How many adjunctions give rise to the same monad?

0.14 MB·English

by  Andrew Salch

HOW MANY ADJUNCTIONS GIVE RISE TO THE SAME MONAD?
6 ANDREWSALCH
1
0
2 Abstract. Given an adjoint pair offunctors F,G, thecompositeGF naturally gets the
structureofamonad. Thesamemonadmayarisefrommanysuchadjointpairsoffunc-
n tors,however.Canonedescribealloftheadjunctionsgivingrisetoagivenmonad?Inthis
u paperwesingleoutaclassofadjunctionswithespeciallygoodproperties,andwedevelop
J methodsforcomputingallsuchadjunctions,uptonaturalequivalence,whichgiverisetoa

9 givenmonad.Todemonstratethesemethods,weexplicitlycomputethefinitaryhomolog-
2 icalpresentationsofthefreeA-modulemonadonthecategoryofsets,forAaDedekind
domain. Wealsoproveacriterion,reminiscentofBeck’smonadicitytheorem,forwhen

] thereisessentially(inaprecisesense)onlyasingleadjunctionthatgivesrisetoagiven
T monad.
C
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t
a
m Contents
[ 1. Introduction. 1

2 2. Homologicalpresentationsof a monadare equivalentto repletereflective
v subcategoriesofitsEilenberg-Moorecategory. 3
9 2.1. Preliminarydefinitions. 3
3
2.2. Replete reflectivesubcategoriespresentinga monadare equivalentto its
7
homologicalpresentations. 7
1
0 2.3. Coordinatizationofthecollectionofhomologicalpresentationsofamonad. 9
. 3. Acriterionforuniquehomologicalpresentabilityofamonad. 10
1
3.1. Preliminarydefinitions. 10
0
5 3.2. Acriterionforuniquehomologicalpresentability. 11
1 4. Explicitexamples:Dedekinddomains. 13
v: References 15
i
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a
1. Introduction.
IfC,DarecategoriesandG :D ÑC afunctorwithaleftadjointF,thenthecomposite
GF gets the structure of a monad. However, given a monad T : C Ñ C, there may be
manycategoriesD andadjointpairsF,GsuchthatGF “T asamonad.Wewillcallsuch
achoiceofD,F,andGapresentationforT.
It haslong been knownthat, amongall presentationsof a givenmonadT, there is an
initialpresentation,theKleislicategoryofT, andaterminalpresentation,theEilenberg-
Moore category of T. Furthermore, Beck’s monadicity theorem gives a necessary and
sufficientconditiononG forthepresentationpD,F,GqtobeequivalenttotheEilenberg-
Moorecategory. (See [5] for a nice expositionof these ideas.) Beck’s resulthas proven
very useful, e.g. in algebraic geometry where, in its dual form for comonads, it is the
foundationforthegeneraltheoryofdescent;see[2].
1
2 ANDREWSALCH
The applications of Beck’s monadicity theorem have made it very clear that, given a
presentation pD,F,Gq of a monad T, it is very useful to be able to tell when pD,F,Gq
istheterminalpresentation(i.e.,theEilenberg-Moorecategory)ofT. However,wehave
long wondered about what to do when one encounters a presentation of a monad which
is notthe terminal(Eilenberg-Moore)presentation, and noteventhe initial (Kleisli) pre-
sentation. Can one describe the collection of all presentationsof a monad? Even better,
can one establish some kind of coordinatesystem on the collectionof presentationsof a
monad,sothatwhenoneencountersapresentationofamonad(oracomonad,e.g.forap-
plicationsindescenttheory),onecangivesomekindof“coordinates”thatdescribewhere
thispresentationsits inrelationtotheinitialandterminalpresentations,andalltheother
presentations,ofthesamemonad?
Inthispaperwestudythecollectionofpresentationsofagivenmonad,butwithare-
strictiononwhatpresentationswearewillingtoconsider.Thisisbecause,givenapresen-
tationpD,F,Gq,onecantriviallyproducemanymorepresentationsbytakingtheCartesian
productof D with any small category. We regardthese presentationsas degenerate, and
wewanttodisregardpresentationswiththiskindofredundantinformationinthem. Con-
sequently,inDefinition2.2,wemakethedefinitionthatapresentationpD,F,Gqissaidto
behomologicalifeveryobjectX ofD canberecoveredfromtheF,G-barconstructionon
X(seeDefinition2.2fortheprecisedefinition).Thisdefinitioneliminatesthe“redundant”
presentationswewantedtoexclude,andhassomeothergoodproperties,describedinRe-
mark2.3. We alsorestrictourattentiontowhatwe call“coequalizable”monads, thatis,
those monadsT for which the Eilenberg-Moorecategoryhas coequalizers; this property
issatisfied inallcasesofinterestwhichweknowof,andinRemark2.9weexplainabit
aboutwhythatis.
Oncethesedefinitionsaremade,wecanprovesomenicetheorems:
‚ In Theorem2.13, we prove that, if T is coequalizable, then the categoryof nat-
ural equivalence classes of homological presentations of T is equivalent to the
partially-ordered collection of reflective replete subcategories of the Eilenberg-
MoorecategoryCT whichcontaintheKleislicategoryC .
T
Thismeansthecategoryofallnaturalequivalenceclassesofhomologicalpre-
sentations of T is always well-behaved in at least one way: it can’t be just any
arbitrarycategory,rather, it is always partially-ordered(i.e., there is atmost one
morphismfromanygivenobjecttoanyothergivenobject).
‚ In Theorem2.20, when T is coequalizableand CT has a biproductand is Krull-
Schmidt,we actuallyconstructa“coordinatesystem” onthe naturalequivalence
classes of homologicalpresentationsof T! Any homologicalpresentation is de-
termineduniquely(uptonaturalequivalence)byspecifyingasubcollectionofthe
collectionofisomorphismclassesofindecomposableobjectsofCT. Insomeprac-
ticalcases, CT isthecategoryoffinitelygeneratedmodulesoveranalgebra,and
thentheverticesoftheAuslander-ReitenquiverofCT actas“coordinates”forthe
collection of natural equivalence classes of homologicalpresentations of T; see
Remark2.22.
‚ In Theorem 3.6, we give a simple and usable criterion for the triviality (up to
natural equivalence) of the collection of homological presentations of T, i.e., a
criterion for when, up to natural equivalence, there exists only one homological
presentationofT (necessarilytheEilenberg-MoorecategoryofT). Asanexam-
ple, in Corollary3.8we showthatthe base-changemonadonmodulecategories
HOWMANYADJUNCTIONSGIVERISETOTHESAMEMONAD? 3
associatedtoafieldextensionhasthispropertyofuniquehomologicalpresentabil-
ity.
‚ InTheorem4.4, we use Theorems2.13andTheorem2.20to explicitlycompute
thecollectionofall(naturalequivalenceclassesof)homologicalfinitarypresenta-
tionsofeachmonadinoneparticularclassofmonads:namely,letAbeaDedekind
domain,andletT denotethemonadonthecategoryofsetswhichsendsasetto
theunderlyingsetofthefree A-moduleitgenerates. We showthatthepartially-
orderedsetofnaturalequivalenceclassesoffinitaryhomologicalpresentationsof
T isequivalenttothesetoffunctions
MaxSpecpAqÑt0,1,2,...,8u
from the set MaxSpecpAq of maximal ideals of A to the set of extended natu-
ral numbers, under the partial ordering in which we let f ď g if and only if
fpmqďgpmqforallmPMaxSpecpAq. SincetheringofintegersZisaDedekind
domain, thisis a veryfundamentalexample. (Here a presentationforT is “fini-
tary”ifitsrightadjointfunctorpreservesfilteredcolimits;thisisaconditionthat,
roughly speaking, guarantees that the data of the presentation is determined by
“finiteinput.” SeeRemark2.5.)
In manycases of interest (e.g. the base-changemonadson modulecategoriesassoci-
ated to mapsof ringsor mapsof schemes), the Eilenberg-MoorecategoryCT is actually
abelian, hence T is coequalizable and CT has a biproduct automatically, and frequently
CT is actually quite computable and understandable. Under those circumstances our re-
sultsseemtobefairlyuseful,andaswehopeTheorem4.4demonstrates,theyareactually
applicableandgiveexplicitnontrivialresultsinconcretesituationsofinterest.
Weareverygratefultotheanonymousrefereeforperspicaciouscommentswhichhelped
ustoimprovethispapergreatly.
2. Homologicalpresentationsofamonadareequivalenttorepletereflective
subcategoriesofitsEilenberg-Moorecategory.
2.1. Preliminary definitions. Throughoutthis paper, when T is a monad, when conve-
nientwewillsometimesalsowriteT fortheunderlyingfunctorofthemonad.
Definition2.1. LetC beacategory,T amonadonC. IfD isacategoryequippedwitha
functorF1 : C Ñ D andarightadjointG1 forF1 suchthattheassociatedmonadG1F1 is
equaltoT,wecallthedatapD,F1,G1qapresentationofT. Sometimesweshalljustwrite
D asshorthandforpD,F1,G1q,whenF1,G1areclearfromcontext.
The collection ofallpresentationsof T forms a large category, whose morphismsare
morphismsofadjunctions(see IV.7of[5] forthe definitionofmorphismsofadjunctions)
which are the identity on C. We call this large category the categoryof presentationsof
T, and for which we will write PrespTq. One also can consider naturaltransformations
between morphisms of adjunctions, and we regard two morphisms as homotopicif there
existsaninvertible-up-to-isomorphismnaturaltransformationbetweenthem;wewillwrite
HopPrespTqq for the category of presentations of T but whose morphisms are homotopy
classesofmorphismsofadjunctions.
We note that PrespTq and HopPrespTqq are not necessarily categories, but are large
categories, because their hom-collectionsare not necessarily hom-sets. The notion of a
category of presentations of a monad appears in VI.5 of [5], but was not there given a
name.
4 ANDREWSALCH
For the purposes of this paper we will mostly be studying presentations of a monad
which have the propertythat there is some minimaldegree of compatibilitybetween the
categoryandthemonad,enoughcompatibilitytoguaranteethate.g.someconstructionsin
homologicalalgebracanbemade.Hereisourdefinition:
Definition2.2. LetC beacategory,T amonad,pD,F1,G1qapresentationofT. Wewill
saythatD isahomologicalpresentationofT if,foreveryT-algebraρ :G1F1X Ñ X,the
coequalizer(inDqofthetwonaturalcounitmaps
(2.1.1) ǫF1X,F1ρ:F1G1F1X ÑF1X
exists,andforeachobjectY ofD,thecanonicalmap
(2.1.2) coeqtǫF1G1Y,F1G1ǫYuÑY
isanisomorphism. WewillwriteHPrespTqforthefulllargesubcategoryofPrespTqgen-
eratedbythehomologicalpresentations.
WewillwriteHopHPrespTqqforthelargecategorywhoseobjectsarehomologicalpre-
sentationsofT,andwhosemorphismsarehomotopyclassesofmorphismsofadjunctions.
Remark2.3. Thereasonforthename“homological”forthiskindofpresentationisthe
following: ifpD,F1,G1qisahomologicalpresentationofamonadandD isabelian,then
eachobjectXinD admitsacanonicalresolution
(2.1.3) 0ÐX ÐF1G1X Ðd0Ý F1G1kerpǫF1G1X ´F1G1ǫXqÐd1Ý...
obtainedbyrepeatedlyapplyingF1G1,formingthecoequalizer2.1.2,andtakingthekernel
ofthecoequalizermap.Thisresolutiongivesusawaytocomputetheleft-derivedfunctors
of any functor on D which is acyclic on every object of the form F1G1X. If pD,F1,G1q
failstobehomological,thenatleastforsomeobjectsXthechaincomplex2.1.3failstobe
exactandhencecannotbeusedtocomputederivedfunctorsinthisway.
Theresolution2.1.3isveryfamiliarandcommonplaceinitsvariousspecialcases. For
example, when C is the category of sets and T the monad given on a set S by taking
the underlying set of the free abelian group generated by S, then the category Ab is a
presentationforT, anditishomological(becauseitistheterminalpresentation,i.e., the
Eilenberg-MoorecategoryofT,whichinCorollary2.14weproveisalwayshomological
foranycoequalizablemonadT).Theresolution2.1.3istheelementaryresolutiononeuses
inafirstcourseinhomologicalalgebratoprovethatfreeresolutionsexistinthecategory
of abelian groups: given an abelian group X, one can form the direct sum ‘ Z, one
xPX
canlet X bethekerneloftheobvioussurjection‘ Z Ñ X,theniteratetoformafree
0 xPX
resolutionofX.
WhenD failstobeabelian,insteadof2.1.3oneformsthesimplicialresolution
... // F1G1F1G1F1G1X // F1G1F1G1X // F1G1X
of X, and one can use this resolution to computemore generalkindsof derivedfunctors
(e.g.ifDhasthestructureofamodelcategory).IneverycasetheconditionthatpD,F1,G1q
ishomologicalisreallytheconditionthatF1,G1 givesusawaytoformacanonicalreso-
lutionofanyobjectin D. In somesense oneshouldthinkofa homologicalpresentation
for a monad T as a category equipped with a way of forming a canonical resolution of
anyobjectbyT-freeobjects,andthatmeansthatthispaperisinsomesensereallyabout
classifyingvariouswaysofformingcanonicalresolutions.
HOWMANYADJUNCTIONSGIVERISETOTHESAMEMONAD? 5
Finally,onemorenoteaboutthemap2.1.2:afterapplyingG1,themapalwaysbecomes
anisomorphism,becausethecofork
G1F1G1F1G1XG1ǫF1G1X ////G1F1G1X //G1X
G1F1G1ǫX
isalwayssplitbytheunitmapηG1X :G1X ÑG1F1G1X,hencethecoforkisasplitcoequal-
izer. Butthemap2.1.2canfailtobeanisomorphismbeforeapplyingG1.
Informally, the general trend is that monads tend to either have a single homological
presentation, up to equivalence, or a truly enormouscollection of homologicalpresenta-
tions,bigenoughtomakeitverydifficulttoexplicitlyclassifythem.However,evenwithin
thehomologicalpresentations,thereisanevenmorerestrictedclassofpresentationsofa
monadwhichwecanreasonablyrestrictourattentionto,namely,thefinitaryhomological
presentations:
Definition2.4. LetC beacategory,T amonad,pD,F1,G1qapresentationofT. Wewill
saythatD isafinitarypresentationofT ifG1 preservesallfilteredcolimitswhichexistin
D.
WewillwriteFinPrespTqforthelargecategoryoffinitarypresentationsofT,FinHPrespTq
forthelargecategoryoffinitaryhomologicalpresentations,andHopFinHPrespTqqforthe
largecategoryoffinitaryhomologicalpresentationsuptonaturalequivalence.
Remark2.5. Inthe categoryofmodulesoveraring,everyobjectisafilteredcolimitof
finitelygeneratedmodules;consequently,whenpD,F1,G1qisafinitarypresentationandD
a categoryof modulesovera ring, thenG1 can be computedon any objectif one knows
how to computeG1 on finitely generated modules. This is actually quite useful; see the
proofofTheorem4.4,forexample.
Remark2.6. ItisnotingeneraltruethateverypresentationofamonadT isfinitary,even
ifT itselfpreservesfilteredcolimits;forexample,letC bethecategoryofsets,letT bethe
monadwhichsendsasetS totheunderlyingsetofthefreeabeliangroupgeneratedbyS,
andletpD,F1,G1qbethepresentationforT inwhichD isthecategoryofreducedabelian
groups, F1 : C Ñ D is the free abelian groupfunctor, andG1 : D Ñ D is the forgetful
functor.Then,foranyprimenumberp,thecolimitofthefiltereddiagram
p p
ZÝÑZÝÑ...
inD iszero,butthecolimitof
G1pZqGÝ1ÑppqG1pZqGÝ1Ñppq...
inC istheunderlyingsetofZr1s.
p
Recall that a subcategory is said to be replete if it contains every object isomorphic
to one of its own objects, and reflective if it a full subcategory and the inclusion of the
subcategoryadmitsaleftadjoint.Weincludefullnessaspartofthedefinitionofareflective
subcategory;thisseemstobe relativelystandard,althoughnotuniversal(seee.g.[5]), in
theliterature.
Definition2.7. LetC beacategory,T amonad,CT theEilenberg-MoorecategoryofT-
algebras. IfD isarepletereflectivesubcategoryofCT,wewillsaythatD presentsT ifD
containsallthefreeT-algebras,i.e.,ifD containstheT-algebraTTX ݵÑX TX forevery
objectXofC.
6 ANDREWSALCH
WewillwriteLocpTqforthepartially-orderedcollectionofallrepletereflectivesubcat-
egoriesCT whichpresentT.
Finally,wewillsaythatanelementD ofLocpTqisfinitaryiftheforgetfulfunctorfrom
D toCT preservesallfilteredcolimitswhichexistinD. We willwrite FinLocpTqforthe
subcollectionofLocpTqconsistingofthefinitaryelements.
Thenotation“LocpTq” ismotivatedbythenotionthatarepletereflectivesubcategory
ofacategoryis,speakingintuitivelyandroughly,akindof“localization”ofthatcategory.
WenotethatLocpTqisnotnecessarilyaset,norevenaclass(wearegratefultoMike
Shulman for pointing out to us that the collection of subcategories of a category is not
necessarily a class!). Sometimes the term “conglomerate” is used for a collection too
large to form a class. In other words, if one wants to use Grothendieck universes, one
mustexpandthe universetwice to go fromsets to conglomerates. In practicalalgebraic,
geometric, and topological situations, however, it seems likely that LocpTq will form a
set. For example, see Corollary 2.16, where we show that mild conditions (a biproduct
condition, a Krull-Schmidtcondition, and a smallness condition)imply that LocpTq is a
set.
Finally, itwill sometimesbeconvenientto havecoequalizersin Eilenberg-Moorecat-
egories. We introduce a definition which describes monads which have this agreeable
property:
Definition 2.8. Let C be a category, T a monad, CT the Eilenberg-Moore category of
T-algebras.WewillsaythatT iscoequalizableifCT hascoequalizers.
Remark2.9. TherearemanyknownconditionsonT whichguaranteethatT iscoequaliz-
able;forexample,inLemmaII.6.6in[3]itisshownthat,ifT preservesreflexivecoequal-
izers,thenT iscoequalizable. Consequently,manyinterestingexamplesofmonadsT are
coequalizable.
For example, suppose R Ñ S is a map of commutativerings. Then the base-change
monad T : ModpRq Ñ ModpRq, i.e., the composite of the extension of scalars func-
tor ModpRq Ñ ModpSq with the restriction of scalars functor ModpSq Ñ ModpRq, is
coequalizable,sinceextensionofscalarsandrestrictionofscalarsarebothrightexact,pre-
serving all coequalizers. If S is finitely generated as an R-module then the base-change
monadfgModpRqÑfgModpRqonthefinitelygeneratedmodulecategoryisalsocoequal-
izable,forthesamereason. ThentheEilenberg-MoorecategoryModpRqT isequivalentto
ModpSq.
More generally,if f : Y Ñ X is a map of schemesand QCModpO q the categoryof
X
quasicoherentO -modules,then the base-changemonad f f˚ is coequalizableif f is an
X ˚
affinemorphism,sinceinthatcase f isrightexact(and f˚isalwaysrightexact,regardless
˚
ofwhether f isaffine).ThentheEilenberg-MoorecategoryQCModpOXqf˚f˚ isequivalent
toQCModpO q,bytheresultsofEGAII.1.4,[4].
Y
Usually(e.g. intheexamplesabove,andinourTheorem4.4)wewillhaveanexplicit
description of the category CT and we will know that it has coequalizers; what will be
interestingandnewwillbethedescriptionoftherestofHPrespTq.
Definition2.10. IfC isacategorywithcoproduct‘,we saythatanobject X ofC isin-
decomposable if X – Y ‘Z implies either Y – 0 or Z – 0. We say that C is weakly
Krull-Schmidt if every object X of C admits a decomposition into a finite coproduct of
indecomposableobjects,andthatdecompositionisuniqueuptopermutationofthesum-
mands.
HOWMANYADJUNCTIONSGIVERISETOTHESAMEMONAD? 7
Definition2.10differsfromtheusualdefinitionofaKrull-Schmidtcategoryinthatwe
donotrequiretheindecomposableobjectstohavelocalendomorphismrings.
2.2. Repletereflectivesubcategoriespresentingamonadareequivalenttoitshomo-
logicalpresentations.
Lemma 2.11. Let C be a category, T a monad on C, pD,F1,G1q a presentationof T. If
D hascoequalizersofallpairsofmapsoftheform2.1.1,thenthecanonicalcomparison
functorK :D ÑCT hasaleftadjoint. Conversely,ifT iscoequalizableandK isfulland
faithfulandhasaleftadjoint,thenD hasallcoequalizers(andinparticular,allpairsof
mapsoftheform2.1.1).
Proof. We will write F : C Ñ CT for the canonicalfunctor andG for its right adjoint.
WhenD hascoequalizersofallparallelpairsoftheform2.1.1,thecomparisonfunctorK
admitsaleftadjointV,definedonobjectsasfollows:ifTX ÝρÑX X isthestructuremapof
aT-algebra,thenV appliedtothatT-algebraisthecoequalizerofthemaps
F1ρX,ǫF1X :F1G1F1X ÑF1X,
usingthefactthatG1F1 “GF “T. (TheresultthatK hasaleftadjointifD hascoequal-
izersisanoldone: itappearsinBeck’sthesis[1],andevenappearsasanexerciseinVI.7
of[5]. Buttheonlycoequalizersoneactuallyneedsaretheonesusedintheconstruction
oftheleftadjoint,i.e.,thoseoftheform2.1.1.)
For the converse: supposeT is coequalizableand K is full and faithfuland has a left
adjointV. SinceleftadjointspreservecolimitsandsincefullnessandfaithfulnessofK is
equivalenttoVK – idD, we cancomputethecoequalizerofanypair f,g : X Ñ Y in D
bycomputingthecoequalizerofKf,KginCT (whichexistssinceT iscoequalizable)and
thenapplyingV. (cid:3)
Lemma 2.12. Let C be a category, T a monad on C, pD,F1,G1q a presentation of T.
SupposeD hascoequalizersofallpairsofmapsoftheform2.1.1. Thenthecomparison
functorK :D ÑCT isfullandfaithfulifandonlyifpD,F1,G1qishomological.
Proof. WeusethesamenotationasintheproofofLemma2.11.ThatKisfullandfaithful
is equivalentto the counitmapVK Ñ idD of the adjunctionbeingan isomorphism. We
recall that K is defined on objects by letting KX be the T-algebra with structure map
G1F1G1X “ TG1X Ñ G1X givenbythecounitnaturaltransformationF1G1 Ñ idD. Now
VKXispreciselythecoequalizerofthetwomaps
ǫF1G1X,F1G1ǫX : F1G1F1G1X ÑF1G1X,
and the map VKX Ñ X is precisely the map 2.1.2. So the conditionthat pD,F1,G1q be
homologicalisequivalenttotheconditionthatVK ÑidD beanisomorphismoffunctors,
i.e.,theconditionthatK befullandfaithful. (cid:3)
Theorem 2.13. Let C be a category, T a coequalizable monad on C. Then the large
homotopycategoryHopHPrespTqqofhomologicalpresentationsofT is equivalentto the
partially-orderedcollectionLocpTqofrepletereflectivesubcategoriesofCT whichpresent
T.
Furthermore, if the forgetful functor CT Ñ C preserves filtered colimits, then this
equivalence restricts to an equivalence between the subcollection HopFinHPrespTqq of
HopHPrespTqqandthesubcollectionFinLocpTqofLocpTq.
8 ANDREWSALCH
Proof. WewriteKlpTqfortheKleislicategoryofT,wewriteF : C ÑCT forthecanon-
ical functor and G for its right adjoint, and we write F2 : C Ñ KlpTq for the canoni-
cal functor and G2 for its right adjoint. The theorem follows almost immediately from
Lemma2.12,whichgivesusthateveryhomologicalpresentationpD,F1,G1qofT hasthe
property that K is faithful and full, hence D is canonically equivalent to a full replete
subcategoryof CT, and Lemma 2.11, which givesus that thatfull replete subcategoryis
reflective. ThatreflectiverepletesubcategorycontainsthefreeT-algebras,i.e.,theKleisli
categoryofT,sincetheKleislicategoryisinitialamongpresentationsofT. Sothatreflec-
tiverepletesubcategoryofCT isanelementofLocpTq. IfwefurthermoreassumethatG
preservesfilteredcolimitsandpD,F1,G1qisfinitary,thensinceG1 “G˝K,foranyfiltered
diagramX inD,wehavethenaturalcommutativediagram
colimG1pXq – //G1pcolimXq
PPPPPP–PPPPPP(( ♥♥♥♥♥♥♥♥♥♥♥♥66
GpcolimKpXqq,
so the map GpcolimKpXqq Ñ G1pcolimXq “ GpKpcolimXqq is an isomorphism, and
sinceGreflectsisomorphisms,Kpreservesfilteredcolimits.
Conversely,ifD isareflectiverepletesubcategoryofCT withinclusion K : D Ñ CT
havingleft adjointV, suppose we write S : KlpTq Ñ D for the inclusionof the free T-
algebras. WeclaimthatthecompositeS ˝F2 :C Ñ D hasrightadjointG˝K : D ÑC,
andthatthecompositemonadG˝K˝S ˝F2 isequaltothemonadT. Thesecondclaim
isveryeasy:thecompositeG˝K˝S isequaltoG2,so
G˝K˝S ˝F2 “G2˝F2 “T.
Thefirstclaimisalsonotdifficult:sinceF “K˝S ˝F2andV ˝K »idD,wehave
V˝F »V ˝K˝S ˝F2 »S ˝F2.
Now V is left adjointto K and F left adjointtoG, so S ˝F2 » V ˝F is left adjointto
G˝K,provingourfirstclaim. ItfollowsthatpD,S ˝F2,G˝KqisapresentationofT.
AllthatremainstobeprovenisthatpD,S ˝F2,G˝Kqisa homologicalpresentation
ofT. Byconstruction,K isfullandfaithful,sobyLemma2.11,D hascoequalizersofall
parallelpairs of the form 2.1.1. So by Lemma 2.12, pD,S ˝F2,G˝Kq is homological.
IfwefurthermoreassumethatG preservesfilteredcolimitsandthatD isfinitary,thenof
coursethecompositeG˝Kpreservesfilteredcolimits,andconsequentlypD,S˝F2,G˝Kq
isalsoafinitaryhomologicalpresentation. (cid:3)
Corollary2.14. IfT iscoequalizable,theEilenberg-MooreadjunctionpCT,F,GqofT isa
homologicalpresentationofT. (And,consequently,theterminalhomologicalpresentation
ofT.)
Corollary2.15. IfT iscoequalizable,thelargehomotopycategoryHopHPrespTqqofho-
mologicalpresentationsofT ispartially-ordered,i.e.,foranyobjectsA,BofHopHPrespTqq,
thereisatmostonemorphismA ÑB. Ifwefurthermoreassumethattheforgetfulfunctor
CT Ñ C preserves filtered colimits, then large homotopy category HopFinHPrespTqq of
finitaryhomologicalpresentationsofT isalsopartially-ordered.
Corollary 2.16. Suppose T is coequalizable and CT is weakly Krull-Schmidt. Suppose
thecollectionofisomorphismclassesofindecomposableobjectsformsaset(notaproper
HOWMANYADJUNCTIONSGIVERISETOTHESAMEMONAD? 9
class!), and suppose that set has cardinality κ. Then HopHPrespTqq is equivalent to a
partially-orderedsetofcardinalitynogreaterthan2ℵκ0.
Proof. The partially-orderedcollection LocpTq, which by Theorem2.13 is equivalentto
HopHPrespTqq, is containedin the collection of subcollectionsof the collection of finite
formalsumsofindecomposableobjects.Thiscollectionis,inturn,containedinthecollec-
tionofsubcollectionsofthecollectionofnot-necessarily-finiteformalsumsofindecom-
posable objects in which each indecomposable object appears only finitely many times.
Thislastcollectionhascardinality2ℵκ0. (cid:3)
WegreatlyimprovethiscardinalityboundinCorollary2.21undertheassumptionthat
CT hasabiproduct.
2.3. Coordinatizationofthecollectionofhomologicalpresentationsofamonad.
Definition2.17. RecallthatacategoryC issaidtohaveabiproductifithasfiniteproducts
andfinitecoproductsand,foreachfinitefamilytXu ofobjectsofC,thecanonicalmap
i iPI
X Ñ X isanisomorphism.
iPI i iPI i
šLemmasś2.18 and 2.19are easy and must be well-known, but we do notknow where
theyalreadyappearintheliterature.
Lemma2.18. SupposeA isarepletereflectivesubcategoryofacategorywithbiproduct.
ThenA hasabiproduct.
Proof. LetC beacategorywithbiproduct,letA bearepletereflectivesubcategoryofC,
letG : A Ñ C denote the inclusion functor, and let F : C Ñ A denote its left adjoint.
GivenobjectsX,Y ofC,weclaimthatFpGpXq‘GpYqqisabiproductforX andY inA.
SinceF isaleftadjoint,itistrivialthatFpGpXq‘GpYqqisacoproductforXandY inA.
ToseethatFpGpXq‘GpYqqisalsoaproductforX andY inA,supposeT isanobjectof
A,andobservethatwehavenaturalbijections
homApT,XqˆhomApT,Yq–homCpGpTq,GpXq GpYqq
–homCpGpTq,GpXqź‘GpYqq
–homCpGpTq,GFpGpXq‘GpYqqq
–homApT,FpGpXq‘GpYqqq,
henceFpGpXq‘GpYqqhastheuniversalpropertyoftheproductinA. (cid:3)
Lemma 2.19. Suppose A is a replete reflective subcategory of a weakly Krull-Schmidt
categoryC withbiproduct‘. IfX –Y‘ZinC,thenXisinA ifandonlyifbothY andZ
areinA.
Proof. WewriteL : C ÑC forthecompositeofthereflectorfunctorC Ñ A withthein-
clusionA ÑC. ByLemma2.18,Ahasabiproduct.SinceLisacompositeofaleftadjoint
(thereflectorfunctor)witharightadjoint(theinclusionfunctor),itpreservesbiproducts,
sincethebiproductisboththefinitecoproductandthefiniteproduct. SoLX – LY ‘LZ,
andifY,Z areinA, thentheunitmapsY Ñ LY andZ Ñ LZ arebothisomorphisms. So
X –Y‘Z ÑLY ‘LZ –LX isanisomorphism.SoXisinA.
Fortheconverse,suppose X isinA. Let X – ‘n X bethedecompositionof X into
i“1 i
indecomposables,givenbytheweakKrull-Schmidtproperty.Thentheunitmap
‘n X –X ÑLX –‘n LX
i“1 i i“1 i
10 ANDREWSALCH
isanisomorphism,anditisthesumofthecomponentmapsX Ñ LX,sowehavesome
i i
permutation
σ:t1,...,nuÑt1,...nu
withthepropertythatLX – X . However,sinceA isarepletereflectivesubcategory,L
i σpiq
isidempotent,soσ˝σ “ σ. Soσmustbetheidentitypermutation. Soeachcomponent
map X Ñ LX isanisomorphism. Henceif X splitsas adirectsum and X isin A, each
i i
summandisalsoinA. (cid:3)
Theorem2.20. (Coordinatization.)LetC beacategory,T acoequalizablemonadonC.
Supposethe Eilenberg-MoorecategoryCT hasa biproductandis weakly Krull-Schmidt.
Write ΓpCTq for the collection of isomorphism classes of indecomposableobjects in CT.
ThenHopHPrespTqqembedsbyanorder-preservingmapintothecollectionofsubcollec-
tionsofΓpCTq.
Proof. ByTheorem2.13,specifyinganelementofHopHPrespTqqisequivalenttospecify-
ingarepletereflectivesubcategoryofCT whichcontainsC ,henceisdetermineduniquely
T
bywhichisomorphismclassesofobjectsinCT arecontainedintherepletereflectivesub-
category. ButbyLemma2.19,arepletereflectivesubcategoryofaweaklyKrull-Schmidt
categorywithbiproductisdetermineduniquelybywhichindecomposablesarecontained
init. (cid:3)
In otherwords: underthe conditionsof Theorem2.20, a homologicalpresentationof
T can be specified by specifying a subcollectionof ΓpCTq (which, as we describe in the
last section of this paper, is actually computable in cases of interest). Since CT is often
computableandunderstandable,Theorem2.20—whenitapplies—givesacoordinatization
ofthecollectionofhomologicalpresentationsofT,asdesired.
Corollary2.21. Suppose T is coequalizableand CT is weakly Krull-Schmidt and has a
biproduct.Supposethecollectionofisomorphismclassesofindecomposableobjectsforms
aset(notaproperclass!),andsupposethatsethascardinalityκ. ThenHopHPrespTqqis
equivalenttoapartially-orderedsetofcardinalitynogreaterthan2κ.
Remark 2.22. Suppose that k is a field and A is a k-algebra. Let C be any category
and T any monad on C whose Eilenberg-Moore category CT is equivalent to the cate-
goryfgModpAq of finitely generated A-modules; for example, we couldlet C be the cat-
egory of B-modules, for some reasonable subalgebra B of A, and we could let T be the
monadassociatedtothefree-forgetfuladjunctionbetween B-modulesand A-modules. In
Auslander-Reitentheory,thesetofisomorphismclassesofindecomposablefinitelygener-
atedA-modulesisexactlythesetΓpfgModpAqqofverticesintheAuslander-Reitenquiver
of CT. So one can regard the vertices of the Auslander-Reiten quiver of CT as “coordi-
nates”forthecollectionofnaturalequivalenceclassesofhomologicalpresentationsofT:
Theorem2.20givesanembeddingofHopPrespTqqintothepartially-orderedsetofsubsets
ofΓpfgModpAqq.
3. Acriterionforuniquehomologicalpresentabilityofamonad.
3.1. Preliminarydefinitions. Somemonadscan(uptonaturalequivalence)onlybeho-
mologically presented in a single way, i.e., HopHPrespTqq is equivalent to a one-object
category.Hereistherelevantdefinition:
Definition3.1. SupposeT isamonad. IfHopHPrespTqq hasonlyasingleelement,then
wesaythatT isuniquelyhomologicallypresentable.