External Spanier-Whitehead duality and homology representation theorems for diagram spaces
aa r X i v : . [ m a t h . K T ] A ug EXTERNAL SPANIER-WHITEHEAD DUALITY ANDHOMOLOGY REPRESENTATION THEOREMS FOR DIAGRAMSPACES
MALTE LACKMANN
Abstract.
We construct a Spanier-Whitehead type duality functor relatingfinite C -spectra to finite C op -spectra and prove that every C -homology theoryis given by taking the homotopy groups of a balanced smash product witha fixed C op -spectrum. We use this to construct Chern characters for certainrational C -homology theories. Introduction
Let C be a small category. A pointed C -space, or diagram space over C , is a functor X : C −→
Top ∗ . The homotopy theory of diagram spaces is studied for various reasons, the perhapsmost fundamental one being Elmendorf’s Theorem [Elm83] which identifies thehomotopy theory of G -spaces for a discrete group G with the homotopy theoryof diagram spaces over the so-called orbit category Or(G). Similarly to classicalhomotopy theory, a major tool to study C -spaces are C -homology theories, whichare collections of functors h C n : Fun( C , Top ∗ ) → Absatisfying the usual Eilenberg-Steenrod axioms, cf. Subsection 5.1. Such theoriescan be constructed by setting h C n ( X ; E ) = π n ( E ∧ C X )where E is a (cofibrant) C op -spectrum. This construction can be traced back to thevery beginning of the theory of spectra in the case that C is the trivial category, andwas first formulated by Davis and L¨uck [DL98] in this general form. It since hasproved useful in many contexts, primarily in work on the Farrell-Jones conjecture[LR05,BLR08,BL12,Weg15,KLR16,R¨up16,Wu16,KUWW18,BB19]. However, thequestion whether every C -homology theory arises in the way described above hadnot yet been addressed. This is answered in the positive by our first theorem, thehomology representation theorem, proved as Theorem 5.2.3: Mathematics Subject Classification.
Primary: 55N91; Secondary: 55M05, 55P42, 18D05.
Key words and phrases. spaces over a category, C -homology theories, representation theorems,external Spanier-Whitehead duality, Chern character.The author was supported by the ERC Advanced Grant ”KL2MG-interactions” (no. 662400)of Wolfgang L¨uck. Theorem A .
Suppose that C is countable. Let h C∗ be any C -homology theory. Thenthere is a C op -spectrum E and a natural isomorphism h C∗ ( − ) ∼ = h C∗ ( − ; E ) . (1) Moreover, every morphism of homology theories h C∗ ( − ; E ) −→ h C∗ ( − ; E ′ ) is induced by a morphism E −→ E ′ in the derived category of C op -spectra. As for the well-known case C = ∗ , the analogous cohomological version of thisstatement is considerably easier to prove, using a Yoneda lemma argument due toBrown [Bro62]. Neeman [Nee01] has vastly generalised this argument to a triangu-lated category setup that is sufficient to treat the case of C -cohomology theories.Specific references for the case of C -spaces are [B´ar14, Lac16].The classical strategy for deducing the homological Theorem A from the coho-mological one is the following: Use Spanier-Whitehead duality to switch betweencohomology and homology, and then use Adams’ version of Brown’s representabilitytheorem to deal with the arising difficulty that the duality functor is only definedon finite spectra. The latter point poses no difficulties, since Adams’ result wasalso generalised by Neeman [Nee97] in a form suitable for our applications.The first point is more difficult. The main innovation here is that the correct notionof duality is not incorporated by a functor D : Fun( C , Sp O ) op −→ Fun( C , Sp O ) , but by a functor D : Fun( C , Sp O ) op −→ Fun( C op , Sp O ) . This is the reason why we called it the external (Spanier-Whitehead) duality func-tor. Note that in the technical sense, the term ”duality” is not justified: It refersto the canonical isomorphism
DDX ∼ = X (2)for dualisable X . However, the two D ’s here are not, as in the classical case, thesame functor, but only formally given by the same construction, applied to C and C op .These two aspects originate from the fact that instead of classical duality theory,which takes place in a monoidal category, the correct framework for us is dualitytheory in a closed bicategory. This was first developped in [MS06, Ch. 16]. Wegive a slightly simplified exposition in Section 4. It is applied to a closed bicate-gory of spectrally enriched categories, derived bimodules and morphisms betweenthese, constructed in Theorem 2.3.1. With the correct setup at hand, the followingstatement, which is our Corollary 4.2.7, may be proved quite analogously to theclassical case. Theorem B .
Every finite C -CW-spectrum is dualisable. For finite groups G , classical genuine G -representation theory takes into accountthe orthogonal representation theory of G . This is a very sophisticated and richtheory. Recently, this approach has been extended to proper equivariant homotopytheory for infinite discrete groups [DHL + ]. We take a different route here, whichuses no representation theory. We want to stress that for (finite or infinite) groups,our results are neither generalisations nor special cases of the genuine results. Werefer the reader to Remark 2.1.5 for a more detailed discussion. OMOLOGY REPRESENTATION THEOREMS FOR DIAGRAM SPACES 3
Our third main result concerns the case of rational C -homology theories. Thesecome from contravariant functors from C to rational spectra, which are identifiedwith rational chain complexes via the stable Dold-Kan correspondence. Note that atthis point we face the problem of upgrading a (weak) monoidal Quillen equivalencebetween two categories of spectra to a Quillen equivalence between diagram spectra,suitably compatible with balanced smash products. This is a quite subtle issue,discussed in Section 3. If the chain complexes we get on the algebraic side split(functorially), this entails the existence of a Chern character, i. e. a decompositionof the rational C -homology theory into a direct sum of shifted Bredon homologytheories, cf. Definition 6.3.4. We can construct such a Chern character in twoinstances: Theorem C .
Let C be arbitrary and assume that h C∗ is a rational C -homologytheory with the property that all coefficient systems h C t are flat as right C -modules.Then there is a Chern character for h C∗ . Theorem D .
Suppose that C = Or( G, F ) where G is finite and all members ofthe family F are cyclic of prime power order. Then a Chern character exists for every C -homology and cohomology theory. Theorem C, which we prove as Corollary 6.3.7, is similar to a theorem of L¨uck[L¨uc02], cf. Remark 6.3.8. It may be applied to G -homology theories whose co-efficient systems have Mackey extensions, cf. Subsection 6.4. Theorem D, provedas Corollary 6.5.4 and Proposition 6.5.5, uses the results of [Li11] on hereditarycategory algebras. Actually, as we prove in Proposition 6.5.1, every C -homologytheory possesses a Chern character if and only if the category algebra Q C , cf. Def-inition 6.2.1, is hereditary. Further directions.
Our results suggest further questions that we find interestingto study. The first refers to the notion of an equivariant homology theory, describedin [LR05, Sec. 6]. This consists of homology theories for all groups at the same time,linked by various induction isomorphisms. These can also be constructed fromsuitable diagram spectra, for instance spectra over the category of small groupoids.
Question . Is there a representation theorem for equivariant homology theories,i. e. does every equivariant homology theory come from a suitable diagram spec-trum?We want to note that all common examples of equivariant homology theories areconstructed using groupoid spectra, except equivariant bordism [L¨uc02, Ex. 1.4]where such a representation is not known to us.
Question . Can Theorem A be generalised to categories enriched in topologicalspaces?Most of our results can be generalised to topological or even spectral categoriessatisfying a certain condition (C), cf. p. 5, but this question refers to the casewhere (C) does not hold, so that the theory can certainly not be built up in itswhole generality.The next question refers to the fact that in the non-equivariant stable category
S H C , the dualisable objects are exactly the finite CW-spectra, and these arealso exactly the compact objects (in the sense that mapping out of them up tohomotopy commutes with direct sums).
MALTE LACKMANN
Question . What is the relation between compact, dualisable and finite objects inthe derived category of C -spectra?The last two questions refer to the case of rational C -homology theories. Question . Find conditions under which the flatness assumption of Theorem C issatisfied.
Question . Can it be characterised for infinite EI categories C when the categoryalgebra Q C is hereditary?This question is the subject of joint work in progress of the author with Liping Li. Organisation of the paper. • Section 2 recalls some background from homotopy theory and constructsthe closed bicategory of spectrally enriched categories. • Section 3 discusses what happens if orthogonal spectra are replaced byanother model category of spectra as the target category of our diagramspectra. • Section 4.1 develops external duality theory in closed bicategories and ap-plies this to C -spectra, proving Theorem B. • Section 5.1 proves Theorem A via the route sketched above. • Section 6 studies the rational case and proves Theorems C and D.
Acknowledgements.
This paper was written during my time as a PhD stu-dent of Wolfgang L¨uck. I was supported by the ERC Advanced Grant ”KL2MG-interactions” (no. 662400). I thank Bertram Arnold, Daniel Br¨ugmann, Em-manuele Dotto, Markus Hausmann, Fabian Henneke, Liping Li, Irakli Patchkoria,Jens Reinhold, and the members of the rubber duck seminar at Bonn for helpfuldiscussions. Special thanks go to Benjamin B¨ohme who has carefully proofread thispaper, and to my office mate Christian Wimmer who patiently explained to me thefoundations of orthogonal spectra.2.
The closed bicategory
DerMod(Sp O )The correct setup for developping external duality theory, as is done in Section 4,is given by the notion of a closed bicategory. This will then be applied to deduceresults about C -spectra. In this first section, we will introduce the actors, i. e.recall the basics about model structures on the category Fun( C , Sp O ) of C -spectrain Subsection 2.1, introduce the notion of a closed bicategory in Subsection 2.2 andshow how Fun( C , Sp O ) can be endowed with this structure, cf. Proposition 2.2.1,and then show that we can preserve this structure when passing to the homotopycategory in Subsection 2.3, especially Theorem 2.3.1. Our closed bicategories will,in the underived, resp. derived case, consist of small spectrally enriched categories(with cofibrant mapping objects), (derived) bimodules over these and morphisms(in the homotopy category) of bimodules.2.1. Recapitulations about the homotopy category of C -spectra. Let Sp O denote the category of orthogonal spectra with the stable model structure, as dis-cussed in [MMSS01], and let C be a small category enriched in Sp O . Let Fun( C , Sp O ) OMOLOGY REPRESENTATION THEOREMS FOR DIAGRAM SPACES 5 denote the category of enriched functors from C to Sp O and enriched natural trans-formations [Bor94b, Def. 6.2.4]. Prominent objects of this category are the repre-sentable functors c = C ( c, ?)for c ∈ Ob( C ), or more generally X ∧ c for some spectrum X , where the smashproduct is meant objectwise.We want to endow Fun( C , Sp O ) with a model structure in which the fibrations andweak equivalences are given by the objectwise fibrations and weak equivalences.This determines the model structure, if it exists, uniquely, justifying that we call it’the’ projective model structure. We want that: • The projective model structure exists. • It is a cofibrantly generated model structure. A class of generating cofibra-tions is given by morphisms of the form X ∧ c → Y ∧ c , where X → Y runsthrough a class of generating cofibrations of Sp O and c through the objectsof C ; a class of generating trivial cofibrations is described similarly. • A cofibration in the projective model structure is objectwise a cofibration.For usual Set-enriched categories C , this is folklore since Sp O is a cofibrantly gener-ated model category [Hir03, Thm. 11.6.1, Prop. 11.6.3]. For spectrally enriched C ,the situation is more subtle. The most general reference we could find is [Shu06]. Theorem 2.1.1 [Shu06, Thm. 24.4].
Suppose that C satisfies (C) The mapping spectra C ( c, d ) are cofibrant for all c, d ∈ C .Then all three items above are satisfied. Shulman’s theorem uses the fact that Sp O satisfies the monoid axiom, as is provedin [MMSS01, Thm. 12.1(iii)]. Because of the above theorem, we assume from now on that our category C satisfies (C) .We denote S H C C = Ho(Fun( C , Sp O ))and use square brackets to indicate that we are talking about morphisms in thehomotopy category:[ X, Y ] C := Hom Ho(Fun( C , Sp O )) ( X, Y ) = Hom
S H C C ( X, Y ) . Fun( C , Sp O ) is a stable model category, so the homotopy category admits a preferred triangulated structure , even in the strong sense of [Hov99, Sec. 7]. We refer to thefact that X f −→ Y −→ Z → Σ X (3)is a distinguished triangle sloppily as Z = C ( f ). Note that this notion makes sensealready in the pointed model category of pointed C -spaces [Hov99, Sec. 6]. If f isa cofibration between cofibrant objects, then C ( f ) = Y /X .It is a well-known fact about triangulated categories that a distinguished triangle (3)induces a long exact sequence . . . −→ [Σ Y, B ] C −→ [Σ X, B ] C −→ [ Cf, B ] C −→ [ Y, B ] C −→ [ X, B ] C −→ . . . (4)and similarly for [ B, − ] C .A triangulated subcategory of S H C C is a full subcategory closed under Σ andΣ − with the property that if it contains a morphism f : X → Y , then also its cone Cf . MALTE LACKMANN
Recall from [MMSS01] that Sp O is inhabited by various spheres F k S n with F S = S and F k ( X ∧ Y ) = ( F k X ) ∧ Y . In the homotopy category, F k S n becomes a k -folddesuspension of S n . The canonical maps F k (S n + ) → F k (D n +1+ ), with k ∈ Z and n ∈ N , define a class of generating cofibrations in Sp O . A class of generatingcofibrations in Fun( C , Sp O ) is thus given by F k (S n + ) ∧ c → F k (D n +1+ ) ∧ c for k ∈ Z , n ∈ N and c ∈ Ob( C ). We will call an object of S H C C a finite C -CW-spectrum if it can be obtained from the trivial functor ∗ by a finite number of gluing stepsusing these generating cofibrations. The C -Spanier-Whitehead category S W C isthe full subcategory of S H C C on the finite C -CW-spectra.The name is justified by the following lemma: Lemma 2.1.2. (a)
S W C is the full subcategory of S H C C on objects of the form Σ N Σ ∞ A for some integer N and some finite pointed C -CW-complex A .(b) If A is a finite C -CW-complex and B is an arbitrary C -CW-complex, then Hom
S W C (Σ N Σ ∞ A, Σ M Σ ∞ B ) ∼ = colim k (cid:8) Σ N + k A, Σ M + k B (cid:9) C , where the curly brackets on the right denote (unstable) homotopy classes of maps of C -spaces. (c) S W C is the smallest triangulated subcategory of S H C C containingthe objects c for all c ∈ Ob( C ) . Note that statement (b) serves as an alternative definition of
S W C , not using S H C C . Proof.
Part (a) is an easy induction. In part (c), the fact that
S W C is triangulatedis clear as well. For the minimality, note that this would be clear inductively if wehad defined finite C -CW-spectra using attaching maps F k (S n ) ∧ c → F k (D n +1 ) ∧ c since D n +1 = C (S n ). Unfortunately, D n +1+ is not the cone of S n + . However,in the homotopy category, we may suspend as often as we want since this is anisomorphism. After one suspension, the basepoint problem vanishes: the inclusionS n + → D n +1+ becomes the inclusion of the boundary B of an ( n + 2)-disk D withtwo boundary points identified (to the basepoint). The cone of the quotient mapS n +1 → B can be identified with D .For part (b), note that it suffices to prove this statement for Σ A and Σ B . Fix B . As in the proof of (c), we only need to show that it holds true for all c andif it is true for A and A ′ , and if f : A → A ′ is a morphism, then it is true for Cf . For corepresentable functors c , the statement boils down to the well-knowncorresponding statement for S H C . Use Theorem 2.3.1 (c) and Lemma 2.3.6 belowto deal with the left-hand side. For the cone argument, first prove that the right-hand side functor (for fixed B ) turns cofibre sequences into long exact sequences,similarly to (4). There is a natural map from the right-hand side to the left-handside which is compatible with these two cone long exact sequences, and thus theclaim follows via induction and the five lemma. (cid:3) Remark . The paper [SS03b] shows that (if spectra are simplicial symmetricspectra) the model categories Fun( C , Sp Σ sSet ) are exactly the simplicial, cofibrantlygenerated, proper, stable model categories with a set of compact generators. Remark . If C = Or( G ) is the orbit category of a group G , then Marc Stephan[Ste16] has shown that Elmendorf’s Theorem holds in orthogonal spectra, i. e. thereis a model structure on naive orthogonal G -spectra ( G -objects in the category oforthogonal spectra) and a Quillen equivalence between this model category and OMOLOGY REPRESENTATION THEOREMS FOR DIAGRAM SPACES 7
Fun( C , Sp O ). However, this may fail in other categories of spectra with the proper-ties discussed in Subsection 3.1 below. For instance, it definitely fails in Ch Q . Thereason is that Stephan’s paper has a cellularity condition that is satisfied by Sp O ,but not by Ch Q . We will not use the spectral Elmendorf Theorem in this paper. Remark . As promised in the introduction, we want to compare our approachto the classical one of classical genuine G -equivariant homotopy theory. Surveyson this topic are [May96], [Sch18, Ch. 3] and [HHR16, Sec. 2,3, App. A,B]. In thiscontext, G is a finite (or compact Lie) group, and usually not Z -graded, but so-called RO ( G )-graded (co-)homology theories are considered and this leads to a stablecategory in which not only S , but all representation spheres S V are invertible withrespect to the smash product, where V runs through all finite subrepresentationsof a so-called universe U . Using Remark 2.1.4 above, one sees that (for Sp O asthe category of spectra) we invert subrepresentations of the trivial universe R ∞ , anapproach sometimes called naive equivariant stable homotopy theory in the genuinecontext.This framework in all its generality breaks down when G becomes an infinite group.Recently, the authors of [DHL + ] developped a generalisation for infinite (or non-compact Lie) groups G with respect to the family of finite (or compact) subgroups.In their setup, smashing with all Thom spaces S ξ , with ξ a G -vector bundle over EG , is inverted. Thus, this gives a different setup than the one we treat here, andin particular does not relate to the Davis-L¨uck construction of homology theoriesoccuring in our homology representation theorem 5.2.3. Also, our theory is moregeneral in that it treats diagram spaces over arbitrary countable categories C .2.2. ∧ C and map C . From now on, the letters A , B and D also refer to spectrallyenriched small categories satisfying (C). The spectrally enriched category A ∧ B op has objects Ob( A ) × Ob( B ) and( A ∧ B op )(( a, b ) , ( a ′ , b ′ )) = A ( a, a ′ ) ∧ B ( b ′ , b ) . An ( A , B )-bimodule is a continuous functor A∧B op → Sp O . The category of ( A , B )-bimodules and ( A , B )-linear morphisms (i. e. natural transformations of enrichedfunctors) we denote by Mod( A , B ) = Fun( A ∧ B op , Sp O ) , with Hom sets denoted by Hom ( A , B ) ( − , − ) and homotopy sets (i. e. Hom sets inthe homotopy category of ( A ∧ B op )-spectra) denoted by [ − , − ] ( A , B ) . ( C , ∗ )- and( ∗ , C )-bimodules are just called left, respectively right C -modules.If X is a right and Y a left C -module, then X ∧ C Y is the spectrumcoequ _ ( c,d ) ∈ Ob( C ) Y ( c ) ∧ C ( c, d ) ∧ X ( d ) ⇒ _ c ∈ Ob( C ) Y ( c ) ∧ X ( c ) . Here, the upper arrow is defined on any ( c, d )-summand via the morphism corre-sponding to X ∗ : C ( c, d ) → map( X ( d ) , X ( c ))under the adjunction between − ∧ X ( d ) and map( X ( d ) , − ). The lower arrow isdefined similarly, using Y instead of X . MALTE LACKMANN
More generally, the balanced smash product X ∧ B Y of an ( A , B )-bimodule X anda ( B , C )-bimodule Y is the ( A , C )-bimodule defined by X ∧ B Y ( a, c ) = X ( a, ?) ∧ B Y (? , c ) . Similarly, the mapping spectrum map C op ( U, X ) between two right C -modules U and X is defined asequ Y c ∈ Ob( C ) map( U ( c ) , X ( c )) ⇒ Y ( c,d ) ∈ Ob( C ) map ( C ( c, d ) , map( U ( d ) , X ( c ))) . More generally, for an ( A , B )-bimodule X and a ( C , B )-bimodule U , we have an( A , C )-bimodule map B op ( U, X ). We can similarly define the mapping spectrumbetween two left C -modules, or between an ( A , B )-bimodule and an ( A , C )-bimodule.We also introduce the ( A , A )-bimodule A defined by( a, a ′ )
7→ A ( a ′ , a ) , this not being a tautology, but referring to the mapping spectra of the category A .The constructions just introduced can not only be defined in Sp O , but in anycosmos V . They are linked in various ways that can be subsumed using the notionof a closed bicategory. Recall that a bicategory A consists of a class of objectsOb( A ), and a small category of 1-morphisms A ( A, B ) between any two objects A and B , together with composition functors that are associative and have unitsup to coherent isomorphisms [Bor94a, Def. 7.7.1]. The morphisms between the 1-morphisms are called 2-morphisms. A bicategory is called closed [MS06, Def. 16.3.1]if for every 1-morphism f : A → B and every object C , the precomposition with f , f ∗ : A ( B, C ) −→ A ( A, C ), as well as the postcomposition with f , f ∗ : A ( C, A ) −→ A ( C, B ), have a right adjoint. Since adjoints are unique up to unique isomorphismif they exist, this is a property of a bicategory, not an additional structure on it.
Proposition 2.2.1.
Let V be a cosmos. Then there is a closed bicategory Mod( V ) in which the objects are given by small V -enriched categories; -morphisms from A to B are ( A , B ) -bimodules, with composition given by balanced product and id A givenby the ( A , A ) -bimodule A ; the -morphisms are given by morphisms of bimodules;if X is an ( A , B ) -bimodule, then the right adjoints of pre- and postcomposition with X are given by map A ( X, − ) and map B op ( X, − ) .Proof. The bicategory structure was first discussed in [B´en73] for V = Set; [HV92,Prop. 2.6] is a classical reference for V = Top, though it omits bicategorical lan-guage. A general reference is [Shu13, Sec. 3, esp. Lemmas 3.25, 3.27]. (cid:3) Remark . In the literature, there are three different names for what we callbimodules here, all of which seem to be common in some circles; the other two aredistributeurs and profunctors. Consequently, the bicategory introduced above issometimes also called Dist( V ) or Prof( V ). Remark . The category Mod( A , B ) can again be jazzed up to a spectrally en-riched category: If we view two ( A , B )-bimodules X and Y as left ( A∧B op )-modules(or right ( A op ∧ B )-modules), we can define a mapping spectrum map ( A , B ) ( X, Y )with underlying set Hom ( A , B ) ( X, Y ). Thus, Mod(Sp O ) is a spectrally enrichedclosed bicategory in the obvious sense. We don’t give further details since we won’tuse this enrichment. OMOLOGY REPRESENTATION THEOREMS FOR DIAGRAM SPACES 9
Example . In addition to V = Set and V = Sp O , another interesting exampleof a cosmos is V = Ab. An Ab-enriched category is usually called a preadditivecategory, and a preadditive category with one element is the same as a ring, witha bimodule in the sense discussed here corresponding to a bimodule in the usualsense (whence the name). Thus, we get as a full sub-bicategory of Mod(Ab) thebicategory of rings, ( R, S )-bimodules and (
R, S )-linear homomorphisms betweenthem, which is sometimes called the Morita category. More generally, you can take V = R − Mod for some commutative ring R . You may also take V = Ch R . ACh R -category is the same as an R -linear dg-category. Suppose that A and B are R -linear categories (concentrated in degree 0), then an ( A , B )-bimodule is the sameas a chain complex of ( A , B )-bimodules over R − Mod. Thus we get as a full sub-bicategory of Mod(Ch R ) the bicategory of R -linear categories and chain complexesof ( A , B )-bimodules. We will study this in detail in the rational case in Section 6.2.3. Deriving ∧ C and map C . We will now derive the whole setup in the sensethat we pass to the homotopy category of every bimodule category Mod( A , B ),and define a derived version of the balanced smash product which allows us toview the collection of all derived bimodule categories as a bicategory, as well asderived versions of the mapping spectra which exhibit this bicategory as closed.Technically, we achieve this by using the notion of a Quillen adjunction of twovariables [Hov99, Sec. 4.1].Throughout the rest of this subsection, let X be an ( A , B )-bimodule, Y a ( B , C )-bimodule, Z a ( C , D )-bimodule, U an ( A , D )-bimodule, and V an ( A , C )-bimodule.(This convention will always be clear from the context.) Theorem 2.3.1.
The following data defines a closed bicategory
DerMod(Sp O ) :objects are small Sp O -enriched categories satisfying (C); -morphisms from A to B are ( A , B ) -bimodules; -morphisms are given by (DerMod( A , B ))( X, Y ) = [
X, Y ] ( A , B ) . The identity -morphism of an object A is the ( A , A ) -bimodule A and the identity -morphism of a -morphism X is id X . The composition of -morphisms and theiradjoints are given by the functors − ∧ L B − : DerMod( A , B ) × DerMod( B , C ) −→ DerMod( A , C ) ,R map C op : DerMod( B , C )) op × DerMod( A , C ) −→ DerMod( A , B ) , and R map A : DerMod( A , B ) op × DerMod( A , C ) −→ DerMod( B , C ) , which are the total derived functors of − ∧ B − , map C op and map A . Explicitly, for Q a functorial cofibrant replacement and R a functorial fibrant replacement, we have X ∧ L B Y ∼ = QX ∧ B QY , R map C op ( Y, V ) ∼ = map C op ( QY, RV ) and R map A ( X, U ) ∼ = map C op ( QX, RU ) . In particular, the closed bicategory structure induces the following natural isomor-phisms: (a)
A ∧ L A X ∼ = X ∼ = X ∧ L B B in DerMod( A , B ) , (b) ( X ∧ L B Y ) ∧ L C Z ∼ = X ∧ L B ( Y ∧ L C Z ) in DerMod( A , D ) , (c) [ X ∧ L B Y, V ] ( A , C ) ∼ = [ X, R map C op ( Y, V )] ( A , B ) ∼ = [ Y, map A ( X, V )] ( B , C ) , (d) R map A ( A , X ) ∼ = X ∼ = R map B op ( B , X ) in DerMod( A , B ) , (e) R map A ( X ∧ L B Y, U ) ∼ = R map B ( Y, R map A ( X, U )) in DerMod( C , D ) , (f) R map D op ( Z, R map A ( X, U )) ∼ = R map A ( X, R map D op ( Z, U )) in DerMod( B , C ) .Proof. Let A , B and C denote small spectrally enriched categories satisfying (C).The closedness of the bicategory Mod(Sp O ) gives natural isomorphisms ϕ l : Hom ( A , C ) ( X ∧ B Y, V ) ∼ = −→ Hom ( B , C ) ( Y, map A ( X, V ))and ϕ r : Hom ( A , C ) ( X ∧ B Y, V ) ∼ = −→ Hom ( A , B ) ( X, map C op ( Y, V )) . The categories Mod( A , B ), Mod( B , C ) and Mod( A , C ) with the quintuple consistingof ∧ B , Hom r = map C op , Hom l = map A and the two isomorphisms ϕ r and ϕ l forman adjunction of two variables in the sense of [Hov99, Def. 4.1.12]. We want toapply [Hov99, Cor. 4.2.5] to show that ∧ B is a Quillen bifunctor.For this we have to check that the pushout product of two generating cofibrations isa cofibration, and that it is a trivial cofibration if one of the factors is a generatingtrivial cofibration. For the definition of the pushout products (cid:3) and (cid:3) B , see [Hov99,Def. 4.2.1]. We check the first statement, the other two being similar. We maychoose the generating cofibrations of the form f ∧ ( a, b ) and g ∧ ( b ′ , c ), where f and g belong to a class of generating cofibrations of Sp O . Up to isomorphism ofmorphisms, we have the identity( f ∧ ( a, b )) (cid:3) B ( g ∧ ( b ′ , c )) ∼ = ( f (cid:3) g ) ∧ B ( b, b ′ ) ∧ ( a, c ) . (5)By the pushout-product axiom for Sp O , f (cid:3) g is a cofibration. Now, B ( b, b ′ ) iscofibrant by (C) and thus ( f (cid:3) g ) ∧ B ( b, b ′ ) is a cofibration, since it is a smashproduct of a cofibration with a cofibrant object. Here we use the pushout-productaxiom for Sp O again. Thus, the right hand side of (5) has the left lifting propertywith respect to all trivial fibrations and is thus a cofibration.Proposition 4.3.1 of [Hov99] then applies to show that we have total derived functorsas in the statement of the theorem and that the quintuple( ∧ L B , R map C op , R map A , Rϕ r , Rϕ l )defines an adjunction of two variables. This gives the isomorphism (c). Isomorphism(b) follows from the explicit description of ∧ L B together with the fact that thebalanced smash product of two cofibrant bimodules is cofibrant, which follows fromthe Quillen bifunctor property.To show that DerMod(Sp O ) is actually a bicategory, we are left to deal with twopoints: Firstly, that there is an associativity isomorphism satisfying a coherencesquare. This follows directly from the corresponding fact for Mod(Sp O ), as in theproof of [Hov99, Prop. 4.3.1 or Prop. 4.3.2]. Secondly, that we have an identity1-morphism at every object. Surprisingly, this is the more difficult part, since theidentity A might be non-cofibrant. However, we may use Corollary 2.3.4 below tosee that A ∧ L A X ∼ = A ∧ A X ∼ = X since A is obviously right flat in the sense of Definition 2.3.3. The coherenceconditions for this unitality isomorphism are readily checked.The fact that the derived mapping functors are right adjoints of the derived smashproducts is part of the adjunction of two variables statement. Summarising, we have OMOLOGY REPRESENTATION THEOREMS FOR DIAGRAM SPACES 11 now proved that DerMod(Sp O ) is a closed bicategory, amounting to isomorphisms(a) to (c).Now the point is that (d) to (f) are valid in any closed bicategory: (d) follows from(a) – if pre- and postcomposition with A is isomorphic to the identity, then thesame has to be true for their adjoints. Similarly, (e) and (f) follow from (b). (cid:3) Proposition 2.3.2. If X is a cofibrant ( A , B ) -spectrum, then X ∧ B − preservesweak equivalences.Proof. We first treat the case where X is F k A ∧ ( a, b ), where A is any pointedCW-complex. Let Y → Y ′ be any weak equivalence of ( B , C )-spectra. Smashingwith F k A ∧ ( a, b ), we get the map F k A ∧ A ( a, − ) ∧ Y ( b, − ) −→ F k A ∧ A ( a, − ) ∧ Y ( b, − ) . Now, A ( a, − ) is objectwise a cofibrant spectrum by (C), and so is F k A . But smash-ing with a cofibrant spectrum preserves weak equivalences by [MMSS01, Prop. 12.3].Now we want to reduce to the general case. By general theory of cofibrantly gener-ated model categories, a cofibrant object is a retract of a cell complex. Since weakequivalences are closed under retracts, we may assume that X is a (transfinite)cell complex, i. e. a transfinite composition (cf. [Hir03, Def. 10.2.2]) of pushoutsalong generating cofibrations. Suppose that the transfinite composition is indexedby some κ and denote the intermediate ’skeleta’ by X α , α < β , where X α +1 can beobtained from X α by a cobase change along a coproduct of generating cofibrations.In particular, X α ֒ → X α +1 is a cofibration in the projective model structure, butthis property is not preserved when smashing (over C ) with an arbitrary spectrum.This is why we have to use the more subtle notion of h -cofibration. This is a con-cept which is not available in an arbitrary model category, but in many topologicalexamples, in particular in Sp O . Our use of h -cofibrations is restricted to this proof.We define a map of C -spectra A → B to be an h -cofibration if B ∧ I + retractsonto A ∧ I + ∪ A B ∧ { } + , cf. [MMSS01, p. 457]. Since the generating cofibrationsare h -cofibrations, the same is true for the inclusions X α → X α +1 . Moreover, h -cofibrations are preserved under balanced smash products by definition.Now we are in shape to prove the proposition for general X by transfinite inductionon β . It is true for the domains and targets of the generating cofibrations by thefirst step of the proof, applied to A = S n + and A = D n + . Thus, if it is true for X α ,then also for X α +1 , using [MMSS01, Thm. 8.12(iv)]. For limit ordinals β , we knowthat X β is the colimit of X α , α < β . This is preserved when smashing with Y and Y ′ . In orthogonal spectra, a stable equivalence is the same as a π ∗ -isomorphism[MMSS01, Prop. 8.7]. Computing the stable homotopy groups commutes withcolimits along h -cofibrations, since these are levelwise closed inclusions. (cid:3) Definition 2.3.3.
An ( A , B )-spectrum F is right flat if F ∧ B − preserves weakequivalences. f : F → X is called a right flat replacement of X if F is right flat and f is a weak equivalence. Corollary 2.3.4.
Let f : F → X be a right flat replacement of X . Then there is anatural isomorphism X ∧ L B Y ∼ = F ∧ B Y for any ( B , C ) -bimodule Y . Proof.
There are weak equivalences X ∧ L B Y = QX ∧ B QY ∼ −→ X ∧ B QY ∼ ←− F ∧ B QY ∼ −→ F ∧ B Y , where the first and second weak equivalence follow from Proposition 2.3.2. (cid:3)
Left flat replacements are defined similarly and the statement of the corollary carriesover mutatis mutandis.
Remark . Proposition 2.3.2 and Corollary 2.3.4 have been proved to show theisomorphism A∧ L A X ∼ = X . The proof are technically much more advanced than therest of the proofs in this section and in particular harder to generalise to other modelcategories of spectra than orthogonal spectra, cf. Section 3. In the understanding ofthe author, this is inevitable for Proposition 2.3.2 since the corresponding statementfor C = ∗ is a subtle point in all treatments he could find, but it would be niceto have a more straightforward proof of the fact that A ∧ L A X ∼ = X , going alonganother route.The one-object Yoneda lemma carries over to the derived setting without troublesince c is a cofibrant C op -spectrum. We state it here for later use. Lemma 2.3.6.
For a C -spectrum X and c ∈ Ob( C ) , there are natural isomorphismsin S H C c ∧ C X ∼ = X ( c ) and R map C ( c, X ) ∼ = X ( c ) . Changing the category of spectra
Throughout the paper hitherto, we investigated C -spectra in the sense of functorsfrom C to the category Sp O of orthogonal spectra. However, the literature alsouses several other model categories of spectra, which are either Quillen equivalentto orthogonal spectra (respecting the smash product in one sense or the other, asdiscussed below), or describe a slightly different version of spectra, e. g. connectivespectra or rational spectra. The purpose of this section is to bring all these othermodels in, in the following two ways: • Firstly, we state conditions under which much of the framework built up sofar can be built up with another category of spectra instead of orthogonalspectra. • Secondly, suppose we have built up the framework for two different modelcategories S and T , and we have a Quillen equivalence between the two.Then we want to compare our constructions, performed in S , can be com-pared with the same constructions, performed in T .The first item will be carried out in Subsection 3.1. We will write down a list ofassumptions on the model category of spectra and then deduce a substantial partof Section 2. Roughly speaking, we generalise enough to write down derived smashproducts and mapping spectra, and prove the various adjunctions between them,cf. Proposition 3.1.1. What we will not prove is the derived Yoneda Lemma A ∧ L A X ∼ = X since the way we proved it used rather specific properties of orthogonal spectra,cf. the proof of Proposition 2.3.2. However, let us emphasise that we pursued aminimalist approach here, proving what we strictly need in the rest of the paperinstead of maximising the generality. We can well imagine that a reader who is,for example, an expert on simplicial homotopy theory will find a way to prove OMOLOGY REPRESENTATION THEOREMS FOR DIAGRAM SPACES 13 the derived Yoneda Lemma for simplicial symmetric spectra, either transferringProposition 2.3.2 or via another route.The second item is dealt with in Subsection 3.2. The Quillen equivalence between S and T has to be compatible with the smash product. The literature knows (atleast) two different ways in which a Quillen equivalence can be compatible withmonoidal structures on its source and target: strong and weak monoidal Quillenequivalences. Their definitions will be recalled below. In many cases, it is possibleto compare two categories of spectra by a strong monoidal Quillen equivalence,and then the comparison result is trivial. For instance, this applies to all pairsof model categories of spectra discussed in [MMSS01]. However, we also need (inSubsection 6.1) the comparison along the more restrictive notion of a weak monoidalQuillen equivalence, which is not trivial any longer, cf. Proposition 3.2.1.Note that the agenda of the first item may be carried out for spectrally enrichedcategories C , while in the second case, we have to restrict to usual Set-enrichedcategories, since it is technically difficult to compare S -enriched with T -enrichedcategories, cf. Remark 3.2.2.The reason that we get into this discussion in detail is twofold: Firstly, it is in-trinsically satisfying to know that our results are independent of the choice of amodel category of spectra. Secondly, and more concretely, our comparison resultswill become crucial in Subsection 6.1, where they are used in the rational case topass from rational spectra to rational chain complexes. Remark . We want to comment the way we intend to apply the comparisonresults of this subsection. Suppose S is a model category of spectra which is Quillenequivalent to orthogonal spectra, and we are interested in Theorem A from theIntroduction for S . Then we will use the result for Sp O , to be proved below, andthen compare the balanced smash product occuring (secretly) on the right handside of (1) to the corresponding balanced smash product in S , using the machinerywe are just about to develop, for instance the isomorphism (6). Similarly, if S is,say, the model category of simplicial symmetric spectra and the homology theory isdefined on simplicial C -sets instead of C -spaces, we may first transfer it to C -spaces(using the Quillen equivalence between simplicial sets and spaces), then apply therepresentation theorem here and translate back to simplicial spaces and simplicialsymmetric spectra.Another strategy would be to develop bicategorical duality theory over S and then prove Theorem A separately for S . Although this is also a totally valid approach, itis not the one we will use here – mainly because of the technical problem mentionedabove that we cannot prove the derived Yoneda Lemma for S and thus do not havea clean bicategory at hand.3.1. Categories of spectra.
We start by distilling properties of Sp O we used toset up the framework of Section 2. Let ( S , ∧ , S ) denote a model category whichalso has a monoidal structure.As already explained in the introduction, our proof of Proposition 2.3.2 is so specificthat we don’t aim at generalising it, or the derived Yoneda lemma. We rather pur-sue a minimalist approach, comprising the following: Set up homotopy categories,Subsection 2.1 up to and including the discussion of the triangulated structure;define balanced smash products and mapping spectra, Subsection 2.2; derive theseas in Theorem 2.3.1 to get ∧ L C and R map C and isomorphisms (b) through (f).To prove these statements, we used the following list of properties of Sp O : • The smash product and mapping spectra furnish Sp O with the structure ofa cosmos, i. e. a closed symmetric monoidal category with all small limitsand colimits. • It has a cofibrantly generated stable [Hov99, Ch. 7] model structure. Thereis a class of generating cofibrations and generating trivial cofibrations whosesources are cofibrant. • The unit of the smash product is cofibrant. • The pushout-product axiom [SS00, Def. 3.1] holds. • The monoid axiom [SS00, Def. 3.3] holds.We have argued that the following ’meta theorem’ holds:
Proposition 3.1.1.
Suppose that a model category ( S , ∧ , S ) of spectra satisfies theabove list of properties. Then the statements of Theorem 2.3.1 hold for S in theplace of Sp O , except that DerMod( S , ) may fail to have identities, thus is not abicategory, and that isomorphism (a) may not hold.Remark . The fact that Sp O is a cosmos (with respect to the smash product)was crucially needed to construct balanced smash products and mapping spectra,and the compatibility with the model structure to derive these, cf. Subsections 2.2and 2.3. The cofibrant generation is needed to construct model structures on C -spectra. The facts that Sp O is a cosmos, the unit is cofibrant and the pushout-product axiom holds imply that it is a monoidal model category in the sense of[Hov99, Def. 4.2.6]. The latter notion is slightly weaker than the three mentionedfacts and would technically also suffice for our purposes. The monoid axiom isneeded for Theorem 2.1.1.The literature in stable homotopy theory contains a plethora of different model cat-egories of spectra. Apart from orthogonal spectra, we will use the category Sp Σ sSet of simplicial symmetric spectra with the stable model structure from [HSS00]. Lemma 3.1.3.
The model category Sp Σ sSet satisfies the above list of properties.Proof. See [HSS00, Thm. 2.2.10, Thm. 3.4.4, Cor. 5.3.8, Cor. 5.5.2]. Note that theauthors of [HSS00] call a monoidal model category what we defined as a modelcategory satisfying the pushout-product axiom. The fact that the unit is cofibrantis remarked on p. 53 of [HSS00]. (cid:3)
Remark . The paper [MMSS01] further treats the model categories of W -spaces and sequential spectra. The treatment of W -spaces and orthogonal spectrais completely analogous, so that all results (even Proposition 2.3.2) will be true for W -spaces, with the same references in [MMSS01] applying. All model categoricalaspects apply to sequential spectra as well, but this is not a closed symmetricmonoidal category and will be treated separately in Subsection 3.3.We will now discuss some model categories of rational spectra originally introducedin [Shi07]. These will be the main actors of Subsection 6.1. The four monoidalmodel categories are: • the category H Q − M od of modules over the monoid H Q in Sp Σ sSet withmodel structure as explained in [SS00, Thm. 4.1(1)]; • the model category of unbounded rational chain complexes [Hov99, Sec. 2.3]; • the category Sp Σ ( s Vect Q ) of symmetric spectra over simplicial Q -vectorspaces [Hov01b]; OMOLOGY REPRESENTATION THEOREMS FOR DIAGRAM SPACES 15 • the category Sp Σ (ch + Q ) of symmetric spectra over non-negatively gradedrational chain complexes [Hov01b].The latter two model structures are constructed following the general construction[Hov99] of a model category of symmetric spectra over a given (nice) monoidalmodel category. It is applied to the categories of simplicial objects in Q -vectorspaces with the model structure from [Qui67, Ch. II.4] and to ch + Q with the projec-tive model structure [DS95, Sec. 7]. Lemma 3.1.5.
The four model categories mentioned above satisfy the list of prop-erties on p. 13.Proof.
The Standing Assumptions 2.4 of [Shi07], proved for our four model cat-egories in Section 3, comprise all our assumptions except the cofibrancy of thesources of the generating (trivial) cofibrations. For H Q − M od , this can be seen asfollows: Generating cofibrations for H Q -modules can be obtained from generatingcofibrations in Sp Σ sSet by smashing with H Q (cf. [SS00, Lemma 2.3]). Since thesehave cofibrant sources and H Q is cofibrant, the smash product is cofibrant in Sp Σ sSet and thus also in H Q −M od since this has less cofibrations. For Ch Q , the sources arecofibrant since they are bounded and (trivially) degreewise projective. For the lat-ter two categories, the stable model structures on symmetric spectra have the samecofibrant objects as the projective model structures introduced [Hov01b, Thm. 8.2]and the generating cofibrations of these have cofibrant sources since this is true for s Vect Q and ch + Q . (cid:3) Comparison between different categories of spectra.
Throughout thissubsection, A , B and C are discrete categories (cf. Remark 3.2.2). Let ( S , ∧ , S )and ( T , ⊗ , T ) denote categories of spectra, i. e., stable model categories satisfyingthe list of assumptions on p. 13. Let F : ( S , ∧ , S ) ⇄ ( T , ⊗ , T ) : G be a Quillen equivalence between two categories of spectra, where F is the leftadjoint. An ( A , B )-bimodule is just a functor in the usual non-enriched sense from A × B op to S , respectively T . We thus have an adjunction F ∗ : Fun( A × B op , S ) ⇄ Fun(
A × B op , T ) : G ∗ which is again a Quillen equivalence [Hir03, Thm. 11.6.5].Recall the definition of weak and strong monoidal Quillen equivalences from [SS03a,Sec. 3.2]: A Quillen equivalence is called strong monoidal if F is strong monoidaland F ( Q S ) → F ( S ) ∼ = T is a weak equivalence for the unit S . It is called weakmonoidal if G is lax monoidal, thus F lax comonoidal, such that the maps ∇ : F ( x ∧ y ) → F ( x ) ⊗ F ( y )are weak equivalences for all cofibrant x and y , and the composite F ( Q S ) → F ( S ) → T is a weak equivalence as well. In our case, the unit S is cofibrant, so this boils downto the fact that F ( S ) → T is a weak equivalence.In the case of a strong monoidal Quillen equivalence (which we face for examplewhen comparing symmetric with orthogonal spectra as our underlying cosmos), as opposed to: enriched everything is straightforward. F ∗ commutes with balanced smash products andthus the same holds for the equivalence of categoriesΦ = Φ ( A , B ) = Ho( F ∗ ) : Ho(Fun( A × B op , S )) → Ho(Fun(
A × B op , T ))and, consequently, its inverse Γ = Ho( G ∗ ). We spell out the natural isomorphisms:Φ( X ∧ L B Y ) ∼ = Φ( X ) ⊗ L B Φ( Y ) , Γ( X ′ ∧ L B Y ′ ) ∼ = Γ( X ′ ) ⊗ L B Γ( Y ′ )(6)as well as R map A (Φ( X ) , Φ( U )) ∼ = Φ( R map A ( X, U )) , R map A (Γ( X ′ ) , Γ( U ′ )) ∼ = Γ( R map A ( X ′ , U ′ ))(7)– these come from the adjunction between balanced smash product and mappingspectrum. Similar isomorphisms hold for R map B .Now we turn to weak monoidal Quillen equivalences. The comonoidal transforma-tion ∇ induces a commutative diagram W b → b ′ F ( X ( b ′ ) ∧ Y ( b )) W b F ( X ( b ) ∧ Y ( b )) W b → b ′ F ( X ( b ′ )) ⊗ F ( Y ( b )) W b F ( X ( b )) ⊗ F ( Y ( b )) ∇ ∇ and thus induces a map on the colimits of the rows. Since F commutes withcolimits, we get ∇ : F ∗ ( X ∧ B Y ) → F ∗ ( X ) ⊗ B F ∗ ( Y ). Proposition 3.2.1. ∇ is a weak equivalence if X and Y are cofibrant.Proof. We first treat the case where X = A ∧ ( a, b ) for some cofibrant spectrum A .But then ∇ is isomorphic to ∇ : F ( A ∧ a ∧ Y ( b, − )) → F ( A ∧ a ) ⊗ F ( Y ( b, − ))which is a weak equivalence since A ∧ a is objectwise cofibrant by discreteness of A and C , and Y ( b, − ) is objectwise cofibrant by [Hir03, Prop. 11.6.3]. Note thenatural isomorphism F ∗ ( A ∧ ( a, b )) ∼ = F ∗ _ b A ∧ a ∼ = _ b F ∗ ( A ∧ a ) ∼ = F ∗ ( A ∧ a ) ⊗ b , since F commutes with colimits.In the general case, X is a retract of a (transfinite) cell complex. We may thusassume that X is itself a cell complex. Arguing by transfinite induction, we haveto show that the property that ∇ is a weak equivalence is preserved under gluingalong coproducts of generating cofibrations and under passage to colimits alongcofibrations.For the first point, we use the first step of the proof and the Cube Lemma [Hov99,Lemma 5.2.6]. The two comparison diagrams consist of cofibrant objects and onecofibration since F ∗ is left Quillen and ∧ B and ⊗ B are Quillen bifunctors, cf. theproof of Theorem 2.3.1.For the second point, suppose that we have a chain of cofibrations of some shape κ . This is a cofibrant diagram in the projective model structure on the functor OMOLOGY REPRESENTATION THEOREMS FOR DIAGRAM SPACES 17 category of κ -sequences: The lifting property can be proved by transfinite induction.Since the colimit is a left Quillen functor [Hir03, Thm. 11.6.8], it preserves weakequivalences between cofibrant objects. (cid:3) With Φ as above, we get a natural isomorphism in Ho(Fun( C , T ))Φ( X ∧ L B Y ) = F ∗ ( QX ∧ B QY ) ∼ = −→ ∇ F ∗ ( QX ) ⊗ B F ∗ ( QY ) ∼ = Φ( X ) ⊗ L B Φ( Y )and we obtain our desired isomorphisms (6) and, by adjointness, (7). Remark . It is important in our discussion that A , B and C are discrete cat-egories. In the case of enriched categories, it is already difficult to define what thecorrect construction of a T -category out of an S -category is [SS03a, Sec. 6]. Wedidn’t succeed to prove comparison results in this case.3.3. Sequential spectra.
Sequential spectra do not form a monoidal model cat-egory, only a model category tensored and cotensored over spaces. The tensor andcotensor structure can be derived by the same Quillen adjunction argument as inTheorem 2.3.1. In this case, even Proposition 2.3.2 may be proved in the same wayas above, relying on the same references in [MMSS01] as this paper treats sequentialand orthogonal spectra uniformly.While it is impossible to formulate duality for sequential spectra over C (using ourmethods), it is possible to write down a homology theory from a sequential C op -spectrum as in (12). For this construction, Theorem 5.2.3 actually holds true aswell. To see this, we compare with a Quillen equivalence to orthogonal C -spectraand only have to show that the balanced smash products are translated into oneanother.Let U ∗ ( Y ) denote the underlying ( A × B op )-sequential spectrum of an ( A × B op )-orthogonal spectrum Y . Let X be an ( A × B op )-space. Then there is a tautologicalisomorphism of sequential spectra U ∗ (Σ ∞ X ∧ Y ) ∼ = X ∧ U ∗ Y inducing the same isomorphism for ∧ B instead of ∧ since U ∗ commutes with col-imits. To pass to the derived functor, it suffices to cofibrantly replace X by anargumentation similar to Corollary 2.3.4. Thus, we get a natural isomorphismHo( U ∗ )(Σ ∞ X ∧ L B Y ) ∼ = X ∧ L B Ho( U ∗ )( Y ) . Similarly, Ho( U ∗ )( R map A (Σ ∞ X, U )) ∼ = R map A ( X, Ho( U ∗ )( U ))where we use in the derivation process that the right adjoint U ∗ preserves fibrantobjects. 4. External Spanier-Whitehead duality
We will now set up an external version of Spanier-Whitehead duality which relatesfinite C -spectra to finite C op -spectra and which allows us to go back and forthbetween homology theories on finite C -spectra and cohomology theories on finite C op -spectra in the proof of Theorem 5.2.3.We will begin by formulating the problem, i. e. by defining the notion of a dualpair. This is carried out in the context of an arbitrary bicategory in Subsection 4.1.There are several equivalent formulations of this notion, the equivalence of which isproved in Proposition 4.1.1. We will later use this formulation of the problem in the bicategory structure on DerMod(Sp O ) discussed in Theorem 2.3.1. The discussionin the end of Subsection 4.1 uses the symmetry of the bicategory DerMod(Sp O )and finally the closedness. The closed structure allows us to write down, in Subsec-tion 4.2, an ansatz for the solution of the above problem: We construct a functor D for which it is plausible that ( X, DX ) is a dual pair. This approach could inprinciple be carried out in any closed bicategory, but we only do this in the exam-ple of DerMod(Sp O ) to simplify the exposition. Finally, we prove in the usual way,using an inductive argument, that finite spectra are dualisable.4.1. Bicategorical duality theory.
The discussion in this subsection is essen-tially equivalent to [MS06, Ch. 16], slightly simplified for our purposes. Also com-pare [LMSM86, Ch. III]. We change our standing notation from the last section:In this section, X will always denote an ( A , B )-bimodule, Y a ( B , A )-bimodule, Z a ( C , A )-bimodule, U a ( B , C )-bimodule, V an ( A , C )-bimodule and W a ( C , B )-bimodule. All morphisms between bimodules are morphisms in the homotopy cat-egory – in other words, we are working in the bicategory DerMod(Sp O ).Given a morphism ε : X ∧ L B Y ( A , A ) −−−−→ A , we may define ε ∗ : [ W, Z ∧ L A X ] ( C , B ) → [ W ∧ L B Y, Z ] ( C , A ) where ε ∗ ( f ) is the composition W ∧ L B Y f ∧ L B Y −−−−→ Z ∧ L A X ∧ L B Y Z ∧ L A ε −−−−→ Z ∧ L A A ∼ = Z .
Similarly, we may define ε ∗ : [ U, Y ∧ L A V ] ( B , C ) → [ X ∧ L B U, V ] ( A , C ) . On the other hand, a morphism η : B ( B , B ) −−−→ Y ∧ L A X yields η ∗ : [ W ∧ L B Y, Z ] ( C , A ) → [ W, Z ∧ L A X ] ( C , B ) and η ∗ : [ X ∧ L B U, V ] ( A , C ) → [ U, Y ∧ L A V ] ( B , C ) . In the following, the letters ε and η are reserved for morphisms with source andtarget as above. The next proposition is the main point of our discussion of dualitysince it shows that the notion of a dual pair can equivalently formulated in termsof ε and η , or only one of them – the other one can be recovered uniquely. It isessentially [LMSM86, Thm. III.1.6] or [MS06, Prop. 16.4.6]. Proposition 4.1.1.
The following data determine one another: (I) morphisms ε and η such that the composition X ∼ = X ∧ L B B X ∧ L B η −−−−→ X ∧ LB Y ∧ L A X ε ∧ L A X −−−−→ A ∧ L A X ∼ = X equals id X and the composition Y ∼ = B ∧ L B Y η ∧ L B Y −−−−→ Y ∧ L A X ∧ L B Y Y ∧ L A ε −−−−→ Y ∧ L A A ∼ = Y equals id Y ; (II) a morphism ε such that ε ∗ is a bijection for all W and Z ; OMOLOGY REPRESENTATION THEOREMS FOR DIAGRAM SPACES 19 (III) a morphism ε such that ε ∗ is a bijection for all U and V ; (IV) a morphism η such that η ∗ is a bijection for all W and Z ; (V) a morphism η such that η ∗ is a bijection for all W and Z .Proof. If ε and η as in (I) are given, then a direct check reveals that ε ∗ and η ∗ areinverse bijections, as are ε ∗ and η ∗ . Thus we recover (II) through (V). We nowshow how to recover (I) from (II), with the proceeding starting from another pointbeing analogous.Suppose that ε ∗ is always a bijection. With C = B , W = B and Z = Y , we get anisomorphism ε ∗ : [ B , Y ∧ L A X ] ( B , B ) → [ B ∧ L B Y, Y ] ( B , A ) . Choosing η as the preimage of the canonical isomorphism B ∧ L B Y ∼ = Y , we getthe second of the two compositions in (I) to equal id Y . Note that we have noother choice for η if we want (I) to hold. Moving on, note that ε ∗ η ∗ is the identityfor all W and Z . Since ε ∗ is a bijection, this exhibits η ∗ as a bijection as welland implies that the other composition η ∗ ε ∗ also equals the identity. Now, thefirst composition in (I), viewed as a morphism X → A ∧ L A X (i. e., forget the lastcanonical isomorphism ϕ X ), equals η ∗ ( ε ), so its image under η ∗ equals ε . But thesame is true for ϕ − X , so the two are equal.It is obvious that the presented constructions are inverse to each other – one way,we forgot about η , and going back, we had a unique choice for η . (cid:3) Remark . Condition (I) says that (
X, Y ) is an adjoint pair in the sense ofadjointness between 1-morphisms in bicategories [Bor94a, Def. 7.7.2].
Definition 4.1.3. ( X, Y ; ε, η ) – equivalently ( X, Y ; ε ) or ( X, Y ; η ) – is called a dualpair of bimodules if the equivalent conditions of Proposition 4.1.1 hold.Note that we can omit one of ε and η from the quadruple ( X, Y ; ε, η ), but not both:for instance, ε is not uniquely determined by X and Y , since we might change itby an automorphism of its source or target. Remark . The discussion above is not symmetric in A and B . We could equallywell have formulated a second kind of duality where we interchanged the role of thesource and target of a 1-morphism, as well as the order of the composition (i. e.balanced smash product) everywhere. This would have given a different notion ofduality with different dual pairs.The bicategory DerMod(Sp O ) has a special kind of symmetry available: By defini-tion, there is a canonical isomorphism of categories between ( A , B )-bimodules and( B op , A op )-bimodules which we denote by X X op . This assignment is involutive, and we have canonical isomorphisms γ : (cid:16) X ∧ L B Y ∼ = −→ (cid:17) op ∼ = −→ Y op ∧ L B op X op of ( A , C )-bimodules, and δ : (id A ) op ∼ = −→ id A op of ( A op , A op )-bimodules. Remark . In the language of [MS06, Sec. 16.2], this refers to the fact thatDerMod(Sp O ) is a symmetric bicategory, with involution A 7→ A op . In this notation, Remark 4.1.4 says that the fact that (
X, Y ; ε, η ) is a dual pair is not equivalent to the fact that ( X op , Y op ; ε ′ , η ′ ) is a dual pair for some ε ′ and η ′ .However, there is the following tautological observation which we will use later: Proposition 4.1.6. ( X, Y ; ε, η ) is a dual pair if and only if the pair ( Y op , X op ; δε op γ − , γη op δ − ) is.Proof. Trivial for condition (I) of Proposition 4.1.1. (cid:3)
Proposition 4.1.7. If ( X, Y ; ε, η ) and ( U, W ; ζ, θ ) are dual pairs, then so is ( X ∧ B U, W ∧ B Y ; ν, ξ ) where ν is the composition X ∧ L B U ∧ L C W ∧ L B Y X ∧ L B ζ ∧ L B Y −−−−−−−→ X ∧ L B B ∧ L B Y ∼ = X ∧ L B Y ε −→ A and ξ is defined similarly.Proof. The proof is trivial for condition (I), cf. [MS06, Thm. 16.5.1]. (cid:3)
The following two propositions use the closedness of DerMod(Sp O ). They are es-sentially Propositions 16.4.13 and 16.4.12 of [MS06]. Proposition 4.1.8. If ( X, Y ; ε ) is a dual pair, then we have the following naturalisomorphisms: Z ∧ L A X ( C , B ) −−−→ ∼ = R map A op ( Y, Z )(8) Y ∧ L A V ( B , C ) −−−→ ∼ = R map A ( X, V ) , (9) and Y ∼ = R map A ( X, A ) . (10) Proof.
For the first two isomorphisms, use condition (II) and Theorem 2.3.1 (c)– and the (usual form of the) Yoneda lemma. Setting C = A and V = A in (9)yields (10). (cid:3) External duality for ( A , B ) -spectra. Considering Equation (10) of Propo-sition 4.1.8 above, we will now reverse the logic, define Y as DX and check whenthis yields a dual pair. Definition 4.2.1.
For an ( A , B )-spectrum X , define the dual of X to be the ( B , A )-spectrum DX = D ( A , B ) X = R map A ( X, A ) . Remark . The notation D ( A , B ) above should draw the reader’s attention tothe fact that the dual of an ( A , B )-spectrum depends on the pair ( A , B ), and notonly on the indexing category A ∧ B op . However, we will only write D from nowon. Remark . If we are sloppy for the moment and ignore the derivation process,we may think of D as given by the formula DX ( c ) = map C ( X ( − ) , C ( c, − )) . We have the evaluation map ε X : X ∧ L B DX ∼ = R map A ( A , X ) ∧ L B R map A ( X, A ) ( A , A ) −−−−→ A . OMOLOGY REPRESENTATION THEOREMS FOR DIAGRAM SPACES 21
Definition 4.2.4. X is called dualisable if ( X, DX ; ε X ) is a dual pair, i. e. if themap ( ε X ) ∗ from Proposition 4.1.1 is a bijection for all W and Z . ε X has the following naturality property: For every morphism f : X → X ′ inDerMod( A , B ), the diagram X ∧ L B DX ′ X ′ ∧ L B DX ′ X ∧ L B DX A f ∧ L B idid ∧ L B Df ε X ′ ε X commutes. It follows that for all W and Z (which we consider fixed from now on),( ε X ) ∗ : [ W, Z ∧ L A X ] ( C , B ) → [ W ∧ L B DX, Z ] ( C , A ) is a natural transformation.Recall that an exact functor between triangulated categories is a functor whichcommutes with the shift functor and sends distinguished triangles to distinguishedtriangles. If S is a triangulated category, then S op becomes a triangulated cate-gory with shift functor the opposite of Σ − , abusively denoted by Σ − again, wherea triangle X → Y → Z → Σ − X is distinguished if and only if Σ − X → Z → Y → X is distinguished in S . Lemma 4.2.5. (a) D : (DerMod( A , B )) op → DerMod( B , A ) is an exact functor.(b) X is dualisable if and only if Σ X is.(c) If X → X ′ → X ′′ → Σ X is a distinguished triangle and X and X ′ are dualis-able, then so is X ′′ .Proof. (a): By Theorem 2.3.1 (e), D (Σ X ) = R map A ( S ∧ L X, A ) ∼ = R map( S , DX ) ∼ = Σ − DX .
To show that D preserves cofiber sequences, we may assume that our cofiber se-quence is of the form X f −→ Y → Cf → Σ X with X and Y cofibrant and f a cofibration. By using the explicit cofibrant modelsand the properties of (underived) mapping spectra, cf. Subsection 2.2, the image ofthe sequence under D is identified with the sequenceΩ DX → hofib( Df ) → DY Df −−→ DX which is a fiber sequence in the sense of [Hov99, Def. 6.2.6]. But fiber and cofibersequences coincide in a stable model category by [Hov99, Thm. 7.1.11]. (b): There is a commutative diagram[ W, Z ∧ L A Σ X ] ( C , B ) [ W ∧ L B D (Σ X ) , Z ] C , A [ W, Σ Z ∧ L A X ] ( C , B ) [ W ∧ L B DX, Σ Z ] C , A ( ε Σ X ) ∗ ∼ = ∼ =( ε X ) ∗ where the vertical arrows are the isomorphisms from Theorem 2.3.1 (b) and (c),and the right one uses in addition the isomorphisms Ω E ∼ = Σ − S ∧ E and R map(Σ − S , F ) ∼ = Σ F in S H C .(c): Fix W and Z . Note that Z ∧ L A − and W ∧ L B − preserve distinguished trianglessince they are left adjoints. By equation (4) on p. 5, the rows of the following ladderare exact: [ W, Z ∧ L A X ] ( C , B ) [ W, Z ∧ L A X ′ ] ( C , B ) [ W, Z ∧ L A X ′′ ] ( C , B ) [ W, Z ∧ L A Σ X ] ( C , B ) [ W, Z ∧ L A Σ X ′ ] ( C , B ) [ W ∧ L B DX, Z ] ( C , A ) [ W ∧ L B DX ′ , Z ] ( C , A ) [ W ∧ L B DX ′′ , Z ] ( C , A ) [ W ∧ L B Σ DX, Z ] ( C , A ) [ W ∧ L B Σ DX ′ , Z ] ( C , A ) . ( εX )1 ∗ ( εX ′ )1 ∗ ( εX ′′ )1 ∗ ( ε Σ X )1 ∗ ( ε Σ X ′ )1 ∗ The statement is now deduced via the five-lemma. (cid:3)
From now on, assume that(FM) The mapping spectra of B are finite CW-spectra.In our applications, B will always be the trivial category ∗ with mapping spectrum S . Lemma 4.2.6.
If condition (FM) holds, then every ( A , B ) -spectrum of the form ( a, b ) is dualisable.Proof. For clarity, denote by a (as usual) the covariant functor corepresented by a , and by a the contravariant functor represented by a during this proof. We firsttreat the case that B is trivial. Note that Da ∼ = a by Lemma 2.3.6 and ε : a ∧ L a ∼ = a ∧ a → A is just the composition in A . It follows that ε ∗ is given by[ W, Z ∧ L A a ] ( C , ∗ ) → [ W ∧ L a, Z ∧ L A a ∧ L a ] ( C , A ) compose −−−−−→ [ W ∧ L a, Z ] ( C , A ) . Lemma 2.3.6 exhibits the source and the target as [
W, Z (? , a )] ( C , ∗ ) . Here, Z ( a, ?)makes sense for a derived module Z because of the definition of weak equivalence.A direct check on elements (assuming that W is cofibrant and Z is fibrant) showsthat the above composition is an isomorphism.In the general case, we have ( a, b ) = a ∧ b ∼ = a ∧ L b . Denote by Db the functor R map( b, S ). This is the dual of b viewed as a ( ∗ , B )-bimodule. This ( ∗ , B )-bimodule is dualisable by condition (FM). By Proposi-tion 4.1.7 and the first part of the proof, ( a, b ) is dualisable with dual D ( a, b ) ∼ = Db ∧ L a . (cid:3) The following corollary summarises the last two sections and comprises Theorem Bfrom the Introduction.
OMOLOGY REPRESENTATION THEOREMS FOR DIAGRAM SPACES 23
Corollary 4.2.7.
Suppose that condition (FM) holds. Then every finite ( A , B ) -CW-spectrum is dualisable. Consequently, for every finite ( A , B ) -spectrum X , any ( A , C ) -spectrum V and any ( C , A ) -spectrum Z , there are natural isomorphisms Z ∧ L A X ∼ = R map A op ( DX, Z ) and DX ∧ L A V ∼ = R map A ( X, V ) ; in particular, there is a natural isomorphism D ( A op , B op ) ( D ( A , B ) X ) op ∼ = X (11) for finite X .Remark . It follows from the proof of Lemma 4.2.6 that if B = ∗ , then thedual of a finite ( A , ∗ )-spectrum is a finite ( ∗ , A )-spectrum. This is false for general B . Remark . In practice, we will refer to (11) sloppily as
DDX ∼ = X . The ’op’in (11) refers to the fact that we have to consider DX as an ( A op , B op )-spectrum,instead of as a ( B , A )-spectrum, which implies that the duality functor is takenwith respect to the (contravariant) A -variance again. Proof of Corollary 4.2.7.
The full subcategory of dualisable objects contains allcorepresentable functors ( a, b ) by Lemma 4.2.6 and is a triangulated subcategory byLemma 4.2.5 (b) and (c). Thus, it contains all finite ( A , B )-spectra by Lemma 2.1.2(c). The two isomorphisms follow from Proposition 4.1.8. The isomorphism X ∼ = DDX follows from the first one by setting C = A and Z = A (or from Proposi-tion 4.1.6). (cid:3) In particular, D constitutes an equivalence of triangulated categories S W C → S W op C op for an arbitrary spectrally enriched category C satisfying (C). Example . Let C be the orbit category of finite subgroups of the integers. Ithas one object and automorphism group Z . We will view C as a spectrally enrichedcategory by adjoining a basepoint and smashing with S . Let X be the Z -space R with the usual translation action. This is a free, thus proper action, so it defines a C op -space X ? that we abusively also denote by X . We want to describe the dual of X + which is a C -spectrum. Suspending once, we get a cofibre sequenceΣ x F −−→ Σ x −→ Σ X + . Here, x denotes the unique object of C and the map F can be described as follows:In the S coordinate, it collapses the antipodal point of the base point to the basepoint. Then it maps the first half of the circle to the circle in the target withthe same x coordinate n , and the second half of the circle to the ( n + 1)-st circle.Dualising and rotating, we thus get a cofibre sequenceΣ − x DF −−−→ Σ − x −→ D ( X + ) . Homological representation theorems
Having established external Spanier-Whitehead duality, we can now prove our ho-mology representation theorem, Theorem 5.2.3, via the route sketched in the In-troduction. Subsection 5.1 first recollect some well-known information about C -homology theories, before Subsection 5.2 uses results of Neeman, as well as theresults of Section 4, to prove the main result. It has the hypothesis that S W C op isa countable category. This turns out to be equivalent to the countability of C itself(up to equivalence of categories), as proved in Subsection 5.3.From now on, C is a discrete index category.5.1. C -homology theories. Recall that a C -homology theory consists of a sequenceof functors h C n : Fun( C , Top ∗ ) → Abfor n ∈ Z , together with natural isomorphisms σ n : h C n (Σ X ) ∼ = h C n − ( X ) such that: • If A f −→ X is a map of pointed C -spaces, then the sequence h C n ( A ) → h C n ( X ) → h C n ( Cf )is exact. • For a collection ( X i ) of pointed C -spaces, the canonical homomorphism M i ∈ I h C n ( X i ) → h C n _ i ∈ I X i ! is an isomorphism. • If f : X → Y is a weak equivalence of C -spaces, then h C n ( f ) is an isomor-phism for all n . C -cohomology theories ( h n ) n are defined similarly, only that they are contravariantfunctors and the wedge axiom has a product instead of a sum.If the functors h C n are only defined on finite C -CW-complexes, then we call h C∗ a homology theory on finite C -CW-complexes . For homology theories, this is the samedatum since the homology of a C -CW-complex is the colimit of the homologies of itsfinite subcomplexes, by a telescope argument well-known from the classical setting.This is, however, not true for cohomology theories. In both cases however, thewedge axiom is void since it follows from the cone axiom for finite wedge sums. Remark . There are variations in this definition which give equivalent notionsof homology theories. For example, the homology theory may only be defined onpointed C -CW-complexes, with the weak equivalence axiom left out (being voidon C -CW-complexes). Such a theory can be extended to all pointed C -spaces via afunctorial CW-approximation. Also, one might define unreduced homology theorieswhich are functors from pairs of (unpointed) C -spaces to abelian groups, satisfyingthe usual Eilenberg-Steenrod axioms. The notions of reduced and unreduced C -homology theories are proved to be equivalent in the classical way [Lac16]. Allcombinations of these two variations occur in the literature.Recall the notion of a (co-)homological functor on a triangulated category from[Nee01, Def. 1.1.7, Rem. 1.1.9]. as opposed to: enriched OMOLOGY REPRESENTATION THEOREMS FOR DIAGRAM SPACES 25
Lemma 5.1.2.
A (co-)homology theory on finite pointed C -CW-complexes is thesame datum as a (co-)homological functor on the triangulated category S W C .Proof. We use the description of
S W C given in Lemma 2.1.2. If H is a homologicalfunctor, then defining h C n ( X ) = H (Σ − n Σ ∞ X )together with the obvious suspension isomorphisms yields a homology theory onfinite C -CW-complexes. Conversely, if h C∗ is such a theory, then Lemma 2.1.2 showsthat H (Σ N Σ ∞ X ) = h C− N ( X )defines a functor on S W C . The short exact cofibre sequence can be turned into along exact sequence by the usual rotation method, showing that H is a homologicalfunctor. It is obvious that these two constructions are inverse to each other. (cid:3) The following construction is classical [DL98, Lemma 4.2]:
Lemma 5.1.3.
Let E : C op → Sp O be a functor. Then h C n ( X ; E ) = π n ( E ∧ L C Σ ∞ X )(12) defines a C -homology theory.Remark . Strictly speaking, in the right-hand side of the above equation, π n ( − )should be [Σ n S , − ] S H C . This coincides with the well-known colimit definition fororthogonal spectra, but not for (all) symmetric spectra, cf. [HSS00, p. 61].5.2.
The homology representation theorem.
Our main result, Theorem 5.2.3,which is Theorem A from the Introduction, can be seen as a converse to Lemma 5.1.3.It shows that every homology theory can be obtained by this construction, in case
S W C op is countable in the following sense. Definition 5.2.1.
A category is called countable if it has countably many objectsand morphisms.
Remark . All our results also apply to categories which are equivalent to count-able categories. We decided to require that they are countable to keep the exposi-tion simple.
Theorem 5.2.3.
Suppose that
S W C op is countable. Let h C∗ be any C -homologytheory. Then there is a C op -spectrum E and a natural isomorphism h C∗ ( − ) ∼ = h C∗ ( − ; E ) . Moreover, every morphism of homology theories h C∗ ( − ; E ) −→ h C∗ ( − ; E ′ ) is induced by a morphism E −→ E ′ in the derived category S H C C op .Remark . The countability of
S W C op is equivalent to the countability of C ,as is proved in Subsection 5.3 below.The morphism in the last statement of Theorem 5.2.3 is in general not unique,already in the case C = ∗ , due to the existence of phantoms. The proof of thetheorem is based on the following two theorems from [Nee97]: Theorem 5.2.5 [Nee97, Thm. 5.1].
Let S be a countable triangulated category.Then the objects of projective dimension ≤ in Fun( S op , Ab) are exactly the ho-mological functors S op → Ab . We cite a second theorem from the same paper. The version in which we state ithere seems to be slightly stronger, but the same proofs apply in our case.In detail: Let T be a triangulated category with arbitrary small coproducts, anddenote by S a triangulated subcategory which • is essentially small, • generates T [Nee97, Def. 2.5], • consists of compact objects [Nee97, Def. 2.2].Neeman insists on S being the category T c of all compact objects (and he requiresthis subcategory to have the other two properties), but this is not really needed. Theorem 5.2.6 [Nee97, Prop. 4.11].
If every homological functor H : S op → Ab has projective dimension ≤ as an object of Fun( S op , Ab) , then the pair ( T , S ) satisfies Brown representability in the sense that the following two assertions hold: (1) Every homological functor H : S op → Ab is naturally isomorphic to a re-striction H ( − ) ∼ = T ( − , X ) ↾ S for some object X of T . (2) Given any natural transformation of functors on S op T ( − , X ) ↾ S → T ( − , Y ) ↾ S , there is a morphism f : X → Y in T inducing the natural transformation.The map f is in general not unique. We apply the two theorems to T = S H C C op and S = S W C op . The generationand compactness hypotheses are trivial. Proof.
Let H be the homological functor on S W C corresponding to (the restrictionof) h C∗ by Lemma 5.1.2. Since D is exact by Lemma 4.2.5 (a), we can define ahomological functor G on S W op C op by G ( Y ) = H ( DY ) . By Theorems 5.2.5 and 5.2.6, there is a fibrant and cofibrant C op -spectrum E thatrepresents G . We thus have natural isomorphisms h C n ( X ) ∼ = H (Σ − n Σ ∞ X ) ∼ = G ( D (Σ − n Σ ∞ X )) ∼ = [ D (Σ − n Σ ∞ X ) , E ] C op ( η X ) ∗ ∼ = [Σ n S , E ∧ L C Σ ∞ X ] ∼ = π n ( E ∧ C Σ ∞ X ) . An arbitrary C -CW-complex X is the colimit of its finite subcomplexes, and bothhomology theories commute with these colimits, so the isomorphism can be pulledover. Finally, an arbitrary C -space can be approximated by a C -CW-complex.The representation of morphisms of homology theories follows analogously frompart (2) of Theorem 5.2.6. (cid:3) C -cohomology theories. A C op -spectrum E defines a cohomology theory via h ∗C ( Y ; E ) = [Σ − n Σ ∞ Y, E ] S H C C op ∼ = π − n ( R map C op (Σ ∞ Y, E )) . If Y is a C -CW-complex and E is fibrant, the R can be omitted. The fact thatevery C -cohomology theory has this form, i. e. the generalisation of the classical OMOLOGY REPRESENTATION THEOREMS FOR DIAGRAM SPACES 27
Brown Representability Theorem, may be obtained by mimicking its original proof[B´ar14, Lac16], or by citing a theorem of Neeman again [Nee01, Thm. 8.3.3]. Notethat the cohomological case is in any way considerably easier than the homologicalcase. It doesn’t need the countability assumption.5.2.2.
Morphisms of C -cohomology theories. These are always represented by mor-phisms in
S H C C op : First, replace the representing spectra E, E ′ by fibrant andcofibrant spectra, and restrict to cofibrant X . Then the n -th degree cohomologytheory is just given by [ X, E n ] C , thus we get various maps E n → E ′ n such thatthe obvious compatibility diagrams commute up to homotopy. Now, rewrite thesediagrams using the structure maps Σ E n → E n +1 and use that these have the ho-motopy extension property since E is cofibrant [MMSS01, Lemma 11.4] to strictifythe diagrams inductively. (This is the argument for sequential spectra; use thearguments presented in Subsection 3.3 to pass to orthogonal spectra.)5.3. Countability considerations.
In practice, it may seem hard to check whether
S W C op is countable for a given category C . However, this turns out to be equivalentof the countability of the category C itself: Proposition 5.3.1.
Let C be a category. Then S W C is equivalent to a countablecategory if and only if C is.Example . If G is a countable group and F is a family of subgroups which iscountable up to conjugation in G , then Or( G, F ) is countable. For instance, F canbe the family of finite subgroups. Example . Let G be a reductive p -adic algebraic group and F the family ofcompact open subgroups. Note that the orbit category is a discrete category, inthe sense that the topology on the morphism spaces is discrete. We now show thatit is countable. Any compact subgroup fixes a vertex in the Bruhat-Tits building,hence is contained in a vertex stabiliser. These are all conjugate to a stabiliser of thevertex of some fundamental chamber, of which there are only finitely many. Let K be a vertex stabiliser. The compact totally disconnected group K has a countablesystem K i of compact open subgroups which form a neighbourhood basis of theidentity. Thus, every subgroup of K lies between some K i and K . But for fixed i , there are only finitely many of these, since they correspond to subgroups ofthe finite group K/K i . To prove that morphism sets are countable, it suffices byLemma 6.4.1 to show that G/K is countable. But
G/K is the orbit of a vertex inthe Bruhat-Tits building, which is countable since the building is a union of ballsand every ball contains only finitely many vertices by local compactness. Witha little more care, one can show that if G is a reductive group over Q p which isabsolutely almost simple and simply-connected, then the morphism sets are evenfinite [Lac]. Lemma 5.3.4.
Let X be a countable pointed CW-complex. (a) For every n , π n ( X ) is countable. (b) Fix a map ∂ D n → X . Then the set [(D n , ∂ D n ) , X ] of homotopy classes ofmaps D n → X rel ∂ D n is countable.Proof. Part (a) is contained in Theorem 6.1 of [LW69]. Part (b) can be provedsimilarly. (cid:3)
Proof of Proposition 5.3.1.
It is obviously necessary that C is equivalent to a count-able category: For any object c , the 0-th singular homology of X ( c ) ∼ = R map C ( c, X )is a well-defined functor H c on S W C . The composition with the Yoneda embed-ding, C op −→ Fun fin . CW ( C , Sp O ) −→ S W C ( H c ) c −−−−→ Fun( C , Ab)is the Yoneda embedding which is fully faithful. It follows that the composition ofthe first two functors sends non-isomorphic objects to non-isomorphic objects andis faithful.For the sufficiency, it is obviously enough to show that the category of finitepointed C -CW-complexes, with homotopy classes of maps, is countable, compareLemma 2.1.2. Note that for a countable C -CW-complex X , all X ( c ) are themselvescountable CW-complexes, because of the condition that C has countable morphismsets.First, we show that there are only countably many homotopy types of objects X ,via induction on the number of cells of X . There are only countably many 0-dimensional CW-complexes since Ob( C ) is countable. Now, we suppose that X isgiven and we want to show that there are only countably many possibilities to attachone further cell. This amounts to choosing an object c (countably many choices)and a based homotopy class of an attaching map S n + ∧ c → X . But these are inbijection with free homotopy classes S n → X ( c ) which is a quotient of π n ( X ( c ))and thus countable by Lemma 5.3.4 (a).The countability of the morphism sets follows similarly from Lemma 5.3.4 (b). (cid:3) The rational case
In this section, for technical reasons, we treat homology theories of C -simplicialsets instead of C -spaces. The results apply to topological spaces, too, since we mayapply the geometric realisation functor objectwise. Our Theorem 5.2.3 holds truealso in this setting, yielding that any homology theory h C∗ is of the form h C∗ ( − ; E )for some E : C op → Sp ΣsSet .Now, suppose that the homology theory h C∗ ∼ = h C∗ ( − ; E ) is rational, i. e. takes valuesin Q -vector spaces. By plugging in corepresentable functors c , it follows that allspectra E ( c ) have rational homotopy groups for all c . Thus the natural map E → H Q ∧ E is a weak equivalence of C -spectra. Note that the right-hand side is not only a func-tor from C to spectra, but to H Q -modules. The stable Dold-Kan correspondence ,discussed in Subsection 6.1, links these to chain complexes.We have thus arrived in a purely algebraic setting. More precisely, we study modulesover a certain category algebra Q C , cf. Subsection 6.2. One of the tools that isavailable here, and was not available in the case of spectra, is the K¨unneth spectralsequence for a tensor product of chain complexes. We use this to prove the existenceof a Chern character in the case of flat coefficients, cf. Corollary 6.3.7. The flatnesshypothesis is true in the homological case if the coefficients extend to Mackeyfunctors, as discussed in Subsection 6.4. Another approach, based on the work ofLiping Li on hereditary category algebras [Li11], is presented in Subsection 6.5.This approach has no hypothesis on the homology theory, but on the category C . OMOLOGY REPRESENTATION THEOREMS FOR DIAGRAM SPACES 29
The stable Dold-Kan correspondence.
We work with the paper [Shi07]which realises the stable Dold-Kan correspondence as a zig-zag of weak monoidalQuillen equivalences (left adjoints on top) H Q −M od Z −←−−−−−−→− U Sp Σ ( s Vect Q ) L ←−−−−−−−−→ φ ∗ N Sp Σ (ch + Q ) D −←−−−−−−→− R Ch Q . (13)The paper constructs these functors over a general ring R (and concentrates on R = Z in some parts of the exposition), but we will only need the special case R = Q . The four model categories used here were introduced in Subsection 3.1.For the definition of the various functors, we refer to [Shi07]. The definitions ofsome of them will be recalled in the proof of Proposition 6.1.1. They have thespecial property that all right adjoints preserve all weak equivalences. This passesto the functor categories and has the consequence that no fibrant replacements arenecessary when the derived functor is computed.For any (Set-enriched) category C , we get Quillen equivalencesFun( C , H Q −M od ) Z ∗ −←−−−−→− U ∗ Fun( C , Sp Σ ( s Vect Q )) L ∗ ←−−−−−−−−−−−−−−→ ( φ ∗ N ) ∗ Fun( C , Sp Σ (ch + Q )) D ∗ −←−−−−→− R ∗ Fun( C , Ch Q ) . By the discussion in Subsection 3.2, we thus get an equivalence of categoriesHo(Fun( C , H Q −M od )) Φ −←−−→− Γ Ho(Fun( C , Ch Q ))where Φ and Γ respect derived balanced smash products and mapping spectra, andare given by Φ = D ∗ Q ( φ ∗ N ) ∗ Z ∗ Q and Γ = U ∗ L ∗ QR ∗ . For a based simplicial set A , let e Q A denote the simplicial Q -vector space which isthe reduced linearisation of A , i. e. it has as a basis in degree n the set of non-basepoint n -simplices A n \ {∗} . Furthermore, let N : s Vect Q → ch + Q denote thenormalised chain complex functor. Proposition 6.1.1. If X is a based simplicial C -set, then there is a natural iso-morphism Φ( H Q ∧ Σ ∞ X ) ∼ = N e Q X .
Proof.
We may assume that X is cofibrant in the projective model structure since N e Q : sSet −→ Ch Q preserves all weak equivalences [GJ09, Prop. 2.14]. We gothrough the construction of Φ step by step. The first cofibrant replacement is notneeded since Σ ∞ X is a cofibrant C -spectrum, and thus H Q ∧ Σ ∞ X is a cofibrant C - H Q -module. The functor Z is given by linearising and then using the canonicalmorphism µ : e Q ( H Q ) → e QS to turn the result into a e QS -module again.We thus have Z ∗ Q ( H Q ∧ Σ ∞ X ) = e QS ⊗ e Q ( H Q ) e Q ( H Q ∧ Σ ∞ X ) ∼ = e QS ⊗ e Q ( H Q ) e Q ( H Q ) ⊗ e QS e Q (Σ ∞ X ) ∼ = e Q (Σ ∞ X ) . Here we used that the functor e Q is strong monoidal and commutes with colimits.Note that ( e Q (Σ ∞ X )) n ∼ = e Q S n ⊗ e Q X , which we refer to as e Q (Σ ∞ X ) = e QS ⊗ e Q X . Next, we apply the functor φ ∗ N objectwise. Here N is the normalised chain com-plex functor as introduced above, which sends e Q (Σ ∞ X ) to a N ( e QS )-module inthe category of symmetric sequences of positive chain complexes. This becomes amodule over Sym( Q [1]) (i. e., a symmetric spectrum) via a ring homomorphism φ : Sym( Q [1]) → N ( e QS )specified on p. 358 of [Shi07]. This ring map is not an isomorphism (it correspondsto a subdivision of a cube into simplices), but a weak equivalence, cf. the proof ofShipley’s Proposition 4.4.Next, we show that N e Q X is a cofibrant C -chain complex. Since N is an equivalenceof categories, it commutes with colimits, and so does e Q . Thus the assertion followsinductively from the fact that N e Q (S n − ) + → N e Q (D n ) + is a cofibration. Thelatter is readily checked since cofibrations of chain complexes over the field Q arejust monomorphisms.A similar inductive argument shows that Sym( Q [1]) ⊗ N e Q X is cofibrant inFun( C , Sp Σ (ch + Q )) and that φ induces a weak equivalence φ ⊗ id : Sym( Q [1]) ⊗ N e Q X −→ N e QS ⊗ N e Q X .
From the right-hand side we go on with the shuffle map of [SS03a, 2.7], appliedlevelwise: ∇ : N e QS ⊗ N e Q X −→ N ( e QS ⊗ e Q X ) . The shuffle map is always a quasi-isomorphism on the level of chain complexes(even a homotopy equivalence with homotopy inverse the Alexander-Whitney map),thus it induces a weak equivalence on each level. To see that it is a morphism ofsymmetric spectra, i. e. Sym( Q [1])-modules, it suffices to show that it is a morphismof N e QS -modules. This is an easy diagrammatic check using the fact that N is alax monoidal transformation [SS03a, p. 256]. Summarising, we have constructed acofibrant replacementSym( Q [1]) ⊗ N e Q X ∇◦ ( φ ⊗ id) −−−−−−→ ∼ φ ∗ N ( e QS ⊗ e Q X ) . The last step is to apply the functor D objectwise to the left-hand side. Butthis is objectwise just the suspension spectrum of N e Q X (?), and D applied tothe suspension spectrum of a chain complex yields just the chain complex itself by[Shi07, Lemma 4.6]. (Suspension spectra are denoted by F in Shipley’s paper.) (cid:3) Rational C -modules and nondegenerate Q C -modules. In the rationalcase, Theorem 5.2.3 says that a C -homology theory always comes from a functor E : C op → Ch Q . Note that this is the same as a chain complex of functors from C op to Q -vector spaces. We now take a closer look at this additive category.Let C be a small category enriched in Q -vector spaces. (Everything holds true overan arbitrary commutative ground ring, though.) Definition 6.2.1.
The category algebra R = Q C of C is given by M c,d ∈C Hom C ( c, d ) , OMOLOGY REPRESENTATION THEOREMS FOR DIAGRAM SPACES 31 with multiplication defined by bilinear extension of the relations g · f = ( gf if g , f are composable0 else . If C happens to be the free Q -linear category on a (Set-enriched) category, we havethe presentation R = Q C ∼ = Q * e f for f : c → d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e g e f = ( e gf if g , f are composable0 else + , where the angle brackets indicate that we take the quotient of the free (non-commutative) algebra over the e f by the said relations. If C has only finitelymany objects, the category algebra R has a unit P c ∈ Ob( C ) id c . For general (i. e. non-object-finite) C , R has only an approximate unit in the sense defined below. Recallthat a net in a set S is a map I → S where I is a directed set, i. e. a partiallyordered set in which any two elements have a common upper bound. Definition 6.2.2.
A ring S has an approximate unit if there is a net ( e i ) i ∈ I ofidempotents in S with the following two properties: • For every s ∈ S , there is some i such that e i s = s = se i . • For i ≤ j , we have e j e i = e i e j = e i .A left S -module M is called non-degenerate if SM = M . Equivalently, if for every m ∈ M there is some i such that e i m = m . The category of non-degenerate left S -modules and S -linear maps is denoted N M od S . Lemma 6.2.3.
Let S be a ring with approximate unit.(a) If M is a non-degenerate left S -module, then there is a natural isomorphism of S -modules S ⊗ S M ∼ = M . (b) A non-degenerate left S -module P which is projective in the category of non-degenerate left S -modules is flat in the sense that − ⊗ S P is an exact from non-degenerate S -modules to abelian groups.Proof. (a) Define an S -linear map f : S ⊗ S M −→ M by s ⊗ m sm . A map g (of sets, say) in the other direction is defined as follows: An element m ∈ M ismapped to e i ⊗ m , where i ∈ I is such that e i m = m . This is well-defined: If j isanother such index, choose k ≥ i, j . Then e i ⊗ m = ( e k e i ) ⊗ m = e k ⊗ ( e i m ) = e k ⊗ m = e j ⊗ m . It is immediate that f ◦ g is the identity. For g ◦ f , use the fact that S has anapproximate unit: Choose e i with e i s = s , then e i sm = sm and g ( f ( s ⊗ m )) = e i ⊗ ( sm ) = s ⊗ m . (b) A non-degenerate S -module is a quotient of a direct sum of left regular repre-sentations S . If it is projective, then it is a direct summand and hence flat by part(a). (cid:3) For our category algebra R , the set I consists of all finite sets of objects of C ,ordered by inclusion, and the approximate unit sends F ∈ I to e F = X c ∈ F id c . The following result is essentially [Mit72, Thm. 7.1].
Proposition 6.2.4.
There is an isomorphism of additive categories
Ξ : Fun( C , Vect Q ) −→ N M od R . There is a similar equivalence between contravariant functors and non-degenerateright modules. If this is also denoted by Ξ , then there are natural isomorphisms of Q -vector spaces Ξ( X ) ⊗ R Ξ( Y ) ∼ = X ⊗ C Y for a right C -module X and left C -module Y , and Hom R (Ξ( X ) , Ξ( Z )) ∼ = Hom C ( X, Z ) for two right C -modules X and Z .Proof. The equivalence is defined as follows: If X : C →
Vect Q is a functor, defineΞ( X ) = M c ∈ Ob( C ) X ( c )with the action of ( f : c → d ) ∈ R on x = ( x c ) c given by( f · x ) d = ( X ( f )( x c ) if d = d . This yields a non-degenerate R -module: Every element lies in some vector subspace L c ∈ F X ( c ), where F is a finite set of objects, and e F acts as the identity on thissubspace.An inverse equivalence Π : N M od R −→ Fun( C , Vect Q )is constructed as follows: If M is a non-degenerate R -module, let(Π( M ))( c ) = id c M .
A morphism f : c → d induces a linear map id c M → id d M since f = id d f .It is easy to check that ΠΞ is the identity. For the other composition, note thatthere is a natural map Ξ(Π( M )) = M c ∈ Ob( C ) id c M → M induced by the inclusions. This will be an injective R -linear map in general sincethe id c are orthogonal idempotents. If M is non-degenerate, it is surjective.The two asserted natural isomorphisms are straightforward. (cid:3) Remark . The categoryCh(
N M od R ) ∼ = Fun( C , Ch Q )can be endowed with a model structure in (at least) two ways. The first one is justthe projective model structure as a functor category, coming from the projectivemodel structure on Ch Q . The second one is the projective model structure on chaincomplexes over N M od R . This model structure (for abelian categories differentfrom modules over a unital ring) is defined in [Hov01a, Sec. 3]. The hypotheses of[Hov01a, Thm. 3.7] are satisfied here since R generates N M od R in the sense that OMOLOGY REPRESENTATION THEOREMS FOR DIAGRAM SPACES 33
N M od R ( R, − ) is faithful. One can easily check that these two model structurescoincide.The discussion of this section allows us, in the rational case, to state Theorem 5.2.3in a completely algebraic way. Corollary 6.2.6. If E is a chain complex of right R -modules, then h C∗ ( X ; E ) = H ∗ ( E ⊗ R X ) . (14) defines a rational reduced C -homology theory. Here X = QN e Q Sing( X ) ∼ = N e Q ( Q (Sing( X ))) denotes a cofibrant replacement of N e Q Sing( X ) and Sing : Top ∗ → sSet ∗ denotesthe singular simplicial complex functor.Conversely, if h C∗ is a rational C -homology theory, then there is a chain complex E ,and a natural isomorphism of homology theories as above.Remark . In the second part of the theorem, if E is cofibrant (as a chain com-plex over N M od R ), one might take N e Q Sing( X ) instead of its cofibrant replacement X . This is due to the fact that in the aformentioned model category, tensoring witha cofibrant chain complexes preserves weak equivalences, by Lemma 6.2.8, so weget an analogue of Corollary 2.3.4. However, we will mainly use (14) in the formwith X since this allows us to manipulate E . Lemma 6.2.8. If X is a cofibrant chain complex over N M od R , then tensoringwith X preserves weak equivalences.Proof. Note that a cofibrant chain complex is degreewise projective by the argu-ment from [Hov99, Lemma 2.3.6]. For positive chain complexes, the assertion thusfollows from the K¨unneth spectral sequence [ML63, Thm. 12.1]. In the generalcase, truncate the chain complexes (the cofibrant one naively, the members of thequasi-isomorphism as in the proof of Corollary 6.3.7 below) and then pass to thecolimit. (cid:3)
Chern characters.
We quickly recall the notion of Bredon homology [DL98,Sec. 3]. Let M be a right R -module. If X is a pointed C -CW-complex, then applyingthe cellular complex objectwise yields a left R -chain complex and the homology ofthe tensor product h C , Br n ( X ; M ) = H n ( M ⊗ R C cell ∗ ( X ; Q ))defines a C -homology theory – use a CW-approximation to extend it to arbitrary C -spaces. Definition 6.3.1.
The coefficient system of a reduced C -homology theory h C∗ is the Z -graded right R -module given by h C n = h C n (S ∧ c ).The Bredon homology with respect to this coefficient system appears in the Atiyah-Hirzebruch spectral sequence h C , Br p ( X ; h C q ) ⇒ h C n ( X ) . (15)It is proved in the same way as in the case C = ∗ [Lac16]. Lemma 6.3.2.
Suppose that the right R -chain complex E is given by a right R -module E = M in degree , and E n = 0 otherwise. Then there is a naturalisomorphism of homology theories H ∗ ( E ⊗ R X ) ∼ = h C , Br ∗ ( X ; M ) . Proof.
The coefficient system of the left-hand homology theory is given by E k in degree k . Since this is 0 in non-zero degrees, the Atiyah-Hirzebruch spectralsequence (15) collapses and gives the above isomorphism. (cid:3) Remark . Alternatively to using the Atiyah-Hirzebruch spectral sequence, onecould also prove Lemma 6.3.2 by using a zig-zag of chain complexes between thesingular and the cellular chain complex which is natural (in cellular maps) andinduces the isomorphism between singular and cellular homology. This then can beupgraded to C -CW-complexes. Such a zig-zag is constructed on p. 121 of [VF04].We now turn to the question of existence of Chern characters, which means forus that the homology theory splits into a direct sum of shifted Bredon homologytheories. By plugging in suspended representable functors S n ∧ c , one sees thatthere is only one choice for the coefficient systems in every degree, yielding thefollowing definition. Definition 6.3.4.
Let h C∗ be a C -homology theory. A Chern character for h C∗ is anisomorphism of C -homology theories h C n ∼ = M s + t = ∗ h C , Br s ( X ; h Ct ) . Chern characters for C -cohomology theories are defined in the exact same way, alsousing direct sums. Lemma 6.3.5.
Let M be a right R -chain complex. Then a Chern character existsfor h C∗ ( − ; M ) if and only if M is isomorphic to a complex with zero differentials inthe derived category of Ch(
N M od R ) .Proof. This follows directly from the second part of Theorem 5.2.3 (representationof morphisms), together with the Dold-Kan correspondence described in Subsec-tion 6.1 and Remark 6.2.5. (cid:3)
Remark . In the case that M is bounded, [Ill02, Sec. 4.5, 4.6] describes howone can find out whether the condition of Lemma 6.3.5 holds, using a sequenceof obstructions living in Ext i ( H p + i − ( M ) , H p ( M )) with i ≥
2. The expositionassumes that R has a unit, but this is not used in the argumentation.We now discuss one approach to construct Chern characters. The following resultwas announced in the Introduction as Theorem C. Proposition 6.3.7.
Suppose that h C∗ is a rational C -homology theory with the prop-erty that all coefficient systems h C t are flat as right C -modules. Then there exists aChern character for h C∗ , which is natural in the homology theory h C∗ .Remark . This result is similar to [L¨uc02, Thm. 4.4] where the case of theproper orbit category of a discrete group is treated, with the additional assumptionthat the homology theory is equivariant, i. e. there are proper homology theoriesfor all discrete groups linked via induction isomorphisms. A technical differenceis that the flatness assumption is not over the orbit category itself, but over a
OMOLOGY REPRESENTATION THEOREMS FOR DIAGRAM SPACES 35 certain category Q Sub( G, FIN ), whereas the homology theories are defined onOr( G, FIN )-spaces as usual. Thus, our theorem does not imply L¨uck’s theoremdirectly.
Remark . Taking into account Lemma 6.3.5, we have proved that whenever achain complex of non-degenerate R -modules has flat homology, then it is isomorphicto a trivial complex in the derived category. For bounded complexes, we may alsosee this as follows: Using the result that over a countable ring, flat modules haveprojective dimension at most 1, we see that all higher Ext i -groups of the homologymodules, i ≥
2, appearing in Remark 6.3.6, vanish.
Proof.
Start with the representation as in (14). First suppose that E is boundedbelow, say positive. We claim that X is degreewise flat. Copying the argumentfrom the proof of [Hov99, Lemma 2.3.6] shows that X is degreewise projective in thecategory N M od R . Note that a fibration is still the same as a degreewise surjectivemap. By Lemma 6.2.3 (b), X is degreewise flat.Having said this, we get a K¨unneth spectral sequence [ML63, Thm. 12.1] E p,q = M s + t = q Tor Rp (H s ( E ) , H t ( X )) ⇒ H p + q ( E ⊗ R X ) . Since the coefficients H s ( E ) are flat, all higher Tor terms vanish and the E pageis concentrated on the line p = 0. It thus degenerates and gives an isomorphism h C n ( X ) ∼ = H n ( E ⊗ R X ) ∼ = M s + t = n H s ( E ) ⊗ R H t ( X ) ∼ = M s + t = n H t (H s ( E ) ⊗ R X ) ∼ = M s + t = n h C , Br t ( X ; H s ( E )) , where we used flatness of H s ( E ) again, and Lemma 6.3.2. Naturality of the K¨unnethspectral sequence shows directly that this isomorphism is natural in X . Naturalityin the homology theory additionally needs the fact that every morphism of homologytheories is induced by a morphism of chain complexes, after possibly replacing E by a fibrant and cofibrant complex, cf. Theorem 5.2.3.For arbitrary E , let τ k E denote the truncations( τ k E ) n = E n , n ≥ k ker( d k ) , n = k , n < k . There are natural injective chain maps τ k E ֒ → E inducing a homology isomorphism in all degrees ≥ k , whereas the homology of τ k E in degrees < k is 0. In particular, H t ( τ k E ) is flat for all t .The maps above exhibit E as the colimit of the sequence τ E ֒ → τ − E ֒ → τ − E ֒ → . . . . We now run the above argument with the truncations τ k E . Note that we need notassume these to be cofibrant, thanks to Lemma 6.2.8. The various isomorphismsH n ( τ k E ⊗ R X ) ∼ = M s + t = n H s (H t ( τ k E ) ⊗ R X ) are natural with respect to the inclusions τ k E ֒ → τ k − E by naturality of theK¨unneth spectral sequence. Passing to the colimit, the right-hand side obviouslygives the desired sum of Bredon homologies. The left-hand side gives H n ( E ⊗ R X )since homology commutes with filtered colimits, and so does − ⊗ R X . (cid:3) A cohomological version can be proved in a very similar way:
Proposition 6.3.10.
Let h ∗C be a rational C -cohomology theory with projective coefficient systems. Then there is a Chern character for h ∗C .Proof. The proof is analogous, using a cohomological version of Corollary 6.2.6 andthe cohomological K¨unneth spectral sequence [Rot79, Thm. 11.34] E p,q = M s + t = q Ext pR (H s ( X ) , H t ( E )) ⇒ H p + q (Hom R ( X, E )) . Note that to formulate Corollary 6.2.6 with Hom R instead of derived tensor product,we also need to replace E fibrantly, since we don’t have a mapping space version ofCorollary 2.3.4 at hand. However, in Ch Q and thus in Fun( C , Ch Q ), all objects arefibrant. (cid:3) Mackey functors.
In this subsection, C is the orbit category of a group G .We will show that the flatness assumption of Corollary 6.3.7 holds if G is finite andthe coefficients can be extended to Mackey functors.Recall that if F is a family of subgroups of G (non-empty and closed under subcon-jugation), then the orbit category Or( G, F ) has as objects the transitive G -spaces G/H for H ∈ F , and as morphisms all G -linear maps. Recall further that an EIcategory is a category in which all endomorphisms are invertible.We will use the following explicit description of the orbit category: Lemma 6.4.1.
Let G be an arbitrary group and F a family of subgroups.(a) For H, K ∈ F , there is an isomorphism φ H,K : K \ Trans G ( H, K ) ∼ = Hom Or( G, F ) ( G/H, G/K ) ,g φ H,K ( g ) with Trans G ( H, K ) = { g ∈ G ; gHg − ⊆ K } and ( φ H,K ( g ))( xH ) = xg − K for x ∈ G . Furthermore, for L ∈ F and g ′ ∈ Trans G ( K, L ) , we have φ K,L ( g ′ ) ◦ φ H,K ( g ) = φ H,L ( g ′ g ) . (b) If F consists of finite groups only, then Or( G, F ) is an EI category. If no confusion can arise, we will only write φ for φ H,K . Proof.
Part (a) is well-known and follows immediately from the fact that the objectsof G are transitive G -spaces. For part (b), note that if H is finite, then gHg − ⊆ H implies by cardinality reasons that gHg − = H and thus g − Hg = H . (cid:3) OMOLOGY REPRESENTATION THEOREMS FOR DIAGRAM SPACES 37
From now on, G is finite and F is the family of all subgroups. Recall that a(rational) Mackey functor assigns to any subgroup H of G a Q -vector space M ( H )and to any inclusion K ⊆ H two homomorphisms I HK : M ( K ) −→ M ( H ) and R HK : M ( H ) −→ M ( K ) , called induction and restriction, and for any g ∈ G conjugation homomorphisms c g : M ( H ) −→ M ( gHg − ) . These have to satisfy certain relations listed for instance in [TW95].Let Ω Q ( G ) denote the Mackey category of G ; we take [TW95, Prop. 2.2] as itsdefinition. It is a category enriched in Q -vector spaces which is not the free Q -linear category on a category. Its objects the finite G -sets. By design, a Mackeyfunctor is just a Q -linear functor Ω Q ( G ) −→ Vect Q . Lemma 6.4.2.
There is a canonical functor I : Or( G ) → Ω Q ( G ) defined by I ( G/H ) = H and I ( φ ( g )) = I KgHg − c g for g ∈ Trans G ( H, K ) .Remark . Since I is injective on objects, it induces a ring homomorphism I onthe category algebras [Xu06, Prop. 3.2.5]. The category algebra of Ω Q ( G ) is called µ Q ( G ), the Mackey algebra. Proof.
Let
H, K, L, g and g ′ be as in Lemma 6.4.1. Calculate: I ( φ ( g ′ ) ◦ φ ( g )) = I ( φ ( g ′ g )) = I Lg ′ gH ( g ′ g ) − c g ′ g = I Lg ′ K ( g ′ ) − I g ′ K ( g ′ ) − g ′ gH ( g ′ g ) − c g ′ c g = I Lg ′ K ( g ′ ) − c g ′ I KgHg − c g = I ( φ ( g ′ )) I ( φ ( g )) . (cid:3) Definition 6.4.4.
A left (or right) rational Or( G )-module M is said to extend toa Mackey functor if it is of the form I ∗ f M for a left (or right) Ω Q ( G )-module f M . Proposition 6.4.5. µ Q ( G ) is a projective left Q Or( G ) -module.Remark . It is not known to us whether the corresponding statement as rightmodules holds. Thus Corollary 6.4.7 cannot be formulated for G - co homology the-ories at the moment. Proof. A Q -basis of µ Q ( G ) is given on the bottom of p. 1875 of [TW95] (cf.Prop. 3.2, 3.3). It consists of all elements I KgLg − c g R HL = I ( φ ( g )) R HL , for L ⊆ H and g ∈ Trans G ( L, K ), up to the following identification: I ( φ ( g )) R HL = I ( φ ( g ′ )) R HL ′ ⇔ ∃ x ∈ H ∩ ( g ′ ) − Kg : L ′ = xLx − . (16)Let P denote a set of representatives of pairs ( H, L ) with L ⊆ H , modulo therelation that for fixed H , L may be conjugated by an element from H : ( H, L ) ∼ ( H, hLh − ). Then we define an Or( G )-linear homomorphism F : M ( H,L ) ∈ P Q Hom
Or( G ) ( G/L, − ) ⊗ Q N G ( L ) Q [ N G ( L ) / ( H ∩ N G ( L ))] −→ µ Q ( G ) ,φ ( g ) ⊗ n I ( φ ( gn )) R HL . We will show that F is an isomorphism, which implies the result since the left-handside is a projective module by the semi-simplicity of all Q N G ( L ).To see that F is surjective, note that by the result cited above, the right-handside has a basis of elements I ( φ ( g )) R HL with L ⊆ H . We only have to achieve( H, L ) ∈ P . For this, choose h ∈ H such that ( H, L ′ ) ∈ P with L ′ = hLh − . For g ′ = gh − , we have h = ( g ′ ) − · · g ∈ ( g ′ ) − Kg ∩ H and thus I ( φ ( g )) R HL = I ( φ ( g ′ )) R HL ′ = F ( φ ( g ) ⊗
1) by (16).Next, we show that F is injective. Fix H and K and consider only morphisms from H to K . Let L be a set of representatives of subgroups of H up to conjugation(in H ). The left-hand side has a basis consisting of all pairs ( L, φ ( g ) ⊗ H, L ) ∈ P and g ∈ K \ Trans G ( L, K ) /N G ( L ). Such an element is mapped to theelement I ( φ ( gn )) R HL on the right-hand side, which is part of the Th´evenaz-Webbbasis. Thus, we only have to show that F is injective when restricted to the basis { ( L, φ ( g ) ⊗ } . Suppose that F ( L, φ ( g ) ⊗
1) = F ( L ′ , φ ( g ′ ) ⊗ . By (16), there exists x ∈ H ∩ ( g ′ ) − Kg such that L ′ = xLx − . In particular, L and L ′ are conjugate in H , i. e. L = L ′ . Then x ∈ N G ( L ). We have g ′ x = kg forsome k ∈ K and consequently φ ( g ) ⊗ φ ( kg ) ⊗ φ ( g ′ x ) ⊗ φ ( g ′ ) ⊗ x = φ ( g ′ ) ⊗ . (cid:3) Corollary 6.4.7.
Let G be finite and h G ∗ a rational G -homology theory with theproperty that all coefficient systems h C t extend to Mackey functors. Then there is aChern character for h G ∗ .Proof. Let M = I ∗ f M . By [TW90, Thm. 9.1], the Mackey algebra (over Q ) issemisimple. Thus, f M is a projective µ Q ( G )-module and hence M is a projective,thus flat, Or( G )-module by Proposition 6.4.5. The existence of the Chern characterthen follows from Corollary 6.3.7. (cid:3) Remark . A similar result was shown by L¨uck [L¨uc02, Thm. 5.2]. His re-sult holds for arbitrary discrete G (with F the family of finite subgroups), butrefers to equivariant homology theories, and the Mackey condition is formulated for Q Sub( G, FIN )-modules, cf. Remark 6.3.8. L¨uck’s definition of Mackey extensionis stronger than our definition given below. Thus his examples, namely equivariantbordism (Ex. 1.4, 6.4) and the equivariant homology theories associated to ratio-nalised algebraic K -theory and rationalised algebraic L -theory of the group ring,as well as rationalised topological K -theory of the reduced group C ∗ -algebra (Ex.1.5, Sec. 8) can also serve as examples for us.In contrast to L¨uck’s result, the argumentation presented here breaks down forinfinite G . While Proposition 6.4.5 still holds true in this case, it is not true anylonger that µ Q ( G ) is semi-simple. We give an example showing that it is not evenvon Neumann regular. Recall from [Goo91] that a ring is called von Neumannregular if every module is flat, and that this is equivalent to the condition that forevery ring element a , there exists a ring element x such that axa = a . Example . Let G = D ∞ = h s, t | s = t = 1 i be the infinite dihedral group,and let Ω Q ( G ) and µ Q ( G ) be defined exactly as above (for finite groups), with the OMOLOGY REPRESENTATION THEOREMS FOR DIAGRAM SPACES 39 difference that the subgroups H and K are restricted to the finite subgroups of G .One can show thatHom Ω Q ( G ) ( h s i , h t i ) = Q h{ I h t i gR h s i ; g ∈ h t i\ G/ h s i}i . Representatives of the ( h t i , h s i )-double cosets are given by ( st ) k for k ∈ Z . Let x k = I h t i ( st ) k R h s i and y k = I h s i ( st ) k R h t i . The y k form a Q -basis of the homomorphismsfrom h t i to h s i similarly.Let a = y = I h s i R h t i ∈ Hom Ω Q ( G ) ( h t i , h s i ). Compute ax k a = I h s i R h t i I h t i ( st ) k R h s i I h s i R h t i = I h s i (1 + t )( st ) k (1 + s ) R h t i = I h s i (( st ) k + t ( st ) k + ( st ) k s + t ( st ) k s ) R h t i = I h s i (( st ) k + st ( st ) k + ( st ) k st + st ( st ) k st ) R h t i = y k + 2 y k +1 + y k +2 . It follows easily that the linear equation axa = a has no solution. Thus, µ Q (D ∞ )is not von Neumann regular.6.5. Hereditary category algebras.
In this subsection we restrict ourselves tofinite categories, so that we only deal with unital rings. Recall that a ring is calledleft hereditary if any submodule of a projective left module is projective. It is calledright hereditary if any submodule of a projective right module is projective.
Proposition 6.5.1.
The following are equivalent for a finite category C :(a) Q C is right hereditary.(b) Every rational C -homology theory possesses a Chern character. The same statement holds for left hereditarity and cohomology.
Proof.
By Lemma 6.3.5, assertion (b) is equivalent to the fact that every chaincomplex of non-degenerate right R -modules is isomorphic to a trivial complex inthe derived category.Over every ring, any (right) chain complex is quasi-isomorphic to a degreewiseprojective one, and it is well-known [Kra07, Sec. 1.6] that these split over righthereditary rings.Conversely, assume that Q C is not right hereditary. One can easily see that thismeans that Ext Q C doesn’t vanish, i. e. there are right Q C -modules M and N such that Ext Q C ( N, M ) = 0. A straightforward triangulated category argument,explained for instance in [Ill02, Sec. 4.5, 4.6], shows how this can be used to con-struct a chain complex L with only nontrivial homology groups H ( L ) ∼ = M and H ( L ) ∼ = N which is not isomorphic to the trivial complex M [0] ⊕ N [ −
1] in thederived category. (cid:3)
For finite EI categories, Liping Li [Li11] has found out when the category algebrais hereditary. Let us first introduce some notation. We call a category C finiteif it has finitely many objects and morphisms. Assume for simplicity that C isconnected. A morphism f is called unfactorisable if it is not an isomorphism,and whenever f = gh , then g or h is an isomorphism. Every morphism can befactored as a composition of unfactorisable morphisms. We now define the uniquefactorisation property which asserts that this factorisation is essentially unique forevery morphism. The definition is [Li11, Def. 2.7], slightly changed since we do notassume that C is skeletal: Definition 6.5.2.
The category C satisfies the unique factorisation property (UFP) if for any two chains x = x α −→ x α −→ . . . α n −−→ x n = y and x = x ′ α ′ −→ x ′ α ′ −→ . . . α ′ n ′ −−→ x ′ n ′ = y of unfactorisable morphisms α i and α ′ i which have the same composition f : x → y ,we have n = n ′ and there are isomorphisms h i : x i → x ′ i for 1 ≤ i ≤ n − h α = α ′ , α ′ n h n − = α n and α ′ i h i − = h i α i for 2 ≤ i ≤ n − , i. e., the following ladder diagram commutes: x x x . . . x n − yx x ′ x ′ . . . x ′ n − y . α id x α h h α α n − α n h n − id y α ′ α ′ α ′ α ′ n − α ′ n Proposition 6.5.3 [Li11, Thm. 5.3, Prop. 2.8]. If C is a finite EI category, then Q C is left hereditary if and only if C satisfies the UFP. Moreover, being left hereditaryand right hereditary is equivalent for Q C .Proof. Note that Li calls hereditary what we call left hereditary. With this in mind,the first statement follows directly from his Proposition 2.8 and Theorem 5.3. Thesecond statement follows from this since ( Q C ) op ∼ = Q ( C op ), and C op satisfies theUFP if and only if C does. (cid:3) Corollary 6.5.4. If C is a finite EI category, then C satisfies the UFP if and only if every rational C -homology theory possesses a Chern character, if and only if every C -cohomology theory does. We finally analyse the case of orbit categories, heading to Theorem D from theIntroduction. Let G be a group and F a family of subgroups of G . Proposition 6.5.5.
Let G be a group and F a family of finite subgroups. Thecategory Or( G, F ) satisfies the UFP if and only if F consists only of cyclic subgroupsof prime power order (where different prime bases may occur in the same family). In particular, if F is the family of all subgroups, then this is the case if and onlyif G is of the form Z /p k for some k . Note the formal similarity of this result toTriantafillou’s results in [Tri83]. Proof. The ’only if ’ part.
Suppose that Or( G, F ) has the UFP. Let F ∈ F . Let H and K be two subgroups of F . Let1 ⊆ H ⊆ H ⊆ . . . H i = H ⊆ H i +1 ⊆ . . . ⊆ H n = F be a chain of subgroups such that H l ⊆ H l +1 is a maximal subgroup, and similarly1 ⊆ K ⊆ K ⊆ . . . K j = K ⊆ K j +1 ⊆ . . . ⊆ K m = F .
Recall the bijection φ from Lemma 6.4.1. We can factor the morphism φ (1) as aproduct of unfactorisables in two ways: Firstly, as G/ φ (1) −−−→ G/H φ (1) −−−→ G/H φ (1) −−−→ . . . φ (1) −−−→ G/H n = G/F
OMOLOGY REPRESENTATION THEOREMS FOR DIAGRAM SPACES 41 and secondly, as G/ φ (1) −−−→ G/K φ (1) −−−→ G/K φ (1) −−−→ . . . φ (1) −−−→ G/K m = G/F .
It follows from the UFP that m = n and that for all l , G/H l and G/K l areisomorphic in Or( G, F ), i. e., H l and K l are conjugate in G . Since H and K werearbitrary, it follows that for any two subgroups of F , one is subconjugate to theother in G . If H and K have the same order, they are thus conjugate in G .This implies that F has to be a p -group for some p . Indeed, suppose that twodifferent primes p and q divide | F | . Then we can choose H of order p and K oforder q . It follows that H = H and K = K , so H and K are conjugate which isabsurd since they have different orders.Next, we prove that F has only normal subgroups. Indeed, let L be minimal non-normal. Then L is different from 1. Let K be a maximal proper subgroup of L which is thus normal in F . Let L = L ⊆ L ⊆ . . . ⊆ L n = F be a chain of subgroups such that L i ⊆ L i +1 is maximal. For f ∈ F ⊆ Trans G ( K, L ),the morphism φ (1) : G/K → G/F has the two factorisations
G/K φ (1) −−−→ G/L φ (1) −−−→ G/L φ (1) −−−→ . . . φ (1) −−−→ G/F and
G/K φ ( f ) −−−→ G/L φ (1) −−−→ G/L φ (1) −−−→ . . . φ (1) −−−→ G/F .
By the UFP, there is g ∈ N G ( L ) such that φ ( g ) = φ ( f ) : G/K → G/L , i. e. g = f modulo L . Thus, f ∈ LN G ( L ) = N G ( L ) and L is normal in F .Finally, we show that F has only one maximal subgroup. For this, consider anymaximal subgroup H of F , and extend it to a chain1 ⊆ H ⊆ H ⊆ . . . H n = H ⊆ F where H i is a maximal subgroup of H i +1 . Let g ∈ Trans G ( H, F ) be arbitrary.Consider the following two factorisations of φ ( g ) : 1 → F : G/ φ (1) −−−→ G/H φ (1) −−−→ G/H φ (1) −−−→ . . . G/H n = G/H φ ( g ) −−−→ G/F and G/ φ ( g ) −−−→ G/H φ (1) −−−→ G/H φ (1) −−−→ . . . G/H n = G/H φ (1) −−−→ G/F .
By the UFP, there is h ∈ N G ( H ) such that φ ( h ) = φ ( g ), i. e. h = g in F \ Trans G ( H, F ).Since H is normal in F , we have F ⊆ N G ( H ) and it follows that N G ( H ) = Trans G ( H, F ) . Now, suppose that H ′ is another maximal subgroup of F . Then H and H ′ areconjugate via some g ∈ G . It follows that g ∈ Trans G ( H, F ) = N G ( H ), so H = H ′ .Thus, F has only one maximal subgroup.We claim that this forces F to be cyclic, and show this claim by induction over theorder of F . Since F is a p -group, it has a non-trivial center C . F/C has only onemaximal subgroup as well, and it follows that
F/C is cyclic. It is an easy exerciseto show that if the quotient of the group by its center is cyclic, the group has tobe abelian. Thus, F is abelian. From the classification of finite abelian groups, F is cyclic. The ’if ’ part.
Now, suppose that F only has cyclic members of prime power order.Given a chain G/H φ ( g ) −−−→ G/H φ ( g ) −−−→ G/H . . . φ ( g n ) −−−→ G/H n of unfactorisable morphisms, we first manipulate it as follows using the equivalencerelation explained in Definition 6.5.2: Substitute H ′ = g − H g , g ′ = 1 and g ′ = g g , i. e. we consider the factorisation G/H φ (1) −−−→ G/H ′ φ ( g g ) −−−−−→ G/H φ ( g ) −−−→ G/H . . . φ ( g n ) −−−→ G/H n with the same composition as before. Repeating this step at positions 2 through n −
1, we arrive at a chain
G/H φ (1) −−−→ G/H ′ φ (1) −−−→ G/H ′ φ (1) −−−→ G/H ′ . . . G/H ′ n − φ ( g ′ ) −−−→ G/H n with composition g ′ modulo H n . Since our replacement algorithm followed thedefinition of UFP, we only need to compare morphisms in such a normal form.Note that since H ′ n − is cyclic of order a power of p , the index [ H ′ i : H ′ i − ] is always p since the morphisms of the chain are unfactorisable. This is true for any otherchain from G/H to G/H n and consequently, the length of such a chain is always n . Let G/H φ (1) −−−→ G/H ′′ φ (1) −−−→ G/H ′′ φ (1) −−−→ G/H ′′ . . . G/H ′′ n − φ ( g ′′ ) −−−→ G/H n be another chain with the same composition, i. e. g ′′ = f g ′ with f ∈ H n . Thisimplies that ( g ′ ) − H n g ′ = ( g ′′ ) − H n g ′′ . Thus, H ′ n − and H ′′ n − are both maximal subgroups of ( g ′ ) − H n g ′ , and since thisis a cyclic group, they coincide: H ′ n − = H ′′ n − . Since g ′′ = f g , we get that φ ( g ′ ) = φ ( g ′′ ). It follows that H ′ i = H ′′ i for all i ≤ n − (cid:3) References [B´ar14] N. B´arcenas,
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