aa r X i v : . [ m a t h . A T ] J a n COMPARING THE ORTHOGONAL AND UNITARY FUNCTOR CALCULI
NIALL TAGGART
Abstract.
The orthogonal and unitary calculi give a method to study functors from the category of real orcomplex inner product spaces to the category of based topological spaces. We construct functors between thecalculi from the complexification-realification adjunction between real and complex inner product spaces.These allow for movement between the versions of calculi, and comparisons between the Taylor towersproduced by both calculi. We show that when the inputted orthogonal functor is weakly polynomial, theTaylor tower of the functor restricted through realification and the restricted Taylor tower of the functoragree up to weak equivalence. We further lift the homotopy level comparison of the towers to a commutativediagram of Quillen functors relating the model categories for orthogonal calculus and the model categoriesfor unitary calculus. Introduction
The orthogonal and unitary calculi allow for the systematic study of functors from either the categoryof real inner product spaces, or the category of complex inner product spaces, to the category of basedtopological spaces. The motivating examples are BO( − ) : V BO( V ), where BO( V ) is the classifyingspace of the orthogonal group of V , and BU( − ) : W BU( W ) where BU( W ) is the classifying space of theunitary group of W . The foundations of orthogonal calculus were originally developed by Weiss in [Wei95],and later converted to a model category theoretic framework by Barnes and Oman in [BO13]. From theunitary calculus perspective, it has long been known to the experts, with the foundations and model categoryframework developed by the author in [Tag19].In this paper we use the similarity between real and complex vector spaces, namely the complexification-realification adjunction to give a formal comparison of the calculi both on the homotopy level by comparingthe towers, and on the model category level by constructing Quillen functors between the model categoriesfor orthogonal calculus and the model categories for unitary calculus. In particular, this paper will allow fora more methodical way of using the calculi together, and to transfer calculations between them.We cover the basic background of the calculi in Section 2. With this background in place, we begin with acomparison of the input functors in Section 3. In particular we construct two Quillen adjunctions between theinput categories with precomposition with realification and precomposition with complexification respectivelybeing right Quillen functors.Denote by E O the category of input functors for orthogonal calculus, that is, Top * -enriched functors fromthe category of real inner product spaces to the category of based spaces, and denote by E U the unitarycalculus analogue. For a full definition of these categories, see Definition 2.1. The realification of complexvector spaces induces a functor r ∗ : E O −→ E U , and the complexification of real vector spaces induces afunctor c ∗ : E U −→ E O , full constructions of such are given in Section 3. These functors behave well withrespect to homogeneous and polynomial functors, see Section 4, where we prove the following as Lemma 4.1and Lemma 4.2. Lemma A. (1) If an orthogonal functor F is n -homogeneous, then r ∗ F is n -homogeneous, where r ∗ : E O −→ E U isprecomposition with the realification functor.(2) If a unitary functor F is n -homogeneous, then c ∗ F is (2 n )-homogeneous, where c ∗ : E U −→ E O isprecomposition with the complexification functor. tilising the Taylor tower for an inputted functor F , and the above result on homogeneous functors, weprove the following. This result appears as Theorem 4.3 and Theorem 4.4 in the text. Theorem B. (1) If an orthogonal functor F is n -polynomial, then r ∗ F is n -polynomial, where r ∗ : E O −→ E U isprecomposition with the realification functor.(2) If a unitary functor F is n -polynomial, then c ∗ F is (2 n )-polynomial, where c ∗ : E U −→ E O isprecomposition with the complexification functor.In Section 4 we also construct Quillen adjunctions between the respective n -polynomial and n -homogeneousmodel structures for the calculi.In [Tag19], the author introduced the notion of weakly polynomial functors. These functors have a goodconnectivity relationship with their polynomial approximations. We show, in Section 5, that for weaklypolynomial functors, the restricted Taylor tower through realification agrees with the Taylor tower for thepre-realified functor. The following is Theorem 5.5. The result does not hold in the complexification inducedcase, since restriction through complexification only picks out even degree polynomial approximations. Theorem C.
Let F be a weakly polynomial orthogonal functor. Then the unitary Taylor tower associatedto r ∗ F is equivalent to the pre-realification of the orthogonal Taylor tower associated to F .We leave the homotopy level comparisons here and turn to comparing the model categories in Section 6.This section introduces the goal for the remainder of the paper. We give a complete diagram, Figure 1, ofQuillen adjunctions between the model categories for the orthogonal and unitary calculi. The remainingsections of the paper are devoted to demonstrating how Figure 1 commutes.We start with the categories of spectra in Section 7, and use the change of group functors of Mandell andMay [MM02], to construct Quillen adjunctions between spectra with an action of O( n ), U( n ) and O(2 n )respectively. We utilise the Quillen equivalence between orthogonal and unitary spectra of [Tag19, Theorem6.4] to show that these change of group functors interact in a homotopically meaningful way with the changeof model functor induced by realification.In Section 8 we move to comparing the intermediate categories. These are categories O( n ) E O n and U( n ) E U n constructed by Barnes and Oman [BO13], and the author [Tag19], which act as an intermediate in the zig-zag of Quillen equivalences for orthogonal and unitary calculus respectively. For this, we introduce two newintermediate categories, O( n ) E U n and U( n ) E O n between the standard intermediate categories. These are thestandard intermediate categories with restricted group actions through the subgroup inclusions O( n ) ֒ → U( n )and U( n ) ֒ → O(2 n ). We exhibit Quillen equivalences between these intermediate categories and the standardintermediate categories, completing the picture using change of group functors from [MM02]. The resultingdiagram of intermediate categories is as follows,O( n ) E O n r ∗ ∼ / / O( n ) E U n U( n ) + ∧ O( n ) ( − ) / / r ! o o U( n ) E U nι ∗ o o c ∗ / / U( n ) E O n O(2 n ) + ∧ U( n ) ( − ) / / c ! ∼ o o O(2 n ) E O n . κ ∗ o o where ∼ denotes a Quillen equivalence.Finally, Section 9 completes the task of showing how Figure 1 commutes by giving commutation results forsub-diagrams of Figure 1 on the homotopy category level. Notation and Conventions.
The use of a superscript O is to denote the orthogonal calculus, and asuperscript U is to denote the unitary calculus. When the superscript is omitted, we mean the statementapplies to both orthogonal and unitary calculus.We will refer to the category of based compactly generated weak Hausdorff spaces as the category of basedspaces and denote this category by Top * . This is a cofibrantly generated model category with weak equiva-lences the weak homotopy equivalences and fibrations the Serre fibrations. The set of generating cofibrationsshall be denoted I , and the set of generating acyclic cofibrations, denoted J . cknowledgements. This work forms part of the authors Ph.D project under the supervision of DavidBarnes, and has benefited greatly from both helpful and encouraging conversations during this supervision.This work has also benefited from enlightening conversations with Greg Arone.2.
The Calculi
In this section we give an overview of the theory of orthogonal and unitary calculi. Throughout let F denoteeither R or C and Aut( n ) = Aut( F n ) denote either O( n ) or U( n ). For full details of the theories, see[Wei95, BO13, Tag19].2.1. Input Functors.
Let J be the category of finite-dimensional F -inner product subspaces of F ∞ , and F -linear isometries. Denote by J the category with the same objects as J and morphism space J ( U, V ) = J ( U, V ) + . These categories are Top * -enriched since J ( U, V ) may be topologised as the Stiefel manifold ofdim F ( U )-frames in V . These categories are the indexing categories for the functors under consideration inorthogonal and unitary calculus. Definition 2.1.
Define E to be the category of Top * -enriched functors from J to Top * .The category E O is category of input functors for orthogonal calculus as studied by Weiss and Barnes andOman [Wei95, BO13]. Moreover E U is the category of input functors for unitary calculus, studied by theauthor in [Tag19]. These input categories are categories of diagram spaces as in [MMSS01] hence they canbe equipped with a projective model structure. Proposition 2.2.
There is a cellular, proper and topological model category structure on the category E , with the weak equivalences and fibrations defined to be the levelwise weak homotopy equivalences andlevelwise Serre fibrations respectively. The generating (acyclic) cofibrations are of the form J ( U, − ) ∧ i where i is a generating (acyclic) cofibration in Top * .2.2. Polynomial functors.
Arguably the most important class of functors in orthogonal and unitary calculiare the n -polynomial functors, and in particular the n -th polynomial approximation functor. Here we givea short overview of these functors, for full details on these functors see [Wei95, BO13, Tag19]. Definition 2.3.
A functor F ∈ E is polynomial of degree less than or equal n or equivalently n -polynomial if the canonical map F ( V ) −→ holim = U ⊆ F n +1 F ( U ⊕ V ) =: τ n F ( V )is a weak homotopy equivalence. Definition 2.4.
The n -th polynomial approximation , T n F , of a functor F ∈ E is defined to be the homotopycolimit of the sequential diagram F ρ / / τ n F ρ / / τ n F ρ / / τ n F ρ / / · · · . Since an n -polynomial functor is ( n + 1)-polynomial, see [Wei95, Proposition 5.4], these polynomial approx-imation functors assemble into a Taylor tower approximating a given input functor. Moreover there is amodel structure on E which captures the homotopy theory of n -polynomial functors. Proposition 2.5 ([BO13, Proposition 6.5], [Tag19, Proposition 2.8]) . There is a cellular proper topologicalmodel structure on E where a map f : E −→ F is a weak equivalence if T n f : T n E −→ T n F is a levelwiseweak equivalence, the cofibrations are the cofibrations of the projective model structure and the fibrationsare levelwise fibrations such that E f / / η E (cid:15) (cid:15) F η F (cid:15) (cid:15) T n E T n f / / T n F s a homotopy pullback square. The fibrant objects of this model structure are precisely the n -polynomialfunctors and T n is a fibrant replacement functor. We call this the n -polynomial model structure and it isdenoted n –poly– E .2.3. Homogeneous functors.
The n -th layer of the Taylor tower satisfies the property that it is both n -polynomial, and its ( n − n -homogeneous. Definition 2.6.
A functor F ∈ E is said to be homogeneous of degree less than or equal n or equivalently n -homogeneous if it is both n -polynomial and has trivial ( n − E which captures the homotopy theory of n -homogeneous functors.Denote by D n F the homotopy fibre of the map T n F −→ T n − F . Proposition 2.7 ([BO13, Proposition 6.9], [Tag19, Proposition 3.13]) . There is a topological model structureon E where the weak equivalences are those maps f such that D n f is a weak equivalence in E , the fibrationsare the fibrations of the n -polynomial model structure and the cofibrations are those maps with the left liftingproperty with respect to the acyclic fibrations. The fibrant objects are n -polynomial and the cofibrant-fibrantobjects are the projectively cofibrant n -homogeneous functors.In [Tag19, §8], the author gave further characterisations of the n -homogeneous model structure. These willprove useful in our comparisons. The results hold true for the orthogonal calculus, with all but identicalproofs. Proposition 2.8 ([Tag19, Proposition 8.3]) . A map f : E −→ F is an acyclic fibration in the n -homogeneousmodel structure if and only if it is a fibration in the ( n − D n -equivalence.This allows us to characterise the acyclic fibrations between fibrant objects. Corollary 2.9 ([Tag19, Corollary 8.4]) . A map f : E −→ F between n -polynomial objects is an acyclicfibration in the n -homogeneous model structure if and only if it is a fibration in the ( n − Lemma 2.10 ([Tag19, Lemma 8.5]) . A map f : X −→ Y is a cofibration in the n -homogeneous modelstructure if and only if it is a projective cofibration and an ( n − Corollary 2.11 ([Tag19, Corollary 8.6]) . The cofibrant objects of the n -homogeneous model structure areprecisely those n -reduced projectively cofibrant objects.2.4. The intermediate categories.
In [Wei95], Weiss constructs a zig-zag of equivalences between thecategory of n -homogeneous functors (up to homotopy) and the homotopy category of spectra with an actionof O( n ). In [BO13], Barnes and Oman put this zig-zag into a model category theoretic framework via a zig-zag of Quillen equivalences between the n -homogeneous model structure on E O , and spectra with an actionof O( n ). This zig-zag moves through an intermediate category, denote O( n ) E O n . In [Tag19], the authorconstructs a similar zig-zag of Quillen equivalences between the unitary n -homogeneous model structure andspectra with an action of U( n ). We give an overview of the construction of these intermediate categoriesand how they relate to spectra and the n -homogenous model structure.Sitting over the space of linear isometries J ( U, V ) the the n -th compliment vector bundle, with total space γ n ( U, V ) = { ( f, x ) : f ∈ J ( U, V ) , x ∈ F n ⊗ F f ( U ) ⊥ } where we have identified the cokernel of f with f ( U ) ⊥ , the orthogonal compliment of f ( U ) in V . Definition 2.12.
Define J n to be the category with the same objects as J and morphism space J n ( U, V )given by the Thom space of the vector bundle γ n ( U, V ).With this, we may define the intermediate categories. efinition 2.13. Define E n to be the category of Top * -enriched functors from J n to Top * , and definethe n -th intermediate category Aut( n ) E n to be the category of Aut( n ) Top * -enriched functors from J n toAut( n ) Top * .Let n S be the functor given by V S nV where nV := F n ⊗ F V . By [BO13, Proposition 7.4] and [Tag19,Proposition 4.2] the intermediate categories are equivalent to a category of n S -modules and hence comeequipped with an n -stable model structure similar to the stable model structure on spectra. The weakequivalences of the n -stable model structure are given by nπ ∗ -isomorphisms. Theses are defined via thestructure maps of objects in Aut( n ) E n , and as such have slightly different forms depending on whether oneis in the orthogonal or unitary setting.For X ∈ O( n ) E O n , nπ k ( X ) = colim q π k + q X ( R q ) , and for Y ∈ U( n ) E U n , nπ k ( Y ) = colim q π k +2 q Y ( C q ) . Proposition 2.14 ([BO13, Proposition 7.4], [Tag19, Proposition 5.6]) . There is a cofibrantly generated,proper, topological model structure on the category Aut( n ) E n , where the weak equivalences are the nπ ∗ -isomorphisms, the cofibrations are those maps with the left lifting property with respect to all levelwiseacyclic fibrations and the fibrations are those levelwise fibrations f : X −→ Y such that the diagram X ( V ) / / (cid:15) (cid:15) Ω nW X ( V ⊕ W ) (cid:15) (cid:15) Y ( V ) / / Ω nW Y ( V ⊕ W ) . is a homotopy pullback square for all V, W ∈ J n .The fibrant objects of the n -stable model structure are called n Ω-spectra and have the property that X ( V ) −→ Ω nW X ( V ⊕ W )is a levelwise weak equivalence. This property can clearly be deduced from the above diagram by consideringthe map X −→ ∗ .To give the Quillen equivalence between these intermediate categories and spectra with an action of Aut( n )we now consider the calculi separately. The constructions are similar for both calculi but it is convenient tohave different notation for the functors involved. We start with the unitary case. Define α n : J U n −→ J U tobe the functor given on objects by α n ( V ) = C n ⊗ C V , and given on morphisms by α n ( f, x ) = ( C n ⊗ C f, x ).This defines a U( n ) Top * -enriched functor when J U is equipped with the trivial U( n )-action. For full details,see [Tag19, Proposition 6.7]. Proposition 2.15 ([Tag19, Theorem 5.8]) . There is a series of Quillen equivalencesU( n ) E U n ( α n ) ! / / Sp U [U( n )] ( α n ) ∗ o o r ! / / Sp O [O( n )] r ∗ o o with ( α n ) ∗ Θ( V ) = Θ( C n ⊗ C V ), and ( α n ) ! is the left Kan extension along α n .The orthogonal case is similar, full details may be found in [BO13, §8]. Proposition 2.16 ([BO13, Proposition 8.3]) . There is a Quillen equivalence( β n ) ! : O( n ) E O n / / Sp O [O( n )] : ( β n ) ∗ o o with ( β n ) ∗ Θ( V ) = Θ( R n ⊗ R V ), and ( β n ) ! is the left Kan extension along β n . .5. The derivatives of a functor.
We now move on to discussing the derivatives of a functor. The deriva-tives are naturally objects in Aut( n ) E n . Their definition comes from constructing an adjunction between E and Aut( n ) E n . The inclusion F m −→ F n onto the first m -coordinates induces a functor i nm : J m −→ J n . Definition 2.17.
Define the restriction functor res n : E n −→ E m to be precomposition with i nm , and definethe induction functor ind nm : E m −→ E n to be the right Kan extension along i nm . In the case m = 0, theinduction functor ind n is called the n -th derivative .Combining this adjunction with a change of group action from [MM02] provides an adjunctionres n / Aut( n ) : Aut( n ) E n / / E : ind n ε ∗ o o . This adjunction is a Quillen equivalence between the n -homogeneous model structure on E and the n -stablemodel structure on Aut( n ) E n . We will refer to the right adjoint as inflation-induction. Proposition 2.18 ([BO13, Theorem 10.1], [Tag19, Theorem 6.5]) . The adjoint pairres n / Aut( n ) : Aut( n ) E n / / n –homog– E : ind n ε ∗ o o is a Quillen equivalence.2.6. Classification of n -homogeneous functors. For a functor F ∈ E U , inflation-induction and the leftadjoint to ( α n ◦ r ) ∗ determine a spectrum Ψ nF with an action of U( n ). That is, Ψ nF = ( α n ◦ r ) ! ind n ε ∗ F .Moreover, for F ∈ E O , inflation-induction and the left adjoint to ( β n ) ∗ defines a spectrum with an action ofO( n ), which we again denote by Ψ nF . Proposition 2.19 ([Wei95, Theorem 7.3],[Tag19, Theorem 7.1]) . Let F ∈ E be n -homogeneous for some n >
0. Then F is levelwise weakly equivalent to the functor defined as U Ω ∞ [( S nU ∧ Ψ nF ) h Aut( n ) ] . Weak Polynomials.
An important class of functors, introduced by the author in [Tag19] are the weakpolynomial functors. These functors have a good connectivity relationship with the polynomial approxima-tions and result in a convergent Taylor tower. We give an overview of the theory here, noting that the proofsprovided by the author in [Tag19, §9] work in the orthogonal setting also.
Definition 2.20.
A map p : F −→ G in E is an order n agreement if there is some ρ ∈ N and b ∈ Z suchthat p U : F ( U ) −→ G ( U ) is (( n + 1) dim R ( U ) − b )-connected for all U ∈ J , satisfying dim F ( U ) ≥ ρ . We willsay that F agrees with G to order n if there is an order n unitary agreement p : F −→ G between them.When two functors agree to a given order, their Taylor tower agree to a prescribed level. The first result inthat direction is the unitary analogue of [Wei98, Lemma e.3]. Lemma 2.21 ([Wei98, Lemma e.3],[Tag19, Lemma 9.5]) . let p : G −→ F be a map in E . Suppose that thereis b ∈ Z such that p U : G ( U ) −→ F ( U ) is (( n + 1) dim R ( U ) − b )-connected for all U ∈ J with dim F ( U ) ≥ ρ .Then τ n ( p ) U : τ n ( G ( U )) −→ τ n ( F ( U ))is (( n + 1) dim R ( U ) − b + 1)-connected for all U ∈ J .Iterating this result, gives the following. Lemma 2.22 ([Wei98, Lemma e.7],[Tag19, Lemma 9.6]) . If p : F −→ G is an order n agreement, then T k F −→ T k G is a levelwise weak equivalence at U ∈ J with dim( U ) ≥ ρ , for k ≤ n .Agreement with the n -polynomial approximation functor for all n ≥ Lemma 2.23 ([Tag19, Lemma 9.10]) . If for all n ≥
0, a unitary functor F agrees with T n F to order n thenthe Taylor tower associated to F converges to F ( U ) at U with dim F ( U ) ≥ ρ . efinition 2.24. A unitary functor F is weakly ( ρ, n ) -polynomial if the map η : F ( U ) −→ T n F ( U ) is anagreement of order n whenever dim F ( U ) ≥ ρ . A functor is weakly polynomial if it is weakly ( ρ, n )-polynomialfor all n ≥ Remark 2.25.
In the above definition of weakly polynomial, we require that the functor is weakly ( ρ, n )-polynomial for all n . Here ρ is permitted to depend on n , i.e. the functor may be weakly ( ρ n , n )-polynomialfor all n , so long as, the sequence ( ρ n ) n ≥ is bounded above, in which case, one may take ρ to the the upperbound of this sequence, hence why we have fixed a ρ in the definition.The following result is useful for identifying weakly polynomial functors. Theorem 2.26 ([Tag19, Theorem 9.14]) . Let
E, F ∈ E are such that there is a homotopy fibre sequence E ( U ) −→ F ( U ) −→ F ( U ⊕ V )for U, V ∈ J . Then(1) If F is weakly ( ρ, n )-polynomial, then E is weakly ( ρ, n )-polynomial; and(2) If E is weakly ( ρ, n )-polynomial and F ( U ) is 1-connected whenever dim F ( U ) ≥ ρ , then F is weakly( ρ, n )-polynomial. 3. Comparing the input functors
Let V ∈ J O , then the complexification of V , C ⊗ R V , is a complex vector space such thatdim C C ⊗ R V = dim R V. Given a R -inner product h− , −i V on V , there is a well defined C -inner product on C ⊗ V , given by h ( a + ib ) ⊗ v, ( c + id ) ⊗ w i = h av, cw i V + h bv, dw i V + i h bv, cw i V − i h av, dw i V where a + ib, c + id ∈ C , and v, w ∈ V. The complexification of an R -linear map T is given by T C = C ⊗ T . Moreover in the finite dimensional casethe matrices representing T and C ⊗ T are equal (corresponding to the inclusion O( n ) ֒ → U( n )) and we getcharacterisations of images and kernels,ker( C ⊗ R T ) = C ⊗ R ker( T ) and im( C ⊗ R T ) = C ⊗ R im( T ) . Given a R -linear isometry, T : V −→ W , C ⊗ T : C ⊗ V −→ C ⊗ W, c ⊗ v c ⊗ T ( v ), is a C -linear isometry,that respects the inner product. It follows that complexification gives a well defined functor c : J O −→ J U .The “opposite operation” to complexification is that of realification. Let W be a complex vector space,then its realification W R is the set W with vector addition and scalar multiplication by reals inheritedunchanged from W and the complex multiplication “forgotten”. If { e , . . . , e n } is a basis for W then { e , . . . , e n , ie , . . . , ie n } is a basis for W R . It follows thatdim R W R = 2 dim C W = dim R W. Up to isomorphism it suffices to check that there is a well defined inner product on the realification of C n induced by the Hermitian inner product on C n . Recall for vectors c = ( c i ) , c ′ = ( c ′ i ) in C n , the Hermitianinner product is given by h c , c ′ i C = n X i =1 c i c ′ i . To obtain a real inner product on R n = ( C n ) R , we realise the vectors c and c ′ as c = a + i b and c ′ = a ′ + i b ,where a , a ′ , b , b ′ ∈ R n . We then define a real inner product on R n as h ( a , b ) , ( a ′ , b ′ ) i R = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 c i c ′ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = |h c , c ′ i| nder the identification c j = a j + ib j , c ′ j = a ′ j + ib ′ j and where ( a , b ) is notation for the vector( a , b , a , b , · · · , a n , b n ) ∈ R n . If T : C k −→ C m is a C -linear map then we may view it as a R -linear map T R : ( C k ) R −→ ( C m ) R . Inparticular, thinking of T as a matrix, we can write T = ( t ij ) i,j , we can rewrite T as a + ib a + ib · · · a k + ib k ... ... ... a m + ib m a m + ib m · · · a mk + ib mk by rewriting each entry as t ij = a ij + ib ij for a ij , b ij ∈ R . Then T R has matrix representation of the form a − b a − b · · · a k − b k b a b a · · · b k a k ... ... ... ... ... ... a m − b m a m − b m · · · a mk − b mk b m a m b m a m · · · b mk a mk where we block decompose each individual entry. It follows thatker( T R ) = (ker( T )) R and im( T R ) = (im( T )) R . If T : V −→ W is a C -linear isometry, then T R : V R −→ W R is a R -linear isometry, and it follows thatrealification gives a well defined functor r : J U −→ J O .3.1. Realification and complexification induce Quillen functors.
For an orthogonal functor F ∈ E O ,precomposition with r , which we call “pre-realification” defines a unitary functor r ∗ F : J U −→ Top * . Hence pre-realification defines a functor r ∗ : E O −→ E U , which has a left adjoint r ! given by the formula,( r ! E )( V ) = Z W ∈ J U E ( W ) ∧ J O ( W R , V ) , i.e. r ! is the left Kan extension along r .Similarly, complexification defines a functor called “pre-complexification”, c ∗ : E U −→ E O , which has leftadjoint c ! given by the left Kan extension along c . These functors are homotopically meaningful when oneconsiders the projective model structures on the categories of input functors. Lemma 3.1.
The adjoint pair r ! : E U / / E O : r ∗ o o is a Quillen adjunction, when both categories are equipped with their projective model structures. Proof.
Let f : E −→ F be a levelwise fibration (resp. levelwise weak equivalence). Then by definition r ∗ f : r ∗ E −→ r ∗ F is a levelwise fibration (resp. levelwise weak equivalence). Hence r ∗ preserves fibrationsand acyclic fibrations. (cid:3) Lemma 3.2.
The adjoint pair c ! : E O / / E U : c ∗ o o is a Quillen adjunction, when both categories are equipped with their projective model structures. Proof.
The proof is all but identical to that of Lemma 3.1. (cid:3) . Comparing the polynomial and homogeneous functors
Homogeneous functors.
Heuristically, the homogeneous functors are the building blocks of the Taylortowers of the calculi. As such we start with a direct comparison between these functors. This comparison isreliant on the classifications of homogeneous functors from orthogonal and unitary calculi.
Lemma 4.1.
If an orthogonal functor F is n -homogeneous, then r ∗ F is n -homogeneous. Proof.
Let F be an n -homogeneous orthogonal functor. Then by the characterisation, Proposition 2.19, F is levelwise weakly equivalent to the functor V Ω ∞ [( S R n ⊗ R V ∧ Ψ nF ) h O( n ) ]where Ψ nF is an orthogonal spectrum with an O ( n )-action. It follows that pre-realification of F is levelwiseweakly equivalent to the functor W Ω ∞ [( S R n ⊗ R W R ∧ Ψ nF ) h O( n ) ] . Using the derived change of group functor, we construct an orthogonal spectrum with an action of U( n ),U( n ) + ∧ L O( n ) Ψ nF := U( n ) + ∧ O( n ) ( E O( n ) + ∧ Ψ nF ) . By the classification of n -homogeneous unitary functors, Proposition 2.19, there is an n -homogeneous functor F ′ associated to the above spectrum, given by W Ω ∞ [( S C n ⊗ C W ∧ (U( n ) + ∧ O( n ) ( E O( n ) + ∧ Ψ nF ))) h U( n ) ] . By [MM02, Proposition V.2.3], F ′ ( W ) is isomorphic toΩ ∞ [U( n ) + ∧ O( n ) (( ι ∗ S C n ⊗ C W ∧ ( E O( n ) + ∧ Ψ nF ))) h (U( n ) ] . The U( n )-action on U( n ) + ∧ O( n ) (( ι ∗ S C n ⊗ C W ∧ ( E O( n ) + ∧ Ψ nF ))) is free (( E O( n ) + is a free O( n )-space),hence taking homotopy orbits equates to taking strict orbits. Hence there is an isomorphism F ′ ( W ) ∼ = Ω ∞ [(U( n ) + ∧ O( n ) (( ι ∗ S C n ⊗ C W ∧ ( E O( n ) + ∧ Ψ nF )))) / U( n )] . The strict U( n )-orbits of the spectrum U( n ) + ∧ O( n ) (( ι ∗ S C n ⊗ C W ∧ ( E O( n ) + ∧ Ψ nF )) are isomorphic to theO( n )-orbits of the spectrum, ι ∗ S C n ⊗ C W ∧ ( E O( n ) + ∧ Ψ nF )), hence F ′ ( W ) is isomorphic toΩ ∞ [( ι ∗ S C n ⊗ C W ∧ ( E O( n ) + ∧ Ψ nF ))) / O( n )] . This last is precisely Ω ∞ [( ι ∗ S C n ⊗ C W ∧ Ψ nF ) h O( n ) ]as homotopy orbits is the left derived functor of strict orbits and smashing with E O( n ) + is a cofibrantreplacement in the projective model structure.Since the action of O( n ) on ι ∗ S C n ⊗ C W is equivalent to the O( n ) action on S R n ⊗ R W R and the one-pointcompactification are isomorphic, the above infinite loop space is isomorphic toΩ ∞ [( S R n ⊗ R W R ∧ Ψ nF ) h O( n ) ] . By the characterisation of n -homogeneous orthogonal functors, we see that this is levelwise weakly equivalentto F ( W R ) = ( r ∗ F )( W ) . (cid:3) Lemma 4.2.
If a unitary functor E is n -homogeneous, then c ∗ E is (2 n )-homogeneous. Proof.
Since E is n -homogeneous, E ( W ) ≃ Ω ∞ [( S C n ⊗ W ∧ Ψ nE ) h U( n ) ] . By definition ( c ∗ E )( V ) = E ( C ⊗ R V ) ≃ Ω ∞ [( S C n ⊗ C C ⊗ R V ∧ Ψ nE ) hU ( n )] . It follows that O(2 n ) + ∧ L U( n ) Ψ nE s an orthogonal spectrum with O(2 n )-action. The classification of homogeneous functors in orthogonalcalculus, Proposition 2.19 gives a (2 n )-homogeneous functor, V Ω ∞ [( S R n ⊗ V ∧ (O(2 n ) ∧ L U( n ) Ψ nE )) h O(2 n ) ] . A similar argument to Lemma 4.1 yields the result. (cid:3)
Polynomial functors.
Using the above results on pre-realification and pre-complexification of homo-geneous functors, we can compare polynomial functors.
Theorem 4.3.
If an orthogonal functor F is n -polynomial, then r ∗ F is an n -polynomial unitary functor,that is, the map r ∗ T O n F −→ T U n ( r ∗ T O n F )is a levelwise weak equivalence for every F ∈ E O . Proof.
We argue by induction on the polynomial degree. The case n = 0 follows by definition. Assume themap r ∗ T O n − F −→ T U n − ( r ∗ T O n − F ) is a levelwise weak equivalence. There is a homotopy fibre sequence T O n F −→ T O n − F −→ R O n F where R O n F is n -homogeneous, [Wei95, Corollary 8.3]. Lemma 4.1 implies that r ∗ R O n F is n -homogeneousin E U , and in particular n -polynomial. As homotopy fibres of maps between n -polynomial objects are n -polynomial, the homotopy fibre of the map r ∗ T O n − F −→ r ∗ R O n F is n -polynomial. Computation of homotopyfibres is levelwise, hence the homotopy fibre in question is r ∗ T O n F , and it follows that r ∗ T O n F −→ T U n ( r ∗ T U n F )is a levelwise weak equivalence. (cid:3) Theorem 4.4.
If an unitary functor E is n -polynomial, then c ∗ F is (2 n )-polynomial, that is, the map c ∗ T U n E −→ T O n ( c ∗ T U n E )is a levelwise weak equivalence for all E ∈ E U . Proof.
The argument follows as in Theorem 4.3 using Lemma 4.2 in place of Lemma 4.1. (cid:3)
Polynomial model structures.
We turn our attention to a model structure comparison. The n -polynomial model structures are left Bousfield localisations, and we use this to our advantage to construct aQuillen adjunction between the n -polynomial model structure for orthogonal calculus and the corresponding n -polynomial model structure for unitary calculus. Lemma 4.5.
The adjoint pair r ! : n –poly– E U / / n –poly– E O : r ∗ o o is a Quillen adjunction. Proof.
Composition of left (resp. right) Quillen functors results in a left (resp. right) Quillen functor, assuch the Quillen adjunction of Lemma 3.1 extends to a Quillen adjunction r ! : E U / / n –poly– E O : r ∗ o o via the Quillen adjunction : E O / / n –poly– E O : o o . Moreover pre-realification sends fibrant objects in n –poly– E O to fibrant objects in n –poly– E U by Theorem4.3, an application of [Hir03, Proposition 3.3.18 and Theorem 3.1.6] yields the result. (cid:3) emma 4.6. The adjoint pair c ! : (2 n ) –poly– E O / / n –poly– E U : c ∗ o o is a Quillen adjunction. Proof.
This follows similarly to Lemma 4.5, using Theorem 4.4 in place of Theorem 4.3. (cid:3)
Homogeneous model structures.
The homogeneous model structures are right Bousfield localisa-tions of the n -polynomial model structures. This fact, together with the further characterisations of thehomogeneous model structure provided by the author in [Tag19], allow for the construction of a Quillenadjunction between the orthogonal n -homogeneous model structure and the unitary n -homogeneous modelstructure. Proposition 4.7.
The adjoint pair r ! : n –homog– E U / / n –homog– E O : r ∗ o o is a Quillen adjunction. Proof.
First suppose that f : E −→ F is a fibration in n –homog– E O . It follows that f is a fibration inthe n -polynomial model structure, and hence r ∗ f is a fibration in n –poly– E U by Lemma 4.5, and hence n –homog– E U .Suppose further that f in an acyclic fibration in n –homog– E O . By Proposition 2.8, it follows that f isan ( n − D O n -equivalence. As above it follows that r ∗ f is a fibration in( n −
1) –poly– E U . In particular the homotopy fibre of r ∗ f is ( n − n –homog– E U hence, since n –homog– E U is stable, r ∗ f is a weak equivalence in n –homog– E U . (cid:3) Proposition 4.8.
The adjoint pair c ! : (2 n ) –homog– E O / / n –homog– E U : c ∗ o o is a Quillen adjunction. Proof.
The proof follows almost verbatim from Proposition 4.7. (cid:3)
Remark 4.9.
Without a clearer understanding on how the pre-realification and pre-complexification functorsbehave with respect to the polynomial approximations, it is not possible to say that they preserve all n -homogeneous equivalences. In particular, if f : X −→ Y is an n -homogeneous equivalence then D n X islevelwise weakly equivalent to D n Y , and there is a diagram of homotopy fibre sequences the form D n X / / ≃ (cid:15) (cid:15) T n X / / (cid:15) (cid:15) T n − X (cid:15) (cid:15) D n Y / / T n Y / / T n − Y, which after applying r ∗ or c ∗ results in diagrams of homotopy fibre sequences (since fibre sequences aredefined levelwise) r ∗ D O n X / / ≃ (cid:15) (cid:15) r ∗ T O n X / / (cid:15) (cid:15) r ∗ T O n − X (cid:15) (cid:15) r ∗ D O n Y / / r ∗ T O n Y / / r ∗ T O n − Y. c ∗ D U n X / / ≃ (cid:15) (cid:15) c ∗ T U n X / / (cid:15) (cid:15) c ∗ T U n − X (cid:15) (cid:15) c ∗ D U n Y / / c ∗ T U n Y / / c ∗ T U n − Y. ince we do not have a useful relation between T U n ( r ∗ X ) and r ∗ T O n ( X ), nor between T O n ( c ∗ X ) and c ∗ T U n ( X ),it is difficult to saying anything meaningful about how the above diagrams relate to the following diagram D U n ( r ∗ X ) / / (cid:15) (cid:15) T U n ( r ∗ X ) / / (cid:15) (cid:15) T U n − ( r ∗ X ) (cid:15) (cid:15) D U n ( r ∗ Y ) / / T U n ( r ∗ Y ) / / T U n − ( r ∗ Y ) . D O n ( c ∗ X ) / / (cid:15) (cid:15) T O n ( c ∗ X ) / / (cid:15) (cid:15) T O n − ( c ∗ X ) (cid:15) (cid:15) D O n ( c ∗ Y ) / / T O n ( c ∗ Y ) / / T O n − ( c ∗ Y ) . Comparing weakly polynomial functors
Agreement.
The notion of agreement plays a central role in the theory or orthogonal and unitarycalculus, for example it is crucial to the proof that the n -th polynomial approximation in n -polynomial, see[Wei98]. The pre-realification and pre-complexification functors behave well with respect to functors whichagree to a certain order. Lemma 5.1. If p : F −→ G in E O is an order n orthogonal agreement, then r ∗ p : r ∗ F −→ r ∗ G in E U is anorder n unitary agreement. Proof.
Since p is an order n orthogonal agreement, there is an integer b ∈ Z , such that p V : F ( V ) −→ G ( V )is ( − b + ( n + 1) dim R V )-connected. It follows by definition that ( r ∗ p ) W = p W R : F ( W R ) −→ G ( W R ) is( − b +( n +1) dim R W R )-connected. Since dim R W R = dim R W , it follows that ( r ∗ p ) W is ( − b +( n +1) dim R W )-connected, and hence r ∗ p is an order n unitary agreement. (cid:3) Lemma 5.2. If p : F −→ G in E U is an order n unitary agreement, then c ∗ p : c ∗ F −→ c ∗ G in E O is anorder 2 n unitary agreement. Proof.
Since p is an order n unitary agreement, there is an integer b , such that p V : F ( V ) −→ G ( V ) is( − b + ( n + 1) dim R V )-connected. It follows by definition that ( c ∗ p ) W = p C ⊗ W : F ( C ⊗ W ) −→ G ( C ⊗ W )is ( − b + ( n + 1) dim R ( C ⊗ W ))-connected. Since dim R C ⊗ W = 2 dim C C ⊗ W = 2 dim R W , it follows that( c ∗ p ) W is ( − b + 2( n + 1) dim R W )-connected, and hence in particular ( − b + (2 n + 1) dim R W )-connected,hence c ∗ p is an order 2 n unitary agreement. (cid:3) Remark 5.3.
In [BE16] Barnes and Eldred give a tower level comparison between Goodwillie calculus andorthogonal calculus. This relies on the functor F from Goodwillie calculus being stably n -excisive, that is,the functor must behave well with respect to (co)Cartesian cubes, see [Goo92, Definition 4.1] for the precisedefinition. The key property gained by a stably n -excisive functor is that the polynomial approximation map p n : F −→ P n F in Goodwillie calculus is an agreement of order n in the Goodwillie calculus setting. Thisallows for a clear comparison between the n -polynomial approximation functors of Goodwillie and orthogonalcalculi. Unfortunately we have been unable to show that, in general, the map F −→ T n F is an agreementof order n in unitary calculus.5.2. Weak Polynomial.
Despite the fact that not all functors agree to a specific order with their polynomialapproximations, there is a large class of functors which do. These functors, which were introduced in [Tag19],are called weakly polynomial and interact meaningfully with the comparisons.
Proposition 5.4. (1) If F ∈ E O is ( ρ, n )-polynomial then the map T U n ( r ∗ η n ) : T U n ( r ∗ F ) −→ T U n ( r ∗ T O n F )is a levelwise weak equivalence for all V with dim( V ) ≥ ρ . Thus, at V with dim( V ) ≥ ρ , the n -thpolynomial approximation of r ∗ F is given by the map r ∗ F −→ r ∗ ( T O n F ).(2) If F ∈ E U is ( ρ, n )-polynomial then T O n ( c ∗ η n ) : T O n ( c ∗ F ) −→ T O n ( c ∗ T U n F ) s a levelwise weak equivalence for all V with dim( V ) ≥ ρ . Thus, at V with dim( V ) ≥ ρ , the (2 n )-thpolynomial approximation of c ∗ F is given by the map c ∗ F −→ c ∗ ( T O n F ). Proof.
We prove part (1), part (2) is similar. There is a commutative diagram r ∗ F / / (cid:15) (cid:15) r ∗ ( T O n F ) (cid:15) (cid:15) T U n ( r ∗ F ) / / T U n ( r ∗ T O n )in which in right hand vertical map is a levelwise weak equivalence by Theorem 4.3. Furthermore, the tophorizontal map is an order n orthogonal agreement, since F is weakly ( ρ, n )-polynomial and pre-realificationpreserves agreements, Lemma 5.1. The lower horizontal map is then a levelwise weak equivalence at V in therequired range by Lemma 2.22. It follows that the map r ∗ F −→ T U n ( r ∗ F ) is a levelwise weak equivalenceat V in the given range if and only if the map r ∗ F −→ r ∗ ( T O n F ) is a levelwise weak equivalence at V in thegiven range. (cid:3) Theorem 5.5.
Let F be a weakly polynomial orthogonal functor. Then the unitary Taylor tower associatedto r ∗ F at V in a given dimensional range, is equivalent to the pre-realification of the orthogonal Taylortower associated to F , at V in the same range, that is, for all n ≥
0, there is a zig-zag of weak equivalencesbetween the top and bottom rows of the following diagram, r ∗ D O n F ( V ) / / ≃ (cid:15) (cid:15) r ∗ T O n F ( V ) / / ≃ (cid:15) (cid:15) r ∗ T O n − F ( V ) ≃ (cid:15) (cid:15) F ( V ) / / T U n ( r ∗ ( T O n F ))( V ) / / T U n − ( r ∗ ( T O n − F ))( V ) D U n ( r ∗ F )( V ) / / ≃ O O T U n ( r ∗ F )( V ) / / / / ≃ O O T U n − ( r ∗ F )( V ) . ≃ O O where F ( V ) is the homotopy fibre of the map T U n ( r ∗ ( T O n F ))( V ) −→ T U n − ( r ∗ ( T O n − F ))( V ). Proof.
There is a commutative diagram r ∗ ( T O n F ) η n / / (cid:15) (cid:15) T U n ( r ∗ ( T O n F )) (cid:15) (cid:15) T U n ( r ∗ F ) T U n ( r ∗ η n ) o o (cid:15) (cid:15) r ∗ ( T O n − F ) η n − / / T U n − ( r ∗ ( T O n − F )) T U n − ( r ∗ F ) T U n − ( r ∗ η n − ) o o where the left hand horizontal maps are both weak equivalences by Theorem 4.3 and the right hand horizontalmaps are both weak equivalences by Proposition 5.4. It follows that r ∗ D O n F is levelwise weakly equivalentto D U n ( r ∗ F ), and the result follows. (cid:3) Combining Theorem 5.5 with Proposition 4.1 we achieve the following corollary.
Corollary 5.6. If F is weakly polynomial and Θ nF is the spectrum associated to the homogeneous functor D O n F , then U( n ) + ∧ L O( n ) Θ nF is the spectrum associated to the homogeneous functor D U n ( r ∗ F ).6. A complete model category comparison
Combining the model categories for orthogonal and unitary calculus produces the following diagram, Figure1, which gives a complete comparison between the orthogonal and unitary calculi. The remained of thispaper is devoted to demonstrating how this diagram commutes. In Section 7 we consider the comparisonsbetween the different categories of spectra used throughout the calculi, and show that the top portion of the iagram commutes. In Section 8, we turn our attention to the intermediate categories for the calculi. Herewe introduce two new categories, which act as intermediate categories between the standard intermediatecategories. With these in place, we demonstrate how the middle portion of Figure 1 commutes. It is thenonly left to describe how the lower two pentagons of Figure 1 commute. We deal with this in Section 9.This is considerably more complex since we are attempting to compose left and right Quillen functors witheach other. Before turning our attention to proving that Figure 1 commutes, we give an example of howthis diagram may be applied in practice. This will utilise the results of Sections 7 and 8, and especially thecommutation results of Section 9. Sp O [O( n )] ( β n ) ∗ ∼ (cid:15) (cid:15) U( n ) + ∧ O( n ) ( − ) / / r ∗ ! ! ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ Sp O [U( n )] r ∗ ∼ (cid:15) (cid:15) O(2 n ) + ∧ U( n ) ( − ) / / ι ∗ o o Sp O [O(2 n )] ( β n ) ∗ ∼ (cid:15) (cid:15) κ ∗ o o κ ∗ | | ①①①①①①①①①①①①①①①①①① Sp U [O( n )] ( γ n ) ∗ ∼ (cid:15) (cid:15) U( n ) + ∧ O( n ) ( − ) / / r ! ∼ a a ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ Sp U [U( n )] ( α n ) ∗ ∼ (cid:15) (cid:15) r ! O O ι ∗ o o r ! ∼ / / Sp O [U( n )] ( δ n ) ∗ ∼ (cid:15) (cid:15) r ∗ o o O(2 n ) + ∧ U( n ) ( − ) < < ①①①①①①①①①①①①①①①①①① O( n ) E O n res n / O( n ) ∼ (cid:15) (cid:15) ( β n ) ! O O r ∗ ∼ / / O( n ) E U n ( γ n ) ! O O U( n ) + ∧ O( n ) ( − ) / / r ! o o U( n ) E U n res n / U( n ) ∼ (cid:15) (cid:15) ( α n ) ! O O ι ∗ o o c ∗ ∼ / / U( n ) E O n O(2 n ) + ∧ U( n ) ( − ) / / c ! o o ( δ n ) ! O O O(2 n ) E O n res n / O(2 n ) ∼ (cid:15) (cid:15) ( β n ) ! O O κ ∗ o o n –homog– E O O ind n ε ∗ O O r ∗ / / n –homog– E U n ε ∗ O O r ! o o c ∗ / / (2 n ) –homog– E O n ε ∗ O O c ! o o Figure 1.
Model categories for orthogonal and unitary calculiThe n -sphere functor interacts well with our comparisons. In fact the following example works just as wellfor J n ( V, − ), V ∈ J O . Example 6.1.
Let n S be the n -sphere from orthogonal calculus, i.e. J O n (0 , − ) ∈ n –homog– E O . Under theQuillen equivalence between n –homog– E O and O( n ) E O n , n S corresponds to O( n ) + ∧ n S in O( n ) E O n , whichunder the Quillen equivalence between Sp O [O( n )] and O( n ) E O n corresponds to O( n ) + ∧ S , that is,O( n ) + ∧ S O( n ) + ∧ n S ✤ L ( β n ) ! o o ✤ L res n / O( n ) / / n S Applying (derived) change of group functor sends O( n ) + ∧ S to U( n ) + ∧ S . As before, this is the stable n -thderivative of n S , i.e. U( n ) + ∧ S U( n ) + ∧ n S ✤ L ( α n ◦ r ) ! o o ✤ L res n / U( n ) / / n S It follows that R r ∗ ( n S ) ∼ = n S in Ho( n –homog– E U ). Applying the (derived) change of group functor U( n ) + ∧ S corresponds to O(2 n ) + ∧ n S in Sp O [O(2 n )]. This is the stable (2 n )-th derivative of (2 n ) S , i.e.O(2 n ) + ∧ S O(2 n ) + ∧ (2 n ) S ✤ L ( β n ) ! o o ✤ L res n / O(2 n ) / / (2 n ) S It follows that c ∗ ( n S ) ∼ = (2 n ) S , in Ho((2 n ) –homog– E O ) and ( rc ) ∗ ( n S ) ∼ = (2 n ) S in Ho((2 n ) –homog– E O ).This is the functor calculus version of complexification followed by realification resulting in a vector spaceof twice the original dimension. . Comparisons of spectra
We have constructed a Quillen adjunction between the orthogonal and unitary n -homogeneous model struc-tures. To give a complete comparison of the theories we must address the comparisons between the othertwo categories in the zig-zag of Quillen equivalences of Barnes and Oman [BO13] and the author [Tag19].We start by addressing the relationship between the categories of spectra. For this, we recall the definitionsand model structures involved. Definition 7.1.
For a compact Lie group G , the category Sp [ G ], is the category of G -objects in Sp and G -equivariant maps, that is, an object in Sp [ G ] is a continuous functor X : J F −→ Top * such that there isa group homomorphism G −→ Aut( X ), where Aut( X ) denotes the group of automorphism of X in Sp . Wewill refer to Sp [ G ] as the category of naÃŕve G -spectra.The category Sp [ G ] comes with a levelwise and a stable model structure induced by the standard levelwiseand stable model structures of Sp . Importantly theses model structures are defined independently of groupactions, we take levelwise weak equivalences, not levelwise weak equivalences on fixed points. Lemma 7.2.
There is a cellular proper and topological model structure on the category Sp [ G ] with the weakequivalences and fibrations defined levelwise. This model structure is called the projective model structure. Lemma 7.3.
There is a cofibrantly generated topological model structure on the category Sp [ G ] with, weakequivalences the π ∗ -isomorphisms, and fibrations the levelwise fibrations f : Θ −→ Ψ such that the diagramΘ( V ) / / (cid:15) (cid:15) Ψ( V ) (cid:15) (cid:15) Ω W Θ( V ⊕ W ) / / Ω W Ψ( V ⊕ W )is a homotopy pullback square for all V, W ∈ J .The π ∗ -isomorphism rely on the model of spectra. For example, they are defined as π k (Θ) = colim q π k + q (Θ( R q ))for orthogonal spectra, and π k (Θ) = colim q π k +2 q (Θ( C q ))for unitary spectra, with the difference coming from the fact that the commutative monoid S for unitaryspectra only takes values on even dimensional spheres.7.1. Change of group.
Let H be a subgroup of a compact Lie group G . Then given a spectrum (in anychosen model) with an action of G , we can restrict through the subgroup inclusion ι : H −→ G to given thespectrum an action of H . Definition 7.4.
For a spectrum Θ with an action of G , let ι ∗ Θ be the same spectrum Θ with an action of H formed by forgetting structure through ι .In detail, let I F be the category of F -inner product subspaces of F ∞ with F -linear isometric isomorphisms.For a spectrum Θ with G -action, the evaluations mapsΘ U,V : I F ( U, V ) −→ Top * (Θ( U ) , Θ( V ))are G -equivariant. We can apply ι ∗ to this, to give a map which is H -equivariant by forgetting structure, ι ∗ Θ U,V : ι ∗ I F ( U, V ) −→ ι ∗ Top * (Θ( U ) , Θ( V )) = Top * ( ι ∗ Θ( U ) , ι ∗ Θ( V )) . This functor has a left adjoint G + ∧ H − : Sp [ H ] −→ Sp [ G ], given on an object Θ of Sp [ H ] by( G + ∧ H Θ)( V ) = G + ∧ H Θ( V ) , compare [MM02, Proposition VI.2.3]. roposition 7.5. The adjoint pair G + ∧ H − : Sp [ H ] / / Sp [ G ] : ι ∗ o o is a Quillen adjunction. Proof.
This follows immediately from noting that the π ∗ -isomorphisms and q-fibrations are defined indepen-dently of the group action. (cid:3) For our particular groups of interest, we achieve the following.
Corollary 7.6.
The adjoint pairU( n ) + ∧ O( n ) − : Sp [O( n )] / / Sp [U( n )] : ι ∗ o o is a Quillen adjunction. Corollary 7.7.
The adjoint pairO(2 n ) + ∧ U( n ) − : Sp [U( n )] / / Sp [O(2 n )] : κ ∗ o o is a Quillen adjunction.7.2. Change of model.
There is also a change of model subtly involved in the theory. This was proven in[Tag19], where the author produced a Quillen equivalence between orthogonal and unitary spectra.
Proposition 7.8 ([Tag19, Theorem 6.4]) . The adjoint pair r ! : Sp U / / Sp O : r ∗ o o is a Quillen equivalence.In [Tag19, Corollary 6.5], the author applied the above Quillen equivalence to a Quillen equivalence betweenthe categories of U( n )-objects in both models for the stable homotopy category. The same is true - withanalogous proof - for O( n )-objects in both model for the stable homotopy category. Corollary 7.9.
The adjoint pair r ! : Sp U [O( n )] / / Sp O [O( n )] : r ∗ o o is a Quillen equivalence.The change of group and change of model are compatible in the following sense. Let H ≤ G act as notationfor either the subgroup inclusion O( n ) ≤ U( n ) or U( n ) ≤ O(2 n ). Lemma 7.10.
The diagram Sp O [ H ] G + ∧ H ( − ) / / r ∗ (cid:15) (cid:15) Sp O [ G ] ι ∗ o o r ∗ (cid:15) (cid:15) Sp U [ H ] G + ∧ H ( − ) / / r ! O O Sp U [ G ] ι ∗ o o r ! O O commutes up to natural isomorphism. Proof.
It suffices to show that the diagram of right adjoints commute. Indeed, ι ∗ (( r ∗ Θ)( V )) = ι ∗ Θ( V R ) = ( ι ∗ Θ)( V R ) = r ∗ (( ι ∗ Θ)( V )) . (cid:3) . Comparing the intermediate categories
To achieve the correct correspondences between U( n ) E U n and O( n ) E O n we introduce two new intermediatecategories via the inclusion of subgroups ι : O( n ) ֒ → U( n ) and κ : U( n ) ֒ → O(2 n ). In our consideration ofthe comparisons between the categories of spectra, the order in which we changed the group and changedthe model was unimportant, since the indexing categories, J O and J U , are equipped with the trivial action.However, for the intermediate categories, the diagram categories J n have a non-trivial action of Aut( n ),hence the order in which one changes group and changes model is important. This section gives the correctmethod for such comparisons. Definition 8.1.
Define O( n ) E U n to be the category of O( n ) Top * -enriched functors from ι ∗ J U n −→ O( n ) Top * where ι ∗ J U n is an O( n ) Top * -enriched category obtained from J U n by forgetting structure through the sub-group inclusion ι : O( n ) −→ U( n ). Similarly define U( n ) E O n to be the category of U( n ) Top * -enrichedfunctors from κ ∗ J O n −→ U( n ) Top * where κ ∗ J O n is an U( n ) Top * -enriched category obtained from J O n byforgetting structure through the subgroup inclusion κ : U( n ) −→ O(2 n ).These categories also come with projective and stable model structures constructed analogously to those ofProposition 2.14. These new intermediate categories will now act as intermediate categories between thestandard intermediate categories of orthogonal and unitary calculus. Further, the new intermediate categoriesequipped with their n -stable model structure are Quillen equivalent to spectra with an appropriate groupaction. The proofs of the following two results follow similarly to [BO13, Proposition 8.3] and [Tag19,Theorem 6.8]. Proposition 8.2.
There is a Quillen equivalence( γ n ) ! : O( n ) E U n / / Sp U [O( n )] : ( γ n ) ∗ o o with ( γ n ) ∗ Θ( V ) = Θ( C n ⊗ C V ), and ( γ n ) ! is the left Kan extension along γ n . Proposition 8.3.
There is a Quillen equivalence( δ n ) ! : U( n ) E O n / / Sp O [U( n )] : ( δ n ) ∗ o o with ( δ n ) ∗ Θ( V ) = Θ( R n ⊗ R V ), and ( δ n ) ! is the left Kan extension along δ n .8.1. Change of group.
Let E ∈ U( n ) E U n , then E is defined by U( n )-equivariant structure maps of theform E U,V : J U n ( U, V ) −→ Top * ( E ( U ) , E ( V )) . Forgetting structure through ι : O( n ) −→ U( n ) yields an O( n )-equivariant map ι ∗ E U,V : ι ∗ J U n ( U, V ) −→ ι ∗ Top * ( E ( U ) , E ( V )) = Top * ( ι ∗ E ( U ) , ι ∗ E ( V )) . This induces a functor ι ∗ : U( n ) E U n −→ O( n ) E U n , which has a left adjoint U( n ) + ∧ O( n ) ( − ) given by(U( n ) + ∧ O( n ) E )( V ) = U( n ) + ∧ O( n ) E ( V ) , with structure maps S nW ∧ U( n ) + ∧ O( n ) E ( V ) ∼ = U( n ) + ∧ O( n ) ( ι ∗ S nW ∧ E ( V )) −→ U( n ) + ∧ E ( V ⊕ W ) , where the isomorphism follows from [MM02, Proposition V.2.3].Completely analogous constructions for the subgroup inclusion i : U( n ) −→ O(2 n ) yields an adjoint pairO(2 n ) + ∧ U( n ) ( − ) : U( n ) E O n / / O(2 n ) E O n : κ ∗ o o . Lemma 8.4.
The adjoint pairU( n ) + ∧ O( n ) ( − ) : O( n ) E U n / / U( n ) E U n : ι ∗ o o . is a Quillen adjunction. roof. The levelwise fibrations, levelwise weak equivalences and nπ ∗ -isomorphisms are defined independentlyof group actions. It follows that ι ∗ preserves these. (cid:3) This results in a square of adjoint functors, Sp U [O( n )] U( n ) + ∧ O( n ) − / / ( γ n ) ∗ (cid:15) (cid:15) Sp U [U( n )] ι ∗ o o ( α n ) ∗ (cid:15) (cid:15) O( n ) E U n U( n ) + ∧ O( n ) − / / ( γ n ) ! O O U( n ) E U n . ι ∗ o o ( α n ) ! O O Lemma 8.5.
The above diagram commutes up to natural isomorphism.
Proof.
Let X be a unitary spectrum with an action of U( n ). Then( ι ∗ α ∗ n )( X )( V ) = ι ∗ X ( nV ) = γ ∗ n ( ι ∗ X )( V ) . The result then follows immediately. Note that the functor ι ∗ restricts the group actions in a compatibleway. The restricted action of O( n ) on X ( nV ) is ι ∗ ( X ( σ ⊗ V ) ◦ X σ ( nV ) ) where σ ∈ U( n ). This is equivalentto the action X ( ι ∗ ( σ ) ⊗ V ) ◦ X ι ∗ ( σ )( nV ) since X ( σ ◦ V ) and X σ ( nV ) commute. This action is precisely theaction we get from first restricting the action and then applying γ ∗ n . (cid:3) Similarly we obtain the following result.
Lemma 8.6.
The adjoint pairO(2 n ) + ∧ U( n ) ( − ) : U( n ) E O n / / O(2 n ) E O n : κ ∗ o o . is a Quillen adjunction.This results is a square of adjoint functors, Sp O [U( n )] O(2 n ) + ∧ U( n ) − / / ( δ n ) ∗ (cid:15) (cid:15) Sp O [O(2 n )] κ ∗ o o ( β n ) ∗ (cid:15) (cid:15) U( n ) E O n O(2 n ) + ∧ U( n ) − / / ( δ n ) ! O O O(2 n ) E O n . κ ∗ o o ( β n ) ! O O Lemma 8.7.
The above diagram commutes up to natural isomorphism.
Proof.
Let X be an orthogonal spectrum with an action of O(2 n ). Then( κ ∗ β ∗ n )( X )( V ) = κ ∗ X ( nV ) = δ ∗ n ( κ ∗ X )( V ) . The result then follows immediately. The group actions restrict in a compatible way as in Lemma 8.5. (cid:3) .2. Change of model through realification.
We define a realification functor r : J U n −→ J O n . Thisfunctor induces a right Quillen functor between O( n ) E U n and O( n ) E O n .On objects, let r be given by forgetting the complex structure, i.e., C k R k . Morphisms in J U n are givenin terms of the Thom space of the vector bundle γ U n ( V, W ) = { ( f, x ) : f ∈ J U ( V, W ) , x ∈ C n ⊗ C f ( V ) ⊥ } over the space of linear isometries J U ( V, W ). We then define realification on a pair ( f, x ) by r ( f, x ) = ( f R , rx )where f R ∈ J O ( V R , W R ) and rx is the image of x under the map C n ⊗ C f ( V ) ⊥ −→ R n ⊗ R ( f R )( V R ) ⊥ . Thismap is the composite C n ⊗ C ( W − f ( V )) ∼ = n M i =1 f ( V ) ⊥ L ni =1 r −−−−−→ n M i =1 ( f R )( V R ) ⊥ ∼ = R n ⊗ R ( W R − ( f R )( V R )) , where r is the standard realification map on vector spaces. It is not hard to check that r gives a well definedmap f ( V ) ⊥ −→ ( f R )( V R ) ⊥ .Restricting the U( n ) action on J U n to an action of O( n ) through the subgroup inclusion ι : O( n ) ֒ → U( n ),induces a functor r : ι ∗ J U n −→ J O n and precomposition defines a functor r ∗ : O( n ) E O n −→ O( n ) E U n . To see that r ∗ is well defined, we check that the map( r ∗ F ) : ι ∗ J U n ( V, W ) −→ Top * (( r ∗ F )( V ) , ( r ∗ F )( V )) = Top * ( F ( V R ) , F ( W R ))is O( n )-equivariant where F ∈ O( n ) E O n . Indeed, let ( f, x ) ∈ ι ∗ J U n ( V, W ) and σ ∈ O( n ),( r ∗ F )( σ ( f, x )) = ( r ∗ F )( f, ι ( σ )( x )) = F ( f R , r ( ι ( σ )( x ))) . For W a complex vector space, the restricted action of U( n ) to O( n ) on C n ⊗ C W is compatible with theO( n )-action on R n ⊗ R rW , hence r ( ι ∗ ( σ )( x )) = σ ( rx ), and the above becomes F ( f R , r ( ι ∗ ( σ )( x ))) = F ( f R , σ ( rx )) = σ ( F ( f R , rx )) = σ (( r ∗ F )( f, x )) . It follows that the required map is O( n )-equivariant and hence r ∗ F is a well defined object of O( n ) E U n .The structure maps of r ∗ F are given by iterating the structure maps of F ; S n ∧ ( r ∗ F )( C k ) = −→ S n ∧ F ( R k ) σ −→ F ( R k +2 ) = −→ ( r ∗ F )( C k +1 ) , where σ : S n ∧ F ( R k ) −→ F ( R k +1 ) is the structure map of F . As r ∗ is defined by precomposition it has anatural left adjoint, r ! given by the left Kan extension along r . Lemma 8.8.
The functor r ∗ : O( n ) E O n −→ O( n ) E U n is a right Quillen functor. Proof.
By definition on objects, r ∗ preserves all levelwise weak equivalences and all levelwise fibrations. Thecompatibility with r ∗ and the structure maps shows that r ∗ preserves fibrant objects. (cid:3) This comparison produces a diagram of adjoint functors Sp O [O( n )] r ∗ / / ( β n ) ∗ (cid:15) (cid:15) Sp U [O( n )] r ! o o ( γ n ) ∗ (cid:15) (cid:15) O( n ) E O n r ∗ / / ( β n ) ! O O O( n ) E U n . r ! o o ( γ n ) ! O O emma 8.9. The above diagram commutes up to natural isomorphism.
Proof.
Consider the diagram of enriched categories, J O J U r o o J O nβ n O O J U nr o o γ n O O It is clear from the definition of these functors that the diagram commutes on objects up to natural isomor-phism. Now on morphisms, take ( f, x ) ∈ J U n . Then r ( γ n ( f, x )) = r (( C n ⊗ f, x ) = ( R n ⊗ f R , rx ) = β n (( f R , rx )) = β n ( r ( f, x )) . It follows that rγ n = β n r . Since the right adjoints in the required diagram are defined in terms of precom-position the result follows. Note that the group actions are also compatible since the unitary γ n has beenrestricted to O( n ) actions. (cid:3) Change of model through complexification.
Define a complexification functor c : J O n −→ J U n , givenon objects by cV = C ⊗ V , and on morphisms by sending ( f, x ) ∈ J O n ( V, W ) to ( C ⊗ f, cx ) ∈ J U n ( cV, cW ),where cx is the image of x under the composition of isomorphisms, R n ⊗ R coker( f ) ϕ −→ ∼ = C n ⊗ R coker( f ) ϕ −→ ∼ = C n ⊗ C C ⊗ R coker( f ) ϕ −→ ∼ = C n ⊗ C coker( C ⊗ f ) . where ϕ ( r , · · · , r n , f ( v )) = ( r + ir , · · · r n − + ir n , f ( v )); ϕ ( c , · · · , c n , f ( v )) = ( c , · · · , c n , , f ( v )); and ϕ ( c , · · · , c n , c, f ( v )) = ( c , · · · , c n , ( C ⊗ f )( c ⊗ v )) . Restricting from O(2 n ) to U( n ) through the subgroup inclusion κ : U( n ) ֒ → O(2 n ) gives a functor c : κ ∗ J O n −→ J U n , and precomposition defines a functor c ∗ : U( n ) E U n −→ U( n ) E O n . This functor is well defined as for X ∈ U( n ) E U n the map c ∗ X : κ ∗ J O n ( V, W ) −→ Top * (( c ∗ X )( V ) , ( c ∗ X )( W )) = Top * ( X ( C ⊗ V ) , X ( C ⊗ W ))is U( n )-equivariant. Indeed, for σ ∈ U( n ),( c ∗ X )( σ ( f, x )) = ( c ∗ X )( f, σx ) = X ( C ⊗ f, c ( κ ∗ ( σ ) x )) = X ( C ⊗ f, σ ( cx ))= X ( C ⊗ f, σx ) = σ (( c ∗ X )( f, x )) . The structure maps of c ∗ X are induced by those of X , i.e., S n ∧ ( c ∗ X )( V ) = −→ S n ∧ X ( C ⊗ V ) σ −→ X (( C ⊗ V ) ⊕ C ) = −→ ( c ∗ X )( V ⊕ R )where σ : S n ∧ X ( W ) −→ ( W ⊕ C ) is the structure map of X ∈ U( n ) E O n .The complexification functor c ∗ has a left adjoint, c ! given by the left Kan extension along c . We obtain asimilar result to the case of realification, Lemma 8.8. Lemma 8.10.
The functor c ∗ : U( n ) E U n −→ U( n ) E O n is a right Quillen functor. his produced a diagram of adjoint functors Sp U [U( n )] r ∗ / / ( α n ) ∗ (cid:15) (cid:15) Sp O [U( n )] r ! o o ( δ n ) ∗ (cid:15) (cid:15) U( n ) E U n c ∗ / / ( α n ) ! O O U( n ) E O n . c ! o o ( δ n ) ! O O Lemma 8.11.
The above diagram commutes up to natural isomorphism.
Proof.
Consider the diagram of enriched categories J U r / / J O J U nα n O O J O n . δ n O O c o o It is clear that this diagram commutes up to natural isomorphism on objects. On morphisms, let ( f, x ) ∈ J O n ( V, W ). Then r ( α n ( c ( f, x ))) ∼ = r ( α n ( C ⊗ f, cx )) = r ( C n ⊗ C C ⊗ R f, cx ) ∼ = r ( C n ⊗ f, cx ) = ( R n ⊗ f R , rcx ) ∼ = δ n ( f, x ) , where rcx = x since if x is of the form ( r , · · · , r n , f ( v )), cx = ( r + ir , · · · r n − + ir n , ( C ⊗ f )(1 ⊗ v ))and hence rcx = r (( r + ir , · · · r n − + ir n , ( C ⊗ f )(1 ⊗ v )))= ( r , r , · · · , r n − , r n , f ( v )) . It follows that rα n c ∼ = δ n . As the right adjoints of the required diagram are defined in terms of precompo-sition, the result follows. The group actions are compatible by a similar argument to Lemma 8.9. (cid:3) A homotopy category level comparison
We have shown previously that all but the bottom pentagons of Figure 1 commute. Moreover, since allof the commutation results for the sub-diagrams - excluding the lower pentagons - involve composing left(resp. right) Quillen functors with left (resp. right) Quillen functors those sub-diagrams commute on thehomotopy category level. Hence, the only sections of Figure 1 left to consider are the lower pentagons. Thecommuting of diagrams of adjoint functors (up to natural isomorphism) means that the respective diagramsof left and right adjoints commute. These pentagons are built from a mixture of left and right adjoints, sowe must address how it commutes in a different manner.
Lemma 9.1.
The diagramO( n ) E O n res n / O( n ) (cid:15) (cid:15) r ∗ / / O( n ) E U n U( n ) + ∧ O( n ) ( − ) / / U( n ) E U n res n / U( n ) (cid:15) (cid:15) n –homog– E O r ∗ / / n –homog– E U commutes up to natural isomorphism. roof. Consider the diagram of enriched categories J O n r / / J U n J O i n O O r / / J U i n O O where i n is the identity on objects and f ( f,
0) on morphisms. This diagram clearly commutes on objectsand morphisms. Let X ∈ O ( n ) E O n . Thenres n (U( n ) + ∧ O( n ) ( r ∗ X )) / U( n ) = (U( n ) + ∧ O( n ) ( X ◦ r ◦ i n )) / U( n ) ∼ = ( X ◦ r ◦ i n ) / O( n ) ∼ = ( X ◦ i n ◦ r ) / O( n )= r ∗ ((res n X ) / O( n ))where the first isomorphism comes from the fact that for any O( n )-space Y , (U( n ) + ∧ O( n ) Y ) / U( n ) ∼ = Y / O( n ),and the second isomorphism follows from the commutation of the above diagram of enriched categories. (cid:3) Lemma 9.2.
The diagramU( n ) E U n c ∗ / / res n / U( n ) (cid:15) (cid:15) U( n ) E O n O(2 n ) + ∧ U( n ) ( − ) / / O(2 n ) E O n res n / O(2 n ) (cid:15) (cid:15) n –homog– E U c ∗ / / (2 n ) –homog– E O commutes up to natural isomorphism. Proof.
This proof follows similarly to the above, starting with the diagram of enriched categories J U n J O nc o o J U i n O O J O c o o i n O O which commutes. (cid:3) These squares are built using alternating left and right adjoints, hence no clean model category commutationis possible. We start with a larger diagram of homotopy categories and then restrict to our required diagram.On the homotopy category level we obtain the following result. emma 9.3. The following diagram of homotopy categoriesHo( Sp O [O( n )]) U( n ) + ∧ L O( n ) ( − ) / / R ( β n ) ∗ (cid:15) (cid:15) Ho( Sp O [U( n )]) R r ∗ / / Ho( Sp U [U( n )]) R ( α n ) ∗ (cid:15) (cid:15) Ho(O( n ) E O n ) L res n / O( n ) (cid:15) (cid:15) Ho(U( n ) E U n ) L res n / U( n ) (cid:15) (cid:15) Ho( n –homog– E O ) R r ∗ / / Ho( n –homog– E U )commutes up to natural isomorphism. Proof.
By the zig-zag of Quillen equivalences, [BO13, Proposition 8.3 and Theorem 10.1] the composite L res n / O( n ) ◦ R ( β n ) ∗ applied to an orthogonal spectrum Θ with an action of O( n ), is levelwise weakly equivalent to the functordefined by the formula V Ω ∞ [( S R n ⊗ V ∧ Θ) h O( n ) ] . The zig-zag of Quillen equivalences from unitary calculus, [Tag19, Theorems 6.8 and 7.5], together withinflating Θ to a spectrum with an action of U( n ) gives a similar characterisation in terms of an n -homogeneousfunctor. The result then follows by Proposition 4.1. (cid:3) A similar result holds true for similar diagram on the right of Figure 1, utilising Lemma 4.2, rather thanProposition 4.1.
Lemma 9.4.
The following diagram of homotopy categoriesHo( Sp U [U( n )]) L r ! / / R ( α n ) ∗ (cid:15) (cid:15) Ho( Sp O [U( n )]) O(2 n ) + ∧ L U( n ) ( − ) / / Ho( Sp O [O(2 n )]) R ( β n ) ∗ (cid:15) (cid:15) Ho(U( n ) E U n ) L res n / U( n ) (cid:15) (cid:15) Ho(O(2 n ) E O n ) L res n / O(2 n ) (cid:15) (cid:15) Ho( n –homog– E U ) R c ∗ / / Ho((2 n ) –homog– E O )commutes up to natural isomorphism. orollary 9.5. The following diagram of homotopy categoriesHo( Sp O [O( n )]) U( n ) + ∧ L O( n ) ( − ) / / Ho( Sp U [U( n )]) R r ∗ / / Ho( Sp U [U( n )]) R ( α n ) ∗ (cid:15) (cid:15) Ho(O( n ) E O n ) L ( β n ) ! O O L res n / O( n ) (cid:15) (cid:15) Ho(U( n ) E U n ) L res n / U( n ) (cid:15) (cid:15) Ho( n –homog– E O ) R r ∗ / / Ho( n –homog– E U )commutes up to natural isomorphism. Proof.
By Lemma 9.3, there is a natural isomorphism R r ∗ ◦ L (res n / O( n )) ◦ R ( β n ) ∗ ∼ = L (res n / U( n )) ◦ R ( α n ) ∗ ◦ R r ∗ ◦ (U( n ) + ∧ L O( n ) ( − )) . By the equivalence of the homotopy categories of Sp O [O( n )] and O( n ) E O n we have that R ( β n ) ∗ ◦ L ( β n ) ! ∼ = .It follows that R r ∗ ◦ L (res n / O( n )) ◦ R ( β n ) ∗ ◦ L ( β n ) ! ∼ = R r ∗ ◦ L (res n / O( n )) ∼ = L (res n / U( n )) ◦ R ( α n ) ∗ ◦ R r ∗ ◦ (U( n ) + ∧ L O( n ) ( − )) ◦ L ( β n ) ! . (cid:3) Corollary 9.6.
The following diagram of homotopy categoriesHo( Sp U [U( n )]) L r ! / / Ho( Sp O [U( n )]) O(2 n ) + ∧ L U( n ) ( − ) / / Ho( Sp O [O(2 n )]) R ( β n ) ∗ (cid:15) (cid:15) Ho(U( n ) E U n ) L ( α n ) ! O O L res n / U( n ) (cid:15) (cid:15) Ho(O(2 n ) E O n ) L res n / O(2 n ) (cid:15) (cid:15) Ho( n –homog– E U ) R c ∗ / / Ho((2 n ) –homog– E O ) Proof.
Using the same argument as Corollary 9.5 using Lemma 9.4 and the equivalence of the homotopycategories of Sp U [U( n )] and U( n ) E U n . (cid:3) By restricting these larger diagrams, we obtain a homotopy category level commutation result for the lowerpentagons of Figure 1. emma 9.7. The diagramHo(O( n ) E O n ) R r ∗ / / L (res n / O( n )) (cid:15) (cid:15) Ho(O( n ) E U n ) U( n ) ∧ L O( n ) ( − ) / / Ho(U( n ) E U n ) L (res n / U( n ) (cid:15) (cid:15) Ho( n –homog– E O ) R r ∗ / / Ho( n –homog– E O )of derived functors commutes up to natural isomorphism. Proof.
By Lemma 7.10, Lemma 8.5 and Lemma 8.9, the composite L (res n / U( n )) ◦ (U( n ) + ∧ L O( n ) ( − )) ◦ R r ∗ is naturally isomorphic to the composite L (res n /U ( n )) ◦ R ( α n ) ∗ ◦ R r ∗ ◦ (U( n ) + ∧ L O( n ) ( − )) ◦ R ( β n ) ! . The result then follows by Corollary 9.5. (cid:3)
Lemma 9.8.
The diagramHo(U( n ) E U n ) R c ∗ / / L (res n / U( n )) (cid:15) (cid:15) Ho(U( n ) E O n ) O(2 n ) ∧ L U( n ) ( − ) / / Ho(O(2 n ) E O n ) L (res n / O(2 n )) (cid:15) (cid:15) Ho( n –homog– E U ) R c ∗ / / Ho((2 n ) –homog– E O )of derived functors commutes up to natural isomorphism. Proof.
We have to show that R c ∗ ◦ L (res n / U( n )) ∼ = L (res n / O(2 n )) ◦ (O(2 n ) ∧ L U( n ) ( − )) ◦ R c ∗ . By Lemma 8.11 and Lemma 8.7 we can replace (up to natural isomorphism) the composite (O(2 n ) ∧ L U( n ) ( − )) ◦ R c ∗ with the composite R ( β n ) ∗ ◦ (O(2 n ) ∧ L U ( n ) ( − )) ◦ R r ∗ ◦ L ( α n ) ! . Corollary 9.6 and the fact that the homotopy categories of Sp U [O( n )] and U( n ) E U n are equivalent yields thatthe composite L (res n / O(2 n )) ◦ R ( β n ) ∗ ◦ (O(2 n ) ∧ L U( n ) ( − )) ◦ R r ∗ ◦ L ( α n ) ! is naturally isomorphic to the composite R c ∗ ◦ L (res n / U( n ))and the result follows. (cid:3) eferences [BE16] D. Barnes and R. Eldred. Comparing the orthogonal and homotopy functor calculi. J. Pure Appl. Algebra ,220(11):3650–3675, 2016.[BO13] D. Barnes and P. Oman. Model categories for orthogonal calculus.
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Mathematical Sciences Research Centre, Queen’s University Belfast, UK
E-mail address : [email protected]@qub.ac.uk