aa r X i v : . [ m a t h . A T ] O c t Genuine equivariant factorization homology
Asaf HorevOctober 17, 2019
Abstract
We construct a genuine G -equivariant extension of factorization homology for G a fi-nite group, assigning a genuine G -spectrum to a manifold with G -action. We show that G -factorization homology is compatible with Hill-Hopkins-Ravenel norms and satisfies equivari-ant ⊗ -excision. Following Ayala-Francis we prove an axiomatic characterization of genuine G -factorization homology. Applications include a description of real topological Hochschildhomology and relative topological Hochschild homology of C n -rings using genuine G -factorizationhomology. Contents ∞ -category theory 10 G -categories . . . . . . . . . . . . . . . . 102.2 Constructing G -categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Parametrized adjoints, colimits, left Kan extensions . . . . . . . . . . . . . . . . . 142.4 G -symmetric monoidal structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 G -manifolds and G -disks 17 G -category of G -manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Representations, G -vector bundles and the G -tangent classifier . . . . . . . . . . 253.3 The G -category of f -framed G -manifolds . . . . . . . . . . . . . . . . . . . . . . . 283.4 G -disjoint union of G -manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5 G -disjoint union of f -framed G -manifolds . . . . . . . . . . . . . . . . . . . . . . 413.6 The G -category of G -disks and the definition of G -disk algebras . . . . . . . . . . 443.7 G -disks as a G -symmetric monoidal envelope . . . . . . . . . . . . . . . . . . . . 463.8 Embedding spaces of G -disks and equivariant configuration spaces . . . . . . . . 483.9 Comparison of the equivariant little disks G -operad and the G - ∞ -operad of V -framed representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 G -factorization homology 56 G -disk algebras and G -factorization homology as a G -functor . 564.2 Extensding G -factorization homology to a G -symmetric monoidal functor . . . . 58 G -factorization homology 66 G -manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 665.2 G - ⊗ -excision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.3 G -sequential unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751 Axiomatic characterization of G -factorization homology theories 777 Equivariant versions of Hochschild homology 80 G -factorization homology . . . . . . . . 807.2 Twisted Topological Hochschild Homology of genuine C n -ring spectra . . . . . . 82 Appendix A The Moore over category 84Appendix B The definition of a G -Symmetric Monoidal category 87Appendix C Mapping spaces in over-categories 92References 94 Factorization homology, introduced by Lurie under the name topological chiral homology ([Lur09b],[Lur]), is an invariant of an E n -algebra and a framed n -dimensional manifold. The factorizationhomology of a framed n -dimensional manifold M with coefficients in an E n -ring spectrum A is aspectrum denoted R M A . If M admits an action of a finite group G then R M A admits an G -actionby functoriality. However, this action is defined only up to coherent homotopy, as R M A is definedby an ∞ -categorical colimit. A fundamental observation of equivariant homotopy theory is thatsuch a “naive” action does not determine the homotopy type of the fixed points. In particularthe action of G on R M A does not define a genuine G -spectrum structure on R M A .The first goal of this paper is to construct and study such a genuine equivariant extensionof factorization homology for a fixed finite group G . We draw on two points of view in order toexplain the expected properties of genuine equivariant factorization homology. Factorization homology as a tensor product
First, according to [Lur09b, rem. 4.1.19]one can intuitively think of R M A as a continuous tensor product ⊗ x ∈ M A indexed by the pointsof M . One should have this intuition in mind when considering the behavior of factorizationhomology with respect to disjoint unions , namely Z M ⊔ M A ≃ Z M A ⊗ Z M A. (1)In order to generalize this behavior to genuine G -spectra we now recall the interaction of thesmash product with the group action. If X is a genuine H -spectrum for H < G a subgroup thenthe smash product ⊗ | G/H | X of G/H copies of X has a naive G -action, induced by the combiningthe action of H on X with the action of G on the indexing set G/H . Hill-Hopkins-Ravenel[HHR16] extended this naive G -action to a genuine G -spectrum, N GH ( X ), the Hill-Hopkins-Ravenel norm of X . More generally they define smash products indexed by finite G -sets as thesmash product of Hill-Hopkins-Ravenel norms (see [HHR16, app. A.3]). Let U be a finite G -set,given by a coproduct of orbits U = ` i ∈ I G/H i with stabilizers H i < G . The U -indexed smashproduct of a family X • = { X i } i ∈ I , where each X i is a genuine H i -spectrum, is the genuine G -spectrum given by the smash product of the norms ⊗ U X • = ⊗ i ∈ I N GH i ( X i ). The indexed smashproduct interacts with smash products and norms as follows. We distinguish between disjoint unions and coproducts since disjoint union is not the categorical coproductin the category of n -dimensional manifolds and open embeddings which we consider below. The indexed smash product takes disjoint unions to smash products: if U ′ , U ′′ are of finite G -sets then the indexed smash product along U ′ ` U ′′ is equivalent to smash product ofthe indexed products, ⊗ U ′ ` U ′′ X • ≃ ( ⊗ U ′ X • ) ⊗ ( ⊗ U ′′ X • ). • The indexed smash product takes topological inductions to norms: given a subgroup
H < G and a finite H -set U , denote the quotient G × H U by ` G/H U . The left action of G onthe first coordinate makes ` G/H U a G -set which we call the topological induction of U from H to G . The norm of an indexed product is given by an indexed product along thetopological induction, ⊗ ` G/H U X • ≃ N GH ( ⊗ U X • ).Note that stating the second property required us to consider tensor products indexed by finite H -sets for all H < G .Interpreting genuine equivariant factorization homology as a tensor product indexed by a G -manifold M , one would expect a similar behavior. To state it, we consider the genuine fac-torization homology of H -manifolds for all subgroups H < G . Namely, for any subgroup
H < G and H -manifold M we expect genuine equivariant factorization homology to assign a genuine H -spectrum R M A ∈ Sp H , which interacts with smash products and norms as follows. • Genuine equivariant factorization homology takes disjoint unions to smash products: if M ′ , M ′′ are n -dimensional G -manifolds then the genuine equivariant factorization homol-ogy along M ′ ⊔ M ′′ is equivalent to the smash products of the genuine equivariant factor-ization homologies along M ′ and M ′′ , Z M ′ ⊔ M ′′ A ≃ ( Z M ′ A ) ⊗ ( Z M ′′ A ) , as genuine G -spectra. • Genuine equivariant factorization homology takes topological inductions to norms: givena subgroup
H < G and an n -dimensional H -manifold M , denote the topological induction G × H M by ⊔ G/H M , with left G -action induced by acting on the first coordinate. The normof genuine equivariant factorization homology along M is equivalent to genuine equivalentfactorization homology along the topological induction ⊔ G/H M , Z ⊔ G/H M A ≃ N GH ( Z M A ) , (2)as genuine G -spectra. Factorization homology as a homology theory
A second point of view on factorizationhomology is given by Ayala-Francis [AF15], where factorization homology is considered as ahomology theory of n -dimensional manifolds. Ayala-Francis start from the observation that fac-torization homology is functorial with respect to open embeddings of framed n -dimensionalmanifolds. Let Mfld frn be the ∞ -category of framed n -dimensional manifolds and framedopen embeddings, and let C be a cocomplete symmetric monoidal ∞ -category. Fixing an E n -algebra A in C , Ayala-Francis consider factorization homology M R M A as a functor of ∞ -categories R − A : Mfld frn → C . Factorization homology extends to a symmetric monoidal functor R − A : Mfld fr, ⊔ n → C ⊗ with respect to disjoint union of manifolds (a functorial version of eq. (1))under mild conditions on C . Namely that the tensor product in C distributes over sifted colimits. Mfld fr, ⊔ n → C ⊗ satisfying ⊗ -excision, and show that R − A is indeed such a homology theory for manifolds.Furthermore, they show that the Eilenberg-Steenrod characterization of generalized homologytheories admits the following generalization. Let H ( Mfld frn , C ) ⊆ Fun ⊗ ( Mfld frn , C ) be the fullsubcategory of symmetric monoidal functors satisfying ⊗ -excision. Theorem 1.0.1 (Ayala-Francis) . There is an equivalence of ∞ -categories Z : Alg E n ( C ) ∼ −→ H ( Mfld frn , C ) , A ( Z − A : Mfld frn → C ) sending an E n -algebra A to factorization homology with coefficients in A . In fact, this theorem holds in greater generality, replacing framed manifolds with B -framedmanifolds and E n -algebras with B -framed n -disk-algebras. The second goal of this paper is toprovide such an axiomatic characterization of genuine equivariant factorization homology (seetheorem 6.0.2). Framed G -manifolds We now describe V -framed G -manifolds, which serve as the geometricinputs of genuine G -factorization homology theories. The notion of V -framed G -manifolds hasalready been studied by [Wee18], though our construction differs from his.Fix a finite group G and n ∈ N . In what follows a G -manifold is an n -dimensional smoothmanifold M with a smooth action of G . We organize G -manifolds and G -equivariant smoothopen embeddings using a topological category Mfld G , which we consider as an ∞ -category bytaking its coherent nerve.Recall that a framing of M is trivialization of its tangent bundle, i.e an isomorphism of tangentbundles T M ∼ = M × R n . In order to define a framing of G -manifolds we consider T M as G -vectorbundle, with G -action induced from the smooth action of G on M by taking differentials. Fixa real n -dimensional G -representation V . A V -framing of M as an isomorphism of G -vectorbundles T M ∼ = M × V over M . The ∞ -category of Mfld G can be enhanced to an ∞ -category Mfld
G,V − fr of V -framed G -manifolds.In fact, we consider genuine equivariant factorization homology theories of G -manifolds withmore general tangential structures (see definition 4.1.2). These tangential structures includeunframed G -manifolds, equivariant orientations in the sense of [CMW01] and manifolds with afree G -action (see section 3.3).We plan to compare this notion of an equivariant tangential structure with the one introducedby [Wee18, sec. 2.2] in future work. Equivariant factorization homology as a single functor of ∞ -categories Viewing fac-torization homology as a homology theory suggests a natural generalization to G -manifolds.Namely, define a G -factorization homology theory as a symmetric monoidal functor Mfld
G,V − fr → C satisfying ⊗ -excision. This is essentially the approach taken by Weelinck in [Wee18], which leadsto a natural generalization of the axiomatic characterization of factorization homology discussedabove. In particular, taking C = Sp G to be the ∞ -category of genuine G -spectra producesinvariants of G -manifolds valued in genuine G -spectra.However, this is not the approach we take in this paper, for two reasons. First, we arelooking for an extension of factorization homology to genuine G -spectra. If M is a G -manifold4nd F : Mfld
G,V − fr → Sp G is a G -factorization homology theory in the sense of [Wee18] thenthe underlying spectrum of F ( M ) ∈ Sp G need not agree with the factorization homology of M .Second, using a single functor Mfld
G,V − fr → C to encode a G -factorization homology theoryprevents us from expressing its expected compatibility with norms described in eq. (2).Our emphasis on the compatibly of equivariant factorization homology with norms impliesthat our notion an equivariant disk algebra, serving as a coefficient system for equivariant factor-ization homology, is different from the one introduced in [Wee18]. For a specific example, compare[Wee18, ex. 1.3] with the description of E σ -algebras in section 7.1. A detailed comparison ofthese two notions will appear in future work. Parametrized ∞ -categories. In order to express both the functoriality of genuine equivariantfactorization homology with respect to equivariant embeddings and the compatibilities of eq. (2)we view genuine factorization homology as a collection of symmetric monoidal functors ∀ H < G : Z − A : Mfld
H,V − fr → Sp H from the ∞ -category of V -framed H -manifolds to the category of genuine H -spectra, coherentlycompatible with restrictions and topological inductions. To make this coherent compatibilities precise we use the theory of parametrized ∞ -categories,developed by Barwick-Dotto-Glasman-Nardin-Shah in [BDG + + , Nar16].Informally, a G - ∞ -category is a diagram of ∞ -categories O opG → C at ∞ indexed contravariantlyby the orbits of G . A G -symmetric monoidal structure encodes a symmetric monoidal structureon each of the ∞ -categories in the diagram together with norm functors and all their expectedcompatibilities. In section 2 we review parametrized ∞ -category theory in more detail.In particular, we use the G - ∞ -category Sp G of genuine G -spectra constructed in [Nar17].As a G - ∞ -category Sp G encodes the ∞ -categories Sp H for all subgroups H < G and therestriction functors relating them. The G -symmetric monoidal structure on Sp G encodes smashproducts and Hill-Hopkins-Ravenel norms. Nardin gives an axiomatic characterization of this G -symmetric monoidal, see [Nar17, cor. 3.28]. This characterization allows us to work withthe Hill-Hopkins-Ravenel norms at a formal level, avoiding the original point set definition of[HHR16]. E V -algebras and V -framed disks. Genuine equivariant factorization homology is an invari-ant of a geometric input, a V -framed G -manifold (described above), and of an algebraic input,an E V -algebra. We now briefly describe this algebraic structure.Conceptually, factorization homology is constructed by gluing local data, given by a coefficientsystem. Such a coefficient system is an algebraic structure indexed by the local geometry ofmanifolds: an n -disk algebra in the case of factorization homology of n -dimensional manifoldsand an E n -algebra in the case of factorization homology of framed n -dimensional manifolds.Similarly, the structure of an E V -algebra is determined by the local structure of V -framed G -manifolds. Let M be a V -framed G -manifold and x ∈ M a point with stabilizer H < G ,then H acts linearly on the tangent space T x M , and the H -representation T x M is isomorphicto V (with the action restricted to H < G ). It follows that x ∈ M has an H -equivariant Here we consider V as an H -representation by restricting the G -action to H < G . In particular, R − A defines a natural transformation between two functors from O opG to symmetric monoidal ∞ -categories. However, such natural transformation does not capture the compatibility of norms with topologicalinductions. To see this, pull the V -framing T M ∼ = M × V along { x } → M . V . Therefore the orbit of x (considered as a G -submanifold of codimension 0) has a G -tubular neighborhood isomorphic to the topologicalinduction ` G/H V = G × H V .Let D V be the G -operad of little V -disks, and E V its genuine operadic nerve (see [Bon19]).We define E V -algebras in Sp G as maps of G - ∞ -operads E V → Sp G . Informally, an E V -algebra A in Sp G assigns to V a genuine G -spectrum A (the “underlying G -spectrum” of A ). The algebraic structure on A is indexed by H -equivariant embeddings for H < G . To an H -embedding V ⊔ V ֒ → V the algebra A assigns a map of genuine H -spectra A ⊗ A → A (a “multiplication map”), and to a H -embedding ⊔ H/K
V ֒ → V the algebra A assigns a map N HK ( A ) → A (a “multiplicative norm map”) from the Hill-Hopkins-Ravenel normof A . All of these maps are coherently compatible with smash products, restrictions of the groupaction and with each other. We use G - ∞ -category theory, and specifically G -symmetric monoidalstructures, to handle these coherent compatibilities.The G - ∞ -operad E V is closely related to Mfld
G,V − fr , as we now explain. Let Disk
G,V − fr be the full G - ∞ -subcategory of Mfld
G,V − fr generated from the G -manifold V by restrictingthe group action, disjoint unions and topological induction (see section 3 for details). By con-struction, the G -symmetric monoidal structure of Mfld
G,V − fr induces a G -symmetric monoidalstructure on Disk
G,V − fr . In section 3.9 we show that Disk
G,V − fr is equivalent to the G -symmetric monoidal envelope of E V . In particular, an E V -algebra in Sp G corresponds to anessentially unique G -symmetric monoidal functor Disk
G,V − fr → Sp G . We call such functors V -framed G -disk algebras in Sp G . Genuine equivariant factorization homology
We encode the functors R − A : Mfld
H,V − fr → Sp H as a single G -symmetric monoidal G -functor Mfld
G,V − fr → Sp G from the G - ∞ -categoryof V -framed G -manifolds to the G - ∞ -category of genuine G -spectra. Given an E V -algebra A in Sp G , let A : Disk
G,V − fr → Sp G denote the corresponding V -framed G -disk algebra. Weconstruct genuine G -factorization homology Z − A : Mfld
G,V − fr → Sp G as the G -left Kan extension of A along the inclusion Disk
G,V − fr ֒ → Mfld
G,V − fr . By work ofShah [Sha18] the genuine G -spectrum R M A has an explicit description as a G -colimit indexedby “little disks in M ”, see definition 4.1.2 and proposition 4.1.4. This construction is indeeda homology theory of G -manifolds, as it extends to a G -symmetric monoidal functor satisfying G - ⊗ -excision.Genuine equivariant factorization homology satisfies the following extension of the Ayala-Francis axiomatic characterization. Choose a G -equivariant Riemannian metric on M use the fact that the exponential map T x M M is H -invariant. compatible with the G -framing heorem 1.0.2. Let H ( Mfld
G,V − fr , Sp G ) ⊂ Fun ⊗ G ( Mfld
G,V − fr , Sp G ) be the full subcategoryof the ∞ -category of G -symmetric monoidal G -functors Mfld
G,V − fr → Sp G which satisfy G - ⊗ -excision and respect sequential unions. Then there is an equivalence of ∞ -categories Z : Alg E V ( Sp G ) ∼ −→ H ( Mfld
G,V − fr , Sp G ) , A ( Z − A : Mfld
G,V − fr → Sp G ) (3) sending an E V -algebra A to G -equivariant factorization homology with coefficients in A . The above result holds in greater generality. First, V -framed G -manifolds can be replacedwith G -manifolds with more general equivariant tangential structures . Second, the G - ∞ -category of genuine G -spectra can be replaced with any presentable G -symmetric monoidal G - ∞ -category C . The general statement is given in theorem 6.0.2, which is the main resultof this paper. Applications
As an application of theorem 6.0.2, we describe two variants of topologicalHochschild homology using genuine G -factorization homology.In section 7.1 we show that the real topological Hochschild homology spectrum of Hesselholt-Madsen [HM13] is equivalent to genuine C -factorization homology over S . Proposition 1.0.3 (proposition 7.1.1) . For A an E σ -algebra in Sp C there is an equivalence ofgenuine C -spectra Z S A ≃ A ⊗ N C e A A. where C acts on S by reflection. By a theorem of ([DMPR17]) it follows that for A a flat ring spectrum with anti-involutionthere is as an equivalence of genuine C -spectra Z S A ≃ T HR ( A ) , where T HR ( A ) is the real topological Hochschild homology of A , see remark 7.1.2.In section 7.2 we show that the “twisted” topological Hochschild homology of a genuine C n -ring spectrum of [ABG +
14, sec. 8] is equivalent to the geometric fixed points of C n -factorizationhomology over S . Proposition 1.0.4 (proposition 7.2.2) . Let A be an E -ring spectrum in Sp C n , and C n y S be the standard action. Then there exists an equivalence of spectra (cid:18)Z S A (cid:19) Φ C n ≃ T HH ( A ; A τ ) . In particular,
T HH ( A ; A τ ) admits a natural circle action. This circle action is equivalent to the circle action on the nerve of the “twisted cyclic barconstruction” of [ABG +
14, sec. 8], which gives an alternative description of the relative norm of[ABG +
14, def. 8.2]. This requires replacing V -framed G -disk algebras with more general G -disk algebras. Conjecturally, this condition can be weakened to distributivity of the tensor product over parametrized siftedcolimits onstruction of V -framed G -disk algebras. Above we gave a rough description of a V -framed G -disk algebra as encoding multiplication maps, multiplicative norm maps and theircoherent compatibilities. Unwinding these compatibilities implied by definition 3.6.11 is usuallya non-trivial task (especially when dim V ≥ V -framed G -disk algebra by specifying multiplication maps, multiplicative norm maps and associated coher-ence data. It is therefore desirable to have some general mechanisms for constructing V -framed G -disk algebras.For example, one would expect to be able to construct a V -framed G -disk algebra from analgebra over the G -operad D V . Such a construction would provide many examples of coefficientsfor genuine equivalent factorization homology of V -framed G -manifolds. More generally, it wouldbe reassuring to have a “rectification” result showing that classical algebras over D V form a modelfor the ∞ -category of V -framed G -disk algebras, in the style of [PS14, thm. 7.10].We leave such constructions for future work. What about compact Lie groups?
It is natural to want to extend genuine equivariantfactorization homology from finite groups to compact Lie groups. There are two different pointsin which one encounters complications.First, we prove theorem 6.0.2 (the axiomatic characterization of genuine G -factorization ho-mology) inductively using equivariant handle bundle decompositions (see [Was69]). We producethese decompositions using equivariant Morse theory, which is more complicated over a compactLie group. Choosing an invariant Morse function gives rise to a handle bundle decomposition,where each handle bundle is an equivariant disk bundle over a critical orbit. However, for acompact Lie group of positive dimension these handle bundles can be non trivial, since criticalorbits are submanifolds of possibly positive dimension.Second, and more fundamental, is the lack of good G - ∞ -category theory for a compact Liegroup G . The source of the problem is the lack of multiplicative norms for subgroups H < G ofnon-finite index. In order to understand the significance of this fact for genuine G -factorizationhomology, consider M = C the complex plane with the standard action of the circle group S = C × . The unit circle is an S -orbit in C , with S -tubular neighborhood given by the openannulus. The embedding of the open annulus in C should induce a “multiplication norm map” ⊗ S A → A of genuine S -spectra, where the tensor product is indexed over the free orbit S .However, we do not have a good definition for the domain of this map as a genuine S -spectrum. Organization
We start by reviewing some parts of parametrized ∞ -category theory in sec-tion 2. We hope this short exposition will assist the reader unfamiliar with the theory of G - ∞ -categories.In section 3 we construct the G - ∞ -categories of G -manifolds and G -disks with equivarianttangential structures, and their G -symmetric monoidal structure which encodes disjoint unionsand topological induction. These constructions provide a bridge between the geometry of G -manifolds and parametrized ∞ -category theory, and enables the construction of genuine G -factorization homology in section 4.2.Our definition of equivariant tangential structures in section 3.3 uses an equivariant version ofthe tangent classifier of [AF15] which may be of independent interest, see section 3.2. While wefocus on framed G -manifolds, our definition is flexible enough to consider more general tangentialstructures such as equivariant orientations, as well as allowing us to restrict our attention tomanifolds with a free G -action.We finish section 3 by studying some aspects of these constructions. In section 3.8 we studythe relation between embedding spaces of G -disks and G -configuration spaces. In section 3.9 use8he work of [Bon19] to show that the G - ∞ -operad encoding V -framed G -disk algebras is closelyrelated to the G -operad of little disks in a representation V .The technical results and constructions of section 3 provide a solid foundation for the use ofabstract theory of parametrized ∞ -categories in the following sections.In section 4 we define framed G -disk algebras and construct G -factorization homology, firstas a G -functor (by G -left Kan extension, see section 4.1) and then as a G -symmetric monoidal G -functor (section 4.2).In section 5 we study the properties of G -factorization homology. In section 5.1 we define G -collar decompositions and construct an “inverse image” functor. We use these in section 5.2,where we define G - ⊗ -excision for a general G -symmetric monoidal functor Mfld G → C , andshow that G -factorization homology satisfies G -tensor excision. In section 5.3 we show that G -factorization homology respects sequential unions.In section 6 we prove our main result, giving an axiomatic characterization of G -factorizationhomology using equivariant Morse theory.In section 7 we describe real topological Hochschild homology using G -factorization homology(section 7.1), and the relative norm of a genuine C n -ring spectrum as the geometric fixed pointsof G -factorization homology (section 7.2).In appendix A we show how to model ∞ -slice categories in the framework of topologicalcategories. For the convenience of the reader we recall the definition of G -symmetric monoidalcategories in appendix B. We collect some Some general statements about mapping spaces inover categories in appendix C. Notation.
In this work we use the quasi-categories as a model ∞ -categories (with the exceptionof remark 2.1.4). We assume the reader is familiar with the theory of ∞ -categories, as developedin [Lur09a] and [Lur]. Explicitly, an ∞ -category is a simplicial set C satisfying the left liftingproperty with respect to inner horns: for every 0 < i < n , any map Λ ni → C admits an extensionto ∆ n → C .All of the manifolds we consider are smooth and n -dimensional for a fixed n ∈ N . We fix afinite group G , and only consider manifolds with actions of subgroups H < G .We frequently construct ∞ -categories from topological categories by taking their coherentnerve (which is called the topological nerve in [Lur09a, def. 1.1.5.5]). We emphasize that thecoherent nerve of a topological category C is a two step construction. First, taking the singularnerve of each mapping space, produces a simplicial category Sing ( C ). Second, applying thesimplicial nerve functor of [Lur09a, def. 1.1.5.5] to Sing ( C ) produces an ∞ -category. We denotethe resulting ∞ -category by N ( C ).We denote parametrized ∞ -categories with an underline, for example C . In general, if C is parametrized over an ∞ -category S we refer to C as an S - ∞ -category. We say that C is a G -category (see definition 2.1.3) if it is parametrized over O opG , where O G is the orbit categoryof G . No other notion of G -categories is used; a G -category is by definition an O opG - ∞ -category. Acknowledgements
This paper is a revised version of my PhD thesis. I wish to thank myadvisors, Emmanuel Farjoun and Yakov Varshavsky, for their continued support throughout thisproject. I am grateful to Tomer Schlank for suggesting this project and for valuable insights.This paper builds on the work of Clark Barwick, Emanuele Dotto, Saul Glasman, DenisNardin and Jay Shah on parametrized higher category theory. I would like to thank ClarkBarwick for helpful discussions and Jay Shah for generously sharing an early draft of [BDG + ].9 Background on parametrized ∞ -category theory In this section we review parametrized ∞ -category theory of Barwick, Dotto, Glassman, Nardinand Shah, developed in [BDG + + ]. We recall the notions of G - ∞ -category theory employed below and fix our notation. We restrict our discussion to the caseof G - ∞ -categories, though nothing substantial would change when working over an arbitraryindexing category.This section contains no original results, all the results of this section are entirely due toBarwick, Dotto, Glassman, Nardin and Shah. G -categories A good starting point to a discussion of G -categories is the Elmendorf-McClure theorem, whichrecasts the equivariant homotopy theory of G -spaces as a presheaf category. Throughout we fixa finite group G . Definition 2.1.1.
The orbit category O G is the full subcategory of G -sets supported by transitive G -sets. Note that every orbit in O G is isomorphic to a quotient of G by some subgroup H < G . Thisisomorphism depends only on the choice of a base point of the orbit, with H the stabilizer ofthe chosen basepoint. We denote the objects of O G either by O by G/H . Despite the suggestivenotation, we try to refrain from a choice of basepoint when possible.Define the ∞ -category of G -spaces Top G as the coherent nerve of the topological categoryof G -CW spaces and G -maps. Theorem 2.1.2 (Elmendorf-McClure, [Elm83]) . There is an equivalence of ∞ -categories Top G ∼ −→ Fun( O opG , S ) , sending a G -space X to its diagram of fixed points, G/H X H . Using straightening/unstraightening ([Lur09a, thm. 2.2.1.2]) we get a third description of a G -space X as the left fibration over O opG classifying the diagram of fixed points of X . Definition 2.1.3. A G - ∞ -category is a coCartesian fibration C ։ O opG . For sake of readability we refer to G - ∞ -categories simply as G -categories. Other notions of G -categories present in the literature are not present in this paper.A G -category C is classified by a diagram of ∞ -categories C • : O opG → C at ∞ sending G/H ∈O opG to the fiber C [ G/H ] of C ։ O opG over G/H . We systematically use the subscript-squarebracket notation C [ G/H ] for the fiber ∞ -category in order to avoid confusion with other subscriptnotations. As above, straightening/unstraightening ([Lur09a, sec. 3.2]) ensures that this is anequivalent description of the G - ∞ -category C . Remark 2.1.4.
Describing C at ∞ as a complete Segal object in the ∞ -category of spaces, wecan use the Elmendorf-McClure theorem to get a third equivalent description of a G -categoryas a complete Segal object in Top G . This follows from following the Segal conditions andcompleteness conditions along the equivalencesFun(∆ op , Top G ) ≃ Fun(∆ op , Fun( O opG , S )) ≃ Fun( O opG , Fun(∆ op , S )) , where the first equivalence is induced by the Elmendorf-McClure theorem.10n particular, categories internal to G -spaces (and to G -sets) are examples of G -categories.Note that [GM17] defines G -categories as categories internal to G -spaces, making them examplesof G -( ∞ -)categories in the sense of [BDG + G -category are good to have in mind, we stick to thedefinition of a G -category as a coCartesian fibration for its explicit nature.When we need more general parametrized ∞ -categories we use the following definition (andthe notation of [Sha18]). Definition 2.1.5.
Let S be an ∞ -category. An S - ∞ -category is a coCartesian fibration C ։ S .We denote the fiber of C over s ∈ S by C [ s ] . We refer to S - ∞ -categories as S -categories. Remark 2.1.6.
Most results recalled in this section hold for general S -categories. One notableexception is the description of Sp S , the S -stabilization of the S -category of S -spaces, using spec-tral Mackey functors. Another exception is the uniqueness of S -symmetric monoidal structureon Sp S . However, these results hold under mild conditions on S . Handling H -categories as G -categories Occasionally we have to consider H -categories forsome subgroup H < G . When doing so we use the slice category
G/H := ( O opG ) ( G/H ) / , theopposite of the category of G -orbits over G/H . The category
G/H is equivalent to the categoryof H -orbits. Moreover, the forgetful functor G/H → O opG is left fibration classified by therepresentable functor Map( − , G/H ) : O opG → S . In particular a G/H -category C ։ G/H is a G -category by postcomposition with the forgetful functor, G/H ։ O opG . Note that this constructionalso avoid a choice of basepoint to G/H . When referring to the fibers of C ։ G/H we adopt thenotation C [ ϕ ] for the fiber over ϕ : G/K → G/H as an object in the slice category
G/H .The G -category G/H has a second role for us, since a G -functor G/H → C corresponds to anobject in the fiber of C ։ O opG over G/H , as we now explain. Under straightening/unstraighteningthe left fibration
G/H ։ O opG corresponds to the representable functor of the orbit G/H , givenby Hom O G ( − , G/H ) : O opG → S , and therefore by the Yoneda lemma ([Lur09a, lem. 5.1.5.2])corresponds to an object of C [ G/H ] 11 . We denote the G -functor corresponding to x ∈ C [ G/H ] by σ x : G/H → C . A more explicit construction is given by choosing a section of the trivial fibration
Arr coCart x → ( C ) ∼ ։ G/H of [Sha18, not. 2.28] and composing with ev : Arr coCart x → ( C ) → C . The G -category of G -spaces The ∞ -categories Fun( G/H, S ) assemble as the fibers of a G -category Top G , the G -category of G -spaces ([BDG + G/H is equivalent to
Top G [ G/H ] ∼ = Fun( G/H, S ) ≃ Fun( O opH , S ) ≃ Top H ,the ∞ -category of H -spaces. By an H -space we always mean an H -CW space. The G -categoryof G -spaces is characterized by the following universal property (see [BDG + G -category C we have an equivalence of ∞ -categories Fun G ( C , Top G ) ≃ Fun( C , S ),i.e Top G is the cofree G -category co-generated by the ∞ -category of spaces.Taking our cue from the Elmendorf-McClure theorem, we think of a G -category as capturingthe notion of a G -action on an ∞ -category. With this intuition in mind one may think of Top G Specifically, they hold for S an atomic orbital ∞ -category. See [BDG + + To make this argument precise we need to replace C with a presheaf of spaces. To achieve that we straighten C ≃ ⊆ C , the maximal G -subgroupoid of C , given as a left fibration by the full maximal sub-simplicial set supportedon the coCartesian edges of C .
11s follows. Imagine that the ∞ -category of spaces admits a non-trivial G -action, whose H -fixedpoints is the ∞ -category of H -spaces for all H < G . Think of
Top G as capturing this imagined G -action. Remark 2.1.7.
In section 3.2 we use the following explicit model for
Top G . Construct anauxiliary topological category O G - Top as follows. An object of O G - Top is G -map X → O wherethe domain X is a G -CW complex and codomain O ∈ O G is a G -orbit. We refer to an object of O G - Top as O G -space, though it should rightfully be called a “ G -space over an orbit”. A mapof O G -spaces is given by a (strictly) commuting squares of G -spaces X (cid:15) (cid:15) / / X (cid:15) (cid:15) O / / O . (4)The mapping spaces of O G - Top are given byMap O G - Top ( X → O , X → O ) = Map G ( X , X ) × Map G ( X ,O ) Map G ( O , O ) , where Map G ( X, Y ) is the space of G -maps X → Y with the compact-open topology.We think of an O G -space X → G/H as representing the H -space given by the fiber X | H of X → G/H over the coset H . On the other hand, given an H -space X we can use topologicalinduction to construct a O G -space G × H X whose fiber over H is X . Note that the O G -space X → O does not represent the G -space X (in fact, choosing an isomorphism O ∼ = G/H forsome
H < G exhibits the G -space X as the topological induction of the H -space represented by X → G/H ).Applying topological nerve construction of [Lur09a, def. 1.1.5.5] produces an ∞ -category N ( O G - Top ). The forgetful N ( O G - Top ) → O G , ( X → O ) O is a Cartesian fibration, and acommuting square (4) describes a coCartesian edge in N ( O G - Top ) if it is a pullback square. Tosee this use [Lur09a, prop. 2.4.1.1 (2)] as in the proof of proposition 3.1.14. The dual coCartesianfibration N ( O G - Top ) ∧ → O opG , described in [BGN14], is a G -category equivalent to Top G . Wecan explicitly describe an object of Top G [ G/H ] in this model as a G -map X → G/H , which weinterpret as the H -space X | eH given by the fiber over the coset eH . A map in Top G is given bya (strictly) commutative diagram of G -spaces X (cid:15) (cid:15) X ′ o o ✤❴ (cid:15) (cid:15) / / Y (cid:15) (cid:15) O O o o = / / O in which the left square is a pullback square. It is a coCartesian edge if and only of the G -map X ′ → Y is a G -homotopy equivalence over O (see [BGN14]). Equivalently, if O = G/H thenthe above edge is coCartesian precisely when the map of fibers X ′ | eH → Y | eH is an H -homotopyequivalence.By definition maps in fiber Top G [ O ] are commutative diagrams as above, with row given by O = ←− O = −→ O . Unwinding the definitions we see that Top G [ O ] is equivalent to N ( Top
G/O ), thecoherent nerve of the topological category of G -CW-spaces over O . If O = G/H then restrictionto the fiber over eH defines an equivalence of topological categories Top
G/G/H ∼ −→ Top H to thetopological category of H -CW-spaces. 12inally, we note that N ( Top
G/O ) ≃ N ( Top G ) /O are equivalent ∞ -categories . We use theMoore over-category of appendix A to see this. By corollary A.0.5 we have N ( Top G ) /O ≃ N (cid:16) ( Top G ) Moore/O (cid:17) . However, since the orbit O is a discrete G -space we see that for every X ∈ Top G the only Moore paths in Map Top G ( X, O ) are constant, so
Top
G/O → ( Top G ) Moore/O is anequivalence of topological categories. Therefore the fiber
Top G [ O ] is equivalent to the slice category N ( Top G ) /O . The mapping spaces of Top G [ O ] ≃ N ( Top
G/O ) will be denoted by Map GO ( X, Y ). The G -category of G -spectra A more interesting example is given by Sp G , the G -categoryof G -spectra, with fiber over G/H is equivalent to Sp G [ G/H ] ≃ Sp H , the ∞ -category genuineorthogonal H -spectra (see [Nar17, thm. 2.40], with origins in [GM11]). For a construction of Sp G as the G -stabilization of Top G see [Nar17, def. 2.35 and thm. 2.36]. G -categories We frequently use the following constructions of G -categories. Construction 2.2.1.
Given two S -categories C , D , the fiber product C × S D is an S -category,the fiberwise product of C and D . If C , D are G -categories, we denote the fiberwise product C × O opG D by C×D . In particular, we use the fiberwise product to restrict a G -category C ։ O opG to a G/H -category C× G/H ։ G/H (“forgetting the G -action on C to get an H -action”). Construction 2.2.2.
Given a G -category C define the fiberwise arrow category Arr G ( C ) asthe fiber product O opG × Fun(∆ , O opG ) Fun(∆ , C ) (see [Sha18, not. 4.29]). Note that Arr G ( C ) isequivalent to the functor G -category Fun G ( O opG × ∆ , C ), where the G -category O opG × ∆ is theconstant G -category on ∆ . More generally, for any S -category C ։ S define the fiberwise arrow S -category Arr S ( C ) as the fiber product S × Fun(∆ ,S ) Fun(∆ , C ). Construction 2.2.3.
Let C be a G -category and x ∈ C [ G/H ] an object over G/H , correspondingto the G -functor σ x : G/H → C . Following [Sha18, not. 4.29], we define the parametrized slice-category C /x ։ G/H by pulling back the coCartsian fibration ev : Arr G ( C ) ։ C along σ x , i.e. C /x := Arr G ( C ) × C G/H . We will also consider C /x ։ G/H as a
G/H -category.Note that the fiber of C /x ։ G/H over ϕ : G/K → G/H is equivalent to the ∞ -over-category( C [ G/K ] ) /ϕ ∗ x , where ϕ ∗ x ∈ C [ G/K ] is determined by choosing a coCartesian lift x → ϕ ∗ x of ϕ . Construction 2.2.4.
For C ։ S an S -category, the fiberwise cone S -category of C is defined asthe parametrized join C ⋆ S S (see [Sha18, not. 4.2] or appendix B). Parametrized functors and parametrized functor categoriesDefinition 2.2.5.
Let C , D be S -categories, i.e. coCartesian fibrations C ։ S, D ։ S . An S -functor is a functor C → D over S which preserves coCartesian edges. Let Fun S ( C , D ) ⊆ Fun /S ( C , D ) be the full subcategory of functors C → D over S which preserve coCartesian edges.When S = O opG we refer to a O opG -functor as a G -functor, and denote the ∞ -category of G -functors by Fun G ( C , D ) . Remark 2.2.6. An S -functor C → D encodes the data of a coherent natural transformation C • ⇒ D • between the S -diagrams C • , D • : S → C at ∞ classified by the coCartesian fibrations C ։ S and D ։ S . 13 emark 2.2.7. Since the left fibration
G/H → O opG is corepresentable by construction, we have C [ G/H ] ≃ Fun G ( G/H, C ).The ∞ -category of G -categories admits an internal hom, a G -category denoted Fun G ( C , D )see [BDG + + G ( C , D ) ։ O opG over G/H admits the following description. Forget the G -action on C , D to an H -action by taking the fiber products C× G/H, D× G/H . The fiber Fun G ( C , D ) [ G/H ] isequivalent to the ∞ -category Fun G/H ( C× G/H, D× G/H ) of
G/H -functors C× G/H → D×
G/H ,(which we think of as modeling “ H -equivariant functors from C to D ”).More generally, for any two S -categories C ։ S, D ։ S there is an S -category of functorsFun S ( C , D ) with fibers Fun S ( C , D ) [ s ] ≃ Fun s ( C × S s, D × S s ) where s = S s/ . The S -category offunctors possesses the universal property of internal hom, from [BDG + Theorem 2.2.8 (Barwick-Dotto-Glasman-Nardin-Shah) . Let C , D , E be S -categories. Thenthere are natural equivalences Fun S ( C , Fun S ( D , E )) ∼ −→ Fun S ( C × S D , E ) , Fun S ( C , Fun S ( D , E )) ∼ −→ Fun S ( C × S D , E ) . Note that if C , D , E are G -categories, then the second equivalence follows from the first byrestricting to the fiber over the orbit [ G/G ], the terminal object of O G . We follow [Nar17], defining parametrized colimits and parametrized left Kan extensions usingparametrized adjoints.
Parametrized adjoints
Let C , D be S -categories. An S -adjunction ([Sha18, def. 8.1]) is arelative adjunction L : C ⇆ D : R over S ([Lur, def. 7.3.2]) where both L and R are S -functors.In particular, for each s ∈ S we have an adjunction L [ s ] : C [ s ] ⇆ D [ s ] : R [ s ] between the fibersover s . When S = O opG we will refer to an O opG -adjunction as a G -adjunction. Parametrized colimits
Let p : I → C be an S -functor, which we think of as an S -diagramin C . The S -colimit of p is an S -object of C , i.e a coCartesian section S − colim −−−→ ( p ) : S → C ofthe structure fibration C ։ S . For a general definition of colim −−−→ ( p ) as the S -initial S -cone under p see [Sha18, def. 5.2]. We define I -shaped S -colimits as the S -left adjoint to the “constant I -diagram” S -functor, following [Nar16, def. 2.1]. This definition is justified by [Sha18, 10.4],since we only take S -colimits in S -cocomplete S -categories.Explicitly, precomposition with the coCartesian fibration I ։ S induces an S -functor ∆ I : C ≃
Fun S ( S, C ) → Fun S ( I, C ), where S is the terminal S -category (given by id : S → S ). If ∆ I ad-mits an S -left adjoint we say that C admits I -indexed S -colimits, and denote the S -left adjointby S − colim −−−→ : Fun S ( I, C ) → C . Note that for every index s ∈ S we have an adjunction of ∞ -categories S − colim −−−→ : Fun s ( I × S s, C × S s ) ⇆ Fun s ( S × S s, C × S s ) ≃ C [ s ] : ∆ I . Particularly, we will use the following type of
G/H -colimit.
Example 2.3.1.
Let C be a G -category, I ։ G/H a G/H -category and p : I → C a G -functor.Since G/H ։ O opG is a left fibration we have Fun G ( I, C ) ≃ Fun
G/H ( I, C× G/H ), under which14 corresponds to a
G/H -functor p : I → C× G/H , or in other words p ∈ Fun
G/H ( I, C× G/H ).Then
G/H − colim −−−→ ( p ) ∈ C [ G/H ] is given by applying the left adjoint of G/H − colim −−−→ : Fun G/H ( I, C× G/H ) ⇆ Fun
G/H ( G/H, C× G/H ) ≃ C [ G/H ] : ∆ I . We say that an S -category C is S -cocomplete if for every s ∈ S the s -category C× s admits I -indexed s -colimits for any s -category I . Parametrized left Kan extensions
We follow [Nar17, def. 2.12] and define S -left Kan exten-sion using the give a global characterization as a left adjoint. For a general definition of pointwiseparametrized left Kan extensions see [Sha18, def. 10.1], which satisfies the global characteriza-tion by [Sha18, 10.4]. We only use the pointwise definition in the proof of proposition 4.2.4, a G -categorical statement independent from the rest of the paper.Let ι : D → M be an S -functor and C an S -category. Restriction along ι induces an S -functor ι ∗ : Fun S ( M , C ) → Fun S ( D , C ). The S -left Kan extension along ι is the S -left adjoint to ι ∗ anddenoted by φ ! .We will use the following propositions from [Sha18]. Proposition 2.3.2. [Sha18, thm. 10.3] Let A : D → C and ι : D → M be S -categories, andsuppose that for every x ∈ M over s ∈ S the s -colimit s − colim −−−→ (cid:16) D /x → D × S s A × S s −−−−→ C × S s (cid:17) exists. Then the S -left Kan extension of A along ι exists (and is essentially unique), and actson x ∈ D by sending it to the s -colimit above, considered as an object in the fiber C [ s ] . Proposition 2.3.3. [Sha18, cor. 10.6] Let C be a S -cocomplete S -category and ι : D → M afully faithful S -functor (i.e fiberwise fully faithful, see [BDG + S -leftKan extension ι ! : M → C exists and is S -fully faithful. When S = O opG we refer to S -left Kan extensions as G -left Kan extensions, which we use todefine G -factorization homology as a G -functor (see proposition 4.1.4). Parametrized Yoneda embedding
Another useful tool available to us is the parametrizedYoneda embedding of [BDG + G -tangent clas-sifier (see construction 3.2.8). Let C be a G -category, and C vop the fiberwise opposite G -category(with fibers ( C vop ) [ G/H ] ∼ = ( C [ G/H ] ) op , see [BDG + + G -functor j : C →
Fun G ( C vop , Top G ), the parametrized Yoneda embedding,which can be informally described as follows. The G -functor j takes x ∈ C [ G/H ] the G/H -functorMap( − , x ) : C vop × G/H → Top G × G/H sending an object y ∈ (( C vop ) × G/H ) [ ϕ ] ∼ = ( C [ G/K ] ) op inthe fiber over ϕ : G/K → G/H to the mapping space Map( y, ϕ ∗ x ) of the ∞ -category C [ G/K ] . G -symmetric monoidal structures The notion of a G -symmetric monoidal structure plays a central role in our presentation of G -factorization homology. In this subsection we give some intuition for G -symmetric monoidalstructure, hopefully making it more approachable. This subsection is expository in nature, theformal definition of a G -symmetric monoidal G -category can be found in [Nar17, sec. 3.1], or inappendix B. 15nformally, the data of a G -symmetric monoidal structure on a G -category C is given by col-lection of symmetric monoidal structures on the fibers C [ G/H ] , together with symmetric monoidalfunctors C [ G/K ] → C [ G/H ] , called norm functors, for each map of orbits G/K → G/H . We havethe following examples in mind. • The coCartesian G -symmetric monoidal structure on Top G , which is given by disjointunions in Top G [ G/H ] ≃ Top H and norm functors ∀ K < H < G : a H/K : Top K → Top H , a H/K X = H × K X, where H × K X is the quotient of G × X by the diagonal action of K . • The Cartesian G -symmetric monoidal structure on Top G , which is given by products of H -spaces and norm functors ∀ K < H < G : Y H/K : Top K → Top H , Y H/K X = Map K ( H, X ) , where Map K ( H, X ) is the space of K -equivariant maps H → X with K acting on H bymultiplication from the right. • The G -category Sp G of G -spectra has a G -symmetric monoidal which is given by smashproducts in Sp G [ G/H ] ≃ Sp H and the Hill-Hopkins-Ravenel norm functors, informally givenby taking X ∈ Sp H to the smash product of | G/H | copies of X with induced G -action.Nardin gave a universal property characterizing this G -symmetric monoidal structure byproving that Sp G admits an essentially unique G -symmetric monoidal structure for whichthe sphere spectrum is the unit, see [Nar17, cor. 3.28].The data of a G -symmetric monoidal structure, along with its coherent compatibility, isencoded by a single coCartesian fibration over the indexing category Fin G ∗ , satisfying certainSegal conditions. In what follows, we try to explain how this technical description is related tothe intuition presented above.We regard the symmetric monoidal structure on each fiber and the norm functors on equalfooting. To that end, consider the G -symmetric monoidal structure as acting on a U -familyof objects, where we index our family be a finite G -set. The members of a U -family x • in a G -category C correspond to the orbits of U , with x W ∈ C [ W ] for each orbit W ∈ Orbit( U ). Givena G -map I : U → G/H , we can use the G -symmetric monoidal structure to construct an element ⊗ I x • ∈ C [ G/H ] . Using the operations ⊗ I we can encapsulate the data G -symmetric monoidalstructure on C .The various operations ⊗ I are subject to certain compatibility conditions, which hold uptocoherent homotopy. In order to encapsulate the compatibility of ⊗ I for various I it is convenientto extend ⊗ I from I : U → G/H to general G -maps of finite G -sets ϕ : U → V . The generalizedoperation ⊗ ϕ takes a U -family to a V -family by acting on the fibers of ϕ , i.e ∀ W ′ ∈ Orbit( V ) : ( ⊗ ϕ x • ) W ′ = ⊗ ϕ − ( W ′ ) (cid:0) x • | ϕ − ( W ′ ) (cid:1) ∈ C [ W ′ ] . Note that we also need to keep track of restrictions taking a U -family x • to a U ′ -family x • | U ′ for each inclusion of G -sets U ′ ֒ → U .All these operations are encoded by a coCartesian fibration over Fin G ∗ , the G -category offinite pointed G -sets (see appendix B), which we think of as our indexing category. Note that16he fiber of Fin G ∗ over G/H is the given by the category of spans of finite G -sets U ← ֓ U ′ → V over G/H , where the wrong way map U ← ֓ U ′ is an inclusion. Restriction to the fiber over eH defines an equivalence ( Fin G ∗ ) [ G/H ] ∼ −→ Fin H ∗ to the category of finite pointed H -sets, describedhere as partly defined H -maps given by spans of finite H -sets ˜ U ← ֓ ˜ U ′ → ˜ V , where the wrongway map is an inclusion of finite H -sets.We end this subsection by briefly sketching how to extract the tensor products and normsfrom a coCartesian fibration p : C ⊗ ։ Fin G ∗ describing a G -symmetric monoidal structure on a G -category C .First we describe the tensor product of two objects x , x ∈ C [ G/H ] . The ∞ -category C [ G/H ] is given as the fiber of p over G/H = −→ G/H . Let U = G/H ` G/H and I ∈ Fin G ∗ given by thefold map I : U → G/H . By the Segal conditions we have an equivalence C ⊗ I ∼ −→ C [ G/H ] × C [ G/H ] from the fiber of p over I . Through this equivalence we identify the ordered pair ( x , x ) ∈C [ G/H ] × C [ G/H ] with an object x • ∈ C ⊗ I (a U -family). Choose a p -coCartesian lift x • → y of thespan U = ←− U I −→ G/H over
G/H . The tensor product x ⊗ x is given by y ∈ C [ G/H ] .Next we describe the norm of an object x ∈ C [ G/K ] along ϕ : G/K → G/H . As before, the ∞ -category C [ G/K ] is the fiber of p over G/K = −→ G/K . Consider the map ϕ : G/K → G/H asan object of
Fin G ∗ . By the Segal conditions we have an equivalence C ⊗ ϕ ∼ −→ C [ G/K ] from the fiberof p over ϕ . Through this equivalence we identify x ∈ C [ G/K ] × C [ G/H ] with an object x • ∈ C ⊗ ϕ (a ϕ -family). Choose a p -coCartesian lift x • → y of the span G/K = ←− G/K ϕ −→ G/H over
G/H .The norm ⊗ ϕ x is given by y ∈ C [ G/H ] . G -manifolds and G -disks Genuine G -factorization homology will be constructed in section 4 using parametrized ∞ -categorytheory. Our goal in this section is to construct and study the G - ∞ -categories needed there. Mostof this section is devoted to the construction of these G - ∞ -categories and their G -symmetricmonoidal structures. These constructions may be of independent interest, as they provide abridge between geometry of manifolds with a finite group action and the theory of parametrized ∞ -categories.In section 3.1 we construct Mfld G , the G -category of G -manifolds. The construction isinspired by the model of Top G described in remark 2.1.7. We then turn to study its relation to G -vector bundles, and construct an equivariant version of the tangent classifier functor of [AF15].This G -tangent classifier is used in section 3.3 to construct framed variants of Mfld G .Next, we turn our attention to G -disjoint unions. In section 3.4 we define a G -symmetricmonoidal structure on Mfld G encoding disjoint unions and topological inductions. The construc-tion is quite explicit, and relies on the unfurling construction Barwick, introduced in [Bar14]. Insection 3.4 we lift G -disjoint unions to a G -symmetric monoidal structures on the framed variantsof Mfld G . Our main tool will be the G -coCartesian structures constructed in [BDG + ].The G -symmetric monoidal structure of G -disjoint unions will be used in section 4 whendefining factorization homology in two ways. First, the expected interaction of genuine G -factorization homology with disjoint unions and topological inductions is expressed by being a G -symmetric monoidal functor from Mfld G . Second, the definition of G -disk algebras relies onthe definition of the G -symmetric monoidal G - ∞ -category of G -disks, defined in section 3.6.Next, we turn to study our constructions. In section 3.7 we show that G -disks are exactlythe G -manifolds generated from linear representations of subgroups H < G by taking disjointunions and topological inductions. In section 3.8 we compare equivariant embeddings of G -diskswith equivariant configurations spaces. The results of this comparison will be used in section 5.217o show that genuine G -factorization homology satisfies ⊗ -excision. In section 3.9 we define the G - ∞ -operad E V of little V -disks, and use the results of section 3.8 to relate E V to V -framed G -disks. G -category of G -manifolds The goal of this subsection is to give an explicit model for the G - ∞ -category Mfld G of n -dimensional G -manifolds.Before going into the details of the construction, let us first recall the construction of the ∞ -category Mfld G of G -manifolds, achieved by a standard procedure. Let M , M be smooth n -dimensional manifolds equipped with a smooth action of a finite group G . The set Emb G ( M , M )of smooth G -equivariant open embeddings M ֒ → M comes with a natural topology, making thecategory Mfld G of n -dimensional G -manifolds into a topological category. We consider Mfld G as an ∞ -category by taking its coherent nerve ([Lur09a, def. 1.1.5.5]).We can extend the construction of Mfld G to construct the G - ∞ -category Mfld G as follows.Consider the ∞ -categories Mfld H of n -dimensional H -manifolds and H -embeddings for all sub-groups H < G . The ∞ -categories Mfld H form a diagram of ∞ -categories, by related by twotypes of functors:1. First, if M is a G -manifold and H < G we can consider M as an H -manifold, which definesa functor of topological categories Mfld G → Mfld H . Similarly we have Mfld H → Mfld K for K < H < G .2. Second, suppose
K, H < G are conjugate subgroups, i.e H = gKg − for some g ∈ G , and M is an H -manifold. We can consider M as a K -manifold by twisting the H -action byconjugation, defining an isomorphism of topological categories conj HK : Mfld H → Mfld K .A standard verification shows that the topological categories Mfld H define a diagram of topolog-ical categories indexed by subgroups H < G , with functors indexed contravariantly by G -maps G/K → G/H . Note that this indexing category is equivalent to the orbit category O G (seedefinition 2.1.1). Composing with the topological nerve we get a diagram of ∞ -categories Mfld • : O opG → C at ∞ , G/H N ( Mfld H ) , which we can unstraighten to a coCartesian fibration U nSt ( Mfld • ) ։ O opG (see [Lur09a, sec.3.2]). The casual reader can use U nSt ( Mfld • ) as the definition of the G -category of G -manifolds,and skip the rest of this subsection.The construction of U nSt ( Mfld • ) is unsatisfying to us in two respects. First, it depends onan implicit choice of an inverse to the inclusion of the full subcategory { G/H } H Definition 3.1.2. An O G -manifold M → O is a smooth n -dimensional manifold M with anaction of G on M by smooth maps, together with a G -map M → O from the underlying G -spaceof the manifold M to a G -orbit O ∈ O G . We always think of an O G -manifold M → G/H as encoding a smooth n -dimensional manifoldwith an action of H , given by the fiber M | H of the G -map M → G/H over the coset H . Notethat a choice of a basepoint o ∈ O induces an isomorphism G/H ∼ = → O, gH g · o , where H < G is the stabilizer of o . We therefore think of an O G -manifold M → O as encoding the smoothaction of H = Stab ( o ) on the fiber M | H . Notation 3.1.3. Suppose M, N are smooth n -dimensional manifolds. Denote by C ∞ ( M, N )the space of smooth maps M → N with the compact-open topology. Definition 3.1.4. Let M → O , M → O be O G -manifolds. For ϕ : O → O a map in O G ,define Emb O G ϕ ( M , M ) ⊂ C ∞ ( M , M ) as the subspace of smooth maps f : M → M such that1. f is a G -map2. f is over ϕ , i.e M (cid:15) (cid:15) f / / M (cid:15) (cid:15) O ϕ / / O (5) is a commutative square of G -spaces.3. the induced map M → O × O M is an embedding.Define the topological space Emb O G ( M , M ) as the coproduct Emb O G ( M , M ) := a ϕ Emb O G ϕ ( M , M ) , (6) where the coproduct is indexed by the set Hom O G ( O , O ) . Notation 3.1.5. When the orbit map ϕ is an identity G/H = −→ G/H we use the notation Emb GG/H ( M , M ) for the space Emb O G ϕ ( M , M ) of G -equivariant embeddings M → M over G/H = −→ G/H . Restriction to the fiber over H defines a homeomorphism from Emb GG/H ( M , M )with the space of H -equivariant embeddings M | H → M | H between the fibers over H . Definition 3.1.6. Let M → O , M → O be O G -manifolds. A G -isotopy over ϕ : O → O isa path in Emb O G ϕ ( M , M ) . When M → O, M → O are over the same orbit we call a path in Emb O G O ( M , M ) a G -isotopy over O . Note that a G -isotopy over G/H is equivalent to an H -equivariant isotopy between two H -equivariant embeddings M | H → M | H . 19 he topological category of O G -manifolds. We now turn to the definition of the topolog-ical category of O G -manifolds. Note that the pullback of smooth embeddings of n -dimensionalmanifolds is a smooth embedding, therefore we have Lemma 3.1.7. Let M → O , M → O , M → O be O G -manifolds. The composition of smoothfunctions defines a continuous map Emb O G ( M , M ) × Emb O G ( M , M ) → Emb O G ( M , M ) , ( g, f ) g ◦ f. Definition 3.1.8. The category of O G -manifolds O G - Mfld is the topological category whoseobjects are a O G -manifolds. The morphism space from M → O to M → O is given by Map O G - Mfld ( M , M ) := Emb O G ( M , M ) .Define a forgetful functor q : O G - Mfld → O G by sending M → O to the orbit O , and thesubspace Emb O G ϕ ( M , M ) ⊂ Emb O G ( M , M ) to ϕ ∈ Hom O G ( O , O ) .By [Lur09a, ex. 1.1.5.12] the topological nerve N ( O G - Mfld ) is an ∞ -category, and by[Lur09a, ex. 1.1.5.8] the topological nerve of O G can be identified with its ordinary nerve, whichwe identify with O G by standard abuse of notation.Applying the topological nerve functor of [Lur09a, 1.1.5.5] to q produces a functor of ∞ -categories N ( q ) : N ( O G - Mfld ) → O G . In particular, an object of the ∞ -category N ( O G - Mfld ) is an O G -manifold M → O , a map isgiven by a commutative square eq. (5) satisfying the conditions of definition 3.1.4, and by [Lur09a,thm. 1.1.5.13] the mapping spaces of N ( O G - Mfld ) are weakly equivalent to the mapping spacesof O G - Mfld . Remark 3.1.9. The fiber of O G - Mfld → O G over an orbit G/H is the topological nerve ofthe topological category whose objects are O G -manifolds M → G/H and morphism spaces are Emb GG/H ( M , M ). This topological category is equivalent to the category Mfld H of H -manifoldsand H -equivariant embeddings by restriction to the fibers over H . Remark 3.1.10. We caution the reader not to pass to ∞ -categories prematurely. One canconstruct the topological category O G - Mfld as a subcategory of the topological arrow category Mfld G ↓ O G . However, the ∞ -category N ( O G - Mfld ) is not a subcategory of the topologicalnerve N ( Mfld G ↓ O G ) in the sense of [Lur09a, sec. 1.2.11]. To see this note that a subcategoryof N ( Mfld G ↓ O G ) is specified by a subcategory of its homotopy category ho N ( Mfld G ↓ O G ),and therefore given by a choosing connected components of each mapping space of Mfld G ↓ O G .On the other hand condition (3) of definition 3.1.8 is not preserved by G -homotopy equivalence,so the subspace Emb GG/H ( M → O , M → O ) ⊂ Map Mfld G ↓O G ( M → O , M → O )is not given by a set of connected components. The same phenomenon exists in the non-equivariant setting. Equivalences of O G -manifolds. Unwinding the definition of equivalence in a nerve of a topo-logical category, we see that a map f : M → M in O G - Mfld is an equivalence in N ( O G - Mfld )if it has a G -isotopy inverse: a map g : M → M in O G - Mfld , together with a G -isotopy over id q ( M ) from g ◦ f to id M and a G -isotopy over id q ( M ) from f ◦ g to id M . Definition 3.1.11. We say that a map f : M → M of O G -manifolds is a G -isotopy equivalence if it is an equivalence in the ∞ -category N ( O G - Mfld ) . f always lies over an isomorphism of orbits q ( f ) : O → O .Using the homeomorphism between the mapping space Emb GG/H ( M , M ) over an orbit G/H and the space of H -equivariant embeddings M | H → M | H we see that a map f : M → M over an orbit G/H is an equivalence in N ( O G - Mfld ) if and only if its restriction to the fibers f | H : M | H → M | H is invertible upto H -isotopy. In particular, f need not induce an equivariantdiffeomorphism. Nonetheless, its existence is enough to ensure that there exists an equivariantdiffeomorphism between underlying manifolds. We learned the following argument from an an-swer of Ian Agol on MathOverflow [ha], which we reproduce here (with addition of a G -action). Proposition 3.1.12. Let M → G/H and M → G/H be two O G -manifolds over G/H . If f ∈ Emb GG/H ( M , M ) and g ∈ Emb GG/H ( M , M ) are G -isotopy inverses over G/H then thereexists a G -equivariant diffeomorphism M ∼ = M over G/H .Proof. We prove the statement by reduction. Since Emb GG/H ( M , M ) is homeomorphic to thespace of H -invariant embeddings between M | H → M | H it is enough to consider the case G = H .Suppose M, N are n -dimensional manifolds with smooth actions of G , and we are given G -equivariant embeddings f : M → N, g : N → M . Consider the direct limit X = colim −−−→ ( M f −→ N g −→ M f −→ N g −→ · · · ) , given by the explicit model M × N ⊔ N × N / ∼ with equivalence relation generated by ( m, k ) ≃ ( f ( m ) , k ) and ( n, k ) ≃ ( g ( n ) , k + 1). Then X is a smooth manifold with an action of G , as asequential union of nested open submanifolds.Since X is G -diffeomorphic to Y = colim −−−→ ( N g −→ M f −→ N g −→ · · · ) (removing the first term ofthe sequence does not change the colimit), it is enough to show that X is G -diffeomorphic to M .Note that X is G -diffeomorphic to colim −−−→ ( M F −→ M F −→ M F −→ · · · ) for F = g ◦ f , and F is G -isotopic to id M . Let F t : M → M, t ∈ [0 , 1] be the G -isotopy from F = id M to F = g ◦ f ,and define X t = colim −−−→ ( M F t −→ M F t −→ · · · ), so that X = X and X = M .Choose a sequence of compact G -submanifolds with boundary K ⊂ K ⊂ K ⊂ · · · M suchthat M = ∪ i K i and F ( K i × [0 , ⊂ int ( K i +1 ). Such a sequence can be chosen inductivelyusing a G -invariant Morse function on M (which exists by [Was69, cor. 4.10]). Define Y t = colim −−−→ ( K F t −→ K F t −→ K F t −→ · · · ) using the restrictions of the F t to the subsets K i . We claimthat Y t = X t , using the standard model for direct limits. Write X t = M × N / ( x, i ) ∼ ( F t ( x ) , i +1),and note that Y t ⊆ X t as the points ( x, i ) with x ∈ K i . We claim that each point x ∈ X t is in Y t .Represent x by ( x, i ) ∈ M × N , then since M = ∪ K i we have x ∈ K j for some j ∈ N . If j ≤ i then K j ⊂ K i , so x ∈ K i , hence ( x, i ) represents an point in Y t . Otherwise ( x, i ) ∼ ( F j − it ( x ) , j ) inrepresents the same point in X t , and since F j − it ( K i ) ⊂ K j we get F j − it ( x ) ∈ K j , so ( F j − it ( x ) , j )represents an element of Y t .We showed that Y t = X t , so it is enough to prove that Y ∼ = M is G -diffeomorphic to Y ∼ = X .By definition we have Y = colim −−−→ ( K ֒ → K ֒ → K ֒ → · · · ) and Y = colim −−−→ ( K F −→ K F −→ K F −→ · · · ), hence it is enough to construct compatible G -diffeomorphims φ i : K i ֒ → K i , i.esatisfying φ i +1 | K i = F ◦ φ i .We now inductively construct G -equivariant maps G i : K i × [0 , → K i such that G = Id K i , ∀ t ∈ [0 , 1] : G t : K i → K i is a diffeomorphism and ∀ x ∈ K i , t ∈ [0 , 1] : F t ◦ G it ( x ) = G i +1 t ( x ), i.e21he diagram K i × [0 , (cid:31) (cid:127) / / G i × Id (cid:15) (cid:15) K i +1 × [0 , G i +1 (cid:15) (cid:15) K i × [0 , F | Ki × [0 , / / K i +1 commutes. We start with setting G t = Id K . Assume that a G i has been constructed. Consider theisotopy K i × [0 , G i × Id −−−−→ K i × [0 , F | Ki × [0 , −−−−−−→ K i +1 . Since K i ⊂ K i +1 is a compact submanifoldand F ( K i ) ⊂ Int ( K i +1 ) the conditions of the isotopy extension theorem [Hir12, ch. 8 thm. 1.3]are satisfied. Therefore there exists a diffeotopy ˜ G i +1 : K i +1 × [0 , → K i +1 which extends theisotopy K i × [0 , G i × Id −−−−→ K i × [0 , F | Ki × [0 , −−−−−−→ K i +1 and satisfies ˜ G i +10 = Id K i +1 , but might notbe G -equivariant. Since K i +1 is compact we can apply [Bre72, thm 3.1], and get a G -equivariantdiffeotopy G i +1 : K i +1 × [0 , → K i +1 with G i +10 = ˜ G i +10 = Id K i +1 and which agrees with ˜ G i +1 on the subset n x ∈ K i +1 | ∀ g ∈ G, t ∈ [0 , 1] : ˜ G i +1 t ( gx ) = g ˜ G i +1 t ( x ) o . In particular, for x ∈ K i we have ˜ G i +1 t ( x ) = F t G it ( x ), so the G -equivariant diffeotopy G i +1 agrees with ˜ G i +1 on K i × [0 , φ i = G i gives the compatible G -diffeomorphisms proving that Y ∼ = M is indeed G -diffeomorphic to Y ∼ = X . Cartesian edges in O G - Mfld. We now identify the Cartesian edges of the forgetful functor N ( O G - Mfld ) → O G , as well as the coCartesian edges over isomorphisms. We start with Lemma 3.1.13. The forgetful functor N ( q ) : N ( O G - Mfld ) → O G is an inner fibration.Proof. For every pair M → O , M → O of O G -manifolds, q induces a Kan fibrationMap Sing ( O G - Mfld ) ( M , M ) → Map Sing ( O G ) ( O , O ) = Hom O G ( O , O ) , because its a map from a Kan simplicial complex and to discrete simplicial set. Therefore by[Lur09a, prop. 2.4.1.10(1)] the functor N ( q ) is an inner fibration.Note that a map M → O from an n -dimensional manifold to a finite set is always a submer-sion, so its pullback along any map of finite sets is an n -dimensional manifold. Proposition 3.1.14. Suppose that ϕ : O → O be a map of orbits, and M → O a O G -manifold.Then the pullback square of topological G -spaces O × O M (cid:15) (cid:15) f / / ❴✤ M (cid:15) (cid:15) O / / O defines a N ( q ) -Cartesian morphism f in O G - Mfld . In particular, N ( q ) is a Cartesian fibration. The map G i is an equivariant diffeotopy in terminology of [Hir12] and an equivariant isotopy starting fromthe identity in the terminology of [Bre72]. roof. Checking that f satisfies the conditions of definition 3.1.8 is immediate.By [Lur09a, prop. 2.4.1.1 (2)] the morphism f is N ( q )-Cartesian if and only if, for every O G -manifold T → O , the square of spaces M ap Sing ( Mfld ) ( T, O × O M ) (cid:15) (cid:15) (cid:15) (cid:15) f ∗ / / M ap Sing ( O G - Mfld ) ( T, M ) (cid:15) (cid:15) (cid:15) (cid:15) Hom O G ( O, O ) q ( f ) ∗ / / Hom O G ( O, O T, O × O M ) ( ( ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ / / Hom( O, O ) × Hom( O,O ) Map( T, M ) t t t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ Hom( O, O )is a homotopy equivalence, or equivalently, if f ∗ induces an equivalence between the fiber overevery τ ∈ Hom( O, O ).Let τ : O → O . Then f ∗ induces a map of fibers over τEmb O G τ ( T, O × O M ) → { τ } × Hom( O,O ) Map( T, M ) = Emb O G q ( f ) ◦ τ ( T, M ) , T (cid:15) (cid:15) g / / O × O M (cid:15) (cid:15) O τ / / O T (cid:15) (cid:15) g / / O × O M (cid:15) (cid:15) f / / ❴✤ M (cid:15) (cid:15) O τ / / O q ( f ) / / O This continuous map is a bijection by the universal property of the pullback. We leave to it tothe reader to verify it is an open map using the definition of the compact-open topology.This gives the following complete description of the Cartesian edges in O G - Mfld . Corollary 3.1.15. A morphism (5) is N ( q ) -Cartesian if and only if it is equivalent to a pullback,i.e. the morphism M (cid:15) (cid:15) / / O × O M (cid:15) (cid:15) O / / O (7) is a G -isotopy equivalence.Proof. Factor the morphism (5) as the composition of (7) and a pullback square. Combiningproposition 3.1.14 and [Lur09a, prop. 2.4.1.7] we see that the morphism (5) is N ( q )-Cartesian ifand only if the map above is N ( q )-Cartesian. Since the morphism (7) lies over an equivalence itis N ( q )-Cartesian if and only if it is an equivalence, by [Lur09a, prop. 2.4.1.5].23 onstruction of the G -category of G -manifolds. The construction of the G -category Mfld G now follows easily from the description of the Cartesian fibration N ( q ) and the explicitconstruction of [BGN14]. Definition 3.1.16. Let p : Mfld G → O opG be the dual of the Cartesian fibration O G - Mfld → O G in the sense of [BGN14, def. 3.5]. Explicitly, Mfld G is the pullback of the effective Burnsidecategory A eff ( O G - Mfld , O G - Mfld × O G O ∼ = G , q-Cart( O G - Mfld )) along the equivalence O opG ∼ ֒ → A eff ( O G , O ∼ = G , O G ) , where O ∼ = G is the maximal subgroupoid of O G and q-Cart( O G - Mfld ) ⊂ O G - Mfld is the subcategory spanned by all objects and morphismswhich are q -Cartesian. By [BGN14, prop. 3.4] the map p : Mfld G → O opG is a coCartesian fibration, and we have anexplicit description of the objects and morphisms of Mfld G . The objects of the total ∞ -category Mfld G are O G -manifolds M → O . A morphism in Mfld G from M → O to M → O is adiagram of the form M (cid:15) (cid:15) M o o (cid:15) (cid:15) / / M (cid:15) (cid:15) O O o o = / / O (8)where the left square is a coCartesian edge in O G - Mfld (in other words, it is equivalent to apullback square, see corollary 3.1.15). This arrow is p -coCartesian exactly when the right squareis a G -isotopy equivalence.Without loss of generality we will represent a morphism in Mfld G by a span (8) where theleft square is a pullback square. Remark 3.1.17. Let H < G be a subgroup. Topological induction defines a functor G × H ( − ) : Mfld H → Mfld G [ G/H ] , G × H M = (( G × M ) /G → ( G × pt ) /H = G/H )where we quotient by the H -action h · ( g, x ) = ( gh − , gx ). Topological induction if a functor oftopological categories, and in fact an equivalence of topological categories Mfld H ∼ −→ Mfld G [ G/H ] ,with inverse ( M → G/H ) M | eH given by restriction to the fiber over eH .Informally, the coCartesian fibration Mfld G → O opG classifies the functor O opG → C at ∞ sending G/H to Mfld H . Notation 3.1.18. We will refer to Mfld G as the G -category of G -manifolds , to stress its con-ceptual role and not its technical construction. We urge the reader to regards the objects of Mfld G not as O G -manifolds (which they are), but as a technical means of encoding manifoldswith an action of a subgroup of G . This naming convention is also compatible with [BDG + Top T is referred to as the T - ∞ -category of T -spaces.By construction, we have a simple description of the fiberwise opposite category ( Mfld G ) vop ,introduced in [BDG + The superscript “vop” stands for taking “vertical opposites”. roposition 3.1.19. Applying the opposite ∞ -category functor ( − ) op to the Cartesian fibration O G - Mfld → O G produces a G -category ( O G - Mfld ) op ։ O opG equivalent to ( Mfld G ) vop ։ O opG .Proof. By [BDG + G -category ( Mfld G ) vop ։ O opG is given by takingthe opposite of the dual Cartesian fibration ( Mfld G ) ∧ ։ O G . The result follows, since takingthe dual coCartesian fibration is homotopy inverse to taking the dual Cartesian fibration (see[BGN14, thm. 1.7]). G -vector bundles and the G -tangent classifier In this subsection we study the relation between G -vector bundles, H -representations of sub-groups H < G and the G -category of G -manifolds, Mfld G , constructed in section 3.1. We dothis by identifying H -representations with G -vector bundles over G/H , which in turn span a full G -subcategory Rep Gn ⊂ Mfld G . An equivariant version of “smooth Kister’s theorem” impliesthat Rep Gn is in fact a G - ∞ -groupoid, which can be identified with the G -space classifying n -dimensional G -vector bundles, BO n ( G ). We use Rep Gn to construct an equivariant version ofthe tangent classifier of [AF15, sec 2.1] (see construction 3.2.8), which will be used in section 3.3to define equivariant tangential structures on G -manifolds. It is worth noting that parametrized ∞ -category theory is essential for construction 3.2.8, which relies on the identification of the G -space BO n ( G ) with a full G -subcategory of Mfld G .We start by recalling the standard definition of G -vector bundles. Definition 3.2.1 (see [Bre72, sect. VI.2], [tD87, ch. I, def. 9.1]) . Let X be a G -space. A G -vector bundle over X is a (real) vector bundle p : E → X together with a G -action on E bybundle maps (i.e linear action on each fiber) such that p is a G -map. We say p : E → X is smooth if E, X are (smooth) G -manifolds and p is a smooth map. Let G − Vect /X denote thecategory of G -vector bundles over X . Note that G -vector bundles are stable under pullback along G -maps, and that a G -vectorbundle over a point is the same as a G -representation. It is useful to keep in mind the corre-spondence between representations of subgroups H < G and G -vector bundles over the orbit G/H : Proposition 3.2.2. [tD87, special case of prop. I.9.2] Let H < G be a subgroup. Restrictionto the fiber over [ eH ] gives an equivalence G − Vect / ( G/H ) ∼ −→ H − Vect /pt ∼ = Rep H fromthe category of G -vector bundles over the orbit G/H to the category of H -representations. Aninverse is given by sending a representation of H on R n to its topological induction G × H R n . The subject of this subsection is the following G -subcategory. Definition 3.2.3. Let Rep Gn ⊂ Mfld G be the full G -subcategory spanned by G -vector bundles ( E → G/H ) , i.e O G -manifolds E → G/H such that E can be endowed with a structure of a G -vector bundle over G/H . Remark 3.2.4. We will use G -vector bundles as a model for “ G -disks”. Specifically, an em-bedding of a G -disk in an O G -manifold M ∈ Mfld G is just a map in Mfld G with target is M and domain in Rep Gn . Genuine G -factorization homology is defined as a parametrized colimitover finite disjoint unions of G -disks in M (see definition 4.1.2). In section 3.6 we organize thesedisjoint unions into a G - ∞ -category Disk G .In order to see the close relation of Rep Gn with representation theory we use the followingequivariant version of the “smooth Kister-Mazur” theorem (see [Kup]).25 roposition 3.2.5. Let V be a finite dimensional real representation of H < G . Let Aut Rep H ( V ) be the automorphism group of V as an H -representation, i.e linear H -equivariant isomorphisms.Let Emb H ( V, V ) denote the subspace of smooth H -equivariant embedding fixing the origin, and Aut H ( V ) ⊂ Emb H ( V, V ) the subspace of H -equivariant diffeomorphisms. Then the inclusions Aut Rep H ( V ) ֒ → Aut H ( V ) ֒ → Emb H ( V, V ) are homotopy equivalences.Proof. The proof of [Kup, thm. 2.4] applies verbatim when restricting to subspaces of H -equivariant maps after checking that the formulas for G (1) s , G (2) s produce H -equivariant homo-topies.The central role played by Rep Gn in what follows stems from the following characterization. Proposition 3.2.6. The G -category Rep Gn is a G - ∞ -groupoid, with fibers ( Rep Hn ) [ G/H ] equiv-alent to the topological groupoid Rep Hn of n -dimensional real representations of H and (lin-ear, H -equivalent) isomorphisms, where the mapping space Iso Rep H ( V , V ) is endowed with thecompact-open topology.Proof. In order to show that Rep Gn is a G - ∞ -groupoid we have to prove that the coCartesianfibration Rep Gn ։ O opG is a left fibration. By [Lur09a, prop. 2.4.2.4] it is enough to show thatthe fibers ( Rep Gn ) [ G/H ] are ∞ -groupoids. The equivalence Mfld G [ G/H ] ∼ = Mfld H of remark 3.1.17takes a G -vector bundle E → G/H to an H -vector bundle E | eH → pt , i.e. an n -dimensionalreal H -representation V = ( H y R n ), so we have to show that for every V , V ∈ Rep Hn theinclusion Aut H ( V , V ) ⊂ Emb H ( V , V ) is a weak equivalence.Let Emb H ( V , V ) ⊂ Emb H ( V , V ) denote the subspace of origin fixing maps. Clearly theinclusion Emb H ( V , V ) ֒ → Emb H ( V , V ) is a homotopy equivalence. By proposition 3.2.5the inclusion Iso Rep H ( V , V ) ֒ → Emb H ( V , V ) is a weak equivalence, so Iso Rep H ( V , V ) ∼ ֒ → Emb H ( V , V ) is a weak equivalence.In other words, the functor Rep Hn → ( Rep Gn ) [ G/H ] is fully faithful. Since by definition it isessentially surjective it is an equivalence of ∞ -categories. In particular ( Rep Gn ) [ G/H ] is equivalentto the (coherent nerve of) the topological groupoid Rep Hn , hence an ∞ -groupoid.By construction of the classifying space of G -vector bundles (see [LR78, Wan80]) we have thefollowing statement. Corollary 3.2.7. The G - ∞ -groupoid Rep Gn corresponds to BO n ( G ) ∈ Top G , the classifying G -space of rank n real G -vector bundles. We can now construct an equivariant version of the tangent classifier of Ayala-Francis (see[AF15, sec. 2.1]). Construction 3.2.8 ( G -tangent classifier) . Let j : Mfld G → Fun G (( Mfld G ) vop , Top G ) be theparametrized Yoneda embedding G -functor of [BDG + Mfld G ) vop ). Define a G -tangent classifier by the composition of G -functors τ : Mfld G j −→ Fun G (( Mfld G ) vop , Top G ) → Fun G (( Rep Gn ) vop , Top G ) ≃ Top G/BO n ( G ) where the last equivalence is given by parametrized straightening/unstraightening.26n order to show that the G -tangent classifier sends a G -manifold M to the G -map classifyingits tangent bundle we will use the following description of the G -slice category Top G/B . Remark 3.2.9. A G -space B defines a G -object B : O opG → Top G (i.e. a coCartesian section, see[BDG + Top G given in remark 2.1.7 we can describe B as B : O opG → Top G , [ G/H ] ( B × G/H → G/H ) . By [AF15, lem] and remark 2.1.7 it follows that the fibers of the parametrized slice category Top G/B are given by (cid:16) Top G/B (cid:17) [ G/H ] ≃ (cid:16) Top G [ G/H ] (cid:17) /B ( G/H ) ≃ (cid:16) Top G/G/H (cid:17) / ( B × G/H → G/H ) ∼ −→ Top G/B × G/H . In particular an object of (cid:16) Top G/B (cid:17) [ G/H ] is given by a G -space over B × G/H , which we consideras an object ( Y → G/H ) ∈ Top G/G/H ≃ Top G [ G/H ] , together with a G -map f : Y → B . We write¯ f : Y → B × G/H for the G -map corresponding to the pair ( Y → G/H, Y f −→ B ).The mapping spaces of the slice category (cid:16) Top G [ G/H ] (cid:17) /B ( G/H ) ≃ Top G/B × G/H will be denotedby Map G/B ( G/H ) ( X, Y ). An explicit description of these mapping spaces is given by the Mooreover category, see appendix A. Proposition 3.2.10. Let ( M → G/H ) be an O G -manifold, and consider the tangent bundle T M → M as a G -vector bundle. Then τ M ∈ (cid:18) Top G/BO n ( G ) (cid:19) [ G/H ] is given by ( M → G/H ) ∈ Top G [ G/H ] together with the G -map τ M : M → BO n ( G ) classifying the tangent bundle of M .Proof. Recall that an O G -manifold M → G/H has an open cover by G -embeddings E α ֒ → M over G/H , where the patches ( E α → G/H ) are G -vector bundles. The mapping spaceMap G ( M, BO n ( G )) is the homotopy limit of Map G ( E α , BO n ( G )), so by the functionality of τ in M we are reduced to verifying the statement for E → G/H a G -vector bundle.By construction the restriction of τ to Rep Gn is given by straightening the functor associatedto the Yoneda embedding Rep Gn ֒ → Fun G (( Rep Gn ) vop , Top G ). Recalling the construction of theparametrized Yoneda embedding ([BDG + τ | Rep Gn is associated to theleft fibration of the parametrized twisted arrow category e O ( Rep Gn / O opG ) ։ ( Rep Gn ) vop × Rep Gn ,end τ E is associated to its pullback P E (cid:15) (cid:15) (cid:15) (cid:15) / / ❴✤ e O ( Rep Gn / O opG ) (cid:15) (cid:15) (cid:15) (cid:15) ( Rep Gn ) vop × G/H id × E / / ( Rep Gn ) vop × Rep Gn . By proposition 3.2.6 this is a pullback square of G - ∞ -groupoids, and using corollary 3.2.7 we27an identify it a homotopy pullback of G -spaces given by the top square of the following diagram G/H ≃ / / = , , P e (cid:15) (cid:15) (cid:15) (cid:15) / / ❴✤ Map(∆ , BO n ( G )) (cid:15) (cid:15) (cid:15) (cid:15) BO n ( G ) × G/H id × e / / proj (cid:15) (cid:15) ❴✤ BO n ( G ) × BO n ( G ) proj (cid:15) (cid:15) G/H e / / BO n ( G ) . Since the bottom square (given by projections to the second coordinate) is a homotopy pull-back square it follows that the outer rectangle is a homotopy limit diagram. Observe thatthe composition of the right vertical maps is an equivalence, and therefore the composition ofthe left vertical maps is an equivalence as well. It follows that τ E is equivalent to the G -map( e, id ) : G/H → BO n ( G ) × G/H ), where e : G/H → BO n ( G ) classifies the G -vector bundle E → G/H .On the other hand the tangent bundle T E is given by fiber product T E ∼ = E × G/H E andtherefore classified by the composition of the bottom maps in T E / / (cid:15) (cid:15) ❴✤ E (cid:15) (cid:15) E ≃ / / G/H e / / BO n ( G ) , which is clearly equivalent to E ≃ −→ G/H ( e,id ) −−−→ BO n ( G ) × G/H proj −−−→ BO n ( G ). G -category of f -framed G -manifolds We now turn to the definition of the G - ∞ -category of G -manifolds with additional tangentialstructure. Our main interest is in the tangential structure defining V -framed G -manifolds, for V a G -representation. However, the definition of equivariant framing on G -manifolds supports otherinteresting tangential structures, including equivariant orientations in the sense of [CMW01],and free G -manifolds (an example not a priori associated with tangential structures).The specific type of G -tangential structure, such as equivariant framing or equivariant ori-entation, is specified by a G -space B and a G -map f : B → BO n ( G ), in the following manner.An f -framing on a G -manifold M is given by a G -map M → B such that the composition M → B f −→ BO n ( G ) classifies the tangent bundle of M . Similarly, if H < G is a subgroup and M is an H -manifold, we say that M is f -framed its tangent bundle is classified by the H -map M → B f −→ BO n ( G ).The ∞ -categories of f -framed H -manifold for H < G can be arranged into an O opG -diagram,encoded by a G - ∞ -category Mfld G,f − fr . We start by giving a precise definition of Mfld G,f − fr and the G -functor Mfld G,f − fr → Mfld G that forgets the tangential structure. Definition 3.3.1. Let B ∈ Top G be a G -space and f : B → BO n ( G ) be a G -map. Define the -categories of f -framed G -manifolds as the pullback Mfld G,f − fr ❴✤ (cid:15) (cid:15) / / Top G/Bf ∗ (cid:15) (cid:15) Mfld G τ / / Top G/BO n ( G ) . Remark 3.3.2. Unwinding the definition, an object of ( Mfld G,f − fr ) [ G/H ] is given by ( M → G/H ) ∈ Mfld G [ G/H ] , a G -map f M : M and a G -homotopy between f ◦ f M exhibiting B f (cid:15) (cid:15) M τ M / / f M : : ✈✈✈✈✈✈✈✈✈✈ BO n ( G )as homotopy coherent diagram of G -spaces.The mapping spaces of Mfld G,f − fr [ G/H ] are given by homotopy pullbacks Emb G,f − frG/H ( M, N ) Map G/B × G/H ( M ¯ f M −−→ B × G/H, N ¯ f N −−→ B × G/H ) Emb GG/H ( M, N ) Map G/BO n ( G ) × G/H ( M ¯ τ M −−→ BO n ( G ) × G/H, N ¯ τ N −−→ BO n ( G ) × G/H ) . p ( f × G/H ) ∗ τ We finish this subsection with some examples of equivariant tangential structures on G -manifolds. We are primarily interested in equivariantly framed G -manifolds, which is our firstexample. Example 3.3.3 ( V -framed G -manifolds) . Let B = pt . A G -map f : pt → BO n ( G ) factorsthrough the space of G -fixed points ( BO n ( G )) G = ` V B Aut Rep Gn ( V ), so choosing f is equivalentto choosing a connected component, i.e a real n -dimensional G -representation V . A V -framingof an H -manifold M is therefore a homotopy lift pt V (cid:15) (cid:15) M τ M / / : : ✉✉✉✉✉✉✉✉✉✉ BAut( V ) , which under proposition 3.2.10 and restriction to fibers over the coset eH is equivalent to a choiceof trivialization T M ∼ = M × V as an H -vector bundle. Example 3.3.4 ( G -manifolds with no tangential structure) . Apply definition 3.3.1 for the G -space B = BO n ( G ) and id : BO n ( G ) → BO n ( G ) constructs Mfld G,id − fr ∼ = Mfld G . Example 3.3.5 ( G -orientated G -manifolds) . Orientations of G -vector bundles were studied byCostenoble, May and Waner in [CMW01] , and used in [CW] to prove equivariant versions ofPoincar´e duality. see [CMW01, def. 2.8] for a precise definition G - n -plane bundle, given by a G -map EO n ( G, S ) → BO n ( G, S ), see [CMW01, thm. 22.4]. Second,there is a G -map f : BO n ( G, S ) → BO n ( G ) representing the forgetful functor from oriented n -plane bundles to G - n -plane bundles. Therefore an orientation on a G -vector bundle is givenby a G -homotopy lift of its classifying map along the G -map f .Applying definition 3.3.1 to B = BO n ( G, S ) and f : BO n ( G, S ) → BO n ( G ) we get a G - ∞ -category Mfld G,or of oriented G -manifolds. Remark 3.3.6. The notion of an oriented G -manifold seems not to agree with the notion oforiented global orbifold (see, for example, [ALR07, p. 34]).Finally, we can use equivariant tangential structures to restrict the class of G -manifolds weconsider, an idea introduced in [AFT17b, rem. 1.1.9]. Example 3.3.7. Applying definition 3.3.1 with B = BO n ( G ) × EG and a G -map given by theprojection pr : BO n ( G ) × EG → BO n ( G ) produces a G - ∞ -category Mfld G,pr − fr . In this ex-ample the forgetful G -functor Mfld G,pr − fr → Mfld G is fully faithful, and exhibits Mfld G,pr − fr as the full G -subcategory of Mfld G spanned by O G -manifolds M → O where M is a free G -manifold. We now give a quick sketch the argument.We consider a manifold M with an action of G , describing an object ( M → G/G ) ∈ Mfld G [ G/G ] (the argument for an O G -manifold M → G/H is similar). A homotopy lift of τ M : M → BO n ( G )along the projection the same as a G -map M → EG . A G -map M → EG exists if and only ifthe action of G on M is free, in which case the space of G -maps Map G ( M, EG ) is contractible.This is easily seen by using the Elmendorf-McClure theorem; the presheaves that represents M and EG send M, EG : O opG → S , M : G/H M H , EG : G/H ( pt, H = e, ∅ H = e, and a map M H → ∅ exists if and only if M H is empty. It follows that a map of O G -presheaves M → EG exists if and only if the action of G on M is free. Finally, if G acts freely on M thenMap Fun( O opG , S ) ( M, EG ) ≃ Map( M, pt ) ≃ pt . G -disjoint union of G -manifolds The goal of this section is to endow the G - ∞ -category Mfld G with a G -symmetric monoidalstructure associated to disjoint unions.Recall that n -dimensional manifolds with G -action and G -equivariant embedding can beorganized into a topological category Mfld G . Despite the fact that Mfld G does not havecoproducts , we can still endow Mfld G with a symmetric monoidal structure by taking dis-joint unions. Therefore the ∞ -category N ( Mfld G ) admits a symmetric monoidal structure N ⊗ ( Mfld G ) ։ Fin ∗ , given by applying the operadic nerve construction of [Lur, def. 2.1.1.23].Similarly, disjoint unions endow the ∞ -category N ( Mfld H ) with a symmetric monoidal struc-ture, making the restriction and conjugation functors symmetric monoidal. We can therefore en-hance Mfld • from a diagram of ∞ -categories to a diagram of symmetric monoidal ∞ -categories N ⊗ ( Mfld • ). However, this construction does not encode the operation of topological inductionand its coherent compatibility with the symmetric monoidal structure and restriction and conju-gation of the action. The main point of this subsection is that all of the structure we are interestedin can be encoded as a G -symmetric monoidal structure on the G -category of G -manifolds (see Note that Emb G ( M ⊔ M , M ) Emb G ( M , M ) × Emb G ( M , M ). G -symmetric monoidal structure by anappropriate variant of the operadic nerve construction, however we are unaware of such con-struction. We therefore define the G -symmetric monoidal structure by explicitly constructing acoCartesian fibration Mfld G, ⊔ ։ Fin G ∗ (see definition 3.4.19).Our construction can be briefly described as follows. The category Fin G ∗ is constructed as acategory of spans in the category of finite G -sets over an orbit, O G - Fin , (see lemma 3.4.2), so itis natural to construct Mfld G, ⊔ as category of spans of an auxiliary ∞ -category O G - Fin - Mfld ,defined over O G - Fin . In definition 3.4.5 we construct O G - Fin - Mfld as a topological categoryover O G - Fin . We want to apply Barwick’s unfurling construction, see [Bar14], to the functor N ( O G - Fin - Mfld ) → O G - Fin , in order to produce a coCartesian fibration Mfld G, ⊔ ։ Fin G ∗ between the respected ∞ -categories of spans. There is a simple criterion, described in [Bar14],that ensures that the unfurled functor is a coCartesian fibration:1. Egressive arrows in O G - Fin , serving as the “wrong way arrows” in the span category Fin G ∗ ,have Cartesian lifts (verified in lemma 3.4.9).2. Ingressive arrows in O G - Fin , serving as the “right way arrows” in the span category Fin G ∗ ,have coCartesian lifts (verified in lemma 3.4.13).3. The pullback squares appearing in the definition of composition in the span category Fin G ∗ satisfies a “Beck-Chevalley condition” (verified in proposition 3.4.15).The resulting “unfurled” ∞ -category Mfld G, ⊔ (see definition 3.4.19) admits an explicit descrip-tion as an ∞ -category of spans. In particular we have a description of the objects, morphismsand coCartesian morphisms of Mfld G, ⊔ . Using the explicit description of Mfld G, ⊔ ։ Fin G ∗ weshow that it satisfies the G -Segal conditions and that its underlying G - ∞ -category is Mfld G (proposition 3.4.21). Construction of the auxiliary category O G - Fin - Mfld In this subsection we define a topological category O G - Fin - Mfld with a functor to the category O G - Fin of finite G -sets over orbits. The topological category O G - Fin - Mfld serves as input to theunfurling construction ([Bar14, sec. 11]), producing a coCartesian fibration Mfld G, ⊔ ։ Fin G ∗ that defines the G -symmetric monoidal structure of G -disjoint union on Mfld G (see defini-tion 3.4.19).We start with a definition of the category O G - Fin , which serves as the base category of theunfurling construction. Definition 3.4.1. The category O G - Fin is the pullback O G - Fin := Fun(∆ , F in G ) × Fun( { } ,F in G ) O G . The category O G - Fin is a full subcategory of the arrow category Fun(∆ , F in G ) , whose ob-jects are arrows U → O in F in G such that O ∈ O G . A morphism in O G - Fin is a summand-inclusion ([Nar16, def. 4.12]) if it factors as an inclusion over orbit-identity followed by a pullbacksquare U (cid:15) (cid:15) (cid:31) (cid:127) / / ϕ ∗ U (cid:15) (cid:15) / / ❴✤ U (cid:15) (cid:15) O / / O ϕ / / O . (9) Note that we the inclusion of G -sets U ֒ → ϕ ∗ U exhibits ϕ ∗ U as the coproduct of the G -sets U and U ′ = ϕ ∗ U \ U . We can therefore identify ϕ ∗ U ∼ = U ` U ′ .Let O G - Fin † ⊂ O G - Fin be the subcategory consisting of all objects while morphisms aresummand-inclusions. 31t is straightforward to see that the G -category Fin G ∗ of [Nar16, def. 4.12] can be defined bythe following unfurling construction. Lemma 3.4.2. The triple ( O G - Fin , O G - Fin × O G O ∼ = G , O G - Fin † ) is an adequate triple in thesense of [Bar14, def. 5.2], and its effective Burnside category fits into a pullback square Fin G ∗ (cid:15) (cid:15) (cid:31) (cid:127) / / ❴✤ A eff ( O G - Fin , O G - Fin × O G O ∼ = G , O G - Fin † ) (cid:15) (cid:15) O G (cid:31) (cid:127) / / A eff ( O G , O ∼ = G , O G ) (10)We now define a topological category of “parametrized O G -manifolds” over O G - Fin . Definition 3.4.3. An O G - Fin -manifold M → U → O is1. a smooth n -dimensional manifold M with an action of G on M by smooth maps,2. together with a G -map M → U from the underlying G -space of the manifold M to a G -finiteset U ∈ F in G ,3. and an arrow U → O in Fin G such that O ∈ O G .An morphism of O G - Fin -manifolds is given by a commuting square of G -spaces M (cid:15) (cid:15) f / / M (cid:15) (cid:15) U (cid:15) (cid:15) ϕ / / U (cid:15) (cid:15) O ϕ / / O , such that the induced map M → O × O M is an embedding. Definition 3.4.4. Let M → U → O , M → U → O be O G - Fin -manifolds and ϕ : I → I a morphism in O G - Fin given by U (cid:15) (cid:15) ϕ / / U (cid:15) (cid:15) O ϕ / / O . Define Emb O G - Fin ϕ ( M , M ) ⊂ C ∞ ( M , M ) asthe subspace of smooth maps f : M → M such that ( f, ϕ, ϕ ) is a morphism of O G - Fin -manifoldsfrom M → U → O to M → U → O . Definition 3.4.5. The Category of O G - Fin -manifolds O G - Fin - Mfld is the topological categorywhose objects are O G - Fin -manifolds. The morphism space from M to M is given by Map O G - Fin - Mfld ( M , M ) := a ϕ Emb O G - Fin ϕ ( M , M ) , where the coproduct is indexed by Hom O G - Fin ( U → O , U → O ) .Define a forgetful functor p : O G - Fin - Mfld → O G - Fin by sending M → U → O to U → O ,and the subspace Emb O G - Fin ϕ ( M , M ) ⊂ M ap ( M , M ) to ϕ ∈ Hom O G - Fin ( U → O , U → O ) . O G - Fin - Mfld for both the topological cate-gory O G - Fin - Mfld , its incarnation as a fibrant simplicial category Sing ( O G - Fin - Mfld ) and itsincarnation as an ∞ -category N ( O G - Fin - Mfld ), distinguishing between these incarnations bycontext. Remark 3.4.6. Note that an equivalence f : M → N in O G - Fin - Mfld is always an embeddingof smooth manifolds, since it lies over an isomorphism of orbits. Moreover, it is G -isotopicto an identity-of-manifolds over the isomorphism p ( f ). On the other hand, if f is G -isotopicto an identity-of-manifolds over an isomorphism of finite G -sets then f is an equivalence in O G - Fin - Mfld , so we have a complete characterization of equivalences in O G - Fin - Mfld . Some Cartesian and coCartesian edges of O G - Fin - Mfld → O G - Fin We characterize p -Cartesian edges of O G - Fin - Mfld over summand-inclusions and p -coCartesianedges over isomorphisms of orbits. We summarize the results of this subsection as follows. Proposition 3.4.7. A morphism f of O G - Fin - Mfld over O G - Fin † is p -Cartesian if and onlyif it is equivalent to a pullback over a summand-inclusion. A morphism g of O G - Fin - Mfld over O G - Fin × O G O ∼ = G is p -coCartesian if and only if it is G -isotopic to an identity-of-manifolds overan orbit-isomorphism. The characterization of p -Cartesian edges is given in corollary 3.4.12, and the characterizationof p -coCartesian edges is given in corollary 3.4.14. Remark 3.4.8. By [Lur09a, prop. 2.4.1.10(1)] the map O G - Fin - Mfld → O G - Fin is an innerfibration. Lemma 3.4.9. Let ϕ ∈ Hom O G - Fin ( U → O , U → O ) be a morphism in O G - Fin given by apullback square U (cid:15) (cid:15) / / ❴✤ U (cid:15) (cid:15) O / / O , and N → U → O a O G - Fin -manifold over its target. Then thepullback M (cid:15) (cid:15) f / / ❴✤ N (cid:15) (cid:15) U (cid:15) (cid:15) / / ❴✤ U (cid:15) (cid:15) O / / O defines a p -Cartesian morphism f in O G - Fin - Mfld lifting ϕ .Proof. According to [Lur09a, prop. 2.4.1.10(2)] we have to show that for every O G - Fin -manifold T → U → O the commutative squareMap( T, M ) (cid:15) (cid:15) (cid:15) (cid:15) f ∗ / / Map( T, N ) (cid:15) (cid:15) (cid:15) (cid:15) Hom O G - Fin ( p ( T ) , p ( M )) p ( f ) ∗ / / Hom O G - Fin ( p ( T ) , p ( N )33s a homotopy pullback. Since the vertical maps are Kan fibrations, this square is a homotopypullback if and only if f ∗ induces an equivalence between the fibers over every vertex of the baseHom O G - Fin ( p ( T ) , p ( M )).Let τ ∈ Hom O G - Fin ( p ( T ) , p ( M )). The functor f ∗ induces a map of the fibers over τ ( f ∗ ) | τ : Emb O G - Fin τ ( T, M ) → { τ } × Hom O G - Fin ( p ( T ) ,p ( M ) Map( T, N ) . Unwinding the definition of the mapping space in O G - Fin - Mfld , we have { τ } × Hom O G - Fin ( p ( T ) ,p ( M ) Map( T, N ) = { τ } × Hom O G - Fin ( p ( T ) ,p ( M ) a ϕ Emb O G - Fin ϕ ( T, N ) ! = Emb O G - Fin p ( f ) ◦ τ ( T, N ) , where the last equality holds since pullback along a fixed map preserve coproducts.Suppose that the O G - Fin -manifold T is given by T → U → O and τ : p ( T ) → p ( N ) is givenby the square U (cid:15) (cid:15) / / U (cid:15) (cid:15) O / / O . Then the map ( f ∗ ) | τ : Emb O G - Fin τ ( T, M ) → Emb O G - Fin p ( f ) ◦ τ ( T, N ) sends h : T → M to f ◦ h :( f ∗ ) | τ : h = T (cid:15) (cid:15) h / / M (cid:15) (cid:15) U (cid:15) (cid:15) / / U (cid:15) (cid:15) O / / O T (cid:15) (cid:15) h / / M (cid:15) (cid:15) f / / ❴✤ N (cid:15) (cid:15) U (cid:15) (cid:15) / / U (cid:15) (cid:15) / / ❴✤ U (cid:15) (cid:15) O / / O / / O . The universal property of the pullback M = N × U U shows that ( f ∗ ) | τ is a continuous bi-jection: injectivity follows from uniqueness of maps to the pullback. Surjectivity: suppose g ∈ Emb O G - Fin p ( f ) ◦ τ ( T, N ), by existence of a map to the pullback we have a candidate map h : T → M over τ such that g = f ◦ g . We have to show that h ∈ Emb O G - Fin τ . Clearly h is a smooth G -map,so we only have to verify condition (3) of definition 3.4.4: h induces an embedding T → O × O M .To see that observe that g induces an embedding T ֒ → O × O N which factors as the map inducedby h followed by the isomorphism O × O M = O × O ( O × O N ) ∼ = O × O N .We leave it as an exercise to the reader to verify that ( f ∗ ) | τ is an open map, and therefore ahomeomorphism.Note that every G -map M → U ` U from a manifold with G -action to a coproduct of G -setsfactors as coproduct of G -maps M = M ` M → U ` U . Lemma 3.4.10. Let ϕ be an inclusion of finite G -sets over id O in O G - Fin , given by the diagram U (cid:15) (cid:15) (cid:31) (cid:127) / / U ` U (cid:15) (cid:15) O / / O , and M ` M → U ` U → O a O G - Fin -manifold over its target. Then he pullback M (cid:15) (cid:15) (cid:31) (cid:127) i / / ❴✤ M ` M (cid:15) (cid:15) U (cid:15) (cid:15) (cid:31) (cid:127) / / U ` U (cid:15) (cid:15) O / / O defines a p -Cartesian morphism i in O G - Fin - Mfld lifting ϕ .Proof. As in lemma 3.4.9, we have to show that for every O G - Fin -manifold T → U → O andevery τ : p ( T ) → p ( M ) the map i ∗ induces equivalence of the fibers Emb O G - Fin τ ( T, M ) → Emb O G - Fin p ( i ) ◦ τ ( T, M a M ) . As above, we use the universal property of the pullback to show this map is a bijection, andleave it to the reader to verify it is an open map.The only part which is different is the verification of condition (3) of definition 3.4.4: g inducesan embedding T ֒ → O × O ( M ` M ), which factors as the composition of the map induced by h , an inclusion and an isomorphism T → O × O M ֒ → O × O M a O × O M ∼ = O × O ( M a M ) . Since the composition is an embedding, the map T → O × O M induced by h is an embedding.Together, the lemmas above show the existence of p -Cartesian lifts over summand-inclusionsand characterizes them. Corollary 3.4.11. Let ϕ ∈ Hom O G - Fin ( U → O , U → O ) be a morphism in O G - Fin † and N → U → O an O G - Fin -manifold over its target. Then the pullback M (cid:15) (cid:15) f / / ❴✤ N (cid:15) (cid:15) U (cid:15) (cid:15) / / U (cid:15) (cid:15) O / / O defines a p -Cartesian morphism f in O G - Fin - Mfld lifting ϕ .Proof. Factor the summand-inclusion ϕ as in (9), apply lemma 3.4.9 and lemma 3.4.10.By [Lur09a, prop. 2.4.1.7 and 2.4.1.5], we have35 orollary 3.4.12. A morphism f of O G - Fin - Mfld over O G - Fin † is p -Cartesian if and only ifit is equivalent to a pullback over a summand-inclusion, i.e the left map in the factorization f = M (cid:15) (cid:15) (cid:31) (cid:127) / / M × U U (cid:15) (cid:15) / / ❴✤ M (cid:15) (cid:15) U (cid:15) (cid:15) = / / U (cid:15) (cid:15) / / U (cid:15) (cid:15) O / / O / / O is an equivalence in O G - Fin - Mfld (a G -isotopy equivalence over U ). Next, we construct p -coCartsian lifts over isomorphism of orbits. Lemma 3.4.13. Let ϕ = U (cid:15) (cid:15) / / U (cid:15) (cid:15) O ∼ = / / O be a morphism of O G - Fin × O G O ∼ = G and M → U → O an O G - Fin -manifold. Then f = M (cid:15) (cid:15) = / / M (cid:15) (cid:15) U (cid:15) (cid:15) / / U (cid:15) (cid:15) O ∼ = / / O is a p -coCartesian lift of ϕ .Proof. By the dual version of [Lur09a, prop. 2.4.1.10(2)] we have to show that for every O G - Fin -manifold T → U → O the square M ap ( M → U → O , T → U → O ) (cid:15) (cid:15) (cid:15) (cid:15) f ∗ / / M ap ( M → U → O , T → U → O ) (cid:15) (cid:15) (cid:15) (cid:15) Hom O G - Fin ( U → O , U → O ) p ( f ) ∗ / / Hom O G - Fin ( U → O , U → O )is a homotopy pullback square. Since the vertical maps are Kan fibrations, this square is a homo-topy pullback if and only if f ∗ induces an equivalence between the fibers. Next, note that the map f ∗ is induced by composition with id M , and the fibers over τ ∈ Hom O G - Fin ( U → O , U → O )and τ ◦ p ( f ) ∈ Hom O G - Fin ( U → O , U → O ) are both subspaces of the space of smooth maps C ∞ ( M, T ): Emb O G - Fin τ ( M, T ) ⊂ C ∞ ( M, T ) , Emb O G - Fin τ ◦ p ( f ) ( M, T ) ⊂ C ∞ ( M, T ) . We finish the proof by observing that these subspaces are equal: conditions (1),(3) of defini-tion 3.4.4 coincide, while the equivalence of condition (2) follows from the commutativity of thesquare M (cid:15) (cid:15) = / / M (cid:15) (cid:15) U / / U , the top square of f . 36e therefore have a characterisation of p -coCartesian edges over orbit isomorphisms. Corollary 3.4.14. A morphism f of O G - Fin - Mfld over an orbit-isomorphism is p -Cartesianif and only if it is equivalent to an identity-of-manifolds, i.e. the right map in the factorization f = M (cid:15) (cid:15) = / / M (cid:15) (cid:15) (cid:31) (cid:127) / / M (cid:15) (cid:15) U (cid:15) (cid:15) / / U (cid:15) (cid:15) = / / U (cid:15) (cid:15) O ∼ = / / O / / O is an equivalence in O G - Fin - Mfld (a G -isotopy equivalence over U ). Construction of the G -symmetric monoidal category Mfld G, ⊔ We now turn to the goal of this subsection, the construction of a G -symmetric monoidal structureon the G -category of G -manifolds. In definition 3.4.19 we use the unfurling construction of [Bar14,sect. 11] to define a coCartesian fibration Mfld G, ⊔ ։ Fin G ∗ , and in proposition 3.4.21 we verifythe Segal conditions, showing that it defines a G -symmetric monoidal structure on Mfld G .We first make sure that the conditions for applying Barwick’s unfurling construction hold.Since Cartesian lifts of egressive morphisms and coCartesian lifts of ingressive morphisms wereconstructed in proposition 3.4.7 it remains to verify the Beck-Chevalley conditions. Proposition 3.4.15. The inner fibration O G - Fin - Mfld → O G - Fin is adequate over the triple ( O G - Fin , O G - Fin × O G O ∼ = G , O G - Fin † ) ([Bar14, def. 10.3]).Proof. Conditions [Bar14, cond. (10.3.1),(10.3.2)] follow from proposition 3.4.7. To verify con-dition [Bar14, cond. (10.3.3)] construct the natural map i ! ◦ q ∗ ( ˜ N ) → q ′∗ ◦ j ! ( ˜ N ) by choosingappropriate p -Cartesian and p -coCartesian lifts, and show that map is the universal map betweentwo models of the same pullback, hence a diffeomorphism over an identity map.Let s q (cid:15) (cid:15) (cid:15) (cid:15) / / i / / ❴✤ s ′ q ′ (cid:15) (cid:15) (cid:15) (cid:15) t / / j / / t ′ be an ambigressive pullback square in O G - Fin , whose objects and morphismsare given by s = ˜ U (cid:15) (cid:15) ˜ O , s ′ = U (cid:15) (cid:15) O , t = ˜ V (cid:15) (cid:15) ˜ O , t ′ = V (cid:15) (cid:15) O , i = ˜ U (cid:15) (cid:15) / / U (cid:15) (cid:15) ˜ O ∼ = / / O ,j = ˜ V (cid:15) (cid:15) / / V (cid:15) (cid:15) ˜ O ∼ = / / O , q = ˜ U (cid:15) (cid:15) / / ˜ V (cid:15) (cid:15) ˜ O / / ˜ O , q ′ = U (cid:15) (cid:15) / / V (cid:15) (cid:15) O / / O And ˜ N = ( ˜ N → ˜ V → ˜ O ) an object in the fiber of p over t . We compute i ! ◦ q ∗ ( ˜ N ) , q ′∗ ◦ j ! ( ˜ N )and the map i ! ◦ q ∗ ( ˜ N ) → q ′∗ ◦ j ! ( ˜ N ) (natural in ˜ N ) by choosing appropriate p -Cartesian and p -coCartesian lifts. 37et ˜ M := ˜ N × ˜ N ˜ U . Since q is a summand-inclusion by corollary 3.4.11 the map˜ M (cid:15) (cid:15) / / ❴✤ ˜ N (cid:15) (cid:15) ˜ U (cid:15) (cid:15) / / ˜ V (cid:15) (cid:15) ˜ O / / ˜ O is p -Cartesian over q , so q ∗ ( ˜ N ) := ( ˜ M → ˜ U → ˜ O ).Since i is over an isomorphism of orbits, by lemma 3.4.13 the map˜ M (cid:15) (cid:15) = / / ˜ M (cid:15) (cid:15) ˜ U (cid:15) (cid:15) / / U (cid:15) (cid:15) ˜ O ∼ = / / O is p -coCartesian over i , so i ! ◦ q ∗ ( ˜ N ) := ( ˜ M → U → O ).Since j is over an isomorphism of orbits, by lemma 3.4.13 the map˜ N (cid:15) (cid:15) = / / ˜ N (cid:15) (cid:15) ˜ V (cid:15) (cid:15) / / V (cid:15) (cid:15) ˜ O ∼ = / / O is p -coCartesian over j , so j ! ( ˜ N ) := ( ˜ N → V → O ).Let M := ˜ N × V U . Since q ′ is a summand-inclusion by corollary 3.4.11 the map M (cid:15) (cid:15) / / ❴✤ ˜ N (cid:15) (cid:15) U (cid:15) (cid:15) / / V (cid:15) (cid:15) O / / O is p -Cartesian over q ′ , so q ′∗ ◦ j ! ( ˜ N ) := ( M → U → O ).Next, we choose a map ξ : q ∗ ( ˜ N ) → q ′∗ ◦ j ! ( ˜ N ) over i by composing the lifts of q and j above38nd using the universal property of the pullback M ˜ M (cid:15) (cid:15) / / ❴✤ ˜ N (cid:15) (cid:15) = / / ˜ N (cid:15) (cid:15) ˜ U (cid:15) (cid:15) / / ˜ V (cid:15) (cid:15) / / V (cid:15) (cid:15) ˜ O / / ˜ O ∼ = / / O ⇒ ˜ M (cid:15) (cid:15) ∃ ! ξ / / ❴❴❴ M (cid:15) (cid:15) / / ❴✤ ˜ N (cid:15) (cid:15) ˜ U (cid:15) (cid:15) / / U (cid:15) (cid:15) / / V (cid:15) (cid:15) ˜ O ∼ = / / O / / O . The map ξ induces the natural map ξ : i ! ◦ q ∗ ( ˜ N ) → q ′∗ ◦ j ! ( ˜ N ) over id s ′ by˜ M (cid:15) (cid:15) = / / ˜ M (cid:15) (cid:15) ∃ ! ξ / / ❴❴❴ M (cid:15) (cid:15) ˜ U (cid:15) (cid:15) / / U (cid:15) (cid:15) = / / U (cid:15) (cid:15) ˜ O ∼ = / / O / / O . In order to verify [Bar14, cond. (10.3.3)] we have to show that ξ is an equivalence in the fiberover s ′ . We show that ξ is a diffeomorphism. Consider the diagram˜ M (cid:15) (cid:15) / / ❴✤ ˜ N (cid:15) (cid:15) ˜ U (cid:15) (cid:15) / / ❴✤ ˜ V (cid:15) (cid:15) U / / V the top square is a pullback square by definition of ˜ M , and the bottom square is a pullbacksquare by assumption. Therefore the outer rectangle is a pullback square. By the universalproperty of M = ˜ N × U V the induced map ξ is a diffeomorphism, as claimed.This ends the proof of proposition 3.4.15.We can now define Mfld G, ⊔ by applying the unfurling construction to O G - Fin - Mfld →O G - Fin . Definition 3.4.16. Define a subcategory ( O G - Fin - Mfld ) † ⊂ O G - Fin - Mfld with the same ob-jects as O G - Fin - Mfld , and with morphisms the p -Cartesian edges over summand-inclusions (i.eover edges over O G - Fin † ). Define a subcategory ( O G - Fin - Mfld ) † ⊂ O G - Fin - Mfld by ( O G - Fin - Mfld ) † := O G - Fin - Mfld × O G - Fin ( O G - Fin × O G O ∼ = G ) ∼ = O G - Fin - Mfld × O G O ∼ = G . Construction 3.4.17. By lemma 3.4.2, proposition 3.4.15 and [Bar14, prop. 11.2] the triple( O G - Fin - Mfld , ( O G - Fin - Mfld ) † , ( O G - Fin - Mfld ) † ) is adequate. This condition ensures we can39orm the ∞ -category of spans A eff ( O G - Fin - Mfld , ( O G - Fin - Mfld ) † , ( O G - Fin - Mfld ) † ). Apply-ing the effective Burnside construction to p : O G - Fin - Mfld → O G - Fin we get a functor A eff ( O G - Fin - Mfld , ( O G - Fin - Mfld ) † , ( O G - Fin - Mfld ) † ) A eff ( O G - Fin , O G - Fin × O G O ∼ = G , O G - Fin † ) , Υ( p ) called the unfurling of p in [Bar14, def. 11.3]. Lemma 3.4.18. The functor Υ( p ) is a coCartesian fibration.Proof. The functor Υ( p ) is an inner fibration by [Bar14, lem. 11.4], and a coCartesian fibrationby [Bar14, lem. 11.5] and proposition 3.4.7. Definition 3.4.19. Define a coCartesian fibration Mfld G, ⊔ ։ Fin G ∗ by pulling Υ( p ) along theinclusion Fin G ∗ ֒ → A eff ( O G - Fin , O G - Fin × O G O ∼ = G , O G - Fin † ) of (10) . Remark 3.4.20. Unwinding the definition of the effective Burnside category, we see that theobjects of Mfld G, ⊔ are O G - Fin -manifolds, and a morphism f : M → M is represented by aspan f = M (cid:15) (cid:15) M o o (cid:15) (cid:15) / / M (cid:15) (cid:15) U (cid:15) (cid:15) U o o (cid:15) (cid:15) / / U (cid:15) (cid:15) O O o o = / / O , where the ’backwards arrow’ is equivalent to a pullback over a summand-inclusion. The morphism f is coCartsian exactly when the ’forward arrow’ is equivalent to an identity-of-manifolds (seeproposition 3.4.7 and [Bar14, lem. 11.5]). Proposition 3.4.21. The coCartesian fibration Mfld G, ⊔ ։ Fin G ∗ of definition 3.4.19 is G -symmetric monoidal category whose underlying G -category is isomorphic to the G -category Mfld G of definition 3.1.16. We call this G -symmetric monoidal structure G -disjoint union of G -manifolds.Proof. By definition B.0.4 the underlying G -category of Mfld G, ⊔ has objects O G - Fin -manifoldsof the form ( M → O = −→ O ) and maps represented by spans of the form M (cid:15) (cid:15) M o o (cid:15) (cid:15) / / M (cid:15) (cid:15) O (cid:15) (cid:15) O o o = (cid:15) (cid:15) = / / O (cid:15) (cid:15) O O o o = / / O with left square equivalent to a pullback. This G -category is isomorphic to Mfld G by the forgetfulfunctor ( M → O = −→ O ) ( M → O ). 40y lemma B.0.10 it is enough to show that for every I = ( U → O ) ∈ Fin G ∗ the inducedfunctor Q ρ W ∗ : Mfld G, ⊔ I → Q W ∈ Orbit( U ) Mfld G [ W ] is an equivalence of ∞ -categories, where ρ W ∗ is induced by the fibration Mfld G, ⊔ ։ Fin G ∗ and the inert edge ρ W = U (cid:15) (cid:15) W ? _ o o = / / = (cid:15) (cid:15) W = (cid:15) (cid:15) O W o o = / / W. , ρ W ∈ Fin G ∗ . Let ( M → U → O ) ∈ Mfld GI be an O G - Fin -manifold. The decomposition U = ` W ∈ Orbit( U ) W into orbits induces a decomposition of M into a disjoint union M = ⊔ W ∈ Orbit( U ) M W . Theaction of ρ W ∗ on ( M → U → O ) is specified by a choice of coCartesian lift over ρ W . By the abovedescription of coCartesian edges we see that M (cid:15) (cid:15) M W ? _ o o (cid:15) (cid:15) = / / M W (cid:15) (cid:15) U (cid:15) (cid:15) W ? _ o o = / / = (cid:15) (cid:15) W = (cid:15) (cid:15) O W o o = / / W. is such a coCartesian edge, therefore the functor Q ρ W ∗ is given by Y ρ W ∗ : Mfld G, ⊔ I → Y W ∈ Orbit( U ) Mfld G [ W ] , Y ρ W ∗ : M (cid:15) (cid:15) U (cid:15) (cid:15) O = F W ∈ Orbit( U ) M W (cid:15) (cid:15) ` W ∈ Orbit( U ) W (cid:15) (cid:15) O M W (cid:15) (cid:15) W = (cid:15) (cid:15) W W ∈ Orbit( U ) which is an equivalence by inspection. G -disjoint union of f -framed G -manifolds In this subsection we lift G -disjoint union of G -manifolds to a G -symmetric monoidal structureon Mfld G,f − fr . Recall that Mfld G,f − fr was defined as the pullback of G - ∞ -categories (seedefinition 3.3.1). We will show that the G -symmetric monoidal structure of Mfld G lifts to Mfld G,f − fr by exhibiting the pullback square of definition 3.3.1 as underlying a pullback squareof G -symmetric monoidal G - ∞ -categories and G -symmetric monoidal functors.In addition to G -disjoint unions of G -manifolds we will use the G -coCartesian structure,constructed in [BDG + ] and given by G -coproducts. In general the G -coCartesian structure ona G -category C is given by a G - ∞ -operad C ∐ . However, we will only use this construction for C with finite G -coproducts, in which case C ∐ is a G -symmetric monoidal G - ∞ -category.We show show that the G -functors in the pulback square of definition 3.3.1 extend to G -symmetric monoidal functors in two steps. By a formal argument these G -functors extend to lax -symmetric monoidal functors. It then remains to verify that these lax G -symmetric monoidalfunctors are in fact G -symmetric monoidal.The following claim allows us to extend G -functors to C from certain G - ∞ -operads to lax G -symmetric monoidal functors. Lemma 3.5.1. Let C be a G -category and O ⊗ a unital G - ∞ -operad. Restriction to the underlying G -category induces an equivalence Alg G ( O, C ) → Fun G ( O, C ) between the ∞ -category of morphisms of G - ∞ -operads from O ⊗ to C ∐ and the ∞ -category of G -functors between the underlying G -categories. Let B ∈ Top G be a G -space and f : B → BO n ( G ) be a G -map. Endow the parametrizedslice G -categories Top G/B , Top G/BO n ( G ) with the G -coCartesian G -symmetric monoidal structure.By lemma 3.5.1 the G -functors f ∗ : Top G/B → Top G/BO n ( G ) , τ : Mfld G → Top G/BO n ( G ) admit an essentially unique lift to lax G -symmetric monoidal functors f ∗ : Top G/B → ( Top G/BO n ( G ) ) ∐ , τ : Mfld G, ⊔ → ( Top G/BO n ( G ) ) ∐ The following description of the G -coCartesian structure ( Top G/B ) ∐ is useful when verifyingthat the lax G -symmetric monoidal functors τ, f ∗ constructed above are in fact G -symmetricmonoidal. Remark 3.5.2. Let I = ( U → G/H ) ∈ Fin G ∗ . Then a U -family x • : U → Top G can bedescribed by a G -map X → U . Moreover, under this description the parametrized coproduct ` I x • : G/H → Top G is given by the G -map X → U → G/H .To see this first construct the left fibration associated to x • , and then notice it is a map of G - ∞ -groupoids and therefore can identified with a map of G -spaces X → U . One should think ofthe family X → U as assigning to each W ∈ Orbit( U ) the G -map ( X | W → W ) ∈ Top G [ W ] , wherewe use the explicit model of remark 2.1.7. In order to see that ` I x • is given by ( X → U → G/H ) ∈ Top G [ G/H ] recall that ` I is given by G -left Kan extension along U → G/H , which by[Sha18, prop. 10.9] is given by (unparametrized) left Kan extension along U → G/H . Applyingstraightening/unstraightening, we see that ` I is let adjoint to pulling back along U → G/H ,and therefore given by post-composition with U → G/H .Let B be a G -space. Combining remark 3.2.9 with the description of U -families in Top G above, we get the following description of G -coproducts in Top G/B . A U -family x • : U → Top G/B is given by a G -map X → U together with a collection of G -maps { X | W → B } indexed by W ∈ Orbit( U ). Equivalently, x • : U → Top G/B is given by a pair of G -maps ( X → U, X → B ).The G -coproduct ` I x • ∈ (cid:16) Top G/B (cid:17) [ G/H ] is given by ( X → U → G/H ) ∈ Top G [ G/H ] togetherwith the G -map X → B . Lemma 3.5.3. The functor τ : Mfld G, ⊔ → ( Top G/BO n ( G ) ) ∐ is a G -symmetric monoidal functor. roof. By proposition 3.2.10 and the Segal conditions we have a concrete description of τ .Namely, if I = ( U → G/H ) ∈ Fin G ∗ and ( M → U → G/H ) ∈ Mfld G, ⊔ I is a O G - Fin -manifold then τ ( M → U → G/H ) ∈ ( Top G/BO n ( G ) ) ∐ is given by ( M → U → G/H ) ∈ Top GI together with the G -map M → BO n ( G ) classifying T M → M . Therefore the G -coproduct ` I τ ( M → U → G/H ) is given by ( M → U → G/H ) ∈ Top G [ G/H ] together with the G -map M → BO n ( G ) classifying T M → M .On the other hand, by remark 3.4.20 the G -disjoint union ⊔ I M ∈ Mfld G [ G/H ] is the O G -manifold given by the composition M → U → G/H , therefore τ ( ⊔ I M ) is given by the O G -manifold ( M → U → G/H ) ∈ Mfld G [ G/H ] together with the G -map M → BO n ( G ) classifying T M → M . Proposition 3.5.4. The G -functor f ∗ : Top G/B → Top G/BO n ( G ) extends to a G -symmetric monoidalfunctor f ∗ : ( Top G/B ) ∐ → ( Top G/BO n ( G ) ) ∐ .Proof. This is an immediate consequence of the description of G -coproducts in Top G/B and Top G/BO n ( G ) : for I = ( U → G/H ) the diagram (cid:16) Top G/B (cid:17) ∐ I f ∗ / / ⊔ I (cid:15) (cid:15) (cid:18) Top G/BO n ( G ) (cid:19) ∐ I ` I (cid:15) (cid:15) (cid:16) Top G/B (cid:17) [ G/H ] f ∗ / / (cid:18) Top G/BO n ( G ) (cid:19) [ G/H ] is commutativity, since remark 3.5.2 implies it is given by( X → U, X → B ) ❴ ⊔ I (cid:15) (cid:15) ✤ f ∗ / / ( X → U, X → B f −→ BO n ( G )) ❴ ` I (cid:15) (cid:15) ( X → U → G/H, X → B ) ✤ f ∗ / / ( X → U → G/H, X → B f −→ BO n ( G )) . It follows that given a G -map f : B → BO n ( G ) over G/H we can endow Mfld G,f − fr with a G -symmetric monoidal structure. Corollary 3.5.5. The G -symmetric monoidal structure of G -disjoint union on Mfld G lifts to a G -symmetric monoidal structure on Mfld G,f − fr , given by the pullback Mfld G,f − fr, ⊔ ❴✤ (cid:15) (cid:15) / / ( Top G/B ) ∐ f ∗ (cid:15) (cid:15) Mfld G, ⊔ τ / / ( Top G/BO n ( G ) ) ∐ . roof. The ∞ -category C at G, ⊗∞ of G -symmetric monoidal categories admits limits, and the for-getful G -functor C at G, ⊗∞ → C at G ∞ sending a G -symmetric monoidal category C ⊗ ։ Fin G ∗ to itsunderlying G -category C = C ⊗ × Fin G ∗ O opG preserves limits. Remark 3.5.6. Informally, we can describe an object of Mfld G,f − fr, ⊔ over ( U → G/H ) ∈ Fin G ∗ as an O G - Fin -manifold ( M → U → G/H ) together an f -framing f M : M → B × G/H . Definition 3.5.7. Let Rep G,f − fr, ⊔ n ⊂ Mfld G,f − fr, ⊔ be the full G -subcategory of Mfld G,f − fr, ⊔ given by the pullback Rep G,f − fr, ⊔ n ❴✤ (cid:15) (cid:15) (cid:31) (cid:127) / / Mfld G,f − fr, ⊔ (cid:15) (cid:15) Rep G, ⊔ n (cid:31) (cid:127) / / Mfld G, ⊔ . It follows that Rep G,f − fr, ⊔ n ⊂ Mfld G,f − fr, ⊔ is the full subcategory of f -framed O G - Fin -manifolds ( E → U → G/H ) where E → U is a G -vector bundle. Note that Rep G,f − fr, ⊔ n ։ Fin G ∗ is a G - ∞ -operad. G -category of G -disks and the definition of G -disk algebras Our next goal is to define the G -symmetric monoidal G - ∞ -category of G -disks Disk G, ⊔ , and itsframed variants Disk G,f − fr, ⊔ . These G - ∞ -categories are the point of contact between equivari-ant algebra and equivariant geometry.On the one hand, we use Disk G, ⊔ to define G -disk algebras, which serve as coefficients forgenuine equivariant factorization homology. In a nutshell, the algebraic structure of a G -diskalgebra is indexed by equivariant embeddings of G -disks.On the other hand, G -disks capture the local geometry of G -manifolds: G -disks are designedto be the G -tubular neighbourhoods of a configuration of orbits in a G -manifold. We will thereforedefine G -disks as a full G -subcategory Disk G ⊂ Mfld G of the G - ∞ -category of G -manifolds.After defining Disk G we show that G -disjoint unions endow it with G -symmetric monoidalstructure (see definition 3.6.5 and corollary 3.6.8). Finally, we construct Disk G,f − fr , a framedversion of the G - ∞ -category of G -disks (see definition 3.6.9) and define f -framed G -disk algebras(see definition 3.6.11). Definition 3.6.1 ( G -disks) . A G -disk is a G -vector bundle E → O rank n , where O ∈ O G isan orbit. Clearly a G -disk is an O G -manifold.Let Disk G ⊂ Mfld G , Disk G ⊂ Mfld G be the full subcategories spanned by O G -manifoldsequivalent to a composition E → U → O of G -vector bundle E → U of rank n over a finite G -set ( U → O ) ∈ Fin G . Remark 3.6.2. The G -subcategory Disk G ⊂ Mfld G is the full G -subcategory generated from G -disks by finite G -disjoint unions. We think of ( E → U → O ) ∈ Mfld G as a G -disjoint union of G -disks: the decomposition U = ⊔ W ∈ Orbit( U ) W into orbits decomposes E into a disjoint unionof G -vector bundles E W → W , and each composition E W → W → O exhibits E W → O as thetopological induction of E W → W along W → O .In fact, Disk G is the free G -category generated from H -representations for H < G , consideredas G -vector bundles over G/H , by disjoint unions and topological induction (see lemma 3.7.2below). 44e first verify that Disk G is a G - ∞ -category. Proposition 3.6.3. The subcategory Disk G ⊂ Mfld G is a G -subcategory stable under equiva-lences.Proof. By [BDG + x → y in Mfld G if x ∈ Disk G then y ∈ Disk G . Recall that an edge M (cid:15) (cid:15) M o o (cid:15) (cid:15) / / M (cid:15) (cid:15) O O ϕ o o = / / O in Mfld G is coCartesian if and only if the left square is equivalent to a pullback square andthe right square is a G -isotopy equivalence. Let ( M → O ) ∈ Disk G , then by definition it isequivalent to E → U → O for U a finite G -set and E → U a G -vector bundle. Pulling backalong ϕ shows that M → O is equivalent to ϕ ∗ E → ϕ ∗ U → O , a G -vector bundle over a finite G -set. Since M → O is equivalent to M → O it follows that ( M → O ) ∈ Disk G . Remark 3.6.4. The coCartesian fibration Disk G ։ O opG is dual to the Cartesian fibration Disk G → O G . G -disjoint union of G -disks We now show (corollary 3.6.8) that G -disjoint union of G -manifolds (see proposition 3.4.21) induces a G -symmetric monoidal structure on Disk G . Definition 3.6.5. Define Disk G, ⊔ ⊂ Mfld G, ⊔ to be the full subcategory spanned by the O G - Fin -manifolds M → U → O equivalent to E → U ′ → U → O where E → U ′ is a G -vector bundleover a finite G -set U ′ . Remark 3.6.6. Note that if M → U → O is equivalent to E → U ′ → U → O where E → U ′ is a G -vector bundle over a finite G -set U ′ , then U ′ = π ( E ) ∼ = π ( M ) is the set of connectedcomponents of M with the induced action. Lemma 3.6.7. The subcategory Disk G, ⊔ ⊂ Mfld G, ⊔ is a G -subcategory stable under equiva-lences.Proof. The proof follows from the characterization of coCartesian edges of Mfld G, ⊔ ։ Fin G ∗ as spans of O G - Fin -manifolds where the ’backwards arrow’ is equivalent to a pullback over asummand-inclusion and the ’forwards arrow’ is equivalent to an identity-of-manifolds, followingthe outline of proposition 3.6.3. Corollary 3.6.8. The operation of G -disjoint union on Mfld G induces a G -symmetric monoidalstructure on the G -subcategory Disk G .Proof. By lemma 3.6.7 it is enough to show that to show that the underlying G -category of Disk G, ⊔ ։ Fin G ∗ is equivalent to Disk G . Indeed, pulling back along the G -functor σ Let B ∈ Top G be a G -space and f : B → BO n ( G ) be a G -map. Define the G -categories of f -framed G -disks as the pullback on the left. Disk G,f − fr ❴✤ (cid:15) (cid:15) (cid:31) (cid:127) / / Mfld G,f − fr (cid:15) (cid:15) Disk G (cid:31) (cid:127) / / Mfld G , Disk G,f − fr, ⊔ ❴✤ (cid:15) (cid:15) (cid:31) (cid:127) / / Mfld G,f − fr, ⊔ (cid:15) (cid:15) Disk G, ⊔ (cid:31) (cid:127) / / Mfld G, ⊔ . The G -symmetric monoidal structure of G -disjoint union on Disk G lifts to a G -symmetricmonoidal structure on Disk G,f − fr , given by the right pullback above. G -disk algebras We define G -disk algebras using G -symmetric monoidal functors. Notation 3.6.10. Let p : C ⊗ ։ Fin G ∗ , q : D ⊗ ։ Fin G ∗ be two G -symmetric monoidal categories.A G -symmetric monoidal functor from C to D is a functor of ∞ -categories f : C ⊗ → D ⊗ over Fin G ∗ that takes p -coCartesian edges to q -coCartesian edges. Denote the ∞ -category of G -symmetricmonoidal functors from C to D by Fun ⊗ G ( C , D ) := Fun Fin G ∗ ( C ⊗ , D ⊗ ) . Definition 3.6.11. Let C ⊗ ։ Fin G ∗ be a G -symmetric monoidal category. A G -disk algebra with values in C is a G -symmetric monoidal functor A : Disk G, ⊔ → C ⊗ (see definition 3.6.5).Denote the ∞ -category of G -disk algebras in C by Fun ⊗ G ( Disk G , C ) .Let f : B → BO n ( G ) a G -map, as in definition 3.3.1. An f -framed G -disk algebra withvalues in C is a G -symmetric monoidal functor A : Disk G,f − fr, ⊔ → C ⊗ (see corollary 3.5.5).Denote the ∞ -category of G -disk algebras in C by Fun ⊗ G ( Disk G,f − fr , C ) . We will use G -disk algebras as coefficients in the definition of G -factorization homology insection 4. Example 3.6.12. Let V : pt → BO n ( G ) be the G -map corresponding to a real n -dimensional G -representation V (see example 3.3.3), and Disk G,V − fr, ⊔ be the G -symmetric monoidal cat-egory of V -framed G -disks. A V -framed G -disk algebra is a G -symmetric monoidal functor Disk G,V − fr, ⊔ → C ⊗ . In corollary 3.9.9 we will see that V -framed G -disk algebras are equivalentto E V -algebras. G -disks as a G -symmetric monoidal envelope There is a close relationship between Disk G and the G - ∞ -category Rep Gn of definition 3.2.3.To state it we first define a G - ∞ -operad Rep G, ⊔ n whose underlying G - ∞ -category is Rep Gn (seedefinition 3.5.7), and then show that Disk G is the G -symmetric monoidal envelope of Rep G, ⊔ n .See [BDG + ] for the construction and universal property of the G -symmetric monoidal enve-lope. Definition 3.7.1. Let Rep G, ⊔ n ⊂ Mfld G, ⊔ be the full G -subcategory on the objects of Rep Gn (using the Segal conditions on the fibers of Mfld G, ⊔ ). Note that Rep G, ⊔ n ։ Fin G ∗ is a G - ∞ -operad. Equivalently, Rep G, ⊔ n is the full subcategory on O G - Fin -manifolds E → U → O where E → U is a G -vector bundle. emma 3.7.2. The G -symmetric monoidal G -category of G -disks, Disk G, ⊔ , is equivalent to Env G ( Rep G, ⊔ n ) , the G -symmetric monoidal envelope of Rep G, ⊔ n .Proof. Recall that Env G ( Rep G, ⊔ n ) is given by the fiber product Rep G, ⊔ n × Fin G ∗ Arr actG ( Fin G ∗ ),where Arr actG ( Fin G ∗ ) ⊂ Arr G ( Fin G ∗ ) is the full subcategory of fiberwise active arrows. Unwindingthe definition, we identify the objects of Env G ( Rep G, ⊔ n ) with E (cid:15) (cid:15) U (cid:15) (cid:15) U o o (cid:15) (cid:15) / / U (cid:15) (cid:15) O O = o o = / / O, where E → U is a G -vector bundle.The inclusion Rep G, ⊔ n ֒ → Mfld G, ⊔ is a morphism of G - ∞ -operads, so by the universal prop-erty of the enveloping G -symmetric monoidal G -category induces a G -symmetric monoidal G -functor Env G ( Rep G, ⊔ n ) → Mfld G, ⊔ , taking an object E (cid:15) (cid:15) U (cid:15) (cid:15) U o o (cid:15) (cid:15) / / U (cid:15) (cid:15) O O = o o = / / O to the O G - Fin -manifold E → U → U → O . Therefore the essential image of Env G ( Rep G, ⊔ n ) → Mfld G, ⊔ is Disk G, ⊔ .We have to show that the G -functor Env G ( Rep G, ⊔ n ) → Disk G, ⊔ is a fully faithful (i.e. thatit is fiberwise fully faithful). However, the mapping spaces of Env G ( Rep G, ⊔ n ) = Rep G, ⊔ n × Fin G ∗ Arr actG ( Fin G ∗ ) are given by homotopy pullbacks of the mapping spaces of Rep G, ⊔ n and Arr actG ( Fin G ∗ )over Fin G ∗ . This follows from the definition of the mapping spaces of Mfld G, ⊔ after decomposingthe mapping spaces of Disk G, ⊔ using the Segal conditions.It follows that G -disk algebras (see definition 3.6.11) are equivalent to algebras over the G - ∞ -operad Rep G, ⊔ n . Corollary 3.7.3. Let C ⊗ be a G -symmetric monoidal category. The ∞ -category Fun ⊗ G ( Disk G , C ) of G -symmetric monoidal functors A : Disk G, ⊔ → C ⊗ is equivalent to the ∞ -category Alg G ( Rep G , C ) of morphisms of G - ∞ -operads Rep G, ⊔ n → C ⊗ , i.e algebras of the G - ∞ -operad Rep G, ⊔ n in C . A similar result holds for f -framed G -disks, for B a G -space and f : B → BO n ( G ) a G -mapas in definition 3.6.9. Proposition 3.7.4. The G -symmetric monoidal category Disk G,f − fr, ⊔ is equivalent to the G -symmetric monoidal envelope of Rep G,f − fr, ⊔ . .8 Embedding spaces of G -disks and equivariant configuration spaces We compare the mapping spaces of f -framed O G -manifolds with equivariant configuration spaces. Notation 3.8.1. Let ( M → G/H ) ∈ Mfld G be an O G -manifold over G/H and ( U → G/H ) ∈ Fin G a finite G -set over G/H . Denote by Conf GG/H ( U ; M ) ⊂ Map GG/H ( U, M ) the space ofinjective G -equivariant functions U → M over G/H with compact-open topology. Remark 3.8.2. The space Conf GG/H ( U ; M ) can be identified with the space of configurations ofdisjoint orbits in the H -manifold M | eH (the fiber of M → G/H over the base coset eH ), wherethe orbits of the configurations are indexed by the orbits of U | eH , with stabilizers specified by Stab ( W ) , W ∈ Orbit( U | eH ).In order to compare equivariant embedding spaces of G -disks in M with equivariant config-uration spaces we first study the equivariant embedding space of a single G -disk. Definition 3.8.3. Let E → U be a G -vector bundle over a finite G -set, and choose a G -equivariant metric on E . For t > define B t ( E ) ⊂ E, B t ( E ) = (cid:8) v ∈ E (cid:12)(cid:12) k v k < t (cid:9) , so B t ( E ) → U is the “open ball of radius t ” subbundle. Define Germ ( E, M ) = colim −−−→ n Emb GG/H ( B n ( E ) , M ) . Lemma 3.8.4. For s < t the restriction map Emb GG/H ( B t ( E ) , M ) → Emb GG/H ( B s ( E ) , M ) is ahomotopy equivalence.Proof. By radial dilation we see that the inclusion B s ( E ) ֒ → B t ( E ) is G -isotopic over G/H to a G -equivariant homeomorphism. Corollary 3.8.5. The restriction map Emb GG/H ( E, M ) → Germ ( E, M ) is a homotopy equiva-lence. Let ( E → U → G/H ) ∈ Disk G be a finite G -disjoint union of G -disks, i.e. E → U a G -vectorbundle, U = π E , and ( M → G/H ) ∈ Mfld G an O G -manifold. Precomposition with the zerosection inclusion U → E defines a fibration c : Emb GG/H ( E, M ) ։ Conf GG/H ( U ; M ) , (11)which we think of as sending a configuration of G -disks in M to the configuration of points whichare in the centers these G -disks.Similarly, for t > Emb GG/H ( B t ( E ) , M ) ։ Conf GG/H ( U ; M ), whose colimitforms a fibration c : Germ ( E, M ) ։ Conf GG/H ( U ; M ).The following corollary is used in the proof of the axiomatic properties of G -factorizationhomology (see the proofs of lemma 5.2.7 and lemma 5.3.4). Corollary 3.8.6. Let ( E → U → G/H ) ∈ Disk G be a finite G -disjoint union of G -disks, i.e. E → U a G -vector bundle, U = π E . Let ( M → G/H ) ∈ Mfld G be an O G -manifold and N ⊂ M an open G -submanifold. Then Emb GG/H ( E, N ) c (cid:15) (cid:15) (cid:15) (cid:15) / / Emb GG/H ( E, M ) c (cid:15) (cid:15) (cid:15) (cid:15) Conf GG/H ( U ; N ) / / Conf GG/H ( U ; M ) is a homotopy Cartesian square of spaces, where the vertical maps are given by (11) . roof. By corollary 3.8.5 the left horizontal maps in the diagram Emb GG/H ( E, N ) Germ ( E, N ) Conf GG/H ( U ; N ) Emb GG/H ( E, M ) Germ ( E, M ) Conf GG/H ( U ; M ) ∼ c ∼ c (12)are homotopy equivalences, so we have to show the right square is a homotopy pullback square.Since the right horizontal arrows are fibrations, it is enough to show that the right square is apullback square.Let x • ∈ Conf GG/H ( U ; N ), given by an injective G -map x • : U → N . For t ∈ R denotethe fiber of Emb GG/H ( B t ( E ) , N ) ։ Conf U ( N ) by Emb GG/H ( B t ( E ) , N ) x • . We have a map offibrations Emb GG/H ( B t ( E ) , N ) x • Emb GG/H ( B t ( E ) , N ) Conf GG/H ( U ; N ) Emb GG/H ( B t ( E ) , M ) x • Emb GG/H ( B t ( E ) , M ) Conf GG/H ( U ; M ) . For small enough t > Top , we see that the rightsquare of diagram (12) is indeed a pullback square.Our goal for the rest of this subsection is to study the framed version of the map (11), and showthat its V -framed variant is an equivalence (example 3.8.10). This fact will be used in section 3.9to compare the G - ∞ -operad Rep G,V − fr, ⊔ n (definition 3.5.7) with the classical G -operad of littledisks in V .We begin by showing that the decomposition of the configuration of G -disks E into orbits of G -disks induces a decomposition on its space of G -embeddings into M . Proposition 3.8.7. Let ( M → G/H ) ∈ Mfld G [ G/H ] be an O G -manifold over G/H and ( E → U → G/H ) ∈ Disk G [ G/H ] . For W ∈ Orbit( U ) let E W ∈ Disk G [ G/H ] denote ( E | W → W → G/H ) ,the restriction of the vector bundle E → U to the orbit W ⊆ U . Then the commutative squareof spaces Emb GG/H ( E, M ) ❴✤ c (cid:15) (cid:15) / / Q W Emb GG/H ( E W , M ) c (cid:15) (cid:15) Conf GG/H ( U ; M ) / / Q W Conf GG/H ( W ; M ) is a homotopy pullback square, where the products are indexed by W ∈ Orbit( U ) , and the verticalmaps are given by (11) .Proof. By corollary 3.8.5 the left horizontal maps in the diagram Emb GG/H ( E, M ) Germ ( E, M ) Conf GG/H ( U ; M ) Q W Emb GG/H ( E W , M ) Q W Germ ( E W , M ) Q W Conf GG/H ( W ; M ) ∼ c ∼ c Proposition 3.8.8. Let M ∈ Mfld G,f − fr [ G/H ] be an f -framed O G -manifold over G/H , given by a O G -manifold M → G/H together with an f -framing f M : M → B lifting τ M : M → BO n ( G ) .Let E ∈ Disk G,f − fr [ G/H ] , given by ( E → U → G/H ) ∈ Disk G [ G/H ] and f -framing f E : E → B . For W ∈ Orbit( U ) denote E W ∈ Disk G,f − fr [ G/H ] denote the restricted G -vector bundle ( E | W → W → G/H ) , with the restricted framing f W : E | W ⊂ E f E −−→ B × G/H .Then the commutative square of spaces Emb G,f − frG/H ( E, M ) ❴✤ c (cid:15) (cid:15) / / Q W Emb G,f − frG/H ( E W , M ) c (cid:15) (cid:15) Conf GG/H ( U ; M ) / / Q W Conf GG/H ( W ; M ) is a homotopy pullback square, where the products are indexed by W ∈ Orbit( U ) , and the verticalmaps are given by precomposition with the zero section.Proof. Recall the notation of remark 3.2.9, B ( G/H ) , BO n ( G )( G/H ) ∈ Top G/G/H ,B ( G/H ) = ( B × G/H → G/H ) , BO n ( G )( G/H ) = ( BO n ( G ) × G/H → G/H ) . Consider the commutative diagram Emb G,f − frG/H ( E, M ) Map G/B ( G/H ) ( E, M ) Q W Emb G,f − frG/H ( E W , M ) Q W Map G/B ( G/H ) ( E W , M ) Emb GG/H ( E, M ) Map G/BO n ( G )( G/H ) ( E, M ) Q W Emb GG/H ( E W , M ) Q W Map G/BO n ( G )( G/H ) ( E W , M ) , whereMap G/B ( G/H ) ( E, M ) = Map G/B ( G/H ) ( E f E −−→ B × G/H, M f M −−→ B × G/H ) , Map G/BO n ( G )( G/H ) ( E, M ) = Map G/BO n ( G )( G/H ) ( E f E −−→ BO n ( G ) × G/H, M f M −−→ BO n ( G ) × G/H ) . (cid:18) E f E −−→ B × G/H (cid:19) = a W (cid:18) E W f W −−→ B × G/H (cid:19) , (cid:16) E τ E −−→ BO n ( G ) × G/H (cid:17) = a W (cid:16) E W τ W −−→ BO n ( G ) × G/H (cid:17) , and in particular the right face is a homotopy pullback square. By [Lur09a, lem. 4.4.2.1] the leftface is a homotopy pullback square.Note that the left face is above diagram the same as the top square of the diagram Emb G,f − frG/H ( E, M ) ❴✤ (cid:15) (cid:15) / / Q W Emb G,f − frG/H ( E W , M ) (cid:15) (cid:15) Emb GG/H ( E, M ) ❴✤ c (cid:15) (cid:15) / / Q W Emb GG/H ( E W , M ) c (cid:15) (cid:15) Conf GG/H ( U ; M ) / / Q W Conf GG/H ( W ; M )By proposition 3.8.7 the bottom square is a homotopy pullback square, hence by [Lur09a, lem.4.4.2.1] so is the outer rectangle. The endomorphism space of a single framed G -disk We identify the endomorphismspace of a single framed G -disk as a loop space. Let E π −→ G/H be a G -vector bundle.Note that as an object of Top G [ G/H ] it is equivalent to the terminal object ( G/H = −→ G/H ),so the G -tangent classifier τ E : E → BO n ( G ) is given by a choice of connected component of( BO n ( G )) H ≃ ` V BAut Rep H ( V ), i.e an H -representation V of dimension n . In particular, wehave an isomorphism E ∼ = V × H G of G -vector bundles over G/H .An f -framing on E is given by a G -map e : E → B lifting V : E → BO n ( G ) up to G -homotopy.Using the equivalence Map G ( E, B ) ≃ Map G ( G/H, B ) ≃ Map H ( pt, B ) ≃ B H we can consider e as a point in B H . Proposition 3.8.9. Let E → G/H be a G -vector bundle with f -framing e : E → B . Then theendomorphism space of ( E → G/H ) ∈ Mfld G,f − fr [ G/H ] is weakly equivalent to the loop space of B H with base point e , Emb G,f − frG/H ( E, E ) ≃ Ω e B H . Proof. The endomorphism space of E is given by the homotopy pullback Emb G,f − frG/H ( E, E ) Map G/B ( G/H ) ( E e −→ B × G/H, E e −→ B × G/H ) Emb GG/H ( E, E ) Map G/BO n ( G )( G/H ) ( E V −→ BO n ( G ) × G/H, E V −→ BO n ( G ) × G/H ) . p ( f × G/H ) ∗ τ (13)We prove our claim by identifying the mapping spaces on the right column with loop spaces andshowing that the horizontal maps are equivalences.51ince Map G/B ( G/H ) ( E e −→ B × G/H, E e −→ B × G/H ) is a mapping space in the slice category (cid:16) Top G [ G/H ] (cid:17) /B ( G/H ) it is equivalent to the homotopy pullbackMap G/B ( G/H ) ( E e −→ B × G/H, E e −→ B × G/H ) Map GG/H ( E → G/H, E → G/H ) ∗ Map GG/H ( E → G/H, B × G/H → G/H ) . p e ∗ e Since ( E → G/H ) ∈ Top G [ G/H ] is terminal we haveMap GG/H ( E → G/H, E → G/H ) ≃ Map GG/H ( G/H = −→ G/H, G/H = −→ G/H ) = ∗ , Map GG/H ( E → G/H, B × G/H → G/H ) ≃ Map GG/H ( G/H = −→ G/H, B × G/H → G/H ) ≃ Map G ( G/H, B ) ∼ = B H , hence Map G/B ( G/H ) ( E e −→ B × G/H, E e −→ B × G/H ) ≃ Ω e B H .Replacing B with BO n ( G ), the same calculation showsMap G/BO n ( G )( G/H ) ( E V −→ BO n ( G ) × G/H, E V −→ BO n ( G ) × G/H ) ≃ Ω V a ρ : H yR n BAut Rep H ( ρ ) = Ω BAut Rep H ( V ) . Identify Emb GG/H ( E, E ) ∼ = Emb GG/H ( V × G/H, V × G/H ) ∼ = Emb H ( V, V ) in the homotopypullback square (13) we get a homotopy pullback square Emb G,f − frG/H ( E, E ) Ω e B H Aut Rep H ( V ) Emb H ( V, V ) Emb H ( V, V ) Ω BAut Rep H ( V ) , p τ where Emb H ( V, V ) is the subspace of H -equivariant self embeddings V ֒ → V that fix the origin.By proposition 3.2.5 the bottom left map is a weak equivalence, and the middle bottomarrow is clearly a homotopy equivalence. Since the composition of the bottom maps is theknown equivalence Aut Rep H ( V ) → Ω BAut Rep H ( V ), we conclude that τ is a weak equivalence,and therefore the top map of the homotopy pullback square is a weak equivalence as well.Finally, we return to the V -framed variant of the map (11). Example 3.8.10. Consider V -framed manifolds for V be a real n -dimensional G -representation(example 3.3.3), and let E ∈ Disk G,V − fr [ G/H ] be given by ( E → U → G/H ) ∈ Disk G [ G/H ] ,with V -framing inducing a trivialization E ∼ = U × V of the G -vector bundle E → U . Con-sider the mapping space from E to ( V × G/H → G/H ) ∈ Disk G,V − fr [ G/H ] . For every orbit W ∈ Orbit( U ) we have E W ∼ = W × V as G -vector bundles over W . Therefore the homo-topy fiber of c : Emb G,V − frG/H ( E W , V × G/H ) → Conf GG/H ( W ; V × G/H ) is equivalent to theloop space of a point (see proposition 3.8.9), hence contractible. It follows that the map52 W Emb G,V − frG/H ( E W , V × G/H ) → Q W Conf GG/H ( W ; V × G/H ) is an equivalence, since its ho-motopy fibers are contractible. By proposition 3.8.8 precomposition with the zero section of E → U induces a homotopy equivalence c : Emb G,V − frG/H ( E, V × G/H ) ∼ −→ Conf GG/H ( U ; V × G/H ) . (14)More generally, for any V -framed O G -manifold M ∈ Mfld G,V − fr [ G/H ] we have c : Emb G,V − frG/H ( E, M ) ∼ −→ Conf GG/H ( U ; M ) , (15)since by proposition 3.8.8 the homotopy fibers are contractible.We will use example 3.8.10 in section 3.9. G -operad and the G - ∞ -operad of V -framed representations Let V be a real n -dimensional representation of G , and Rep G,V − fr, ⊔ the G - ∞ -operad of defi-nition 3.5.7. In this subsection we define the G - ∞ -operad E V of little G -disks (definition 3.9.5)using the genuine operadic nerve construction of Bonventre, and show that it is equivalent to Rep G,V − fr, ⊔ (proposition 3.9.8), hence E V -algebras are equivalent to V -framed G -disk algebras.We first review the relevant details of Bonventre’s construction. This construction is bestunderstood in the light of [BP17, thm. III] which gives a (right) Quillen equivalence i ∗ : sOp G → sOp G between the G -graph model structure on simplicial G -operads (where weak equivalences is de-tected on graph-subgroup fixed points) and the projective model structure on genuine G -operads. Construction 3.9.1 (The genuine equivariant category of operators, see [Bon19, def. 4.1]) . Let P ∈ sOp G be a genuine G -operad. Define a simplicial category P ⊗ as follows. The objects of P ⊗ are objects of Fin G ∗ , i.e. G -maps U → G/H from a finite G -set to a G -orbit. The simplicialspace of maps P ⊗ ( U → G/H, U → G/K ) is given byMap P ⊗ (cid:18) U ↓ G/H , U ↓ G/K (cid:19) = a ϕ Y W ∈ Orbit( U ) P (cid:18) f − ( W ) ↓ W (cid:19) , where the coproduct is indexed by ϕ ∈ Map Fin G ∗ (cid:18) U ↓ G/H , U ↓ G/K (cid:19) . Composition in P ⊗ is definedusing coproducts of the composition maps of the genuine G -operad P . Theorem 3.9.2 ([Bon19, thm. 4.10]) . Let P ∈ sOp G be a genuine G -operad, and N ⊗ ( P ) thecoherent nerve of the P ⊗ . If P ∈ sOp G is locally fibrant, then N ⊗ ( P ) is a G - ∞ -operad. We call N ⊗ ( O ) as the genuine operadic nerve of O . Corollary 3.9.3 ([Bon19, cor 6.3]) . Let O ∈ sOp G be a graph-fibrant simplicial G -operad witha single color. Then i ∗ O ∈ sOp G is locally fibrant, and thus there exists a G - ∞ -operad N ⊗ ( O ) associated to O . In particular, the genuine coherent nerve construction associates a G - ∞ -operad to the equiv-ariant little disk operad. 53 xample 3.9.4 ([Bon19, ex. 6.5]) . Let V be a real orthogonal n -dimensional G -representation,and D ( V ) the open unit disk of V . For H < G and U a finite H -set let Emb Aff,H ( U × D ( V ) , D ( V )) denote the space of H -equivariant affine embeddings U × D ( V ) ֒ → D ( V ). Let D V be the little V -disks operad (see e.g [GM17, def. 1.1] or [BH15, def. 3.11(ii)]). Applyingthe functor Sing to the spaces ( D V ) n we get a locally fibrant simplicial G -operad, hence anassociated G - ∞ -operad N ⊗ ( D V ).The mapping spaces of N ⊗ ( D V ) are given byMap N ⊗ ( D V ) (cid:18) U ↓ G/H , U ↓ G/K (cid:19) = a ϕ Y Gx ∈ Orbit( U ) Emb Aff,Stab ( x ) ( f − ( x ) × D ( V ) , D ( V )) . Definition 3.9.5. Fix a real orthogonal G -representation V , and let D V denote the G -operad oflittle V -disks. Let E ⊗ V denote the genuine operadic nerve N ⊗ ( D V ) of [Bon19, ex. 6.5]. Before defining E V -algebras we recall the definition of a G - ∞ -operad map. Notation 3.9.6. Let P ⊗ → Fin G ∗ , Q ⊗ → Fin G ∗ be G - ∞ -operads (see [Nar17, def. 3.1]). A mapof G - ∞ -operads from P ⊗ to Q ⊗ is a map of simplicial sets f : P ⊗ → Q ⊗ such that1. The diagram P ⊗ Q ⊗ Fin G ∗ f commutes.2. The functor f carries coCartesian edges over inert morphisms to coCartesian edges. Definition 3.9.7. Let C ⊗ ։ Fin G ∗ be a G -symmetric monoidal category. An E V -algebra in C is a map of G - ∞ -operads A : E V → C ⊗ . Let Alg E V ( C ) ⊆ Fun / Fin G ∗ ( E ⊗ V , C ⊗ ) denote the fullsubcategory spanned by E V -algebras. Comparison with Rep G,V − fr, ⊔ . We can now easily compare the G - ∞ -operads E V of defini-tion 3.9.5 and Rep G,V − fr, ⊔ of definition 3.5.7.We start with some observations. Fix a V -framed G -diffeomorphism D ( V ) ∼ = V . Let H < G and U ′ a finite H -set and U = G × H U ′ its topological induction. Then the topological inductionof U ′ × D ( V ) from H to G is U × D ( V ). Note that the induced map U × D ( V ) → G/H is G -vector bundle equivalent to U × V → U by our chosen diffeomorphism, and hence V -framed.Let Emb Aff,GG/H ( U × D ( V ) , G/H × D ( V )) denote the space of affine G -embeddings over G/H .Note that restriction to the fiber over eH defines a homeomorphism Emb Aff,H ( U ′ × D ( V ) , D ( V )) ∼ = Emb Aff,GG/H ( U × D ( V ) , G/H × D ( V )) . On the other hand, affine G -embeddings U × D ( V ) ֒ → G/H × D ( V ) over G/H are clearly V -framed (using the chosen G -diffeomorphism D ( V ) ∼ = V . Therefore we have a map Emb Aff,GG/H ( U × D ( V ) , G/H × D ( V )) ֒ → Emb G,V − frG/H ( U × D ( V ) , G/H × D ( V )) . F : E V → Rep G,V − fr, ⊔ over Fin G ∗ as follows. For every finite G -set U define F ( U → G/H ) = ( U × D ( V ) → U → G/H ). Define F on mapping spaces by theembeddingsMap E V (cid:18) U ↓ G/H , U ↓ G/K (cid:19) = a ϕ Y Gx ∈ Orbit( U ) Emb Aff,Stab ( x ) ( f − ( x ) × D ( V ) , D ( V )) ֒ → a ϕ Y W ∈ Orbit( U ) Emb G,V − frW ( f − ( W ) × D ( V ) , G/H × D ( V ))= Map Rep G,V − fr, ⊔ ( U × D ( V ) , U × D ( V )) . Proposition 3.9.8. The functor F : E V → Rep G,V − fr, ⊔ is an equivalence of G - ∞ -operads.Proof. By construction F is a functor over Fin G ∗ , therefore it is enough to show that F is anequivalence of ∞ -categories. Clearly F is essentially surjective, since any V -framed G -vectorbundle over a finite G -set U is equivariant to U × V ∼ = U × D ( V ). We therefore have to showthat F is fully faithful.By the Segal conditions it is enough to show that F induces an equivalence of spaces F : Map E V (cid:18) U ↓ G/H , G/H ↓ G/H (cid:19) → Map Rep G,V − fr, ⊔ U × D ( V ) ↓ U ↓ G/H , G/H × D ( V ) ↓ G/H ↓ G/H on the mapping spaces over ϕ ∈ Map Fin G ∗ (cid:18) U ↓ G/H , G/H ↓ G/H (cid:19) . By example 3.9.4 we haveMap E V (cid:18) U ↓ G/H , G/H ↓ G/H (cid:19) ≃ Y W ∈ Orbit( G/H ) Emb Aff,Stab ( W ) ( f − ( W ) × D ( V ) , D ( V )) , and since Rep G,V − fr, ⊔ ⊂ Mfld G,V − fr, ⊔ is a full G -subcategory we haveMap Rep G,V − fr, ⊔ U × D ( V ) ↓ U ↓ G/H , G/H × D ( V ) ↓ G/H ↓ G/H = Emb G,V − frG/H ( U × D ( V ) , G/H × D ( V )) ∼ = Emb G,V − frG/H ( U × V, G/H × V ) . Consider the commutative diagram Q W ∈ Orbit( U ) Emb Aff,Stab ( W ) ( f − ( W ) × D ( V ) , D ( V )) c (cid:15) (cid:15) F / / Emb G,V − frG/H ( U × V, G/H × V ) c (cid:15) (cid:15) Q W ∈ Orbit( U ) Inj Stab ( W ) ( f − ( W ) , V ) / / Conf GG/H ( U ; V × G/H )where the vertical map is given by taking the centers of disks, and the right vertical map is givenby precomposition with the zero section. We wish to prove that the top horizontal map is anequivalence. The left vertical map is known to be an equivalence (see [BH15, prop. 4.19] and[GM17, lem 1.2]). The right vertical map is an equivalence by example 3.8.10, and the bottomhorizontal map is a homeomorphism by inspection.55e immediately see that E V -algebras are equivalent to V -framed G -disk algebras. Corollary 3.9.9. There is an equivalence of ∞ -categories Alg E V ( C ) ≃ Fun ⊗ G ( Disk G,V − fr , C ) .Proof. Precomposition with the equivalence F : E V ∼ −→ Rep G,V − fr, ⊔ of proposition 3.9.8 in-duces an equivalence Alg E V ( C ) ∼ −→ Alg Rep G,V − fr ( C ) . By proposition 3.7.4 the G -symmetric monoidal envelope of Rep G,V − fr, ⊔ is equivalent to Disk G,V − fr, ⊔ ,so by its universal property we have Alg Rep G,V − fr ( C ) ≃ Fun ⊗ G ( Disk G,V − fr , C ) . G -factorization homology In this section we use the G -categories Mfld G,f − fr and Disk G,f − fr to define genuine equivariantfactorization homology. We define G -factorization homology, first as a parametrized colimit(definition 4.1.2), then as a G -functor (proposition 4.1.4) and finally as a G -symmetric monoidalfunctor (definition 4.2.3). G -disk algebras and G -factorization homology asa G -functor In this subsection we define equivariant factorization homology (see proposition 4.1.4). This isan smooth equivariant version of the factorization homology of [AF15] and of topological chiralhomology of [Lur, 7.5.2].In order to define genuine G -factorization homology using parametrized ∞ -colimits we first re-call the definition of a parametrized over-category from [Sha18]. The parametrized over-categoryplays the role of an indexing category in the G -colimit defining factorization homology below(see definition 4.1.2), and more generally in the G -colimit formula for G -left Kan extensions (see[Sha18, thm. 10.3]).Let C be a G -category and x ∈ C [ G/H ] an object over G/H , classified by the G -functor σ x : G/H → C . Define the parametrized over-category C /x ։ G/H (see [Sha18, not. 4.29]) as thefiber product Arr G ( C ) × C G/H , considered as a G/H -category by pulling back the coCartesianfibration ev : Arr G ( C ) → C along σ x : G/H → C . Note that the fiber of C /x ։ G/H over ϕ : G/K → G/H is equivalent to the ∞ -over-category ( C [ G/K ] ) /ϕ ∗ x , where ϕ ∗ x ∈ C [ G/K ] isdetermined by choosing a coCartesian lift x → ϕ ∗ x of ϕ .If C ′ ⊆ C is a full G -subcategory we abuse notation and write C ′ /x for the restricted G -over-category, given by the fiber product C ′ × C C /x .We now return to the definition of genuine G -factorization homology.Let A ∈ Fun ⊗ G ( Disk G,f − fr , C ) be an f -framed G -disk algebra with values in C , and M ∈ Mfld G,f − fr [ G/H ] an f -framed O G -manifold. In the following definition we use the parametrizedover-category Disk G,f − fr/M associated to M ∈ Mfld G,f − fr [ G/H ] and Disk G,f − fr ⊂ Mfld G,f − fr . Remark 4.1.1. Note that Disk G/M → G/H is the coCartesian fibration dual to the Cartesianfibration ( Disk G ) /M → ( O G ) / [ G/H ] (see [Lur09a, prop 2.4.3.1], compare [Sha18, prop. 4.31]),and therefore can be modeled by the topological Moore over category (see appendix A).56onstruct a G -functor over G/H by composing Disk G,f − fr/M ( ( ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ / / Disk G,f − fr × G/H (cid:15) (cid:15) (cid:15) (cid:15) A × id / / C× G/H u u u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ G/H. (16)Consider the functor (16) as an G/H -diagram in the G/H -category C× G/H . Note that the G/H -colimit of the above diagram is a coCartesian section of C× G/H ։ G/H , or equivalentlya G -functor G/H → C and that a G -functor G/H → C represents an object of C over [ G/H ]. Definition 4.1.2. Let M ∈ Mfld G,f − fr [ G/H ] be an f -framed O G -manifold, and A an f -framed G -diskalgebra. Define the G -factorization homology of M with coefficients in A by the parametrizedcolimit Z M A ∈ C , Z M A := G/H − colim −−−→ (cid:18) Disk G,f − fr/M → Disk G,f − fr × G/H A × id −−−→ C× G/H (cid:19) . (17)In what follows, assume that C is a G -cocomplete G -category (i.e C has all G/H -colimits forevery H < G , see [Sha18, def. 5.12]), so that all the parametrized colimits of proposition 4.1.4exist. Next we show that the assignment M R M A extends to a G -functor R − A : Mfld G,f − fr →C , and that the G -functors R − A are in turn functorial in A (proposition 4.1.4). Construction 4.1.3. Let ι : Disk G,f − fr ֒ → Mfld G,f − fr denote the inclusion of the full G -subcategory of finite G -disjoint unions of G -disks and C be a cocomplete G -symmetric monoidalcategory. The inclusion G -functor ι induces a restriction G -functor ι ∗ : Fun G ( Mfld G,f − fr , C ) → Fun G ( Disk G,f − fr , C ). By [Sha18, cor. 10.6] (proposition 2.3.3) the restriction G -functor has afully faithful left G -adjoint ι ! : Fun G ( Disk G,f − fr , C ) ⇆ Fun G ( Mfld G,f − fr , C ) : ι ∗ . In particular, define ι ! to be the fully faithful left adjoint of ι ! : Fun G ( Disk G,f − fr , C ) ⇆ Fun G ( Mfld G,f − fr , C ) : ι ∗ , (18)the adjunction of ∞ -categories between the fibers over the terminal orbit [ G/G ] Proposition 4.1.4. Let C be a cocomplete G -symmetric monoidal category. Then the functor Fun ⊗ G ( Disk G,f − fr , C ) → Fun G ( Disk G,f − fr , C ) ι ! −→ Fun G ( Mfld G,f − fr , C ) , ( A : Disk G,f − fr D → C ⊗ ) ( A : Disk G,f − fr → C ) ( ι ! A : Mfld G,f − fr → C ) sends a G -disk algebra A to a G -functor ι ! A : Mfld G,f − fr → C , , M ( ι ! A )( M ) = Z M A. Proof. By [Sha18, thm. 10.4] for every G -disk algebra A ∈ Fun G ( Mfld G,f − fr , C ) the left G -adjoint ι ! ( A ) : Mfld G,f − fr → C ) is given by left G -Kan extension of A along ι . By [Sha18, thm10.3] applying the ι ! ( A ) to M ∈ Mfld G,f − fr [ G/H ] is given by the G/H -colimit( ι ! A )( M ) = G/H − colim −−−→ (cid:18) Disk G,f − fr/M → Disk G,f − fr × H/G A × id −−−→ C× G/H (cid:19) = Z M A. .2 Extensding G -factorization homology to a G -symmetric monoidalfunctor In this subsection we prove (proposition 4.2.2) that G -factorization homology with values in apresentable G -symmetric monoidal category extends to a G -symmetric monoidal functor (seedefinition 4.2.3). Definition 4.2.1. Let C ⊗ ։ Fin G ∗ be a G -symmetric monoidal category. We say C ⊗ is apresentable G -symmetric monoidal category if the underlying G -category is presentable and forevery active map α : I → J in Fin G ∗ the G -functor ⊗ α : C ⊗ → C ⊗ Let C ⊗ ։ Fin G ∗ be a presentable G -symmetric monoidal category.Then the adjunction eq. (18) lifts to an adjunction ( ι ⊗ ) ! : Fun ⊗ G ( Disk G,f − fr , C ) (cid:15) (cid:15) / / Fun ⊗ G ( Mfld G,f − fr , C ) : ( ι ⊗ ) ∗ o o (cid:15) (cid:15) ι ! : Fun G ( Disk G,f − fr , C ) Fun G ( Mfld G,f − fr , C ) : ι ∗ p p (19) where ι ⊗ : Disk G,f − fr, ⊔ → Mfld G,f − fr. ⊔ is the inclusion of the subcategory of f -framed indexeddisks (see corollary 3.5.5). Note that since ι ! is fully faithful the Segal conditions imply that ( ι ⊗ ) ! : Fun ⊗ G ( Disk G,f − fr , C ) → Fun ⊗ G ( Mfld G,f − fr , C ) is fully faithful. Definition 4.2.3. For C ⊗ ։ Fin G ∗ , A : Disk G,f − fr, ⊔ → C ⊗ as in proposition 4.2.2, denote the G -symmetric monoidal functor ( ι ⊗ ) ! by R − A : Mfld G,f − fr. ⊔ ' ' ❖❖❖❖❖❖❖❖❖❖❖❖ / / C ⊗ } } ④④④④④④④④ Fin G ∗ . Commutativity of the diagram (19) shows that R − A extends the G -functor ι ! A : Mfld G,f − fr → C of eq. (18) which sends an O G -manifold ( M → G/H ) to its G -factorization homology ι ! A ( M ) = R M A (proposition 4.1.4) to a G -symmetric monoidal functor. We call the G -symmetric monoidalfunctor R − A : Mfld G,f − fr, ⊔ → C ⊗ the G -factorization homology functor with coefficients in A . In the remainder of this subsection we prove proposition 4.2.2. The proof has two parts,the first is a general G -categorical lemma, lemma 4.2.5, giving conditions ensuring that a G -leftKan extension lifts to a G -symmetric monoidal functor, and the second is a verification of theseconditions.We start with by recalling the notion of a G -lax monoidal functor and stating a usefulproposition from [BDG + ]. Let D ⊗ , C ⊗ be G -symmetric monoidal categories. Recall that alax G -symmetric monoidal G -functor F from D to C is a functor F : D ⊗ → C ⊗ over Fin G ∗ which preserves inert edges (i.e. coCartesian edges over inert morphisms). Let Alg ( D , C ) ⊂ Fun / Fin G ∗ ( D ⊗ , C ⊗ ) be the full subcategory of functors over Fin G ∗ which are lax G -symmetricmonoidal. 58 roposition 4.2.4. Let C ⊗ ։ Fin G ∗ be a presentable G -symmetric monoidal category, let M ⊗ ։ Fin G ∗ be a small G -symmetric monoidal category and ι ⊗ : D ⊗ ֒ → M ⊗ an inclusionof a full G -symmetric monoidal subcategory. Denote by ι : D → M the induced G -functor on theunderlying categories.Then the restriction along ι ⊗ has a left adjoint ( ι ⊗ ) ! : Alg ( D , C ) → Alg ( M , C ) . Moreover,the adjunction ( ι ⊗ ) ! : Alg ( D , C ) ⇆ Alg ( M , C ) : ( ι ⊗ ) ∗ restricts to the adjunction ι ! : Fun( D , C ) ⇆ Fun( M , C ) : ι ∗ , where ι ! : Fun( D , C ) → Fun( M , C ) is left adjoint to the restriction along ι .In particular we have a commuting square of ∞ -categories Alg ( D , C ) (cid:15) (cid:15) ( ι ⊗ ) ! / / Alg ( M , C ) (cid:15) (cid:15) Fun G ( D , C ) ι ! / / Fun G ( M , C ) . We will prove proposition 4.2.2 by applying the following G -categorical lemma (a G -categoricalversion of [AFT17a, lem. 2.16]). Lemma 4.2.5. Let C ⊗ ։ Fin G ∗ be a presentable G -symmetric monoidal category, let D ⊗ , M ⊗ ։ Fin G ∗ be small G -symmetric monoidal categories and ι ⊗ : D ⊗ ֒ → M ⊗ be an inclusion of a full G -symmetric monoidal subcategory. Denote by ι : D → M the induced G -functor on the underlyingcategories.If for every active morphism ψ : I → J in a fiber ( Fin G ∗ ) [ G/H ] and every coCartesian lift x → y of ψ to M ⊗ the G/H -functor ⊗ ψ : ( D ⊗ ) /x → ( D ⊗ Fun ⊗ G ( D , C ) (cid:15) (cid:15) ( ι ⊗ ) ! / / Fun ⊗ G ( M , C ) (cid:15) (cid:15) Fun G ( D , C ) ι ! / / Fun G ( M , C ) commutes, where ( ι ⊗ ) ! and ι ! the left adjoins to the restrictions along ι ⊗ and ι , respectively.Proof. Applying proposition 4.2.4 we have:( ι ⊗ ) ! : Alg ( D , C ) (cid:15) (cid:15) Alg ( M , C ) : ( ι ⊗ ) ∗ p p (cid:15) (cid:15) ι ! : Fun G ( D , C ) Fun G ( M , C ) : ι ∗ p p We need to show that the adjunction ( ι ⊗ ) ! : Alg ( D , C ) ⇆ Alg ( M , C ) : ( ι ⊗ ) ∗ restricts to an ad-junction between the full subcategoriesFun ⊗ G ( D , C ) ⊂ Alg ( D , C ) , Fun ⊗ G ( M , C ) ⊂ Alg ( M , C ) . Clearly precomposition with the G -symmetric monoidal functor ι ⊗ : D ⊗ → M ⊗ takes G -symmetricmonoidal functors to G -symmetric monoidal functors, so the right adjoint restricts to a functor( ι ⊗ ) ∗ : Fun ⊗ G ( M , C ) → Fun ⊗ G ( D , C ) . F ⊗ : D ⊗ → C ⊗ be a G -symmetric monoidal functor, with F : C → D the induced G -functor on the underlying categories. Applying the left adjoint ( ι ⊗ ) ! to F ⊗ we get a lax G -symmetric monoidal functor ( ι ⊗ ) ! F ⊗ : M ⊗ → C ⊗ , in other words ( ι ⊗ ) ! F ⊗ preserves coCartesianedges over inert morphisms. We have to show that ( ι ⊗ ) ! F ⊗ preserves all coCartesian edges.Using the inert-fiberwise active factorization system on M ⊗ (which exists on any G - ∞ -operad,see [BDG + ]), we are reduced to showing that ( ι ⊗ ) ! F ⊗ preserves fiberwise active coCartesianedges. By the Segal conditions it is enough to show ( ι ⊗ ) ! F ⊗ preserves arrows over maps I → J in Fin G ∗ with J = ( G/H = −→ G/H ).Before showing that ( ι ⊗ ) ! F ⊗ preserve these coCartesian edges, let us first recall how thefunctor ( ι ⊗ ) ! F ⊗ acts on morphisms.By definition ι ! : M → C is a left G -Kan extension. Using the construction of [Sha18, def.10.1] we have a G -functor ( D × M Arr G ( M )) ⋆ M M → C which is an M -parametrized G -colimit diagram, where Arr G ( M ) = O opG × Fun(∆ , O opG ) Fun(∆ , M ) ≃ Fun G ( O opG × ∆ , M )is the fiberwise arrow category (see [Sha18, not. 4.29]). Note that by definition the restrictionto the first coordinate D × M Arr G ( M ) → (cid:0) D × M Arr G ( M ) (cid:1) ⋆ M M → M factors as D × M Arr G ( M ) π D −−→ D F −→ C . and the restriction to the second coordinate is the left G -Kan extensionfunctor ι ! F , i.e ι ! F : M → ( D × M Arr G ( M )) ⋆ M M → C .Let x ∈ M ⊗ be an object over I = ( U → G/H ) and ψ : I → J be an active morphism in thefiber ( Fin G ∗ ) [ G/H ] with target J = ( G/H = −→ G/H ), given by the span ψ = U f (cid:15) (cid:15) U = o o f (cid:15) (cid:15) f / / G/H = (cid:15) (cid:15) G/H G/H = o o = / / G/H . Denote the G -functor classified by x by x • : U → M (see remark B.0.8). Pulling back thecoCartesian fibration ( D × M Arr G ( M )) ⋆ M M ։ M along x • we get a U -parametrized G -colimit diagram ( D × M Arr G ( M ) × M U ) ⋆ U U → ( D × M Arr G ( M )) ⋆ M M → C (implicitlyusing [Sha18, lem. 4.4]), and therefore a U -colimit diagram p : ( D × M Arr G ( M ) × M U ) ⋆ U U → C× U . Denote the U -category indexing the colimit diagram above by D /x • := D × M Arr G ( M ) × M U .Note that by definition the restriction of p to D /x • factors as the U -functor D /x • → D× U F × U −−−→ C× U and the restriction to U is the U -functor ι ! F ( x • ) : U x • × U −−−−→ M× U ι ! F × U −−−−→ C× U . Since C ⊗ is a presentable G -symmetric monoidal category the tensor product functor ⊗ ψ : Y I C× U → C× G/H 60f definition B.0.11 is a distributive G/H -functor (see [Nar17, def. 3.15]). Therefore the U -colimitdiagram p induces a G/H -colimit diagram Y I D /x • ! ⋆ G/H G/H → Y I (cid:16) D /x • ⋆ U U (cid:17) Q ψ p −−→ Y I ( C× U ) ⊗ ψ −−→ C× G/H (20)exhibiting the G/H -object ⊗ ψ Y I ι ! F ( x • ) ! : G/H ≃ −→ Y I U Q I ι ! F ( x • ) −−−−−−−→ Y I C× U ⊗ ψ −−→ C× G/H as the G/H -colimit of p : Y I D /x • → Y I D× U → Y I C× U ⊗ ψ −−→ C× G/H. First, note that we can express the G/H -colimit ⊗ ψ ( Q I ι ! F ( x • )) of (20) in simpler terms.Since ( ι ⊗ ) ! F ⊗ : D ⊗ → C ⊗ is a lax G -symmetric monoidal functor we have G/H ≃ (cid:15) (cid:15) x / / ( ι ⊗ ) ! F ⊗ ( x ) ) ) M ⊗ ≃ (cid:15) (cid:15) (( ι ⊗ ) ! F ⊗ ) / / C ⊗ ≃ (cid:15) (cid:15) Q I U Q I x • × U / / Q I ι ! F ( x • ) Q I M× U Q I ι ! F × U / / Q I C × U , therefore ⊗ ψ ( Q I ι ! F ( x • )) ≃ ⊗ ψ (( ι ⊗ ) ! F ⊗ ( x )).On the other hand, we can also express the G/H -diagram p in simpler terms. To see thisobserve the commutative diagram( D ⊗ ) /x ≃ (cid:15) (cid:15) / / D ⊗ ≃ (cid:15) (cid:15) F ⊗ / / C ⊗ ≃ (cid:15) (cid:15) Q I D /x • ⊗ ψ (cid:15) (cid:15) / / Q I D× U ⊗ ψ (cid:15) (cid:15) Q I F × U / / Q I C× U ⊗ ψ (cid:15) (cid:15) D / ⊗ ψ x ∼ = (cid:16) D× G/H (cid:17) / ⊗ ψ x / / D× G/H F × G/H / / C× G/H (21)where the left vertical column is induced by taking the G/H -limit of the rows of the following61iagram of G/H -categories: D ⊗ ≃ (cid:15) (cid:15) / / M ⊗ ≃ (cid:15) (cid:15) Arr G/H ( M ⊗ ) o o ≃ (cid:15) (cid:15) / / M ⊗ ≃ (cid:15) (cid:15) G/H x o o ≃ (cid:15) (cid:15) Q I D× U / / ⊗ ψ (cid:15) (cid:15) Q I M× U ⊗ ψ (cid:15) (cid:15) Arr G/H ( Q I M× U ) o o ⊗ ψ (cid:15) (cid:15) / / Q I M× U ⊗ ψ (cid:15) (cid:15) Q I U Q I x • o o ≃ (cid:15) (cid:15) D× G/H / / M× G/H Arr G/H ( M× G/H ) o o / / M× G/H G/H. ⊗ ψ x o o Note that p is the composition of the middle row of diagram (21) followed by the lower rightvertical G/H -functor ⊗ ψ . Therefore p is equivalent to the composition of the left vertical columnof diagram (21) followed by the bottom row:( D ⊗ ) /x ⊗ ψ −−→ D / ⊗ ψ x → D× G/H F × G/H −−−−−→ C× G/H. Finally, by the assumption of the lemma the G/H -functor ⊗ ψ : ( D ⊗ ) /x → (cid:16) D× G/H (cid:17) / ⊗ ψ x is G/H -cofinal, therefore ⊗ ψ (cid:0) ( ι ⊗ ) ! F ⊗ ( x ) (cid:1) ≃ G/H − colim −−−→ (cid:18) ( D ⊗ ) /x ⊗ ψ −−→ D / ⊗ ψ x → D× G/H F × G/H −−−−−→ C× G/H (cid:19) ∼ −→ G/H − colim −−−→ (cid:18) D / ⊗ ψ x → D× G/H F × G/H −−−−−→ C× G/H (cid:19) ≃ ι ! F ( ⊗ ψ x ) , so we have a coCartesian edge e : ( ι ⊗ ) ! F ⊗ ( x ) → ι ! F ( ⊗ ψ x ) in C ⊗ over ψ .We can now show that ( ι ⊗ ) ! F ⊗ : M ⊗ → C ⊗ preserves coCartesian edges over ψ : I → J asabove. Let e ′ : x → y be a coCartesian edge in M ⊗ over ψ . By definition of ⊗ ψ this coCartesianedge factors as x → ⊗ ψ x ∼ −→ y over I ψ −→ J = −→ J (See [Lur09a, rem. 2.4.1.4 and prop. 2.4.1.5]).Applying ( ι ⊗ ) ! F ⊗ we get ( ι ⊗ ) ! F ⊗ ( e ′ ) : ( ι ⊗ ) ! F ⊗ ( x ) → ( ι ⊗ ) ! F ⊗ ( y ) = ι ! F ( y ), and we need to show( ι ⊗ ) ! F ⊗ ( e ′ ) is a coCartesian lift of ψ . However, we already have a coCartesian lift of ψ , the edge e we constructed above. Therefore ( ι ⊗ ) ! F ⊗ ( e ′ ) factors through e as ( ι ⊗ ) ! F ⊗ ( x ) e −→ ι ! F ( ⊗ ψ x ) → ι ! F ( y ). Note that the morphism ι ! F ( ⊗ ψ x ) → ι ! F ( y ) is induced from ⊗ ψ x ∼ −→ y , and thereforean equivalence. Hence ( ι ⊗ ) ! F ⊗ ( e ′ ) is coCartesian as a composition of a coCartesian edge and anequivalence.This ends the proof of lemma 4.2.5.We can now prove proposition 4.2.2 by verifying the cofinality conditions of lemma 4.2.5. Infact, we prove the cofinality of the maps in lemma 4.2.5 by showing that they are equivalences.We rely on the following result to reduce our calculations to the non-framed case B = BO n ( G ). Proposition 4.2.6. Let f : B → BO n ( G ) be a G -map as in definition 3.3.1, and M ∈ Mfld G,f − fr [ G/H ] an f -framed O G -manifold over G/H . Then the G/H -functor ( Mfld G,f − fr ) /M → Mfld G/M s an equivalence of G/H -categories.In particular, every G -submanifold N ⊆ M has an essentially unique lift to N ∈ Mfld G,f − fr/M (informally, M induces an f -framing of N ).Proof. We show that for every ϕ ∈ G/H, ϕ : G/K → G/H the induced functor on the fibersover ϕ , (cid:16) ( Mfld G,f − fr ) /M (cid:17) [ ϕ ] → (cid:16) Mfld G/M (cid:17) [ ϕ ] , is an equivalence. By construction the fibers ofthe parametrized over category are equivalent to the over categories (cid:16) ( Mfld G,f − fr ) /M (cid:17) [ ϕ ] ≃ (cid:16) Mfld G,f − fr [ G/K ] (cid:17) /ϕ ∗ M , (cid:16) ( Mfld G ) /M (cid:17) [ ϕ ] ≃ (cid:16) Mfld G [ G/K ] (cid:17) /ϕ ∗ M . By definition 3.3.1 the fiber Mfld G,f − fr [ G/K ] is given by the pullback of ∞ -categories Mfld G,f − fr [ G/K ] (cid:15) (cid:15) / / ❴✤ ( Top G [ G/K ] ) /B ( G/K ) (cid:15) (cid:15) Mfld G [ G/K ] / / ( Top G [ G/K ] ) /BO n ( G )( G/K ) , where B ( G/K ) = ( B × G/K → G/K ) , BO n ( G )( G/K ) = ( BO n ( G ) × G/K → G/K ) , see remark 2.1.7. We can simplify the pullback square above using the equivalences of re-mark 3.2.9:( Top G [ G/K ] ) /B ( G/K ) ∼ −→ Top G/B × G/K , ( Top G [ G/K ] ) /BO n ( G )( G/K ) ∼ −→ Top G/BO n ( G ) × G/K . We can now express the slice category ( Mfld G,f − fr [ G/K ] ) /ϕ ∗ M as a pullback of slice categories( Mfld G,f − fr [ G/K ] ) /ϕ ∗ M (cid:15) (cid:15) / / ❴✤ (cid:16) Top G/B × G/K (cid:17) / ( ϕ ∗ M → B × G/K ) (cid:15) (cid:15) ( Mfld G [ G/K ] ) /ϕ ∗ M / / (cid:16) Top G/BO n ( G ) × G/K (cid:17) / ( ϕ ∗ M → BO n ( G ) × G/K ) (cid:15) (cid:15) Top G/ϕ ∗ M . By [AF15, lem. 2.5] both the bottom right vertical arrow and the composition of the rightvertical arrows are equivalences of ∞ -categories. By the two-out-of-three property we see thatthe top vertical arrow is an equivalence of ∞ -categories, and therefore the left vertical arrow isalso an equivalence, as claimed. Proof of proposition 4.2.2. By the Segal conditions and proposition 4.2.6 the G/H -functors( Disk G,f − fr, ⊔ ) /M i → ( Disk G, ⊔ ) /M i , i = 1 , G/H -categories, therefore it is enough to prove the non-framed case.Let ψ : I → J be an active morphism in the fiber ( Fin G ∗ ) [ G/H ] . Without loss of generality, ψ is represented by the span ψ = U (cid:15) (cid:15) U o o (cid:15) (cid:15) / / U (cid:15) (cid:15) G/H G/H = o o = / / G/H . By remark 3.4.20 a coCartesian lift f : M → M of ψ is represented by a span f = M (cid:15) (cid:15) M o o (cid:15) (cid:15) ∼ / / M (cid:15) (cid:15) U (cid:15) (cid:15) U o o (cid:15) (cid:15) / / U (cid:15) (cid:15) G/H G/H = o o = / / G/H . By lemma 4.2.5 it is enough to show that the G/H -functor ( Disk G, ⊔ ) /M → ( Disk G, ⊔ Disk G, ⊔ ) /M i := Disk G, ⊔ × Mfld G, ⊔ ( Mfld G, ⊔ ) /M i , i = 1 , , and therefore the fibers over ϕ are given by (cid:16) ( Disk G, ⊔ ) /M i (cid:17) [ ϕ ] = (cid:16) Disk G, ⊔ (cid:17) [ ϕ ] × ( Mfld G, ⊔ ) [ ϕ ] (cid:16) ( Mfld G, ⊔ ) /M i (cid:17) [ ϕ ] , i = 1 , . By the definition of parametrized slice category [Sha18, not. 4.29] we have (cid:16) ( Mfld G, ⊔ ) /M i (cid:17) [ ϕ ] ∼ = (cid:16) ( Mfld G, ⊔ ) [ ϕ ] (cid:17) /ϕ ∗ M i , i = 1 , , where ϕ ∗ M i , i = 1 , M i → U i → G/H along ϕ : G/K → G/H .Next, note that ( Mfld G, ⊔ ) [ ϕ ] ∼ = ( Mfld G, ⊔ ) ϕ ∗ I i is the fiber of Mfld G, ⊔ ։ Fin G ∗ over ϕ ∗ I i =( U i × G/H G/K → G/H ) ∈ Fin G ∗ .However, using the definition of the coCartesian fibration Mfld G, ⊔ ։ Fin G ∗ (definition 3.4.19)and the definition of the unfurling construction (see [Bar14, prop. 11.6] and the description of thefibers following it) we see that ( Mfld G, ⊔ ) [ ϕ ] is equivalent to ( O G - Fin - Mfld ) ϕ ∗ I i , (the coherentnerve of) the full topological subcategory of O G - Fin -manifolds, O G - Fin - Mfld (definition 3.4.5)spanned by O G - Fin -manifolds over ϕ ∗ I i . It follows that the ∞ -category (cid:16) ( Mfld G, ⊔ ) /M i (cid:17) [ ϕ ] isequivalent to the slice category (( O G - Fin - Mfld ) ϕ ∗ I i ) /ϕ ∗ M i , modeled by the coherent nerve ofthe Moore over category (( O G - Fin - Mfld ) ϕ ∗ I i ) Moore /ϕ ∗ M i . Therefore, the fiber (cid:16) ( Disk G, ⊔ ) /M i (cid:17) [ ϕ ] is64quivalent to the full subcategory of (( O G - Fin - Mfld ) ϕ ∗ I i ) Moore /ϕ ∗ M i spanned by objects representedby morphisms ( E → U ′ → U i × G/H G/K → G/H ) → ( ϕ ∗ M i → U i × G/H G/K → G/H ) over φ ∗ I i , where E → U ′ is a G -vector bundle. Recall that U ′ = π ( E ) (remark 3.6.6). Unwinding thedefinition of morphisms in O G - Fin - Mfld over ϕ ∗ I i = U i × G/H G/K , we see that such morphismsare represented by commutative diagrams E (cid:15) (cid:15) E ′∼ o o (cid:15) (cid:15) (cid:31) (cid:127) / / M i (cid:15) (cid:15) π ( E ) (cid:15) (cid:15) π ( E ) = o o (cid:15) (cid:15) ϕ ∗ I i (cid:15) (cid:15) ϕ ∗ I i = o o (cid:15) (cid:15) = / / ϕ ∗ I i (cid:15) (cid:15) G/K G/K = o o = / / G/K, or equivalently, by a G -equivariant embedding E ֒ → M i over ϕ ∗ I i .With this concrete description of the fibers at hand, the induced functor between the fibers (cid:16) ( Disk G, ⊔ ) /M (cid:17) [ ϕ ] → (cid:16) ( Disk G, ⊔ ) /M (cid:17) [ ϕ ] is given by composition with ϕ ∗ f = ϕ ∗ M (cid:15) (cid:15) ϕ ∗ M o o (cid:15) (cid:15) ∼ / / ϕ ∗ M (cid:15) (cid:15) ϕ ∗ I (cid:15) (cid:15) ϕ ∗ I o o (cid:15) (cid:15) / / ϕ ∗ I (cid:15) (cid:15) G/K G/K = o o = / / G/K . By inspection the induced functor( ϕ ∗ f ) ◦ − : (( O G - Fin - Mfld ) ϕ ∗ I ) Moore /ϕ ∗ M → (( O G - Fin - Mfld ) ϕ ∗ I ) Moore /ϕ ∗ M E (cid:15) (cid:15) (cid:31) (cid:127) / / ϕ ∗ M (cid:15) (cid:15) π ( E ) $ $ ❍❍❍❍❍❍❍❍❍ ϕ ∗ I (cid:15) (cid:15) G/K E (cid:15) (cid:15) (cid:31) (cid:127) / / ϕ ∗ M (cid:15) (cid:15) ϕ ∗ M o o (cid:15) (cid:15) ∼ / / ϕ ∗ M (cid:15) (cid:15) π ( E ) $ $ ❍❍❍❍❍❍❍❍❍ ϕ ∗ I (cid:15) (cid:15) ϕ ∗ I o o (cid:15) (cid:15) / / ϕ ∗ I (cid:15) (cid:15) G/K G/K = o o = / / G/K is an equivalence of topological categories. 65 Properties of G -factorization homology In this subsection we prove two properties of G -factorization homology: it satisfies G - ⊗ -excision(proposition 5.2.3) and respects G -sequential colimits (proposition 5.3.3). G -manifolds We define G -collar decompositions of G -manifolds and construct inverse image functors (con-struction 5.1.4). In the next subsection we use these constructions to define G - ⊗ -excision andprove that G -factorization homology satisfies G - ⊗ -excision (proposition 5.2.3).We begin with an equivariant version of collar-gluing, see [AF15, def. 3.13]. The samedefinition is given in [Wee18, def. 4.20]. Definition 5.1.1. Let M ∈ Mfld G be an n -dimensional G -manifold. A G -collar decomposition of M is a smooth G -invariant function f : M → [ − , to the closed interval for which therestriction f | ( − , : M | ( − , → ( − , is a manifold fiber bundle, with a choice of trivialization M | ( − , ∼ = M × ( − , . Here M ( − , = f − ( − , , M = f − (0) . For such a decomposition,denote M + := f − ( − , , M − := f − [ − , .A G -collar decomposition of an f -framed O G -manifold M ∈ Mfld G,f − fr [ G/H ] is a G -collar de-composition of the underlying G -manifold M . Remark 5.1.2. Note that M | ( − , is a tubular neighborhood of the codimension one G -submanifold M , and that M splits M into two G -manifolds, i.e. there exists a continuous G -invariant function M \ M → [ − , \ { } → {− , } to the set with two elements. On theother hand, a G -submanifold M ⊂ M of codimension one that splits M into two G -manifoldshas an equivariant tubular neighbourhood T ⊂ M equivalent to the total space ν ( M ) of thenormal bundle of M (compatible with the M ⊂ M and the zero section M → ν ( M ) ). Byassumption, the normal bundle of M is a trivial vector bundle of rank 1 with trivial G action.A choice of G -diffeomorphisms T ∼ = ν ( M ) ∼ = M × R ∼ = M × ( − , 1) (compatible with M )determines a G -collar decomposition of M . Remark 5.1.3. A G -collar decomposition f : M → [ − , 1] defines a decomposition of M intoa union of open G -submanifolds M = M − ∪ M + with a chosen isomorphism M − ∩ M + ∼ = M × ( − , M = U ∪ V of M as a union of two open G -submanifolds. We will seethat G -equivariant homology is compatible with G -collar decompositions (definition 5.2.2 andproposition 5.2.3). This should be compared with Bredon homology, which is compatible withall decompositions M = U ∪ V into two equivariant open subsets (the equivariant Mayer-Vietorisproperty).Next we construct an “inverse image” functor f − : Mfld ∂,or/ [ − , → ( Mfld G,f − fr [ G/H ] ) /M fromthe ∞ -category of 1-dimensional oriented manifolds with boundary over the interval [ − , 1] (see[AF15]). By proposition 4.2.6 we have an equivalence of ∞ -categories( Mfld G,f − fr [ G/H ] ) /M ∼ −→ ( Mfld G [ G/H ] ) / ( M → G/H ) , (22)so it is enough to construct f − : Mfld ∂,or/ [ − , → ( Mfld G [ G/H ] ) / ( M → G/H ) for ( M → G/H ) ∈ Mfld G [ G/H ] the underlying O G -manifold of M ∈ Mfld G,f − fr [ G/H ] .Note that both the domain and the codomain of the functor f − can be described usingcoherent nerve of the Moore over categories (see appendix A), since the ∞ -categories Mfld bnd,or , Mfld G [ G/H ] ∼ = Mfld G [ G/H ] Construction 5.1.4. Let ( M → G/H ) be an O G -manifold with a collar decomposition f : M → [ − , Mfld ∂,or ) Moore / [ − , → ( Mfld G [ G/H ] ) Moore / ( M → G/H ) between theMoore over categories:1. Send an object of ( Mfld ∂,or ) Moore / [ − , given by oriented embedding ϕ : V ֒ → [ − , 1] to itsinverse image, f − ( ϕ ) : f − V ֒ → M given by the pullback of ϕ along f . Since the function f is G -invariant the embedding f − ( ϕ ) is G -equivariant. The composition f − V (cid:31) (cid:127) f − ( ϕ ) / / M / / G/H makes f − V a G -manifold over G/H , hence f − ( ϕ ) is a point in the topological space Emb GG/H ( f − V, M ), i.e an object of the Moore over category ( Mfld G [ G/H ] ) Moore / ( M → G/H ) .2. Let ϕ : V ֒ → [ − , 1] and ϕ ′ : V ′ ֒ → [ − , 1] be two objects of the Moore over category( Mfld ∂,or ) Moore / [ − , . Let ( h, ( r, γ )) be a point in Map ( Mfld ∂,or ) Moore / [ − , ( ϕ, ϕ ′ ), where h : V ֒ → V ′ is an oriented embedding and ( r, γ ) ∈ [0 , ∞ ) × (cid:0) Emb ∂,or ( V, [ − , (cid:1) [0 , ∞ ) is a Moore pathfrom ϕ to ϕ ′ ◦ h . Define a continuous function f − : Map ( Mfld ∂,or ) Moore / [ − , ( ϕ, ϕ ′ ) → Map ( Mfld G [ G/H ] ) Moore / ( M → G/H ) (cid:0) f − (Im ϕ ) ⊂ M, f − (Im ϕ ′ ) ⊂ M (cid:1) ,f − ( h, ( r, γ )) := ( f − ( h ) , ( r, α )) , f − V ❴✤ (cid:15) (cid:15) (cid:31) (cid:127) f − ( h ) / / f − ❴✤ (cid:15) (cid:15) (cid:31) (cid:127) f − ( ϕ ′ ) / / M f (cid:15) (cid:15) V (cid:31) (cid:127) ϕ / / V (cid:31) (cid:127) h / / [ − , f − ( h ) is given by the pullback f − V ❴✤ (cid:15) (cid:15) (cid:31) (cid:127) f − ( h ) / / f − ❴✤ (cid:15) (cid:15) (cid:31) (cid:127) f − ( ϕ ′ ) / / M f (cid:15) (cid:15) V (cid:31) (cid:127) ϕ / / V (cid:31) (cid:127) h / / [ − , α : [0 , ∞ ) → Emb G ( f − ( V ) , M ) is the Moore path of length r defined as follows. If x ∈ M | ( − , ∼ = M × ( − , 1) corresponds to ( y, s ) ∈ M × ( − , 1) define α t ( x ) = ( y, γ t ◦ ϕ − ( s )) ∈ M × ( − , ∼ = M | ( − , , otherwise (i.e. f ( x ) = ± 1) define α t ( x ) = x . Verification that ( r, α ) is a smooth G -equivariant isotopy depending continuously on γ is left to the reader.Clearly f − preserve disjoint unions. Remark 5.1.5. More generally, one can try to define an inverse image functor along a generalsmooth invariant map M → N to a oriented manifold with boundary N . However, not everymap f will do. First, in order to define the isotopy lift α assume that the restrictions of f to f − ( N \ ∂N ) and f − ( ∂N ) are smooth fiber bundles, and use G -equivariant parallel transport67etween the fibers. The connections on the fiber bundles need to be compatible in order for α to be continuous and smooth. However, such parallel transport defines functions which are onlycontinuous in the C -topology on Emb G ( f − V, M ), since they depend on the time derivative ofthe isotopy γ . Nevertheless, if the connections chosen are flat then parallel transport dependsonly on the end points, and therefore defines a continuous function relative to the compact-opentopology. All these conditions can be can be captured together by assuming that f : M → N isa G -invariant flat complete Riemannian submersion. This condition implies that the restrictionsto N \ ∂N and ∂N are flat fiber bundles, with compatibly chosen flat G -equivariant Ehresmannconnections (i.e. a constructible fiber bundle relative to the boundary stratification). G - ⊗ -excision We define an equivariant version of ⊗ -excision of [AF15, def. 3.15] (see definition 5.2.2), andprove it is satisfied by G -factorization homology (proposition 5.2.3).Given a G -symmetric monoidal functor F : Mfld G,f − fr → C and a G -collar decomposition ofan f -framed O G -manifold M ∈ Mfld G,f − fr [ G/H ] we construct a comparison map F ( M − ) ⊗ F ( M × ( − , F ( M + ) → F ( M ) in C [ G/H ] . This construction depends on the “inverse image” functor of con-struction 5.1.4. Construction 5.2.1. Let F : Mfld G,f − fr → C be a G -symmetric monoidal functor. Let M ∈ Mfld G,f − fr [ G/H ] with underlying O G -manifold ( M → G/H ) ∈ Mfld G [ G/H ] , and f : M → [ − , 1] a G -collar decomposition. Consider the Disk ∂,or/ [ − , -shaped diagram in C [ G/H ] given by the functor Disk ∂,or/ [ − , → Mfld ∂/ [ − , f − −−→ ( Mfld G [ G/H ] ) / ( M → G/H ) ≃ ( Mfld G,f − fr [ G/H ] ) /M F −→ ( C [ G/H ] ) /F ( M ) (23)where the first functor is the embedding of disks in manifolds followed by the functor forgettingorientation (see [AF15, def. 2.18]), the second functor is the inverse image functor defined inconstruction 5.1.4, followed by the equivalence of eq. (22), and the third functor is induced by theaction of F on the over categories. By [AF15, lem. 3.11] there is a cofinal map ∆ op → Disk ∂,or ,therefore the colimit of eq. (23) in C [ G/H ] ) /F ( M ) is given by colim −−−→ ( · · · →→→ F ( M − ) ⊗ F ( M × ( − , ⊗ F ( M + ) ⇒ F ( M − ) ⊗ F ( M + )) (cid:15) (cid:15) F ( M ) ∈ ( C [ G/H ] ) /F ( M ) , known as the two sided bar construction. Assume that C [ G/H ] admits sifted colimits and thatthe tensor product functor of C [ G/H ] preserves sifted colimits separately in each variable (i.e thecoCartesian fibration C ⊗ [ G/H ] → Fin ∗ is compatible with sifted colimits in the sense of [Lur, def.3.1.1.18]). Then the relative tensor product F ( M − ) ⊗ F ( M × ( − , F ( M + ) can be identified withthe colimit of this two sided bar construction (see [Lur, thm. 4.4.2.8]). Hence we identify thecolimit of the diagram eq. (23) with F ( M − ) ⊗ F ( M × ( − , F ( M + ) (cid:15) (cid:15) F ( M ) ∈ ( C [ G/H ] ) /F ( M ) . (24)68 efinition 5.2.2. A G -symmetric monoidal functor F : Mfld G,f − fr → C satisfies G - ⊗ -excision if for every M ∈ Mfld G,f − fr with underlying O G -manifold ( M → G/H ) together with a G -collardecomposition f : M → [ − , the morphism (24) is an equivalence in C [ G/H ] . The main result of this subsection is Proposition 5.2.3. Let A : Disk G,f − fr, ⊔ → C ⊗ be an f -framed G -disk algebra. Then the G -factorization homology functor R A : Mfld G,f − fr, ⊔ → C ⊗ of definition 4.2.3 satisfies G - ⊗ -excision. Remark 5.2.4. We view the proof of proposition 5.2.3 as an instance of “equivariant push-forward”. We conjecture that the pushforward paradigm of [AF15, sec. 3.4] and [AFT17a,sec. 2.5] has an equivariant generalization to a smooth constructible G -fiber bundle betweenequivariantly-framed O G -manifolds with boundary. However, the definition of equivariantlyframed O G -manifolds with boundary is beyond the scope of this work.Instead, we are able to prove proposition 5.2.3 without these definitions because the actionof G on the oriented manifold [ − , 1] is trivial.We could have followed a slightly more general approach, considering a G -constructible bundle M → N where N has boundary and trivial G -action. To do this, note that since G acts triviallyon N , any G -embedding V ֒ → N must have a trivial G -action as well, so the slice category of G -disks over N is a constant G -diagram. This allows us to harness the definition of (nonequivariant)framing given in [AF15] to construct a replacement for the expected “ G -slice category of f -framed G -embeddings V ֒ → N ” needed to preform pushforward. We chose not to prove thisgeneralization since we do not currently need it, and we believe it would further obfuscate theproof.In order to prove proposition 5.2.3 we need the following auxiliary construction. Construction 5.2.5. Let M → G/H be an O G -manifold and f : M → [ − , 1] be a G -collardecomposition of M . Define a G/H -category X f → G/H and G/H -functors ev : X f → Disk G,f − fr/M , ev : X f → G/H × Disk ∂,or/ [ − , by the taking the limit of the following diagram of G/H -categories. Disk G,f − fr/M Fun G/H ( G/H × ∆ , Mfld G,f − fr/M ) G/H × Disk ∂,or/ [ − , Mfld G,f − fr/M Mfld G,f − fr/M f − where f − is the inverse image functor of construction 5.1.4. Remark 5.2.6. Using proposition 4.2.6 and unwinding the definitions shows that the fiber of X f → G/H over ϕ : G/K → G/H is given by the limit of( Disk G [ G/K ] ) /ϕ ∗ M Fun(∆ , ( Disk G [ G/K ] ) /ϕ ∗ M ) Disk ∂,or/ [ − , ( Mfld G [ G/K ] ) /ϕ ∗ M ( Mfld G [ G/K ] ) /ϕ ∗ M , f − ∞ -over categories ( Disk G [ G/H ] ) /ϕ ∗ M , ( Mfld G [ G/H ] ) /ϕ ∗ M can be modeled as the coherentnerve of the Moore over category (see appendix A). Explicitly, an object of ( X f ) [ ϕ ] is given by( g : V ֒ → [ − , , h : E ֒ → ϕ ∗ M, h ′ : E ֒ → f − V, γ ) where • V is a finite disjoint union of 1-dimensional oriented disks with boundary, i.e oriented openintervals equivalent to R and oriented half open intervals equivalent to [0 , 1) or (0 , • g is an orientation preserving embedding of V into the closed interval [ − , • E → U → G/K is a finite G -disjoint union of G -disks (i.e E → U a G -vector bundle overa finite G -set), • h is a G -equivariant embedding over G/K of E into the pullback of M → G/H along ϕ , • h ′ is a G -equivariant embedding over G/K of E into the preimage f − V • γ is a Moore path in Emb G [ G/K ] ( E, ϕ ∗ M ) from h to E h ′ −→ f − V f − ( g ) −−−−→ ϕ ∗ M .The functor ev sends the object ( g : V ֒ → [ − , , h : E ֒ → ϕ ∗ M, h ′ : E ֒ → f − V, γ ) describedabove to ( h : E ֒ → ϕ ∗ M ) ∈ ( Mfld G [ G/K ] ) /ϕ ∗ M , while the functor ev sends it to ( g : V ֒ → [ − , ∈ Disk ∂,or/ [ − , .By [Lur09a, prop. 2.4.7.12] it follows that for every ϕ ∈ G/H the functor( ev ) [ ϕ ] : ( X f ) [ ϕ ] → ( Disk G [ G/H ] ) /ϕ ∗ M is a Cartesian fibration (and therefore that ev is a G/H -Cartesian fibration, see [Sha18, def.7.1]).The following lemma is the main ingredient in the proof of proposition 5.2.3. Lemma 5.2.7. The G/H -functor ev : X f → Disk G,f − fr/M is G/H -cofinal. The following proof is an adaptation of [Lur, thm. 5.5.3.6], [AF15, lem. 3.21] and [AFT17a,lem. 2.27] to the equivariant setting. Proof of lemma 5.2.7. By proposition 4.2.6 we have to prove that ev : X f → Disk G/M is G/H -cofinal. By [Sha18, thm. 6.7, def. 6.8] the G/H functor ev is G/H -cofinal if and only if for each( ϕ : G/K → G/H ) ∈ G/H the functor ( ev ) [ ϕ ] : ( X f ) [ ϕ ] → ( Disk G/M ) [ ϕ ] is cofinal.By replacing f : M → [ − , 1] with ϕ ∗ M → M f −→ [ − , 1] we reduce to ϕ = ( G/H = −→ G/H ) ∈ G/H : it is enough to prove that ( ev ) [ G/H ] is cofinal.By remark 5.2.6 the functor ( ev ) [ G/H ] is a Cartesian fibration, therefore by [Lur09a, prop.4.1.3.2] it is enough to show that for each ( E ֒ → M ) ∈ ( Disk G [ G/H ] ) /M the fiber ( ev ) − ( E ֒ → M )is weakly contractible.Note that the category ( ev ) − ( E ֒ → M ) has a functor to Disk ∂,or/ [ − , by construction:( ev ) − ( E ֒ → M ) Disk ∂,or/ [ − , ∼ = { E ֒ → M } × Disk ∂,or/ [ − , ( X f ) [ G/H ] ( Disk G [ G/H ] ) /M × Disk ∂,or/ [ − , F un (∆ , ( Mfld G [ G/H ] ) /M ) ( Mfld G [ G/H ] ) /M × ( Mfld G [ G/H ] ) /M p (( ev ) [ G/H ] , ( ev ) [ G/H ] ) p ( ι,f − ) ev ) − ( E ֒ → M ) → Disk ∂,or/ [ − , is pullback of a left fibration, sinceit can be written as( ev ) − ( E ֒ → M ) Disk ∂,or/ [ − , (cid:16) ( Mfld G [ G/H ] ) /M (cid:17) ( E֒ → M ) / ( Mfld G [ G/H ] ) /M F un (∆ , ( Mfld G [ G/H ] ) /M ) ( Mfld G [ G/H ] ) /M × ( Mfld G [ G/H ] ) /M , p f − p ( { E֒ → M } ,id ) where the middle horizontal arrow is a left fibration by [Lur09a, cor. 2.1.2.2].The left fibration ( ev ) − ( E ֒ → M ) → Disk ∂,or/ [ − , classifies the functor( Mfld G [ G/H ] ) /M → S , ( V ֒ → [ − , Map ( Mfld G [ G/H ] ) /M ( E ֒ → M, f − V ֒ → M ) , and by [Lur09a, 3.3.4.5] we have to show that the colimit colim −−−→ ( V ֒ → [ − , ∈ Disk ∂,or [ − , Map ( Mfld G [ G/H ] ) /M ( E ֒ → M, f − V ֒ → M )is weakly contractible.Let Disk ∂,or ([ − , Disk ∂,or/ [ − , andsets of morphisms given by forgetting the topology of the mapping spaces of Disk ∂,or/ [ − , (see[AF15, def. 2.8]). Note that the category Disk ∂,or ([ − , V ( [ − , 1] for V a finite disjoint union of intervals in [ − , − , 1, after excluding the whole interval [ − , ⊆ [ − , Disk ∂,or ([ − , → Disk ∂,or/ [ − , is cofinal, hence it isenough to show that the homotopy colimit hocolim −−−−−→ ( V ( [ − , ∈ Disk ∂,or ([ − , Map ( Mfld G [ G/H ] ) /M ( E ֒ → M, f − V ֒ → M )is contractible.Using observation 1 we see that the space Map ( Mfld G [ G/H ] ) /M ( E ֒ → M, f − V ֒ → M ) is thehomotopy fiber of Emb G [ G/H ] ( E, f − V ) → Emb G [ G/H ] ( E, M ), hence by [Lur09a] it is enough toshow that the map hocolim −−−−−→ ( V ( [ − , ∈ Disk ∂,or ([ − , Emb G [ G/H ] ( E, f − V ) → Emb G [ G/H ] ( E, M )is an equivalence.By [Lur09a, thm. 6.1.0.6] colimits in S are universal, therefore by corollary 3.8.6 it is enoughto prove that the map hocolim −−−−−→ ( V ( [ − , ∈ Disk ∂,or ([ − , Conf GG/H ( U ; f − V ) → Conf GG/H ( U ; M )is an equivalence. Since n Conf GG/H ( U ; f − V ) o ( V ( [ − , ∈ Disk ∂,or ([ − , is a complete open coverof Conf GG/H ( U ; M ) it follows from [DI04, cor. 1.6] that the above map is an equivalence.71e will also need a simple cofinality lemma. Assume we have a coCartesian fibration p : D ։ C between S -categories C , D (i.e. an S -coCartesian fibration, see [Sha18, rem. 7.3]), and an S -object x : S → C . Let p − ( x ) := S × C D be the pullback of p along x . Since p − ( x ) ։ S is acoCartesian fibration we can considered p − ( x ) as an S -category, which we denote by p − ( x ). Lemma 5.2.8. Let C , D be S -categories and p : D ։ C be a coCartesian fibration, and x : S → C an S -object of C . Then the S -functor p − ( x ) → D /x is S -cofinal.Proof. By [Sha18, thm. 6.7] we have to show that for each s ∈ S the functor p − ( x ) [ s ] → ( D x ) [ s ] between the fibers of p − ( x ) → D /x over s is cofinal. Since p [ s ] : D [ s ] ։ C [ s ] is a coCartesianfibration it follows that ( p − s ] ( x ( s )) → ( D [ s ] ) /x ( s ) is cofinal. The result now follows from theequivalence p − s ] ∼ = ( p [ s ] ) − ( x ( s )).With lemma 5.2.7 at hand we turn to the proof of proposition 5.2.3. The proof follows theoutline of the proof of [AF15, prop. 3.23] (“pushforward”). Proof of proposition 5.2.3. By eq. (17) and lemma 5.2.7 we have Z M A := G/H − colim −−−→ ( Disk G,f − fr/M → Disk G,f − fr × G/H A × G/H −−−−−→ C× G/H )= G/H − colim −−−→ ( X f ev −−→ Disk G,f − fr/M → Disk G,f − fr × G/H A × G/H −−−−−→ C× G/H ) . Using the characterization of parametrized Kan extensions as parametrized left adjoints (see[Sha18, thm. 10.4], and also [Nar16, def. 2.10 and def. 2.12]) we can express the above G/H -colimit as a left G/H -Kan extension of L : X f → C× G/H along the structure map X f → G/H ,where L is the G/H -functor given by the composition L : X f ev −−→ Disk G,f − fr/M → Disk G,f − fr × G/H A × G/H −−−−−→ C× G/H. (25)Equivalently the G/H -colimit over X f is given by the left G/H -adjoint to restriction along thestructure map X f → G/H , G/H − colim −−−→ : Fun G/H ( X f , C× G/H ) ⇆ Fun G/H ( G/H, C× G/H ) ≃ C× G/H. By construction ev : X f → G/H × Disk ∂,or/ [ − , is a G/H category, therefore the structure map X f → G/H factors as X f ev −−→ G/H × Disk ∂,or/ [ − , → G/H . We can now extend L along X f → G/H in two steps, again using [Sha18, thm. 10.4], as the composition of left G/H -adjoints( ev ) ! : Fun G/H ( X f , C× G/H ) Fun G/H ( G/H × Disk ∂,or/ [ − , , C× G/H ) : ( ev ) ∗ ,G/H − colim −−−→ : Fun G/H ( G/H × Disk ∂,or/ [ − , , C× G/H ) Fun G/H ( G/H, C× G/H ) ≃ C× G/H, where ( ev ) ! is the left G/H -Kan extension of (25) along ev . In particular restricting to fibersover ( G/H = ←− G/H ) ∈ G/H we get composition of unparametrized left adjoints (see [Lur, prop.7.3.2.6] and [Sha18, def. 8.1]):( ev ) ! : Fun G/H ( X f , C× G/H ) Fun G/H ( G/H × Disk ∂,or/ [ − , , C× G/H ) : ( ev ) ∗ , /H − colim −−−→ : Fun G/H ( G/H × Disk ∂,or/ [ − , , C× G/H ) Fun G/H ( G/H, C× G/H ) ≃ C [ G/H ] . Applying both left adjoints to the G/H -functor L : X f → C× G/H of (25) produces the G -factorization homology R M A = G/H − colim −−−→ ( L : X f → C× G/H ). Let L ′ := ( ev ) ! ( L ) ∈ Fun G/H ( G/H × Disk ∂,or/ [ − , , C× G/H ) be the left G/H -Kan extension of L along ev . Thenthe G/H -colimit of L ′ is G/H − colim −−−→ ( L ′ ) = G/H − colim −−−→ (( ev ) ! ( L )) ≃ G/H − colim −−−→ ( L ) = Z M A. Next, note that the G/H -colimit over the constant diagram G/H × Disk ∂,or/ [ − , is equivalentto an unparametrized colimit over Disk ∂,or/ [ − , . To see this, use the equivalenceFun G/H ( G/H × Disk ∂,or/ [ − , , C× G/H ) ∼ −→ Fun( Disk ∂,or/ [ − , , C [ G/H ] ) ,L ′ L ′ | n G/H = ←− G/H o × Disk ∂,or/ [ − , and the global definition of a colimit as a left adjoint colim −−−→ : Fun( Disk ∂,or/ [ − , , C [ G/H ] ) ⇆ C [ G/H ] . Therefore, we have Z M A ≃ G/H − colim −−−→ ( L ′ ) ≃ colim −−−→ (cid:18) L ′ | n G/H = ←− G/H o × Disk ∂,or/ [ − , (cid:19) , which we write as Z M A ≃ colim −−−→ ( V ֒ → [ − , ∈ Disk ∂,or/ [ − , L ′ ( G/H = −→ G/H, V ֒ → [ − , . Out next goal is to calculate L ′ ( y ) for y = ( G/H = −→ G/H, V ֒ → [ − , V ֒ → [ − , ∈ Disk ∂,or/ [ − , is an oriented embedding. We claim that L ′ ( y ) ≃ R f − V A ∈ C [ G/H ] isthe G -factorization homology of f − ( V ) ∈ Mfld G,f − fr [ G/H ] . After asserting our claim we use thecofinal map ∆ op → Disk ∂,or/ [ − , of [AF15, lem. 3.11] to deduce R M A is equivalent to the colimitof the simplicial diagram ∆ op → n G/H = ←− G/H o × Disk ∂,or/ [ − , L ′ −→ C [ G/H ] . Since the functor L ′ ( V ֒ → [ − , ≃ R f − V A takes disjoint unions over [ − , 1] to tensor product in C [ G/H ] (seeproposition 4.2.2), and using the equivalence of oriented open and half open intervals over [ − , Disk ∂,or/ [ − , ) we see that R M A ∈ C [ G/H ] is equivalent to the realization of the twosided bar construction · · · →→→ Z M − A ⊗ Z M × ( − , A ⊗ Z M − A ⇒ Z M − A ⊗ Z M − A, [ n ] Z M − A ! ⊗ Z M × ( − , A ! ⊗ n ⊗ Z M − A ! . By [Lur, 4.4.2.8-11] we see that G -factorization homology of M is equivalent to the relative tensorproduct R M A ≃ ( R M − A ) ⊗ ( R M × ( − , A ) ( R M − A ). Therefore it is enough to prove our claim that L ′ ( y ) ≃ R f − V A ∈ C [ G/H ] . 73ince L ′ is the left G/H -Kan extension of L : X f → C× G/H along ev , it is given by thefollowing G/H -colimit L ′ ( y ) = G/H − colim −−−→ (cid:16) ( X f ) /y → X f L −→ C× G/H (cid:17) = G/H − colim −−−→ (cid:18) ( X f ) /y → X f ev −−→ Disk G,f − fr/M → Disk G,f − fr × G/H A × G/H −−−−−→ C× G/H (cid:19) Next we replace the G/H -category X f indexing the above colimit by a G/H -category which ismore closely related to G -disks in f − V . Note that ev : X f → G/H × Disk ∂,or is a coCartesianfibration, and let ( ev ) − ( y ) denote the pullback of X f along the G/H -functor G/H → G/H × Disk ∂,or , ( G/K → G/H ) ( G/K → G/H, V ֒ → [ − , y = ( G/H = −→ G/H, V ֒ → [ − , ∈ G/H × Disk ∂,or . By lemma 5.2.8 the G/H -functor ( ev ) − ( y ) → ( X f ) /y is G/H -cofinal, hence L ′ ( y ) is the G/H -colimit of the G/H -diagram( ev ) − ( y ) → ( X f ) /y → X f ev −−→ Disk G,f − fr/M → Disk G,f − fr × G/H A × G/H −−−−−→ C× G/H. Since ev : X f → G/H × Disk ∂,or factors through ( Disk G,f − fr/M ) × G/H (cid:16) G/H × Disk ∂,or/ [ − , (cid:17) wecan express ( ev ) − ( y ) as the iterative pullback( ev ) − ( y ) X f Disk G,f − fr/M × G/H G/H ( Disk G,f − fr/M ) × G/H (cid:16) G/H × Disk ∂,or/ [ − , (cid:17) G/H − y G/H × Disk ∂,or/ [ − , . p ( ev ,ev ) p id × y On the other hand we can express X f is the pullback X f Fun G/H ( G/H × ∆ , Mfld G,f − fr/M )( Disk G,f − fr/M ) × G/H (cid:16) G/H × Disk ∂,or/ [ − , (cid:17) Mfld G,f − fr/M × G/H Mfld G,f − fr/M . p ( ev ,ev ) ι × f − Notice that the composition Disk G,f − fr/M × G/H G/H ( Disk G,f − fr/M ) × G/H (cid:16) G/H × Disk ∂,or/ [ − , (cid:17) Mfld G,f − fr/M × G/H Mfld G,f − fr/Mid × yι × f − 74s equivalent to Disk G,f − fr/M × G/H G/H ( ι,f − ( y )) −−−−−−→ Mfld G,f − fr/M × Mfld G,f − fr/M , and therefore that ( ev ) − ( y ) ∼ = (cid:16) Disk G,f − fr/M (cid:17) / ( f − V ֒ → M ) ≃ Disk G,f − fr/f − V (compare [AF15, lem. 2.1]). Finally, since the diagram( ev ) − ( y ) ≃ (cid:15) (cid:15) / / ( X f ) /y / / X f (cid:15) (cid:15) Disk G,f − fr/f − V / / Disk G,f − fr/M / / Disk G,f − fr × G/H A × G/H / / C× G/H commutes, we get L ′ ( y ) ≃ G/H − colim −−−→ (cid:18) Disk G,f − fr/f − V → Disk G,f − fr × G/H A × G/H −−−−−→ C× G/H (cid:19) . Therefore, by the definition of left G/H -Kan extension we see that indeed L ′ ( y ) ≃ R f − V A . G -sequential unions Definition 5.3.1. Let M be a G -manifold. A G -sequential union of M is a sequence of open G -submanifolds M ⊂ M ⊂ · · · ⊂ M with M = ∪ ∞ i =1 M i . A G -sequential union of an f -framed O G -manifold M ∈ Mfld G,f − fr [ G/H ] is a G -sequential union of its underlying G -manifold. If F : Mfld G,f − fr → C is a G -symmetric monoidal functor and M = ∪ ∞ i =1 M i is a G -sequentialunion of M ∈ Mfld G,f − fr [ G/H ] , then we have a comparison morphism colim −−−→ F ( M i ) → F ( M ) in C [ G/H ] . Definition 5.3.2. We say that G -symmetric monoidal functor F : Mfld G,f − fr → C respects G -sequential unions if for every G -sequential union M = ∪ ∞ i =1 M i the comparison morphism colim −−−→ F ( M i ) → F ( M ) is an equivalence in C [ G/H ] . Proposition 5.3.3. Let C ⊗ ։ Fin G ∗ be a G -symmetric monoidal G -category and A be an f -framed G -disk algebra with values in C . Then G -factorization homology R − A : Mfld G,f − fr → C of definition 4.2.3 respects G -sequential unions. The proof of proposition 5.3.3 relies on the following lemma. Lemma 5.3.4. Let M ∈ Mfld G,f − fr [ G/H ] be an f -framed O G -manifold over G/H , and M = ∪ ∞ i =1 M i a G -sequential union of M . Then the G/H -functor colim −−−→ Disk G,f − fr/M i ∼ −→ Disk G,f − fr/M is anequivalence of G/H -categories. roof. By proposition 4.2.6 it is enough to prove that the G/H -functor colim −−−→ Disk G/M i → Disk G/M is a fiberwise equivalence. Without loss of generality we show that the functor be-tween the fiber over ( G/H = ←− G/H ) ∈ G/H is an equivalence. Since colimits of G -categoriesare computed fiberwise, we have to show that colim −−−→ i ( Disk G [ G/H ] ) /M i → ( Disk G [ G/H ] ) /M is anequivalence of ∞ -categories.In order to show that this functor is fully faithful we first show that Emb GG/H ( E, M ) isequivalent to the homotopy colimit hocolim −−−−−→ i Emb GG/H ( E, M i ). Let ( E → U → G/H ) ∈ Disk G bea finite G -disjoint union of G -disks, i.e. E → U a G -vector bundle, U = π E . By corollary 3.8.6the square Emb GG/H ( E, M i ) (cid:15) (cid:15) / / Emb GG/H ( E, M ) (cid:15) (cid:15) Conf GG/H ( U ; M i ) / / Conf GG/H ( U ; M )is a homotopy pullback square for each i ∈ N . Since filtered homotopy colimits preserves homo-topy pullbacks, the square hocolim −−−−−→ i Emb GG/H ( E, M i ) (cid:15) (cid:15) / / Emb GG/H ( E, M ) (cid:15) (cid:15) hocolim −−−−−→ i Conf GG/H ( U ; M i ) / / Conf GG/H ( U ; M )is also a homotopy pullback square. However, n Conf GG/H ( U ; M i ) o i ∈ N is a complete open coverof Conf GG/H ( U ; M ), so by [DI04, cor. 1.6] the bottom map is a weak equivalence. Therefore themap hocolim −−−−−→ i Emb GG/H ( E, M i ) ∼ −→ Emb GG/H ( E, M ) is a weak equivalence.We now show that colim −−−→ i ( Disk G [ G/H ] ) /M i → ( Disk G [ G/H ] ) /M is fully faithful.Let ( E ′ → U ′ → G/H ) , ( E ′′ → U ′′ → G/H ) ∈ Disk G [ G/H ] and f ′ : E ′ ֒ → M i ′ f ′′ : E ′′ ֒ → M i ′′ )be two G -embeddings over G/H , representing two objects in colim −−−→ i ( Disk G [ G/H ] ) /M i . For i greater then i ′ and i ′′ they represent objects of the same slice category( f ′ i : E ′ f ′ −→ M i ′ ⊆ M i ) , ( f ′′ i : E ′′ f ′′ −−→ M i ′′ ⊆ M i ) ∈ ( Disk G [ G/H ] ) /M i , with mapping space Map ( Disk G [ G/H ] ) /Mi ( f ′ i : E ′ ֒ → M i , f ′′ i : E ′′ ֒ → M i ) given by the homotopy fiberof ( f ′′ i ) ∗ : Emb GG/H ( E ′ , E ′′ ) → Emb GG/H ( E ′ , M i ) over f ′ i ∈ Emb GG/H ( E ′ , M i ). Homotopy fibersare preserved by filtered homotopy colimits, so the homotopy fiber of the map Emb GG/H ( E ′ , E ′′ ) → hocolim −−−−−→ i Emb GG/H ( E ′ , M i ) ≃ Emb GG/H ( E ′ , M )induced by post composition with f ′′ : E ′′ ֒ → M i ⊂ M over f ′ : E ′ ֒ → M i ⊆ M is equivalentto hocolim −−−−−→ i Map ( Disk G [ G/H ] ) /Mi ( f ′ i : E ′ ֒ → M i , f ′′ i : E ′′ ֒ → M i ). On the other hand, this homotopyfiber is equivalent to the mapping space of the slice category ( Disk G [ G/H ] ) /M , hence colim −−−→ i Map ( Disk G [ G/H ] ) /Mi ( f ′ i : E ′ ֒ → M i , f ′′ i : E ′′ ֒ → M i )76s homotopy equivalent toMap ( Disk G [ G/H ] ) /M ( f ′ : E ′ ֒ → M, f ′′ : E ′′ ֒ → M ) , so the functor colim −−−→ i ( Disk G [ G/H ] ) /M i → ( Disk G [ G/H ] ) /M is fully faithful.It remains to show that colim −−−→ i ( Disk G [ G/H ] ) /M i → ( Disk G [ G/H ] ) /M is essentially surjective.Let ( E → U → G/H ) ∈ Disk G [ G/H ] , ( f : E ֒ → M ) ∈ ( Disk G [ G/H ] ) /M for E → U a G -vectorbundle. Choose t > f to the open ball of radius t bundle, B t ( E ) ֒ → E f −→ M , factors through some M i ⊆ M . By radial dilation we see that theinclusion ( B t ( E ) → G/H ) → ( E → G/H ) is an equivalence in Disk G [ G/H ] . Postcomposition with f : E ֒ → M induces an equivalence ( f : E ֒ → M ) ≃ ( B t ( E ) ֒ → E f −→ M ) of objects in the slicecategory ( Disk G [ G/H ] ) /M . On the other hand, since ( B t ( E ) ֒ → E f −→ M ) factors through M i thisobject is clearly in the image of the functor colim −−−→ i ( Disk G [ G/H ] ) /M i → ( Disk G [ G/H ] ) /M , showingthe functor is indeed essentially surjective.We now show that G -factorization homotopy respects sequential colimits. Proof of proposition 5.3.3. Let M ∈ Mfld G,f − fr [ G/H ] be an f -framed O G -manifold and M = ∪ ∞ i =1 M i a G -sequential union of M . The assembly map colim −−−→ i R M i A → R M A factors as a sequence ofequivalences colim −−−→ i Z M i A = colim −−−→ i (cid:18) G/H − colim −−−→ (cid:18) Disk G,f − fr/M i → Disk G,f − fr × G/H A × id −−−→ C× G/H (cid:19)(cid:19) ≃ G/H − colim −−−→ (cid:18) colim −−−→ i (cid:18) Disk G,f − fr/M i → Disk G,f − fr × G/H A × id −−−→ C× G/H (cid:19)(cid:19) ∼ −→ G/H − colim −−−→ (cid:18) Disk G,f − fr/M → Disk G,f − fr × G/H A × id −−−→ C× G/H (cid:19) = Z M A, where the second equivalence is induced by the equivalence colim −−−→ Disk G,f − fr/M i ∼ −→ Disk G,f − fr/M of lemma 5.3.4. G -factorization homologytheories In this subsection we give an axiomatic characterization of G -factorization homology theorieswith values in a presentable G -symmetric monoidal G -category (definition 4.2.1), as G -symmetricmonoidal functors that satisfy G - ⊗ -excision (definition 5.2.2) and respects G -sequential unions(definition 5.3.2). Definition 6.0.1. Let C ⊗ ։ Fin G ∗ be a G -symmetric monoidal category and B → BO n ( G ) a G -map, as in definition 3.3.1. An equivariant homology theory of G -manifolds is a G -symmetricmonoidal functor F : Mfld G,f − fr, ⊔ → C ⊗ which satisfies G - ⊗ -excision and respects G -sequentialunions. We denote the full subcategory of equivariant homology theories by H ( Mfld G,f − fr , C ) ⊂ Fun ⊗ G ( Mfld G,f − fr , C ) . The main result in this subsection is the following characterization of G -factorization homol-ogy. 77 heorem 6.0.2. Let C ⊗ ։ Fin G ∗ be a presentable G -symmetric monoidal category. Then thefull subcategory H ( Mfld G,f − fr , C ) ⊂ Fun ⊗ G ( Mfld G,f − fr , C ) is spanned by objects for which thecounit map of the adjunction ( ι ⊗ ) ! : Fun ⊗ G ( Disk G,f − fr , C ) / / Fun ⊗ G ( Mfld G,f − fr , C ) : ( ι ⊗ ) ∗ o o of (19) is an equivalence. In particular, the adjunction restricts to an equivalence ( ι ⊗ ) ! : Fun ⊗ G ( Disk G,f − fr , C ) ∼ −→ H ( Mfld G,f − fr , C ) , A Z − A sending an f -framed G -disk algebra A to G -factorization homology with coefficients in A .Proof. Let A be a G -disk algebra. By proposition 5.2.3 and proposition 5.3.3 the functor( ι ⊗ ) ! : Fun ⊗ G ( Disk G,f − fr , C ) → Fun ⊗ G ( Mfld G,f − fr , C )factors though the full G -subcategory H ( Mfld G,f − fr , C ) ⊂ Fun ⊗ G ( Mfld G,f − fr , C ).On the other hand, let F ∈ H ( Mfld G,f − fr , C ) be an equivariant homology theory of G -manifolds. Denote by A : Disk G,f − fr, ⊔ → C ⊗ the restriction of F along ι ⊗ . We have to showthat the counit R − A → F is an equivalence. Since F, R − A are G -symmetric monoidal functorsit is enough to show that for every f -framed O G -manifold M ∈ Mfld G,f − fr the counit map R M A → F ( M ) is an equivalence in C . We proceed by induction.For k = 0 , , . . . , n let F ≤ k ⊆ Mfld G,f − fr be the full G -subcategory of f -framed O G -manifoldswhose underlying O G -manifold is of the form ( M × G/H D → G/H ) where G/H ∈ O G is a G -orbit, M → G/H is a k -dimensional O G -manifold and ( D → G/H ) is a finite G -disjoint unionof ( n − k )-dimensional G -disks, i.e. equivalent to D → U → G/H where U is a finite G -set and D → U is a G -vector bundle of rank n − k (and therefore U = π ( D )).We now prove that the counit map is an equivalence on objects of F ≤ k by induction on k .For k = 0 the underlying O G -manifold of M ∈ F ≤ is simply a finite G -disjoint union of G -disks, ( D → G/H ) ∈ Disk G , therefore M ∈ Disk G,f − fr and R M A ≃ A ( M ) = F ( M ), since ι ! is fully faithful A is the restriction of F along ι .For k ≥ 1, let N ∈ F ≤ k with underlying O G -manifold ( M × G/H D → G/H ). We showthat the counit map R N A → F ( N ) is an equivalence using equivariant Morse theory. In whatfollows we only consider G -submanifolds of M × G/H D → G/H , which by proposition 4.2.6have an essentially unique f -framing induced from the inclusion into N . Therefore we omit theidentification of such G -submanifolds with their f -framed lift to Mfld G,f − fr .Choose a G -equivariant Morse function f : M → R with f − ( −∞ , r ] a compact G -submanifoldfor every r ∈ R (see [Was69, thm. 4.10]). Choose an increasing sequence of regular values r < r < r < · · · such that f − ( −∞ , r ) = ∅ , the interval ( r i , r i +1 ) contains a single criticalvalue and r i → ∞ .Let M i := f − (( −∞ , r i )) , then M = ∞ S i =0 M i and therefore M × G/H D ∼ = (cid:18) ∞ S i =0 M i (cid:19) × G/H D ∼ = ∞ S i =0 (cid:0) M i × G/H D (cid:1) is a G -sequential union of ( M × G/H D → G/H ) (definition 5.3.1). Sinceboth F ∈ H ( Mfld G,f − fr , C ) and R − A respect G -sequential unions (definition 5.3.2 and propo-sition 5.3.3) we have F ( M × G/H D ) ≃ colim −−−→ F ( M i × G/H D ) , R M × G/H D A ≃ colim −−−→ R M i × G/H D A .Therefore it is enough to prove that the counit map R M i × G/H D A → F ( M i × G/H D ) is an equiv-alence, which we prove by induction on i . 78et M i := f − ( −∞ , r i ]. Since M i is compact M i +1 \ M i has only a finite number of criticalorbits, x j : W j ֒ → M, j = 1 , . . . , s . Note that the tangent bundle T x j M → W j over the criticalorbit x j is a G -vector bundle which decomposes as a direct sum of two G -bundles T x j ∼ = P j ⊕ E j on which the Hessian is negative definite (called the index E j ) and positive definite (called theco-index P j ).By [Was69, thm. 4.6] M i +1 is equivariantly diffeomorphic to M i with s handle-bundles N , . . . , N s disjointly attached, where the handle-bundle N j := D ( P j ) × W j D ( E j ) is the fiberwiseproduct of the closed unit disk bundles D ( P j ) → W j , D ( E j ) → W j , attached to M i along D ( P j ) × W j S ( E j ) where S ( E j ) → W j is the unit sphere bundle of the negative definite G -subbundle (the index).Since the handle-bundles are attached disjointly and F, R − A are G -symmetric monoidal wecan reduce to the case of a single handle-bundle by attaching one handle-bundle at a time.Therefore we assume that there is a single critical orbit x : W ֒ → M in M i +1 \ M i with T x M ∼ = P ⊕ E , and M i +1 ∼ = M i S D ( P ) × W S ( E ) (cid:0) D ( P ) × W D ( E ) (cid:1) .Let A ( E ) → W denote the unit annulus bundle of E , i.e. the open unit disk bundle minusthe zero section. Note that A ( E ) is a G -tubular neighbourhood of S ( E ), therefore M i +1 ∼ = M i S D ( P ) × W A ( E ) (cid:0) D ( P ) × W A ( E ) (cid:1) a union of k -dimensional G -manifolds with boundary along a k -dimensional manifold with boundary.Discarding boundary points we see that the M i +1 is equivariantly diffeomorphic to the unionof M i with the G -manifold D ( P ) × W D ( E ) along the G -manifold D ( P ) × W A ( E ). After takingfibered product with the fibration map D → G/H we have M i +1 × G/H D ∼ = ( M i × G/H D ) [ ( ( D ( P ) × W A ( E )) × G/H D ) (cid:0) ( D ( P ) × W D ( E )) × G/H D (cid:1) . (26)This decomposition has the following properties:1. The decomposition of eq. (26) is in fact a G -collar decomposition. Intuitively, the codi-mension one G -submanifold ( D ( P ) × W S ( E )) × G/H D splits the handle bundle of eq. (26)to two G -submanifolds, M i and the handle bundle. Explicitly, construct a G -collar decom-position by defining a G -invariant smooth G -invariant function M i +1 → [ − , 1] for whichthe restriction to the open interval ( − , 1) is a manifold bundle as follows. Compose the G -diffeomorphism of eq. (26) with the restriction of the Morse function f : M → R to thehandle-bundle of eq. (26), followed by a smooth function Ψ : R → [ − , 1] such that(a) it sends the closed interval ( −∞ , a + ǫ ] to − ǫ > c − ǫ, ∞ ) to 1 for c the unique critical value of f in the interval [ a, b ].(c) it has a positive derivative in the open interval ( a + ǫ, c − ǫ ).Note that the fibers of M i +1 → [ − , 1] over ( − , 1) are (cid:0) ( D ( P ) × W S ( E, r )) × G/H D (cid:1) where S ( E, r ) is the radius- r -sphere bundle, for various radii r .2. The induced handle-bundle (cid:0) ( D ( P ) × W D ( E )) × G/H D → G/H (cid:1) ∈ Disk G is a finite G -disjoint union of G -disks, since the open unit disk bundle of a G -vector bundle is equivalentto the entire vector bundle.We now distinguish between two cases, according to the rank of the bundle E → W .1. If the critical orbit x has zero index, i.e. the Hessian is positive definite on T x M , then E → W is a rank zero G -vector bundle, and its unit annulus A ( E ) = ∅ is empty. In this79ase the G -collar decomposition of eq. (26) is a disjoint union M i +1 × G/H D ∼ = ( M i × G/H D ) ⊔ (cid:0) ( D ( P ) × W D ( E )) × G/H D (cid:1) . Since F, R − A are G -symmetric monoidal functors we have Z M i +1 A ≃ (cid:18)Z M i A (cid:19) ⊗ Z ( ( D ( P ) × W D ( E )) × G/H D ) A ! ,F ( M i +1 ) ≃ F ( M i ) ⊗ F (cid:0) ( D ( P ) × W D ( E )) × G/H D (cid:1) where (cid:0) ( D ( P ) × W D ( E )) × G/H D (cid:1) ≃ (cid:0) ( P × W E ) × G/H D (cid:1) ∈ Disk G is a finite G -disjointunion of G -disks. Therefore R M i +1 A ∼ −→ F ( M i +1 ) by induction on i .2. Otherwise the critical orbit x has positive index, i.e. rank( E ) > 0. In this case, A ( E ) ∼ = S ( E ) × ( − , 1) where G acts trivially on the open interval ( − , f is G -invariant. It follows that( D ( P ) × W A ( E )) × G/H D ∼ = A ( E ) × W ( P × G/H D ) ∼ = S ( E ) × W (( − , × P × G/H D ) , hence ( S ( E ) → W → G/H ) is a G -manifold of dimensiondim S ( E ) = rank( E ) − ≤ dim M − k − , so we have ( D ( P ) × W A ( E )) × G/H D ∈ F k − . It follows by induction on k that the counitmap R ( D ( P ) × W A ( E )) × G/H D A ∼ −→ F (( D ( P ) × W A ( E )) × G/H D ) is an equivalence.The G -functor R − A satisfies G - ⊗ -excision by proposition 5.2.3 and F satisfies G - ⊗ -excisionby assumption, therefore applying F, R − A to the G -collar decomposition of eq. (26) we get F ( M i × G/H D ) ⊗ F ( ( D ( P ) × W A ( E )) × G/H D ) F (cid:0) ( D ( P ) × W D ( E )) × G/H D (cid:1) ∼ −→ F ( M i +1 ) , Z ( M i × G/H D ) A ! ⊗ (cid:18)R ( D ( P ) × W A ( E )) × G/H D A (cid:19) Z ( D ( P ) × W D ( E )) × G/H D ! A ∼ −→ Z M i +1 A and by induction on i the map R M i +1 A ∼ −→ F ( M i +1 ) is an equivalence. As an application of the G - ⊗ -excision property (proposition 5.2.3) we describe two variants oftopological Hochschild homology using G -factorization homology. G -factorization homology Let C denote the cyclic group of order two and let σ be its one dimensional sign representation. The structure of an E σ -algebra in Sp C . Let us first describe the algebraic structure of an E σ -algebra A in Sp C . We will use this description in the proof of proposition 7.1.1.80 By corollary 3.9.9 we have an equivalence Alg E σ ( Sp C ) ≃ Fun ⊗ G ( Disk C ,σ − fr , Sp C ) , so A corresponds to a C -symmetric monoidal functor A : Disk C ,σ − fr → Sp C . In par-ticular the G -symmetric monoidal functor A restricts to symmetric monoidal functors A [ C /C ] : Disk C ,σ − fr [ C /C ] → Sp C , A [ C /C ] : Disk C ,σ − fr [ C /e ] → Sp . (27) • By abuse of notation, we use A to denote the “underlying” genuine C -spectrum, A [ C /C ] ( R σ ) ∈ Sp C , where R σ ∈ Disk C ,σ − fr is the one dimensional sign representation of C , considered as a σ -framed C -manifold. • Unwinding the definitions we see that Disk C ,σ − fr [ C /e ] is equivalent to the ∞ -category Disk fr of [AF15, rem 2.10]. Since A : Disk C ,σ → Sp C is a G -functor it is compatible with theforgetful functors Res C e : Disk C ,σ [ C /C ] → Disk C ,σ [ C /e ] ≃ Disk fr , Res C e : Sp C → Sp , therefore A [ C /e ] ( R ) = A [ C /e ] ( Res C e R σ ) ≃ Res C e A [ C /C ] ( R σ ) = Res C e A . • Observe that Res C e A is endowed with a structure of an E -sing spectrum. To see this,recall that R ∈ Disk fr is an E -algebra in Disk fr , which induces an equivalence betweenthe symmetric monoidal envelope of E and Disk fr (see [AFT17a, prop. 2.12]). • Let ⊔ C R ∈ Disk C ,σ − fr [ C /C ] , ⊔ C R = C × R denote the topological induction of R ∈ Disk fr . The compatibility of the G -symmetricmonoidal functor A : Disk C ,σ − fr → Sp C with with topological induction and the Hopkins-Hill-Ravenel norm, ⊔ C : Disk fr ≃ Disk C ,σ [ C /C ] → Disk C ,σ [ C /e ] , N C e : Sp → Sp C , implies that A [ C /C ] ( ⊔ C R ) ≃ N C e A . • Note that N C e A is an E -algebra in Sp C , since N C e : Sp → Sp C is a symmetric monoidalfunctor and Res C e A is an E -ring spectrum. • The “underlying” C -spectrum A has the structure of a module over N C A . To see thisstructure, note that an equivariant oriented embeddings (cid:0) ⊔ C R (cid:1) ⊔ R σ ֒ → R σ induces a map N C e A ⊗ A → A. Proposition 7.1.1. For A an E σ -algebra in Sp C there is an equivalence of genuine C -spectra Z S A ≃ A ⊗ N C e A A. where C acts on S by reflection. roof. Consider the C -collar gluing S = R σ ∪ ⊔ C R R σ into two hemispheres, where eachhemisphere is reflected onto itself by the action of C . Note that the intersection ⊔ C R consists oftwo segments interchanged by the action of C . Applying proposition 5.2.3 we get an equivalenceof genuine C -spectra Z S A ≃ (cid:18)Z R σ A (cid:19) ⊗ (cid:18)R ⊔ C R A (cid:19) (cid:18)Z R σ A (cid:19) ≃ A ⊗ N C e A A. Remark 7.1.2. The tensor product A ⊗ N C e A A appearing in proposition 7.1.1 is equivalent tothe derived smash product A ∧ L N C e A A of left and right N C e -modules. Dotto, Moi, Patchkoriaand Reeh ([DMPR17]) show that for A a flat ring spectrum with anti-involution there is a stableequivalence of genuine C -spectra T HR ( A ) ≃ A ∧ LN C e A A, where T HR ( A ) is the B¨okstedt model for real topological Hochschild homology.By [DMPR17, def. 2.1] we can interpret a ring spectrum with anti-involution as an algebraover an operad Ass σ in C -sets. Direct inspection shows Ass σ is equivalent to G -operad D σ ofthe little σ -disks , whose genuine operadic nerve is E σ . Regarding a flat ring spectrum withanti-involution A as an E σ -algebra in Sp C , we can reinterpret proposition 7.1.1 as an equivalence Z S A ≃ T HR ( A )of genuine C -spectra. C n -ring spec-tra We start with a general lemma relating trivially framed G -disk algebras to E n -algebras. Let G be a finite group acting trivially on R n , and Mfld G, R n − fr the G -category of trivially framed G -manifolds. Lemma 7.2.1. Let C ⊗ ։ Fin G ∗ be a G -symmetric monoidal ∞ -category. The ∞ -category Fun ⊗ G ( Disk G, R n − fr , C ) of trivially framed G -disk algebras in C is equivalent to the ∞ -category Alg E n ( C [ G/G ] ) of E n -algebras in the fiber C [ G/G ] . The structure of a trivially framed C n -disk algebra Let C n the cyclic group of order n and C = Sp C n , the C n - ∞ -category of genuine C n -spectra. We will use the following anexplicit description of the trivially framed C n -disk algebra corresponding to A . The C n -functor A : Disk C n , R n − fr, ⊔ → Sp C n sends ∀ H < C n : A [ C n /H ] : ⊔ C n /H R N C n H ( A ) ∈ Sp H , where N C n H ( A ) denotes the Hill-Hopkins-Ravenel norm applied to the restriction of the genuine C n -spectrum A ∈ Sp C n to Sp H . In particular, A : Disk C n , R n − fr, ⊔ → Sp C n sends R n with trivial C n -action to A ∈ Sp C n and the topological induction ⊔ C n R = C n × R ∈ to N C n e ( A ) ∈ Sp C n . This also follows from a direct analysis of the mapping spaces of Rep C ,σ − fr, ⊔ , which are homotopicallydiscrete. 82e will need some notation for our next statement. Let A be an E n -ring spectrum in Sp C n .Define an A − A op -bimodule structure on A ∈ Sp C n with “twisted” left multiplication, given byfirst acting on the scalar by the generator τ ∈ C n : A ⊗ A τ ⊗ A → A τ , x ⊗ a ⊗ y τ x · a · y. We denote this “twisted” A − A -bimodule by A τ . Let T HH ( A ; A τ ) denote the topologicalHochschild homology of A with coefficients in A τ . Proposition 7.2.2. Let A be an E -ring spectrum in Sp C n , and C n y S be the standardaction. Then there exists an equivalence of spectra (cid:18)Z S A (cid:19) Φ C n ≃ T HH ( A ; A τ ) . In particular, T HH ( A ; A τ ) admits a natural circle action.Proof. Consider S as the n -fold covering space p : S → S , with the standard C n -action givenby deck transformations. Let S = U ∪ U ∩ V V be the standard collar decomposition of the base S by hemispheres. Construct a C n -collar decomposition S = p − ( U ) ∪ p − ( U ∩ V ) p − ( V ) of thecovering space by taking preimages. Observe that the pieces of this C n -collar decomposition aregiven by topological induction, p − ( U ) = ⊔ C n U ∼ = ⊔ C n R , p − ( V ) = ⊔ C n V ∼ = ⊔ C n R ,p − ( U ∩ V ) = ⊔ C n ( U ∩ V ) ∼ = ⊔ C n ( R ⊔ R ) = ( ⊔ C n R ) ⊔ ( ⊔ C n R ) . Therefore by C n - ⊗ -excision Z S A ≃ Z p − ( U ) A ! O R p − U ∩ V ) A Z p − ( V ) A ! ≃ Z ⊔ Cn R A ! O R ( ⊔ Cn R ⊔ ( ⊔ Cn R A Z ⊔ Cn R A ! ≃ ( N C n e A ) O ( N Cne A ) ⊗ ( N Cne A ) op ( N C n e A ) τ . Let us pause and explain the superscript decorations in the last term. The (cid:16)R p − ( U ∩ V ) A (cid:17) -module structure of R p − ( U ) A is induced by the inclusion p − ( U ∩ V ) ֒ → p − ( U ). When weidentify p − ( U ∩ V ) ∼ = ( ⊔ C n R ) ⊔ ( ⊔ C n R ) the module structure on R p − ( U ) A is naturally identifiedwith an ( N C n e A ) − ( N C n e A )-bimodule structure, or equivalently a right ( N C n e A ) ⊗ ( N C n e A ) op -module structure. Similarly, R p − ( V ) A is naturally a left N C n e ( A ) − N C n e ( A ) op -module. Howeverthe left module structure is induced by an embedding ⊔ C n R ֒ → p − V which defers from thestandard embedding (the topological induction of R ֒ → V ) by a deck transformation. Thereforethe left multiplication is “twisted”, i.e. given by first acting on the scalar by the generator τ ∈ C n . In order to remember this twist in the module structure of the right hand side we addthe superscript τ .Next we take geometric fixed points of R S A . Since the geometric fixed points functor( − ) Φ C n : Sp C n → Sp is symmetric monoidal and preserve homotopy colimits (cid:18)Z S A (cid:19) Φ C n ≃ ( N C n e A ) Φ C n O ( N Cne A ) Φ Cn ⊗ (( N Cne A ) op ) Φ Cn (( N C n e A ) τ ) Φ C n ≃ A O A ⊗ A op A τ . T HH ( A ; A τ ) of A ∈ Sp with coefficients in the A − A -bimodule A τ .Finally, we describe the natural circle action on (cid:0)R S A (cid:1) Φ C n . Note that the automorphismspace of S ∈ Mfld C n , − fr [ C n /C n ] acts on S , so by functoriality it induces a natural action on (cid:0)R S A (cid:1) Φ C n . The endomorphism space of S ∈ Mfld C n , − fr [ C n /C n ] is the space of C n -equivariantoriented embeddings Emb C n ( S , S ). In particular the endomorphism space S ∈ Mfld C n , − fr [ C n /C n ] includes rotations of S , therefore the circle group acts on R S A by rotations, and by functorialityon (cid:0)R S A (cid:1) Φ C n . Remark 7.2.3. The inclusion of the circle group into Emb C n ( S , S ) is in fact a deformationretract. Remark 7.2.4. This theorem can be seen as an instance of a more general principle: factoriza-tion homology with local coefficients on a manifold M can be constructed as the fixed points of G -factorization homology on a cover of M . Relation to the relative norm construction The spectrum T HH ( A ; A τ ) and its circleaction have been used to define the relative norm in [ABG + + U a complete universe of the circle group (in the sense of orthogonal spectra), and definea complete C n -universe ˜ U = ι ∗ C n U . Let R be an associative ring orthogonal C n -spectrum indexedon the universe ˜ U . Let I R ∞ ˜ U , I U R ∞ denote the “change of universe” functors. The relative norm N S C n R of [ABG + 14, def. 8.2] is the genuine S -spectra defined as I U R ∞ (cid:12)(cid:12)(cid:12) N cyc,C n ∧ ( I R ∞ ˜ U R ) (cid:12)(cid:12)(cid:12) , where N cyc,C n ∧ ( − ) is the “twisted cyclic bar construction” of [ABG + 14, def. 8.1].Note that the geometric realization | N cyc,C n ∧ ( I R ∞ ˜ U R ) | is equivalent to T HH ( R ; R σ ), computedusing the standard bar resolution. By proposition 7.2.2 there exists an equivalence of spectra (cid:18)Z S R (cid:19) Φ C n ≃ (cid:12)(cid:12)(cid:12) N cyc,C n ∧ (cid:16) I R ∞ ˜ U R (cid:17)(cid:12)(cid:12)(cid:12) , where one the left hand side we consider R as an E -algebra in Sp C n .Moreover, by inspection the above equivalence respects the circle action, hence after applyingthe change of universe functor I U R ∞ we get an equivalence of genuine S -spectra N S C n R ≃ I U R ∞ (cid:18)Z S R (cid:19) Φ C n ! . Appendix A The Moore over category Let C be a topological category and x ∈ C an object. Denote by N ( C ) ∈ C at ∞ the coherentnerve of C , and by N ( C ) /x ∈ C at ∞ the over category. Note that N ( C ) /x is not equivalent to thecoherent nerve of C /x , the topological over category: both have the same objects, but a pointin Map C /x ( y f −→ x, y f −→ x ) is an given by a map h ∈ Map C ( y , y ) satisfying f = f ◦ h ,while a point in Map N ( C ) /x ( y f −→ x, y f −→ x ) is given by a map h ∈ Map C ( y , y ) together84ith a path in Map C ( y , x ) from f to f ◦ h . Nevertheless, it is useful to have a topologicalcategory whose coherent nerve is equivalent to N ( C ) /x . Of course, this could be achieved byapplying homotopy coherent realization to N ( C ) /x , but unwinding the construction one seesthat an explicit description of topological category involves a lot of simplicial combinatorics. Inwhat follows we construct a topological category C Moore /x whose coherent nerve is equivalent to N ( C ) /x , which avoids simplicial combinatorics.An obvious candidate for the mapping space Map C Moore /x ( y f −→ x, y f −→ x ) is the space ofmaps h : y → y in C together with a path from f to f ◦ f in Map C ( y , x ), formally givenby the fiber product Map C ( y , y ) × Map C ( y ,x ) P (Map C ( y , x )). However, one runs into troublewhen trying to define composition functions which are strictly associative, since the compositionaction uses concatenation of paths. The problem of defining strictly associative concatenation ofpaths has a classical solution, namely replacing the space of paths with the homotopy equivalentspace of Moore paths. Defining the mapping space Map C Moore /x ( y f −→ x, y f −→ x ) using Moorepaths leads to a simple construction of a topological category C Moore /x , the Moore over category(definition A.0.1), whose coherent nerve is equivalent to N ( C ) /x (corollary A.0.5).We first recall the definition of the Moore path space and concatenation of Moore paths. Let X be a topological space. The Moore path space of X is the subspace M ( X ) ⊂ [0 , ∞ ) × X [0 , ∞ ) , M ( X ) = (cid:8) ( r, γ ) | the restriction γ | [ r, ∞ ) is a constant function (cid:9) , where X [0 , ∞ ) is the space of functions [0 , ∞ ) → X endowed with the compact-open topology.The “starting point” and “finishing point” fibrations α, ω : M ( X ) ։ X are the given by α ( r, γ ) = γ (0) , ω ( r, γ ) = γ ( r ). Moreover, the “ends points” map ( α, ω ) : M ( X ) ։ X × X is also a Serrefibration. Concatenation of Moore paths is defined by ∗ : M ( X ) × X M ( X ) → M ( X ) , ( r , γ ) ∗ ( r , γ ) = r + r , t ( γ ( t ) t ≤ r γ ( t − r ) t ≥ r ! . It is straightforward to verify that concatenation of paths is associative, i.e.(( r , γ ) ∗ ( r , γ )) ∗ ( r , γ ) = ( r , γ ) ∗ (( r , γ ) ∗ ( r , γ )) . For x ∈ X a point, the “constant instant Moore path” (0 , t x ) ∈ M ( X ) is a neutral elementfor concatenation.With the definition of Moore paths at hand, we can define the Moore path category. Definition A.0.1. Let C be a topological category and x ∈ C an object. Define a topologicalcategory C Moore /x with objects arrows f : y → x , i.e pairs ( y, f ) where y ∈ C , f ∈ Map C ( y, x ) , andmorphism spaces Map C Moore /x ( y f −→ x, y f −→ x ) given by the fiber products { f } × Map C ( y ,x ) M (Map C ( y , x )) × Map C ( y ,x ) , ( f ◦ C ( − )) Map C ( y , y )= { (( r, γ ) , h ) | γ (0) = f , γ ( r ) = f ◦ C h } . Define composition in C Moore /x by ◦ : Map C Moore /x ( y f −→ x, y f −→ x ) × Map C Moore /x ( y f −→ x, y f −→ x ) → Map C Moore /x ( y f −→ x, y f −→ x )((( r, γ ) , h ) , (( r ′ , γ ′ ) , h ′ )) (( r, γ ) ∗ ( r ′ , γ ′ ◦ C h ) , h ′ ◦ C h ) and identity of f : y → x by ((0 , t f ) , id y ) ∈ Map C Moore /x ( y f −→ x, y f −→ x ) , using the constantinstant Moore path at f . We call C Moore /x the Moore over category of C over x . bservation 1. The mapping space Map C Moore /x is the homotopy fiber of f ◦ ( − ) : Map C ( y , y ) → Map C ( y , x ) . Remark A.0.2. If the mapping spaces Map C ( y, x ) of C has a smooth structure one can replacethe Moore spaces of continuous Moore paths by spaces of piecewise smooth Moore paths, withoutchanging the ∞ -category represented by N ( C Moore /x ). Lemma A.0.3. The coherent nerve of the Moore category C Moore /x has a terminal object ( x = −→ x ) ∈ C Moore /x .Proof. For every object ( y f −→ x ) ∈ C Moore /x the mapping space Map C Moore /x ( y f −→ x, x = −→ x ) is thespace of Moore paths in Map C ( y, x ) starting at f , a contractible space.Define a functor of topological categories U : C Moore /x → C sending U : ( y f −→ x ) y and U : Map C /x ( y f −→ x, y f −→ x ) → Map C ( y , y ) , U : (( r, γ ) , h ) h on mapping spaces by projection. Lemma A.0.4. The induced map of coherent nerves N ( U ) : N ( C Moore /x ) → N ( C ) is a rightfibration.Proof. First we observe that N ( U ) is an inner fibration. For each pair of objects ( y f −→ x ) , ( y f −→ x ) ∈ C Moore /x the map U : Map C /x ( y f −→ x, y f −→ x ) → Map C ( y , y ) is a pull-back of the “end points” fibration ( α, ω ) along { f } × ( f ◦ C ( − )), and therefore a fibration. By[Lur09a, prop. 2.4.1.10 (1)] it follows that N ( U ) is an inner fibration.By [Lur09a, prop. 2.4.2.4] we need to show that every morphism (( r, γ ) , h ) : ( y f −→ x ) → ( y f −→ x ) in C Moore /x is U -Cartesian. By [Lur09a, prop. 2.4.1.10 (2)] we have to show that forevery ( y f −→ x ) in C Moore /c the diagramMap C Moore /x ( f, f ) U (cid:15) (cid:15) (cid:15) (cid:15) (( r,γ ) ,h ) ◦− / / Map C Moore /x ( f, f ) U (cid:15) (cid:15) (cid:15) (cid:15) Map C ( y, y ) h ◦ C − / / Map C ( y, y )is homotopy Cartesian. We show that the induced map between the fibers is a homotopy equiv-alence. For every point ( h ′ : y → y ) ∈ Map C ( y, y ), the fiber over h ′ is the space of Moore pathsin Map C ( y, x ) starting at f and ending at f ◦ h , the fiber over h ◦ h ′ is the space of Moore pathsin Map C ( y, x ) starting at f and ending at f ◦ h ′ ◦ h , and the map between the fibers is givenby concatenation with the Moore path ( r, γ ◦ h ′ ) starting at f ◦ h ′ and ending at f ◦ h ◦ h ′ , ahomotopy equivalence. Corollary A.0.5. Let C be a topological category and x ∈ C an object. The coherent nerve N ( C Moore /x ) is equivalent to the ∞ -over category N ( C ) /x .Proof. The right fibration N ( U ) : N ( C Moore /x ) → N ( C ) takes the terminal object ( x = −→ x ) ∈ N ( C Moore /x ) to x ∈ C . By [Lur09a, prop. 4.4.4.5] the right fibrations N ( U ) : N ( C Moore /x ) → N ( C )and N ( C ) /x ։ N ( C ) are equivalent fibrant objects of the contravariant model structure on sSet /N ( C ) (both right fibrations classify the representable functor Map( − , x ) : C op → S ), andclaim the follows. 86 ppendix B The definition of a G -Symmetric Monoidalcategory This appendix contains no original results or definitions. The notion of G -symmetric monoidal ∞ -category, developed by Barwick, Dotto, Glasman, Nardin and Shah, is central to our treatmentof G -factorization homology. For the convenience of the reader we include the definition here(see definition B.0.7), which is equivalent to the definition given in [Nar17]. Parametrized join First we recall the parametrized version of the join construction. Definition B.0.1. Let S be an ∞ -category. Restricting along S × ∂ ∆ → S × ∆ defines a functor sSet /S × ∆ → sSet /S × ∂ ∆ ∼ = sSet /S ×{ } × sSet /S ×{ } which carries coCartesian fibrations over S × ∆ to coCartesian fibrations over S × ∂ ∆ = S × { } ` S × { } . This functor has a rightadjoint which is called the S -parametrized join and denoted by sSet /S × ∆ ⇆ sSet /S ×{ } × sSet /S ×{ } : ⋆ S . By [Sha18, prop. 4.3], if C ։ S, D ։ S are coCartesian fibrations (i.e S -categories), then C ⋆ S D ։ S is a coCartesian fibration.It follows from [Sha18, thm. 4.16] that the parametrized join carries coCartesian fibrationsover S × ∂ ∆ to inner fibrations over S × ∆ with coCartesian lifts over S × ∂ ∆ .The parametrized join X ⋆ S Y → S × ∆ of X → S, Y → S can be informally described asfollows (see [Sha18, lem. 4.4]): its restriction to S × { } is X → S , its restriction to S × { } is Y → S , and for each s ∈ S its restriction to { s } × ∆ is the join X [ s ] ⋆ Y [ s ] , where X [ s ] , Y [ s ] arethe fibers of X → S, Y → S over s ∈ S .Fact: for the case Y = S one gets a coCartesian fibration X ⋆ S S ։ S . Finite pointed G -sets We denote by Fin G ∗ the G -category of finite pointed G -sets of [Nar16,def. 4.12]. An object I ∈ Fin G ∗ over the orbit G/H is a G -equivariant map I = ( U → G/H )from a finite G -set U . A morphism in Fin G ∗ over ϕ : G/K → G/H is a span of the form U (cid:15) (cid:15) U ′ o o (cid:15) (cid:15) / / V (cid:15) (cid:15) G/H G/K ψ o o = / / G/K where the left square is a summand inclusion, i.e it induces an inclusion of U ′ into the pullback ψ ∗ U = G/K × G/H U . The span above is a coCartesian edge if the left square is Cartesian andthe map U ′ → V is an isomorphism of finite G -sets ([Nar16, lem. 4.9, def. 4.12]). We call thespan above inert if U ′ → V is an isomorphism. Notation B.0.2. Let G/K ∈ O opG be an orbit. Denote by I + ( G/K ) = ( G/K = −→ G/K ) ∈ Fin G ∗ the finite pointed set given by the identity map of G/K . Definition B.0.3. Let I ∈ Fin G ∗ , I = ( U → G/H ) be a finite pointed G -set over G/H . Recallthat the left fibration G/H = ( O opG ) [ G/H ] / → O opG , ( G/H ← G/K ) G/K classifies the representable functor Hom( − , G/H ) : O opG → Set (see [BDG + ( Fin G ∗ ) [ G/H ] of finite G -sets over G/H is in bijection with the set of atural transformations N at (cid:16) Hom( − , G/H ) , ( Fin G ∗ ) ( − ) (cid:17) , which in turn is in bijection with theset of G -functors G/H → Fin G ∗ . Define σ : G/H → Fin G ∗ as the G -functor correspondingto I under the bijection above. Explicitly, σ acts on objects by σ : ( G/H ϕ ←− G/K ) ( ϕ ∗ U → G/K ) . The underlying G -categories of the G -diagram classified by C ⊗ ։ Fin G ∗ By straight-ening/unstraightening for G -categories ([BDG + C ⊗ ։ Fin G ∗ corresponds to a G -functor Fin G ∗ → C at ∞ ,G , which we can interpret as a Fin G ∗ -shaped G -diagram in C at ∞ ,G . The functor Fin G ∗ → C at ∞ ,G assigns to each I ∈ ( Fin G ∗ ) [ G/H ] an object of( C at ∞ ,G ) [ G/H ] = Fun( G/H, C at ∞ ) ≃ ( C at ∞ ) coCart/G/H (see [BDG + G/H , which can be constructed as follows. Definition B.0.4. Let C ⊗ ։ Fin G ∗ be a coCartesian fibration, and I ∈ Fin G ∗ a G -set over G/H .Define a coCartesian fibration C ⊗ ։ G/H by pulling back C ⊗ along σ , C ⊗ (cid:15) (cid:15) (cid:15) (cid:15) / / ❴✤ C ⊗ (cid:15) (cid:15) (cid:15) (cid:15) G/H σ / / Fin G ∗ . In particular, for I + ( G/G ) = ( G/G = −→ G/G ) , the terminal object of Fin G ∗ , denote by C := C ⊗ the underlying G -category of C ⊗ . An inert diagram in Fin G ∗ Let I = ( U → G/H ) be a finite pointed G -set over G/H ,as before. Applying the parametrized join construction for S = G/H and the left fibrations G/H = −→ G/H, U ։ G/H we get a coCartesian fibration U ⋆ G/H G/H ։ G/H × ∆ , whichwe can consider as a G -category by composing with the coCartesian fibration G/H × ∆ ։ G/H and the left fibration G/H ։ O opG .For each I ∈ Fin G ∗ we construct a G -functor Φ : U ⋆ G/H G/H → Fin G ∗ (a G -diagram in Fin G ∗ ): Definition B.0.5. Let I = ( U → G/H ) be a finite pointed G -set over G/H . We define a G -functor Φ : U ⋆ G/H G/H (cid:15) (cid:15) (cid:15) (cid:15) / / Fin G ∗ (cid:15) (cid:15) (cid:15) (cid:15) G/H × ∆ / / / / G/H / / / / O opG by specifying its restrictions to U ։ G/H × { } and G/H ։ G/H × { } , together with its actionon morphisms over ( id, → ∈ G/H × ∆ : We think of a coCartesian fibration over G/H as representing an H -category, since the category G/H =( O opG ) / [ G/H ] is equivalent to O opH . Since Fin G is a category, a map of finite G -sets U → V induces a G -functor U → V . By comparison, a map f : x → y in an ∞ -category C induces a span x = C /x ∼ ←− C /f → C /y = y where both arrows are left fibrationsand the left arrow is an equivalence of ∞ -categories. . The G -functor U → Fin G ∗ is the composition U / / / / G/H σ / / Fin G ∗ , where the firstmap is the left fibration induced by U → G/H .2. The G -functor G/H → Fin G ∗ is the composition G/H / / / / G/G σ / / Fin G ∗ ,where the first map is the structure map G/H ։ O opG = G/G and the second map is the G -functor corresponding to I + ( G/G ) (in fact, the composition is just σ ).3. Let ( G/H ψ ←− G/K ) ∈ G/H , then the fiber of U ⋆ G/H G/H → G/H × ∆ over ( { ψ } , → is (cid:16) U ⋆ G/H G/H (cid:17) ψ = U ψ ⋆ { ψ } , a co-cone diagram on the finite set of maps ϕ : G/H → U such that G/K ϕ / / ψ ❋❋❋❋❋❋❋❋ U (cid:15) (cid:15) G/H commutes. Therefore, morphisms of U ⋆ G/H G/H ։ G/H × ∆ over ( id ψ , → ∈ G/H × ∆ are in bijection to ϕ : U → G/H making the above diagramcommute. Let ¯ ϕ : G/K → ψ ∗ U be the unique map given by G/K = (cid:31) (cid:31) ∃ ! ¯ ϕ ●●●● ϕ % % ψ ∗ U / / (cid:15) (cid:15) ❴✤ U (cid:15) (cid:15) G/K ψ / / G/H The functor Φ sends the morphism over ( id ψ , → corresponding to ϕ : U → G/H to the span of finite pointed G -sets ψ ∗ U (cid:15) (cid:15) G/K ¯ ϕ o o = / / = (cid:15) (cid:15) G/K = (cid:15) (cid:15) G/K G/K = o o = / / G/K. Using the fact that O G is atomic (i.e orbits have no non-trivial retracts) one can check thatthe left square is a summand inclusion.Steps 1 and 2 define Φ on every morphism over G/H × (0 → , since every such morphismuniquely decomposes as a morphism in U followed by a morphism over ( { ψ } , → for some ψ ∈ G/H . Verifying that Φ is well defined is a straightforward calculation, using the fact thatevery morphism of U ⋆ G/H G/H can be uniquely decomposed as a morphism over ( { ψ ′ } , → followed by a morphism in G/H . Construction of Segal maps and definition of a G -symmetric monoidal ∞ -category For any coCartesian fibration over Fin G ∗ we construct ’Segal maps’: Definition B.0.6. Let C ⊗ ։ Fin G ∗ be a coCartesian fibration and I = ( U → G/H ) a finitepointed G -set over G/H . Construct a G -functor over G/H by the following steps: . Pulling C ⊗ along Φ produces a coCartesian fibration (Φ ) ∗ C ⊗ ։ U ⋆ G/H G/H , whichwe can consider as a coCartesian fibration over G/H × ∆ by the composition (Φ ) ∗ C ⊗ ։ U ⋆ G/H G/H ։ G/H × ∆ . (28) 2. The restriction of the coCartesian fibration (28) to G/H × { } is given by U × G/H C ⊗ ։ U ։ G/H × { } , as it is the pullback of C ⊗ ։ Fin G ∗ along U / / / / G/H σ / / Fin G ∗ .3. The restriction of the coCartesian fibration (28) to G/H × { } is given by G/H ×C ։ G/H = −→ G/H × { } , as it is the pullback of C ⊗ ։ Fin G ∗ along G/H / / / / G/G σ / / Fin G ∗ .4. Therefore, the coCartesian fibration (28) classifies a G -functor over G/HU × G/H C ⊗ ' ' ' ' PPPPP / / G/H ×C y y y y ssss G/H, which by [BDG + G -functor over G/Hφ : C ⊗ ' ' ' ' ◆◆◆◆◆ / / Fun G/H ( U , G/H ×C ) u u u u ❦❦❦❦❦❦ G/H. (29) We call (29) the Segal map of I . We can now give the definition of a G -symmetric monoidal G -category. Definition B.0.7. A G -symmetric monoidal G -category is a coCartesian fibration C ⊗ ։ Fin G ∗ such that for every finite pointed G -set I = ( U → G/H ) the Segal map φ of eq. (29) is anequivalence of G/H -categories. Remark B.0.8. Let C ⊗ → Fin G ∗ be a G -symmetric monoidal G -category. The Segal conditionsimply that an object x ∈ C ⊗ over I = ( U → G/H ) ∈ Fin G ∗ classifies a G -functor x • : U → C . Tosee this, first note that by Yoneda’s lemma x defines a G/H object σ x : G/H → C ⊗ . Since x ∈ C ⊗ is over I ∈ Fin G ∗ the composition G/H σ x −→ C ⊗ → Fin G ∗ is equivalent to σ : G/H → Fin G ∗ ,so σ x factors as σ x : G/H → C ⊗ → C ⊗ . Therefore we can regard σ x as a G/H -object of C ⊗ .Using the Segal conditions we identify σ x with a G/H -object of Fun G/H ( U , C× G/H ). Finallywe use the equivalenceFun G/H ( G/H, Fun G/H ( U , C× G/H ) ≃ Fun G/H ( U , C× G/H ) ≃ Fun G ( U , C ) . to identify σ x : G/H → Fun G/H ( U , C× G/H ) with a G -functor x • : U → C .90 emark B.0.9. The codomain of the above Segal map is equivalent to a parametrized product:The “internal hom” G/H -functor Fun G/H ( U , − ) : C at G/H ∞ → C at G/H ∞ is right adjoint to thecomposition C at G/H ∞ / / C at U ∞ / / C at G/H ∞ , ( D ։ G/H ) ✤ / / ( D × G/H U ։ U ) ✤ / / ( X × G/H U ։ U → G/H ) . Therefore, it decomposes as the composition of the right adjoints:Under this equivalence, the Segal map of I = ( U → G/H ) is given by φ : C ⊗ & & & & ▼▼▼▼▼▼ / / Q I C× U z z z z ✉✉✉✉ G/H. (30)In particular, we can identify an object x ∈ C ⊗ over I with a G/H -object of Q I C× U as follows.Since Fin G ∗ ։ O opG , I [ G/H ] the object x belongs to the fiber C ⊗ [ G/H ] , and by Yoneda’s lemmais classified by a G -functor σ x : G/H → C ⊗ . Since x ∈ C ⊗ is over I ∈ Fin G ∗ , the G -functor G/H → C ⊗ ։ Fin G ∗ classifies I ∈ Fin G ∗ , and is therefore equivalent to σ . Therefore itinduces a G/H -functor G/H → C ⊗ . Post-composing with the Segal map of eq. (30) we getour desired G/H -object G/H → C ⊗ ∼ −→ Q I C× U , which by abuse of notation we also denoteby σ x : G/H → Q I C× U .Unpacking the construction of the Segal maps (29) in definition B.0.7 gives the followingfiberwise characterization of G -symmetric monoidal categories, which is easier to verify. Lemma B.0.10. A coCartesian fibration C ⊗ ։ Fin G ∗ is a G -symmetric monoidal category(definition B.0.7) if and only if for each finite pointed G -set J = ( V → G/K ) ∈ Fin G ∗ thefunctor C ⊗ J → Y W ∈ Orbit( ψ ∗ U ) C [ W ] is an equivalence of ∞ -categories, where C ⊗ J is the fiber of C ⊗ ։ Fin G ∗ over J = ( V → G/K ) and the above functor is the product of C ⊗ J → C [ W ] associated to the Fin G ∗ edges ∀ W ∈ Orbit( V ) : V (cid:15) (cid:15) W o o = / / = (cid:15) (cid:15) W = (cid:15) (cid:15) G/K W o o = / / W. Proof. The Segal condition of G -symmetric monoidal G -categories states that the Segal map φ is a parametrized equivalence, i.e for each ( G/H ψ ←− G/K ) ∈ G/H , the Segal map φ induces an equivalence between the fibers( C ⊗ ) [ ψ ] → Fun G/H ( U , G/H ×C ) [ ψ ] . C ⊗ over ψ is the fiber of C ⊗ ։ Fin G ∗ over the finite pointed G -set J := ( ψ ∗ U → G/K ). The fiber of Fun G/H ( U , G/H ×C ) over ψ is the ∞ -category of G -functors Fun O opG ( ψ ∗ U , C ).Decomposing the finite G -set ψ ∗ U = ` W ∈ Orbit( ψ ∗ U ) W into orbits we haveFun O opG ( ψ ∗ U , C ) ∼ = Fun O opG ( a W , C ) ≃ Y W Fun O opG ( W , C ) ≃ Y W ∈ Orbit( ψ ∗ U ) C [ W ] . Since both sides depend only on J = ( ψ ∗ U → G/K ) ∈ Fin G ∗ the result follows.We end this appendix with the definition of parametrized tensor product functors in a G -symmetric monoidal category. Definition B.0.11. Let C ⊗ ։ Fin G ∗ be a G -symmetric monoidal category. Let I = ( U → G/H ) , J = ( V → G/H ) ∈ Fin G ∗ be two object over the orbit G/H , and f : I → J a morphismin ( Fin G ∗ ) [ G/H ] , given by U (cid:15) (cid:15) U = o o (cid:15) (cid:15) f / / V (cid:15) (cid:15) G/H G/H = o o = / / G/H. The morphism f corresponds to a functor ∆ → ( Fin G ∗ ) [ G/H ] , or equivalently to a G -functor σ Appendix C Mapping spaces in over-categories We prove some simple properties of mapping spaces in over categories. Lemma C.0.1. Consider the over category C /b for b an object in an ∞ -category C . Let x → b, y → b, y → b be objects in C /b , and a morphism ϕ in C /b from y → b to y → b . Then Map C /b ( x → b, y → b ) / / ϕ ∗ (cid:15) (cid:15) Map C ( x, y ) ϕ ∗ (cid:15) (cid:15) Map C /b ( x → b, y → b ) / / Map C ( x, y )92 s a homotopy pullback square.Proof. The mapping space Map C /b ( x → b, y → b ) is the homotopy fiber of the postcompositionmap Map C ( x, y ) → Map C ( x, b ). Therefore the lower square and outer rectangle in the diagramMap C /b ( x → b, y → b ) / / ϕ ∗ (cid:15) (cid:15) Map C ( x, y ) ϕ ∗ (cid:15) (cid:15) Map C /b ( x → b, y → b ) / / (cid:15) (cid:15) Map C ( x, y ) ( y → b ) ∗ (cid:15) (cid:15) ∗ x → b / / Map / C ( x, b )are homotopy pullback diagram. It follows that the top square is a homotopy pullback square. Lemma C.0.2. Let f : b → b ′ be a morphism in an ∞ -category C , and consider the postcompo-sition functor f ∗ : C /b → C /b ′ . Let x → b, y → b, y → b be objects in C /b , and a morphism ϕ in C /b from y → b to y → b . Then Map C /b ( x → b, y → b ) f ∗ / / ϕ ∗ (cid:15) (cid:15) Map C /b ′ ( x → b f −→ b ′ , y → b f −→ b ′ ) ϕ ∗ (cid:15) (cid:15) Map C /b ( x → b, y → b ) f ∗ / / Map C /b ′ ( x → b f −→ b ′ , y → b f −→ b ′ ) is a homotopy pullback square.Proof. Consider the commutative diagramMap C /b ( x → b, y → b ) f ∗ / / ϕ ∗ (cid:15) (cid:15) Map C /b ′ ( x → b f −→ b ′ , y → b f −→ b ′ ) / / ϕ ∗ (cid:15) (cid:15) Map C ( x, y ) ϕ ∗ (cid:15) (cid:15) Map C /b ( x → b, y → b ) f ∗ / / Map C /b ′ ( x → b f −→ b ′ , y → b f −→ b ′ ) / / Map C ( x, y ) . By lemma C.0.1 the right square and the outer rectangle are homotopy pullback squares, hencethe left square is a homotopy pullback square.Next, let f : b → b ′ be a morphism in an ∞ -category C as before, and T : M → C /b ′ a functorof ∞ -categories. Define an ∞ -category M T as the pullback M Tu (cid:15) (cid:15) / / C /bf ∗ (cid:15) (cid:15) M T / / C /b ′ . emma C.0.3. Let C , f : b → b ′ , T : M → C /b ′ and M T be as above. Let X, Y , Y be objects in M T and Φ : Y → Y be morphism in M T . Then Map M T ( X, Y ) Φ ∗ / / (cid:15) (cid:15) Map M T ( X, Y ) (cid:15) (cid:15) Map M ( u ( X ) , u ( Y )) u ( Φ ) ∗ / / Map M ( u ( X ) , u ( Y )) is a homotopy pullback square.Proof. Denote the images of X, Y , Y ∈ M F in C /b by x → b, x → y , x → y , and the image of Φ by ϕ . Using the equivalences u ( X ) ≃ ( x → b f −→ b ′ ) , u ( Y ) ≃ ( y → b f −→ b ′ ) , u ( Y ) ≃ ( y → b f −→ b ′ ) in C /b ′ we can identify the mapping spacesMap C /b ′ ( T u ( X ) , T u ( Y i )) ≃ Map C /b ′ ( x → b f −→ b ′ , y i f −→ b ′ ) , i = 1 , . Under these identifications we have a commutative diagramMap M T ( X, Y ) Map M ( u ( X ) , u ( Y ))Map C /b ( x → b, y → b ) Map C /b ′ ( x → b f −→ b ′ , y → b f −→ b ′ )Map C /b ( x → b, y → b ) Map C /b ′ ( x → b f −→ b ′ , y f −→ b ′ ) . Tϕ ∗ f ∗ ϕ ∗ f ∗ The top square is a homotopy pullback square by the definition of M T as a pullback, and thebottom square is a homotopy pullback square by lemma C.0.2. Therefore the outer rectangle isa homotopy pullback square. 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