Equivariant nonabelian Poincaré duality and equivariant factorization homology of Thom spectra
Jeremy Hahn, Asaf Horev, Inbar Klang, Dylan Wilson, Foling Zou
EEquivariant nonabelian Poincar´e duality and equivariantfactorization homology of Thom spectra
Asaf Horev, Inbar Klang, and Foling ZouAppendix by Jeremy Hahn and Dylan WilsonJune 25, 2020
Abstract
In this paper, we study genuine equivariant factorization homology and its interactionwith equivariant Thom spectra, which we construct using the language of parametrizedhigher category theory. We describe the genuine equivariant factorization homology of Thomspectra, and use this description to compute several examples of interest. A key ingredientfor our computations is an equivariant nonabelian Poincar´e duality theorem, in which weprove that factorization homology with coefficients in a G -space is given by a mappingspace. We compute the Real topological Hochschild homology (THR) of the Real bordismspectrum MU R and of the equivariant Eilenberg–MacLane spectra H F and H Z (2) , as wellas factorization homology of the sphere S σ with coefficients in these Eilenberg–MacLanespectra. In the appendix, Jeremy Hahn and Dylan Wilson compute THR( H Z ). In this paper, we study the equivariant factorization homology of Thom spectra. Factorizationhomology has emerged as a fruitful topic of research in recent years; its roots lie in the studyof configuration spaces and their relation to mapping spaces, but it has also proven valuablein studying topological field theories, and as a unified way to treat Hochschild homology the-ories. Here we primarily take the axiomatic perspective on factorization homology, introducedby Ayala–Francis [AF15]. Ayala–Francis describe factorization homology with coefficients in an E n -algebra A , (cid:82) − A , as a homology theory for n -manifolds: it satisfies a version of the Eilenberg–Steenrod axioms, including functoriality and excision, and is determined by these axioms.In the case n = 1, (cid:82) S A agrees with Hochschild homology of a ring A . Furthermore, If A is a commutative ring spectrum, the factorization homology (cid:82) M A agrees with the Loday con-struction, which gives higher Hochschild homology [Pir00] for M = S n and iterated Hochschildhomology for M = T n . As such, it is reasonable to expect that factorization homology mightbe of use in understanding invariants related to algebraic K-theory, and recently, Ayala–Mazel-Gee–Rozenblyum used factorization homology to obtain a new description of the cyclotomic tracefrom K-theory to topological cyclic homology (see, e.g. [AMGR17].) In this paper, we considergenuine equivariant factorization homology, introduced by the first named author [Hor19] in histhesis; other definitions of equivariant factorization homology have also been introduced inde-pendently by Weelinck [Wee18] and by the third named author [Zou20]. The paper [Hor19] usesparametrized higher category theory to define equivariant factorization homology axiomaticallyas a homology theory for G -manifolds, where G is a finite group. For 1-dimensional manifolds,1 a r X i v : . [ m a t h . A T ] J un his construction recovers Real topological Hochschild homology [DMPR17] and C n -relative topo-logical Hochschild homology [ABG + n -fold loop map is an E n -ring spectrum.Blumberg–Cohen–Schlichtkrull [BCS10] showed that the Thom spectrum functor respects thecyclic bar construction, and used this to describe the topological Hochschild homology of Thomspectra and compute several examples. Schlichtkrull [Sch11] generalized this to higher Hochschildhomology of commutative ring spectra. The second named author [Kla18] showed that the Thomspectrum functor respects factorization homology, and used this to describe the factorizationhomology and E n topological Hochschild cohomology of Thom spectra and to compute examples.In this paper, we apply this philosophy to equivariant Thom spectra and equivariant factor-ization homology. We give a construction of equivariant Thom spectra reminiscent of that ofAndo–Blumberg–Gepner–Hopkins–Rezk [ABG + Theorem.
Let X be a pointed G -space and Ω V f : Ω V X → Pic( Sp G ) be a map of E V -algebras.Then for every V -framed G -manifold M , there is an equivalence of genuine G -spectra (cid:90) M Th (Ω V f ) (cid:39) Th (cid:18)(cid:90) M Ω V X (Ω V f ) ∗ −−−−−→ (cid:90) M Pic( Sp G ) → Pic( Sp G ) (cid:19) . Here, Th : Top G/ Pic( Sp G ) → Sp G is the parametrized Thom G -functor in Construction 3.10.Thus we can leverage knowledge of equivariant factorization homology on the level of spaces todetermine equivariant factorization homology of Thom spectra. This is made particularly usefulby the equivariant nonabelian Poincar´e duality theorem of the third named author [Zou20], whichwe improve here in the axiomatic context. This is Theorem 2.2: Theorem.
For M a V -framed G -manifold and X a pointed G -space satisfying connectivityhypotheses, there is a natural equivalence of G -spaces (cid:90) M Ω V X (cid:39) Map ∗ ( M + , X ) . Here, M + is the one-point-compactification of M .This theorem generalizes the nonabelian Poincar´e duality theorem of Salvatore, Lurie, andAyala–Francis. It describes equivariant factorization homology of an equivariant algebra in thecategory of G -spaces as a compactly supported mapping space. From our equivariant nonabelianPoincar´e duality theorem, we recover equivariant Atiyah duality and equivariant Poincar´e dualityfor V -framed G -manifolds.We use our structural results on the equivariant Thom spectrum functor, along with theequivariant nonabelian Poincar´e duality theorem, to make several computations of interest. Forexample, we compute the factorization homology of representation spheres with coefficients inthe Real bordism spectrum M U R , the Real topological Hochschild homology of H F and H Z (2) ,2nd the equivariant factorization homology of the representation spheres S σ with coefficients in H F and H Z (2) . The appendix, written by Jeremy Hahn and Dylan Wilson, computes the Realtopological Hochschild homology of H Z .The computations in this paper rely on the two main theorems quoted above: equivariantnonabelian Poincar´e duality (Theorem 2.2) and the behavior of the Thom spectrum functorunder equivariant factorization homology (Theorem 5.20). The reader mainly interested in com-putations can keep these theorems in mind while focusing on Section 6 and the appendix. Structure of the paper.
In Section 2, we prove the equivariant nonabelian Poincar´e dualitytheorem, using results from Section 5, and recover equivariant Atiyah duality for V -framed G -manifolds. In Section 3, we study the G -Thom spectrum functor, with preliminaries onparametrized ∞ -category theory given in Section 7. In Section 4, we show that our G -Thomspectrum functor respects equivariant factorization homology. In Section 5, we explain how theThom spectrum of a V -fold loop map gives rise to an E V -algebra, and give a description of theequivariant factorization homology of the G -Thom spectrum of a V -fold loop map. In Section 6,we give computations of the equivariant factorization homology of certain Thom spectra usingresults from the previous sections. The appendix, by Jeremy Hahn and Dylan Wilson, gives acomputation of THR( H Z ). Notation.
We use Joyal’s quasi-categories as a theory of ∞ -categories, developed in [Lur09]and [Lur12]. We make extensive use of the theory of paramaretrized- ∞ -categories of Barwick–Dotto–Glasman–Nardin–Shah, developed in [BDG + + Top G for the G - ∞ -category of G -spaces and Fin G ∗ for the G - ∞ -category of finite pointed G -spaces. Following [Lur12], we write Sp for the ∞ -category of spectra, and use the symbol ⊗ todenote the smash product symmetric monoidal structure on Sp . We use the notation E ⊗ Σ ∞ + X or E ⊗ X for the smash product of a spectrum E and a space X (exhibiting Sp as tensored overspaces). We denote the wedge product of spectra by ⊕ . We write Top G for the ∞ -categoryFun( O opG , S ) of presheaves on the orbit category O G , and refer to its objects as G -spaces. Weuse the notation Sp G for the ∞ -category of genuine G -spectra, and denote smash products ofgenuine G -spectra by ⊗ . Finally, we denote the smash product of E ∈ Sp G and X ∈ Top G ,traditionally written E ∧ X + , by E ⊗ Σ ∞ + X or E ⊗ X . Acknowledgments.
We would like to thank Jeremy Hahn and Dylan Wilson for their con-tribution to this paper, as well as for many helpful conversations on these topics. We wouldalso like to thank Peter Bonventre for sharing a draft of his paper, Mike Hill for helpful re-marks on Snaith splittings, and Mona Merling for an illuminating discussion during an earlypart of this project. The authors would like to thank the Isaac Newton Institute for Mathemat-ical Sciences for support and hospitality during the programme “Homotopy harnessing higherstructures”, when work on this project was started. This work was supported by EPSRC grantnumber EP/R014604/1. The first author acknowledges support by ERC-2017-STG 75908 to D.Petersen, and by ISF grant 87590021 to I. Dan-Cohen.
Let G be a finite group and V a finite-dimensional real representation of G . In this section weprove the equivariant version of nonabelian Poincar´e duality regarding equivariant factorization3omology (Theorem 2.2). Our approach is similar to the one taken by Ayala–Francis (see [AF15,sec. 4]). Definition 2.1. A V -framing of a smooth G -manifold is an equivariant isomorphism of vectorbundles T M ∼ = M × V. Let M be a V -framed G -manifold, and let A be an E V algebra in Top G . The equivariantfactorization homology of M with coefficients in A , denoted as (cid:90) M A , is a homotopy colimit of adiagram of G -spaces indexed by V -disks with V -framed embeddings in M , (cid:90) M A = colim −−−→ (cid:16) Disk
G,V − fr/M → Disk
G,V − fr A −→ Top G (cid:17) . The construction also works when A is an E V algebra in Sp G . In that case, (cid:82) M A is a G -spectrumrather than a G -space. See [Hor19, def. 3.9.7] for the definition of E V -algebras and [Hor19, def.4.1.2] for the construction of equivariant factorization homology. Theorem 2.2 (Equivariant nonabelian Poincar´e duality) . For a V -framed G -manifold M and X ∈ Top G ∗ such that π k ( X H ) = 0 for all subgroups H < G and k < dim( V H ) , there is a naturalequivalence of G -spaces (cid:90) M Ω V X (cid:39) Map ∗ ( M + , X ) . Here, M + is the one-point-compactification of M . We now give an outline of the proof of Theorem 2.2, using some constructions in Section 5and lemmas proven in the rest of this section. The idea is to use the uniqueness of genuineequivariant factorization homology theories in [Hor19].
Outline of proof.
We first explain the functors in the statement. In Remark 5.10, we constructthe ∞ -functor Map ∗ (( − ) + , X ) as the underlying functor of the G -symmetric monoidal ∞ -functorMap ∗ (( − ) + , X ) ∈ Fun ⊗ G ( Mfld G, (cid:116) , Top G ∗ )at the fiber over G/G . Precomposing with the forgetful G -functor Mfld
G,V − fr, (cid:116) → Mfld G, (cid:116) ,we get Map ∗ (( − ) + , X ) ∈ Fun ⊗ G ( Mfld
G,V − fr, (cid:116) , Top G ∗ ) , (2.3)whose underlying functor over G/G we still denote by Map ∗ (( − ) + , X ). Furthermore, restrictingMap ∗ (( − ) + , X ) to V -framed V -disks gives rise to an E V -algebra in Top G , which is exactly Ω V X as defined in Eq. (5.9).We claim that the functor in Eq. (2.3) is a G -factorization homology theory of V -framed G -manifolds. That is, it satisfies G - ⊗ -excision in the sense of [Hor19, Definition 5.2.2] andrespects G -sequential unions in the sense of [Hor19, Definition 5.3.2]. We prove this later inProposition 2.13 and Proposition 2.20.By the axiomatization [Hor19, Theorem 6.0.2], we can recover a G -factorization homologytheory F by F (cid:39) (cid:90) − (( ι ⊗ ) ∗ F ) , See also Definition 4.1 for an equivalent definition. The formula above is compatible with [Hor19, def. 4.1.2]. To see this, use the colimit formula of [Wee18, def.4.14], and combine [Wee18, thm. 4.33] with [Hor19, prop. 5.2.3, 5.3.3] to compare the two constructions. ι ⊗ : Disk
G,V − fr → Mfld
G,V − fr is the inclusion of G - ∞ -categories. We take the G -factorization homology theory F to be the functor Map ∗ (( − ) + , X ) in Eq. (2.3). We can identifythe coefficient ( ι ⊗ ) ∗ (Map ∗ (( − ) + , X )) with Ω V X as defined in Eq. (5.9) and get an equivalenceof G -functors Map ∗ (( − ) + , X ) (cid:39) (cid:90) − Ω V X. The conclusion follows from taking the underlying functor over the orbit
G/G .We will prove the claims used in the outline soon. Before that, we take a detour to deducefrom Theorem 2.2 a version of the equivariant Atiyah duality theorem (Corollary 2.4) and theequivariant Poincar´e duality theorem (Corollary 2.7) for V -framed G -manifolds. Atiyah dualityfor G -manifolds has previously been studied in [LMSM86, III.5]. Corollary 2.4.
Suppose M is a V -framed G -manifold and E is G -spectrum such that π Hk (Σ V E ) =0 for k < dim( V H ) . Then there is a G -equivalence: Ω ∞ (Σ ∞ + M ⊗ E ) (cid:39) Map ∗ ( M + , Ω ∞− V E ) . In particular, taking E = Σ ∞ S to be the G -sphere spectrum, we recover Atiyah duality for M : Ω ∞ (Σ ∞ + M ) (cid:39) Map ∗ ( M + , Ω ∞ Σ ∞ S V ) . Proof.
We consider the G - infinite loop space of the G -spectrum E , Ω ∞ E . Thus, we have anequivalence of G - E ∞ spaces Ω ∞ E ∼ = Ω V colim W Ω W E V + W The two are equivalent as G -infinite loop spaces, and in particular as E V -algebras. Denote colim W Ω W E V + W by X . Thus Ω ∞ E (cid:39) Ω V X as E V -algebras. Note that X (cid:39) Ω ∞ (Σ V E ), andwe have π Hk ( X ) ∼ = π Hk (Σ V E ) for k ≥
0. Therefore by assumption, X satisfies the connectivityhypotheses in Theorem 2.2, and we obtain (cid:90) M Ω ∞ E (cid:39) Map ∗ ( M + , X ) . (2.5)We claim that there is a G -equivalence: (cid:90) M Ω ∞ E (cid:39) Ω ∞ (Σ ∞ + M ⊗ E ) . (2.6)First, Ω ∞ (Σ ∞ + ( − ) ⊗ E ) is a G -factorization homology on V -framed G -manifolds, as we showlater in Lemma 2.8. Second, the factorization homology theories (cid:82) − Ω ∞ E and Ω ∞ (Σ ∞ + ( − ) ⊗ E )have the same coefficients: their coefficients are the E V -algebras Ω ∞ and Ω ∞ (Σ ∞ + V ⊗ E ). Themap which contracts V to a point, Ω ∞ (Σ ∞ + V ⊗ E ) → Ω ∞ E , is a map of E V -algebras, as it isa map of G -infinite loop spaces. It is an equivalence of G -spaces, and therefore an equivalenceof E V -algebras. So the two factorization homology theories in Eq. (2.6) agree and we obtain theequivalence.The desired equivalence follows from combining (2.5) and (2.6). To apply to E = Σ ∞ S , wecheck when k < dim( V H ): π Hk (Σ ∞ S V ) ∼ = π Map G ( G/H + ∧ S k , Ω ∞ Σ ∞ S V ) ∼ = π Map Sp G (Σ ∞ + ( G/H ) ⊗ Σ ∞ S k , Σ ∞ S V ) ∼ = π Map Sp H (Σ ∞ S k , Σ ∞ S V ) ∼ = π Map Sp (Σ ∞ S k , Σ ∞ S V H ) = 0 . orollary 2.7. Suppose M is a V -framed G -manifold and B is a Mackey functor. Then H (cid:63) ( M ; B ) ∼ = (cid:101) H V − (cid:63) ( M + ; B ) . In particular, if M is closed, then H (cid:63) ( M ; B ) ∼ = H V − (cid:63) ( M ; B ) .Proof. We can give S V an H -CW decomposition with the lowest cells other than the base point indimension dim( V H ). So we have π Hk (Σ V H B ) ∼ = ˜ H Hk ( S V ; B ) = 0 when k < dim( V H ). Thereforewe can take E in Corollary 2.4 to be the Eilenberg–MacLane spectrum H B and getΩ ∞ (Σ ∞ + M ⊗ H B ) (cid:39) Map ∗ ( M + , K ( B, V )) . The desired Poincar´e duality follows from taking homotopy groups on both sides and identifying: π (cid:63) Ω ∞ (Σ ∞ + M ⊗ H B ) ∼ = H (cid:63) ( M, B ); π (cid:63) (Map ∗ ( M + , K ( B, V ))) ∼ = (cid:101) H V − (cid:63) ( M + ; B ) . Lemma 2.8.
Let E be a G -spectrum. Then Ω ∞ (Σ ∞ + ( − ) ⊗ E ) is a G -factorization homologytheory on V -framed G -manifolds.Proof. One can express Ω ∞ (Σ ∞ + ( − ) ⊗ E ) as the composition of G -functorsΩ ∞ (Σ ∞ + ( − ) ⊗ E ) : Mfld
G,V − fr → Mfld
G fgt −−→
Top G Σ ∞ + −−→ Sp G −⊗ E −−−→ Sp G Ω ∞ −−→ Top G . Each G -functor in the composition extends to a G -symmetric monoidal functor:1. The G -functor Mfld
G,V − fr → Mfld G , forgetting the V -framing, is G -symmetric monoidalby construction.2. The functor f gt : Mfld G → Top G is G -symmetric monoidal, as it can be defined bythe following construction. As a functor of topological categories, the forgetful functor Mfld n → Top is symmetric monoidal (takes disjoint unions to coproducts). Construct theforgetful functor f gt : Mfld G → Top G by applying the genuine operadic nerve construction(see also Section 7.8).3. The G -functor Σ ∞ + : Top G → Sp G is a G -left adjoint, hence strongly prereserves G -colimits. In particular, it extends to a G -symmetric monoidal functor with respect tothe G -coCartesian monoidal structure on both categories.4. Similarly, Sp G −⊗ E −−−→ Sp G strongly preserves G -colimits, and therefore extends to a G -symmetric monoidal functor.5. The G -functor Sp G Ω ∞ −−→ Top G is a G -right adjoint, and therefore extends to a G -symmetricmonoidal functor with respect to the G -Cartesian monoidal structures.6. Finally, since Sp G is a G -semi-additive G - ∞ -category, the G -Cartesian and G -coCartesianmonoidal structure are canonically equivalent.In fact, this decomposition also shows that Ω ∞ (Σ ∞ + ( − ) ⊗ E ) preserves sifted colimits fiberwise.We now verify the ⊗ -excision axiom. Let M = M (cid:48) ∪ M × R M (cid:48)(cid:48) be a G -collar decompositionof V -framed G -manifolds. After applying the forgetful functor Mfld
G,V − fr → Mfld
G fgt −−→
Top G G -space M = M (cid:48) (cid:96) M × R M (cid:48)(cid:48) is equivalent to the geometric realization (cid:12)(cid:12) M (cid:48) (cid:96) M (cid:48)(cid:48) M (cid:48) (cid:96) M × R (cid:96) M (cid:48)(cid:48) M (cid:48) (cid:96) M × R (cid:96) M × R (cid:96) M (cid:48)(cid:48) · · · (cid:12)(cid:12) . Since Ω ∞ (Σ ∞ + ( − ) ⊗ E ) preserves geometric realizations, we have an equivalenceΩ ∞ (Σ ∞ + ( M ) ⊗ E ) (cid:39) B (cid:0) Ω ∞ (Σ ∞ + ( M (cid:48) ) ⊗ E ) , Ω ∞ (Σ ∞ + ( M × R ) ⊗ E ) , Ω ∞ (Σ ∞ + ( M (cid:48) ) ⊗ E ) (cid:1) , hence Ω ∞ (Σ ∞ + ( − ) ⊗ E ) satisfies ⊗ -excision. A similar approach verifies Ω ∞ (Σ ∞ + ( − ) ⊗ E ) respects G -sequential unions, hence it is a G -factorization homology theory.In the remainder of this section, we complete the proof outline of Theorem 2.2 by showingthat Map ∗ (( − ) + , X ) in Eq. (2.3) respects G -sequential unions (Proposition 2.13) and satisfies G - ⊗ -excision (Proposition 2.20). Notice that both properties are fiberwise for the fiber over theorbit G/H . Without loss of generality we may work with H = G . We think of Map ∗ (( − ) + , X )as a topological functor between topological categories. Remark 2.9.
Examining the proof of [Hor19, thm. 6.0.2], we see that it is enough to verifythe axioms of a G -factorization homology theory for an equivariant handle decomposition ofa V -framed G -manifold M arising from a G -invariant Morse function. It is therefore enoughto verify ⊗ -excision under the assumption that M is the interior of a compact manifold withboundary ∂M . Similarly, when verifying the G -sequential union property we may assume thatthe sequential union given by a sequence of regular values of a G -equivariant Morse function on M . We start with proving the G -sequential union property. We find it convenient to replace thespace Map ∗ ( M + , X ) with the space of compactly supported maps. Definition 2.10.
For a G -manifold M and a based G -space X , let Map c ( M, X ) = { f ∈ Map(
M, X ) | supp( f ) is compact } be the space of compactly supported maps. Here, the support of a map f is the closure of thepreimage of the compliment of the base point. Remark 2.11.
Let M ⊂ M be an open inclusion of G -manifolds. Extending f ∈ Map c ( M , X )by the base point on M − M gives a G -map Map c ( M , X ) → Map c ( M , X ). There is a G -equivalence Map c ( M, X ) ∼ −→ Map ∗ ( M + , X ), which is natural for the variable M .The following a lemma is a geometric observation. Lemma 2.12.
Let f : M → R be an equivariant Morse function and t < s be two regularvalues. Denote M = f − ( −∞ , t ) , M = f − ( −∞ , s ) , and let X be any based G -space. Then Map c ( M , X ) → Map c ( M , X ) is a G -cofibration.Proof. Assume t = 0 without loss of generality. We show that Map c ( M , X ) → Map c ( M , X ) isan G -NDR (neighborhood deformation retract) pair, thus a G -cofibration. Assume (cid:15) > − (cid:15), (cid:15) ) are all regular values for f . We prepare several functions to constructthe G -NDR data ( h, u ).Let u (cid:48) : Map c ( M , X ) → [0 , s ] be u (cid:48) ( − ) = sup { f (supp( − )) , } u : Map c ( M , X ) → [0 ,
1] be u ( − ) = min { u (cid:48) ( − ) /(cid:15), } . Then Map c ( M , X ) = u − (0) and u − ( t ) = Map c ( f − ( −∞ , (cid:15)t ) , X ) for t ∈ (0 , u isequivariant because f is.Since ( − (cid:15), (cid:15) ) are all regular values of the Morse function, we can construct an equivariantflow F ∈ Map( I, Diff( M , M )) such that F (0) = id M and F ( t )( M ) = f − ( −∞ , t(cid:15) ) . Now take h : Map c ( M , X ) × I → Map c ( M , X ) to be h ( − , t ) = − ◦ F ( u ( − ) t ) . It is easy to verify that ( h, u ) represents Map c ( M , X ) → Map c ( M , X ) as a G -NDR pair. Proposition 2.13.
Given G -manifolds M ⊂ M ⊂ · · · ⊂ M with M = ∪ i M i , there is a G -equivalence Map ∗ ( M + , X ) (cid:39) hocolim i Map ∗ (( M i ) + , X ) . Proof.
By Remark 2.9 we may assume that there is an equivariant Morse function f : M → R such that for each s ∈ R , f − ( −∞ , s ] is compact, and that M i = f − ( −∞ , s i ) for regular values s < s < · · · . Then M i ⊂ f − ( −∞ , s i ] ⊂ M i +1 , soMap c ( M, X ) (cid:39) colim i Map c ( M i , X ) . Since Map c ( M i , X ) → Map c ( M i +1 , X ) is a G -cofibration (Lemma 2.12),colim i Map c ( M i , X ) (cid:39) hocolim i Map c ( M i , X ) . Via the functorial identification Map c ( M, X ) (cid:39) Map ∗ ( M + , X ), we haveMap ∗ ( M + , X ) (cid:39) hocolim i Map ∗ (( M i ) + , X ) . Next, we prove the G - ⊗ -excision property. We begin by fixing a G -collar decomposition. Notation 2.14.
In the rest of this section, X is a pointed G -space and M is an n -dimensional G -manifold. We fix a G -collar decomposition M = M (cid:48) (cid:83) M × R M (cid:48)(cid:48) , where M (cid:48) , M (cid:48)(cid:48) are open G -submanifolds and M is a closed G -submanifold of codimension 1. We abuse notation to write M − M (cid:48) for M − ( M (cid:48) − M × [0 , + ∞ )) and M − M (cid:48)(cid:48) for M − ( M (cid:48)(cid:48) − M × ( −∞ , M − M (cid:48) and M − M (cid:48)(cid:48) have boundaries M × { } ∼ = M . Lemma 2.15.
The diagram
Map ∗ ( M + , X ) (cid:47) (cid:47) (cid:15) (cid:15) Map ∗ (( M − M (cid:48) ) + , X ) (cid:15) (cid:15) Map ∗ (( M − M (cid:48)(cid:48) ) + , X ) (cid:47) (cid:47) Map ∗ ( M +0 , X ) is a homotopy pullback diagram of G -spaces. roof. All of the embeddings of submanifolds are proper, therefore the natural maps M +0 → ( M − M (cid:48) ) + , ( M − M (cid:48) ) + → M + (and similarly with M (cid:48)(cid:48) )are defined and are pointed maps. The maps in the diagram are restrictions along those. M is a closed submanifold, so M +0 is a closed subspace of ( M − M (cid:48) ) + and of ( M − M (cid:48)(cid:48) ) + , and isthe intersection of the two. Similarly, ( M − M (cid:48) ) + and ( M − M (cid:48)(cid:48) ) + are closed subspaces of M + .Thus, defining a pointed map M + → X is equivalent to defining pointed maps ( M − M (cid:48) ) + → X and ( M − M (cid:48)(cid:48) ) + → X which agree on the overlap, M +0 . This shows that the square is apullback square. It is also a homotopy pullback since the restriction maps to Map ∗ (( M ) + , X )are G -fibrations, as shown in the following Lemma 2.16. Lemma 2.16.
The restriction map
Map ∗ (( M − M (cid:48) ) + , X ) → Map ∗ (( M ) + , X ) is a G -fibrationwith fiber Map ∗ (( M (cid:48)(cid:48) ) + , X ) . The corresponding statement in which M (cid:48) and M (cid:48)(cid:48) are switchedalso holds.Proof. For the first part, it suffices to show that M +0 → ( M − M (cid:48) ) + is a G -cofibration in theHurewicz sense, as mapping out of it would give a Hurewicz fibration, which is in particular aSerre fibration.By Remark 2.9 we may assume that either M is closed or M is the interiorof a compact manifold with boundary ∂M . In the first case denote ∂M = ∅ . We further assume that M is embedded in some orthogonal G -representation W (This is possible by [Mos57]). Since both M and M − M (cid:48) are submanifolds of M which are closed, ∂M and D = ∂ ( M − M (cid:48) ) ∩ ∂M are both submanifolds of ∂M which are closed. (See Fig. 1 forillustration.)All of ∂M , M , D, ∂ ( M − M (cid:48) ) , M − M (cid:48) are also close submanifolds of W , consequently equivariantly embedded as a retract of an open sub-space of W ([IK00, Theorem 1.4]), showing that they are all G -ENRs(Euclidean neighborhood retract). Figure 1: IllustrationBy [LMSM86, III.4], an inclusion of G -ENRs is a G -cofibration. So all maps in the followingpushout square are G -cofibrations: ∂M M D ∂ ( M − M (cid:48) ) M − M (cid:48) Therefore, M +0 = M /∂M −→ ∂ ( M − M (cid:48) ) /D −→ M − M (cid:48) /D = ( M − M (cid:48) ) + is a composite of G -cofibrations. This proves the first part.To find the fiber, we take the preimage of the constant map in Map ∗ (( M ) + , X ). Because ∂ ( M − M (cid:48) ) = M ∪ ∂M D , a map from ( M − M (cid:48) , D ) to ( X, ∗ ) that map M to the base point ∗ is the same as a map from ( M − M (cid:48) , ∂ ( M − M (cid:48) )) to ( X, ∗ ). So the fiber can be identified withMap ∗ ( M − M (cid:48) /∂ ( M − M (cid:48) ) , X ) ∼ = Map ∗ ( M (cid:48)(cid:48) /∂M (cid:48)(cid:48) , X ) (cid:39) Map ∗ (( M (cid:48)(cid:48) ) + , X ).Let B be a based G -space. To set up for a bar construction, we use the Moore path spaceand loop space of B , such that Ω B is a monoid and acts on P B : P B = { ( l, α ) ∈ R ≥ × Map( R ≥ , B ) | α (0) = ∗ , α ( t ) = α ( l ) for t ≥ l } , Ω B = { ( l, α ) ∈ R ≥ × Map( R ≥ , B ) | α (0) = ∗ , α ( t ) = ∗ for t ≥ l } ⊂ P B. l . Given two path elements ( l , α ) and ( l , α ) in P B ,the concatination of them is defined to ( l + l , α .α ) where( α .α )( t ) = (cid:40) α ( t ) , ≤ t ≤ l ; α ( t − l ) , l < t. This strictifies Ω B to a strict monoid and gives a strict left action of Ω B on P B by concatenation.The unit for Ω B is (0 , ∗ ) where ∗ ( t ) = ∗ and a homotopy inverse is the reverse of loops. Thereverse of ( l, α ) ∈ Ω B is defined to be ( l, ¯ α ) where¯ α ( t ) = (cid:40) α ( l − t ) , ≤ t ≤ l ; ∗ , l < t. The reverse of loops is a monoid homomorphism Ω B → (Ω B ) op . Composing it with the leftaction gives the right action of Ω B on P B . There is also the evaluation map that records theendpoint of a path ev : P B → B, ( l, α ) (cid:55)→ α ( l ) . Lemma 2.17.
Suppose
Map ∗ (( M ) + , X ) is G -connected. Then there is an equivalence betweenthe bar construction B (Map ∗ (( M (cid:48) ) + , X ) , ΩMap ∗ (( M ) + , X ) , Map ∗ (( M (cid:48)(cid:48) ) + , X )) and the homotopy pullback Map ∗ (( M − M (cid:48)(cid:48) ) + , X ) × Map ∗ (( M ) + ,X ) Map ∗ (( M − M (cid:48) ) + , X ) . Proof.
For brevity, we write B for Map ∗ (( M ) + , X ), B (cid:48)(cid:48) for Map ∗ (( M − M (cid:48)(cid:48) ) + , X ) and B (cid:48) forMap ∗ (( M − M (cid:48) ) + , X ). Then B is a based G -space with base point the constant map to thebase point of X and G acts by conjugation.We first describe the bar construction in the statement. Denote by × B the homotopypullback of spaces over B . Consider the restriction B (cid:48) = Map ∗ (( M − M (cid:48) ) + , X ) → Map ∗ (( M ) + , X ) = B . By Lemma 2.16, we have G -equivalences between the fiber and the homotopy fiber of this re-striction: Map ∗ (( M (cid:48)(cid:48) ) + , X ) (cid:39) B (cid:48) × B P B , and similarly, Map ∗ (( M (cid:48) ) + , X ) (cid:39) B (cid:48)(cid:48) × B P B . We have a right action of Ω B on Map ∗ (( M (cid:48) ) + , X ) and a left action on Map ∗ (( M (cid:48)(cid:48) ) + , X ) byΩ B acting on P B through these equivalences. Geometrically, this action is by gluing together amap in ΩMap ∗ (( M ) + , X ) ∼ = Map ∗ (( M × R > ) + , X ) and a map in Map ∗ (( M (cid:48) ) + , X ) via a choiceof identification of M (cid:48) ∪ M M × R > ∼ = M (cid:48) . The two sided bar construction in the statment isequivalent to the geometric realization of the simplicial G -space B ( B (cid:48)(cid:48) × B P B , Ω B , B (cid:48) × B P B ) . G -equivalence: B (Map ∗ (( M (cid:48) ) + , X ) , Ω B , Map ∗ (( M (cid:48)(cid:48) ) + , X )) (cid:39) B ( B (cid:48)(cid:48) × B P B , Ω B , B (cid:48) × B P B ) (cid:39) B (cid:48)(cid:48) × B B ( P B , Ω B , P B ) × B B (cid:48) . We claim that there is an equivalence B ( P B , Ω B , P B ) (cid:39) B as G -spaces when B is G -connected. Moreover, it is an equivalence as G -spaces over B on both sides. Here, the two aug-mentation maps of B are identity on both sides; the two augmentation maps of B ( P B , Ω B , P B )over B are induced by ev : P B → B on either of the P B in the bar construction, which wedenote them by ev l and ev r . These two maps are actually G -homotopic, as one can construct anexplicit homotopy ev l (cid:39) ev r by evaluating along the concatenated paths from the left endpoint tothe right endpoint in each simplicial level. We skip the details here. Now, we take the geometricrealization of the simplicial levelwise G -fibration: B ∗ (Ω B , Ω B , P B ) → B ∗ ( P B , Ω B , P B ) ev l → B and to obtain a sequence ∗ (cid:39) B (Ω B , Ω B , P B ) → B ( P B , Ω B , P B ) ev l → B . (2.18)It suffices to show that ev l (equivalently, ev r ) is a G -weak equivalence. This is because for eachsubgroup H < G , when ( B ) H is connected, Eq. (2.18) is known to be a quasifibration aftertaking H -fixed points.Consequently, we have B (Map ∗ (( M (cid:48) ) + , X ) , Ω B , Map ∗ (( M (cid:48)(cid:48) ) + , X )) (cid:39) B (cid:48)(cid:48) × B B × B B (cid:48) (cid:39) B (cid:48)(cid:48) × B B (cid:48) . Lemma 2.19.
Let M be a smooth G -manifold and N be a closed sub- G -manifold. Let X be a based G -space such that X H is dim( M H ) -connected for all subgroups H < G . Then
Map ∗ ( M/N, X ) is G -connected. Here, since M is a smooth G -manifold, M H is also a manifold, but possibly empty or withcomponents of different dimensions. We define dim( ∅ ) = − M H ) to be the biggestdimension of the components. Proof.
Take a triangulation of (
M, N ), which exists by [Ill78, Theorem 3.6]. It gives (
M, N ) arelative G -CW structure. Denote M − = N and S − = ∅ . Then Map ∗ ( M − /N, X ) = ∗ is G -connected. We induct on the G -CW skeleton of M . For k ≥
0, we have the pushout: (cid:96) H i G/H i × S k − M k − (cid:96) H i G/H i × D k M kf It gives a cofiber sequence: M k − /N → M k /N → (cid:95) H i ( G/H i ) + ∧ S k . X gives a fiber sequence: (cid:89) H i Map ∗ (( G/H i ) + ∧ S k , X ) → Map ∗ ( M k /N, X ) → Map ∗ ( M k − /N, X ) . For any subgroups H and H i , by the double coset formula, G/H i ∼ = (cid:96) j H/K ij as H -sets,where each of the K ij is H intersecting some conjugate of H i . Therefore,Map ∗ (( G/H i ) + ∧ S k , X ) H ∼ = Map ∗ (( (cid:95) j ( H/K ij ) + ) ∧ S k , X ) H ∼ = (cid:89) j Map ∗ ( S k , X K ij ) . Since k ≤ dim ( M H i ) ≤ dim ( M K ij ), X K ij is k -connected by assumption. So Map ∗ ( S k , X K ij )is connected. By long exact sequence of homotopy groups and induction, Map ∗ ( M k /N, X ) H isconnected for all k .This implies that Map ∗ ( M/N, X ) is G -connected. Proposition 2.20.
Let M be a V -framed G -manifold with the collar decomposition in Nota-tion 2.14, and let X be a G -space as in Theorem 2.2. Then there is a G -equivalence: Map ∗ (( M (cid:48) ) + , X ) ⊗ Map ∗ (( R × M ) + ,X ) Map ∗ (( M (cid:48)(cid:48) ) + , X ) → Map ∗ ( M + , X ) . Proof.
Note that ( R × M ) + (cid:39) Σ( M +0 ). The homotopy coherent quotient on the left-hand side,over Map ∗ (( R × M ) + , X ) (cid:39) ΩMap ∗ ( M +0 , X ), can be identified with the bar construction B (Map ∗ (( M (cid:48) ) + , X ) , ΩMap ∗ ( M +0 , X ) , Map ∗ (( M (cid:48)(cid:48) ) + , X )) . Since M × R is V -framed, ( M × R ) H ∼ = M H × R is either empty or a manifold of dimension V H , as we can find local charts using the exponential maps. So dim( M H )+1 ≤ dim( V H ). By ourassumption on the connectivity of X , Lemma 2.19 applied to the pair ( M, N ) = ( M , ∅ ) showsthat Map ∗ ( M +0 , X ) is G -connected. So we can use Lemma 2.17 to identify the bar constructionwith the homotopy pullback Map ∗ (( M − M (cid:48)(cid:48) ) + , X ) × Map ∗ (( M ) + ,X ) Map ∗ (( M − M (cid:48) ) + , X ), thenuse Lemma 2.15 to identify it with Map ∗ ( M + , X ). In this section, we will define the G -Thom spectrum functor, and show that it respects G -colimitsand is G -symmetric monoidal. This will allow us to conclude, in the next section, that it respectsequivariant factorization homology.We first recall the construction of Thom spectra according to [ABG + ∞ -groupoids. The Thom spectrum of astable spherical fibration E over X is defined as the colimit of X E −→ Pic( Sp ) → Sp . (3.1)Here, Pic( Sp ) is the Picard space of the ∞ -category of spectra, that is, the classifying space oflocal systems of invertible spectra; Pic( Sp ) → Sp is the inclusion of a sub- ∞ -groupoid; the map X E −→ Pic( Sp ) is the classification map of the spherical fibration E , and the colimit of Eq. (3.1)is indexed by X , considered as an ∞ -groupid. 12ogether these Thom spectra assemble to a colimit preserving functor from spaces overPic( Sp ) to spectra: S / Pic( Sp ) (cid:39) Psh(Pic( Sp )) → Sp . By the universal property of the presheaf category, this functor is characterized by its restrictionalong the Yoneda embedding Pic( Sp ) → Psh(Pic( Sp )), see [ABG + Sp ) S / Pic( Sp ) Psh(Pic( Sp )) Sp . (cid:39) We apply a similar approach to construct a G -equivariant Thom spectrum. The goal of thissection is the following theorem. Theorem 3.2.
There exists a G -symmetric monoidal functor Th : Top G/ Pic( Sp G ) (cid:39) Psh G (Pic( Sp G )) → Sp G that strongly preserves G -colimits. Moreover, let E ∈ Sp G be an invertible genuine G -spectrumand e : X → Pic( Sp G ) be a G -map from a G -space X such that e is G -homotopic to the constantmap with value E . Then the functor Th takes e ∈ (cid:18) Top G/ Pic( Sp G ) (cid:19) [ G/G ] to the genuine G -spectrum E ⊗ X ∈ Sp G . Let us briefly explain the notation used in Theorem 3.2. Since Sp is the fiber of the G - ∞ -category Sp G over G/e , we can endow Pic( Sp ) with a G -action whose H -fixed points areequivalent to the Picard space of genuine H -spectra. We call the resulting G -space the Picard G -space of Sp G , and think of it as an object in the category of G -spaces, Pic( Sp G ) ∈ Top G (see Section 3.1 for details). Since Top G is the fiber of the G - ∞ -category Top G over G/G , theobject Pic( Sp G ) ∈ Top G [ G/G ] defines a G -functor O opG → Top G by forgetting the G -action ofPic( Sp G ). Finally, the G - ∞ -category Top G/ Pic( Sp G ) is the parametrized slice category of Top G :its fiber (cid:18) Top G/ Pic( Sp G ) (cid:19) [ G/H ] is equivalent to the slice ∞ -category Top H/ Pic( Sp G ) of H -spaces over Pic( Sp G ). In particular, the fiber (cid:18) Top G/ Pic( Sp G ) (cid:19) [ G/G ] is equivalent to the category of G -spaces over Pic( Sp G ), and its objects are given by maps of G -spaces e : X → Pic( Sp G ). Remark 3.3.
For Theorem 3.2 and its applications, one could also work with p -local genuine G -spectra. Let Sp G ( p ) ⊂ Sp G be the G -subcategory of (fiberwise) p -local spectra. Note that the G -symmetric monoidal structure of Sp G induces a G -symmetric monoidal structure on Sp G ( p ) , asthe essential image of the G -localization − ⊗ S ( p ) : Sp G → Sp G , One should think of Sp G as defining a nontrivial G -action on Sp . Throughout this paper, an H -space means a space with an action of the subgroup H < G . + ar, thm. 3.2 and rem. 3.3]. Replacing Sp G with Sp G ( p ) , we obtain a p -local Thomspectrum functor Th : Top G/ Pic( Sp G ( p ) ) (cid:39) Psh G (Pic( Sp G ( p ) )) Th (cid:48) −−→ Sp G ( p ) . The entire section as well as Section 4 and Section 5 hold mutatis mutandis. In particular, the p -local G -Thom spectrum of a G -map X → Pic( S ( p ) ) is given by Th (cid:16) X → Pic( S ( p ) ) → Pic( Sp G ( p ) ) (cid:17) , where the second map is the inclusion Pic( S ( p ) ) ⊂ Pic( Sp G ( p ) ). G -space We define the Picard G -space of a G -symmetric monoidal G - ∞ -category and show that it inheritsa G -symmetric monoidal structure.Let p : C ⊗ (cid:16) Fin G ∗ be a G -symmetric monoidal G - ∞ -category and let C (cid:16) O opG be itsunderlying G - ∞ -category. Recall that each fiber C [ G/H ] of the underlying G - ∞ -category isendowed with a symmetric monoidal structure, defined by the pull back of p along Fin ∗ → Fin G ∗ , I (cid:55)→ ( I × G/H → G/H ) . Definition 3.4.
An object x in the G - ∞ -category C which is over G/H is invertible if x isan invertible object in the ∞ -category C [ G/H ] , that is, the object x ∈ C [ G/H ] is dualizable andthe evaluation map x ⊗ x ∧ → is an equivalence. The Picard G -space Pic( C ) is the maximal G - ∞ -groupoid of C spanned by invertible objects. Remark 3.5.
By construction, Pic( C ) is a G - ∞ -groupoid, that is, the map Pic( C ) → O opG is aleft fibration. Since the ∞ -category of left fibrations over O opG is equivalent to the ∞ -category of G -spaces, there is a G -space corresponding to the G - ∞ -groupoid Pic( C ). By abuse of notation,we write Pic( C ) ∈ Top G for this G -space. Remark 3.6.
The fiber of Pic( C ) over G/H can be identified with the Picard space Pic (cid:16) C [ G/H ] (cid:17) . The G -symmetric monoidal structure of the Picard G -space. In Lemma 7.12, it isshown that the left fibration C ⊗ coCart (cid:16) Fin G ∗ endows the maximal G - ∞ -groupoid C (cid:39) ⊆ C with a G -symmetric monoidal structure. This G -symmetric monoidal structure further restricts to a G -symmetric monoidal structurePic( C ) ⊗ (cid:16) Fin G ∗ on Pic( C ) ⊆ C (cid:39) , as we now explain.Let I = ( U → G/H ) ∈ Fin G ∗ be an object. As shown in Lemma 7.12, the G -Segal map for C ⊗ coCart gives an equivalence ( C ⊗ coCart ) [ I ] ∼ −→ (cid:81) W ∈ Orbit( U ) C (cid:39) [ W ] . Construction 3.7.
Let Pic( C ) ⊗ ⊆ C ⊗ coCart be the full subcategory whose fiber over the object I = ( U → G/H ) ∈ Fin G ∗ , (cid:0) Pic( C ) ⊗ (cid:1) [ I ] ⊆ (cid:0) C ⊗ coCart (cid:1) [ I ] (cid:39) (cid:89) W ∈ Orbit U C (cid:39) [ W ] ,
14s spanned by tuples of invertible objects( x [ W ] ) ∈ (cid:89) W ∈ Orbit U Pic( C [ W ] ) . Here, Pic( C [ W ] ) ⊆ ( C [ W ] ) (cid:39) = C (cid:39) [ W ] is the Picard space of the fiber C [ W ] . Lemma 3.8.
The restriction of C ⊗ coCart (cid:16) Fin G ∗ to Pic( C ) ⊗ ⊆ C ⊗ coCart defines a G -symmetricmonoidal structure on Pic( C ) .Proof. Let I = ( U → G/H ) ∈ Fin G ∗ be an object. By construction, we have(Pic( C ) ⊗ ) [ I ] (cid:39) (cid:89) W ∈ Orbit U Pic( C [ W ] ) . We show that the restriction p : Pic( C ) ⊗ ⊆ C ⊗ coCart p (cid:48) (cid:16) Fin G ∗ is a left fibration. In the language of [BDG + C ) ⊗ is a Fin G ∗ -subcategory. By [BDG + p (cid:48) -coCartesian edge x → y lies in Pic( C ) ⊗ just in case x does. These are true since in eachfiber Pic( C ) [ G/H ] ⊂ C (cid:39) [ G/H ] , invertible objects are closed under tensor products, and for every ϕ : G/K → G/H , the norm functor ⊗ ϕ : C [ G/K ] → C [ G/H ] is symmetric monoidal, and in partic-ular preserves invertible objects.The G -Segal conditions for C ⊗ coCart restrict to give the G -Segal conditions for Pic( C ) ⊗ . Remark 3.9.
By Example 7.16, we can think of Pic( C ) as a G -commutative algebra in ( Top G ) × . G -functor We are now ready to construct the G -functor of Theorem 3.2. Construction 3.10.
Let Th (cid:48) : Psh G (Pic( Sp G )) → Sp G be the G -left Kan extension of the inclu-sion Pic( Sp G ) → Sp G along the parametrized Yoneda embedding ι : Pic( Sp G ) → Psh G (Pic( Sp G )).Define Th : Top G/ Pic( Sp G ) (cid:39) Psh G (Pic( Sp G )) Th (cid:48) −−→ Sp G to be Th (cid:48) precomposed with an inverse of the natural equivalence of Corollary 7.5. Proposition 3.11. A G -functor Top G/ Pic( Sp G ) → Sp G is equivalent to Th if and only if itpreserves colimits and its restriction along the parametrized Yoneda embedding Pic( Sp G ) (cid:44) → Psh G (Pic( Sp G )) (cid:39) Top G/ Pic( Sp G ) is equivalent to the inclusion Pic( Sp G ) → Sp G .Proof. This follows from [Sha18, thm. 11.5].
Corollary 3.12.
The restriction of the G -functor Th : Top G/ Pic( Sp G ) → Sp G to the fiber over G/e ∈ O opG is equivalent to the Thom spectrum functor S / Pic( Sp ) (cid:39) Psh(Pic( Sp )) → Sp of [ABG + roof. It suffices to show the restriction of the Thom spectrum G -functor Th satisfies the uni-versal property of the Thom spectrum functor as in [ABG + G/e of the corresponding G - ∞ -categories as (cid:18) Top G/ Pic( Sp G ) (cid:19) [ G/e ] (cid:39) (cid:16) Top G [ G/e ] (cid:17) / Pic( Sp G ) [ G/e ] (cid:39) S / Pic( Sp ) , (cid:16) Sp G (cid:17) [ G/e ] (cid:39) Sp . By construction, the functor S / Pic( Sp ) (cid:39) (cid:18) Top G/ Pic( Sp G ) (cid:19) [ G/e ] Th [ G/e ] −−−−−→ (cid:16) Sp G (cid:17) [ G/e ] (cid:39) Sp preserves colimits and its restriction alongPic( Sp ) (cid:39) Pic( Sp G ) [ G/e ] → Psh G (Pic( Sp G )) [ G/e ] (cid:39) S / Pic( Sp ) is equivalent to the canonical inclusion Pic( Sp ) (cid:39) Pic( Sp G ) [ G/e ] → ( Sp G ) [ G/e ] (cid:39) Sp .The Thom spectrum of a stably trivial sphere bundle over X is given by a smash product S n ⊗ Σ ∞ + X with an invertible spectrum S n ∈ Sp . Our next goal is an equivariant version of thisfact.We will use the following notation when discussing G -Thom spectra. Notation 3.13.
Consider Pic( Sp G ) as a G -space by Remark 3.5. An invertible H -spectrum E ∈ (cid:16) Pic( Sp G ) (cid:17) [ G/H ] (cid:39) Pic( Sp H ) is then an H -fixed point of the G -space Pic( Sp G ), whichcorresponds to an H -equivariant map ∗ → Pic( Sp G ). We denote by f E : ∗ → Pic( Sp G ) the H -map corresponding to E . Proposition 3.14.
The G -functor Th sends the map of H -spaces f E : ∗ → Pic( Sp G ) to itscorresponding invertible H -spectrum E ∈ Pic( Sp G ) [ G/H ] .Proof. The parametrized Yoneda embedding ι : Pic( Sp G ) → Psh G (Pic( Sp G )) is fully faithful([BDG + Th (cid:48) ◦ ι is equivalent tothe inclusion Pic( Sp G ) → Sp G , so Th (cid:48) ( ι ( E )) = E . We will finish the proof by showing that f E : ∗ → Pic( Sp G ) corresponds to ι ( E ) ∈ Top G/ Pic( Sp G ) under the equivalencePsh G (Pic( Sp G )) (cid:39) Top G/ Pic( Sp G ) in Corollary 7.5.The equivalence of Corollary 7.5 sends the representable presheaf ι ( E ) ∈ Psh G (Pic( Sp G )) [ G/H ] to a G/H -functor of
G/H - ∞ -groupoids (cid:16) Pic( Sp G ) /E (cid:16) Pic( Sp G ) × G/H (cid:17) ∈ (cid:16) Top
G/H (cid:17) / (Pic( Sp G ) × G/H ) (cid:39) (cid:18) Top G/ Pic( Sp G ) (cid:19) [ G/H ] . Sp G ) /E ) are all contractible as slices of ∞ -groupoids, so the natural G/H -functor σ E : G/H → (Pic( Sp G )) /E is a G/H -equivalence. It follows that the
G/H -functorPic( Sp G ) /E (cid:16) Pic( Sp G ) × G/H is equivalent to the composition
G/H σ E −−→ (Pic( Sp G )) /E (cid:16) Pic( Sp G ) × G/H.
This composition is precisely the
G/H -object E : G/H → Pic( Sp G ) × G/H associated to E ,which corresponds to the H -map f E : ∗ → Pic( Sp G ) under the isomorphismFun G/H ( G/H,
Pic( Sp G ) × G/H ) ∼ = Fun G ( G/H,
Pic( Sp G )) ∼ = Map H ( ∗ , Pic( Sp G )) . Using Proposition 3.14 we can calculate the equivariant Thom spectrum of G -nullhomotopicmaps. Proposition 3.15.
Let X ∈ Top G (cid:39) Top G [ G/G ] be a G -space and E ∈ Pic( Sp G ) (cid:39) Pic( Sp G ) [ G/G ] be an invertible G -spectrum, then Th ( X → ∗ f E −−→ Pic( Sp G )) (cid:39) E ⊗ Σ ∞ + X. Proof.
The point ∗ ∈
Top G (cid:39) Top G [ G/G ] is the terminal G -space. Express the G -space X as G - colim −−−→ X ( ∗ ), the G -colimit of the constant G -diagram X → G/G → Top G , where the second functor corresponds to the terminal G -space. Postcomposition with f E inducesa G -functor ( f E ) ∗ : Top G (cid:39) Top G/ ∗ → Top G/ Pic( Sp G ) ,X (cid:55)→ ( X → ∗ ) (cid:55)→ ( X → ∗ f E −−→ Pic( Sp G )) , and this G -functor strongly preserves G -colimits. Since ( f E ) ∗ ( ∗ ) = (cid:16) f E : ∗ → Pic( Sp G ) (cid:17) wehave ( f E ) ∗ ( X ) = ( f E ) ∗ (cid:16) G - colim −−−→ X ( ∗ ) (cid:17) (cid:39) G - colim −−−→ X ( f E ) ∈ (cid:18) Top G/ Pic( Sp G ) (cid:19) [ G/G ] . We can now apply the G -functor Th of Construction 3.10 to ( f E ) ∗ ( X ). By Proposition 3.11 andProposition 3.14 we have Th (( f E ) ∗ ( X )) (cid:39) Th (cid:16) G - colim −−−→ X ( f E ) (cid:17) (cid:39) G - colim −−−→ X ( Th ( f E )) (cid:39) G - colim −−−→ X ( E ) . On the other hand, E ⊗ Σ ∞ + X (cid:39) E ⊗ Σ ∞ + (cid:16) G - colim −−−→ X ( ∗ ) (cid:17) (cid:39) G - colim −−−→ X (cid:0) E ⊗ Σ ∞ + ∗ (cid:1) (cid:39) G - colim −−−→ X ( E ) . Together we have Th (cid:16) X → ∗ f E −−→ Pic( Sp G ) (cid:17) (cid:39) E ⊗ Σ ∞ + X, as claimed. 17e end this section by extending Th to a G -symmetric monoidal functor.We first describe the G -symmetric monoidal structure on ( Top G ) / Pic( Sp G ) . We have seen inSection 3.1 that the G -symmetric monoidal structure of Sp G induces a G -symmetric monoidalstructure on the G - ∞ -groupoid Pic( Sp G ), and that we can consider Pic( Sp G ) as a G -commutativealgebra in ( Top G ) × . Therefore, we can endow the parametrized slice category ( Top G ) / Pic( Sp G ) with the parametzied slice G -symmetric monoidal structure of Section 7.5. Proposition 3.16.
The G -functor Th of Construction 3.10 extends to a G -symmetric monoidalfunctor Th ⊗ : (cid:18) Top G/ Pic( Sp G ) (cid:19) ⊗ → ( Sp G ) ⊗ , where (cid:18) Top G/ Pic( Sp G ) (cid:19) ⊗ is the slice G -symmetric monoidal structure of Section 7.5.Proof. By Proposition 7.8, the G -left Kan extension Th (cid:48) extends to a G -symmetric monoidalfunctor. The result follows from the fact that the equivalence of Corollary 7.5 extends to a G -symmetric monoidal equivalence, see Theorem 7.20. Corollary 3.17.
Let X be a G -space, let f : Y → Pic( Sp G ) be a G -map and A = Th ( Y f −→ Pic( Sp G )) its G -Thom spectrum. Then we have an equivalence of genuine G -spectra Th ( X × Y pr −→ Y f −→ Pic( Sp G )) (cid:39) A ⊗ Σ ∞ + X. Proof.
The tensor product in (
Top G ) / Pic( Sp G ) admits the following description (cid:16) Y f −→ Pic( Sp G ) (cid:17) ⊗ (cid:16) X → ∗ S −→ Pic( Sp G ) (cid:17) (cid:39) (cid:16) X × Y → ∗ × Y f S × f −−−→ Pic( Sp G ) × Pic( Sp G ) ⊗ −→ Pic( Sp G ) (cid:17) , where S ∈ Sp G is the G -sphere spectrum. Since the map f S : ∗ → Pic( Sp G ) is constant on S ,the unit of Pic( Sp G ) = Pic( Sp G ) [ G/G ] , the composition Y = ∗ × Y f S × f −−−→ Pic( Sp G ) × Pic( Sp G ) ⊗ −→ Pic( Sp G )is equivalent to f : Y → Pic( Sp G ). Therefore (cid:16) Y f −→ Pic( Sp G ) (cid:17) ⊗ (cid:16) X → ∗ S −→ Pic( Sp G ) (cid:17) (cid:39) (cid:16) X × Y pr −→ Y f −→ Pic( Sp G ) (cid:17) . The G -Thom functor is G -symmetric monoidal so Th (cid:16) X × Y pr −→ Y f −→ Pic( Sp G ) (cid:17) (cid:39) Th (cid:16) Y f −→ Pic( Sp G ) (cid:17) ⊗ Th (cid:16) X → ∗ S −→ Pic( Sp G ) (cid:17) (cid:39) A ⊗ (cid:0) S ⊗ Σ ∞ + X (cid:1) (cid:39) A ⊗ Σ ∞ + X, where the second equivalence follows from Proposition 3.15.18 Parametrized Thom spectrum and genuine equivariantfactorization homology
In this section, we use the results of Section 3 to prove that our G -Thom spectrum functorrespects equivariant factorization homology.We will use both V -framed G -disk algebras and E V -algebras, whose definitions we now recall;these definitions are in fact equivalent.Let V be a real n -dimensional representation of G . The representation V defines a G -map pt → BO n ( G ) to the classifying G -space of rank n real G -vector bundles (see [Hor19, cor. 3.2.7,ex. 3.3.3]), which defines a notion of V -framed G -manifolds and V -framed G -disks (see [Hor19,ex. 3.3.3]). Let C be a G -symmetric monoidal G - ∞ -category. Definition 4.1. A V -framed G -disk algebra (see [Hor19, def. 3.6.11, ex. 3.6.12]) in C is a G -symmetric monoidal functor Disk
G,V − fr, (cid:116) → C ⊗ . One can also consider the notion of an E V -algebra, that is, a map of G - ∞ -operads E ⊗ V → C ⊗ , where E ⊗ V is the G - ∞ -operad of [Bon19, ex. 6.5]. It is the geunine operadic nerve of the V -littledisks G -operad. In fact, these two notions are equivalent, and we will use them interchangeablyfor the rest of this paper. Proposition 4.2 ([Hor19, cor. 3.9.9]) . There is an equivalence of ∞ -categories Alg E V ( C ) (cid:39) Fun ⊗ G ( Disk
G,V − fr , C ) between the ∞ -category of E V -algebras in C and the ∞ -category Fun ⊗ G ( Disk
G,V − fr , C ) of V -framed G -disk algebras in C . With these definitions at hand we study the compatibility of the G -functor Th and genuineequivariant factorization homology. We’ll use the following equivariant version of [AF15, lem.3.25]. Lemma 4.3.
Let C ⊗ , D ⊗ be presentable G -symmetric monoidal G - ∞ -categories, and let F : C ⊗ → D ⊗ be a G -symmetric monoidal G -functor whose restriction to the underlying G - ∞ -categories stronglypreserves G -colimits (see [Sha18, def. 11.2]). Let A ∈ Alg E V ( C ) be an E V -algebra in C . Thenthe composition F ◦ (cid:90) − A : Mfld
G,V − fr → C → D is equivalent to (cid:82) − F ( A ) : Mfld
G,V − fr → D , where F ( A ) is the E V -algebra in D correspondingthe V -framed G -disk algebra Disk
G,V − fr, (cid:116) A −→ C ⊗ F −→ D ⊗ under Proposition 4.2. roof. The G -functor F ◦ (cid:82) − A extends to a G -symmetric monoidal G -functor Mfld
G,V − fr, (cid:116) (cid:82) − A −−−→ C ⊗ F ⊗ −−→ D ⊗ . The G -symmetric monoidal functor (cid:82) − A satisfies G - ⊗ -excision and respects G -sequential unions(see [Hor19, prop 5.2.3, prop. 5.3.3]). Since F strongly preserves G -colimits the functors Mfld
G,V − fr [ G/H ] (cid:82) − A −−−→ C [ G/H ] F −→ D [ G/H ] preserve colimits, so F ◦ (cid:82) − A also satisfies G - ⊗ -excisionand respects G -sequential unions. By [Hor19, thm. 6.0.2] we have an equivalence H ( Mfld
G,V − fr , D ) ∼ −→ Fun ⊗ G ( Disk
G,V − fr , D )from the ∞ -category of G -symmetric monoidal functors which satisfy G - ⊗ -excision and re-spect G -sequential unions to the ∞ -category of V -framed G -disk algebras, given by restric-tion to Disk
G,V − fr ⊂ Mfld
G,V − fr . It follows that F ◦ (cid:82) − A is equivalent to a genuine G -factorization homology Mfld
G,V − fr, (cid:116) → D ⊗ with coefficients given by the restriction of F ◦ (cid:82) − A to Disk
G,V − fr, (cid:116) . Since Disk
G,V − fr, (cid:116) → Mfld
G,V − fr, (cid:116) (cid:82) − A −−−→ C ⊗ corresponds to the E V -algebra A , the coefficients of F ◦ (cid:82) − A correspond to F ( A ). Proposition 4.4. If A is an E V -algebra in Top G/ Pic( Sp G ) , then we have a natural equivalence (cid:90) − Th ( A ) (cid:39) Th (cid:18)(cid:90) − A (cid:19) . Proof.
By Construction 3.10 and Proposition 3.16, the G -Thom spectrum Th : Top G/ Pic( Sp G ) → Sp G strongly preserves G -colimits, and extends to a G -symmetric monoidal G -functor. The claimnow follows from Lemma 4.3 with C = Top G/ Pic( Sp G ) and D = Sp G . V -fold loop spaces and E V -algebras The coefficients for (cid:90) M − , where M is a V -framed G -manifold, are E V -algebras as in Defini-tion 4.1. The E V -algebras we consider in this paper typically arise in two ways: as Thom spectraof V -fold loop maps, or as G -commutative algebras. In this section, we will explain how weconsider each of those as an E V -algebra, and give a description of the equivariant factorizationhomology of the G -Thom spectrum of a V -fold loop map. G -commutative algebras as E V -algebras The first examples of E V -algebras are given by G -commutative algebras. Recall ([Nar17, ex. 3.3])that a G -commutative algebra is a map of G - ∞ -operads from the terminal G - ∞ -operad to a G -symmetric monoidal G - ∞ -category C ⊗ → Fin G ∗ . Since the terminal G - ∞ -operad Fin G ∗ → Fin G ∗ is itself a G -symmetric monoidal G - ∞ -category, a G -commutative algebra A ∈ CAlg G ( C ) is a lax G -symmetric monoidal functor A : Fin G ∗ → C ⊗ .20ote that the structure map Disk
G,V − fr, (cid:116) → Fin G ∗ can itself be considered as a G -symmetricmonoidal functor. Therefore we can consider any G -commutative algebra A ∈ CAlg G ( C ) as an E V -algebra by precomposition with the structure map Disk
G,V − fr, (cid:116) → Fin G ∗ A −→ C ⊗ . For C ⊗ = Top G, × the notion of G -commutative algebra agrees with a G -symmetric monoidalstructure on a G -space, considered as a G - ∞ -groupoid. V -fold loop spaces as E V -algebras In this subsection, we explain how to establish V -fold loop spaces as G -symmetric monoidalfunctors Ω V X : Disk
G,V − fr, (cid:116) → Top G, × . This is in Construction 5.8.First, we upgrade the one point compactification to a G -symmetric monoidal functor. Werely on the fact that one point compactification defines a functor of topological categories. Construction 5.1.
Let M ∈ Mfld n be an n -dimensional manifold, and denote its one pointcompactification by M + ∈ Top ∗ . Since morphisms in Mfld n are open embeddings of manifolds,one point compactification defines a functor ( − ) + : Mfld n → ( Top ∗ ) op . Furthermore, ( − ) + takesdisjoint unions to wedge sums, so it defines a symmetric monoidal functor( − ) + : ( Mfld n , (cid:116) ) → (( Top ∗ ) op , ∨ ) . We consider both categories
Mfld n and Top ∗ as topological symmetric monoidal categories usingthe compact open topology. By [hc], this one point compactification is a functor of topologicalcategories.From a symmetric monoidal topological category, one can consider the G -objects and applythe genuine operadic nerve construction of [Bon19] to obtain a G -symmetric monoidal G - ∞ -category. For details of this procedure, see Section 7.8. This gives another way to construct the G - ∞ -category Mfld G, (cid:116) . Lemma 5.2.
Let ( Mfld n , (cid:116) ) be the topological symmetric monoidal category of [AF15, def. 2.1].The G -symmetric monoidal G - ∞ -category of topological G -objects in ( Mfld n , (cid:116) ) is equivalent tothe G -symmetric monoidal G - ∞ -category Mfld G, (cid:116) defined in [Hor19, sec. 3.4]. Similarly, one can use the genuine operadic nerve construction to construct the G - ∞ -category ofpointed G -spaces with the G -coCartesian G -symmetric monoidal structure (see [BDG + ar, sec. B -coCartesian operads]). Lemma 5.3.
The G -symmetric monoidal G - ∞ -category of topological G -objects in (( Top ∗ ) op , ∨ ) is equivalent to the G - ∞ -category ( Top G, ∨∗ ) vop , where the G - ∞ -category of pointed G -spaces Top G ∗ is endowed with the G -coCartesian G -symmetric monoidal structure. Since the construction of the G -symmetric monoidal G - ∞ -category of topological G -objectsis functorial, we can apply it to the one point compactification functor, and get a G -symmetricmonoidal functor ( − ) + : Mfld G, (cid:116) → ( Top G, ∨∗ ) vop . (5.4)Next, we describe the G -symmetric monoidal functor Map ∗ ( − , X ).21 onstruction 5.5. Let X ∈ ( Top G ∗ ) [ G/G ] be a pointed topological G -space. Applying theparametrized Yoneda embedding of [BDG + X we get a G -functorMap ∗ ( − , X ) : ( Top G ∗ ) vop → Top G . By [Sha18, cor. 11.9] the functor Map ∗ ( − , X ) : ( Top G ∗ ) vop → Top G preserves G -limits. Sincethe G -symmetric monoidal structures on ( Top G ∗ ) vop and Top G are G -Cartesian it follows thatMap ∗ ( − , X ) extends to a G -symmetric monoidal functorMap ∗ ( − , X ) : ( Top G, ∨∗ ) vop → Top G, × . (5.6)Composing Eq. (5.4) and Eq. (5.6), we get a G -symmetric monoidal functorMap ∗ (cid:0) ( − ) + , X (cid:1) : Mfld G, (cid:116) ( − ) + −−−→ ( Top G, ∨∗ ) vop Map ∗ ( − ,X ) −−−−−−−→ Top G, × . (5.7) Construction 5.8.
Fixing an n -dimensional representation V , we can precompose Eq. (5.7)with the forgetful map from V -framed G -manifolds to G -manifolds. We can further restrict to V -framed G -disks and obtain Disk
G,V − fr, (cid:116) ⊂ Mfld
G,V − fr, (cid:116) → Mfld G, (cid:116) ( − ) + −−−→ ( Top G, ∨∗ ) vop Map ∗ ( − ,X ) −−−−−−−→ Top G, × . (5.9)We denote the composite by Ω V X , and it is an E V algebra in Top G . The underlying G -spaceof Ω V X is given by evaluating the functor Ω V X at V ∈ Disk
G,V − fr [ G/G ] , which is the G -spaceMap ∗ ( S V , X ) of pointed maps from the representation sphere S V = V + to X . Note that the G -space Map ∗ ( S V , X ) is equivalent to the V -fold loop space of X , which justifies the name Ω V X . Remark 5.10.
Restricting the G -symmetric monoidal functor of Eq. (5.7) to the fiber over G/G ∈ O opG defines a symmetric monoidal functor of ∞ -categoriesMap ∗ (cid:0) ( − ) + , X (cid:1) : Mfld G, (cid:116) ( − ) + −−−→ ( Top G, ∨∗ ) op Map ∗ ( − ,X ) −−−−−−−→ Top G, × . Note that, to obtain this symmetric monoidal functor alone, we can simply apply the operadicnerve construction of [Lur12, 2.1.1.23] to the corresponding symmetric monoidal topological cat-egories and functors. Indeed, we can identify the ∞ -categories and functors above as follows: • The symmetric monoidal ∞ -category Mfld G, (cid:116) is equivalent to the operadic nerve of thetopological category of smooth n -dimensional G -manifolds and G -equivariant smooth em-beddings, with symmetric monoidal structure given by disjoint unions. • The symmetric monoidal ∞ -category Top G, ∨∗ is equivalent to the operadic nerve of topo-logical category of pointed G -CW spaces with the coCartesian monoidal structure (givenby the wedge of pointed G -spaces). The operadic nerve of the opposite topological categoryis equivalent to (cid:16) Top G, ∨∗ (cid:17) op . • The symmetric monoidal ∞ -category Top G, × is equivalent to the operadic nerve of topo-logical category of G -CW spaces with the Cartesian monoidal structure (given by theproducts of G -spaces). • The symmetric monoidal functor ( − ) + : Mfld G, (cid:116) → ( Top G, ∨∗ ) op can be identified with theoperadic nerve of the one point compactification functor, as in Construction 5.1.22 The symmetric monoidal functor Map ∗ ( − , X ) : ( Top G, ∨∗ ) op → Top G, × can be identifiedwith the operadic nerve of the topological functor sending a pointed G -space Y to thespace of pointed maps Map ∗ ( Y, X ), with G -action given by conjugation.We can therefore identify the G -space Map ∗ ( M + , X ) with the space of pointed maps M + → X ,with G acting by conjugation. Remark 5.11.
Note that Construction 5.8 is functorial in X ∈ Top G, ∗ (cid:39) ( Top G ∗ ) [ G/G ] . Theparametrized Yoneda embedding j : Top G ∗ → Psh G ( Top G ) of [BDG + G -functor, and in particular defines a map between the fibers over G/G ∈ O opG , which is a functorof ∞ -categories Top G, ∗ → Fun G (( Top G ∗ ) vop , Top G ) , X (cid:55)→ Map ∗ ( − , X ) : ( Top G ∗ ) vop → Top G . This functor factors through the full subcategory of Fun G (( Top G ∗ ) vop , Top G ) spanned by G -functors preserving finite G -products. Let Fun × G (( Top G ∗ ) vop , Top G ) denote the ∞ -category of G -symmetric monoidal G -functors with respect to the G -Cartesian G -symmetric monoidal struc-tures. Since Fun × G (( Top G ∗ ) vop , Top G ) ⊂ Fun G (( Top G ∗ ) vop , Top G ) is equivalent to the full sub-category described above, the functor Map ∗ ( − , X ) lifts to Top G, ∗ → Fun × G (( Top G ∗ ) vop , Top G ) , X (cid:55)→ Map ∗ ( − , X ) : ( Top G ∗ ) vop → Top G . Precomposition with the G -symmetric monoidal functor Disk
G,V − fr, (cid:116) ⊂ Mfld
G,V − fr, (cid:116) → Mfld G, (cid:116) ( − ) + −−−→ ( Top G, ∨∗ ) vop defines a functor of ∞ -categories Top G, ∗ → Fun × G (( Top G ∗ ) vop , Top G ) → Fun ⊗ G ( Disk
G,V − fr , Top G ) , X (cid:55)→ Ω V X. V -fold loop maps as E V -algebras Let f : X → Y be a map of pointed G -spaces. From the functoriality of Construction 5.8 we getΩ V f : Ω V X → Ω V Y , which is a map of E V -algebras in Top G . Suppose Ω V Y is a G -commutativealgebra. That is, there is a G -commutative algebra A such that the following construction inSection 5.1 Disk
G,V − fr, (cid:116) → Fin G ∗ A −→ Top G is equivalent as a G -symmetric monoidal functor toΩ V Y : Disk
G,V − fr, (cid:116) → Top G . In this case, the map Ω V f can be considered as an E V -algebra in Top
G/A . Definition 5.12.
Let C ⊗ be a G -symmetric monoidal G - ∞ -category and A : Fin G ∗ → C ⊗ be a G -commutative algebra. Define C ⊗ /A → Fin G ∗ by applying the construction [Lur12, def. 2.2.2.1]for K = ∆ , S = Fin G ∗ and S × K → S the identity map. The following result on G -symmetric monoidal structure on parametrized slice categories isknown. 23 emma 5.13. The map C ⊗ /A → Fin G ∗ defines a G -symmetric monoidal G - ∞ -category, withunderlying G - ∞ -category equivalent to the parametized slice C /A → O opG of [Sha18, not. 4.29]. Lemma 5.14.
Let O ⊗ → Fin G ∗ be a G - ∞ -operad. The ∞ -category Alg O ( C ⊗ /A ) of O -algebras in C is equivalent to the slice ∞ -category Alg O ( C ) /A of O -algebras over A . In particular, taking C ⊗ = ( Top G ) × and A = Pic( Sp G ), we have an equivalence of ∞ -categories Alg E V (cid:16) ( Top G ) / Pic( Sp G ) (cid:17) (cid:39) Alg E V ( Top G ) / Pic( Sp G ) , by which we can consider Ω V f : Ω V X → Pic( Sp G ) as an E V -algebra in Top G/ Pic( Sp G ) . ApplyingProposition 4.4 with coefficients A = Ω V f , we get a natural equivalence (cid:90) − Th (Ω V f ) (cid:39) Th (cid:18)(cid:90) − Ω V f (cid:19) (5.15)of G -functors Mfld
G,V − fr → Sp G . In this subsection we describes the interaction between equivariant Thom spectra and equiv-ariant factorization homology. Our main result is Theorem 5.20, which describes the genuine G -factorization homology theory (cid:90) − Ω V f : Mfld
G,V − fr → Top G/ Pic( Sp G ) appearing at the right hand side of Eq. (5.15).In fact, this description works when Pic( Sp G ) is replaced by any G -commutative algebra B in G -spaces. For the next propositions we fix B ∈ CAlg G ( Top G ) to be a G -commutative algebrain G -spaces and Ω V f : Ω V X → B to be a map of E V -algebras. By Section 5.3, Ω V f can beconsidered as an E V -algebra in Top
G/B .We first state some properties of the forgetful G -functor Top
G/B → Top G . Lemma 5.16.
The forgetful G -functor fgt : Top
G/B → Top G preserves G -colimits. Lemma 5.17.
Let ( Top G ) × denote the G -Cartesian G -symmetric monoidal structure on Top G .The forgetful G -functor fgt : Top
G/B → Top G extends to a G -symmetric monoidal functorfgt : ( Top G ) × /B → ( Top G ) × . Note that the composition
Disk
G,V − fr, (cid:116) Ω V f −−−→ ( Top G ) × /B fgt −−→ ( Top G ) × is equivalent toΩ V X . Combining these two lemmas with Lemma 4.3, we get: Proposition 5.18.
Let Ω V f : Ω V X → B be a V -fold loop map. Then we have a natural equiv-alence fgt (cid:18)(cid:90) − Ω V f (cid:19) (cid:39) (cid:90) − Ω V X of G -functors Mfld
G,V − fr → Top G .
24e therefore know that for a V -framed G -manifold M ∈ Mfld
G,V − fr [ G/G ] , (cid:90) M Ω V f ∈ ( Top
G/B ) [ G/G ] (cid:39) ( Top G ) /B is given by a map of G -spaces (cid:90) M Ω V X → B. (5.19)Our next task is to describe this map. Consider id B as an object of Alg E V ( Top G ) /B ,and observe that the map Ω V f : Ω V X → B can be considered as a map (cid:15) : Ω V f → id B inAlg E V ( Top G ) /B (cid:39) Alg E V ( Top
G/B ). This map of E V -algebras induces a natural transformation (cid:15) ∗ : (cid:90) − Ω V f → (cid:90) − id B . Composing this with the forgetful functor fgt : Top
G/B → Top G , we get f gt ( (cid:15) ∗ ) = (Ω V f ) ∗ : (cid:90) − Ω V X → (cid:90) − B .In particular, for a V -framed G -manifold M , the morphism (cid:15) ∗ : (cid:90) M Ω V f → (cid:90) M id B is givenby the map of G -spaces (Ω V f ) ∗ : (cid:90) M Ω V X → (cid:90) M B over B . It follows that the G -map ofEq. (5.19) factors as (cid:90) M Ω V X (Ω V f ) ∗ −−−−−→ (cid:90) M B → B, where (cid:90) M B → B is given by (cid:90) M id B . Specializing to the G -commutative algebra B = Pic( Sp G )and combining with Eq. (5.15), we have therefore shown Theorem 5.20.
Let X be a pointed G -space and Ω V f : Ω V X → Pic( Sp G ) be a map of E V -algebras. Then for every V -framed G -manifold M , there is an equivalence of genuine G -spectra (cid:90) M Th (Ω V f ) (cid:39) Th (cid:18)(cid:90) M Ω V X (Ω V f ) ∗ −−−−−→ (cid:90) M Pic( Sp G ) → Pic( Sp G ) (cid:19) . Here, Th : Top G/ Pic( Sp G ) → Sp G is the parametrized Thom G -functor in Construction 3.10. Assume that G is a finite group and V is a finite dimensional G -representation. In this section,we prove Theorem 6.1, which deals with factorization homology when the algebra A is a Thomspectrum of a more highly commutative map than E V ; it is as commutative as a representationthat M embeds in. We apply Theorem 6.1 to compute the genuine equivariant factorizationhomology of certain Thom spectra. In Corollary 6.5, we compute the factorization homology ofthe Real bordism spectrum, M U R . In Corollary 6.6, we treat Eilenberg–MacLane spectra; seethe appendix by Hahn–Wilson for a computation of THR( H Z ). In Corollary 6.7, we compute C -relative THH (see [ABG + C ( H F ). Theorem 6.1.
Let A be the G -Thom spectrum of an E V ⊕ W -map, Ω V ⊕ W f : Ω V ⊕ W X → Pic( Sp G ) , ith π k ( X H ) = 0 for all subgroups H < G and k < dim(( V ⊕ W ) H ) . Let M be a G -manifoldof the same dimension as the representation V . Suppose that M × W embeds equivariantly in V × W , and that there is an equivariant embedding V (cid:44) → M (call its image D ). Then (cid:90) M × W A (cid:39) A ⊗ Σ ∞ + Map ∗ ( M + − D, Ω W X ) . Remark 6.2.
Recall that we use ⊗ to denote the smash product of ( G -)spectra, and Map ∗ todenote the G -space of non-equivariant based maps.A particularly useful corollary is obtained by setting W = R and M = S V . Corollary 6.3.
Let A be the G -Thom spectrum of an E V ⊕ R -map Ω V ⊕ R X → Pic( Sp G ) with π k ( X H ) = 0 for all subgroups H < G and k < dim( V H ) + 1 . Then (cid:90) S V × R A (cid:39) A ⊗ Σ ∞ + (Ω X ) . Proof of Theorem 6.1.
Denote the equivariant embedding by emb : M × W (cid:44) → V × W . Let M × W be ( V ⊕ W )-framed as a submanifold. Denote by f : X → B V ⊕ W Pic( Sp G ) the mapwhose Ω V ⊕ W -looping is Ω V ⊕ W f : Ω V ⊕ W X → Pic( Sp G ).Consider the following commutative diagram, where the first horizontal map is an equivalenceby Theorem 2.2. The right hand column uses the homeomorphism ( M × W ) + ∼ = Σ W ( M + ). (cid:82) M × W Ω V ⊕ W X ∼ (cid:47) (cid:47) (Ω V ⊕ W f ) ∗ (cid:15) (cid:15) Map ∗ (Σ W ( M + ) , X ) f ∗ (cid:15) (cid:15) (cid:82) M × W Pic( Sp G ) (cid:47) (cid:47) emb ∗ (cid:15) (cid:15) Map ∗ (Σ W ( M + ) , B V ⊕ W Pic( Sp G )) emb ∗ (cid:15) (cid:15) (cid:82) V × W Pic( Sp G ) (cid:47) (cid:47) ∼ (cid:15) (cid:15) Map ∗ ( S V ⊕ W , B V ⊕ W Pic( Sp G )) ∼ (cid:15) (cid:15) Pic( Sp G ) = (cid:47) (cid:47) Pic( Sp G )By Theorem 5.20, (cid:90) M × W A is the equivariant Thom spectrum of the left hand vertical compos-ite; thus it is equivalent to the equivariant Thom spectrum of the right hand vertical composite.Note that this composite is also equal toMap ∗ (Σ W ( M + ) , X ) emb ∗ (cid:47) (cid:47) Map ∗ ( S V ⊕ W , X ) f ∗ (cid:47) (cid:47) Map ∗ ( S V ⊕ W , B V ⊕ W Pic( Sp G )) (cid:39) Pic( Sp G )The map emb ∗ above is induced by the embedding emb : M × W (cid:44) → V × W , equivalently bythe Pontryagin-Thom collapse map associated to it, S V ⊕ W → Σ W ( M + ). We have an inclusionof a small disk D ∼ = V in M , and the cofiber sequenceΣ W ( M + − D ) Σ W i −→ Σ W ( M + ) −→ Σ W S V ∼ = S V ⊕ W is split (up to homotopy) by this Pontryagin-Thom collapse map, as the composite collapse( V × W ) + → ( M × W ) + → ( D × W ) + ∼ = ( V × W ) + is homotopic to the identity. Thus we havean equivalence 26 emb ∗ , i ∗ ) : Map ∗ (Σ W ( M + ) , X ) ∼ (cid:47) (cid:47) Map ∗ ( S V ⊕ W , X ) × Map ∗ (Σ W ( M + − D ) , X ) ∼ (cid:15) (cid:15) Map ∗ ( S V ⊕ W , X ) × Map ∗ ( M + − D, Ω W X )Furthermore, this equivalence fits in the following commutative diagram:Map ∗ (Σ W ( M + ) , X ) ( emb ∗ ,i ∗ ) (cid:47) (cid:47) ∼ (cid:47) (cid:47) emb ∗ (cid:15) (cid:15) Map ∗ ( S V ⊕ W , X ) × Map ∗ ( M + − D, Ω W X ) pr (cid:15) (cid:15) Map ∗ ( S V ⊕ W , X ) = (cid:47) (cid:47) f ∗ (cid:15) (cid:15) Map ∗ ( S V ⊕ W , X ) f ∗ (cid:15) (cid:15) Map ∗ ( S V ⊕ W , B V ⊕ W Pic( Sp G )) = (cid:47) (cid:47) Map ∗ ( S V ⊕ W , B V ⊕ W Pic( Sp G ))We have shown that (cid:90) M × W A is equivalent to the equivariant Thom spectrum of the left handvertical composite, thus it is also equivalent to the equivariant Thom spectrum of the right handvertical composite, which, by Corollary 3.17, is equivalent to A ⊗ Σ ∞ + Map ∗ ( M + − D, Ω W X ).Our first application computes the factorization homology of M U R . The Real bordism spec-trum M U R is the Thom spectrum of a map of C - E ∞ spaces BU R → Pic( Sp C ) (for example,as in Remark 13 of [HL18]). Since ( BU R ) e (cid:39) BU and ( BU R ) C (cid:39) BO , the C -space BU R is C -connected. Lemma 6.4. If X is a G -connected E V -algebra, then π k ( B V X ) = 0 for k ≤ dim( V H ) . Thus,the connectivity condition in Theorem 6.1 or Corollary 6.3 is satisfied when X is G -connected.Proof. We say that a G -space X is V -connected if π k ( X ) = 0 for k ≤ dim( V H ). The V -folddelooping can be computed by the monadic bar construction B V X = B (Σ V , D V , X ), whereD V is the monad associated to the little V -disk operad. Since fixed points commutes withgeometric realization, it suffices to show that each Σ V D V X is V -connected. This follows fromthat (Σ V D V X ) H ∼ = Σ V H (D V X ) H and that D V X is G -connected for a G -connected X (for theproof, see [Zou20, Lemma 8.4]).Corollary 6.3 and Lemma 6.4 combine to give Corollary 6.5. (cid:90) S V × R M U R (cid:39) M U R ⊗ Σ ∞ + ( B V BU R ) In particular,
THR(
M U R ) (cid:39) M U R ⊗ Σ ∞ + ( B σ BU R )We now use Theorem 6.1 and Corollary 6.3, along with theorems of Behrens–Wilson [BW18]and Hahn–Wilson [HW18] which show that certain equivariant Eilenberg–MacLane spectra areThom spectra, to compute equivariant factorization homology with coefficients in these spectra.Take G = C . Let σ be its sign representation, ρ ∼ = σ + 1 its 2-dimensional regular representa-tion, and λ ∼ = 2 σ its two-dimensional rotation representation. Let THR denote Real topologicalHochschild homology [DMPR17], which is equivalent to (cid:82) S σ by [Hor19, remark 7.1.2].27 orollary 6.6. We have(1)
THR( H F ) (cid:39) H F ⊗ Σ ∞ + (Ω S ρ +1 ) (cid:39) H F ⊗ Σ ∞ + (Ω σ S λ +1 ) (2) THR( H Z (2) ) (cid:39) H Z (2) ⊗ Σ ∞ + (Ω σ ( S λ +1 (cid:104) λ + 1 (cid:105) )) (3) (cid:82) S λ H F (cid:39) H F ⊗ Σ ∞ + S λ +1 (4) (cid:82) S λ H Z (2) (cid:39) H Z (2) ⊗ Σ ∞ + ( S λ +1 (cid:104) λ + 1 (cid:105) ) Here, S λ +1 (cid:104) λ + 1 (cid:105) is the fiber of the unit map S λ +1 → K ( Z , λ + 1) = Ω ∞ Σ λ +1 H Z .Proof. By Theorem 1.2 of [BW18], the Eilenberg–MacLane spectrum H F is equivariantly theThom spectrum of a ρ -fold loop map Ω ρ S ρ +1 → BO C . As the inclusion BO C → Pic( Sp C )is a map of G -symmetric monoidal G -spaces, H F is also the Thom spectrum of a ρ -fold loopmap Ω ρ S ρ +1 → Pic( Sp C ). Thus Corollary 6.3 yields the first equivalence of (1), with V = σ and W = R . Furthermore, Hahn and Wilson [HW18] have shown that H F is equivariantly theThom spectrum of a ( λ + 1)-fold loop map Ω λ S λ +1 → Pic( S (2) ), and that H Z (2) is equivariantlythe Thom spectrum of a ( λ + 1)-fold loop map Ω λ ( S λ +1 (cid:104) λ + 1 (cid:105) ) → Pic( S ( p ) ). Corollary 6.3 withRemark 3.3 yields (3) and (4), with V = λ and W = R .For the second equivalence of (1) and for (2), there is an isomorphism λ + 1 ∼ = 2 σ + 1 and anequivariant embedding S σ × R (cid:44) → σ + 1, thus an equivariant embedding S σ × ρ (cid:44) → λ + 1. Weintend to use Theorem 6.1 with M = S σ , V = σ , W = ρ and X = B λ +1 Ω λ S λ +1 or X = B λ +1 Ω λ ( S λ +1 (cid:104) λ + 1 (cid:105) ) respectively.To check the assumptions, by Lemma 6.4 it suffices to show that Ω λ S λ +1 and Ω λ ( S λ +1 (cid:104) λ + 1 (cid:105) )are C -connected. This is true by Lemma 2.19, since dim(( S λ ) e ) = 2 and dim(( S λ ) C ) = 0; itcan also be verified that S λ +1 and S λ +1 (cid:104) λ + 1 (cid:105) are C -connected and underlying 2-connected.So, from Theorem 6.1 we obtain (cid:90) S σ × ρ H F (cid:39) H F ⊗ Σ ∞ + Map ∗ ( σ + , Ω ρ B λ +1 Ω λ S λ +1 ); (cid:90) S σ × ρ H Z (2) (cid:39) H Z (2) ⊗ Σ ∞ + Map ∗ ( σ + , Ω ρ B λ +1 Ω λ ( S λ +1 (cid:104) λ + 1 (cid:105) )) . To simplify, we have Ω ρ B λ +1 Ω λ S λ +1 (cid:39) B σ Ω λ S λ +1 (cid:39) Ω σ S λ +1 , since Ω σ S λ +1 is C -connected.As σ is equivariantly contractible, Map ∗ ( σ + , Ω σ S λ +1 ) (cid:39) Ω σ S λ +1 . The second equivalence issimilar.From either of the two descriptions of THR( H F ) in Corollary 6.6, one can use the Snaithsplitting to compute THR( H F ) as an H F -module. From either Σ ∞ + ΩΣ S ρ (cid:39) ⊕ k ≥ S kρ orΣ ∞ + Ω σ Σ σ S ρ (cid:39) ⊕ k ≥ S kρ , we haveTHR( H F ) (cid:39) H F ⊗ Σ ∞ + Σ ∞ + ΩΣ S ρ (cid:39) ⊕ k ≥ Σ kρ H F . This recovers the additive part of THR( H F ) in [DMPR17, Theorem 5.18]. They use thismodule structure and the fact that THR( H F ) is an associative H F -algebra to promote this toan equivalence of C -ring spectra. In particular, π (cid:70) THR( H F ) ∼ = π (cid:70) ( H F )[ x ρ ] . We can also compute THH C ( H F ). 28 orollary 6.7. THH C ( H F ) (cid:39) H F ⊗ Σ ∞ + (Ω S )Note that this only computes the underlying (non-equivariant) spectrum of THH C ( H F ).Section 5 of [AGH +
20] uses Theorem 3.2 from our paper in a somewhat different approach tocompute THH C ( H F ) as a C -spectrum. Proof.
Let g denote the generator of C , and for a C -space X , let L g X denote the twisted freeloop space { γ : I → X | γ (1) = gγ (0) } . Let S rot denote the circle, with C acting by rotation.Note that S rot is a R -framed C -manifold.By [Hor19, proposition 7.2.2], for A a C n -ring spectrum, THH C n is given by the C n -geometricfixed points of (cid:82) S rot A . Using the description of H F in the proof of Corollary 6.6 and Theo-rem 5.20, we have (cid:90) S rot H F (cid:39) Th (cid:32)(cid:90) S rot Ω ρ S ρ +1 → Pic( Sp C ) (cid:33) . By Theorem 2.2, we can identify the G -spaces: (cid:90) S rot Ω ρ S ρ +1 (cid:39) Map( S rot , Ω σ S ρ +1 ) (6.8)Moreover, the Thom spectrum (cid:82) S rot H F has an H F -orientation given by the composite H F ⊗ (cid:90) S rot H F id ⊗ i (cid:47) (cid:47) H F ⊗ H F mult (cid:47) (cid:47) H F Here, the map i : (cid:82) S rot H F → H F exists because H F is commutative. For example, we cantake i to be induced on factorization homology by the embedding S rot × R → λ . By the Thomisomorphism and Eq. (6.8), we have H F ⊗ (cid:90) S rot H F (cid:39) H F ⊗ Σ ∞ + Map( S rot , Ω σ S ρ +1 )Upon passage to geometric fixed points, we obtainΦ C ( H F ) ⊗ THH C ( H F ) (cid:39) Φ C ( H F ) ⊗ Σ ∞ + Map C ( S rot , Ω σ S ρ +1 )Because Φ C ( H F ) (cid:39) H F [ t ], where t is in degree 1, we have that H F ⊗ THH C ( H F ) (cid:39) H F ⊗ Σ ∞ + ( L g Ω σ S ρ +1 )By Corollary 16 of [KK10], H F ⊗ Σ ∞ + ( L g Ω σ S ρ +1 ) (cid:39) H F ⊗ Σ ∞ + ( L Ω S ) (cid:39) H F ⊗ H F ⊗ Σ ∞ + (Ω S )Note that THH C ( H F ) and H F ⊗ Σ ∞ + (Ω S ) are both H F -modules (the former is in factan algebra over H F , as H F is commutative.) They are equivalent after smashing with H F ,therefore they are equivalent. 29 Some results in parametrized category theory
In this section we gather the results used in Section 3, with partial proofs. Much of this sectionhas to do with parametrized symmetric monoidal structures. However, a complete treatment ofthis subject is beyond the scope of this paper. We will therefore consider only G - ∞ -categoriesand G -symmetric monoidal structures (with the exception of Section 7.1). In this subsection, we state the result that we need about parameterized straightening/unstraightening.The results are stated for S - ∞ -categories, i.e., coCartesian fibrations over a fixed ∞ -category S .Taking S = O opG recovers the notion of G - ∞ -categories, used throughout this paper. Definition 7.1. An S -fibration X (cid:16) C (see [Sha18, def. 7.1]) is an S -right fibration if X [ s ] (cid:16) C [ s ] is a right fibration for every s ∈ S . Definition 7.2.
Suppose C is an S - ∞ -category. Let ( C at ∞ ,S ) / C denote the S -slice category (see[Sha18, not. 4.29]). Let ( C at ∞ ,S ) S − right/ C ⊆ ( C at ∞ ,S ) / C denote the full subcategory spanned by s -right fibrations ( X (cid:16) C × S s ) ∈ (cid:0) C at ∞ ,s (cid:1) / C × S s (cid:39) (cid:0) ( C at ∞ ,S ) / C (cid:1) [ s ] . Theorem 7.3.
Suppose C (cid:16) S is an S -category. Then there is a natural equivalence of S -categories Y : Psh S ( C ) ∼ −→ ( C at ∞ ,S ) S − right/ C . If x ∈ C [ s ] then Y sends the representable presheaf ι ( x ) to the s -right fibration (cid:0) C /x (cid:16) C × S s (cid:1) ∈ (cid:0) C at ∞ ,s (cid:1) s − right/ C × S s (cid:39) (cid:16) ( C at ∞ ,S ) S − right/ C (cid:17) [ s ] . For the following statements, let B ∈ Top S be an S - ∞ -groupoid and ι : B → Psh S ( B ) itsparametrized Yoneda embedding ( [BDG + Lemma 7.4.
Suppose X (cid:16) B is an S -right fibration of S - ∞ -categories. Then X is an S - ∞ -groupoid.Proof. We have to show that the coCartesian fibration X (cid:16) S is a left fibration. By [Lur09,prop. 2.4.2.4] it is enough to show that each fiber X [ s ] is a Kan complex. [Lur09, prop. 2.4.2.4]also guarantees that B [ s ] is a Kan complex. Since X [ s ] → B [ s ] is a right fibration over a Kancomplex, we deduce that X [ s ] is indeed a Kan complex. Corollary 7.5.
There is a natural equivalence of S -categories Y : Psh S ( B ) ∼ −→ ( Top S ) /B . If x ∈ B [ s ] , then Y sends the representable presheaf ι ( x ) to (cid:0) B /x (cid:16) B × S s (cid:1) ∈ (cid:0) Top s (cid:1) /B × S s (cid:39) (cid:16) ( Top S ) /B (cid:17) [ s ] . .2 Parametrized presheaves and Day convolution Let C be a G - ∞ -category. If C has a G -symmetric monoidal structure C ⊗ , then Psh G ( C ) has a G -symmetric monoidal structure Psh G ( C ) ⊗ → Fin G ∗ given by the G -Day convolution of [BDG + ar]with respect to G -symmetric monoidal structure on C and the Cartesian G -symmetric monoidalstructure on Top G . Our goal in this subsection is Proposition 7.8; informally, it states thatparametrized left Kan extension along the Yoneda embedding j : C → Psh G ( C ) takes a G -symmetric monoidal functor from C to a G -symmetric monoidal functor from Psh G ( C ).We will need the following statement, which currently does not appear in the literature. Lemma 7.6.
The parametrized Yoneda embedding j : C → Psh G ( C ) extends to a G -symmetricmonoidal G -functor j ⊗ : C ⊗ → Psh G ( C ) ⊗ . We use the notion of a G -cocomplete G - ∞ -category from [Sha18, def. 5.12], and the notionof a distributive G -symmetric monoidal G - ∞ -category from [BDG + ar] . Note that the essen-tially unique G -symmetric monoidal structure of Sp G of [Nar17, cor. 3.28] is distributive byconstruction.Let F ⊗ : C ⊗ → E ⊗ be a G -symmetric monoidal functor, with underlying G -functor F : C → E .If the underlying G - ∞ -category E is G -cocomplete, then F ⊗ admits a G -operadic left Kanextension along j ⊗ , ( j ⊗ ) ! F ⊗ : Psh G ( C ) ⊗ → E ⊗ , constructed in [BDG + ar]. By construction, ( j ⊗ ) ! F ⊗ is a lax G -symmetric monoidal functor. Inorder to show that ( j ⊗ ) ! F ⊗ is in fact G -symmetric monoidal, we use the following propositionfrom [BDG + ar]: Proposition 7.7 ([BDG + ar]) . Let F : C ⊗ → E ⊗ , p ⊗ : C ⊗ → D ⊗ be lax G -symmetric monoidalfunctors, and ( p ⊗ ) ! F ⊗ : D ⊗ → E ⊗ the G -operadic left Kan extension of F ⊗ along p ⊗ . Assumethat the G -symmetric monoidal structure of E ⊗ is distributive. Then the underlying G -functorof ( p ⊗ ) ! F ⊗ is equivalent to the G -left Kan extension of F : C → E along p : C → D . If follows that if the G -symmetric monoidal structure of E ⊗ is distributive, then the G -functor j ! F : Psh G ( C ) → E can be extended to a lax G -symmetric monoidal functor ( j ! F ) ⊗ : Psh G ( C ) ⊗ → E ⊗ given by the G -operadic left Kan extension ( j ⊗ ) ! F ⊗ . Proposition 7.8.
Let F ⊗ : C ⊗ → E ⊗ be a G -symmetric monoidal functor from a small G -symmetric monoidal ∞ -category C ⊗ to a distributive G -symmetric monoidal G - ∞ -category E ⊗ ,with G -cocomplete underlying G - ∞ -category E . Then ( j ! F ) ⊗ : Psh G ( C ) ⊗ → E ⊗ is a G -symmetricmonoidal functor. In the course of the proof we use the following notation.
Notation 7.9.
For C ⊗ → Fin G ∗ a G -symmetric monoidal G - ∞ -category and I ∈ Fin G ∗ :1. Let C ⊗ I denote the fiber of C ⊗ → Fin G ∗ over I .2. For I = ( U → G/H ) let C ⊗ denote the G/H -category constructed by pulling back along σ : G/H → Fin G ∗ , see [Hor19, def. B.0.4]. See [Nar17, def. 3.15] for a definition of a distributive parametrized functor. roof. We have to check that the lax G -symmetric monoidal functor ( j ! F ) ⊗ : Psh G ( C ) ⊗ → E ⊗ is G -symmetric monoidal. The idea of the proof is simple: reduce to the case of parametrizedrepresentable presheaves, where the claim is clear. The argument is a bit convoluted due to theinvolved definition of a distributive G -symmetric monoidal structure.Let I ∈ Fin G ∗ , I = ( U → G/H ). Parametrized Day convolution defines a distributive G -symmetric monoidal structure on Psh G ( C ), so the G/H -functor ⊗ I : Psh G ( C ) ⊗ (cid:39) (cid:89) I Psh G ( C ) × U → Psh G ( C ) × G/H of [Hor19, def. B.0.11] is distributive (see [Nar17, def. 3.15]). Here (cid:81) I : C at U ∞ → C at G/H ∞ is theright adjoint of ( − × G/H U ) : C at G/H ∞ → C at U ∞ , and the equivalence is homotopy inverse to theparametrized Segal map of [Hor19, rem. B.0.9].We have to show that for every X ∈ Psh G ( C ) ⊗ I (cid:39) (cid:0) Psh G ( C ) ⊗ (cid:1) [ G/H ] the lax structure map ⊗ I ( j ! F ⊗ ( X )) → j ! F ( ⊗ I X ) (7.10)is an equivalence. We first reduce to representable presheaves. By Corollary 7.23 we can write X as a U -colimit X (cid:39) U − colim ( jχ ) for some U -diagram χ : K → C × U . Inspect the followingdiagram: ⊗ I ( j ! F ⊗ ( X )) j ! F ( ⊗ I ( X )) ⊗ I ( j ! F ⊗ ( U - colim −−−→ ( jχ ))) j ! F ( ⊗ I ( U - colim −−−→ ( jχ ))) ⊗ I ( U - colim −−−→ ( j ! F ⊗ ( jχ ))) j ! F ( G/H - colim −−−→ ( ⊗ I ( jχ ))) G/H - colim −−−→ ( ⊗ I ( j ! F ⊗ ( jχ ))) G/H - colim −−−→ ( j ! F ⊗ ( ⊗ I ( jχ ))) . ∼ ∼∼ (1) ∼ (2) ∼ (2) ∼ (1) The commutativity of the diagram follows from the naturality of the lax structure map (7.10).The equivalences marked (1) follow from the fact that j ! F strongly preserves G -colimits, and theequivalences marked (2) follow from the distributivity of G -Day convolution.By naturality of the lax structure map (7.10) it is therefore enough to show that the laxstructure map ⊗ I ( j ! F ⊗ ( j ⊗ X )) → j ! F ( ⊗ I j ⊗ X ) is an equivalence for X ∈ C ⊗ I . This follows frominspecting the following diagram ⊗ I j ! F ( j ⊗ X ) ⊗ I F ⊗ ( X ) j ! F ( ⊗ I ( j ⊗ X )) j ! F ( j ( ⊗ I X )) F ( ⊗ I X ) , (1) (3) ∼ (4)(2) ∼ as we now explain. We wish to show that the diagonal map marked (1) is an equivalence.Since the parametrized Yoneda embedding j is G -symmetric monoidal, its lax structure map ⊗ I ( j ⊗ X ) → j ( ⊗ I X ) is an equivalence. It follows that it is still an equivalence after applying j ! F , showing that the map (2) is also an equivalence. Therefore it is enough to show that themap marked (3) is an equivalence. Note that the map marked (3) is the lax structure map of the32omposition j ! F ⊗ ◦ j ⊗ . We now use the fact that the parametrized Yoneda embedding is fullyfaithful ([BDG + F → j ! F ◦ j is a natural equivalence. It follows that the left-pointinghorizontal maps in the diagram are equivalences (the square commutes by naturality). Hence itis enough to prove that the map marked (4) is an equivalence, which is clear since it is the laxstructure map of a G -symmetric monoidal functor F ⊗ . G - ∞ -subgroupoid and G -symmetric monoidal structures We recall the definition of the maximal G - ∞ -subgroupoid of an G - ∞ -category C , and verify thata G -symmetric monoidal structure on C induces a G -symmetric monoidal on its maximal G - ∞ -subgroupoid. Recall that a G - ∞ -groupoid, or a G -space, is a G - ∞ -category G (cid:16) O opG in whichevery edge is coCartesian ([BDG + G (cid:16) O opG is a left fibration.Let C (cid:16) O opG be a G - ∞ -category. The maximal G -subgroupoid of C is the subcategory C (cid:39) ⊂ C spanned by all objects and all coCartesian edges. By construction, C (cid:39) ⊂ C is the maximal G - ∞ -subcategory ([BDG + G - ∞ -groupoid. Note that for every orbit W ∈ O opG ,the morphisms in the fiber ( C (cid:39) ) [ W ] are coCartesian edges in C over id W , which by [Lur09, prop.2.4.1.5] are exactly equivalences over id W . Hence we have ( C (cid:39) ) [ W ] = ( C [ W ] ) (cid:39) as subsets of C [ W ] . Construction 7.11.
Suppose C ⊗ (cid:16) Fin G ∗ is a G -symmetric monoidal G - ∞ -category. Define C ⊗ coCart ⊂ C ⊗ as the full subcategory spanned by the coCartesian morphisms over Fin G ∗ . Lemma 7.12.
The composition C ⊗ coCart ⊂ C ⊗ (cid:16) Fin G ∗ is a coCartesian fibration which definesa G -symmetric monoidal structure on the full G - ∞ -subgroupoid of C . During the proof we use the notation C ⊗ I for the fiber of C ⊗ over I ∈ Fin G ∗ , see Notation 7.9. Proof.
The map C ⊗ coCart → Fin G ∗ is a left fibration by [Lur09, cor. 2.4.2.5]. Pulling back C ⊗ coCart ⊂ C ⊗ → Fin G ∗ over the G -functor σ
Let Sp G (cid:16) O opG be the G - ∞ -category of genuine G -spectra, see [Nar17]. Thereis an essentially unique G -symmetric monoidal structure on Sp G with unit the sphere spec-trum, see [Nar17, cor. 3.28]. By construction, ( Sp G ) ⊗ is a distributive G -symmetric monoidal33 - ∞ -category (in other words, parametrized smash products distribute over parametrized colim-its). Informally, this G -symmetric monoidal structure encodes smash products and Hill-Hopkin-Ravenel norms. The G -symmetric monoidal structure on Sp G induces a G -symmetric monoidalstructure on its maximal subgroupoid ( Sp G ) (cid:39) . G -symmetric monoidal categories and G -commutative algebras Recall that a G -symmetric monoidal category is an G -commutative monoid in C at G ∞ (see [Nar17,sec. 3.1]). Theorem 7.14 ([Nar17, thm. 2.32]) . There is an equivalence of ∞ -categories CMon G ( C at G ∞ ) (cid:39) CAlg G (( C at G ∞ ) × ) between the ∞ -category CMon G ( C at G ∞ ) of G -symmetric monoidal G - ∞ -categories and the ∞ -category CAlg G (( C at G ∞ ) × ) of G -commutative algebras in C at G ∞ , with respect to the G -Cartesian G -symmetric monoidal structure. Lemma 7.15.
The equivalence of [Nar17, thm. 2.32] restricts to an equivalence
CMon G ( Top G ) (cid:39) CAlg G (( Top G ) × ) between the ∞ -category CMon G ( Top G ) of G -symmetric monoidal G - ∞ -groupoids and the ∞ -category CAlg G (( Top G ) × ) of G -commutative algebras in Top G . Example 7.16.
The Picard G -space Pic( C ) admits G -symmetric monoidal structure (see Sec-tion 3.1), and therefore defines a G -commutative monoid in Top G . Applying the previous lemmawe can consider Pic( C ) as a G -commutative algebra in ( Top G ) × . G -symmetric monoidal categories Definition 7.17.
Let C ⊗ be a G -symmetric monoidal G - ∞ -category and A : Fin G ∗ → C ⊗ a G -commutative algebra. Define C ⊗ /A → Fin G ∗ by applying the construction [Lur12, def. 2.2.2.1]for K = ∆ , S = Fin G ∗ and S × K → S the identity map. The following statement is a result of [Lur12, sec. 2.2.2] together with the G -Segal conditionsof a G -symmetric monoidal ∞ -category. Proposition 7.18.
The map C ⊗ /A → Fin G ∗ defines a G -symmetric monoidal G - ∞ -category, withunderlying G - ∞ -category equivalent to the parametized slice C /A → O opG of [Sha18, not. 4.29]. We will use the following description of O -algebras in C ⊗ /A . Proposition 7.19.
Let O ⊗ → Fin G ∗ be a G - ∞ -operad. The ∞ -category Alg O ( C ⊗ /A ) of O -algebrasin C is equivalent to the slice ∞ -category Alg O ( C ) /A of O -algebras over A . Theorem 7.20.
Suppose B ⊗ (cid:16) Fin S ∗ is an S -symmetric monoidal S - ∞ -groupoid. Then thenatural equivalence of Corollary 7.5 extends to an S -symmetric monoidal equivalence Psh S ( B ) ⊗ ∼ −→ ( Top S ) ⊗ /B , where the S -symmetric monoidal structure on the right hand side is given by Section 7.5 and the S -symmetric monoidal structure on the left hand side is given by S -Day convolution ([BDG + ar]). .7 Parametrized presheaves and parametrized colimits In this subsection we state some properties of the parametrized presheaf category, defined in[BDG + G ( C ) = Fun G ( C vop , Top G ) denote the parametrized presheaf G - ∞ -category of a small G - ∞ -category C → O opG , and let j : C (cid:44) → Psh G ( C ) be the parametrized Yoneda embedding of[BDG + G -functors out of Psh G ( C ) using parametrized G -left Kan extension (see[Sha18, sec. 10] and [Nar16, def. 2.12]). Specifically, we will use [Sha18, thm. 11.5]. LetFun LG ( C , D ) ⊆ Fun G ( C , D ) denote the full subcategory of G -functors which strongly preserve G -colimits ([Sha18, def. 11.2]). Theorem 7.21 (Shah) . Let C be a G - ∞ -category and let E be a G -cocomplete G - ∞ -category.Then restriction along the G -Yoneda embedding j : C → Psh G ( C ) defines an equivalence of ∞ -categories Fun LG (Psh G ( C ) , E ) ∼ −→ Fun G ( C , E ) with inverse given by G -left Kan extension along j . Unsurprisingly, every parametrized presheaf is equivalent to a parametrized colimit of rep-resentable presheaves. Before giving a formal statement we recall the relevant definition ofparametrized colimits in Psh G ( C ). Let G/H ∈ O opG be an orbit and I → G/H be a
G/H -category. Keeping in mind the equivalencePsh G ( C ) [ G/H ] (cid:39) Fun G ( G/H,
Psh G ( C )) (cid:39) Fun
G/H ( G/H,
Psh G ( C ) × G/H ) , we define a G/H -functor∆ I : Psh G ( C ) [ G/H ] (cid:39) Fun
G/H ( G/H,
Psh G ( C ) × G/H ) → Fun
G/H ( I, Psh G ( C ) × G/H ) , induced by precomposition with the structure map I → G/H . By definition
G/H -colimits inPsh G ( C ) along I -shaped diagrams are given by the left adjoint G/H - colim −−−→ : Fun G/H ( I, Psh G ( C ) × G/H ) (cid:28) Psh G ( C ) [ G/H ] : ∆ I . See [Nar17, def. 1.15] for details.
Lemma 7.22.
Let X ∈ Psh G ( C ) be a presheaf over G/H ∈ O opG . Then X is equivalent to a G/H -colimit of a diagram of representable presheaves K χ −→ C × G/H j × G/H −−−−−→
Psh G ( C ) × G/H for some
G/H -functor χ : K → C × G/H .Proof.
By the G -Yoneda lemma, [Sha18, lem. 11.1], the identity functor Id : Psh G ( C ) → Psh G ( C )is a G -left Kan extension of j along itself. By [Sha18, thm. 10.4] we can express the value ofthis G -left Kan extension on X as a G/H -colimit X = Id ( X ) (cid:39) G/H - colim −−−→ (cid:18) C /X → C × G/H j × G/H −−−−−→
Psh G ( C ) × G/H (cid:19) , where C /X = C × Psh G ( C ) Psh G ( C ) /X is the pullback of the G -slice category Psh G ( C ) /X ([Sha18,not. 4.29]) along the G -Yoneda embedding j . 35 orollary 7.23. Let U be a finite G -set and let X : U → Psh G ( C ) × U be U -functor. Then thereexists a U -functor χ : K → C × U , such that the U -colimit of K χ −→ C × U j × U −−−→ Psh G ( C ) × U is equivalent to X .Proof. Decompose U = (cid:96) W ∈ Orbit( U ) W into orbits. The result follows from Lemma 7.22 andthe equivalence (cid:89) W C at W ∞ ∼ −→ C at U ∞ , ( C W (cid:16) W ) W ∈ Orbit( U ) (cid:55)→ (cid:32)(cid:97) W C W (cid:16) (cid:97) W W = U (cid:33) , where coproducts and products are indexed over W ∈ Orbit( U ). G - ∞ -category of topological G -objects In this subsection, we review the genuine operadic nerve construction of [Bon19], for the case ofa symmetric monoidal topological category (considered as a multi-colored topological operad).Let C be a topological category, i.e., a category enriched in Top, the category of compactlygenerated (weak) Hausdorff topological spaces. Let ⊗ be an enriched symmetric monoidal struc-ture on C with unit I . We refer to such an enriched symmetric monoidal category as a symmetricmonoidal topological category .The main construction of this section, Construction 7.27, associates a topological category C ⊗ over Fin G ∗ to a symmetric monoidal topological category C . The main theorem of this subsectionallows us to quickly construct G -symmetric monoidal G - ∞ -categories as N ( C ⊗ ), the coherentnerve of C ⊗ (see [Lur09, def. 1.1.5.5]). Theorem 7.24.
For C be a symmetric monoidal topological category, let N ( C ⊗ ) denote thecoherent nerve of C ⊗ , the topological category of Construction 7.27. Then N ( C ⊗ ) → Fin G ∗ is a G -symmetric monoidal G - ∞ -category. We start with some preliminaries needed for Construction 7.27.Let I be a small category. Make Fun( I, C ) into a topological category by endowing the setNat( F, G ) of natural transformations between
F, G : I → C with the topology of the equalizerof (cid:81) i ∈ I Map C ( F i, Gi ) ⇒ (cid:81) φ : i → i (cid:48) Map C ( F i, Gi (cid:48) ), with one map induced by precomposition with F ( φ ) and the other by postcomposition with G ( φ ).Recall that a covering map p : I → J induces a monoidal pushforward functor p ⊗∗ : Fun( I, C ) → Fun( J, C ), see [Rub17, sec. 8.5]. Proposition 7.25.
The monoidal pushforward functor p ⊗∗ : Fun( I, C ) → Fun( J, C ) is a topolog-ical functor. See the following mathoverflow post: https://mathoverflow.net/questions/51783/enriched-monoidal-categories [hm]. In Kelly’s book as linked in the post, the tensor product of V -enriched categories is definedon page 12. roof. Let
F, G ∈ Fun( I, C ) be functors. The mapping space Nat( p ⊗∗ F, p ⊗∗ G ) is obtained as theequalizer of (cid:89) j ∈ J Map C ( ⊗ pi = j F i, ⊗ pi = j Gi ) ⇒ (cid:89) φ : j → j (cid:48) Map C ( ⊗ pi = j F i, ⊗ pi (cid:48) = j (cid:48) Gi (cid:48) ) . Because C is a symmetric monoidal topological category, the maps (cid:89) pi = j Map C ( F i, Gi ) → Map C ( ⊗ pi = j F i, ⊗ pi = j Gi )are continuous, and similarly with i (cid:48) , j (cid:48) . Thus Nat( F, G ) → Nat( p ⊗∗ F, p ⊗∗ G ) is continuous.Let U be a G -set. The action groupoid of U , denoted B U G , has objects x ∈ U and morphismsHom( x, y ) = { g ∈ G | gx = y } . A map of G -sets f : U → V induces a functor on the actiongroupoids, which we denote by Bf : B U G → B V G .Note that the action groupoid of a pullback of G -sets is isomorphic to the strict pullback ofaction groupoids, P ∼ = X × Z Y ⇒ B P G ∼ = B X G × B Z G B Y G. The G -category of finite pointed G -sets. Let
Fin G ∗ be the G - ∞ -category of finite pointed G -sets, see [Nar17, def. 2.14]. We will use the following model of Fin G ∗ : an object of Fin G ∗ isgiven by a map of finite G -sets U → O where O is an orbit (a transitive G -set). A morphism ψ : I → I in Fin G ∗ from I = ( U → O ) to I = ( U → O ) is given by a span of arrows (adiagram of G -sets) of the form U (cid:15) (cid:15) U (cid:48) (cid:15) (cid:15) f (cid:111) (cid:111) p (cid:47) (cid:47) U (cid:15) (cid:15) O O (cid:47) (cid:47) ϕ (cid:111) (cid:111) = (cid:47) (cid:47) O (7.26)where the induced map U (cid:48) → ϕ ∗ U is injective (or equivalently there exists another G -set U (cid:48)(cid:48) with a G -map U (cid:48)(cid:48) → ϕ ∗ U that induces an isomorphism U (cid:48) (cid:96) U (cid:48)(cid:48) ∼ = −→ ϕ ∗ U ). Constructing a topological category over Fin G ∗ . Our goal is to construct a topologicalcategory C ⊗ with a functor C ⊗ → Fin G ∗ , whose coherent nerve ([Lur09, def. 1.1.5.5]) will be the G -symmetric monoidal G - ∞ -category of topological G -objects in C . Construction 7.27.
Let C be a symmetric monoidal topological category. We construct atopological category C ⊗ over Fin G ∗ as follows. • An object x ∈ C ⊗ over I ∈ Fin G ∗ , I = ( U → O ) is a functor x : B U G → C . • Let x ∈ C ⊗ be an object over I = ( U → O ) and ψ : I → I a morphism of Fin G ∗ given bythe diagram (7.26). Denote the composition B U (cid:48) G Bf −−→ B U F x −→ C by f ∗ x ∈ Fun( B U (cid:48) G, C ),and its monoidal pushforward along p : U (cid:48) → U by p ⊗∗ f ∗ x : B U G → C .Suppose we are also given y ∈ C ⊗ over I = ( U → O ). Define the space of morphisms of C ⊗ from x to y over ψ to be Map ψ C ⊗ ( x, y ) = Nat( p ⊗∗ f ∗ x, y ).37 Define the mapping space in the topological category C ⊗ as Map C ⊗ ( x, y ) = (cid:96) ψ Map ψ C ⊗ ( x, y ),where the coproduct is indexed over all ψ ∈ Hom
Fin G ∗ ( I , I ). • Let x , x , x ∈ C ⊗ be object over I , I , I ∈ Fin G ∗ . In what follows we construct acontinuous maps Map ψ C ⊗ ( x , x ) × Map ψ C ⊗ ( x , x ) → Map ψ ψ C ⊗ ( x , x ) , for each ψ : I → I , ψ : I → I in Fin G ∗ . This allows us to define the composition mapMap C ⊗ ( x , x ) × Map C ⊗ ( x , x ) → Map C ⊗ ( x , x )as the coproduct of these maps. In other words, we make sure that C ⊗ → Fin G ∗ respectscompositions by definition.We first choose an explicit description of the composition ψ ψ : I → I . Let I = ( U → O ) , I = ( U → O ) , I = ( U → O ) and ψ = U (cid:15) (cid:15) U (cid:48) (cid:15) (cid:15) f (cid:111) (cid:111) p (cid:47) (cid:47) U (cid:15) (cid:15) O O (cid:47) (cid:47) ϕ (cid:111) (cid:111) = (cid:47) (cid:47) O , ψ = U (cid:15) (cid:15) U (cid:48) (cid:15) (cid:15) f (cid:111) (cid:111) p (cid:47) (cid:47) U (cid:15) (cid:15) O O (cid:47) (cid:47) ϕ (cid:111) (cid:111) = (cid:47) (cid:47) O . The composition ψ ψ is given by ψ ψ = U (cid:15) (cid:15) U (cid:48) (cid:15) (cid:15) f f (cid:111) (cid:111) p p (cid:47) (cid:47) U (cid:15) (cid:15) O O (cid:47) (cid:47) ϕ ϕ (cid:111) (cid:111) = (cid:47) (cid:47) O , U (cid:48) p (cid:47) (cid:47) f (cid:15) (cid:15) U (cid:48) f (cid:15) (cid:15) p (cid:47) (cid:47) U U (cid:48) p (cid:47) (cid:47) f (cid:15) (cid:15) U U (7.28)where the maps f : U (cid:48) → U (cid:48) , p : U (cid:48) → U (cid:48) are given by the pullback square in the diagramof finite G -spaces on the right of (7.28). Note that the pullback square of diagram (7.28)induces a strict pullback square of action groupoids in the following diagram of groupoids B U (cid:48) G p (cid:47) (cid:47) f (cid:15) (cid:15) B U (cid:48) G f (cid:15) (cid:15) p (cid:47) (cid:47) B U GB U (cid:48) G p (cid:47) (cid:47) f (cid:15) (cid:15) B U GB U G. By [HHR16, prop. A.31] it follows that the following diagram commutes up to natural38somorphism (given by the symmetric monoidal structure of C )Fun( B U (cid:48) G, C ) ( p ) ⊗∗ (cid:47) (cid:47) Fun( B U (cid:48) G, C ) ( p ) ⊗∗ (cid:47) (cid:47) Fun( B U G, C )Fun( B U (cid:48) G, C ) ( p ) ⊗∗ (cid:47) (cid:47) ( f ) ∗ (cid:79) (cid:79) Fun( B U G, C ) ( f ) ∗ (cid:79) (cid:79) Fun( B U G, C ) . ( f ) ∗ (cid:79) (cid:79) In particular, for x ∈ Fun( B U G, C ) we get a natural isomorphism( p ) ⊗∗ ( f ) ∗ ( p ) ⊗∗ ( f ) ∗ x ∼ = ( p ) ⊗∗ ( p ) ⊗∗ ( f ) ∗ ( f ) ∗ x ∼ = ( p p ) ⊗∗ ( f f ) ∗ x , b (7.29)where the second isomorphism is given by [HHR16, prop. A.29]. Note that the mappingspaces of the topological functor categories in (7.29) are the spaces of natural transfor-mations, so the functor ( p ) ⊗∗ ( f ) ∗ : Fun( B U G, C ) → Fun( B U G, C ) induces a continuousmap Nat (cid:0) ( p ) ⊗∗ ( f ) ∗ x , x (cid:1) → Nat (cid:0) ( p ) ⊗∗ ( f ) ∗ ( p ) ⊗∗ ( f ) ∗ x , ( p ) ⊗∗ ( f ) ∗ x (cid:1) (7.30)We now define the map Map ψ C ⊗ ( x , x ) × Map ψ C ⊗ ( x , x ) → Map ψ ψ C ⊗ ( x , x ) as the com-position Nat (( p ) ⊗∗ ( f ) ∗ x , x ) × Nat (( p ) ⊗∗ ( f ) ∗ x , x ) (cid:15) (cid:15) Nat (( p ) ⊗∗ ( f ) ∗ ( p ) ⊗∗ ( f ) ∗ x , ( p ) ⊗∗ ( f ) ∗ x ) × Nat (( p ) ⊗∗ ( f ) ∗ x , x ) ◦ (cid:15) (cid:15) Nat (( p ) ⊗∗ ( f ) ∗ ( p ) ⊗∗ ( f ) ∗ x , x ) ∼ = (cid:15) (cid:15) Nat (cid:0) ( p p ) ⊗∗ ( f f ) ∗ x , x (cid:1) , (7.31)where the first map is given by (7.30) on the first coordinate and the identity on the second,the second map is the composition in Fun( B U G, C ) and the last isomorphism is inducedby (7.29).Associativity of the composition in C ⊗ follows from [HHR16, prop. A.29].In order to prove Theorem 7.24 we show that N ( C ⊗ ) → Fin G ∗ is a coCartesian fibration andverify the G -Segal conditions. Checking that C ⊗ → Fin G ∗ is a coCartesian fibration. Our next goal is to show that thefunctor C ⊗ → Fin G ∗ we constructed is a coCartesian fibration (see [Lur09, def. 2.4.2.1]).39 emma 7.32. Let ψ : I → I be a morphism of Fin G ∗ given by (7.26) and x ∈ C ⊗ over I , i.e.,a functor x : B U G → C . Define y : B U G → C over I by setting y = ( p ) ⊗∗ ( f ) ∗ x , and define ψ ∈ Map ψ C ⊗ ( x, y ) as the identity natural transformation ( p ) ⊗∗ ( f ) ∗ x = −→ ( p ) ⊗∗ ( f ) ∗ x = y . Thenfor every ψ : I → I in Fin G ∗ and t ∈ C ⊗ over I the continuous map ( ψ ) ∗ : Map ψ C ⊗ ( y, t ) → Map ψ ψ C ⊗ ( x, t ) as defined in (7.31) is an isomorphism.Proof. Suppose ψ , ψ are given by (7.28). Then, explicitly, ( ψ ) ∗ is the compositeNat (cid:0) ( p ) ⊗∗ ( f ) ∗ y, z (cid:1) ψ ◦ → Nat (cid:0) ( p ) ⊗∗ ( f ) ∗ ( p ) ⊗∗ ( f ) ∗ x, t (cid:1) ∼ = → Nat (cid:0) ( p p ) ⊗∗ ( f f ) ∗ x, t (cid:1) . The first map is an isomorphism by the definition of y and the second map is an isomorphisminduced by (7.29) as before. Corollary 7.33.
The map N ( C ⊗ ) → Fin G ∗ is a coCartesian fibration.Proof. The lemma above implies that the squareMap C ⊗ ( y, t ) (cid:15) (cid:15) (cid:15) (cid:15) ψ ∗ (cid:47) (cid:47) Map C ⊗ ( x, t ) (cid:15) (cid:15) (cid:15) (cid:15) Hom
Fin G ∗ ( I , I ) ψ ∗ (cid:47) (cid:47) Hom
Fin G ∗ ( I , I )is homotopy Cartesian, since it induces weak equivalences on the homotopy fibers of the verticalmaps. Therefore by [Lur09, prop. 2.4.1.10] the morphism ψ : x → y in C ⊗ is a coCartesian liftof ψ : I → I in Fin G ∗ . We have showed that every ψ : I → I and x ∈ C ⊗ over I has acoCartesian lift ψ . Passing to the coherent nerve of C (see [Lur09, def. 1.1.5.5]) we know thatthe map N ( C ⊗ ) → Fin G ∗ is an inner fibration (again by [Lur09, prop. 2.4.1.10]), so we haveshowed that it is coCartesian fibration by verifying [Lur09, def. 2.4.2.1]. G -Segal conditions. In order to prove that N ( C ⊗ ) (cid:16) Fin G ∗ is a G -symmetric monoidalcategory (see the head of [Nar17, sec. 3.1] for a definition) we have to verify the G -Segalconditions. Notation 7.34.
For I ∈ Fin G ∗ , I = ( U → O ), let C ⊗ I be the fiber of C ⊗ → Fin G ∗ over I .In other words, C ⊗ I is the topological category with objects given by functors x : B U G → C and mapping spaces Map C ⊗ I ( x, y ) = Map id I C ⊗ ( x, y ) = Nat( x, y ). Notation 7.35.
For W a G -orbit, let C ⊗ [ W ] be the fiber of C ⊗ → Fin G ∗ over ( W = −→ W ). Remark 7.36.
It is easy to see that if W ∼ = G/H then B W G (cid:39) BH , hence C ⊗ [ W ] is equivalentto the topological category Fun( BH, C ) of H -objects in C .If I ∈ Fin G ∗ , I = ( U → O ) and W ∈ Orbit( U ), then consider the following morphism in Fin G ∗ , U (cid:15) (cid:15) W (cid:15) (cid:15) f (cid:111) (cid:111) = (cid:47) (cid:47) = (cid:15) (cid:15) W = (cid:15) (cid:15) O W = (cid:47) (cid:47) (cid:111) (cid:111) = (cid:47) (cid:47) W, f : W → U is just an incusion of the orbit W into U . Since C ⊗ → Fin G ∗ is a coCartesianfiberation it induces a functor between the coherent nerves of C ⊗ I and C ⊗ [ W ] , but our choice ofcoCartesian edges above implies that this functor is just the coherent nerve of f ∗ : C ⊗ I → C ⊗ [ W ] , ( x : B U G → C ) (cid:55)→ ( f ∗ x : B W G Bf −−→ B U G x −→ C ) . Taking the product over all orbits W ∈ Orbit( U ) we get a functor of topological categories C ⊗ I → (cid:81) W ∈ Orbit( U ) C ⊗ [ W ] whose coherent nerve is equivalent to the G -Segal map. Checking the G -Segal conditions amounts to proving Lemma 7.37.
The functor C ⊗ I → (cid:81) W ∈ Orbit( U ) C ⊗ [ W ] is an equivalence.Proof. The orbit decomposition U = (cid:96) W ∈ Orbit( U ) W induces C ⊗ I = Fun( B U G, C ) = Fun( (cid:97) W B W G, C ) ∼ −→ (cid:89) W Fun( B W G, C ) = (cid:89) W C ⊗ [ W ] , which is the functor described above. Proof of Theorem 7.24 .
The map N ( C ⊗ ) → Fin G ∗ is a coCartesian fibration by Corollary 7.33,and by Lemma 7.37 it satisfies the G -Segal conditions.41 ppendix A The Real topological Hochschild homology of H Z , by Jeremy Hahn and Dylan Wilson In this appendix we explain how the results of the main body of the paper allow one to calculatethe Real topological Hochschild homology of the Eilenberg–MacLane Mackey functor H Z . Inparticular, we deduce the following theorem, which verifies a conjecture of Dotto, Moi, Patchko-ria, and Reeh [DMPR17, p. 63]. Theorem A.1.
There is an equivalence of H Z -module spectra THR( H Z ) (cid:39) H Z ⊕ (cid:77) k ≥ Σ kρ − H Z /k. Dotto, Moi, Patchkoria, and Reeh were able to prove that Theorem A.1 holds after localizationat any odd prime, and so also after localization away from 2 [DMPR17, Theorem 5.27 & Corollary5.28]. However, they did not have methods to calculate THR( H Z ) (2) (cid:39) THR( H Z (2) ). On theother hand, the main body of this paper provides methods to calculate the THR of Thomspectra, and the authors of this appendix previously constructed H Z (2) as a C -equivariantThom spectrum in [HW18]. These results were combined in Corollary 6.6(ii) of the main bodyto prove that THR( H Z (2) ) (cid:39) H Z (2) ⊗ Σ ∞ + Ω σ ( S λ +1 (cid:104) λ + 1 (cid:105) ) . The main contribution of the appendix is to observe that this can be made more explicit:
Lemma A.2.
There is an equivalence of H Z (2) -module spectra H Z (2) ⊗ Σ ∞ + Ω σ ( S λ +1 (cid:104) λ + 1 (cid:105) ) (cid:39) H Z (2) ⊕ (cid:77) k ≥ Σ kρ − H Z /k (2) . We deduce Lemma A.2 from the non-equivariant calculation of THH( H Z ) together with thefollowing C -equivariant fact, which we prove before 2-localization: Lemma A.3.
There is a cofiber sequence of H Z -module spectra H Z ⊗ Σ ∞ + Ω σ S λ +1 (cid:104) λ + 1 (cid:105) → (cid:77) k ≥ Σ kρ H Z → (cid:77) s ≥ Σ sρ H Z . Proof of Lemma A.3.
Applying Ω σ to the definition of S λ +1 (cid:104) λ + 1 (cid:105) yields a fiber sequence of C -equivariant spaces Ω σ S λ +1 (cid:104) λ + 1 (cid:105) → Ω σ S λ +1 → Ω σ K( λ + 1 , Z ) , where Ω σ K( λ + 1 , Z ) = Ω σ K(2 σ + 1 , Z ) (cid:39) K( σ + 1 , Z ) (cid:39) CP ∞ R . In particular, since CP ∞ R classifies Real line bundles, it follows that the cofiber of the mapΩ σ S λ +1 (cid:104) λ + 1 (cid:105) → Ω σ S λ +1 is the Thom space of a Real line bundle L over Ω σ S λ +1 . Using thefact that H Z is Real oriented, we conclude that there is a cofiber sequence of H Z -modules H Z ⊗ Σ ∞ + Ω σ S λ +1 (cid:104) λ + 1 (cid:105) → H Z ⊗ Σ ∞ + Ω σ S λ +1 → H Z ⊗ Σ ∞ + (cid:0) Ω σ S λ +1 (cid:1) L (cid:39) Σ ρ H Z ⊗ Σ ∞ + Ω σ S λ +1 . By [Hil17, Theorem 4.3], there is a James splittingΣ ∞ + Ω σ S λ +1 (cid:39) Σ ∞ + Ω σ Σ σ S ρ (cid:39) S ⊕ S ρ ⊕ S ρ ⊕ S ρ ⊕ · · · .
42n particular, there is a cofiber sequence of H Z -modules H Z ⊗ Σ ∞ + Ω σ S λ +1 (cid:104) λ + 1 (cid:105) → (cid:77) k ≥ Σ kρ H Z → (cid:77) s ≥ Σ sρ H Z , as desired. Proof of Lemma A.2.
By 2-localizing the result of Lemma A.3, we learn thatTHR( H Z (2) ) (cid:39) H Z (2) ⊗ Σ ∞ + Ω σ S λ +1 (cid:104) λ + 1 (cid:105) may be calculated as the fiber of a map f of H Z (2) -module spectra f : (cid:77) k ≥ Σ kρ H Z (2) → (cid:77) s ≥ Σ sρ H Z (2) . Since the domain of f is a direct sum of free H Z (2) -module spectra, f is determined by asequence of elements f k ∈ π kρ (cid:16)(cid:76) s ≥ Σ sρ H Z (2) (cid:17) . The RO ( C )-graded homotopy groups of H Z ,as nicely displayed for example in [Gre17, p.6], show that there are no classes in π kρ (Σ sρ H Z (2) )unless k = s . Furthermore, one sees that any such class is determined by its underlying non-equivariant class in π k (Σ k H Z (2) ) ∼ = Z (2) , and in particular the map f is entirely determinedby its underlying non-equivariant map f underlying : (cid:77) k ≥ Σ k H Z (2) → (cid:77) s ≥ Σ s H Z (2) . The fiber of f underlying must agree with the known non-equivariant calculationTHH( H Z (2) ) (cid:39) H Z (2) ⊕ (cid:77) s ≥ Σ s − ( H Z /s ) (2) , which determines the map f underlying well enough to determine the fiber of f up to equivalence. Proof of Theorem A.1.
As with any C -equivariant spectrum, there is a pullback squareTHR( H Z ) THR( H Z )[ ]THR( H Z ) (2) THR( H Z ) ⊗ H Q . The 2-local Lemma A.2 allows to calculate the lower left corner of the square, while the result[DMPR17, Corollary 5.28] of Dotto, Moi, Patchkoria, and Reeh calculates the upper right. Fromthese results, we learn that the square is a direct sum of squares H Z H Z [1 / H Z (2) H Q , k ≥
1, Σ kρ − H Z /k Σ kρ − H Z /k [1 / kρ − H Z /k (2) . eferences [ABG + ∞ -categorical approach to R -line bundles, R -module Thom spec-tra, and twisted R -homology. J. Topol. , 7(3):869–893, 2014.[ABG + arXiv preprintarXiv:1401.5001 , 2014.[AF15] David Ayala and John Francis. Factorization homology of topological manifolds. Journal of Topology , 8(4):1045–1084, 2015.[AGH +
20] Katharine Adamyk, Teena Gerhardt, Kathryn Hess, Inbar Klang, and Hana JiaKong. Computational tools for twisted topological Hochschild homology of equiv-ariant spectra. arXiv preprint arXiv:2001.06602 , 2020.[AMGR17] David Ayala, Aaron Mazel-Gee, and Nick Rozenblyum. The geometry of the cyclo-tomic trace. arXiv preprint arXiv:1710.06409 , 2017.[BCS10] Andrew J Blumberg, Ralph L Cohen, and Christian Schlichtkrull. TopologicalHochschild homology of Thom spectra and the free loop space.
Geometry & Topol-ogy , 14(2):1165–1242, 2010.[BDG + arXiv preprint arXiv:1608.03654 , 2016.[BDG + arXiv preprint arXiv:1608.03657 , 2016.[BDG + ar] Clark Barwick, Emanuele Dotto, Saul Glasman, Denis Nardin, and Jay Shah.Parametrized and equivariant higher algebra. To appear.[Bon19] Peter Bonventre. The genuine operadic nerve. arXiv preprint arXiv:1904.01465 ,2019.[BW18] Mark Behrens and Dylan Wilson. A C -equivariant analog of Mahowald´s Thomspectrum theorem. Proceedings of the American Mathematical Society , 2018.[DMPR17] Emanuele Dotto, Kristian Moi, Irakli Patchkoria, and Sune Precht Reeh. Realtopological Hochschild homology. arXiv preprint arXiv:1711.10226 , 2017.[Gre17] J.P.C. Greenlees. Four approaches to cohomology theories with reality. arXivpreprint arXiv:1705.09365 , 2017.[hc] Federico Cantero (https://mathoverflow.net/users/9809/federico can-tero). Continuity of taking collapse maps. MathOverflow.URL:https://mathoverflow.net/questions/189313/continuity-of-taking-collapse-maps (version: 2014-12-12).[HHR16] Michael A Hill, Michael J Hopkins, and Douglas C Ravenel. On the nonexistenceof elements of Kervaire invariant one.
Annals of Mathematics , 184(1):1–262, 2016.45Hil17] Michael A Hill. On the algebras over equivariant little disks. arXiv preprintarXiv:1709.02005 , 2017.[HL18] Michael J Hopkins and Tyler Lawson. Strictly commutative complex orientationtheory.
Mathematische Zeitschrift , 290(1-2):83–101, 2018.[hm] Fernando Muro (https://mathoverflow.net/users/12166/fernando muro). Enrichedmonoidal categories. MathOverflow. URL:https://mathoverflow.net/q/51783 (ver-sion: 2011-01-11).[Hor19] Asaf Horev. Genuine equivariant factorization homology. arXiv preprintarXiv:1910.07226 , 2019.[HW18] Jeremy Hahn and Dylan Wilson. Eilenberg-Maclane spectra as equivariant Thomspectra. arXiv preprint arXiv:1804.05292 , 2018.[IK00] S¨oren Illman and Marja Kankaanrinta. Three basic results for real analytic properG-manifolds.
Mathematische Annalen , 316(1):169–183, Jan 2000.[Ill78] S¨oren Illman. Smooth equivariant triangulations of G -manifolds for G a finite group. Math. Ann. , 233(3):199–220, 1978.[KK10] Daisuke Kishimoto and Akira Kono. On the cohomology of free and twisted loopspaces.
Journal of Pure and Applied Algebra , 214(5):646–653, 2010.[Kla18] Inbar Klang. The factorization theory of Thom spectra and twisted nonabelianPoincar´e duality.
Algebr. Geom. Topol. , 18(5):2541–2592, 2018.[LMSM86] L Gaunce Lewis, Jr., J Peter May, Mark Steinberger, and James E McClure. Equiv-ariant stable homotopy theory.
Lecture notes in mathematics , 1213:x+538, 1986.With contributions by J. E. McClure.[Lur09] Jacob Lurie.
Higher topos theory , volume 170. Princeton University Press, 2009.[Lur12] Jacob Lurie. Higher algebra.
Preprint, available at , 2012.[May72] J. P. May.
The geometry of iterated loop spaces . Springer-Verlag, Berlin-New York,1972. Lectures Notes in Mathematics, Vol. 271.[Mos57] George Daniel Mostow. Equivariant embeddings in Euclidean space.
Annals ofMathematics , pages 432–446, 1957.[Nar16] Denis Nardin. Parametrized higher category theory and higher algebra: Expos´e IV–Stability with respect to an orbital ∞ -category. arXiv preprint arXiv:1608.07704 ,2016.[Nar17] Denis Nardin. Stability and distributivity over orbital ∞ -categories . PhD thesis,Massachusetts Institute of Technology, 2017.[Pir00] Teimuraz Pirashvili. Hodge decomposition for higher order Hochschild homology. Ann. Sci. ´Ecole Norm. Sup. (4) , 33(2):151–179, 2000.[Rub17] Jonathan Rubin. Normed symmetric monoidal categories. arXiv preprintarXiv:1708.04777 , 2017. 46Sch11] Christian Schlichtkrull. Higher topological Hochschild homology of Thom spectra.
Journal of Topology , 4(1):161–189, 2011.[Sha17] Jay Shah.
Parametrized higher category theory . PhD thesis, Massachusetts Instituteof Technology, 2017.[Sha18] Jay Shah. Parametrized higher category theory and higher algebra: Expos´e II–Indexed homotopy limits and colimits. arXiv preprint arXiv:1809.05892 , 2018.[Wee18] T.A.N. Weelinck. Equivariant factorization homology of global quotient orbifolds. arXiv preprint arXiv:1810.12021 , 2018.[Zou20] Foling Zou.