃ is strictly exact; cf. [8, Def. 1.5.(i, ii)].

(2) An n-pretriangle U ∈ Ob \underline{C^{+}(Δ̄_{n}^{#})} is an n-triangle with respect to (C, T, ϑ) if and only if it is an n-triangle with respect to (C̃, T̃, ϑ̃).

Proof. Ad (1). We have to show that given m, n ⩾ 0 and a periodic monotone map Δ̄_{n} ^{\overline{}^{p}} Δ̄_{m}, we have \underline{p}^{#} ⋆ϑ̃_{m} =^{!} ϑ̃_{n} ⋆\underline{p}^{#}. Let us verify this at X ∈ Ob \underline{C̃^{+}(Δ̄_{n}^{#})} . Choose a split monomorphism X ^{\overline{}^{i}} Z with Z ∈ Ob \underline{C^{+}(Δ̄_{n}^{#})} by Remark 9. It suffices to show that (X\underline{p}^{#})ϑ̃_{m}[(i\underline{p}^{#})^{+1}] =^{!} (Xϑ̃_{n})\underline{p}^{#}[(i\underline{p}^{#})^{+1}]. In fact, we obtain

\begin{aligned}(X\underline{p}^{#})ϑ̃_{m}[(i\underline{p}^{#})^{+1}] & = & [i\underline{p}^{#}]^{+1}(Z\underline{p}^{#})ϑ_{m} & = & [i\underline{p}^{#}]^{+1}(Zϑ_{n})\underline{p}^{#} \\ & = & ([i]^{+1}Zϑ_{n})\underline{p}^{#} & = & (Xϑ̃_{n}[i^{+1}])\underline{p}^{#} & = & (Xϑ̃_{n})\underline{p}^{#}[(i\underline{p}^{#})^{+1}] . \end{aligned}

We have to show that given n ⩾ 0, we have \underline{𝔣}_{n} ⋆ ϑ̃_{n+1} =^{!} ϑ̃_{2n+1} ⋆ \underline{𝔣}_{n}. Let us verify this at X ∈ Ob \underline{C̃^{+}(Δ̄_{2n+1}^{#})} . Choose a split monomorphism X ^{\overline{}^{i}} Z with Z ∈ Ob \underline{C^{+}(Δ̄_{2n+1}^{#})} by Remark 9. It suffices to show that (X\underline{𝔣}_{n})ϑ̃_{n+1}[(i\underline{𝔣}_{n})^{+1}] =^{!} (Xϑ̃_{2n+1})\underline{𝔣}_{n}[(i\underline{𝔣}_{n})^{+1}]. In fact, we obtain

\begin{aligned}(X\underline{𝔣}_{n})ϑ̃_{n+1}[(i\underline{𝔣}_{n})^{+1}] & = & [i\underline{𝔣}_{n}]^{+1}(Z\underline{𝔣}_{n})ϑ_{n+1} & = & [i\underline{𝔣}_{n}]^{+1}(Zϑ_{2n+1})\underline{𝔣}_{n} \\ & = & ([i]^{+1}Zϑ_{2n+1})\underline{𝔣}_{n} & = & (Xϑ̃_{2n+1}[i^{+1}])\underline{𝔣}_{n} & = & (Xϑ̃_{2n+1})\underline{𝔣}_{n}[(i\underline{𝔣}_{n})^{+1}] . \end{aligned}

The inclusion functor C [figure] C̃ is strictly exact since it strictly commutes with shift by construction, since it is subexact because the induced functor on the Freyd categories is an equivalence, and since Zϑ̃_{n} = Zϑ_{n} for Z ∈ Ob \underline{C^{+}(Δ̄_{n}^{#})} .

Now suppose that both ϑ̃ and ϑ̃′ are Heller triangulations on (C̃, T̃) such that C [figure] C̃ is strictly exact. Suppose given n ⩾ 0 and X ∈ Ob \underline{C̃^{+}(Δ̄_{n}^{#})} . We have to show that Xϑ̃_{n} =^{!} Xϑ̃′_{n}. Choose a split monomorphism X ^{\overline{}^{i}} Z with Z ∈ Ob \underline{C^{+}(Δ̄_{n}^{#})} ; cf. Remark 9. It suffices to show that

Xϑ̃_{n}[i^{+1}] =^{!} Xϑ̃′_{n}[i^{+1}]. In fact,

Xϑ̃_{n}[i^{+1}] = [i]^{+1}Zϑ̃_{n} = [i]^{+1}Zϑ_{n} = [i]^{+1}Zϑ̃′_{n} = Xϑ̃′_{n}[i^{+1}] .

Ad (2). Suppose given an n-pretriangle U ∈ Ob \underline{C^{+}(Δ̄_{n}^{#})} . Now U is an n-triangle with respect to (C, T, ϑ) if and only if Uϑ_{n} = 1 , and with respect to (C̃, T̃, ϑ̃) if and only if Uϑ̃_{n} = 1 ; cf. [8, Def. 1.5.(ii)]. Since Uϑ_{n} = Uϑ̃_{n}, these assertions are equivalent. Cf. also [8, Lem. 3.8].

2.3 Functoriality of the Karoubi hull

We shall prove the universal property of the Karoubi hull directly, without making recourse to Remark 43 and Lemma 44. We will make use of the universal property and the abelianness of the Freyd category, however.

Proposition 13 Suppose given Heller triangulated categories (C, T, ϑ), (C′, T′, ϑ′). Call the strictly exact inclusion functors K : C \overline{}C̃ and K′ : C′ \overline{}C̃′.

(1) Suppose given an exact functor (F, a) from C to C′. We may construct an exact functor (F̃, ã) from C̃ to C̃′ such that

\xymatrix{C \ar[r]^{(F,a)} \ar[d]_{K} & C′ \ar[d]^{_{K_{′}}} \\ C̃ \ar[r]^{(F̃,ã)} & C̃′}

commutes, i.e. such that (F, a)(K′, 1) = (K, 1)(F̃, ã), i.e. such that F ⋆ K′ = K ⋆ F̃ and a ⋆ K′ = K ⋆ ã, i.e. such that uF̃ = uF for u ∈ Mor C and Zã = Za for Z ∈ Ob C.

The functor F̃ and the condition a⋆K′ = K ⋆ ã uniquely determines ã. If a = 1, then ã = 1.

(2) Given two exact functors (F̃_{1} , ã_{1}) and (F̃_{2} , ã_{2}) such that (F, a)(K′, 1) = (K, 1)(F̃_{1} , ã_{1}) = (K, 1)(F̃_{2} , ã_{2}), there exists a unique isotransformation F̃_{1} \overline{}^{φ}_{∼} F̃_{2} such that K ⋆ φ = 1, i.e. such that Zφ = 1 for Z ∈ Ob C. This isotransformation φ is periodic.

(3) Suppose given exact functors (F, a) and (G, b) from C to C′. Suppose given a periodic transformation s from F to G.

Construct (F̃, ã) and (G̃, b̃) as in (1).

There exists a unique periodic transformation s̃ from F̃ to G̃ such that K ⋆ s̃ = s ⋆ K′, i.e. such that Zs̃ = Zs for Z ∈ Ob C.

Proof. Given X ∈ Ob C̃, we choose X ^{\overline{}^{i_{X}}} Z_{X} ^{\overline{}^{p_{X}}} X in C̃ such that i_{X} · p_{X} = 1_{X} and such that Z_{X} ∈ Ob C.

Moreover, choose these objects and morphisms in such a way that Z_{XT̃} = Z_{X} T, i_{XT̃} = i_{X} T̃ and p_{XT̃} = p_{X} T̃ for X ∈ Ob C̃.

Furthermore, if X ∈ Ob C, then choose Z_{X} = X and i_{X} = 1_{X} and p_{X} = 1_{X} .

Given X ^{\overline{}^{u}} Y in C̃, we let Z_{X} ^{\overline{}^{z_{u}}} Z_{Y} be defined by z_{u} := p_{X} · u · i_{Y} ; cf. Remark 39.

Ad (1). Since F is subexact, F̂ is exact. Since W is a summand of an object in C, also W F̂ is a summand of an object in C′, hence bijective. So F̃ := F̂|C̃′ ; C̃ is welldefined.

We want to show that the functor F̃ preserves weak kernels and is therefore subexact; cf. Lemma 41. In fact, given W ^{\overline{}^{w}} B ^{\overline{}^{f}} C in C̃ such that w is a weak kernel of f, we get a factorisation w = w′i, where K [figure: i] B is a kernel of f in Ĉ. Considering an epimorphism P ^{\overline{}^{p}} K in Ĉ with P ∈ Ob C̃, we obtain a factorisation pi = p′w = p′w′i, whence p = p′w′, whence w′ is epic. Since w′F̂ is epic and iF̂ is a kernel of fF̂, we obtain that wF̂ = wF̃ is a weak kernel of fF̂ = fF̃.

The universal property of the Freyd construction yields a transformation â : T̂F̂ \overline{}F̂ T̂′. We let the transformation ã : T̃F̃ \overline{}F̃ T̃′ be defined on X ∈ Ob C̃ ⊆ Ob Ĉ as Xã := Xâ. In particular, Zã = Za for Z ∈ Ob C.

Given n ⩾ 0, it remains to be shown that \underline{F̃^{+}(Δ̄_{n}^{#})} ⋆ ϑ̃′_{n} =^{!} (ϑ̃_{n} ⋆ \underline{F̃^{+}(Δ̄_{n}^{#})}) · \underline{ã^{+}(Δ̄_{n}^{#})}; cf. Definition 1. Let us verify this at X ∈ Ob \underline{C̃^{+}(Δ̄_{n}^{#})}. Let X ^{\overline{}^{i}} Z be a split monomorphism with Z ∈ Ob \underline{C^{+}(Δ̄_{n}^{#})}, existent by Remark 9. It suffices to show that

(X\underline{F̃^{+}(Δ̄_{n}^{#})})ϑ̃′_{n} · [(i\underline{F̃^{+}(Δ̄_{n}^{#})})^{+1}] =^{!} (Xϑ̃_{n})\underline{F̃^{+}(Δ̄_{n}^{#})} · X\underline{ã^{+}(Δ̄_{n}^{#})} · [(i\underline{F̃^{+}(Δ̄_{n}^{#})})^{+1}] .

In fact, we obtain

(X\underline{F̃^{+}(Δ̄_{n}^{#})})ϑ̃′_{n} · [(i\underline{F̃^{+}(Δ̄_{n}^{#})})^{+1}] = [i\underline{F̃^{+}(Δ̄_{n}^{#})}]^{+1} · (Z\underline{F̃^{+}(Δ̄_{n}^{#})})ϑ̃′_{n}
= [i\underline{F̃^{+}(Δ̄_{n}^{#})}]^{+1} · (Z\underline{F^{+}(Δ̄_{n}^{#})})ϑ′_{n}
=^{(F, a) ex.} [i\underline{F̃^{+}(Δ̄_{n}^{#})}]^{+1} · (Zϑ_{n})\underline{F^{+}(Δ̄_{n}^{#})} · Z\underline{a^{+}(Δ̄_{n}^{#})}
= [i\underline{F̃^{+}(Δ̄_{n}^{#})}]^{+1} · (Zϑ̃_{n})\underline{F̃^{+}(Δ̄_{n}^{#})} · Z\underline{ã^{+}(Δ̄_{n}^{#})}
= [i]^{+1}\underline{F̃^{+}(Δ̄_{n}^{#})} · (Zϑ̃_{n})\underline{F̃^{+}(Δ̄_{n}^{#})} · Z\underline{ã^{+}(Δ̄_{n}^{#})}
= (Xϑ̃_{n})\underline{F̃^{+}(Δ̄_{n}^{#})} · [i^{+1}]\underline{F̃^{+}(Δ̄_{n}^{#})} · Z\underline{ã^{+}(Δ̄_{n}^{#})}
= (Xϑ̃_{n})\underline{F̃^{+}(Δ̄_{n}^{#})} · X\underline{ã^{+}(Δ̄_{n}^{#})} · [(i\underline{F̃^{+}(Δ̄_{n}^{#})})^{+1}] .

If a = 1, then â = 1, so ã = 1. It remains to show that ã is uniquely determined by F̃ and the condition a ⋆ K′ = K ⋆ ã. In fact, given X ∈ Ob C̃, we have

Xã · i_{X} F̃T̃′ = i_{X} T̃F̃ · Z_{X} ã = i_{X} T̃F̃ · Z_{X}a ,

and i_{X} F̃T̃′ is monic. Ad (2). Define φ : F̃_{1} ^{\overline{}^{∼}} F̃_{2} at X ∈ Ob C̃ by

\xymatrix{Z_{X}F \ar[r]^{p_{X} F̃_{1}} \ar[d]_{1} & XF̃_{1} \ar[r]^{i_{X} F̃_{1}} \ar[d]^{≀}_{Xφ} & Z_{X}F \ar[d]^{1} \\ Z_{X}F \ar[r]^{p_{X} F̃_{2}} & XF̃_{2} \ar[r]^{i_{X} F̃_{2}} & Z_{X}F ;}

cf. Remark 40. The tuple φ = (Xφ)_{X∈Ob C̃} is actually a transformation from F̃_{1} to F̃_{2} , for given X ^{\overline{}^{u}} Y in C̃, we obtain

p_{X} F̃_{1} · uF̃_{1} · Y φ · i_{Y} F̃_{2} = p_{X} F̃_{1} · uF̃_{1} · i_{Y} F̃_{1}
= p_{X} F̃_{1} · (u · i_{Y})F̃_{1}  = p_{X} F̃_{1} · (i_{X} · z_{u})F̃_{1}
= p_{X} F̃_{1} · i_{X} F̃_{1} · z_{u}F̃_{1}  = ((p_{X} · i_{X}) · z_{u})F
= (z_{u} · (p_{Y} · i_{Y}))F  = z_{u}F̃_{2} · p_{Y} F̃_{2} · i_{Y} F̃_{2}
= (z_{u} · p_{Y})F̃_{2} · i_{Y} F̃_{2}  = (p_{X} · u)F̃_{2} · i_{Y} F̃_{2}
= p_{X} F̃_{2} · uF̃_{2} · i_{Y} F̃_{2}  = p_{X} F̃_{1} · Xφ · uF̃_{2} · i_{Y} F̃_{2} ,

and p_{X} F̃_{1} is epic and i_{Y} F̃_{2} is monic. Note that commutativity of the diagram above is also necessary, for we require K ⋆ φ = 1. This ensures uniqueness of φ. It remains to show that φ is a periodic transformation from (F̃_{1} , ã_{1}) to (F̃_{2} , ã_{2}). In fact, given X ∈ Ob C̃, we get

Xã_{1} · XφT̃′ · i_{X} F̃_{2}T̃′ = Xã_{1} · i_{X} F̃_{1}T̃′
= i_{X} T̃F̃_{1} · Z_{X} ã_{1}  = i_{XT̃}F̃_{1} · Z_{X} ã_{1}
= i_{XT̃}F̃_{1} · Z_{X}a  = i_{XT̃}F̃_{1} · Z_{X} ã_{2}
= XT̃φ · i_{XT̃}F̃_{2} · Z_{X} ã_{2} = XT̃φ · i_{X} T̃F̃_{2} · Z_{X} ã_{2}
= XT̃φ · Xã_{2} · i_{X} F̃_{2}T̃′ ,

and i_{X} F̃_{2}T̃′ is monic. Ad (3). Define s̃: F̃ \overline{}G̃ at X ∈ Ob C̃ by

\xymatrix{Z_{X}F \ar[r]^{p_{X} F̃} \ar[d]_{Z_{X}s} & XF̃ \ar[r]^{i_{X} F̃} \ar[d]^{≀}_{Xs̃} & Z_{X}F \ar[d]^{Z_{X}s} \\ Z_{X}G \ar[r]^{p_{X} G̃} & XG̃ \ar[r]^{i_{X} G̃} & Z_{X}G ;}

cf. Remark 40. The tuple s = (Xs)_{X∈Ob C̃} is actually a transformation from F̃ to G̃, for given X ^{\overline{}^{u}} Y in C̃, we obtain

p_{X} F̃ · uF̃ · Y s̃ · i_{Y} G̃ = p_{X} F̃ · uF̃ · i_{Y} F̃ · Z_{Y} s
= p_{X} F̃ · (u · i_{Y})F̃ · Z_{Y} s = p_{X} F̃ · (i_{X} · z_{u})F̃ · Z_{Y} s
= p_{X} F̃ · i_{X} F̃ · z_{u}F̃ · Z_{Y} s = ((p_{X} · i_{X}) · z_{u})F · Z_{Y} s
= (z_{u} · (p_{Y} · i_{Y}))F · Z_{Y} s = Z_{X}s · (z_{u} · (p_{Y} · i_{Y}))G
= Z_{X}s · z_{u}G̃ · p_{Y} G̃ · i_{Y} G̃ = Z_{X}s · (z_{u} · p_{Y})G̃ · i_{Y} G̃
= Z_{X}s · (p_{X} · u)G̃ · i_{Y} G̃ = Z_{X}s · p_{X} G̃ · uG̃ · i_{Y} G̃
= p_{X} F̃ · Xs̃ · uG̃ · i_{Y} G̃ ,

and p_{X} F̃ is epic and i_{Y} G̃ is monic. Note that commutativity of the diagram above is also necessary, for we require K ⋆ s̃ = s ⋆ K′. This ensures uniqueness of s. It remains to show that s̃ is a periodic transformation from (F̃, ã) to (G̃, b̃). In fact, given X ∈ Ob C̃, we get

Xã · Xs̃T̃′ · i_{X} G̃ T̃′ = Xã · i_{X} F̃ T̃′ · Z_{X}sT̃′
= i_{X} T̃F̃ · Z_{X} ã · Z_{X}sT̃′ = i_{XT̃}F̃ · Z_{X} ã · Z_{X}sT̃′
= i_{XT̃}F̃ · Z_{X}a · Z_{X}s T′ = i_{XT̃}F̃ · Z_{X} T s · Z_{X}b
= i_{XT̃}F̃ · Z_{X} T̃s · Z_{X}b̃ = i_{XT̃}F̃ · Z_{XT̃}s · Z_{X}b̃
= XT̃s̃ · i_{XT̃}G̃ · Z_{X}b̃ = XT̃s̃ · i_{X} T̃G̃ · Z_{X}b̃
= XT̃s̃ · Xb̃ · i_{X} G̃ T̃′ ,

and i_{X} G̃ T̃′ is monic.

2.4 Closed Heller triangulated categories

Recall that given a Heller triangulated category (C, T, ϑ), its Karoubi hull C̃ is Heller triangulated, too; cf. Proposition 12.(1). More precisely, (C̃, T̃, ϑ̃) is Heller triangulated, where T̃ and ϑ̃ are as in §2.2.

Definition 14

A Heller triangulated category (C, T, ϑ) is called closed if whenever (X, Y, Z̃) is a 2-triangle in C̃ and X, Y ∈ Ob C, then Z̃ is isomorphic to an object of C.

Cf. [8, Def. 1.5.(i, iii)].

I do not know an example of a non-closed Heller triangulated category.

As usual, we will call Z̃ the cone of X \overline{}Y, being unique up to isomorphism. Thus we may rephrase that by definition, (C, T, ϑ) is closed if it is closed under taking cones in the Karoubi hull C̃.

Remark 15 The Heller triangulated category (C, T, ϑ) is closed if and only if given X ^{\overline{}^{f}} Y in C, there exists a 2-triangle X ^{\overline{}^{f}} Y \overline{}Z \overline{}X^{+1} in C.

Cf. [9, Def. A.6].

Proof. If (C, T, ϑ) is closed, then given X ^{\overline{}^{f}} Y in C, there exists a 2-triangle X ^{\overline{}^{f}} Y \overline{}Z̃ \overline{}X^{+1} in C̃ by [8, Lem. 3.1], and we may substitute Z̃ isomorphically by an object Z in Ob C, so we are done by [8, Lem. 3.4.(4)].

Conversely, if we dispose of this existence property, and if we are given a 2-triangle (X, Y, Z̃) in C̃ with X, Y ∈ Ob C, then there exists a 2-triangle (X, Y, Z) with Z ∈ Ob C, too, and we may apply [8, Lem. 3.4.(6)] to conclude that Z ≃ Z̃. So (C, T, ϑ) is closed.

Remark 16 If idempotents split in C, then (C, T, ϑ) is closed.

Proof. If idempotents split in C, then C = C̃.

Remark 17 Suppose given Heller triangulated categories (C, T, ϑ), (C′, T′, ϑ′) and a full and faithful strictly exact functor C ^{\overline{}^{F}} C′. Furthermore, suppose that whenever given a 2-triangle (XF, Y F, Z′) in C′, where X, Y ∈ Ob C, then there exists Z ∈ Ob C such that Z′ ≃ ZF.

Suppose that C′ is closed. Then C is closed.

Proof. Suppose given X ^{\overline{}^{f}} Y in C. There exists a 2-triangle XF ^{\overline{}^{fF}} Y F \overline{}Z′ \overline{}XF^{+1} in C′. By assumption, there exists Z ∈ Ob C such that ZF ≃ Z′. By isomorphic substitution and fullness of F, we obtain a 2-triangle XF ^{\overline{}^{fF}} Y F ^{\overline{}^{gF}} ZF ^{\overline{}^{hF}} XF^{+1} in C′; cf. [8, Lem. 3.4.(4)]. Since

(X, Y, Z)ϑ_{2}\underline{F^{+}(Δ̄_{2}^{#})} = (X, Y, Z)\underline{F^{+}(Δ̄_{2}^{#})}ϑ′_{2} = (XF, Y F, ZF)ϑ′_{2} = 1 ,

we conclude by faithfulness of \underline{F^{+}(Δ̄_{2}^{#})} that (X, Y, Z)ϑ_{2} = 1 ; cf. Remark 8, [8, Def. 1.5.(ii)]. So we are done by Remark 15.

Remark 18 A closed Heller triangulated category is Verdier triangulated.

Proof. Its Karoubian hull is Verdier triangulated [8, Prop. 3.6]. An additive shift-closed subcategory of a Verdier triangulated category that is closed under forming cones is Verdier triangulated.

Definition 19 Suppose given a closed Heller triangulated category (C, T, ϑ).

Suppose given n ⩾ 0 and Y ∈ Ob C(Δ̇_{n}) and X ∈ Ob C^{+, ϑ=1}(Δ̄_{n}^{#}) such that X|Δ̇_{n} = Y.

Then Y is called the base of the n-triangle X.

Lemma 20 Suppose given a closed Heller triangulated category (C, T, ϑ) and n ⩾ 0. The re-

striction functor C^{+, ϑ=1}(Δ̄_{n}^{#}) ^{\overline{}^{(−)|Δ̇n}} C(Δ̇_{n}) is strictly dense, i.e. surjective on objects. In other words, each object Y ∈ Ob C(Δ̇_{n}) is the base of an n-triangle.

So weakening the assumption in [8, Lem. 3.1] that idempotents be split in C to the assumption that C be closed, we nonetheless obtain the conclusion of loc. cit.

Proof. Suppose given Y ∈ Ob C(Δ̇_{n}). By [8, Lem. 3.1], we obtain an n-triangle X̃ ∈ Ob C^{+, ϑ=1}(Δ̄_{n}^{#}) such that X̃|Δ̇_{n} = Y.

By [8, Lem. 3.4.(1)], we have a triangle (X̃_{α/0} , X̃_{β/0} , X̃_{β/α}) whenever 0 < α < β < 0^{+1}. Since C is closed, X̃_{β/α} is isomorphic to an object of C. Isomorphic substitution, which is permitted without leaving C̃^{+, ϑ=1}(Δ̄_{n}^{#}) by [8, Lem. 3.4.(4)], yields an n-triangle in C^{+, ϑ=1}(Δ̄_{n}^{#}) that restricts to Y on Δ̇_{n}; cf. Proposition 12.(2).

3 Heller triangulated subcategories

Definition 21 Given a Heller triangulated category (C′, T′, ϑ′), a full subcategory C ⊆ C′ is called a full Heller triangulated subcategory of C′ if there exist T and ϑ such that (C, T, ϑ) is a Heller triangulated category and such that the inclusion functor C [figure] C′ is strictly exact.

We remark that in this case, the automorphism T and the tuple of transformations ϑ are uniquely determined by (C′, T′, ϑ′) as respective restrictions; cf. [8, Def. 1.5.(iii)], Remark 8.

Example 22 Let (C, T, ϑ) be a Heller triangulated category. Let C̃ be the Karoubi hull of C, and let (C̃, T̃, ϑ̃_{n}) be the Heller triangulated category from Construction 11. By Proposition 12.(1), C is a Heller triangulated subcategory of C̃.

Lemma 23 Suppose given a closed Heller triangulated category (C′, T′, ϑ′), and a full subcategory C ⊆ C′ such that the following conditions (1, 2) hold.

(1) C T′ = C.

(2) Given a 2-triangle (X, Y, Z′) in C′ with X, Y ∈ Ob C, then Z′ is isomorphic to an object of C.

Then C, equipped with the shift T and the tuple ϑ obtained by restriction from T′ and ϑ′, respectively, is a Heller triangulated subcategory of C′. Moreover, (C, T, ϑ) is closed.

Proof. Let T denote the restriction of T′ to an automorphism of C, which exists by assumption (1). Write C [figure: i] C′ for the inclusion functor. Since C′ is closed, assumption (2) allows to conclude that C is a full additive subcategory of C′, and moreover, that C is weakly abelian such that i is subexact; cf. Lemma 41. Given n ⩾ 0 and X ∈ Ob \underline{C^{+}(Δ̄_{n}^{#})}, we define, by restriction, ([X]^{+1} ^{\overline{}^{Xϑ_{n}}} [X^{+1}]) := ([X]^{+1} ^{\overline{}^{Xϑ′_{n}}} [X^{+1}]). Since \underline{C^{+}(Δ̄_{n}^{#})} ^{\overline{}^{\underline{i}}} \underline{C^{′+}(Δ̄_{n}^{#})} is full and faithful by Remark 8, this is a welldefined transformation satisfying ϑ_{n} ⋆ \underline{i} = \underline{i} ⋆ ϑ′_{n}. Given m, n ⩾ 0 and a periodic monotone map Δ̄_{n} ^{\overline{}^{p}} Δ̄_{m}, we have \underline{p}^{#} ⋆ \underline{i} = \underline{i} ⋆ \underline{p}^{#}, whence

\underline{p}^{#} ⋆ ϑ_{m} ⋆ \underline{i} = \underline{p}^{#} ⋆ \underline{i} ⋆ ϑ′_{m} = \underline{i} ⋆ \underline{p}^{#} ⋆ ϑ′_{m} = \underline{i} ⋆ ϑ′_{n} ⋆ \underline{p}^{#} = ϑ_{n} ⋆ \underline{i} ⋆ \underline{p}^{#} = ϑ_{n} ⋆ \underline{p}^{#} ⋆ \underline{i} , so that we may conclude that \underline{p}^{#} ⋆ ϑ_{m} = ϑ_{n} ⋆ \underline{p}^{#}, for \underline{i} is faithful. Given n ⩾ 0, we have \underline{𝔣}_{n}⋆ \underline{i} = \underline{i} ⋆ \underline{𝔣}_{n}, whence \underline{𝔣}_{n} ⋆ ϑ_{n+1} ⋆ \underline{i} = \underline{𝔣}_{n} ⋆ \underline{i} ⋆ ϑ′_{n+1} = \underline{i} ⋆ \underline{𝔣}_{n} ⋆ ϑ′_{n+1} = \underline{i} ⋆ ϑ′_{2n+1} ⋆ \underline{𝔣}_{n} = ϑ_{2n+1} ⋆ \underline{i} ⋆ \underline{𝔣}_{n} = ϑ_{2n+1} ⋆ \underline{𝔣}_{n} ⋆ \underline{i} , so that we may conclude that \underline{𝔣}_{n} ⋆ ϑ_{n+1} = ϑ_{2n+1} ⋆ \underline{𝔣}_{n}, for \underline{i} is faithful. Hence ϑ is a Heller triangulation on (C, T); cf. [8, Def. 1.5.(i)]. By construction, C [figure: i] C′ is strictly exact. By (2) and Remark 17, the Heller triangulated category (C, T, ϑ) is closed.

4 Functors are exact if and only if they are compatible with n-triangles

Suppose given Heller triangulated categories (C, T, ϑ) and (C′, T′, ϑ′). Concerning the notion of n-triangles in a Heller triangulated category, cf. [8, Def. 1.5.(ii)]. For n ⩾ 0, an object Y in C(Δ̄_{n}^{#}) is called periodic if [Y]^{+1} = [Y^{+1}]. Suppose given an additive functor C ^{\overline{}^{F}} C′ and an isomorphism T F \overline{}^{a}_{∼} F T′.

For z ∈ Z, we let T^{z} F \overline{}^{a^{(z)}}_{∼} F T^{′z} be defined by

a^{(0)} := 1_{F}
a^{(z+1)} := (T ⋆ a^{(z)}) · (a ⋆ T^{′z})  for z ⩾ 0
a^{(z−1)} := (T^{−} ⋆ a^{(z)}) · (T^{−} ⋆ a^{−} ⋆ T^{′z−1}) for z ⩽ 0

Then (T^{z} ⋆ a^{(w)}) · (a^{(z)} ⋆ T^{′w}) = a^{(z+w)} : T^{z+w} F ^{\overline{}^{∼}} F T^{′z+w} for z, w ∈ Z. Given a periodic n-pretriangle X ∈ Ob C^{+, periodic}(Δ̄_{n}^{#}), for sake of brevity we denote in this section by

Y := X(F (Δ̄_{n}^{#})) ∈ Ob C′(Δ̄_{n}^{#})

the diagram obtained by pointwise application of F to X. We have

Y |Δ̇_{n}^{+1} = X|Δ̇_{n}((T F)(Δ̇_{n}))  \overline{}^{X|Δ̇n(a(Δ̇_{n}))}_{∼}  X|Δ̇_{n}((F T′)(Δ̇_{n})) = (Y |Δ̇_{n})^{+1} .

Isomorphic substitution along this isomorphism turns Y |Δ̄_{n}^{△▽} into a diagram Y̆ |Δ̄_{n}^{△▽} for a periodic object Y̆ ∈ Ob C(Δ̄_{n}^{#}) thus defined. We have an isomorphism Y \overline{}^{ă}_{∼} Y̆ in C′(Δ̄_{n}^{#}) that at (β/α)^{+z} for 0 ⩽ α ⩽ β ⩽ n and z ∈ Z is given by

(Y_{(β/α)^{+z}}  \overline{}^{ă_{(β/α)^{+z}}}_{∼}  Y̆_{(β/α)^{+z}}) := (X_{β/α} T^{z} F  \overline{}^{X_{β/α} a^{(z)}}_{∼}  X_{β/α}F T^{′z} ) .

In fact, given 0 ⩽ α ⩽ n and z ∈ Z, we obtain a commutative quadrangle

\xymatrix{X_{n/α} T^{z} F \ar[r]^{x T^{z} F} \ar[d]_{X_{n/α} a^{(z)}}^{≀} & X_{α/0} T^{z+1} F \ar[d]^{X_{α/0} a^{(z+1)}}_{≀} \\ X_{n/α}F T^{′z} \ar[r]^{(xF T^{′z})(X_{α/0} a T^{′z})} & X_{α/0}F T^{′z+1} ,}

for (X_{n/α} a^{(z)})(xF T^{′z})(X_{α/0} a T^{′z}) = (x T^{z} F)(X_{α/0} T a^{(z)})(X_{α/0} a T^{′z}) = (x T^{z} F)(X_{α/0} a^{(z+1)}) .

The remaining commutativities required for the naturality of Y \overline{}^{ă}_{∼} Y̆ follow by naturality of a^{(z)}. We remark that ă|Δ̇_{n} = 1_{F (Δ̇_{n})} . If F is subexact, then Y is an n-pretriangle and Y̆ is a periodic n-pretriangle.

Lemma 24 Suppose given an exact functor (F, a). Then for each n-triangle X of C, i.e. X ∈ Ob C^{ϑ=1,+}(Δ̄_{n}^{#}), the object Y̆ of C′(Δ̄_{n}^{#}) defined by (1) and (2) is an n-triangle of C′, i.e. Y̆ ∈ Ob C^{′ ϑ=1,+}(Δ̄_{n}^{#}).

(1) We have [Y̆]^{+1} = [Y̆^{+1}].

(2) On Δ̄_{n}^{△▽}, the object Y̆ |Δ̄_{n}^{△▽} arises from Y := X(F (Δ̄_{n}^{#}))|Δ̄_{n}^{△▽} by isomorphic substitution along Y |Δ̇_{n}^{+1} = X|Δ̇_{n}(T(Δ̇_{n}))(F (Δ̇_{n})) \overline{}^{a(Δ̇_{n})}_{∼} X|Δ̇_{n}(F (Δ̇_{n}))(T′(Δ̇_{n})) = (Y |Δ̇_{n})^{+1}.

Cf. [8, Lem. 3.8] for the case of a strictly exact functor.

Proof. Suppose given n ⩾ 0 and an n-triangle X ∈ Ob C^{+, ϑ=1}(Δ̄_{n}^{#}). By construction, Y̆ is periodic. We have to show that Y̆ ϑ′_{n} =^{!} 1_{[Y̆]^{+1}} . We obtain

Y ϑ′_{n} = X (\underline{F^{+}(Δ̄_{n}^{#})} ⋆ ϑ′_{n})
= X ((ϑ_{n} ⋆ \underline{F^{+}(Δ̄_{n}^{#})}) · \underline{a^{+}(Δ̄_{n}^{#})})
= X (ϑ_{n} ⋆ \underline{F^{+}(Δ̄_{n}^{#})}) · X\underline{a^{+}(Δ̄_{n}^{#})}
= X\underline{a^{+}(Δ̄_{n}^{#})} .

In particular, Y ϑ′|Δ̇_{n} = X|Δ̇_{n} \underline{a(Δ̇_{n})}. Hence, restricting the stably commutative quadrangle

\xymatrix{[Y]^{+1} \ar[r]^{[ă]^{+1}} \ar[d]_{Y ϑ′_{n}} & [Y̆]^{+1} \ar[d]^{Y̆ ϑ′_{n}} \\ [Y^{+1}] \ar[r]^{[ă^{+1}]} & [Y̆^{+1}]}

to Δ̇_{n}, we obtain the stably commutative quadrangle

\xymatrix{Y |Δ̇_{n}^{+1} \ar[r]^{X|Δ̇n \underline{a(Δ̇_{n})}} \ar[d]_{X|Δ̇n \underline{a(Δ̇_{n})}} & (Y |Δ̇_{n})^{+1} \ar[d]^{Y̆ ϑ′_{n}|Δ̇n} \\ (Y |Δ̇_{n})^{+1} \ar[r]^{1} & (Y |Δ̇_{n})^{+1} .}

whence Y̆ ϑ′_{n}|Δ̇_{n} = 1_{(Y |Δ̇n)^{+1}} . Since the functor from \underline{C^{+}(Δ̄_{n}^{#})} to \underline{C(Δ̇_{n})} induced by restriction is an equivalence by [8, Prop. 2.6], this implies that Y̆ ϑ′_{n} = 1_{[Y̆]^{+1}} .

Proposition 25 Suppose C to be closed.

The pair (F, a) is an exact functor if and only if for each n-triangle X of C, the object Y̆ of C′(Δ̄_{n}^{#}) defined by (1, 2) is an n-triangle of C′.

(1) We have [Y̆]^{+1} = [Y̆^{+1}].

(2) On Δ̄_{n}^{△▽}, the object Y̆ |Δ̄_{n}^{△▽} arises from Y := X(F (Δ̄_{n}^{#}))|Δ̄_{n}^{△▽} by isomorphic substitution

along Y |Δ̇_{n}^{+1} = X|Δ̇_{n}(T(Δ̇_{n}))(F (Δ̇_{n})) \overline{}^{a(Δ̇_{n})}_{∼} X|Δ̇_{n}(F (Δ̇_{n}))(T′(Δ̇_{n})) = (Y |Δ̇_{n})^{+1}.

Proof. In view of Lemma 24, it suffices to show that if each n-triangle X in C yields an n-triangle Y̆ in C′ by (1, 2), then (F, a) is exact.

We claim that F is subexact. By Lemma 41, it suffices to show that given a morphism S ^{\overline{}^{p}} T in C, there exists a weak cokernel of p that is mapped by F to a weak cokernel. Since C is a closed Heller triangulated category, a weak cokernel of p is contained in the the completion of S ^{\overline{}^{p}} T to a 2-triangle X by Lemma 20. We form the corresponding 2-triangle Y̆ defined by (1, 2). Since it contains a weak cokernel of SF ^{\overline{}^{pF}} T F, and since Y̆ is isomorphic, in C^{′+}(Δ̄_{n}^{#}), to X(F^{+}(Δ̄_{2}^{#})), the image under F of the weak cokernel of p that is contained in the 2-triangle X is in fact a weak cokernel of pF. This proves the claim.

We claim that

(ϑ_{n} ⋆ \underline{F^{+}(Δ̄_{n}^{#})}) · \underline{a^{+}(Δ̄_{n}^{#})} = \underline{F^{+}(Δ̄_{n}^{#})} ⋆ ϑ′_{n}

for all n ⩾ 0. Suppose given X ∈ Ob C^{+}(Δ̄_{n}^{#}). Since C is a closed Heller triangulated category, there exists an n-triangle X′ such that X′|Δ̇_{n} = X|Δ̇_{n}; cf. Lemma 20. By [8, Prop. 2.6], we have an isomorphism X \overline{}^{f}_{∼} X′ in \underline{C^{+}(Δ̄_{n}^{#})} that restricts to the identity on Δ̇_{n}. We dispose of a commutative diagram

\xymatrix{[X]^{+1} \ar[r]^{Xϑ_{n}} \ar[d]_{[f]^{+1}} & [X^{+1}] \ar[d]^{[f^{+1}]} \\ [X′]^{+1} \ar[r]^{X′ϑ_{n}} & [X^{′+1}]}

in \underline{C^{+}(Δ̄_{n}^{#})}. Since, by construction, X′ϑ_{n} = 1, we have Xϑ_{n} = [f]^{+1} · [f^{+1}]^{−} in \underline{C^{+}(Δ̄_{n}^{#})}.

Likewise, we have a commutative quadrangle

\xymatrix{[X\underline{F^{+}(Δ̄_{n})}]^{+1} \ar[r]^{X\underline{F^{+}(Δ̄_{n})}ϑ′_{n}} \ar[d]_{[f\underline{F^{+}(Δ̄_{n})}]^{+1}} & [(X\underline{F^{+}(Δ̄_{n})})^{+1}] \ar[d]^{[(f\underline{F^{+}(Δ̄_{n})})^{+1}]} \\ [X′\underline{F^{+}(Δ̄_{n})}]^{+1} \ar[r]^{X′\underline{F^{+}(Δ̄_{n})}ϑ′_{n}} & [(X′\underline{F^{+}(Δ̄_{n})})^{+1}] ,}

in \underline{C^{′+}(Δ̄_{n}^{#})}. We want to calculate its lower arrow. Since X′ is an n-triangle, we have an isomorphism Y′ \overline{}^{ă′}_{∼} Y̆′ formed as above, where Y̆′ϑ′_{n} = 1. The stably commutative quadrangle

\xymatrix{[Y′]^{+1} \ar[r]^{[ă′]^{+1}} \ar[d]_{Y′ϑ′_{n}} & [Y̆′]^{+1} \ar[d]^{Y̆′ϑ′_{n} = 1} \\ [Y^{′+1}] \ar[r]^{[ă^{′+1}]} & [Y̆^{′+1}]}

yields by restriction to Δ̇_{n} the commutative diagram

[figure: X′|Δ̇_{n} \underline{(T F)(Δ̇_{n})}; X′|Δ̇n \underline{a(Δ̇_{n})}; X′|Δ̇_{n} \underline{(F T′)(Δ̇_{n})}; [Y′]^{+1}|Δ̇_{n}; [ă]^{+1}|Δ̇n; Y′ϑ′_{n}|Δ̇n; [Y̆′]^{+1}|Δ̇_{n}; 1; [Y^{′+1}]|Δ̇_{n}; 1; [Y̆^{′+1}]|Δ̇_{n} ,]

whence Y′ϑ′_{n}|Δ̇_{n} = X′|Δ̇_{n} \underline{a(Δ̇_{n})} = X′\underline{a^{+}(Δ̄_{n}^{#})}|Δ̇_{n}. Since the functor from \underline{C^{′+}(Δ̄_{n}^{#})} to \underline{C′(Δ̇_{n})} induced by restriction is an equivalence by [8, Prop. 2.6], this implies that

X′\underline{F^{+}(Δ̄_{n})}ϑ′_{n} = Y′ϑ′_{n} = X′\underline{a^{+}(Δ̄_{n}^{#})} .

So we can conclude that

X\underline{F^{+}(Δ̄_{n})}ϑ′_{n} = [f\underline{F^{+}(Δ̄_{n})}]^{+1} · X′\underline{F^{+}(Δ̄_{n})}ϑ′_{n} · [(f\underline{F^{+}(Δ̄_{n})})^{+1}]^{−}
= [f\underline{F^{+}(Δ̄_{n})}]^{+1} · X′\underline{a^{+}(Δ̄_{n}^{#})} · [(f\underline{F^{+}(Δ̄_{n})})^{+1}]^{−}
= [f\underline{F^{+}(Δ̄_{n}^{#})}]^{+1} · X′\underline{a^{+}(Δ̄_{n}^{#})} · (f\underline{(F T′)^{+}(Δ̄_{n}^{#})})^{−}
= [f\underline{F^{+}(Δ̄_{n}^{#})}]^{+1} · (f\underline{(T F)^{+}(Δ̄_{n}^{#})})^{−} · X\underline{a^{+}(Δ̄_{n}^{#})}
= ([f]^{+1} · [f^{+1}]^{−})\underline{F^{+}(Δ̄_{n}^{#})} · X\underline{a^{+}(Δ̄_{n}^{#})}
= X(ϑ_{n} ⋆ \underline{F^{+}(Δ̄_{n}^{#})}) · X\underline{a^{+}(Δ̄_{n}^{#})} .

This proves the claim.

Corollary 26 Suppose (C, T, ϑ) to be closed. Suppose given an additive functor C ^{\overline{}^{F}} C′ such that T F = F T′. Then F is strictly exact if and only if for each n ⩾ 0 and each n-triangle X ∈ Ob C^{+, ϑ=1}(Δ̄_{n}^{#}), the diagram X(F (Δ̄_{n}^{#})) ∈ Ob C′(Δ̄_{n}^{#}), obtained by pointwise application of F, is an n-triangle.

Proof. In this case, we have a = 1_{T F} = 1_{F T′} and Y̆ = Y = X(F (Δ̄_{n}^{#})). Since (F, 1) is exact if and only if F is strictly exact, the assertion follows by Proposition 25.

5 Adjoints

5.1 Adjoints and shifts

Suppose given categories A and A′. Suppose given an endofunctor T of A. Suppose given an endofunctor T′ of A′. Suppose given functors A [figure: F; G] A′ such that F ⊣ G via unit 1 ^{\overline{}^{ε}} F G and counit GF ^{\overline{}^{η}} 1, i.e. (Gε)(ηG) = 1_{G} and (εF)(F η) = 1_{F} . Suppose given T F ^{\overline{}^{α}} F T′. Let

(GT ^{\overline{}^{β}} T′G) := (GT ^{\overline{}^{GT ε}} GT F G ^{\overline{}^{GαG}} GF T′G ^{\overline{}^{ηT′G}} T′G) .

So we have the commutative diagram

\xymatrix{GT \ar[r]^{β} \ar[d]_{GT ε} & T′G \\ GT F G \ar[r]^{GαG} & GF T′G . \ar[u]_{ηT′G}}

Lemma 27

(1) We have the commutative diagram

\xymatrix{T F \ar[r]^{α} \ar[d]_{εT F} & F T′ \\ F GT F \ar[r]^{F βF} & F T′GF . \ar[u]_{F T′η}}

(2) We have the commutative quadrangle

\xymatrix{T \ar[r]^{εT} \ar[d]_{T ε} & F GT \ar[d]^{F β} \\ T F G \ar[r]_{αG} & F T′G .}

(2^{∘}) We have the commutative quadrangle

\xymatrix{GT F \ar[r]^{Gα} \ar[d]_{βF} & GF T′ \ar[d]^{ηT′} \\ T′GF \ar[r]_{T′η} & T′ .}

(3) Suppose that T and T′ are autofunctors. Write G′ = T′GT^{−}. If α is an isotransformation,

then so is β, where (T′G ^{\overline{}^{β^{−}}} GT) = (T′G = T′GT^{−}T ^{\overline{}^{T′GT^{−}εT}} T′GT^{−}F GT = T′GT^{−}F T′T^{′−}GT ^{\overline{}^{T′GT^{−}α^{−}T^{′−}GT}} T′GT^{−}T F T^{′−}GT = T′GF T^{′−}GT ^{\overline{}^{T′ηT^{′−}GT}} T′T^{′−}GT = GT ) .

(3^{∘}) Suppose that T and T′ are autofunctors. If β is an isotransformation, then so is α, where (F T′ ^{\overline{}^{α^{−}}} T F) = (F T′ = T T^{−}F T′ ^{\overline{}^{T εT^{−}F T′}} T F GT^{−}F T′ = T F T^{′−}T′GT^{−}F T′ ^{\overline{}^{T F T^{′−}β^{−}T^{−}F T′}} T F T^{′−}GT T^{−}F T′ = T F T^{′−}GF T′ ^{\overline{}^{T F T^{′−}ηT′}} T F T^{′−}T′ = T F ) .

Proof. Ad (2). We have

(εT)(F β) = (εT)(F GT ε)(F GαG)(F ηT′G)
= (T ε)(εT F G)(F GαG)(F ηT′G)
= (T ε)(αG)(εF T′G)(F ηT′G)
= (T ε)(αG) .

Ad (1). We have

(εT F)(F βF)(F T′η) =^{(2)} (T εF)(αGF)(F T′η)
= (T εF)(T F η)α
= α .

Ad (3). We have

β · (T′GT^{−}εT)(T′GT^{−}α^{−}T^{′−}GT)(T′ηT^{′−}GT)
= (βT^{−}T)(T′GT^{−}εT)(T′GT^{−}α^{−}T^{′−}GT)(T′ηT^{′−}GT)
= (GT T^{−}εT)(βT^{−}F GT)(T′GT^{−}α^{−}T^{′−}GT)(T′ηT^{′−}GT)
= (GεT)(βT^{−}F T′T^{′−}GT)(T′GT^{−}α^{−}T^{′−}GT)(T′ηT^{′−}GT)
= (GεT)(GT T^{−}α^{−}T^{′−}GT)(βT^{−}T F T^{′−}GT)(T′ηT^{′−}GT)
= (GεT)(Gα^{−}T^{′−}GT)(βF T^{′−}GT)(T′ηT^{′−}GT)
=^{(2^{∘})} (GεT)(Gα^{−}T^{′−}GT)(GαT^{′−}GT)(ηT′T^{′−}GT)
= (GεT)(ηGT)
= 1

and

(T′GT^{−}εT)(T′GT^{−}α^{−}T^{′−}GT)(T′ηT^{′−}GT) · β
= (T′GT^{−}εT)(T′GT^{−}α^{−}T^{′−}GT)(T′ηT^{′−}GT)(T′T^{′−}β)
= (T′GT^{−}εT)(T′GT^{−}α^{−}T^{′−}GT)(T′GF T^{′−}β)(T′ηT^{′−}T′G)
= (T′GT^{−}εT)(T′GT^{−}α^{−}T^{′−}GT)(T′GT^{−}T F T^{′−}β)(T′ηG)
= (T′GT^{−}εT)(T′GT^{−}F T′T^{′−}β)(T′GT^{−}α^{−}T^{′−}T′G)(T′ηG)
= (T′GT^{−}εT)(T′GT^{−}F β)(T′GT^{−}α^{−}G)(T′ηG)
=^{(2)} (T′GT^{−}T ε)(T′GT^{−}αG)(T′GT^{−}α^{−}G)(T′ηG)
= (T′Gε)(T′ηG)
= 1 .

5.2 An adjoint of an exact functor is exact

The Verdier triangulated version of the following proposition is due to Margolis [11, App. 2, Prop. 11], and, in a more general form, to Keller and Vossieck [5, 1.6]. Proposition 28 Suppose given Heller triangulated categories (C, T, ϑ) and (C′, T′, ϑ′). Suppose given an exact functor (F, a) from C to C′; cf. Definition 1. Suppose given a functor C ^{\overline{}^{G}} C′. So C [figure: F; G] C′ and T F \overline{}^{a}_{∼} F T′ .

(1) If F ⊣ G, then there exists an isomorphism T′ G \overline{}^{b}_{∼} G T such that (G, b) is an exact functor from C′ to C.

Choose a unit 1_{C′} ^{\overline{}^{ε}} F G and a counit GF ^{\overline{}^{η}} 1_{C}. Then, more precisely, we may choose

(T′ G ^{\overline{}^{b}} G T) := (G T ^{\overline{}^{G T ε}} G T F G ^{\overline{}^{GaG}} GF T′ G ^{\overline{}^{η T′ G}} T′ G)^{−} .

(1^{∘}) If G ⊣ F, then there exists an isomorphism T′ G \overline{}^{b}_{∼} G T such that (G, b) is an exact functor from C′ to C. Choose a unit 1_{C′} ^{\overline{}^{ε}} GF and a counit F G ^{\overline{}^{η}} 1_{C}. Then, more precisely, we may choose

(T′ G ^{\overline{}^{b}} G T) := (T′ G ^{\overline{}^{ε T′ G}} GF T′ G ^{\overline{}^{Ga^{−}G}} G T F G ^{\overline{}^{G T η}} G T) .

Proof. Ad (1). By Lemma 42.(1^{∘}), G is subexact. Lemma 27.(3) yields the isotransformation b^{−} := (G T ε)(GaG)(η T′ G). Suppose given n ⩾ 0. We shall make use of the abbreviation \underline{G} = \underline{G^{+}(Δ̄_{n}^{#})} , etc. We have to show that

(ϑ′_{n} ⋆ \underline{G}) · \underline{b} =^{!} \underline{G} ⋆ ϑ_{n} ,

i.e. that

(\underline{G} ⋆ ϑ_{n}) · \underline{b}^{−} =^{!} ϑ′_{n} ⋆ \underline{G} ,

i.e. that

(\underline{G} ⋆ ϑ_{n}) · (\underline{G} ⋆ \underline{T} ⋆ \underline{ε}) · (\underline{G} ⋆ \underline{a} ⋆ \underline{G}) · (\underline{η} ⋆ \underline{T′} ⋆ \underline{G}) =^{!} ϑ′_{n} ⋆ \underline{G}

Recall that [−]^{+1} denotes the outer shift, that [−^{+1}] denotes the inner shift and that ϑ_{n} : [−]^{+1} ^{\overline{}^{∼}} [−^{+1}] on \underline{C^{+}(Δ̄_{n}^{#})} ; similarly on \underline{C^{′+}(Δ̄_{n}^{#})} . We obtain

(\underline{G} ⋆ ϑ_{n}) · (\underline{G} ⋆ \underline{T} ⋆ \underline{ε}) · (\underline{G} ⋆ \underline{a} ⋆ \underline{G}) · (\underline{η} ⋆ \underline{T′} ⋆ \underline{G})
= (\underline{G} ⋆ ϑ_{n}) · (\underline{G} ⋆ [−^{+1}] ⋆ \underline{ε}) · (\underline{G} ⋆ \underline{a} ⋆ \underline{G}) · (\underline{η} ⋆ \underline{T′} ⋆ \underline{G})
= (\underline{G} ⋆ [−]^{+1} ⋆ \underline{ε}) · (\underline{G} ⋆ ϑ_{n} ⋆ \underline{F} ⋆ \underline{G}) · (\underline{G} ⋆ \underline{a} ⋆ \underline{G}) · (\underline{η} ⋆ \underline{T′} ⋆ \underline{G})
= (\underline{G} ⋆ [−]^{+1} ⋆ \underline{ε}) · (\underline{G} ⋆ ((ϑ_{n} ⋆ \underline{F}) · \underline{a}) ⋆ \underline{G}) · (\underline{η} ⋆ \underline{T′} ⋆ \underline{G})
=^{(F, a) ex.} (\underline{G} ⋆ [−]^{+1} ⋆ \underline{ε}) · (\underline{G} ⋆ \underline{F} ⋆ ϑ′_{n} ⋆ \underline{G}) · (\underline{η} ⋆ \underline{T′} ⋆ \underline{G})
= (\underline{G} ⋆ [−]^{+1} ⋆ \underline{ε}) · (\underline{G} ⋆ \underline{F} ⋆ ϑ′_{n} ⋆ \underline{G}) · (\underline{η} ⋆ [−^{+1}] ⋆ \underline{G})
= (\underline{G} ⋆ [−]^{+1} ⋆ \underline{ε}) · (\underline{η} ⋆ [−]^{+1} ⋆ \underline{G}) · (ϑ′_{n} ⋆ \underline{G})
= (\underline{G} ⋆ \underline{ε} ⋆ [−]^{+1}) · (\underline{η} ⋆ \underline{G} ⋆ [−]^{+1}) · (ϑ′_{n} ⋆ \underline{G})
= ϑ′_{n} ⋆ \underline{G} .

Ad (1^{∘}). Cf. Lemma 27.(1).

Example 29 Suppose we are in the situation of Proposition 28.(1). Then ε and η are periodic; cf. Definition 4. Ad ε : 1 \overline{}F G. The functor (F, a)(G, b) = (F G , aG · F b) is exact; cf. Remark 3. The functor (1_{C} , 1) is exact. The quadrangle

\xymatrix{T \ar[r]^{T ε} \ar[d]_{1} & T F G \ar[d]^{aG·F b} \\ T \ar[r]^{εT} & F G T}

commutes by Lemma 27.(2).

Ad η : GF \overline{}1. The functor (G, b)(F, a) = (GF , bF · Ga) is exact; cf. Remark 3. The functor (1_{C′} , 1) is exact. The quadrangle

\xymatrix{T′ GF \ar[r]^{T′ η} \ar[d]_{bF ·Ga} & T′ \ar[d]^{1} \\ GF T′ \ar[r]^{η T′} & _{T_{′}}}

commutes by Lemma 27.(2^{∘}).

5.3 A functor shiftcompatibly adjoint to a strictly exact functor is strictly exact

Suppose given closed Heller triangulated categories (C, T, ϑ) and (C′, T′, ϑ′)

Recall that an additive functor F : C \overline{}C′ is strictly exact if and only if (F, 1) is exact; cf. [8, Def. 1.5.(iii)], Definition 1.

Corollary 30

Suppose given a strictly exact functor C ^{\overline{}^{F}} C′.

Suppose given a functor C ^{\overline{}^{G}} C′.

(1) If F ⊣ G, with unit ε : 1 \overline{}F G and counit η : GF \overline{}1 such that (G T ε)(η T′ G) = 1, then G is strictly exact.

(1^{∘}) If G ⊣ F, with unit ε : 1 \overline{}F G and counit η : GF \overline{}1 such that (ε T′ G)(G T η) = 1, then G is strictly exact.

Proof. Ad (1). In the notation of Proposition 28.(1), we have a = 1, and, consequently, b = (G T ε)(η T′ G) = 1. Hence by loc. cit., (G, 1) is exact, i.e. G is strictly exact.

6 Localisation

We prove that the localisation C//N of a Heller triangulated category C at a thick subcategory N is Heller triangulated in such a way that the localisation functor C ^{\overline{}^{L}} C//N is strictly exact; cf. [8, Def. 1.5]. There is considerable overlap with the classical localisation theory of Verdier triangulated categories, due to Verdier [13], which we include for sake of self-containedness.

Let (C, T, ϑ) be a closed Heller triangulated category; cf. Definition 14.

Definition 31 A full additive subcategory N ⊆ C is called thick if the conditions (1, 2, 3) are satisfied; cf. [12, Prop. 1.3]

(1) We have N^{+1} = N (closed under shift).

(2) Given a 2-triangle (X, Y, Z) in C with X and Y in Ob N, then Z ∈ Ob N (closed under taking cones).

(3) Given X, Y ∈ Ob C with X ⊕ Y in Ob N, then X ∈ Ob N (closed under taking summands).

Let N be a thick subcategory of C. By Lemma 23, conditions (1) and (2) of Definition 31 yield that N is a Heller triangulated subcategory of C.

Let M(N) := \{(X ^{\overline{}^{f}} Y) ∈ C : the cone of f is in Ob N \}. An element of M(N) is called an M(N)-isomorphism or often just an N-isomorphism (not to be confused with “an isomorphism in N ”). If N is unambiguous, then an N-isomorphism is denoted by X ⟹ Y. For instance, X ⟹ 0 if and only if 0 ⟹ X if and only if X ∈ Ob N.

Lemma 32 The subset M(N) of N-isomorphisms in C is a multiplicative system in C in the sense of Definition 45.

Proof. Ad (Fr 2). Suppose given X_{1/0} ^{\overline{}^{x}} X_{2/0} ^{\overline{}^{x}} X_{3/0} ^{\overline{}^{x}} X_{4/0} such that X_{1/0} ^{\overline{}^{x}} X_{3/0} and X_{2/0} ^{\overline{}^{x}} X_{4/0} are M-isomorphisms. We complete to a 4-triangle X ∈ Ob C^{+, ϑ=1}(Δ̄_{n}^{#}) using closedness of C; cf. Lemma 20. By [8, Lem. 3.4.(1, 6)], we have X_{3/1} , X_{4/2} ∈ Ob N. We have to show that X_{2/1} , X_{3/2} , X_{4/3} , X_{4/1} ∈ Ob N. Let the periodic monotone map Δ̄_{5} ^{\overline{}^{p}} Δ̄_{4} be defined by 0p := 1, 1p := 1, 2p := 2, 3p := 3, 4p := 4 and 5p := 4. The 2-triangle Xp^{#}𝔣_{2} ∈ Ob C^{+, ϑ=1}(Δ̄_{2}^{#}) is given by

[figure: 0; 0; X_{2^{+1}/4}; 0; X_{1^{+1}/2}; −x; +; X_{1^{+1}/4}; x; 0; X_{4/3}; _{(_{1 0})}; +; X_{4/3}⊕X_{1^{+1}/2}; _{(\begin{matrix}x \\ −x \end{matrix})}; _{(\begin{matrix}0 \\ 1 \end{matrix})}; +; X_{1^{+1}/3}; x; 0; X_{3/1}; x; +; X_{4/1}; x; x; +; X_{4/2}; _{(_{x x})}; +; 0 ,]

cf. [8, Lem. 3.4.(1, 2), §1.2.1.2, §1.2.2.2.]. Since X_{3/1} , X_{4/2} ∈ Ob N, and since N is closed under cones, we have X_{4/3} ⊕ X_{1^{+1}/2} ∈ Ob N. Since N is closed under summands and under shift, we obtain X_{4/3} , X_{2/1} ∈ Ob N. Since N is closed under cones and under shift, X_{4/1} ∈ Ob N ensues. Considering X again, since N is closed under cones, we finally obtain X_{3/2} ∈ Ob N.

Ad (Fr 3). Let X ^{\overline{}^{f}} Y be a morphism in C such that there exists Y ⟹^{s} Z with fs = 0. We obtain a factorisation (X ^{\overline{}^{f}} Y) = (X ^{\overline{}^{u}} N ^{\overline{}^{v}} Y) with N ∈ Ob N. Completing (T_{1/0} ^{\overline{}^{t}} T_{2/0} ^{\overline{}^{t}} T_{3/0}) := (X ^{\overline{}^{u}} N ^{\overline{}^{v}} Y) to a 3-triangle by Lemma 20, we obtain T_{1/2^{−1}} ⟹^{t} T_{1/0} , which composes to zero with (T_{1/0} ^{\overline{}^{t}} T_{3/0}) = (X ^{\overline{}^{f}} Y).

Ad (Fr 4). Suppose given

\xymatrix{X′ \\ X \ar@{=>}[r] \ar[u] & Y}

in C. Prolonging X \overline{}X′ to a 2-triangle (X′′, X, X′), then completing X′′ \overline{}X ⟹ Y to a 3-triangle using Lemma 20, we obtain, by [8, Lem. 3.4.(6)], a 3-triangle T with (T_{2/0} ^{\overline{}^{t}} T_{3/0}) = (X ⟹ Y) and (T_{2/0} ^{\overline{}^{t}} T_{2/1}) = (X \overline{}X′). Then T_{3/2} ∈ Ob N, whence T_{2/1} ⟹ T_{3/1} . The weak square (T_{2/0} , T_{3/0} , T_{2/1} , T_{3/1}) is a completion as sought.

Note that if X ∈ (Ob C) ∖ (Ob N), then (0, 0, 0, X) is a weak square in which 0 \overline{}0 is an N-isomorphism, but 0 \overline{}X is not.

The localisation of C at M(N), defined as in §A.4, is also called the localisation of C at N, and also written C//N := C_{M(N)}. Concerning the localisation functor C ^{\overline{}^{L}} C//N, we refer to §A.4.

Recall that an additive functor between weakly abelian categories is called subexact if it induces an exact functor on the Freyd categories; cf. [8, §1.2.1.3]; cf. also Lemma 41.

Lemma 33 The category C//N is weakly abelian. The functor C ^{\overline{}^{L}} C//N is subexact.

Proof. By Remark 51, the category C//N is additive, and the localisation functor L : C \overline{}C//N is additive. We claim that L maps weak kernels to weak kernels. Let X ^{\overline{}^{f}} Y be a weak kernel of Y ^{\overline{}^{g}} Z in C. We claim that it remains a weak kernel in C//N. Suppose given a morphism T ^{\overline{}^{t}} Y in C such that tg = 0 in C//N, which we, by isomorphic replacement, may assume given. Let T′ ⟹^{s} T be such that stg = 0 in C; cf. Remark 47. Since f is a weak kernel of g in C, we have a factorisation st = uf. Hence t = (s^{−1}u)f is a factorisation of t over f in C//N.

Substituting isomorphically in C//N and using duality, for C//N to be weakly abelian, it suffices to show that each morphism X ^{\overline{}^{f}} Y has a weak kernel resp. is a weak kernel in C//N. But by the property of L just shown, we may use a weak kernel of f in C resp. a morphism f is a weak kernel of in C.

Remark 34 The category C//N carries a shift automorphism C//N ^{\overline{}^{T}} C//N, f/t \overline{}f^{+1}/t^{+1}. We have L T = T L.

Proof. This functor is welldefined since N, and hence M(N), is closed under shift in C. Likewise, its inverse f/t \overline{}f^{−1}/t^{−1} is welldefined.

Lemma 35 Suppose given a Heller triangulated category (D, T, θ).

Suppose given a weakly abelian category D′ and an automorphism D′ ^{\overline{}^{T′}} D′. Suppose given a subexact additive functor D ^{\overline{}^{G}} D′ strictly compatible with shift, i.e. G T′ = T G. Suppose that D(Δ̇_{n})^{\overline{}^{G(Δ̇_{n})}} D′(Δ̇_{n}) is 1-epimorphic for n ⩾ 0.

Then the functor \underline{D^{+}(Δ̄_{n}^{#})} ^{\overline{}^{\underline{G^{+}(Δ̄_{n}^{#})}}} \underline{D^{′+}(Δ̄_{n}^{#})} is 1-epimorphic.

Moreover, there exists a unique Heller triangulation θ′ on (D′, T′) such that D ^{\overline{}^{G}} D′ is strictly exact; cf. [8, Def. 1.5].

Proof. Given n ⩾ 0. Since the residue class functors D(Δ̇_{n}) \overline{}\underline{D(Δ̇_{n})} and D′(Δ̇_{n}) \overline{}\underline{D′(Δ̇_{n})} are full and dense, they are 1-epimorphic by [8, Cor. A.37] ; concerning notation, cf. [8, §2.4]. The commutative quadrangle

\xymatrix{D(Δ̇_{n}) \ar[r]^{G(Δ̇_{n})} \ar[d] & D′(Δ̇_{n}) \ar[d] \\ \underline{D(Δ̇_{n})} \ar[r]^{\underline{G(Δ̇_{n})}} & \underline{D′(Δ̇_{n})}}

shows that \underline{G(Δ̇_{n})} is 1-epimorphic. Restriction induces equivalences \underline{D^{+}(Δ̄_{n}^{#})} ^{\overline{}^{(−)|Δ̇n}} \underline{D(Δ̇_{n})} and

\underline{D^{′+}(Δ̄_{n}^{#})} ^{\overline{}^{(−)|Δ̇n}} \underline{D′(Δ̇_{n})} by [8, Prop. 2.6]. Therefore, the commutative quadrangle

\xymatrix{\underline{D(Δ̇_{n})} \ar[r]^{\underline{G(Δ̇_{n})}} & \underline{D′(Δ̇_{n})} \\ \underline{D^{+}(Δ̄_{n}^{#})} \ar[u]^{(−)|Δ̇n} \ar[r]^{\underline{G^{+}(Δ̄_{n}^{#})}} & \underline{D^{′+}(Δ̄_{n}^{#})} \ar[u]_{(−)|Δ̇n}}

shows that \underline{G^{+}(Δ̄_{n}^{#})} is 1-epimorphic; concerning notation, cf. [8, §1.2.1.1, §1.2.1.3]. Therefore, we may define a transformation θ′_{n} for D′ by the requirement that

[figure: \underline{D^{+}(Δ̄_{n}^{#})}; \underline{G^{+}(Δ̄_{n}^{#})}; [−]^{+1}; [−^{+1}]; \underline{D^{′+}(Δ̄_{n}^{#})}; [−]^{+1}; [−^{+1}]; \underline{D^{+}(Δ̄_{n}^{#})}; \underline{G^{+}(Δ̄_{n}^{#})}; \underline{D^{′+}(Δ̄_{n}^{#})}; θ_{n}; θ′_{n}]

be commutative, i.e. that θ_{n} ⋆ \underline{G^{+}(Δ̄_{n}^{#})} = \underline{G^{+}(Δ̄_{n}^{#})} ⋆ θ′_{n}. In other words, there exists a unique θ′_{n} making this diagram commutative.

Let θ′ := (θ′_{n})_{n⩾0} , where for n = 0, we make use of \underline{D^{′+}(Δ̄_{0}^{#})} = 0. We claim that θ′ is a Heller triangulation on (D′, T′), i.e. that (D′, T′, θ′) is a Heller triangulated category. Once this is proven, we see that by construction, D ^{\overline{}^{G}} D′ is strictly exact; cf. [8, Def. 1.5.(iii)].

Suppose given m, n ⩾ 0 and a periodic monotone map Δ̄_{n} ^{\overline{}^{p}} Δ̄_{m}. To prove that \underline{p}^{#} ⋆ θ′_{m} =^{!} θ′_{n} ⋆ \underline{p}^{#}, we may precompose with the 1-epimorphic functor \underline{G^{+}(Δ̄_{n}^{#})} to obtain

\underline{G^{+}(Δ̄_{n}^{#})} ⋆ \underline{p}^{#} ⋆ θ′_{m} =  \underline{p}^{#} ⋆ \underline{G^{+}(Δ̄_{m}^{#})} ⋆ θ′_{m} = \underline{p}^{#} ⋆ θ_{m} ⋆ \underline{G^{+}(Δ̄_{m}^{#})}

=^{(D,T,θ)} Heller triangulated θ_{n} ⋆ \underline{p}^{#} ⋆ \underline{G^{+}(Δ̄_{m}^{#})} = θ_{n} ⋆ \underline{G^{+}(Δ̄_{n}^{#})} ⋆ \underline{p}^{#} = \underline{G^{+}(Δ̄_{n}^{#})} ⋆ θ′_{n} ⋆ \underline{p}^{#} .

Suppose given n ⩾ 0. To prove that \underline{𝔣}_{n} ⋆ θ′_{n+1} =^{!} θ′_{2n+1} ⋆ \underline{𝔣}_{n} , we may precompose with the 1-epimorphic functor \underline{G^{+}(Δ̄_{2n+1}^{#})} to obtain

\underline{G^{+}(Δ̄_{2n+1}^{#})} ⋆ \underline{𝔣}_{n} ⋆ θ′_{n+1} =  \underline{𝔣}_{n} ⋆ \underline{G^{+}(Δ̄_{n+1}^{#})} ⋆ θ′_{n+1}
=  \underline{𝔣}_{n} ⋆ θ_{n+1} ⋆ \underline{G^{+}(Δ̄_{n+1}^{#})}

=^{(D,T,θ)} Heller triangulated θ_{2n+1} ⋆ \underline{𝔣}_{n} ⋆ \underline{G^{+}(Δ̄_{n+1}^{#})}

=  θ_{2n+1} ⋆ \underline{G^{+}(Δ̄_{2n+1}^{#})} ⋆ \underline{𝔣}_{n} = \underline{G^{+}(Δ̄_{2n+1}^{#})} ⋆ θ′_{2n+1} ⋆ \underline{𝔣}_{n} .

Proposition 36 Recall that (C, T, ϑ) is a closed Heller triangulated category, and that N is a thick subcategory of C.

There exists a unique Heller triangulation θ on (C//N , T) such that C ^{\overline{}^{L}} C//N is strictly exact; cf. [8, Def. 1.5].

Then (C//N , T, θ) is a closed Heller triangulated category; cf. Definition 14.

Proof. By Lemma 33, the category C//N is weakly abelian, and L : C \overline{}C//N is subexact. By Remark 34, C//N carries a shift automorphism, and L is compatible with the shift automorphisms on C and on C//N. By Lemma 50, the functor C(Δ̇_{n})^{\overline{}^{L(Δ̇_{n})}} (C//N)(Δ̇_{n}) is 1-epimorphic for n ⩾ 0. Therefore, existence and uniqueness of θ follow by Lemma 35.

It remains to be shown that C//N is closed. By isomorphic substitution, it suffices to show that each morphism in the image of L has a cone in C//N ; cf. [8, Lem. 3.4.(6)]. But this follows from C being closed and from L being strictly exact.

An object (X ^{\overline{}^{x}} X′) of the Freyd category Ĉ is called N-zero if x factors over an object of N ; concerning Ĉ, cf. [8, §A.6.3]. Note that an object of Ĉ that is isomorphic to a summand of an N-zero object is itself N-zero.

Remark 37 A morphism in C is an N-isomorphism if and only if its kernel and its cokernel, taken in Ĉ, are N-zero.

Note that this criterion does not make reference to the Heller triangulated structure on C, but only to the fact that C is weakly abelian. One might ask for conditions on N that only use weak abelianess of C, and that nonetheless suffice to turn C_{M(N)} into a weakly abelian category – where now M(N) is the subset of morphisms of C defined by the criterion given in Remark 37.

Proof of Remark 37. Suppose that X ^{\overline{}^{f}} Y is an N-isomorphism in C. Then it has a weak kernel N and a weak cokernel M in Ob N. By construction of the kernel in Ĉ, it is of the form (N \overline{}X). Dually, the cokernel is of the form (Y \overline{}M); cf. [8, §A.6.3].

Conversely, suppose that the kernel and the cokernel of the morphism X ^{\overline{}^{f}} Y, taken in Ĉ, are N-zero. Consider the exact functor Ĉ ^{\overline{}^{L̂}} (C//N)ˆ that prolongs L on the level of Freyd categories. It maps f to an isomorphism, since in the abelian category (C//N)ˆ , the image of f has zero kernel and zero cokernel. Since C//N \overline{}(C//N)ˆ is full and faithful, the image of f under L in C//N is an isomorphism, too. Hence f is an N-isomorphism in C; cf. Remark 46.

Proposition 38 (universal property) Recall that (C, T, ϑ) is a closed Heller triangulated category, and that N is a thick subcategory of C.

Let θ be the unique Heller triangulation on (C//N , T) such that the localisation functor C ^{\overline{}^{L}} C//N is strictly exact; cf. Proposition 36. Suppose given a Heller triangulated category (C′, T′, ϑ′).

Recall that we write C, C′ _{ex} for the category of exact functors and periodic transformations from C to C′; cf. Definition 6.

Write C, C′ _{ex, N} ⊆ C, C′ _{ex} for the full subcategory consisting of exact functors (F, a) such that NF ≃ 0 for all N ∈ Ob N.

Recall that we write C, C′ _{st ex} for the category of strictly exact functors and periodic transformations from C to C′; cf. Definition 6.

Write C, C′ _{st ex, N} ⊆ C, C′ _{st ex} for the full subcategory consisting of strictly exact functors F such that NF ≃ 0 for all N ∈ Ob N.

(1) We have a strictly dense equivalence

C, C′ _{ex, N} ^{\overline{}^{L ⋆ (−)}} C//N , C′ _{ex}
(L ⋆ G , L ⋆ b) = (L, 1) ⋆ (G, b) \overline{}(G, b) .

(2) We have a strictly dense equivalence

C, C′ _{st ex, N} ^{\overline{}^{L ⋆ (−)}} C//N , C′ _{st ex}
L ⋆ G \overline{}G .

Proof.

Ad (1). Welldefinedness of the functor L ⋆ (−) follows from L being strictly exact and exact functors being stable under composition; cf. Proposition 36, Remark 3.

We make use of the universal property of the localisation to the extent stated in Remark 51.

Suppose given exact functors C [figure: (F,a); (G,b)] C′ and a periodic transformation F ^{\overline{}^{u}} G.

Let F̆ : C//N \overline{}C′ be defined by L ⋆ F̆ := F. Let Ğ : C//N \overline{}C′ be defined by L ⋆ Ğ := G.

Recall that the shift on C//N is, abusively, also denoted by T, so that T ⋆ L = L ⋆ T. Let the transformations ă and b̆ be defined by

L ⋆ (T ⋆ F̆ ^{\overline{}^{ă}} F̆ ⋆ T′) := (T ⋆ F ^{\overline{}^{a}} F ⋆ T′)
L ⋆ (T ⋆ Ğ ^{\overline{}^{b̆}} Ğ ⋆ T′) := (T ⋆ G ^{\overline{}^{b}} G ⋆ T′) .

Let the transformation F̆ ^{\overline{}^{ŭ}} Ğ be defined by

L ⋆ (F̆ ^{\overline{}^{ŭ}} Ğ) := (F ^{\overline{}^{u}} G) .

We have to show that (F̆, ă) is exact and that ŭ is periodic.

Ad F̆ exact. Since Xă = X L ă = Xa is an isomorphism for X ∈ Ob C//N = Ob C, the transformation a is an isotransformation.

To show that F̆ is subexact, by Lemma 41, it suffices to show that given a morphism f in C//N, it has a weak cokernel that is preserved by F̆. By isomorphic substitution, we may assume that f = f′ L for some morphism f′ in C. Let (f′, g′, h′) be a 2-triangle in C; cf. Lemma 20. Since L is strictly exact, the 2-triangle (f, g′ L, h′ L) results. In particular, g′ L is a weak cokernel of f. Since F is subexact, g′F = g′ L F̆ is a weak cokernel of f′F = f′ L F̆ = fF̆.

Suppose given n ⩾ 0. We shall make use of the abbreviation \underline{F} = \underline{F^{+}(Δ̄_{n}^{#})} , etc. It remains to show that

(θ_{n} ⋆ \underline{F̆}) · \underline{ă} =^{!} \underline{F̆} ⋆ ϑ′_{n} .

Since \underline{L} = \underline{L^{+}(Δ̄_{n}^{#})} is 1-epimorphic by Lemmata 50 and 35, it suffices to show that

\underline{L} ⋆ ((θ_{n} ⋆ \underline{F̆}) · \underline{ă}) =^{!} \underline{L} ⋆ \underline{F̆} ⋆ ϑ′_{n} .

In fact,

\underline{L} ⋆ ((θ_{n} ⋆ \underline{F̆}) · \underline{ă}) = (\underline{L} ⋆ θ_{n} ⋆ \underline{F̆}) · (\underline{L} ⋆ \underline{ă})
=^{L ex.} (ϑ_{n} ⋆ \underline{L} ⋆ \underline{F̆}) · (\underline{L} ⋆ \underline{ă})
= (ϑ_{n} ⋆ \underline{F}) · \underline{a}
=^{(F, a) ex.} \underline{F} ⋆ ϑ′_{n}
= \underline{L} ⋆ \underline{F̆} ⋆ ϑ′_{n} .

Ad ŭ periodic. We have to show that

(T ⋆ ŭ) · b̆ =^{!} ă · (ŭ ⋆ T′)

as transformations from T ⋆ F̆ to Ğ ⋆ T′. By Remark 51, it suffices to show that

L ⋆ ((T ⋆ ŭ) · b̆) =^{!} L ⋆ (ă · (ŭ ⋆ T′)) .

In fact,

L ⋆ ((T ⋆ ŭ) · b̆) = (L ⋆ T ⋆ ŭ) · (L ⋆ b̆)
= (T ⋆ L ⋆ ŭ) · (L ⋆ b̆)
= (T ⋆ u) · b
=^{u per.} a · (u ⋆ T′)
= (L ⋆ ă) · (L ⋆ ŭ ⋆ T′)
= L ⋆ (ă · (ŭ ⋆ T′)) .

Ad (2). Welldefinedness of the functor L ⋆ (−) follows from L being strictly exact and strictly exact functors being stable under composition; cf. Proposition 36, Remark 3.

Keep the notation of the proof of (1). Given an exact functor (F, a) from C to C′, we infer from a = 1, using L ⋆ ă = a, that ă = 1.

A Some general assertions

This appendix serves as a tool kit consisting of known results and folklore lemmata. We do not claim originality.

A.1 Remarks on coretractions and retractions

Remark 39 Let A be a category. Suppose given X, Z in Ob A, and morphisms X ^{\overline{}^{i}} Z ^{\overline{}^{p}} X such that ip = 1_{X} . Suppose given Y, W in Ob A, and morphisms Y ^{\overline{}^{j}} W ^{\overline{}^{q}} Y such that jq = 1_{Y} . Suppose given X ^{\overline{}^{u}} Y in A. Let Z ^{\overline{}^{v}} W be defined by v := puj. Then vq = pu and iv = uj.

\xymatrix{Z \ar[r]^{p} \ar[d]_{v} & X \ar[r]^{i} \ar[d]_{u} & Z \ar[d]^{v} \\ W \ar[r]^{q} & Y \ar[r]^{j} & W}

Proof. We have vq = pujq = pu and iv = ipuj = uj.

Remark 40 Let A be a category. Suppose given Z, X, Z′, W, Y, W′ ∈ Ob A. Suppose given morphisms X ^{\overline{}^{i}} Z ^{\overline{}^{p}} X such that ip = 1_{X} . Suppose given morphisms X ^{\overline{}^{i′}} Z′ ^{\overline{}^{p′}} X such that i′p′ = 1_{X} . Suppose given morphisms Y ^{\overline{}^{j}} W ^{\overline{}^{q}} Y such that jq = 1_{Y} . Suppose given morphisms Y ^{\overline{}^{j′}} W′ ^{\overline{}^{q′}} Y such that j′q′ = 1_{Y} . Suppose given Z ^{\overline{}^{v}} W and Z′ ^{\overline{}^{v′}} W′ such that pi′v′ = vqj′. Then there exists a unique morphism X ^{\overline{}^{u}} Y in A such that vq = pu and i′v′ = uj′.

\xymatrix{Z \ar[r]^{p} \ar[d]_{v} & X \ar[r]^{i′} \ar[d]_{u} & Z′ \ar[d]^{v′} \\ W \ar[r]^{q} & Y \ar[r]^{j′} & W′}

If v and v′ are isomorphisms, so is u.

Proof. Uniqueness follows from p being epic and j′ being monic. For existence, we let u := ivq = i′v′q′, the latter equality holding because of pivqj′ = pipi′v′ = pi′v′ = vqj′ = vqj′q′j′ = pi′v′q′j′, using p epic and j′ monic. Then pu = pi′v′q′ = vqj′q′ = vq and uj′ = ivqj′ = ipi′v′ = i′v′. If v and v′ are isomorphisms, then let u′ := jv^{−}p = j′v^{′−}p′ to get uu′ = ivqj′v^{′−}p′ = ipi′v′v^{′−}p′ = 1 and u′u = jv^{−}p i′v′q′ = jv^{−}vqj′q′ = 1, so that u′ = u^{−}. In particular, u is an isomorphism.

A.2 Two lemmata on subexact functors

Suppose given weakly abelian categories A and A′; cf. e.g. [8, Def. A.26.(3)]. Suppose given an additive functor F : A \overline{}A′. Recall that F is called subexact if the induced functor F̂ : Â \overline{}Â′ on the Freyd categories is exact; cf. [8, §1.2.1.3].

Lemma 41 The following assertions (1, 2, 3, 3^{∘}, 4, 4^{∘}) are equivalent.

(1) The functor F is subexact.

(2) The functor F preserves weak kernels and weak cokernels.

(3) The functor F preserves weak kernels.

(3^{∘}) The functor F preserves weak cokernels.

(4) For each morphism X ^{\overline{}^{t}} Y in A, there exists a weak kernel W ^{\overline{}^{w}} X such that wF is a weak kernel of tF.

(4^{∘}) For each morphism X ^{\overline{}^{t}} Y in A, there exists a weak cokernel Y ^{\overline{}^{w′}} W′ such that w′F is a weak cokernel of tF.

Proof. Ad (1) ⇒ (4). Suppose given a morphism X ^{\overline{}^{t}} Y in A. Let K [figure: i] X be a kernel of t in Â. Choose

A ^{\overline{}^{b}} K with A ∈ Ob A. Since F̂ is exact, AF̂ ^{\overline{}^{(bi)F̂}} XF̂ ^{\overline{}^{tF̂}} Y F̂ is exact at XF̂. So (bi)F̂ = (bi)F is a weak kernel of tF̂ = tF in A′.

Ad (4) ⇒ (3). Given a morphism X ^{\overline{}^{t}} Y in A and a weak kernel W ^{\overline{}^{w}} X, a morphism V ^{\overline{}^{v}} X is a weak kernel of t if and only if both (w factors over v) and (v factors over w). So if wF is a weak kernel of tF, so is vF. Consequently, if F preserves a single weak kernel of t, it preserves all of them.

Ad (3) ⇒ (2). This follows by [8, Rem. A.27].

Ad (2) ⇒ (1). Using duality and uniqueness of the kernel up to isomorphism, it suffices to show that F̂ maps a chosen kernel of a given morphism to a kernel of its image under F̂. Since F preserves weak kernels, this follows by construction of a kernel; cf. e.g. [8, §A.6.3, item (1) before Rem. A.27].

Lemma 42 Suppose that C ^{\overline{}^{F}} C′ is subexact. Suppose given a functor C ^{\overline{}^{G}} C′.

(1) If G ⊣ F, then G is subexact.

(1^{∘}) If F ⊣ G, then G is subexact.

Proof. Ad (1). As an adjoint functor between additive categories, G is additive.

Let 1 ^{\overline{}^{ε}} GF be a unit and F G ^{\overline{}^{η}} 1 a counit of the adjunction G ⊣ F.

By Lemma 41, it suffices to show that G preserves weak cokernels. Suppose given X′ ^{\overline{}^{u}} X ^{\overline{}^{v}} X′′ such that v is a weak cokernel of u. We have to show that Gv is a weak cokernel of Gu. Suppose given t : XG \overline{}T such that uG · t = 0. Then

u · Xε · tF = X′ε · uGF · tF = X′ε · (uG · t)F = 0 .

Since v is a weak cokernel of u, we obtain a morphism s : X′′ \overline{}T F such that v · s = Xε · tF. Then

vG · (sG · T η) = XεG · tF G · T η = XεG · XGη · t = t .

A.3 Karoubi hull

The construction of the Karoubi hull is due to Karoubi; cf. [6, III.II].

Suppose given an additive category A. The Karoubi hull Ã has

Ob Ã := { (A, e) : A ∈ Ob A, e ∈ _{A}(A, A) with e^{2} = e }

and, given (A, e), (B, f) ∈ Ob Ã,

A((A, e), (B, f)) := { u ∈ _{A}(A, B) : e · u · f = u } .

Then Ã is an additive category, in which all idempotents are split.

Composition is inherited from A. We have a full and faithful additive functor

\begin{matrix}A & ^{\overline{}^{K}} & Ã \\ (X ^{\overline{}^{u}} Y) & \overline{} & ((X, 1) ^{\overline{}^{u}} (Y, 1)) , \end{matrix}

which we often consider as an inclusion of a full subcategory.

Suppose given an additive category B in which all idempotents are split.

Remark 43 Write A, B _{add} for the category of additive functors and transformations between such from A to B. The induced functor A, B ^{\overline{}^{K ⋆ (−)}} Ã, B restricts to a strictly dense equivalence

A, B _{add} ^{\overline{}^{K ⋆ (−)}} Ã, B _{add} .

Lemma 44 Suppose given an additive functor A ^{\overline{}^{I}} A′ to an additive category A′ in which all idempotents split. By Remark 43, we obtain a functor J : Ã \overline{}A′, unique up to isomorphism, such that the following triangle of functors commutes.

\xymatrix{A \ar[d]_{K} \ar[r]^{I} & A′ \\ Ã \ar[ur]_{J}}

If I is full and faithful, and if every object of A′ is a direct summand of an object in the image of I, then J is an equivalence.

By abuse of notation, in the situation of Lemma 44, we also write Ã = A′ and consider I to be an inclusion of a full subcategory.

A.4 Multiplicative systems

The construction of the quotient category of a Verdier triangulated category is due to Verdier; cf. [13].

Suppose given a category C.

Definition 45 A set M of morphisms of C is called a multiplicative system in C if (Fr 1-4) are satisfied. An element of M is called an M-isomorphism and denoted by X ⟹ Y.

(Fr 1) Each identity in C is an M-isomorphism.

(Fr 2) Suppose given X ^{\overline{}^{f}} Y ^{\overline{}^{g}} Z ^{\overline{}^{h}} W in C such that fg and gh are M-isomorphisms.

Then f, g, h and f · g · h are M-isomorphisms.

(Fr 3) Suppose given X [figure: f; g] Y in C. There exists an M-isomorphism s such that sf = sg if and only if there

exists an M-isomorphism t such that ft = gt.

(Fr 4) Given \xymatrix{X \ar@{=>}[r] \ar[d] & Y \\ X′} in C, there exists a completion to a commutative quadrangle \xymatrix{X \ar@{=>}[r] \ar[d] & Y \ar[d] \\ X′ \ar@{=>}[r] & Y′}.

Dually, given \xymatrix{ & Y \ar[d] \\ X′ \ar@{=>}[r] & Y′} in C, there exists a completion to a commutative quadrangle \xymatrix{X \ar@{=>}[r] \ar[d] & Y \ar[d] \\ X′ \ar@{=>}[r] & Y′}.

Cf. [13, §2, no. 1].

Suppose given a multiplicative system M in C. Using (Fr 2), we note that in the first assertion of (Fr 4), if X \overline{}X′ is an M-isomorphism, then there exists a commutative completion with Y \overline{}Y′ being an M-isomorphism. And dually.

The category C_{M} , called localisation of C at M, is defined as follows. Let Ob C_{M} := Ob C. A morphism from X to Y is a double fraction, which is an equivalence class of diagrams of the following form.

\xymatrix{ & X′ \ar@{=>}[dl]_{s} \ar[r]^{f} & Y′ \\ X &  &  & Y \ar@{=>}[ul]_{t}}

The diagrams (s, f, t) and (s′s, s′ft′, tt′) are declared to be elementarily equivalent, provided s′ and t′ are M-isomorphisms. To form double fractions, we take the equivalence relation generated by elementary equivalence.

The equivalence class of the diagram (s, f, t) is written s\ f/t. So s\ f/t = s̃\ f̃/t̃ if and only if there exist M-isomorphisms u, ũ, v, ṽ such that us = ũs̃ and tv = t̃ṽ and ufv = ũf̃ṽ.

[figure]

Write f/t := 1\ f/t, called a right fraction, and s\f := s\ f/1, called a left fraction. Using (Fr 4), each morphism in C_{M} can be represented both by a left fraction and by a right fraction. Given right fractions f/t and f̃/t̃, they are equal if there exist M-isomorphisms u, v and ṽ such that ufv = uf̃ṽ and tv = t̃ṽ. By (Fr 3), this implies the existence of M-isomorphisms v, ṽ and u′ such that f(vu′) = f̃(ṽu′) and t(vu′) = t̃(ṽu′). Dually for left fractions.

So double fractions are a self-dual way to represent morphisms in C_{M} . Right or left fractions are more efficient in many arguments.

The composite of two double fractions s\ f/t and u\ g/v is defined, using (Fr 4) for the commutative diagram

[figure: s; f; g′; X; Y; t; Z ,; v; f′; x; u; g; y]

to be equivalently s\ fg′/vy or xs\ f′g/v. By (Fr 4, 2, 3), this definition is independent of the chosen completion with g′ and y, and, likewise, of the chosen completion with x and f′.

Independence of the choice of the representative s\ f/t is seen considering an elementary equivalence and using (Fr 4, 2), thus obtaining an elementary equivalence of the two possible representatives of the composite. Likewise independence of the representative u\ g/v.

Associativity follows using right fractions and a commutative diagram constructed by means of (Fr 4),

[figure: X; Y; Z; W .]

Given f ∈ Mor C, we also write 1\ f/1 =: f in C_{M}, by abuse of notation. Note that in C_{M} , we have s\ f/t = s^{−}f t^{−}.

Remark 46 A double fraction s\ f/t represents an isomorphism in C_{M} if and only if f is an M-isomorphism.

Sketch. First, using (Fr 2), we reduce to the case of a right fraction g/u. For a right fraction in turn, the assertion follows applying (Fr 2) to an associativity diagram as above.

Remark 47 Given X [figure: f; g] Y in C, we have f = g in C_{M} if and only if there exists an M-isomorphism t such

that ft = gt in C, or, equivalently, if and only if there exists an M-isomorphism s such that sf = sg in C.

Remark 48 We have a functor C ^{\overline{}^{L}} C_{M} , f \overline{}1\ f/1 = f, called localisation functor.

Given a category T , we let C, T _{M} be the full subcategory of C, T consisting of functors that send all M-isomorphisms in C to isomorphisms in T . The induced functor

C , T _{M} ^{\overline{}^{L ⋆ (−)}} C_{M} , T

is a strictly dense equivalence, i.e. it is surjective on objects, full and faithful.

Sketch. Given a functor F ∈ Ob C , T _{M} , we may define F̆ on C_{M} by letting XF̆ := XF for X ∈ Ob C_{M} = Ob C and by (s\ f/t)F̆ := (sF)^{−} · (fF) · (tF)^{−}. Then L ⋆ F̆ = F.

Given a transformation (F ^{\overline{}^{u}} G) ∈ Mor C , T _{M} , we may define F̆ ^{\overline{}^{ŭ}} Ğ by setting Xŭ := Xu for X ∈ Ob C_{M} = Ob C. Then L ⋆ ŭ = u.

Lemma 49 Given n ⩾ 0, the functor

C(Δ̇_{n}) ^{\overline{}^{L(Δ̇_{n})}} C_{M}(Δ̇_{n}) ,

given by pointwise application of L, is dense.

Proof. We may assume n ⩾ 1.

Suppose given X ∈ Ob C_{M}(Δ̇_{n}). To prove that for i ∈ [1, n − 1] there exists an X′ ∈ Ob C_{M}(Δ̇_{n}) isomorphic to X such that X′_{j} ^{\overline{}^{x′}} X′_{j+1} is in the image of L for j ∈ [1, i − 1], we proceed by induction on i ⩾ 1. Suppose the

assertion to be true for i. Let us prove the assertion for i + 1. Write X′_{i} ^{\overline{}^{x′}} X′_{i+1} as a right fraction f/s. If i = n − 1, we replace X′_{i+1} by the target of f, and f/s by f. If i ⩽ n − 2, we write X′_{i+1} ^{\overline{}^{x′}} X′_{i+2} as a right fraction g/u and construct the following commutative diagram using (Fr 4).

[figure: X′′_{i+1}; g′; s′; X′_{i}; f; X′_{i+1}; s; g; X′_{i+2}; u]

We replacing the object X′_{i+1} by X′′_{i+1} , the morphism f/s by f and the morphism g/u by g′/us′.

In both cases, we obtain a diagram isomorphic to X′ that conincides with X′ on [1, i] and whose morphism from i to i + 1 is in the image of L.

Lemma 50 Given n ⩾ 0, the functor

C(Δ̇_{n}) ^{\overline{}^{L(Δ̇_{n})}} C_{M}(Δ̇_{n})

is 1-epimorphic.

Proof. We shall apply [8, Lem. A.35]. By Lemma 49, L(Δ̇_{n}) is dense.

Suppose given X, Y ∈ Ob C(Δ̇_{n}) and a morphism X L(Δ̇_{n}) ^{\overline{}^{g}} Y L(Δ̇_{n}) in C_{M}(Δ̇_{n}). Let g_{i} be represented by a right fraction f_{i}/s_{i} for i ∈ [1, n].

We claim that for i ∈ [1, n], we can find representatives f′_{j}/s′_{j} for j ∈ [1, i] such that there exist h_{j} with s′_{j}h_{j} = ys′_{j+1} and f′_{j}h_{j} = xf′_{j+1} in C for j ∈ [1, i−1]. Let f′_{1} := f_{1} and s′_{1} := s_{1} . Proceeding by induction on i, we have to write the right fraction f_{i+1}/s_{i+1} suitably as f′_{i+1}/s′_{i+1} . First of all, by (Fr 4), we find an M-isomorphism σ and a morphism ξ such that yσ = s′_{i}ξ in C. We have f′_{i}ξσ^{−} = xf_{i+1}s_{i+1}^{−} in C_{M} . Using (Fr 4) and (Fr 2), we find M-isomorphisms s′ and σ′ such that σs′ = s_{i+1}σ′ in C. Hence

f′_{i}ξs′ = f′_{i}ξs′s^{′−}σ^{−}s_{i+1}σ′ = f′_{i}ξσ^{−}s_{i+1}σ′ = xf_{i+1}s_{i+1}^{−}s_{i+1}σ′ = xf_{i+1}σ′

in C_{M} . Composing with a further M-isomorphism, we may assume that f′_{i}ξs′ = xf_{i+1}σ′ in C; cf. Remark 47. We take h_{i} := ξs′ and s′_{i+1} := s_{i+1}σ′ and f′_{i+1} := f_{i+1}σ′.

[figure: X_{i}; x; f′_{i}; X_{i+1}; f_{i+1}; ξ; s′; σ′; Y_{i}; s′_{i}; y; Y_{i+1}; σ; s_{i+1}]

This proves the claim, in particular for i = n.

[figure: X_{1}; x; f′_{1}; X_{2}; f′_{2}; ⋯; X_{n−1}; x; f′_{n−1}; X_{n}; f′_{n}; h_{1}; ⋯; h_{n−1}; Y_{1}; y; s′_{1}; Y_{2}; s′_{2}; ⋯; Y_{n−1}; y; s′_{n−1}; Y_{n}; s′_{n}]

Condition (C) of loc. cit. is satisfied letting the epizigzag have length 0, letting the monozigzag be the single backwards diagram morphism consisting of the morphisms s′_{i}, and letting the required diagram morphism in the image of L(Δ̇_{n}) consist of the morphisms f′_{i}.

Remark 51 Suppose the category C to be additive.

(1) An object X is isomorphic to 0 in C_{M} if and only if X ⟹ 0, or, equivalently, if and only if 0 ⟹ X.

(2) The category C_{M} is additive, and the functor L : C \overline{}C_{M} is additive.

(3) Given an additive category T , the strictly dense equivalence

C , T _{M} ^{\overline{}^{L ⋆ (−)}} C_{M} , T

restricts to a strictly dense equivalence from the category of additive functors from C_{M} to T to the category of additive functors from C to T that sends all M-isomorphisms to isomorphisms, written

C , T _{add, M} ^{\overline{}^{L ⋆ (−)}} C_{M} , T _{add} .

Sketch. Ad (1). If X is isomorphic to 0, then X ⟹ X′ ⟸ 0 ; cf. Remark 46. By (Fr 4), we conclude that 0 ⟹ X. Ad (2). Given X, Y ∈ Ob C, the direct sum X ⊕ Y, together with X ^{\overline{}^{(^{1 0} )}} X ⊕ Y and Y ^{\overline{}^{(^{0 1} )}} X ⊕ Y, remains a coproduct in C_{M} . For existence of an induced morphism from the coproduct, we use (Fr 4, 2) to produce a common denominator of two right fractions. To prove uniqueness of the induced morphism, we suppose given (\begin{matrix}f \\ g \end{matrix} )/s and (\begin{matrix}f′ \\ g′ \end{matrix} )/s, without loss of generality with common denominator, such that f/s = (^{1 0}) · (\begin{matrix}f \\ g \end{matrix} )/s = (^{1 0}) · (\begin{matrix}f′ \\ g′ \end{matrix} )/s = f′/s and g/s = (^{0 1}) · (\begin{matrix}f \\ g \end{matrix} )/s = (^{0 1}) · (\begin{matrix}f′ \\ g′ \end{matrix} )/s = g′/s in C_{M} . So there exists an M-isomorphism u such that fu = f′u, and an M-isomorphism v such that gv = g′v, both in C. By (Fr 4, 2), we obtain a common M-isomorphism w such that fw = f′w and

gw = g′w in C. Hence (\begin{matrix}f \\ g \end{matrix} ) w = (\begin{matrix}f′ \\ g′ \end{matrix} ) w in C. Therefore (\begin{matrix}f \\ g \end{matrix} )/s = (\begin{matrix}f′ \\ g′ \end{matrix} )/s in C_{M} .

Moreover, the automorphism (\begin{matrix}1 & 0 \\ 1 & 1 \end{matrix}) of X ⊕ X remains an automorphism in C_{M} . Ad (3). Since L is additive, L ⋆ (−) sends additive functors to additive functors. Conversely, given an additive functor F : C \overline{}T , the functor F̆ as constructed in the proof of Remark 48 is additive.

References

[1] Balmer, P., Schlichting, M., Idempotent completion of triangulated categories, J. Alg. 236 (2), 819–834, 2001.

[2] Bernstein, J.; Beilinson, A.A.; Deligne, P., Faisceaux pervers, Astérisque 100, 1982.

[3] Happel, D., Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, LMS LN 119, 1988.

[4] Heller, A., Stable homotopy categories, Bull. Am. Math. Soc. 74, 28–63, 1968.

[5] Keller, B., Vossieck, D., Sous les catégories dérivées, C. R. Acad. Sci. Paris, Série I, 225–228, 1987.

[6] Karoubi, M., K-théorie, Conférences au séminaire d’été de Montréal No. 36, Presses de Montréal, 1969.

[7] Künzer, M., On derived categories, diploma thesis, 1996.

[8] Künzer, M., Heller triangulated categories, Homol. Homot. Appl. 9 (2), 233–320, 2007.

[9] Künzer, M., Nonisomorphic Verdier octahedra on the same base, preprint, arxiv, 0708.0151, 2007.

[10] Maltsiniotis, G., Catégories triangulées supérieures, Pré-preprint,

    www.math.jussieu.fr/~maltsin/ps/triansup.ps, 2005.

[11] Margolis, H.R., Spectra and the Steenrod algebra, North-Holland, 1983.

[12] Rickard, J., Derived categories and stable equivalence, J. Pure Appl. Alg. 61, 303–317, 1989.

[13] Verdier, J.L., Catégories Derivées, published in SGA 4 1/2, SLN 569, 262–311, 1977 (written 1963).

Matthias Künzer University of Stuttgart Fachbereich Mathematik

Pfaffenwaldring 57 D-70569 Stuttgart

kuenzer@mathematik.uni-stuttgart.de

w5.mathematik.uni-stuttgart.de/fachbereich/Kuenzer/