aa r X i v : . [ m a t h . A T ] A ug Model Categories for Orthogonal Calculus
David Barnes Peter OmanAugust 24, 2018
Abstract
We restate the notion of orthogonal calculus in terms of model categories. This pro-vides a cleaner set of results and makes the role of O ( n )–equivariance clearer. Thuswe develop model structures for the category of n –polynomial and n –homogeneousfunctors, along with Quillen pairs relating them. We then classify n –homogeneousfunctors, via a zig-zag of Quillen equivalences, in terms of spectra with an O ( n )–action. This improves upon the classification theorem of Weiss. As an application,we develop a variant of orthogonal calculus by replacing topological spaces withorthogonal spectra.Mathematics Subject Classification: 55P42, 55P91 and 55U35 Orthogonal calculus is a beautiful tool for calculating homotopical properties of a func-tor from the category of finite dimensional inner product spaces and linear isometriesto the category of based spaces. Interesting examples of such functors abound andinclude classical objects in algebraic and geometric topology: the classifying space ofthe orthogonal group, F ( V ) = BO ( V ); the classifying space of the group of home-omorphisms of V , F ( V ) = B Top( V ); and the space of Euclidean embeddings for afixed manifold M , F ( V ) = Emb( M, V ). We call the category of such functors andnatural transformations between them E . Orthogonal calculus is based on the notionof n –polynomial functors, which are well-behaved functors in E . These functors sat-isfy an extrapolation condition, which allows one to identify the value at some vectorspace from the values at vector spaces of greater dimension, see section 5. In geometricterms, orthogonal calculus approximates a functor (locally around R ∞ ) via polynomialfunctors and attempts to reconstruct the global functor from the associated “infinitesi-mal” information. More concretely, the calculus splits a functor F in E into a tower offibrations, the n th –fibration of this tower consists of a map from the n –polynomial ap-proximation of F to the ( n − F . The homotopy fibreof this map is then an n –homogeneous functor and is classified by an O ( n )–spectrumup to homotopy.A question about an input functor can thus be translated into a question about thespectra and the fibration sequences they are part of, hopefully making the answer easier1o find. Indeed, from a computational perspective, the orthogonal tower provides aspectral sequence the inputs of which have more structure and more constraints than theoriginal functor. A particularly interesting example of how useful orthogonal calculuscan be is [ALV07]. In that paper, orthogonal calculus is used to prove that the rationalhomology type of certain spaces of embeddings of a manifold is determined by therational homology type of the manifold. In addition, the homogeneous approximationsthemselves typically correspond to spectra of intrinsic interest. For example, in [Wei95],Weiss determines that the 1-homogeneous approximation of the functor BO ( − ) is thesphere spectrum and the 1-homogeneous approximation of B Top( − ) is Waldhausen’sA-theory of a point.In the construction of the orthogonal calculus, [Wei95], there are numerous times thatone wishes to study a functor ‘up to homotopy’ for some (usually non-standard) notionof homotopy. Furthermore, the classification of the n th –fibre of the tower is givenin terms of an equivalence of homotopy categories. We replace ‘up to homotopy’ bysome appropriate notion of weak equivalence within a model structure. This allows usto replace an equivalence of homotopy categories by the stronger and more structurednotion of a Quillen equivalence: exhibiting an equivalence of ‘homotopy theories’ ratherthan merely an equivalence of derived categories.In [Wei95], Weiss implicitly constructs a localisation of the standard projective modelstructure, capturing the homotopy theory of n –polynomial functors. One would expectthat such a localization can be realized in model theoretic language and indeed this isthe case. Via a homotopy-idempotent functor, we can left Bousfield localize to createa (right proper) model structure whose fibrant objects are the n –polynomial functors.We call this model category n –poly– E . However, subtleties arise when attempting toconstruct a model structure capturing the homotopy theory of n –homogeneous functorsand the associated classification by O ( n )-spectra.A priori, the n –homogeneous structure is a right localisation of the n –polynomial struc-ture. Since we not working stably, this requires the full Bousfield localisation machineryin the sense of Hirschhorn [Hir03]. Indeed, in order to know that the n –polynomialmodel structure is well-behaved enough to admit a right localisation (specifically, thatit is a cellular model category), we must make use of this general machinery to providea second construction of n –poly– E .These two constructions show that n –poly– E is right proper cellular model category.Thus we are able to perform a right Bousfield localisation to obtain a new model cat-egory n –homog– E whose cofibrant–fibrant objects are precisely the n –homogeneousfunctors. The task is now to identify this complicated model structure as somethingsimpler. With the work of Weiss, one would hope to prove a Quillen equivalence betweenthis model category and the category of O ( n )–objects in spectra; however, we show O ( n )–objects in spectra are not the most natural models for homogeneous functors.By carefully examining the derived equivalence in [Wei95], we construct a non–standardmodel of the (free) O ( n )–equivariant stable category which we call O ( n ) E n . Thiscategory captures the appropriate structure for differentiation, see section 3. Indeed,we exhibit a left Quillen equivalence from O ( n ) E n to the category of orthogonal spectrawith an O ( n )–action. We then show that homogeneous functors are classified, via2ifferentiation as a right Quillen functor, by objects of O ( n ) E n . Thus we recover thederived equivalence of Weiss via a zig-zag of Quillen equivalences and gain a muchbetter understanding of the role of equivariance within the theory. In addition, it maybe of technical interest that this is an example of a twofold (left and right) localisationof a cellular model category where one has a nice description of the resulting homotopytheory; in general, one would expect such a thing to be quite unwieldy.In more detail, the category O ( n ) E n is best thought of as a variant of the theory of O ( n )–equivariant orthogonal spectra, where a universe with trivial O ( n )-action is usedand the sphere spectrum is replaced by the O ( n )–equivariant object which at V takesvalue S nV , the one-point compactification of R n ⊗ V . From such an object one canobtain an object of E by neglect of structure and taking orbits, we call this compositefunctor res n /O ( n ) and we examine it in more detail in sections 4 and 9. When n > O ( n ) E n that is a variation of the usual modelstructure on spectra, adjusted to account for our non-standard ‘sphere spectrum’, seesection 7.The functor V S nV was already known to occur in orthogonal calculus. From aspectrum with O ( n )–action X , one can construct an n –homogeneous functor by thefollowing rule V EO ( n ) + ∧ O ( n ) [Ω ∞ ( X ∧ S nV )]where the term X ∧ S nV is equipped with the diagonal action. We are able to replacethis construction with the pair of Quillen equivalences as below, see sections 8 and 10.The first category is the model category of n –homogeneous functors from vector spacesand linear isometries to based spaces, the second our new category and the third themodel category of O ( n )–objects in spectra. n –homog– E n ε ∗ / / O ( n ) E πn res n /O ( n ) o o ( − ) ∧ J n J / / O ( n ) IS α ∗ n o o The functor α ∗ n takes a spectrum X to the object of O ( n ) E n which at V takes value X ( nV ) with O ( n )–action given by first applying the O ( n )–action of X and then the O ( n )–action induced by that on nV = R n ⊗ V .These Quillen equivalences give a much cleaner statement than the homotopy classifica-tion of [Wei95, Section 7], which was greatly hampered by the need to work exclusivelywith Ω –spectra. They also make the role of the functor V S nV clearer. Indeed,much of the difficulty of this work was to establish the correct categories and functorsfor the above.The sets of maps at which we left localise and the set of objects at which we right localiseare intrinsically bound into the definition of O ( n ) E n and the notion of differentiation(which is embodied by the functor ind n ε ∗ ). We find these intricate relations quitegratifying to see and perhaps illuminate what kind of definitions one would need tocreate a new calculus.Equally, it would be interesting to see how much of this work can be replicated forthe other notions of calculus that use homotopy theory. For example, the Goodwillie3alculus of functors [Goo90, Goo92, Goo03] was one of the inspirations for orthogonalcalculus. It has long been known that they are strongly related, despite their verydifferent inputs, thus it should not be surprising that [BCR07] follows a similar patternto our work.We conclude this paper with an application, because of the way we have used modelcategories to develop the theory, we immediately obtain a stable variant of orthogonalcalculus by replacing topological spaces with orthogonal spectra, see section 11.Finally, we believe that other extensions and alterations of orthogonal calculus will bemuch easier to create now that we have a good model category foundation. For example,one could study functors into some localisation of spaces or spectra. Alternatively,one could repeat this work for the unitary calculus, where O ( n ) is replaced by U ( n ).Indeed, this project began as an attempt to perform an equivariant version of orthogonalcalculus. It is now clear that any such attempt will need to have clear and precisecategorical constructions, along with well-behaved model categories. Organisation
We begin in section 2 with a brief introduction to spaces with a group action andenriched functors. This provides the language we need section 3 to define the categories O ( n ) E n . We show how differentiation relates these categories in section 4.The notions of n –polynomial and n –homogeneous functors are introduced in section 5.The next task, completed in section 6, is to find a model category in which the fibrantobjects are n –polynomial functors and another where the cofibrant–fibrant objects andthe n –homogeneous functors.Staying with model structures, in section 7 we produce an n –stable model structureon O ( n ) E n . This will be an intermediary model structure sitting between the n –homogeneous model structure on E and the category of O ( n )–equivariant objects inthe category of orthogonal spectra. We prove that O ( n ) E n with the n –stable modelstructure is Quillen equivalent to the model category of O ( n )–objects in orthogonalspectra in section 8.In section 9 we prove that the differentiation functor is a right Quillen functor from the n –polynomial model category to O ( n ) E n equipped with the stable model structure.We use this to show that O ( n ) E n is in fact Quillen equivalent to the n –homogeneousmodel category in section 10. As a consequence, we recover the statement that thetower has the desired form and that the homotopy type of the n th –fibre is determinedby a spectrum with an O ( n )–action. We conclude the paper by developing a stablevariant of orthogonal calculus in section 11.We have tried to make this paper largely self–contained and hence we have reproduceda fair amount of Weiss’s work, often with improvements in the proofs or descriptionsdue to our use of model categories. There are some areas that we have not been able toimprove upon, most notably the homotopy colimit calculation of [Wei95, Theorem 4.1],the properties of the functor T n , from Theorem 6.3 and the errata, and the calculationsof Examples 5.7 and 6.4. We find this to be acceptable, as the aim of this paper is not4o replace [Wei95] but to put it into the modern language of model categories. Since equivariance is vital to our approach, we briefly introduce discuss spaces with agroup action and functors enriched over spaces with a group action. The following is asummary of [MM02, Section III.1].In this section G will be a compact Lie group. A based G –space X is a topologicalspace with a continuous action of G on the space X . We require that this action beassociative and unital and that the basepoint of X be fixed by the action of G . Acontinuous map f : X → Y between two G –spaces is said to be an equivariant map if f ( gx ) = gf ( x ) for any g ∈ G and x ∈ X . We write G Top for the category of based G –spaces and equivariant mapsThe category G Top is a closed symmetric monoidal category, whose monoidal prod-uct is given by the smash product of based spaces equipped with the diagonal actionof G . We note that whenever a smash product of two G –spaces appears in [Wei95],it is also given the diagonal action. The corresponding internal function object isthe space of non–equivariant maps, which has a G –action defined by conjugation. For G –spaces X and Y , we denote this space by Top( X, Y ) and we see that g ∗ f = gf g − for f ∈ Top(
X, Y ) and g ∈ G . In particular, Top( X, Y ) G is precisely the space ofequivariant maps from X to Y .This category has a cofibrantly generated proper model structure where the fibrationsand weak equivalences are those equivariant maps f : X → Y whose underlying mapof spaces i ∗ f : i ∗ X → i ∗ Y is a fibration or weak homotopy equivalence of spaces. Thegenerating cofibrations of this model structure are given by the standard boundaryinclusions: ( G × S n − ) + → ( G × D n ) + for n >
0, with the sphere and disc both given the trivial G –action. The generatingacyclic cofibrations are given by the maps( G × D n ) + → ( G × D n × [0 , + where ( g, x ) ( g, x,
0) and n >
0. Using [MM02, Lemma IV.6.6] it is easy to checkthat these definitions give us a symmetric monoidal model structure on the category of G –spaces. Hence the smash product and internal function object have derived functorson the homotopy category.We will also need the language of functors enriched over G -spaces, so we give anintroduction here. We take our definitions from [MM02, Section II.1].Following the usual convention we call a space–enriched functor from a topologicalcategory D to spaces a D –space . A map of D –spaces f : E → F is then a collectionof continuous maps f ( d ) : E ( d ) → F ( d ) such that for any element α ∈ D ( d, e ) we have5 commutative square E ( d ) E ( α ) / / f ( d ) (cid:15) (cid:15) E ( e ) f ( e ) (cid:15) (cid:15) F ( d ) F ( α ) / / F ( e )If the category D is enriched over G –spaces, then we can also consider continuousfunctors E from D to G –spaces such that the map E d,e : D ( d, e ) −→ Top( E ( d ) , E ( e ))is G –equivariant. We call such a functor a G –equivariant D –space . Such functorsare precisely the G Top –enriched functors from D to G Top . We then define a mapof G –equivariant D –spaces , f : E → F , to be a collection of equivariant maps f ( d ) : E ( d ) → F ( d ) such that for any element α ∈ D ( d, e ) we have a commutativesquare as before. It is important to note that we ask for this diagram to commutefor any α , even though E ( α ) or F ( α ) are not necessarily equivariant maps. Thecategory of G –equivariant D –spaces and maps of G –equivariant D –spaces will bedenoted G D Top .Our final piece of business in this section is to note that the category of G –equivariant D –spaces is itself enriched over G –spaces. We will use this in section 4 to definedifferentiation. To describe this enrichment we need to use the notion of enriched ends,details can be found in [Bor94]. Definition 2.1
For E and F in G D Top , define the G –equivariant space of maps from E to F to be the following enriched end (which is constructed in the category G Top ). Nat G D Top ( E, F ) = Z d ∈D Top( E ( d ) , F ( d ))It is routine to show that Nat G D Top ( E, F ) G is the space of maps of G –equivariant D –spaces.For model category purposes and constructions it is helpful to also have a tensor andcotensor. For E ∈ G D Top and A a topological space with G –action, there is anotherobject A ⊗ E ∈ G D Top , which at U takes value A ∧ E ( U ). The structure maps ofthis new object are given by D ( d, e ) E d,e −→ Top( E ( d ) , E ( e )) A ∧− −→ Top( A ∧ E ( d ) , A ∧ E ( e )) . There is also a cotensor product, Hom(
A, E ) which at U takes value Top( A, E ( U )).The structure map for this object is given below. D ( d, e ) E d,e −→ Top( E ( d ) , E ( e )) Top( A, − ) −→ Top(Top(
A, E ( d )) , Top(
A, E ( e ))) . Let G D Top(
E, F ) denote the set of maps in the category G D Top between two objectsof that category. Similarly, let G Top(
A, B ) denote the set of maps in the category6
Top between two objects A and B . Then we can relate the enrichment, tensor andcotensor by the natural isomorphisms G D Top( E, Hom(
A, F )) ∼ = G D Top( A ⊗ E, F ) ∼ = G Top( A, Nat G D Top ( E, F ))Thus we see that G D Top is a closed module over G Top , in the sense of [Hov99, Section4.1]. One can repeat these constructions with the category of D –spaces and maps of D –spaces to see that it is a closed module over Top .For a fixed G –topological category D we would like an adjunction comparing G D Topand G Top . It is clear that any G –equivariant D –space gives a Top –enriched functorfrom D to based topological spaces by forgetting the G –actions. Hence there is aforgetful functor from G D Top to D Top . We call this functor i ∗ just as the forgetfulfunctor from G Top to Top is called i ∗ .There is a left adjoint to i ∗ , which we write as G + ∧ − . Let E ∈ D Top , then at d wedefine ( G + ∧ E )( d ) = G + ∧ E ( d )The structure map is then defined as follows( G + ∧ E ( d )) ∧ D ( d, e ) → G + ∧ ( E ( d ) ∧ i ∗ D ( d, e )) → G + ∧ E ( e )where the first map is an isomorphism given by ( g, x, y ) ( g, x, g − y ) for g ∈ G , x ∈ E ( d ) and y ∈ D ( d, e ). The second map is then the action map of E . It is routineto check that we have an adjoint pair as claimed. O ( n ) E n Our primary objects of study are continuous functors from the category of finite di-mensional real inner product spaces and linear isometries to based spaces. We callthis category E . These functors are the input to orthogonal calculus, the output is atower of fibrations, whose fibres are continuous functors like the input but have morestructure. The main theorem of orthogonal calculus is that these fibres can be classifiedin terms of spectra with an O ( n )–action. On the way to this classification we needan intermediate category O ( n ) E n . In this section we introduce the categories E and O ( n ) E n . Since they are both defined in terms of enriched functors, we will need toconstruct a collection of enriched categories J n for n > Definition 3.1
Let U be a real inner product space with countably infinite dimension.For V and W finite dimensional inner product subspaces of U , we define Mor(
V, W ) to be the Stiefel manifold of linear isometries from V to W .We then define J to be the topological category with objects the class of finite dimen-sional real inner product subspaces of U . The morphism space of maps from V to W ,written Mor ( V, W ) , is defined to be Mor(
V, W ) + (the space Mor(
V, W ) with a disjointbasepoint added). e define I to be the topological category with same objects as J but with morphismsgiven by the space of linear isometric isomorphisms from V to W (with a disjointbasepoint added). We write I ( V, W ) + for this space. Our input functors can then be described as J –spaces. Such an object F consists of acollection of spaces F ( V ), one for each finite dimensional real inner product subspace,along with continuous maps F ( V ) ∧ Mor ( V, W ) → F ( W )that satisfy an evident associativity condition and a unit condition when V = W .Recall that the fibres of the tower will be classified using the category O ( n ) E n . Toconstruct this category we must define an enriched category J n , which we build outof some O ( n )–equivariant vector bundles. The category J n is constructed in [Wei95,Section 1] and is given an O ( n )–action in [Wei95, Section 3].For each pair V and W we define an O ( n )–equivariant vector bundle γ n ( V, W ) on thespace Mor(
V, W ). The total space of this vector bundle is given by { ( f, x ) | f ∈ Mor(
V, W ) and x ∈ R n ⊗ ( W − f ( V )) } where W − f ( V ) denotes the orthogonal complement of the image of f . The group O ( n ) acts on R n via linear isometries and we extend this action to R n ⊗ ( W − f ( V ))by letting σ ∈ O ( n ) act by σ ⊗ Id. Now we let O ( n ) act on γ n ( V, W ) by the rule σ ( f, x ) = ( f, σ ⊗ Id( x )). Definition 3.2
For each n > , the n th –jet category J n is an O ( n ) –topologicalcategory with the same class of objects as J . The space of morphisms from V to W is the Thom space of γ n ( V, W ) and is denoted Mor n ( V, W ) . Note that the point atinfinity is fixed under the O ( n ) -action. We also see that when n = 0 this definitionagrees with our earlier description of J .The composition rule for Mor n is induced by the map of spaces γ n ( V, W ) × γ n ( U, V ) −→ γ n ( U, W ) given by the formula (( f, x ) , ( g, y ) ( f g, x + f ∗ ( y )) , where f : V → W , g : U → V , f ∗ = Id ⊗ f , x ∈ R n ⊗ ( W − f ( V )) and y ∈ R n ⊗ ( V − g ( U )) . Definition 3.3
We define E n to be J n Top , the category of J n –spaces and maps of J n -spaces.We define O ( n ) E n to be O ( n ) J n Top , the category of O ( n ) –equivariant J n –spaces andmaps of O ( n ) –equivariant J n –spaces. When n = 0, O ( n ) E n is the same as E , our category of input functors.We will later use the categories O ( n ) E n to classify the fibres of the orthogonal tower.For this result it is essential that we use the O ( n )–topological enrichment. In [Wei95],only E n is used, but using the adjunction of the previous section we see that any objectof O ( n ) E n defines an object of E n by forgetting the O ( n )-actions.8 Differentiation
Differentiation is a method of taking a functor in E and making a functor in O ( n ) E n .This process is central to orthogonal calculus because the n th –fibre of the tower fora functor E ∈ E will be determined by the n th –derivative of E (up to homotopy).We will obtain an adjoint pair between E and O ( n ) E n . The (derived) counit of thisadjunction will then describe how to include the n th –fibre of the tower into the n th –term.For the sake of completeness, we consider a more general version of this adjunction.We define differentiation as a functor O ( m ) E m → O ( n ) E n for m n .Let i nm : R m → R n be the map x ( x, m n . This map induces a grouphomomorphism O ( m ) → O ( n ), where O ( m ) acts on the first m coordinates and leavesthe rest unchanged. This makes i nm : R m → R n a map of O ( m )–equivariant objects.We can also use i nm to induce a functor of O ( m )–topological categories J m → J n . Todo so, we apply the Thom space construction to the map of O ( m )–equivariant spaces:( i nm ) U,V : γ m ( U, V ) −→ γ n ( U, V )( f, x ) ( f, i nm ⊗ Id( x ))We thus have a series of maps of enriched categories. J i −→ J i −→ J i −→ . . . i nn − −→ J n i n +1 n −→ . . . We now study how these maps induce functors between the categories E n for varying n . By adding in change of groups functors, we also achieve adjoint pairs between thecategories O ( n ) E n for varying n . Definition 4.1
Define the restriction functor res nm : E n → E m as precompositionwith i nm : J m → J n , where m n .Similarly define the restriction–orbit functor res nm /O ( n − m ) : O ( n ) E n → O ( m ) E m as the functor which sends X to ( X ◦ i nm ) /O ( n − m ) an O ( m ) –topological functor from J m to based O ( m ) –spaces. When discussing the composition of restriction and evaluation, we sometimes omitnotation for restriction. On a vector space V , ( X ◦ i nm ) /O ( n − m )( V ) = X ( V ) /O ( n − m ),which is an O ( m )–space. These restriction functors both have right adjoints. The firststep is to identify the right adjoint of the orbit functor. Lemma 4.2
There is an adjoint pair ( − ) /O ( n − m ) : O ( n ) Top −−→←− O ( m ) Top : CI nm The right adjoint is defined as the composite of two functors. The first takes an O ( m )–space A and considers it as an O ( m ) × O ( n − m )–space by letting the O ( n − m )–factoract trivially, this is called ε ∗ A . The second functor takes ε ∗ A and sends it to the9opological space of O ( m ) × O ( n − m )–maps from O ( n ) to ε ∗ A . We can thereforewrite CI nm A = F O ( m ) × O ( n − m ) ( O ( n ) + , ε ∗ A ) Lemma 4.3
There is a right adjoint to res nm , called induction , defined as (ind nm X )( V ) = Nat E m (Mor n ( V, − ) , X ) where the right hand side is the topological space of maps between two objects of E m .There is a right adjoint to res nm /O ( n − m ) , called inflation–induction , which wewrite as ind nm CI X , it is defined as (ind nm CI X )( V ) = Nat O ( m ) E m (Mor n ( V, − ) , CI nm X )When m = 0 we usually replace CI n with ε ∗ , as here CI n is simply equipping X withthe trivial O ( n )–action.Now we may define differentiation. The motivation for this definition comes from twolemmas. Firstly, Lemma 4.6, which can be thought of as describing differentiation asa measure of a ‘rate of change’. Secondly, Lemma 5.5, which can be thought of asdescribing differentiation as a measure of how far a functor is from being ‘polynomial’. Definition 4.4
Let E ∈ E , then the n th –derivative of E is ind n ε ∗ E ∈ O ( n ) E n . For F ∈ E n we also talk of ind n +1 n F as being the derivative of F .Note that i ∗ ind n ε ∗ E ∈ E n is equal to ind n E , so ind n E ( V ) has an O ( n )–actionfor each V . Indeed, [Wei95, Proposition 3.1] uses this fact to say (using our newlanguage) that ind n E can be thought of as an object of O ( n ) E n . That object isprecisely ind n ε ∗ E .We are most interested in the pair (res n /O ( n ) , ind n ε ∗ ), though we will need the non-equivariant functor ind nm for some calculations. We will sometimes omit res n from ournotation, provided that no confusion can occur.We can give another relation between induction and the categories J n for varying n .The following is [Wei95, Proposition 1.2], which shows how one can construct J n +1 from J n . Proposition 4.5
For all V and W in J and all n > there is a natural homotopycofibre sequence Mor n ( R ⊕ V, W ) ∧ S n → Mor n ( V, W ) → Mor n +1 ( V, W ) Proof
Identifying S n as the closure of the subspace ( i, x ) ∈ γ n ( V, R ⊕ V ), where i isthe standard inclusion, the composition mapMor n ( R ⊕ V, W ) ∧ Mor n ( V, R ⊕ V ) → Mor n ( V, W )10estricts to a morphism Mor n ( R ⊕ V, W ) ∧ S n → Mor n ( V, W ). The homotopy cofibreof the restriction is then a quotient of [0 , ∞ ] × γ n ( R ⊕ V, W ) × R n . The desired home-omorphism, away from the base point, is induced by the association below. Consider aquadruple ( t ∈ [0 , ∞ ] , f ∈ R ⊕ V, W ) , y ∈ R n ⊗ ( W − f ( R ⊕ V )) , z ∈ R n )we send this to the element ( f | V , x ) ∈ Mor n +1 ( V, W ). Where x = y + ( f | R ∗ )( z ) + tω ( f | R ∗ (1)), and ω : W → R n +1 ⊗ W identifies W ∼ = ( R n ⊗ W ) ⊥ ⊂ R n +1 ⊗ W .From this cofibre sequence we can make a fibre sequence by applying the functorNat E n ( − , F ) for F ∈ E n . The following result is [Wei95, Proposition 2.2]. Lemma 4.6
For all V ∈ J n and F ∈ E n , there is a natural homotopy fibre sequence res n +1 n ind n +1 n F ( V ) −→ F ( V ) −→ Ω n F ( R ⊕ V ) n -polynomial functors We want to study a well-behaved collection of functors in E : those whose derivativesare eventually trivial. By analogy with functions on the real numbers, we call thesefunctors polynomial. In this section we introduce this class of functors and examinehow they relate to differentiation. Definition 5.1
For vector spaces V and W in J , let Sγ n +1 ( V, W ) be the total spaceof the unit sphere vector bundle of γ n +1 ( V, W ) . We can think of Sγ n +1 ( − , − ) + as a continuous functor from J op × J to based spaces,we can use this to define a functor from E to itself. Definition 5.2
For E ∈ E , define τ n E ∈ E by ( τ n E )( V ) = Nat E ( Sγ n +1 ( V, − ) + , E ) We also have a natural transformation of self-functors on E : ρ n : Id → τ n This natural transformation comes from the map Sγ n +1 ( V, W ) + → Mor ( V, W ) andthe Yoneda lemma.There is another description of Sγ n +1 ( − , − ), by [Wei95, Proposition 4.2] it is a homo-topy colimit: Sγ n +1 ( V, A ) + ∼ = hocolim = U ⊂ R n +1 Mor ( U ⊕ V, A )where the right hand side is the Bousfield-Kan formula for the homotopy colimit ofthe functor U → Mor ( U ⊕ V ) as U varies over the topological category of non-zero11ubspaces of R n +1 and inclusions. This construction is described in great detail in[Lin10, Appendix A]. Thus we see that τ n E ( V ) = holim = U ⊂ R n +1 E ( U ⊕ V )We choose to define τ n in terms of Sγ n +1 ( − , − ) + and we then define polynomialfunctors in terms of τ n . Thus the definition below is [Wei95, Proposition 5.2]. Definition 5.3
A functor E from J to based spaces is said to be polynomial ofdegree less than or equal to n if and only if ( ρ n ) E : E → τ n E is an objectwise weak equivalence of J –spaces. We sometimes say that such an E is n –polynomial . The value of an n –polynomialfunctor E at V is determined, up to homotopy, by the values E ( U ⊕ V ) (and themaps between them) for non-zero subspaces of U of R n +1 . Hence we can think ofan n –polynomial functor as one where it is possible to extrapolate the information of E ( U ) from the spaces E ( U ⊕ V ) (and maps between them).The homotopy fibre of ρ n : E → τ n E measures how far E is from being n –polynomial,thus it would be helpful to be able to identify this fibre. The following lemma, from[Wei95, Section 5], does so and shows the fundamental relation between differentiationand n –polynomial functors. Proposition 5.4
The topological space
Mor n +1 ( V, A ) is the mapping cone (cofibre) ofthe projection Sγ n +1 ( V, A ) + → Mor ( V, A ) . This statement is natural in V and A . Proof
The mapping cone is the pushout of the diagram below, where we use [0 , ∞ ] =[0 , ∞ ) c (with basepoint ∞ ) instead of the unit interval. This helps in identifying thepushout. Sγ n +1 ( V, A ) + / / (cid:15) (cid:15) Mor ( V, A ) + (cid:15) (cid:15) Sγ n +1 ( V, A ) + ∧ [0 , ∞ ] / / P The top horizontal map is the projection, the left vertical map sends a point x to( x, V can be written as a unit vector times some length: S ( V ) × [0 , ∞ ) ∼ = V . Thus writing S V for the one-point compactification of V , wesee that S ( V ) × [0 , ∞ ] ∼ = S V , where any vector of ’infinite length’ is identified with thepoint at infinity in S V .The pushout consists of points ( f, x, t ), where t ∈ [0 , ∞ ] and ( f, x ) ∈ Sγ n +1 ( V, A ),modulo the relations ( f, x, ∞ ) = ( f ′ , x ′ , ∞ ) and ( f, x,
0) = ( f, x ′ , n +1 ( V, W ), it sends any point of form ( f, x, ∞ )to the basepoint and sends ( f, x, t ) to ( f, xt ) for all other t . It is clear that this is awell-defined map; indeed, it is a homeomorphism.12 emma 5.5 For any n ∈ N , V ∈ J and E ∈ E , there exists a natural fibrationsequence res n +10 ind n +10 E ( V ) → E ( V ) → τ n E ( V ) Proof
We have the natural cofibre sequence Sγ n +1 ( V, A ) + → Mor ( V, A ) + → Mor n +1 ( V, A )which is natural in V and A with respect to J . This assembles to give a cofibresequence of J –spaces: Sγ n +1 ( V, − ) + → Mor ( V, − ) → Mor n +1 ( V, − )Now consider the induced maps of spacesNat E ( Sγ n +1 ( V, − ) + , E ) ← Nat E (Mor ( V, − ) , E ) ← Nat E (Mor n +1 ( V, − ) , E )We can identify the above with( τ n E )( V ) ← E ( V ) ← (res n ind n +10 E )( V )which is a fibre sequence for all V . Corollary 5.6
Let E a functor from J to based spaces that is n –polynomial. Then ind n +10 E (and hence ind n +10 ε ∗ E ) is objectwise acyclic. As one would hope from the words used, any ( n − E is n –polynomial. That result is [Wei95, Proposition 5.4], which we reproduce later asproposition 6.7.Our goal is to construct a tower relating the n and ( n − E of E and classify the fibres of this tower. Any such fibre will be n –polynomial and be T n − –contractible, hence we make the following definition, see[Wei95, Definition 7.1]. Definition 5.7
An object E ∈ E is n –homogeneous if it is polynomial of degree atmost n and T n − E is weakly equivalent to a point. We will construct model structures that capture the notion of n –polynomial or n –homogeneous functors in their homotopy theory. To do so, we will need some moretechnical information on n –polynomial functors.A routine exercise in using the long exact sequence of a fibration and the five lemmagives [Wei95, Lemma 5.3], which is stated below. Lemma 5.8
Let g : E → F be a map in E , assume that ind n +10 F is objectwise con-tractible and E is n –polynomial. Then the homotopy fibre of g is an n –polynomialfunctor.
13n particular, this proves that the homotopy fibre of a map between n –polynomialobjects is n –polynomial. We now need [Wei95, Definition 5.9], this condition oftencrops up. The following lemma is an example of why this notion is useful. Definition 5.9
We say that a functor E ∈ E is connected at infinity if the space hocolim k E ( R k ) is connected. The following result is [Wei95, Proposition 5.10] and we follow that proof.
Lemma 5.10
Let g : E → F be a morphism in E between n –polynomial objects suchthat the homotopy fibre of g is objectwise acyclic and F is connected at infinity. Then g is an objectwise weak equivalence. Proof
The problem lies in the fact that at each stage V , the homotopy fibre is definedvia a fixed choice of basepoint in F ( V ), but we need an isomorphism of homotopygroups between E ( V ) and F ( V ) for all choices of basepoints. Let F b ( V ) be thesubspace of F ( V ) consisting of only the basepoint component of F ( V ).We prove that F b → F is a equivalence after applying the functor T n = hocolim k τ kn .Note that since E and F are n –polynomial, the maps E → T n E and F → T n F areobjectwise weak equivalences. Consider the maphocolim k τ kn F b −→ hocolim k τ kn F For each choice of basepoint, the homotopy fibre of τ kn F b → τ kn F is either empty orcontractible. If C is some component in F ( V ) ≃ τ kn F ( V ), then because f is connectedat infinity, there is some l such that the image of C in τ ln F ( V ) is in the basepointcomponent. This holds since τ ln F ( V ) is defined using only the terms F ( V ⊕ U ) for U of dimension greater than or equal to l . Hence C is contained in τ ln F b ( V ) and therecan be no empty fibres.We thus have objectwise weak equivalences T n F b → T n F Consider the map T n E ( V ) → T n F ( V ) and choose some basepoint x in T n F ( V ), thenwe see that x ∈ τ kn F ( V ) for some k . As k increases, eventually x is in the samecomponent as the canonical basepoint of τ kn F ( V ). Hence by our assumptions, thehomotopy fibre for this choice x is contractible. So T n E → T n F is a objectwise weakequivalence and it follows that E → F is a objectwise weak equivalence.Now we turn to [Wei95, lemma 5.11] to show that for τ m preserves n –polynomialfunctors. The proof is simply that homotopy limits commute, (so τ n τ m = τ m τ n ) andthat homotopy limits preserve weak equivalences. Lemma 5.11 If E is an n –polynomial object of E , then so is τ m E for any m > . We will need an improved version of corollary 5.6, specifically, we need [Wei95, Corollary5.12], which we give below. We will see later that this result implies that ind n ε ∗ takesfibrant objects of the n –polynomial model structure on E to fibrant objects of the n –stable model structure on O ( n ) E n . 14 roposition 5.12 If E is an n –polynomial object of E , then for any V ∈ J , wehave a weak equivalence of spaces ind n E ( V ) → Ω n ind n E ( V ⊕ R ) Proof
When n = 0 there is nothing to prove, so let n >
0. By corollary 5.6 we seethat ind n +10 E is objectwise contractible, but this space also appears in the homotopyfibration sequence of lemma 4.6.res n +1 n ind n +10 E ( V ) −→ ind n E ( V ) −→ Ω n ind n E ( V ⊕ R )We claim that ind n E and the functor F defined by the rule V Ω n ind n E ( V ⊕ R )are both n –polynomial. Furthermore we claim that F is connected at infinity.Once we have shown this, we will be able to apply lemma 5.10 to see that we have aweak equivalence for all V ∈ J ind n E ( V ) −→ Ω n ind n E ( V ⊕ R )Our first claim was that ind n E is an n –polynomial object of E . We know that E is n –polynomial and lemma 5.11 tells us that τ n − E is n –polynomial. The homotopyfibre of E → τ n − E is ind n E and by by lemma 5.8 we see that it is n –polynomial.Our second claim was that F was n –polynomial. We know that the functor ind n E is n –polynomial by the first claim, Ω preserves n –polynomial functors, thus Ω n ind n E is n –polynomial. Now we use the fact that if A is n –polynomial, then the functor V A ( V ⊕ R ) is n –polynomial, and thus F is n –polynomial.Our third claim was that F is connected at infinity, this follows since it is the restrictionof an object of E n . As with calculus in the smooth setting, we wish to approximate a functor in E byan n –polynomial functor. This is done by iterating τ n to construct a functorial n –polynomial replacement. From this we can create a new model structure on E , wherethe fibrant objects are n –polynomial. We show that this model structure can alsobe created using a left Bousfield localisation. Combining these two methods tells usmore about the n –polynomial model structure, in particular, we see that it is rightproper and cellular and hence can undergo a right Bousfield localisation. By a carefulchoice of localisation we construct a model category whose cofibrant-fibrant objects areprecisely the n –homogeneous functors of E . We start by introducing the projectivemodel structure model structure on E .When equipped with this model structure, E is a topological model category, in thesense of [Hov99, Definition 4.2.18]. Hence we see that the enrichment, tensor productand cotensor product are well behaved with respect to the model structure. This isanalogous to simplicial model categories, but with topological spaces taking the placeof simplicial sets. 15ecall the notion of cellular model categories from [Hir03, Section 12], this is a strongerversion of cofibrant generation. For example, based topological spaces are cellular.This condition is a requirement for our localisations to exist. The following result is[MMSS01, Theorem 6.5] combined with a routine, but technical, argument to see thatthe resulting model structure is cellular. Lemma 6.1
There is a proper, cellular model structure on the category E where thefibrations and weak equivalences are defined objectwise. This is known as the projectivemodel structure and we simply write E for this model structure. The generatingcofibrations have form Mor ( V, − ) ∧ S n − → Mor ( V, − ) ∧ D n + and the generating acyclic cofibrations have form Mor ( V, − ) ∧ D n + → Mor ( V, − ) ∧ ( D n × [0 , + for V ∈ J and n > . Let [ − , − ] denote maps in the homotopy category of E . Lemma 6.2
The functors Sγ n ( V, − ) + and Mor n ( V, − ) are cofibrant objects of E , for n > . Proof
The homotopy limit used to construct τ n preserves objectwise fibrations andacyclic fibrations by [Hir03, Theorem 18.5.1]. It follows that Sγ n ( V, − ) + is cofibrant.Since Mor n +1 ( V, − ) is the mapping cone of a map between two cofibrant objects, Sγ n ( V, − ) + → Mor ( V, − ), it is also cofibrant. Definition 6.3
Define T n : E → E to be T n E = hocolim E ρ E / / τ n E τ n ρ E / / τ n E τ n ρ E / / . . . The inclusion map ( η n ) E : E → T n E is a natural transformation. Definition 6.4
A map f ∈ E is said to be an T n –equivalence if T n f is an objectwiseweak equivalence. We use the functor T n : E → E to construct a new model structure on E . Themethod is known as Bousfield-Friedlander localisation with respect to T n . Specificallywe apply [Bou01, Theorem 9.3] which is an updated and improved version of [BF78,Theorem A.7]. We will shortly obtain this model structure via a different method, butwe need this version to see that the new model structure is right proper. Proposition 6.5
There exists a proper model structure on E such that a map f isa weak equivalence if and only if f is an T n –equivalence. The cofibrations are thecofibrations of the projective model structure on E . The fibrant objects are preciselythe n –polynomial objects. A map f : X → Y is a fibration if and only if it is an bjectwise fibration and the diagram below is a homotopy pullback in the projectivemodel structure. X f / / ρ (cid:15) (cid:15) Y ρ (cid:15) (cid:15) T n X T n f / / / / T n Y We call this the n –polynomial model structure on E and denote it by n –poly– E . Proof
We need to show the following axioms: • (A1) if f : X → Y is an objectwise weak equivalence, then so is T n f ; • (A2) for each X ∈ E , the maps η, T n η : T n X → T n X are weak equivalences; • (A3) for a pull back square V k / / g (cid:15) (cid:15) X f (cid:15) (cid:15) W h / / / / Y in E , if f is a fibration of fibrant objects such that η : X → T n X, η : Y → T n Y ,and T n h : T n W → T n Y are weak equivalences, then T n k : T n V → T n X is a weakequivalence.Weiss proves axioms A1 and A2 in [Wei95, Theorem 6.3] and [Wei98]. For A3, recallthat finite homotopy limits commute with directed homotopy colimits, and therefore T n preserves finite homotopy limits and A3 follows.It follows immediately that the fibrations of n –poly– E are precisely those objectwisefibrations that are also n –polynomial maps, as defined in [Wei95, Definition 8.1]. Sim-ilarly an n –homogeneous map is an n –polynomial map that is a weak equivalence in( n −
1) –poly– E . Thus we are justified in saying that these model categories containthe homotopy theory that Weiss studies.This new model structure turns the identity functor into the left adjoint of a Quillenpair from the projective model structure on E to the n –polynomial model structure.Id : E −−→←− n –poly– E : IdIf we let [ − , − ] np denote maps in the homotopy category of n –poly– E then we seethat for X and Y in E [ X, T n Y ] ∼ = [ X, Y ] np There is another way to obtain the n –polynomial model structure, here we use theleft Bousfield localisations of [Hir03]. This technique is more modern than that of the T n –localisation above. It has the major advantage that we can now conclude that themodel category n –poly– E is cellular, which we will need shortly.17 roposition 6.6 The model category n –poly– E is the left Bousfield localisation of E with respect to the collection below and hence is cellular and topological. S n = { Sγ n +1 ( V, − ) + → Mor ( V, − ) | V ∈ J } Proof
We know that E is left proper and cellular, hence [Hir03, Theorem 4.1.1] appliesand we see that the localisation L S n E exists and is left proper, cellular and topological.The cofibrations are those from E , the fibrant objects and weak equivalences aredefined in terms of homotopy mapping objects as we describe below. Note that a weakequivalence between fibrant objects is precisely an objectwise weak equivalence.If we can show that n –poly– E has the same weak equivalences as L S n E , then we willknow that these model structures are precisely the same and hence cellular and proper.Since E is enriched over topological spaces and all objects are fibrant, the homotopymapping object from A to B is given by the enrichment Nat E ( b cA, B ) where b c denotescofibrant replacement. See [Hir03, Example 17.2.4] for more details.The domains and codomains of S n are cofibrant by lemma 6.2, hence the fibrant objectsof L S n E are those X such that( ρ n ) X ( V ) X ( V ) = Nat E (Mor ( V, − ) , X ) → Nat E ( Sγ n +1 ( V, − ) + , X ) = ( τ n X )( V )is a weak homotopy equivalence of based spaces for all V . So we see that the fibrantobjects are precisely the n –polynomial objects. The weak equivalences of L S n E arethose maps f : X → Y such that the inducedNat E ( b cY, Z ) → Nat E ( b cX, Z )gives a weak homotopy equivalence of spaces whenever Z is n –polynomial. Recall that[ − , − ] denotes maps in the homotopy category of E and [ − , − ] np denotes maps in thehomotopy category of n –poly– E . Since E is a topological model category we canrelate the homotopy groups of Nat E ( − , − ) to maps in the homotopy category of E : π n Nat E ( b cX, Z ) ∼ = [ X, Ω n Z ] ∼ = [ X, Ω n Z ] np where Z is n –polynomial. Since X → T n X is a T n equivalence it follows that π ∗ Nat E ( b cX, Z ) ∼ = π ∗ Nat E ( b cT n X, Z )Hence the map η : X → T n X is a weak equivalence of L S n E and so the collection of S n –equivalences is precisely the class of T n –equivalences.Now that we know the S n –equivalences are the weak equivalences for the n –polynomialmodel structure, we can give a proof that an ( n − n –polynomial.The following result is a reproduction of [Wei95, Proposition 5.4]. We note that the‘ E –substitutions’ of the reference are precisely S n − –equivalences and hence are T n − –equivalences. Proposition 6.7 If E ∈ E is polynomial of degree at most n − , then it is polynomialof degree at most n . roof What we will actually show is that any S n –equivalence is an S n − –equivalence.Thus we must prove that Sγ n +1 ( V, W ) → Mor ( V, − ) is an S n − –equivalence for any V . By the two-out-of-three property we can reduce this to proving that the map α below is an S n − –equivalence.The standard inclusion R n → R n +1 induces a map of vector bundles γ n ( V, W ) → γ n +1 ( V, W ) and hence a map of their respective unit sphere bundles. α : Sγ n ( V, − ) + → Sγ n +1 ( V, − ) + We can write Sγ n +1 ( V, − ) + as the fibrewise product over Mor ( V, − ) (denoted ⊠ )of Sγ n ( V, − ) + and Sγ ( V, − ) + . Thus we can write Sγ n +1 ( V, − ) + as the homotopypushout of the diagram Sγ n ( V, − ) + p ←− Sγ n ( V, − ) + ⊠ Sγ ( V, − ) + p −→ Sγ ( V, − ) + Writing ǫ n for the n –dimensional trivial bundle, we see that there is a pullback square( ǫ n ⊕ γ n ( R ⊕ V, − )) + / / (cid:15) (cid:15) γ n ( V, − ) + (cid:15) (cid:15) Mor ( R ⊕ V, − ) / / Mor ( V, − )The map p can then be identified as the projection map S ( ǫ n ⊕ γ n ( R ⊕ V, − )) + −→ Mor ( R ⊕ V, − )Hence the vector bundle Sγ n +1 ( V, − ) + is the homotopy pushout of Sγ n ( V, − ) + ←− S ( ǫ n ⊕ γ n ( R ⊕ V, − )) + p −→ Mor ( R ⊕ V, − )If p is an S n − –equivalence, then so is its homotopy pushout, which is α . The unitsphere of a Whitney sum of vector bundles is equal to the fibrewise join of the unitsphere bundles. Hence the domain of p can be written as the homotopy pushout Sγ n ( R ⊕ V, − ) + ←− S n − ∧ Sγ n ( R ⊕ V, − ) + δ −→ S n − ∧ Mor ( R ⊕ V, − )The map δ is an S n − –equivalence, hence the top map in the commutative diagrambelow is an S n − –equivalence. Sγ n ( R ⊕ V, − ) + / / * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ S ( ǫ n ⊕ γ n ( R ⊕ V, − ) + p (cid:15) (cid:15) Mor ( R ⊕ V, − )Since the diagonal map is an element of S n − , it follows that p is an S n − –equivalence,as desired. Corollary 6.8
The identity functor is the left adjoint of a Quillen pair from the n –polynomial model structure to the m –polynomial model structure, for m < n . Id : n –poly– E −−→←− m –poly– E : Id19o define a homotopy theory for n –homogeneous functors, we construct a right Bous-field localisation of the n –polynomial structure. Proposition 6.9
There is a topological model structure on E whose cofibrant-fibrantobjects are precisely the class of n –homogeneous objects that are cofibrant in the pro-jective model structure in E . We denote this model structure n –homog– E . There isa Quillen pair Id : n –homog– E −−→←− n –poly– E : Id The fibrations of n –homog– E are the same as those for n –poly– E . The weak equiva-lences of n –homog– E are those maps f such that res n ind n T n f is an objectwise weakequivalence in E . Proof
We produce this model category by a right Bousfield localisation of n –poly– E at the collection of objects K n = { Mor n ( V, − ) | V ∈ J } Since n –poly– E is right proper and cellular, we are able to use [Hir03, Theorem 5.1.1]to see that this localisation, R K n ( n –poly– E ), exists.The weak equivalences of this model category are called K n –cellular equivalences andare defined in terms of homotopy mapping objects. By [Hir03, Example 17.2.4] we seethat if A is cofibrant in E then the homotopy mapping object from A to X is given byNat E ( A, T n X ). This is different from the homotopy mapping object used previously,as we now want a homotopy mapping object for n –poly– E .A map f : X → Y is a K n –cellular equivalence in R K n ( n –poly– E ) if and only if(ind n T n X )( V ) = / / Nat E (Mor n ( V, − ) , T n X ) T n f ∗ (cid:15) (cid:15) (ind n T n Y )( V ) = / / Nat E (Mor n ( V, − ) , T n Y )is a weak equivalence for all V . Hence we have proven that the weak equivalences ofthis new model structure are those maps f such that res n ind n T n f is an objectwiseweak equivalence of E . Furthermore, any ( n − A is cofibrant in R K n ( n –poly– E ) if and only if it is cofibrant in E and forany K n –cellular equivalence X → Y , the induced mapNat E ( A, T n X ) −→ Nat E ( A, T n Y )is a weak equivalence of spaces. The general theory of right localisations tell us that a K n –cellular equivalence between cofibrant objects is a T n –equivalence.Now we show that the cofibrant–fibrant objects are n –homogeneous. Let A be cofibrantand fibrant in R K n ( n –poly– E ). Then ∗ → T n − A is a K n –cellular equivalence, andhence we have a weak equivalence of spacesNat E ( A, ∗ ) −→ Nat E ( A, T n − A )20hus we have isomorphisms0 = [ A, T n − A ] ∼ = [ A, T n − A ] ( n − p ∼ = [ T n − A, T n − A ] ( n − p ∼ = [ T n − A, T n − A ]This tells us that T n − A is objectwise contractible. Since A is fibrant, we know it is n –polynomial and we now know that A is n –homogeneous.Now we must show that if B is a cofibrant object of E which is n –homogeneous, thenit is cofibrant and fibrant in n –homog– E . We immediately see that B is fibrant, asit is n –polynomial. So consider the cofibrant replacement of B in the n –homogeneousmodel structure f : b cB → B . Since the codomain is fibrant, so is the domain and thuswe know that res n ind n f is an objectwise weak equivalence. We must prove that f itself is an objectwise weak equivalence. We do so by adapting some of the argumentof [Wei95, Corollary 5.13].The homotopy fibre of f , which we call D , is n –polynomial and res n ind n D is ob-jectwise contractible. Consequently, the map D → τ n − D has trivial fibre and bothdomain and codomain are n –polynomial. If we can show that τ n − D is connected atinfinity, then we can conclude that D is n − τ n − commutes with sequential homotopy colimits thennoting that D is connected (in fact contractible) at infinity as D is n –homogeneous.This commutation result is simply a calculation and occurs as [Wei95, Lemma 5.14].The functor D is n –homogeneous as the sequential homotopy colimit used to define T n − will commute with the homotopy pullback used to define D . Hence T n − D isobjectwise contractible. Since D is ( n − D is levelwise equivalent to T n − D . Thus D is objectwise contractible. Another application of lemma 5.10 showsthat f is a objectwise weak equivalence as desired. Remark 6.10
The weak equivalences for the n –homogeneous model structure are de-termined by the functor ind n . So is reasonable to ask if a weak equivalence of n –polynomial objects is determined by the functors ind k for k n . The best ap-proximation to this idea is [Wei95, Theorem 5.15], which describes T n –equivalences interms of something akin to parametrised spectra. It would be valuable and interestingto see if the techniques of [MS06] can be applied to expand upon this result. It is fascinating to see how closely the sets S n and K n are related to orthogonal spectra.The set S n comes from the cofibre sequence of proposition 5.4. Inverting S n has theeffect of killing the objects of K n +1 . As we will see shortly, the set of generatingcofibrations for the n –stable model structure on O ( n ) E n is also a localisation, whichhas also killed the elements of K n +1 . Except that in this case we will use the cofibresequence of proposition 4.5.The case n = 0 is much simpler than the rest. In particular, the 0–homogeneousmodel structure is equal to the 0–polynomial model structure. A fibrant object inthe 0–polynomial model structures is a homotopically constant J –space: the maps X ( V ) → X ( V ⊕ W ) are weak equivalences of spaces for all V and W . A T –equivalenceis a map f ∈ E such that hocolim k f ( R k ) is a weak homotopy equivalence of spaces.A thorough study of this kind of model category appears in [Lin10, Section 15]. The21iscussion at the start of [Wei95, Section 7] and [Lin10, Theorem 1.1] tell us that thehomotopy theory of n –poly– E is the homotopy theory of based spaces. Thus weunderstand the n = 0 case very well.For n >
0, we want to understand the n –homogeneous objects, so we try to find amore structured model category that captures this homotopy theory, while having weakequivalences that are simpler to understand. n -stable model structure We produce a model structure on the category O ( n ) E n so that it is Quillen equivalentto E equipped with the n –homogeneous model structure. For this section, we are onlyinterested in the case n >
0, as the 0–homogeneous model structure on E is the sameas the 0–polynomial model structure.We begin by relating O ( n ) E n to orthogonal spectra, the primary difference being thatan object X in O ( n ) E n has structure maps of form X ( V ) → Ω nW X ( V ⊕ W ). Thusthe model structure that we produce, which we call the n –stable model structure, isa variation of the usual stable model structure to account for the unusual structuremaps.To compare O ( n ) E n with orthogonal spectra, we apply the method of [MM02] and[MMSS01] for describing diagram spectra as diagram spaces. We want to reverse thisprocess and describe J n –spaces as diagram spectra. Definition 7.1
For V a vector space, the one-point compactification of V will bedenoted by S V , hence S R n = S n . Definition 7.2
For each n > , consider the functor nS from I to based spaces,which on an object V takes value nS ( V ) = S nV = S R n ⊗ V . A map f : V → W acts onthe one-point compactification as id ⊗ f : R n ⊗ V → R n ⊗ W . We see that nS has an O ( n )–action by acting on the R n factor. Note that 0 S ( V ) = S for any V . Lemma 7.3
For each n > , nS is a commutative monoid in the category of I –spaces. Proof
The multiplication is Z A,B I ( A ⊕ B, V ) + ∧ S nA ∧ S nB ∼ = Z A,B I ( A ⊕ B, V ) + ∧ S n ( A ⊕ B ) → S nV where the last map is induced by the I –action map of nS .We also see that this multiplication nS ∧ nS → nS is O ( n )–equivariant, with thediagonal action on the domain. 22 roposition 7.4 For each n > , the category E n is equivalent to the category of nS –modules in I –spaces. Similarly the category O ( n ) E n is the category of nS –modules in O ( n ) –equivariant I –spaces. Proof An nS –module in the category of I –spaces is a topological functor X from I to based spaces with an action map X ∧ nS → X .Applying [MMSS01] to this data gives a topological category J n such that J n –spacesare precisely nS –modules in I –spaces. The category J n has the same objects as I and the morphism spaces are given by J n ( U, V ) = nS –mod( U ∗ ∧ nS, V ∗ ∧ nS )where U ∗ is the I –space I ( U, − ) + . This expression can be reduced to Z A ∈I I ( A ⊕ U, V ) + ∧ S nA from which one can see that J n is isomorphic to J n . A specific isomorphism is inducedby the map below I ( A ⊕ U, V ) + ∧ S nA −→ Mor n ( U, V )( f, x ) ( f | U , ( R n ⊗ f ) x )For the equivariant case, an nS –module in the category of O ( n )–equivariant I –spacesis an O ( n )–topological functor Y from I (with trivial action) to O ( n )–spaces with anaction map Y ∧ nS → Y , that is levelwise a map of O ( n )–spaces. The calculation thatthe associated diagram category is isomorphic to J n is similar to the non-equivariantcase.Now we know that we have a category of diagram spectra, we can apply the rest of[MMSS01] to obtain a stable model structure. We now restrict ourselves to the case n > O ( n )–spaces.Here a map is a weak equivalence or fibration if the underlying space map is so. Thegenerating (acyclic) cofibrations have form O ( n ) + ∧ i where i is a generating (acyclic)cofibration for the model category of spaces.There is a levelwise model structure on the categories O ( n ) E n for all n >
0, where theweak equivalences are the levelwise weak equivalences of underlying non-equivariantspaces.
Definition 7.5
In the category O ( n ) E n , a map f : E → F is said to be a levelwisefibration or a levelwise weak equivalence if each f ( V ) : E ( V ) → F ( V ) is a fibrationor weak homotopy equivalence of underlying spaces for each V ∈ I . A cofibration isa map that has the left lifting property with respect to all maps which are levelwisefibrations and levelwise weak equivalences. Let I Top and J Top be the generating sets for the weak homotopy equivalence modelstructure on spaces. The following lemma, which is an application of [MMSS01, Theo-rem 6.5], gives the generating sets for the levelwise model structure on O ( n ) E n .23 emma 7.6 The collections of cofibrations, levelwise fibrations and levelwise weakequivalences form a cellular, proper, topological model structure on the category O ( n ) E n .We denote this model category by O ( n ) E ln . The generating sets are given below. I = { Mor n ( V, − ) ∧ O ( n ) + ∧ i | V ∈ I , i ∈ I Top } J = { Mor n ( V, − ) ∧ O ( n ) + ∧ j | V ∈ I , i ∈ J Top } We continue to follow the pattern for diagram spectra and construct the kinds ofhomotopy groups we need to consider in order to define an n –stable model structure.It is useful to note that a map f ∈ O ( n ) E n is a levelwise fibration or weak equivalenceif and only if i ∗ f ∈ E n is. Similarly, our weak equivalences for the n –stable modelstructure will be defined in terms of underlying non-equivariant spaces. Definition 7.7
The n –homotopy groups of an object X of O ( n ) E n are defined as nπ k ( X ) = colim q π nq + k ( X ( R q )) A map is said to be an nπ ∗ –equivalence if it induces isomorphisms on n –homotopygroups for all integers k . Lemma 7.8
A levelwise weak equivalence of O ( n ) E n is an nπ ∗ –equivalence. Now that we have the set of weak equivalences we are interested in, we should identifythe fibrant objects of the n –stable model structure. The defining property of thesefibrant objects should be that a nπ ∗ –isomorphism between two fibrant objects is alevelwise weak equivalence. Definition 7.9
An object of O ( n ) E n is an n Ω –spectrum if the adjoints of its struc-ture maps X ( V ) → Ω nW X ( V ⊕ W ) are weak homotopy equivalences. Lemma 7.10 An nπ ∗ –equivalence between n Ω –spectra is a levelwise weak equivalence. Proof
We want to show that π k ( f ) : π k ( X ( V )) → π k ( Y ( V )) is an isomorphism for all V and all k >
0. Choose an isomorphism R a → V , then π q ( X ( V )) ∼ = π q ( X ( R a )) ∼ = colim b π q (Ω nb X ( R a + b ))Which is isomorphic to nπ q − na ( X ). Since f is an nπ ∗ –equivalence, it induces anisomorphism π q X ( V ) ∼ = nπ q − na ( X ) → nπ q − na Y ∼ = π q Y ( V )Hence f ( V ) : X ( V ) → Y ( V ) is a weak equivalence of spaces.We will use a left Bousfield localisation to create the n –stable model structure. Thuswe need to identify a class of maps which we will invert to create the n –stable modelstructure from the projective model structure. We will then need to show that this classis generated by a set and that this class coincides with the nπ ∗ –isomorphisms. Thegeneral theory of localisations tells us that the class of weak equivalences is determinedby the fibrant objects, in the sense of the following definition.24 efinition 7.11 We say that a map f : X → Y is an n –stable equivalence if theinduced map on levelwise homotopy categories f ∗ : [ Y, E ] l → [ X, E ] l is an isomorphismfor all n Ω –spectra E . Recall from proposition 4.5 that, for inner-product spaces V and W , there is a map S nW → Mor n ( V, V ⊕ W ) induced by sending x ∈ nW to ( i, x ) ∈ γ n ( V, V ⊕ W ), where i : V → V ⊕ W is the standard inclusion. By the Yoneda lemma, we can turn this intoa map λ nV,W : Mor n ( V ⊕ W, − ) ∧ S nW −→ Mor n ( V, − ) Lemma 7.12
The maps λ nV,W are n –stable equivalences and nπ ∗ –isomorphisms. Proof
These maps have been chosen so that( λ nV,W ) ∗ : Nat O ( n ) E n (Mor n ( V, − ) , X ) −→ Nat O ( n ) E n (Mor n ( V ⊕ W, − ) ∧ S nW , X )is precisely the adjoint of the structure map of X . Thus they are automatically n –stable equivalences. To see that they are nπ ∗ –isomorphisms, we follow the usual cal-culation and see that the result holds as λ nV,W ( U ) gets more and more connected asthe dimension of U increases.If we fix a particular linear isometry V ⊕ W → U (the choice is unimportant), we canuse our description of Mor n ( V, U ) in proposition 7.4 as a coend to write λ nV,W ( U ) : O ( U ) + ∧ O ( U − V − W ) S n ( U − V ) −→ O ( U ) + ∧ O ( U − V ) S n ( U − V ) By the definition of nπ ∗ , it is easy to see that this map is a nπ ∗ –isomorphism if andonly if its suspension by nV is. So we only need study Σ nV λ nV,W ( U ). The advantageof doing so is that O ( U ) will act on the sphere term and we can rewrite it asΣ nV λ nV,W ( U ) : O ( U ) /O ( U − V − W ) + ∧ S nU −→ O ( U ) /O ( U − V ) + ∧ S nU This new map is ( n + 1) dim( U ) − dim( V ) − dim( W )–connected. So when we lookat the nπ k –homotopy groups, we are looking at maps from S nU + k to a space that is( n + 1) dim( U ) − dim( V ) − dim( W )–connected. Since the dimension of U increases inthe colimit, it is clear that we have a isomorphism of the colimits.The collection of λ nV,W is the set of maps we wish to invert. We now need to turn thesemaps into cofibrations. Then we can them to the generating acyclic cofibrations for theprojective model structure to make a generating set for the n –stable model structure.Recall that the pushout product f (cid:3) g of two maps f : A → B and g : X → Y , is definedto be the map f (cid:3) g : A ∧ Y a A ∧ X B ∧ X −→ B ∧ Y Definition 7.13
Let
M λ nV,W be the mapping cylinder of λ nV,W (which is homotopyequivalent to the codomain). Let k nV,W : Mor n ( V ⊕ W, − ) ∧ S nW → M λ nV,W be theinclusion into the top of the cylinder. Now define J ′ = J ∪ { i (cid:3) k nV,W | i ∈ I Top
V, W ∈ I} roposition 7.14 There is a cofibrantly generated, proper, topological model structureon the category O ( n ) E n , called the n –stable model structure . The cofibrations areas for the levelwise model structure, the weak equivalences are the nπ ∗ –isomorphismsand the fibrant objects are the n Ω –spectra. This model category is written O ( n ) E πn . The proof of this result is all but identical to [MMSS01] or [MM02]. As an illustration,we will identify the fibrations of this model structure. But first, we want to justify theuse of the term stable with the following lemma, which is proved in the same manneras for other categories of diagram spectra.
Lemma 7.15
A map f in O ( n ) E n is an nπ ∗ –equivalence if and only if Σ f is an nπ ∗ –equivalence. Lemma 7.16
A map f : E → B has the right-lifting-property with respect to J ′ if andonly if f is an levelwise fibration and the diagram below is always a homotopy pullback. E ( V ) / / (cid:15) (cid:15) Ω nW E ( V ⊕ W ) (cid:15) (cid:15) B ( V ) / / Ω nW B ( V ⊕ W ) Thus the fibrant object are precisely the n Ω –spectra. Proof
Assume that f has the right-lifting-property with respect to J ′ , then it iscertainly an levelwise fibration. So we must check that f has the right-lifting-propertywith respect to i (cid:3) k V,W . This is equivalent to checking that O ( n ) E n ( k ∗ V,W , p ∗ ) is anacyclic fibration of spaces. We know that k V,W is a cofibration of J n –spaces and p is a fibration, so it suffices to show that O ( n ) E n ( k ∗ V,W , p ∗ ) is a weak equivalence. Bythe way we have constructed k V,W all we need to show is that O ( n ) E n (( λ nV,W ) ∗ , p ∗ ) isa weak equivalence. Writing out what this means is precisely the statement that thediagram of the lemma is a homotopy pullback. Carefully reading this argument showsthat the converse is also true. Corollary 7.17
A map f in O ( n ) E n is an nπ ∗ –equivalence if and only if it is an n –stable equivalence. O ( n ) E n and O ( n ) I S
We prove that the category O ( n ) E n is Quillen equivalent to the category of O ( n )–objects in the category of orthogonal spectra. The basic idea for this section (and themodel structure of the previous section) comes from [Wei95, Section 3]. In that sectionWeiss constructs a spectrum Θ E from the data of a ‘symmetric’ object E in E n (suchan object is precisely an object of O ( n ) E n ). His notion of equivalence of spectra thencorresponds to our notion of nπ ∗ –isomorphism.The essential concept is that if one has a spectrum X with an action of O ( n ), one canmake an object O ( n ) E n , which at V takes value X ( R n ⊗ V ).26 efinition 8.1 The category of O ( n ) –equivariant objects in orthogonal spectra, writ-ten as O ( n ) IS , is the category of O ( n ) –objects in E and O ( n ) –equivariant maps.This category has a cofibrantly generated, proper, stable model structure where a map f is a weak equivalence or fibration if and only if it is so when considered as a non-equivariant map in the stable model structure on orthogonal spectra. Thus an object X ∈ O ( n ) IS is a continuous functor X from J to based topologicalspaces such that there is a group homomorphism from O ( n ) into E ( X, X ). It followsthat each space X ( V ) has a group action and the structure maps S W ∧ X ( V ) → X ( V ⊕ W ) are O ( n )–equivariant ( S W has the trivial action). We can also describe X as a functor X : ε ∗ J → O ( n ) Top of categories enriched over O ( n ) Top . Hence wehave a continuous map X ( V, W ) : J ( V, W ) → Top( X ( V ) , X ( W )) O ( n ) We want to find a map of enriched categories α n : J n → J so that any object X of E can be made into an object X ◦ α n of E n . We will then deal with equivariance andshow that the O ( n )–action on X turns X ◦ α n into an object of O ( n ) E n . Definition 8.2
There is a map of topological categories α n : J n → J which sends theobject U to R n ⊗ U = nU and on morphism spaces acts as α n ( U, V ) : J n ( U, V ) −→ J ( nU, nV )( f, x ) ( R n ⊗ f, x )Now consider some X ∈ O ( n ) IS , we have X ◦ α n , a continuous functor from J n tobased spaces. The space X ( nU ) has two different O ( n )–actions on it, the first comesfrom the fact that for any V , X ( V ) is an O ( n )–space. For σ ∈ O ( n ) we denote thisself-map of X ( V ) as σ X ( V ) : X ( V ) → X ( V ). The second action comes from thinkingof an element σ of O ( n ) as a map σ ⊗ U : R n ⊗ U → R n ⊗ U Thus we have an element ( σ ⊗ U, ∈ J ( nU, nU ) applying X to this gives a self-mapof X ( nU ), which we call X ( σ ⊗ U ). By the definition of X , these two actions commute.Now we are ready to make α ∗ n X ∈ O ( n ) E n . Ignoring equivariance, it is just thecomposite functor X ◦ α n . Hence at V , ( α ∗ n X )( V ) = X ( nV ). Now we must equip itwith an O ( n )–action, we let σ ∈ O ( n ) act on ( α ∗ n X )( V ) by X ( σ ⊗ U ) ◦ σ X ( U ) . Thus α ∗ n X takes values in O ( n )–spaces. We must now check that the map α ∗ n X : J n ( U, V ) −→ Top( α ∗ n X ( U ) , α ∗ n X ( V ))is an O ( n )–equivariant map. This will imply that the structure map below is O ( n )–equivariant, where the domain has the diagonal action. Hence α ∗ n X will be an objectof O ( n ) E n . S nV ∧ ( α ∗ n X )( U ) −→ ( α ∗ n X )( U ⊕ V )27t is routine to check that the following diagram commutes.Mor n ( U, V ) α n / / σ (cid:15) (cid:15) Mor ( nU, nV ) X / / ( σ − ⊗ U ) ∗ ◦ ( σ ⊗ V ) ∗ (cid:15) (cid:15) Top( X ( nU ) , X ( nV )) ( X ( σ − ⊗ U )) ∗ ◦ ( X ( σ ⊗ V )) ∗ (cid:15) (cid:15) Mor n ( U, V ) α n / / Mor ( nU, nV ) X / / Top( X ( nU ) , X ( nV ))Applying X ◦ α n to any pair ( g, y ) ∈ Mor ( W, W ′ ) gives an O ( n )–equivariant map X ( nW ) → X ( nW ′ ), it follows that the two expressions below are equal. X ( σ ⊗ W ′ ) ◦ X ( R n ⊗ g, y ) ◦ X ( σ − ⊗ W ) σ X ( W ) ◦ X ( σ ⊗ W ′ ) ◦ X ( R n ⊗ g, y ) ◦ X ( σ − ⊗ W ) ◦ σ − X ( W ′ ) We now describe a left adjoint to α ∗ n . We can think of J ( nU, V ) as a topological spacewith a left action of J and a right action of J n . We can use this to put a J actiononto an object of O ( n ) E n . So let Y ∈ O ( n ) E n , then (using enriched coends) we define( Y ∧ J n J )( V ) = Z U ∈J n Y ( U ) ∧ J ( nU, V )To see that this is the left adjoint of α ∗ n is a formal exercise in manipulating coends. Proposition 8.3
The adjoint pair ( − ) ∧ J n J : O ( n ) E n −−→←− O ( n ) IS : α ∗ n is a Quillen equivalence between O ( n ) E n equipped with the n –stable model structureand O ( n ) IS . Proof
It is routine to check that α ∗ n preserves levelwise fibrations, levelwise trivialfibrations and takes fibrations in O ( n ) E πn to fibrations in O ( n ) IS .A straightforward argument using cofinality shows that a map f ∈ O ( n ) IS is a π ∗ –isomorphism if and only if α ∗ n f is an nπ ∗ –isomorphism. All that remains is to showthat the derived unit is an nπ ∗ –isomorphism. Since our categories are stable, it sufficesto do so for the generator of O ( n ) E n , which is O ( n ) + ∧ nS . Writing nS as Mor n (0 , − ),it is easy to see that O ( n ) + ∧ Mor n (0 , − ) ∧ J n J = O ( n ) + ∧ Mor (0 , − )and that α ∗ n of this is O ( n ) + ∧ Mor n (0 , − ).We note that α ∗ n also has a right adjoint, given in terms of an enriched end. Howeverwe have not been able to show that this right adjoint is part of a Quillen pair, hence wecannot pass directly from E to O ( n ) IS . Instead, we have a zig-zag of Quillen pairs,which is commonplace when working with model categories and no real disadvantage.Now that we fully understand the n –stable model structure on O ( n ) E n , we shouldcompare it with the n –polynomial and n –homogeneous model structures on E .28 Inflation–induction as a Quillen functor
In this section we show that inflation–induction and restriction–orbits form a Quillenpair between the n –stable model structure on O ( n ) E n and E equipped with either the n –polynomial or the n –homogeneous model structures. Lemma 9.1
For n > , there is a Quillen adjunction res n /O ( n ) : O ( n ) E ln −−→←−E : ind n ε ∗ Proof
A generating cofibration takes form Mor n ( V, − ) ∧ O ( n ) + ∧ i , where i is agenerating cofibration for the model category of based spaces. Applying the left adjointto this gives Mor n ( V, − ) ∧ i . By lemma 6.2, Mor n ( V, − ) is a cofibrant object of E . Itfollows that the left adjoint preserves cofibrations. The case of acyclic cofibrations isidentical. Lemma 9.2
For n > , there is a Quillen adjunction res n /O ( n ) : O ( n ) E πn −−→←− n –poly– E : ind n ε ∗ Proof
The identity functor is a left Quillen functor from E to n –poly– E . Sores n /O ( n ) is a left Quillen functor from O ( n ) E n to the n –polynomial model struc-ture on E . To check that this Quillen functor passes to the n –stable model structurewe apply [Hir03, Theorem 3.1.6 and Proposition 3.1.18]. We must show that ind n ε ∗ takes n –polynomial objects of E to n –stable objects of O ( n ) E n . We have done so inproposition 5.12.Composing this adjunction with the change of model structures adjunction between n –poly– E and m –poly– E , for n > m , gives a Quillen pair between O ( n ) E πn and m –poly– E . If X is an m –polynomial functor then, by proposition 6.7, we know thatit is ( n − n ε ∗ X is levelwise contractible by corollary 5.6.Therefore, on homotopy categories, the derived functor of ind n ε ∗ sends every objectof Ho( m –poly– E ) to the terminal object of Ho( O ( n ) E πn ). Lemma 9.3
The left derived functor of res n /O ( n ) is levelwise weakly equivalent to EO ( n ) + ∧ O ( n ) res n ( − ) . Proof
Let X be an object of O ( n ) E πn , then b cX → X is a levelwise acyclic fibrationand b cX is levelwise free. Hence the two maps below are objectwise weak equivalencesof E . EO ( n ) + ∧ O ( n ) res n ( b cX ) −→ res n ( b cX ) /O ( n ) EO ( n ) + ∧ O ( n ) res n ( b cX ) −→ EO ( n ) + ∧ O ( n ) res n X Lemma 9.4
For n > there is a Quillen adjunction res n /O ( n ) : O ( n ) E πn −−→←− n –homog– E : ind n ε ∗ roof We know that a map f is a weak equivalence between fibrant objects in the n –homogeneous model structure if and only if ind n ε ∗ f is a levelwise weak equivalence.Hence this adjunction is a Quillen pair by [Hir03, Proposition 3.1.18].We draw these Quillen pairs to show how they are related, left adjoints will be eitheron the top, or on the left of a pair. O ( n ) E ln res n /O ( n ) / / (cid:15) (cid:15) E n ε ∗ o o / / n –poly– E o o (cid:15) (cid:15) O ( n ) E πn res n /O ( n ) / / O O n –homog– E n ε ∗ o o O O
10 The classification of n -homogeneous functors Now we prove that the homotopy category of O ( n ) E πn is the homotopy category of n –homogeneous functors. Throughout this section we will keep the diagram of Quillenfunctors below in mind. n –homog– E n ε ∗ / / O ( n ) E πn res n /O ( n ) o o ( − ) ∧ J n J / / O ( n ) IS α ∗ n o o Theorem 10.1
For n > , the Quillen adjunction res n /O ( n ) : O ( n ) E πn −−→←− n –homog– E : ind n ε ∗ is a Quillen equivalence Proof
The derived functor of the right Quillen ind n ε ∗ reflects equivalence in the n –homogeneous structure. Thus the only thing to show is that the derived unit is anequivalence in the stable structure.Given some cofibrant X ∈ O ( n ) E n , by proposition 8.3, we have an nπ ∗ –isomorphisms X −→ X f −→ α ∗ n b f ( X ∧ J n J )where b f denotes fibrant replacement in the stable model category on O ( n ) IS . LetΨ denote b f X ∧ J n J , this is an Ω –spectrum with an action of O ( n ). Let b c denotecofibrant replacement in O ( n ) E πn , then one can construct a commutative square X / / (cid:15) (cid:15) b cα ∗ n Ψ (cid:15) (cid:15) ind n ε ∗ T n res n X/O ( n ) / / ind n ε ∗ T n res n ( b cα ∗ n Ψ) /O ( n )We want to know that map 2 is an n –stable equivalence. Map 1 is a stable equivalence,as is 4, since derived functors preserve all weak equivalences. Thus we must show that30ap 3 is a stable equivalence. Lemma 9.3 tells us that the codomain of map 3 islevelwise weakly equivalent toind n ε ∗ T n ( EO ( n ) + ∧ O ( n ) res n ( α ∗ n Ψ)The object T n ( EO ( n ) + ∧ O ( n ) res n ( α ∗ n Ψ) ∈ E is identified in [Wei95, Example 6.4]. Itis shown to be equivalent to the object F Ψ of E defined by V hocolim k Ω nk [ EO ( n ) + ∧ O ( n ) (Ψ( R k ) ∧ S nV )]where θ ∧ S nV has the diagonal action. This calculation is performed via a connectivityargument based on the interplay between homotopy–orbits and the functor Ω ∞ Σ ∞ onequivariant spaces.The derived n –stable equivalence then follows from the calculation of ind n ε ∗ F Ψ , asgiven in [Wei95, Example 5.7]. This example proves that ind m ε ∗ F Ψ is given by V hocolim k Ω nk [ EO ( n − m ) + ∧ O ( n − m ) (Ψ( R k ) ∧ S nV )]We are interested in the case m = n , where we see that hocolim k Ω nk (Ψ( R k ) ∧ S nV ) isweakly equivalent to ( α ∗ n Ψ)( V ).We wish to remark that despite extensive efforts, the authors have been unable toimprove upon the two examples quoted above. It is interesting to note how closely thederivatives of F Ψ are related to the generalised restriction–orbit functors res nm Ψ /O ( n − m ).Composing the right derived functor of ind n ε ∗ with the left derived functor of ( − ) ∧ J n J recovers the classification in [Wei95, Theorem 7.3]. Corollary 10.2
There is an equivalence of homotopy categories:
Ho( O ( n ) IS ) / / Ho( n –homog– E ) o o Now we can show how an n –polynomial object X is made from an ( n − n –homogeneous object. The following is [Wei95, Theorem 9.1]. For X ∈ E , inflation–induction and the left adjoint of α ∗ n determine an object of O ( n ) IS ,denote this object by Ψ nX . Theorem 10.3
For any X ∈ E , n > and V ∈ J , there is a homotopy fibrationsequence Ω ∞ [ EO ( n ) + ∧ O ( n ) (Ψ nX ∧ S nV )] −→ ( T n X )( V ) −→ ( T n − X )( V )31e now give the picture of the tower for X ∈ E . (cid:15) (cid:15) T X (cid:15) (cid:15) Ω ∞ [ EO (3) + ∧ O (3) (Ψ X ∧ S V )] o o T X (cid:15) (cid:15) Ω ∞ [ EO (2) + ∧ O (2) (Ψ X ∧ S V )] o o T X (cid:15) (cid:15) Ω ∞ [ EO (1) + ∧ O (1) (Ψ X ∧ S V )] o o X / / ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ < < ②②②②②②②②②②②②②②②②②② B B ✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆ T X We have completed our task: the constructions of [Wei95] are now realised as Quillenfunctors on model categories, we have classified homogeneous functors via a Quillenequivalence, the relation between homogenous functors and O ( n )–spectra has beenmade precise and orthogonal calculus is ready to be generalised to equivariant or stablesettings.
11 Stable Orthogonal Calculus
As an application, we outline a stable variant of orthogonal calculus, which replacesbased topological spaces with orthogonal spectra. As Σ ∞ is a symmetric monoidalfunctor, [MM02, Lemma II.4.8], we can think of the categories J n as enriched overspectra; in detail, we make a new category K n , with the same objects as J n , whosehom-objects are given by K n ( U, V ) = Σ ∞ Mor n ( U, V )We define the category F n to be the category of O ( n ) IS –enriched functors from K n to O ( n ) IS . Notice this is the same category as taking O ( n ) Top –enriched functorsfrom J n to O ( n ) IS .The required adjunctions which define the notion of differentiation are simply enrichedKan extension. Thus changing the codomain to spectra causes no additional complica-tions. In particular, inflation–induction ind nm : O ( m ) F m → O ( n ) F n is defined as(ind nm CI F )( V ) = Nat O ( m ) F m ( K n ( V, − ) , CI nm F )where CI nm is given by the adjunction( − ) /O ( n − m ) : O ( n ) IS−−→←− O ( m ) IS : CI nm For fibrant E ∈ F , define τ n E ∈ F by( τ n E )( V ) = Nat F (Σ ∞ Sγ n +1 ( V, − ) + , E )32t this point we note that Σ ∞ : Top → IS is left Quillen. In particular, this impliesthe alternative description of τ n still holds for fibrant E ∈ F : τ n E ( V ) = holim = U ⊂ R n +1 E ( U ⊕ V ) Definition 11.1
A functor E from K to spectra is said to be polynomial of degreeless than or equal to n if and only if each spectrum E ( V ) is an Ω -spectrum and ( ρ n ) E : E → τ n E is a levelwise stable equivalence. This second condition is equivalent to asking that foreach V , E ( V ) → holim = U ⊂ R n +1 E ( U ⊕ V ) is a weak equivalence of spectra. In addition, since Σ ∞ : Top → IS is left Quillen, we retain the homotopy cofibresequences below. K n ( R ⊕ V, W ) ∧ S n −→ K n ( V, W ) −→ K n +1 ( V, W )Σ ∞ Sγ n +1 ( V, W ) + −→ K ( V, W ) −→ K n +1 ( V, W )In particular, note that E ∈ F is n –polynomial if and only if ind n +10 E ( V ) is π ∗ –isomorphic to a point, for each V ∈ K .For i >
0, we build the fibrant replacement functor T i from τ i exactly as in definition6.3. Definition 11.2
An object E ∈ E is n –homogeneous if it is polynomial of degreeat most n and T n − E is weakly equivalent to the terminal object. Since O ( n ) IS , n >
0, is a spectra-enriched (or topologically enriched) cellular modelcategory, we may apply our previous work and obtain all the necessary model structures.Specifically, we obtain the n –polynomial and n –homogeneous model structures on F ,as well as the n –stable model structure on O ( n ) F n , in precisely the same manner.We thus have a tower of fibrations relating T n E to T n − E for varying n . The utilityof the tower comes from the identification of the homotopy fibres in terms of inflation–induction, so that is our next task. Theorem 11.3
For n > , the adjoint functors res n /O ( n ) : O ( n ) F πn −−→←− n –homog– F : ind n ε ∗ determine a Quillen equivalence.
33n particular, by simply replacing spectra for spaces, we obtain a zig-zag of Quillenequivalences, where O ( n )( I × I ) S is the category of O ( n )–objects in F . n –homog– F n ε ∗ / / O ( n ) F πn res n /O ( n ) o o ( − ) ∧ K n K / / O ( n )( I × I ) S α ∗ n o o We can easily go further by noting that this category is the category of O ( n )–objectsin orthogonal bi-spectra. Following [Jar97, Corollary 2.30], the “diagonal” functor d : ( I × I ) S → IS as a right adjoint, defines an equivalence of the associated homotopy categories. Inaddition, the diagonal preserves the tensor product of a spectrum with a space, so theequivalence lifts to an equivalence of homotopy categories d ∗ : Ho ( O ( n )( I × I ) S ) ≃ −→ Ho ( O ( n ) IS )It should be routine to extend this to a Quillen equivalence, provided one takes careover the categorical foundations.In summary, we have the following identification of homogeneous functors. Corollary 11.4
There is an equivalence of homotopy categories:
Ho( O ( n ) IS ) / / Ho( n –homog– F ) o o For F ∈ F , we denote the corresponding object in the homotopy category of O ( n ) IS as Φ nF . Having identified the fibres via a Quillen equivalence and in terms of orthogonal spectra,we have obtained the desired stable variant of orthogonal calculus.
Theorem 11.5
A functor F ∈ F determines a tower of fibrations: (cid:15) (cid:15) T F (cid:15) (cid:15) [ EO (3) + ∧ O (3) (Φ F ∧ S V )] o o T F (cid:15) (cid:15) [ EO (2) + ∧ O (2) (Φ F ∧ S V )] o o T F (cid:15) (cid:15) [ EO (1) + ∧ O (1) (Φ F ∧ S V )] o o F / / ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ < < ②②②②②②②②②②②②②②②②②② B B ✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆ T F One can now study interactions between orthogonal calculus and stable homotopytheory, particularly interactions with various localizations of spectra, which the authorsbelieve to be intractable without the current work. For example, let X be an objectof F such that T X is objectwise contractible. If we look at the derivatives of X rationally, then we know by [GS11] that the n th derivative is given by a torsion moduleover the twisted group ring H ∗ (B SO ( n ))[ C ].34 eferences [ALV07] Gregory Arone, Pascal Lambrechts, and Ismar Voli´c. Calculus of functors,operad formality, and rational homology of embedding spaces. Acta Math. ,199(2):153–198, 2007.[BCR07] Georg Biedermann, Boris Chorny, and Oliver R¨ondigs. Calculus of functorsand model categories.
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