Suspension spectra of matrix algebras, the rank filtration, and rational noncommutative CW-spectra
aa r X i v : . [ m a t h . A T ] J a n Suspension spectra of matrix algebras, the rankfiltration, and rational noncommutativeCW-spectra
Gregory Arone ∗ Stockholm [email protected] Ilan Barnea † Haifa [email protected] M. Schlank ‡ Hebrew [email protected] 27, 2021
Abstract
In a companion paper [ABS1] we introduced the stable ∞ -categoryof noncommutative CW-spectra, which we denoted NSp . Let M denotethe full spectrally enriched subcategory of NSp whose objects are the non-commutative suspension spectra of matrix algebras. In [ABS1] we provedthat
NSp is equivalent to the ∞ -category of spectral presheaves on M .In this paper we investigate the structure of M , and derive some conse-quences regarding the structure of NSp .To begin with, we introduce a rank filtration of M . We show thatthe mapping spectra of M map naturally to the connective K -theoryspectrum ku , and that the rank filtration of M is a lift of the classicalrank filtration of ku . We describe the subquotients of the rank filtrationin terms of complexes of direct-sum decompositions which also arose inthe study of K -theory and of Weiss’s orthogonal calculus. We prove thatthe rank filtration stabilizes rationally after the first stage. Using thiswe give an explicit model of the rationalization of NSp as presheaves ofrational spectra on the category of finite-dimensional Hilbert spaces andunitary transformations up to scaling. Our results also have consequencesfor the p -localization and the chromatic localization of M . Contents ∗ Supported in part by the Swedish Research Council, grant number 2016-05440 † Supported by ISF 786/19 ‡ Supported by ISF 1588/18 and BSF 2018389 The functors G k,l and their stabilization 93 The rank filtration 114 The restriction of G k,l to finite sets 145 G k,l is a cofibrant Γ -space 186 Connection with topological K -theory 257 Subquotients of the rank filtration 308 Connection with the complex of direct-sum decompositions 359 Some calculations of S k,l p -localization of M In our previous paper [ABS1] we introduced the ∞ -category of noncommutativeCW-spectra, which we denoted NSp . This is the stabilization of the ∞ -categoryof noncommutative CW-complexes, denoted NCW , which in turn is the ind com-pletion of the ∞ -category of finite noncommutative CW-complexes, denoted NCW f . The latter is defined as opposite of the topological nerve of the topo-logical category whose objects are the C ∗ -algebras which are noncommutativeCW-complexes in the sense of [ELP] and whose hom-spaces are given by takingthe topology of pointwise norm convergence on the sets of ∗ -homomorphisms.The main result of [ABS1] says that NSp is equivalent to the ∞ -categoryof spectral presheaves over a small spectrally enriched ∞ -category M . Thespectral ∞ -category M is defined to be the full spectral subcategory of NSp ,whose objects are noncommutative suspension spectra of matrix algebras. Inthis paper we analyze the category M in considerable detail. We introduce arank filtration of M , describe the subquotients of the rank filtration and usethis to give an explicit model for the rationalization of NSp . Remark . In [ABS1] we made extensive use of Hinich’s theory of enriched ∞ -categories (see [Hin2, Hin3]). In this paper we also use this theory andterminology on a few occasions. The interested reader is referred also to [ABS1,Section 3] for a summery of the parts of this theory relevant to us.Given an integer k ≥
1, let M k be the C ∗ -algebra of k × k matrices over C .We have the stabilization functorΣ ∞ NC : NCW → NSp and the set of objects of M is { Σ ∞ NC M k | k ≥ } . Thus the objects of M are inone to one correspondence with the positive integers. Given two integers k, l ,2e denote the spectral mapping object in M by S k,l := Hom M (Σ ∞ NC M k , Σ ∞ NC M l ) ≃ Hom
NSp (Σ ∞ NC M k , Σ ∞ NC M l ) ∈ Sp , where Sp is the ∞ -category of (ordinary) spectra.Our goal in this paper is to analyze the category M . For this, it will beconvenient to use a model category that models the ∞ -category of spectra Sp .Analyzing M means, firstly, that we want to describe, for each k and l , thehomotopy type of S k,l . For this purpose it is adequate to use the simple modelfor spectra, as a sequence of pointed spaces with structure maps. But we alsowant to model the composition maps, S k,l ∧ S j,k → S j,l . (1)To do this properly, we need to use a more sophisticated model for spectra,which incorporates a smash product.To be more explicit, in [ABS1] we defined a “strict” model of M . That is, wedefined a category M s strictly enriched in a certain monoidal model categoryof spectra Sp M , such that the enriched ∞ -localization of M s is equivalent to M .The model category Sp M is the category of pointed continuous functors frompointed finite CW-complexes to pointed topological spaces, endowed with thestable model structure. (By a topological space in this paper we always mean acompactly generated weak Hausdorff space.) Day convolution turns Sp M into asymmetric monoidal model category. See [Lyd2, MMSS, Lyd1] for more detailson this model structure.Recall that any relative category, that is a pair ( C , W ) consisting of a cate-gory C an a subcategory W ⊆ C , has a canonically associated ∞ -category C ∞ ,obtained by formally inverting the morphisms in W in the infinity categoricalsense. There is also a canonical localization functor C → C ∞ satisfying a uni-versal property. We refer the reader to [Hin1] for a thorough account, and alsoto the discussion in [BHH, Section 2.2]. We refer to C ∞ as the ∞ -localizationof C (with respect to W ). If C is a model category or a (co)fibration category,we always take W to be the set of weak equivalences in C .We have a canonical equivalence of symmetric monoidal ∞ -categories Sp M ∞ ≃ Sp . We will identify the two ∞ -categories above through this equivalence. Similarly,if Top is the category of pointed topological spaces endowed with the Quillenmodel structure [Qui], then we have a canonical equivalenceTop ∞ ≃ S ∗ where S ∗ is the ∞ -category of pointed spaces. Again, we identify the two ∞ -categories above through this equivalence.We denote the localization functor from Sp M to Sp M ∞ = Sp by ∂ ∂ : Sp M → Sp . G ∈ Sp M is a pointed continuous functors from pointed finite CW-complexesto pointed topological spaces, then ∂ G is the spectrum corresponding to thesequence of spaces { G ( S ) , G ( S ) , . . . } (where we identify a pointed topologicalspace with its image in Top ∞ = S ∗ ). In other words we can write ∂ G ≃ hocolim n Σ − n Σ ∞ G ( S n ) , where by hocolim here we mean ∞ -colimit in Sp . This is known as the stabi-lization, or the first derivative of the functor G .The way we define M s in [ABS1] is by letting for every k, l Hom M s ( M k , M l ) := G k,l ∈ Sp M , where G k,l ( X ) := SC ∗ (C ( X, M l ) , M k )(see also Definition 2.2). Here C ( X, M l ) is the space of pointed map from X to M l , considered as a C ∗ -algebra, and SC ∗ ( − , − ) denotes the space of C ∗ -algebramaps with the topology of pointwise norm convergence. Since M s is a modelfor M we have that G k,l is a model for S k,l , or in other words, the stabilizationof G k,l is S k,l : S k,l ≃ ∂ G k,l ≃ hocolim n Σ − n Σ ∞ G k,l ( S n ) . Remark . In this paper we use both M and M s . For convenience of notationwe denote both categories by M trusting that it is clear from the context whichis meant.We proceed to investigate the homotopy type of S k,l . It is not hard to showthat if l > k then S k,l is contractible (see corollary 5.8), so we generally assumethat k ≥ l . To analyze S k,l further, we introduce a natural filtration of M inSection 3, which we call the rank filtration . More precisely, for each k and l , wedefine a sequence of subfunctors G k,l,i ∈ Sp M G k,l, ⊂ G k,l, ⊂ · · · ⊂ G k,l, ⌊ kl ⌋ = G k,l, ⌊ kl ⌋ +1 = · · · = G k,l . Upon stabilization, we obtain a sequence of spectra S k,l, ֒ → S k,l, ֒ → · · · ֒ → S k,l, ⌊ kl ⌋ = S k,l . We think of this sequence as defining a filtration of S k,l . This filtration ismultiplicatively compatible with composition. This means that there are maps,compatible with (1) S k,l,m ∧ S j,k,n → S j,l,mn . One can say that the category M is filtered by the multiplicative monoid ofpositive integers.The functors G k,l,m are, by definition, functors from the category of pointedfinite CW-complexes to pointed topological spaces. But it turns out that they4re determined by their restriction to the category of pointed finite sets. Specif-ically, we show in Theorem 5.7 that G k,l,m is equivalent to both the strict andthe derived left Kan extension of its restriction to pointed finite sets. In otherwords, G k,l,m are Γ-spaces.In Section 4 we describe explicitly the restriction of G k,l to pointed finitesets (Proposition 4.5). In Section 5 we show that the Γ-spaces G k,l,m are infact cofibrant in a generalized Reedy model structure considered by Bousfield-Friedlander [BF], Lydakis [Lyd1] and Berger-Moerdijk [BM].Proposition 4.5 indicates that G k,l is similar to the Γ-space that models theK-theory spectrum ku . This is not a coincidence. The inclusion of matrix alge-bras M k → M k +1 that sends a matrix a to a ⊕ G k,l → G k +1 ,l , which in turn induces a map of spectra S k,l → S k +1 ,l . We provein Section 6 that for each fixed l , hocolim k →∞ S k,l ≃ ku (thus all the mappingspectra S k,l are naturally spectra over ku ). The case l = 1 of this observationgoes back to Segal [Se2] and was exploited extensively by Dadarlat with variouscollaborators (for example see [DM]). Furthermore, it turns out that the functor l hocolim k →∞ S k,l takes values in module spectra over ku . This allows for anatural way do define bivariant connective K -theory on NSp , which generalizesthe definition of Dadarlat and McClure [DM] to the noncommutative setting.We also observe in this section that the rank filtration of M is a lift of the clas-sical rank filtration of ku . Later in the paper we prove that the map S k,l → ku induces an isomorphism on π (Lemma 9.4).The results of Sections 4 and 5 are useful for the homotopic analysis of S k,l and of the rank filtration. One of our main results is an explicit description of theassociated graded filtration. Thus we describe, for each k, l, m , the homotopycofiber spectrum S k,lm := S k,l,m / S k,l,m − , as well as the induced maps S k,lm ∧ S j,kn → S j,lmn . Our description features certain U ( m )-complexes L ⋄ m . Roughly speaking L ⋄ m is the space of direct-sum decompositions of C m (see Definition 8.6). Thesecomplexes have some remarkable homotopical properties, which we will reviewbelow. But first let us state the result describing the rank filtration in termsof L ⋄ m . When U and V are unitary vector spaces, let Inj( U, V ) denote thespace of (necessarily injective) linear transformations of U into V that preservethe unitary product. Note for future reference that there is a homeomorphismInj( C m , C n ) ∼ = U ( n ) /U ( n − m ). The following is Theorem 8.8 in the text: Theorem 1.3.
There is an equivalence of spectra S k,lm ≃ Σ ∞ L ⋄ m ∧ U ( m ) Inj( C lm , C k ) + , where U ( m ) acts through the identification C lm ∼ = C m ⊗ C l . The compositionmap S k,ln ∧ S j,km → S j,lmn is determined by the map L ⋄ n ∧ L ⋄ m → L ⋄ mn defined bytensor product of decompositions, and the obvious composition map Inj( C ln , C k ) × Inj( C km , C j ) → Inj( C lmn , C km ) × Inj( C km , C j ) → Inj( C lmn , C j ) . L ⋄ m were first introduced in [Ar1], and were studied in detailin [BJL+] and [AL3]. They play a role in describing the subquotients of therank filtration of K -theory [AL1, AL2]. Therefore it is perhaps not surprisingthat they play a similar role in the rank filtration of M , given the connectionbetween M and ku .Next proposition lists some relevant facts about the complexes L ⋄ m (Propo-sition 9.1 in the paper). Proposition 1.4. L ⋄ = S .2. The complex L ⋄ m is rationally contractible for all m > .3. The complex L ⋄ m is contractible unless m is a prime power.4. If m = p k where p is a prime and k > then L ⋄ p k is p -local.5. The complex L ⋄ p k has chromatic type k . Here are some consequences of Theorem 1.3 and Proposition 1.4. To beginwith, we have a simple description of the endomorphisms in M . The endomor-phism spectrum of Σ ∞ NC M k is the group ring spectrum of the projective unitarygroup P U ( k ). Corollary 1.5. S k,k ≃ Σ ∞ P U ( k ) + . But our main application of Theorem 1.3 and Proposition 1.4(2) is to give asimplified description of the rational homotopy type of S k,l , and consequently ofthe rationalization of NSp . Recall that the composition maps S j,k ∧ S k,l → S j,l restrict to maps of the form S j,k, ∧ S k,l, → S j,l, . It follows that the spectra S k,l, assemble to a spectral category, that we denote M , which has the sameobjects as M , and equipped with a functor M → M that is the identity onobjects. Informally speaking, M is the first stage of the rank filtration of M .It follows from Theorem 1.3 that there is an equivalence S k,l, = S k,l ≃ Σ ∞ Inj( C l , C k ) / U (1)+ . It is worth noting that S k,l, is a suspension spectrum . Let P Inj be the topo-logically enriched symmetric monoidal category of finite positive dimensionalHilbert spaces and embeddings up to scalar. That is, up to isomorphism theobjects of P Inj are given by C k for k ≥ P Inj( C l , C k ) = Inj( C l , C k ) / U (1) = U ( k ) / ( U (1) × U ( k − l )) . P Inj is a symmetric monoidal category, with the monoidal structure given bythe tensor product. Since the functor Σ ∞ + , from topological spaces to our modelof spectra Sp M , is symmetric monoidal, we can define a category enriched in Sp M ,which we denote P Inj Sp , by applying Σ ∞ + to the mapping spaces of P Inj. Weshow in Section 5.1 that M ≃ ( P Inj Sp ) op . The following is an easy consequence of Proposition 1.4(2)6 orollary 1.6.
The natural map Σ ∞ Inj( C l , C k ) / U (1)+ ≃ S k,l, ≃ Q −−→ S k,l is a rational homotopy equivalence.Remark . If one lets k go to ∞ in corollary 1.6, one obtains the classical factthat the canonical map Σ ∞ C P ∞ + → ku is a rational equivalence. So corollary 1.6can be thought of as a lift of this fact.Corollary 1.6 says that the functor M → M is a rational equivalence ofspectral categories. Using this, we can give a rather explicit description ofthe rationalization of NSp . We discuss the general construction of rational lo-calization and p -localization of a stable, monoidal, ∞ -category in Section 10.Let NSp Q denote the rational localization of the ∞ -category of noncommutativespectra NSp and let Sp Q denote the rational localization of the usual ∞ -categoryof spectra Sp . It is well known that Sp Q is a symmetric monoidal presentable ∞ -category, and the rationalization functor L Q : Sp → Sp Q is symmetric monoidal.Let P Inj ∞ denote the topological nerve of the topological category P Injdefined above. Applying the symmetric monoidal functor L Q ◦ Σ ∞ + : S → Sp Q to the mapping spaces of P Inj ∞ we obtain an ∞ -category enriched in Sp Q which we denote by P Inj Sp Q ∞ . Let P Sp Q (( P Inj Sp Q ∞ ) op ) denote the ∞ -category of Sp Q -enriched functors from P Inj Sp Q ∞ to Sp Q . The following theorem summarizesthe results of Section 10 about Sp Q : Theorem 1.8 (Theorem 10.5) . There are equivalences of symmetric monoidal ∞ -categories NSp Q ≃ P Sp Q (( P Inj Sp Q ∞ ) op ) ≃ Fun( P Inj ∞ , Sp Q ) . Remark . Note that the expression on the right in Theorem 1.8 does not useenriched ∞ -categories. This is the usual ∞ -category of functors from P Inj ∞ to Sp Q . Note also that the mapping spaces in P Inj are all finite connected CW-complexes (manifolds, even). It is natural to wonder if one can given a moredirect algebraic model of
NSp Q as a dg-category. p -local and chromatic picture Now instead of rationalizing, suppose we fix a prime p and localize everything at p . One can obtain further information about the p -localization of M . It followsfrom Proposition 1.4 parts (3) and (4) that the filtration is p -locally constant7xcept at powers of p . Therefore it is natural to regrade the filtration of S k,l asfollows S k,l, ֒ → S k,l,p ֒ → S k,l,p ֒ → · · · ֒ → S k,l,p i · · · With this grading, (the p -localization) of M is a filtered category in the usualsense, that composition adds degrees. Furthermore, we have Corollary 1.10.
Fix a prime p and localize everything at p . The map S k,l,p n → S k,l induces an isomorphism on Morava K ( i ) -theory for i ≤ n . The last corollary may have consequences for “noncommutative chromatichomotopy theory”, but we will not pursue it here.
Section by section outline of the paper
In Section 2 we set the stage by recalling some relevant definitions from [ABS1].We introduce functors G k,l , where k, l are positive integers. The functors G k,l encode all the information about morphisms in M . More precisely, the stabi-lization of G k,l is the spectral mapping object from k to l in M . In Section 3we introduce a natural filtration of the functors G k,l , which we call the rankfiltration.In Section 4 we give an explicit description of the restriction of G k,l to pointedfinite sets. In Section 5 we establish various properties of the restriction of G k,l to finite sets. Most importantly, we show that the functor G k,l is a Γ-space, inthe sense that it is determined by its values on finite sets, and we also observethat the restriction of G k,l to finite sets is cofibrant in the Reedy model structureon Γ-spaces.In Section 6 we show that all the mapping spectra in M are equipped witha natural map to the connective K -theory spectrum ku . We observe that therank filtration of M is a lift of the classical rank filtration of ku . We alsodiscuss how one can use our models to represent the K -theory functor on non-commutative complexes. We make connection with some work of Dadarlat andMcClure [DM].In Sections 7 and 8 we describe the subquotients of the rank filtration interms of complexes of direct-sum decompositions that arose earlier in the studyof the rank filtration of ku . Since complexes of direct-sum decompositions arewell-studied, we obtain interesting consequences about M and NSp . In Section 9we use the results of preceding sections to calculate the mapping spectra in M in some cases. In Section 10 we use those results to give an explicit modelfor the rationalization of M and NSp . Rationally,
NSp is equivalent to the ∞ -category of presheaves of rational spectra on the ∞ -category whose objects arefinite-dimensional Hilbert spaces, and whose hom-spaces are linear embeddingsmodulo scalars. We also point out some consequences that our models have forthe p -localization and potentially chromatic localization of M and NSp .8 cknowledgements We would like to thank Jeffrey Carlson for fruitful correspondences during theearly stages of our work. We are grateful to Vladimir Hinich for explaining tous his theory of enriched infinity categories and its relevance to our work. G k,l and their stabilization In this section we recall the construction of M as a category strictly enrichedin Sp M (see Remark 1.2). We will introduce certain functors G k,l ∈ Sp M , whichwill represent the mapping spectra in M .To begin with, let SC ∗ denote the category of (non-unital) separable C ∗ -algebras and ∗ -homomorphisms. Note that the matrix algebras { M k | k =1 , , . . . } are objects of SC ∗ . Consider SC ∗ as a topologically enriched category,where for every A, B ∈ SC ∗ we endow the set of ∗ -homomorphisms SC ∗ ( A, B )with the topology of pointwise norm convergence. It is well-known that SC ∗ is cotensored over the category of pointed finite CW-complexes [AG]. For afinite pointed CW-complex X and a C ∗ -algebra A we denote the contensoringby C ( X, A ).Next, we want to use SC ∗ to define M as a category enriched in Sp M . Re-call that the underlying category of Sp M is the category of pointed continuousfunctors from pointed finite CW-complexes to pointed topological spaces. Definition 2.1.
Let G k,l : CW f ∗ → Top be the functor defined as follows G k,l ( X ) = Map SC ∗ (C ( X, M l ) , M k ) . We will consider G k,l to be an object of Sp M . Notice that for all k, l, m , thereis a natural map G k,l ( X ) ∧ G l,m ( Y ) → G k,m ( X ∧ Y ), defined as a compositionof the following maps.Map SC ∗ (C ( X, M l ) , M k ) ∧ Map SC ∗ (C ( Y, M m ) , M l ) →→ Map SC ∗ (C ( X, M l ) , M k ) ∧ Map SC ∗ (C ( X ∧ Y, M m ) , C ( X, M l )) →→ Map SC ∗ (C ( X ∧ Y, M m ) , M k ) . Here the second map is composition, and the first map is induced by the coten-soring Map SC ∗ (C ( Y, M m ) , M l ) → Map SC ∗ (C ( X ∧ Y, M m ) , C ( X, M l )) . This map induces natural maps G k,l ∧ G l,m → G k,m (2)where ∧ denotes internal smash product (aka Day convolution).9 efinition 2.2. Let M be the following Sp M -enriched category. The objects of M are positive integers. Given two integers k, l , the mapping spectrum from k to l is given by G k,l . The composition law in M is defined by the structuremaps like in (2), for all k, l, m . Remark . Recall that in the infinity categorical picture, M is the full spectralsubcategory of NSp whose objects are { Σ ∞ NC M k | k = 1 , . . . } . We show in [ABS1]that the enriched coherent nerve of M from Definition 2.2 is equivalent to M defined above. The objects of M provide a set of compact generators of NSp .The main result of [ABS1] says that the ∞ -category NSp is equivalent to the ∞ -category P Sp ( M ) of spectral presheaves on M . In this paper we investigatethe category M , with the eventual goal in mind of understanding NSp . In viewof the results of [ABS1] in this paper we identify
NSp with P Sp ( M ). In theremaining part of the paper we will study the category M mainly using theexplicit model provided in Definition 2.2, but will state the results also in theinfinity categorical picture.Recall from the introduction that we identify Sp M ∞ = Sp and we denote by ∂ : Sp M → Sp , the localization functor. If G ∈ Sp M is a pointed continuous func-tors from pointed finite CW-complexes to pointed topological spaces, then ∂ G is the spectrum corresponding to the sequence of spaces { G ( S ) , G ( S ) , . . . } , orin other words ∂ G ≃ hocolim n Σ − n Σ ∞ G ( S n ) . This is known as the stabilization, or the first derivative of the functor G . Wehave shown in [ABS1] that for all natural numbers k, l , we have S k,l := Hom NSp (Σ ∞ NC M k , Σ ∞ NC M l ) ≃ ∂ G k,l ≃ hocolim n Σ − n Σ ∞ G k,l ( S n ) . Example 1.
Let us consider the case k = l = 1 . The functor G , is given asfollows G , ( X ) = Map SC ∗ (C ( X, M ) , M ) = Map SC ∗ (C ( X, C ) , C ) . By the Gelfand-Naimark theorem, it follows that G , ( X ) ∼ = X , and therefore S , = Σ ∞ S is the ordinary sphere spectrum.Remark . Let us interpret S , = Σ ∞ S as the endomorphism spectrumEnd NSp (Σ ∞ NC M ). We have identified NSp with the ∞ -category of spectral presheaveson M . In this picture, the ∞ -category of spectral presheaves on the full Sp -enriched subcategory of M consisting of the object Σ ∞ NC M can be identified withthe ∞ -category Sp of “commutative” or “ordinary” spectra. There is an “inclu-sion” functor of Sp -tensored categories Sp → NSp which in terms of presheaves isdefined by an Sp -enriched left Kan extension (weighted colimit) from { Σ ∞ NC M } to M . The inclusion functor has a right adjoint NSp → Sp , a kind of “abelian-ization” functor, defined by restriction of presheaves. We will say a little moreabout it in Section 6. 10 The rank filtration
In this section we introduce the rank filtration of G k,l , which induces a rankfiltration of the spectral category M . In later sections we will see that the rankfiltration of M is a lift of the classical rank filtration of the connective K -theoryspectrum ku .Let l, k ≥
1, let X be a finite pointed CW-complex and let f ∈ G k,l ( X ) = SC ∗ (C ( X, M l ) , M k ) , be a map. Let A f ⊆ M k be the image of f . A f acts as non-unital C ∗ -Algebraon the Hilbert space C k and thus we get an orthogonal decomposition C k =Ker A f ⊕ A f · C k . Denote V f := A f · C k ⊆ C k . We shall filter the space G k,l ( X ) according to the dimension of V f . The following theorem is useful inthat analysis. Theorem 3.1.
Let X be a pointed compact metrizable space and let l ≥ .There is a bijection between the closed subsets of X \ {∗} and closed two-sidedideals of C ( X, M l ) , defined by the following correspondence F I F := { f ∈ C ( X, M l ) | ∀ x ∈ F. f ( x ) = 0 } . Proof.
The case l = 1 is well-known. We will show that it implies the rest. Two C ∗ -algebras A and B are called strongly Morita equivalent if they are relatedby a B - A -imprimitivity bimodule in the sense of [Rie]. Let K be the algebraof compact operators on an infinite dimensional separable Hilbert space. It isshown in [BGR] that if A and B are separable, then A and B are stronglyMorita equivalent iff A ⊗ K ∼ = B ⊗ K . It is not hard to see that C ( X, M l ) ∼ = C ( X ) ⊗ M l and M l ⊗ K ∼ = K , so wehaveC ( X, M l ) ⊗ K ∼ = (C ( X ) ⊗ M l ) ⊗ K ∼ = C ( X ) ⊗ ( M l ⊗ K ) ∼ = C ( X ) ⊗ K . Thus, C ( X, M l ) and C ( X ) are strongly Morita equivalent. By [Zet], we havean isomorphism between the sets of closed two-sided ideals of C ( X, M l ) and ofC ( X ). Lemma 3.2.
Let X ∈ CW f ∗ be pointed finite CW-complex and let f ∈ G k,l ( X ) = SC ∗ (C ( X, M l ) , M k ) . Then f admits a unique factorization of the following form C ( X, M l ) ։ C ( F f ∪ {∗} , M l ) f ′ M k (3) where the first map is the surjective restriction to a finite subset F f ⊂ X \{∗} andthe second map f ′ : C ( F f ∪ {∗} , M l ) M k is a monomorphism. In particularwe have V f = V f ′ . roof. First, note that ker( f ) is a closed two sided ∗ -ideal of C ( X, M l ). ByTheorem 3.1, there exists a closed subset F f of X \ {∗} such thatker( f ) = I F f = { g ∈ C ( X, M l ) | g | F f = 0 } . Notice that ker( f ) = I F f is also the kernel of the restriction homomorphismC ( X, M l ) → C ( F ∪ {∗} , M l ) . We claim that the restriction homomorphism is surjective. This amounts toshowing that any map from a closed subset of X to M l can be extended to amap from X to M l . This in turn follows immediately from the Tietze extensiontheorem.It follows that f admits a unique factorization of the following formC ( X, M l ) ։ C ( F f ∪ {∗} , M l ) f ′ M k (4)where the first map is restriction to a subset F f and the second map f ′ : C ( F f ∪{∗} , M l ) → M k is a monomorphism. Moreover F f is the minimal subset of X \ {∗} for which the map f factors through C ( F f ∪ {∗} , M l ). Notice thatIm ( f ′ ) is finite-dimensional as a vector space over C . This implies that F f isfinite. Lemma 3.3. let f ∈ G k,l ( X ) = SC ∗ (C ( X, M l ) , M k ) , we have l | dim V f and l · | F f | ≤ dim V f ≤ k Proof.
By lemma 3.2 the function f can be factored as a surjection followed byan injection C ( X, M l ) → M F f l f ′ → M k Thus we have an isomorphism M F f l ∼ = A f . 1 ∈ M F f l now acts on C k as aprojection onto V f and thus we get a unital action of M F f l on V f . Now for x ∈ F f denote by W x the unital M F f l module obtained by the canonical actionon C l via map M F f l → M { x } l = M l . Every finite dimensional unital M F f l module is a direct sum of finitely many copies of the W x ’s. We thus get that V f = M x ∈ F f = W e x x . The injectivity of the map M F f l f ′ → M k implies that e x ≥ x ∈ F f .Since dim V f = l P e x and V f ⊆ C k we get the claim.12 efinition 3.4. Let l, k ≥
1, let X be a finite pointed CW-complex and let f ∈ G k,l ( X ) = SC ∗ (C ( X, M l ) , M k ) . We define the rank of f to be the non-negative integerrank( f ) := dim V f l ∈ Z ≥ . Suppose we have a map α : X → Y in CW f ∗ . By functoriality, it inducesa map G k,l ( X ) → G k,l ( Y ). Suppose f ∈ G k,l ( X ). By definition, f is a ∗ -homomorphism f : C ( X, M l ) → M k , and the image of f in G k,l ( Y ) is thecomposite homomorphismC ( Y, M l ) α ∗ −−→ C ( X, M l ) f −→ M k . Therefore the rank of the image of f in G k,l ( Y ) is at most the rank of f . Becauseof this, the following definition really does describe a functor. Definition 3.5.
Let k, l ≥ m ≥
0. Define the functors G k,l,m : CW f ∗ → Topas follows G k,l,m ( X ) = { f ∈ G k,l ( X ) | rank( f ) ≤ m } ⊆ G k,l ( X ) . Similarly, define S k,l,m to be the stabilization of G k,l,m . Explicitly, S k,l,m is thespectrum { G k,l,m ( S ) , G k,l,m ( S ) , . . . } . Remark . Note that for all X ∈ CW f ∗ and k, l ≥ G k,l, ( X ) = ∗ .Additionally by lemma 3.3 for m ≥ ⌊ kl ⌋ we get G k,l,m = G k,l .We have defined a filtration of G k,l by sequence of subfunctors ∗ = G k,l, ⊂ G k,l, ⊂ · · · ⊂ G k,l, ⌊ kl ⌋ = G k,l . We call this filtration the rank filtration . Now recall that the functors G k,l represent mapping spectra in M and that composition in M is determined bymaps of the following form G k,l ( X ) ∧ G l,m ( Y ) → G k,m ( X ∧ Y ) . The following proposition tells how the rank filtration interacts with composi-tion.
Proposition 3.7.
For all r and s , the composition map above restricts to anatural map G k,l,r ( X ) ∧ G l,m,s ( Y ) → G k,m,rs ( X ∧ Y ) Proof.
Recall that G k,l ( X ) = SC ∗ (C ( X, M l ) , M k ). Written in these terms, thecomposition map has the following form SC ∗ (C ( X, M l ) , M k ) ∧ SC ∗ (C ( Y, M m ) , M l ) →→ SC ∗ (C ( X, M l ) , M k ) ∧ SC ∗ (C ( X ∧ Y, M m ) , C ( X, M l )) →→ SC ∗ (C ( X ∧ Y, M m ) , M k ) . f ∈ SC ∗ (C ( X, M l ) , M k ) and g ∈ SC ∗ (C ( Y, M m ) , M l ) have ranks r and s respectively. Let f ⊙ g denote the image of f ∧ g in SC ∗ (C ( X ∧ Y, M m ) , M k ).Our goal is to show that f ⊙ g has rank rs .By Lemma 3.2, there exist finite subsets F f ⊂ X r {∗} , F g ⊂ Y r {∗} suchthat f factors as C ( X, M l ) ։ C ( F f ∪ {∗} , M l ) f ′ M k , and there is a similarfactorization of g . It follows that f ⊙ g factors as followsC ( X ∧ Y, M m ) ։ C ( F f × F g ∪ {∗} , M m ) f ′ ⊙ g ′ M k where the second map is itself the following composite( M F g m ) F f g ′× Ff −−−−→ M F f l f ′ −→ M k . Here the first map is the cartesian product of | F f | copies of the g ′ with itself.This map determines an action of M F g × F f m on C k . Our goal is to show that M F g × F f m · C k has dimension rsm .Recall that A f = A f ′ is the image of f ′ . Since rank( f ) = r , A f · C k hasdimension rl . If B ⊂ M l is a C ∗ -subalgebra such that B · C l has dimension d , and we let B F f act on C k via the map f ′ , then B F f · C k has dimension rd .Now take B to be the image of g ′ . Since rank( g ) = s , B · C l has dimension sm ,so finally we conclude that M F g × F f m · C k has dimension rsm . This means that f ⊙ g has rank rs . Remark . Proposition 3.7 can be intepreted as follows: the rank filtration isa filtration of the category M by the multiplicative monoid of natural numbers. G k,l to finite sets It will turn out that the functor G k,l , whose domain is the category of pointedfinite CW-complexes, is determined by its restriction to the category of pointedfinite sets. In this section we give an explicit description of the restriction of G k,l to finite sets.Let us begin with a definition, which also serves to establish some notation. Definition 4.1.
For a natural number i , let [ i ] = { , , . . . , i } , considered asa pointed set with basepoint 0. Let Fin ∗ be the category whose objects are { [0] , [1] , . . . , [ k ] , . . . } and whose morphisms are basepoint-preserving functions.For k ≥
0, let
Fin ≤ k ∗ denote the full subcategory of Fin ∗ spanned by the objects { [0] , [1] , . . . , [ k ] } . We will also use the notation i for the unpointed set { , . . . , i } . Remark . The category
Fin ∗ is denoted Γ in some sources, and Γ op in someother sources. We find the notation Fin ∗ to be more descriptive. But followingthe tradition established by Segal [Se1], we call pointed functors Fin ∗ → TopΓ -spaces . 14e will now examine the restriction of G k,l to Fin ∗ . For a finite pointedset [ t ], G k,l ([ t ]) is the space of non-unital C ∗ -algebra homomorphisms from M tl to M k . Spaces of such homomorphisms are well-understood. We want todescribe them in a way that makes the functoriality in [ t ] explicit. We need afew definitions. Definition 4.3.
The category of pointed multisets is defined as follows. Theobjects are ordered t -tuples ( m , . . . , m t ) of natural numbers. The possibility t = 0 is included, in which case the tuple is empty. A morphism ( m , . . . , m t ) → ( n , . . . , n s ) consists of a pointed function α : [ t ] → [ s ] such that n j = Σ i ∈ α − ( j ) m i for all 1 ≤ j ≤ s . In particular, if j is not in the image of α then n j = 0. Notethat there are no restrictions on m i for i ∈ α − (0).Given a pointed multiset ( m , . . . , m t ) and a pointed function of sets α : [ t ] → [ s ] we define α ∗ ( m , . . . , m t ) to be the multiset ( n , . . . , n s ) with n j = Σ i ∈ α − ( j ) m i for all 1 ≤ j ≤ s . Example 2.
Let α : [3] → [2] be the function defined by α (0) = α (1) = 0 , α (2) = α (3) = 1 . Then α ∗ (4 , ,
3) = (5 , . Suppose we have a multiset ( m , . . . , m t ) and a natural number l . We willmake much use of the unitary vector space C ( m + ··· + m t ) l . We identify this vectorspace with C m + ··· + m t ⊗ C l ∼ = C m ⊗ C l ⊕ · · · ⊕ C m t ⊗ C l . Notice that there are commuting actions of U ( m ) × · · · × U ( m t ) and U ( l ) on C ( m + ··· + m t ) l . It follows that these groups act on any space obtained by applyinga continuous functor to this vector space.Now suppose we have a morphism of pointed multisets α : ( m , . . . , m t ) → ( n , . . . , n s ), so ( n , . . . , n s ) = α ∗ ( m , . . . , m t ). Choose unitary isomorphisms C n j ∼ = → C Σ i ∈ α − j ) m i for all 1 ≤ j ≤ s . The function α together with theseisomorphisms determine an inner-product-preserving inclusion C ( n + ··· + n s ) l → C ( m + ··· + m t ) l . This inclusion in turn defines a map of spaces (where k is anothernatural number)Inj( C ( m + ··· + m t ) l , C k ) → Inj( C ( n + ··· + n s ) l , C k )A different choice of isomorphisms C n j ∼ = → C Σ i ∈ α − j ) m i will change the map byprecomposition with a unitary automorphism of C ( n + ··· + n s ) l that is induced byautomorphisms of C n , . . . , C n s . Therefore we get a well-defined (i.e., indepen-dent of choices of isomorphisms) mapInj( C ( m + ··· + m t ) l , C k ) → Inj( C ( n + ··· + n s ) l , C k ) / Q sj =1 U ( n j ) Moreover, it is easy to see that the map passes to a well-defined map betweenquotientsInj( C ( m + ··· + m t ) l , C k ) / Q ti =1 U ( m i ) → Inj( C ( n + ··· + n s ) l , C k ) / Q sj =1 U ( n j ) (5)The upshot is that we have defined a functor from the category of pointed multi-sets to spaces that sends ( m , . . . , m t ) to Inj( C ( m + ··· + m t ) l , C k ) / Q ti =1 U ( m i ) .15 emark . Here is a slightly different way to think of the map (5). Let m = m + · · · + m t . There are homeomorphismsInj( C ml , C k ) / Q ti =1 U ( m i ) ∼ = U ( k ) / t Y i =1 U ( m i ) × U ( k − ml )and similarlyInj( C nl , C k ) / Q sj =1 U ( n j ) ∼ = U ( k ) / s Y j =1 U ( n j ) × U ( k − nl ) . A morphism of multisets α : ( m , . . . , m t ) → ( n , . . . , n s ) gives a canonical wayto conjugate Q ti =1 U ( m i ) × U ( k − ml ) into a subgroup of Q sj =1 U ( n j ) × U ( k − nl ),and therefore gives rise to a U ( k )-equivariant map U ( k ) / t Y i =1 U ( m i ) × U ( k − ml ) → U ( k ) / s Y j =1 U ( n j ) × U ( k − nl ) . Now we can describe the functor G k,l on finite sets. The following propositionis essentially due to Bratelli [Br]. Proposition 4.5.
There is a homeomorphism G k,l ([ t ]) ∼ = _ ( m ,...,m t ) Inj( C ( m + ··· + m t ) l , C k ) / Q ti =1 U ( m i )+ (6) The wedge sum on the right is indexed on non-zero ordered t -tuples ( m , . . . , m t ) of non-negative integers (the zero tuple corresponds to the basepoint). The func-toriality on the right hand side is defined as follows. A pointed map α : [ t ] → [ s ] ,induces a map _ ( m ,...,m t ) Inj( C ( m + ··· + m t ) l , C k ) / Q ti =1 U ( m i )+ →→ _ ( n ,...,n s ) Inj( C ( n + ··· + n s ) l , C k ) / Q sj =1 U ( n j )+ that sends the wedge summand corresponding to ( m , . . . , m t ) to the wedge sum-mand corresponding to α ∗ ( m , . . . , m t ) by the map (5) , assuming α ∗ ( m , . . . , m t ) is not a tuple of zeros. If α ∗ ( m , . . . , m t ) consists just of zeros, then α sendsthe corresponding wedge summand to the basepoint.Proof. By definiton 2.1, there is a homeomorphism G k,l ([ t ]) ∼ = SC ∗ ( M tl , M k ) . For every multi-set ( m , . . . , m t ), we define a mapInj( C ( m + ··· + m t ) l , C k ) / Q ti =1 U ( m i ) → SC ∗ ( M tl , M k ) (7)16s follows. Suppose we have a unitary isometric inclusion C ( m + ··· + m t ) l ֒ → C k .From this, we get a unitary isomorphism (determined up to an automorphismof C k − ml ) C m ⊗ C l ⊕ · · · ⊕ C m t ⊗ C l ⊕ C k − ml ∼ = → C k (8)Having fixed such an isomorphism, we associate with it a C ∗ -algebra homomor-phism M tl → M k as follows: the i -th factor M l of M tl acts on C m i ⊗ C l byidentity on C m i and by the standard action on C l . Note that the action of M tl on C k − ml is multiplication by zero.Automorphisms of C k − ml commute with the action of M tl on C m ⊗ C l ⊕· · ·⊕ C m t ⊗ C l ⊕ C k − ml . It follows that changing isomorphism 8 by an automorphismof C k − ml does not change the resulting algebra homomorphism from M tl to M k .It follows in turn that we have a well-defined mapInj( C ( m + ··· + m t ) l , C k ) → SC ∗ ( M tl , M k ) . It follows from elementary representation theory (Schur Lemma) that two ele-ments of Inj( C ( m + ··· + m t ) l , C k ) induce the same algebra homomorphism if andonly if they differ by an action of U ( m ) × · · · × U ( m t ). Therefore we get awell-defined injective map in (7).Taking union over multi-sets of the form ( m , . . . , m t ) with fixed t , we obtaina map _ ( m ,...,m t ) Inj( C ( m + ··· + m t ) l , C k ) / Q ti =1 U ( m i )+ ∼ = → SC ∗ ( M tl , M k ) (9)which we claim is a homeomorphism. Indeed, we already know that it is injec-tive. Next we need to show that the map (9) is surjective. Let f : M tl → M k be a C ∗ -algebra homomorphism. Let I , . . . , I t be the identity elements of the t factors M l of M tl . Then f ( I ) , . . . , f ( I t ) are pairwise commuting hermitianidempotents in M k . It follows that for i = 1 , . . . , t , f ( I i ) is hermitian projec-tion onto U i , where U , . . . , U t are pairwise orthogonal subspaces of C k . Nowsuppose that 1 ≤ i ≤ t and A i is an element of the i th factor M l of M tl . Then f ( A i ) = f ( A i ) f ( I i ) = f ( I i ) f ( A i ). Thus f ( A i ) commutes with the hermitianidempotent f ( I i ). It follows that f ( A i ) leaves invariant U i and the orthogonalcomplement of U i . Moreover, since f ( A i ) = f ( A i ) f ( I i ) it follows that f ( A i )is the composition of projection onto U i and a linear transformation of U i . Itfollows that the restriction of f to the i th factor of M tl defines a unital repre-sentation of the algebra M l on U i . Since M l is Morita equivalent to C , U i isisomorphic to a sum of copies of the standard representation of M l . This meansthat we can write U i ∼ = C m i ⊗ C l , where m , . . . , m t are some non-negative in-tegers. With this identification the i -th M l acts on U i via standard action, andit follows that f is in the image of the map (9)We have shown that the map (9) is a bijection. To show that it is a home-omorphism, observe that U ( k ) acts continuously on the source and the target.Moreover, both the source and the target are topologized as the disjoint unionof U ( k )-orbits. This is true by definition for the source. To see this for the tar-get, notice that the map that associates to an algebra morphism f : M tl → M k m , . . . , m t ) is continuous and therefore locally constant, and U ( k )acts transitively on the preimage of any t -tuple of integers. Thus the map (9)is a U ( k )-equivariant bijection between disjoint unions of orbits of a continuousaction U ( k ). It follows that it is a homeomorphism.The statement about functoriality follows by straightforward diagram-chasing.In the previous section we defined the rank filtration of G k,l . Unwindingthe definitions, we find that if f ∈ G k,l ([ t ]) belongs to the wedge summandcorresponding to ( m , . . . , m t ) in Proposition 4.5, thenrank( f ) = m + · · · + m t . Thus, on finite sets the rank filtration is given by the following formula: G k,l,m ([ t ]) = _ { ( m ,...,m t ) | m ··· + mt ≤ m } Inj( C ( m + ··· + m t ) l , C k ) / Q ti =1 U ( m i )+ . (10) G k,l is a cofibrant Γ -space In this section we observe that for all k, l, m the restriction of the functor G k,l,m from finite complexes to finite sets is cofibrant, in a certain well-known modelstructure on Γ spaces. This implies that the strict smash product between thesefunctors is equivalent to the derived smash product. We also show that the valueof the functor G k,l,m on pointed finite CW-complexes is equivalent to both thestrict and the derived left Kan extension of the restriction of G k,l,m to thecategory of pointed finite sets. Furthermore the Γ-space G k,l,m is min( ⌊ kl ⌋ , m )-skeletal. This implies that G k,l,m is determined by its restriction to the categoryof sets of cardinality at most min( ⌊ kl ⌋ , m ).For any fixed [ t ], there is a canonical mapcolim U G k,l,m ( U + ) → G k,l,m ([ t ]) (11)where U ranges over the poset of proper subsets of t = { , . . . , t } and U + = U ∪ { } . Recall once again that there is an isomorphism G k,l,m ([ t ]) = _ { ( m ,...,m t ) | m ··· + mt ≤ m } Inj( C ( m + ··· + m t ) l , C k ) / Q ti =1 U ( m i )+ . (12)With this isomorphism in mind, the following lemma is proved by routinemanipulations of colimits Lemma 5.1.
There is an isomorphism colim U ( t G k,l,m ( U + ) ∼ = _ { ( m ,...,m t ) | m i =0 for some ≤ i ≤ t and m + ··· + m t ≤ m } Inj( C ( m + ··· + m t ) , C k ) / Q ti =1 U ( m i )+ he map colim U ( t G k,l ( U + ) → G k,l ([ t ]) corresponds, under this isomorphism, to inclusion of the wedge sum of all sum-mands indexed by tuples ( m , . . . , m t ) where m i = 0 for at least one i . It follows that the map (11) is an inclusion of a union of path components.Furthermore, the action of Σ t on the quotient space of this inclusion is free.This means that (11) is a Σ t -equivariant cofibration, and this in turn meansthat as a functor on Fin ∗ , G k,l,m is cofibrant in the model structure defined forΓ-spaces in [BF, Lyd1] (technically, there references work with Γ-simplicial sets,but an analogous structure exists for Γ-spaces). This model structure is alsodiscussed in [BM] as an example of a generalized Reedy model structure. Wewill refer to this model structure simply as the Reedy model structure.Let us note that the quotient space of (11) is given by the wedge sum _ { ( m ,...,m t ) | m ,...,m t ≥ and m + ··· + m t ≤ m } Inj( C ( m + ··· + m t ) l , C k ) / Q ti =1 U ( m i )+ . Notice that the quotient is trivial for t > m and for t > ⌊ kl ⌋ . In the terminol-ogy of [BF, Lyd1], this means that the Γ-space G k,l,m is min( m, ⌊ kl ⌋ )-skeletal.This implies that G k,l,m is determined on Fin ∗ , via left Kan extension, by itsrestriction to the subcategory of sets of cardinality at most min( m, ⌊ kl ⌋ ).We want to verify that G k,l is equivalent, as a functor on CW f ∗ , to both thestrict and the derived left Kan extension of the restriction of G k,l,m to Fin ∗ , andeven to Fin ≤ min( m, ⌊ kl ⌋ ) ∗ , where Fin ≤ j ∗ ⊂ Fin ∗ is the full subcategory consistingof [0] , . . . , [ j ]. Left Kan extension can be described as a coend. Let us introducesome notation.Let Top denote the category of pointed compactly generated weak Hausdorffspaces with the standard model structure of Quillen [Qui]. Every object in Topis fibrant, and every CW-complex is cofibrant. Suppose C is a small category,and we have a pair of functors F : C op → Top and G : C →
Top. We denote thecoend of F and G by R C s F ∧ G , or R x ∈C s F ( x ) ∧ G ( x ). The subscript s is thereto indicate that this is a strict coend, as opposed to the homotopy coend. Letus recall the definition and construction of the latter. Definition 5.2.
For a functor F , let Q o F denote an objectwise cofibrant re-placement of F and let Q p F denote a cofibrant replacement in a projectivemodel structure. If C is a generalized Reedy category in the sense of [BM], thenlet Q r F denote a cofibrant replacement in the Reedy model structure.The following lemma is standard Lemma 5.3.
There are natural equivalences Z C s Q p F ∧ Q p G ≃ Z C s Q o F ∧ Q p G ≃ Z C s Q p F ∧ Q o G ≃ Z C s Q r F ∧ Q r G roof. Let us prove, for example, the equivalence R C s Q p F ∧ Q o G ≃ R C s Q r F ∧ Q r G . The other equivalences are similar. It is enough to prove that for allspaces Z there is an equivalence, natural in Z Map
Top Z C s Q p F ∧ Q o G, Z ! ≃ Map
Top Z C s Q r F ∧ Q r G, Z ! . Note that these are derived mapping spaces as the source is cofibrant and thetarget fibrant. For any covariant/contravariant pair of functors
F, G , there is ahomeomorphismMap
Top Z C s F ∧ G, Z ! ∼ = Nat( F, Map(
G, Z )) . Therefore it is enough to show that there is a natural equivalenceNat ( Q p F, Map( Q o G, Z )) ≃ Nat ( Q r F, Map( Q r G, Z )) . (13)The key observation is that if Q r G ( − ) is Reedy cofibrant then the functorMap( Q r G ( − ) , Z ) is Reedy fibrant. The two functors Map( Q o G ( − ) , Z ) andMap( Q r G ( − ) , Z ) are weakly equivalent functors from C op to Top. They arefibrant in the projective and the Reedy model structure respectively. On theother hand, the functors Q p F and Q r F are weakly equivalent functors that arecofibrant in the projective and the Reedy model structure. It follows that thetwo sides of (13) are the derived mapping spaces from F to Map( Q o ( G ) , Z ) inthe projective and the Reedy model structure respectively. Since the two modelstructures have the same week equivalences, the two derived mapping spacesare equivalent. Definition 5.4.
The homotopy coend of F and G , denoted by R C h F ∧ G , isdefined to be any one of the equivalent coends in Lemma 5.3. Remark . As mentioned in the introduction, we identify the ∞ -localizationof Top with the ∞ -category of pointed spaces Top ∞ = S ∗ . Let F ∞ : C op → S ∗ and G ∞ : C → S ∗ be the compositions of F and G with the localization functorTop → Top ∞ . Then it is known (see for example [BHH, Proposition 2.5.6])that the image of the homotopy coend of F and G under the localization functorTop → Top ∞ is equivalent to the ∞ -coend of F ∞ and G ∞ . In the sequel we willsometimes abuse notation and identify the homotopy coend with the ∞ -coendand more generally homotopy colimits with ∞ -colimits.The case we are interested in is of the covariant functor G k,l,m : Fin ∗ → Topand the contravariant functor X − : Fin op ∗ → Top, where X is a pointed finiteCW complex. The strict coend R [ t ] ∈ Fin ∗ s X t ∧ G k,l,m ([ t ]) is a model for the strict(continuous) Kan extension of G k,l,m from Fin ∗ to Top. The homotopy coendof same functors is a model for the homotopy Kan extension. We saw abovethat G k,l,m is cofibrant in the Reedy model structure. The functor X − is also20eedy cofibrant if X is a CW complex. This amounts to saying that for all t , theinclusion of the fat diagonal ∆ t X into X t is a Σ t -equivariant cofibration. Sinceboth functors are Reedy cofibrant, their homotopy coend is in fact equivalentto the strict coend. Lemma 5.6.
Let X be a pointed CW complex and j any integer satisfying ∞ ≥ j ≥ min( m, ⌊ kl ⌋ ) . All the maps in the following diagram are equivalences R [ t ] ∈ Fin ≤ j ∗ s X t ∧ G k,l,m ([ t ]) → R [ t ] ∈ Fin ∗ s X t ∧ G k,l,m ([ t ]) ↓ ↓ R [ t ] ∈ Fin ≤ j ∗ h X t ∧ G k,l,m ([ t ]) → R [ t ] ∈ Fin ∗ h X t ∧ G k,l,m ([ t ]) Proof.
The vertical maps are equivalences because the functors X − and G k,l,m are each Reedy cofibrant. The top map is an equivalence because G k,l,m is j -skeletal.Now comes the main result of this section: G k,l,m is equivalent to both thestrict and the derived left Kan extension of its restriction to the category offinite sets of size at most min( m, ⌊ kl ⌋ ). Theorem 5.7.
For all k, l, m, and ∞ ≥ j ≥ min( m, ⌊ kl ⌋ ) , the functor G k,l,m isequivalent to both the strict and the derived left Kan extension of G k,l,m | Fin ≤ j ∗ along the inclusion Fin ≤ j ∗ ⊆ CW ∗ .Proof. By lemma 5.6, the map from the derived left Kan extension to the strictone is an equivalence. So it is enough to prove the statement for strict Kanextension. So throughout this proof, R stands for the strict coend. There is anatural assembly map Z [ t ] ∈ Fin ∗ X t ∧ G k,l,m ([ t ]) → G k,l,m ( X ) . (14)We will prove that it is a homeomorphism, if X is a finite CW complex. Thisis enough for proving the theorem. Thus we need to prove that (14) is bijectiveand bi-continuous.First we prove surjectivity. Let f ∈ G k,l,m ( X ) ⊆ SC ∗ (C ( X, M l ) , M k ) . Factor f using lemma 3.2. Denote t = | F f | and choose a pointed bijection[ t ] ∼ = −→ F f ∪ {∗} . It follows that f admits a factorizationC ( X, M l ) → M tl f ′ → M k . Note that since V f = V ′ f we have f ′ ∈ G k,l,m ([ t ]) This means that f is in theimage of the map X t ∧ G k,l,m ([ t ]) → G k,l,m ( X ). Thus f is in the image of theassembly map. Since f was an arbitrary element of G k,l,m ( X ), we have provedsurjectivity of the assembly map. 21ext we show that the assembly map is injective. Suppose that ( α, g ) ∈ X t ∧ G k,l,m ([ t ]) and ( α , g ) ∈ X t ∧ G k,l,m ([ t ]) represent two elements of R [ t ] ∈ Fin ∗ X t ∧ G k,l ([ t ]) that are mapped to the same element of G k,l,m ( X ). Wehave to show that ( α, g ) and ( α , g ) represent the same element of R [ t ] ∈ Fin ∗ X t ∧ G k,l ([ t ]). Without loss of generality we may assume that the functions α : [ t ] → X and α : [ t ] → X are injective. Indeed, suppose for example that α isnot injective. Then α can be factored as a surjection followed by injection, say[ t ] ։ [ t ′ ] α ′ ֒ → X . Let g ′ be the image of g under the map G k,l,m ([ t ]) → G k,l,m ([ t ′ ]).Then ( α ′ , g ′ ) represents the same element as ( α, g ) in the coend.Assuming α and α are injective, let f be the common image of ( α, g ) and( α , g ) in G k,l,m ( X ). Then there is a unique subset F ⊂ X \ {∗} such that f factors as in lemma 3.2. By the minimality of F , F is contained in the image of α and in the image of α . Choose a pointed bijection α ′ : [ t ] ∼ = → F ∪ {∗} . Thisbijection factors through α and α . By slight abuse of notation, let use α ′ todenote also the element of X t that is the composed map [ t ] α ′ → F ∪ {∗} ֒ → X .Putting it all together we obtain a commutative diagramC ( X, M l ) α ∗ ✲ M tl M t l α ∗ ❄ ✲ M t l ❄ ⊂ f ′ ✲ α ′ ∗ ✲✲ M k f ✲ . It follows that ( α, g ) and ( α , g ) are both mapped to the same element ( α ′ , f ′ )of the coend R [ t ] ∈ Fin ∗ X t ∧ G k,l,m ([ t ]).We have shown that the assembly map (14) is a bijection. Its domain iscompact and its codomain is Hausdorff, so it is a homeomorphism. Corollary 5.8. If l > k then S k,l ≃ ∗ .Proof. It follows immediately from Proposition 4.5 that if l > k then the re-striction of G k,l to Fin ∗ is trivial, i.e., G k,l ([ t ]) ∼ = ∗ for every finite pointed set[ t ]. By Theorem 5.7 G k,l is equivalent to the homotopy left Kan extension ofthe restriction of G k,l to Fin ≤ k ∗ , so G k,l ≃ ∗ . Since S k,l is the stabilization of G k,l , it follows that S k,l ≃ ∗ as well. The functor G k,l, is especially well behaved. The following lemma is an easyconsequence of Theorem 5.7 taken with j = 1. We also give a direct proof Lemma 5.9.
Let l, k ≥ and let X be a finite pointed CW-complex. Then theassembly map a X : X ∧ G k,l, ( S ) → G k,l, ( X )22 s a homeomorphism.Proof. Since X ∧ G k,l, ( S ) is compact and G k,l, ( X ) is Hausdorff it is enough toshow that a X is a bijection. Since M l is simple we have G k,l ( S ) = SC ∗ inj ( M l , M k ) + where SC ∗ inj ( M l , M k ) ⊂ SC ∗ ( M l , M k ) is the space of injective C ∗ -algebra maps.similarly we get G k,l, ( S ) = SC ∗ , ( M l , M k ) + where SC ∗ , ( M l , M k ) ⊂ SC ∗ inj ( M l , M k )is the space of maps f with rank( f ) = 1. We get that ( X ∧ G k,l, ( S )) r {∗} =( X r {∗} ) × SC ∗ , ( M l , M k ) and the map a X ( x , f ) ∈ G k,l, ( X ) ⊂ G k,l ( X ) is thecomposition C ( X, M l ) x ∗ −→ M l f −→ M k . The injectivity now follows from the uniqueness of the factorisation in lemma 3.2and the surjectivity from the existence in lemma 3.2 and lemma 3.3.
Corollary 5.10.
Let l, k ≥ , the natural map Σ ∞ G k,l, ( S ) → S k,l, is an equivalence. Let P Inj be the topologically enriched symmetric monoidal category of finitepositive dimensional Hilbert spaces and isometric embeddings up to scalar. Thatis, up to isomorphism objects are given by C k for k ∈ Z ≥ and P Inj( C l , C k ) = U ( k ) / ( U (1) × U ( k − l )) . The symmetric monoidal structure is given by the tensor product. We have atopologically enriched symmetric monoidal functorEnd : P Inj → SC ∗ , that sends the Hilbert space V to the C ∗ -algebra End( V ) of linear maps V → V and the embedding i : V → W is sent to the ∗ -homomorphism End( V ) → End( W ) sending A ∈ End( V ) to i ◦ A ◦ i − ◦ p ∈ End( W ), where p : W → Im ( i )is the orthogonal projection. The monoidal coherence maps of End are givenby the natural isomorphismsEnd( V ) ⊗ End( V ) ∼ −→ End( V ⊗ V )and C ∼ −→ End( C ) . Lemma 5.11.
The map P Inj( C l , C k ) + → SC ∗ ( M l , M k ) = G k,l ( S ) induced by the functor End is an embedding with image G k,l, ( S ) . roof. We first show that the map is surjective on G k,l, ( S ). Let i : C l → C k bean isometric embedding. The map f i = End( i ) ∈ SC ∗ ( M l , M k ) clearly satisfies V f = Im( i ) ⊆ C k and thus rank( f ) = dim V fi l = ll = 1. On the other hand if f i ∈ G k,l, ( S ), then either f i = 0 and thus is in the image of the base pointor dim V f i = 1. In the case dim V f i = 1 we get that the map f factors as anisomorphism followed by an injection. M l ∼ −→ End( V f i ) → M k subjectivity now follows from the fact that every automorphism of M l is inner.We now prove injectivity. First since V f i = Im( i ), f i determines Im( i ). Weare thus reduced to show that if two embeddings i, j : C l → C k have the sameimage V and the induced maps M l → End( V ) are the same then i and j differby a scalar. This follows from the fact that the center of M l is exactly the scalarmatrices. We thus get a continuous bijection P Inj( C l , C k ) + → G k,l, ( S ). Sinceit has compact domain and Hausdorff target, it is a homeomorphism.Applying the topological nerve to P Inj we get a symmetric monoidal ∞ -category P Inj ∞ . We thus get that End induces a symmetric monoidal functor g End : P Inj op ∞ → NCW . Composing with the symmetric monoidal functor Σ ∞ : NCW → NSp we get asymmetric monoidal functorΣ ∞ ◦ g End : P Inj op ∞ → NSp . For a closed symmetric monoidal ∞ -category M denote by Cat ⊗M the ∞ -category of symmetric monoidal M -enriched ∞ -categories. If M and N areclosed symmetric monoidal ∞ -categories and a : M → N is a symmetric monoidalfunctor which admits a right adjoint b , then we have an induced adjunction a ! : Cat ⊗M ⇄ Cat ⊗N : b ! . We shall especially use the case where a = Σ ∞ + : S → Sp . Using the identifi-cation Cat ⊗S ∼ = Cat ⊗ , we obtain an adjunction(Σ ∞ + ) ! : Cat ⊗ ⇆ Cat ⊗ Sp : (Ω ∞ ) ! Given a symmetric monoidal ∞ -category C ∈
Cat ⊗ we denote C Sp := (Σ ∞ + ) ! ( C ) ∈ Cat ⊗ Sp . As a stable ∞ -category NSp is naturally Sp -enriched. Thus we have naturalisomorphismsMap Cat ⊗ Sp (( P Inj Sp ∞ ) op , NSp ) ≃ Map
Cat ⊗ Sp ((Σ ∞ + ) ! ( P Inj op ∞ ) , NSp ) ≃ Map
Cat ⊗ ( P Inj op ∞ , (Ω ∞ ) ! NSp ) ≃ Map
Cat ⊗ ( P Inj op ∞ , NSp ) .
24e denote the mate of Σ ∞ ◦ g End ∈ Map
Cat ⊗ ( P Inj op ∞ , NSp ) under this adjunctionby e E : ( P Inj Sp ∞ ) op → NSp . This is a symmetric monoidal Sp -enriched functor. Lemma 5.12.
We get the following commutative diagram for every k, l ≥ ( P Inj Sp ∞ ) op ( C k , C l ) ∼ (cid:15) (cid:15) ˜E / / Map
NSp (Σ ∞ M k , Σ ∞ M k ) ∼ (cid:15) (cid:15) Σ ∞ + Map P Inj ∞ ( C l , C k ) ∼ (cid:15) (cid:15) End / / Σ ∞ Map
NCW ( M k , M l ) ∼ (cid:15) (cid:15) ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ Σ ∞ G k,l, ( S ) ∼ * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ / / Σ ∞ G k,l ( S ) / / ∂ ( G k,l ) ∂ ( G k,l, ) ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ Proof.
The commutation of the lower triangle and the right trapezoid is clear.The left square is a consequence of lemma 5.11. To see the commutation of thetop trapezoid consider the digram in
Cat ⊗ Sp .( P Inj Sp ∞ ) op ' ' ❖❖❖❖❖❖❖❖❖❖❖❖ / / (Σ ∞ ) ! ( NCW ) (cid:15) (cid:15) NSp . K -theory In this section we show that for every fixed l , the spectrum S ∞ ,l := colim k →∞ S k,l ,is equivalent to the connective K -theory spectrum ku . We will use this obser-vation to show how the spectra S ∞ ,l together represent K -theory of noncom-mutative CW-complexes.Recall that for a pointed finite set [ t ], G k,l ([ t ]) = SC ∗ ( M tl , M k ) ∼ = _ ( m ,...,m t ) Inj( C ( m + ··· + m t ) l , C k ) / Q ti =1 U ( m i )+ . The inclusions of algebras M k → M k +1 that send a matrix a to a ⊕ l , · · · → G k,l → G k +1 ,l → · · · . efinition 6.1. For each fixed l , define the functor G ∞ ,l := colim k →∞ G k,l .Here by colim we mean strict rather than homotopy colimit. Lemma 6.2. G ∞ ,l is equivalent to the homotopy colimit hocolim k →∞ G k,l .Moreover there is a homeomorphism (where as usual the wedge sum is indexedon the set of non-zero t -tuples of non-negative integers) G ∞ ,l ([ t ]) = _ ( m ,...,m t ) Inj( C ( m + ··· + m t ) l , C ∞ ) / Q ti =1 U ( m i )+ and a homotopy equivalence G ∞ ,l ([ t ]) ≃ _ ( m ,...,m t ) BU ( m ) × · · · × BU ( m t ) + . Proof.
The mapInj( C ( m + ··· + m t ) l , C k ) / Q ti =1 U ( m i ) → Inj( C ( m + ··· + m t ) l , C k +1 ) / Q ti =1 U ( m i ) is an inclusion of a submanifold, and in particular a cofibration. It follows thatthe colimit is equivalent to the homotopy colimit. As k goes to ∞ , the colimitof G k,l is, by definition, homeomorphic to _ ( m ,...,m t ) Inj( C ( m + ··· + m t ) l , C ∞ ) / Q ti =1 U ( m i )+ . Note that Inj( C ( m + ··· + m t ) l , C ∞ ) is a contractible space with a free action of Q ti =1 U ( m i ), and moreover the quotient mapInj( C ( m + ··· + m t ) l , C ∞ ) → Inj( C ( m + ··· + m t ) l , C ∞ ) / Q ti =1 U ( m i ) is a fiber bundle. It follows that the quotient space is equivalent to the classifyingspace BU ( m ) × · · · × BU ( m t ).Recall S k,l is the stabilization of G k,l . We define S ∞ ,l accordingly. Definition 6.3.
The spectrum S ∞ ,l is defined as follows S ∞ ,l := colim k →∞ S k,l . Since stabilization commutes with homotopy colimits of pointed functors, S ∞ ,l is the stabilization of G ∞ ,l . It is worth noting that the homotopy typeof S ∞ ,l is independent of l . Indeed, the following statement is an immediateconsequence of Lemma 6.2. Corollary 6.4.
A choice of an inclusion C ֒ → C l induces an equivalence of Γ -spaces G ∞ ,l → G ∞ , . In fact, the spectrum S ∞ ,l is homotopy equivalent to the connective K -theoryspectrum ku for each l . We record this observation in a lemma.26 emma 6.5. For each l the spectrum S ∞ ,l is homotopy equivalent to ku .Proof. The Γ-space G ∞ , is equivalent to the Γ-space constructed by Segalin [Se1, Section 2]. It is, in Segal’s terminology, a special Γ-space. This meansthat for any two pointed finite sets [ s ] , [ t ], the natural map G ∞ , ([ s ] ∨ [ t ]) → G ∞ , ([ s ]) × G ∞ , ([ t ]) is an equivalence. It follows from Lemma 6.2 that G ∞ , ([1]) ∼ = ∞ _ m =1 BU ( m ) + . So S ∞ , is the spectrum associated with the group-completion of W ∞ m =1 BU ( m ) + ,which is well-known to be ku . By corollary 6.4, it follows that S ∞ ,l ≃ ku for all l . It follows that for each k, l there is a natural map S k,l → S ∞ ,l ≃ ku . We willshow later, after we analyze the subquotients of the rank filtration, that thismap induces an isomorphism on π (Lemma 9.4). In particular, it is not trivial. K -theory of noncommutative CW-complexes. We will now discuss how the spectra S ∞ ,l can be used to represent the K theoryfunctor on the category NSp . In this subsection we consider M as an ∞ -categoryand use the ∞ -categorical picture.Consider the enriched Yoneda embedding of M op (which is an Sp rev -functor): Y : M op → Fun Sp ( M , Sp ) . The sequence of algebras · · · → M k → M k +1 → · · · gives rise to a directsequence in M op : · · · → Σ ∞ NC M k → Σ ∞ NC M k +1 → · · · Applying the Yoneda embedding Y to this sequence, we obtain a sequenceof spectral functors Y (Σ ∞ NC M k ) ∈ Fun Sp ( M , Sp ), that are characterized by thefollowing property Y (Σ ∞ NC M k )(Σ ∞ NC M l ) ≃ Hom M (Σ ∞ NC M k , Σ ∞ NC M l ) = S k,l . The homotopy colimit of this sequence is a spectral functor that will representconnective K -theory. Let us give this functor a name: Definition 6.6.
We define the spectrally enriched functor ku ∈ Fun Sp ( M , Sp )to be the following colimit: ku = colim k →∞ Y (Σ ∞ NC M k ) . Note that for every l we have the following equivalences, the last of whichfollows from Lemma 6.5. The last equivalence justifies the notation ku for thisfunctor. ku (Σ ∞ NC M l ) := colim k →∞ Hom
NSp (Σ ∞ NC M k , Σ ∞ NC M l ) = S ∞ ,l ≃ ku.
27e know by the main result of [ABS1] that we can identify
NSp with P Sp ( M ),the category of contravariant spectrally enriched functors from M to Sp . Thisimplies that covariant enriched functors from M to Sp , such as ku , can be usedto define homology theories on NSp , and therefore also on
NCW .Indeed suppose h : M → Sp is a covariant Sp -functor. Then h determinesan Sp -tensored functor h ∧ M ( − ) : NSp → Sp by the universal property of the enriched Yoneda embedding, such that there isa natural equivalence h ( k ) ∧ k ∈M S k,l ≃ h (Σ ∞ NC M l ) . The universal property also tells us that this induces an equivalence between the ∞ -category of Sp -tensored functors NSp → Sp and the ∞ -category of Sp -functors M → Sp . By taking homotopy groups π ∗ ( h ∧ M ( − )) one obtains a generalizedhomology theory on noncommutative CW-spectra (see [BJM, Definition 4.1]).In particular, let us take h = ku . We interpret the property that ku (Σ ∞ NC M l ) ≃ ku for all l as saying that ku represents a version of connective K -theory. Wehave a further enhancement of this fact. Lemma 6.7.
The functor ku takes values in the category of ku -module spectra.Proof. The monoidal structure on M gives rise to maps S k ′ ,l ′ ∧ S k,l → S kk ′ ,ll ′ .Fixing l ′ = 1 we obtain maps S k ′ , ∧ S k,l → S kk ′ ,l , natural in l . Taking limits as k, k ′ → ∞ we obtain maps, still natural in Σ ∞ NC M l ku (Σ ∞ NC M ) ∧ ku (Σ ∞ NC M l ) → ku (Σ ∞ NC M l ) . Identifying ku (Σ ∞ NC M ) with ku , this map endows ku (Σ ∞ NC M l ) with the structureof a ku -module, functorial in Σ ∞ NC M l .The ∞ -category of ku -module spectra ku − mod is stable and thus Sp -tensored.In light of Lemma 6.7, we may now view ku as an Sp -functor ku : M → ku − mod.Thus, by the universal property of the enriched Yoneda embedding, ku deter-mines an Sp -tensored functor ku ∧ M ( − ) : NSp → ku − mod . This allows for the following definition:
Definition 6.8.
For
X, Y ∈ NSp we define kk ( X, Y ) := [ ku ∧ M X, ku ∧ M Y ] ku − mod ,kk ∗ ( X, Y ) = π ∗ Map ku − mod ( ku ∧ M X, ku ∧ M Y ) = kk (Σ ∗ X, Y ) . It follows from Lemma 6.7 that for all k, l ≥ kk ∗ (Σ ∞ NC M k , Σ ∞ NC M l ) ≃ π ∗ Map ku − mod ( ku, ku ) ≃ π ∗ ku. Sp -enriched subcategory of M containing the object Σ ∞ NC M . A commutative spectrum is made noncom-mutative by means of an Sp -enriched left Kan extension. Suppose that X is acommutative spectrum, which we also may consider as a noncommutative spec-trum by Kan extension. Standard adjunction implies that there is a naturalequivalence ku ∧ M X ≃ ku ∧ X, where the symbol ∧ on the right hand side denotes the usual smash product ofspectra. It follows that if X and Y are commutative spectra then there is anequivalence kk ∗ ( X, Y ) ≃ π ∗ Map Sp ( X, ku ∧ Y ) . In the case when X and Y are suspension spectra of pointed finite CW-complexes,this is essentially [DM, Proposition 3.1]. Thus kk defined above is a natural ex-tension for the connective bivariant K -theory of Dadarlat and McClure. Thisalso suggests that some results from [loc. cit.] may be generalized from finiteCW-complexes to finite noncommutative CW-complexes. This will be addressedin future papers.We conclude by remarking that since the functor ku takes values in modulesover ku , one can invert the Bott element and get a functor representing non-connective K -theory. ku It follows immediately from equation (10), that the filtration of G k,l by G k,l,m interacts well with the stabilization map G k,l → G k +1 ,l that we considered inthe previous section. To be more specific, there is a commutative diagram · · · → G k,l,m → G k,l,m +1 → · · · → G k,l ↓ ↓ · · · ↓· · · → G k +1 ,l,m → G k +1 ,l,m +1 → · · · → G k +1 ,l ↓ ↓ · · · ↓ ... ... · · · ... · · · → G ∞ ,l,m → G ∞ ,l,m +1 → · · · → G ∞ ,l . On finite sets, G ∞ ,l,m is given, at least up to homotopy, by the following formula G ∞ ,l,m ([ t ]) ≃ _ { ( m ,...,m t ) | m ··· + mt ≤ m } BU ( m ) × · · · × BU ( m t ) + We conclude that the rank filtration of G k,l induces a compatible rank filtrationof G ∞ ,l . Upon passing to stabilization, the rank filtration of G ∞ ,l induces afiltration of ku by a sequence of spectra: · · · → S ∞ ,l,m → S ∞ ,l,m +1 → · · · → S ∞ ,l ≃ ku. K -theory spectrum ku studied, for example, in [AL2] (we remark that ku is denoted bu in [loc. cit]).Thus the rank filtration of S k,l is a lift of the classical rank filtration of ku . In this section we investigate the subquotients of the rank filtration. We showthat on the level of the functors G k,l the subquotients of the rank filtrationhave a presentation as a homotopy coend over the category Epi of finite setsand surjections, as opposed to the category
Fin ∗ of pointed sets and all pointedfunctions. Definition 7.1.
Let G mk,l : CW ∗ → Top be the quotient functor G mk,l := G k,l,m /G k,l,m − . Similarly, let S k,lm be the homotopy cofiber of the map S k,l,m − → S k,l,m .It is not hard to check that for any X ∈ CW ∗ the map G k,l,m − ( X ) → G k,l,m ( X ) is a cofibration in Top. Thus, G mk,l the levelwise homotopy cofiberof the map G k,l,m − → G k,l,m in the category of pointed continuous functorsFun ∗ ( CW ∗ , Top). It follows from [ADL, Lemma 2.16] that G mk,l is also the homo-topy cofiber of the map G k,l,m − → G k,l,m in the projective model structure onFun ∗ ( CW ∗ , Top), and thus also in the stable model structure Sp M .Since strict (homotopy) left Kan extension commutes with strict (homotopy)cofiber, we get from Theorem 5.7 that the functor G mk,l is equivalent to both thestrict and the derived left Kan extension of its restriction to Fin ≤ j ∗ for any ∞ ≥ j ≥ min( m, ⌊ kl ⌋ ). Thus, for any finite pointed CW-complex X , we havethe folowing formula: G mk,l ( X ) ∼ = Z [ t ] ∈ Fin ≤ j ∗ s X t ∧ G mk,l ([ t ]) ≃ Z [ t ] ∈ Fin ≤ j ∗ h X t ∧ G mk,l ([ t ]) . (15)Since stabilization commutes with homotopy cofibers, S k,lm is equivalent to thestabilization of G mk,l .It follows immediately from equation (10) that on objects G mk,l is given asfollows G mk,l ([ t ]) = _ { ( m ,...,m t ) | m ··· + mt = m } Inj( C ( m + ··· + m t ) l , C k ) / Q ti =1 U ( m i )+ (16)Which also can be written as G mk,l ([ t ]) = _ { ( m ,...,m t ) | m ··· + mt = m } Inj( C ml , C k ) / Q ti =1 U ( m i )+
30o understand the effect of G mk,l on morphisms, let α : [ t ] → [ s ] be a pointedfunction. If | α ∗ ( m , . . . , m t ) | < m , then G mk,l ( α ) takes the summand of G mk,l ([ t ])corresponding to ( m , . . . , m t ) to the basepoint. If | α ∗ ( m , . . . , m t ) | = m ,then G mk,l ( α ) takes the corresponding summand of G mk,l ([ t ]) to the summandof G mk,l ([ s ]) indexed by α ∗ ( m , . . . , m t ) by the map (5).One attractive property of G mk,l is that it has a more compact coend formulathan the one given in equation 15 (see Proposition 7.5). In this formula thecategory Fin ∗ is replaced with the smaller category Epi of non-empty unpointedsets and surjections. For k ≤ ∞ let Epi ≤ k be the category of non-empty finitesets of cardinality at most k and epimorphisms between them. In the case k = ∞ , this is the category of all non-empty finite sets and surjections, and wedenote it simply Epi .If α : t ։ s is a morphism in Epi , and ( m , . . . , m t ) is a multiset, thenwe may define α ∗ ( m , . . . , m t ) = ( n , . . . , n s ) in the usual way, by saying that n j = P i ∈ α − ( j ) m i . Note that in this case there is an equality m + · · · + m t = n + · · · + n s . Note also that if m i > i then n j > j . Definition 7.2.
Let Top u be the category of unpointed topological spaces. Let G mk,l : Epi → Top u be the following functor. On objects, it is defined by thefollowing formula G mk,l ( t ) = a { ( m ,...,m t ) | mi> ,m ··· + mt = m } Inj( C ml , C k ) / Q ti =1 U ( m i ) On morphisms, G mk,l is defined similarly to G k,l and G mk,l . Given a surjection α : t ։ s , the summand indexed by ( m , . . . , m t ) is mapped to the summandindexed by α ∗ ( m , . . . , m t ) by the same map as in (5).We will make much use of the functor G mk,l + : Epi → Top, which is obtainedby adding a disjoint basepoint to G mk,l . We introduced an unpointed version ofthe functor because it will be convenient to have it at some point. Remark . For a multi-set ( m , . . . , m t ) define its support to be the set A = { i | m i > } ⊆ t . Notice that for any subset A ⊆ t there is a natural wayto identify G mk,l ( A ) + with a wedge summand of G mk,l ([ t ]). Namely, G mk,l ( A ) + isidentified, on the right hand side of (16) as the wedge sum corresponding toindices ( m , . . . , m t ) whose support is exactly A . With this identification, thereis a homeomorphism G mk,l ([ t ]) ∼ = _ A ⊆ t G mk,l ( A ) + Moreover, the functoriality in [ t ] is defined on the right hand side as follows.Suppose α : [ t ] → [ s ] is a pointed function. Suppose A ⊆ t . If α sends someelement of A to the basepoint of [ s ], then the corresponding summand G mk,l ( A )is sent to the basepoint. Otherwise, this summand is sent to G mk,l ( α ( A )) usingthe surjection A ։ α ( A ) defined by α .31onsider the functor Fin ∗ × Epi op → Top([ t ] , u ) → [ t ] ∧ u One can think of this functor as a
Fin ∗ − Epi -bimodule. It is often used toestablish connections between categories of
Fin ∗ -modules and Epi -modules. Wehave the following proposition:
Proposition 7.4.
There is a homeomorphism and an equivalence, natural in [ t ] ranging over Fin ∗ : G mk,l ([ t ]) ∼ = Z u ∈ Epi s [ t ] ∧ u ∧ G mk,l ( u ) + ≃ Z u ∈ Epi h [ t ] ∧ u ∧ G mk,l ( u ) + Proof.
First of all, let us construct a natural map. Fix a surjective function u ։ u . One has a map[ t ] ∧ u ∧ G mk,l ( u ) + → G mk,l ([ t ]) . The map is defined as follows. First, the surjection u ։ u induces a map G mk,l ( u ) ։ G mk,l ( u ), and therefore [ t ] ∧ u ∧ G mk,l ( u ) + → [ t ] ∧ u ∧ G mk,l ( u ) + .Second, there is a bijection of sets [ t ] ∧ u ∼ = t u + , so a non basepoint of this setis a map f : u → t . This defines a map G mk,l ( u ) → G mk,l ( f ( u )). Finally, thereis an inclusion of G mk,l ( f ( u )) + as a wedge summand of G mk,l ([ t ]), as in Remark7.3.The map is natural in the variable [ t ] (exercise for the reader).The following diagram commutes because G mk,l is functorial with respect tosurjections. [ t ] ∧ u ∧ G mk,l ( u ) + → [ t ] ∧ u ∧ G mk,l ( u ) + ↓ ↓ [ t ] ∧ u ∧ G mk,l ( u ) + → G mk,l ([ t ])It follows that there is a natural transformation of functors of [ t ] (recall that R s denotes strict coend) Z u ∈ Epi s [ t ] ∧ u ∧ G mk,l ( u ) + → G mk,l ([ t ]) . (17)We claim that this map is in fact an isomorphism. To see this notice that foreach fixed t there is an isomorphism of functors of u [ t ] ∧ u = t u + ∼ = _ A ⊆ t Epi ( u, A ) + (18)For each A , the functor u Epi ( u, A ) is a representable functor Epi op → Top.By coYoneda lemma, there is an isomorphism Z u ∈ Epi s [ t ] ∧ u ∧ G mk,l ( u ) + ∼ = _ A ⊆ t G mk,l ( A ) + G mk,l ([ t ]), again as in Remark 7.3.It follows that the map (17) is in fact an isomorphism. On the other hand,Equation (18) shows that the functor u [ t ] ∧ u is cofibrant in the projectivemodel structure on the functor category [ Epi op , Top]. It follows that the naturalmap from the homotopy coend to the strict coend is an equivalence: Z u ∈ Epi h [ t ] ∧ u ∧ G mk,l ( u ) + ≃ → Z u ∈ Epi s [ t ] ∧ u ∧ G mk,l ( u ) + . As a consequence, we have a simplified coend formula for G mk,l ( X ) where X is a CW complex. Proposition 7.5.
Let X be a finite pointed CW-complex. There is a homeo-morphism and an equivalence, natural in XG mk,l ( X ) ∼ = Z u ∈ Epi s X ∧ u ∧ G mk,l ( u ) + ≃ Z u ∈ Epi h X ∧ u ∧ G mk,l ( u ) + . The statement remains true if
Epi is replaced with
Epi ≤ k .Proof. We prove the part for the homotopy coend, and the proof of the strictpart is identical. By the standard coend formula (see equation 15), there is anequivalence G mk,l ( X ) ≃ Z [ t ] ∈ Fin ∗ h X [ t ] ∧ G mk,l ([ t ]) . By Proposition 7.4, there is an equivalence G mk,l ([ t ]) ≃ Z u ∈ Epi h [ t ] ∧ u ∧ G mk,l ( u ) + . It follows that there is an equivalence G mk,l ( X ) ≃ Z [ t ] ∈ Fin ∗ h X [ t ] ∧ (cid:18)Z u ∈ Epi h [ t ] ∧ u ∧ G mk,l ( u ) + (cid:19) . By associativity of coend (“Fubini theorem”), the right hand side is equivalentto Z u ∈ Epi h Z [ t ] ∈ Fin ∗ h X [ t ] ∧ [ t ] ∧ u ! ∧ G mk,l ( u ) + . It remains to show that there is a natural equivalence Z [ t ] ∈ Fin ∗ h X [ t ] ∧ [ t ] ∧ u ≃ X ∧ u . This is elementary. The argument goes as follows. The set [ t ] ∧ u is equivalentto the total homotopy cofiber of the cubical diagram A Fin ∗ ( A + , [ t ]), where33 ranges over subsets of u , and the maps are induced by collapsing the com-plement of a subset to the basepoint. By the coYoneda lemma, R [ t ] ∈ Fin ∗ h X [ t ] ∧ Fin ∗ ( A + , [ t ]) ≃ X A . It follows that R [ t ] ∈ Fin ∗ h X [ t ] ∧ [ t ] ∧ u is equivalent to the totalhomotopy cofiber of the cubical diagram A X A , where A ranges over subsetsof u . The total cofiber is equivalent to X ∧ u .Our next step is to use Proposition 7.5 to describe the subquotient spectra S k,lm . Let I : Epi op → Top be the (unique) functor defined by I ( t ) = (cid:26) S t = 1 ∗ t = 1 Lemma 7.6.
There are equivalences, where
Epi can be replaced with
Epi ≤ k S k,lm ≃ Σ ∞ Z t ∈ Epi h I ∧ G mk,l + ≃ Z t ∈ Epi h Σ ∞ I ∧ G mk,l + . Proof.
By definition, S k,lm is the stabilization of the functor X Z u ∈ Epi h X ∧ u ∧ G mk,l ( u ) + The right hand side is a weighted homotopy colimit of reduced functors from CW f ∗ → Top. Since stabilization commutes with such homotopy colimits, itfollows that there is an equivalence S k,lm ≃ Z u ∈ Epi h ∂ ( X ∧ u ) ∧ G mk,l ( u ) + where ∂ ( X ∧ u ) denotes the stabilization of the functor X X ∧ u . Observethat ∂ ( X ∧ u ) is equivalent to Σ ∞ S if | u | = 1, and is equivalent to ∗ if | u | > u ∂ ( X ∧ u ) is equivalent to Σ ∞ I as a functor Epi op → Sp . The lemma follows.Recall once again that G mk,l + is defined by the following formula, G mk,l ( t ) + = _ { ( m ,...,m t ) | mi> , Σ mi = m } Inj( C ml , C k ) / Q ti =1 U ( m i )+ ∼ = ∼ = _ { ( m ,...,m t ) | mi> , Σ mi = m } U ( k ) / t Y i =1 U ( m i ) × U ( k − lm ) ! + . In the special case l = 1 , k = m we get that G mm, ( t ) + = _ { ( m ,...,m t ) | mi> , Σ mi = m } U ( m ) / t Y i =1 U ( m i ) + . (19)34nd in general, there are equivalences G mk,l ( t ) + ≃ _ { ( m ,...,m t ) | mi> , Σ mi = m } U ( k ) /U ( k − lm ) + ∧ U ( m ) U ( m ) / t Y i =1 U ( m i ) + ≃≃ U ( k ) /U ( k − lm ) + ∧ U ( m ) G mm, ( t ) + . (20)It is easily checked that the last equivalence is functorial in t and therefore wehave an equivalence of functors Epi → Top G mk,l + ≃ U ( k ) /U ( k − lm ) + ∧ U ( m ) G mm, . And upon applying lemma 7.6 we obtain an equivalence S k,lm ≃ U ( k ) /U ( k − lm ) + ∧ U ( m ) S m, m = Inj( C lm , C k ) + ∧ U ( m ) S m, m . (21)We remind the reader that U ( m ) is considered a subgroup of U ( k ) via thediagonal map U ( m ) ֒ → U ( lm ) followed by the inclusions U ( lm ) ֒ → U ( lm ) × U ( k − lm ) ֒ → U ( k ). Alternatively, U ( m ) acts on Inj( C lm , C k ) through its obviousaction on C lm = C l ⊗ C m . Equation (21) reduces the problem of describing S k,lm for general k, l, m to de-scribing S m, m for all m . In this section we use Lemma 7.6 to show that S m, m isequivalent to the suspension spectrum of the complex of direct-sum decomposi-tions of C m , which we denote L ⋄ m . This leads to a complete description of S k,lm in terms of the complexes L ⋄ m (Theorem 8.8).The complexes L ⋄ m were first introduced in [Ar1], and were studied in detailin [BJL+] and [AL3]. They play a role in orthogonal calculus, and also indescribing the subquotients of the rank filtration of K -theory [AL1, AL2]. Theyhave some remarkable homotopical properties, that we will recall in the nextsection (Proposition 9.1).Our proof that S m, m is equivalent to the suspension spectrum of L ⋄ goesthough an intermediate complex, which we call the complex of ordered direct-sum decompositions. Let us give the formal definition. Definition 8.1.
Let D o m be the following category objects in topological spaces.Its objects are ordered tuples ( E , . . . , E t ) of pairwise orthogonal proper, non-trivial vector subspaces of C m , whose direct sum is C m . A morphism ( E , . . . , E t ) → ( F , . . . , F s ) consists of a surjective function α : { , . . . , t } ։ { , . . . , s } such thatfor each 1 ≤ i ≤ t , E i ⊆ F α ( i ) .We call the category D o m the category of proper, ordered direct-sum decom-positions of C m . The set of objects and the set of morphisms both have a35opology. There is a natural action of U ( m ) on D o m , and object and morphismsets of are topologized as unions of U ( m )-orbits.There is a convenient presentation of D o m as the Grothendieck constructionapplied to the functor G mm, of Definition 7.2. Let us recall the definition of (aversion of) the Grothendieck construction. Definition 8.2.
Suppose C is a small category and F : C →
Top u is a functor.The Grothendieck construction on F (a.k.a the wreath product of C and F ) isthe following pointed topological category, denoted C ≀ F . The objects of C ≀ F arepairs ( c, x ) where c is an object of C , and x ∈ F ( c ). A morphism ( c, x ) → ( d, y )in C ≀ F is a morphism α : c → d in C such that F ( α )( x ) = y . The space of objectsof C ≀ F is topologized as the disjoint union ` c F ( c ) indexed by the objects of C , and the space of morphisms is topologized as the disjoint union ` c → d F ( c ),indexed by morphisms of C .The following well-known lemma can be thought of as a topological analogueof Thomason’s homotopy colimit theorem. Lemma 8.3.
Suppose C is a small category and F : C →
Top u is a functor.There is a natural equivalence hocolim C F ≃ |C ≀ F | . Proof.
It is easy to see that the simplicial nerve of
C ≀ F is isomorphic , as asimplicial space, to Bousfield and Kan’s simplicial model for hocolim C F . Infact, both simplicial spaces are given in simplicial degree k by the space a c →···→ c k F ( c ) . The i -th face map d i is defined by dropping c i and, if i = 0, using the func-toriality of F to map F ( c ) to F ( c ). The degeneracy map s i is defined byduplicating c i .Now recall that we have a functor G mm, : Epi → Top u (Definition 7.2). Let Epi > be the full subcategory of Epi consisting of sets of cardinality greaterthan 1. By slight abuse of notation we denote the restriction of G mm, to Epi > also by G mm, . The following lemma is straightforward from the definitions: Lemma 8.4.
There is an isomorphism of topological categories
Epi > ≀ G mm, ∼ = D o m . Given a space X , let X ⋄ denote the unreduced suspension of X . We havethe following connection between S m, m and D o m . Proposition 8.5.
There is a natural equivalence S m, m ≃ Σ ∞ |D o m | ⋄ . roof. We saw in lemma 7.6 that S m, m ≃ Σ ∞ Z Epi h I ∧ G mm, where I : Epi op → Top is the functor that sends 1 to S and sends all otherobjects to ∗ . So we need to show that there is an equivalence of pointed spaces Z Epi h I ∧ G mm, ≃ |D o m | ⋄ . Let S : Epi op → Top be the constant functor S ( t ) ≡ S . Let S > : Epi op → Top be the functor S > (1) = ∗ and S > ( t ) ≡ S for t >
1. Then there is ahomotopy cofibration sequence of functors S > → S → I . It follows that thereis a homotopy cofibration sequence of coends Z Epi h S > ∧ G mm, → Z Epi h S ∧ G mm, → Z Epi h I ∧ G mm, It is a standard fact that Z Epi h S ∧ G mm, ≃ hocolim Epi ∗ ( G mm, ) ∼ = (hocolim Epi G mm, ) + (here hocolim ∗ denotes pointed homotopy colimit, while hocolim denotes un-pointed homotopy colimit). Since Epi has a final object 1, it follows thathocolim
Epi ∗ G mm, ≃ G mm, (1) = S . On the other hand, since 1 is the initial object of
Epi op , and S > (1) = ∗ itfollows easily that S > is equivalent to the functor obtained by restricting S tothe subcategory Epi > of sets of cardinality greater than 1, and then takingderived left Kan extension back to Epi op . By standard adjunctions, it followsthat there are equivalences Z Epi h S > ∧ G mm, ≃ Z Epi > h S ∧ G mm, ≃ (hocolim Epi > G mm, ) + It follows that there is a homotopy cofibration sequence(hocolim
Epi > G mm, ) + → S → Z t ∈ Epi h I ∧ G mm, By lemma 8.4 D o m is the Grothendieck construction on G mm, . It follows byLemma 8.3 that hocolim Epi > G mm, ≃ |D o m | . So we have a homotopy cofibration sequence |D o m | + → S → Z t ∈ Epi h I ∧ G mm, This implies that R t ∈ Epi h I ∧ G mm, ≃ |D o m | ⋄ .37ur next step is to show that D o m can be replaced with a smaller category,which we call the poset of unordered direct-sum decompositions. First, thedefinition. Definition 8.6.
Let D m be the following category objects in topological spaces.Its objects are unordered sets { E i | i ∈ I } of pairwise orthogonal proper, non-trivial vector subspaces of C m , whose direct sum is C m . There is a uniquemorphism { E i | i ∈ I } → { F j | j ∈ J } if for each i ∈ I there is a (necessarilyunique) j ∈ J such that E i ⊆ F j . In keeping with recent literature, the geo-metric realization of D m will be denoted L m , and its unreduced suspension istherefore L ⋄ m .As with D o m , there is a natural action of U ( m ) on D m and both the sets ofobjects and morphisms of D m are topologized as unions of U ( m )-orbits. Wenote that for any two objects P, Q of D o m there is at most one morphism from P to Q . In other words, D o m is a topological preorder . By contrast, the category D m is a topological poset: it is the poset of isomorphism classes of D o m . Thecategory D m will be referred to as the category, or poset, of proper, unordered direct-sum decompositions of C m .There is a topological functor q : D o m → D m , which forgets the order of thecomponents. Proposition 8.7.
The natural functor q : D o m → D m induces an equivalence ofgeometric realizations |D o m | ≃ −→ |D m | Proof.
We are going to use Quillen’s theorem A. We need a version of it thatis valid for topological categories. There are several such versions scattered inthe literature, we will use [EbRW, Theorem 4.7]. According to this theorem, itis enough if we prove the following1. For every object Λ of D m , the classifying space of the over category q/ Λis contractible.2. The map from the morphism space of D o m to the object space of D o m ,that sends every morphism to its target, is a fibration (in the languageof [EbRW], this means that D o m is right fibrant).3. The map from the space of objects of the over category q/ D m to the spaceof objects of D m , that sends a morphism to its target, is a fibration. Here q/ D m is the category of arrows in D m of the form q (Θ) → Λ, where Θ isan object of D o m .For part (1), let Λ = { F i | i ∈ I } be an object of D m , i.e., an unordered collectionof pairwise orthogonal non-trivial subspaces of C m whose direct sum is C m . Let t be the number of elements of I and choose a bijection I ∼ = { , . . . , t } . Then e Λ = ( F , . . . , F t ) is a choice of lift of Λ to an object of D o m . An object of q/ Λconsists of an object Θ = ( E , . . . , E s ) such that each E i is a subspace of F j for some (necessarily unique) j . It follows that there exists a unique surjection α : { , . . . , s } ։ { , . . . , t } such that E i ⊂ F α ( i ) for all i . This means that38here is a unique morphism from Θ to e Λ in q/ Λ. Thus the category q/ Λ hasa (not necessarily unique) terminal object, and therefore its classifying space iscontractible.For part (2), using the identification of D o m with the Grothendieck construc-tion Epi > ≀ G mm, (Lemma 8.4), the map from the space of morphisms of D o m to the space of objects of D o m which sends each morphism to its target, has thefollowing form a s ։ t ∈ Epi > a { ( m ,...,m s ) | mi> , Σ mi = m } U ( m ) / s Y i =1 U ( m i ) → a t ∈ Epi > a { ( n ,...,n t ) | ni> , Σ nj = m } U ( m ) / t Y j =1 U ( n j )(22)where for every surjective function s ։ t , the space U ( m ) / Q si =1 U ( m i ) is sentto U ( m ) / Q sj =1 U ( n j ), where for each j = 1 , . . . , t , n j = Σ i ∈ α − ( j ) m i , by thecanonical quotient map associated with the sub-conjugation of Q si =1 U ( m i ) into Q sj =1 U ( n j ) induced by α . The map (22) is clearly a fibration. Indeed, it is a U ( m ) equivariant map between disjoint union of U ( m )-orbits, and such a mapis necessarily a fibration.Finally, the proof of part (3) is similar to that of part (2). Since D m is theposet of isomorphism classes of the pre-order D o m , the space of objects of thecategory q/ D m is the quotient of the space of morphisms of D o m by the action ofthe groupoid of isomorphisms of the target. Similarly, the space of objects of D m is the quotient of the space of objects of D o m by the groupoid of isomorphisms.This means that we have the following map a s ։ t a { ( m ,...,m s ) | mi> , Σ mi = m } U ( m ) / s Y i =1 U ( m i ) Iso( t ) → a t a { ( n ,...,n t ) | ni> , Σ nj = m } U ( m ) / t Y j =1 U ( n j ) Iso( t ) The action of the groupoid of isomorphisms of the variable t respects the actionof U ( m ). It follows that the resulting map is still a U ( m )-equivariant mapbetween disjoint union of orbits, and therefore is a fibration.Propositions 8.5 and 8.7, together with equation (21) give us an equivalence S k,lm ≃ U ( k ) /U ( k − lm ) + ∧ U ( m ) Σ ∞ L ⋄ m . (23)We also want to describe the composition morphisms S k,lm ∧ S j,kn → S j,lmn . Webegin by observing that tensor product induces a natural map L ⋄ m ∧ L ⋄ n → L ⋄ mn as follows. Suppose that E = { E i | i ∈ I } and F = { F j | j ∈ J } are direct-sumdecompositions of C m and C n respectively. Then E ⊗ F := { E i ⊗ F j | ( i, j ) ∈ I × J } is a direct-sum decomposition of C m ⊗ C n ∼ = C mn . Note that if at leastone of E , F is a proper decomposition (i.e., has more than one component) then E ⊗ F is a proper decomposition as well. This means that the tensor productinduces a map L ⋄ m ∧ L ⋄ n → L ⋄ mn as desired. Note that this map is equivariantwith respect to the tensor product homomorphisms U ( m ) × U ( n ) → U ( mn ).39ext, we extend it to a map (cid:0) U ( k ) /U ( k − lm ) + ∧ U ( m ) L ⋄ m (cid:1) ∧ (cid:0) U ( j ) /U ( j − kn ) + ∧ U ( n ) L ⋄ n (cid:1) →→ U ( j ) /U ( j − lnm ) + ∧ U ( nm ) L ⋄ nm . Now recall that U ( k ) /U ( k − lm ) ∼ = Inj( C lm , C k ) and U ( j ) /U ( j − kn ) ∼ = Inj( C kn , C j ).Given homomorphisms g ∈ Inj( C lm , C k ) and f ∈ Inj( C kn , C j ), we may form thehomomorphism f ◦ g n : C lmn ֒ → C j . Clearly, this defines a map U ( k ) /U ( k − lm ) × U ( j ) /U ( j − kn ) → U ( j ) /U ( j − lnm ). This map is equivariant with respectto the tensor product homomorphism U ( m ) × U ( n ) → U ( mn ). Combining itwith the map L ⋄ m ∧ L ⋄ n → L ⋄ mn defined earlier, we obtain the desired map.We are ready to state the main theorem of the paper. Theorem 8.8.
There is an equivalence S k,lm ≃ Σ ∞ U ( k ) /U ( k − lm ) + ∧ U ( m ) L ⋄ m ∼ = Σ ∞ Inj( C ml , C k ) + ∧ U ( m ) L ⋄ m . Under this equivalence, the composition product S k,lm ∧ S j,kn → S j,lmn correspondsto the map (cid:16) Inj (cid:0) C ml , C k (cid:1) + ∧ U ( m ) L ⋄ m (cid:17) ∧ (cid:16) Inj (cid:0) C nk , C j (cid:1) + ∧ U ( n ) L ⋄ n (cid:17) →→ Inj (cid:0) C nml , C j (cid:1) + ∧ U ( nm ) L ⋄ nm . that were defined above.Proof. We already proved the formula for S k,lm (equation 23). It remains tocheck the statement about the composition product. Recall that S k,lm is thestabilization of the functor G mk,l . The composition product is determined by thenatural transformation G mk,l ( v ) ∧ G nj,k ( u ) → G mnj,l ( u × v ). An analysis of thiscomposition map shows that it is induced by disjoint union of maps of the formInj( C ( m + ··· + m t ) l , C k ) / Q tj =1 U ( m j ) × Inj( C ( n + ··· + n s ) k , C j ) / Q si =1 U ( n i ) →→ Inj( C ( P i = s,j = ti =1 ,j =1 n i m j ) l , C j ) / Q i = s,j = ti =1 ,j =1 U ( n i m j ) that sends ( q, p ) to p ◦ q n , where n = n + · · · + n s . Note that the decompositionof C mn associated with the target of this map is the tensor product of the givendecompositions of C m and C n , just as was claimed. This induces the claimedmap of spectra. S k,l In this section we calculate the spectra S k,l in some cases, and also prove thatthe map S k,l → ku is an isomorphism on π . Our main tool is Theorem 8.8,which expresses the subquotients of the rank filtration in terms of the complexes L ⋄ m . To use it, we need to know something about the complexes L ⋄ m . So let us40egin by reviewing some of the rather remarkable properties of these complexesthat were uncovered in [Ar1, AL1, BJL+, AL3]. The following proposition liststhe relevant facts. Proposition 9.1.
1. The space L ⋄ m is rationally contractible for m > .2. The space L ⋄ m is (integrally) contractible unless m is a prime power.3. If m = p k with p a prime and k > , then L ⋄ p k is p -local, and has chromatictype k .Proof. Except for the statement about the chromatic type, this is [AL1, propo-sition 9.6], which in turn relies on [Ar1]. The statement about the chromatictype is part of [Ar2, Theorem 2.2].The proofs in [Ar2] are based on a rather deep connection between L ⋄ m and the calculus of functors. Since part (1) plays a prominent role in ourapplications, we indicate an independent, more direct way to prove this part.The space L ⋄ m is equivalent to the total homotopy cofiber of the following m − U = { i , . . . , i k } ⊆ { , . . . , m } , with i > . . . > i k . Let X ( U ) be the space of chains of decompositions of C m ofthe form (Λ < · · · < Λ k ) where each Λ j has i j -components. If U is emptythen X ( U ) = ∗ . Note that in general X ( U ) is a disjoint union of U ( m )-orbits.The assignment U
7→ X ( U ) defines a diagram indexed on the opposite of theposet of subsets of { , . . . , m } , i.e., an m − L ⋄ m is equivalent to the total homotopy cofiber ofthe cube X . For example, in the case m = 3, X is the following square of U (3)-orbits. U (3) / Σ ≀ U (1) × U (1) → U (3) /U (2) × U (1) ↓ ↓ U (3) / Σ ≀ U (1) → U (3) /U (3) (24)Here the upper right corner is X ( { } ), the space of decompositions of C with 2components, the lower left corner is X ( { } ), the space of decompositions with 3components, and the upper left corner is X ( { , } ), the space of morphisms froma decomposition with 3 components to a decomposition with 2 components.Each one of the horizontal maps in (24) is a map of U (3)-orbits, induced bysubgroup inclusions Σ ≀ U (1) × U (1) → U (2) × U (1) and Σ ≀ U (1) → U (3).Note that in both of these cases, the subgroup that is being included is thenormalizer of a maximal torus. It follows that each one of the horizontal mapsis a rational equivalence, and therefore the total cofiber of (24) is trivial inrational homology. Since it is simply connected, it is also trivial in rationalhomotopy.More generally suppose m > i and consider the map X ( { m, i , . . . , i k } ) →X ( { i , . . . , i k } ). This map is a disjoint union of maps between U ( m )-orbits. Foreach path component of X ( { m, i , . . . , i k } ), the isotropy group is the normal-izer of a maximal torus of the isotropy group of a corresponding component of X ( { i , . . . , i k } ). It follows that the map X ( { m, i , . . . , i k } ) → X ( { i , . . . , i k } ) is41lways a rational equivalence, and therefore the total homotopy cofiber of X ,which is L ⋄ m , is rationally trivial. Corollary 9.2.
The map S k,l, → S k,l . is a rational equivalence.Proof. To see this, consider the filtration ∗ = S k,l, → S k,l, → S k,l, → · · · → S k,l, ⌊ kl ⌋ = S k,l It follows from Theorem 8.8 and part 1 of Proposition 9.1 that for all m > S k,lm of the map S k,l,m − → S k,l,m is rationally trivial. Itfollows that the map S k,l, → S k,l is a rational equivalence.Here is an explicit description of L ⋄ m for some values of m . Proposition 9.3. L ⋄ ∼ = S L ⋄ ∼ = Σ R P .3. More generally, if p is a prime then L ⋄ p is a union of p − shifted copiesof the mod p Moore space.
As a first application of Theorem 8.8 and Proposition 9.1, let us prove thatthe map S k,l → ku induces an isomorphism on π . Lemma 9.4.
Assume that k ≥ l . The map S k,l → S ∞ ,l ≃ ku induces anisomorphism on π .Remark . We remind the reader that if k < l , S k,l ≃ ∗ . Proof.
We saw in Section 3 that the mapping spectra S k,l are filtered by asequence of spectra ∗ = S k,l, → S k,l, → S k,l, → · · · → S k,l, ⌊ kl ⌋ = S k,l Consider the commutative diagram S k,l, → S k,l ↓ ↓ S ∞ ,l, → S ∞ ,l We will prove that the left, top and bottom maps in this diagram induce anisomorphism on π . It then follows that the right map induces an isomorphismon π , which is what we want to prove.By Theorem 8.8, S k,l, ≃ S k,l ≃ Σ ∞ Inj( C l , C k ) /U (1) + , S ∞ ,l, ≃ Σ ∞ Inj( C l , C ∞ ) /U (1) + . Since Inj( C l , C k ) /U (1) and Inj( C l , C ∞ ) /U (1) are path-connected spaces, themap S k,l, → S ∞ ,l, induces on π the isomorphism from Z to itself.To analyze the map S k,l, → S k,l recall, again from Theorem 8.8, thatthe subquotient S k,l,m / S k,l,m − is equivalent to the suspension spectrum ofInj( C ml , C k ) + ∧ U ( m ) L ⋄ m . For m > L m is path-connected, so L ⋄ m is simply-connected. It follows that S k,l,m / S k,l,m − is 1-connected for m > S k,l, → S k,l is 1-connected, and in particular it inducesan isomorphism on π . The same argument applies in the case k = ∞ , whichcompletes the proof.Since the rank filtration of S k,l has length ⌊ kl ⌋ , we can conclude that if k < l then S k,l, is in fact equivalent to S k,l . Lemma 9.6. If l ≤ k ≤ l − then S k,l is integrally equivalent to Σ ∞ U ( k ) /U ( k − l ) × U (1) + . In particular, for k = l the spectrum S k,k = End NSp (Σ ∞ NC M k ) is equivalent to Σ ∞ P U ( k ) + : the group ring spectrum of the projective unitary group.
10 On the rationalization and p -localization of M Let C be a stable presentable closed symmetric monoidal ∞ -category. Denoteby ⊗ the tensor product, by 1 C the unit and by Hom( • , • ) the internal hom. Forevery n ∈ Z and an object X ∈ C there is a natural multiplication by n map[ n ] : X → X . We will say that an object X ∈ C is rational if for every n = 0 themap [ n ] : X → X is an isomorphism. Simlarly, we will say that X is p -local fora prime p , if [ n ] : X → X is an isomorphism for every n that is not divisible by p . We denote the collection of rational objects in C by C Q and the collection of p -local objects by C ( p ) . If C = C Q (resp. C = C ( p ) ) then we say that C is rational(resp. p -local). The naturality of [ n ] implies that C Q and C ( p ) are closed in C under all small limits and colimits and that for every X ∈ C and Y ∈ C Q (resp. Y ∈ C ( p ) ) we have X ⊗ Y, Hom(
X, Y ) ∈ C Q (resp. X ⊗ Y, Hom(
X, Y ) ∈ C ( p ) ). Wethus get that C Q and C ( p ) are themselves stable presentable closed symmetricmonoidal ∞ -categories. Further the inclusion i C Q : C Q ⊂ C admits a symmetric monoidal left adjoint called rationalization L C Q : C → C Q . Same holds for p -localization. 43urther, the left adjoints are given by the following formulas L Q ( X ) = L Q (1 C ) ⊗ X = colim (cid:20) X [1] −→ X [2] −→ X [3] −→ X · · · (cid:21) L ( p ) ( X ) = L p (1 C ) ⊗ X = colim (cid:20) X [ p ′ ] −−→ X [ p ′ ] −−→ X [ p ′ ] −−→ X · · · (cid:21) where p ′ , p ′ , . . . is the list of integers not divisible by p .Since NSp is left-tensored over Sp , we have that NSp Q (resp. NSp ( p ) ) is left-tensored over Sp Q (resp. Sp ( p ) ). Let M Q (resp. M ( p ) ) to be the full Sp Q -enriched (resp. Sp ( p ) -enriched) subcategory of NSp Q (resp. NSp ( p ) ) spanned by L NSp Q (Σ ∞ NC M n ) (cid:16) resp. L NSp ( p ) (Σ ∞ NC M n ) (cid:17) for n ∈ N . Lemma 10.1.
For all k, l ∈ N there are equivalences Hom M Q ( L NSp Q (Σ ∞ NC M k ) , L NSp Q (Σ ∞ NC M l )) ≃ L Sp Q Hom
NSp (Σ ∞ NC M k , Σ ∞ NC M l ) and Hom M ( p ) ( L NSp ( p ) (Σ ∞ NC M k ) , L NSp ( p ) (Σ ∞ NC M l )) ≃ L Sp ( p ) Hom
NSp (Σ ∞ NC M k , Σ ∞ NC M l ) Proof.
We will go over the (very straightforward) proof of the rational case.The proof of the p -local case is practically identical.Hom M Q ( L NSp Q (Σ ∞ NC M k ) , L NSp Q (Σ ∞ NC M l )) = Hom NSp Q ( L NSp Q (Σ ∞ NC M k ) , L NSp Q (Σ ∞ NC M l )) == Hom NSp (Σ ∞ NC M k , L NSp Q (Σ ∞ NC M l )) == Hom NSp (Σ ∞ NC M k , colim (cid:20) Σ ∞ NC M l [1] −→ Σ ∞ NC M l [2] −→ Σ ∞ NC M l [3] −→ · · · (cid:21) ) == colim (cid:20) Hom
NSp (Σ ∞ NC M k , Σ ∞ NC M l ) [1] −→ Hom
NSp (Σ ∞ NC M k , Σ ∞ NC M l ) [2] −→ [2] −→ Hom
NSp (Σ ∞ NC M k , Σ ∞ NC M l ) [3] −→ · · · (cid:21) = L Sp Q Hom
NSp (Σ ∞ NC M k , Σ ∞ NC M l ) . Here the first equality is by definition, the second equality is using the adjunction L NSp Q ⊢ i NSp Q , the third equality is the formula for L NSp Q , the forth equality is bythe compactness of Σ ∞ NC M k and the fifth equality uses the formula for L Sp Q .The main theorems of [ABS1] have rational and p -local analogs with com-pletely analogous proofs. Theorem 10.2.
Let D be a symmetric monoidal cocomplete rational (resp. p -local) ∞ -category. Suppose that there is a small set C of compact objects in D ,that generates D under colimits and desuspentions. Assume that D ∈ C and C s closed under tensor product. Thinking of D as left-tensored over Sp Q (resp. Sp ( p ) ), we let C be the full Sp Q -enriched (resp. Sp ( p ) -enriched) subcategory of D spanned by C . Then we have a natural symmetric monoidal functor of categoriesleft-tensored over Sp Q (resp. Sp ( p ) ) P Sp Q ( C ) ∼ −→ D (cid:16) resp. P Sp ( p ) ( C ) ∼ −→ D (cid:17) , which is an equivalence of the underlying ∞ -categories and sends each repre-sentable presheaf Y ( c ) to c ∈ C . Theorem 10.3.
The Sp Q -enriched (resp. Sp ( p ) -enriched) category M Q (resp. M ( p ) ) acquires a canonical symmetric monoidal structure, the category of presheaves P Sp Q ( M Q ) (resp. P Sp ( p ) ( M ( p ) ) ) acquires a canonical symmetric monoidal left Sp Q -tensored (resp. Sp ( p ) -tensored) structure and we have a natural symmetricmonoidal left Sp Q -tensored (resp. Sp ( p ) -tensored) functor P Sp Q ( M Q ) ∼ −→ NSp Q (cid:16) resp. , P Sp ( p ) ( M ( p ) ) ∼ −→ NSp ( p ) (cid:17) which is an equivalence of the underlying ∞ -categories. An explicit presentation of M Q and NSp Q We can use our results to give a very explicit description of M Q , and thereforeof the noncommutative rational stable homotopy category.The rationalization functor L Sp Q : Sp → Sp Q is a symmetric monoidal leftadjoint. As explained in Section 5.1, we have an induced adjunction( L Sp Q ) ! : Cat ⊗ Sp ⇆ Cat ⊗ Sp Q : i ! . Composing adjunctions we obtain( L Sp Q ◦ Σ ∞ + ) ! : Cat ⊗ ⇆ Cat ⊗ Sp Q : (Ω ∞ ◦ i ) ! . We denote P Inj Sp Q ∞ := ( L Sp Q ◦ Σ ∞ + ) ! ( P Inj ∞ ) ∈ Cat ⊗ Sp Q and we denote the mate of L Sp Q ◦ Σ ∞ ◦ g End ∈ Map
Cat ⊗ ( P Inj op ∞ , NSp Q ) under thisadjunction by e E Q : ( P Inj Sp Q ∞ ) op → NSp Q ∈ Cat ⊗ Sp Q . Proposition 10.4.
The functor e E Q : ( P Inj Sp Q ∞ ) op → NSp Q ∈ Cat ⊗ Sp Q is fully faithful as an Sp Q -enriched functor with essential image M Q ⊆ NSp Q . roof. The statement about the essential image is not effected by changingenrichment and is thus clear from the description of the functor End. Thefully-faithfulness follows from lemma 5.12 and corollary 9.2.In view of theorem 10.3 and proposition 10.4. We get the following result:
Theorem 10.5.
We have a sequence of equivalences of symmetric monoidal ∞ -categories Fun( P Inj ∞ , Sp Q ) ∼ = P Sp Q (( P Inj Sp Q ∞ ) op ) ∼ = NSp Q . Proof.
The second equivalence is an immediate corollary of Theorem 10.3 andProposition 10.4. For the first equivalence note that P Sp Q (( P Inj Sp Q ∞ ) op ) = Fun Sp Q ( P Inj Sp Q ∞ , Sp Q ) , where Fun Sp Q stands for Sp Q -enriched functors. But we have an induced adjunc-tion ( L Sp Q ◦ Σ ∞ + ) ! : Cat ⇆ Cat Sp Q : (Ω ∞ ◦ i ) ! , so we have natural equivalencesFun Sp Q ( P Inj Sp Q ∞ , Sp Q ) ≃ Fun Sp Q (( L Sp Q ◦ Σ ∞ + ) ! ( P Inj ∞ ) , Sp Q ) ≃ Fun( P Inj ∞ , (Ω ∞ ◦ i ) ! Sp Q ) ≃ Fun( P Inj ∞ , Sp Q ) , and are done. p -local and chromatic picture Now instead of rationalizing, suppose we fix a prime p and localize everythingat p . One can obtain further information about the p -localization of M . Itfollows from Proposition 1.4 parts (3) and (4) that the rank filtration of M ( p ) isconstant except at powers of p . Therefore it is natural to regrade the filtrationof S k,l as follows S k,l, ֒ → S k,l,p ֒ → S k,l,p ֒ → · · · ֒ → S k,l,p i · · · Let us loosely refer to S k,l,p i as morphisms of filtration i . We can say that the p -localization of M is a filtered category in the sense that the composition of amorphism of filtration i and a morphism of filtration j has filtration i + j . Withthis grading, (the p -localization) of M is a graded category in the usual sense,that composition adds degrees.Lastly, let us mention that the last part of Proposition 9.1 implies the fol-lowing Corollary 10.6.
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