On good morphisms of exact triangles
aa r X i v : . [ m a t h . A T ] S e p ON GOOD MORPHISMS OF EXACT TRIANGLES
J. DANIEL CHRISTENSEN AND MARTIN FRANKLAND
Abstract.
In a triangulated category, cofibre fill-ins always exist. Neeman showed that there isalways at least one “good” fill-in, i.e., one whose mapping cone is exact. Verdier constructed a fill-inof a particular form in his proof of the 4 × Contents
1. Introduction 12. Good morphisms 33. Verdier good morphisms 54. Obstruction theory 75. Inclusions and projections 106. Homotopy cartesian squares 147. Maps of triangles with one or two zero components 228. Maps between rotations of a triangle 26References 291.
Introduction
In this paper, we study various classes of maps between triangles in a triangulated category T ,and use our results to prove lifting criteria for commutative squares, a pasting lemma for homotopycartesian squares, and a variety of results comparing these classes of maps.In order to explain our results, we make a few definitions. A map of triangles is a commutativediagram X u / / f (cid:15) (cid:15) Y g (cid:15) (cid:15) v / / Z h (cid:15) (cid:15) w / / Σ X Σ f (cid:15) (cid:15) X ′ u ′ / / Y ′ v ′ / / Z ′ w ′ / / Σ X ′ in which the rows are (exact) triangles, i.e., those specified by the triangulation. We often write( f, g, h ) for such a map.Following [Nee91], we say that a map ( f, g, h ) of triangles is middling good if it can be extendedto a 4 × good if its mapping cone (Definition 2.5) is exact.Neeman showed that every good map of triangles is middling good, but that there exist maps whichare middling good but not good, as well as maps which are not middling good.If f and g are given, a fill-in is a map h making the rest of the diagram commute. The fill-in is (middling) good if the resulting map ( f, g, h ) of triangles is. Date : September 2, 2020.2020
Mathematics Subject Classification.
Primary 18G80.
Key words and phrases. triangulated category, exact triangle, cofibre sequence, octahedral axiom, enhancement.
In this paper, we introduce a new condition: a map ( f, g, h ) of triangles is
Verdier good if h is constructed as in Verdier’s proof that middling good fill-ins exist; see Definition 3.1. As aresult, every Verdier good morphism is middling good. Lemma 3.4 provides a more conceptualcharacterization of Verdier good maps that was suggested to us by Haynes Miller.A natural question is whether either of good or Verdier good implies the other. While we do notanswer this question in general, we show that it holds in a number of special situations. For example,for maps of triangles with at most one non-zero component, we show that being good, Verdier good,middling good, and nullhomotopic are equivalent, and also show that these are equivalent to a certainToda bracket containing zero (see Proposition 7.13).The case where a map of triangles has at most two non-zero components is more interesting, andleads to our new lifting criteria. For maps of the form (0 , g, h ) and ( f, , h ), we show that beinggood, Verdier good and nullhomotopic are equivalent (Proposition 7.4). Using this, we prove thefollowing lifting criteria: Theorem (Corollary 7.7) . Given a solid arrow commutative square X u / / f (cid:15) (cid:15) Y g (cid:15) (cid:15) k } } ⑤ ⑤ ⑤ ⑤ X ′ u ′ / / Y ′ , choose extensions of the rows to cofibre sequences with cofibres Z and Z ′ , respectively. The followingare equivalent.(1) There exists a lift k : Y → X ′ .(2) The map Z → Z ′ is a fill-in and the map ( f, g, of triangles is good.(3) The map Z → Z ′ is a fill-in and the map (0 , f, g ) of rotated triangles is Verdier good. These criteria can be used to define obstruction classes as subsets of [
Z, Z ′ ], which contain zeroif and only if the lift exists. This is described after Theorem 4.3 and Corollary 7.7, and is relatedto the work in [CDI04].The rotation is needed in last case of the theorem because Proposition 7.4 made no claimsabout maps of the form ( f, g, f, g, ∞ -category; see Example 7.10.Continuing the theme mentioned earlier, we give other classes of maps of triangles for which good-ness and Verdier goodness agree. Proposition 5.1 shows that this holds for certain split inclusions orsplit projections. Proposition 3.7 shows that when the domain or codomain triangle is contractible,every map is both good and Verdier good. Goodness is invariant under chain homotopy, but we donot know whether the same is true for Verdier goodness. In fact, we do not know whether everynullhomotopic map is Verdier good.Proposition 8.1 studies certain maps from a triangle to its rotation, and again shows that goodnessand Verdier goodness agree on this class. We use this to prove Corollary 8.4, which says: there existmaps of triangles which are not middling good; there exist maps of triangles which are middling goodbut neither good nor Verdier good; and middling goodness is not invariant under chain homotopy.We give explicit examples in the same section.The final family of maps we consider are maps of the form (1 , g, h ), where we have that X = X ′ .In Proposition 6.9, we show that such a map is good if and only if the center square is homotopycartesian (Definition 6.1) with a specific differential, reformulating a result of Neeman [Nee01]. Wealso characterize when the center square is homotopy cartesian. The main result of this section,Proposition 6.11, states that the pasting of two homotopy cartesian squares is again homotopy N GOOD MORPHISMS OF EXACT TRIANGLES 3 cartesian. While the statement does not mention good morphisms, the proof makes essential useof them. We then use this to strengthen a result of [Vak01] about a natural candidate triangleassociated to the pasting. We also characterize which maps of the form (1 , g, h ) are Verdier good,and show that in this case good implies Verdier good.
Open questions. (1) Is Verdier goodness invariant under chain homotopy? In particular, is every nullhomotopicmap Verdier good?(2) Is Verdier goodness invariant under rotation?(3) Do either of good or Verdier good imply the other? In particular, given f and g , is there afill-in h that is simultaneously good and Verdier good? Organization.
We begin in Section 2 by giving background on maps of triangles and defining thenotion of a good map of triangles. In Section 3, we introduce the notion of a Verdier good map oftriangles, give a characterization due to Haynes Miller, and prove some first results about Verdiergood maps. In Section 4, we give our first lifting criterion, in terms of good fill-ins. The remainingsections are organized by the family of maps that they study. In Section 5, we study various splitinclusions and split projections, and show that every map of triangles is a composite of two mapsthat are good and Verdier good. We study maps of the form (1 , g, h ) as well as homotopy cartesiansquares in Section 6. This section contains our results about pasting such squares, and also hasexamples illustrating the subtlety in the choice of differential. In Section 7 we focus on maps thathave at most two non-zero components. Our final section, Section 8, deals with certain maps froma triangle to its rotation, which allows us to produce a number of interesting examples.
Conventions.
We work in a fixed triangulated category T throughout, with the self-equivalencedenoted Σ : T ≃ −→ T . We suppress the natural isomorphisms ΣΣ − ∼ = 1 and Σ − Σ ∼ = 1 from thenotation. A candidate triangle is a diagram in T of the form X u / / Y v / / Z w / / Σ X satisfying vu = 0, wv = 0, and (Σ u ) w = 0 [Nee01, Definition 1.1.1]. A candidate triangle thatbelongs to the class specified by the triangulated structure is often called a distinguished triangle ;we will say exact triangle , or simply triangle for short. Acknowledgments.
We thank Haynes Miller, Amnon Neeman, and Marius Thaule for helpfuldiscussions.We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada(NSERC). Cette recherche a ´et´e financ´ee par le Conseil de recherches en sciences naturelles et eng´enie du Canada (CRSNG). RGPIN-2016-04648 (Christensen), RGPIN-2019-06082 (Frankland).2.
Good morphisms
In this section, we provide some background on the notion of a “good morphism” between trian-gles, which was first introduced by Neeman in [Nee91].
Definition 2.1. A map of candidate triangles is a commutative diagram X u / / f (cid:15) (cid:15) Y g (cid:15) (cid:15) v / / Z h (cid:15) (cid:15) w / / Σ X Σ f (cid:15) (cid:15) X ′ u ′ / / Y ′ v ′ / / Z ′ w ′ / / Σ X ′ (2.2)in which the rows are candidate triangles. We often write ( f, g, h ) for such a map. We will referto (2.2) many times in this paper. J. DANIEL CHRISTENSEN AND MARTIN FRANKLAND
Definition 2.3.
A map of triangles as in (2.2) is middling good if it extends to a 4 × X f (cid:15) (cid:15) u / / Y g (cid:15) (cid:15) v / / Z h (cid:15) (cid:15) w / / Σ X Σ f (cid:15) (cid:15) X ′ f ′ (cid:15) (cid:15) u ′ / / Y ′ g ′ (cid:15) (cid:15) v ′ / / Z ′ h ′ (cid:15) (cid:15) w ′ / / Σ X ′ Σ f ′ (cid:15) (cid:15) X ′′ f ′′ (cid:15) (cid:15) u ′′ / / Y ′′ g ′′ (cid:15) (cid:15) v ′′ / / Z ′′ -1 h ′′ (cid:15) (cid:15) w ′′ / / Σ X ′′ Σ f ′′ (cid:15) (cid:15) Σ X Σ u / / Σ Y Σ v / / Σ Z Σ w / / Σ X where the first three rows and columns are exact, and the bottom-right square anticommutes asindicated. Remark . Verdier showed that given a commutative diagram X u / / f (cid:15) (cid:15) Y g (cid:15) (cid:15) v / / Z w / / Σ X Σ f (cid:15) (cid:15) X ′ u ′ / / Y ′ v ′ / / Z ′ w ′ / / Σ X ′ in which the rows are exact, there exists a fill-in h : Z → Z ′ making ( f, g, h ) into a middling goodmap of triangles. We recall the proof and characterize Verdier’s extensions in Section 3. Definition 2.5.
The mapping cone of a map of triangles (2.2) is the sequence X ′ ⊕ Y h u ′ g − v i / / Y ′ ⊕ Z h v ′ h − w i / / Z ′ ⊕ Σ X h w ′ Σ f − Σ u i / / Σ X ′ ⊕ Σ Y, where we think of elements of direct sums as column vectors. The map (2.2) is good if the mappingcone is a triangle.As an example, the zero map of triangles is always good, since triangles are closed under directsum.In [Nee91, Theorem 2.3], Neeman shows that every good map of triangles is middling good. Healso shows that there are maps which are middling good but not good. We will prove a slightlystronger claim in Corollary 8.4. Definition 2.6.
Let ( f, g, h ) be a map of candidate triangles as in (2.2) and let ( f ′ , g ′ , h ′ ) beanother map between the same candidate triangles. A chain homotopy from ( f, g, h ) to ( f ′ , g ′ , h ′ )consists of maps F : Y → X ′ , G : Z → Y ′ and H : Σ X → Z ′ such that f ′ − f = F u + Σ − ( w ′ H ), g ′ − g = Gv + u ′ F and h ′ − h = Hw + v ′ G , as illustrated in the diagram X (cid:15) (cid:15) (cid:15) (cid:15) u / / Y F } } ⑤⑤⑤⑤⑤⑤⑤⑤ (cid:15) (cid:15) (cid:15) (cid:15) v / / Z G ~ ~ ⑤⑤⑤⑤⑤⑤⑤⑤ (cid:15) (cid:15) (cid:15) (cid:15) w / / Σ X H | | ②②②②②②②②② (cid:15) (cid:15) (cid:15) (cid:15) X ′ u ′ / / Y ′ v ′ / / Z ′ w ′ / / Σ X ′ . When such a chain homotopy exists, we say that ( f, g, h ) is chain homotopic to ( f ′ , g ′ , h ′ ).A map of candidate triangles ( f, g, h ) is nullhomotopic if it is chain homotopic to the zero map(0 , , X → Y → Z → Σ X is contractible if its identity map (1 X , Y , Z )is nullhomotopic (which makes it automatically exact [Nee01, Proposition 1.3.8]). For example,every split triangle is contractible. N GOOD MORPHISMS OF EXACT TRIANGLES 5
Remark . Neeman observes that if ( f, g, h ) is chain homotopic to ( f ′ , g ′ , h ′ ), then they haveisomorphic mapping cones, so are either both good or both not good. In particular, any map oftriangles which is nullhomotopic is good. It is also immediate that being good is invariant underrotation of triangles. 3. Verdier good morphisms
In this section we introduce the notion of a Verdier good morphism of triangles, give a character-ization of Verdier good morphisms due to Haynes Miller, and give a simple situation in which goodmorphisms and Verdier good morphisms agree.
Definition 3.1.
A map of triangles as in (2.2) is
Verdier good if h can be constructed as inVerdier’s proof of the 4 × X u −→ Y g −→ Y ′ : X u / / Y g (cid:15) (cid:15) v / / Z α (cid:15) (cid:15) ✤✤✤ w / / Σ XX gu / / Y ′ g ′ (cid:15) (cid:15) e v / / A β (cid:15) (cid:15) ✤✤✤ e w / / Σ XY ′′ g ′′ (cid:15) (cid:15) Y ′′ γ (cid:15) (cid:15) ✤✤✤ Σ Y Σ v / / Σ Z, and an octahedron for the composite X f −→ X ′ u ′ −→ Y ′ : X f / / X ′ u ′ (cid:15) (cid:15) f ′ / / X ′′ α (cid:15) (cid:15) ✤✤✤ f ′′ / / Σ XX u ′ f = gu / / Y ′ v ′ (cid:15) (cid:15) e v / / A β (cid:15) (cid:15) ✤✤✤ e w / / Σ XZ ′ w ′ (cid:15) (cid:15) Z ′ γ (cid:15) (cid:15) ✤✤✤ Σ X ′ Σ f ′ / / Σ X ′′ , and that h : Z → Z ′ is given by h = β ◦ α . Remark . The property of h being a Verdier good fill-in does not depend on the choice of trianglesextending the five maps u , u ′ , f , g , and gu = u ′ f . These five triangles may be fixed, and hence weredrawn with solid arrows. Similarly, the notion of Verdier good morphism ( f, g, h ) is invariant underpre- and post-composing with isomorphisms of triangles. One can readily check that the identitymorphism of triangles is Verdier good (by taking α = 1 Z and β = 1 Z ), so it follows that everyisomorphism of triangles is Verdier good. The notion of good morphism is also invariant under pre-and post-composing with isomorphisms of triangles, and every isomorphism of triangles is good,since the mapping cone is contractible. (See [Nee91, §
1, Case 1].)
Remark . It is not clear whether being Verdier good is invariant under rotation of triangles.Since being good is invariant under rotation, our later results that show that being Verdier good is
J. DANIEL CHRISTENSEN AND MARTIN FRANKLAND equivalent to being good in certain situations provide mild evidence in favour. It is also not clearwhether the notion of Verdier goodness is self-dual. Given a map of triangles ( f, g, h ) as in (2.2),the dual map is (Σ f, h, g ), viewed as a map of triangles in T op from the bottom row to the toprow. The original map ( f, g, h ) is Verdier good if and only if the rotation (Σ g, Σ f, h ) of the dualmap is Verdier good in T op . In fact, Verdier goodness is self-dual if and only if Verdier goodness isinvariant under rotation of triangles.Verdier’s argument says that any commutative square admits a Verdier good fill-in. A Verdiergood morphism in particular extends to a 4 × Lemma 3.4.
A map of triangles as in (2.2) is Verdier good if and only if it extends to a × diagram X f (cid:15) (cid:15) u / / Y g (cid:15) (cid:15) v / / Z h (cid:15) (cid:15) w / / Σ X Σ f (cid:15) (cid:15) X ′ f ′ (cid:15) (cid:15) u ′ / / Y ′ g ′ (cid:15) (cid:15) v ′ / / Z ′ h ′ (cid:15) (cid:15) w ′ / / Σ X ′ Σ f ′ (cid:15) (cid:15) X ′′ f ′′ (cid:15) (cid:15) u ′′ / / ❴❴❴ Y ′′ g ′′ (cid:15) (cid:15) v ′′ / / ❴❴❴ Z ′′ -1 h ′′ (cid:15) (cid:15) w ′′ / / ❴❴❴ Σ X ′′ Σ f ′′ (cid:15) (cid:15) Σ X Σ u / / Σ Y Σ v / / Σ Z Σ w / / Σ X (i.e., it is middling good) and there is an object A together with maps satisfying the following threehexagonal diagrams: X gu = u ′ f / / Y ′ e v (cid:4) (cid:4) ✟✟✟✟✟✟✟✟ v ′ (cid:27) (cid:27) ✼✼✼✼✼✼✼✼ Z (cid:0)(cid:7)(cid:1)(cid:6)(cid:2)(cid:5)(cid:3)(cid:4) ✡✡✡✡ w D D ✡✡✡✡ α / / ❴❴❴❴❴ A (cid:0)(cid:7)(cid:1)(cid:6)(cid:2)(cid:5)(cid:3)(cid:4) ✹✹✹✹ e w Z Z ✹✹✹✹ β (cid:5) (cid:5) ✡ ✡ ✡ ✡ β / / ❴❴❴❴❴ Z ′ (cid:0)(cid:7)(cid:1)(cid:6)(cid:2)(cid:5)(cid:3)(cid:4) ✟✟✟✟ (Σ f ′ ) w ′ (cid:3) (cid:3) ✟✟✟✟ Y ′′ (cid:0)(cid:7)(cid:1)(cid:6)(cid:2)(cid:5)(cid:3)(cid:4) ✹✹✹ (Σ v ) g ′′ Y Y ✹✹✹✹✹ X ′′ , u ′′ o o ❴ ❴ ❴ ❴ α Z Z ✺ ✺ ✺ ✺ where each triangular shape is either commutative or exact (as indicated by the degree shifts), andeach of the three diameters composes to the corresponding vertical map in the × diagram; and Σ X Σ f z z ttttt Σ u $ $ ■■■■■ Σ X ′ Σ YA β z z t t t e w O O β $ $ ■■■ Z ′ w ′ O O h ′ $ $ ❏❏❏ Y ′′ v ′′ z z ✈ ✈ ✈ g ′′ O O Z ′′ Y ′ e v (cid:15) (cid:15) Y v (cid:15) (cid:15) g sssssss X ′ u ′ f f ▲▲▲▲▲▲▲ f ′ (cid:15) (cid:15) AZ α : : ✉✉✉✉ − X ′′ α e e ❑ ❑ ❑ ❑ Σ − Z ′′ , Σ − h ′′ c c ❍ ❍ ❍ Σ − w ′′ : : ✉✉✉ where all squares commute, except for the indicated square which anticommutes. N GOOD MORPHISMS OF EXACT TRIANGLES 7
Proof.
Let us recall Verdier’s argument. The two octahedra from Definition 3.1 produce maps X ′′ α −→ A β −→ Y ′′ , to which we apply the octahedral axiom: X ′′ α / / A β (cid:15) (cid:15) β / / Z ′ α (cid:15) (cid:15) ✤✤✤ γ / / Σ X ′′ X ′′ β α / / Y ′′ γ (cid:15) (cid:15) v ′′ / / ❴❴❴ Z ′′ β (cid:15) (cid:15) ✤✤✤ w ′′ / / ❴❴❴ Σ X ′′ Σ Z − Σ α (cid:15) (cid:15) Σ Z γ (cid:15) (cid:15) ✤✤✤ Σ A Σ β / / Σ Z ′ . A straightforward listing of all equations (and exactness conditions) shows that the three octahedraare equivalent to the three hexagonal diagrams plus two of the equations in the 4 × × × (cid:3) Definition 3.5.
Given two solid triangles X u / / Y v / / Z w / / Σ X θ t t ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ X ′ u ′ / / Y ′ v ′ / / Z ′ w ′ / / Σ X ′ , a map h : Z → Z ′ is called a lightning flash if it can be expressed as a composite h = v ′ θw forsome map θ : Σ X → Y ′ .The following is proved in [Nee91, Lemma 2.1] but stated differently. Lemma 3.6.
Let ( f, g, h ) be a Verdier good map of triangles and let v ′ θw : Z → Z ′ be a lightningflash. Then ( f, g, h + v ′ θw ) is also Verdier good. (cid:3) Proposition 3.7.
Every map of triangles whose source or target is contractible is good and Verdiergood.Proof.
Such a map of triangles is nullhomotopic, hence good, by Remark 2.7.Let ( f, g, h ) be the given map of triangles, and let h be a Verdier good fill-in for the square formedusing f and g . By [Nee91, Lemma 2.2] (or its dual in case the target is contractible), the difference h − h is a lightning flash. So by Lemma 3.6, the fill-in h is also Verdier good. (cid:3) Obstruction theory
In this section, we give a lifting criterion in terms of good fill-ins. Given a commutative square X f / / u (cid:15) (cid:15) X ′ u ′ (cid:15) (cid:15) Y g / / Y ′ , J. DANIEL CHRISTENSEN AND MARTIN FRANKLAND one can look for obstructions to the existence of a lift X f / / u (cid:15) (cid:15) X ′ u ′ (cid:15) (cid:15) Y g / / k > > ⑥⑥⑥⑥ Y ′ making both triangles commute. We’ll transpose this picture, to make it fit naturally with thenotation for chain homotopies: X u / / f (cid:15) (cid:15) Y g (cid:15) (cid:15) k } } ④ ④ ④ ④ X ′ u ′ / / Y ′ . A naive way to form an obstruction class is to extend the rows to triangles and consider thepossible fill-in maps X u / / f (cid:15) (cid:15) Y g (cid:15) (cid:15) v / / Z h (cid:15) (cid:15) ✤✤✤ w / / Σ X Σ f (cid:15) (cid:15) X ′ u ′ / / Y ′ v ′ / / Z ′ w ′ / / Σ X ′ (4.1)making both squares commute. The collection of such maps is a subset of [ Z, Z ′ ]. Note that if alift k : Y → X ′ exists, then v ′ g = v ′ u ′ k = 0 and (Σ f ) w = (Σ k )(Σ u ) w = 0, so the zero map Z → Z ′ is one of the fill-ins. One might hope that the converse holds, i.e., if the fill-in can be chosen to bezero, then the lift k exists, but the following example shows that this is not true in general. Example 4.2.
In the derived category of the integers D ( Z ), let A [ k ] denote the chain complexhaving the abelian group A concentrated in degree k . For an integer n ≥
2, let ǫ n ∈ Ext Z ( Z /n, Z ) ∼ = Z /n denote the canonical generator, sitting in a triangle Z [0] n / / Z [0] q n / / Z /n [0] ǫ n / / Z [1] , where q n : Z → Z /n denotes the quotient map. In the morphism of triangles Z q n / / (cid:15) (cid:15) Z /n ǫ n / / ǫ n (cid:15) (cid:15) k | | ③ ③ ③ ③ Z [1] (cid:15) (cid:15) − n / / Z [1] (cid:15) (cid:15) Z /n ǫ n / / Z [1] − n / / Z [1] − q n / / Z /n [1] , the fill-in map is zero, but no lift k exists.In [CDI04], an obstruction theory for such lifts is developed, working in a model category. In thecase of a stable model category, it does specialize to giving an obstruction lying in the group [ Z, Z ′ ],which is zero if and only if the lift exists. This is possible because the obstructions are producedusing a strictly commutative starting square. Fill-in maps produced in this way are better thangeneral fill-ins. We use this observation to give an obstruction class in any triangulated category. Theorem 4.3 (Lifting criterion) . Given a solid arrow commutative square X u / / f (cid:15) (cid:15) Y g (cid:15) (cid:15) k } } ⑤ ⑤ ⑤ ⑤ X ′ u ′ / / Y ′ , N GOOD MORPHISMS OF EXACT TRIANGLES 9 choose extensions of the rows to cofibre sequences as in (4.1) . Then a lift k exists if and only if thezero map Z → Z ′ is a good fill-in. Define the obstruction class to be the subset of [
Z, Z ′ ] consisting of the good fill-in maps.Then the theorem says that a lift k exists if and only if the obstruction class contains zero. Notethat [Nee91, Theorem 1.8] says that the obstruction class is non-empty.In Corollary 7.7 we extend this result to show that Verdier good maps can also be used to detectlifting. Proof.
If the lift k exists, then the triple ( f, g,
0) is a map of triangles. Moreover, it is chainhomotopic to the zero map, using k as the only non-zero component of the chain homotopy. Thezero map of triangles is always good, since the coproduct of triangles is a triangle. Therefore, ouroriginal map ( f, g,
0) is good, and so 0 is in the obstruction class.Conversely, suppose that ( f, g,
0) is good. After rotating, the mapping cone of this map isΣ − Z ′ ⊕ X h − Σ − w ′ − f u i / / X ′ ⊕ Y h − u ′ − g v i / / Y ′ ⊕ Z h − v ′ w i / / Z ′ ⊕ Σ X, which we are assuming is a triangle. Since the rightmost map is diagonal, this triangle must beisomorphic to the sum of the two original triangles (with the first rotated). That is, there is amatrix (cid:20) a bc d (cid:21) of maps making the following diagram commute:Σ − Z ′ ⊕ X h − Σ − w ′ − f u i / / X ′ ⊕ Y h − u ′ − g v i / / Y ′ ⊕ Z h − v ′ w i / / Z ′ ⊕ Σ X Σ − Z ′ ⊕ X h − Σ − w ′ u i / / X ′ ⊕ Y h − u ′ v i / / h a bc d i O O Y ′ ⊕ Z h − v ′ w i / / Z ′ ⊕ Σ X. (4.4)The left square implies that bu = − f and du = u. (4.5)The middle square implies that − u ′ b − gd = 0 . (4.6)Consider the following composite of morphisms of triangles: include the unprimed triangle intothe bottom row of (4.4), map upwards using (4.4), and then project onto the unprimed triangle.This gives a map of triangles X u / / Y v / / Z w / / Σ XX u / / O O Y v / / d O O Z w / / O O Σ X, O O which shows that d is an isomorphism.Now let k = − bd − . By (4.5), ku = − bd − u = − bu = f, and by (4.6) u ′ k = − u ′ bd − = g. Thus k is the required lift. (cid:3) Corollary 4.7.
In the above situation, if [ Z, Z ′ ] = 0 , then a lift always exists. Even this special case seems non-trivial. The proof relies on a tricky argument due to Neeman,combined with Theorem 4.3.
Proof.
As mentioned above, [Nee91, Theorem 1.8] shows that there is always at least one good fill-inmap Z → Z ′ . But if all such maps are zero, it follows that the zero map must be a good fill-in, soTheorem 4.3 applies. (cid:3) Example 4.8.
Consider the following lifting problem:0 / / (cid:15) (cid:15) Y g (cid:15) (cid:15) k } } ④ ④ ④ ④ X ′ u ′ / / Y ′ . In this case, there is a unique fill-in h = v ′ g , which is therefore good. So we see that g lifts through u ′ if and only if v ′ g = 0, which recovers the usual exactness property for mapping into a triangle.When we take Y ′ = 0 instead of X = 0, we get a dual result.5. Inclusions and projections
In this section, we give more situations in which good morphisms and Verdier good morphismsagree. The situations involve maps of triangles in which some of the component morphisms aresplit inclusions or split projections. We also show that every map of triangles is a composite of twomaps that are good and Verdier good.
Proposition 5.1. (1) For a map of triangles of the form X inc (cid:15) (cid:15) u / / Y inc (cid:15) (cid:15) v / / Z h (cid:15) (cid:15) w / / Σ X inc (cid:15) (cid:15) X ⊕ X ′′ [ u c u ′′ ] / / Y ⊕ Y ′′ [ v ′ v ′ ] / / Z ′ (cid:20) w ′ w ′ (cid:21) / / Σ X ⊕ Σ X ′′ the following are equivalent:(a) The map is good.(b) The map is Verdier good.(c) The fill-in h : Z → Z ′ makes the following diagram exact: X ′′ [ vcu ′′ ] / / Z ⊕ Y ′′ [ h v ′ ] / / Z ′ w ′ / / Σ X ′′ . When the matrix entry c is zero (so that the second row is isomorphic to a direct sum oftriangles), these three conditions hold for any h .(2) Similarly, a map of triangles of the form X ⊕ X ′′ proj (cid:15) (cid:15) [ u c u ′′ ] / / Y ⊕ Y ′′ proj (cid:15) (cid:15) [ v ′ v ′ ] / / Z ′ h (cid:15) (cid:15) (cid:20) w ′ w ′ (cid:21) / / Σ X ⊕ Σ X ′′ proj (cid:15) (cid:15) X ′′ u ′′ / / Y ′′ v ′′ / / Z ′′ w ′′ / / Σ X ′′ is good if and only if it is Verdier good if and only if the fill-in h : Z ′ → Z ′′ makes thefollowing diagram exact: Z ′ h w ′ h i / / Σ X ⊕ Z ′′ [ Σ u (Σ c ) w ′′ ] / / Σ Y Σ v ′ / / Σ Z ′ , N GOOD MORPHISMS OF EXACT TRIANGLES 11 and these conditions hold when c is zero.Proof. We will prove the statement involving inclusions. The statement involving projections isproved similarly.We begin by showing that conditions (1a) and (1c) are equivalent. A straightforward computationwith shears shows that the mapping cone( X ⊕ X ′′ ) ⊕ Y (cid:20) u c u ′′
00 0 − v (cid:21) / / ( Y ⊕ Y ′′ ) ⊕ Z (cid:20) v ′ v ′ h − w (cid:21) / / Z ′ ⊕ Σ X w ′ w ′ − Σ u / / (Σ X ⊕ Σ X ′′ ) ⊕ Σ Y is isomorphic to the direct sum of the three (candidate) triangles X / / / / Σ X / / Σ XY / / Y / / / / Σ YX ′′ h u ′′ vc i / / Y ′′ ⊕ Z [ v ′ h ] / / Z ′ w ′ / / Σ X ′′ . (5.2)Therefore, the mapping cone is exact if and only if the bottom row of (5.2) is exact.We next prove that (1b) implies (1c). An octahedron for the composite X u −→ Y inc −−→ Y ⊕ Y ′′ isof the form: X u / / Y inc (cid:15) (cid:15) v / / Z α = [ ϕa ] (cid:15) (cid:15) ✤✤✤ w / / Σ XX [ u ] / / Y ⊕ Y ′′ proj (cid:15) (cid:15) [ v
00 1 ] / / Z ⊕ Y ′′ β =[ b (cid:15) (cid:15) ✤✤✤ [ w / / Σ XY ′′ (cid:15) (cid:15) Y ′′ γ =0 (cid:15) (cid:15) ✤✤✤ Σ Y Σ v / / Σ Z, satisfying the equations ϕv = v, a v = 0 , wϕ = w and b v = 0 . The equations ϕv = v and wϕ = w force ϕ to be an isomorphism. The third column is exact if andonly if b ϕ + a = 0 holds. Note that picking ϕ = 1 Z , a = 0, and b = 0 yields a valid choice.An octahedron for the composite X inc −−→ X ⊕ X ′′ u ′ −→ Y ⊕ Y ′′ is of the form: X inc / / X ⊕ X ′′ [ u c u ′′ ] (cid:15) (cid:15) proj / / X ′′ α = [ vcu ′′ ] (cid:15) (cid:15) ✤✤✤✤ / / Σ XX [ u ] / / Y ⊕ Y ′′ [ v ′ v ′ ] (cid:15) (cid:15) [ v
00 1 ] / / Z ⊕ Y ′′ β =[ b v ′ ] (cid:15) (cid:15) ✤✤✤✤ [ w / / Σ XZ ′ (cid:20) w ′ w ′ (cid:21) (cid:15) (cid:15) Z ′ γ = w ′ (cid:15) (cid:15) ✤✤✤✤ Σ X ⊕ Σ X ′′ proj / / Σ X ′′ , satisfying the equations b v = v ′ , w ′ b = w and w ′ b = 0 . Those are precisely the equations for b : Z → Z ′ being a fill-in. Verdier good fill-ins are those ofthe form β ◦ α = b ϕ + v ′ a. Using the equations ϕv = v and a v = 0, we obtain isomorphisms of rows X ′′ [ vcu ′′ ] / / Z ⊕ Y ′′∼ = h ϕ −
00 1 i (cid:15) (cid:15) [ b v ′ ] / / Z ′ w ′ / / Σ X ′′ X ′′ [ vcu ′′ ] / / Z ⊕ Y ′′∼ = h − a i (cid:15) (cid:15) [ b ϕ v ′ ] / / Z ′ w ′ / / Σ X ′′ X ′′ [ vcu ′′ ] / / Z ⊕ Y ′′ [ b ϕ + v ′ a v ′ ] / / Z ′ w ′ / / Σ X ′′ . Since the top row is exact, the bottom row is exact, proving the exactness property of Verdier goodfill-ins.To see that (1c) implies (1b), let h : Z → Z ′ be a fill-in satisfying (1c). Then the choices ϕ = 1 Z , a = 0, b = 0, and b = h exhibit h as Verdier good.When c = 0, we verify condition (1c). Without loss of generality, we may assume that the bottomrow is a direct sum of triangles, so the map we are considering is of the form X inc (cid:15) (cid:15) u / / Y inc (cid:15) (cid:15) v / / Z h = h h h i (cid:15) (cid:15) w / / Σ X inc (cid:15) (cid:15) X ⊕ X ′′ h u u ′′ i / / Y ⊕ Y ′′ h v v ′′ i / / Z ⊕ Z ′′ h w w ′′ i / / Σ X ⊕ Σ X ′′ . Note that h is necessarily an isomorphism and w ′′ h = 0. The following isomorphism of triangles X ′′ h u ′′ i / / Y ′′ ⊕ Z h h v ′′ h i / / Z ⊕ Z ′′ [ 0 w ′′ ] / / Σ X ′′ X ′′ h u ′′ i / / Y ′′ ⊕ Z h v ′′ i / / Z ⊕ Z ′′ [ 0 w ′′ ] / / ∼ = h h h i O O Σ X ′′ shows that the top row is isomorphic to a direct sum of exact triangles, and is therefore exact,proving (1c). (cid:3) Lemma 5.3.
Let ( f, g, h ) be a good map of triangles.(1) The inclusion of the target of ( f, g, h ) into its mapping cone is a good map of triangles.(2) The projection X ′ ⊕ Y proj (cid:15) (cid:15) h u ′ g − v i / / Y ′ ⊕ Z proj (cid:15) (cid:15) h v ′ h − w i / / Z ′ ⊕ Σ X proj (cid:15) (cid:15) h w ′ Σ f − Σ u i / / Σ X ′ ⊕ Σ Y proj (cid:15) (cid:15) Y − v / / Z − w / / Σ X − Σ u / / Σ Y. N GOOD MORPHISMS OF EXACT TRIANGLES 13 from the mapping cone onto the translation of the source of ( f, g, h ) is a good map of trian-gles.Proof. We will prove the first statement; the second statement is dual. Using the notation from (2.2),the inclusion of the target of ( f, g, h ) into its mapping cone is X ′ inc (cid:15) (cid:15) u ′ / / Y ′ inc (cid:15) (cid:15) v ′ / / Z ′ inc (cid:15) (cid:15) w ′ / / Σ X ′ inc (cid:15) (cid:15) X ′ ⊕ Y h u ′ g − v i / / Y ′ ⊕ Z h v ′ h − w i / / Z ′ ⊕ Σ X h w ′ Σ f − Σ u i / / Σ X ′ ⊕ Σ Y. By Proposition 5.1, that map is good if and only if the candidate triangle Y h v ′ g − v i / / Z ′ ⊕ Z h h − w i / / Z ′ ⊕ Σ X [ 0 − Σ u ] / / Σ Y is exact. By the isomorphism of rows Y h v ′ g − v i / / Z ′ ⊕ Z ∼ = h h i (cid:15) (cid:15) h h − w i / / Z ′ ⊕ Σ X [ 0 − Σ u ] / / Σ YY h − v i / / Z ′ ⊕ Z h − w i / / Z ′ ⊕ Σ X [ 0 − Σ u ] / / Σ Y, the top row is indeed exact. (cid:3) Lemma 5.4.
Let ( f, g, h ) be a map of triangles as in (2.2) and ( f , g, h ) a map of triangles fromthe same source triangle, as in the diagram X f (cid:15) (cid:15) u / / Y g (cid:15) (cid:15) v / / Z h (cid:15) (cid:15) w / / Σ X Σ f (cid:15) (cid:15) X u / / Y v / / Z w / / Σ X. Assume that one of the two maps is nullhomotopic. Then both maps are good if and only if the mapof triangles X (cid:20) ff (cid:21) (cid:15) (cid:15) u / / Y h gg i (cid:15) (cid:15) v / / Z h hh i (cid:15) (cid:15) w / / Σ X (cid:20) Σ f Σ f (cid:21) (cid:15) (cid:15) X ′ ⊕ X h u ′ u i / / Y ′ ⊕ Y h v ′ v i / / Z ′ ⊕ Z h w ′ w i / / Σ X ′ ⊕ Σ X is good.The analogous statement for two maps of triangles to the same target triangle holds.Proof. Assume that the map ( f , g, h ) is nullhomotopic (in particular good). Then the map( h ff i , (cid:2) gg (cid:3) , (cid:2) hh (cid:3) ) is chain homotopic to ( (cid:2) f (cid:3) , [ g ] , (cid:2) h (cid:3) ). The latter has as mapping cone the mappingcone of ( f, g, h ) direct sum with X u −→ Y v −→ Z w −→ Σ X . (cid:3) The following uses a variation on an argument of Neeman [Nee91, before Remark 1.10].
Proposition 5.5.
Every map of triangles is a composite of two maps that are good and Verdiergood.
In particular, a composite of Verdier good maps need not be Verdier good, and a composite ofmiddling good maps need not be middling good.
Proof.
Let ( f, g, h ) be a map of triangles as in (2.2). Consider the commutative diagram X inc (cid:15) (cid:15) u / / Y inc (cid:15) (cid:15) v / / Z inc (cid:15) (cid:15) w / / Σ X inc (cid:15) (cid:15) X ⊕ X ′ h f i ∼ = (cid:15) (cid:15) h u u ′ i / / Y ⊕ Y ′ h g i ∼ = (cid:15) (cid:15) h v v ′ i / / Z ⊕ Z ′ h h i ∼ = (cid:15) (cid:15) h w w ′ i / / Σ X ⊕ Σ X ′ h f i ∼ = (cid:15) (cid:15) X ⊕ X ′ proj (cid:15) (cid:15) h u u ′ i / / Y ⊕ Y ′ proj (cid:15) (cid:15) h v v ′ i / / Z ⊕ Z ′ proj (cid:15) (cid:15) h w w ′ i / / Σ X ⊕ Σ X ′ proj (cid:15) (cid:15) X ′ u ′ / / Y ′ v ′ / / Z ′ w ′ / / Σ X ′ , which consists of three maps of triangles whose composite is ( f, g, h ). By Proposition 5.1, the topand the bottom maps are good and Verdier good (as is the middle map, by Remark 3.2). ByRemark 3.2, the composite of the top two (or bottom two) maps is good and Verdier good. (cid:3) Homotopy cartesian squares
In this section, we recall the notion of a homotopy cartesian square and build on work of Neemanshowing a relationship between such squares and good morphisms of the form (1 , g, h ). We thenprove a pasting lemma for homotopy cartesian squares. We also characterize the Verdier goodmorphisms of the form (1 , g, h ), and deduce that every good morphism of the form (1 , g, h ) isVerdier good.We will use [Nee01, Definition 1.4.1, Lemma 1.4.3], but different preprint versions of the bookhave different sign conventions. Let us recall the sign convention and check the details of the proof.
Definition 6.1.
A commutative square Y g (cid:15) (cid:15) f / / Z g ′ (cid:15) (cid:15) Y ′ f ′ / / Z ′ (6.2)is called homotopy cartesian if there is an exact triangle Y h gf i / / Y ′ ⊕ Z [ f ′ − g ′ ] / / Z ′ ∂ / / Σ Y (6.3)for some map ∂ : Z ′ → Σ Y , called the differential .Note that exactness of (6.3) is equivalent to exactness of Y h g − f i / / Y ′ ⊕ Z [ f ′ g ′ ] / / Z ′ ∂ / / Σ Y, via the automorphism − Z : Z → Z . Remark . The horizontal and vertical directions play asymmetric roles in the sign convention.The square (6.2) is homotopy cartesian with differential ∂ : Z ′ → Σ Y if and only if its diagonal N GOOD MORPHISMS OF EXACT TRIANGLES 15 reflection Y f (cid:15) (cid:15) g / / Y ′ f ′ (cid:15) (cid:15) Z g ′ / / Z ′ is homotopy cartesian with differential − ∂ : Z ′ → Σ Y .The following is stated in [Nee01, Remark 1.4.5]. Here we fill in some details of the proof. Lemma 6.5.
Consider a homotopy cartesian square Y g (cid:15) (cid:15) v / / Z h (cid:15) (cid:15) Y ′ v ′ / / Z ′ with given differential ∂ : Z ′ → Σ Y . Then the square extends to a good map of triangles of the form (1 , g, h ) , as illustrated in the diagram X u / / Y g (cid:15) (cid:15) v / / Z h (cid:15) (cid:15) w / / Σ XX u ′ / / Y ′ v ′ / / Z ′ w ′ / / Σ X, satisfying ∂ = (Σ u ) w ′ .Moreover, we may prescribe the top row or the bottom row.Proof. The map is constructed in [Nee01, Lemma 1.4.4]. Let us recall Neeman’s proof and checkthat the resulting map is good.Assume given a triangle X u −→ Y v −→ Z w −→ Σ X , prescribed as top row. Let ϕ : Z ′ → Σ X be a goodfill-in in the diagram Y [ gv ] / / Y ′ ⊕ Z proj (cid:15) (cid:15) [ v ′ − h ] / / Z ′ ϕ (cid:15) (cid:15) ✤✤✤ ∂ / / Σ YY v / / Z w / / Σ X − Σ u / / Σ Y. (6.6)A straightforward computation shows that its mapping cone is isomorphic to the direct sum of thethree summands Z / / Z / / / / Σ ZY / / / / Σ Y / / Σ YY ′ v ′ / / Z ′ − ϕ / / Σ X − (Σ g )(Σ u ) / / Σ Y ′ . Via the isomorphism, the inclusion of the bottom row of (6.6) into the mapping cone has as com-ponent into the third summand Y g (cid:15) (cid:15) v / / Z h (cid:15) (cid:15) w / / Σ X − Σ u / / Σ Y Σ g (cid:15) (cid:15) Y ′ v ′ / / Z ′ − ϕ / / Σ X − (Σ g )(Σ u ) / / Σ Y ′ , (6.7) which is the desired map (up to rotation), taking w ′ := − ϕ and u ′ := gu . By Lemma 5.3, theinclusion of the bottom row of (6.6) into the mapping cone is good. By Lemma 5.4, its thirdcomponent (6.7) is also good. Furthermore, the differential satisfies ∂ = (Σ u )( − ϕ ) = (Σ u ) w ′ .Now assume given a triangle X u ′ −→ Y ′ v ′ −→ Z ′ w ′ −→ Σ X , prescribed as bottom row. Applying anappropriate automorphism of X to the map (1 , g, h ) found above will adjust the bottom row to theprescribed one, modify the top row, and keep the differential ∂ as it is. (cid:3) We next give a characterization of the good maps of the form (1 , g, h ). Definition 6.8.
In a candidate triangle X u −→ Y v −→ Z w −→ Σ X , the map u is replaceable with replacing map b u if X b u / / Y v / / Z w / / Σ X is exact. Replaceability of v or w is defined similarly. The candidate triangle is replaceably exact if its three maps are replaceable [Vak01, Definition 1.3]. Proposition 6.9.
Consider a map of triangles (1 , g, h ) of the form X u / / Y g (cid:15) (cid:15) v / / Z h (cid:15) (cid:15) w / / Σ XX u ′ / / Y ′ v ′ / / Z ′ w ′ / / Σ X and the candidate triangle Y [ gv ] / / Y ′ ⊕ Z [ v ′ − h ] / / Z ′ (Σ u ) w ′ / / Σ Y. (6.10) (1) The map is good if and only if the middle square is homotopy cartesian with differential ∂ = (Σ u ) w ′ : Z ′ → Σ Y (i.e., (6.10) is exact).(2) The middle square is homotopy cartesian if and only if the candidate triangle (6.10) isreplaceably exact. Note that if the cofibre fill-in h : Z → Z ′ were missing, we could choose a good one, and if thefibre fill-in g : Y → Y ′ were missing, we could choose a good one. Proof.
Part (1) is a slightly stronger formulation of [Nee01, Lemma 1.4.3], based on the same proof,which we recall here. The mapping cone of (1 , g, h ) is X ⊕ Y h gu g − v i / / Y ′ ⊕ Z h v ′ h − w i / / Z ′ ⊕ Σ X h w ′ − Σ u i / / Σ X ⊕ Σ Y. A straightforward computation shows that it is isomorphic to the direct sum of the candidatetriangle (6.10) and the triangle X / / / / Σ X / / Σ X. (2) By definition, the middle square is homotopy cartesian if and only if (6.10) is replaceable onthe right. Assuming that the middle square is homotopy cartesian, we will now show that (6.10) isreplaceable in the middle. Applying Lemma 6.5 to the middle square and prescribing the top row,there is a good map of triangles X u / / Y g (cid:15) (cid:15) v / / Z h (cid:15) (cid:15) w / / Σ XX u ′ / / Y ′ v ′ / / Z ′ b w ′ / / Σ X. N GOOD MORPHISMS OF EXACT TRIANGLES 17
Now postcompose with the isomorphism of triangles X u ′ / / Y ′ v ′ / / Z ′ θ ∼ = (cid:15) (cid:15) ✤✤✤ b w ′ / / Σ XX u ′ / / Y ′ v ′ / / Z ′ w ′ / / Σ X, where θ : Z ′ → Z ′ is any fill-in. By part (1), Y [ gv ] / / Y ′ ⊕ Z [ v ′ − θh ] / / Z ′ (Σ u ) w ′ / / Σ Y is a triangle. Dualizing the argument, prescribing the bottom row, there is an automorphismΓ : Y ∼ = −→ Y such that Y [ g Γ v ] / / Y ′ ⊕ Z [ v ′ − h ] / / Z ′ (Σ u ) w ′ / / Σ Y is a triangle. This shows that (6.10) is replaceable on the left. (cid:3) We illustrate the usefulness of our results by proving the following fact about homotopy cartesiansquares, whose statement does not involve the notion of good morphism.
Proposition 6.11 (Pasting Lemma) . Consider a diagram X f (cid:15) (cid:15) ϕ / / Y g (cid:15) (cid:15) ψ / / Z h (cid:15) (cid:15) X ′ ϕ ′ / / Y ′ ψ ′ / / Z ′ . If the two squares are homotopy cartesian, then so is their pasting, the big rectangle.Moreover, given differentials ∂ L : Y ′ → Σ X and ∂ R : Z ′ → Σ Y of the left square and right squarerespectively, there exists a differential ∂ P : Z ′ → Σ X for the pasted rectangle satisfying ( (Σ ϕ ) ∂ P = ∂ R ∂ P ψ ′ = ∂ L . Proof.
We claim that the square X h fψϕ i (cid:15) (cid:15) ϕ / / Y h gψ i (cid:15) (cid:15) X ′ ⊕ Z h ϕ ′
00 1 i / / Y ′ ⊕ Z is homotopy cartesian with differential [ ∂ L ]. This follows from the commutative diagram X " f ϕ / / X ′ ⊕ Z ⊕ Y s ∼ = (cid:15) (cid:15) h ψ ′ − g i / / Y ′ ⊕ Z [ ∂ L / / Σ XX " fψϕϕ / / X ′ ⊕ Z ⊕ Y (cid:20) ψ ′ − g − ψ (cid:21) / / Y ′ ⊕ Z [ ∂ L / / Σ X involving the shear isomorphism s with matrix ψ .The top row is exact since it is a direct sum of the triangle that shows that the left square ishomotopy cartesian with a trivial triangle involving the identity map on Z . Therefore, the bottomrow is exact, showing that the square is homotopy cartesian.Keeping that specific choice of differential and applying Lemma 6.5, there exists a (good) mapof triangles X ϕ (cid:15) (cid:15) h fψϕ i / / X ′ ⊕ Z h ϕ ′
00 1 i (cid:15) (cid:15) [ ψ ′ ϕ ′ − h ] / / Z ′ ∂ P / / Σ X Σ ϕ (cid:15) (cid:15) Y h gψ i / / Y ′ ⊕ Z [ ψ ′ − h ] / / Z ′ ∂ R / / Σ Y (6.12)satisfying [ − ∂ L ] = − ∂ P [ ψ ′ − h ] = [ − ∂ P ψ ′ ] . (The minus sign on the left-hand side comes from the reflection, as in Remark 6.4.) Here weprescribed the bottom row, coming from the fact that the original right square was homotopycartesian with given differential ∂ R : Z ′ → Σ Y . The top row of (6.12) exhibits the big rectangleas homotopy cartesian with differential ∂ P : Z ′ → Σ X . The chosen differential ∂ P satisfies ∂ R =(Σ ϕ ) ∂ P and ∂ L = ∂ P ψ ′ , as desired. (cid:3) Remark . [LZ19, Lemma 2.7] is a form of pasting lemma. However, in the case n = 3 (which isthe case we are considering, of usual triangles), their statement only allows pasting an isomorphismof arrows, i.e., our diagram with ψ and ψ ′ being isomorphisms. Corollary 6.14.
Consider a composite of good maps of triangles of the form X u / / Y g (cid:15) (cid:15) v / / Z h (cid:15) (cid:15) w / / Σ XX u / / Y g (cid:15) (cid:15) v / / Z h (cid:15) (cid:15) w / / Σ XX u / / Y v / / Z w / / Σ X. Then the candidate triangle Y [ g g v ] / / Y ⊕ Z v − h h ] / / Z u ) w / / Σ Y (6.15) is replaceably exact.Proof. By Proposition 6.11, the pasting of the top middle and bottom middle squares is homotopycartesian. The claim then follows from Proposition 6.9 (2). (cid:3)
One might hope that the differential for the pasted rectangle could be chosen to be (Σ u ) w ,or, in other words, that the candidate triangle (6.15) is exact. In Example 6.24, we will show thatthis need not be the case. In [Vak01, after Proposition 1.16], Vaknin shows that (6.15) is “virtual”,which is weaker than being replaceably exact.We now give a characterization of when a map (1 , g, h ) of triangles is Verdier good. N GOOD MORPHISMS OF EXACT TRIANGLES 19
Proposition 6.16.
A map of triangles of the form X u / / Y g (cid:15) (cid:15) v / / Z h (cid:15) (cid:15) w / / Σ XX u ′ / / Y ′ v ′ / / Z ′ w ′ / / Σ X is Verdier good if and only if it extends to an octahedron, i.e., it appears as the top two rows of anoctahedron.Proof. An octahedron for the composite X u −→ Y g −→ Y ′ is of the form: X u / / Y g (cid:15) (cid:15) v / / Z α (cid:15) (cid:15) ✤✤✤ w / / Σ XX gu = u ′ / / Y ′ g ′ (cid:15) (cid:15) v ′ / / Z ′ β (cid:15) (cid:15) ✤✤✤ w ′ / / Σ XY ′′ g ′′ (cid:15) (cid:15) Y ′′ γ (cid:15) (cid:15) ✤✤✤ Σ Y Σ v / / Σ Z. (6.17)An octahedron for the composite X −→ X u ′ −→ Y ′ is of the form: X / / X u ′ (cid:15) (cid:15) / / (cid:15) (cid:15) / / Σ XX u ′ = gu / / Y ′ v ′ (cid:15) (cid:15) v ′ / / Z ′ β ∼ = (cid:15) (cid:15) ✤✤✤ w ′ / / Σ XZ ′ w ′ (cid:15) (cid:15) Z ′ (cid:15) (cid:15) Σ X / / , with the automorphism β satisfying β v ′ = v ′ and w ′ β = w ′ .( ⇐ =) If the map of triangles (1 , g, h ) extends to an octahedron, then that octahedron is a validchoice for the octahedron (6.17), in particular with α = h . Picking β = 1 Z ′ yields a valid choicefor the second octahedron, exhibiting h = 1 Z ′ h as Verdier good. (= ⇒ ) Assume that h is Verdier good, exhibited by two octahedra as above. Using the automor-phism β : Z ′ ∼ = −→ Z ′ to modify the first octahedron (6.17) yields an octahedron X u / / Y g (cid:15) (cid:15) v / / Z β α = h (cid:15) (cid:15) w / / Σ XX u ′ / / Y ′ g ′ (cid:15) (cid:15) β v ′ = v ′ / / Z ′ β β − (cid:15) (cid:15) w ′ β − = w ′ / / Σ XY ′′ g ′′ (cid:15) (cid:15) Y ′′ γ (cid:15) (cid:15) ✤✤✤ Σ Y Σ v / / Σ Z that extends the map (1 , g, h ). (cid:3) The proof of [Nee01, Proposition 1.4.6] then implies the following.
Corollary 6.18.
If a map of triangles of the form (1 , g, h ) is good, then it is Verdier good. (cid:3) We don’t know whether Verdier good implies good for such maps of triangles. The discussion in[May01, Remark 3.7] suggests that this is not the case, without providing an explicit counterexam-ple.The following examples illustrate the difference between a map (1 , g, h ) being good and merelyhaving a homotopy cartesian middle square. They also show that we may not always prescribeboth rows in Lemma 6.5. Furthermore, they provide examples of maps of the form (1 , g, h ) that aremiddling good but not Verdier good.
Example 6.19.
Let C be the cyclic group of order 4, with generator g , and consider the groupalgebra R = F C ∼ = F [ x ] /x , with x := g −
1. We will work in the stable module categoryStMod( R ), as in [CF17, Appendix A]. See also [Ben98, § §
5] for background on stablemodule categories. For r ∈ R , let µ r : R/x i → R/x j denote the R -module map sending 1 to r , whenit is defined. In the map of triangles R/x ⊕ R/x h µ x i / / R/x ⊕ R/x µ (cid:15) (cid:15) [ 0 µ ] / / R/x [ aµ x µ x ] (cid:15) (cid:15) h µ x i / / R/x ⊕ R/x R/x ⊕ R/x µ / / R/x [ µ x ] / / R/x ⊕ R/x h µ
00 1 i / / R/x ⊕ R/x , (6.20)a general fill-in has the stated form, with a ∈ F . By Lemma 5.4, the map (6.20) is good if andonly if its restriction and projection R/x (cid:15) (cid:15) µ x / / R/x (cid:15) (cid:15) µ / / R/x aµ x (cid:15) (cid:15) µ x / / R/x (cid:15) (cid:15) R/x µ / / R/x µ x / / R/x µ / / R/x is good. By [Nee91, §
1, Case 2], this holds if and only if aµ x is a lightning flash, which holds if andonly if a = 0 holds. Starting with the non-good fill-in [ µ x µ x ] and using the shear automorphism[ ] : R/x ⊕ R/x ∼ = −→ R/x ⊕ R/x
2N GOOD MORPHISMS OF EXACT TRIANGLES 21 in the bottom row, we obtain a map of triangles
R/x ⊕ R/x h µ x i / / R/x ⊕ R/x µ (cid:15) (cid:15) [ 0 µ ] / / R/x h µ x i (cid:15) (cid:15) h µ x i / / R/x ⊕ R/x R/x ⊕ R/x µ / / R/x [ µ x ] / / R/x ⊕ R/x [ µ µ ] / / R/x ⊕ R/x (6.21)which is not good and has the same (homotopy cartesian) middle square as the good map (6.20) inthe case a = 0. Alternately, one can check the (non-)goodness claims using Proposition 6.9 (1).A straightforward calculation shows that the map (6.21) does not extend to an octahedron, so itis not Verdier good, by Proposition 6.16. Indeed, (6.21) does extend to a 4 × R/x ⊕ R/x h µ x i / / R/x ⊕ R/x µ (cid:15) (cid:15) [ 0 µ ] / / R/x h µ x i (cid:15) (cid:15) h µ x i / / R/x ⊕ R/x R/x ⊕ R/x µ / / R/x [ µ x ] (cid:15) (cid:15) [ µ x ] / / R/x ⊕ R/x h µ i (cid:15) (cid:15) [ µ µ ] / / R/x ⊕ R/x R/x ⊕ R/x h µ cµ i (cid:15) (cid:15) R/x ⊕ R/x [ 0 µ x ] (cid:15) (cid:15) R/x ⊕ R/x [ 0 µ x ] / / R/x with c ∈ F , and, up to isomorphism, every extension is of this form. But this diagram does notsatisfy the additional equation required of an octahedron: (cid:0) Σ (cid:2) µ x (cid:3)(cid:1) [ µ µ ] = (cid:2) µ (cid:3) [ µ µ ] = (cid:2) µ µ µ (cid:3) = h µ cµ i (cid:2) µ (cid:3) = (cid:2) µ cµ µ (cid:3) : R/x ⊕ R/x → R/x ⊕ R/x.
So (6.21) is middling good, but not Verdier good. This also provides an alternate proof that themap (6.21) is not good, via Corollary 6.18.
Example 6.22.
Consider the derived category of the integers D ( Z ), with the notation as in Ex-ample 4.2. Let n ≥ Z [0] / / Z [0] q (cid:15) (cid:15) q / / Z / ] (cid:15) (cid:15) ǫ / / Z [1] Z [0] q / / Z / h q − ǫ i / / Z / ⊕ Z [1] [ ǫ / / Z [1]is good. In fact, [ ] is the unique good fill-in here. However, the map of triangles with the samemiddle square Z [0] / / Z [0] q (cid:15) (cid:15) q / / Z / ] (cid:15) (cid:15) ǫ / / Z [1] Z [0] q / / Z / h q − ǫ i / / Z / ⊕ Z [1] [ ǫ − / / Z [1] (6.23) is not good. In fact, the unique good fill-in is (cid:2) ǫ (cid:3) .As in Example 6.19, the map (6.23) extends to a 4 × not to an octahedron, so itis middling good but not Verdier good. Example 6.24.
Let us return to the stable module category StMod( R ) from Example 6.19. Con-sider the diagram with exact rows R/x ⊕ R/x ⊕ R/x (cid:20) µ x (cid:21) / / R/x ⊕ R/x ⊕ R/x h µ i (cid:15) (cid:15) [ 0 0 µ ] / / R/x (cid:20) (cid:21) (cid:15) (cid:15) (cid:20) µ x (cid:21) / / R/x ⊕ R/x ⊕ R/x R/x ⊕ R/x ⊕ R/x h µ µ x i / / R/x ⊕ R/x
3[ 1 0 ] (cid:15) (cid:15) " µ x
00 00 µ / / R/x ⊕ R/x ⊕ R/x (cid:20) µ x (cid:21) (cid:15) (cid:15) " µ µ x / / R/x ⊕ R/x ⊕ R/x R/x ⊕ R/x ⊕ R/x µ / / R/x (cid:20) µ x (cid:21) / / R/x ⊕ R/x ⊕ R/x (cid:20) µ µ (cid:21) / / R/x ⊕ R/x ⊕ R/x . By Lemma 5.4 and Proposition 5.1 (1), the top part is a good map of triangles. By Lemma 5.4and Proposition 5.1 (2), the bottom part is a good map of triangles. By the same argument as inExample 6.19, the composite map of triangles from the top row to the bottom row is not good.7.
Maps of triangles with one or two zero components
In this section we study maps of triangles with at most two non-zero components and show thatthe properties of being good, Verdier good, and nullhomotopic agree for such maps in most cases.This leads to a lifting condition expressed in terms of Verdier good fill-ins.
Lemma 7.1.
A map (0 , g, h ) of triangles is nullhomotopic if and only if it is nullhomotopic via achain homotopy (0 , G, . Rotating the triangles gives similar statements for any map with at mosttwo non-zero components.Proof. Assume that the map (0 , g, h ) is nullhomotopic via a chain homotopy (
F, G, H ), as illustratedin the diagram X (cid:15) (cid:15) u / / Y F } } ⑤⑤⑤⑤⑤⑤⑤⑤ g (cid:15) (cid:15) v / / Z G ~ ~ ⑤⑤⑤⑤⑤⑤⑤⑤ h (cid:15) (cid:15) w / / Σ X H | | ②②②②②②②②② (cid:15) (cid:15) X ′ u ′ / / Y ′ v ′ / / Z ′ w ′ / / Σ X ′ . In particular, the map 0 : Σ X → Σ X ′ satisfies 0 = w ′ H + Σ( F u ). Hence, the following solid diagramcommutes: X Σ − H (cid:15) (cid:15) u / / Y F (cid:15) (cid:15) v / / Z e G (cid:15) (cid:15) ✤✤✤ w / / Σ X H (cid:15) (cid:15) Σ − Z ′ − Σ − w ′ / / X ′ u ′ / / Y ′ v ′ / / Z ′ . Let e G : Z → Y ′ be a fill-in. Then (0 , G + e G,
0) is a nullhomotopy of (0 , g, h ):( G + e G ) v + u ′ Gv + u ′ F = g w + v ′ ( G + e G ) = v ′ G + Hw = h. (cid:3) Corollary 7.2.
A map of triangles with only one non-zero component, i.e., of the form ( f, , or (0 , g, or (0 , , h ) , is nullhomotopic if and only if that component is a lightning flash. (cid:3) Remark . By considering the difference of two maps of triangles, Lemma 7.1 (resp. Corollary 7.2)characterizes the chain homotopies that modify only two components (resp. only one component).
N GOOD MORPHISMS OF EXACT TRIANGLES 23
Proposition 7.4.
For a map of triangles with at most two non-zero components, the following areequivalent.(1) The map is nullhomotopic.(2) The map is good.If the map has the form (0 , g, h ) or ( f, , h ) , then those conditions are further equivalent to:(3) The map is Verdier good. We don’t know whether being middling good is equivalent to the above conditions. We considermaps of the form ( f, g,
0) in Proposition 7.11.
Proof. (1) ⇐⇒ (2). Since goodness and the property of being nullhomotopic are invariant underrotation, it suffices to treat one of the three cases, say, ( f, g, f, g, F, , ⇐⇒ (1) in separate cases. Case (0 , g, h ) . An octahedron for the composite X u −→ Y g −→ Y ′ is of the form: X u / / Y g (cid:15) (cid:15) v / / Z α = [ a w ] (cid:15) (cid:15) ✤✤✤ w / / Σ XX / / Y ′ g ′ (cid:15) (cid:15) inc / / Y ′ ⊕ Σ X β =[ g ′ b ] (cid:15) (cid:15) ✤✤✤ proj / / Σ XY ′′ g ′′ (cid:15) (cid:15) Y ′′ γ =(Σ v ) g ′′ (cid:15) (cid:15) ✤✤✤ Σ Y Σ v / / Σ Z, (7.5)with a v = g and g ′′ b = Σ u . An octahedron for the composite X −→ X ′ u ′ −→ Y ′ is of the form: X / / X ′ u ′ (cid:15) (cid:15) inc / / X ′ ⊕ Σ X α = h u ′ a i (cid:15) (cid:15) ✤✤✤ proj / / Σ XX / / Y ′ v ′ (cid:15) (cid:15) inc / / Y ′ ⊕ Σ X β =[ v ′ b ] (cid:15) (cid:15) ✤✤✤ proj / / Σ XZ ′ w ′ (cid:15) (cid:15) Z ′ γ = h w ′ i (cid:15) (cid:15) ✤✤✤ Σ X ′ inc / / Σ X ′ ⊕ Σ X, (7.6)with w ′ b = 0. Verdier good fill-ins are the maps h = β ◦ α = v ′ a + b w. For such a fill-in h , the map of triangles (0 , g, h ) is nullhomotopic via the chain homotopy (0 , a , b ),as illustrated in the diagram X (cid:15) (cid:15) u / / Y } } ⑤⑤⑤⑤⑤⑤⑤⑤ g (cid:15) (cid:15) v / / Z a ~ ~ ⑤⑤⑤⑤⑤⑤⑤⑤ h (cid:15) (cid:15) w / / Σ X (cid:15) (cid:15) b | | ②②②②②②②②② X ′ u ′ / / Y ′ v ′ / / Z ′ w ′ / / Σ X ′ . Conversely, assume that (0 , g, h ) is nullhomotopic. By Lemma 7.1, we may assume that thenullhomotopy is of the form (0 , a , h = v ′ a holds. We will show that thegiven a and b = 0 arise as valid choices in the octahedra above. By Proposition 6.9, there existsa fill-in in the diagram Y v / / Z a (cid:15) (cid:15) w / / Σ X − b (cid:15) (cid:15) ✤✤✤ − Σ u / / Σ YY g / / Y ′ g ′ / / Y ′′ g ′′ / / Σ Y making the middle square homotopy cartesian with differential ∂ = (Σ v ) g ′′ , i.e., making the follow-ing triangle exact: Z [ a w ] / / Y ′ ⊕ Σ X [ g ′ b ] / / Y ′′ (Σ v ) g ′′ / / Σ Z. This is the third column of (7.5). Moreover b satisfies g ′′ b = Σ u , as required.For (7.6), picking a = 0 and b = 0 yields a valid choice. It follows that h = v ′ a is a Verdiergood fill-in. Case ( f, , h ) . The argument is the same as the previous case, using a good fibre fill-in inProposition 6.9. (cid:3)
Now we can refine Theorem 4.3.
Corollary 7.7 (Lifting Criterion) . Given a solid arrow commutative square X u / / f (cid:15) (cid:15) Y g (cid:15) (cid:15) k } } ⑤ ⑤ ⑤ ⑤ X ′ u ′ / / Y ′ , choose extensions of the rows to cofibre sequences as in (4.1) . The following are equivalent.(1) There exists a lift k : Y → X ′ .(2) The map Z → Z ′ is a fill-in and the map ( f, g, of triangles is good.(3) The map Z → Z ′ is a fill-in and the map (0 , f, g ) of rotated triangles is Verdier good. (cid:3) We need to rotate the triangles in the last case because Proposition 7.4 does not include the case( f, g, Z → Z ′ such that the rotated map (Σ − h, f, g ) is Verdier good. A lift exists if and onlyif this obstruction class contains zero.Next, we tackle the remaining case ( f, g, N GOOD MORPHISMS OF EXACT TRIANGLES 25
Definition 7.8.
A triangulated category T admits weakly functorial octahedra if for everydiagram X ϕ (cid:15) (cid:15) f / / X ϕ (cid:15) (cid:15) f / / X ϕ (cid:15) (cid:15) Y g / / Y g / / Y (7.9)in T , there exist octahedra X and Y based on the top and bottom rows of (7.9) respectively, anda map of octahedra ϕ : X → Y extending the given map between their bases ( ϕ , ϕ , ϕ ). Example 7.10.
If a triangulated category T admits a 3-pretriangulated enhancement in the senseof Maltsiniotis [Mal06], then it admits weakly functorial octahedra. Indeed, pick distinguished octahedra (i.e., 3-triangles) X and Y based on X → X → X and Y → Y → Y respectively.Then any map ( ϕ , ϕ , ϕ ) between their bases as in (7.9) extends to a map of octahedra ϕ : X → Y .This is stronger than the condition in Definition 7.8, where the two octahedra may depend on thegiven map between bases.The homotopy category of a stable model category (or more generally of a complete and cocom-plete stable ∞ -category) admits a canonical n -triangulation for every n ; see [Mal06] and [GˇS16, The-orem 13.6, Examples 13.8]. See also related work on higher triangulations in [K¨un07]. Proposition 7.11.
Assume that the triangulated category T admits weakly functorial octahedra. Ifa map of triangles ( f, g, is good, then it is Verdier good.Proof. Consider the diagram: X u / / Y F (cid:15) (cid:15) g / / Y ′ X f / / X ′ u ′ / / Y ′ . (7.12)By assumption, we may pick octahedra based on X u −→ Y g −→ Y ′ and X f −→ X ′ u ′ −→ Y ′ that admita map of octahedra ϕ extending the map between their bases (7.12). Keeping the notation asin Definition 3.1, denote the new components of the natural transformation ϕ by ϕ Z : Z → X ′′ , ϕ A : A → A , and ϕ Y ′′ : Y ′′ → Z ′ . Due to the map of triangles X gu / / Y ′ e v / / A ϕ A (cid:15) (cid:15) e w / / Σ XX u ′ f / / Y ′ e v / / A e w / / Σ X, the map ϕ A is an isomorphism. Moreover, the maps Z ϕ A α −−−→ A β ϕ − A −−−−→ Y ′′ remain valid choices forthe first octahedron. These choices yield a Verdier good fill-in h = β ( ϕ A α ) = β α ϕ Z = 0 ϕ Z = 0 , where the second equality uses the naturality of ϕ . (cid:3) We now study a particularly simple family of maps of triangles.
Proposition 7.13.
For a map of triangles of the form X (cid:15) (cid:15) u / / Y (cid:15) (cid:15) v / / Z h (cid:15) (cid:15) w / / Σ X (cid:15) (cid:15) X ′ u ′ / / Y ′ v ′ / / Z ′ w ′ / / Σ X ′ , the following are equivalent:(1) The map (0 , , h ) is good.(2) The map (0 , , h ) is Verdier good.(3) The map (0 , , h ) is middling good.(4) The Toda bracket h w ′ , h, v i ⊆ T (Σ Y, Σ X ′ ) contains zero.(5) The map h is a lightning flash.Moreover, analogous statements hold for maps of the form ( f, , and (0 , g, . Since the otherconditions are rotation invariant, it follows that Verdier goodness is rotation invariant for mapswith at most one non-zero component.Proof. The equivalence of (1), (2) and (5) follows from Corollary 7.2 and Proposition 7.4. (Neemanalso showed that (1) is equivalent to (5) in [Nee91, §
1, Case 2].) That (2) implies (3) is exactlyVerdier’s argument, which we summarized in Lemma 3.4.We will show the implications (3) = ⇒ (4) = ⇒ (5).To show that (3) implies (4), assume that (0 , , h ) is middling good. A 4 × , , h ) may be assumed to be of the form X (cid:15) (cid:15) u / / Y (cid:15) (cid:15) v / / Z h (cid:15) (cid:15) w / / Σ X (cid:15) (cid:15) X ′ inc (cid:15) (cid:15) u ′ / / Y ′ inc (cid:15) (cid:15) v ′ / / Z ′ h ′ (cid:15) (cid:15) w ′ / / Σ X ′ inc (cid:15) (cid:15) X ′ ⊕ Σ X proj (cid:15) (cid:15) h u ′ a u i / / ❴❴❴❴ Y ′ ⊕ Σ Y proj (cid:15) (cid:15) [ h ′ v ′ b ] / / ❴❴❴❴ Z ′′ -1 h ′′ (cid:15) (cid:15) h c − (Σ w ) h ′′ i / / ❴❴❴❴ Σ X ′ ⊕ Σ X proj (cid:15) (cid:15) Σ X Σ u / / Σ Y Σ v / / Σ Z Σ w / / Σ X with h ′′ b = Σ v and ch ′ = w ′ . These equations exhibit the composite c ( − b ) ∈ h w ′ , h, v i as beingin the fibre-cofibre Toda bracket [CF17, Definition 3.1]. Exactness of the third row implies that cb = 0. Hence, 0 ∈ h w ′ , h, v i .To show that (4) implies (5), assume that h w ′ , h, v i contains zero. Using the iterated cofibre Todabracket [CF17, Definition 3.1], this means that there exists a map ϕ making the diagram Y v / / Z w / / Σ X ϕ (cid:15) (cid:15) − Σ u / / Σ Y (cid:15) (cid:15) Y v / / Z h / / Z ′ w ′ / / Σ X ′ commute. The condition w ′ ϕ = 0 implies that ϕ lifts as ϕ = v ′ e ϕ for some e ϕ : Σ X → Y ′ . So we have h = ϕw = v ′ e ϕw , which is a lightning flash. (cid:3) It follows that we can characterize when an arbitrary 3-fold Toda bracket contains zero: for anycomposable maps Y v −→ Z h −→ Z ′ w ′ −→ W , the Toda bracket h w ′ , h, v i contains zero if and only if h = v ′ θw for some θ , where v ′ is the fibre of w ′ and w is the cofibre of v .8. Maps between rotations of a triangle
In this final section, we study a special situation in which we again find that good morphismsand Verdier good morphisms agree, and use this to present some interesting examples.
N GOOD MORPHISMS OF EXACT TRIANGLES 27
Proposition 8.1.
Consider a map of triangles of the form X u (cid:15) (cid:15) u / / Y v (cid:15) (cid:15) v / / Z h (cid:15) (cid:15) w / / Σ X Σ u (cid:15) (cid:15) Y v / / Z w / / Σ X − Σ u / / Σ Y, where the second row is the displayed rotation of the first row, and the first two vertical maps aretaken from the triangle. Then the following are equivalent:(1) The map ( u, v, h ) is good.(2) The map ( u, v, h ) is Verdier good.(3) The Toda bracket h− Σ u, h, v i ⊆ T (Σ Y, Σ Y ) contains zero.(4) The map h is a lightning flash.When h = w , these conditions are also equivalent to the given triangle being contractible.Proof. A chain homotopy consisting of the identity map 1 Y and zero elsewhere shows that ( u, v, h )is chain homotopic to (0 , , h ). Since goodness is invariant under chain homotopy (Remark 2.7)condition (1) is equivalent to (0 , , h ) being good. By Proposition 7.13, this is equivalent to condi-tions (3) and (4). In the case that h = w , Neeman showed in [Nee91, §
1, Case 2] that this is alsoequivalent to the triangle being contractible. We will show that (2) is equivalent to (4).We begin by assuming that ( u, v, h ) is Verdier good. Every octahedron for the composite X u −→ Y v −→ Z is isomorphic to one of the form: X u / / Y v (cid:15) (cid:15) v / / Z α = [ aw ] (cid:15) (cid:15) ✤✤✤ w / / Σ XX / / Z w (cid:15) (cid:15) inc / / Z ⊕ Σ X β =[ w b ] (cid:15) (cid:15) ✤✤✤ proj / / Σ X Σ X − Σ u (cid:15) (cid:15) Σ X γ =0 (cid:15) (cid:15) ✤✤✤ Σ Y Σ v / / Σ Z. (8.2)The top-middle square gives the equation av = v , that is, ( a − v = 0. Therefore, a = 1 + ϕw forsome ϕ : Σ X → Z . The staircase equation from Z ⊕ Σ X to Σ Y is (Σ u ) ◦ proj = ( − Σ u ) ◦ (cid:2) w b (cid:3) ,or equivalently, (Σ u )( b + 1) = 0. Therefore, b = − wψ for some ψ : Σ X → Z .Making such choices for two octahedra, every Verdier good fill-in will be of the form h = β α = (cid:2) w b (cid:3) (cid:20) a w (cid:21) = wa + b w = w (1 + ϕ w ) + ( − wψ ) w = w ( ϕ + ψ ) w, which is a lightning flash, proving (4).To prove that (4) implies (2), assume that h is a lightning flash. By Lemma 3.6, it is enough toshow that ( u, v,
0) is Verdier good. One can check that choosing a = 1 and b = 1 in (8.2) gives anoctahedron. Making these choices for both octahedra, the Verdier fill-in is β α = (cid:2) w − (cid:3) (cid:20) w (cid:21) = 0 . So ( u, v,
0) is Verdier good. (cid:3)
Example 8.3.
As mentioned in the proof of Proposition 8.1, Neeman shows in [Nee91, §
1, Case 2]that the map of triangles X (cid:15) (cid:15) u / / Y (cid:15) (cid:15) v / / Z w (cid:15) (cid:15) w / / Σ X (cid:15) (cid:15) Y v / / Z w / / Σ X − Σ u / / Σ Y is good if and only if the given triangle is contractible. By Proposition 7.13, this is equivalent tothe map (0 , , w ) being middling good. However, as we observed in the proof of Proposition 8.1,this map of triangles is chain homotopic to the map X u (cid:15) (cid:15) u / / Y v (cid:15) (cid:15) v / / Z w (cid:15) (cid:15) w / / Σ X Σ u (cid:15) (cid:15) Y v / / Z w / / Σ X − Σ u / / Σ Y. This map ( u, v, w ) is always middling good, but is good if and only if it is Verdier good if and onlyif the given triangle is contractible, by Proposition 8.1.As a consequence, we deduce the following result.
Corollary 8.4. (1) There exist maps of triangles which are not middling good.(2) There exist maps of triangles which are middling good but neither good nor Verdier good.(3) Middling goodness is not invariant under chain homotopy.Proof.
Consider a non-contractible triangle in Example 8.3. (We will give some non-contractibletriangles in Examples 8.5 and 8.6.) Then (0 , , w ) is not middling good, proving (1). And ( u, v, w )is middling good, but is neither good nor Verdier good, proving (2). These two maps are chainhomotopic, proving (3). (cid:3) Example 8.5.
In the derived category of the integers D ( Z ), consider the rotation of the map oftriangles in Example 4.2: Z [0] (cid:15) (cid:15) n / / Z [0] (cid:15) (cid:15) q n / / Z /n [0] ǫ n (cid:15) (cid:15) ǫ n / / Z [1] (cid:15) (cid:15) Z [0] q n / / Z /n [0] ǫ n / / Z [1] − n / / Z [1] . All maps Z [1] → Z /n [0] are zero for degree reasons, so the fill-in ǫ n is not a lightning flash. Itfollows that this triangle is not contractible, that (0 , , ǫ n ) is not middling good, etc. One can alsoshow directly that (0 , , ǫ n ) does not extend to a 4 × Example 8.6.
In the stable homotopy category, consider the mod n Moore spectrum M ( n ) for n ≥
2, sitting in an exact triangle S n / / S q / / M ( n ) δ / / S . By the long exact sequence of homotopy groups, the connecting map δ induces0 = π ( δ ) : π M ( n ) → π S . It follows that δ is not a lightning flash when regarded as part of a map (0 , , δ ) of triangles. Again,this implies that this triangle is not contractible, etc. N GOOD MORPHISMS OF EXACT TRIANGLES 29
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