Brown representability for space-valued functors

BR O WN REPRESENT ABILITY F OR SP A CE-V ALUED FUNC TORS BORIS CHORNY Abstra ct. In this pap er we pro ve t wo theorems whic h resemble the classical cohomo- logical and homological Brown represen tability theorems. The main difference is that our results class ify contra v arian t functors from s paces to spaces up to w eak equiv alence of functors. In more detail, we sho w that every contra v ariant functor from sp aces t o spaces which takes coprodu cts to p rod ucts up to homotop y , and takes homotopy p ushouts to homotopy pullbacks is natu rally w eekly equiv alent to a representable f unct or. The second representabilit y th eorem states: every con trav aria nt contin uous functor from the category of finite simplicial sets to simplicia l sets taking homotopy pushouts to homoto py p ullbacks is equ ival ent to the restriction of a represen table functor. This theorem may b e considered as a contra v ariant analog of Goo dwillie’s classification of linear functors [14]. 1. Introduction The cla ssical Brown r epresen tabilit y t h eorem [4] classifies c ontra v a riant functors from the homotop y category of p oint ed connecte d C W -complexes to the category of sets satisfying Milnor’s wedge axiom (W) and Ma yer-Vie toris p r op erty (MV). (W): F ( ` X i ) = Q F ( X i ); (MV): F ( D ) → F ( B ) × F ( A ) F ( C ) is surjectiv e for ev ery homotop y pushout square A / /   B   C / / D . In this p ap er we addr ess a similar classification pr ob lem, but the functors we classify are th e homot opy f unctors from spaces t o spaces, sat isfyin g (h W) and (hMV), the higher homotop y versions of (W) and (MV). (hW): F ( ` X i ) ≃ Q F ( X i ); (hMV): F ( D ) / /   F ( B )   F ( C ) / / F ( A ) is a homotop y pullbac k for every homoto py p ushout square A / /   B   C / / D . Homotop y fu nctors F : S op → S s atisfying (hW) and (h MV) are called c ohomol o gic al in this pap er. Our main result app earing in Theorem 4.1 b elo w is that suc h a fu nctor is naturally weakly equiv a lent to repr esen table fu nctor. W e should mention right a wa y , that b y sp ac es w e alw a ys mean simplicial sets in this pap er. I t is w ell kno wn that the homotop y cat egory of the un p ointe d s p aces fails to satisfy Date : August 11, 2021. 1 2 BORIS CHO R NY Bro wn representabilit y [15, Prop. 2.1]. The enric hed framework is more forgiving. Our results are form ulated for the unp oint ed spaces, bu t th ey remain v alid in the p ointe d situation to o. Note ho wev er, that n either our theorem implies Bro wn represen tabilit y , n or the con verse. W e assume s tr onger (higher homotop y) conditions ab out the fu nctor, bu t we also obtain an enric hed represen tabilit y result. Nev ertheless, o ur result has a natural predecessor from the Calculus of homotop y func- tors. Go o dwillie’s classification of linear functors [1 4] is related to the c lassical homological Bro wn represen tabilit y in the same w ay as our r ep resen tabilit y theorem related to the cohomologi cal Bro wn represen tabilit y . The second cla ssifi cation resu lt prov ed in this p ap er is “essenti ally equiv alen t” to Go o d- willie’s classification of finitary linear functors. The difference is that we p r o v e a higher homotop y v ersion of the homologi cal Brown representabilit y representabilit y in its con- tra v ariant form. Recall [2] that ev ery cohomological f unctor from the category of compact sp ectra to abelian groups is a restriction of a represent able functor. W e pro ve a non-stable enric hed v ersion o f this statemen t: ev ery con tra v ariant homot opy fu nctor from finite spaces to spaces satisfying (hMV) is equ iv alen t to a restriction of a repr esentable functor. Such functors are c alled h omolo gic al . Although there is n o direct imp lications b etw een o ur t h eorem and Go o dwillie’s classifica- tion of linear functors, there is an additional feature that our results share. In b oth cases ev- ery small functor ma y b e approximate d by an initial, up to homotop y , repr esen table/linear functor, i.e., b oth constructions ma y b e view ed as homoto pical lo calizations in some model catego ry of functors. Ho we ver the collectio n of all functors from spaces to sp aces do es not form a lo cally small category (natural transf ormations b et ween functors need not form a small s et in general). Ou r remedy to this pr oblem is to co ns id er only smal l functors, i.e., the functors obtained as left Kan extensions of functors defined on a small full sub category of spaces. The metho d o f pro of of o u r results deserves a commen t. The Y oneda em b edding Y : S → S op of sp aces (=simplicial s ets) into the catego ry of sm all contra v arian t fu nctors has a left adjoin t Z = ev ∗ . In this pap er we introd uce a lo calization on the category of small con tra v ariant fu nctors suc h that this pair of a d j oin t fu nctors b ecomes a Quillen equiv alence, while th e lo cal ob jects are equiv alent to the representa ble f unctors. In other words, w e ha v e a new mo d el for spaces, wher e ev ery homotop y type is r epresen ted by a compact (i.e., fi nitely presenta ble) ob ject, the represen table fu nctor. Unfortunately our new mo del of spaces is not class-cofibran tly generated, therefore we can not immediately apply it to the theory of homotopical lo calizations in spaces. Instead w e apply it to th e study of represent abilit y conditions for s m all functors. W e express the prop ert y for con trav arian t functors to satisfy (hW) and (hMV) as a local condition, i.e., suc h fu nctors become l o cal ob jects with resp ect to certain cla ss of maps. W e iden tify this class precisely and argue that the class of ob ject lo cal with resp ect to those maps is exactly the class of fu nctor equiv alen t to the representable functors, therefore th e lo calizat ion we constructed is the lo calization with resp ect to the class of m aps ensurin g that the local ob jects are the cohomologica l functors. BRO WN REPRESENT ABILITY FOR SP ACE-V ALUED FUNCTO R S 3 Therefore, to b e equiv alent to a rep r esen table functor is the same as sa tisfy the conditions (hW) and (hMV), moreo ve r, ev ery functor has the univ ersal, up to homotop y , approxima- tion b y a cohomologica l functor – the fibran t replacemen t in the lo calized mo d el cate gory , see Remark 4.2 for more details. W e finish our pap er w ith a n argument that t h e new mod els of spac es, app earing as local- izations of class-cofibran tly generated mo del categories, are not class-cofibrantly generated. This conclusion is quite unexp ected, b ecause the lo calizatio n of a com binatorial mo del cat - egory is alwa ys a com b inatorial model category (a t least under V opˇ enk a’s principle) [5]. 1.1. Ac knowledgmen t. W e thank Amnon Neeman for numerous h elpful con versati on, whic h led to th e results in this pap er . W e also thank T om Go o dwillie and the anonymous referee for h elpful remarks a b out the early v ersion of this pap er. 2. Model ca t egories of sma ll functor s and their localiza tion The ob j ect of study of this pap er is homotop y theory of con tra v ariant f u nctors from the catego ry of spaces S to S . The to talit y of th ese functors does not form a ca tegory in the usual sense, since the natural transformations b etw een t wo functors need not form a set in general, but rather a prop er class. W e c h o ose to tr eat a suffi cien tly la rge su b collection of functors, includin g all interesting fun ctors and forming a lo cally small catego ry . T h e next definition describ es elemen ts o f a reasonably large su b collection. Definition 2.1. Let D b e a (not necessarily small) simplicial category . A functor X e : D → S is r epr e se ntable if there is an ob j ect D ∈ D suc h that X e is naturally equiv alen t to R D , wher e R D ( D ′ ) = hom D ( D , D ′ ). A fun ctor X e : C → S is called smal l if X e is a small w eigh ted colimit of representables. R emark 2 .2 . G.M. Kelly [19] calls small functors ac c essible and weig hted colimit s in- dexe d . He prov es that small functors form a simplicial category w h ic h is closed und er small (we ighte d) colimits [19, Prop. 5.34 ]. In order to do h omotop y th eory we n eed to work in a category which is not on ly co- complete, bu t also complete (at least un d er finite limits). F ortunately , there is a simp le sufficien t co nd ition in th e s ituation of s m all functors. Theorem 2.3. If D is c o c omplete, then the c ate gory S D of smal l fu nctors D → S is c omplete. The m ain te chnical tool used in the pr o v e of the classification theorem is the theory of homotop y lo calizat ions. More sp ecifically , we apply certain homotop y lo calizatio ns in the catego ry o f small con tra v arian t f unctors S S op , or in a Quillen equiv alen t mo del category of maps of spaces with the equiv ariant mo del stru cture [12, 10]. Let us briefly r ecall the definitions and basic p rop erties of the inv olv ed mod el ca tegories. The pro j ective mo del structure on the small con tra v ariant fun ctors w as constru cted in [10]. The wea k equ iv alences and fibr ations in this mo del category are ob ject wise. This 4 BORIS CHO R NY mo del structure i s generated b y the cl asses of generating co fib rations and generating trivial cofibrations I = { R A ⊗ ∂ ∆ n ֒ → R A ⊗ ∆ n | A ∈ S , n ≥ 0 } , J = { R A ⊗ Λ n k ֒ → R A ⊗ ∆ n | A ∈ S , n ≥ k ≥ 0 } . The classes I and J satisfy the conditions of the generalized small ob ject argument [8], therefore w e refer to this model catego ry as class-c ofibr antly gener ate d , s ee [8, Definition 1.3] for the detailed definition and d iscu ssion. Note that the r epresen table f u nctors are cofibran t ob jects and the rest of cofibr an t ob jects are obtained as retracts if I -cellular ob jects. Another example of a class-cofibrantly generated mo del category is giv en by the equi- v arian t mo del structure on the maps of s p aces S [2] eq . Th e central concept of th e equiv ariant homotop y theory is the cate gory of orbits . In the category of m ap s of sp aces the sub- catego ry of orb its O [2] is th e full sub catego ry of S [2] consisting of d iagrams of the form T e =  X ↓ ∗  , X ∈ S . Motiv ation of this terminology and fu rther generalizatio n of th e con- cept of orb it can b e found in [13]. Equiv arian t homotop y and homology theories w ere dev elop ed in [11]. The theory of equiv arian t homotopical lo calizations was in tro d uced in [7]. W eak equiv alences and fibr ations in the equiv ariant mo del category are determined b y the follo wing rule: a map f : X e → Y e is a w eak equiv alence or a fibration if for ev ery T e ∈ O [2] the ind uced map of sp aces hom( T e , f ) : hom( T e , X e ) → h om( T e , Y e ). The catego ries of maps of spaces and sm all con tra v ariant f unctors are related by th e functor O : S [2] → S S op , called the orbit-p oint functor (generalizing the fixed-p oin t f unctor from the equiv arian t homotop y theory with r esp ect to a group action), whic h is defin ed b y the form u la ( X e ) O ( Y ) = hom  Y ↓ ∗ , X e  , for all Y ∈ S . Orb it-p oin t functor has a left adjoin t called the r e alization functor | − | [2] : S S op → S [2] . The main result of [10] is that this pair of fun ctors is a Quillen equiv alence. Before p r o ving the main classificatio n result, w e suggest the follo wing alternativ e char- acterizat ion of functors satisfying (hW) and (hMV) as local ob jects with resp ect to s ome class of m aps. Homotop y functors as lo cal ob jects. By defin ition ev ery cohomolog ical fu nctor F is a homotop y functor, i.e., F ( f ) : F ( B ) → F ( A ) is a w eak equiv alence for eve ry wea k equiv alence f : A → B . Denote by F 1 the class of maps b etw een representable functors induced by w eak equiv alences: F 1 = { f ∗ : R A → R B | f : A → B is a w.e. } , where R A denotes the represen table fun ctor R A = S ( − , A ). Y oneda’s lemma implies that F 1 -lo cal functors are precisely the fibrant h omotop y fu nc- tors. Cohomology functors as lo cal ob jects. Giv en a homotop y functor F , it suffices to demand tw o additional prop er ties f or the fu nctor F to be cohomological: F must con ve rt BRO WN REPRESENT ABILITY FOR SP ACE-V ALUED FUNCTO R S 5 copro ducts to p r o ducts u p to homotop y and it also m ust con ve rt homotop y push outs to homotop y pu llb ac k s . Y oneda’s lemma and the standard comm utation rules of v arious (ho)(co)li mits with hom( − , − ) implies that b oth prop erties are lo cal w ith resp ect to the follo w ing c lasses of maps: F 2 = n a R X i → R ` X i    ∀{ X i } i ∈ I ∈ S I o and F 3 =    ho colim   R A / /   R C R B   − → R D       A / /   C   B / / D – homotop y pushout in S    . Ob jects whic h are lo cal with resp ect to F = F 1 ∪ F 2 ∪ F 3 are precisely the fibrant homotop y f unctors. Lemma 2.4. Any functor F : S op → S satisfying (hMV) is a homotopy functor, i.e., for any we ak e quivalenc e f : A → B , the map F ( f ) : F ( B ) → F ( A ) is a we ak e qui v alenc e. Pr o of. Giv en a w eak equiv alence f : A → B the follo wing comm utativ e squ are is a homo- top y p ushout: A A f   A f / / B . Applying F we obtain: F ( B ) F ( f ) / / F ( f )   F ( A ) F ( A ) F ( A ) . The later squ are is a h omotop y p ullbac k iff F ( f ) is a wea k equiv alence. Therefore, an y functor satisfying ( h MV) is automatically a homotop y functor.  W e conclude that it suffices to inv ert F = F 2 ∪ F 3 . R emark 2.5 . The indexing category I u sed to describ e F 2 is a completely a rb itrary s m all discrete catego ry . In particular I can b e empty . Th is imp lies that the map ∅ → R ∅ is in F 2 . In other w ords, if F is a cohomological functor, then F ( ∅ ) = ∗ . This p rop erty is analogous to th e r equirement that ev ery lin ear fun ctor is reduced in homoto py calculus. R emark 2.6 . S ince homologica l functors (see a b rief explanation on p . 2 or an official Definition 4.5) are defined on the category of finite simplicial sets, we need to adjust the definition of F 3 . F ′ 3 =    ho colim   R A / /   R C R B   − → R D       A / /   C   B / / D – homotop y pushout, A, B , C, D ∈ S fin    . 6 BORIS CHO R NY Then the reduced h omologica l functors in S S op fin (with the pro jectiv e m o del structure) are precisely the functors whic h are lo cal with respect to F ′ = F ′ 3 ∪ {∅ → R ∅ } 2.1. Lo calization. Represen ting the class of cohomology functors as lo cal ob jects do es not cont rib ute m uc h to their understanding. Our next goal is to mak e sure that there exists a localization of the mo del structure with resp ect to F and the class of ob jects w e a willing to classify will b e r epresen ted, up to homotop y , b y the elemen ts of the homotop y catego ry of the localized mo del category . After w e ac h iev e this, we ha ve a c han ce to find a simpler mod el catego ry , Q uillen equiv alen t to the lo calized mo d el cate gory , hence classi- fying the ob jects of the homotopy categ ory . In addition the lo calization approac h to the classification problem supplies us with an appr o ximation to ol, n amely the fib ran t rep lace- men t in the lo calized category , so that ev ery fu nctor ma y b e turned into a cohomologic al functor in a functorial wa y and su c h approxima tion is initial with r esp ect to m aps in to other cohomological fu nctors. Lo calizatio n pr o cedure is not alw ays a routine. F or example, the existence of lo cal- ization of spaces with resp ect to the class of cohomologica l equiv alences is still an op en problem (assuming V op ˇ enk a’s pr inciple in add ition to the standard axioms this question w as p ositiv ely settled [6]). I n our situation no curren tly existing general lo calizatio n ma- c hine ma y b e immediately ap p lied, since F is a p rop er cla ss of maps and the cate gory of small functors is not cofibran tly generated. W e will implement an ad hoc approac h to this lo calizatio n p roblem. Namely , relying on th e intuitio n stemmin g out of the classical Bro wn represen tabilit y we assume that the lo calized mod el category will b e equiv alen t to the category of sp aces, construct such lo calization disregarding F , and afterwards p ro ve that this lo calization is precisely the localization w ith resp ect to F . The basic idea b ehind this lo calization is to turn the adju nction ev ∗ : S S op ⇄ S : Y in to a Quillen equiv alence (to see th at th is is in d eed an adjun ction note that ev ∗ ( F ) = F ⋆ I d S ). F or this purp ose we will u se the derived version of the unit of this adjunction: F → Y ev ∗ F . W e need to turn q = Y ev ∗ : S S op → S S op in to a h omotop y functor. Since ev ∗ is a h omotop y functor in the pro j ectiv e mo del structure and Y p reserv es w eak equiv alences of fibrant simp licial sets, the derived v ersion of q ma y b e c hosen to b e the comp osition Q = Y c ev ∗ , wh ere d ( − ) is a functorial fibr ant replacemen t in simplicial sets. Q is equipp ed with a coaugmen tation η : Id → Q , defined as a co mp osition of the unit of adjunction with the application of Y on the n atural map of simplicial sets ev ∗ ( F ) → \ ev ∗ ( F ) . The ca tegory S [2] eq is relate d to the category of co ntra v arian t f unctors b y the Qu illen equiv alence [1 0]: (1) | − | [2] : S S op ⇄ S [2] eq : ( − ) O . W e w ould lik e to lo calise simultaneo u s ly the mo d el category S [2] eq , so that the adju nction (1) w ould remain Quillen equiv alence. In order to construct the required lo calizatio n of S [2] eq w e will tak e the deriv ed v ersion of the un it of the adju nction (2) L : S 2 eq ⇆ S : R, BRO WN REPRESENT ABILITY FOR SP ACE-V ALUED FUNCTO R S 7 where L  A ↓ B  = A and R ( A ) = A ↓ ∗ . W e defin e Q ′  A ↓ B  = ˆ A ↓ ∗ , and notice that the un it of the adjunction (2), comp osed with the application of R on the natural map L  A ↓ B  → \ L  A ↓ B  , pro vides Q ′ with a coaugment ation η ′ : Id → Q ′ . It turn s out th at the lo calizati on of the mo del category S [2] eq with resp ect to Q ′ is precisely the lo calizatio n of S [2] eq with resp ect to the class of maps |F | [2] = |F 1 | [2] ∪ |F 2 | [2] ∪ |F 3 | [2] , where |F 1 | [2] =  A ↓ ∗ − → B ↓ ∗     A → B is a w .e. in S  , |F 2 | [2] = ( a X i ↓ ∗ − → ` X i ↓ ∗      ∀{ X i } i ∈ I ∈ S I ) , and |F 3 | [2] =              ho colim        A ↓ ∗ / /   C ↓ ∗ B ↓ ∗        − → D ↓ ∗            A / /   C   B / / D is a homotopy pushout in S              . R emark 2.7 . The realiza tion fun ctor | − | [2] ma y b e viewe d as a coend Inc ⊗ S − , where Inc : S = O [2] ֒ → S [2] is the fully-faithfu l em b edding of the su b category of orbits [10]. Therefore, computing the realization of the repr esen table functors is ju st th e ev aluation of Inc at the represen ting ob ject, since the dual of the Y oneda lemma app lies. The main tec hn ical ac h iev emen t of this pap er, wh ic h is b ehind the pro of of the repre- sen tabilit y theorem is th e f ollo win g. Theorem 2.8. Ther e exist lo c alizations of the pr oje ctive mo del structur e on S S op with r esp e c t to Q and of the e quivariant mo del structur e on S [2] with r e sp e ct to Q ′ , so that al l adjunctions in the fol lowing triangle b e c ome Quil len e quivalenc es. S R   Y   S S op |−| [2] , , ev ∗ ? ? S [2] ( − ) O l l L T T Pr o of. The existence of localization follo ws from the Bousfield-F riedlander theorem [3, A.7]. W e ha ve to v erify that 8 BORIS CHO R NY (1) Q and Q ′ preserve w eak equiv alences; (2) Q and Q ′ are coaugmen ted, homotop y idemp oten t f unctors; (3) Pull bac k of a Q ( Q ′ )-equiv alence along a Q ( Q ′ )-fibration is a Q ( Q ′ )-equiv alence again (the resu lting lo calized c ategory becomes rig ht prop er). Q and Q ′ are constructed in s u c h a w a y that conditions (1) and (2) are s atisfied. The v erification is a routine. In order to v erify (3) n otice that a map in S S op ( S [2] ) is a Q ( Q ′ )-equiv alence iff the m ap induced b et wee n the v alues of the fun ctors in ∗ ∈ S (0 ∈ [2]) is a w eak equiv alence. Since S is righ t prop er, an y pull bac k of suc h map along a lev elwise fibration w ill ha ve the same prop erty . Certainly any Q ( Q ′ )-fibration is a lev elwise fi bration, hence the conditions of Bousfield-F riedlander theorem are satisfied. It remains to sho w that the adjunctions in the triangle ab ov e b ecame Quillen equiv a- lences. Note that the comp osition of the righ t adjoin ts of the right edge and of the b ase of the triangle equals to the righ t adj oin t of the left edge ( R ( − )) O = Y ( − ), so it suffices to v erify only that the right edge and the b ase of the triangle are Q u illen equiv alences. The adjun ction of the right edge is a Qu illen pair, since the left adjoint L preserv es cofibrations and trivial cofibrations. It r emains to show th at A = L  A ↓ B  → X is a w eak equiv alence iff  A ↓ B  → R ( X ) =  X ↓ ∗  is a Q ′ -equiv alence, w h ic h is clea r. The adju nction in the base of the triangle is a Qu illen p air by Dugger’s lemma [16, 8.5.4], since the right adj oin t preserv es fibrations of fib ran t ob jects (in the categ ory of maps Q ′ - fibrant o b ject are weakl y equiv alen t to orb its, hen ce their orbit p oin ts are we akly equ iv alen t to repr esen table fun ctors, i.e., Q -fibrant in the category of con tr a v ariant f unctors, but fibrations of Q -lo cal ob jects are Q -fibrations), and also tr ivial fi brations (since those d o not c h an ge under localization). It remains to sho w that for a ll cofibran t F ∈ S S op and for all Q ′ -fibrant  A ↓ B  ≃  A ↓ ∗  a m ap f : F →  A ↓ ∗  O = R A is a Q -equiv alence iff the adjoint map f ♯ : | F | [2] →  A ↓ ∗  is a Q ′ -equiv alence. The ‘only if ’ direction follo ws by applying the realizatio n functor on f , since | R A | [2] =  A ↓ ∗  and realizatio n preserves w eak equiv alence of cofibran t ob jects. Th e ‘i f ’ d irection follo ws from computation of the v alue F ( ∗ ): u s ing the comp osition of t wo left adj oints L ( | F | [2] ) = ev ∗ ( F ), we find out that F ( ∗ ) is equiv alent to the d omain of | F | [2] ).  R emark 2.9 . Theorem 2.8 pr o vides us with t wo mod el of spaces with the follo wing pr op erty: ev ery ob ject is w eakly equiv alen t to an ℵ 0 -small ob ject. This conclusion s eems con tra- in tuitiv e in view of Ho v ey’s pro of that every cofibrant and ℵ 0 -small, relativ e to cofibrations, ob ject in a p oin ted finitely generated mo d el category C is ℵ 0 -small in Ho( C ) [17, 7.4 .3]. Ho w ev er, there is no cont radiction with our result, s ince the lo calized mo del categories S [2] eq or S S op are v ery far from b eing fin itely generated. BRO WN REPRESENT ABILITY FOR SP ACE-V ALUED FUNCTO R S 9 It is tempting to try to app ly these mo dels to the problem of lo calizatio n of s p aces with resp ect to some prop er class of maps, whic h w e co u ld not do before d ue to set theoretical difficulties (the cardinalit y of d omains and co domains of these maps would n ot b e b ound ed b y an y fi xed cardinal). Ho wev er, there is still an obstacle p rev enting an immediate ap p lica- tion of these mo d els to lo calizat ion problems in S . The Bousfield-F r iedlander localization mac hinery u sed to prov e Th eorem 2.8 do es not p ro vide the lo calized mo del catego r ies with a class of generati ng trivial cofibr ations that is necessary for c onstru ction of new localiza- tions. In fact the new m o del cat egories fail to b e (c lass-)cofibrantly ge nerated, as we will sho w in Secti on 5. Our next goal is to sh o w that Q -localization is precisely the localization with resp ect to F and Q ′ -lo calization is precisely the lo calization with resp ect to |F | [2] . 3. Tech nical preliminaries Recall that we are going to pro ve t w o more theorems in this p ap er. T heorem 4.1 classifies cohomologi cal functors and Theorem 4.9 classifies homological fu nctors. Ho wev er the tec h - nicalities b ehind the pro ofs are v ery similar. Therefore, while w e a re heading to wards th e pro of of Theorem 4.1 fi rst, w e indicate little adju stmen ts required to adapt the argument for the p ro of of Th eorem 4.9. The Q -lo cal ob j ects are p r ecisely th e fu nctors (lev elwise) weakly equiv alen t to th e rep re- sen table functors R A with A fib rant . W e need to s h o w that ev ery ob ject in S S op is F -lo cal equiv alen t to a representable functor. Ev ery s m all con trav ariant functor may b e app ro ximated b y an I -cellular diagram, up to a (lev elwise) w eak equiv alence [10], where I =  ∂ ∆ n ↓ ∆ n ⊗ R A     A ∈ S  . Therefore, it suffices to sho w that eve ry I -cellular diagram is F -equiv alen t to a repr esentable functor. W e a re going to pr o v e it by cellular induction, but w e precede the p r o of with the follo w ing lemma, wh ic h says that th e basic buildin g b lo c ks of cellular complexes are F - equiv alen t to represent able functors. Lemma 3.1. F or every A ∈ S , n ≥ 0 , ther e exists A ′ ∈ S such that ∂ ∆ n ⊗ R A F ≃ R A ′ . Pr o of. W e will pro v e the statemen t with A ′ ≃ ∂ ∆ n ⊗ A . The proof is b y induction on n . F or n = 0 we hav e ∂ ∆ 0 ⊗ R A = ∅ ⊗ R A = ∅ F ≃ R ∅ = R ∂ ∆ 0 ⊗ A , since the map ∅ → R ∅ is in F b y Remark 2.5. Alternativ ely , if one is willing to exclude F 2 from F , then for th e base of ind uction it suffices to assu me that the cohomology functor F is reduced, i.e., F ( ∅ ) = ∗ ; cf. Remark 2.6. In other w ords the basis for induction holds f or F ′ equiv alences a s w ell. 10 BORIS CHO R NY Supp ose the state ment is true for n , i. e., ∂ ∆ n ⊗ R A F ≃ R ∂ ∆ n ⊗ A ; w e need to sho w it for n + 1. ∂ ∆ n +1 ⊗ R A ≃ colim     ∂ ∆ n ⊗ R A   / /  _   ∆ n ⊗ R A ∆ n ⊗ R A     F ≃ ho colim     R ∂ ∆ n ⊗ A / /   R A R A     ≃ ho colim     R ∂ ∆ n ⊗ A / /   R ∆ n ⊗ A R ∆ n ⊗ A     F ≃ R colim     ∂ ∆ n ⊗ A   / /  _   ∆ n ⊗ A ∆ n ⊗ A     ≃ R (∆ n ` ∂ ∆ n ∆ n ) ⊗ A ≃ R ∂ ∆ n +1 ⊗ A , where the fir st F -equiv alence is ind uced b y the F -equiv alence in the upp er left v ertex of the diagram (by indu ction h yp othesis) and in the other t w o ve rtices w e ha v e lev elwise w eak equiv alences. (If we w ill map b oth homotopy pu shouts int o an arb itrary F -lo cal ob ject W f , w e will obtain a lev elwise w eak equiv alence of homoto py pullbac k squares of spaces). The second F -equiv alence is indu ced by th e map from F 3 ⊂ F corresp onding to the h omotop y pushout square: ∂ ∆ n ⊗ A   / /  _   ∆ n ⊗ A   ∆ n ⊗ A / / (∆ n ` ∂ ∆ n ∆ n ) ⊗ A. The ab o ve argu m en t applies for all finite A if w e consider F ′ instead of F , as we did not use an y equiv alences induced by an elemen t of F 2 .  W e will need to use the f ollo win g s tandard result Lemma 3.2. The fol lowing c ommutative squar e is a pushout squar e A f / / g   B g ′   C f ′ / / D if and o nly if the squar e A ` A ∇ / / f ` g   A g ′ f   B ` C / / D is a p ushout squar e. BRO WN REPRESENT ABILITY FOR SP ACE-V ALUED FUNCTO R S 11 Pr o of. Represen t t h e tw o pushout diagrams a s th e co equalizers: A ` A ∇ / / f ` g / / A ` B ` C / / D and A ` A ` A ` A ∇ 2 / / ( f ` g ) ` ( f ` g ) / / A ` B ` C / / D There exist natural maps in b oth d irections b etw een the co equ alizer diagrams, sho wing that their colimits co incide.  Lemma 3.3. L et M b e a class-c ofibr antly gener ate d mo del c ate gory, such tha t the class of gener ating c ofibr ations I has ℵ 0 -smal l do mains with r e sp e ct to the c ofibr atons. Then every I -c el lular c omplex X ∈ M may b e de c omp ose d into an ω - indexe d c olimit X = colim n X n such that for every n ∈ N ther e is a pushout squar e (3) A / /  _ f   X n  _   B / / X n +1 , wher e the map A ֒ → B is a c opr o duct of a set of maps fr om I . Pr o of. Ev ery I -c ellular complex X h as a decomp osition in to a colimit indexed by a cardinal λ : X = colim a<λ ( X 0 , 0 → · · · → X a, 0 → X a +1 , 0 → · · · ) , where X 0 , 0 = ∅ , X a, 0 is obtained fr om X a − 1 , 0 b y att ac hin g a cell g ∈ I : (4) C / /  _ g   X a − 1 , 0  _   D / / X a, 0 , and X a, 0 = colim b k + 1, then pu t X n,a = X n − 1 ,a ` X n − 1 ,a − 1 X n,a − 1 . W e ha ve X a, 0 = colim n<ω X n,a , sin ce in the comm utativ e diagram C / /  _ g   X k ,a − 1   / / X k +1 ,a − 1   / / . . . / / X a − 1 , 0 = colim n<ω X n,a − 1   D / / X ′ k ,a / / X k +1 ,a / / . . . / / X a, 0 = colim n<ω X n,a all squares comp osing the ladder are pushouts by definition, so is t h e ou ter square. It remains to show that X k +1 ,a is obtained from X k ,a as in a push out of the form (3). F or other v alues of the first index this is clear. It suffi ces to sho w that the square A ` C  _   / / X k ,a  _   X k ,a − 1 B ` D / / X k +1 ,a is a pu shout. First let us split it in to t wo squares (5) A ` C  _   / / A ` X k ,a − 1  _   / / X k ,a  _   B ` D / / B ` X ′ k ,a / / X k +1 ,a and then sh o w that these t wo squ ares a re p ushouts. The left square is a push out as a copro du ct of tw o pushout sq u ares. It remains to sho w that the righ t squ are of (5) is a pushout. Let us start w ith the f ollo win g pushout square: A / /  _   X ′ k ,a  _   B / / X k +1 ,a . Lemma 3.2 imp lies that th e s quare A ` A   / / B ` X ′ k ,a   A / / X k +1 ,a BRO WN REPRESENT ABILITY FOR SP ACE-V ALUED FUNCTO R S 13 is also a pushout. No w w e ca n split it into t wo squares again A ` A / /  _   A ` X k ,a − 1  _   / / B ` X ′ k ,a   A / / X k ,a / / X k +1 ,a , where the right square is exactly the r igh t square of (5 ) an d the left squ are is a push ou t b y Lemm a 3.2 since the squ are A / / X k ,a − 1 A / / X k ,a is a pushout. Therefore th e right s q u are is also a p ushout, which is what we n eeded to sho w.  Prop osition 3.4. Every I -c el lular c omplex X e ∈ S S op is F -e quivalent to a r epr esentable functor R A for some A . Pr o of. By definition, ev ery I -cellular complex X has a decomp osition in to a colimit indexed b y a cardinal λ starting from th e in itial ob ject and on eac h stage one elemen t of I is attac hed. By Lemma 3.3, there is an alternativ e decomp osition of X : X = colim a<ω ( X 0 → · · · → X a → X a +1 → · · · ) , where X 0 = ∅ a nd X a +1 is obtained fr om X a b y attac hing a small co llection cells: ` A ( ∂ ∆ n ⊗ R A ) / /  _   X a  _   ` A (∆ n ⊗ R A ) / / X a +1 . Note for the b asis of induction, that X 1 is F -equiv alen t to R ` A A , sin ce ∆ n ⊗ R A ≃ R A . Assuming, by induction, that X a is F -equiv alen t to a repr esen table f unctor R C a , we n otice, b y Lemma 3.1, that ` A ( ∂ ∆ n ⊗ R A ) = ∂ ∆ n × ` A R A is F -equiv alen t to R ∂ ∆ n ⊗ ` A A = R ` A ( ∂ ∆ n ⊗ A ) , and ` A (∆ n ⊗ R A ) ≃ R ` A A , so all the v ertices of th e h omotopy pushout ab o ve are F -equiv alen t to rep r esen table functors R A ′ for s ome A ′ . W e conclude that X a +1 is F -equiv alen t to a representa ble functor R C a +1 , where C a +1 is the h omotop y p ushout ( ` A A ← ` A ( ∂ ∆ n ⊗ A ) → C a ), similarly t o the argu m en t o f Lemma 3.1. W e obtain the follo win g counta ble comm utativ e ladder: X 0   / / O F   · · ·   / / X a   / / O F   X a +1   / / O F   · · · R C 0 / / · · · / / R C a / / R C a +1 / / · · · . 14 BORIS CHO R NY T aking homotop y colimit of the upp er and the lo wer ro w s w e find that X F ≃ ho colim a<ω R C a , since if w e will map b oth homotop y colimits in to an arb itrary F -lo cal functor W , w e will obtain a weak equiv alence betw een the homotop y in verse limits. Finally , ho colim a<ω R C a = ho colim a<ω  R C 0 f 0 − → · · · − → R C a f a − → R C a +1 f a +1 − → · · ·  ma y b e repr esen ted as a h omotop y push out as follo ws: ho colim a<ω R C a ≃ ho colim       ( ` R C a ) ` ( ` R C a ) 1 ` f / / ∇   ` R C a ` R C a       , where f = ` a<ω f a is the sh ift map and ∇ is the co diagonal. Observe th at the h omotop y pushout ab ov e is w eakly equiv alen t to the infinite telescop e construction. All v ertices of the homotop y pu s hout ab o ve are F -equiv alen t to certain representable functors through the resp ectiv e F -equiv alences fr om F 2 . T esting by mapp ing in to an arbitrary F -l o cal functor W , we fi nd th at the homotop y p ushout ab o ve is F -equiv alen t to the homotopy pushout of the r esp ectiv e represen table functors. The latter pushout is F -equiv alen t to an represent able functor R A through an F - equiv alence from F 3 .  4. Repr esent ability theorems W e are rea d y no w to pro ve the represen tabilit y theorems. Theorem 4.1. L et F : S op → S b e a smal l, homotopy functor c onverting c opr o ducts to pr o ducts, up to homotopy, and homotop y pushouts to homotopy pul lb acks. Then ther e exists a fibr ant sim plicial set Y , such tha t F ( − ) ≃ S ( − , Y ) . The value of Y may b e c ompute d by substituting ∗ into F and applying the fibr ant r eplac ement: Y = [ F ( ∗ ) Pr o of. W e h a v e pro ven so far t h at that the Q -lo calization constructed in 2.1 is essen tially the localization with resp ect to F : ev ery e lement of F is a Q -equiv alence, hence Q -fibrant ob jects are F -local and the in ve rs e inclusion follo ws f r om Prop osition 3.4, wh ic h says, in particular, that ev ery F -local ob ject is also Q -fibrant, hence an y Q -equiv alence is also an F -e quiv alence. Giv en a small functor F satisfying the cond itions of the theorem, consider its fibr an t replacemen t in the p r o jectiv e mo del structure F ˜ ֒ → ˆ F , then ˆ F is F -l o cal a n d th erefore also Q -fibrant, h ence the fibrant r ep lacemen t of ˆ F in the Q -lo cal m o del stru cture is a pro jectiv e w eak equiv alence F ≃ ˆ F ˜ →S ( − , [ F ( ∗ )). Th erefore it suffi ces to ta ke Y = [ F ( ∗ ) to prov e th e first statemen t of the representabilit y theorem. T o construct an approximat ion by a cohomol ogical functor for a f u nctor G consid er the factorizat ion of th e map G → ∗ into a trivial cofibration follo we d by a fi bration in the Q -lo cal mo del structure: G ˜ ֒ → ˆ G ։ ∗ . Th en th e map γ : G ˜ ֒ → ˆ G is initial, up to homoto py , b eneath maps o f G in to other fibran t cohomologica l fu nctors  BRO WN REPRESENT ABILITY FOR SP ACE-V ALUED FUNCTO R S 15 R emark 4.2 . Actually , we hav e pro ven a little bit m ore: for ev ery fu nctor G : S op → S there exists an approximat ion of G by a un iv ersal, up to homotop y , cohomolo gical functor, i.e., there exists a natural t ran s formation γ : G → ˆ G , w h ere ˆ G is c ohomological, suc h that for ev ery fib r an t cohomological functor H , any m ap G → H factors thr ou gh γ and the factorizat ion is un ique up to simp licial homotopy . R emark 4.3 . There is a different, simpler, approac h to the cla ssifi cation of co homological functors, wh ic h also do es n ot use the assu mption that th e functor is small: give n a simp licial cohomologi cal functor G : S op → S , consider the natural map q : G ( X ) → S ( X , G ( ∗ )) obtained b y adju n ction fr om the natural map X = S ( ∗ , X ) → S ( G ( X ) , G ( ∗ )), whic h exists, in turn, since G is simp licial. Th e map q is a n equiv alence if X = ∗ , wh ic h giv es a basis for induction on the cellular structur e of X similar to Prop osition 3.4. This appr oac h is simp ler, and more general (w orks for all functors, not necessarily sm all), but it does not giv e the b enefit of representing, cohomolo gical functors as fibr an t ob jects in a mo d el catego ry on sm all functors. W e o we this remark to T. Go o dwillie. R emark 4.4 . A similar representabilit y result w as obtained by J.F. Jardine [18]. His en- ric hed rep r esen tabilit y th eorem applies to fairly general mo del categorie s satisfying th e conditions analogous to the defin ition of a well- generated triangulated catego r y , but the conditions demanded from the fun ctor in this work are muc h more restrictiv e then ours: comm utation with arbitrary homotop y colimits. The fact that we restricted these condi- tions only to copro d ucts and h omotopy pushout allo ws us to call it the enriched Br own represent abilit y . Our metho d can b e extended to other mo del categ ories a s w ell, including those that d o not satisfy the conditions of Jardine’s th eorem. In [9] w e pro ve a similar represent abilit y theorem in th e dual category of s p ectra. Homologi cal Bro wn representabilit y fo r space-v alued fun ctors is essen tially Go o d w illie’s classification of linear functors. W e choose, ho we ver, to discuss the con trav arian t version of this theorem in our w ork (our result is relate d to Goo d w illie’s theorem in the same w a y as Adams’ repr esen tabilit y theorem [2] related to G.W. Whitehead’s [20] classification of generalize d homology theories). Eve n though p hilosophically th e tw o v ersions are the same, in ord er to obtain an implicati on b et w een them, w e would ha ve to w ork out a stable analogue of our theo rem and then use S -dualit y . W e lea ve it to the in terested reader. Definition 4.5. Simp licial f unctor F : S op → S is called homolo gic al if F con v erts homo- top y p ushouts of fin ite simplicial sets to homotop y pullbac ks. Example 4.6. Any fun ctor of the form H X,Y ( − ) = X × S ( − , Y ) is h omologic al; w e would lik e to distinguish homologica l functors of the form H ∗ ,Y , hence the next definition. Definition 4.7. A homol ogical functor F is r e duc e d if F ( ∅ ) ≃ ∗ . Similarly to Lemma 3. 1 w e ha ve Lemma 4.8. L et F b e a r e duc e d homolo gic al func tor, then for al l n ≥ 0 ther e is a we ak e quivalenc e F ( ∂ ∆ n ) ≃ S ( ∂ ∆ n , Y ) , wher e Y is a fibr ant simplicial set we akly e quivalent to F ( ∗ ) . 16 BORIS CHO R NY Pr o of. The statemen t is pro v ed b y induction on n . F or n = 0 there is a w eak equiv alence F ( ∂ ∆ 0 ) = F ( ∅ ) ≃ ∗ = S ( ∅ , [ F ( ∗ )) = S ( ∂ ∆ 0 , [ F ( ∗ )). Supp ose that the statemen t is true for n , then ∂ ∆ n +1 ≃ ∆ n ` ∂ ∆ n ∆ n , hence F ( ∂ ∆ n +1 ) ≃ holim( F (∆ n ) → F ( ∂ ∆ n ) ← F (∆ n ). Lemma 2.4 implies that F is a homotop y fu nctor, hence F ( ∂ ∆ n +1 ) ≃ holim( F ( ∗ ) → S ( ∂ ∆ n , d F ( ∗ )) ← F ( ∗ )) (inductiv e a ssu mption) ≃ holim( S ( ∗ , [ F ( ∗ )) → S ( ∂ ∆ n , [ F ( ∗ )) ← S ( ∗ , [ F ( ∗ ))) ( ∗ is a u nit in S ) ≃ S (ho colim( ∗ ← ∂ ∆ n → ∗ ) , [ F ( ∗ )) ≃ S ( ∂ ∆ n +1 , [ F ( ∗ ))  Theorem 4.9. L et F b e a r e duc e d homolo gic al functor F : S op → S , then for al l finite simplicial sets K ∈ S ther e is a we ak e quivalenc e F ( K ) ≃ S ( K, Y ) , wher e Y is a fibr ant simplicial set we akly e q u ivalent to F ( ∗ ) . Pr o of. It is p ossible to p ro ve this theorem along th e lines of th e pr o of of Th eorem 4.1, but the mo d el catego ries app earing on the wa y are all com bin atorial and the required lo calizat ions are all with r esp ect to sets of maps, so the mo d el theoretical part of this result is standard and not so in teresting. Instead we c h ose t o use the approac h of Re mark 4.3. Since F is a simplicial functor, similarly to Remark 4.3 there is a natural map F ( X ) → S ( X, F ( ∗ )), whic h is a w eak equiv alence if X = ∗ . This is the b ase for cellular induction. Let X b e a fi nite simplicial set, i.e., there is a finite c hain of inclusions ∅ = X 0 → X 1 . . . X a → X a +1 → . . . X k = X , so that X a +1 is obtained from X a b y attac hing a cell: ∂ ∆ n / /  _   X a   ∆ n / / X a +1 . Applying F we obtain a homot opy pullbac k F ( ∂ ∆ n ) F ( X a ) o o F (∆ n ) O O F ( X a +1 ) o o O O Assuming, by induction, that F ( X a ) = S ( F ( X a ) , [ F ( ∗ )) and using Lemm a 4.8 we obtain: F ( X a +1 ) ≃ holim( S ( ∗ , [ F ( ∗ )) → S ( ∂ ∆ n , d F ( ∗ )) ← S ( F ( X a ) , [ F ( ∗ ))) ≃ S (ho colim( ∗ ← ∂ ∆ n → X a ) , [ F ( ∗ )) ≃ S ( X a +1 , [ F ( ∗ )) . After k steps we obtain F X ≃ S ( X , [ F ( ∗ )).  BRO WN REPRESENT ABILITY FOR SP ACE-V ALUED FUNCTO R S 17 5. An ex ample of a non-class -cofibrantl y genera ted model ca tegor y The mo del of sp aces on the category of small contra v ariant fun ctors, which we con- structed in Secti on 2, h as a v ery nice prop erty: ev ery ob ject in it is wea kly equiv alen t to an ℵ 0 -small ob j ect — the repr esentable functor. Our initial motiv ation f or lo oking into this mo del category w as to u se this prop ert y in order to construct some homotopical lo- calizat ions with resp ect to ce rtain classes of maps, since the set-theoretical difficulties do not constitute an ob s truction in our mo d el. Ho wev er, another difficulty came up and w e could not ov ercome it so far: the lo calized mo d el catego ry on S S op is not class cofibrantly generated, h ence th e standard metho ds for constructing lo calizations are n ot applicable. On the other hand , this is the first example of a non-class-cofibran tly generated model catego ry arising in the top ological context. Examples of m o del categories featurin g similar prop erties, b u t taking origin in abstract categ ory theory app eared in [1]. There are t wo sligh tly differen t v ersions of the definition of the class-cofibrantly gen- erated m o del catego ries. The first one d emands that the domains and the cod omains of the generating (trivial) cofibrations are λ -present able, a nd th e second one in more general demanding only that the (co)domains are λ -small with resp ect to cofibr ations. T h is con- fusion p robably has its origin in the difference b et wee n the com bin atorial mo d el cat egories b y J. S mith and the cellular mo d el categories by P . Hir s c hhorn. F or example, the pro- jectiv e mo d el structur e on S S op is class-cofibran tly generated of the fi rst kin d, wh ile the equiv arian t mo del structure on the maps of spaces S [2] is cl ass-cofibrantly generate d only of the second kind. Th e r esp ectiv e lo calizatio n s of th ese mo del categories constructed in this pap er are not class-cofibrantly generated In o rd er to see that our m o del category is not class-cofibran tly generated w e form ulate a simple Prop osition 5.1. L et M b e a class-c ofibr antly gener ate d mo del c ate gory such that the domains and the c o domains of the gener ating trivial c ofibr ations ar e λ - pr esentable f or some c ar dinal λ . Then the fibr ations ar e close d in the c ate gory mor( M ) under se quential λ -filter e d c olimits, in p articular the fibr ant obje cts ar e close d in M under se quential c olimits. If the (c o)domains of the gener ating trivial c ofibr ations ar e λ -smal l with r esp e ct to c ofibr ations only, then the same c onclusion holds for se quential c olimits with c ofibr ations as b onding maps. The pro of is left to the r eader. If the lo calized mo del category on S S op w ould b e class-cofibrant ly generated, then the fibrant ob jects would b e closed u nder sequential λ -filtered colimits b y Prop osition 5.1. But it is easy to see that the represen table fu n ctors are not closed under sequential colimits of an y cardin alit y , hence the lo calized mo d el categ ory is n ot class-cofibrantly generated, at least by the first defin ition. Ev en more int eresting example is the lo calization of the equiv ariant mo del category on S [2] . The fibr an t ob j ects (i.e., the diagrams equiv arian tly homotop y equiv alen t to the orbits) are not clo sed under sequen tial co limits ev en if t he b onding maps are cofibrations. 18 BORIS CHO R NY Consider, for example, the follo win g colimit: colim n<ω [ n ] ↓ ∗ ! =  ℵ 0 ↓ ∗  , where [ n ] = ` n ∗ . It is quite surprisin g, but if w e rep lace all the b ondin g maps b y cofibrations, this colimit will b e no longer equ iv alen t to the orbit  ℵ 0 ↓ ∗  . colim             • • • • •   • • • •   • ... • • • • • •     / /   / /   / /             = • . . . • • • • • • • • ,   since if w e try to map the orbit  ℵ 0 ↓ ∗  in to the last colimit, then suc h map must factor through one of the finite sta ges, an d n o map corresp onds to the connected component of the identit y map on ℵ 0 ↓ ∗ in the mappin g space hom  ℵ 0 ↓ ∗ , ℵ 0 ↓ ∗  . The same argument generalize s to s equen tial colimits of any cardinalit y , hen ce w e can conclude that t h e lo calized model category on maps of spaces is not class-co fib ran tly gen- erated of t h e s econd kind . Referen ces [1] J. Ad´ amek, H. Herrlich, J. Rosick´ y, and W. Tholen. W eak factorization systems and top ological functors. Appl. Cate g. Struct ur es , 10(3):23 7–249, 2002 . 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Dep ar tment of Ma thema tics, The Unive rsity of Haif a a t Oranim, Tivon 36006, Israel E-mail addr ess : chorny@math.h aifa.ac.il