Equivariant infinite loop space theory, I. The space level story
aa r X i v : . [ m a t h . A T ] A p r EQUIVARIANT INFINITE LOOP SPACE THEORY, I.THE SPACE LEVEL STORY
J. PETER MAY, MONA MERLING, AND ANG´ELICA M. OSORNO
Abstract.
We rework and generalize equivariant infinite loop space theory,which shows how to construct G -spectra from G -spaces with suitable structure.There is a naive version which gives naive G -spectra for any topological group G , but our focus is on the construction of genuine G -spectra when G is finite.We give new information about the Segal and operadic equivariant infiniteloop space machines, supplying many details that are missing from the litera-ture, and we prove by direct comparison that the two machines give equivalentoutput when fed equivalent input. The proof of the corresponding nonequivari-ant uniqueness theorem, due to May and Thomason, works for naive G -spectrafor general G but fails hopelessly for genuine G -spectra when G is finite. Evenin the nonequivariant case, our comparison theorem is considerably more pre-cise, giving a direct point-set level comparison.We have taken the opportunity to update this general area, equivariant andnonequivariant, giving many new proofs, filling in some gaps, and giving somecorrections to results in the literature. Contents
1. Introduction and preliminaries 2Acknowledgements 61.1. Preliminaries about G -spaces and Hopf G -spaces 61.2. Preliminaries about G -cofibrations and simplicial G -spaces 71.3. Categorical preliminaries and basepoints 91.4. Spectrum level preliminaries 111.5. Equivariant infinite loop space machines 122. The simplicial and conceptual versions of the Segal machine 132.1. Definitions: the input of the Segal machine 132.2. The simplicial version of the Segal machine 172.3. The conceptual version of the Segal machine 182.4. A factorization of the conceptual Segal machine 203. The homotopical version of the Segal machine 233.1. The categorical bar construction 233.2. The naive homotopical Segal machine 263.3. The genuine homotopical Segal machine 283.4. Change of groups and compact Lie groups 304. The generalized Segal machine 324.1. G -categories of operators D over F and D - G -spaces 324.2. G -categories of operators D G over F G and D G - G -spaces 33 Mathematics Subject Classification.
Primary 55P42, 55P43, 55P91;Secondary 18A25, 18E30, 55P48, 55U35. D , T G ) and Fun( D G , T G ) 344.4. Comparisons of D - G -spaces and E - G -spaces for ν : D −→ E G -operads to G -categories of operators 415.1. G -categories of operators associated to a G -operad C G E ∞ G -operads and E ∞ G -categories of operators 436. The generalized operadic machine 446.1. The Steiner operads 456.2. The classical operadic machine 466.3. The monads D and D G associated to the G -categories D and D G π I + ∧ ( S V ) • on D G,V ι ω D X D X is Reedy cofibrant 698.3. The proof that D preserves F • -equivalences 719. Proofs of technical results about the Segal machine 789.1. Combinatorial analysis of A • ⊗ F X G -cofibrations 90References 921. Introduction and preliminaries
Equivariant homotopy theory is much richer than nonequivariant homotopy the-ory. Equivariant generalizations of nonequivariant theory are often non-trivial andoften admit several variants. Nonequivariantly, symmetric monoidal (or equiva-lently permutative) categories and E ∞ spaces give rise to spectra. There are sev-eral “machines” that take such categorical or space level input and deliver spectraas output, and there are comparison theorems showing that all such machines areequivalent [29, 30, 38, 51].Equivariantly, there are different choices of G -spectra that can be taken as outputof infinite loop space machines. One choice is naive G -spectra, which are simplyspectra with a G -action. They can be defined for any topological group G . Theweak equivalences between them are the G -maps that induce nonequivariant weakequivalences on fixed point spectra. As we shall indicate, it is quite straightforwardto generalize infinite loop space theory so as to accept naive input, such as G -spaceswith actions by nonequivariant E ∞ operads, and deliver naive G -spectra. Moreover, QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 3 the nonequivariant comparisons generalize effortlessly to this context. As we shallsee, this much all works for any topological group G .Naive G -spectra really are naive. For example, one cannot prove any version ofPoincar´e duality in the cohomology theories they represent. They can be indexedon the natural numbers, and n should be thought of geometrically as a stand-infor R n or S n , with trivial G -action. When G is a compact Lie group, we alsohave genuine G -spectra, which are indexed on representations or, more precisely,real vector spaces (better, inner product spaces) V with an action of G . Theirone-point compactifications are the representation spheres, denoted S V , and thesespheres are inverted in the genuine G -stable homotopy category.The weak equivalences between genuine G -spectra are again nonequivariantequivalences on fixed point spectra. However, since suspending and looping withrespect to the spheres S V does not commute with passage to H -fixed points whenthe action of G on V is nontrivial, the H -fixed spectra retain homotopical infor-mation about the representations of G . That information is encoded in the notionof equivalence of genuine G -spectra. One can also restrict attention to subclassesof representations and obtain a plethora of kinds of G -spectra intermediate be-tween the naive ones (indexed only on trivial representations) and the genuine ones(indexed on all finite dimensional representations).Our focus is on finite groups G and equivariant infinite loop space machineswhose inputs are G -spaces X with extra structure and whose outputs are genuineΩ- G -spectra whose zeroth spaces are equivariant group completions of X . The twononequivariant machines in most common use are those of Segal [51] and the seniorauthor [27], and we call these the Segal and operadic machines. The Segal machinewas generalized equivariantly in [54] and the operadic machine was generalizedequivariantly in [10] and more recently in [12], which can be viewed in part as aprequel to this paper. We study these equivariant generalizations of the Segal andoperadic infinite loop space machines and prove that when fed equivalent data theyproduce equivalent output.Due to their very different constructions, the two machines have different ad-vantages and disadvantages. Whereas the operadic machine is defined only forfinite groups, the Segal machine can be used to construct genuine G -spectra forany compact Lie group G . These spectra are unfortunately not Ω- G -spectra unless G is finite, but their restrictions to finite subgroups H are Ω- H -spectra. The Segalmachine also works simplicially and is likely to be the machine of choice in motiviccontexts, if and when such a motivic theory is developed; it has yet to be developedeven nonequivariantly. The operadic machine generalizes directly to give machinesthat manufacture intermediate types of G -spectra from intermediate types of inputdata, but we do not know a Segal type analogue. Due to its more topological flavor,the operadic machine was used to produce genuine G -spectra from categorical datain [12], where the machine was used to give categorical proofs of topological results.We have several ways to generalize the Segal machine equivariantly, and we havecomparisons among them. We develop the one closest to Segal’s original version in §
2, highlighting the role of its inductive simplicial definition in proving the groupcompletion property. This version of the machine, which has not previously beendeveloped equivariantly, starts from F - G -spaces, namely functors from the category F of finite sets, the opposite of Segal’s category Γ, to the category of based G -spaces. It produces naive G -spectra for any topological group G . We shall use it J. PETER MAY, MONA MERLING, AND ANG´ELICA M. OSORNO to prove the equivariant group completion property for our other versions of theSegal machine. Following Segal [51] nonequivariantly, we compare that machine toa conceptual version that is defined by categorical prolongation of functors definedon F to functors defined on the category W G of G -CW complexes. However, thehomotopical conditions needed to make the conceptual machine useful are seldomsatisfied by the examples that arise in nature.We recall the homotopically well-behaved machine in §
3, which is based on thetwo-sided bar construction. This version of the Segal machine was first definednonequivariantly by Woolfson [59], but the equivariant generalization of his defini-tion starting from F - G -spaces fails to be well-behaved homotopically. FollowingShimakawa [54, 55, 56], we instead focus on an equivariant generalization that startsfrom the category of F G - G -spaces, which are functors from the category F G of fi-nite G -sets to the category of based G -spaces. As we reprove, these categories ofinput data are equivalent. The comparison of equivalences and the relevant “spe-cialness” conditions shed considerable light on the underlying homotopy theory. Toexplain ideas without technical clutter, we defer the longer proofs about the Segalmachine from § § § F -spaces and E ∞ spaces, and G -categories of operators serve the samepurpose equivariantly. We have E ∞ G -categories of operators D over F and E ∞ G -categories D G over F G , and we have algebras over each; F = D and F G = D G are special cases. Ignoring operads, we develop and compare Segal machines thatproduce genuine G -spectra from such algebras in § § G -category ofoperators D ( C G ) over F and a G -category of operators D G ( C G ) over F G froman operad C G . Our focus in this paper is on E ∞ operads C G , and we prove that D ( C G ) and D G ( C G ) are E ∞ G -categories of operators when C G is an E ∞ operad.Letting C G be an E ∞ G -operad, we have algebras over C G , D ( C G ), and D G ( C G ).In §
6, we generalize the operadic machine to accept such generalized input. Start-ing from C G -algebras, the equivariant operadic machine was first developed byCostenoble and Waner [10] and is given a thorough modern redevelopment in [12].Therefore we focus on the generalization and on comparisons of the machines start-ing from these three kinds of operadic input. This is conceptually the same asin May and Thomason [38], but the key proof equivariantly is considerably moreintricate and is deferred to §
8. A curious feature is that, in contrast to the Segalmachine, there is no particular need to consider F G rather than F when developingthe operadic machine, although use of F G is convenient for purposes of comparison.The comparison of the Segal and operadic machines starting from the same inputis given in §
7. It seems quite amazing to us. Even nonequivariantly, it is far moreprecise than the comparison given in [38]. There is a family of operads, called theSteiner operads [58] (see also [12]). They are variants of the little cubes and littlediscs operads that share the good properties and lack the bad properties of eachof those, as explained in [36, § In part they follow extensive unpublished work of Hauschild, May, and Waner.
QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 5 that they mediate between the Segal and operadic infinite loop space machines justas if they had been invented for that purpose. That is wholly unexpected and trulyuncanny.Some background may help explain why we find such a precise point-set levelcomparison so surprising. In the Segal machine, higher homotopies are encoded inthe specialness property of the structure map δ : X n −→ X n relating the n th spaceto the n th power of the first space of an F - G -space. In the operadic machine, theyare encoded in the structure given by the action maps C G ( n ) × X n −→ X of a C G -algebra. Actions by the G -category of operators D G ( C G ) encode both sourcesof higher homotopies in the same structure and yet, up to equivalence, carry nomore information than either does alone.The difference is perhaps illuminated by thinking about the commutativity op-erad N with N ( j ) = ∗ . The G -category D ( N ) over F is just F itself. An N - G -space X = X is the same thing as an F - G -space with X n = X n . Bothjust give X the structure of a commutative monoid in G T , and that is far too re-stricted to give the domain of an infinite loop space machine: the fixed point spacesof the infinite loop G -spaces resulting from such input are equivalent to productsof Eilenberg-MacLane spaces. From the point of view of the Segal machine, weare replacing N - G -spaces with homotopically well-behaved F - G -spaces as input.From the point of view of the operadic machine, we are replacing N - G -spaces by C G -spaces for any chosen E ∞ operad C G .Consideration of categorical input is conspicuous by its absence in this paperand is crucial to the applications to equivariant algebraic K -theory that we have inmind. Both for application to the most natural input data and for the multiplicativetheory, it is essential to work 2-categorically with lax functors, or at least pseudo-functors, rather than just with categories and functors, and it is desirable to workwith symmetric monoidal G -categories rather than just the (genuine) permutative G -categories defined in [12]. That requires quite different categorical underpinningsthan are discussed in this paper or in [12] and will be treated in detail in [15, 16].We note that while permutative G -categories and their symmetric monoidal gener-alization are defined operadically in the cited papers, they are processed using theequivariant version of the Segal machine that we develop in this paper.Model categorical interpretations may also be conspicuous by their paucity. Allrelevant model structures are developed in the papers [18, 24, 26, 44], and it is nothard to interpret some of our work, but not the essential parts, model theoretically.One point is that the group completion property, which is central to the theory andnearly all of its applications, is invisible to the relevant model categories. Another isthat model categorical cofibrant and/or fibrant approximation might obscure whatis intrinsically a quite intricate collection of very precisely interrelated notions. Per-haps unfashionably, we are interested in preserving as much point-set level structureas possible, which we find illuminating, and of course that is precisely what moreabstract frameworks are designed to avoid.We complete this section with some preliminaries that give common backgroundfor the various machines, fixing notations and definitions that are used throughoutthe paper. While our main interest is in finite groups, unless otherwise specifiedwe let G be any topological group here and in § §
3. Subgroups of G areunderstood to be closed and homomorphisms are understood to be continuous. Werestrict to finite groups starting in § J. PETER MAY, MONA MERLING, AND ANG´ELICA M. OSORNO
Acknowledgements.
This work was partially supported by Simons Foundationgrant No. 359449 (Ang´elica Osorno) and the Woodrow Wilson Career EnhancementFellowship held by the third-named author, and an AMS Simons Travel grant heldby the second-named author. An NSF RTG grant supported a month long visitof the third-named author and a week long visit of the second-named author toChicago. Part of the writing took place while the second-named author was inresidence at the Hausdorff Research Institute for Mathematics, and she is thankfulfor their hospitality.We would like to thank the following people for many insightful discussions thathelped shape ideas during the long course of this project: Clark Barwick, AndrewBlumberg, Anna Marie Bohmann, Emanuele Dotto, Bertrand Guillou, Nick Gurski,Lars Hesselholt, Mike Hill, Niles Johnson, Cary Malkiewich, Kristian Moi, IrakliPatchkoria, Emily Riehl, Jonathan Rubin, and Stefan Schwede.1.1.
Preliminaries about G -spaces and Hopf G -spaces. We let U be the cat-egory of compactly generated spaces weak Hausdorf spaces, let U ∗ be the categoryof based spaces, and let T be its subcategory of nondegenerately based spaces. Welet G U , G U ∗ and G T be the categories of G -spaces, based G -spaces, and nonde-generately based G -spaces, with left action by G ; G acts trivially on basepoints.Maps in these categories are G -maps. We write G T ( X, Y ) for the based space ofbased G -maps X −→ Y , with basepoint the trivial map.We let T G be the category whose objects are the nondegenerately based G -spaces,but in contrast to G T , whose morphisms are all based maps, not just the G -maps.Then G acts on maps by conjugation: for a map f : X −→ Y , ( gf )( x ) = g · f ( g − · x ).We write T G ( X, Y ) for the based G -space of based maps X −→ Y , with G actingby conjugation. Thus T G ( X, Y ) is a G -space with fixed point space G T ( X, Y ).We can view T G as a G -category such that G acts trivially on objects, and then G T can be viewed as the fixed point category of T G . In our treatment of theSegal machine, we will also need to consider the full subcategories G W ⊂ G T and W G ⊂ T G of based G -CW complexes. Properties of G -spaces are very often defined by passage to fixed point spaces.For example, a G -space X is said to be G -connected if X H is (path) connected forall H ⊂ G . Definition 1.1.
Let f : K −→ L be a map of G -spaces. We say that f is a weak G -equivalence if f H : K H −→ L H is a weak equivalence for all H ⊂ G . A family ofsubgroups of G is a set of subgroups closed under subconjugacy. For a family F ofsubgroups of G , we say that f is a weak F -equivalence if f is a weak H -equivalencefor all H ∈ F . We often omit the word weak, taking it to be understood throughout.The following families are central to equivariant bundle theory and to the analysisof equivariant infinite loop space machines. They will be used ubiquitously. Let Σ n denote the n th symmetric group. Definition 1.2.
For a subgroup H of G and a homomorphism α : H −→ Σ n , letΛ α be the subgroup { ( h, α ( h )) | h ∈ H } of G × Σ n . All subgroups Λ of G × Σ n suchthat Λ ∩ Σ n = { e } are of this form. Let F n denote the family of all such subgroups.Taking α to be trivial, we see that H ∈ F n for all n and all H ⊂ G . For our purposes, we need not restrict to essentially small full subcategories.
QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 7
Remark 1.3.
Since our homomorphisms are continuous, any α : G −→ Σ n factorsthrough a homomorphism π ( G ) −→ Σ n . In particular, there are no non-trivialhomomorphisms if G is connected. There are also infinite discrete groups G thatadmit no non-trivial homomorphisms to a finite group. Therefore, although thefamilies F n appear in general, they are only of real interest when G is finite.When G is finite, we adopt the following conventions on finite G -sets. Notation 1.4.
Let n denote the based set { , , . . . , n } with basepoint 0. For afinite group G , a homomorphism α : G −→ Σ n determines the based G -set ( n , α )specified by letting G act on n by g · i = α ( g )( i ) for 1 ≤ i ≤ n . Conversely, a based G -action on n determines a G -homomorphism α by the same formula. Every basedfinite G -set with n non-basepoint elements is isomorphic to one of the form ( n , α )for some α . We understand based finite G -sets to be of this form throughout.We also need some preliminaries about H -spaces, which we call Hopf spaces toavoid confusion with subgroups of G . Recall that a Hopf space is a based space X with a product such that the basepoint is a two-sided unit up to homotopy. Forsimplicity, we assume once and for all that our Hopf spaces are homotopy associativeand homotopy commutative , since that holds in our examples. We say that a Hopfspace is grouplike if, in addition, π ( X ) is a group, necessarily abelian. Definition 1.5.
A Hopf map f : X −→ Y is a group completion if Y is grouplike, f ∗ : π ( X ) −→ π ( Y ) is the Grothendieck group of the commutative monoid π ( X ),and for every every field of coefficients, f ∗ : H ∗ ( X ) −→ H ∗ ( Y ) is the algebraiclocalization obtained by inverting the elements of the submonoid π ( X ) of H ∗ ( X ). Definition 1.6.
A Hopf G -space is a based G -space X with a product G -mapsuch that its basepoint e is a two-sided unit element, in the sense that left orright multiplication by e is a weak G -equivalence X −→ X . Then each X H is aHopf space, and we assume as before that each X H is homotopy associative andcommutative. A Hopf G -space X is grouplike if each X H is grouplike. A Hopf G -map f : X −→ Y is a group completion if Y is grouplike and the fixed pointmaps f H are all nonequivariant group completions. Clearly a group completion ofa G -connected Hopf G -space is a weak G -equivalence.1.2. Preliminaries about G -cofibrations and simplicial G -spaces. Since G -cofibrations play an important role in our work, we insert some standard remarksabout them. Remark 1.7.
A map is a G -cofibration if it satisfies the G -homotopy extensionproperty and a basepoint ∗ ∈ X is nondegenerate if the inclusion ∗ −→ X is a G -cofibration. Since we are working in G U , a G -cofibration is an inclusion with closedimage [33, Problem 5.1]. By [6, Proposition A.2.1] (or [33, p. 43]), if i : A −→ B is a closed inclusion of G -spaces, then i is a G -cofibration if and only if ( B, A ) is a G -NDR pair. Using this criterion, we see that i is then also an H -cofibration forany (closed) subgroup H of G and that passage to orbits or to fixed points over It would suffice to assume that left and right translation by any element are homotopic. Segal [51, §
4] describes the notion of group completion a bit differently, in a form less amenableto equivariant generalization, and he makes several reasonable restrictive hypotheses in his proofof the group completion property. In particular, he assumes that X is a topological monoid andthat π ( X ) contains a cofinal free abelian monoid. They are part of the h -model structure on G U , as in [37] nonequivariantly. J. PETER MAY, MONA MERLING, AND ANG´ELICA M. OSORNO H gives a cofibration. Moreover, just as nonequivariantly, a pushout of a map of G -spaces along a G -cofibration is a G -cofibration.The Segal and operadic infinite loop space machines are both constructed usinggeometric realizations of simplicial G -spaces X q . Such realizations are only well-behaved when X q is Reedy cofibrant. Definition 1.8.
Let X q be a simplicial G -space with G -space of n -simplices X n .The n th latching space of X is given by L n X = n − [ i =0 s i ( X n − ) . It is a G -space, and the inclusion L n X −→ X n is a G -map. We say that X q isReedy cofibrant if this map is a G -cofibration for each n .With different nomenclature, this concept was studied nonequivariantly in theearly 1970’s (e.g. [27, § Lemma 1.9.
A simplicial G -space X q is Reedy cofibrant if all degeneracy operators s i are G -cofibrations.Proof. The nonequivariant statement is proven by an inductive application of Lil-lig’s union theorem stating that the union of cofibrations is a cofibration [23] (or[27, Lemma A.6]). The proof can be found in [51, proof of A.5] or [22, proof of2.4.(b)]. The equivariant proof is the same, using the equivariant version of Lillig’stheorem, which is a particular case of [7, Theorem A.2.7]. (cid:3)
The converse to Lemma 1.9 is proved in [45, Proposition 4.11], but we shall notuse it.
Theorem 1.10.
Let f q : X q −→ Y q be a map of Reedy cofibrant simplicial G -spacessuch that each f n is a weak G -equivalence. Then the realization | f q | : | X q | −→ | Y q | is a weak G -equivalence.Proof. Nonequivariantly this is in [28, Theorem A.4], and the equivariant versionfollows by application of the nonequivariant case to fixed point spaces, noting thatgeometric realization commutes with taking fixed points. (cid:3)
The following result is well-known, but since we could not find a proof in thepublished literature, we provide one in § Theorem 1.11.
Let f q : X q −→ Y q be a map of Reedy cofibrant simplicial G -spacessuch that each f n is a G -cofibration. Then the realization | f q | : | X q | −→ | Y q | is a G -cofibration. Remark 1.12.
Without exception, every simplicial G -space used in this paper isReedy cofibrant. In each case, we can check from the definitions and the fact that weare working with nondegenerately based G -spaces that all s i are G -cofibrations. Forthe examples appearing in the Segal machine, the verifications are straightforward.For the examples appearing in the operadic machine, the verifications follow thosein [27, Proposition A.10] and elaborations of the arguments there. We were inspired by [45], an unpublished masters thesis, which gives a detailed exposition ofsimplicial spaces. However, its statement of Theorem 1.11 is missing a necessary hypothesis.
QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 9
Categorical preliminaries and basepoints.
Some familiarity with enrichedcategory theory, especially in equivariant contexts, may be helpful. A more thor-ough treatment of the double enrichment present here is given in [17]. We needsome general definitions that start with a closed symmetric monoidal category V with unit object U and product denoted by ⊗ . Closed means that we have internalfunction objects V ( V, W ) in V giving an adjunction V ( X ⊗ Y, Z ) ∼ = V ( X, V ( Y, Z ))in V . We assume that V is complete and cocomplete.A V -category E is a category enriched in V . This means that for each pair ( m, n )of objects of E there is an object E ( m, n ) of V and there are unit and compositionmaps I : U −→ E ( m, m ) and C : E ( n, p ) ⊗ E ( m, n ) −→ E ( m, p ) satisfying theidentity and associativity axioms. It would be more categorically precise to write E ( m, n ), saving E ( m, n ) for the underlying set of morphisms m −→ n . A V -functor F : E −→ Q between V -categories is a functor enriched in V . This means that foreach pair ( m, n ), there is a map F : E ( m, n ) −→ Q ( F ( m ) , F ( n ))in V , and these maps are compatible with the unit and composition of E and Q .A V -transformation η : F −→ F ′ is given by maps η m : U −→ Q ( F ( m ) , F ′ ( m )) in V such that the evident naturality diagram commutes in V . E ( m, n ) F / / F ′ (cid:15) (cid:15) Q ( F ( m ) , F ( n )) ( η n ) ∗ (cid:15) (cid:15) Q ( F ′ ( m ) , F ′ ( n )) ( η m ) ∗ / / Q ( F ( m ) , F ′ ( n ))We are especially interested in the cases V = T and V = G T . In this paper,topological G -categories are understood to mean categories enriched in G T . Whenwe enrich in spaces, the hom set is obtained from the hom space just by forgettingthe topology, and we omit the underline. Thus when thinking of enrichment in T ,we write T ( X, Y ) for the based space of based maps X −→ Y when there is no G -action in sight and we write G T ( X, Y ) for the based space of based G -maps X −→ Y when X and Y are G -spaces.We previously defined T G to be the category of based G -spaces and nonequiv-ariant maps, with G acting by conjugation on the hom spaces T G ( X, Y ). From thepoint of view of enriched category theory, T G just gives another name for the homobjects that give the enrichment. That is, G T ( X, Y ) = T G ( X, Y ) , and similarly for U . We speak in general of G T -categories, but we distinguishnotationally by writing G T -functors as X : E −→ G T when G acts trivially on E and X : E −→ T G in general. In the first case, we are enriching just in T , usingspaces of G -maps, but this is a G -trivial special case of equivariant enrichment.There is considerable confusion in the literature concerning the handling of base-points. The category T has two symmetric monoidal products, ∧ with unit S and × with unit ∗ . When enriching in T , we must use ∧ since we must use the closedstructure given by the spaces F ( X, Y ) of based maps, with basepoint given by In the sequels [15, 16], we work more generally with categories internal to G U . X → ∗ → Y . However, we sometimes use the forgetful functors T −→ U and G T −→ G U to forget basepoints in our enrichments, and implicitly we are thenthinking about × .In all variants and generalizations of the Segal machine, we start with a category E enriched over T or G T . It has a zero object 0, so that there are unique maps0 → n and n → n ∈ E . Then E ( n,
0) and E (0 , n ) are each a pointand the map m → → n is the basepoint of E ( m, n ), which must be nondegenerateto have the cited enrichment. We are concerned with G T -functors defined on E ;by neglect of G -action, they are also T -functors. The following trivial observationhas been overlooked since the start of this subject. Lemma 1.13.
For any T -functor X : E −→ T , X (0) = ∗ .Proof. The unique map 0 → E (0 ,
0) is both the basepoint and the identity; X : E (0 , −→ T ( X (0) , X (0)) must send it to both the trivial map X (0) −→ X (0)that sends all points to the basepoint and the identity map. This can only happenif X (0) is a point. (cid:3) Remark 1.14.
For a G U -category J and a G T -category Q , we can add disjointbasepoints to the hom objects of J to form a G T -category J + or we can forgetbasepoints to regard Q as a G U -category U Q . Via the adjunction betwen ( − ) + and U , G U -functors J −→ U Q can be identified with G T -functors J + −→ Q . Remark 1.15.
Let E be a G T -category with a zero object. A G U -functor X : U E −→ U G is said to be reduced if X (0) is a point. If X is reduced themap X (0) −→ X ( n ) induced by 0 −→ n gives each X ( n ) a basepoint, and com-position preserves basepoints. Moreover, if these basepoints are nondegenerate, X will give a T -functor E −→ T G , as X then sends the zero map m → → n to thetrivial map X ( m ) → ∗ = X (0) → X ( n ).Let E be a G T -category with a zero object 0, and let X and Y be respectivelycovariant and contravariant G T -functors E −→ T G . Then the tensor product offunctors Y ⊗ E X is defined as the coequalizer of the diagram _ m,n Y n ∧ E ( m, n ) ∧ X m / / / / _ n Y n ∧ X n , where the arrows are given by the action of E on X and on Y , respectively.We obtain G U -functors U X , U Y : U E −→ U G by forgetting basepoints. Thetensor product of functors U Y ⊗ U E U X is the coequalizer of the diagram a m,n Y n × E ( m, n ) × X m / / / / a n Y n × X n . The following result shows that the difference between these two constructionsis only apparent. Later on, we shall sometimes use wedges and smash products andsometimes instead use disjoint unions and products, whichever seems convenient.
Lemma 1.16.
With E , X , and Y as above, there is a natural isomorphism Y ⊗ E X ∼ = U Y ⊗ U E U X. Proof.
We will show that the quotient of ` n Y n × X n given by the coequalizerencodes the required basepoint identifications. Let ( y, ∗ n ) ∈ Y n × X n . As noted QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 11 above, the basepoint in X n is given by ∗ n = (0 ,n ) ∗ ( ∗ ), where ∗ is the uniquepoint in X and 0 ,n is the unique map 0 −→ n in E . Then( y, ∗ n ) = ( y, (0 ,n ) ∗ ( ∗ )) ∼ (0 ∗ ,n ( y ) , ∗ ) = ( ∗ , ∗ ) , the last equation following from the fact that Y is a singleton. A similar argumentshows that ( ∗ n , x ) ∼ ( ∗ , ∗ ). (cid:3) Spectrum level preliminaries.Definition 1.17.
A naive G -spectrum, which we prefer to call a G -prespectrum,is a sequence of based G -spaces { T n } n ≥ and based G -maps σ n : Σ T n −→ T n +1 .It is a naive Ω- G -spectrum if the adjoint maps ˜ σ n : T n −→ Ω T n +1 are weak G -equivalences. It is a positive Ω- G -spectrum if the ˜ σ n are weak G -equivalences for n ≥
1. We let G P denote the category of G -prespectra. We call zeroth spaces T of naive Ω- G -spectra naive infinite loop G -spaces.When we restrict to compact Lie groups, our preferred category of (genuine) G -spectra will be the category G S of orthogonal G -spectra. Orthogonal G -spectraand their model structures are studied in [24], to which we refer the reader fordetails and discussion of the following definition. We use Remark 1.14. Definition 1.18.
Let G be a compact Lie group and let I G be the G U -category offinite dimensional real G -inner product spaces and linear isometric isomorphisms,with G acting on morphism spaces by conjugation. Note that I G is symmetricmonoidal under ⊕ . An I G - G -space is a G U -functor I G −→ T G or, equivalentlyby Remark 1.14, a G T -functor I G + −→ T G . The sphere I G - G -space S is givenby S ( V ) = S V . The external smash product X ⊼ Y : I G × I G −→ T G of I G - G -spaces X and Y is the G U -functor given by( X ⊼ Y )( V, W ) = X ( V ) ∧ Y ( W ) . A (genuine orthogonal) G -spectrum is an I G - G -space E : I G −→ T G together witha G U -transformation E ⊼ S −→ E ◦ ⊕ between G U -functors I G × I G −→ T G .Thus we have G -spaces E ( V ), morphism G -maps I G ( V, V ′ ) −→ U G ( E ( V ) , E ( V ′ ))and structure G -maps σ : E ( V ) ∧ S W −→ E ( V ⊕ W ) . natural in V and W . Note in particular that I G ( V, V ) is the orthogonal group O ( V ), with G acting by conjugation, so that E ( V ) is both a G -space and an O ( V )-space and σ is a map of both G -spaces and O ( V ) × O ( W )-spaces. A G -spectrum E is an Ω- G -spectrum if the adjoint maps˜ σ : E ( V ) −→ Ω W E ( V ⊕ W )are weak G -equivalences. It is a positive Ω- G -spectrum if these maps are weak G -equivalences when V G = 0. We let G S denote the category of G -spectra. Wecall zeroth spaces E (0) of Ω- G -spectra genuine infinite loop G -spaces, or simplyinfinite loop G -spaces. The Ω- G -spectra are the fibrant objects in the stable model structure on G S and the positive Ω- G spectra are the fibrant objects in the positive stable modelstructure. The identity functor is a left Quillen equivalence from the stable modelstructure to the positive stable model structure. We have the following change ofuniverse functor, which is a right Quillen adjoint. Definition 1.19.
The forgetful functor i ∗ : G S −→ G P sends a G -spectrum X to the (naive) G -prespectrum with n th space X n = X ( R n ). Remark 1.20. If V G = 0, we can write V = R ⊕ W and thus S V = S ∧ S W andΩ V = ΩΩ W . Then ˜ σ : X −→ Ω V X ( V ) factors as the composite X σ / / Ω X σ / / ΩΩ W X ( R ⊕ W ) = Ω V X ( V ) . If X is a positive Ω- G -spectrum, then the second arrow is a weak G -equivalence.Therefore, if X −→ Ω X is a group completion, then so is X −→ Ω V X ( V ) for all V such that V G = 0.1.5. Equivariant infinite loop space machines.
In this section, we give a veryquick overview of equivariant infinite loop space machines. The details are workedout in the next few sections.Nonequivariantly, there are several recognition principles that one can apply tospaces to determine whether they become infinite loop spaces after group comple-tion. One is the operadic approach developed by the first author in [27] and anotheris the approach using Γ-spaces developed by Segal in [51]. The opposite categoryof Segal’s Γ is the category F of finite sets, and we shall call Γ-spaces F -spaces.Infinite loop space machines take some appropriate input Y , part of which is anunderlying Hopf space X , and construct from it an Ω-spectrum E Y together with agroup completion X −→ E Y . This form was taken as the definition of an infiniteloop space machine in [38]. If G acts on the input data Y through maps that arecompatible with the structure, then both the operadic machine and the Segal ma-chine generalize immediately to give infinite loop space machines landing in naiveΩ- G -spectra.The genuine equivariant theory is much harder since genuine infinite loop G -spaces have deloopings not only with respect to all spheres S n , but also with respectto all representation spheres S V for all finite dimensional G -representations V . Forfinite G , the genuine equivariant generalization of the operadic approach was firstworked out in [10] and is worked out more fully in [12]. The genuine equivariantversion of Segal’s approach was first worked out in [54] and is worked out more fullyhere. Both machines are generalized here to forms which accept the same input,and then they are proven to give equivalent output when fed the same input.When G is a finite group, genuine equivariant infinite loop space machines E G take appropriate input Y with underlying Hopf G -spaces X to genuine Ω- G -spectra E Y . They restrict to give underlying naive G -spectra, and the group completion X −→ E Y is seen by the underlying naive G -spectrum. As we shall see, themachines as they appear most naturally do not take precisely this form, and wethen have to tweak them into the form just specified. There are general featurescommon to any equivariant infinite loop space machine, and we refer the readerto [12, § QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 13 The simplicial and conceptual versions of the Segal machine
There are several variants of the Segal infinite loop space machine, as originallydeveloped by Segal [51] and Woolfson [59]. Later sources include Bousfield andFriedlander [8], working simplicially, and, much later, Mandell, May, Schwede, andShipley [25]. Equivariant versions appear in Shimada and Shimakawa [53, 54, 55]and, later, Blumberg [5].We here give a simplicial variant and two equivalent conceptual variants, onestarting from finite sets and the other starting from finite G -sets. Of course, theuse of finite G -sets is mainly of interest when G is finite, but it applies in general.We defer consideration of our preferred homotopical variant to the next section.The simplicial variant is the equivariant version of Segal’s original definition[51]. As far as we know, his paper is the only source in the literature that actuallyproves the crucial group completion property, and his proof makes essential use ofhis original simplicial definition. This version does not directly generalize to givegenuine G -spectra when G is a compact Lie group or even a finite group, and itdoes not appear in the equivariant literature. Therefore, even at this late date,there is no published account of the equivariant Segal machine that proves thegroup completion property. Just as nonequivariantly, this property is central to theapplications, especially to algebraic K -theory.In fact, we do not know a direct proof of the group completion property startingfrom the conceptual or homotopical variants treated in [8, 25, 53, 59] and, equivari-antly, [5, 54]. Rather, we derive it for the conceptual variants from their equivalencewith the simplicial variant. To give the group completion property equivariantly,to differentiate the theory for varying types of groups, and to prepare for a com-parison with the operadic machine, we give a fully detailed exposition of the Segalmachine in all its forms. This may also be helpful to the modern reader since evennonequivariantly the original sources make for hard reading and are sketchy in someessential respects.2.1. Definitions: the input of the Segal machine.Definition 2.1.
Let F be the opposite of Segal’s category Γ. It is the categoryof finite based sets n = { , , . . . , n } with 0 as basepoint. The morphisms are thebased maps, and the unique morphism that factors through is a nondegeneratebasepoint for F ( m , n ). Let Π ⊂ F be the subcategory with the same objectsand those morphisms φ : m −→ n such that φ − ( j ) has at most one element for1 ≤ j ≤ n ; these are composites of projections, injections, and permutations. LetΣ ⊂ Π be the subgroupoid with the same objects and the elements of the symmetricgroups Σ n , regarded as based isomorphisms n −→ n , as morphisms.Since the composition in F factors through the smash product, we can view F as a category enriched in T , with the discrete topology on the based hom sets. Definition 2.2. An F -space is a T -functor X : F −→ T , written n X n ; Π-spaces are defined similarly. A map of F -spaces or of Π-spaces is a T -naturaltransformation. We are referring to simplicial spaces, not simplicial sets, here. His proof imposes some unnecessary restrictive hypotheses that generally hold in practice. As in [25] and elsewhere, we use the notation F to avoid confusion between Γ and Γ op = F . Definition 2.3.
The
Segal maps δ i : n −→ in F send i to 1 and j to 0 for j = i . These maps are all in Π. For an F -space X , the Segal map δ : X n −→ X n has coordinates induced by the δ i . If n = 0, we interpret δ as the terminal map X −→ ∗ . The “multiplication map” φ n : n −→ , which is not in Π, sends j to 1for 1 ≤ j ≤ n . It induces an “ n -fold multiplication” X n −→ X on an F -space X . Definition 2.4. An F - G -space is a T -functor X : F −→ G T ; Π- G -spaces aredefined similarly. A map of F - G -spaces or Π- G -spaces is a G T -natural transfor-mation. Since Σ ⊂ Π ⊂ F , X n and X n are ( G × Σ n )-spaces and δ : X n −→ X n isa map of G × Σ n -spaces. Remark 2.5.
It is usual, starting in [51, Definition 1.2], to define an F -space tobe a (non-enriched) functor F −→ T , requiring X to be contractible, and to saythat X is reduced if X is a point. This led to mistakes and confusion, as explainedin [35]. As we observed in Lemma 1.13, our requirement that X be a T -functorforces X to be a point for trivial reasons; compare Remark 1.15.Recall from Definition 1.2 that for a homomorphism α : G −→ Σ n , Λ α is thesubgroup { ( g, α ( g )) | g ∈ G } of G × Σ n . Definition 2.6.
Let
X, Y be Π- G -spaces, and f : X −→ Y be a map of Π- G -spaces.(i) The Π-space X is F • - special if δ : X n −→ X n is a weak Λ α -equivalence for all n ≥ α : G −→ Σ n (where Σ = { e } = Σ ).(ii) The Π-space X is special if each δ : X n −→ X n is a weak G -equivalence.(iii) The map f is an F • -level equivalence if each f n : X n −→ Y n is a weak Λ α -equivalence for all homomorphisms α : G −→ Σ n .(iv) The map f is a level G -equivalence if each f n : X n −→ Y n is a weak G -equivalence.An F - G -space X is F • -special or special if its underlying Π- G -space is so. Aspecial F - G -space X is grouplike or, synonymously, very special if π ( X H ) is agroup (necessarily abelian) under the induced product for each H ⊂ G . A map f : X −→ Y of F - G -spaces is an F • -level equivalence or level equivalence if it is soas a map of Π- G -spaces.When there are no non-trivial homomorphisms G −→ Σ n , for example when G is connected, a Π- G -space is F • -special if and only if it is special. In fact, the notionof an F • -special Π- G -space is only of substantial interest when G is finite. However, § Lemma 2.7.
Let X be a G -space, and let Λ = { ( h, α ( h )) | h ∈ H } ⊂ G × Σ n ,where H ⊂ G and α : H −→ Σ n is a homomorphism. Then there is a naturalhomeomorphism ( X n ) Λ ∼ = Y X K i , where the product is taken over the orbits of the H -set ( n , α ) and the K i ⊂ H arethe stabilizers of chosen elements in the corresponding orbit.Proof. The Λ-action on X n is given by( h, α ( h ))( x , . . . , x n ) = ( hx α ( h − )(1) , . . . , hx α ( h − )( n ) ) . QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 15
The partition of n into H -orbits decomposes n as the wedge of finite subsets, eachwith a transitive set of shuffled indices, so it is enough to consider each H -orbitseparately. Thus we may as well assume that the H -action on ( n , α ) is transitive.Note that this reduction is natural in X .Let K ⊂ H be the stabilizer of 1 ∈ n . We claim that projection onto the firstcoordinate induces the required natural homeomorphism π : ( X n ) Λ −→ X K . If( x , . . . , x n ) ∈ X n is a Λ-fixed point, then x is a K -fixed point since kx = kx α ( k − )(1) = x for k ∈ K , the second equality holding because ( x , . . . , x n ) is fixed by Λ.To construct π − : X K −→ ( X n ) Λ , for 1 ≤ j ≤ n choose h j ∈ H such that α ( h j )(1) = j . This choice amounts to choosing a system of coset representatives for H/K , and the map j [ h j ] gives a bijection of H -sets between ( n , α ) and H/K .We claim that the map : X −→ X n that sends x to the n -tuple ( h x, . . . , h n x )restricts to the required inverse π − . This map is clearly continuous. We first showthat if x ∈ X K , then ( h x, . . . , h n x ) is fixed by Λ. Let h ∈ H and note that α ( hh α ( h − )( j ) )(1) = α ( h )( α ( h α ( h − )( j ) )(1)) = α ( h )( α ( h − )( j )) = j. In view of our bijection between ( n , α ) and H/K , there exists k ∈ K such that hh α ( h − )( j ) = h j k . The j th coordinate of ( h, α ( h )) · ( h x, . . . , h n x ) is given by hh α ( h − )( j ) x = h j kx = h j x, the second equality holding because x ∈ X K . Thus ( h x, . . . , h n x ) is fixed by Λ.Since K is the stabilizer of 1, h ∈ K . Thus the first coordinate of ( h x, . . . , h n x )is x itself, and π ◦ π − = id. If ( x , . . . , x n ) is fixed by Λ, then x j = h j x forall j . By the definition of h j , the j th coordinate of ( h j , α ( h j )) · ( x , . . . , x n ) is h j x α ( h − j )( j ) = h j x . Since( x , . . . , x n ) is a Λ-fixed point, this shows that x j = h j x ,hence π − ◦ π = id. (cid:3) Definition 2.8.
For a based G -space X , let R X denote the Π- G -space with n th G -space X n . The Π-space structure is given by basepoint inclusions, projections,and permutations. Lemma 2.9. If f : X −→ Y is a weak equivalence of based G -spaces, then theinduced map R f : R X −→ R Y is an F • -level equivalence of Π - G -spaces.Proof. This is immediate from Lemma 2.7. (cid:3)
Lemma 2.10.
Let f : X −→ Y be an F • -level equivalence of Π - G -spaces. Then X is F • -special if and only if Y is F • -special. Similarly, if f is a level G -equivalence,then X is special if and only if Y is special.Proof. Consider the commutative diagram X n f n / / δ (cid:15) (cid:15) Y nδ (cid:15) (cid:15) X n f n / / Y n . For the first statement, the horizontal arrows are weak Λ α -equivalences by assump-tion and Lemma 2.9, so one of the vertical arrows is a weak Λ α -equivalence if andonly if the other one is. The proof of the second statement is similar but simpler. (cid:3) We defined F • -special Π- G -spaces in terms of those subgroups Λ in the family F n (see Definition 1.2) which are defined by homomorphisms α : G −→ Σ n , ignoringthose which are defined by homomorphisms β : H −→ Σ n for proper subgroups H of G . The following result shows that when G is finite we obtain the same notionif we instead use all of the groups in F n . The result implicitly relates F • -specialΠ- G -spaces to equivariant covering space theory and relates F • -special F - G -spacesto the operadic approach to equivariant infinite loop space theory. It thereforeexplains and justifies the terms F • -special and F • -level equivalence. Lemma 2.11.
Assume that G is finite. Then a Π - G -space X is F • -special if andonly if the Segal maps δ : X n −→ X n are weak F n -equivalences for all n ≥ .Similarly, a map f : X −→ Y of Π - G -spaces is an F • -level equivalence if and onlyif f n is a weak F n -equivalence for all n ≥ .Proof. If δ : X n −→ X n is a weak F n -equivalence, then it is a weak Λ β -equivalencefor all H ⊂ G and all homomorphisms β : H −→ Σ n . Restricting to those homo-morphisms with domain G , this condition for all n implies that X is F • -special.Conversely, assume that X is F • -special. We must prove that each δ is a weak F n -equivalence.Thus consider a subgroup Λ β ⊂ G × Σ n , β : H −→ Σ n . We will show that δ : X n −→ X n is a weak Λ β -equivalence by displaying it as a retract of a suitableweak equivalence. The homomorphism β gives rise to an H -set B = ( n , β ). Embed B as a subset of the G -set A = G + ∧ H B and observe that, as an H -set, A splits as B ∨ C , where C = ( A \ B ) + . Let p = | A \ B | and q = n + p . Use the given orderingof B and an ordering of C to identify A with ( q , α ). Here q = n ∨ p and α is ahomomorphism G −→ Σ q which when restricted to H is of the form β ∨ γ . Thatis, ( q , α | H ) = ( n , β ) ∨ ( p , γ | H ). Let ι : ( n , β ) −→ ( q , α | H ) and π : ( q , α | H ) −→ ( n , β )be the inclusion that sends i to i for 0 ≤ i ≤ n and the projection that sends i to i for 0 ≤ i ≤ n and i to 0 for i > n . Then the following diagram displays a retraction.Its bottom arrows are the evident inclusion and projection. X nδ (cid:15) (cid:15) ι ∗ / / X qδ (cid:15) (cid:15) π ∗ / / X nδ (cid:15) (cid:15) X n / / X q / / X n Since X is F • -special, the middle vertical arrow δ is a weak Λ α -equivalence andthus a weak Λ α | H -equivalence. Therefore the left arrow δ is a weak Λ β -equivalence.Similarly, if f : X −→ Y is an F • -level equivalence, we have a retract diagram X nf n (cid:15) (cid:15) ι ∗ / / X qf q (cid:15) (cid:15) π ∗ / / X nf n (cid:15) (cid:15) Y n ι ∗ / / Y q π ∗ / / Y n in which f q is a weak Λ α -equivalence and therefore f n is a weak Λ β -equivalence. (cid:3) QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 17
The simplicial version of the Segal machine.
Let ∆ be the usual sim-plicial category. A simplicial object is a contravariant functor defined on ∆, and acosimplicial object is a covariant functor. Regarding F as a full subcategory of thecategory of based sets, we may regard the simplicial circle S s = ∆[1] /∂ ∆[1] as acontravariant functor F : ∆ op −→ F . By pullback along F , an F - G -space X canbe viewed as a simplicial G -space, and it has a geometric realization | X | = | X ◦ F | ;we use the standard realization, taking degeneracies into account. The evident G -map X × I −→ | X | factors through a natural G -map Σ X −→ | X | with adjoint η : X −→ Ω | X | . We shall give the proof of the following result in § § Proposition 2.12. If X is a special F - G -space, then the G -map η : X −→ Ω | X | is a group completion of Hopf G -spaces. From here, the Segal machine in its first avatar is constructed as follows [51, § Remark 2.13.
We have the smash product ∧ : F × F −→ F . It sends ( m , n ) to mn and is strictly associative and unital using lexicographic ordering. The unit is . We also have the wedge sum ∨ : F × F −→ F which sends ( m , n ) to m + n .It is also strictly associative and unital, with unit , and F is bipermutative underthis sum and product. Definition 2.14.
Let X be a special F - G -space. We have the functor X ◦ ∧ : F × F −→ G T . For each q , let X [ q ] be the F - G -space that sends p to X ( p ∧ q ); thus X [0] = ∗ and X [1] = X . Following Segal, define the classifying F - G -space B X to be the F - G -space whose q th G -space is the realization | X [ q ] | . Iterating, with B X = X ,define B n +1 X = B ( B n X ) for n ≥
0. The F - G -spaces B n X for n ≥ B n X ) is G -connected, they are also grouplike. Notation 2.15.
Let S NG X denote the resulting naive G -prespectrum with n th space( S NG X ) n = ( B n X ) for n ≥
0. Thus its 0th G -space is X and, by Proposition 2.12,its structure map X −→ Ω( S NG X ) is a group completion and the structure maps( S NG X ) n −→ Ω( S NG X ) n +1 for n ≥ G -equivalences.With this definition, S NG X is a positive Ω- G -spectrum. Varying the definitionby taking the 0th G -space to be Ω( B X ) , the nonequivariant Segal machine playsa special role. As proven in [38], any infinite loop space machine that takes F -spaces, or appropriate more general input, to Ω-spectra and has a natural groupcompletion map from X to its zeroth space is equivalent to the Segal machine. Theproof makes essential use of the fact that the Segal machine produces F F -spaces,namely functors F −→ F T .We emphasize that this construction of the Segal machine works for any topo-logical group G . Moreover, the uniqueness proof for infinite loop space machinesin [38] works verbatim to compare any other infinite loop space machine landingin naive G -spectra to the Segal machine. However, even when G is finite, this con-struction does not work to construct genuine G -spectra from F - G -spaces: there isno evident way to build in deloopings by non-trivial representations of G . Perversely, [38] takes ∆ to be the opposite of the category every other reference calls ∆.
The conceptual version of the Segal machine.
Returning to F - G -spaces,the more conceptual variants of the nonequivariant Segal machine generalize to givegenuine G -spectra when G is compact Lie. These variants do not make use of thefunctor F : ∆ op −→ F . That is, underlying simplicial G -spaces play no role intheir construction. We follow the nonequivariant exposition of [25]; [5] gives somerelevant equivariant details. While [5] dealt with compact Lie groups, much of itapplies equally well to general topological groups G . We introduce notation for thecategories of enriched functors that we shall be using. Recall § Notation 2.16.
For a (small) G T -category D , let Fun( D , T G ) denote the categoryof G T -functors D −→ T G and G T -natural transformations between them. When G acts trivially on D , as is the case of F , a G T -functor defined on D take valueson morphisms in the fixed point spaces T G ( X, Y ) G = G T ( X, Y ) of based G -maps X −→ Y . We therefore use the alternative notation Fun( D , G T ) in that case. Definition 2.17. A W G - G -space Y is a G T -functor W G −→ T G . A map of W G - G -spaces is a G T -natural transformation between them. Regarding F as a G -trivial G -category, it is both a full subcategory of G W and a G -trivial full G -subcategoryof W G . We have the functor categoriesFun( F , G T ) = Fun( F , T G )of F - G -spaces and Fun( W G , T G ) of W G - G -spaces. The inclusion F ⊂ W G inducesa forgetful functor U : Fun( W G , T G ) −→ Fun( F , T G ) . We say that a W G - G -space Y is F • -special, special, or grouplike if the F - G -space U Y is so.As a matter of elementary category theory (see e.g. [25]), the functor U has aleft adjoint prolongation functor P : Fun( F , T G ) −→ Fun( W G , T G ) . The study of model structures on W G - G -spaces given in [5] when G is a compactLie group applies verbatim when G is any topological group. Remark 2.18.
Let Y be any W G - G -space, such as Y = P X for an F - G -space X .For G -spaces A, B ∈ G W , the adjoint B −→ W G ( A, A ∧ B ) of the identity map on A ∧ B can be composed with Y to obtain a G -map B −→ T G ( Y ( A ) , Y ( A ∧ B )) . Its adjoint is a G -map(2.19) Y ( A ) ∧ B −→ Y ( A ∧ B ) . Letting A = S n and B = S with trivial G -action, these maps give the structuremaps Σ Y ( S n ) −→ Y ( S n +1 )of a naive G -prespectrum U G P Y .When G is a compact Lie group we can define an orthogonal G -spectrum givenat level V by Y ( S V ). The composites I G ( V, V ′ ) / / W G ( S V , S V ′ ) Y / / T G ( Y ( S V ) , Y ( S V ′ )) QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 19 of Y and the map induced by one-point compactification of maps V −→ V givea G U -functor I G −→ T G or, equivalently and more sensibly here, a G T -functor I G + −→ T G . Just as in the nonequivariant case [25], letting A = S V and B = S W in (2.19) for representations V and W , we obtain the structure G -mapsΣ W Y ( S V ) −→ Y ( S V ⊕ W )of an orthogonal G -spectrum U G S Y such that i ∗ U G S Y = U G P Y .More generally, as in [25], we have forgetful functors U C : Fun( W G , T G ) −→ C , where C can be the category of G -prespectra, symmetric G -spectra, or orthogonal G -spectra. Of course, nonequivariantly, Segal took C to be prespectra. We choosenaive G -prespectra, G P , for general topological groups G and genuine orthogonal G -spectra, G S , for compact Lie groups G . Definition 2.20.
For a general topological group G , the conceptual Segal machineon F - G -spaces is the composite U G P ◦ P : Fun( F , G T ) = Fun( F , T G ) −→ G P . For compact Lie groups G , the conceptual Segal machine on F - G -spaces is theanalogous composite U G S ◦ P : Fun( F , G T ) −→ G S . Its composite with i ∗ : G S −→ G P is U G P ◦ P .The functor P is a left Kan extension that is best viewed as a tensor productof functors. For A ∈ G W , we have the contravariant G T -functor A • : F −→ G T . Conceptually, it is the represented functor that sends n to the function space W G ( n , A ) ∼ = A n with its induced action by G . By definition,(2.21) ( P X )( A ) = A • ⊗ F X. Taking A = n , the unit η : X −→ UP X of the adjunction sends x ∈ X n to (id n , x );by Yoneda, η is a natural isomorphism. For a W G - G -space Y : W G −→ T G , thecounit ε : PU Y −→ Y is given on A ∈ G W by the composites W G ( n , A ) ∧ Y ( n ) Y ∧ id / / T G ( Y ( n ) , Y ( A )) ∧ Y ( n ) eval / / Y ( A ) . For an F - G -space X : F −→ G T , we write X n for X ( n ) as before, but wefollow the usual convention of abbreviating notation by writing ( P X )( A ) = X ( A )for general A ∈ G W . The following result is a variant of Segal’s [51, Proposition3.2 and Lemma 3.7]. Proposition 2.22.
The naive G -prespectrum U G P P X is naturally isomorphic tothe G -prespectrum S NG X (of Notation 2.15). Thus, if X is special, then U G P P X isa positive Ω - G -prespectrum with bottom structural map a group completion of X . Segal’s proof is very briefly sketched in [51, § § U G P P X transparent. Proposition 2.23.
For F - G -spaces X , there is a natural G -homeomorphism | X | −→ ( S ) • ⊗ F X = ( P X )( S ) . A factorization of the conceptual Segal machine.
The previous sectionsapply to general topological groups. We continue in that generality. The resultsof this section are illuminating in general, but they are only really useful when G is finite. Here we consider the G -category F G of finite based G -sets ratherthan just the category F of finite based sets. Use of F G in tandem with F isessential to our work. In practice, input arises most often as F - G -spaces but, bya result of Shimakawa [55] that we shall reprove with different details, these areinterchangeable with F G - G -spaces. Definition 2.24.
Let F G be the G -category of finite based G -sets and all basedfunctions, with G acting by conjugation on function sets. For convenience andprecision, we restrict the objects of F G to be the finite G -sets A = ( n , α ), asin Notation 1.4. Let Π G be the G -subcategory with the same objects and thosemorphisms φ : ( m , α ) −→ ( n , β ) such that φ − ( j ) has at most one element for1 ≤ j ≤ n . We obtain inclusions F ⊂ F G and Π ⊂ Π G by restricting to the trivialhomomorphisms ε n : G −→ Σ n .As with F , we view F G as a category enriched in G T , with the discrete topologyon the based hom sets of maps, on which G acts by conjugation. The basepoint of F G (( m , α ) , ( n , β )) is the unique map that factors through . When G is finite, afinite G -set is evidently a G -CW complex and for a general G we can enlarge W G if necessary to obtain an inclusion F G ⊂ W G . Definition 2.25. An F G - G -space Y is a G T -functor Y : F G −→ T G ; a Π G - G -space Y is a G T -functor Y : Π G −→ T G . Morphisms are G T -natural transforma-tions. We write Y ( A ) for the value of Y on A = ( n , α ), writing Y n for the value of Y on ( n , ε n ). We let ( Y n ) α denote Y n with the G -action g ( y , . . . , y n ) = ( gy α ( g − )(1) , . . . , gy α ( g − )( n ) ) . It can be identified with the G -space T G ( A, Y ). Definition 2.26.
For A = ( n , α ) and a Π G - G -space Y , define a based G -map ε : A ∧ A −→ = S by the Kronecker δ function: ( i, j ) i = j and to 0 if i = j . Its adjoint is a G -map A −→ F G ( A, ). Composing with Y : F G ( A, ) −→ T G ( Y ( A ) , Y )and adjointing, we obtain a G -map ∂ A : A ∧ Y ( A ) −→ Y . Thus ∂ A ( j, y ) = ( δ j ) ∗ ( y )for 1 ≤ j ≤ n , where δ j is induced by the j th projection ( n , α ) −→ ( , ε ). The Segal map δ A : Y ( A ) −→ T G ( A, Y ) ∼ = ( Y n ) α is the adjoint of ∂ A ; we usually abbreviate δ A to δ . Note that δ is a G -map, althoughit components δ j are usually not. Definition 2.27.
A Π G - G -space Y is special if the δ A are weak G -equivalencesfor all A = ( n , α ). A map f : Y −→ Z of Π G - G -spaces is a level G -equivalence ifeach f : Y ( A ) −→ Z ( A ) is a weak G -equivalence. We say that an F G - G -space isspecial if its underlying Π G - G -space is so and that a map of F G - G -spaces is a level G -equivalence if its underlying map of Π G - G -spaces is so. The starting point of [55] came from conversations during a long and mutually profitablevisit Shimakawa made to the first author.
QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 21
The inclusion F ֒ → F G induces a restriction functor U : Fun( F G , T G ) −→ Fun( F , G T )which has a left adjoint prolongation functor P : Fun( F , G T ) −→ Fun( F G , T G ) . The adjunction ( P , U ) of Definition 2.17 factors as the composite of the analogousadjunctions given by the functorsFun( F , G T ) P / / Fun( F G , T G ) P / / Fun( W G , T G )Fun( W G , T G ) U / / Fun( F G , T G ) U / / Fun( F , G T ) . The units of these adjunctions are isomorphisms since the forgetful functors U areinduced by the full and faithful inclusions F ֒ → F G and F G ֒ → W G . Notation 2.28.
For A ∈ W G , we now write A • ambiguously for both the restrictionto F G ⊂ T G and the restriction to F ⊂ F G ⊂ T G of the represented functor T G ( − , A ) : T opG −→ T G .Then the factorization of P as PP takes the explicit form(2.29) A • ⊗ F X ∼ = A • ⊗ F G ( F G ⊗ F X ) = A • ⊗ F G P X. While our main interest is in F G - G -spaces and F - G -spaces, we will also usethe analogous forgetful and prolongation functors relating Π G - G -spaces and Π- G -spaces.Observe that we have no analogue for Π G - G -spaces (or for F G - G -spaces) of thedichotomies between F • -special and special and between F • -level equivalences andlevel G -equivalences that we had for Π- G -spaces (and thus for F - G -spaces). Thefollowing result shows that the notions defined in Definition 2.27 for Π G - G -spacescorrespond to the F • -notions for Π- G -spaces. That should help motivate the latter,which may at first sight have seemed unnatural. Theorem 2.30.
The adjoint pairs of functors
Fun(Π , G T ) P / / Fun(Π G , T G ) U o o and Fun( F , G T ) P / / Fun( F G , T G ) U o o specify equivalences of categories. Moreover, the following statements hold.(i) A Π G - G -space Y is special if and only if the Π - G -space U Y is F • -special.(ii) A map f : Y −→ Z of Π G - G -spaces is a level G -equivalence if and only if themap U f : U Y −→ U Z of Π - G -spaces is an F • -level equivalence.(iii) A Π - G -space X is F • -special if and only if the Π G - G -space P X is special.(iv) A map f of Π - G -spaces is an F • -level equivalence if and only if the map P f of Π G - G -spaces is a level G -equivalence.All of these statements remain true with with Π and Π G replaced by F and F G . Proof.
For a Π- G -space X and a finite G -set A = ( n , α ),( P X )( A ) = A • ⊗ Π X, where A • : Π −→ G T is the functor that sends m to Π G ( m , A ). Recall that theunderlying set of Π G ( m , A ) is just Π( m , n ) with G -action induced by the action of G on n given by α . The action of G on A • ⊗ Π X is induced by the diagonal action.For n ≥
0, the unit η : X n −→ Π( − , n ) ⊗ Π X is the G -map given by η ( x ) = (id n , x ). It is a G -homeomorphism with inverse givenby η − ( µ, x ) = µ ∗ ( x ) for µ : m −→ n in Π and x ∈ X m . Clearly η − is well-defined, η − η = id, and ηη − = id since ( µ, x ) ∼ (id n , µ ∗ x ). Since η − is inverse to a G -map,it is a G -map.We must show that the counit ε : PU Y −→ Y is an isomorphism for a Π G - G -space Y . Again let A = ( n , α ) ∈ Π G . Then ε : ( PU Y )( A ) = A • ⊗ Π ( U Y ) −→ Y ( A )is the G -map given by ε ( µ, y ) = µ ∗ y for µ : m −→ A and y ∈ Y m , where µ ∗ : Y m −→ Y ( A ). It is a G -homeomorphism with inverse given by ε − ( y ) = ( ι − , ι ∗ y ) for y ∈ Y ( A ), where ι ∈ Π G ( A, n ) is the morphism whose underlying function on n is the identity. Clearly εε − = id, and ε − ε = id since ( µ, y ) ∼ ( ι − , ι ∗ µ ∗ y ). Theidentification uses the morphism ι ◦ µ in Π. Again, since ε − is inverse to a G -map,it is a G -map. The proof with Π and Π G replaced by F and F G is the same.To prove (i) and (ii), we describe more explicitly how a Π G - G -space Y is re-constructed from its underlying Π- G -space. For a finite G -set A = ( n , α ), let Y αn denote Y n with a new action · α of G specified in terms of α and the original actionof G by g · α y = α ( g ) ∗ ( g · y ). In effect, ε − identifies Y ( A ) with the G -space Y αn .Consider Λ α = { ( g, α ( g )) } . Projection onto the first coordinate gives an isomor-phism Λ α −→ G . The Λ α -action on Y n obtained by restriction of the action of G × Σ n is given by ( g, α ( g )) · y = α ( g ) ∗ ( g · y ) . Thus it coincides with the G -action that we used to define Y αn . This immediatelyimplies (ii). Similarly, the Λ α -action on Y n obtained by restriction of the action of G × Σ n given by the diagonal action of G and the permutation action of Σ n is g ( y , . . . , y n ) = ( gy α ( g − )(1) , . . . , gy α ( g − )( n ) ) . Thus it coincides with the G -action that we used to define ( Y n ) α . Therefore ε − identifies the Segal G -map δ : Y ( n , α ) −→ ( Y n ) α with the Λ α -map δ : Y n −→ Y n .This immediately implies (i), and (iii) and (iv) follow formally from (i) and (ii)since η : id −→ UP is an isomorphism.Since statements (i)-(iv) for F - G -spaces and F G - G -spaces depend only on theirunderlying Π- G -spaces and Π G - G -spaces, they follow immediately. (cid:3) We record analogues for F G - G -spaces of Lemmas 2.9 and 2.10 for F - G -spaces.While they could be proven directly, we just observe that the first follows immmedi-ately from Theorem 2.30(iii), and the second follows as in the proof of Lemma 2.10. Definition 2.31.
For a based G -space X , let R G X denote the Π G - G -space with( n , α )th G -space ( X n ) α . Conceptually, it is obtained by prolonging the Π- G -space R X from Definition 2.8 to a Π G - G -space. QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 23
Lemma 2.32. If f : X −→ Y is a weak equivalence of based G -spaces, then theinduced map R G f : R G X −→ R G Y is a level G -equivalence of Π G - G -spaces. Lemma 2.33. If f : X −→ Y is a level G -equivalence of Π G - G -spaces, then X isspecial if and only if Y is special. The factorization of the Segal machine holds for any topological group G . Sinceactions of G on finite G -sets factor through actions of π ( G ), it is clear that W G - G -spaces generally incorporate much more information than F - G -spaces do.3. The homotopical version of the Segal machine
For the moment, we continue to work with a general topological group G . How-ever, our interest is to understand the Segal machine homotopically when G is finiteor compact Lie. While U G S P ( X ) gives the most conceptually natural equivariantversion of the Segal machine on F - G -spaces X , it is by itself of negligible use sincethe functor P does not enjoy good homotopical properties before some kind of ho-motopical approximation of X . We define a naive homotopical Segal machine in § § §
9. In § § The categorical bar construction.
We here define the variant of the barconstruction used in the homotopical Segal machine. We begin with some gen-eral definitions that start with a closed symmetric monoidal category V and a V -category E , as in § Y be a contravariant and X be a covariant V -functor E −→ V . They are given by objects Y n and X n in V and maps in V Y : E ( m, n ) −→ V ( Y n , Y m ) and X : E ( m, n ) −→ V ( X m , X n )that are compatible with composition and identity. These have adjoint evaluationmaps E Y : E ( m, n ) ⊗ Y n −→ Y m and E X : E ( m, n ) ⊗ X m −→ X n . Associated to the triple ( Y, E , X ) we have a categorical two-sided bar construc-tion B q ( Y, E , X ). It is a simplicial object in V . Its q -simplex object in V is thecoproduct(3.1) B q ( Y, E , X ) = a ( n ,...,n q ) Y n q ⊗ E ( n q − , n q ) ⊗ · · · ⊗ E ( n , n ) ⊗ X n , where ( n , . . . , n q ) runs over the ( q + 1)-tuples of objects of E . Its faces d i for0 ≤ i ≤ q are induced by the evaluation maps of Y and X and by composition in E , and its degeneracies s i for 0 ≤ i ≤ q are induced by the unit maps of E . In moredetail, the d i are induced by E ( n , n ) ⊗ X n E X / / X n if i = 0, E ( n i , n i +1 ) ⊗ E ( n i − , n i ) C / / E ( n i − , n i +1 ) if 0 < i < q , Y n q ⊗ E ( n q − , n q ) E Y / / Y n q − if i = q. The s i are induced by U I / / E ( n i , n i ) if 0 ≤ i ≤ q .When V is cartesian closed, so that ⊗ = × , B q ( Y, E , X ) is the nerve of an internal“Grothendieck category of elements” C ( Y, E , X ). The objects and morphisms ofthis category are both objects of V . The object of objects is the coproduct C = a n Y n × X n , where n runs over the objects of E . The object of morphisms is the coproduct C = a m,n Y n × E ( m, n ) × X m , where ( m, n ) runs over the pairs of objects of E . We have source, target, andidentity maps S , T , and I given as follows: S and T are given on components bythe evaluation maps of Y and X . S = E Y × id : Y n × E ( m, n ) × X m −→ Y m × X m T = id × E X : Y n × E ( m, n ) × X m −→ Y n × X n ; I is induced by the identity maps U −→ E ( n, n ) of E .The composition C : C × C C −→ C is induced by the composition in E . The point is that when ⊗ = × we have theidentification C × C C ∼ = a ( m,n,p ) Y p × E ( n, p ) × E ( m, n ) × X m . where ( m, n, p ) runs over the triples of objects of E .When V has a suitable covariant simplex functor ∆ −→ V so that we havea “geometric realization” functor from simplicial objects in V to V , we define B ( Y, E , X ) to be the realization of B q ( Y, E , X ). Remark 3.2.
The two-sided bar construction goes back to [40, § V = G U for any G , where we take ⊗ = × . It also applies with V = G T ,where we take ⊗ = ∧ . Given a based triple ( Y, E , X ), so taking V = G T andusing ⊗ = ∧ everywhere, specialization of our general construction gives a barconstruction B ∧ ( Y, E , X ). Alternatively, forgetting about basepoints, we can take V = G U and use ⊗ = × everywhere to get a bar construction B × ( Y, E , X ).Neither is right for our purposes. With B ∧ , a key later proof, that of the wedgeaxiom in § B × , we could not enrich over G T , as we nowexplain.As in § E has a zero object 0 such that each E (0 , n ) and E ( n,
0) is a point. Then each E ( m, n ) has the basepoint m → → n . For eachobject n of E , we have the represented functor E n = E ( − , n ). In particular, E is QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 25 the constant functor ∗ at a point. As n varies, the bar constructions B × ( E n , E , X )give a covariant functor B × ( E , E , X ) : E −→ G T . It is a G U -enriched functor but it is not G T -enriched because the map E ( m, n ) −→ T G (cid:0) B × ( E m , E , X ) , B × ( E n , E , X ) (cid:1) does not send the basepoint of the source to the basepoint (zero map) of the target.Let ε : B × ( E , E , X ) −→ X be the canonical map of (non-enriched) functors E −→ G T , constructed at level n by passing to realization from the map of sim-plicial G -spaces ε n : B × q ( E n , E , X ) −→ ( X n ) q that is given by composition in E and the action of E on X . Here ( X n ) q denotesthe constant simplicial G -space at X n . Each ε n is a G -homotopy equivalence withhomotopy inverse η n : X n −→ B × ( E n , E , X )given by sending x ∈ X n to the zero simplex (id n , x ) ∈ E ( n, n ) × X n . Then B × ( ∗ , E , X ) is contractible (since X = ∗ and E = ∗ ).We have an inclusion ∗ −→ Y given by the basepoints of the Y n and we define(3.3) B ( Y, E , X ) = B × ( Y, E , X ) /B × ( ∗ , E , X ) . The inclusions of basepoints ∗ −→ Y n are G -cofibrations. Since our bar construc-tions are Reedy cofibrant simplicial G -spaces, so that the geometric realization ofa level G -cofibration is also a G -cofibration (see Theorem 1.11), these inclusionsinduce G -cofibrations B × ( ∗ , E , X ) −→ B × ( Y, E , X ). Therefore the quotient map(3.4) B × ( Y, E , X ) −→ B ( Y, E , X )is a G -homotopy equivalence. With Y = E n , this gives the following result. Proposition 3.5. B ( E , E , X ) is a G T -functor E −→ T G , and ε induces a level-wise G -homotopy equivalence B ( E , E , X ) −→ X of such functors with level inversesinduced by the η n . Remark 3.6.
Again using Reedy cofibrancy, we see that B ( Y, E , X ) is the geo-metric realization of the simplicial based G -space whose space of q -simplices is B q ( Y, E , X ) /B q ( ∗ , E , X ) . This can be rewritten as the wedge of half-smash products _ n Y n ∧ B × q − ( E n , E , X ) + . As explained in complete categorical generality in [57, Lemma 19.7], we can com-mute realization and ⊗ E to obtain the isomorphism B ( Y, E , X ) ∼ = Y ⊗ E B ( E , E , X ) . One proof uses a direct comparison of definitions on the level of q -simplices for each q , but the result is also an application of the (enriched) categorical Fubini theorem.More generally, if D and E are both as above, ν : D −→ E is a G T -functor, Y is a contravariant G T -functor E −→ T G , and X is a covariant G T -functor D −→ T G , there is an isomorphism B ( ν ∗ Y, D , X ) ∼ = Y ⊗ E B ( E , D , X ) , where ν ∗ Y = Y ◦ ν and, similarly, each E n is viewed as a contravariant G T -functor D −→ T G by precomposition with ν . Remark 3.7.
The homotopical Segal machine is obtained by using examples oftwo-sided bar constructions to construct W G - G -spaces. The structure maps of the G -spectra they give by restricting to spheres must come from comparison maps ofthe form(3.8) B ( Y, E , X ) ∧ C −→ B ( Y ∧ C, E , X ) . We have such maps with our definition of the bar construction, but we would nothave them if we tried to use B × . The point is that (2.19) in Remark 2.18 does notapply if we use B × due to the difference between unbased and based enrichment.To make this more precise, consider the relationship between smash products andproducts. For based spaces A , B , and C , there is no natural map( A × B ) ∧ C −→ ( A ∧ C ) × B. Under the natural isomorphism ( A × B ) × C ∼ = ( A × C ) × B , we collapse out differentsubspaces to construct the source and target, and neither is contained in the other.Explicitly, writing a, b, c for the basepoints of A, B, C and x, y, z for general pointsof
A, B, C , we identify all points ( x, y, c ) and ( a, b, z ) with ( a, b, c ) in the source,but we identify all points ( a, z, y ) and ( x, c, y ) with the point ( a, c, y ) in the target.Our interest is in the case where E is F or F G or the more general categories ofoperators D and D G to be introduced later. Note that although F is topologicallydiscrete and G -trivial, we still view it as a category enriched in T G .3.2. The naive homotopical Segal machine.
Specializing from the previoussection, let Y : F −→ G T be a contravariant G T -functor and X : F −→ G T be a covariant G T -functor. We then have the bar construction B ( Y, F , X ). Theaction of G on it is induced diagonally by the actions on the Y n and X n .For G T -functors Y : F opG −→ T G and X : F G −→ T G , we have the resultingtwo-sided bar construction B ( Y, F G , X ). For its construction, we must rememberthe action of G on the finite G -sets F G (( m , α ) , ( n , β )). While we are interested ingeneral X , in both cases we are only interested in particular Y , namely those ofthe form Y = A • , as in Notation 2.28.Nonequivariantly, Woolfson [59] constructed a homotopical Segal machine byrestricting B ( A • , F , X ) to spheres A = S n . Equivariantly, we can apply thesame construction, taking G to be any topological group, X to be an F - G -space,and A to be in G W . For reasons we now explain, this construction fails to lead togenuine Ω- G -spectra when G is finite, even when X is F • -special.When A = n , A • is the represented functor F n = F ( − , n ), and as n varieswe obtain the F - G -space B ( F , F , X ) whose n th G -space is B ( F n , F , X ). Wehave an implicit and important action of Σ n on source and target; Σ n ⊂ F ( n , n )acts from the left on F n by postcomposition in F , and that induces the action on B ( F n , F , X ). Observe that B q ( F n , F , X ) is a simplicial ( G × Σ n )-space and ε n isthe geometric realization of a map B q ( F n , F , X ) −→ ( X n ) q of simplicial ( G × Σ n )-spaces. Proposition 3.5 specializes to give the following result. This is revisionist. He was writing before the two-sided bar construction was formally defined.
QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 27
Proposition 3.9.
Let X be an F - G -space X . Then the map ε : B ( F , F , X ) −→ X of F - G -spaces is a level G -equivalence, hence X is special if and only if B ( F , F , X ) is special. Warning 3.10.
While the map ε n is a map of ( G × Σ n )-spaces, the map η n : X n −→ B ( F n , F , X )is not Σ n -equivariant since the action of Σ n occurs on F ( − , n ) in the target and on X n in the source, so that η ( σx ) = (id n , σx ) while ση ( x ) = ( σ, x ). Thus there is noreason to expect ε to be a level F • -equivalence and no reason to expect B ( F , F , X )to be F • -special even if X is so.By Propositions 2.22 and 3.9, B ( F , F , X ) and X can be used interchangeablywhen passing to naive G -prespectra. Recall that we also have the two-sided barconstruction with F replaced by F G . We elaborate our comparison to include U B ( F G , F G , P X ). Here B ( F G , F G , Y ) is defined at level ( n , α ) by replacing theleft variable F G by the functor F G ( − , ( n , α )) : F opG −→ T G represented by ( n , α ).This comparison will also pave the way towards the construction of genuine G -spectra when G is a compact Lie group.Still letting G be any topological group and using Lemma 2.33, we have thefollowing analogue of Proposition 3.9. Proposition 3.11.
Let Y be an F G - G -space. Then ε : B ( F G , F G , Y ) −→ Y is alevel G -equivalence, hence Y is special if and only if B ( F G , F G , Y ) is special. We view U B ( F G , F G , P X ) as a genuine homotopical approximation to X inview of the following corollary, which is immediate from Theorem 2.30. Note thatwe can identify X with UP X via the unit isomorphism. Corollary 3.12.
For any F - G -space X , the map U ε : U B ( F G , F G , P X ) −→ X isan F • -level equivalence, hence X is F • -special if and only if U B ( F G , F G , P X ) is F • -special. Remark 3.13.
We compare bar constructions along the adjoint equivalence ( P , U )between F - G -spaces and F G - G -spaces. For F - G -spaces X , we have P B ( F , F , X ) = F G ⊗ F B ( F , F , X ) ∼ = B ( F G , F , X ) . The inclusion ι : F −→ F G induces a natural map of F G - G -spaces ι ∗ : P B ( F , F , X ) −→ B ( F G , F G , P X )such that the following diagrams commute; the second is obtained from the first byapplying U . P B ( F , F , X ) ι (cid:15) (cid:15) P ε / / P XB ( F G , F G , P X ) ε ♣♣♣♣♣♣♣♣♣♣♣ B ( F , F , X ) U ι (cid:15) (cid:15) ε / / X U B ( F G , F G , P X ) U ε ♣♣♣♣♣♣♣♣♣♣♣♣ In the first, the diagonal arrow ε is a level G -equivalence, but we cannot expect ι and P ε to be level G -equivalences since that would imply that all three arrows inthe second diagram are F • -level equivalences, contradicting Warning 3.10. In the second diagram, U ε is an F • -level equivalence and the other two arrows are level G -equivalences.Using Proposition 2.22, we see that this comparison implies the following com-parison of naive G -prespectra. Proposition 3.14.
Let X be a special F - G -space. Then the positive naive Ω - G -prespectra obtained by prolonging X , B ( F , F , X ) , and U B ( F G , F G , P X ) to W G - G -spaces and then restricting to spheres S n are level G -equivalent. Their bottomstructural maps are compatible group completions of G -spaces equivalent to X .Proof. For any A , functoriality of prolongation applied to the second diagram ofRemark 3.13 gives a commutative diagram B ( A • , F , X ) (cid:15) (cid:15) / / X ( A ) .B ( A • , F G , P X ) ♣♣♣♣♣♣♣♣♣♣♣ Here we have used the factorization of prolongation from F - G -spaces to W G - G -spaces through F G - G -spaces and the isomorphism PU ∼ = Id, where P is prolonga-tion from F - G -spaces to F G - G -spaces. By Lemma 2.10 and Remark 3.13, all threeof our F - G -spaces are special. Restricting to spheres S n , we can apply Proposi-tion 2.22 to each of them. Taking A = S , we see compatible G -equivalences with X , and taking A = S , we see that the bottom structure maps are compatiblegroup completions. That implies that we have weak G -equivalences at level 1.In turn, since we are comparing Ω- G -prespectra, that implies that we have weak G -equivalences at all levels n . (cid:3) The genuine homotopical Segal machine.
Let G be a compact Lie group.We cannot expect to construct genuine Ω- G -spectra from special F - G -spaces, butwe show here how to construct a genuine G -spectrum S G X from an F • -special F - G -space X . Equivalently, we construct a genuine G -spectrum S G Y from a special F G - G -space Y . We think of Y = P X or, equivalently, X = U Y . When G is finite, S G X is a (genuine) positive Ω- G -spectrum whose bottom structural G -map is agroup completion of X . Prolongation of B ( F , F , X ) to a W G - G -space does notgive a positive Ω- G -spectum; prolongation of B ( F G , F G , Y ) to a W G - G -space does.The following definition gives a modernized version of Shimakawa’s equivariantSegal machine [54]. The strange looking notation id ∗ anticipates a later general-ization. Recall Notation 2.28. Definition 3.15.
Write id ∗ Y = B ( F G , F G , Y ) for an F G - G -space Y ; thus id ∗ isa functor Fun( F G , T G ) −→ Fun( F G , T G ). For a compact Lie group G , the Segalmachine S G on F G - G -spaces is the compositeFun( F G , T G ) id ∗ / / Fun( F G , T G ) P / / Fun( W G , T G ) U G S / / G S . More explicitly, taking A = S V , S G ( Y )( V ) = B (( S V ) • , F G , Y ) = ( S V ) • ⊗ F G id ∗ Y. Orthogonal G -spectra had not been developed when [54] was written; he worked with Lewis-May G -spectra. QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 29
The Segal machine S G on F - G -spaces X is defined by S G X = S G P X. The definition makes sense for any X . When X is special, Proposition 3.14 showsthat the underlying naive G -prespectrum of S G X is equivalent to S NG X hence is apositive Ω- G -prespectrum with bottom structural map a group completion of X . Remark 3.16.
The group completion property is not easy to see directly from thedefinition of S G . Shimakawa’s strategy [56, p 357], not fully detailed, was to showthat for H ⊂ G , Woolfson’s version of the nonequivariant Segal machine S ( Y H )is equivalent to ( S G Y ) H , where Y H is the composite of restriction to F and the H -fixed point functor, so that ( Y H ) n = Y ( n ) H , and then to quote the equivalenceof Woolfson’s version with Segal’s original version. With our proof, the equivalenceon fixed points follows formally from the group completion property, as is shownquite generally in [12, Theorem 2.20].We are primarily interested in understanding S G X when G is finite and X is F • -special. However, the following variant of the standard notion of a linear functor(compare [5, 25]) makes sense for any topological group G . Recall that G W and W G are the categories of based G -CW complexes whose respective morphisms arebased G -maps and all based maps, with G acting by conjugation. Definition 3.17. A W G - G -space Z is positive linear if for any G -connected A andany G -map f : A −→ B in G W , Z ( A ) f ∗ / / Z ( B ) i ∗ / / Z ( Cf )is a fibration sequence of based G -spaces, where i : B −→ Cf is the cofiber of f . That is, the induced map from Z ( A ) to the homotopy fiber of i ∗ is a weak G -equivalence.The “positive” refers to the assumption that A is G -connected.In § G and is of independent interest. It is perhaps surprising that we only need X tobe special, not F • -special, for the first statement and that we do not know howto derive either statement from the other. However, we will only make use of thesecond statement in this paper. Theorem 3.18.
Let G be a topological group. If X is a special F - G -space, thenthe W G - G -space that sends A to B ( A • , F , X ) is positive linear. If Y is a special F G - G -space, such as P X for an F • -special F - G -space X , then B ( A • , F G , Y ) ispositive linear. Now let G be finite. We want to understand the structure maps of the genuine G -spectrum S G X . The following result is closely related to nonequivariant resultsin [8, 25, 46, 51, 59] and equivariant results of Segal [52] and Shimakawa [54]. Ourformulation is a slight variant of the specialization to finite groups G of a result ofBlumberg [5, Theorem 1.2] about W G - G -spaces for compact Lie groups G . Theorem 3.19.
Let G be finite and let Z be a positive linear W G - G -space suchthat the restriction of Z to F G is a special F G - G -space. Then U G S Z is a positive Ω - G -spectrum. Remark 3.20.
This result depends on the Wirthm¨uller isomorphism, and theproof of its specialization to finite groups can be simplified quite a bit by use ofthe simplified proof of that result in [34]. In turn, that depends implicitly onAtiyah duality for finite based G -sets and, as explained in [13, § ε ofDefinition 2.26 plays a central role in that. This ties the proof of Theorem 3.19 tothe Segal map Y ( G/H ) −→ T G ( G/H, Y ); compare [5, Remark 3.18]. Remark 3.21.
Clearly Theorem 3.19 applies to Z = P X when X is F • -special.Despite Theorem 3.18, we have no such conclusion when X is only special.Here now is the fundamental theorem about the Segal machine for finite groups. Theorem 3.22.
Let G be finite and let X be an F • -special F - G -space. Then S G X is a positive Ω - G -spectrum. Moreover, if V G = 0 , then the composite X −→ B ( F G , F G , P X ) = ( S G X )( S ) −→ Ω V ( S G X )( S V ) of η and the structure G -map is a group completion.Proof. Let Z be the W G - G -space B (( − ) • , F G , P X ). Then Z is positive linear byTheorem 3.18 and its restriction to F is an F • -special F - G -space by Corollary 3.12.Therefore Theorem 3.19 implies the first statement, and the second follows fromProposition 3.14 and Remark 1.20. (cid:3) Change of groups and compact Lie groups.
We summarize our conclu-sions. For any topological group G , we have a functor S NG that takes F - G -spacesto G -prespectra. It has four variants. The first uses Segal’s original simpliciallydefined inductive machine. The second is the conceptual machine and the thirdand fourth are composites that first take F - G -spaces to W G - G -spaces by one oftwo choices of a bar construction and then take W G - G -spaces to G -prespectra.We restrict attention to special F - G -spaces X , reduced as always, for clarityin this digressive section. Then the four choices are equivalent and the functor S NG assigns a positive naive Ω- G -spectrum to X , together with a group completion η : X ≃ ( S NG X ) −→ Ω( S NG X ) .If H is a subgroup of G and we write ι : H −→ G for the inclusion, then we havevarious functors ι ∗ that restrict given G actions to H actions. By inspection, thesefunctors commute with all constructions in sight. Therefore ι ∗ S NG X ∼ = S NH ι ∗ X . Infact, these functors ι ∗ are Quillen right adjoints with respect to the various modelstructures in [5, 24, 25] on our categories. Moreover, they also commute with thegroup completion maps η , so that ηι ∗ = ι ∗ η : ι ∗ X −→ ( S NH ι ∗ X ) .For finite groups G , if X is F • -special rather than just special, we have a posi-tive genuine Ω- G -spectrum S G X with underlying naive Ω- G -spectrum S NG X . For-mally we have a forgetful functor from genuine orthogonal G -spectra indexed on acomplete G -universe to naive G -spectra indexed on the trivial universe, and thisfunctor is a Quillen right adjoint with respect to to the various model structuresin [5, 24, 25]. Restriction to subgroups works the same way on the level of genuine G -spectra as it does on the level of naive G -spectra. Note that if H is a finitesubgroup of a topological group G and X is an F - G -space which is F • -special asan F - H -space, then ι ∗ S NG X ∼ = S NH ι ∗ X is the underlying naive Ω- H -spectrum of agenuine Ω- H -spectrum.Now let G be a compact Lie group. Just assuming that X is special, ourconstruction still gives a genuine G -spectrum S G X whose underlying naive G -prespectrum is the Ω- G -spectrum S NG X , and we still have the group completion QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 31 η : X −→ Ω( S G X ) . However, in contrast to the case of finite groups, even if X is F • -special, S G X need not be a positive genuine Ω- G -spectrum. Let R denote fibrantapproximation in the positive stable model structure on G -spectra. We agree toreplace S G X by its fibrant approximation RS G X , and we regard the new S G X asthe genuine G -spectrum constructed from the F - G -space X . Its underlying naive G -spectrum is positive fibrant, and its structure maps ( S G X ) −→ Ω V ( S G X )( V )are group completions when V G = 0. However, they are not group completions ofthe originally given X , as we now explain in a model theoretic framework. Never-theless, even though we know little about how its homotopy type relates to X , weview the new S G as the best equivariant infinite loop space machine we can hopefor when G is a compact Lie group.The adjoint pair relating W G - G -spaces to orthogonal G -spectra is a Quillen equiv-alence even when G is a compact Lie group, as noted in [5, Theorem A.13]. However,consider the prolongation functor P from F - G -spaces to W G - G -spaces and its rightadjoint U . We have noted that the unit of the adjunction is the identity. When G is finite, we have a complementary result about the counit. It is best expressedmodel theoretically, and we digress to summarize relevant background.The absolute stable model structure on the category Fun( W G , T G ) is definedin [5, § A.4]. It starts with the absolute level model structure in which a map Y −→ Z is a fibration or weak equivalence if each Y ( A ) −→ Z ( A ) is a Serre G -fibration or a weak G -equivalence. The absolute stable model structure has thesame cofibrations, but a map is a weak equivalence if it is a stable equivalence inthe sense that the underlying map of G -prespectra is a π ∗ -isomorphism. The fibrant W G - G -spaces Z are those for which the maps Z ( A ) −→ Ω V Z ( A ∧ S V ) of (I.3.4)are weak G -equivalences for all A ∈ W G and all G -representations V . There is aQuillen equivalent stable model structure with the same weak equivalences, and itis Quillen equivalent to the category of orthogonal G -spectra.It has long been understood that the categories Fun( F , T G ) and Fun( F G , T G )admit stable model structures such that the three pairs ( P , U ) in sight are Quillenadjunctions. The proofs are similar to those of [25, §
18] and [50]. More recentrelevant expositions are in [44, 48, 49]. We start with the level model structures,which are defined in the same way as for Fun( W G , T G ). The stable equivalences arethe maps f such that P f is a stable equivalence of W G - G -spaces. The fibrant objectsare the grouplike F • -special F - G -spaces or the grouplike special F G - G -spaces. Proposition 3.23.
Let G be finite. Let Y be a positive linear W G - G -space whoseunderlying F G - G -space is special. Let λ : X −→ U Y be a bifibrant approximation ofthe underlying F - G -space U Y in the stable model structure on F - G -spaces. Thenthe composite of P λ and the counit ε is a stable equivalence P X −→ PU Y −→ Y .Proof. This is proven nonequivariantly in [25, Lemma 18.10], and we can mimicthe argument indicated there. As in [25, Proposition 18.8], P X is positive linear.Therefore, since G is finite, Theorem 3.19 implies that, after applying U G S to thecomposite, we obtain a map of connective orthogonal Ω- G -spectra which is a weak G -equivalence on 0th spaces and is therefore a stable equivalence. (cid:3) As in [25, Theorem 0.10], this implies that ( P , U ) is a connective Quillen equiv-alence between F - G -spaces and W G - G -spaces, that is, a Quillen adjoint pair that The paper [5] deferred exposition, which would have been digressive there. Note that the group completion property is invisible from the model theoretic perspective. induces an equivalence between the respective homotopy categories of connectiveobjects. These conclusions do not generalize to compact Lie groups G . As wasnoted by Blumberg [5, Appendix C], following Segal [52], they already fail when G = S . In fact, taking G = S , Blumberg concludes that no reasonable conditionon an F - G -space can imply that U G S P X is a positive Ω- G -spectrum. However,there was no reason to expect any such result. From the point of view of con-structing genuine G -spectra with good properties from naturally occurring spacelevel data, our construction works as well as can be expected. However, unlikethe case of finite groups, there is no reasonable specification of space level dataon F - G -spaces sufficient to construct all connective G -spectra. We cannot evenexpect to construct suspension G -spectra since we do not have an analog of theBarratt-Priddy-Quillen theorem for compact Lie groups.4. The generalized Segal machine
Unless otherwise specified, we take our group G to be finite from here on out.The input of the Segal infinite loop space machine looks nothing like the inputof the operadic machine. To compare them, we must generalize the natural inputof both to obtain common input to which generalizations of both machines apply.We explain the generalized Segal machine in this section, postponing considerationof operads to the next. We define two equivariant versions of the categories ofoperators introduced in [38]. One version has finite sets as objects, the other finite G -sets, generalizing F and F G respectively. As in § G -category of operators to be an E ∞ G -category of operators, we generalize the homotopical version of the Segal machineby generalizing its input from F - G -spaces to D - G -spaces, where D is any E ∞ G -category of operators over F . We compare the D and D G machines to the F and F G machines by proving that they have equivalent inputs and that they produceequivalent output when fed equivalent input. Thus the increased generality is moreapparent than real. The point of the generalization is that operadic data feednaturally into the D - G -space rather than the F - G -space machine. We reiteratethat the categorical input data of the sequels [15, 16] is intrinsically operadic.4.1. G -categories of operators D over F and D - G -spaces.Definition 4.1. A G -category of operators D over F , abbreviated G - CO over F ,is a category enriched in G T whose objects are the based sets n for n ≥ G -functors Π ι / / D ξ / / F such that ι and ξ are the identity on objects and ξ ◦ ι is the inclusion. Here G actstrivially on Π and F . We say that D is reduced if D ( m , n ) is a point if either m = 0 or n = 0, and we restrict attention to reduced G - COs over F henceforward.A morphism ν : D −→ E of G - CO s over F is a G T -functor over F and under Π. Remark 4.2.
We have omitted cofibration conditions that will be added later,since what we need is a bit different for the Segal and the operadic machines. SeeRemark 4.16 for the Segal and Remark 6.15 for the operadic machine. The purposeof these conditions is to ensure that all bar constructions in sight are realizationsof Reedy cofibrant simplicial G -spaces, as claimed in Remark 1.12. QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 33
Since we have morphism G -spaces D ( m , n ) such that composition is given byequivariant maps, we have G -fixed identity elements, and we have mapsΠ( m , n ) −→ D ( m , n ) −→ F ( m , n )whose composite is the inclusion. Note that Π( m , n ) is contained in D ( m , n ) G . Ofcourse, ξ : D −→ F is a map of G - COs over F for any G - CO D over F .Recall that T G denotes the G T -category of nondegenerately based G -spaces andall based G -maps, with G -acting by conjugation. Definition 4.3. A D - G -space X is a G T -functor X : D −→ T G . A map of D - G -spaces is a G T -natural transformation. Composing X with ι : Π −→ D gives X an underlying Π- G -space. We say that X is F • -special if its underlying Π- G -spaceis F • -special. We say that a map of D - G -spaces is an F • -level equivalence if itsunderlying map of Π- G -spaces is an F • -level equivalence. Let Fun( D , T G ) denotethe category of D - G -spaces.4.2. G -categories of operators D G over F G and D G - G -spaces. We can furthergeneralize the input, following § Definition 4.4. A G -category of operators D G over F G , abbreviated G - CO over F G , is a category enriched in G T with objects the based G -sets ( n , α ) for n ≥ α : G −→ Σ n , together with G -functorsΠ G ι G / / D G ξ G / / F G such that ι G and ξ G are the identity on objects and ξ G ◦ ι G is the inclusion. Wesay that D G is reduced if D G (( m , α ) , ( n , β )) is a point if either m = 0 or n = 0,and we restrict attention to reduced G - COs over F G henceforward. A morphism ν G : D G −→ E G of G - CO s over F G is a G T -functor over F G and under Π G . Remark 4.5.
As in Remark 4.2, we have omitted the cofibration condition specifiedin Remark 4.16 and the evident analogue of the cofibration condition specified inRemark 6.15.Of course, G acts non-trivially on the morphism sets of Π G and F G . We havemorphism G -spaces D G (( m , α ) , ( n , β )) such that composition is given by G -maps,we have G -fixed identity elements, and we have G -mapsΠ G (( m , α ) , ( n , β )) −→ D G (( m , α ) , ( n , β )) −→ F G (( m , α ) , ( n , β ))whose composite is the inclusion. Again, ξ G : D G −→ F G is a map of G - COs over F G for any G - CO D G over F G .Regarding sets n as G -trivial G -sets, we have the following observation. Lemma 4.6.
The full subcategory D with objects n of a G -category of operators D G over F G is a G -category of operators over F . Conversely, just as we constructed Π G and F G from Π and F , we can constructa G - CO D G over F G from any G - CO D over F . We shall only be interested inthose D G that are constructed in this fashion. Construction 4.7.
Let D be a G -category of operators over F . We define a G -category of operators D G over F G whose full subcategory of objects n is D .The morphism G -space D G (( m , α ) , ( n , β )) is the space D ( m , n ), with G -action induced by conjugation and the original G -action on D ( m , n ). Explicitly, for f ∈ D G (( m , α ) , ( n , β )), g · f = β ( g ) ◦ ( gf ) ◦ α ( g − );We check that g · ( h · f ) = ( gh ) · f using that G acts trivially on permutations sincethey are in the image of Π. Composition and identity maps are inherited from D and are appropriately equivariant.A routine verification shows the following. Lemma 4.8.
The inclusion D ֒ → D G makes the following diagram of G T -categoriescommute. (4.9) Π / / (cid:127) _ (cid:15) (cid:15) D / / (cid:127) _ (cid:15) (cid:15) F (cid:127) _ (cid:15) (cid:15) Π G / / D G / / F G . Moreover, a map ν : D −→ E of G -COs over F induces a map ν G : D G −→ E G of G -COs over F G , which is compatible with the inclusions. Definition 4.10. A D G - G -space Y is a G T -functor Y : D G −→ T G . A map of D G - G -spaces is a G T -natural transformation. Composing Y with ι G : Π G −→ D G gives Y an underlying Π G - G -space. We say that Y is special if its underlying Π G - G -space is special. We say that a map of D G - G -spaces is a level G -equivalence if itsunderlying map of Π G - G -spaces is a level G -equivalence. Let Fun( D G , T G ) denotethe category of D G - G -spaces.4.3. The equivalence between
Fun( D , T G ) and Fun( D G , T G ) . We can now gen-eralize § D - G -spaces and D G - G -spaces. The forgetfulfunctor U : Fun( D G , T G ) −→ Fun( D , T G )has a left adjoint prolongation functor P : Fun( D , T G ) −→ Fun( D G , T G ) . Explicitly,( P X )( n , α ) = D G ( − , ( n , α )) ⊗ D X = _ m D G ( m , ( n , α )) ∧ X m / ∼ , where ( f, φ ∗ x ) ∼ ( f φ, x ) for a map φ : k −→ m in D , an element x ∈ X k , and amap f : m −→ ( n , α ) in D G ( m , ( n , α )). (We have written out this coequalizer of G -spaces explicitly to facilitate checks of details). The following result generalizesTheorem 2.30 from F to an arbitrary G - CO over F . Theorem 4.11.
The adjoint pair of functors
Fun( D , T G ) P / / Fun( D G , T G ) U o o specifies an equivalence of categories. QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 35
The proof is very similar to that of the special case D = F dealt with inTheorem 2.30. Only points of equivariance require comment, and the followinglemma is the key to understanding the relevant G -actions. It identifies the G -space( P X )( n , α ) with the space X n with a new G -action induced by α and the original G -action on X n . We denote this G -space X αn . Lemma 4.12.
For a D - G -space X , the G -space ( P X )( n , α ) is G -homeomorphic tothe G -space X αn , namely X n with the G -action · α specified by g · α x = α ( g ) ∗ ( gx ) .Via this homeomorphism, the evaluation maps D G (( n , α ) , ( p , β )) ∧ ( P X )( n , α ) −→ ( P X )( p , β ) are given on the underlying spaces by the corresponding maps for X , D ( n , p ) ∧ X n −→ X p . Proof.
Modulo equivariance, this is an application of the Yoneda lemma. Writeid α : n −→ ( n , α ) for id : n −→ n regarded as an element of D G ( n , ( n , α )). Define F : X αn −→ D G ( − , ( n , α )) ⊗ D X by sending x ∈ X n to the equivalence class of (id α , x ). Then F is a G -map since F ( g · α x ) = (id α , g · α x ) = (id α , α ( g ) ∗ ( gx )) ∼ (id α ◦ α ( g ) , gx ) = ( g id α , gx ) . Define an inverse map F − : D G ( − , ( n , α )) ⊗ D X −→ X αn by sending the equivalence class of ( f, x ) ∈ D G ( m , ( n , α )) × X m to f ∗ ( x ), wherewe think of f as a map m −→ n in D and interpret f ∗ ( x ) to mean X ( f )( x ) ∈ X n .Note that F − is well defined. We have F − F ( x ) = F − (id α , x ) = id ∗ x = x and F F − ( f, x ) = F ( f ∗ x ) = (id α , f ∗ x ) ∼ ( f, x ) , hence F and F − are inverse homeomorphisms. Note that F − is automatically a G -map since it is inverse to the G -map F . The compatibility with the D G - G -spacestructure is clear. (cid:3) Using this, we mimic the proof of Theorem 2.30 to prove the equivalence of thecategories of D - G -spaces and D G - G -spaces. Proof of Theorem 4.11.
Clearly, since the inclusion D −→ D G is full and faithful,the unit X −→ UP X of the adjunction is an isomorphism for any D - G -space X . Let Y be a D G - G -space. We must show that the counit PU Y −→ Y of theadjunction is an isomorphism. A check of definitions shows that the counit G -map( PU Y )( n , α ) −→ Y ( n , α ) agrees under the isomorphism of Lemma 4.12 with themap, necessarily a G -map, id α ∗ : Y αn −→ Y ( n , α )induced by the morphism id α ∈ D G ( n , ( n , α )). Writing α id : ( n , α ) −→ n forid : n −→ n regarded as an element of D G (( n , α ) , n ), we see that α id induces theinverse homeomorphism α id ∗ : Y ( n , α ) −→ Y αn to id α ∗ . Again, α id ∗ is automatically a G -map since it is inverse to a G -map. (cid:3) Just as for F - G -spaces in § D - G -space X has two Π G - G -spaces associatedto it. We can either apply P to its underlying Π- G -space or we can apply P to X and take its underlying Π G - G -space. The proof of Theorem 4.11 implies that thesetwo Π G - G -spaces coincide. Therefore the four statements about Π and Π G that arelisted in Theorem 2.30 also hold for D and D G . We record them in the followingtwo corollaries. Corollary 4.13. A D - G -space X is F • -special if and only if the D G - G -space P X isspecial. A D G - G -space Y is special if and only if the D - G -space U Y is F • -special. Corollary 4.14.
A map f of D - G -spaces is an F • -level equivalence if and only P f is a level G -equivalence of D G - G -spaces. A map f of D G - G -spaces is a level G -equivalence if and only if U f is an F • -level equivalence of D - G -spaces. Comparisons of D - G -spaces and E - G -spaces for ν : D −→ E . Let X bean F - G -space. Then X and the D - G -space ξ ∗ X = X ◦ ξ : D −→ T G have the sameunderlying Π- G -space, hence one is F • -special or special if and only if the other isso. The left adjoint of ξ ∗ : Fun( F , T G ) −→ Fun( D , T G ) is given by the evident leftKan extension along ξ : D −→ F . Following [38, Theorem 1.8] nonequivariantly,we expect the bar construction to give a homotopically well-behaved variant. With F replaced by D , the analogue of Proposition 3.9 holds and admits the same proof.To implement this strategy, we start with an F • -special D - G -space Y and con-struct from it an F • -special F - G -space ξ ∗ Y together with a zigzag of F • -equivalencesbetween Y and ξ ∗ ξ ∗ Y . We shall use this to construct a Segal machine whose inputis an F • -special D - G -space Y and whose output is equivalent to S G ξ ∗ Y .As in [38], we work more generally here, starting from a map ν : D −→ E of G -COs over F and comparing D - G -spaces and E - G -spaces. We are mainly interestedin the case ν = ξ . We write ν G : D G −→ E G for the induced map of G -COs over F G . Focus on ν G rather than ν allows us to focus on G -equivalence rather than F • -equivalence. For clarity, we sometimes write U D and P D instead of U and P forthe adjunction between Fun( D , T G ) and Fun( D G , T G ), and similarly for E . Definition 4.15.
For Z ∈ Fun( D G , T G ), define ν G ∗ Z ∈ Fun( E G , T G ) by ν G ∗ Z = B ( E G , D G , Z ) . Here the target is defined levelwise by replacing E G by the composite E G ( − , ( n , α )) ◦ ν G : D op −→ T G , of ν G and the G T -functor E opG −→ T G represented by ( n , α ). For Y ∈ Fun( D , T G ),define ν ∗ Y ∈ Fun( E , T G ) by ν ∗ Y = U E ν G ∗ P D Y = U E B ( E G , D G , P D Y ) . Remark 4.16.
To ensure that the bar constructions we use are geometric realiza-tions of Reedy cofibrant simplicial G -spaces, we require the unit maps ∗ −→ D G (( n , α ) , ( n , α ))of G -COs over F G to be G -cofibrations, and similarly for G -COs over F . Thisholds when D G is constructed from a G -operad C such that id : ∗ −→ C (1) is a G -cofibration, as is true in our examples. QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 37
Definition 4.17.
A map ν G : D G −→ E G of G - CO s over F G is a G -equivalence ifeach map ν G : D G (( m , α ) , ( n , β )) −→ E G (( m , α ) , ( n , β ))is a weak G -equivalence. A map ν : D −→ E of G - CO s over F is an F • -equivalenceif the associated map ν G : D G −→ E G of G - CO s over F G is a G -equivalence.Recall the notion of an F • -level equivalence of Π- G -spaces from Definition 2.6and Lemma 2.11. Recall too that a map of Π- G -spaces is an F • -level equivalence ifand only if its associated map of Π G - G -spaces is a level G -equivalence; see Defini-tion 2.27 and Theorem 2.30(iv). These definitions and results are inherited by D and D G (as in Corollary 4.14). The following definition recalls notation from § Definition 4.18.
For a G -CO D over F and a fixed n , let D n be the corepre-sented D - G -space specified by D n ( p ) = D ( n , p ), with the action of D given bycomposition. Analogously, we have corepresented D G - G -spaces D G ( n ,α ) . Lemma 4.19. If ν : D −→ E is an F • -equivalence of G -COs over F , then for each n , ν restricts to an F • -level equivalence of D - G -spaces D n −→ ν ∗ E n .Proof. One can check that the cited restriction is a map of D - G -spaces. Moreover,an easy comparison of definitions shows that P D n can be identified with D G n .Therefore the conclusion follows from Corollary 4.14 and our definition of an F • -equivalence of G -COs over F , which of course was chosen in order to make thisand cognate results true. (cid:3) Theorem 4.20.
Let ν G : D G −→ E G be a G -equivalence of G -COs over F G .(i) Let X be an E G - G -space and Y a D G - G -space. Then there are natural zigzagsof level G -equivalences between ν G ∗ ν ∗ G X and X and between ν G ∗ ν G ∗ Y and Y .(ii) Y is a special D G - G -space if and only if ν G ∗ Y is a special E G - G -space.(iii) A map f of D G - G -spaces is a level G -equivalence if and only if ν G ∗ f is a level G -equivalence of E G - G -spaces.Proof. Abbreviate notation here by writing α for a finite G -set ( n , α ) and X α for X ( n , α ) when X is an E G -space. Starting with X , we have the natural maps(4.21) ( ν G ∗ ν G ∗ X ) α = B ( E G , D G , ν G ∗ X ) α B (id ,ν G , id) / / B ( E G , E G , X ) α ε / / X α . Starting with Y we have the natural maps(4.22) ( ν G ∗ ν G ∗ Y ) α = ν G ∗ B ( E G , D G , Y ) α B ( D G , D G , Y ) α ε / / B ( ν G , id , id) o o Y α . The maps ε with targets X α and Y α are G -equivalences, with the usual inverseequivalences η , as in the proof of Proposition 3.9(ii). At each level α , the othertwo maps are realizations of levelwise simplicial G -equivalences, and the Reedycofibrancy of the simplicial bar construction ensures that these realizations arethemselves G -equivalences, by Theorem 1.10. Note that we do not need X or Y tobe special to prove (i).By Lemma 2.33, (i) implies (ii). Indeed, Y is special if and only if ν G ∗ ν G ∗ Y isspecial and, since ν G ∗ ν G ∗ Y and ν G ∗ Y have the same underlying Π- G -space, one isspecial if and only if the other is so. Part (iii) follows from the naturality of the G -equivalences in (4.22). (cid:3) As in Lemma 4.8, the following diagram of G T -categories commutes.Π (cid:127) _ (cid:15) (cid:15) / / D ν / / (cid:127) _ (cid:15) (cid:15) E (cid:127) _ (cid:15) (cid:15) ξ / / F (cid:127) _ (cid:15) (cid:15) Π G / / D G ν G / / E G ξ G / / F G . Therefore ν ∗ U E = U D ν ∗ G , as we shall use in the proof of the following analogue ofTheorem 4.20 for G -COs over F . Theorem 4.23.
Let ν : D −→ E be an F • -equivalence of G -COs over F .(i) Let X be an E - G -space and Y be a D - G -space. Then there are natural zigzagsof F • -equivalences between ν ∗ ν ∗ X and X and between ν ∗ ν ∗ Y and Y .(ii) Y is an F • -special D - G -space if and only if ν ∗ Y is an F • -special E - G -space.(iii) A map f of D - G -spaces is an F • -level equivalence if and only if ν ∗ f is an F • -level equivalence of E - G -spaces.Proof. Recall from Theorem 4.11 that ( P D , U D ) is an adjoint equivalence of cate-gories, and similarly for E . Note that we have the following sequence of naturalisomorphisms of D G - G -spaces P D ν ∗ X ∼ = P D ν ∗ U E P E X = P D U D ν ∗ G P E X ∼ = ν ∗ G P E X. To prove (i), write ≃ to indicate a zigzag of F • -level equivalences. Recall fromCorollary 4.14 that P takes F • -level equivalences to level G -equivalences and U takes level G -equivalences to F • -level equivalences, while ν G ∗ preserves level G -equivalences by Theorem 4.20(iii). Therefore, by Theorem 4.20(i), we have thezigzags ν ∗ ν ∗ X = U E ν G ∗ P D ν ∗ X ∼ = U E ν G ∗ ν ∗ G P E X ≃ U E P E X ∼ = X and ν ∗ ν ∗ Y = ν ∗ U E ν G ∗ P D Y = U D ν ∗ G ν G ∗ P D Y ≃ U D P D Y ∼ = Y of level G -equivalences. Using Corollary 4.14, (ii) and (iii) follow as in the proof ofTheorem 4.20. (cid:3) Comparisons of inputs and outputs of the generalized Segal machine.Definition 4.24.
We say that a G - CO D G over F G is an E ∞ G - CO if the map ξ G : D G −→ F G is a G -equivalence. We say that a G - CO D over F is an E ∞ G - CO if its associated D G is an E ∞ G - CO over F G .The term “ E ∞ ” is convenient, but it is a little misleading, as will become clearwhen we turn to operads.We assume throughout this section that D is an E ∞ G - CO over F , and wespecialize the results of the previous section to ξ : D −→ F and ε G : D G −→ F G .The following results are just specializations of Theorems 4.20 and 4.23. Theorem 4.25.
The following conclusions hold.(i) Let X be an F G - G -space and Y a D G - G -space. Then there are natural zigzagsof level G -equivalences between ξ G ∗ ξ ∗ G X and X and between ξ ∗ G ξ G ∗ Y and Y .(ii) Y is a special D G - G -space if and only if ξ G ∗ Y is a special F G - G -space.(iii) A map f of D G - G -spaces is a level G -equivalence if and only if ξ G ∗ f is a level G -equivalence of F G - G -spaces. QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 39
Theorem 4.26.
The following conclusions hold.(i) Let X be an F - G -space and Y a D - G -space. Then there are natural zigzagsof F • -equivalences between ξ ∗ ξ ∗ X and X and between ξ ∗ ξ ∗ Y and Y .(ii) Y is an F • -special D - G -space if and only if ξ ∗ Y is an F • -special F - G -space.(iii) A map f of D - G -spaces is an F • -level equivalence if and only if ξ ∗ f is an F • -level equivalence of F - G -spaces. Therefore the pair of functors ξ ∗ G : Fun( F G , T G ) −→ Fun( D G , T G ) and ξ G ∗ : Fun( D G , T G ) −→ Fun( F G , T G )induce inverse equivalences between the homotopy categories obtained by invertingthe respective level G -equivalences, and this remains true if we restrict to specialobjects. Similarly, the pair of functors ξ ∗ : Fun( F , T G ) −→ Fun( D , T G ) and ξ ∗ : Fun( D , T G ) −→ Fun( F , T G )induce inverse equivalences between the homotopy categories obtained by invertingthe respective F • -level equivalences, and this remains true if we restrict to F • -specialobjects. We conclude that the four input categories for Segal machines displayedin the square of the following diagram are essentially equivalent.(4.27) Fun( F , T G ) ξ ∗ (cid:15) (cid:15) Fun( F G , T G ) U o o ξ ∗ G (cid:15) (cid:15) Fun( W G , T G ) U o o U G S / / G S Fun( D , T G ) Fun( D G , T G ) U o o Here ξ ∗ U = U ξ ∗ G . We regard the four categories in the square as possible domaincategories for generalized Segal infinite loop space machines.The functors in the square are right adjoints. By Theorems 2.30 and 4.11, theinclusions of F in F G and D in D G induce equivalences of categories, denoted U in the diagram. Their left adjoints P give inverses, and the functors U and P preserve the relevant special objects and levelwise equivalences. Theorems 4.25and 4.26 show that the vertical arrows ξ ∗ and ξ ∗ G become equivalences of homotopycategories with inverses ξ ∗ and ξ G ∗ (not the left adjoints) after inverting the relevantequivalences. Consider the following diagram of functors.(4.28) Fun( D , T G ) ξ ∗ (cid:15) (cid:15) P / / Fun( D G , T G ) ξ G ∗ (cid:15) (cid:15) Fun( F , T G ) P / / Fun( F G , T G ) P / / Fun( W G , T G ) U G S / / G S The isomorphism PU ∼ = id on F G - G -spaces and the definitions of ξ ∗ and ξ G ∗ imply that the square commutes up to natural isomorphism. The composite in thebottom row is our original conceptual Segal machine. We can specialize by letting ξ = id : F −→ F and ξ G = id : F G −→ F G . According to Definition 3.15, thecomposite U G S P id ∗ starting at Fun( F G , T G ) is the Segal machine S G on F G - G -spaces Y and the composite U G S P id ∗ P is the Segal machine on F - G -spaces X .That is(4.29) S G Y = U G S P id ∗ Y and S G X = U G S P id ∗ P X ∼ = U G S PP id ∗ X. We regard the composites obtained by replacing the functors id ∗ with ξ ∗ and ξ G ∗ as generalized Segal machines S G defined on D - G -spaces X and D G - G -spaces Y . Explicitly, we define the corresponding orthogonal G -spectra to be(4.30) S G Y = U G S P ξ G ∗ Y and S G X = U G S P ξ G ∗ P X ∼ = U G S PP ξ ∗ X. Clearly the machines starting with F - G -spaces or F G - G -spaces are equivalentand similarly for D and D G . Theorem 4.32 below will show that the machines start-ing with F - G -spaces or D - G -spaces and the machines starting with F G - G -spacesor D G - G -spaces are equivalent. Thus the four machines in sight are equivalent un-der our equivalences of input data. That is, we obtain equivalent output by startingat any of the four vertices of the square, converting input data from the other threevertices to that one, and taking the relevant machine S G . We use the followinginvariance principle in the proof of Theorem 4.32. Proposition 4.31.
The following conclusions about homotopy invariance hold.(i) If f : X −→ Y is a level G -equivalence of D - G -spaces, then f : B ( A • , D , X ) −→ B ( A • , D , Y ) is a weak G -equivalence for all A ∈ G W .(ii) If f : X −→ Y is a level G -equivalence of D G - G -spaces, then the induced map f : B ( A • , D G , X ) −→ B ( A • , D G , Y ) is a weak G -equivalence for all A ∈ G W .(iii) If f : X −→ Y is an F • -level equivalence of D - G -spaces, then the induced map f : B ( A • , D G , P X ) −→ B ( A • , D G , P Y ) is a weak G -equivalence for all A ∈ G W .Proof. By Remark 1.12 (see also Remark 4.16), our bar constructions are all geo-metric realizations of Reedy cofibrant simplicial G -spaces, hence Theorem 1.10gives the first conclusion. The second statement follows similarly. The third fol-lows from the second using that P f is a level G -equivalence of D G - G -spaces byCorollary 4.14. (cid:3) The limitations of the first part and need for the second are clear from the factthat B ( F , D , X ) is only level G -equivalent, not F • -level equivalent, to X . Theorem 4.32.
The following four equivalences of outputs hold.(i) If X is a D - G -space, then the G -spectra S G X and S G ξ ∗ X are equivalent.(ii) If Y is a D G - G -space, then the G -spectra S G Y and S G ξ G ∗ Y are equivalent.(iii) If X is an F - G -space, then the G -spectra S G X and S G ξ ∗ X are equivalent.(iv) If Y is an F G - G -space, then the G -spectra S G Y and S G ξ G ∗ Y are equivalent.Proof. We first prove (ii), which is the hardest part, and then show the rest. Thuslet Y be a D G - G -space. We claim that the W G - G -spaces P ξ G ∗ Y and P id ∗ ξ G ∗ Y arelevel equivalent. Applying U G S , this will give (ii). Thus let A ∈ W G . Then( P ξ G ∗ Y )( A ) = A • ⊗ F G B ( F G , D G , Y ) ∼ = B ( A • , D G , Y )and ( P id ∗ ξ G ∗ Y )( A ) = A • ⊗ F G B ( F G , F G , ξ G ∗ Y ) ∼ = B ( A • , F G , ξ G ∗ Y ) . QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 41
In both cases, the isomorphism comes from Remark 3.6. By Theorem 4.25(i) thereis a natural zigzag of level G -equivalences between ξ ∗ G ξ G ∗ Y and Y . By Proposi-tion 4.31(ii), this gives a zigzag of level G -equivalences of D G - G -spaces B ( A • , D G , Y ) ≃ B ( A • , D G , ξ ∗ G ξ G ∗ Y ) . Since ξ G : D G −→ F G is a G -equivalence of G -categories of operators, B (id , ξ G , id)induces an equivalence at the level of q -simplices of the bar constructions B ( A • , D G , ξ ∗ G ξ G ∗ Y ) −→ B ( A • , F G , ξ G ∗ Y ) . Again, since these bar constructions are geometric realizations of Reedy cofibrantsimplicial G -spaces, we get a weak G -equivalence on geometric realizations by The-orem 1.10. This proves the claim and thus proves (ii).To prove (i), let X be a D - G -space. Applying (ii) to Y = P X and using (4.29)and (4.30), we see that S G X ∼ = S G P X ≃ S G ξ G ∗ P X ∼ = S G P ξ ∗ X ∼ = S G ξ ∗ X. To prove (iv), let Y be an F G - G -space. We claim that the W G - G -spaces P id ∗ Y and P ξ G ∗ ξ ∗ G Y are level equivalent. Here( P id ∗ Y )( A ) = A • ⊗ F G B ( F G , F G , Y ) ∼ = B ( A • , F G , Y )and ( P ξ G ∗ ξ ∗ G Y )( A ) = A • ⊗ F G B ( F G , D G , ξ ∗ G Y ) ∼ = B ( A • , D G , ξ ∗ G Y ) . The isomorphisms again come from Remark 3.6. Just as before, B (id , ξ G , id) in-duces a weak G -equivalence B ( A • , D G , ξ ∗ G Y ) −→ B ( A • , F G , Y ) . Finally, (iii) follows by application of (iv) to Y = P X . (cid:3) Use of the generalized homotopical Segal machine will be convenient when wecompare the Segal and operadic machines, but it is logically unnecessary. We couldjust as well replace D - G -spaces Y by the F - G -spaces ξ ∗ Y and apply the Segal ma-chine S G on them. We have just shown that we obtain equivalent outputs from thesetwo homotopical variants of the Segal machine. We conclude that all homotopicalSegal machines in sight produce equivalent output when fed equivalent input. Theresulting G -spectra are equivalent via compatible natural zigzags. We concludethat all of our machines are essentially equivalent, and they are all equivalent toour preferred machine S G on F • -special F - G -spaces.5. From G -operads to G -categories of operators We show here how operadic input, like Segalic input, can be generalized to G -categories of operators. This section, like the previous one, is based on thenonequivariant theory developed in [38], but considerations of equivariance requirea little more work. We show how to construct G -categories of operators from G -operads in § E ∞ G -operads to E ∞ G -categories of operators, which is not obvious equivariantly, in § G -categories of operators associated to a G -operad C G . We assumethat the reader is familiar with operads, as originally defined in [27]. More recentbrief expositions can be found in [31, 32]. Operads make sense in any symmetricmonoidal category. Ours will be in the cartesian monoidal category G U . Weassume once and for all that our operads C are reduced, meaning that C (0) is apoint. We have a slight clash of notation since we follow [12] in writing C for anoperad in U , regarded as a G -trivial G -operad, whereas we write C G for a general G -operad. This clashes with the dichotomy between D and D G . Definition 5.1.
Let C G be an operad of G -spaces. We construct a G -CO over F , which we denote by D ( C G ), abbreviated D when there is no risk of confusion.Similarly, we write D G = D G ( C G ) for the associated G -CO over F G . The morphism G -spaces of D are D ( m , n ) = a φ ∈ F ( m , n ) Y ≤ j ≤ n C G ( | φ − ( j ) | )with G -action induced by the G -actions on the C G ( n ). Write elements in the form( φ, c ), where c = ( c , . . . , c n ). For ( φ, c ) : m −→ n and ( ψ, d ) : k −→ m , define( φ, c ) ◦ ( ψ, d ) = ( φ ◦ ψ, Y ≤ j ≤ n γ ( c j ; Y φ ( i )= j d i ) σ j ) . Here γ denotes the structural maps of the operad. The d i with φ ( i ) = j are orderedby the natural order on their indices i and σ j is that permutation of | ( φ ◦ ψ ) − ( j ) | letters which converts the natural ordering of ( φ ◦ ψ ) − ( j ) as a subset of { , . . . , k } to its ordering obtained by regarding it as ` φ ( i )= j ψ − ( i ), so ordered that elementsof ψ − ( i ) precede elements of ψ − ( i ′ ) if i < i ′ and each ψ − ( i ) has its naturalordering as a subset of { , . . . , k } .The identity element in D ( n , n ) is ( id, id n ), where id on the right is the unitelement in C G (1). The map ξ : D −→ F sends ( φ, c ) to φ . The inclusion ι : Π −→ D sends φ : m −→ n to ( φ, c ), where c i = id ∈ C G (1) if φ ( i ) = 1 and c i = ∗ ∈ C G (0)if φ ( i ) = 0. This makes sense since Π is the subcategory of F with morphisms φ such that | φ − ( j ) | ≤ ≤ j ≤ n .Observe that D is reduced as a G -CO over F since C G is reduced as an operad. Observation 5.2.
Since we will need it later and it illustrates the definition, wedescribe explicitly how composition behaves when the point c or d in one of themaps is of the form (id , . . . , id) ∈ C (1) × · · · × C (1).For φ : m −→ n and a permutation τ : m −→ m ,( φ, c , . . . , c n ) ◦ ( τ, id , . . . id) = ( φ ◦ τ, Y j ( γ ( c j , id , . . . , id) σ j )= ( φ ◦ τ, c σ , . . . , c n σ n ) , where c j ∈ C ( | φ − ( j ) | ), and σ j ∈ Σ ( | ( φ ◦ τ ) − ( j ) | = Σ | φ − ( j ) | . Note that σ j dependsonly on φ and τ and not on c .For ψ : m −→ n and a permutation τ : n −→ n ,( τ, id , . . . , id) ◦ ( ψ, d , . . . , d n ) = ( τ ◦ ψ, Y j ( γ (id , d τ − ( j ) ) σ j ))= ( τ ◦ ψ, d τ − (1) , . . . , d τ − ( n ) ) For this and related reasons, we do not adopt the original notation ˆ C from [38]. QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 43 since each σ j is the identity (because there is only one i such that τ ( i ) = j ). Example 5.3.
The commutativity operad N has n th space a point for all n . Wethink of it as a G -trivial G -operad. Then F = D ( N ), again regarded as G -trivial.For any operad C G , an F - G -space Y can be viewed as the D ( C G )- G -space ξ ∗ Y .Thus D ( C G )- G -spaces give a generalized choice of input to the Segal machine. Aswe shall discuss in § C G on a G -space X gives rise to an action of D ( C G )on the Π- G -space R X with ( R X ) n = X n .5.2. E ∞ G -operads and E ∞ G -categories of operators. Let C G be a (reduced)operad of G -spaces, or G -operad for short. We say that C G is an E ∞ G -operad if C G ( n ) is a universal principal ( G, Σ n )-bundle for each n . This means that C G ( n )is a Σ n -free ( G × Σ n )-space such that C G ( n ) Λ ≃ ∗ if Λ ⊂ G × Σ n and Λ ∩ Σ n = e (that is, if Λ ∈ F n ) . Since C G ( n ) is Σ n -free, C G ( n ) Λ = ∅ if Λ / ∈ F n . We call G -spaces with an actionby an E ∞ G -operad E ∞ G -spaces. As we recall from [10, 12] in the next section,they provide input for an infinite loop space machine that sends an E ∞ G -space toa genuine Ω- G -spectrum whose zeroth space is a group completion of X . We provethe following theorem. Theorem 5.4. If C G is an E ∞ G -operad, then D = D ( C G ) is an E ∞ G -CO over F or, equivalently, D G is an E ∞ G -CO over F G . Consider the trivial map of G -operads ξ : C G −→ N that sends each C G ( n ) tothe point N ( n ). Of course, N is not an E ∞ G -operad, but it is clear from thedefinitions that F = D ( N ) is an E ∞ G -CO over F . The map ξ : D −→ F is D ( ξ ).The following result with C ′ = N has Theorem 5.4 as an immediate corollary. Wegive pedantic details of equivariance since this is the crux of the comparison ofinputs of the Segal and operadic machines. Theorem 5.5.
Let ν : C G −→ C ′ G be a map of G -operads such that the fixed pointmap ν Λ : C G ( n ) Λ −→ C ′ G ( n ) Λ is a weak equivalence for all n and all Λ ∈ F n . Thenthe induced map ν G : D G −→ D ′ G of G -COs over F G is a G -equivalence.Proof. We must prove that the map ν G : D G (( m , α ) , ( n , β )) −→ D ′ G (( m , α ) , ( n , β ))is a G -equivalence for all ( m , α ) and ( n , β ). Recall that D G (( m , α ) , ( n , β )) is just D ( m , n ) with the G -action given by g · f = β ( g ) ◦ ( gf ) ◦ α ( g − ).Let H be a subgroup of G . We claim that there is a homeomorphism (cid:2) D G (( m , α ) , ( n , β )) (cid:3) H ∼ = a φ Y i C ( q i ) Λ i , where φ runs over the H -equivariant maps ( m , α ) −→ ( n , β ), i runs over the H -orbits of ( n , β ), and q i and Λ i depend only on φ and H , with Λ i in F q i . Thishomeomorphism is moreover compatible with the map ν G . It follows from theassumption that the map( ν G ) H : (cid:2) D G (( m , α ) , ( n , β )) (cid:3) H −→ (cid:2) D ′ G (( m , α ) , ( n , β )) (cid:3) H is a weak equivalence for all subgroups H , as wanted. To prove the claim, recall that D ( m , n ) = a φ : m −→ n Y ≤ j ≤ n C ( | φ − ( j ) | ) . Using Definition 5.1 and in particular Observation 5.2, the new action on D ( m , n )is given by g · ( φ ; x , . . . , x n ) = ( β ( g ) φα ( g − ); gx β ( g − )(1) σ β ( g − )(1) ( g − ) , . . . , gx β ( g − )( n ) σ β ( g − )( n ) ( g − )) , where σ j ( g − ) ∈ Σ | ( φ ◦ α ( g − )) − ( j ) | = Σ | φ − ( j ) | is that permutation of | ( φ ◦ α ( g − )) − ( j ) | letters which converts the natural ordering of ( φ ◦ α ( g − )) − ( j ) as a subset of { , . . . , m } to its ordering obtained by regarding it as ` φ ( i )= j α ( g )( i ), so orderedso that α ( g )( i ) precedes α ( g )( i ′ ) if i < i ′ .Note that the component corresponding to φ is nonempty in the H -fixed pointsif and only if φ is H -equivariant. In what follows we fix such a φ . Careful analysisof the definition of σ j ( h ) shows that for j ∈ { , . . . , n } , and h, k ∈ H , we have(5.6) σ j ( hk ) = σ j ( h ) σ β ( h − )( j ) ( k ) . The H -action shuffles the indices within each H -orbit of ( n , β | H ), so it is enoughto consider each H -orbit separately. We can assume then that the H -action on( n , β | H ) is transitive. The rest of the proof is analogous to the proof of Lemma 2.7.Since φ is H -equivariant and the action is transitive, all the sets φ − ( j ) have thesame cardinality, say q . Let K be the stabilizer of 1 ∈ n under the action of H . By(5.6), σ restricted to K is homomorphism, and thusΛ = { ( k, σ ( k ) | k ∈ K } ⊆ G × Σ q is a subgroup that belongs to F q . To complete the proof of the claim, we note thatthe projection to the first coordinate induces a homeomorphism (cid:0) C ( q ) × · · · × C ( q ) (cid:1) H −→ C ( q ) Λ . One can easily check that if ( x , . . . , x n ) ∈ C ( q ) n is an H -fixed point, then x is aΛ-fixed point, since for all k ∈ K we have that( k, σ ( k )) · x = kx σ ( k − ) = kx β ( k − )(1) σ β ( k − )(1) ( k − ) = x , the last equality being true by the assumption that ( x , . . . , x n ) was fixed by H .To construct an inverse, for every j , choose h j ∈ H such that β ( h j )(1) = j . Notethat choosing these amounts to choosing a system of coset representatives for H/K .Consider the map C ( q ) −→ C ( q ) n that sends x to the n -tuple with j th coordinate x j = h j xσ ( h − j ) . We leave it to the reader to check that this map restricts to the fixed points and isinverse to the projection. (cid:3) The generalized operadic machine
Having redeveloped the Segal infinite loop space machine equivariantly, we nowreview and generalize the equivariant operadic infinite loop space machine. Sincethe prequel [12] of Guillou and the first author also reviews that machine, in partfollowing the earlier treatment of Costenoble and Waner [10], and since the equi-variant generalization of the basic definitions of [27] is entirely straightforward, weshall be brief, focusing on the material that is needed here and is not treated in[12]. In particular, again following the nonequivariant work of Thomason and the
QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 45 first author [38], we develop a generalization of the equivariant operadic machineanalogous to our generalization of the equivariant Segal machine. We compare theinputs and outputs of the classical and generalized machines and show that theyare equivalent. Starting from an E ∞ G -operad, the generalized input is the sameas the generalized input to the Segal machine that we saw in § The Steiner operads.
The advantages of the Steiner operads over the littlecubes or little discs operads are explained in detail in [36, § C n work well nonequivariantly, but are too square for multiplicative andequivariant purposes. The little discs operads are too round to allow maps of op-erads D n −→ D n +1 that are compatible with the natural map Ω n X −→ Ω n +1 Σ X .The Steiner operads are more complicated to define, but they enjoy all of the goodproperties of both the little cubes and little discs operads. Some such family ofoperads must play a central role in any version of the operadic machine, but thespecial features of the Steiner operads will play an entirely new and unexpectedrole in our comparison of the operadic and Segal machines.We review the definition and salient features of the equivariant Steiner operadsfrom [12, § V be a finite dimensional real inner product space and let G act on V throughlinear isometries. We are only interested in the group action when G is finite; formore general groups G , we only use these operads to construct naive G -spectra,taking the action to be trivial. Define a Steiner path to be a continuous map h from I to the space of distance-reducing embeddings V −→ V such that h (1) is theidentity map; thus | h ( t )( v ) − h ( t )( w ) | ≤ | v − w | for all v, w ∈ V and t ∈ I . Define π ( h ) : V −→ V by π ( h ) = h (0) and define the “center point” of h to be the value of0 ∈ V under the embedding h (0), that is c ( h ) = π ( h )(0) ∈ V . Crossing embeddings V −→ V with id W sends Steiner paths in V to Steiner paths in V ⊕ W .For j ≥
0, define K V ( j ) to be the G -space of j -tuples ( h , . . . , h j ) of Steinerpaths such that the embeddings π ( h r ) = h r (0) for 1 ≤ r ≤ j have disjoint images; G acts by conjugation on embeddings and thus on Steiner paths and on j -tuplesthereof. Pictorially (albeit imprecisely), one can think of a point in the Steineroperad as a continuous deformation of V into a point in the little disks operad. Thesymmetric group Σ j permutes j -tuples. We take K V (0) = ∗ and let id ∈ K V (1)be the constant path at the identity V −→ V . Compose Steiner paths pointwise,( ℓ ◦ h )( t ) = ℓ ( t ) ◦ h ( t ) : V −→ V . Define the structure maps γ : K V ( k ) × K V ( j ) × · · · × K V ( j k ) −→ K V ( j + · · · + j k )by sending ( h ℓ , . . . , ℓ k i ; h h , , . . . , h ,j i , . . . , h h k, , . . . , h k,j k i )to h ℓ ◦ h , , . . . , ℓ ◦ h ,j , . . . , ℓ k ◦ h k, , . . . , ℓ k ◦ h k,j k i . Note that K is the trivial operad, K (0) = ∗ , K (1) = { id } and K ( j ) = ∅ for j ≥
2. Via π , the operad K V acts on Ω V X for any G -space X in the same waythat the little cubes operad acts on n -fold loop spaces or the little discs operad actson V -fold loop spaces.Define ζ : K V ( j ) −→ Conf ( V, j ), where
Conf ( V, j ) is the configuration G -spaceof ordered j -tuples of distinct points of V , by sending h h , . . . , h j i to ( c ( h ) , . . . , c ( h j )).The original argument of Steiner [58] generalizes without change equivariantly to prove that ζ is a ( G × Σ j )-deformation retraction. With G finite, we may take thecolimit over V in a complete G -universe U to obtain an E ∞ G -operad K U .6.2. The classical operadic machine.
Recall the definition of an E ∞ G -operadfrom § C G be a fixed chosen E ∞ G -operad throughout this section.A C G -algebra ( X, θ ) is a G -space X together with ( G × Σ j )-maps θ j : C G ( j ) × X j −→ X for j ≥ §
1] or [31, 32] commute. We call analgebra over C G a C G -space. Since C G (0) = {∗} , the action determines a basepointin X , and we assume that it is nondegenerate. Several examples of E ∞ operads C G and C G -spaces appear in [12, 14]. As explained in [36, § G is finite, this machine is a functor E G from C G -spaces to (genuine)orthogonal G -spectra. We summarize the construction of E G , following [12].An operad C G determines a monad C G on based G -spaces such that the category C G [ G T ] of C G -algebras is isomorphic to the category C G [ G T ] of C G -algebras.Nonequivariantly, this motivated the definition of operads [27, 31, 32], and it isproven equivariantly in [10, 12]. Intuitively, C G X is constructed as the quotientof ` j ≥ C G ( j ) × Σ j X j by basepoint identifications. Formally, it is the categoricaltensor product of functors C G ⊗ I X • , where I is the category of finite sets j = { , . . . , j } and injections. To be explicit, for 1 ≤ i ≤ j , let σ i : j − −→ j bethe ordered injection that skips i in its image. Note that the morphisms in I are generated by the maps σ i and the permutations. Then C G is regarded asa functor I op −→ G U via the right action of Σ j and the “degeneracy maps” σ i : C G ( j ) −→ C G ( j −
1) specified in terms of the structure map γ of C G by(6.1) σ i ( c ) = γ ( c ; id i − × × id j − i )where 0 ∈ C G (0) and id ∈ C G (1); X • is the covariant functor I −→ G U that sends j to X j and uses the left action of Σ j and the injections σ i : X j − −→ X j given byinsertion of the basepoint in the i th position.There are several choices that can be made in the construction of the machine E G ,as discussed in [12]. We use the construction landing in orthogonal G -spectra. It ismore natural topologically to land in Lewis-May or EKMM G -spectra [11, 21, 24]since all such G -spectra are fibrant and the relationship of E G to the equivariantBarratt-Priddy-Quillen theorem is best explained using them. That variant of theoperadic machine is discussed and applied in [12], and we shall say nothing moreabout it here.Regardless of such choices, the construction of E G is based on the two-sidedmonadic bar construction defined in [27, § G -space B ( F, C G , X ) for a monad C G in G T , a C G -algebra X , and a C G -functor F = ( F, λ ). Here F : G T −→ G T is a functor and λ : F C G −→ F is anatural transformation such that λ ◦ F η = Id : F −→ F and λ ◦ F µ = λ ◦ λ C G : F C G C G −→ F. There results a simplicial G -space B q ( F, C G , X ) with q -simplices F C qG X . Its geo-metric realization is B ( F, C G , X ). We emphasize that while the bar constructioncan be specified in sufficiently all-embracing generality that both the categorical I was often denoted Λ in the 1970’s and is nowadays often denoted F I . QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 47 version used in the Segal machine and the monadic version used in the operadicmachine are special cases [41, 42, 43, 57], these constructions look very different.To incorporate the relationship to loop spaces encoded in the Steiner operads,we define C V = C G × K V . We view C G -spaces as C V -spaces via the projection to C G , and we view G -spaces Ω V X as C V -spaces via the projection to K V . We write C V for the monad on G T associated to C V . Theorem 6.2.
The composite of the map C V X −→ C V Ω V Σ V X induced by theunit of the adjunction (Σ V , Ω V ) and the action map C V Ω V Σ V X −→ Ω V Σ V X specifies a natural map α : C V X −→ Ω V Σ V X which is a group completion if V contains R . These maps specify a map of monads C V −→ Ω V Σ V . The second statement is proven by the same formal argument as in [27, Theorem5.2]. The first statement is discussed and sharpened to the case V ⊃ R in [12]. Theadjoint ˜ α : Σ V C V −→ Σ V of α is an action of the monad C V on the functor Σ V and we have the monadic bar construction( E G X )( V ) = B (Σ V , C V , X )for C G -spaces X . An isometric isomorphism V −→ V ′ in I G induces naturaltransformations Σ V −→ Σ V ′ and C V −→ C V ′ , which in turn induce a map B (Σ V , C V , X ) −→ B (Σ V ′ , C V ′ , X ) . These maps assemble to make E G X into an I G - G -space. Smashing with G -spacescommutes with realization of based simplicial G -spaces, by the same proof as in thenonequivariant case [27, Proposition 12.1], and inclusions V −→ W induce maps ofmonads C V −→ C W . This gives the structural maps σ : Σ W − V E G X ( V ) ∼ = B (Σ W , C V , X ) −→ B (Σ W , C W , X ) = E G Y ( W )which are compatible with the I G - G -space structure, so E G X is an orthogonalspectrum. The structure maps of E G X and their adjoints are closed inclusions.As explained in [12], and as goes back to [27] nonequivariantly and to Costenobleand Waner equivariantly [10], we have the following theorem, which gives the basichomotopical property of the infinite loop space machine E G . Theorem 6.3.
There are natural maps
X B ( C V , C V , X ) ε o o B ( α, id , id) / / B (Ω V Σ V , C V , X ) ζ / / Ω V B (Σ V , C V , X ) of C V -spaces. The map ε is a G -homotopy equivalence with a natural G -homotopyinverse ν (which is not a C V -map), the map B ( α, id , id) is a group completion when V contains R , and the map ζ is a weak G -equivalence.Proof. The first statement is a standard property of the bar construction that worksjust as well equivariantly as nonequivariantly [27, Proposition 9.8] or [57, Lemma9.9]. The second statement is deduced from Theorem 6.2 by passage to fixed pointspaces and use of the same argument as in the nonequivariant case [28, Theorem2.3(ii)]. The last statement is an equivariant generalization of [27, Theorem 12.7]that is proven carefully in [10, Lemmas 5.4, 5.5]. See [12] for further discussion andvariants of the construction. (cid:3)
Define ξ = ζ ◦ B ( α, id , id) ◦ ν : X −→ Ω V E G X ( V ) . Then ξ is a natural group completion when V ⊃ R and is thus a weak G -equivalencewhen X is grouplike. The following diagram commutes, where ˜ σ is adjoint to σ . X ξ y y ssssssssss ξ ' ' PPPPPPPPPPPPP Ω V E G X ( V ) Ω V ˜ σ / / Ω V ⊕ W E G X ( V ⊕ W ) . Therefore Ω V ˜ σ is a weak equivalence if V contains R . For general topologicalgroups, everything works exactly the same way provided that we restrict to those V with trivial G -action. However, even if G = S , the group completion propertyof α fails if V is a non-trivial representation of G , as was first noticed by Segal [52].A proof can be found in [5, Appendix B]. We restrict G to be finite from now on. Remark 6.4.
With G finite, it is harmless to think of E G X ( V ) as an Ω- G -spectrum. If we set E ′ G X ( V ) = Ω E G X ( V ⊕ R ), then the maps ˜ σ : E G X ( V ) −→ E ′ G X ( V ) specify an equivalence from E G X to an Ω- G -spectrum, giving a simpleand explicit fibrant approximation whose zeroth space is a group completion of X .6.3. The monads D and D G associated to the G -categories D and D G . Recall that the category C G [ G T ] of algebras over an operad C G is isomorphic tothe category C G [ G T ] of algebras over the associated monad C G . Let D = D ( C G )be the G -CO over F associated to C G and let D G be the associated G -CO over F G .As worked out nonequivariantly in [38, § D on the category ofΠ- G -spaces and D G on the category of Π G - G -spaces. The commutative diagram ofinclusions of categories Π / / i (cid:15) (cid:15) Π Gi G (cid:15) (cid:15) D / / D G gives rise to a commutative diagram of forgetful functorsFun(Π , T G ) Fun(Π G , T G ) U o o Fun( D , T G ) i ∗ O O Fun( D G , T G ) . i ∗ G O O U o o Categorical tensor products then give left adjoints making the following diagramcommute up to natural isomorphism.Fun(Π , T G ) P / / D (cid:15) (cid:15) Fun(Π G , T G ) D G (cid:15) (cid:15) Fun( D , T G ) P / / Fun( D G , T G )Here D and D G are the left adjoints of i ∗ and i ∗ G , respectively. By a standard abuseof notation, we write D and D G for the resulting endofunctors i ∗ D on Fun(Π G , T G )and i ∗ G D G on Fun(Π G , T G ). Explicitly, the monads D and D G are defined as(6.5) ( D X )( n ) = D ( − , n ) ⊗ Π X QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 49 for a Π- G -space X , where D ( − , n ) is the represented functor induced by i , and(6.6) ( D G Y )( n , α ) = D G ( − , ( n , α )) ⊗ Π G Y for a Π G - G -space Y , where D G ( − , ( n , α )) is the represented functor induced by i G .The units η of the adjunctions ( D , i ∗ ) and ( D G , i ∗ G ) give the unit maps of themonads, and the action maps of the D - G -spaces D X and D G - G -spaces D G Y givethe products µ . More concretely, µ : DD −→ D and µ : D G D G −→ D G are derivedfrom the compositions in D and D G , respectively, and thus from the structuremaps γ of C G . The unit maps η are derived from the identity morphisms in thesecategories and thus from the unit element id ∈ C G (1).As shown nonequivariantly in [38, § D , T G ) of D - G -spaces to the category D [Fun(Π , G T )] of algebrasover the monad D and from the category Fun( D G , T G ) of D G - G -spaces to thecategory D G [Fun(Π G , G T )] of algebras over the monad D G . Proposition 6.7.
The categories
Fun( D , T G ) and D [Fun(Π , T G )] are isomorphic.The categories Fun( D G , T G ) and D G [Fun(Π G , T G )] are isomorphic. Therefore thecategories D [Fun(Π , T G )] and D G [Fun(Π G , T G )] are equivalent.Proof. The proof for D is a comparison of action maps D ( m , n ) ∧ X ( m ) −→ X ( n )and ( D X )( n ) −→ X ( n ); it is entirely analogous to the original argument for alge-bras over operads in [27, Proposition 2.8]. A similar proof works for D G . The resultthere can also be derived from the result for D , using D G P ∼ = PD and Theorem 4.11,and that result also implies the last statement. (cid:3) Since the categories of D -algebras and of D G -algebras are equivalent, the monads D and D G can be used interchangeably. This contrasts markedly with the Segalmachine, where considerations of specialness led us to focus on D G rather than D .We need some homotopical and some formal properties of the monads D and D G ,following [38]. We first establish the formal properties, whose proofs are identical tothose in [38]. We first write the following results in terms of D and D for simplicity.With attention to enrichment, the parallel results for D G work in exactly the sameway, and they can also be derived from the results for D by use of the isomorphism PD ∼ = D G P and Proposition 6.7.Recall that we have the functor R : G T −→ Fun(Π , G T ) given by ( R X ) n = X n .Define L : Fun(Π , G T ) −→ G T by L Y = Y . Then ( L , R ) is an adjoint pair suchthat LR = Id. On a Π- G -space Y , the unit δ : Id −→ RL of the adjunction is givenby the Segal maps. Letting C G [ G T ] denote the category of C G -spaces, we showthat ( L , R ) induces an adjunction between that category and Fun( D , T G ). Proposition 6.8.
The adjunction ( L , R ) between Fun(Π , G T ) and G T inducesan adjunction between Fun( D , T G ) and C G [ G T ] such that LR = Id and the unit δ : Id −→ RL is given by the Segal maps. A D - G -space with underlying Π - G -space R X determines and is determined by a C G -space structure on LR X = X .Proof. The nonequivariant proof of [38, Lemma 4.2] applies verbatim. (cid:3)
We require an analyis of the behavior of the monad D with respect to the ad-junction ( L , R ). Proposition 6.9.
Let X be a G -space and Y be a Π - G -space.(i) The G -space LDR X = ( DR X ) is naturally G -homeomorphic to C G X . (ii) The Π - G -space DR X is naturally isomorphic to the Π - G -space RC G X .(iii) The following diagram is commutative for each n. ( D Y ) n ( D δ ) n / / δ (cid:15) (cid:15) ( DRL Y ) n ∼ = ( C G L Y ) n ∼ = δ (cid:15) (cid:15) ( D Y ) n D δ ) n / / ( DRL Y ) n ∼ = ( C G L Y ) n (iv) The functor RC G L on Π - G -spaces is a monad with product and unit inducedfrom those of C G via the composites RC G LRC G L = RC G C G L R µ L / / RC G L and Id δ / / RL R η L / / RC G L . (v) The natural transformation D δ : D −→ DRL ∼ = RC G L is a morphism of mon-ads in the category Fun(Π , G T ) of Π - G -spaces.(vi) If ( F, λ ) is a C G -functor in V , then F L : Fun(Π , G T ) −→ V is an RC G L -functor in V with action λ L : F LRC G L = F C G L −→ F L . Therefore, bypullback, F L is a D -functor in V with action the composite F LD F LD δ / / F LDRL ∼ = F LRC G L = F C G L λ L / / F L . Proof.
Nonequivariantly, these results are given in [38, §
6] and the equivarianceadds no complications. The proofs are inspections of definitions and straightforwarddiagram chases. (cid:3)
Using the adjunction ( P , U ) and the isomorphism D G P ∼ = PD , we derive theanalogue for D G . As in Definition 2.31, we write R G = PR : C G [ G T ] −→ Fun( D G , T G )and L G = LU : Fun( D G , T G ) −→ C G [ G T ] . With D , R , and L replaced by D G , R G , and L G , we then have the following analogueof Proposition 6.9. Proposition 6.10.
Let X be a G -space and Y be a Π G - G -space.(i) The G -space L G D G R G X is naturally G -homeomorphic to C G X .(ii) The Π G - G -space D G R G X is naturally isomorphic to R G C G X .(iii) The following diagram is commutative for each ( n , α ) . ( D G Y )( n , α ) D G δ / / δ (cid:15) (cid:15) ( D G R G L G Y )( n , α ) ∼ = ( C G L G Y ) ( n ,α ) ∼ = δ (cid:15) (cid:15) ( D G Y ) ( n ,α )1 ( D G δ ) ( n ,α )1 / / ( D G R G L G Y ) ( n ,α )1 ∼ = ( C G L G Y ) ( n ,α ) (iv) The functor R G C G L G on Π G - G -spaces is a monad with product and unitinduced from those of C G via the composites R G C G L G R G C G L G = R G C G C G L G R G µ L G −−−−−→ R G C G L G and Id δ −−−−→ R G L G R G η L G −−−−→ R G C G L G . QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 51 (v) The natural transformation D G δ : D G −→ D G R G L G ∼ = R G C G L G is a mor-phism of monads in the category Fun(Π G , T G ) of Π G - G -spaces.(vi) If ( F, λ ) is a C G -functor in V , then F L G : Fun(Π G , T G ) −→ V is an R G C G L G -functor in V with action λ L G : F L G R G C G L G = F C G L G −→ F L G . There-fore, by pullback, F L G is a D G -functor in V with action the composite F L G D GF L G D G δ / / F L G D G R G L G ∼ = F L G R G C G L G = F C G L G λ L G / / F L G . We now turn to the homotopical properties of the monads D and D G and theiralgebras. The proofs of the homotopical properties are similar to those in [38], butconsiderably more difficult, so some will be deferred to § D rather than D G . Our interest is in E ∞ operads,but we allow more general operads until otherwise indicated.Reedy cofibrancy of Π and F -spaces can be defined as in [4], but we shall beinformal about the former and not use the latter (see Remark 8.5). We give a quickdefinition, which mimics Definition 1.8. Definition 6.11.
For a Π- G -space X , the ordered injections σ i : n − −→ n induce maps σ i : X n − −→ X n . A point of X n in the image of some σ i is said tobe degenerate. The n th latching space of X is the set of degenerate points in X n : L n X = n [ i =1 σ i ( X n − ) . It is a ( G × Σ n )-space, and the inclusion L n X −→ X n is a ( G × Σ n )-map. We saythat X is Reedy cofibrant if this map is a ( G × Σ n )-cofibration for each n .Observe that L n R X is the subspace of X n consisting of those points at least onecoordinate of which is the basepoint. By our standing assumption that basepointsare nondegenerate, R X is Reedy cofibrant. Remark 6.12.
Just as nonequivariantly [38, Definition 1.2], we can impose acofibration condition on general Π- G -spaces X which ensures that they are Reedycofibrant. Given an injection φ : m −→ n in Π, we let Σ φ be the subgroup of Σ n consisting of those permutations τ such that τ (im φ ) = im φ . Then Σ φ acts on theset m and φ is a Σ φ -map. If the map φ ∗ : X m −→ X n is a ( G × Σ φ )-cofibrationfor all injections φ , then a direct application of [7, Theorem A.2.7] shows that X isReedy cofibrant.Following [27] nonequivariantly, we say that a G -operad C G is Σ-free if the actionof Σ j on C G ( j ) is free for each j . Surprisingly, we only need that much structure toprove the following result. It is the equivariant generalization of [38, Lemma 5.6],which implicitly used Reedy cofibrancy of Π-spaces via the remark above. Theorem 6.13.
Let C G be a Σ -free G -operad.(i) If f : X −→ Y is an F • -level equivalence of Reedy cofibrant Π - G -spaces, then D f : D X −→ D Y is an F • -level equivalence.(ii) If X is an F • -special Reedy cofibrant Π - G -space, then D X is F • -special.(iii) For any Reedy cofibrant Π - G -space X , D X is a Reedy cofibrant Π - G -space.Proof. The proof of (i) requires quite lengthy combinatorics about the structureof D X and its fixed point subspaces, so we defer it to § δ : X −→ RL X , the map D δ is an F • -level equivalence. Since its target DRL X ∼ = RC G L X is F • -special, Lemma 2.10 implies that D X is also F • -special. Aswe explain in § D X . (cid:3) By Theorem 4.11 and Corollaries 4.13 and 4.14, the isomorphism PD ∼ = D G P and Theorem 6.13 imply the following analogue of that result. Proposition 6.14.
Assume that each C G ( j ) is Σ j -free.(i) If f : X −→ Y is an F • -level equivalence of Reedy cofibrant Π - G -spaces, then D G P f : D G P X −→ D G P Y , is a level G -equivalence.(ii) If X is a F • -special Reedy cofibrant Π - G -space, then D G P X is a special Π G - G -space. Comparisons of inputs and outputs of the operadic machine.
We havethree equivalent ways to construct G -spectra from D G - G -spaces. We can convert D G - G -spaces to D - G -spaces via Proposition 6.7, and we can convert those to C G -spaces by Proposition 6.18 below. We can then apply the original machine, or wecan generalize the machine to both D - G -spaces and D G - G -spaces. All three makeuse of the two-sided monadic bar construction of [27], starting from the formalitiesof Propositions 6.9 and 6.10. Remark 6.15.
Just as nonequivariantly [38, Addendum 1.7], we impose an addi-tional cofibration condition on D to ensure that our bar constructions are given byReedy cofibrant simplicial G -spaces. As in Remark 6.12, for an injection φ : m −→ n , let Σ φ ⊂ Σ n be the subgroup of permutations τ such that τ (im φ ) = im φ .Then the map D ( q , m ) −→ D ( q , n ) induced by φ is a ( G × Σ φ )-map, and we re-quire it to be a ( G × Σ φ )-cofibration. This holds for our categories of operators D = D ( C ) since we are assuming that the inclusion of the identity ∗ −→ C G (1) isa G -cofibration.In fact, we really only have two machines in view of the following result. Weemphasize how different this is from the Segal machine, where the D and D G barconstructions are not even equivalent, let alone isomorphic. Proposition 6.16.
Let C G be a G -operad in G T with category of operators D .For any D -space X and any C G -functor F : G T −→ G T , there is a natural iso-morphism of G -spaces B ( F L , D , X ) ∼ = B ( F L G , D G , P X ) . Proof.
Since Id ∼ = UP , PD ∼ = D G P and L G = LU , we have isomorphisms of q -simplices F LD q X ∼ = F L G D qG P X. Formal checks show that these isomorphisms commute with the face and degeneracyoperators. The conclusion follows on passage to geometric realization. (cid:3)
For variety, and because that is what we shall use in the next section, we focuson D G rather than D in this section. The following result compares the inputs tomachines given by D G - G -spaces and C G -spaces. Definition 6.17.
For a D G - G -space Y , define a C G -space X ( Y ) by X ( Y ) = B ( C G L G , D G , Y ) . QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 53
Here C G is regarded as a functor G T −→ C G [ G T ], and the construction makessense since the realization of a simplicial C G -space is a C G -space, exactly as nonequiv-ariantly [27, Theorem 12.2]. Proposition 6.18.
For special D G - G -spaces Y whose underlying Π -spaces areReedy cofibrant, there is a zigzag of natural level G -equivalences of D G - G -spacesbetween Y and R G X ( Y ) . For C G -spaces X , there is a natural G -equivalence of C G -spaces from X ( R G X ) to X .Proof. Proposition 6.14 implies that D G δ : D G Y −→ D G R G L G Y ∼ = R G C G L G Y isa level G -equivalence of D G - G -spaces. The realization of simplicial Π G - G -spacesis defined levelwise, and since realization commutes with products of G -spaces, wehave the right-hand isomorphism in the diagram Y B ( D G , D G , Y ) ε o o B ( D G δ, id , id) / / B ( R G C G L G , D G , Y ) ∼ = R G B ( C G L G , D G , Y ) = R G X ( Y ) . By standard properties of the bar construction, as in [27] nonequivariantly, ε isa level G -equivalence of D G - G -spaces. Since the bar constructions are geometricrealizations of Reedy cofibrant simplicial G -spaces (see Remark 1.12), it followsfrom Theorem 1.10 that B ( D G δ, id , id) is a level G -equivalence of D G - G -spaces.For the second statement, we apply L G to the level G -equivalence of D G - G -spaces R G X ( R G X ) ∼ = B ( R G C G L G , D G , R G X ) ∼ = B ( R G C G L G , R G C G L G , R G X ) ε / / R G X, where the second isomorphism follows from Proposition 6.10 and inspection. (cid:3) Therefore, after inverting the respective equivalences, the functors R G and X induce an equivalence of categories between C G -spaces and special D G - G -spaceswhose underlying Π- G -spaces are Reedy cofibrant. We conclude that, for an E ∞ -operad C G , the input categories for operadic machines given by C G -spaces and by D G - G -spaces are essentially equivalent.To generalize the machine from C G -spaces to D G - G -spaces, we again use theproduct operads C V = C G × K V , where K V is the V th Steiner operad. We write D G,V for the monad associated to the resulting category of operators D G,V over F G . Then a D G -space is a D G,V -space for any representation V by pullback alongthe projection D G,V −→ D G . Definition 6.19.
For a D G - G -space Y , define the V th space of the orthogonal G -spectrum E G Y to be the monadic two-sided bar construction(6.20) E G ( Y )( V ) = B (Σ V L G , D G,V , Y ) . The right action of D G,V on Σ V L G is obtained from the projection D G,V −→ K V and the action of K V on Σ V , via Proposition 6.9(vi). The I G - G -space structure isgiven as follows. For an isometric isomorphism V −→ V ′ in I G , the map B (Σ V L G , D G,V , Y ) −→ B (Σ V ′ L G , D G,V ′ , Y )is the geometric realization of maps induced at all simplicial levels by S V −→ S V ′ and K V −→ K V ′ . Similarly, since smashing commutes with geometric realization,the structure maps B (Σ V L G , D G,V , Y ) ∧ S W −→ B (Σ V ⊕ W L G , D G,V ⊕ W , Y )are induced from the maps of monads D G,V −→ D G,V ⊕ W . Just as we used S G for all variants of the Segal machine, we are using E G for allvariants of the operadic machine. To justify this, we must show that the machine E G on D G - G -spaces does indeed generalize the machine E G on C G -spaces X . To seethat, observe that Proposition 6.10(ii) implies that we have a natural isomorphism D G,V R G X ∼ = R G C V X . Since L G R G = Id, that gives us a natural isomorphism(6.21) B (Σ V L G , D G,V , R G X ) ∼ = B (Σ V , C V , X ) , where we regard R G X as a D G - G -space via Proposition 6.8. Together with Propo-sition 6.18, this gives the following comparison of outputs of our machines. Corollary 6.22.
For C G -spaces X , E G X is naturally isomorphic to E G R G X . Forspecial D G - G -spaces Y whose underlying Π - G -space is Reedy cofibrant, there is azigzag of natural equivalences connecting E G Y to E G R G X ( Y ) ∼ = E G X ( Y ) . Thus the machines E G on C G -spaces and on special D G - G -spaces are essentiallyequivalent. Properties of the machine on special D G - G -spaces are essentially thesame as properties of the machine on C G -spaces, as can either be proven directlyor read off from the equivalence of machines.7. The equivalence between the Segal and operadic machines
We give an explicit comparison between the generalized Segal and generalizedoperadic infinite loop space machines. The comparison is needed for consistency andbecause each machine has significant advantages over the other. That was alreadyclear nonequivariantly, and it seems even more true equivariantly. As in [12, 36],in the previous section we used the Steiner operads rather than the little cubesoperads that were used in [27, 38]. That change made equivariant generalizationeasy, and [36] gave other good reasons for the change. However, nothing like thepresent comparison was envisioned in earlier work. As we have recalled, the Steineroperad is built from paths of embeddings. We shall see that these paths give riseto a homotopy that at one end relates to the generalized Segal machine and at theother end relates to the generalized operadic machine. That truly seems uncanny.7.1.
The statement of the comparison theorem.
To set the stage, we reca-pitulate some of what we have done. We fix an E ∞ operad C G of G -spaces. Wethen have an E ∞ G -CO D = D ( C G ) over F and an E ∞ G -CO D G over F G . Ourprimary interest here is in infinite loop space machines defined either on special F G - G -spaces or on C G -spaces. The Segal machine is defined on the former and theoperadic machine is defined on the latter. We have generalized both machines sothat they accept special D G - G -spaces as input. Moreover, we have compared inputsand shown that both special F G - G -spaces and C G -spaces are equivalent to special D G - G -spaces and therefore to each other. Further, we have compared outputs.We have shown that application of the generalized Segal machine to D G - G -spacesis equivalent to application of the original homotopical Segal machine to F G - G -spaces, and that application of the generalized operadic machine to D G - G -spacesis equivalent to application of the original operadic machine to C G -spaces.In more detail, F • -special F - G -spaces, F • -special D - G -spaces, special F G - G -spaces, and special D G - G -spaces are all equivalent by Theorems 4.25 and 4.26,and the Segal machines on all four equivalent inputs give equivalent output byTheorem 4.32. We may therefore focus on the Segal machine S G defined on special QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 55 D G - G -spaces Y . Similarly, C G -spaces, F • -special D - G -spaces, and special D G - G -spaces are equivalent by Propositions 6.8 and 6.18, and the operadic machine on C G -spaces is a special case of the operadic machine on D G - G -spaces by Corollary 6.22.Thus we may again focus on the operadic machine E G defined on special D G - G -spaces Y .Thus, fixing an E ∞ operad C G with associated category of operators D G over F G , we consider special D G - G -spaces Y . The V th G -space of S G Y is( S G Y )( V ) = B (( S V ) • , D G , Y ) , where, as before, ( S V ) • denotes the composite G T -functor D opG ξ / / F opG / / T G that sends the object ( n , α ) to the cartesian power ( S V ) ( n,α ) = T G (( n , α ) , S V ).With ⋆ thought of as a place holder for the representation V in the V th level of thespectrum, we adopt the notation S G Y = B (( S ⋆ ) • , D G , Y ) . Let C V be the product operad C G × K V and let D G,V = D G ( C V ) with associatedmonad D G,V on the category of Π G - G -spaces. The V th G -space of E G Y is( E G Y )( V ) = B (Σ V L G , D G,V , Y ) . Again using ⋆ as a place holder for the representation V of the V th level of thespectrum, we adopt the notation E G Y = B (Σ ⋆ L G , D G,⋆ , Y ) . Note the different uses of the ⋆ notation. In both machines, it is a placeholder forrepresentations V . However, in the Segal machine, we are using cartesian powersof G -spheres S V to obtain functors ( S V ) • : F opG −→ T G , whereas in the operadicmachine we are using the suspension functor Σ V associated to S V together with theSteiner operad K V . While a two-sided bar construction is used in both machines,the similarity of notation hides how different these bar constructions really are: theuse of categories and contravariant and covariant functors in one is quite differentfrom the use of monads, (right) actions on functors, and (left) actions on objectsin the other. Nevertheless, our goal is to give a constructive proof of the followingcomparison theorem. Theorem 7.1.
For special D G - G -spaces Y , there is a natural zigzag of equivalencesof orthogonal G -spectra between S G Y and E G Y . The proof of the comparison theorem.
We display the zigzag and thenfill in the required constructions and proofs in subsequent sections. In addition tousing ⋆ as a placeholder for representations V , we use • as a placeholder for finite G -sets ( n , α ). (7.2) S G Y B (( S ⋆ ) • , D G , Y ) B (( S ⋆ ) • , D G,⋆ , Y ) π O O i (cid:15) (cid:15) B ( I + ∧ ( S ⋆ ) • , D G,⋆ , Y ) B (( S ⋆ ) • , D G,⋆ , Y ) i O O B ( • ( S ⋆ ) , D G,⋆ , Y ) ι O O ω (cid:15) (cid:15) B (Σ ⋆ L G , D G,⋆ , Y ) E G Y. We shall construct the intermediate orthogonal G -spectra and maps in this zigzagand prove directly that all of the maps except ω are stable equivalences.Recall that the homotopy groups of a pointed G -space X are π Hq ( X ) = π q ( X H )and the homotopy groups of an orthogonal G -spectrum T are π Hq ( T ) = colim V π Hq (Ω V T ( V ))for q ≥
0, where the colimits are formed using the adjoint structure maps of T ; our G -spectra are all connective, so that their negative homotopy groups are zero. Amap T −→ T ′ is a stable equivalence if its induced maps of homotopy groups areisomorphisms. That depends only on large V . Thus we may focus on those V thatcontain R , so that the group completions of Theorem 6.2 are available. ApplyingΩ V to the V th spaces implicit in (7.2), we obtain a diagram of G -spaces under Y .By completely different proofs, both maps Y −→ Ω V S G Y ( V ) and Y −→ Ω V E G Y ( V )are group completions. Therefore, once we prove that the arrows other than ω are stable equivalences, it will follow that ω is also a stable equivalence. Indeed,arranging as we may that our outputs are Ω- G -spectra and using that they areconnective, ω is a stable equivalence if and only if the map ω it induces on 0th G -spaces is a G -equivalence. The displayed group completions imply that ω inducesa homology isomorphism on fixed point spaces. Since these spaces are Hopf spaces,hence simple, it follows that ω induces an isomorphism on homotopy groups, sothat ω is a G -equivalence.Since wedges taken over G -sets ( n , α ) play a significant role in our arguments,we introduce the following convenient notation. Notation 7.3.
For a based space A , let n A denote the wedge sum of n copies of A .Similarly, for a G -set ( n , α ) and a based G -space A , let ( n ,α ) A denote the wedge sumof n copies of A with G -acting on A , but also interchanging the wedge summands. QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 57
We write ( j, a ) ∈ ( n ,α ) A for the element a in the j th summand. The G -action isgiven explicitly by g · ( j, a ) = ( α ( g )( j ) , g · a ). Remark 7.4.
In constructing the diagram, we shall encounter an annoying butminor clash of conventions. There is a dichotomy in how one chooses to define thefaces and degeneracies of the categorical bar construction. We made one choice in § q -simplices, we agree to replacethe previous d i and s i by d q − i and s q − i , respectively. With the new convention, d is given by the evaluation map of the left (contravariant) variable in the categoricaltwo-sided bar construction, rather than the right variable.7.3. Construction and analysis of the map π . Turning to the diagram (7.2),we first define the top map π . We start by defining its source orthogonal G -spectrum B (( S ⋆ ) • , D G,⋆ , Y ). The V th space, as the notation indicates, is defined by pluggingin V for ⋆ ; it is the bar construction B (( S V ) • , D G,V , Y ), as defined in § q -simplices given by thewedge over all sequences ( n q , α q ) , . . . , ( n , α ) of the G -spaces ( S V ) ( n q ,α q ) ∧ (cid:0) D G,V (( n q − , α q − ) , ( n q , α q )) ×· · ·× D G,V (( n , α ) , ( n , α )) × Y ( n , α ) (cid:1) + . We have implicitly composed Y with the evident projections D G,V −→ D G toregard Y as a G T -functor defined on each D G,V , and we have composed the ( S V ) • with the composite D G,V −→ D G −→ F G to regard the ( S V ) • as functors definedon D G,V . Note that B (( S ⋆ ) • , D G,⋆ , Y ) is not the restriction of a W G - G -space, butit is an I G - G -space. For an isometric isomorphism V −→ V ′ in I G , the map B (( S V ) • , D G,V , Y ) −→ B (( S V ′ ) • , D G,V ′ , Y )is the geometric realization of the map induced at each simplicial level by the maps K V −→ K V ′ and S V −→ S V ′ . Geometric realization commutes with ∧ , andthe structure maps of the orthogonal G -spectrum B (( S ⋆ ) • , D G,⋆ , Y ) are geometricrealizations of levelwise simplicial maps given by the maps j : D G,V −→ D G,V ⊕ W induced by the inclusions K V −→ K V ⊕ W and the maps(7.5) i : ( S V ) ( n ,α ) ∧ S W −→ ( S V ⊕ W ) ( n ,α ) defined by i (cid:0) ( v , . . . , v n ) ∧ w (cid:1) = ( v ∧ w, . . . , v n ∧ w ) . An alternative construction is to use Remark 2.18 and (3.8) in Remark 3.7 to obtain G -maps B (( S V ) • , D G,V ⊕ W , Y ) ∧ S W −→ B (( S V ⊕ W ) • , D G,V ⊕ W , Y )and to precompose with the G -map B (( S V ) • , D G,V , Y ) −→ B (( S V ) • , D G,V ⊕ W , Y )induced by j : D G,V −→ D G,V ⊕ W . One can easily check that these maps do indeedgive maps of bar constructions that specify the structure maps for an orthogonal G -spectrum. The projections D G,V −→ D G induce the top map π of orthogonal G -spectra in (7.2).Recall that colim V K V ( j ) = K U ( j ), so that colim V ( C G × K V ) is the product C G × K U , which is an E ∞ G -operad since it is the product of two such operads.Therefore the projection ( C G × K U )( j ) −→ C G ( j ) is a Λ-equivalence for all Λ ∈ F j and, by Theorem 5.5, the map D G ( C G × K U ) −→ D G ( C G ) is a G -equivalence of G -COs over F G . The projection map π : B (( S ⋆ ) • , D G,⋆ , Y ) −→ B (( S ⋆ ) • , D G , Y )is not a level G -equivalence, but a direct comparison of colimits shows that π is astable equivalence. In more detail, in computing π on homotopy groups, we startfrom the commutative diagramsΩ V B (( S V ) • , D G,V , Y ) / / π (cid:15) (cid:15) Ω W B (( S W ) • , D G,W , Y ) π (cid:15) (cid:15) Ω V B (( S V ) • , D G , Y ) / / Ω W B (( S W ) • , D G , Y ) , where V ⊂ W . We then take H -fixed points and their homotopy groups. Since theinclusions D G,V −→ D G,W become isomorphisms on homotopy groups in increasingranges of dimensions, by inspection of the homotopy types of the G -spaces com-prising the Steiner operads in [12, § π is a stable equivalence.7.4. The contravariant functors I + ∧ ( S V ) • on D G,V . In the notation I + ∧ ( S ⋆ ) • in (7.2), ⋆ is again a place holder for V , and the notation stands for G T -functors I + ∧ ( S V ) • : ( D G,V ) op −→ T G that are given on objects by sending ( n , α ) to I + ∧ ( S V ) ( n ,α ) , where I is the unitinterval and we have adjoined a disjoint basepoint and taken the smash productin order to have domains for based homotopies. The crux of our comparison is tospecify the functors on I + ∧ ( S V ) • on morphisms in terms of homotopies that arededuced from the paths that comprise the Steiner operads.Note that ( S V ) ( n ,α ) is just ( S V ) n with the G -action · α specified by g · α ( x , . . . , x n ) = ( gx α ( g ) − (1) , . . . , gx α ( g ) − ( n ) ) = α ( g ) ∗ ( gx , . . . , gx n ) . Therefore, by Theorem 4.11 and Lemma 4.12, it is enough to instead define G T -functors I + ∧ ( S V ) • : ( D V ) op −→ T G given on objects by sending n to I + ∧ ( S V ) n and then apply the functor P to obtainthe desired functors on D G,V . We choose to do this in order to make the definitionsa little less cumbersome.We construct the required maps on hom objects as composites D V ( m , n ) / / D ( K V )( m , n ) ˜ H / / T G ( I + ∧ ( S V ) n , I + ∧ ( S V ) m ) . The first map is the evident projection, and we shall use the same letter for maps andtheir composites with that projection. To define ˜ H , we shall construct a homotopy(7.6) H : I + ∧ ( S V ) n ∧ D ( K V )( m , n ) −→ ( S V ) m and then set(7.7) ˜ H ( f )( t, v ) = ( t, H ( t, v, f )) , where t ∈ I , v ∈ ( S V ) n , and f ∈ D ( K V )( m , n ). We have written variables in theorder appropriate to thinking of the homotopies H as the core of the evaluation QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 59 maps of the contravariant functor I + ∧ ( S V ) • : D V −→ T G . Note that such eval-uation maps, after prolongation to D G,V , give the zeroth face operation d in thesimplicial G -spaces whose realizations give the central bar constructions in (7.2).Writing H t for H at time t , H will relate to the evaluation maps of the rep-resented functor ( S V ) • used in the left variable of the Segal machine and H willrelate to the maps that define the action of the monad K V on the functor Σ V thatis used in the left variable of the operadic machine.The following construction is the heart of the matter. Recall that a Steiner pathin V is a map h : I −→ R V such that h (1) = id, where R V is the space of distancereducing embeddings V −→ V . The space K V ( s ) of the Steiner operad is the spaceof s -tuples of Steiner paths h r such that the h r (0) have disjoint images. We definea homotopy γ : I × S V × K V ( s ) −→ ( S V ) s with coordinates γ r by letting γ r ( t, v, h h , . . . , h s i ) = (cid:26) w if h r ( t )( w ) = v ∗ if v / ∈ im( h r ( t )) . If t = 1, this is just the diagonal map S V −→ ( S V ) s , which is relevant to theSegal machine. If t = 0, this map lands in the s -fold wedge s ( S V ) of copies of S V since the conditions v ∈ im( h r (0)) as r varies are mutually exclusive; that is, thereis at most one r such that v ∈ im( h r (0)). This is relevant to the operadic machinesince the action map˜ α : Σ V K V X = K V X ∧ S V −→ X ∧ S V = Σ V X is given by˜ α (( h h , . . . , h s i , x , . . . , x s ) , v ) = (cid:26) ( x r , w r ) if h r (0)( w r ) = v ∗ if v / ∈ im( h r (0)) for 1 ≤ r ≤ s. Remember that we understand Σ V A to be A ∧ S V for a based G -space A , butwe write the V coordinate on the left when looking at the evaluation maps of thefunctor I + ∧ ( S V ) • .We now define the homotopy H of (7.6). Recall from Definition 5.1 that D ( K V )( m , n ) = a φ : m −→ n Y ≤ j ≤ n K V ( s j ) , where s j = | φ − ( j ) | . Let f = ( φ ; k , . . . , k n ) ∈ D ( K V )( m , n ), where φ ∈ F ( m , n )and k j ∈ K V ( s j ). For 1 ≤ i ≤ m , define the i th coordinate H i of H as follows. If φ ( i ) = j , 1 ≤ j ≤ n , and i is the r th element of φ − ( j ) with its natural ordering asa subset of m , then H i ( t, v , . . . , v n , f ) = γ r ( t, v j , k j ) , where γ r is the r th coordinate of γ : I × S V × K V ( s j ) −→ ( S V ) s j . If φ ( i ) = 0, then H i is the trivial map.It requires some combinatorial inspection to check that these maps do indeedspecify a G T -functor I + ∧ ( S V ) • : D opV −→ T G , but we leave that to the reader.Prolonging these functors using Lemma 4.12, we obtain the G T -functors I + ∧ ( S V ) • : D opG,V −→ T G , ( n , α ) I + ∧ ( S V ) ( n ,α ) , needed to define the two-sided bar constructions B ( I + ∧ ( S V ) • , D G,V , Y ). Just asin § V B ( I + ∧ ( S V ) • , D G,V , Y ) gives an I G - G -space, and wecan construct structure maps that make it into an orthogonal G -spectrum.We denote by ( S V ) • the restrictions of the functors I + ∧ ( S V ) • to t = 1. Note thatthe ( S V ) • are just the functors ( S V ) • used to define B (( S V ) • , D G,V , Y ). Similarly,we denote by ( S V ) • the restrictions of the functors I + ∧ ( S V ) • to t = 0. For anybased G -space A , let i and i denote the inclusions of the top and bottom copy of A into the cylinder I + ∧ A , where G acts trivially on the interval I . Note that i and i are G -homotopy equivalences.The functors ( S V ) • and ( S V ) • from D opG,V to T G are restrictions of I + ∧ ( S V ) • ,hence they commute with the face d , which is given by the evaluation maps ofthese functors. It is clear that the maps i and i commute with all other faces anddegeneracies. Since they are levelwise G -equivalences of Reedy cofibrant simplicial G -spaces, their realizations(7.8) B (( S V ) • , D G,V , Y ) i −→ B ( I + ∧ ( S V ) • , D G,V , Y ) i ←− B (( S V ) • , D G,V , Y ) , are G -equivalences. Therefore the maps i and i in (7.2) are level equivalences oforthogonal G -spectra.7.5. Construction and analysis of the map ι . To define the map of orthogonal G -spectra labeled ι in (7.2), we must look more closely at ( S V ) • . Note that H sendsan element indexed on φ : m −→ n to an element of the product over 1 ≤ j ≤ n of the wedge sums s j ( S V ), where | φ − ( j ) | = s j . If we restrict the domain of H to n ( S V ) ∧ D ( K V )( m , n ) ⊂ ( S V ) n ∧ D ( K V )( m , n ), we land in m ( S V ) since for anyelement ( v , . . . , v n , f ) in the domain of H such that all but one of the factors v j is ∗ , only those i such that φ ( i ) = j can contribute a non-basepoint image. Therefore˜ H restricts to a G T -functor • ( S V ) : ( D V ) op −→ T G that on objects sends n to n ( S V ) and on morphism spaces is the adjoint of the restriction of H from productsto wedges. On wedges, we have the composite n ( S V ) ∧ D V ( m , n ) −→ n ( S V ) ∧ D ( K V )( m , n ) −→ m ( S V )of projection and the evident map obtained by unravelling the definition of H .Upon applying prolongation P , we obtain a G T -functor • ( S V ) : ( D G,V ) op −→ T G . It is defined on objects by sending ( n , α ) to ( n ,α ) ( S V ), and it is a subfunctor of( S V ) • : ( D G,V ) op −→ T G . Note that the map i : ( S V ) n ∧ S W −→ ( S V ⊕ W ) n of (7.5)restricts to the canonical identification of n ( S V ) ∧ S W with n ( S V ⊕ W ), and this worksjust as well when the twisted action of α is taken into account. Just as in § G -spectrum B ( • ( S ⋆ ) , D G,⋆ , Y ). The inclusions of wedges into products give the inclusions of barconstructions that together specify the map of G -spectra labeled ι in (7.2). It isworth pausing to say what is going on philosophically before showing that ι is astable equivalence of orthogonal G -spectra. The contravariant functor ( S V ) • from D G,V to T G is purely categorical since it factors through F G and applies just aswell to give a functor A • for any A . The action of F G on ( S V ) • does not restrictto an action on the system of subspaces • ( S V ). Use of the Steiner operad in effectgives a new and more geometric functor ( S V ) • . It is again defined on products, but QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 61 it depends on the geometry encoded in the Steiner operads and it does restrict toa functor defined on • ( S V ). That is, we have commutative diagrams ( n ,β ) ( S V ) ∧ D G,V (( m , α ) , ( n , β )) ι ∧ id (cid:15) (cid:15) / / ( m ,α ) ( S V ) ι (cid:15) (cid:15) ( S V ) ( n ,β ) ∧ D G,V (( m , α ) , ( n , β )) / / ( S V ) ( m ,α ) where the horizontal arrows are evaluation maps of the functors • ( S V ), and ( S V ) • ,respectively, and the vertical arrows are given by inclusions of wedges in products.The diagram displays the essential part of the map d in the simplicial bar con-structions that ι compares; ι is the identity on all factors in D G,V or Y of the G -spaces of q -simplices in these bar constructions. From here, it is routine to checkthat these maps of bar constructions specify a map of orthogonal G -spectra.We claim that ι is a stable equivalence, and we spend the rest of this subsectionproving this. Note that both the source and target of ι are orthogonal G -spectrawhich at level V are geometric realizations of simplicial G -spaces. Before passageto geometric realization, we have functors ∆ op × I G −→ T G . It is not hard tosee that the simplicial structure commutes with the spectrum structure maps, sothat we can view these as simplicial orthogonal G -spectra. Geometric realizationof simplicial objects can be done in any bicomplete category that is tensored overspaces, using the usual coend definition. See for example [11, Definition X.1.1] for adiscussion in the case of EKMM spectra. In the case of orthogonal G -spectra, sincecolimits and tensoring with spaces is done levelwise, we see that for a simplicialorthogonal G -spectrum T q , the V -th level of the geometric realization is given by | T q | ( V ) = | T q ( V ) | . Therefore, the source and target of ι can be viewed equivalently as geometric real-izations in orthogonal G -spectra of simplicial orthogonal G -spectra. In light of thiswe can instead think of ι as a map of simplicial orthogonal G -spectra.First, we claim that ι gives a stable equivalence of the q th orthogonal G -spectrumfor each q . This means that ι induces isomorphismscolim V π H ∗ (cid:0) Ω V B q ( • ( S V ) , D G,V , Y ) (cid:1) −→ colim V π H ∗ (Ω V B q (( S V ) • , D G,V , Y )) . To prove this, recall that it is standard that finite wedges are finite products inthe stable category [1, Proposition III.3.11]; the same proof works equivariantly.The maps j : D G,V −→ D G,V ⊕ W of Steiner operads also induce isomorphisms onhomotopy groups in increasing dimensions. The claim follows by inspection ofcolimits.Next, we claim that the geometric realization of the map of simplicial orthogonal G -spectra ι is also a stable equivalence. This follows from Proposition 7.9 below,but that requires a notion of Reedy cofibrancy for simplicial orthogonal G -spectrawhich we now explain.There is a definition of latching objects L n T for simplicial objects T in anycocomplete category as a certain colimit (see, for example, [20, Definition 15.2.5],[47]), of which Definition 1.8 is a specialization. Again, since colimits of orthogonalspectra are computed levelwise, we have that ( L n T )( V ) = L n ( T ( V )) for a simplicialorthogonal G -spectrum T . A map of orthogonal G -spectra A −→ X is an h -cofibration if it satisfies thehomotopy extension property (see [19, § A.5], [24, § I.4], [25, §
5] for more details).We say that a simplicial orthogonal G -spectrum is Reedy h -cofibrant if for all n ≥ L n T −→ T n is an h -cofibration of orthogonal G -spectra. We claimthat the simplicial orthogonal G -spectra that are the source and target of ι areReedy h -cofibrant. Since for each V , the simplicial G -spaces are bar constructions,they are Reedy cofibrant, so it remains to show that the homotopy extensionscan be made compatibly with the orthogonal G -spectrum structure. To see this,note that the latching maps are constructed from the cofibration ∗ −→ C V (1) = C G (1) × K V (1) that includes the identity element of the operad. That map isa cofibration because of the assumption of Remark 4.16 that ∗ −→ C G (1) is acofibration and the fact that { id } ֒ → K V (1) is the inclusion of a deformation retract.The explicit retraction from K V (1) onto { id } is easily seen to be compatible withthe I G - G -space structure and with the inclusions j : K V (1) −→ K V ⊕ W (1), so it iscompatible with the structure maps. This in turn implies the compatibility of theretracts for the latching maps of the bar constructions as V varies.To finish the proof, we apply the following result to the map ι in (7.2). Proposition 7.9.
Let f q : T q −→ T ′ q be a map of simplicial Reedy h -cofibrant orthog-onal G -spectra that is a stable equivalence at each simplicial level. Then the map oforthogonal G -spectra | f q | obtained by geometric realization is a stable equivalence.Proof. The proof is the same as the space level analogue, using the constructionof the filtration on geometric realization via pushouts (see [45, Theorem 4.15] and[11, Theorem X.2.4]). The key facts we need about h -cofibrations of orthogonal G -spectra are that they are stable under cobase change [19, Proposition A.62], thatthey satisfy the analogue of [11, Lemma X.2.3] (which is proven in exactly thesame way), that the gluing lemma for h -cofibrations and stable equivalences holds(see [19, Corollary B.21], [24, Theorem I.4.10 (iv)]), and that the filtered colimitalong h -cofibrations of a sequence of stable equivalences is a stable equivalence [19,Proposition B.17]. (cid:3) We note that the above result holds generally in any good model category ten-sored over spaces. A recent treatment is offered in [47, see Corollary 10.6.]. Wehave not quoted that result since there is no published proof that there is a modelstructure on the category of orthogonal G -spectra in which the cofibrations are the h -cofibrations. We believe that the methods of [3] (and [2, § Construction of the map ω . To construct the map ω in (7.2) and thus tocomplete the proof of Theorem 7.1, we must define maps B ( • ( S V ) , D G,V , Y ) −→ B (Σ V L G , D G,V , Y ) . Both source and target are realizations of simplicial (based) G -spaces, and we define ω as the realization of a map of simplicial G -spaces. On the spaces of 0-simpliceswe define ω : B ( • ( S V ) , D G,V , Y ) = _ ( n ,α )( n ,α ) S V ∧ Y ( n , α ) + −→ Σ V Y = B (Σ V L G , D G,V , Y ) to be the wedge sum of the composites of the quotient maps S V ∧ Y ( n , α ) + −→ S V ∧ Y ( n , α ) QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 63 with the composites ( n ,α ) S V ∧ Y ( n , α ) id ∧ δ / / ( n ,α ) S V ∧ Y ( n ,α )1 ν / / S V ∧ Y τ / / Y ∧ S V = Σ V Y . Here, for based spaces A and B , define ν : n A ∧ B n −→ A ∧ B by ν (( i, a ) , ( b , . . . , b n )) = ( a, b i )where ( i, a ) denotes the element a ∈ A of the i th wedge summand of n A and b j ∈ B ,1 ≤ j ≤ n . We check explicitly that ν is G -equivariant: g · (( i, a ) , ( b , . . . , b n )) = (cid:0) ( α ( g )( i ) , g · a ) , ( g · b α ( g ) − (1) , . . . , g · b α ( g ) − ( n ) ) (cid:1) ( g · a, g · b i )since the α ( g )( i ) position is α ( g ) − α ( g )( i ) = i . The map τ : A ∧ B −→ B ∧ A is theusual twist. Then all of the maps in the definition of ω are equivaraint. Notation 7.10.
For q >
0, we may write the space of q -simplices of B ( • ( S V ) , D G,V , Y )as the wedge over pairs ( m , α ) , ( n , β ) of the spaces ( n ,β ) S V ∧ (cid:0) D G,V (( m , α ) , ( n , β )) × Z ( m , α ) (cid:1) + , where Z ( m , α ) is the wedge over sequences (( m , α ) , . . . , ( m q − , α q − )) of thespaces D G,V (cid:0) ( m q − , α q − ) , ( m , α ) (cid:1) × · · · × D G,V (cid:0) ( m , α ) , ( m , α ) (cid:1) × Y ( m , α ) . We write ¯ Z ( m , α ) for the quotient of Z ( m , α ) obtained by replacing × by ∧ here.Recall the definition of ( D G,V Y ) from (6.6). The G -space ( D G,V Y )( n , β ) is aquotient of the wedge over all ( m , α ) of the G -spaces D G,V (( m , α ) , ( n , β )) ∧ Y ( m , α ) . Therefore the space Σ V L G D qG,V Y of q -simplices of B (Σ V L G , D G,V , Y ) is a quo-tient of the wedge over all ( m , α ) of the spaces( D G,V (( m , α ) , ) ∧ ¯ Z ( m , α )) ∧ S V . Define ω q by passage to wedges over ( m , α ) and to quotients from the composites ( n ,β ) S V ∧ (cid:0) D G,V (( m , α ) , ( n , β )) × Z ( m , α ) (cid:1) + (cid:15) (cid:15) ( n ,β ) S V ∧ D G,V (( m , α ) , ( n , β )) ∧ ¯ Z ( m , α ) id ∧ δ ∧ id (cid:15) (cid:15) ( n ,β ) S V ∧ D G,V (( m , α ) , ) ( n ,β ) ∧ ¯ Z ( m , α ) ν ∧ id (cid:15) (cid:15) S V ∧ D G,V (( m , α ) , ) ∧ ¯ Z ( m , α ) τ (cid:15) (cid:15) D G,V (( m , α ) , ) ∧ ¯ Z ( m , α ) ∧ S V , where the first map is the evident quotient map. Since we know that these maps are G -equivariant, to check commutative di-agrams we may drop the α ’s and β ’s from the notation and only consider theunderlying nonequivariant spaces. On underlying spaces, the composite above is n S V ∧ (cid:0) D V ( m , n ) × Z m (cid:1) + (cid:15) (cid:15) n S V ∧ D V ( m , n ) ∧ ¯ Z m id ∧ δ ∧ id (cid:15) (cid:15) n S V ∧ D V ( m , ) n ∧ ¯ Z mν ∧ id (cid:15) (cid:15) S V ∧ D V ( m , ) ∧ ¯ Z mτ (cid:15) (cid:15) D V ( m , ) ∧ ¯ Z m ∧ S V . We must show that these maps ω q specify a map of simplicial spaces. Againrecall Remark 7.4. In both the categorical and monadic bar constructions, the facemaps d i for i > D V and the action of D V on Y .Since the ω q are defined using the quotient maps Z m −→ ¯ Z m , commutation withthese face maps is evident. Similarly, commutation with the degeneracy maps s i for i > s , d , and d . An essentialpoint is that the Segal maps are in Π G , and we are taking the categorical tensorproduct over Π G in the target. First consider s on zero simplices. For ( i, v ) in n S V , that is, v in the i th summand, and y ∈ Y n , ωs (( i, v ) , y ) = ω (( i, v ) , id n , y )= ( δ i , y ) ∧ v = (id , δ i ( y )) ∧ v = s ω (( i, v ) , y ) . Here id n ∈ D V ( n , n ), the third equation uses δ i = id ◦ δ i and the equivalencerelation defining D V Y , and the last equation uses that ω (( i, v ) , y ) = δ i ( y ) ∧ v . Thecommutation of ω and s on q -simplices for q > ω with d and d on 1-simplices, and the argument for q -simplices for q > L G Y = Y . Again, we only writethe maps of underlying spaces, dropping from the notation the indices that indicate G -actions since the G -action is not relevant to checking that the diagrams commute.The top left corners of the following two diagrams are canonically isomorphic, butthey are written differently to clarify the top horizontal arrows. QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 65 n ( S V ) ∧ D V ( m , n ) + ∧ ( Y m ) + (cid:15) (cid:15) H ∧ id / / m ( S V ) ∧ ( Y m ) + (cid:15) (cid:15) n ( S V ) ∧ D V ( m , n ) ∧ Y m id ∧ δ ∧ id (cid:15) (cid:15) H ∧ id / / m ( S V ) ∧ Y m id ∧ δ (cid:15) (cid:15) n ( S V ) ∧ D V ( m , ) n ∧ Y mν ∧ id (cid:15) (cid:15) m ( S V ) ∧ ( Y ) mν (cid:15) (cid:15) S V ∧ D V ( m , ) ∧ Y m H ∧ id ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ (cid:15) (cid:15) S V ∧ Y (cid:15) (cid:15) Σ V LD V Y ∼ = / / Σ V C V L Y ˆ α / / Σ V Y n ( S V ) ∧ (cid:0) D V ( m , n ) × Y m (cid:1) + (cid:15) (cid:15) id ∧ µ / / n ( S V ) ∧ ( Y n ) + (cid:15) (cid:15) n ( S V ) ∧ D V ( m , n ) ∧ Y m id ∧ δ ∧ id (cid:15) (cid:15) id ∧ µ / / n ( S V ) ∧ Y n id ∧ δ (cid:15) (cid:15) n ( S V ) ∧ D V ( m , ) n ∧ Y mν ∧ id (cid:15) (cid:15) n ( S V ) ∧ ( Y ) nν (cid:15) (cid:15) S V ∧ D V ( m , ) ∧ Y m id ∧ µ / / (cid:15) (cid:15) S V ∧ Y (cid:15) (cid:15) Σ V LD V Y Σ V L µ / / Σ V Y Here µ denotes the action of D V on Y , which is given by the adjoints of the G -maps Y : D V ( m , n ) −→ T G ( Y m , Y n ). Both top pieces of the diagrams commuteby formal inspection, the first lower rectangle commutes by the definitions of H and ˆ α , as recalled in the previous section, and the second lower rectangle commutesby definition. It is not hard to check that the maps ω are maps of I G - G -spacesand that they are compatible with the structure maps, so that they give a map oforthogonal G -spectra.8. Proofs of technical results about the operadic machine
We prove Theorem 6.13(i) and (iii) in this section. Thus let C G be a Σ-free G -operad and let D be the monad on Π- G -spaces associated to the category ofoperators D = D ( C G ). Part (i) asserts that D preserves F • -equivalences, and itsproof is the hardest equivariant work we face. It involves a detailed combinatoricalanalysis of the structure of the monad D . The structure of D X . We first discuss the structure of D X for a Π- G -space X . This entails combinatorial analysis of Π and F that will also be relevant to thetechnical proofs for the Segal machine in § n . Then the definition of ( D X ) n given in (6.5) implies that it is the quotient (cid:18) _ q D ( q , n ) ∧ X q (cid:19) / ( ∼ )where ∼ is the equivalence relation specified by( ψ ∗ d ; x ) ∼ ( d ; ψ ∗ x )for d ∈ D ( q , n ), ψ ∈ Π( p , q ) and x ∈ X p . By Lemma 1.16, we can replace wedgesand smash products by disjoint unions and products, that is, ( D X ) n is the quotient (cid:18) a q D ( q , n ) × X q (cid:19) / ( ∼ ) . Recall that the morphism space D ( q , n ) is given by the disjoint union of com-ponents indexed on all φ : q −→ n in FD ( q , n ) = a φ ∈ F ( q , n ) Y ≤ j ≤ n C G ( j φ ) , where j φ = | φ − ( j ) | . The basepoint is the component indexed on φ = 0 q,n . Wewrite a non-basepoint morphism as ( φ ; c ), where c = ( c , . . . , c n ) with c j ∈ C G ( j φ ).For a morphism ψ : p −→ q in Π, write ψ ∗ : Y ≤ j ≤ n C G ( j φ ) −→ Y ≤ j ≤ n C G ( j φψ )for the map D ( q , n ) −→ D ( p , n ) induced by ψ from the component of φ to thecomponent of φ ◦ ψ , and write ψ ∗ : X p −→ X q for the induced morphism giving thecovariant functoriality of X . Then the equivalence relation ∼ takes the form(8.1) ( φ ◦ ψ ; ψ ∗ c ; x ) ∼ ( φ ; c ; ψ ∗ x )for ( φ ; c ) ∈ D ( q , n ) and x ∈ X p . We shall use the identifications induced by ∼ tocut down on the number of components that need be considered. To this end, wefirst describe the structure of Π and F , partially following [38, § Definition 8.2.
Recall that Π is the subcategory of F with the same objects andthose maps φ : m −→ n such that | φ − ( j ) | ≤ ≤ j ≤ n . A map π ∈ Πis a projection if | π − ( j ) | = 1 for 1 ≤ j ≤ n . A map ι ∈ Π is an injection if ι − (0) = { } . The permutations are the maps in Π that are both injections andprojections. A projection or injection is proper if it is not a permutation. Recallthat I is the subcategory of injections in Π. Definition 8.3.
A map φ ∈ F is ordered (or more accurately monotonic) if i < j implies φ ( i ) ≤ φ ( j ); note that this does not restrict the ordering of those i suchthat φ ( i ) = k for some fixed k . A map ε ∈ F is effective if ε − (0) = { } , and aneffective map ε is essential if it is surjective, that is, if j ε ≥ ≤ j ≤ n .Observe that every morphism of Π is a composite of proper projections, properinjections, and permutations, and that F is generated under wedge sum and com-position by Π and the single product morphism φ : −→ . QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 67
Lemma 8.4.
A map φ : m −→ n in F factors as the composite ι ◦ ε ◦ π of aprojection π , an essential map ε and an injection ι , uniquely up to permutation.That is, given two such decompositions of φ , there are permutations σ and τ makingthe following diagram commute. q ε / / σ (cid:15) (cid:15) r ι (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ τ (cid:15) (cid:15) m π > > ⑥⑥⑥⑥⑥⑥⑥⑥ π ′ ❆❆❆❆❆❆❆❆ nq ε ′ / / r ι ′ ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ Proof.
The projection π is determined up to order by which i ≥ m are mappedto 0 in n . The injection ι is determined up to order by which j ≥ n are not in theimage of φ . Up to order, ε is the wedge sum in F of the product maps φ s j : s j −→ ,where s j = | φ − ( j ) | for those j such that 1 ≤ j ≤ n and φ − ( j ) is nonempty. Upto permutation, these s j run through the numbers | ε − ( j ) | , 1 ≤ j ≤ r . (cid:3) Remark 8.5.
We remark that Π and F are dualizable Reedy categories, as definedby Berger and Moerdijk [4, Definition 1.1 and Example 1.9(b)]. They write F + forthe monomorphisms and F − for the epimorphisms in F . We have factored epi-morphisms into composites of projections and essential maps to make the structureclearer. We say that an F - G -space is Reedy cofibrant if its underlying Π- G -spaceis so (see Definition 6.11). In discarded drafts, we proved that all bar construction F - G -spaces used in the Segal machine are Reedy cofibrant.A map φ : q −→ n in F is ineffective if and only if it factors as a composite q π / / p ζ / / n , where p = q − φ , π is the proper ordered projection such that π ( k ) = 0 if andonly if φ ( k ) = 0, and ζ is an effective morphism. Then j ζ = j φ for j ≥ π ∗ ( c ) = c for any c ∈ Q j C G ( j ζ ). Therefore( φ ; c ; x ) = ( ζπ ; π ∗ ( c ); x ) ∼ ( ζ ; c ; π ∗ ( x )) . This says both that we may restrict to those wedge summands that are indexed onthe effective morphisms of F , ignoring the ineffective ones, and that we can ignorethe proper projections in Π, restricting further analysis to ∼ applied to morphismsof I ⊂
Π. Here we must start paying attention to permutations.
Lemma 8.6. If ε : p −→ n is an effective morphism in F , there is a permutation ν ∈ Σ p such that ε ◦ ν is ordered; ν is not unique, but the ordered morphism ε ◦ ν is. Applying ∼ to the permutations ν , we can further restrict to components indexedon ordered effective morphisms. We abbreviate notation. Notation 8.7.
We say that an ordered effective morphism in F is an OE -function.If ε is effective and ε ◦ ν is ordered, we call it the OE -function associated to ε . Welet E ( p , n ) denote the set of all OE -functions p −→ n . Definition 8.8.
Let ε : p → n be an OE -function. Note that the sum of the j ε is p and define Σ( ε ) = Σ ε × · · · × Σ n ε ⊂ Σ p , where the inclusion is determined by identifying p with ε ∨ · · · ∨ n ε . In otherwords, we partition { , . . . , p } into n blocks of letters, as dictated by ε . Lemma 8.9. If ε : p → n is an OE -function and ν ∈ Σ p , then ε ◦ ν is ordered(and hence equal to ε ) if and only ν is in the subgroup Σ( ε ) . The equivalence relation ∼ is defined in terms of precomposition of morphismsof F with morphisms of Π, while the action of Π on D X is defined in terms ofpostcomposition of morphisms of F with morphisms of Π. Especially for permuta-tions, these are related. We discuss composition with permutations on both sidesin the following three remarks. Remark 8.10.
Let ε : p −→ n be an OE -function and let σ ∈ Σ n . Define τ ( σ ) ∈ Σ p to be the permutation that permutes the n blocks of letters { ε , . . . , n ε } as σ permutes n letters. Then σ ◦ ε ◦ τ ( σ ) − is again ordered, and it is the OE -function associated to σ ◦ ε . Moreover, the function τ = τ ε : Σ n −→ Σ p is ahomomorphism. Inspecting our identifications and using Observation 5.2, we seethat postcomposition with σ sends a point with representative ( ε ; c , . . . , c n ; x ) tothe point with representative(8.11) (cid:0) σετ ( σ ) − ; c σ − (1) , . . . , c σ − ( n ) ; τ ( σ ) ∗ x (cid:1) . Remark 8.12.
One can think of an OE -function ε : p −→ n as an ordered partitionof the set { , . . . , p } into n ordered subsets. Observe that ε is essential if and only if j ε > ≤ j ≤ n , so that our n subsets are all nonempty. Define the signatureof ε to be the unordered set of numbers { ε , . . . , n ε } . The action of Σ n permutespartitions with the same signature (and thus the same p ). The OE -functions ε and σ ◦ ε ◦ τ ( σ ) − have the same signature, and any two ordered partitions with thesame signature are connected this way. Remark 8.13. If ε : p −→ n is an OE -function and ρ ∈ Σ p , then( ε ◦ ρ ; c ; x ) ∼ ( ε ; ( ρ − ) ∗ c ; ρ − ∗ x ) . We have now accounted for ∼ applied to all proper projections and to all permu-tations. It remains to consider proper injections ι . For any such ι : p −→ q , thereis a permutation ν ∈ Σ p such that ι ◦ ν is ordered. Recall that we have the orderedinjections σ i : p − −→ p that skip i , 1 ≤ i ≤ p . Every proper ordered injectionis a composite of such σ i , so it remains to account for ∼ applied to the σ i . Thesegive an equivalence relation on a q a ε ∈ E ( q , n ) (cid:18) Y ≤ j ≤ n C G ( j ε ) (cid:19) × Σ( ε ) X q whose quotient is ( D X ) n . The component with q = 0 gives the basepoint.The monad D , like the monad C G , is filtered. Its p th filtration at level n , denoted F p ( D X ) n , is the image of the components indexed on q ≤ p . We can think of thequotient as given by filtration-lowering “basepoint identifications”, namely(8.14) ( ε ; c , . . . , c n ; ( σ i ) ∗ x ) ∼ ( ε ◦ σ i ; c , . . . , c i − , σ r c i , c i +1 , . . . , c n ; x ) , QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 69 for some i = 1 , . . . , q . Here r is the position of i within its block of j ε letters, where j = ε ( i ), and σ r : C G ( j ε ) −→ C G ( j ε −
1) is the map from (6.1). In equivalentabbreviated notation, we write this as(8.15) ( ε ; c ; ( σ i ) ∗ x ) ∼ ( ε ◦ σ i ; σ ∗ i c ; x ) Definition 8.16.
Fixing an ε ∈ E ( p , n ), write [ c ; x ] for an element of Y ≤ j ≤ n C G ( j ε (cid:1) × Σ( ε ) X p , meaning that ( c ; x ) is a representative element for an orbit [ c ; x ] under the actionof Σ( ε ). Recall from Definition 6.11 that x is degenerate if x ∈ L p X , that is, if x = ( σ i ) ∗ y for some y ∈ X p − and some i . Say that ( ε ; [ c ; x ]) is degenerate if x is degenerate. Since τ ◦ σ i is a proper injection for any τ ∈ Σ( ε ) and any i , thecondition of being degenerate is independent of the orbit representative ( c ; x ).By use of (8.15), we reach the following description of the elements of ( D X ) n . Lemma 8.17.
A point of ( D X ) n has a unique nondegenerate representative ( ε ; [ c ; x ]) . Just as nonequivariantly ([38, p. 218]), we have pushouts of ( G × Σ n )-spaces(8.18) ` ε ∈ E ( p , n ) (cid:0) Q ≤ j ≤ n C G ( j ε ) (cid:1) × Σ( ε ) L p X ν / / (cid:15) (cid:15) F p − ( D X ) n (cid:15) (cid:15) ` ε ∈ E ( p , n ) (cid:0) Q ≤ j ≤ n C G ( j ε ) (cid:1) × Σ( ε ) X p / / F p ( D X ) n With notation as in (8.15), the map ν sends a point with orbit representative( ε ; c ; ( σ i ) ∗ x ) to the point with orbit representative ( ε ◦ σ i ; σ ∗ i c ; x ).Recall that X is Reedy cofibrant if the inclusion L p X −→ X p is a ( G × Σ p )-cofibration for each p . When X is Reedy cofibrant, each component of the leftvertical map is a ( G × Σ p )-cofibration before taking the quotient by Σ( ε ), so themap on Σ( ε ) quotients is a G -cofibration by [7, Lemma A.2.3]. Since Σ n acts bypermuting components and the factors of the displayed products, it follows thatthe left vertical map is a ( G × Σ n )-cofibration, hence so is the right vertical arrow.8.2. The proof that D X is Reedy cofibrant. Let T ⊂ n \ { } . We use thenotation Σ T for the subgroup of Σ n of those permutations τ such that τ ( T ) = T .Note that Σ T consists of those permutations that act separately within T and itscomplement. As a group, Σ T it is isomorphic to Σ | T | × Σ n −| T | . Notation 8.19.
We denote by σ T : n − | T | −→ n the ordered injection that missesthe elements of T . It can be written as σ T = σ i k · · · σ i where i < · · · < i k arethe elements of T . If T is empty we use the convention that σ T = id (which makessense as the empty composition). Note that for a Π- G -space X , σ T X n −| T | = \ i ∈ T σ i X n − . For the case T = ∅ this matches the intuition that an empty intersection of subsetsof X n should be X n . For consistency of notation here, we might have written σ ∗ r instead of σ r , as appears in (6.1). Note that any ordered injection is of the form σ T . By Remark 6.12, it sufficesto show that for all subsets T , the maps(8.20) σ T : ( D X ) n −| T | −→ ( D X ) n are ( G × Σ T )-cofibrations. Note that the action of Σ T on ( D X ) n −| T | is given byrestricting to the action on the block n \ T and identifying it with the set n − | T | .Consider the cube obtained by mapping the pushout square of (8.18) for ( D X ) n −| T | to the pushout square (8.18) for ( D X ) n . Write σ T for the maps from the four cor-ners of the first square to the four corners of the second square. We will prove byinduction on p that the map(8.21) σ T : F p ( D X ) n −| T | −→ F p ( D X ) n is a ( G × Σ T )-cofibration, and we assume this for the map with p replaced by p − D X n −| T | are ( G × Σ n −| T | )-cofibrations,so they are ( G × Σ T )-cofibrations via the action described above. The vertical mapsin the diagram (8.18) for D X n are ( G × Σ n )-cofibrations, so in particular they arealso ( G × Σ T )-cofibrations.The map σ T on the left corners of the diagram is given by( ε, [( c , . . . , c n −| T | ) , x )] ( σ T ◦ ε, [( d, x )]) , where d is the n -tuple given by d j = ( ∈ C G (0) if j ∈ Tc σ − T ( j ) if j T It is not hard to see that for the left entries of the pushout diagram, the map σ T is the inclusion of those components labeled by maps ε : p −→ n that miss theelements of T . The groups Σ n −| T | and Σ n act on the source and target, respectively,as stated in Remark 8.10. In particular, both actions shuffle components, hence theinclusion of components is a ( G × Σ T )-cofibration. By the induction hypothesis,the map connecting the top right corners of the cube is also a ( G × Σ T )-cofibration.It follows from Proposition 10.1 that the map (8.21) connecting the bottom rightcorners of the cube is a ( G × Σ T )-cofibration, as claimed, noting that σ T a ε ∈ E ( p , n −| T | ) (cid:0) n −| T | Y j =1 C G ( j ε ) (cid:1) × Σ( ε ) X p ! ∩ a ε ∈ E ( p , n ) (cid:0) n Y j =1 C G ( j ε ) (cid:1) × Σ( ε ) L p X ! is equal to σ T a ε ∈ E ( p , n −| T | ) (cid:0) n −| T | Y j =1 C G ( j ε ) (cid:1) × Σ( ε ) L p X ! . To complete the proof that the map (8.20) is a ( G × Σ T )-cofibration, we useProposition 10.2 to conclude that the map of colimits σ T : ( D X ) n −| T | = colim p F p ( D X ) n −| T | −→ colim p F p ( D X ) n = ( D X ) n is a ( G × Σ T )-cofibration. We must check that the intersection condition σ T ( F p ( D X ) n −| T | ) ∩ F p − ( D X ) n = σ T ( F p − ( D X ) n −| T | )of Proposition 10.2 is satisfied. One inclusion is obvious. For the other, take anelement ( ε, [ c, x ]) ∈ F p ( D X ) n −| T | \ F p − ( D X ) n −| T | ; in particular, x ∈ X p \ L p X .Then σ T ( ε, [ c, x ]) = ( σ T ε, [ σ ∗ T c, x ]) ∈ F p ( D X ) n \ F p − ( D X ) n , as required. QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 71
The proof that D preserves F • -equivalences. We assume given an F • -equivalence f : X −→ Y between Reedy cofibrant Π- G -spaces. Theorem 6.13(i)says that D f : D X −→ D Y is an F • -equivalence. We shall prove it by proving byinduction on p that f induces an F n -equivalence F p ( D X ) n −→ F p ( D Y ) n for each n and each p ≥
0, there being nothing to prove when p = 0. By the usual gluing lemmaon pushouts, proven equivariantly in [6, Theorem A.4.4] (but also a model theoreticformality), it suffices to prove that the maps induced by f on the source and targetof the left vertical arrow in (8.18) induce equivalences on Λ-fixed point spaces,where Λ ⊂ G × Σ n and Λ ∩ Σ n = { e } . We have Λ = { ( h, α ( h )) | h ∈ H } for somesubgroup H of G and homomorphism α : H −→ Σ n , and we regard n as a based H -set via α . Fixing Λ for the rest of the section, we shall prove Theorem 6.13(i)by analyzing Λ-fixed points.We first consider the target, that is the lower left corner of the diagram. Toclarify the argument, we separate out some of its combinatorics before proceeding. Definition 8.22.
Let ε : p −→ n be an OE -function, let τ : Σ n −→ Σ p be thehomomorphism determined by ε as defined in Remark 8.10, and define β : H −→ Σ p to be the composite homomorphism τ α . Say that ε is Λ-fixed if α ( h ) ε = εβ ( h ) forall h ∈ H . Note that this implies that j ε = k ε if j and k are in the same H -orbit.Define E ( p , n ) Λ to be the set of all Λ-fixed OE -functions p −→ n . Fix ε ∈ E ( p , n ) Λ . Say that a function γ = ( γ , . . . , γ n ) : H −→ Σ( ε )is admissible , or admissible with respect to α , if(8.23) γ j ( hk ) = γ j ( h ) γ α ( h ) − ( j ) ( k )for h, k ∈ H and 1 ≤ j ≤ n . For any function γ : H −→ Σ( ε ), define a function γ · β : H −→ Σ p by ( γ · β )( h ) = γ ( h ) β ( h ) . We leave the combinatorial proof of the following lemma to the reader. Whenthe action of H on n \ has a single orbit, there is a conceptual rather thancombinatorial proof using wreath products. Lemma 8.24.
Fix ε ∈ E ( p , n ) Λ . A function γ : H −→ Σ( ε ) is admissible withrespect to α if and only if γ · β is a homomorphism H −→ Σ p . The following result is the central step of the proof of Theorem 6.13(i). Itidentifies the Λ-fixed points of the bottom left corner of the pushout diagram (8.18).
Proposition 8.25.
Let
Λ = { ( h, α ( h )) } and assume that the action of H on n \{ } defined by α is transitive. Then there is a natural homeomorphism (cid:16) ` ε ∈ E ( p , n ) (cid:0)(cid:0) Q ≤ j ≤ n C G ( j ε ) (cid:1) × Σ( ε ) X p (cid:1)(cid:17) Λ ω (cid:15) (cid:15) ` ε ∈ E ( p , n ) Λ (cid:16) ` γ : H −→ Σ( ε ) C G (1 ε ) Λ γ × X Λ γ p (cid:17) / Σ( ε ) . In the target, the second wedge runs over all admissible functions γ = ( γ , . . . , γ n ) : H −→ Σ( ε ); We use the notation · since we often use juxtaposition to mean composition in this section. the groups Λ γ and Λ γ are specified by Λ γ = { ( h, ( γ · β )( h )) | h ∈ H } ⊂ G × Σ p and Λ γ = { ( k, γ ( k )) | k ∈ K } ⊂ G × Σ ε , where K ⊂ H is the isotropy group of under the action of H on n given by α .Proof. Since α ( k )(1) = 1 for k ∈ K , γ is a homomorphism K −→ Σ ε by special-ization of (8.23). In the target, we pass to orbits from the Σ( ε )-action defined onthe term in parentheses by ρ ( γ ; c ; x ) = ( ρ ∗ γ ; cρ − ; ρ ∗ x ) , Here ρ = ( ρ , . . . , ρ n ) is in Σ( ε ), γ is admissible, c ∈ C G (1 ε ) Λ γ , and x ∈ X Λ γ p . The j th coordinate of ρ ∗ γ is defined by(8.26) ( ρ ∗ γ ) j ( h ) = ρ j γ j ( h ) ρ − α ( h ) − ( j ) . A quick check of definitions shows that( ρ ∗ γ ) · β = ρ ( γ · β ) ρ − , which also implies that ρ ∗ γ is admissible since γ is admissible. Similarly, cρ − isfixed by Λ ρ ∗ γ since c is fixed by Λ γ and ρ ∗ ( x ) is fixed by Λ ρ ∗ γ since x is fixed byΛ γ . Thus the action makes sense. Moreover, as we shall need later, this action isfree. If ρ ( γ ; c ; x ) = ( γ ; c ; x ), then cρ − = c and thus ρ = 1 since Σ ε acts freelyon C G (1 ε ). Also, ρ ∗ γ = γ and thus ρ − j γ j ( h ) ρ α ( h ) − ( j ) = γ j ( h ) for all h . Taking j = 1, this implies that ρ α ( h ) − (1) = 1 for all h . Since we are assuming the actionof H induced by α on n \ { } is transitive, this implies that ρ = 1 ∈ Σ( ε ).We turn to the promised homeomorphism. By (8.11), for a point z representedby ( ε ; c , . . . , c n ; x ), c j ∈ C G ( j ε ) and x ∈ X p , ( h, α ( h )) z is represented by (cid:0) α ( h ) εβ ( h ) − ; hc α ( h ) − (1) , . . . , hc α ( h ) − ( n ) ; β ( h ) ∗ ( hx ) (cid:1) where, as before, β = τ α . Assume that z is fixed by Λ. Then we must have α ( h ) εβ ( h ) − = ε , so that α ( h ) ε = εβ ( h ) and thus ε ∈ E ( p , n ) Λ . We must also have( c , . . . , c n ; x ) ∼ (cid:0) hc α ( h ) − (1) , . . . hc α ( h ) − ( n ) ; β ( h ) ∗ ( hx ) (cid:1) , so that for each h ∈ H there exists γ ( h ) = γ ( h ) × · · · × γ n ( h ) ∈ Σ( ε ) such that(8.27) c j γ j ( h ) = hc α ( h ) − ( j ) and x = γ ( h ) ∗ β ( h ) ∗ ( hx ) . Note that for any given n -tuple ( c , . . . , c n ), γ ( h ) is unique since the action of Σ( ε )on Q ≤ j ≤ n C G ( j ε ) is free. For h, k ∈ H and 1 ≤ j ≤ n , c j γ j ( hk ) = ( hk ) c α ( hk ) − ( j ) = h ( kc α ( k ) − ( α ( h ) − ( j )) )= hc α ( h ) − ( j ) γ α ( h ) − ( j ) ( k )= c j γ j ( h ) γ α ( h ) − ( j ) ( k ) . Since the action of Σ j ε on C G ( j ε ) is free, this implies that (8.23) holds, so that γ isadmissible. Note that we have not yet used that the action given by α is transitive.Now the map ω is defined by ω ( ε ; c , . . . , c n ; x ) = ( ε ; γ ; c ; x ) . QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 73
We see from (8.27) that x is in X Λ γ p and that c is in C G (1 ε ) Λ γ , the latter usingthe fact that K is the isotropy group of 1 ∈ n \ { } . We must check that our mapis well-defined. Thus suppose that( ε ; c , . . . , c n ; x ) ∼ ( ε ; d , . . . , d n ; y ) . Then there exists ρ ∈ Σ( ε ) such that c j = d j ρ j and y = ρ ∗ x . Using (8.26) and(8.27), hd α ( h ) − ( j ) = hc α ( h ) − ( j ) ρ − α ( h ) − ( j ) = c j γ j ( h ) ρ − α ( h ) − ( j ) = d j ρ j γ j ( h ) ρ − α ( h ) − ( j ) = d j ( ρ ∗ γ ( h )) j Thus, comparing with (8.27) for ( d , . . . , d n ; y ), and using the freeness of the action,we see that ω ( ε ; d , . . . , d n ; y ) = ( ε ; ρ ∗ γ, d , y ) = ( ε ; ρ ∗ γ, c ρ − , ρ ∗ x ) = ρ ( ε ; γ ; c ; x ) , so that the targets of our equivalent elements are equivalent.Clearly ω is continuous since it is obtained by passage to orbits from a (discon-nected) cover by restriction to subspaces of the projection that forgets the coordi-nates ( c , . . . , c n ).To define ω − , first choose coset representatives for H/K where K is the isotropygroup of 1, that is, choose h j ∈ H such that α ( h j )(1) = j for 1 ≤ j ≤ n , taking h = e . Then define ω − by ω − ( ε ; γ ; c ; x ) = ( ε ; c , . . . , c n ; x )where c j = h j cγ j ( h j ) − . Note that c = c and that the map does not depend onthe choice of coset representatives. Here, ε is Λ-fixed, γ : H −→ Σ( ε ) is admissible, c ∈ C G (1 ε ) Λ γ and x ∈ X Λ γ p . We must show that ω − ( ε ; γ ; c ; x ) is fixed by Λ. First,note that ( h, α ( h )) sends ε to α ( h ) εβ ( h ) − = ε , since ε ∈ E ( p , n ) Λ . Omitting ε from the notation for readability,( h, α ( h ))( c , . . . , c n , x ) = ( hc α ( h ) − (1) , . . . , hc α ( h ) − ( n ) , β ( h ) ∗ ( hx ))= ( hc α ( h ) − (1) , . . . , hc α ( h ) − ( n ) , γ ( h ) − ∗ ( x )) ∼ ( hc α ( h ) − (1) γ ( h ) − , . . . , hc α ( h ) − ( n ) γ n ( h ) − , x ) . We claim that hc α ( h ) − ( j ) γ j ( h ) − = c j . The definition of c α ( h ) − ( j ) gives us thefollowing identification. hc α ( h ) − ( j ) γ j ( h ) − = h (cid:0) h α ( h ) − ( j ) cγ α ( h ) − ( j ) ( h α ( h ) − ( j ) ) − (cid:1) γ j ( h ) − Now note that α ( hh α ( h ) − ( j ) )(1) = α ( h )( α ( h ) − ( j )) = j, thus hh α ( h ) − ( j ) is in the coset represented by h j . So there exists a k ∈ K suchthat hh α ( h ) − ( j ) = h j k . Since γ satisfies equation (8.23), we get the following: γ j ( h ) γ α ( h ) − ( j ) ( h α ( h ) − ( j ) ) = γ j ( hh α ( h ) − ( j ))= γ j ( h j k )= γ j ( h j ) γ α ( h j ) − ( j ) ( k )= γ j ( h j ) γ ( k ) Thus hc α ( h − )( j ) γ j ( h ) − = h j kcγ ( k ) − γ j ( h j ) − = h j cγ j ( h j ) − = c j as claimed. Thus the map really does land in the Λ-fixed points.To show that ω − is well-defined, note that if ( ε ; γ ; c ; x ) ∼ ( ε, ρ ∗ γ ; cρ − , ρ ∗ x )for some ρ ∈ Σ( ε ), we have that ω − sends the latter to ( ε ; d , . . . , d n ; ρ ∗ x ), where d j = h j cρ − ( ρ j γ j ( h j ) ρ − α ( h j ) − ( j ) ) − = h j cρ − ( ρ j γ j ( h j ) ρ − ) − = h j cρ − ρ γ j ( h j ) − ρ − j = c j ρ − j Thus ( d , . . . , d n , ρ ∗ x ) = ρ · ( c , . . . , c n , x ) , so the map is well-defined. This map is clearly continuous.It is easy to see that the map forward and the map backward composed in eitherorder are the identity, hence we get the claimed homeomorphism. (cid:3) The restriction to transitive action by α in the previous result serves only tosimplify the combinatorics. The following remark indicates the changes that areneeded to deal with the general case. Remark 8.28.
When the action of H on n \ { } is not transitive, we argue anal-ogously to Lemma 2.7 and Theorem 5.5 to obtain an anaologous homeomorphism.We break the H -set n \ { } given by α into a disjoint union of orbits H/K a of size n a = | H/K a | , where P a n a = n and K a is the isotropy group of the initial element,denoted 1 a , in its orbit in n \ { } . That breaks n into the wedge of subsets n a and breaks Σ( ε ) into a product of subgroups Σ( ε ( a )) = Q j ∈ H/K a Σ( j ε ). Payingattention to the ordering, the product of the C G ( j ε ) in the source of ω breaks intothe product over a of those C G ( j ε ) such that j is in the a th orbit of n \ { } . Togeneralize the target of ω accordingly, define subgroupsΛ a γ = { ( k, γ a ( k )) | k ∈ K a } ⊂ G × Σ (1 a ) ε and replace C G (1 ε ) Λ γ by the product over a of the C G ((1 a ) ε ) Λ aγ . With thesechanges of source and target and just a bit of extra bookkeeping, it is straightfor-ward to state and prove the general analogue of Proposition 8.25.In the single orbit case, observe that if f : X p −→ Y p is a Λ γ -equivalence, itinduces an equivalence on the target of ω before passage to quotients under theaction of Σ( ε ). Since the Σ( ε ) action is free, the equivalence passes to the quotients.Generalizing to the multi-orbit case, this concludes our proof that we have a Λ-equivalence in the lower left corner of the pushout diagram (8.18).We next consider the upper left corner of (8.18). Precisely the same argument asthat just given, but with X p replaced by L p X , identifies the Λ-fixed subspace of theupper left corner in terms of appropriate fixed point subspaces of L p X . Thereforethe same argument as that just given shows that the following result implies that f induces an equivalence on the upper left corner, as needed to complete the proofof Theorem 6.13(i). Proposition 8.29. If f : X −→ Y is an F • -equivalence of Reedy cofibrant Π - G -spaces, then f : L n X −→ L n Y is an F n -equivalence for each n . QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 75
To prove this, we need more combinatorics to identify ( L n X ) Λ , where Λ ∈ F n .We again take Λ = { ( h, α ( h )) | α : H −→ Σ n } ⊂ G × Σ n and again view n \ { } asan H -set via α . For a subset U of n \ { } with u elements, recall from Notation 8.19that σ U : n − u −→ n denotes the ordered injection that skips the elements in U . Itis a composite of degeneracies σ U = σ i k · · · σ i where i < · · · < i k are the elementsof U . Given 1 ≤ i ≤ n , let π i : n −→ n − be the ordered projection that sends i to 0. Similarly, define π U : n −→ n − u to be the ordered projection that sends theelements of U to 0; explicitly, π U = π i · · · π i k . Note that π U σ U is the identity, and σ U π U ( i ) = ( i if i U i ∈ U. Remark 8.30.
The maps σ i correspond to the degeneracies in ∆ op via the inclusion F : ∆ op −→ F , except there is a shift since we are indexing on the non-zero elementsof n . The maps π i are mostly invisible to ∆. The collection of maps { σ, π } satisfiesthe following subset of the simplicial relations, as can be easily checked. π i π j = π j − π i if i < jσ j σ i = σ i σ j − if i < jπ i σ j = σ j − π i if i < jπ i σ i = id . Now assume that U ⊂ n \{ } is a H -subset of n \{ } and note that its complementis also a H -subset of n \ { } . For h ∈ H , define α U ( h ) = π U α ( h ) σ U : n − u −→ n − u . This is essentially the restriction of α ( h ) to n \ U , but using the ordered inclusion σ U to identify that set with n − u . Note that α U ( e ) = id and that(8.31) α ( h ) σ U = σ U π U α ( h ) σ U = σ U α U ( h )since σ U π U is the identity on n \ U and 0 on U and since α ( h ) σ U is 0 on U andtakes n \ U to itself. This implies that α U is a homomorphism H −→ Σ n − u since α U ( h ) α U ( k ) = π U α ( h ) σ U π U α ( k ) σ U = π U α ( h ) α ( k ) σ U = π U α ( hk ) σ U = α U ( hk ) , where the second equality uses (8.31) with h replaced by k . Thus we can defineΛ U = { ( h, α U ( h )) | h ∈ H } . We have the following identification of ( L n X ) Λ . Henceforward we abbreviate no-tation for the action of F on X , writing σ U for σ U ∗ and so forth. Lemma 8.32. ( L n X ) Λ = [ σ U (cid:18) ( X n − u ) Λ U (cid:19) where the union runs over the H -orbits U ⊂ n \ { } .Proof. For U ⊂ n \ { } and z ∈ ( X n − u ) Λ U , σ U z is a Λ-fixed point since( h, α ( h )) · ( σ U z ) = α ( h )( hσ U z ) = α ( h ) σ U ( hz ) = σ U α U ( h )( hz ) = σ U z, for h ∈ H ; the next to last equality holds by (8.31) and the last holds since z is aΛ U -fixed point. This gives one inclusion. For the other inclusion, we first note that the action of Σ n on L n X can beexpressed as follows. Let ρ ∈ Σ n and x ∈ L n X , so that x = σ i y for some i ,1 ≤ i ≤ n , and some y ∈ X n − . Then ρx = ρσ i y = σ ρ ( i ) ˜ ρy where ˜ ρ = π ρ ( i ) ρσ i is a permutation in Σ n − , as is easily checked. Now supposethat x is a Λ-fixed point and let U be the H -orbit of i in n \{ } . Then x ∈ σ j ( X n − )for all j ∈ U since x = ( h, α ( h )) · x = α ( h )( hx ) = α ( h )( hσ i y ) = α ( h ) σ i ( hy ) = σ α ( h )( i ) g α ( h )( hy )for h ∈ H . It follows that x = σ U z , where z = π U x ∈ X n − u . We claim that z is aΛ U -fixed point. Indeed σ U z = x = α ( h )( hx ) = α ( h ) σ U ( hz ) = σ U α U ( h )( hz ) , and the claim follows since σ U is injective. This gives the other inclusion. (cid:3) Next we consider the intersection of the subspaces corresponding to two suchsubsets U . Lemma 8.33.
Let U and V be disjoint H -subsets of the action of H on n \ { } given by α . Let U have u elements and V have v elements. Then σ U (cid:18) ( X n − u ) Λ U (cid:19) ∩ σ V (cid:18) ( X n − v ) Λ V (cid:19) = σ U ∪ V (cid:18) ( X n − u − v ) Λ U ∪ V (cid:19) = σ U σ ˜ V (cid:18) ( X n − u − v ) (Λ U ) e V (cid:19) , where e V , also with v elements, is the subset of n \ ( U ∪ { } ) that σ U maps onto thesubset V of n \ { } .Proof. The notation obscures the fact that ˜ V depends on U , but we always use itdirectly following the U to which it pertains. Note that the last equality followsfrom the facts that σ U ∪ V = σ U σ ˜ V and Λ U ∪ V = (Λ U ) ˜ V . Suppose that x is in theintersection. Then x = σ i y i for all i ∈ U and also for all i ∈ V , hence x = σ U ∪ V z ,where z = π U ∪ V x ∈ X n −| U ∪ V | . By the same argument as in the proof of theprevious lemma, z is a Λ U ∪ V -fixed point.For the opposite inclusion, let x = σ U ∪ V z , where z ∈ ( X n −| U ∪ V | ) Λ U ∪ V . Then x = σ U y U where y U = σ ˜ V z is a Λ U -fixed point by the same argument as in theprevious proof. Indeed, its first part works for all H -subsets, not just orbits, to givethat x is a Λ-fixed point, and then its second part gives that y U is a Λ U -fixed point.The symmetric argument gives that x = σ V y V where y V is a Λ V -fixed point. (cid:3) Finally, we use these lemmas to prove Proposition 8.29.
Proof of Proposition 8.29.
We first observe that an argument similar to the proofof Lemma 8.32 shows that for all orbits (in fact, all H -subsets) U of n , σ U (cid:0) ( X n − u ) Λ U (cid:1) = (cid:0) σ U ( X n − u ) (cid:1) Λ = σ U ( X n − u ) ∩ X Λ n . Since σ U is a closed inclusion, this shows that σ U (cid:0) ( X n − u ) Λ U (cid:1) is closed in X Λ n .Let U , . . . , U m be the orbits of the H -set n \ { } , with corresponding cardinalities QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 77 u , . . . , u m . For 1 ≤ k ≤ m , we have σ U k ( X Λ Uk n − u k ) ∩ (cid:18) k − [ i =1 σ U i ( X Λ Ui n − u i ) (cid:19) = k − [ i =1 σ U k ( X Λ Uk n − u k ) ∩ σ U i ( X Λ Ui n − u i )= k − [ i =1 σ U k σ ˜ U i ( X (Λ Uk ) ˜ Ui n − u k − u i )= σ U k (cid:0) k − [ i =1 σ ˜ U i ( X (Λ Uk ) ˜ Ui n − u k − u i ) (cid:1) where the next to last equality holds by Lemma 8.33 and the others are standardset manipulations. We therefore have inclusions which give the following pushoutdiagram.(8.34) σ U k (cid:0) S k − i =1 σ ˜ U i ( X (Λ Uk ) ˜ Ui n − u k − u i ) (cid:1) (cid:15) (cid:15) / / S k − i =1 σ U i ( X Λ Ui n − u i ) (cid:15) (cid:15) σ U k ( X Λ Uk n − u k ) / / S ki =1 σ U i ( X Λ Ui n − u i )This diagram is clearly a pushout of sets. It is a pushout of spaces since the lowerhorizontal and the right vertical arrows are closed inclusions by our first observation,so that their target has the topology of the union. By Lemma 8.32, the lower rightcorner is ( L n X ) Λ when k = m .We claim that the left vertical arrow and therefore the right vertical arrow is acofibration for each k ≤ m , the assertion being vacuous if k = 1. We prove thisby induction on n . Thus suppose it holds for all values less than n . In particular,assume that it holds for each n − u k . Note that the orbits of n \ ( U k ∪ { } ) are˜ U , . . . , ˜ U k − , ˜ U k +1 , . . . , ˜ U m . Since σ U k (or any restriction of it to a subspace) is ahomeomorphism onto its image, it suffices to prove that the left vertical map is acofibration before application of σ U k . With n replaced by n − u k and with each ˜ U i referring to U k , the induction hypothesis applied to right vertical arrows gives that k − [ i =1 σ ˜ U i ( X (Λ Uk ) ˜ Ui n − u k − u i ) −→ [ i =1 ,...,k − ,k +1 σ ˜ U i ( X (Λ Uk ) ˜ Ui n − u k − u i )and each map [ i =1 ,...,k − ,k +1 ,...,k + j − σ ˜ U i ( X (Λ Uk ) ˜ Ui n − u k − u i ) −→ [ i =1 ,...,k − ,k +1 ,...,k + j σ ˜ U i ( X (Λ Uk ) ˜ Ui n − u k − u i ) , ≤ j ≤ m , is a cofibration. When j = m , the target of the last map is( L n − u k X ) Λ Uk , and ( L n − u k X ) Λ Uk −→ ( X n − u k ) Λ Uk is a cofibration since X is Reedy cofibrant. Application of σ U k to the composite ofthese cofibrations gives the left vertical arrow, completing the proof of our claim.This allows us to prove by induction on k that we have a weak equivalence k [ i =1 σ U i ( X Λ Ui n − u i ) −→ k [ i =1 σ U i ( Y Λ Ui n − u i ) for any k ≤ m . Since the σ U i are homeomorphisms onto their images, the base case k = 1 holds by our assumption on f . The inductive step is an application of thegluing lemma to the pushout diagram (8.34). For k = m , this gives that( L n X ) Λ −→ ( L n Y ) Λ is a weak equivalence, as required. (cid:3) Proofs of technical results about the Segal machine
We prove Propositions 2.22 and 2.23 in § § § W G - G -spaces satisfy the wedge axiom formulated in Definition 9.9, andwe prove that in § Combinatorial analysis of A • ⊗ F X . Let X be an F - G -space. We firstcompare the functor A • ⊗ F X with geometric realization. Recall that the objectsof ∆ are the ordered finite sets [ n ] = { , , . . . , n } and its morphisms are the non-decreasing functions. As in § F denote the simplicial circle S s = ∆[1] /∂ ∆[1]viewed as a functor ∆ op −→ F . Take the topological circle to be S = I/∂I . Remark 9.1.
The functor F sends the ordered set [ n ] to the based set n . For a map φ : [ n ] −→ [ m ] in ∆ and 1 ≤ j ≤ n , F φ : m −→ n sends i to j if φ ( j − < i ≤ φ ( j )and sends i to 0 if there is no such j . Thus( F φ ) − ( j ) = { i | φ ( j − < i ≤ φ ( j ) } for 1 ≤ j ≤ n. If δ i : [ n − −→ [ n ] and σ i : [ n + 1] −→ [ n ], 0 ≤ i ≤ n , are the standard face anddegeneracy maps that skip or repeat i in the target, then F δ i = d i : n −→ n − is the ordered surjection that repeats i for i < n but sends n to 0 if i = n , and F σ i = s i : n −→ n + is the ordered injection that skips i + 1. Note in particularthat F δ = φ : −→ , which sends 1 and 2 to 1. In F , we also have orderedprojections π i : n −→ n − , used in § π i sends i to 0 and it sends j to j if j < i and to j − j > i .To prove Proposition 2.23, we must compare | X | = X ⊗ ∆ ∆ with X ( A ) := P ( X )( A ) = A • ⊗ F X when A = S . To aid in the comparison, we rewrite | X | as ∆ ⊗ ∆ op X . Here ∆ onthe left is the covariant functor ∆ −→ U that sends [ n ] to the topological simplex∆ n = { ( t , . . . , t n ) | ≤ t ≤ · · · ≤ t n ≤ } . Nowadays it is more usual to use tuples ( s , s , . . . , s n ) such that 0 ≤ s i ≤ P i s i = 1, but the formulae s i = t i +1 − t i and t i = s + · · · + s i − translate betweenthe two descriptions. For 0 ≤ i ≤ n , the face map δ i : ∆ n − −→ ∆ n and thedegeneracy map σ i : ∆ n +1 −→ ∆ n are given by δ i ( t , . . . , t n − ) = (0 , t , . . . , t n − ) if i = 0( t , . . . t i − , t i , t i , t i +1 , . . . t n − ) if 0 < i < n ( t , . . . , t n − ,
1) if i = nσ i ( t , . . . , t n +1 ) = ( t , . . . , t i , t i +2 , . . . , t n +1 ) . In § § σ i +1 as a map in the category I of finite sets and injections. QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 79
Map ∆ n to ( S ) n by sending ( t , . . . , t n ) ∈ ∆ n to ( t , . . . , t n ) ∈ ( S ) n . Lookingat the definition of the functor F , we see that this defines a map ξ : ∆ −→ ( S ) • of cosimplicial spaces, where ( S ) • is a cosimplicial space by pullback along F .Therefore ξ induces a natural map ξ ∗ : | X | = ∆ ⊗ ∆ op X −→ ( S ) • ⊗ F X = X ( S ) . Recall that every point of | X | is represented by a unique point ( u, x ) such that u ∈ ∆ p is an interior point and x ∈ X p is a nondegenerate point [39, Lemma 14.2].Said another way, | X | is filtered with strata F p | X | \ F p − | X | = (∆ p \ ∂ ∆ p ) × ( X p \ L p X ) , where L p X , the p th latching space, is the union of the subspaces s i ( X p − ) (seeDefinition 1.8). The construction of F p | X | from F p − | X | is summarized by theconcatenated pushout diagrams(9.2) ∂ ∆ p × L p X / / (cid:15) (cid:15) ∆ p × L p X (cid:15) (cid:15) ∂ ∆ p × X p / / ∆ p × L p X ∪ ∂ ∆ p × X p (cid:15) (cid:15) / / F p − | X | (cid:15) (cid:15) ∆ p × X p / / F p | X | We shall describe X ( A ) similarly for all A ∈ G W , and we shall specialize to A = S to see that ξ ∗ is a natural homeomorphism, using results about the structureof F recorded in § Remark 9.3.
The functor F is a map of generalized Reedy categories in the senseof [4]. Recall that the latching G -space L p X ⊂ X p of an F - G -space X is definedto be the latching space of its underlying Π- G -space, as defined in Definition 6.11.The F - G -space X also has a latching space when regarded as a simplicial G -spacevia F . Direct comparison of definitions shows that these two latching spaces arethe same.By Lemma 1.16, the G -space X ( A ) is the quotient of ` n ≥ A n × X n obtained byidentifying ( φ ∗ ( a ) , x ) with ( a, φ ∗ ( x )) for all φ : m −→ n in F , a ∈ A n , and x ∈ X m .Here φ ∗ ( a , . . . , a n ) = ( b , . . . , b m ) where b i = a φ ( i ) , with b i = ∗ if φ ( i ) = 0, and φ ∗ ( x ) is given by the covariant functoriality of X . The image of ` n ≤ p A n × X n istopologized as a quotient and denoted F p X ( A ), and X ( A ) is given the topology ofthe union of the F p X ( A ). Notation 9.4.
For an unbased G -space U , the configuration space Conf ( U, p ) isthe G -subspace of X p of points ( u , . . . , u p ) such that u i = u j for i = j . For a based G -space A , the based fat diagonal δA p ⊂ A p is the G -subspace of points ( a , . . . , a p )such that either some a i is the basepoint or a i = a j for some i = j . Observe that A p \ δA p = Conf ( A \ {∗} , p ) . Lemma 9.5. F X ( A ) = ∗ × X . For p ≥ , the stratum Conf ( A \ {∗} , p ) × Σ p ( X p \ L p X ) . Warning: we are thinking of both source and target as cosimplicial unbased spaces.
The construction of F p X ( A ) from F p − X ( A ) is summarized by the concatenatedpushout diagrams δA p × Σ p L p X / / (cid:15) (cid:15) A p × Σ p L p X (cid:15) (cid:15) δA p × Σ p X p / / A p × Σ p L p X ∪ δA p × Σ p X p (cid:15) (cid:15) / / F p − X ( A ) (cid:15) (cid:15) A p × Σ p X p / / F p X ( A ) Proof.
Using projections in F , every point of ` n ≥ A n × X n is equivalent to a point( a, x ) such that either n = 0 or no coordinate of a is the basepoint of A . Usingpermutations and canonical maps φ i : i −→ when i coordinates of a are equal,every point is equivalent to a point ( a, x ) such that a has no repeated coordinates.We must take orbits under the action of Σ p as stated to avoid double counting ofelements. Using injections, every point is equivalent to a point ( a, x ) such that x is nondegenerate. Taking care of the order in which the cited operations are taken,using Lemma 8.4, the conclusion follows. (cid:3) It is now easy to see that ξ : | X | −→ X ( S ) is a homeomorphism. Proof of Proposition 2.23.
As noted in Remark 9.3, the latching subspaces L p X for X as a ∆ op - G -space and as an F - G -space agree. We consider the strata on thefiltrations for | X | and X ( S ) and find that ξ defines homeomorphisms(∆ p \ ∂ ∆ p ) × ( X p \ L p X ) −→ Conf ( S \ {∗} , p ) × Σ p ( X p \ L p X ) . To see this, identify
Conf ( S \{∗} , p ) with Conf ( I \{ , } , p ). Then ξ sends a point(( t , . . . , t p ) , x ) in the domain with 0 < t < · · · < t p < t , . . . , t p ) , x ) in the target. Note that (( t , . . . , t p ) , x ) is the unique representativeof its class such that the coordinates t i are in increasing order. (cid:3) Proof of Proposition 2.22.
Recall that we have the classifying F - G -space B X with q th space | X [ q ] | and that S n X = ( B n X ) . We must prove that ( B n X ) is home-omorphic to X ( S n ), and Proposition 2.23 shows that | X [ q ] | ∼ = X [ q ]( S ). Forany A ∈ G W , we have an F - G -space A ⊗ X with q -th space ( X [ q ])( A ). Thus B X ∼ = S ⊗ X . We claim that S n ⊗ X is isomorphic to B n X ; evaluating at q = 1,this gives the desired homeomorphism. Since S n = S ∧ S n − , the claim is an im-mediate induction from the following result, which is essentially Segal’s [51, Lemma3.7]. (cid:3) Proposition 9.6.
For
A, B ∈ G W , A ⊗ ( B ⊗ X ) and ( A ∧ B ) ⊗ X are naturallyisomorphic.Proof. Recall that X [ q ] has j th space X jq , that is X ( j ∧ q ). Thus X [ q ]( B ) is aquotient of ` j B j × X ( j ∧ q ). We can write it schematically as B • ⊗ F X ( • ∧ q ).Writing ⋆ for another schematic variable, we can write the q -th space of A ⊗ ( B ⊗ X )as A ⋆ ⊗ F ( B • ⊗ F X ( • ∧ ⋆ ∧ q )) . QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 81
It is a quotient of ` i,j A i × B j × X jiq . We define a map A ⊗ ( B ⊗ X ) −→ ( A ∧ B ) ⊗ X by passage to coequalizers from the maps that send(( a , . . . , a i ) , ( b , . . . , b j ) , x ) to (( a r ∧ b s ) , x )where the a r ’s are in A , the b s ’s are in B , and x ∈ X jiq . Here ( a r ∧ b s ) meansthe set of a r ∧ b s in reverse lexicographic order. Indeed, since j ∧ i is orderedlexicographically, we must order the a r ∧ b s to match that. However, r runs throughindices in i and s runs through indices in j , so the reverse lexicographical order isrequired. In the other direction, given a t ∧ b t for 1 ≤ t ≤ k and x ∈ X kq we map(( a ∧ b , . . . , a k ∧ b k ) , x ) to (( a , . . . , a k ) , ( b , . . . , b k ) , ∆ ∗ ( x )) , where ∆ : k ∧ q −→ k ∧ k ∧ q is ∆ ∧ id. Following Segal [51, p. 304], “we shall omitthe verification that the two maps are well-defined and inverse to each other”. It canbe seen in terms of the explicit description of the filtration strata in Lemma 9.5. (cid:3) The proof of the group completion property.
Let X be a special F - G -space, where G is any topological group. Then the Segal maps δ H : X Hn −→ ( X n ) H = ( X H ) n are weak equivalences and X H is a nonequivariant special F -space. We emphasizethat we only need this naive condition: we do not require X to be F • -special.It is convenient but not essential to modify the definition of a special F - G -spaceby requiring the Segal maps δ to be G -homotopy equivalences rather than just weak G -equivalences, and then their fixed point maps δ H are also homotopy equivalences.We can make this assumption without loss of generality since we are free to replace X by Γ X , where Γ is a cofibrant approximation functor on G -spaces. We give X a Hopf G -space structure by choosing a G -homotopy inverse to δ when n = 2 andusing φ . Then X and each X H are homotopy associative and commutative, asin our standing conventions about Hopf G -spaces in § G -spaces, but doing so explicitly only obscures the exposition.We must prove that the canonical map η : X −→ Ω | X | is a group completionin the sense of Definition 1.6. Passage to H -fixed point spaces commutes withrealization, as we see by inspection of elements of | X | H in nondegenerate form | x, u | where x is a nondegenerate n -simplex and u is an interior point of ∆ n forsome n : x must be H -fixed. It also commutes with taking loops since G actstrivially on S . Thus the equivariant case of Proposition 2.12 follows directly fromthe nonequivariant case. We therefore take G = e and ignore equivariance in therest of this section.If M is a topological monoid, we use its product to define a simplicial space B q M with B n M = M n . Then | B q M | is just the classical classifying space BM . When M is commutative, B q M is the simplicial space obtained by pullback of the evidentspecial F -space with n th space M n . When X is a special F -space its first space X plays the role of M . Since X = ∗ , X has a specified unit element e . Spacesof the form X for a special F -space X give the Segal version of an E ∞ -space.It makes sense to ask that a simplicial space X be reduced and special, sincewe can use iterated face maps to define Segal maps X n −→ X n . The Segal mapsof F -spaces are the images under F of these more general Segal maps. Then X is a Hopf space with product induced by a homotopy inverse to the second Segalmap and the map d : X −→ X . When X is an F -space, φ = F d . Spacesof the form X for a reduced special simplicial space X give the Segal version of an A ∞ -space. The group completion theorem for (reduced) special F -spaces is aspecial case of the following more general group completion theorem. Theorem 9.7. If X is a reduced special simplicial space such that X and Ω | X | arehomotopy associative and commutative, then η : X −→ Ω | X | is a group completion. Just as for classical A ∞ -spaces, one can prove that X is equivalent as a Hopfspace to a topological monoid. Then the theorem can be derived from the resultfor topological monoids. However, as we indicate briefly, the result as stated, withhomotopy commutativity weakened to the assumption that left and right multipli-cation by any element are homotopic, was proven but not stated in [40, § X n = G n for atopological monoid G but the proof is given in the generality of Theorem 9.7. Wesummarize the argument, referring to [40] for details.Since π ( | X | ) = π (Ω | X | ) is an abelian group, it is isomorphic to H ( | X | ; Z ).Using the K¨unneth theorem and inspection, we can check that π (Ω | X | ) is thegroup completion of π ( X ) by an easy chain level argument given in [40, Lemma15.2]. We just replace G and BG there with X and Ω | X | here.For the rest, the proof of the homological part of the group completion property isthe same as in [40, § T, S )relating the category of reduced special simplicial spaces, denoted S + T there, tothe category T . The functor T = | − | is geometric realization. The functor S isa reduced version of the total singular complex. For a based space K , S p K is theset of p -simplices ∆ p −→ K that map all vertices to the basepoint. In particular, S K = Ω K . Let φ : T SK −→ K and ψ : X −→ ST X be the counit and unit of theadjunction. Then [40, Proposition 15.5] gives the following result.
Proposition 9.8. If K ∈ T is path connected, then φ : T SK −→ K is a weakequivalence. For any X ∈ S + T , T ψ : T X −→ T ST X is a weak equivalence.
From here, the main tool is the standard homology spectral sequence of thefiltered space
T X = | X | . We take coefficients in a field R . Then, using the K¨unneththeorem and the fact that X is special, we see that E X is the algebraic barconstruction on H ∗ ( X ), so that E p,q X = Tor H ∗ ( X ) p,q ( R, R ). Clearly E , X = R and E ,q X = 0 for q >
0. The spectral sequence converges to H ∗ ( | X | ). We havethe analogous spectral sequence for ST X . The idea is to apply an appropriateversion of the comparison theorem for spectral sequences, [40, Lemma 15.6], to themap of spectral sequences induced by the map of simplicial spaces ψ : X −→ ST X .On 1-simplices, ψ = η : X −→ Ω | X | and therefore E ψ = Tor η ∗ (id , id). The map { E r ψ } of spectral sequences converges to the weak equivalence T X −→ T ST X .Therefore E ∞ ψ is an isomorphism.Write A = H ∗ ( X ) and let ι : A −→ A be its localization at the monoid π ( X ).Write B = H ∗ (Ω | X | ) and let ζ : A −→ B be the map of R -algebras such that ζ ◦ ι = η ∗ ; it is given by the universal property of localization. We must prove that ζ is an isomorphism. It is a classical algebraic result [9, Proposition VI.4.1.1] thatTor ι (id , id) : Tor A ( R, R ) −→ Tor A ( R, R ) QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 83 is an isomorphism in our situation. Therefore we can identify E ψ withTor ζ (id , id) : Tor A ( R, R ) −→ Tor B ( R, R ) . The rest of the argument is given in detail in [40, p. 93]. Both A and B arethe tensor products of their identity components with the group ring R [ π (Ω | X | )].The cited version of the comparison theorem shows how to prove that E , ∗ ψ isan isomorphism and E , ∗ ψ is an epimorphism. For connected graded algebras A ,Tor A , ∗ and Tor A , ∗ compute the generators and relations of A . Now the detailedargument of [40, p. 93] proves by induction on degree that ζ is an isomorphismbetween the identity components of A and B and is therefore an isomorphism.9.3. The positive linearity theorem.
We prove Theorem 3.18 in this section andthe next. In both, let G be any topological group and let X be an F - G -space. Fora based G -CW complex A , write Y ( A ) ambiguously for either B ( A • , F , X ), where X is special, or B ( A • , F G , P X ), where X is F • -special; the latter case is equivalentto B ( A • , F G , Z ), where Z is a special F G - G -space. Under either assumption,write B q ( A ; X ) for the simplicial bar construction whose realization is Y ( A ). Thesenotations remain fixed throughout these two sections. Recall § F and F G , B q ( A ; X ) is Reedy cofibrant in the simplicial sense.To prove that the W G - G -space Y is positive linear, as defined in Definition 3.17,we first isolate properties that together imply positive linearity. Definition 9.9. A W G - G -space Z satisfies the wedge axiom if for all A and B in G W the natural map π : Z ( A ∨ B ) −→ Z ( A ) × Z ( B )induced by the canonical G -maps A ∨ B −→ A and A ∨ B −→ B is a weak G -equivalence. Definition 9.10.
Let Z be a W G - G -space and consider simplicial based G -CWcomplexes A q .(i) Z commutes with geometric realization if the natural G -map | Z ( A q ) | −→ Z ( | A q | )is a homeomorphism.(ii) Z preserves Reedy cofibrancy if the simplicial G -space Z ( A q ) is Reedy cofi-brant when A q is Reedy cofibrant. Definition 9.11. A W G - G -space Z preserves connectivity if Z ( A ) is G -connectedwhen A is G -connected.Our W G - G -space Y satisfies all of the properties above. We record the resultshere, with the proofs delayed to later. Proposition 9.12.
The W G - G -space Y satisfies the wedge axiom. Proposition 9.13.
The W G - G -space Y commutes with realization and preservesReedy cofibrancy. Lemma 9.14.
The W G - G -space Y preserves connectivity. Granting these results, Theorem 3.18 is an application of the following theorem.
Theorem 9.15.
Let Z be a W G - G -space that satisfies the wedge axiom, commuteswith geometric realization, preserves Reedy cofibrancy, and preserves connectivity.Then Z is positive linear. The proof centers around the following construction of cofiber sequences in termsof wedges and geometric realizations; it is a corrected version and equivariant gen-eralization of a construction due to Woolfson [59].
Construction 9.16.
Let f : A −→ B be a map in G W with cofiber i : B −→ Cf ,where A is G -connected. We give an elementary simplicial description of Cf interms of wedges. Let q A denote the wedge of q copies of A , labelling the i th wedgesummand as A i and setting A = ∗ . We define a simplicial G -space W q ( B, A ) whosespace of q -simplices W q ( B, A ) is B ∨ q A . Define face and degeneracy operators d i and s i with domain W q ( B, A ) for 0 ≤ i ≤ q as follows. • All d i and s i map B onto B by the identity map. • d maps A to B via f and maps A j to A j − by the identity map if j > • d i , 0 < i < q , maps A j to A j by the identity map if j < i and maps A j to A j − by the identity map if j > i . • d q maps A j to A j by the identity map if j < q and maps A q to ∗ . • s i maps A j to A j by the identity map if j ≤ i and maps A j to A j +1 by theidentity map if j > i .The simplicial identities are easily checked. Note that there is not much choice:the s i must be inclusions with s i s j = s j +1 s i for i ≤ j and they must satisfy d i s i = id = d i +1 s i . If we specify the s i as stated, then the d i must be as statedexcept in the exceptional cases noted for d and d q . Lemma 9.17.
The realization | W q ( B, A ) | is homeomorphic to Cf . In particular,the realization | W q ( ∗ , A ) | is homeomorphic to Σ A .Proof. Clearly every point of W q ( B, A ) for q ≥ with I , the realization is the quotient of B × {∗} ∐ ( B ∨ A ) × I obtainedby the identifications( b, t ) ∼ ( b, ∗ ) for b ∈ B and t ∈ I since s b = b ( a, ∼ ( f ( a ) , ∗ ) for a ∈ A since d ( a ) = f ( a )( a, ∼ ( ∗ , ∗ ) for a ∈ A since d ( a ) = ∗ It is simple to verify that the result is homeomorphic to Cf = B ∪ f ( A ∧ I ). (cid:3) We record the following result, which is the equivariant generalization of [27,Theorem 12.7]. It is proven by applying that result to H -fixed point simplicialspaces for all closed subgroups H of G . A G -map f is a G -quasifibration if each f H is a quasifibration. Similarly, a map p of simplicial based G -spaces is a simplicialbased Hurewicz G -fibration if each p H is a simplicial based Hurewicz fibration inthe sense of [27, Definition 12.5]). Theorem 9.18.
Let E q and B q be simplicial based G -spaces and let p q : E q −→ B q a simplicial based Hurewicz G -fibration with fiber F q = p − q ( ∗ ) . If each B q is G -connected and B q is Reedy cofibrant, then the realization | p q | : | E q | −→ | B q | is a G -quasifibration with fiber | F q | . They fail with the erroneous specification of faces in [59, Lemma 1.10].
QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 85
Proof of Theorem 9.15.
Let f : A −→ B be a map in G W , where A is G -connected.We must show that application of Z to the cofiber sequence A f / / B i / / Cf gives a fiber sequence.Let p q : W q ( B, A ) −→ W q ( ∗ , A ) be the map of simplicial G -spaces given by send-ing B to the basepoint and let i q : B q −→ W q ( B, A ) be the inclusion, where B q denotes the constant simplicial space at B . On passage to realization, these givethe canonical maps B i / / Cf p / / Σ A .
We apply our given W G - G -space Z to these maps to obtain the sequence Z ( B ) Z ( i ) / / Z ( Cf ) Z ( p ) / / Z (Σ A ) . We claim that it is a fiber sequence. Since Z commutes with realization and since Z ( B ) ∼ = | Z ( B q ) | , this sequence is G -homeomorphic to the sequence | Z ( B q ) | | Z ( i q ) | / / | Z ( W q ( B, A )) | | Z ( p q ) | / / | Z ( W q ( ∗ , A )) | . On q -simplices, before realization, we have the commutative diagram Z ( B ) / / Z ( B ∨ q A ) / / (cid:15) (cid:15) Z ( q A ) (cid:15) (cid:15) Z ( B ) / / Z ( B ) × Z ( A ) q / / Z ( A ) q where the horizontal arrows are given by the evident inclusions and projections. Thevertical arrows are the canonical maps and are G -equivalences by the wedge axiom.The simplicial G -spaces W q ( B, A ) and W q ( ∗ , A ) are trivially Reedy cofibrant, henceso are Z ( W q ( B, A )) and Z ( W q ( ∗ , A )). The nondegeneracy of basepoints implies that Z ( B ) × Z ( A ) • and Z ( A ) • are also Reedy cofibrant. Therefore the vertical arrowsbecome G -equivalences after passage to realization. Since A and therefore Z ( A ) is G -connected, Theorem 9.18 applies to show that the realization of the bottom rowis a fibration sequence up to homotopy. Thus we have the fiber sequence Z ( B ) −→ Z ( Cf ) −→ Z (Σ A )and therefore also the fiber sequenceΩ Z (Σ A ) −→ Z ( B ) −→ Z ( Cf ) . Specializing to the map id : A −→ A , we also have a fiber sequence Z ( A ) −→ Z ( CA ) −→ Z (Σ A ) . Since Z ( CA ) is G -contractible, we therefore have a G -equivalence Z ( A ) −→ Ω Z (Σ A )and therefore the desired fiber sequence Z ( A ) −→ Z ( B ) −→ Z ( Cf ) . (cid:3) Proof of Lemma 9.14.
Let A ∈ G W be G -connected, so that A H is connected forall H . Using that passage to fixed points commutes with pushouts one leg of whichis a closed inclusion, it is easily checked that geometric realization and the barconstruction commute with passage to H -fixed points. Using this and Remark 3.6, we see that Y ( A ) H is the geometric realization of a simplicial space with 0-simplicesgiven by(9.19) _ n ( A H ) n ∧ ( X Hn ) + or(9.20) _ ( n ,α ) ( A ( n ,α ) ) H ∧ (( P X )( n , α ) H ) + , depending on whether Y ( A ) is B ( A • , F , X ) or B ( A • , F G , P X ) for an F - G -space X . We claim that the space of 0-simplices in either case is connected, so that thegeometric realization Y ( A ) H is also connected. In the first case, it is clear thatthe space (9.19) is connected since we assume that A H is connected, and in thesecond case, the space (9.20) is connected because ( A ( n ,α ) ) H is connected. Indeed,note that ( A ( n ,α ) ) H ∼ = ( A n ) Λ where Λ = { ( h, α ( h )) | h ∈ H } ⊂ G × Σ n , thus byLemma 2.7, ( A ( n ,α ) ) H ∼ = Y A K i , where the product is taken over the orbits of the H -set ( n , α | H ) and the K i ⊂ H are the stabilizers of elements in the corresponding orbit. Again, by our assumptionthat the A K i are connected it follows that ( A ( n ,α ) ) H is connected. (cid:3) The following result will apply to show that Y commutes with realization. Lemma 9.21.
Let Z be a W G - G -space such that Z ( A ) is naturally isomorphic to | Z q ( A ) | for some functor Z q from W G to simplicial based G -spaces. If the natural G -map | Z q ( A q ) | −→ Z q ( | A q | ) is a G -homeomorphism for all simplicial based G -spaces A q and all q ≥ , then thenatural map | Z ( A q ) | −→ Z ( | A q | ) is a G -homeomorphism.Proof. For a bisimplicial G -space, realizing first in one direction and then the othergives a space that is G -homeomorphic to the one obtained by realizing in the op-posite order. Let A q be a simplicial G W -space. Then Z ( A p ) = | q Z q ( A p ) | ,so | Z ( A q ) | = | p
7→ | q Z q ( A p ) || . By the assumption on Z q , Z ( | A q | ) = | q Z q ( | p A p | ) | ∼ = | q
7→ | p Z q ( A p ) || . The result follows. (cid:3)
Proof of Proposition 9.13.
We first prove that Y commutes with realization. Let A • be a simplicial G W -space. In view of Lemma 9.21, it suffices to prove that thenatural map | Y q ( A • ) | −→ Y q ( | A • | )is a G -homeomorphism for all q ≥
0. Using the description of Y q in Remark 3.6and the commutation of realization with products and half-smash products, thisfollows from the definition of Y .Now assume that A q is a Reedy cofibrant simplicial G -space. We must provethat Y ( A q ) is Reedy cofibrant. Let Y = B (( − ) • , F , X ); the proof in the case QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 87 Y = B (( − ) • , F G , P X ) is the same. By Lemma 1.9, it suffices to show that all ofthe degeneracy maps S i : B (( A n − ) • , F , X ) B (( s i ) • , id , id) −−−−−−−−−→ B (( A n ) • , F , X ) , are G -cofibrations. Using a shorthand notation, the maps S i are geometric realiza-tions of maps of Reedy cofibrant simplicial G -spaces that are given on q -simplicesby the G -cofibrations( s i ) n q ∧ (id) : _ n ,...,n q ( A n − ) n q ∧ ( − ) −→ _ n ,...,n q ( A n ) n q ∧ ( − ) . By Theorem 1.11, the S i are therefore G -cofibrations. (cid:3) The proof that the wedge axiom holds.
We must prove Proposition 9.12,which says that the W G - G -space Y satisfies the wedge axiom. Proof of Proposition 9.12.
We write the proof for Y ( A ) = B ( A • , F G , P X ), where X is F • -special. The proof for Y ( A ) = B ( A • , F , X ), where X is (naively) special,is essentially the same, but simpler since it is simpler to keep track of equivariancein that case. For convenience of notation, we write a, b, c, d, e, f for based finite G -sets, that is objects of F G . We write X ( a ) for the value of P X on a and A a forthe based G -space T G ( a, A ), with G acting by conjugation.Recall from § B ( A • , F G , P X ) = B × ( A • , F G , P X ) /B × ( ∗ , F G , P X ), where B × is the usual categorical bar construction defined using the cartesian product.Since the map B × ( A • , F G , P X ) −→ B ( A • , F G , P X )of (3.4) is a G -equivalence, it suffices to prove the result for Y ( A ) = B × ( A • , F G , P X )instead. To do so, we shall construct a G -homotopy commutative diagram of G -spaces(9.22) Z ( A, B ) F x x rrrrrrrrrr Q ' ' ◆◆◆◆◆◆◆◆◆◆◆ Y ( A ∨ B ) P / / Y ( A ) × Y ( B )in which F and Q are weak G -equivalences and P is the canonical map induced bythe projections π A : A ∨ B −→ A and π B : A ∨ B −→ B . This will prove that P isa weak G -equivalence.In this section, we abbreviate notation by writing C ( A ; X ) for the category inter-nal to G U whose nerve is B × q ( A, F G , P X ) and whose classifying G -space is there-fore Y ( A ). Recall from § G -spaces of C ( A ; X )are a a A a × X ( a )and a a,c A c × F G ( a, c ) × X ( a ) . Its source and target G -maps S and T are induced from the evaluation maps ofthe contravariant G T -functor A • and the covariant G T -functor X from F G to It is irrelevant here that the new Y is not a W G - G -space since it is not a G T -functor, asdiscussed in § T G . The identity and composition G -maps I and C are induced from identitymorphisms and composition in F G .Analogously, we define Z ( A, B ) to be the classifying G -space of the categoryinternal to G U C ( A, B ; X ) whose nerve is the simplicial bar construction B × q ( A • × B ⋆ ; F G × F G ; ( P X )( • ∨ ⋆ )) . Its object and morphism G -spaces are a a,b ( A a × B b ) × X ( a ∨ b )and a a,b,c,d ( A c × B d ) × F G ( a, c ) × F G ( b, d ) × X ( a ∨ b ) . Here S and T are induced from the evaluation maps of A • × B ⋆ , and the compositeof ∨ : F G × F G −→ F G with the evaluation maps of P X . Again, identity andcomposition maps are induced from identity morphisms and composition in F G .Using categories, functors, and natural transformations to mean these notionsinternal to G U in what follows, we shall define functors giving the following diagramof categories and shall prove that it is commutative up to natural transformation.(9.23) C ( A, B ; X ) F w w ♥♥♥♥♥♥♥♥♥♥♥♥ Q ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ C ( A ∨ B ; X ) P / / C ( A ; X ) × C ( B ; X )Passing to classifying G -spaces, this will give the diagram (9.22).To define F and Q , it is convenient to write elements of A a as based maps µ : a −→ A , with G acting by conjugation on maps. The functor F sends an object( µ, ν, x ) to ( µ ∨ ν, x ) and sends a morphism ( µ, ν, φ, ψ, x ) to ( µ ∨ ν, φ ∨ ψ, x ). Thefunctor Q sends an object ( µ, ν, x ) to ( µ, x a ) × ( ν, x b ) where x a and x b are obtainedfrom x ∈ X ( a ∨ b ) by using the G -maps induced by the projections π a : a ∨ b −→ a and π b : a ∨ b −→ b . It sends a morphism ( µ, ν, φ, ψ, x ) to the morphism ( µ, φ, x a ) × ( ν, ψ, x b ). As in (9.22), P is induced by the projections π A and π B . Noting that π a and π b are G -fixed morphisms of F G and that π A ◦ ( µ ∨ ν ) = µ ◦ π a and π B ◦ ( µ ∨ ν ) = ν ◦ π b , we see that the morphisms( µ, π a , x ) × ( ν, π b , x ) : ( π A ◦ ( µ ∨ ν ) , x ) × ( π B ◦ ( µ ∨ ν ) , x ) −→ ( µ, x a ) × ( ν, x b )give a natural transformation P ◦ F −→ Q in diagram (9.23) that induces a G -homotopy P ◦ F −→ Q in diagram (9.22).While Q need not be an equivalence of categories of any sort, we see from theassumption that P X is special and our use of the projections π a and π b that Q gives a level weak equivalence of simplicial G -spaces on passage to nerves. Reedycofibrancy of the bar constructions then implies that the induced map Q in (9.22)is a weak equivalence of classifying G -spaces. To complete the proof, we shallconstruct a functor F − : C ( A ∨ B ; X ) −→ C ( A, B ; X ) and natural transformationsId −→ F − ◦ F and Id −→ F ◦ F − . Passing to classifying G -spaces, this will implythat F in (9.22) is a G -homotopy equivalence. QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 89
For ω : f −→ A ∨ B , define ω A = π A ◦ ω and ω B = π B ◦ ω . Define σ ω : f −→ f ∨ f ,called the splitting of ω , by σ ω ( j ) = j in the first copy of f if ω ( j ) ∈ A \ ∗ j in the second copy of f if ω ( j ) ∈ B \ ∗∗ if ω ( j ) = ∗ Observe that ω factors as the composite f σ ω / / f ∨ f ω A ∨ ω B / / A ∨ B. Define F − on objects by F − ( ω, x ) = ( ω A , ω B , ( σ ω ) ∗ ( x )) . For a morphism ( ω, φ, x ), ω : f −→ A ∨ B , φ : e −→ f , and x ∈ X ( e ), observe that( ω ◦ φ ) A = ω A ◦ φ and ( ω ◦ φ ) B = ω B ◦ φ. Define F − on morphisms by F − ( ω, φ, x ) = ( ω A , ω B , φ, φ, ( σ ω ◦ φ ) ∗ ( x ) . A check of definitions using the commutative diagram e φ / / σ ω ◦ φ (cid:15) (cid:15) f σ ω (cid:15) (cid:15) ω / / A ∨ Be ∨ e φ ∨ φ / / f ∨ f ω A ∨ ω B : : ✈✈✈✈✈✈✈✈✈✈ shows that S ◦ F − = F − ◦ S and T ◦ F − = F − ◦ T , and F − is clearly compatiblewith composition and identities. It is easily checked that F − is continuous onobject and morphism G -spaces, but equivariance is a little more subtle. We firstclaim that σ g · ω = g · σ ω . Since ( g · ω )( j ) = gω ( g − j ), σ g · ω ( j ) = j in the first copy of f if gω ( g − j ) ∈ A \ ∗ j in the second copy of f if gω ( g − j ) ∈ B \ ∗∗ if gω ( g − j ) = ∗ . On the other hand, using the definition of σ ω and the fact that gg − = 1, g · σ ω ( j ) = gσ ω ( g − j ) = j in the first copy of f if ω ( g − j ) ∈ A \ ∗ j in the second copy of f if ω ( g − j ) ∈ B \ ∗ g ∗ = ∗ if ω ( g − j ) = ∗ . Observing that gz ∈ A \ ∗ if and only if z ∈ A \ ∗ and similarly for B , we see thatthese agree, proving the claim. Since π A is a G -map, we also have g · ω A = g · ( π A ◦ ω ) = gπ A ωg − = π A gωg − = π A ◦ g · ω = ( g · ω ) A , and similarly for B . Putting these together gives F − ( g · ( ω, x )) = F − ( g · ω, gx ) = (cid:0) ( g · ω ) A , ( g · ω ) B , ( σ g · ω ) ∗ ( gx ) (cid:1) = (cid:0) g · ω A , g · ω B , ( g · σ ω ) ∗ ( gx ) (cid:1) . Since the evaluation map F G ( a, b ) ∧ X ( a ) −→ X ( b ) is a G -map, this is equal to (cid:0) g · ω A , g · ω B , g · (( σ ω ) ∗ x ) (cid:1) = g · F − ( ω, x ) . This shows that F − is a G -map on objects. Following the same argument andusing that g · ( ω ◦ φ ) = ( g · ω ) ◦ ( g · φ ), we see that F − is a G -map on morphisms.Now consider the composite F ◦ F − . It sends the object ( ω, x ) in C ( A ∨ B ; X )to the object ( ω A ∨ ω B , ( σ ω ) ∗ x ). Here ( ω A ∨ ω B , σ ω , x ) is a morphism from ( ω, x )to ( ω A ∨ ω B , ( σ ω ) ∗ x ). We claim that this morphism is the component at ( ω, x ) of anatural transformation Id −→ F ◦ F − . These morphisms clearly give a continuousmap from the object G -space of C ( A ∨ B ; X ) to its morphism G -space, and it isnot hard to check naturality using the diagram just above. To show equivariance,if g ∈ G then g · ( ω A ∨ ω B , σ ω , x ) = ( g · ( ω A ∨ ω B ) , g · σ ω , gx ) = (( g · ω ) A ∨ ( g · ω ) B , σ g · ω , gx ) , which is precisely the component at g · ( ω, x ) = ( gω, gx ).The composite F − ◦ F sends an object ( µ, ν, x ) in C ( A, B ; X ) to the object( π A ◦ ( µ ∨ ν ) , π B ◦ ( µ ∨ ν ) , ( σ µ ∨ ν ) ∗ x ) = ( µ ◦ π a , ν ◦ π b , ( σ µ ∨ ν ) ∗ x ) . Note that σ µ ∨ ν is given by ˜ σ µ ∨ ˜ σ ν , where ˜ σ µ : a −→ a ∨ b is the inclusion, exceptthat if µ ( j ) = ∗ , then ˜ σ µ ( j ) = ∗ , and similarly for ˜ σ ν : b −→ a ∨ b . Here (cid:0) π A ◦ ( µ ∨ ν ) , π B ◦ ( µ ∨ ν ) , ˜ σ µ , ˜ σ ν , x (cid:1) is a morphism in C ( A, B ; X ) from ( µ, ν, x ) to (cid:0) π A ◦ ( µ ∨ ν ) , π B ◦ ( µ ∨ ν ) , ( σ µ ∨ ν ) ∗ x (cid:1) .To see that the source of this morphism is as claimed, observe that µ ◦ π a ◦ ˜ σ µ = µ since π a ◦ ˜ σ µ = id except on those j such that µ ( j ) = ∗ , and similarly for ν . Thenaturality follows from the equations˜ σ µ ◦ φ = ( φ ∨ ψ ) ◦ ˜ σ µ ◦ φ and ˜ σ ν ◦ ψ = ( φ ∨ ψ ) ◦ ˜ σ ν ◦ ψ , which are easily checked. The continuity of the assignment is also easily verified.For g ∈ G , a verification similar to that for F ◦ F − shows that g acting on thecomponent of our natural transformation at ( µ, ν, x ) is the component of the trans-formation at g · ( µ, ν, x ) = ( gµ, gν, gx ). (cid:3) Appendix: Levelwise realization of G -cofibrations In this section we prove Theorem 1.11. We will need the following two stan-dard results about G -cofibrations. Recall that we are working in the category ofcompactly generated weak Hausdorff spaces, so all cofibrations are closed inclu-sions. Throughout this section we use the convention of identifying the domain ofa cofibration with its image. Proposition 10.1.
Consider the following diagram in G -spaces. A / / i (cid:15) (cid:15) α ❆❆❆❆❆❆❆ C (cid:15) (cid:15) γ ❆❆❆❆❆❆❆❆ A ′ / / i ′ (cid:15) (cid:15) C ′ (cid:15) (cid:15) B / / β ❆❆❆❆❆❆❆ D ❆❆❆❆❆❆❆ B ′ / / D ′ QUIVARIANT INFINITE LOOP SPACE THEORY, I. THE SPACE LEVEL STORY 91
Assume that the front and back faces are pushouts, α , β , γ , i and i ′ are G -cofibrations, and A ′ ∩ B = A (as subsets of B ′ ). Then the map D −→ D ′ isalso a G -cofibration.Proof. The nonequivariant result is [22, Proposition 2.5] and, as pointed out there,the equivariant proof goes through the same way. (cid:3)
The following result is given in [7, Proposition A.4.9] and [22, Lemma 3.2.(a)].
Proposition 10.2.
Let A −→ A −→ A −→ · · · and B −→ B −→ B −→ · · · be diagrams of G -cofibrations and let f i : A i −→ B i be a map of diagrams such thateach f i is a G -cofibration. Assume moreover that for every i ≥ , A i − = A i ∩ B i − .Then the induced map colim i A i −→ colim i B i is a G -cofibration. The following observation is the key to using these results to prove Theorem 1.11.
Lemma 10.3.
Let f q : X q −→ Y q be a map of simplicial G -spaces that is levelwiseinjective. Then for all n ≥ and all ≤ i ≤ n − , s i ( Y n − ) ∩ f n ( X n ) = f n s i ( X n − ) and L n Y ∩ f n ( X n ) = f n ( L n X ) . Proof.
For the first statement, one of the inclusions is obvious. For the other in-clusion, take y = s i ( y ′ ) = f n ( x ). Then y ′ = d i s i ( y ′ ) = d i f n ( x ) = f n − d i ( x ). Thus, y = s i f n − d i ( x ) = f n s i d i ( x ) ∈ f n s i ( X n − ), as wanted. The second statement isobtained from the first by taking the union over all i . (cid:3) Proposition 10.4.
Let X q and Y q be Reedy cofibrant simplicial G -spaces, and let f q : X q −→ Y q be a level G -cofibration. Then for all n ≥ , L n X −→ L n Y is a G -cofibration.Proof. We proceed by induction on n . Note that when n = 1, L X = s X whichis homeomorphic to X , so there is nothing to prove. For n >
1, let k ≤ n −
1. Justas nonequivariantly, we have a pushout square(10.5) s k ( S k − i =0 s i ( X n − )) (cid:15) (cid:15) / / S k − i =0 s i ( X n − ) (cid:15) (cid:15) s k ( X n − ) / / S ki =0 s i ( X n − ) . We show by induction on n and k that the right vertical map is a G -cofibration.By the inductive hypothesis for n − X q is Reedy cofibrant, wehave that the composite k − [ i =0 s i ( X n − ) −→ k [ i =0 s i ( X n − ) −→ . . . −→ n − [ i =0 s i ( X n − ) = L n − X −→ X n − is a G -cofibration. Since s k is a G -homeomorphism onto its image, the left handmap in the square is a G -cofibration. Therefore, the righthand map in (10.5) is alsoa G -cofibration by cobase change.We apply Proposition 10.1 to the pushout in (10.5) to deduce that the maps k [ i =0 s i ( X n − ) −→ k [ i =0 s i ( Y n − ) are G -cofibrations. In particular, by induction on n and k , L n X −→ L n Y is a G -cofibration. The required intersection condition follows from Lemma 10.3. (cid:3) We are now ready to prove Theorem 1.11.
Proof of Theorem 1.11.
Recall from § | X q | is the colimit of its filtrationpieces F p | X | , and that these can be built by the iterated pushout squares (9.2).Note that all of the vertical maps in diagram (9.2) are G -cofibrations. There is asimilar diagram for | Y q | and a map of diagrams induced by f q . The maps betweenthe three corners of the upper squares are G -cofibrations. By Proposition 10.1 andLemma 10.3, the map between the pushouts of the upper squares is a G -cofibration.We assume by induction that the map F p − | X | −→ F p − | Y | is a G -cofibration,so again we have that all three maps between the corners of the lower pushoutsquare for X and the one for Y are G -cofibrations. Lemma 10.3 implies again thatthe intersection condition necessary to apply Proposition 10.1 holds, and we candeduce by induction that the map F p | X | −→ F p | Y | l is a G -cofibration.Lastly, we check the intersection condition of Proposition 10.2, namely that f ( F p | X | ) ∩ F p − | Y | = f ( F p − | X | ). One inclusion is obvious. To see the otherinclusion, take an element ( u, f p x ) in f ( F p | X | ) \ f ( F p − | X | ) = (∆ p \ ∂ ∆ p ) × f p ( X p \ L p X ). Since f p is injective, we can again use Lemma 10.3 to see that( u, f p x ) ∈ F p Y \ F p − Y . Thus an element in the intersection must be in F p − | X | .By Proposition 10.2, we conclude that the map | f q | : | X q | −→ | Y q | is a G -cofibration. (cid:3) References [1] J. F. Adams.
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