The Adams isomorphism revisited

We establish abstract Adams isomorphisms in an arbitrary equivariantly presentable equivariantly semiadditive global category. This encompasses the well-known Adams isomorphism in equivariant stable homotopy theory, and applies more generally in the settings of $G$-Mackey functors, $G$-global homotopy theory, and equivariant Kasparov categories.


Introduction
In [Ada84], Adams proved a surprising result in equivariant stable homotopy theory: given a finite group G and a finite pointed G-CW-complex on which G acts freely away from the basepoint, there is an equivalence between the G-orbit spectrum and the (derived) G-fixed spectrum of the suspension spectrum Σ ∞ X.This sits in stark contrast to the unstable or naïvely stable situation, where the G-fixed points of X are just trivial unless G = e.In [LMS86], Lewis, May, and Steinberger extended this equivalence to all so-called G-free genuine G-spectra, and called it the Adams isomorphism.Since then, it has become an indispensable tool in equivariant homotopy theory, for example allowing one to calculate the G-equivariant stable homotopy groups of a free G-CW complex X simply as the non-equivariant stable homotopy groups of its G-orbits X/G.
The goal of the present article is to show that the Adams isomorphism is a purely formal consequence of equivariant semiadditivity, an analogue of semiadditivity in the context of parametrized higher category theory [BDG + 16] introduced by the authors [CLL23a] building on work of Nardin [Nar16].Our treatment can be viewed as a highly abstract version of the classical proof of the Adams isomorphism, which is a clever reduction to an application of the Wirthmüller isomorphism for equivariant spectra.The notion of equivariant semiadditivity is designed to provide an abstract formulation of the Wirthmüller isomorphism, and the key observation of this paper is that the Adams isomorphism and its proof admit similar abstract formulations.This in particular provides analogues of the Adams isomorphism in various new settings, like that of G-global homotopy theory, G-Mackey functors, and equivariant Kasparov categories.
The categorical framework we will work in is that of global categories:1 families of categories C : Glo op → Cat indexed by the (2, 1)-category Glo of finite groups, group homomorphisms, and conjugations.We will require C to be equivariantly presentable in the sense of [CLL23b], meaning that C factors through the subcategory Pr L → Cat of presentable categories and that for all injective group homomorphisms i : H → G the restriction functors i * : C(G) → C(H) admit left adjoints i !: C(H) → C(G) satisfying a base change condition.We will further require C to be equivariantly semiadditive: each category C(G) is semiadditive in the usual sense and for every injection i : H → G a certain norm map Nm i : i !→ i * to the right adjoint i * of i * is invertible.
To state the abstract Adams isomorphism in this context, consider a normal subgroup N of a finite group G.As a consequence of equivariant presentability of C, the category C(G) is canonically tensored over G-spaces, and in analogy with the usual definition for genuine G-spectra we say that an object X ∈ C(G) is N -free if the canonical map X ⊗ EF N → X ⊗ 1 = X is an equivalence.Here EF N is the universal G-space for the family F N of subgroups H ⩽ G such that H ∩ N = e (see Definition 5.1).On the N -free G-objects, one can define an N -orbit functor −/N : C(G) N -free → C(G/N ), which is a partial left adjoint of the inflation functor C(G/N ) → C(G) associated to the quotient morphism G → G/N .The inflation functor moreover admits a right adjoint (−) N : C(G) → C(G/N ), called the N -fixed point functor, and our main result is the following comparison between these adjoints: Main Theorem (Abstract Adams isomorphism for global categories, Theorem 5.2).Let C be an equivariantly presentable and equivariantly semiadditive global category.Let N G be a normal subgroup of a finite group G, and assume that the N -fixed point functor (−) N : C(G) → C(G/N ) preserves colimits.Then there is a natural equivalence Nm : X/N ∼ − − → X N in C(G/N ) for every N -free object X ∈ C(G) N -free .
Specializing the theorem, we obtain abstract Adams isomorphisms for various global categories of interest: (1) For C(G) = Sp G , the category of genuine G-spectra (Example 5.3), this precisely recovers the classical Adams isomorphism [LMS86, Theorem II.7.1].
(2) For C(G) = Sp gl G , the category of G-global spectra (Example 5.4), this provides a global Adams isomorphism refining the classical Adams isomorphism and generalizing the result for G = N appearing in [Tig22].
(3) There are also 'non-group completed' versions of the previous two examples, yielding Adams isomorphism for special G-equivariant or G-global Γ-spaces, see Example 5.6.(4) If E is any semiadditive category, the global category of Mackey functors in E, given by C(G) = Mack E (G) := Fun × (Span(Fin G ), E), satisfies the assumptions of the theorem, see Example 5.7.In particular, taking E to be the 1-category of abelian groups, this yields Adams isomorphisms for classical Mackey functors, while for E the derived category of abelian groups we obtain an Adams isomorphism for Kaledin's derived Mackey functors [Kal11].(5) The theorem further applies to C(G) = KK G , the G-Kasparov K-theory category defined by Bunke, Engel, and Land [BEL21], which we recall in Example 5.10.
In fact, we prove a more general statement in which the category Glo is replaced by a small category T and the role played by the wide subcategory Orb ⊆ Glo of injective homomorphisms in the definitions of equivariant presentability and semiadditivity is taken by a so-called atomic orbital subcategory P ⊆ T , see Theorem 4.8.
Related work.Given that the Adams isomorphism is both a fundamental and yet somewhat surprising feature of equivariant stable homotopy theory, it is natural to ask for the key properties necessary to establish Adams isomorphisms in other contexts.The quest for such formal Adams isomorphisms goes back at least to [May03], where May pointed out the formal similarity of the Adams isomorphism to the Wirthmüller isomorphism, but noted that unlike the latter it so far resisted an axiomatic treatment.
A first step in this direction was taken by Hu [Hu03], who provided a very general duality result in parametrized homotopy theory encompassing both the Wirthmüller isomorphism as well as the Adams isomorphism in equivariant stable homotopy theory for compact Lie groups.While her treatment provided part of the original inspiration for our approach, the nature of her argument is ultimately geometric rather than categorical, employing a parametrized version of the Pontryagin-Thom construction.An analogue of her argument in the context of equivariant motivic homotopy theory was indicated by Hoyois [Hoy17, Section 1.4.1].
An entirely different approach to formal Adams isomorphisms was given by Sanders [San19], based on the notion of the compactness locus of a geometric functor f * between rigidly compactly generated tensor triangulated categories.This setup is somewhat orthogonal to ours: while Sanders' approach works for a single functor f * without need for a parametrized category in the background, we on the other hand do not require stability or any sort of monoidal structure.In fact, several of our key examples do not fit into Sanders' framework despite having natural symmetric monoidal structures: the tt-category of global spectra is not rigid (Remark B.6), while examples like G-equivariant Γ-spaces or G-Mackey functors are not even stable.Nevertheless, we will show in Appendix B that our abstract Adams isomorphism agrees with that of Sanders whenever the two frameworks overlap.
In addition to these general approaches, also the classical Adams isomorphism for genuine G-spectra has been revisited several times.In particular, Reich and Varisco [RV16] first lifted the construction of the classical Adams isomorphism from the homotopy category to the ∞-categorical level, by giving an explicit and natural model of the map in the 1-category of orthogonal spectra.
A treatment in the language of parametrized category theory was first given by Quigley and Shah [QS21, Remark 4.29], who introduced a parametrized Tate construction for genuine equivariant spectra, with the classical Adams isomorphism as a special case.Similarly to our approach below, their construction is based on the norm maps of Hopkins and Lurie [HL13].
A treatment of the Adams isomorphism in equivariant motivic homotopy theory was given by Gepner and Heller [GH23].
Outline.In Section 2 we recall relevant background material on parametrized higher category theory.Section 3 introduces and studies torsion objects in parametrized categories, generalizing the notion of N -free G-spectra.Using this language, we then state and prove our general abstract Adams isomorphism in Section 4. In Section 5 we specialize this result to the Adams isomorphism for global categories stated above, and we discuss various examples.
The paper ends with three short appendices.In Appendix A we prove a general statement about cocontinuity of the right adjoints to restrictions for several universal examples of parametrized higher categories, which is in particular used in establishing the Adams isomorphisms for global and equivariant Γ-spaces.In Appendix B, we compare our abstract Adams isomorphism to that of Sanders [San19].
Finally, Appendix C collects some aspects of the calculus of mates used throughout the paper.

Recollections on parametrized higher categories
In this article, we will freely use the language of parametrized higher category theory, as established by [BDG + 16, Sha23, Nar16] and, from the perspective of categories internal to ∞-topoi, by [Mar21, MW24, MW22].We will follow our conventions from [CLL23a].Throughout this section, we fix a small category T .Remark 2.2.As a category of functors into a Cartesian closed category, Cat T is again Cartesian closed.We write Fun T for the internal hom objects.
We will refer to Glo-categories as global categories throughout.
Example 2.4.We write Orb ⊆ Glo for the wide subcategory spanned by the injective group homomorphisms.Restricting along the inclusion, every global category has an underlying Orb-category.
Any T -category can be equivalently viewed as a limit preserving functor PSh(T ) op → Cat; in particular, we will frequently evaluate T -categories at general presheaves over T .
Example 2.5 (cf.[CLL23a, Example 2.1.11and Remark 2.1.16]).We have a limit preserving functor Spc T = PSh(T ) /• : PSh(T ) op → Cat sending a presheaf X to PSh(T ) /X , with contravariant functoriality via pullback.As a functor T op → Cat this can be equivalently described as Spc T (A) ≃ PSh(T /A ), with functoriality via restriction along the postcomposition maps Example 2.6.For every finite group G, the slice category Orb /G is equivalent to the 1-category Orb G of transitive G-sets (via sending an injective homomorphism φ : H → G to G/im(φ)).By Elmendorf's theorem [Elm83], we can therefore identify each individual Spc Orb (G) with the ∞-category of G-spaces.In fact, as explained in [CLL23b] we can bundle up these equivalences to give an explicit model S of Spc Orb as follows: We have a functor Fun(B(−), SSet) : Glo op → Cat sending a group G to the category Fun(BG, SSet) of G-objects in the 1-category of simplicial sets, with the obvious functoriality.For every such G, we can equip Fun(BG, SSet) with the G-equivariant weak equivalences, i.e. those maps whose geometric realization is a G-equivariant homotopy equivalence.The restrictions are then clearly homotopical, so this defines a functor from Glo op into the category of relative categories.Postcomposing with ∞-categorical localization, we therefore obtain a global category S, and [CLL23b, Theorem 5.5] provides the desired equivalence S ≃ Spc Orb .
Definition 2.7 ([CLL23a, Definition 4.2.2,Definition 4.2.11]).Let P ⊆ T be a wide subcategory.We say that P is orbital if for every pullback diagram in PSh(T ), with A, B, B ′ ∈ T and p : A → B in P , the morphism p ′ : A ′ → B ′ can be decomposed as a disjoint union (p i ) n i=1 : If P is orbital, we let F P T ⊆ Spc T = PSh(T ) /• denote the full parametrized subcategory spanned at B ∈ T by those morphisms A → B in PSh(T ) which are equivalent to disjoint unions (p i ) n i=1 : n i=1 A i → B for morphisms p i : A i → B in P .Orbitality of P guarantees that this is well-defined.

We say a map
Example 2.8.The (2, 1)-category Glo is orbital (as a subcategory of itself): the finite coproduct completion F Glo := F Glo Glo ⊆ PSh(Glo) is simply the (2, 1)-category of finite groupoids, so it has all pullbacks, and as it moreover contains all representables, limits in F Glo are already limits in PSh(Glo).
Example 2.9.Remark 2.18.Already fiberwise presentability ensures that the restriction functors f * admit right adjoints f * .If C is T -presentable, then these right adjoints again satisfy a Beck-Chevalley condition, i.e.C is also T -complete.We caution the reader, however, that the Beck-Chevalley condition for the right adjoints does not hold for general P -presentable T -categories.
Specializing the above, we get notions of Orb-presentable and Glo-presentable global categories; below, we will refer to these as equivariantly presentable and globally presentable, respectively.
Example 2.19.For any finite group G, we define Sp G , the (∞-)category of genuine G-spectra, as the localization of the 1-category Fun(BG, Sp Σ ) of symmetric spectra with G-action at the G-stable weak equivalences of [Hau17, Definition 3.25].These fit together into a global category Sp, with structure maps given as left derived functors of the evident restriction functors between the 1-categorical models.
In more detail, G → Fun(BG, Sp Σ ) becomes a global 1-category via precomposition.For every G, we can consider the full subcategory Fun(BG, Sp Σ ) cof of those Gsymmetric spectra that are cofibrant in the equivariant projective model structure of [Hau17,Theorem 4.8]; the restriction maps then preserve these subcategories and are homotopical in the above weak equivalences when restricted to them by [CLL23b, Lemma 9.3], so that we obtain a functor Fun(−, Sp Σ ) cof from Glo into the 2-category of relative 1-categories.Postcomposition with the localization functor then yields the global category Sp.
We call this the global category of equivariant spectra; it is equivariantly presentable and equivariantly stable: in fact, [CLL23b,Theorem 9.4]  Example 2.21.For any finite group G, we can consider the (∞-)category ΓS spc * (G) of special Γ-G-spaces in the sense of Shimakawa [Shi89,Shi91], defined as a certain localization of the category of reduced functors from the category of finite pointed sets into the 1-category of G-simplicial sets.For varying G, these again fit together into a global category ΓS spc * , see [CLL23b, paragraph after Lemma 7.14] for details.By Theorem 7.17 of op.cit., this is the free equivariantly presentable and equivariantly semiadditive global category.

Torsion and complete objects
In the equivariant stable homotopy theory of a finite group G, there are two frequently used categories of equivariant spectra: the category Sp BG of Borel Gspectra and the category Sp G of genuine G-spectra.The forgetful functor Sp G → Sp BG admits fully faithful left and right adjoints Sp BG → Sp G .The essential image of the left adjoint consists of the Borel-free G-spectra: those X such that the canonical map X ⊗ EG → X is an equivalence; here EG is a free G-space whose underlying space is contractible and − ⊗ EG denotes the tensoring of a G-spectrum with the G-space EG.Equivalently, a G-spectrum X is Borel-free if and only the H-geometric fixed points Φ H (X) are trivial for every nontrivial subgroup H ⩽ G, which explains the use of the word 'free'.Dually, the essential image of the right adjoint consists of the Borel-complete G-spectra: those X such that the canonical map X → X EG is an equivalence, or equivalently those X such that genuine and homotopy H-fixed points agree for every subgroup H ⩽ G.
These two notions of Borel G-spectra have generalizations to an arbitrary family F of subgroups of G, that is, a collection of subgroups closed under conjugacy and passing to smaller subgroups.Let EF denote the universal G-space for F,3 characterized by Following [MNN17], we say that a G-spectrum X is F-torsion if the map X ⊗EF → X is an equivalence; other names used in the literature are F-spectra [LMS86, Definition II.2.1], F-objects [MM02, II.6.1], and The Borel-free and Borel-complete G-spectra are recovered by letting F consist of only the trivial subgroup.More generally, if N G is normal, taking F to be those subgroups H ⩽ G such that H ∩ N = e gives the notion of N -free G-spectra discussed in the introduction.
The goal of this section is to provide a general definition of torsion and complete objects in a parametrized category.Throughout this section, we let P be an arbitrary small (∞-)category.
Definition 3.1.For an object A ∈ P , a P -family over A is a (non-empty) sieve of P /A , i.e. a (non-empty) full subcategory F ⊆ P /A satisfying the property that for every morphism in Example 3.2.The whole subcategory Aℓℓ A := P /A is always a P -family over A. Example 3.5.Assume that P is a wide subcategory of some other small ∞category T , and let f : A → B be a morphism in T .Then there is a P -family F f over A containing those morphisms p : C → A in P for which the composite f p : C → B is also in P .
Definition 3.6.Let F ⊆ P /A be a P -family over A. We define EF ∈ PSh(P /A ) as the unique presheaf with where the first functor is the map classifying the sieve F and the second one picks out the morphism of spaces ∅ → 1.
Under the canonical identification PSh(P /A ) ≃ − → PSh(P ) /A , we will generally think of EF as a presheaf on P equipped with a morphism i : EF → A. Note that i is a monomorphism, as EF ∈ PSh(P /A ) is (−1)-truncated.
Remark 3.7.By Kan's pointwise formula [Cis19, Proposition 6.4.9 op ] and the sieve property, EF is the image of the terminal object under the left Kan extension PSh(F) → PSh(P /A ).Writing the terminal object in PSh(F) as a colimit of representables as usual, we therefore get an equivalence in PSh(P /A ).Applying the equivalence PSh(P /A ) ≃ PSh(P ) /A , we see that the map i : EF → A of P -presheaves is just the tautological map colim Example 3.8.If F = P /A is the maximal family, we get EF = A.
Example 3.9.Let F ⊆ Orb G = Orb /G be a family of subgroups of a finite group G. Then EF agrees (up to the equivalence between G-spaces and Orb G -presheaves provided by Elmendorf's theorem) with the usual universal G-space for the family F as recalled in the introduction of this section.
Given a presentable P -category C and a family F over A ∈ P , we may now define the notion of F-torsion and F-complete objects.
Construction 3.10.Recall from [MW22, Proposition 8.2.9] that C is canonically tensored over the P -category Spc P of P -spaces.More precisely, there is a unique P -functor − ⊗ − : C × Spc P → C (2) such that − ⊗ 1 : C → C is the identity and which preserves P -colimits in each variable separately in the following sense: for each A ∈ P the functor obtained by evaluating (2) in degree A preserves (non-parametrized) colimits in both variables, and ⊗ satisfies the projection formula, i.e. for every f : A → B in P the canonical maps In a similar way, we obtain a cotensoring (−) (−) : C(A) × PSh(P ) op /A → C(A) which preserves limits in both variables.
Example 3.11.Recall the Orb-category S of equivariant spaces from Example 2.6, giving an explicit model of Spc Orb .
We will now make the tensoring of the underlying Orb-category of Sp explicit in this model: by the uniqueness statement recalled in the previous construction, it will suffice to construct some Orb-functor Sp × S → Sp that preserves colimits in each variable separately and that restricts to the identity when we plug in the terminal object in the second variable.We claim that descends to the required functor.In particular for each fixed G, the parametrized tensoring agrees with the usual tensoring of G-spectra by G-spaces.
To verify the claim, we first observe that [Hau17, Proposition 2.29 and Lemma 4.5] show that for every fixed G the smash product is a left Quillen bifunctor with respect to the usual equivariant model structure on Fun(BG, SSet) from [Ste16, Proposition 2.16] (whose weak equivalences are the G-equivariant weak equivalences and whose cofibrations are the levelwise injections).This shows that (3) is welldefined and homotopical for every fixed G (as every G-simplicial set is cofibrant), and that the induced functor Sp G × S G → Sp G preserves colimits in each variable.
For every injective homomorphism f : H → G the left Kan extension functors f !: Fun(BH, SSet) → Fun(BG, SSet) and f !: Fun(BH, Sp Σ ) → Fun(BG, Sp Σ ) are left Quillen (see [Len20, Proposition 1.1.18]and [Hau17, Section 5.2], respectively), and so the projection formula can be checked on the level of models again, where this is a trivial computation.
Altogether, we see that the induced functor Sp × S → Sp preserves colimits in each variable.As (3) restricts to the identity for the terminal G-simplicial sets, this then completes the proof of the claim.
As an upshot we can in particular simply compute the above abstract tensoring of Sp G in terms of the levelwise smash product-in fact, for fixed G there isn't even any need to derive as smashing with a fixed G-simplicial set also clearly preserves level weak equivalences.
Example 3.12.In the same way, one can identify the above tensoring for global spectra as well as equivariant or global Γ-spaces as the functor induced by the usual smash product.
Definition 3.13 (F-torsion and F-complete objects).Let C be a P -presentable P -category, let A ∈ P , and let F ⊆ P /A be a P -family over A.
We denote by the full subcategories of F-torsion and F-complete objects, respectively.
Example 3.14.Let P = Orb and let C = Sp| Orb be the Orb-category of genuine equivariant spectra.Combining Examples 3.9 and 3.11, we see that for a family F of subgroups of a finite group G and a genuine G-spectrum X ∈ Sp G , the map X⊗EF → X from the previous definition agrees with the familiar one, showing that it recovers the usual notion of F-torsion recalled in the beginning of this section.
Passing to adjoints, we then deduce the same statement for the F-complete objects.
The F-torsion G-spectra admit a well-known interpretation in terms of geometric fixed points: X is F-torsion if and only Φ H (X) = 0 for all H / ∈ F. Indeed, as the geometric fixed point functors Φ H : Sp G → Sp are symmetric monoidal and detect equivalences, this is immediate from the definition of EF.
Example 3.15.Let us describe torsion objects in the underlying Orb-category of the global category Sp gl from Example 2.20: if F is any family of subgroups of some finite group G, then Example 3.12 shows that a G-global spectrum X (represented as a symmetric spectrum with G-action) is F-torsion if and only if the collapse map X ∧EF + → X is a G-global stable weak equivalence, i.e. if and only if Let us now define Φ φ X := Φ H (φ * X) as the H-equivariant geometric fixed points of the H-spectrum underlying φ * X.As usual geometric fixed points are compatible with restriction to subgroups, we then immediately conclude from the above characterization of torsion G-spectra that the G-global spectrum X is F-torsion if and only if Φ φ X = 0 for all φ : H → G with im(φ) / ∈ F.
We have the following useful description of the F-torsion and F-complete objects: In particular, while C(A) F -tors and C(A) F -comp are typically distinct subcategories of C(A), they are abstractly equivalent, both being equivalent to the category C(EF).
Proof.By P -presentability of C, i * has a left adjoint i !, and the Beck-Chevalley condition for the pullback diagram in PSh(P ) shows that the unit id → i * i ! is an equivalence.Thus, [Lur24, Tag 02EX] implies that i ! is fully faithful with essential image those X ∈ C(A) for which the counit map i !i * X → X is an equivalence.But since the tensoring by P -spaces on C preserves P -colimits in both variables, the counit is conjugate to the map The argument for i * is dual.□ Corollary 3.17.In the above situation, the map C(A) → lim (p : B→A)∈F op C(B) induced by the restrictions p * : C(B) → C(A) restricts to equivalences For the underlying Orb-category of Sp, this result appears as [MNN17, Theorem 6.27], expressing the category of F-complete G-spectra as a limit of H-spectra.Proof.The first two claims follow from the identification of C(A) F -tors with the essential image of i !, as i !preserves colimits and every morphism p : C → A in F factors through i : EF → A (see Remark 3.7), so that p ! factors through i ! .For the final claim, we write EF ≃ colim (p : C→A)∈F p ! C as in Remark 3.7 and consider for every F-torsion object X the composite equivalence where the penultimate equivalence uses the projection formula.□

The abstract Adams isomorphism
Throughout this section, we fix a small category T and an atomic orbital subcategory P ⊆ T .The goal of this section is to prove an abstract Adams isomorphism in a P -presentable P -semiadditive T -category C: under suitable assumptions on a map f : A → B in T , we will construct a map Nm f : f fr !(X) → f * (X) from the partially defined left adjoint f fr ! of f * to its right adjoint f * , and show that it is an equivalence.One subtlety here is that the functor f * : C(B) → C(A) need not have a fully defined left adjoint.Instead, we will show that it at least exists on a certain class of f -free objects: Definition 4.1.For a morphism f : A → B in T , recall from Example 3.5 the P -family F f ⊆ P /A consisting of those maps p : C → A in P such that also the composite f p : C → B is in P .We will denote by the full subcategory of F f -torsion objects, in the sense of Definition 3.13 applied to the underlying P -category, and refer to its objects as f -free objects.
Example 4.2.If f is itself a morphism in P , then F f = P /A and thus every object We now construct the promised partial left adjoint: Lemma 4.4.Let C be P -presentable and let f : A → B be a map in T .There exists a unique functor f fr !: of mapping spaces for X ∈ C(A) f -free and Y ∈ C(B).
Proof.It suffices to show that the left hand side of (4) is corepresentable for every f -free X.As corepresentable functors are closed under limits, we may assume by Corollary 3.19 that X = p ! X ′ for some (p : . By definition of F f , the map f p : C → B lies in P , so (f p) !X is the desired corepresenting object.□ We now come to the comparison map f fr !→ f * , which is a slight modification of the norm construction of Hopkins and Lurie [HL13, Construction 4.1.8].One obstacle we will encounter here is that the right adjoints do not satisfy base change in general (cf.Remark 2.18).However, we at least have the following weaker statement: Lemma 4.5.Let C be a P -presentable T -category and let Proof.Let p : C → A be a map in P such that f p belongs to P .Then in the iterated pullback the Beck-Chevalley maps associated to the left hand square and to the total rectangle are equivalences by P -presentability.Thus, 2-out-of-3 implies that the Beck-Chevalley map of the right hand square is inverted by p * .□ Construction 4.6.Let f : A → B be a map in PSh(T ) and consider the pullback Assume that the diagonal ∆ : A → A× B A is a map in F P T .Let C be P -semiadditive, and assume that the required right adjoints exist in the following zig-zag: (5) (a) We write N : id → pr 1 * pr * 2 for the composite of the first three maps in (5).(b) If C is in addition P -presentable, then Lemma 4.5 shows that the Beck-Chevalley map BC * : f * f * → pr 1 * pr * 2 becomes an equivalence after applying i * : C(A) → C(EF f ).Applying i * to (5) thus gives a map (c) By restricting Nm fr f along i ! and passing to adjoints, we obtain a map , which we may equivalently view as a natural transformation Nm fr f : We are now ready to state our main result: Theorem 4.8 (Abstract Adams isomorphism).Let C be a P -semiadditive Ppresentable T -category and let f : A → B be a map in T such that the diagonal A → A × B A lies in F P T .Then: (1) For any p : C → A in P such that also f p is a map in P and every X ∈ C(A) in the essential image of p ! , the map Nm fr f : While the proof will be given below after some preparations, let us already note the following consequence in the stable setting: Corollary 4.9.Let C be P -stable and P -presentable, and let f : A → B be a map in T whose diagonal belongs to F P T .Then Nm fr f : f fr !X → f * X is an equivalence for every compact f -free object X.
Proof.By Corollary 3.19, any X ∈ C(A) f -free can be written as colim k∈K p k! X k for some category K, maps p k : A k → A in P with f p k in P , and objects X k ∈ C(A k ).Filtering K by finite categories, we can hence write X as a filtered colimit of objects of the form colim k∈K p k! X k with p k , X k as above and such that K is in addition finite.If X is now also compact, it is therefore a retract of such a finite colimit.

By the theorem, Nm fr
) is an equivalence for every k.The claim follows as f ! and f * preserve retracts and finite colimits (by stability). □ We now turn to the proof of Theorem 4.8.We begin with a T -presentable version of the first statement: Lemma 4.10.Let C be T -presentable and P -semiadditive, let p : C → A be a map in P , and let f : A → B be a map in T such that also f p belongs to P .Then Nm agrees with Nm f p .As both Nm f p and Nm p are equivalences by P -semiadditivity, the claim follows by 2-out-of-3.□ We will now deduce the theorem by reducing to the above lemma via an embedding trick. Lemma As the total rectangles only differ up to the naturality equivalences F f * ≃ f * F and F pr * 2 ≃ pr * 2 F , the first statement now follows from Corollary C.6 op .For the second statement we consider the diagram Here the unmarked squares commute by naturality, while the subdiagram ( ‡) commutes by Lemma C.2.The square ( †) commutes before inverting the horizontal equivalences by [CLL23a, Lemma 4.4.8op ]; since also the vertical maps are equivalences by the semiadditivity assumption on F , we conclude that also ( †) commutes.
where all unlabelled equivalences come from the compatibility of F with restrictions.By the coherence conditions for the latter, the composite of the middle row is then just induced by the equivalence pr * 2 F ≃ F pr * 2 ; with this established, all squares simply commute by naturality.The top right path F → pr 1 * F pr * 2 literally agrees with the bottom composite in the previous diagram; on the other hand, the left hand column spells out the definition of the map N : F → pr 1 * pr * 2 F .Thus, this diagram witnesses the desired identity, completing the proof of the proposition.□ Proof of Theorem 4.8.We will prove the first statement, the second one will then follow from this via Corollary 3.19, similarly to the proof of Corollary 4.9.
For this, let i again denote the monomorphism EF f → A. The claim then translates to demanding that the composite be an equivalence for every X ∈ ess im(p ! ) and Y ∈ C(B); similarly, Lemma 4.10 translates to saying that the analogously defined map is an equivalence in every P -semiadditive T -presentable D.
Passing to a larger universe, we may assume that C is small.We now employ Lemma 4.11 to obtain a T -presentable P -semiadditive D together with a fully faithful P -semiadditive functor C → D, and we consider the diagram where we have written [ , ] for Hom and have suppressed the naturality constraints of F for simplicity.Here the top row spells out (7) while the bottom row spells out the analogously defined map for D.
The two rectangles marked ( * ) commute by Proposition 4.13 while all other rectangles commute by naturality.By P -semiadditivity, F (X) lies in the essential image of p ! : D(C) → D(A), so Lemma 4.10 shows that the bottom composite On the other hand, Lemma 4.12 together with P -semiadditivity shows that the lower left vertical map BC * : ] is an equivalence.As F is fully faithful, also all top vertical maps are equivalences, so the claim now follows by 2-out-of-3.□

Adams isomorphisms in equivariant mathematics
Our main interest of the abstract Adams isomorphism from Theorem 4.8 is in the case where T = Glo and P = Orb.In this section, we will spell out the resulting Adams isomorphism in this setting, and provide a range of examples.
Definition 5.1 (N -free G-objects).Let N be a normal subgroup of a finite group G. Specializing Example 3.5 to the quotient map f : G → G/N , we get EF N := EF f ∈ Spc Orb (G), which we can equivalently view as the G-space satisfying For any equivariantly presentable global category C, we then define the subcategory C(G) N -free ⊆ C(G) of N -free G-objects as the full subcategory of (f : G → G/N )free objects in the sense of Definition 3.13, i.e. the subcategory of those objects X ∈ C(G) for which the map X ⊗ G EF N → X is an equivalence.
We then obtain the following specialization of the abstract Adams isomorphism: Theorem 5.2 (Abstract Adams isomorphism for global categories).Let C : Glo op → Cat be an equivariantly presentable equivariantly semiadditive global category and let N G be a normal subgroup of a finite group G. Assume that the N -fixed point functor (−) N : C(G) → C(G/N ) preserves (non-parametrized) colimits.Then there is a natural equivalence Proof.This is an instance of Theorem 4.8, applied to the quotient morphism f : G → G/N .The diagonal of f is the map G → G × G/N G, which is injective as it admits a retraction, and thus lies in Orb.□ In the remainder of this section, we discuss a wide range of examples of this theorem.
Example 5.3 (Equivariant homotopy theory).Let C = Sp be the global category of genuine equivariant spectra from Example 2.19.For any finite group H, the suspension spectra of transitive H-sets form a set of compact generators for Sp H , see [Hau17, Proposition 4.9(3)].Given a quotient map q : G → G/N , we immediately see that the inflation functor q * : Sp G/N → Sp G sends these compact generators to compact objects, so that its right adjoint (−) N preserves colimits.
Thus, we may apply the above theorem to Sp.The resulting equivalence X/N ≃ X N for N -free genuine G-spectra is precisely the classical Adams isomorphism.
Example 5.4 (Global equivariant homotopy theory).Similarly, consider the global category Sp gl of global spectra (Example 2.20).Given any quotient map q : G → G/N , the functor (−) N = q * : Sp gl G → Sp gl G/N is cocontinuous as a special case of Theorem A.1 in the appendix.Let us also give a direct argument for this: For each H, the objects φ !S for φ running through all homomorphisms into H form a set of compact generators of Sp gl H , see [CLL23a,Theorem 7.1.12].Taking H = G/N , the Beck-Chevalley condition shows that q * φ !S is equivalent to (G× G/N φ) !S, hence in particular again compact; as before, we then conclude that the right adjoint (−) N preserves colimits.
As a consequence of the theorem, we therefore get equivalences of G/N -global spectra X/N ≃ X N for any N -free G-global spectrum X, i.e. those X such that Φ φ X = 0 whenever im(φ) ∩ N ̸ = e, see Example 3.15.In the special case G = N an alternative approach to the Adams isomorphism for G-global spectra, based on the usual proof of the equivariant Adams isomorphism, appears in [Tig22].
Warning 5.5.As the global category Sp gl is even globally presentable, the restriction q * : C(G/N ) → C(G) has an honest left adjoint −/N , and the norm construction provides a comparison map X/N → X N for every G-global spectrum X, not just the N -free ones.Beware, however, that this map is typically far from being an equivalence.As a concrete example, let G = N = C 2 and X = S be the is the suspension spectrum of the global classifying space of C 2 , so its underlying non-equivariant spectrum is the suspension spectrum of the usual classifying space BC 2 .On the other hand, the underlying spectrum of X C2 is given by the categorical C 2 -fixed points of X, so the tom Dieck splitting identifies it with S ∨ Σ ∞ + BC 2 .Thus, already the zeroth non-equivariant homotopy groups of X/C 2 and X C2 differ.
Example 5.6.We further recall the global categories ΓS spc * and ΓS gl, spc * of equivariant and global special Γ-spaces, respectively (Example 2.21).Given any quotient map q : G → G/N , Theorem A.1 shows that the right adjoints q * = (−) N are cocontinuous, so the theorem provides us with Adams isomorphisms for N -free G-equivariant and G-global special Γ-spaces.In light of [Ost16, Theorem 5.9] and [Len20, Theorem 3.4.21],these can be viewed as a 'non-group completed' versions of the Adams isomorphisms discussed in the previous two examples, and in particular provide a further refinement of them in the connective case.
Example 5.7 (Mackey functors).Let E be a semiadditive category, and define a global category Mack E by Mack E (G) = Fun × (Span(Fin G ), E), the category of G-Mackey functors in E. We will show that Mack E admits abstract Adams isomorphisms.
Note that for an arbitrary finite groupoid A, viewed as a presheaf on Glo, one has One can show that for any fully faithful functor α : A → B between finite groupoids, the induction-restriction adjunction α !: Fin A ⇄ Fin B : α * satisfies the assumptions of [BH21, Corollary C.21], and so we obtain a pair of functors α !: Span(Fin A ) ⇄ Span(Fin B ) : α * such that α * is both left and right adjoint to α ! .The (co)unit of the adjunction is induced by the (co)unit of the original adjunction, and so the fact that these functors satisfy the appropriate base change can be deduced from the case of F Orb Glo .Applying [CLL23a, Remark 4.3.12],we conclude that Span(F Orb Glo ) is equivariantly semiadditive, and by 2-functoriality of Fun × (−, E) we deduce the same for Mack E .
It remains to show that the fixed point functors (−) N : Mack E (G) → Mack E (G/N ) preserve colimits.This is clear for sifted colimits, as these are computed pointwise in Mack E (G) [Lur09, Proposition 5.5.8.10(4)] and (−) N is given by precomposition.Because finite coproducts in Mack E (G) are also products, it follows that (−) N commutes with all colimits.We conclude that Mack E admits abstract Adams isomorphisms.
Example 5.8 (Representation theory).Let E be a presentable semiadditive category.Then one may consider the global category C(G) := Fun(BG, E), which is equivariantly presentable and equivariantly semiadditive by Proposition 4.3.3 and Proposition 4.4.9 of [HL13], respectively.For example: In this case, we do not in general have an abstract Adams isomorphism as the fixed point functors do not need to preserve colimits: as a concrete example, the functor (−) C2 : Fun(BC 2 , Vect F2 ) → Vect F2 does not preserve the exact sequence The following lemma gives a complete characterization for when C has abstract Adams isomorphisms, in terms of the notion of 1-semiadditivity from [HL13, Definition 4.4.2]: Lemma 5.9.Let E be a semiadditive category.The global category C(G) := Fun(BG, E) admits abstract Adams isomorphisms if and only if E is 1-semiadditive.
Proof.Since the category Fun(BG, E) is generated under colimits by objects induced from the trivial group, [HL13, Lemma 4.3.8], it follows that every object X ∈ C(G) is N -free for N ⩽ G.In particular, C admits abstract Adams isomorphisms if and only if the N -fixed point functors (−) N = (−) hN : C(G) → C(G/N ) preserve colimits.Since any morphism of groups factors as a quotient map followed by an injection, this is equivalent to the condition that f * preserves colimits for any morphism f : H → G of finite groups.But by [HL13, Proposition 4.3.9],this is equivalent to the condition that E is 1-semiadditive.□ Example 5.10 (Equivariant Kasparov categories).For a finite group G, we may consider the G-equivariant Kasparov category KK G = Ind(KK G sep ) from [BEL21].We claim that these categories assemble into an equivariantly presentable and equivariantly semiadditive global category KK, which we will call the global Kasparov category.We will further see that KK satisfies the conditions of Theorem 5.2, thus providing formal Adams isomorphisms for equivariant Kasparov categories.
We construct KK using the following four steps: (1) We start by considering the global category C * Alg nu sep : Glo op → Cat given by the assignment G → Fun(BG, C * Alg nu sep ), where C * Alg nu sep denotes the 1category of separated non-unital C * -algebras.
(2) For every finite group G, there is a class of morphisms in Fun(BG, C * Alg nu sep ) called the kk We will now show that KK is both equivariantly presentable and equivariantly semiadditive.Fiberwise semiadditivity is clear, as KK is even fiberwise stable by Theorem 1.3 of op.cit.Given an inclusion H ⩽ G of finite groups, the adjunction Res G H ⊣ Ind G H at the C * -algebra level4 descends by Theorem 1.22 of op.cit. to the level of equivariant Kasparov categories.Since the Beck-Chevalley conditions are satisfied at the level of C * -algebras, they descend to the level of equivariant Kasparov categories, showing that the global Kasparov category is equivariantly complete.Furthermore, Theorem 1.23(1) of op.cit.shows that the functor Ind G H : KK H → KK G is also a left adjoint to Res G H , as exhibited by an explicit unit transformation id → Res G H Ind G H constructed at the level of C * -algebras.One can deduce from [CLL23a,Lemma 4.3.11]that this map agrees with the dual adjoint norm map Nm : id → Res G H Ind G H constructed in [CLL23a, Construction 4.3.8],and using fiberwise semiadditivity of KK it then follows from [CLL23a, Lemma 4.5.4] that KK is even equivariantly semiadditive.Finally, equivariant presentability of KK is a consequence of fiberwise presentability and equivariant semiadditivity.
Since the inflation functor KK G/N → KK G , given by restriction along the quotient map G → G/N , preserves compact objects by construction of KK, its right adjoint (−) N : KK G → KK G/N preserves colimits; it thus follows from Theorem 5.2 that KK admits abstract Adams isomorphisms.
Remark 5.11.In the previous example, the abstract Adams isomorphism is not optimal: it is proven in [BEL21, Theorem 1.23] that the left and right adjoint to the inflation functor KK → KK G agree on all objects in KK G , not just the G-free ones, and this is expected to hold more generally for the inflation functor KK G/N → KK G for N G.

Appendix A. Cocontinuity for the universal examples
Consider an orbital category T and an atomic orbital subcategory P ⊆ T .In [CLL23b], the authors constructed the universal example of a P -semiadditive (resp.P -stable) P -presentable T -category, denoted by CMon P P ▷T (resp.Sp P P ▷T ).Similarly, [CLL23a] constructs the universal examples CMon P T and Sp P T of T -presentable Psemiadditive resp.P -stable T -categories.
The goal of this appendix is to show that the cocontinuity condition of Theorem 4.8 is always satisfied for these universal T -categories: Theorem A.1.Let f : A → B be any map in T .Then each of the functors As recalled in 2, for Orb ⊆ Glo these universal examples can be explicitly described via global or equivariant Γ-spaces and global or equivariant spectra, respectively, and this provides the cocontinuity used in Examples 5.4 and 5.6.
For the proof of the theorem, we recall that [CLL23a, Definition 4.8.1]constructs CMon P T as a certain subcategory of the internal hom Fun T (F P T, * , Spc T ), where F P T, * denotes the T -category of pointed objects in F P T .We therefore begin with a statement about general T -categories of copresheaves: Proposition A.2. Let I be any T -category and let f : A → B lie in F T .Then f * : Fun T (I, Spc T )(A) → Fun T (I, Spc T )(B) preserves sifted colimits.
Proof.We will prove the proposition by treating successively more general cases.
Step 1.We first treat the special case that I is terminal and the presheaves A and B are representable, so that f is a map in T .Identifying Spc T (A) = PSh(T ) /A ≃ PSh(T /A ) with Fun × (F T (A) op , Spc) and Spc T (B) ≃ Fun × (F T (B) op , Spc), the structure map f * : Spc T (B) → Spc T (A) is given by restriction along the opposite of . However, f ! has a right adjoint f * (given by pulling back along f ) by orbitality of T , and this preserves finite coproducts as the corresponding map of presheaf categories is even a left adjoint.Thus, under the chosen identifications f * : Spc T (A) → Spc T (B) is simply given by restricting along the opposite of f * : F T (B) → F T (A).
We now observe that Fun × (F T (A) op , Spc) ⊆ PSh(F T (A)) is closed under sifted colimits as sifted colimits in spaces commute with finite products, and similarly for Fun × (F T (B) op , Spc).Thus, sifted colimits on both sides can be computed pointwise and the claim follows immediately.
Step 2. Next, we consider the case that I is terminal and B is representable, so that f decomposes as a finite coproduct (f i ) : Then f * is simply given as the product of the individual f i * , so the claim follows from the previous step, using again that sifted colimits commute with products.
Step 3. Now we can treat the case that I is terminal and f : A → B is a general map in F T .Writing B as a colimit of representables, we see that the restrictions t * : Spc T (B) → Spc T (C) for all maps t : C → B with C ∈ T are jointly conservative.As they are moreover left adjoints, it will be enough to show that t * f * preserves sifted colimits for each such t.For this we consider the pullback diagram By the Beck-Chevalley condition for the T -complete category Spc T , we get t * f * ≃ g * u * .On the other hand, g is a map in F T , so the previous step shows that g * preserves sifted colimits.As u * is a left adjoint, the claim follows.
Step 4. Now assume I is a T -presheaf.Then f * : Fun T (I, C)(B) → Fun T (I, C)(A) agrees up to the equivalences from [CLL23a, Corollary 2.2.9] with the restriction (f × I) * : C(B × I) → C(A × I).As f × I is a pullback of f , it again lies in F T , so (f × I) * preserves sifted colimits by the previous step.
Step 5. Now we treat the general case.For each T -functor t : J → I, the functor t * : Fun T (I, C) → Fun T (J, C) preserves both T -limits and colimits by [MW24, Proposition 4.3.1].In particular, the previous step shows that preserves sifted colimits whenever J is a T -presheaf.As the functors t * for t running through all maps from T -presheaves to I are jointly conservative by [Mar21, Corollary 4.7.17], the claim follows.□ In order to apply this to CMon P T we will use: Lemma A.3.Let A ∈ PSh(T ).Then the full subcategory CMon P T (A) ⊆ Fun T (F P T, * , Spc T )(A) is closed under sifted colimits.
Proof.Decomposing A into representables and using that restriction functors are cocontinuous, we may assume that A itself is representable.
We write F P T, * × A for the cocartesian unstraightening of the functor T op → Cat, B → F P T, * (B) × Hom(B, A), and we will as usual denote objects in this unstraightening by triples (B, X + , f ) with B ∈ T op , X + ∈ F P T, * (B), and f : B → A. Then [CLL23a, Remark 2.2.14] provides an equivalence Fun T (F P T, * , Spc T )(A) ≃ Fun( F P T, * × A, Spc) and Remark 4.9.9 of op.cit.shows that CMon P T (A) corresponds under this equivalence to the full subcategory of all F : F P T × A → Spc satisfying the following conditions: (1) For every f : B → A in T the restriction of F to the non-full subcategory F P T (B) × {f } is semiadditive in the usual, non-parametrized sense.(2) For every p : B → A in P and f : A → A ′ in T a certain natural Segal map Obviously, the first property is stable under sifted colimits, while the second property is even stable under arbitrary colimits.□ Remark A.4.In fact, both the above lemma as well as its proof do not need any orbitality assumption on T .
Proof of Theorem A.1.For CMon P T , we observe that as f * is a right adjoint, it in particular preserves finite products.Since both source and target are semiadditive, we conclude that f * also preserves finite coproducts, so it only remains to show that it also preserves sifted colimits.The latter is however immediate by combining Proposition A.2 with Lemma A.3, finishing the argument for CMon P T .For CMon P P ▷T , we recall from [CLL23b, Theorems 6.18 and 6.19] that there exists a fully faithful T -functor ι !: CMon P P ▷T → CMon P T , such that ι !: CMon P P ▷T (X) → CMon P T (X) sits in a sequence of adjoint functors ι !⊣ ι * ⊣ ι * for every X ∈ T .The total mate of the structure equivalence ι !f * ≃ f * ι ! then provides an equivalence f * ι * ≃ ι * f * .Combining this with full faithfulness of ι !, we then conclude that f * : CMon P P ▷T (A) → CMon P P ▷T (B) agrees with the composite In [San19, Theorem 2.9], Sanders provides an alternative treatment of formal Adams isomorphisms.In this appendix, we discuss the relationship between his approach and ours.
Sanders' theorem is formulated in the context of geometric functors f * : D → C between rigidly-compactly generated tensor-triangulated categories.In the language of higher category theory, this corresponds to the condition that D and C are compactly generated stable symmetric monoidal ∞-categories whose compact objects are precisely the dualizable objects, and f * is a symmetric monoidal colimitpreserving functor.This implies that f * admits a right adjoint f * satisfying the projection formula, which in turn admits a further right adjoint f ! .
Consider now a full subcategory B ⊆ C, and suppose that the inclusion i !: B → C fits in an adjoint triple i !⊣ i * ⊣ i * , so that B is in particular closed under colimits in C. In total we have where the dotted maps indicate that the adjoints do not necessarily exist.Sanders proves the following theorem: Theorem B.1 ([San19, Theorem 2.9]).Consider the previous situation, and assume further that B is generated under colimits by the thick tensor ideal The composite i * f * : D → B admits a left adjoint, which we will denote by and there is a canonical equivalence which we call the Wirthmüller isomorphism associated to f * and B.
In order to be able to compare the Wirthmüller isomorphism ( (3) There exists a preferred equivalence i * ω f ≃ 1 C(EF f ) such that the composite In part (3), we have identified C(A) f -free with C(EF f ) in light of Lemma 3.16.
Proof.Part (1) follows immediately from the fact that f * is by assumption a geometric functor between rigidly compactly generated categories.Next we show part (2).For this we further note that the inclusion i !: B = C(A) f -free → C(A) admits a right adjoint i * which admits a further right adjoint i * by Lemma 3.16.It remains to check that B is generated under colimits by the subcategory By Corollary 3.19, B is generated under colimits by the objects of the form i !X for i : C → A in F ⊆ P /A and X ∈ C(C).Since C(C) is compactly generated, we may in fact restrict to compact objects X ∈ C(C).It thus remains to show that i !X is contained in I for every compact object X ∈ C(C).In other words, we have to show that for every c ∈ C(A) c , the object f * (c ⊗ i !X) is again compact.But this is a consequence of the equivalences using the fact that the functors i * and (f i) !preserves compact objects (as their right adjoints preserve colimits) and the fact that compact objects in C(C) are closed under tensor product (as they agree with the dualizable objects).This finishes the proof of part (2).
It remains to show that under this equivalence, the norm map Nm f agrees with the composite where the last equivalence is the projection formula for i ! .Equivalently, by passing to right adjoints, it will suffice to show that the equivalence i * f * ∼ − − → i * f !agrees under this equivalence with the map where the last equivalence is the closed monoidal structure on i * , obtained by adjunction from the projection formula for i ! .By the definition of the maps ψ and ψ ad in [San19, Remark 2.15], the latter composite is adjoint to the map and hence it will suffice to show that the following diagram commutes: Observe that this is a special case of the statement that the equivalence is C(B)-linear, in the sense that for all objects X, Y ∈ C(B) the following diagram commutes: This statement is a somewhat tedious exercise in calculus of mates: one needs to show that all the Beck-Chevalley transformations used in the definition Nm fr f are compatible with the projection formula equivalences.We will only sketch the proof.Looking at the zig-zag (5) on page 15, one observes that that the maps ε and BC * are compatible with the projection formula equivalences, and it thus remains to show the map Nm ∆ is also compatible.More generally, we will argue that for every morphism p : A → B in F P T the norm equivalence Nm p : p ! ∼ − − → p * is compatible with the projection formula equivalences.By adjunction, it again suffices to show this for the map Nm p : id → p * p * from [CLL23a, Construction 4.3.8],which is defined using the analogous zig-zag (5).We thus obtain a further reduction to the statement for Nm ∆p : (∆ p ) ! Proof.This is immediate from Proposition B.2, using the observation that every object in C(A) is f -free (Example 4.2).□ Remark B.6.One can show that the usual smash product of spectra makes Sp gl into a symmetric monoidal global category such that the tensor product even preserves Glo-colimits in each variable.However, in Sp gl G the dualizable objects do not coincide with the compacts, so that the global Adams isomorphism is not an instance of Sanders' framework: If p : G → e denotes the unique homomorphism, then p * Σ ∞ + B gl C 2 ≃ p * (S/C 2 ) is compact (Example 5.4), but its underlying non-equivariant spectrum is Σ ∞ + BC 2 , which is not compact and hence not dualizable.As the forgetful functor from Gglobal spectra to spectra is symmetric monoidal, it preserves dualizable objects, and accordingly the G-global spectrum p * Σ ∞ + B gl C 2 is not dualizable either.

Appendix C. The calculus of mates
In this final appendix we recall some general identities involving Beck-Chevalley transformations (or mates).
Throughout, we work in a fixed ambient (2, 2)-category U, which we for simplicity assume to be strict; the example we will apply these results to in the main text is the strict (2, 2)-category of (∞-)categories, functors, and homotopy classes of natural transformations.Recall once more that for a square in which the two vertical maps admit left adjoints f !, g !, the Beck-Chevalley transformation BC != BC !(σ) : g !u * → v * f ! is defined as the composite It will be convenient for us to write this composite in a more diagrammatic fashion as the pasting We point out that any such pasting diagram indeed has an unambiguous composite (i.e. the resulting natural transformation is independent of the order in which we paste the individual 2-cells); see [Pow90] for a precise statement and proof.
Remark C.1.If u * and v * admit right adjoints u * , v * , there is a dually defined Beck-Chevalley transformation BC * : f * v * → u * g * .Below we will restrict to the case of the map BC !, the other case being formally dual.
Lemma C.2.Consider a square (8) such that f * and g * admit left adjoints f ! and g !, respectively.Then the following diagrams of 2-cells commute: Proof.We will prove the first statement (which is the only one used in the main text); the proof of the second statement is similar.
Plugging in the definition of BC !, the lower left composite is given by the pasting where the vertical equivalences are induced by the given equivalences between the right adjoints.
Proof.Applying the previous lemma iteratively to agrees with the Beck-Chevalley map for the total rectangle, when we take the left adjoints of f * h * and g * k * to be h !f ! and k !g !, respectively, with the induced unit and counit.□ As in Corollary C.4 (or by direct inspection), we in particular conclude: Corollary C.6.Assume once more we are given a diagram (8) whose vertical maps admit left adjoints, and assume we are further given equivalences of 1-morphisms u ′ * ≃ u * and v ′ * ≃ v * .Then we have a commutative diagram Definition 2.1.A T -category is a functor C : T op → Cat.If C and D are Tcategories, then a T -functor F : C → D is a natural transformation from C to D. The category Cat T of T -categories is defined as the functor category Cat T := Fun(T op , Cat).
Lemma 3.16.Let C, A, and F be as in Definition 3.13.Then the restriction functor i * : C(A) → C(EF) along the map i : EF → A in PSh(P ) admits fully faithful left and right adjoints i !, i * : C(EF) → C(A) whose essential images are C(A) F -tors and C(A) F -comp , respectively.
Example 4.3.Let T = Glo, P = Orb, and let C = Sp be the global category of equivariant spectra from Example 2.19.If f is the projection G → G/N for a normal subgroup N G, then Example 3.14 shows that a G-spectrum X ∈ Sp G is f -free if and only if it is N -free in the sense of the introduction.
Remark 4.7.If C is P -semiadditive and T -presentable, the Beck-Chevalley map f * f * → pr 1 * pr * 2 is already an equivalence before applying i * , and so (5) determines a map Nm f : idC(A) → f * f * with i * Nm f = Nmfr f .Moreover, f * admits a globally defined left adjoint f !, and Nm f adjoints to a map Nm f : f !→ f * restricting to Nm fr f on C(A) f -free .The maps Nm f and Nm f are then literally instances of (the dual of) the norm construction of Hopkins and Lurie [HL13, Construction 4.1.8].

Lemma 4. 12 .
Let C, D be T -categories with finite P -products and let F : C → D preserve finite P -products.Let f : B → C be a map in T such that both f * : C(C) → C(B) and f * : D(C) → D(B) admit right adjoints f * , and let p : A → B be a map in P such that also f p belongs to P .Then the Beck-Chevalley map F f * X → f * F X is an equivalence whenever X ∈ C(A) lies in the essential image of p * : C(C) → C(A).Proof.We may assume without loss of generality that X = p * Y with Y ∈ C(A).By the compatibility of Beck-Chevalley maps with composition (Lemma C.3 op ), the composite F f * p * Y → f * F p * Y → f * p * F Y of Beck-Chevalley maps then agrees with the Beck-Chevalley map F f * p * Y → f * p * F Y , which we can further identify up to conjugation by the obvious equivalences with the Beck-Chevalley map F (f p) * Y → (f p) * F (see Corollary C.4 op ).As p and f p are maps in P and F preserves finite P -products, the claim now simply follows by 2-out-of-3.□ Proposition 4.13.Let F : C → D be a P -semiadditive functor of P -semiadditive T -categories.Let f : A → B be a map in T such that the diagonal ∆ : A → A × B A lies in F P T and such that all the requisite adjoints exist to define Nm fr f for C and D. Then the diagrams G 0 -equivalences by op.cit..By [BEL21, Theorem 1.22], the restriction functor Res G H : Fun(BG, C * Alg nu sep ) → Fun(BH, C * Alg nu sep ) for a group homomorphism H → G sends kk G 0 -equivalences to kk H 0 -equivalences.In particular, C * Alg nu sep refines to a functor C * Alg nu sep : Glo op → RelCat, where RelCat denotes the category of relative categories.(3) Composing C * Alg nu sep with the Dwyer-Kan localization functor L : RelCat → Cat gives a global category KK sep : Glo op → Cat.By definition its value at G is the category KK G sep of [BEL21, Definition 1.2].(4) Finally, we define the global category KK as the composite KK : Glo op KK sep − −−− → Cat Ind − − → Pr L .It is given on objects by G → KK G and on morphisms by sending a group homomorphism H → G to the restriction functor Res G H : KK G → KK H constructed in [BEL21, Theorem 1.22].
CMon P T (B), each of which is cocontinuous.Next, we recall [CLL23b, Definition 8.9] that Sp P T is obtained from CMon P T by pointwise taking the tensor product in Pr L with the category Sp of spectra.By the above, f * : CMon P T (B) ⇄ CMon P T (A) : f * is an internal adjunction in Pr L .By 2-functoriality of Sp ⊗ -, we can therefore identify f * : Sp P T (B) → Sp P T (A) with the cocontinuous functor Sp ⊗ f * : Sp ⊗ CMon P T (A) → Sp ⊗ CMon P T (B).The argument for Sp P P ▷T is analogous.□ Appendix B. Comparison with Sanders' approach where C c ⊆ C and D c ⊆ D denote the subcategories of compact (equivalently: dualizable) objects.Define ω f := f ! 1 D .Then: (a) The composite i * f !: D → B has a right adjoint; (b) There is a canonical equivalence i * (ψ) : i Sanders to our abstract Adams isomorphism, let us briefly summarize the proof of Theorem B.1: (1) Part (a) follows from the fact that left adjoint f * i !: B → C of i * f !sends the generating set of compact objects I ⊆ B to compact objects in D, by definition of I. (2) Part (b) is equivalent to the statement that the lax D-linear functor i * f ! is in fact D-linear; since it preserves colimits by (a) and its left adjoint f * i ! is D-linear, this is an instance of a general fact: for D rigidly compactly generated, the right adjoint G : N → M of a D-linear left adjoint F : M → N between D-linear categories is again D-linear as soon as G preserves colimits.(3) To construct the left adjoint in part (d), it suffices to do so objectwise on the generators b ∈ I ⊆ B. There one can show that we have b ∨ ∈ I, and a computation shows that the desired object (f i) !(b) is given by f * (b ∨ ) ∨ .(4) Finally, checking that the canonical comparison maps in (c) and (d) are equivalences is quite intricate, see [San19, Proposition 2.31].However, when there is an equivalence i * ω f ≃ 1 B (as will be the case in the context we are interested in) the equivalences in (c) and (d) are direct consequences of the equivalence of (b).We now have the following comparison result: Proposition B.2.Let C be a symmetric monoidal T -category, that is, a functor C : T op → CMon(Cat).Assume that C(C) is rigidly compactly generated for all C ∈ T .Let P ⊆ T be an atomic orbital subcategory and assume that C is Ppresentable and P -semiadditive, and that the tensor product − ⊗ − : C × C → C preserves P -colimits in each variable, in the sense of [MW22, Definition 8.1.1].Let f : A → B be a morphism in T whose diagonal is in F P T .Then: (1) The functor f * : C(A) → C(B) preserves colimits, and so Theorem 4.8 provides an abstract Adams isomorphism Nm fr f : f fr !∼ − − → f * on f -free objects.(2) The functor f * : C(B) → C(A), equipped with the subcategory B := C(A) f -free ⊆ C(A) of f -free objects, satisfies all the assumptions of Theorem B.1, thus giving a Wirthmüller isomorphism ϖ For part (3), consider the abstract Adams isomorphism Nmf : f fr !∼ − − → f * i !(−) fromTheorem 4.8.Passing to right adjoints gives an equivalence (Nm f ) ad : i * f * ∼ − − → i * f !, and evaluating at 1 C(B) provides an equivalence By passing to left adjoints, this is equivalent to the norm map Nm fr f : f fr !∼ − − → f * i !being C(B)-linear, which in turn is equivalent to the map Nm fr f : i * → i * f * f * being C(B)-linear, i.e. for all X ∈ C(A) and Y ∈ C(B) the following diagram commutes: i

∼
− − → (∆ p ) * .Since ∆ p is a disjoint summand inclusion in PSh(T ), the statement here follows from the uniqueness statement from[CLL23a,  Lemma 4.1.4].□ Remark B.3.In the language of [San19], we have in particular shown that the category C(A) f -free of f -free objects is contained in the compactness locus of the functor f * .In general the compactness locus may be larger, for example in the case of the inflation functor Sp G/N → Sp G , see Theorem 6.2 of op.cit.Remark B.4.In [BDS16, Theorem 1.7], Balmer, Dell'Ambrogio, and Sanders prove a formal Wirthmüller isomorphism in the context of rigidly-compactly generated tensor-triangulated categories: given a geometric functor f * : D → C whose right adjoint f * : D → C preserves compact objects, the functor f * admits a left adjoint f ! and there is an ur-Wirthmüller isomorphism ϖ : f !(−) ∼ − − → f * (ω f ⊗ −).Since the theorem by Sanders mentioned above reduces to this theorem by taking B = C, it follows that Proposition B.2 provides an equivalence between the ur-Wirthmüller isomorphism from op. cit.and our norm equivalences: Corollary B.5.In the situation of Proposition B.2, assume the map f : A → B is a morphism in P .Then we have B = C(A), and there exists a preferred equivalence ω f ≃ 1 C(A) such that the composite f !(−) ϖ − → f * (ω f ⊗ −) ≃ f * agrees with the norm map Nm f : f !→ f * .
identity for the unit and counit of the adjunction f !⊣ f * , the pasting of the shaded subdiagram is the identity transformation f * ⇒ f * .Thus, the pasting (10) simplifies to the pasting of the unshaded portion on the right, which precisely gives the other composite in the diagram (9).□Finally, let us explain in which way Beck-Chevalley maps compose.All of the following statements are instances of [KS74, Proposition 2.2], and they are explicitly verified in loc.cit.using similar pasting arguments to the one presented above.We begin with horizontal compositions of Beck-Chevalley transformations:Lemma C.3.Assume that all the vertical maps in the diagram admit left adjoints.Then the compositeh !w * u * x * g !u * x * v * f !BC !(τ ) BC !(σ)of the Beck-Chevalley maps for the two little squares agrees with the Beck-Chevalley map of the total rectangle.□Using this, we can establish one half of a 'homotopy invariance' property of Beck-Chevalley transformations:Corollary C.4.Assume once more we are given a diagram (8) whose vertical maps admit left adjoints, and assume we are further given equivalences of 1-morphisms f ′ * ≃ f * and g ′ * ≃ g * (in particular, f ′ * and g ′ * again admit left adjoints).Then the Beck-Chevalley transformations fit into a commutative diagram immediately yields the claim.□We can also compose Beck-Chevalley transformations vertically:Lemma C.5.Assume that all vertical maps in the diagram The wide subcategory Orb ⊆ Glo from Example 2.4 is an orbital subcategory of Glo, see [CLL23a, Example 4.2.5].Example 2.10.A T -category C is said to admit finite P -coproducts if it admits F P Tindexed colimits in the sense of [CLL23a, Definition 2.3.8].Explicitly, this means that the restriction functor p * : C(B) → C(A) admits a left adjoint p ! : C(A) → C(B) for every morphism p : A → B in F P T , and for every pullback square (1) the Beck-Chevalley transformation 2 T -functor F : C → D preserves finite P -coproducts if for everyp : A → B in F P T the Beck-Chevalley map p ! F A → F B p ! (the mate of the naturality constraint F A f * ≃ f * F B ) is an equivalence.If C is in addition fiberwise stable, i.e. it factors through the category of stable categories and exact functors, then we will call it P -stable.We say that a T -category C is P -presentable if it admits finite Pcoproducts and is fiberwise presentable in the sense that the functor C : T op → Cat factors through the subcategory Pr L ⊆ Cat.Definition 2.16.We say that a T -category C is T -presentable if it is fiberwise presentable and T -cocomplete, i.e. for every f : A → B in T (or equivalently in PSh(T )) the restriction f * admits a left adjoint f !satisfying base change.
makes precise that it is the free equivariantly presentable and stable global category on one generator.Example 2.20.We can also localize Fun(BG, Sp Σ ) at the G-global stable weak equivalences of [Len20, Definition 3.1.28],i.e. those maps f such that the underived restriction φ * f is an H-stable weak equvialence for every φ : H → G, yielding the category Sp gl G of G-global spectra.These again fit together into a global category Sp gl , which we call the global category of global spectra.It is the free globally presentable equivariantly stable global category on one generator, see [CLL23a, Theorem 7.3.2].Its underlying category Sp gl e is the usual category of global spectra (with respect to finite groups) in the sense of [Sch18], see [Hau19, Theorem 5.3].
On the other hand, specializing it to the underlying Orb-category of Sp gl we get a G-global refinement of this result in the form of equivalences Proof of Corollary 3.17.This follows immediately by combining the lemma with the descent equivalence C(EF) ≃ lim B∈F op C(B) induced by the equivalence EF ≃ colim B∈F B from Remark 3.7.□Remark3.18.The lemma in particular tells us that i * is a localization in the above situation.More generally, for any P -category C, we will refer to maps f in C(A) for which i * f is an equivalence as F-weak equivalences.Using descent once more, we see this is equivalent to p * f being an equivalence for every p ∈ F.As another application of the lemma we obtain the following decomposition result:Corollary 3.19.The subcategory C(A) F -tors ⊆ C(A) is closed under colimits and contains all objects of the form p ! Y for (p : C → A) ∈ F and Y ∈ C(C).Conversely, every F-torsion object X in C(A) can be written as a colimit of objects of this form.
To see that also the final remaining subdiagram ( * ) commutes, we first note that by another application of Lemma C.3 op the right hand vertical composite F pr 1 * ∆ * ∆ * pr * 2 → pr 1 * ∆ * F ∆ * pr * 2 is induced by the Beck-Chevalley map F pr 1 * ∆ * → pr 1 * ∆ * F for the adjunction ∆ * pr * 1 ⊣ pr 1 * ∆ * .By Corollary C.4 op we can therefore rewrite the top right composite as F ≃ pr 1 * ∆ * F ≃ pr 1 * ∆ * F ∆ * pr * 2 .As this further agees with the bottom left composite by naturality, this completes the proof that ( * ) and hence the whole diagram commutes.
The top row spells out the definition of F (N ) : F → F pr 1 * pr * 2 ; it therefore remains to show that the lower composite F → pr 1 * F pr * 2 agrees with the composite F → pr 1 * pr * 2 F ≃ pr 1 * F pr * 2 .For this we contemplate the diagram