The morphism axiom for n-angulated categories

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by  Emilie Arentz-Hansen

THEMORPHISMAXIOMFORn-ANGULATEDCATEGORIES
EMILIEARENTZ-HANSEN,PETTERANDREASBERGH,ANDMARIUSTHAULE
ABSTRACT. Weshowthatthemorphismaxiomforn-angulatedcategoriesisredundant.
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2 1. INTRODUCTION

n In[3],Geiss, Keller andOppermannintroducedn-angulatedcategories. Theseare
a generalizations of triangulated categories, in the sense that triangles are replaced by
J
n-angles,thatis,morphismsequencesoflengthn.Thusa3-angulatedcategoryispre-

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ciselyatriangulatedcategory.
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Therearebynownumerousexamples ofsuchgeneralizedtriangulated categories.

] They appear for example as certain cluster tilting subcategories of triangulated cate-
T gories.Recently,in[4],Jassointroducedthenotionofalgebraicn-angulatedcategories.
C SuchacategoryisbydefinitionequivalenttoastablecategoryofaFrobeniusn-exact
. category,thelatterbeingagenralizationofanexactcategory;exactsequencesarere-
h
t placedbyso-calledn-exactsequences. Algebraicn-angulatedcategoriesaretherefore
a
naturalgeneralizationsofalgebraictriangulatedcategories,whicharedefinedascate-
m
goriesequivalenttostablecategoriesofFrobeniusexactcategories. By[4,Section6.5],
[
then-angulatedcategoriesthatariseasclustertiltedsubcategoriesofalgebraictrian-

1 gulated categories are themselves algebraic. Actually, the only examples so far of n-
v angulatedcategoriesthatarenotalgebraicaretheratherexoticonesin[1]. Atthetime
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ofwriting,thereisnonotionoftopologicaln-angulatedcategories,exceptinthetrian-
1
gulatedcase.
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7 Theaxiomsdefiningn-angulatedcategoriesaregeneralizedversionsoftheaxioms
0 defining triangulated categories. Among these axioms, themorphism axiom and the
. generalizedoctahedralaxiomaretheonesthatguaranteeaninterestingtheory. Apart
1
0 fromtheexistenceaxiom,whichstatesthateverymorphismispartofann-angle,the
6 otheraxiomsare“bookkeepingaxioms,”toborrowPaulBalmer’sterminology.Thegen-
1 eralized octahedral axiom was introduced in [2], but in this paper we present it in a
:
v muchmorecompactandreadableform.
i Thetopic ofthis paper isthe morphism axiom, whichstates that a morphism be-
X
tweenthebasesoftwon-anglescanbeextendedtoamorphismofn-angles. Weshow
r
a thatthisaxiomisredundant;itfollowsfromthegeneralizedoctahedralaxiomandthe
existenceaxiom.Fortriangulatedcategories,thiswasprovedbyMayin[5].
2. THEAXIOMSFORn-ANGULATEDCATEGORIES
Inthissection, werecalltheaxiomsforn-angulated categories. Wefixanadditive
categoryC withanautomorphismΣ:C →C,andanintegerngreaterthanorequalto
three.
AsequenceofobjectsandmorphismsinC oftheform
A −α→1 A −α→2 ···−α−n−−→1 A −α−→n ΣA
1 2 n 1
Date:January27,2016.
2010MathematicsSubjectClassification. 18E30.
Keywordsandphrases. Triangulatedcategories,n-angulatedcategories,morphismaxiom.
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2 EMILIEARENTZ-HANSEN,PETTERANDREASBERGH,ANDMARIUSTHAULE
iscalledann-Σ-sequence. Whenever convenient, weshalldenotesuchsequences by
A•,B•etc.TheleftandrightrotationsofA•arethetwon-Σ-sequences
A −α→2 A −α→3 ···−α−→n ΣA −(−−−1)−n−Σ−α→1 ΣA
2 3 1 2
and
Σ−1An−(−−−1−)n−Σ−−−1−α→n A1−α→1 ···−α−n−−→2 An−1−α−n−−→1 An
respectively,andatrivialn-Σ-sequenceisasequenceoftheform
A→−1 A→0→···→0→ΣA
oranyofitsrotations.
A morphism A• −→ϕ B• of n-Σ-sequences is a sequence ϕ=(ϕ1,ϕ2,...,ϕn) of mor-
phismsinC suchthatthediagram
A α1 A α2 A α3 ··· αn−1 A αn ΣA
1 2 3 n 1
ϕ1 ϕ2 ϕ3 ϕn Σϕ1
B β1 B β2 B β3 ··· βn−1 B βn ΣB
1 2 3 n 1
commutes. Itisanisomorphismifϕ ,ϕ ,...,ϕ areallisomorphismsinC,andaweak
1 2 n
isomorphismifϕi andϕi+1 areisomorphisms forsome1≤i ≤n (withϕn+1:=Σϕ1).
Notethatthecompositionoftwoweakisomorphismsneednotbeaweakisomorphism.
Also,notethatiftwon-Σ-sequences A•andB•areweaklyisomorphicthroughaweak
isomorphism A•−→ϕ B•,thentheredoesnotnecessarilyexistaweakisomorphismB•→
A•intheoppositedirection.
Let N be a collection of n-Σ-sequences in C. Then the triple (C,Σ,N ) is an n-
angulatedcategoryifthefollowingfouraxiomsaresatisfied:
(N1) (a) N isclosedunderdirectsums,directsummandsandisomorphismsofn-
Σ-sequences.
(b) ForallA∈C,thetrivialn-Σ-sequence
A→−1 A→0→···→0→ΣA
belongstoN .
(c) Foreachmorphismα: A →A inC,thereexistsann-Σ-sequenceinN
1 2
whosefirstmorphismisα.
(N2) Ann-Σ-sequencebelongstoN ifandonlyifitsleftrotationbelongstoN .
(N3) Giventhesolidpartofthecommutativediagram
A α1 A α2 A α3 ··· αn−1 A αn ΣA
1 2 3 n 1
ϕ1 ϕ2 ϕ3 ϕn Σϕ1
B β1 B β2 B β3 ··· βn−1 B βn ΣB
1 2 3 n 1
with rows in N , the dotted morphisms exist and give a morphism of n-Σ-
sequences.
(N4) Giventhesolidpartofthediagram
A1 α1 A2 α2 A3 α3 A4 α4 ··· αn−2 An−1 αn−1 An αn ΣA1
ϕ2 ϕ3 ψ4 ϕ4 ϕn−1 ψn ϕn
A1 β1 B2 β2 B3 β3 B4 β4 ··· βn−2 Bn−1 βn−1 Bn βn ΣA1
α1 θ3 θ4 θn−1 θn Σα1
A2 ϕ2 B2 γ2 C3 γ3 C4 γ4 ··· γn−2 Cn−1 γn−1 Cn γn ΣA2
THEMORPHISMAXIOMFORn-ANGULATEDCATEGORIES 3
withcommutingsquaresandwithrowsinN ,thedottedmorphismsexistsuch
thateachsquarecommutes,andthen-Σ-sequence
−α4 0 −α5 0 0
α3 ϕ4−β3 −ϕ5−β4 0
A −−ϕ−3→A ⊕B −"−−ψ−4−−−θ−3→# A ⊕B ⊕C −"−−ψ−5−−−θ−4−γ−3→#
3 h i 4 3 5 4 3
−α5 0 0 −α6 0 0
−ϕ5−β4 0 ϕ6−β5 0
−"−−ψ−5−−−θ−4−γ−3→# A ⊕B ⊕C −"−−ψ−6−−−θ−5−γ−4→# ···
6 5 4
−αn−1 0 0
(−1)n−1ϕn−1 −βn−2 0 (−1)nϕn −βn−1 0
···−"−−−−ψ−n−−−1−−−−−θn−−−2−γ−n−−−3→# An⊕Bn−1⊕Cn−2−·−−−ψ−n−−−−−θn−−−1−γ−n−−−2→¸
(−1)nϕn −βn−1 0
−·−−−ψ−n−−−−−θn−−−1−γ−n−−−2→¸ Bn⊕Cn−1−[−θ−n−γ−n−−−1→] Cn−Σ−α−2−◦−γ→n ΣA3
belongstoN .
Inthiscase,thecollectionN isann-angulationofthecategoryC (relativetothe
automorphism Σ), and the n-Σ-sequences in N are n-angles. If (C,Σ,N ) satisfies
(N1), (N2)and (N3), then N isapre-n-angulation, andthe triple(C,Σ,N ) isapre-
n-angulatedcategory.
Remark2.1. (1)In[3],thefourthdefiningaxiomisthe“mappingconeaxiom”,which
says that in the situation of (N3), the morphisms ϕ ,...,ϕ can be chosen in such a
3 n
waythatthemappingconeof(ϕ ,ϕ ,...,ϕ )belongstoN . Whenn=3,thatis,inthe
1 2 n
triangulated case, itwasshownbyNeemanin [6,7]thatthis mapping cone axiomis
equivalenttotheoctahedralaxiom,whentheotherthreeaxiomshold.
Whenn=3,thediagraminouraxiom(N4)ispreciselytheoctahedraldiagram,and
theaxiomistheoctahedralaxiom.Thus(N4)canbethoughtofasahigheranalogueof
theoctahedralaxiom.Itwasshownin[2]thatwhen(N1),(N2)and(N3)hold,then(N4)
andthemappingconeaxiomsareequivalent.1 Thusourdefinitionofann-angulated
categoryisequivalenttothatofGeiss,KellerandOppermannin[3].
(2)LetBbeanobjectinC.Anyn-Σ-sequence A•inducesaninfinitesequence
···→(B,Σ−1A )−(−Σ−−−1α−n−)→∗ (B,A )−(−α−1)→∗ (B,A )→···→(B,A )−(−α−n−)→∗ (B,ΣA )→···
n 1 2 n 1
ofabeliangroupsandhomomorphisms, where(B,A)denotesHomC(B,A). Ifthisse-
quenceisexactforallB, then A• iscalledanexact n-Σ-sequence. By[3, Proposition
2.5],everyn-angleinapre-n-angulatedcategoryisexact.
Now modify axiom (N1) into a new axiom (N1*), whose parts (b) and (c) are un-
changed, but with the following part (a): if A• →B• is a weak isomorphism of exact
n-Σ-sequences with A• ∈N , then B• belongs to N . Moreover, let (N2*) be the fol-
lowingweakversionofaxiom(N2): theleftrotationofevery n-Σ-sequenc inN also
belongstoN .Thenby[2,Theorem3.4],thefollowingareequivalent:
(i) (C,Σ,N )satisfies(N1),(N2),(N3),
(ii) (C,Σ,N )satisfies(N1*),(N2),(N3),
(iii) (C,Σ,N )satisfies(N1*),(N2*),(N3).
Itisnotknownwhether onlyaxiom(N2)canbereplacedby(N2*), without replacing
(N1)by(N1*).
1WethankGustavoJassoforshowingusthecompactdiagramwehaveusedinaxiom(N4).Thisaxiomis
notstrictlythesameasaxiom(N4*)in[2].However,itfollowsfromtheproofsin[2,Section4]thatthetwoare
equivalent.
4 EMILIEARENTZ-HANSEN,PETTERANDREASBERGH,ANDMARIUSTHAULE
3. AXIOM(N3)ISREDUNDANT
Inthissectionweshowthatthemorphismaxiom,i.e.axiom(N3),followsfromthe
other axioms. In light of Remark2.1(2), it would perhaps seem natural to ask which
axiomsoneshouldusetoprovethis.Inotherwords,shouldoneuseaxioms(N1),(N2),
(N4),axioms(N1*),(N2),(N4),oraxioms(N1*),(N2*),(N4)? Theansweristhatitdoes
notmatter.Infact,axiom(N3)isaconsequenceof(N1)(c)and(N4).
Theorem3.1. Axiom(N3)followsfrom(N1)(c)and(N4).
Proof. Assumethatweareinthesituationofaxiom(N3),i.e.thatwearegiventhesolid
partofthecommutativediagram
A α1 A α2 A α3 ··· αn−1 A αn ΣA
1 2 3 n 1
ϕ1 ϕ2 ϕ3 ϕn Σϕ1
B β1 B β2 B β3 ··· βn−1 B βn ΣB
1 2 3 n 1
withrowsinN .Wemustprovethatdottedmorphismsϕ ,...,ϕ existandgiveamor-
3 n
phism(ϕ ,ϕ ,ϕ ,...,ϕ )ofn-Σ-sequences.
1 2 3 n
Byassumption,theequalityϕ ◦α =β ◦ϕ holds;wedefineγ tobethismorphism,
2 1 1 1 1
i.e.γ =ϕ ◦α =β ◦ϕ .Nowapplyaxiom(N1)(c)tothethreemorphisms
1 2 1 1 1
γ : A →B , ϕ : A →B , ϕ : A →B
1 1 2 2 2 2 1 1 1
inC,andobtainthreen-Σ-sequences
A −γ→1 B −γ→2 C −γ→3 ···−γ−n−−→1 C −γ→n ΣA ,
1 2 3 n 1
A −ϕ−→2 B −δ→2 D −δ→3 ···−δ−n−−→1 D −δ→n ΣA
2 2 3 n 2
and
A −ϕ−→1 B −ǫ→2 E −ǫ→3 ···−ǫ−n−−→1 E −ǫ→n ΣA
1 1 3 n 1
inN .Next,considerthetwodiagrams
A1 α1 A2 α2 A3 α3 A4 α4 ··· αn−2 An−1 αn−1 An αn ΣA1
ϕ2 ρ3 ρ4 ρn−1 ρn
A1 γ1 B2 γ2 C3 γ3 C4 γ4 ··· γn−2 Cn−1 γn−1 Cn γn ΣA1
α1 θ3 θ4 θn−1 θn Σα1
A2 ϕ2 B2 δ2 D3 δ3 D4 δ4 ··· δn−2 Dn−1 δn−1 Dn δn ΣA2
and
A1 ϕ1 B1 ǫ2 E3 ǫ3 E4 ǫ4 ··· ǫn−2 En−1 ǫn−1 En ǫn ΣA1
β1 ψ3 ψ4 ψn−1 ψn
A1 γ1 B2 γ2 C3 γ3 C4 γ4 ··· γn−2 Cn−1 γn−1 Cn γn ΣA1
ϕ1 η3 η4 ηn−1 ηn Σϕ1
B1 β1 B2 β2 B3 β3 B4 β4 ··· βn−2 Bn−1 βn−1 Bn βn ΣB1
Notethatalltherowsinthesetwodiagramsaren-Σ-sequencesinN .Moreover,itfol-
lowsfromthedefinitionofthemorphismγ thatthesolidpartsofthediagramscom-
1
mute.Wemaythereforeapplyaxiom(N4);thereexistdottedmorphismssuchthateach
THEMORPHISMAXIOMFORn-ANGULATEDCATEGORIES 5
squarecommutes.Axiom(N4)alsogivesothermorphisms,thediagonalones,butthese
arenotneededhere.
Thetwodiagramssharethesamemiddlerow.Wemaythereforecombinetheupper
partofthefirstdiagramandthelowerpartofthesecond,andobtainthecommutative
diagram
A1 α1 A2 α2 A3 α3 A4 α4 ··· αn−2 An−1 αn−1 An αn ΣA1
ϕ2 ρ3 ρ4 ρn−1 ρn
A1 γ1 B2 γ2 C3 γ3 C4 γ4 ··· γn−2 Cn−1 γn−1 Cn γn ΣA1
ϕ1 η3 η4 ηn−1 ηn Σϕ1
B1 β1 B2 β2 B3 β3 B4 β4 ··· βn−2 Bn−1 βn−1 Bn βn ΣB1
Finally,byomittingthemiddlerow,weobtainthecommutativediagram
A1 α1 A2 α2 A3 α3 A4 α4 ··· αn−2 An−1 αn−1 An αn ΣA1
ϕ1 ϕ2 η3◦ρ3 η4◦ρ4 ηn−1◦ρn−1 ηn◦ρn Σϕ1
B1 β1 B2 β2 B3 β3 B4 β4 ··· βn−2 Bn−1 βn−1 Bn βn ΣB1
Nowforeach3≤k≤n,defineamorphismϕ byϕ =η ◦ρ .Then(ϕ ,ϕ ,ϕ ,...,ϕ )
k k k k 1 2 3 n
isamorphismofn-Σ-sequences,andthiscompletestheproof. (cid:3)
Remark3.2. (1)Asnotedintheproof,notallofaxiom(N4)isneeded. Thediagonal
morphismsprovidedbytheaxiomplaynorole,and,consequently,neitherdoesthen-
Σ-sequenceinvolvingthesemorphisms.
(2)Notethatifn=3,thatis,inthetriangulatedcase,theresultrecoversthatofMay
in[5],butwithaslightlydifferentsetup.Indeed,May’sresultwastheinspirationforthe
theorem.
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DEPARTMENTOFMATHEMATICALSCIENCES,NTNU,NO-7491TRONDHEIM,NORWAY
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