Monoidal algebraic model structures

0.46 MB·

by  Emily Riehl

MONOIDAL ALGEBRAIC MODEL STRUCTURES
EMILYRIEHL
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Abstract. Extendingpreviouswork,wedefinemonoidalalgebraicmodelstructuresand
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give examples. The main structural component is what we call an algebraic Quillen
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two-variable adjunction; the principal technical work is to develop the category theory
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necessary to characterize them. Our investigations reveal an important role played by
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“cellularity”—loosely,thepropertyofacofibrationbeingarelativecellcomplex,notsim-

plyaretractofsuch—whichweparticularlyemphasize.Amainresultisasimplecriterion
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whichshowsthatalgebraicQuillentwo-variableadjunctionscorrespondpreciselytocell
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structuresonthepushout-productsofgenerating(trivial)cofibrations. Asacorollary,we
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discoverthatthefamiliarmonoidalmodelstructuresoncategoriesandsimplicialsetsadmit
. thisextraalgebraicstructure.
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m Contents
[ 1. Introduction 2

2 2. Doublecategories,mates,parameterizedmates 6
v 2.1. Doublecategoriesandmates 6
3 2.2. Parameterizedmates 8
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3. Preliminariesonalgebraicweakfactorizationsystems 10
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3.1. Weakfactorizationsystems 10
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. 3.2. Algebraicweakfactorizationsystems 12
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3.3. Thealgebraicsmallobjectargument 12
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4. Morphismsofalgebraicweakfactorizationsystems 13
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1 4.1. Morphisms 13
: 4.2. Colaxmorphisms,laxmorphisms,andadjunctions 14
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i 4.3. Bicolaxmorphisms,bilaxmorphisms,andtwo-variableadjunctions 15
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5. Thecompositioncriterion 18
r 5.1. Doublecategoricalaspectsofalgebraicweakfactorizationsystems 18
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5.2. Adjunctionsarisingfromtwo-variableadjunctions 20
5.3. Compositioncriterion 23
6. Thecellularityanduniquenesstheorems 27
6.1. Cellularityandadjunctionsofawfs 27
6.2. Thecellularitytheorem 28
6.3. Extendingtheuniversalpropertyofthesmallobjectargument 32
7. AlgebraicQuillentwo-variableadjunctions 34
7.1. Algebraicmodelstructures 35
7.2. AlgebraicQuillenadjunctions 35
7.3. AlgebraicQuillentwo-variableadjunctions 38
8. Monoidalalgebraicmodelstructures 40
References 47
Date:June7,2012.
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2 EMILYRIEHL
1. Introduction
Algebraic modelstructures, introducedin [22], are a structuralextensionof Quillen’s
modelcategories[21] in which cofibrationsand fibrationsare “algebraic,” i.e., equipped
with specified retractionsto their left or rightfactors which can be used to solve all lift-
ing problems. The factorizationsthemselvesare much more than functorial: the functor
mappinganarrowtoitsrightfactorisa monadandthefunctormappingtoitsleftfactor
isacomonadonthearrowcategory.Inparticular,thedataofanalgebraicmodelcategory
determinesafibrantreplacementmonadandacofibrantreplacementcomonad.
Despite the stringentstructuralrequirementsof this definition, algebraic modelstruc-
turesare quiteabundant. A modifiedsmallobjectargument,dueto Richard Garnerpro-
ducesan algebraicmodelstructurein placeof an ordinarycofibrantlygeneratedone[6].
Thedifferenceisthatthemaincomponentsofamodelstructure—theweakfactorization
systems(C∩W,F)and(C,F∩W)—arereplacedwithalgebraicweakfactorizationsystems
(C,F)and(C,F),whicharecategoricallybetterbehaved.
t t
Wefindtheweakfactorizationsystemperspectiveonmodelcategoriesclarifying. The
overdeterminationofthemodelcategoryaxiomsandtheclosurepropertiesoftheclassesof
cofibrationsandfibrationsareconsequencesofanalogouscharacteristicsoftheconstituent
weakfactorizationsystems. Quillen’ssmallobjectargumentis reallya constructionof a
functorialfactorizationfor a cofibrantly generated weak factorization system; the model
structure context is beside the point. Also, the equivalence of various definitions of a
Quillen adjunction has to do with the separate interactions between the adjunction and
eachweakfactorizationsystem.
Moreprecisely,analgebraicmodelstructureonacategoryMwithweakequivalences
W consists of two algebraic weak factorization systems (henceforth, awfs for both the
singularandtheplural)togetherwithamorphismξ: (C,F)→(C,F)betweenthemsuch
t t
thattheunderlyingweakfactorizationsystemsformamodelstructureintheusualsense.
HereC andCarecomonadsandF andFaremonadsonthearrowcategoryM2thatsend
t t
anarrowtoitsappropriatefactorwithrespecttothefunctorialfactorizationsofthemodel
structure. We write R,Q: M2 ⇒ M for the functors that assign to an arrow the object
through which it factors. The notation is meant to evoke fibrant/cofibrant replacement:
slicingoverthe terminalobjector undertheinitialobjectdefinesthe fibrantreplacement
monadandcofibrantreplacementcomonad,alsodenotedRandQ.
The natural transformation ξ, which we call the comparison map, plays a number of
roles.Itscomponents
Cf
(1.1) domf // Qf
Ctf (cid:15)(cid:15) ✇✇✇ξ✇f✇✇✇✇✇;; (cid:15)(cid:15) Ftf
Rf //codf
Ff
arenaturalsolutionstotheliftingproblem(1.1)thatcomparesthetwofunctorialfactoriza-
tionsof f ∈M2. Additionally,ξmustsatisfytwopentagons:oneinvolvingthecomultipli-
cationsofthecomonadsandoneinvolvingthemultiplicationsofthemonads.Underthese
hypotheses,ξdeterminesfunctorsoverM2
(1.2) ξ : C-coalg→C-coalg ξ∗: F-alg→F-alg
∗ t t
betweenthecategoriesofcoalgebrasforthecomonadsandbetweenthecategoriesofalge-
brasforthemonads.
MONOIDALALGEBRAICMODELSTRUCTURES 3
Elementsof,e.g.,thecategoryF-algarecalledalgebraicfibrations;theirimagesunder
theforgetfulfunctortoM2 areinparticularfibrationsinthemodelstructure. Thealgebra
structureassociatedtoanalgebraicfibrationdeterminesacanonicalsolutiontoanylifting
problem of that arrow against an algebraic trivial cofibration. Naturality of ξ together
with the functors (1.2) imply that there is also a single canonical solution to any lifting
problemofanalgebraictrivialcofibrationagainstanalgebraictrivialfibration:thesolution
constructedusingξ andtheawfs(C,F)agreeswiththesolutionconstructedusingξ∗ and
∗ t
theawfs(C,F).
t
Forcertainliftingproblems,thesecanonicalsolutionsthemselvesassembleintoanat-
ural transformation. For instance, the natural solution to the usual lifting problem that
comparesthetwofibrant-cofibrantreplacementsofanobjectdefinesanaturaltransforma-
tionRQ⇒QRthatturnsouttobeadistributivelawofthefibrantreplacementmonadover
the cofibrantreplacementcomonad. Itfollowsthat Q lifts to a comonadon the category
R-alg of algebraic fibrant objects, and dually R lifts to a monad on Q-coalg. The coal-
gebrasfortheformerandalgebrasforthelatercoincide,definingacategoryofalgebraic
fibrant-cofibrantobjects.
Any ordinary cofibrantly generated model structure gives rise to an algebraic model
structure using a modified form of Quillen’s small object argumentdue to Richard Gar-
ner. As a result, this algebraic structure is much more commonthat might be supposed.
Wheneverthe category permits the small object argument, any small category of arrows
generates an algebraic weak factorization system that satisfies two universal properties,
bothofwhichwefrequentlyexploit[5,6].
Awfswereintroducedtoimprovethecategoricalpropertiesofordinaryweakfactoriza-
tion systems [8]. One feature of awfs is that the left and right classes are closed under
colimitsand limits, respectively, in the followingprecise sense. By standardmonadicity
results,theforgetfulfunctorsC-coalg → M2, F-alg → M2 createallcolimitsandlimits,
respectively,existinginM2. Inthecontextofalgebraicmodelstructures,thisgivesanew
recognitionprincipleforcofibrationsconstructedascolimitsandfibrationsconstructedas
limits. Familiarly,acolimit(inthearrowcategory)ofcofibrationsisnotnecessarilyacofi-
bration. Butifthecofibrationsadmitcoalgebrastructuresthatarepreservedbythemaps
inthediagram,thenthecolimitiscanonicallyacoalgebraandhenceacofibration.
Whenthemodelstructureiscofibrantlygenerated,allfibrationsandalltrivialfibrations
arealgebraic,i.e.,admitalgebrastructures;interestinglythedualstatementsdonothold.
Transfinite composites of pushouts of coproducts of generating cofibrations i ∈ I—the
class of maps denoted I-cell in the classical literature [10, 9]—are necessarily algebraic
cofibrations.Accordingly,wecalltheclassofcofibrationsthatadmitaC-coalgebrastruc-
turethecellularcofibrations;acofibrationiscellularifandonlyifitcanbemadealgebraic.
Allcofibrationsareatleastretractsofcellularones.Cellularitywillplayaninterestingand
importantroleinthenewresultsthatfollow.
For example, if A is a commutative ring, the arrow 0 → A generates an awfs on the
categoryof A-moduleswhose rightclass isthe epimorphismsandwhose leftclassis the
injections with projective cokernel. Each epimorphism M ։ N admits (likely many)
algebrastructures: an algebrastructureisa section N → M, notassumed to bea homo-
morphism. Thecellular maps—thatis, those arrowsadmittingcoalgebrastructures—are
thoseinjectionswhichhavefreecokernel.
The basic theory of algebraic model structures is developedin [22]; referencesto re-
sultsthereinwillhavetheformI.x.x.Inparticular,thatpaperdefinesanalgebraicQuillen
4 EMILYRIEHL
adjunction,whichisanordinaryQuillenadjunctionsuchthattherightadjointliftstocom-
mutingfunctorsbetweenthealgebraic(trivial)fibrationsandtheleftadjointliftstocom-
mutingfunctorsbetweenthecategoriesofalgebraic(trivial)cofibrations. Thisshouldbe
thoughtof an algebraization of the usual conditionthat the right adjointpreserves fibra-
tions and trivial fibrations and left adjoint preserves cofibrationsand trivial cofibrations.
We also ask that the lifts of one adjoint determine the lifts of the other in a sense made
precisebelow, a conditionthatmirrorsthe classical factthata Quillenadjunctioncan be
detectedbyexaminingtheleftorrightadjointalone. AlgebraicQuillenadjunctionsexist
in an important class of examples: when a cofibrantly generated algebraic model struc-
tureisliftedalonganadjunction,theresultingQuillenadjunctioniscanonicallyalgebraic.
Examples include the geometric realization–total singular complex adjunction between
simplicialsetsandspaces,theadjunctionbetweenG-spacesandspace-valuedpresheaves
on the orbit category for a groupG, and the adjunctionsestablishing a projective model
structure.
A classical categorical result characterizes lifted functors of algebraic fibrations, i.e.,
functorsbetweenthecategoriesofalgebrasforthemonads,ascertainnaturaltransforma-
tionssometimescalledlaxmonadmorphisms,butthisconditionalonefailstocapturethe
symmetryof the classical situation where a right adjointpreservesfibrations if and only
ifitsleftadjointpreservestrivialcofibrations. Therearetwowaystodescribethedesired
additionalhypothesis. Oneistoaskthatthemateofthenaturaltransformationcharacter-
izingtheliftedfunctorofalgebraicfibrationsdefinestheliftedfunctorofalgebraictrivial
cofibrations. An equivalent condition is that the lifted functor of algebraic fibrations is
in fact a lifted doublefunctor between doublecategoriesof algebraicfibrations, suitably
defined.
Inthispaper,weextendtheseresultsinordertodefinemonoidalandeventuallyenriched
algebraic model structures. The technical work in this paper puts the latter definition in
immediatereach;however,wepostponeittoafuturepaperwhichwillhavespacetofully
explore examples. Much of the structure of a closed monoidal category or a tensored
and cotensoredenrichedcategoryis encodedin a two-variable adjunction. For enriched
categories,theconstituentbifunctorsarecommonlydenoted
V×M −⊙− //M Vop×M {−,−} // M Mop×M hom(−,−)//V
andcomeequippedwithhom-setisomorphisms
(1.3) M(V⊙M,N)(cid:27)M(M,{V,N})(cid:27)V(V,hom(M,N))
naturalinallthreevariables.Fixinganyonevariable,two-variableadjunctionsgiveriseto
parameterizedfamiliesofordinaryadjunctions,e.g.,−⊙M ⊣hom(M,−).
The monoidal case necessarily precedes the enriched one but also inherits all of its
complexity. Aclosedmonoidalcategorywithanalgebraicmodelstructureisamonoidal
algebraicmodelcategory if the canonicalcomparisonbetween a cofibrantobjectand its
tensor with the cofibrant replacementof the monoidal unit is a weak equivalence and if
the closed monoidal structure is an algebraic Quillen two-variable adjunction. Such an
adjunctionconsistsofthreefunctorsliftingtheso-called“pushout-product”
C-coalg×C-coalg // C-coalg
t t
C-coalg×C-coalg //C-coalg
t t
C-coalg×C-coalg //C-coalg
MONOIDALALGEBRAICMODELSTRUCTURES 5
suchthatthematesofthecharacterizingnaturaltransformationsdeterminesimilarliftsof
theleftandrightclosures. Inthebestcases,thesefunctorssatisfythreeevidentcoherence
conditionswhich say that various canonicalcoalgebra structures agree, but we shall see
thatsuchcoherenceistoomuchtoaskforingeneral.
OnecouldalsodefineaweakernotionofanalgebraicQuillenbifunctorinthecontext
of monoidalor enriched model categories in which some of the adjoint bifunctorsdon’t
exist. Thisisless categoricallychallengingthanthetheorypresentedhere,so the details
maybesafelylefttothereader.
ThreemaintechnicaltheoremsallowustoidentifyalgebraicQuillentwo-variablead-
junctions in practice. The first describes a composition criterion that identifies when a
liftedbifunctorispartofatwo-variableadjunctionofawfs,theappropriatenotionofalge-
braicQuillentwo-variableadjunctionforcategoriesequippedwithasingleawfsinplace
of a full algebraic model structure. The other two results, which we call the cellular-
ity and uniqueness theorems, combine to characterize two-variable adjunctions of awfs
in the case when the awfs are cofibrantly generated. The cellularity theorem says that a
two-variableadjunctionofawfsarisesfromanyassignmentofcoalgebrastructurestothe
pushout-productofthegenerators;hence,suchstructuresexistifandonlyifthepushout-
productofthegeneratorsiscellular.Theuniquenesstheoremsaysthatsuchanassignment
completelydeterminestheliftedfunctors,soatmostonetwo-variableadjunctionofawfs
canbeobtainedinthisway.
Several new categorical results were necessary to make all of this precise. Of most
generalcategoricalinterestisthetheoryofparameterizedmates,introducedin§2below.
Thistheorydescribestherelationshipbetweenthe naturaltransformationscharacterizing
the lifts of the three bifunctors constituting a two-variable adjunction of awfs and their
interactionswithordinaryadjunctionsofawfs.
Other results appearing below are designed to deal with complications arising in the
proofsof the cellularity and uniquenesstheorems. The main technical difficulty is quite
simplyaccountedfor: in [22], the onlyadjunctionsconsideredbetweenarrowcategories
were those of the form T2: M2 ⇄ K2: S2, i.e., defined pointwise by an ordinary ad-
junctionT: M ⇄ K: S betweenthebasecategories. However,theadjunctionsonarrow
categoriesarisingfromtwo-variableadjunctionsonthebasesnolongerhavethisformand
inparticulardon’tpreservecomposabilityofarrows. Thus,thedoublecategoricalcompo-
sitioncriterionweusetogreateffectinthepreviouspapertocharacterizethoseliftedleft
adjointsthatdetermineliftsofrightadjointsmusttakeonanewform.
In§2,weintroducedoublecategories,mates,andparameterizedmatesandprovesome
elementarylemmas which will be quoted frequently. In §3, we review lifting properties
and functorialfactorizations, wfs and awfs, and the algebraic small object argument. In
§4,wepresentavarietyofnotionsofmorphismbetweenawfsondifferentcategoriesand
definethenewnotionoftwo-variableadjunctionofawfs. In§5,weprovethecomposition
criterionwhichallowsustorecognizewhenagivenliftedbifunctorofawfs(co)algebras
determinesa two-variableadjunctionsofawfs. In§6, weusethisresulttoprovethecel-
lularitytheorem.WethenextendtheuniversalpropertyofGarner’ssmallobjectargument
andusethistoprovetheuniquenesstheorem. In§7,weapplytheseresultstomodelcat-
egories,introducinganotionofalgebraicQuillentwo-variableadjunctions. Finallyin§8,
wedefinemonoidalalgebraicmodelstructuresanddescribeexamples.
6 EMILYRIEHL
2. Doublecategories,mates,parameterizedmates
Thecalculusofmateswillplayanimportantconceptualandcalculationalroleinwhat
follows.Tostreamlinelaterproofs,wetakeafewmomentsin§2.1tooutlinetheimportant
featureswithoutgettingmiredintechnicaldetails. Thecanonicalreferenceis[16];wealso
like[23].
Bifunctors, meaning functorswhose domain is the productof two categories, are de-
termined by the collection of single-variable functors obtained when one object is fixed
togetherwith the naturaltransformationsbetweensuch functorsarising frommorphisms
in that category. This fact is often expressed by saying that category CAT is cartesian
closed. For this simple reason, the classical theory of mates extends to a new theory of
parameterizedmates,introducedin§2.2.
2.1. Doublecategoriesandmates. AdoublecategoryDisacategoryinternaltoCAT:
dom
◦ //
D1×D1 //D1 oo id //D0
D0 cod
TheobjectsandarrowsofD arecalledobjectsandhorizontalarrowsofDwhiletheobjects
0
andarrowsofD arecalledverticalarrowsandsquares.Viathefunctorsdom,cod: D ⇒
1 1
D , the sources and targets of vertical arrows are objects of D, and likewise the squares
0
can be depicted in the way their name suggests. Squares can be composed horizontally
usingcompositioninD andverticallyusingthefunctor◦,whosedomainisthepullback
1
ofdomalongcod. Asaconsequenceoffunctorialityof◦,theorderinwhichverticaland
horizontalcompositesaretakeninapastingdiagramofsquaresdoesnotmatter. Werefer
toD asthecategoryofverticalarrows; thiscategoryforgetsthecompositionofvertical
1
arrowsandremembersonlythehorizontalcompositionofsquares.
Example2.1. AcategoryMgivesrisetoadoublecategorySq(M)
dom
M3 (cid:27)M2×M2 ◦ // M2 oo id ////M
M
cod
whoseobjectsareobjectsofM,horizontalandverticalarrowsaremorphismsofM, and
squares are commutative squares. The category of vertical arrows is usually called the
arrowcategoryandplaysanessentialroleinthispaper.
Given categories, functors, and adjunctions, as displayed below, there is a bijection
betweennaturaltransformationsinthesquareinvolvingtheleftadjointsandnaturaltrans-
formationsinthesquareinvolvingtherightadjoints
(2.2) · H // · · H // · · H //·
OO OO OO OO
T ⊣ S T′ ⊣ S′ T λw T′ ! S uρ S′
(cid:15)(cid:15)· //(cid:15)(cid:15)· ·(cid:15)(cid:15) //·(cid:15)(cid:15) · //·
K K K
givenbytheformulas
(2.3) ρ=S′Kǫ·S′λ ·ι and λ=ν ·T′ρ ·T′Hη,
S HS KT T
where η and ǫ are the unit and counit for T ⊣ S and ι and ν are the unit and counit for
T′ ⊣S′. Correspondingλandρarecalledmates.
Example2.4. AnaturaltransformationH ⇒Kisitsownmatewithrespecttotheidentity
adjunctions.
MONOIDALALGEBRAICMODELSTRUCTURES 7
Example2.5. Write1fortheterminalcategory.Adjunctarrows f♯: Tm→k∈K, f: m→
Sk ∈ M correspondingunder the adjunctionT: M ⇄ K: S are mates in the following
squares
1 m // M 1 m //M
OO OO
1 f♯w T ! 1 uf S
(cid:15)(cid:15) (cid:15)(cid:15)
1 //K 1 //K
k k
Example 2.6. If M has a left-closed monoidalstructure and f: m′ → m ∈ M, then the
inducednaturaltransformations
M 1 // M M 1 // M
OO OO
m⊗− f⊗−w m′⊗− ! hom(m,−) uhom(f,−) hom(m′,−)
(cid:15)(cid:15) (cid:15)(cid:15)
M //M M //M
1 1
aremates.Analogouscorrespondencesholdforanyparameterizedadjunction[18,IV.7.3].
There are double categories Ladj and Radj whose objects are categories, horizontal
arrows are functors, vertical arrows are adjunctions in the direction of the left adjoint,
andwhosesquaresare naturaltransformationsas displayedin the middleand right-hand
squaresof (2.2), respectively. The mates correspondenceis natural, or, more accurately,
functorial,inthefollowingprecisesense.
Theorem2.7(Kelly-Street[16,§2]). Thematescorrespondencegivesanisomorphismof
doublecategoriesLadj(cid:27)Radj.
ThissaysthatanaturaltransformationobtainedbypastingsquaresinLadjeitherverti-
callyorhorizontallyisthemateofthenaturaltransformationobtainedbypastingthemates
ofthesesquaresinRadj. The“calculusofmates”referstothisfact,which,whenusedin
conjunctionwithExamples2.4–2.6,impliesthatmatessatisfy“dual”diagrams.
Forinstance,supposethefunctorsH andK of(2.2)aremonads(H,η,µ),(K,η,µ)and
supposeT = T′ and S = S′. A pair(S,ρ) as in the rightsquare of(2.2) is a lax monad
morphismif
HSK
S✼ H♦ρ♦♦♦77 ❖❖❖ρ❖K''
✞ ✼ HHS SKK
(2.8) η(cid:3)(cid:3)✞✞S✞✞✞✞ ρ ✼✼✼S✼✼η(cid:27)(cid:27) and µ✴S✴✴✴✴✴ ✎✎✎S✎✎µ
HS //SK ✴(cid:23)(cid:23) ρ (cid:7)(cid:7)✎✎
HS // SK
commute. Thedefinitionsin§4takeseveralequivalentformsonaccountofthefollowing
result.
Lemma2.9(Appelgate[11]). Alaxmonadmorphism(S,ρ)determinesandisdetermined
byaliftofS toafunctorK-alg→H-alg.
Proof. TheH-algebrastructureassignedtotheimageofaK-algebrat: Kx → xunderS
is
HSx ρx //SKx St // Sx (cid:3)
Thedualnotion,acolaxmonadmorphism,isapair(S,ρ)satisfyingdiagramsanalogous
to(2.8)butwiththedirectionofρreversed.
8 EMILYRIEHL
Lemma2.10. Suppose(S,ρ)isalaxmonadmorphism,T ⊣S,andλisthemateofρwith
respecttothisadjunction.Then(T,λ)isacolaxmonadmorphism.
Proof. We show (T,λ) satisfies the pentagon and leave the triangle as an exercise. The
pentagonfor(S,ρ)saysthattheleftpastedsquares
· HH // · · H // · H //· · HH // · · H // · H // ·
OO OO OO OO OO
1 µu 1 S ρu S ρu S 1 µw 1 T λw T λw T
· // · = · //· // · ·(cid:15)(cid:15) // ·(cid:15)(cid:15) = ·(cid:15)(cid:15) //·(cid:15)(cid:15) // ·(cid:15)(cid:15)
OO H OO OO K K OO H K K
S ρu S 1 µu 1 T λw T 1 µw 1
· //· · //· ·(cid:15)(cid:15) //·(cid:15)(cid:15) ·(cid:15)(cid:15) //·(cid:15)(cid:15)
K K K K
areequalinRadj.ByTheorem2.7thepastedcompositesoftheirmatesinLadj,displayed
ontherightabove,alsoagree. (cid:3)
Of course, analogousresultshold with any 2-categoryin place of CAT; Theorem2.7
assertsthatthefunctorsLAdj,RAdj: 2-CAT⇒DblCATareisomorphic.
2.2. Parameterizedmates. Bya lemmabelow,in the contextofa two-variableadjunc-
tion, or more generally a parameterized adjunction, the mates correspondences for the
adjunctionsobtainedbyfixingtheparameterarenaturalintheparameter. Thismeansthat
thetwosetsofmatesassembleintonaturaltransformationsoftwovariables. Wesaythat
natural transformations correspondingin this way are parameterized mates. We are not
awareifthiscorrespondencehasbeenstudiedbefore,butitisessentialtodescribethein-
teractionsbetweenawfsandtwo-variableadjunctions.Thefollowinglemmasestablishthe
barebonesofthistheory.
First, we provethatif we fix oneof the variablesin a naturaltransformationbetween
bifunctorswhicharepointwiseadjointsandthentakemates,theresultingpointwisemates
assembletogiveanaturaltransformationbetweentheappropriatebifunctors.
Lemma 2.11. Suppose given a pair of left-closed bifunctors ⊗,⊗′; ordinary functors
K,M,N;andanaturaltransformationλ : Kk⊗′ Mm→N(k⊗m)asdisplayed
k,m
K×M K×M // K′×M′
⊗ λw ⊗′
(cid:15)(cid:15) (cid:15)(cid:15)
N // N
N
Letρ denotethemateofthenaturaltransformationλ withrespecttotheadjunctions
k,− k,−
k⊗− ⊣ hom(k,−) and Kk⊗′ − ⊣ hom′(Kk,−). Thenthe ρ are also naturalin K and
k,−
assembleintoanaturaltransformationρ : Mhom(k,n)→hom′(Kk,Nn)
k,n
M M //M′
OO OO
hom uρ hom′
Kop×N //K′op×N′
K×N
MONOIDALALGEBRAICMODELSTRUCTURES 9
Proof. NaturalityofλinKsaysthatforany f: k′ →kinK,thepastedcomposites
M 1 // M M // M′ = M M // M′ 1 // M′
k⊗− f⊗−w k′⊗− λk′,−w Kk′⊗′− k′⊗− λk,−w Kk⊗′− Kf⊗′−w Kk′⊗′−
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
N //N // N′ N // N′ // N′
1 N N 1
areequal.ByTheorem2.7,thepastedcomposites
M 1 //M M //M′ = M M //M′ 1 // M′
OO OO OO OO OO OO
uhom(f,−) uρk′,− uρk,− uhom′(Kf,−)
hom(k,−) hom(k′,−) hom′(Kk′,−) hom(k,−) hom′(Kk,−) hom′(Kk′,−)
N // N //N′ N //N′ //N′
1 N N 1
arealsoequal,whichsaysthattheρ arenaturalinK. (cid:3)
k
Thefollowinglemmaestablishestheparameterizedmatescorrespondence.
Lemma2.12. Giventwo-variableadjunctions(⊗,hom ,hom ),(⊗′,hom′,hom′)andfunc-
ℓ r ℓ r
torsK,M,Nasbelow,thereisabijectivecorrespondencebetweennaturaltransformations
K×M K×M//K′×M′ M M //M′ K K //K′
OO OO OO OO
⊗ λw ⊗′ homℓ uρℓ hom′ℓ homr uρr hom′r
(cid:15)(cid:15) (cid:15)(cid:15)
N //N Kop×N // K′op×N′ Mop×N // M′op×N′
N Kop×N Mop×N
obtainedbyapplyingthepointwisematescorrespondencetoeithervariable.
Proof. By symmetry, it suffices to show that if we fix K and takes pointwise mates to
defineρℓ fromλandthenfixNandtakepointwisematestodefineρr fromρℓ,theresultis
thesameasfixingM andtakingpointwisematestodefineρr fromλ. Thisfollowsfrom
theformulas(2.3),thecompatiblehom-setisomorphisms(1.3) andadiagramchase. We
leavethedetailsasanexercisetothereaderwiththefollowinghint:wheninasequenceof
composablearrows,oneseestheunitfollowedbyarrowsintheimageoftherightadjoint,
thisassertsthatthecompositeisadjuncttowhateverremainswhentheunitandtheright
adjointareerased. Wemadefrequentuseofthisobservationanditsdual. (cid:3)
Acarefulstatementofthe“multi-functoriality”oftheparameterizedmatescorrespon-
dence, the appropriate analog of Theorem 2.7, involves category objects in the category
ofmulticategoriesequippedwithcertainadditionalstructure. Thisresultwillappearina
separatepaper[3],jointworkwithEugeniaChengandNickGurski.Forpresentpurposes,
weonlyneedapreliminarylemmainthisdirection.
Lemma2.13. Compositionofparameterizedmatesinanyofthethreevariableswithor-
dinarymatespointingincompatibledirectionsiswell-defined.
10 EMILYRIEHL
Proof. Supposeλ,ρℓ,ρr areparameterizedmatesasinLemma2.12,andsupposeαandβ
aremateswithrespecttothetopsquaresofthefollowingdiagraminLadj(cid:27)Radj.
J J //J′ J J //J′
OO OO OO OO
T ⊣ S T′ ⊣ S′ T ⊣ S T′ ⊣ S′
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
K //K′ K // K′
OO K OO OO K OO
−⊗m ⊣ homr(m,−) homℓ(−,n) ⊣ homr(−,n)
−⊗′Mm ⊣ hom′r(Mm,−) hom′ℓ(−,Nn) ⊣ hom′r(−,Nn)
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
N //N′ Mop //M′op
N M
ApplyingTheorem2.7andLemma2.11totheleft-handrectangle,weseethat
T′J⊗′ M α⊗′1 // KT ⊗′ M λT,1 // N(T ⊗−) and
JShom (−,−) βhomr //S′Khom (−,−) S′ρr // S′hom′(M,N)
r r r
aremates;fromright-handrectangle,weseethatthissecondnaturaltransformationand
Mhom (T,−) ρℓT,1 //hom′(KT,N)hom′ℓ(α,N)//hom′(T′J,N)
ℓ ℓ ℓ
aremates. ByLemma2.12,thethreecompositenaturaltransformationsareparameterized
mates. (cid:3)
Asaconsequence,algebraicQuillentwo-variableadjunctionspointinginthedirection
of the left adjoints can be composed in any of their variableswith algebraic Quillen ad-
junctionspointingalsointhedirectionoftheleftadjoints;seeLemma7.14.
3. Preliminariesonalgebraicweakfactorizationsystems
Webrieflyreviewafewkeytopics:liftingproperties,weakfactorizationsystems,func-
torialfactorizations,algebraicweakfactorizationsystems, andthealgebraicsmallobject
argument.
3.1. Weakfactorizationsystems. Wewrite1,2,3,4forthecategoriesassignedtothese
ordinals;e.g.,2isthe“walkingarrow”category,3isthefreecategorycontainingacom-
posablepairofarrows,andsoon. ThefunctorcategoryM2 isthecategorywhoseobjects
arearrowsinM,depictedvertically,andwhosemorphisms(u,v): f ⇒garecommutative
squares
(3.1) · u // ·
f g
·(cid:15)(cid:15) // ·(cid:15)(cid:15)
v
Any such square presentsa lifting problem of f againstg; a solution would be an arrow
fromthebottomlefttotheupperrightsuchthatbothresultingtrianglescommute.Ifevery
liftingproblempresentedbyamorphism f ⇒ ghasasolution,wesaythat f hastheleft
liftingpropertyagainstgand,equivalently,thatghastherightliftingpropertyagainst f.
Definition 3.2 (I.2.3, I.2.4). A weak factorization system (L,R) on M consists of two
classesofmorphismssuchthat