Homology and cohomology via enriched bifunctors

HOMOLOGY AND COHO MOLOGY VIA ENRICHED BIFUNCTORS KAZUHI S A SHIMAKA W A, K OHEI YOSHIDA, AND T ADA YUKI HAR AGUCHI T o the memory of Pr ofesso r Nobuo Shimada Abstra ct. Let T op and Diff b e the categories of top ological and dif- feologi cal spaces, respectively . By using an adj unction b etw een T op and Diff we show that th e full sub category NG of T op consisting of numer- ically generated spaces is complete, cocomplete, and cartesian closed. In fact, NG can b e embedded in to Diff as a cartesian closed full subcat- egory . It follo ws then th at the category NG 0 of numerically generated p ointed spaces is complete, co complete, and monoidally closed with re- sp ect to th e smash prod uct. These features of NG 0 are used to establish a simple b u t flexible method for constructing generalized h omology and cohomology theories by using t he notion of en ric hed bifunctors. 1. Introduction On the catego ry of p oin ted top ological spaces, generalized homolo gy and cohomology theories are usually constructed b y using sp ectra. Ho w ev er, the definition of a spectrum is not suitable for m ore general t yp e of mo del catego ries, e.g. the category of equiv arian t spaces, b ecause it is to o tigh tly connected to the framewo rk of cla ssical stable homotop y theory . In this pap er, w e present an alternativ e and more catego ry theoretical app r oac h to homology and cohomology theories wh ic h is based on the n otion of a linear enric hed functor ins tead of a sp ectrum. Our strategy is as follo ws: W e fi rst replace th e category T op 0 of p oin ted top ological spaces by a conv enien t fu ll sub cate gory NG 0 whic h fu lfills ou r requirement s. More explicitly , we call a top olo gical space X n umerically generated if it has the fin al top olo gy with resp ect to its singu lar simplexes, and defin e NG 0 to b e th e full su b category of T op 0 consisting of numerically generated spaces. Then N G 0 is complete, cocomplete, and is monoidally closed in the sense that there is an inte rnal h om Z Y satisfying a natural bijection hom NG 0 ( X ∧ Y , Z ) ∼ = hom NG 0 ( X, Z Y ) . 2000 Mathematics Subje ct Classific ation. Primary 55N20; Secondary 18G55, 18B30. 1 2 K. SHIMAKA W A, KOHEI YOSHIDA, AND T ADA YUKI HARAGUCHI Moreo v er, there exist a r eflector ν : T op 0 → NG 0 suc h th at the natural map ν X → X is a we ak equiv alence, and a sequence of w eak equiv alences Y X ← ν map 0 ( X, Y ) → map 0 ( X, Y ) where map 0 ( X, Y ) is th e set of p oin ted maps from X to Y equipp ed w ith the compact-op en to p olog y . Th us NG 0 is eligible, from th e viewp oin t of homotop y theory , as a conv enien t replacemen t for T op 0 . A functor T : NG 0 → NG 0 is called enric hed if the corresp ondence w hic h maps f : X → Y to T f : T X → T Y induces a morp hism Y X → T Y T X b e- t w een internal homs, and is called linear if it conv erts a cofibr ation sequen ce in to a homotopy fibration sequence. Theorem 6.4 states that ev ery linear enric hed functor T d efi nes a generalized homology theory X 7→ { h n ( X ; T ) } suc h that h n ( X ; T ) is isomorp h ic to π n T X for n ≥ 0, and to π 0 T (Σ − n X ) for n < 0. W e see that ev ery homology th eory represented b y a sp ectrum comes from a linear enriched f u nctor, and vice ve rsa. In fact, the func- tor whic h assigns to a linear enriched functor its deriv ativ e, in the sense of Go o dwillie [2], at S 0 induces a Quillen equiv ale nce b et w een the mo d el cate- gories of linear enric hed fu nctors and s p ectra. Th us the homotop y category of linear enric hed functors is equiv alen t to the stable category . A b enefit of our approac h is its flexibilit y: F or example, the enric hedness of T implies the orthogonalit y , h ence the symmetricit y , of its deriv ativ es { T (Σ n X ) | n ≥ 0 } . Our metho d for constructing cohomology theories is based on the notion of a bifunctor NG op 0 × NG 0 → NG 0 , con tra v arian t with r esp ect to the first argumen t and co v arian t with r esp ect to the second argument. There are bi- v ariant v ersion of the notions of enric hedness and linearit y , and Th eorem 7.3 states th at ev ery bifunctor F which is b iv ariantly enric hed and linear de- termines a pair of a generalized h omology theory X 7→ { h n ( X ; F ) } and a cohomology theory X 7→ { h n ( X ; F ) } su c h that h n ( X ; F ) ∼ = π 0 F ( S n + k , Σ k X ) , h n ( X ; F ) ∼ = π 0 F (Σ k X, S n + k ) hold wh en ev er k and n + k are non-negativ e. J ust as a sp ectrum r epre- sen ts b oth homology and cohomology th eories, eve ry linear enriched fu nctor T : NG 0 → NG 0 pro du ces homology and cohomology theories through the bilinear fu n ctor F giv en by the formula F ( X, Y ) = ( T Y ) X . How ev er, there exist enriched bifunctors not of the form as ab o v e bu t induce in teresting biv arian t homology-cohomolo gy theories. (Cf. Remark after Th eorem 7.3.) Since our approac h is based solely on the standard category theoretical notions such as enric hedness and lin earit y , it can b e easily extended to other mo del categories consisting of sim p licial sets, diffeologica l spaces, equiv ariant (top ologica l or diffeologic al) spaces, etc . Some of suc h extensions will b e discussed in a subsequent pap er. HOMOLOGY A ND COHOM OLOGY VIA ENRICHED BIFUNCTORS 3 2. Diffeological sp aces T o deriv e the pr op erties of the category of numerically generated spaces w e use an adjunction b et we en the categories of top ological and diffeologi cal spaces, r esp ectiv ely . Recall fr om [3] that a diffeologic al space consists of a set X together with a family D of maps fr om op en subsets of Eu clidean spaces into X satisfying the follo wing conditions: Co v ering. An y constant p arametrization R n → X b elongs to D . Lo calit y. A parametrization σ : U → X b elongs to D if ev ery p oint u of U has a neigh b orhoo d W suc h that σ | W : W → X b elongs to D . Smo oth compatibilit y. If σ : U → X b elongs to D , then so do es the comp osite σ f : V → X for any smo oth map f : V → U b et w een op en subsets of Euclidean spaces. W e call D a diffeology of X , and eac h member of D a plot of X . A map b et wee n diffeologica l spaces f : X → Y is called smooth if for an y plot σ : U → X of X the composite f σ : U → Y is a plot of Y . In particular, if D and D ′ are d iffeologies on a set X then the ident it y map ( X, D ) → ( X, D ′ ) is smo oth if and only if D is con tained in D ′ . In that case, w e sa y that D is fin er than D ′ , or D ′ is coarser than D . Clearly , the class of Diffeologica l spaces and smo oth maps form a category Diff . Theorem 2.1. The c ate gory Diff is c omplete, c o c omplete, and c artesian close d. A category is complete if it has equalizers and small pro ducts, and is co- complete if it has co equalizers and small copro ducts. Therefore, the theorem follo ws f rom the basic constructions give n b elo w. Pro ducts. Given diffeologic al spaces X j , j ∈ J , their pro duct is giv en b y a pair ( Q j ∈ J X j , D ), wh ere D is the set of parametrizations σ : U → Q j ∈ J X j suc h that eve ry its comp onent σ j : U → X j is a plot of X j . Copro ducts. The copro duct of X j , j ∈ J , is give n by ( ` j ∈ J X j , D ), wh ere D is the set of parametrizations σ : U → ` j ∈ J X j whic h can b e written lo cally as the comp osite of the inclusion X j → ` j ∈ J X j with a plot of X j . Subspaces. Any subset A of a diffeologica l space X is itself a diffeologica l space with p lots giv en b y those parametrizations σ : U → A suc h th at the p ost comp osition with the in clusion A → X is a plot of X . Quotien ts. Let p : X → Y b e a surjection fr om a diffeologic al sp ace X to a set Y . Then Y inherits from X a diffeolog y consisting of those parametriza- tions σ : U → Y which lifts lo cally , at ev ery p oin t u ∈ U , along p . Exp onen tials. Giv en diffeological sp aces X and Y , the set hom Diff ( X, Y ) has a diffeology D X,Y consisting of those σ : U → hom Diff ( X, Y ) such that 4 K. SHIMAKA W A, KOHEI YOSHIDA, AND T ADA YUKI HARAGUCHI for eve ry plot τ : V → X of X , th e comp osite U × V ( σ ,τ ) − − − → hom Diff ( X, Y ) × X ev − → Y is a plot of Y . Putting it differen tly , D X,Y is the coarsest diffeology suc h that the ev aluation map hom Diff ( X, Y ) × X → Y is smo oth. Let u s denote by C ∞ ( X, Y ) the diffeolog ical space (hom Diff ( X, Y ) , D X,Y ). Then ther e is a natural map α : C ∞ ( X × Y , Z ) → C ∞ ( X, C ∞ ( Y , Z )) giv en b y the formula: α ( f )( x )( y ) = f ( x, y ) for x ∈ X and y ∈ Y . T he follo wing exp onenti al la w implies th e cartesian closedness of Diff . Theorem 2.2 ([3, 1.60]) . The map α induc es a smo oth isomorphism C ∞ ( X × Y , Z ) ∼ = C ∞ ( X, C ∞ ( Y , Z )) . 3. Numericall y genera t ed sp aces Giv en a top olog ical space X , let D X b e the diffeological space with the same underlyin g set as X and w ith all conti nuous maps from op en sub sets of Eu clidean spaces into X as p lots. Clearly , a conti nuous map f : X → Y induces a smo oth m ap D X → D Y . Hence th ere is a fun ctor D : T op → Diff whic h maps a top ological space X to the diffeologica l sp ace D X . On the con trary , an y d iffeological space X determines a top ological space T X ha ving th e same underlying set as X and is equipp ed with the final top ology with resp ect to the p lots of X . An y smo oth map f : X → Y induces a contin uous map T X → T Y , hence we hav e a f unctor T : Diff → T op . Prop osition 3.1. The functor T is a left adjoint to D . Pr o of. Let X b e a diffeologi cal space and Y a top ologic al space. Then a m ap f : T X → Y is con tin uous if and only if the comp osite f ◦ σ is con tin uous for ev ery plot σ of X . But this is equiv alent to sa y that f : X → D Y is smo oth. Thus the n atural m ap hom T op ( T X, Y ) → h om Diff ( X, D Y ) is bijectiv e for ev ery X ∈ Diff and Y ∈ T op .  Prop osition 3.2. A top olo gic al sp ac e X is numeric al ly gener ate d i f and only if the c ounit of the adjunction T D X → X is a home omorphism. Pr o of. The condition T D X = X h olds if and only if X has the final top ology with resp ect to all th e contin uous maps from an op en su bset of a Eu clidean space into X . B ut this is equiv alen t to sa y that X h as the final top ology with resp ect to the singular simplexes of X .  Let us write ν = T D , so that X is n umerically generated if and on ly if ν X = X holds. HOMOLOGY A ND COHOM OLOGY VIA ENRICHED BIFUNCTORS 5 Lemma 3.3. F or any top olo gic al sp ac e X we have ν ( ν X ) = ν X . Pr o of. Ev ery plot σ : U → X of X lifts to a p lot of ν X , since ν U = U h olds for an y op en subset U of R n . Thus ν X has the same plots as X , and hence ν ( ν X ) h as the same top ology as ν X .  It follo ws that NG is reflectiv e in T op , and the corresp ondence X 7→ ν X induces a reflector ν : T op → N G . Prop osition 3.4. The c ate gory NG is c omplete and c o c omp lete. F or every smal l diagr am F : J → NG , we have lim J F ∼ = T (lim J D F ) ∼ = ν (lim J I F ) colim J F ∼ = T (colim J D F ) ∼ = colim J I F wher e I denotes the inclusion fu nctor NG → T op . Pr o of. Since Diff is complete, the diagram D F : J → Diff has a limiting cone { φ j : lim J D F → D F ( j ) } . W e shall s h o w that the cone { T φ j : T (lim J D F ) → T D F ( j ) = F ( j ) } is a limiting cone to F . Let { ψ j : X → F ( j ) } b e an arbitrary cone to F . Then { D ψ j : D X → D F ( j ) } is a cone to D F , hence there is a uniqu e morphism u : D X → lim J D F suc h that Dψ j = φ j ◦ u holds. But then T u : X = T D X → T (lim J D F ) is a uniqu e morph ism su c h that ψ j = T φ j ◦ T u holds . Hence { T φ j } is a limiting cone to F . Sin ce the righ t adjoin t functor D preserve s limits, w e hav e T (lim J D F ) ∼ = T D (lim J I F ) = ν (lim J I F ) . Similar argumen t sho ws that T (colim J D F ) is a colimit of F . But in this case we ha v e T (colim J D F ) ∼ = colim J ν F = colim J I F , b ecause th e left adjoin t functor T p reserv es colimits.  4. Exponentials in N G A map f : X → Y betw een top ologica l spaces is said to b e n umerically con tin uous if th e comp osite f ◦ σ : ∆ n → Y is con tin uous for ev ery s ingular simplex σ : ∆ n → X . C learly , we ha ve the f ollo wing. Prop osition 4.1. L et f : X → Y b e a map b etwe en top olo gic al sp ac es. Then the fol lowing c onditions ar e e quivalent: (1) f : X → Y is numeric al ly c ontinuous. (2) f ◦ σ : U → Y is c ontinuous for any c ontinuous map σ : U → X fr om an op e n subset U of a Euclide an sp ac e into X . (3) f : ν X → Y i s c ontinuous. (4) f : D X → D Y is smo oth. 6 K. SHIMAKA W A, KOHEI YOSHIDA, AND T ADA YUKI HARAGUCHI Let us denote by map ( X , Y ) the set of con tin uous maps from X to Y equipp ed with the compact-op en top ology , and let smap ( X , Y ) b e the set of n umerically con tin uous maps equ ipp ed with the initial top ology w ith resp ect to the maps σ ∗ : smap ( X, Y ) → map (∆ n , Y ) , σ ∗ ( f ) = f ◦ σ, where σ : ∆ n → X runs through singular simp lexes of X . More explicitly , the space map ( X, Y ) has a sub b ase consisting of those subsets W ( K, U ) = { f | f ( K ) ⊂ U } , where K is a compact sub set of X and U is an op en su bset of Y . O n the other h and, smap ( X , Y ) has a su bbase consisting of th ose su b sets W ( σ , L, U ) = { f | f ( σ ( L )) ⊂ U } , where σ : ∆ n → X is a singular simp lex, L a compact sub s et of ∆ n , and U an op en sub s et of Y . As w e ha v e W ( σ , L, U ) ∩ map ( X , Y ) = W ( σ ( L ) , U ) , the inclusion map map ( X, Y ) → smap ( X, Y ) is contin uous. Prop osition 4.2. The inclusion map map ( X, Y ) → smap ( X, Y ) is bij e c- tive for al l Y if, and only if, X is numeric al ly gener ate d. Pr o of. If X is n umerically generated then a n umerically con tin uous map f : X → Y is automatically con tin uous. Hence map ( X , Y ) → smap ( X , Y ) is su rjectiv e for all Y . Con v ersely , if map ( X, ν X ) → smap ( X, ν X ) is sur - jectiv e, then the unit of the adjunction X → ν X is cont inuous, implying that X is numerically generated.  Prop osition 4.3. Supp ose that X is a CW-c omplex. Then the inclusion map map ( X , Y ) → smap ( X , Y ) is a home om orphism for any Y . Pr o of. Since X h as the w eak top ology w ith resp ect to the f amily of closed cells, a map f from X to Y is con tin uous if and on ly if it is numerically con tin uous. Hence map ( X , Y ) → smap ( X , Y ) is a bijection. T o pro ve the con tin uit y of its inv erse, w e ha ve to sh o w that ev ery subset of the form W ( K, U ) is op en in smap ( X , Y ). Since X is closure-finite, K is conta ined in a finite complex, sa y A . Let { e 1 , . . . , e k } b e the set of cells of A , and let L i = ψ − 1 i ( e i ∩ K ) ⊂ ∆ n i , where ψ i : ∆ n i → X is a c haracteristic map for e i . Then we ha v e W ( K, U ) = W ( ψ 1 , L 1 , U ) ∩ · · · ∩ W ( ψ k , L k , U ) . Hence W ( K, U ) is op en in smap ( X , Y ).  HOMOLOGY A ND COHOM OLOGY VIA ENRICHED BIFUNCTORS 7 The prop osition ab o v e implies th at any CW-complex X is numerically generated. T h us we ha ve the f ollo wing. Corollary 4.4. The c ate gory NG c ontains al l CW- c om plexes. R emark. If X and Y are CW-complexes then so is the pro d uct X × Y in NG . In fact, if { e j | j ∈ J } and { e ′ k | k ∈ K } are the sets of closed cells of X and Y , r esp ectiv ely , then X × Y h as the weak top ology with resp ect to the set of closed cells { e j × e ′ k | ( j, k ) ∈ J × K } . Thus the sub categ ory of NG consisting of CW-complexes is closed under fi n ite pro d ucts. F or numerically generated sp aces X and Y , let us denote Y X = ν smap ( X, Y ) . Then there is a map α : Z X × Y → ( Z Y ) X whic h assigns to f : X × Y → Z the map α ( f ) : X → Z Y giv en by the formula α ( f )( x )( y ) = f ( x, y ) for x ∈ X and y ∈ Y . Theorem 4.5. The natur al map α : Z X × Y → ( Z Y ) X is a home omorphism. This clearly implies th e follo wing. Corollary 4.6. The c ate gory NG is a c artesian close d c ate gory. T o prov e the theorem, we use the relationship b etw ee n Y X and the ex- p onentia ls in Diff . Prop osition 4.7. F or any top olo gic al sp ac es X and Y , we have D smap ( X , Y ) = C ∞ ( D X, D Y ) . Pr o of. Let σ : U → smap ( X, Y ) b e a map f rom an op en sub set U ⊂ R n . Then σ is a plot of D smap ( X, Y ) if and only if the comp osite U σ − → smap ( X, Y ) τ ∗ − → map (∆ m , Y ) is con tin uous for ev ery singular simplex τ : ∆ m → X . But τ ∗ σ corresp onds to the the comp osite U × ∆ m σ × τ − − → smap ( X, Y ) × X ev − → Y , under th e homeomorphism map ( U × ∆ m , Y ) ∼ = map ( U, map (∆ m , Y )) . Th us σ is a p lot of D smap ( X, Y ) if and only if ev( σ, τ ) is con tin uous for ev ery τ . wh ic h is equiv ale nt to sa y that σ is a plot of C ∞ ( D X, D Y ).  W e are no w ready to prov e Th eorem 4.5. 8 K. SHIMAKA W A, KOHEI YOSHIDA, AND T ADA YUKI HARAGUCHI Pr o of of The or em 4.5 . The map α is a h omeomorphism b ecause it coincides with the comp osite of homeomorph isms Z X × Y = ν smap ( X × Y , Z ) = T C ∞ ( D ( X × Y ) , D Z ) = T C ∞ ( D X × D Y , D Z ) ∼ = T C ∞ ( D X, C ∞ ( D Y , D Z )) (1) = T C ∞ ( D X, D smap ( Y , Z )) = ν smap ( X , smap ( Y , Z )) ∼ = ν smap ( X, ν smap ( Y , Z )) = ( Z Y ) X (2) in whic h (1) f ollo ws from Th eorem 2.2, and (2) is ind uced b y the numerical isomorphism smap ( Y , Z ) → ν smap ( Y , Z ).  5. The sp a ce of bas epoint p reser ving m aps Let T op 0 and Diff 0 b e the categories of p oin ted ob jects in T op and Diff , resp ectiv ely . Evidentl y , th e adjunction ( T , D ) b etw een T op and Diff induces an adjunction ( T 0 , D 0 ) b et ween T op 0 and Diff 0 . Th us the category NG 0 of p oin ted ob jects in NG can b e iden tified with a fu ll sub categ ory of T op 0 consisting of those p ointed spaces ( X , x 0 ) such that ν X = X holds. Clearly , NG 0 is complete and cocomplete. Giv en p oin ted s paces X and Y , let map 0 ( X, Y ) and smap 0 ( X, Y ) de- note, resp ecti ve ly , the sub space of map ( X , Y ) and smap ( X , Y ) consisting of basep oin t preservin g maps. T hen Pr op osition 4.3 implies the follo wing. Prop osition 5.1. If X is a p ointe d CW-c omplex, then the inclusion map map 0 ( X, Y ) → smap 0 ( X, Y ) is a home omorphism for any p ointe d sp ac e Y . As b efore, let us d enote Y X = ν smap 0 ( X, Y ). By taking the constan t map as basep oint, Y X is r egarded as an ob ject of NG 0 . Recall that the smash pro duct X ∧ Y of p oin ted spaces X = ( X, x 0 ) and Y = ( Y , y 0 ) is defined to b e the quotien t of X × Y by its sub space X ∨ Y = X × { y 0 } ∪ { x 0 } × Y . W e no w define a p ointe d map α 0 : Z X ∧ Y → ( Z Y ) X to b e the comp osite Z X ∧ Y p ∗ − → ( Z, z 0 ) ( X × Y ,X ∨ Y ) α ′ − → ( Z Y ) X , where the mid - dle term ( Z , z 0 ) ( X × Y ,X ∨ Y ) denotes the sub space of Z X × Y consisting of th ose maps f : X × Y → Z su c h that f ( X ∨ Y ) = { z 0 } holds, p ∗ is indu ced b y the natural map p : X × Y → X ∧ Y , and α ′ is the restriction of the homeomorphism ν smap ( X × Y , Z ) ∼ = ν smap ( X, ν smap ( Y , Z )) . HOMOLOGY A ND COHOM OLOGY VIA ENRICHED BIFUNCTORS 9 Since p : X × Y → X ∧ Y is unive rsal among con tinuous maps f : X × Y → Z satisfying f ( X ∨ Y ) = { z 0 } , the ind uced map p ∗ is bijectiv e, hence so is α 0 . Prop osition 5.2. The map α 0 : Z X ∧ Y → ( Z Y ) X is a home omorphism for any X , Y , Z ∈ N G 0 . Pr o of. Giv en p airs of top ological spaces ( X, A ), ( Y , B ), w e ha v e ( Y , B ) ( X,A ) = T C ∞ (( D X, D A ) , ( D Y , D B )) , where C ∞ (( D X, D A ) , ( D Y , D B )) is the subspace of C ∞ ( D X, D Y ) consist- ing of th ose smo oth maps f : D X → D Y satisfying f ( D A ) ⊂ D B . Thus, to pro v e that α 0 is a homeomorphism, we need only sh o w that for an y p ointe d diffeologic al spaces A , B and C , the bijection p ∗ : C ∞ (( A ∧ B , ∗ ) , ( C, c 0 )) → C ∞ (( A × B , A ∨ B ) , ( C , c 0 )) induced b y the natural map p : A × B → A ∧ B = A × B / A ∨ B is a smo oth isomorphism. Here ∗ = p ( A ∨ B ) is a basep oin t of A ∧ B , and c 0 is a basep oin t of C . T o s ee th at ( p ∗ ) − 1 is sm o oth, let u s tak e a plot σ : U → C ∞ (( A × B , A ∧ B ) , ( C, c 0 )) and show that e σ = ( p ∗ ) − 1 · σ is a plot of C ∞ (( A ∧ B , ∗ ) , ( C, c 0 )). By d efinition, this is the case if for any p lot τ : V → A ∧ B the comp osite U × V e σ × τ − − → C ∞ (( A ∧ B , ∗ ) , ( C, c 0 )) × ( A ∧ B ) ev − → C is a plot of C . But for an y v ∈ V there exist a n eigh b orho o d W of v and a plot e τ : W → A × B suc h that p e τ = τ | W hold. Therefore, th e comp osite ev( e σ × τ ) coincides, on U × W , with the plot U × W e σ × e τ − − → C ∞ (( A × B , A ∨ B ) , ( C , c 0 )) × ( A × B ) ev − → C. This implies that ev( e σ × τ ) is lo cally , hence global ly , a plot of C . T h us e σ = ( p ∗ ) − 1 · σ is a plot of C ∞ (( A ∧ B , ∗ ) , ( C, c 0 )).  The prop ositio n implies a n atural bijection hom NG 0 ( X ∧ Y , Z ) ∼ = hom NG 0 ( X, Z Y ) . Hence we ha ve the f ollo wing. Corollary 5.3. The c ate gory NG 0 is a symmetric monoidal close d c ate gory with tensor pr o duct ∧ and internal hom of the form Z Y . Prop osition 5.4. (1) F or every p ointe d sp ac e X , the c ounit of the adjunc- tion ε : ν X → X is a we ak homotopy e quivalenc e. (2) If X ∈ NG 0 , then the b ije ction ι : map 0 ( X, Y ) → smap 0 ( X, Y ) is a we ak homotopy e quivalenc e. 10 K. SHIMAKA W A, KOHEI YOSHIDA, AND T ADA YUKI HARAGUCHI Pr o of. (1) Since S n is a CW-complex, w e ha v e π n ( X, x ) = π 0 map (( S n , e ) , ( X, x )) ∼ = π 0 smap (( S n , e ) , ( X, x )) for every x ∈ X . Therefore, th e numerical ly con tinuous map η : X → ν X induces th e inv erse to ε ∗ : π n ( ν X , x ) → π n ( X, x ). (2) W e ha v e a commutat ive d iagram map 0 ( S n , map 0 ( X, Y )) ι ∗ / / map 0 ( S n , smap 0 ( X, Y )) smap 0 ( S n , smap 0 ( X, Y )) map ( S n ∧ X , Y ) ∼ = / / α 0 O O smap ( S n ∧ X , Y ) , ∼ = O O whic h sho ws th at ι ∗ : map 0 ( S n , map 0 ( X, Y )) → map 0 ( S n , smap 0 ( X, Y )) is bijectiv e. Thus smap 0 ( X, Y ) h as the same n -lo ops as map 0 ( X, Y ). More- o v er, a similar d iagram as ab ov e, but S n replaced by I + ∧ S n , sho ws that smap 0 ( X, Y ) h as the same h omotop y classes of n -lo ops as map 0 ( X, Y ). It follo ws th at ι ∗ : π n ( map 0 ( X, Y ) , f ) ∼ = π n ( smap 0 ( X, Y ) , f ) is an isomorphism for ev ery f ∈ map 0 ( X, Y ) and n ≥ 0.  Corollary 5.5. F or al l X , Y ∈ NG 0 , the sp ac e Y X is we akly e quivalent to the sp ac e of maps map 0 ( X, Y ) e quipp e d with the c omp act-op en top olo gy. 6. Homology th eories v ia enr iched functor s Let C 0 b e a fu ll sub cate gory of NG 0 . Th en C 0 is an enric hed category o v er NG 0 with hom-ob jects F 0 ( X, Y ) = Y X . A co v ariant fun ctor T fr om C 0 to NG 0 is called en ric hed if th e corresp onden ce F 0 ( X, Y ) → F 0 ( T X, T Y ), whic h maps f to T f , is a morp hism in NG 0 , th at is, a basep oin t p reserving con tin uous m ap. Similarly , a contra v arian t functor T form C 0 to NG 0 is called enric hed if the map F 0 ( X, Y ) → F 0 ( T Y , T X ) is a morphism s in NG 0 . Let T : C 0 → NG 0 b e an enr iched fun ctor. Then for any p oin ted map f : X ∧ Y → Z such that Y and Z are ob jects of C 0 the comp osite X α 0 ( f ) − − − → F 0 ( Y , Z ) T − → F 0 ( T Y , T Z ) induces, b y adjunction, a p oin ted map X ∧ T Y → T Z . S imilarly , an enriched cofunctor T : C op 0 → NG 0 assigns X ∧ T Z → T Y as an adjunct to the comp osite X → F 0 ( Y , Z ) → F 0 ( T Z, T Y ). HOMOLOGY A ND COHOM OLOGY VIA ENRICHED BIFUNCTORS 11 Prop osition 6.1. Enriche d functors and c ofunctors f r om C 0 to N G 0 pr e- serve homotopies. Pr o of. Let h : I + ∧ X → Y be a p oin ted homotop y b et w een h 0 and h 1 . Then an enr iched functor T : C 0 → NG 0 induces a h omotop y I + ∧ T X → T Y b et we en T h 0 and T h 1 . Similarly , an enr ic hed cofunctor T ind uces a homotop y I + ∧ T Y → T X b et w een T h 0 and T h 1 .  Corollary 6.2. If T : C 0 → NG 0 is an enriche d f u nctor then a homotopy e q uivalenc e f : X → Y induc e s isomorph isms T f ∗ : π n T X ∼ = π n T Y for n ≥ 0 . Si milarly, if T is an enriche d c ofunctor then f induc es isomorphisms T f ∗ : π n T Y ∼ = π n T X for n ≥ 0 . F rom n o w on, we assu me that C 0 satisfies the follo wing conditions: (i) C 0 con tains all fin ite CW-complexes. (ii) C 0 is closed under finite w edge sum. (iii) If A ⊂ X is an inclusion of ob jects in C 0 then its cofib er X ∪ C A b elongs to C 0 ; in particular, C 0 is closed under the susp ension fu nctor X 7→ Σ X . The category FCW 0 of finite CW-complexes is a t ypical example of such a catego ry . Giv en a cont inuous map f : X → Y , let E ( f ) = { ( x, l ) ∈ X × map ( I , Y ) | f ( x ) = l (0) } b e th e m ap p ing trac k of f . Then the map p : E ( f ) → Y , p ( x, l ) = l (1), has the h omotopy lifting prop ert y for all spaces, and hence ind uces a bijection p ∗ : π n +1 ( E ( f ) , F ( f )) → π n +1 Y for all n ≥ 0, where F ( f ) den otes the fib er of p at the basep oint of Y . A sequen ce of p oin ted maps Z i − → X f − → Y is called a homotop y fib ration sequence if there is a homotop y of p ointed maps from f ◦ i to the constant map suc h that the indu ced map Z → F ( f ) is a w eak h omotop y equ iv alence. Definition 6.3. An enriched f unctor T : C 0 → NG 0 is called linear if for ev ery pair of ob jects ( X , A ) with A ⊂ X , the sequence T A → T X → T ( X ∪ C A ) , induced b y the cofibration sequence A ⊂ X ⊂ X ∪ C A , is a homotop y fibration sequence with resp ect to the null h omotop y of T A → T ( X ∪ C A ) coming fr om the cont raction of A within the red u ced cone C A . Lik ewise, an enric hed cofunctor T : C op 0 → NG 0 is called linear if the indu ced sequence T ( X ∪ C A ) → T X → T A is a homotop y fibration s equ ence. 12 K. SHIMAKA W A, KOHEI YOSHIDA, AND T ADA YUKI HARAGUCHI If T is a linear functor then eve ry p air ( X , A ) giv es r ise to an exact sequence of p ointed sets · · · → π n +1 T ( X ∪ C A ) ∆ − → π n T A T i ∗ − − → π n T X T j ∗ − − → π n T ( X ∪ C A ) → · · · terminated at π 0 T ( X ∪ C A ). Here i and j denote the inclusions A ⊂ X and X ⊂ X ∪ C A , resp ectiv ely , and ∆ is the comp osite π n +1 T ( X ∪ C A ) p − 1 ∗ − − → π n +1 ( E ( T j ) , F ( T j )) ∂ − → π n F ( T j ) ν − 1 ∗ − − → π n T A. Similarly , a linear cofunctor T in duces an exact sequence · · · → π n +1 T A ∆ − → π n T ( X ∪ C A ) T j ∗ − − → π n T X T i ∗ − − → π n T A → · · · terminated at π 0 T A . Theorem 6.4. F or every line ar functor T : C 0 → NG 0 , ther e exists a gen- er alize d homo lo gy the ory X 7→ { h n ( X ; T ) } define d on C 0 such that h n ( X ; T ) is isomorphic to π n T X if n ≥ 0 , and to π 0 T (Σ − n X ) otherwise. Pr o of. W e first sh o w that the map T ( X ∨ Y ) → T X × T Y induced by the pro jections of X ∨ Y ont o X and Y is a we ak equiv alence . This means that the functor Γ → NG 0 , whic h maps a p oin ted finite set k = { 0 , 1 , . . . , k } to T ( X ∧ k ) = T ( X ∨ · · · ∨ X ), is sp ecia l in th e sense that the n atural map T ( X ∧ k ) → T ( X ) k is a w eak equiv ale nce for all k ≥ 0. Hence π n T X is an ab elian monoid with resp ect to the m ultiplication π n T X × π n T X ∼ = π n T ( X ∨ X ) T ∇ ∗ − − − → π n T X induced by the folding map ∇ : X ∨ X → X . This multiplica tion coincides with the standard m ultiplication of π n T X b eca use they are compatible with eac h other. In particular, π 1 T X is an ab elian group for all X . Moreo v er, an y p ointe d map f : X → Y indu ces a natural transformation T ( X ∧ k ) → T ( Y ∧ k ), hence a h omomorp hism of ab elian monoids T f ∗ : π n T X → π n T Y for all n ≥ 0. T o see that T ( X ∨ Y ) → T X × T Y is a w eak equiv alence, it suffi ces to sho w that the s equence T X → T ( X ∨ Y ) → T Y , in duced by the inclusion X → X ∨ Y and the p ro jection X ∨ Y → Y , is a homotopy fibration sequence. But T X → T ( X ∨ Y ) → T Y is homotop y equiv alent to the homotop y fibration sequence T X → T ( X ∨ Y ) → T ( C X ∨ Y ) through the homotop y equiv alence T ( C X ∨ Y ) ≃ T Y induced by the retraction C X ∨ Y → Y . Next consider the homotop y exact sequ ence asso ciated with the sequence T X → T ( C X ) → T (Σ X ). As T ( C X ) is w eakly contrac tible, we obtain for ev ery n ≥ 0 a short exact sequ ence of ab elian monoids 0 → π n +1 T (Σ X ) ∆ − → π n T X → 0 . HOMOLOGY A ND COHOM OLOGY VIA ENRICHED BIFUNCTORS 13 Since π n +1 T (Σ X ) is an ab elia n group, the homomorphism ∆ is injectiv e, hence an isomorphism. But this in turn means that π n T X is an ab elian group for n ≥ 0. Therefore, h n ( X ; T ) is an ab elian group for all n ∈ Z . F or a p oin ted m ap f : X → Y , w e define h n ( f ) : h n ( X ; T ) → h n ( Y ; T ) to b e the homomorph ism induced b y T f : T X → T Y for n ≥ 0, and T (Σ − n f ) for n < 0. It is easy to see th at the fun ctor X 7→ { h n ( X ; T ) } together with a natural isomorphism h n ( X ; T ) ∼ = h n +1 (Σ X ; T ), give n by ∆ − 1 for n ≥ 0 and the iden tit y for n < 0, satisfies the homotop y and exact- ness axioms.  Example 6.5. (1) Let AG ( X ) denote the top ological f ree ab elian group generated b y X with basep oint ident ified with 0. Then th e corresp ondence X 7→ AG ( X ) defines a linear enric hed fun ctor NG 0 → NG 0 . By the Dold- Thom theorem, h n ( X, AG ) is isomorphic to the s in gular homology group for a CW-complex X . But this is not the case for general X . In fact, the singular homology theory cannot b e repr esented b y a linear enric hed functor T : NG 0 → NG 0 , b ecause the singular h omology do es not satisfy the wedge axiom (cf. [1, 6.11]), contradicti ng to the fact th at the homology theory giv en by a linear enr ic hed fu nctor must satisfy th e axiom. (2) Lab eled confi gu r ation spaces also give r ise to linear enric hed fu nctors. Let C M ( R ∞ ) b e the configuration space of finite p oin ts in R ∞ with lab els in a p artial ab elian monoid M . Then Ω C Σ X ∧ M ( R ∞ ) is a linear enric hed fu nc- tor of X , and hence defines a connectiv e h omology theory . W e ha ve shown in [5] that the classical homology , the stable h omotop y , and th e connectiv e K -homology theories can b e obtained in this w a y . If T : C 0 → NG 0 is an enr ic hed fun ctor then for ev ery X ∈ C 0 the spaces T (Σ n X ) together w ith the maps S 1 ∧ T (Σ n X ) → T (Σ n +1 X ) in duced b y the homeomorphism S 1 ∧ Σ n X ∼ = Σ n +1 X form a presp ectrum ∂ T X . F ollo wing Go o dwillie [2], w e call ∂ T X the deriv ativ e of T at X . Let L ( ∂ T X ) b e th e sp ectrification of ∂ T X (cf. [4]). Then its zeroth sp ace L ( ∂ T X ) 0 is an infi- nite lo op space and the corresp ondence X 7→ L ( ∂ T X ) 0 defines an enric hed functor LT : C 0 → NG 0 . If, moreo v er, T is linear then Theorem 6.4 implies that the natural map T X → L T X is a we ak equiv alence f or ev ery X ; h en ce LT defines the same homology theory as T . An enr iched functor T is called stable if the natural map T X → LT X is a homeomorphism for ev ery X . In particular, LT is stable for any T . L et SLEF b e the category of stable lin ear enric hed functors C 0 → NG 0 with enric hed n atural trans formations as morph isms. Then there is a fu nctor D from SL EF to the category Sp ec of sp ectra wh ic h m aps a stable functor 14 K. SHIMAKA W A, KOHEI YOSHIDA, AND T ADA YUKI HARAGUCHI T to its deriv ative ∂ T S 0 at S 0 . W e h a v e sh o wn that the homology the- ory h • ( − ; T ) indu ced by a linear enriched fu n ctor T is rep r esen ted b y the sp ectrum D ( LT ) = L ( ∂ T S 0 ). Con v ersely , any homology theory represented by a sp ectrum comes from a linear enriched functor. In fact, D has a left adjoint I : Sp ec → SLE F defined as follo ws: F or a sp ectrum E = { E n } , I E is the enric hed fu nctor whic h maps X to the zeroth space L ( E ∧ X ) 0 of the sp ectrification of the presp ect rum E ∧ X = { E n ∧ X } . The unit E → D I E and the counit I D T → T of th e adjunction are weak equiv ale nces give n by the m aps E n → Ω n ( E n ∧ S n ) → Ω ∞ ( E ∞ ∧ S n ) = L ( E ∧ S n ) 0 = I E ( S n ) = D I E n , I D T X = L ( ∂ T S 0 ∧ X ) 0 Lµ − − → L ( ∂ T X ) 0 = LT X ∼ = T X, where µ : ∂ T S 0 ∧ X → ∂ T X is a map of presp ectra consisting of the maps T ( S n ) ∧ X → T (Σ n X ) induced by the iden tit y of S n ∧ X . Let us regard Sp ec and SLE F as a m o del category with resp ect to the classes of w eak equiv alences, fibrations, and cofibrations consist, resp ec- tiv ely , of leve l wea k equiv alences, lev el fi brations, and morph isms that ha v e the left lifting prop ert y with r esp ect to the class of trivial fi b rations. Then w e ha v e the f ollo wing. Prop osition 6.6. The functor D : SLEF → Sp ec is a right Quil len e q uiv- alenc e, and henc e induc es an e quivalenc e b etwe en the homotopy c ate gories. Corollary 6.7. The homotop y c ate gory of SLEF is e quivalent to the stable c ate gory. R emark. Let T : C 0 → NG 0 b e an enriched fun ctor. Th en for every real inner p ro duct space V the standard orthogonal group action on V ind uces an orthogonal group action on the one-p oin t compactification S V of V , and hence on T (Σ V X ) where Σ V X = S V ∧ X . With resp ect to the orthog- onal group action on eac h T (Σ V X ), the presp ectrum { T (Σ V X ) } ind exed b y real inn er pro du ct spaces V form an orthogonal p resp ectrum . T h us the deriv ati ve ∂ T X = { T (Σ n X ) } at X extends to an orthogonal sp ectrum, and consequen tly is a symmetric sp ectrum with resp ect to the symmetric group action on T (Σ n X ) ind uced by the em b edding of the symmetric group into the orthogonal group as the subgroup of co ord inate p erm utations. 7. Biv ariant homology-coh omology theo ries W e n o w in tro d u ce the notion of a bilinear fu nctor, and describ e a passage from b ilinear fu nctors to generalized cohomology theories. In fact, w e shall sho w that a bilinear functor giv es rise to a pair of generalized h omology and HOMOLOGY A ND COHOM OLOGY VIA ENRICHED BIFUNCTORS 15 cohomology theories, or in other words, a biv aria nt h omology-co homology theory . Let F : C 0 op × C 0 → NG 0 b e a biv arian t functor wh ic h is con tra v arian t with resp ect to the first argument , and is co v ariant with resp ect to the second argum ent. W e say that F is enric hed (o v er NG 0 ) if f or all p oin ted spaces X , X ′ , Y , and Y ′ , the map F 0 ( X ′ , X ) × F 0 ( Y , Y ′ ) → F 0 ( F ( X, Y ) , F ( X ′ , Y ′ )) , ( f , g ) 7→ F ( f , g ) is contin uous and is p oin ted in the sense that if either f or g is constan t then so is F ( f , g ). Definition 7.1. An enriched bifu nctor F : C 0 op × C 0 → NG 0 is called a bilinear fu n ctor if for all ( X , A ) and ( Y , B ) the s equ ences F ( X ∪ C A, Y ) → F ( X, Y ) → F ( A, Y ) , F ( X, B ) → F ( X , Y ) → F ( X, Y ∪ C B ) are h omotop y fi bration sequences. Example 7.2. Giv en a linear fun ctor T : C 0 → NG 0 , ther e is a bilinear functor F T giv en by the formula F T ( X, Y ) = F 0 ( X, T Y ). Suc h a bilinear functor is said to b e of the normal form. Conv ersely , a bilinear functor F determines a linear functor T : C 0 → NG 0 b y p utting T X = F ( S 0 , X ), called the co v ariant part of F . Th er e is a n atural transform ation F → F T suc h that F ( X, Y ) → F T ( X, Y ) = F 0 ( X, F ( S 0 , Y )) m aps every ξ ∈ F ( X , Y ) to the map X → F ( S 0 , Y ) , x 7→ F ( x, 1 Y )( ξ ). Here we identi fy x ∈ X with the evident map S 0 → X . Theorem 7.3. F or every biline ar functor F : C 0 op × C 0 → NG 0 , ther e exist a gener alize d homolo gy the ory X 7→ { h n ( X ; F ) } and a g e ner alize d c ohomol - o gy the ory X 7→ { h n ( X ; F ) } such that (7.1) h n ( X ; F ) ∼ = π 0 F ( S n + k , Σ k X ) , h n ( X ; F ) ∼ = π 0 F (Σ k X, S n + k ) hold whenever k , n + k ≥ 0 . Mor e over, h n ( X ; F ) is natur al ly isomorphic to the n - th homolo gy gr oup h n ( X ; T ) given by the c ovariant p art T of F . Pr o of. Since F ( X, Y ) is linear with resp ect to Y , π n F ( X, Y ) is an ab elian group f or all X , Y and n ≥ 0. C learly , this ab elian grou p structur e is natural with r esp ect to b oth X and Y . Moreo ver, the bilinearit y of F implies natural isomorphisms π n F ( X, Y ) ∼ = π n +1 F ( X, Σ Y ) , π n F (Σ X, Y ) ∼ = π n +1 F ( X, Y ) Consequent ly , there is a n atural isomorphism π 0 F ( X, Y ) ∼ = π 0 F (Σ X, Σ Y ), called the susp ension isomorphism. 16 K. SHIMAKA W A, KOHEI YOSHIDA, AND T ADA YUKI HARAGUCHI F or ev ery p ointed space X and every inte ger n , let us defin e h n ( X ; F ) = colim k →∞ π 0 F ( S n + k , Σ k X ) , h n ( X ; F ) = colim k →∞ π 0 F (Σ k X, S n + k ) where the colimits are tak en with r esp ect to the susp ens ion isomorp hisms. Clearly (7.1) h olds , and we hav e h n ( X ; F ) ∼ = h n ( X ; T ) where T is the co- v ariant part of F . Thus the functor X 7→ { h n ( X ; F ) } together with the eviden t natural isomorphism h n ( X ; F ) ∼ = h n +1 (Σ X ; F ) d efi nes a generalized homology theory . Similarly , the co v ariant functor X 7→ { h n ( X ; F ) } together with th e natural isomorph ism h n +1 (Σ X ; F ) ∼ = h n ( X ; F ) defines a general- ized cohomolog y theory , b ecause it satisfies the homotop y and exa ctness axioms.  Example 7.4. (1) Let F : NG 0 op × NG 0 → NG 0 b e the b ilinear functor giv en by the form ula F ( X, Y ) = AG ( Y ) X . Then h n ( X, F ) is the singular cohomology group for a C W-complex X . But there exists no b ilinear f unctor represent ing the singular cohomology groups of all X . (Cf. Example 6.5 (2).) (2) The second author constructs in [6] a b ilinear functor F suc h that h n ( X ; F ) is isomorph ic to the ˇ Cec h cohomology group for all X , and h n ( X ; F ) is isomorph ic to the S teenro d homology group f or any compact metrizable space X . Referen ces [1] A . Dold and R. Thom. Quasifaserungen und unendliche symmetrische pro dukte. Ann. Math. , 67:239–281, 1958. [2] T. Goo dwillie. Calculus i: The fi rst deriv ative of pseudoisotopy theory . K -The ory , 4:1–27, 1990. [3] P . Iglesias-Zemmour. D iffeology . (working d ocument). [4] G. Lewis, J. P . May , and M. Steinberger. Equivariant Stable Homotopy The ory , volume 1213 of L e ctur e Notes i n Math. Springer-V erlag , 1986. [5] K . Shimak a wa . Configuration sp aces with partially summable lab els and homology theories. Math. J. Okayama Univ. , 43:43–7 2, 2001. [6] K . Y oshida. A contin uous bifunctor representing ˇ Cec h cohomology . p rep rint. Gradua te School of Na tural Science and Technolo gy, Oka y ama Uni ver- sity, Oka y ama 700-853 0 Jap an Gradua te School of Na tural Science and Technolo gy, Oka y ama Uni ver- sity, Oka y ama 700-853 0 Jap an Gradua te School of Na tural Science and Technolo gy, Oka y ama Uni ver- sity, Oka y ama 700-853 0 Jap an