Theory and Applications of Categories, Vol. 5, No. 9, 1999, pp. 202–250.

WHEN PROJECTIVE DOES NOT IMPLY FLAT, AND OTHER HOMOLOGICAL ANOMALIES

L. GAUNCE LEWIS, JR.

Transmitted by Ronald Brown

ABSTRACT. If 𝔐 is both an abelian category and a symmetric monoidal closed category, then it is natural to ask whether projective objects in 𝔐 are flat, and whether the tensor product of two projective objects is projective. In the most familiar such categories, the answer to these questions is obviously yes. However, the category 𝔐_{G} of Mackey functors for a compact Lie group G is a category of this type in which projective objects need not be so well-behaved. This category is of interest since good equivariant cohomology theories are Mackey functor valued. The tensor product on 𝔐_{G} is important in this context because of the role it plays in the not yet fully understood universal coefficient and Künneth formulae. This role makes the relationship between projective objects and the tensor product especially critical. Unfortunately, if G is, for example, O(n), then projectives need not be flat in 𝔐_{G} and the tensor product of projective objects need not be projective. This misbehavior complicates the search for full strength equivariant universal coefficient and Künneth formulae. The primary purpose of this article is to investigate these questions about the interaction of the tensor product with projective objects in symmetric monoidal abelian categories. Our focus is on functor categories whose monoidal structures arise in a fashion described by Day. Conditions are given under which such a structure interacts appropriately with projective objects. Further, examples are given to show that, when these conditions aren’t met, this interaction can be quite bad. These examples were not fabricated to illustrate the abstract possibility of misbehavior. Rather, they are drawn from the literature. In particular, 𝔐_{G} is badly behaved not only for the groups O(n), but also for the groups SO(n), U(n), SU(n), Sp(n), and Spin(n). Similar misbehavior occurs in two categories of global Mackey functors which are widely used in the study of classifying spaces of finite groups. Given the extent of the homological misbehavior in Mackey functor categories described here, it is reasonable to expect that similar problems occur in other functor categories carrying symmetric monoidal closed structures provided by Day’s machinery.

Contents

1 Compatibility questions for symmetric monoidal closed abelian categories 205

2 Day’s symmetric monoidal closed structures 208

3 Positive results on functor categories 213

Received by the editors 1999 May 4 and, in revised form, 1999 September 6. Published on 1999 September 16. 1991 Mathematics Subject Classification: Primary: 18D10, 18D15, 18G05, 19A22, 55M35, 55N91, 57S15; Secondary: 18E10, 55P91, 57S10. Key words and phrases: closed category, symmetric monoidal category, Mackey functor, projective, flat, compact Lie group. \begin{matrix}c \\ ◯ \end{matrix} L. Gaunce Lewis, Jr. 1999. Permission to copy for private use granted.

4 An introduction to (G, U)-Mackey functors 217

5 An introduction to globally defined Mackey functors 219

6 A bestiary of symmetric monoidal closed abelian categories 221

7 Mackey functors for incomplete universes and the proof of Theorem 6.1 224

8 Mackey functors for compact Lie groups and the proof of Theorem 6.5 230

9 Mackey functors for compact Lie groups and the proof of Theorem 6.9 235

10 S^{1}-Mackey functors and the proof of Theorem 3.8 239

11 Globally defined Mackey functors and the proof of Theorem 6.10 243

Introduction

Let 𝔐 be a symmetric monoidal closed abelian category. The tensor product of two objects M and N in 𝔐 is denoted M □ N. The closed structure of 𝔐 is provided by an internal hom functor adjoint to the tensor product; the result of applying this functor to M and N is an object of 𝔐 denoted ⟨M, N⟩. In contrast, the abelian group of morphisms from M to N in 𝔐 is denoted 𝔐(M, N).

It is often important to know how well the symmetric monoidal closed and abelian category structures on 𝔐 interact. Two obvious questions about this interaction are whether projective objects are flat and whether the tensor product of two projectives is projective. In the classical examples, these questions are answered by the dicta “projective implies flat” and “the tensor product of projectives is projective”. Unfortunately, these familiar dicta fail wildly in some less well known examples of symmetric monoidal closed abelian categories.

One of the primary purposes of this article is to describe some examples of this homological misbehavior which arise in the study of Mackey functors. This context yields three families of symmetric monoidal closed abelian categories in which projectivity does not imply flatness. In one of these families, the tensor product of two projectives need not be projective. These examples were not contrived merely to demonstrate the theoretical possibility of this sort of misbehavior. One of these families has appeared repeatedly in the literature in both representation theory and stable homotopy theory [1, 2, 4, 5, 7, 9, 10, 12– 14, 24–26, 28, 29]. A second family, the one in which both homological anomalies occur, provides one of the very few reasonable definitions of a Mackey functor for a compact Lie group [21]. It is unlikely that the other definitions are better behaved in this regard. The misbehavior of projectivity with respect to tensor products is especially serious in this category because of the important role these two notions play in universal coefficient and Künneth theorems. The third family of categories arises naturally in the study of the equivariant Hurewicz and suspension homomorphisms in equivariant homotopy theory [18, 19].

Our two questions about the behavior of projective objects in symmetric monoidal closed abelian categories fit naturally into a larger group of six interrelated questions about the interaction of the monoidal closed and abelian structures on 𝔐. The other four questions in this group are less frequently discussed. Like our question about the relation between projectivity and flatness, each of these four deals with a pair of functors from the list

⟨M,?⟩ : 𝔐 [figure]𝔐 ⟨?, M⟩ : 𝔐^{op} [figure]𝔐
𝔐(M,?) : 𝔐 [figure]Ab 𝔐(?, M) : 𝔐^{op} [figure]Ab
M□? : 𝔐 [figure]𝔐,

and asks whether the exactness of one of the functors in the pair implies the exactness of the other.

Some sense of the importance of one of the four is easily conveyed. This question asks whether the exactness of the functor ⟨M,?⟩ implies the exactness of the functor 𝔐(M,?). Assume that R is a commutative ring, G is a finite group, and R[G] is the group ring of G over R. The categories of R-modules and of R[G]-modules are the classic examples of symmetric monoidal closed abelian categories. In the category of R-modules, the exactness of ⟨M,?⟩ implies the exactness of 𝔐(M,?) because the two functors differ only in that 𝔐(M, N) is the abelian group underlying the R-module ⟨M, N⟩. However, if M is taken to be R with trivial G-action in the category of R[G]-modules, then the functor ⟨M,?⟩ is exact for formal reasons, whereas the failure of the exactness of 𝔐(M,?) is the source for group cohomology. Thus, the relation between the exactness of ⟨M,?⟩ and 𝔐(M,?) is not obvious and can have far-reaching consequences. A second purpose of this article is to call attention to these four infrequently discussed questions, and to describe their relation to the two more familiar questions in our group of six.

All of the symmetric monoidal closed abelian categories considered here are functor categories, and their monoidal structures are defined using a general procedure introduced by Day [6]. Roughly speaking, Day provides a symmetric monoidal closed structure on a functor category whose domain category O is a full subcategory of a symmetric monoidal category S. The behavior of the projective objects in these functor categories can be related to various properties of O. In particular, if O is closed under the product operation on S, then the tensor product of projective objects in the associated functor category is projective (see Proposition 3.2). Further, if S is a symmetric monoidal closed category and O is closed under the function object construction on S, then projective objects in the functor category are flat (see Proposition 3.3). A third purpose of this article is to introduce these sufficient conditions for the good behavior of functor categories.

When the domain category O does not satisfy these simple closure conditions, the interaction between the monoidal closed and abelian structures on the associated functor category can be extremely bad. In particular, if G is a compact Lie group which contains the orthogonal group O(2) as a subquotient, then there are non-flat projective objects in the category of G-Mackey functors (see Example 6.7). Moreover, in the category of O(n)-Mackey functors, for n ≥ 2, there are pairs of projective objects whose tensor product is not projective (see Theorem 6.9). Badly behaved examples like these appear so ubiquitously in the context of Mackey functors that it is only prudent to assume that the interaction is bad in any case where our closure conditions fail. One is almost tempted to conjecture that our sufficient conditions are also necessary. The behavior of functor categories is not, however, this predictable. One would expect badly behaved projective objects in the category of S^{1}-Mackey functors since the appropriate domain category fails to satisfy our closure conditions. However, in this functor category, projective implies flat, and the tensor product of a pair of projective objects is projective (see Theorem 3.8).

The tensor product and internal hom functors which provide Day’s symmetric monoidal closed structures are, to say the least, usually not easily computed. The final purpose of this article is to describe techniques which can be used to manipulate these functors.

In the first section, we give the precise statements of our six homological questions, and make a few observations about relations among them. The next section contains a review of Day’s monoidal closed structures, and an introduction to the relevant examples of categories carrying these structures. The focus of the third section is on our positive results guaranteeing that a functor category carrying one of Day’s monoidal structures satisfies our various compatibility conditions. The fourth and fifth sections contain more detailed descriptions of our families of examples. Section six is devoted to the statements of our negative results indicating how badly behaved the tensor product can be in a functor category to which the results of section three are inapplicable. The remaining sections provide the proofs of various results stated in sections three and six. In particular, our negative results about the category of Mackey functors associated to an incomplete indexing universe are proven in section seven. Sections eight and nine contain the proofs of our results about the homological misbehavior of the category of Mackey functors for a compact Lie group. Section ten provides the proofs of our positive results on the category of S^{1}-Mackey functors. Our results about the homological misbehavior of globally defined Mackey functors are proven in the last section.

1. Compatibility questions for symmetric monoidal closed abelian categories

Here, the compatibility questions of interest to us are phrased in terms of six axioms which a symmetric monoidal closed abelian category 𝔐 may satisfy. Certain fundamental connections between these axioms are also explored. Four of these six axioms are clearly satisfied by both the category R-Mod of modules over a commutative ring R and the category R[G]-Mod of modules over the group ring R[G] of a finite group G over the ring R. The other two axioms are obviously satisfied by R-Mod, but are typically not satisfied by R[G]-Mod. The nature, and significance, of the misbehavior of R[G]-Mod with respect to these two axioms is therefore discussed here.

1.1. Definition.

(a) Since an object M of 𝔐 is said to be projective if the functor 𝔐(M,?) : 𝔐 [figure]Ab is exact, an object M is said to be internally projective if the functor ⟨M,?⟩ : 𝔐 [figure]𝔐 is exact.

(b) Since M is injective if the functor 𝔐(?, M) : 𝔐^{op} [figure]Ab is exact, M is said to be internally injective if the functor ⟨?, M⟩ : 𝔐^{op} [figure]𝔐 is exact.

Our 6 axioms which may be satisfied by a symmetric monoidal closed abelian category 𝔐 are:

PiF If an object M of 𝔐 is projective, then it is also flat.

PiIP If an object M of 𝔐 is projective, then it is internally projective.

IiII If an object M of 𝔐 is injective, then it is internally injective.

IPiP If an object M of 𝔐 is internally projective, then it is projective.

IIiI If an object M of 𝔐 is internally injective, then it is injective.

TPPP If the objects M and N of 𝔐 are projective, then so is M □ N.

In the introduction, it was noted that IPiP holds in the category R-Mod of modules over a commutative ring R, but fails in the category R[G]-Mod of modules over the group ring R[G] of a finite group G over R. The argument for the failure of IPiP in R[G]-Mod made use of R considered as an R[G]-module with trivial G-action. The significance of the R[G]-module R in this context is that it is the unit for the tensor product on R[G]-Mod. The question of whether 𝔐 satisfies the axioms IPiP and IIiI is closely tied to the behavior of the unit for the tensor product on 𝔐, which we denote by • (or •_{𝔐}). In particular, the canonical unit isomorphism ⟨• , M⟩ ≅ M of 𝔐 implies that • is internally projective. Thus, IPiP can’t be satisfied unless • is projective.

1.2. Lemma.

(a) The category 𝔐 satisfies IPiP if and only if the unit • for the tensor product on 𝔐 is projective.

(b) The category 𝔐 satisfies IIiI if the unit • for the tensor product on 𝔐 is projective.

Proof. We have already noted that • must be projective if 𝔐 satisfies IPiP. The rest of part (a) and part (b) both follow easily from the natural isomorphism 𝔐(• , ⟨M, N⟩) ≅ 𝔐(M, N).

The question of whether the unit • for 𝔐 is projective seems to have an significance far beyond this lemma. There is a theory of localization for well-behaved symmetric monoidal closed abelian categories which generalizes the classical theory of localization for a commutative ring and its modules. This theory can be developed using techniques drawn from the theory of localization for noncommutative rings [27]. The various localizations of a category 𝔐 provided by this theory are all symmetric monoidal closed abelian categories. In the most favorable cases, such a localization of 𝔐 can can be identified as the category of modules over the localization of the unit • of 𝔐. However, as one might expect from the theory of localization of noncommutative rings, the typical localization of 𝔐 is a more subtle construction. The simplest indicator of whether the localized category is of this more subtle kind is that the more subtle construction occurs precisely when the unit of the localized category fails to be projective in that category.

The hom/tensor adjunction which provides the closed structure on 𝔐 implies that the remaining 4 axioms fall into two pairs of roughly equivalent conditions. In categories of functors into Ab, which always have enough projectives and injectives, the conditions in each pair are, in fact, equivalent.

1.3. Proposition.

(a) If 𝔐 satisfies PiIP, then it also satisfies TPPP. Moreover, if 𝔐 has enough projectives, then TPPP implies PiIP.

(b) If 𝔐 has enough projectives, then PiF implies IiII. If 𝔐 has enough injectives, then IiII implies PiF.

Proof. Let f : L [figure]M and g : M [figure]N be a monomorphism and an epimorphism in 𝔐, respectively. Both implications in part (a) follow from the equivalence of the lifting problems represented by the two diagrams

\xymatrix{ & P □ Q \ar[d]^{h} \ar@{-->}[dl]_{H} &  & P \ar[d]^{h̃} \ar@{-->}[dl]_{H̃} \\ M \ar[r]_{g} & N & ⟨Q, M⟩ \ar[r]_{g_{∗}} & ⟨Q, N⟩,}

in which P and Q are projective objects in 𝔐. Both implications in part (b) follow from the equivalence of the lifting problems represented by the two diagrams

\xymatrix{I &  &  & P \ar[d]^{k̃} \ar@{-->}[dl]_{K̃} \\ P □ L \ar[r]_{1□f} \ar[u]^{k} & P □ M \ar@{-->}[ul]_{K} & ⟨M, I⟩ \ar[r]_{f^{∗}} & ⟨L, I⟩,}

in which P and I are projective and injective in 𝔐, respectively.

2. Day’s symmetric monoidal closed structures

The categories of greatest interest in this article are functor categories with symmetric monoidal closed structures of the sort described by Day in [6]. However, only a rather simple case of Day’s general approach is needed for our examples. In this section, that case is reviewed, and our basic examples of functor categories carrying Day’s structures are introduced.

Let S be a symmetric monoidal Ab-category whose tensor product ∧ is bilinear, and O be a skeletally small full subcategory of S. The relevant case of Day’s approach applies to any reasonably well-behaved subcategory O, and provides a symmetric monoidal closed structure on the category 𝔐_{O} of additive functors from O into Ab. In fact, Day’s machinery applies equally well to functors into the category R-Mod of modules over a commutative ring R. However, restricting attention to functors into Ab somewhat simplifies our notation.

Our entire discussion of 𝔐_{O} is plagued by an annoying problem with the variance of our functors. The general observations in the next section about the relation between Day’s structures and our axioms are most easily presented in terms of the category of covariant functors out of O. However, some of our examples are drawn from geometric sources, like cohomology theories, which naturally yield contravariant functors. Most of the time, this difference in the preferred variance is nothing more than a notational nuisance. The preference for covariant functors in the next section arises from Proposition 3.3, which necessarily applies only to covariant functors. All of the other results in that section apply equally well to covariant and contravariant functors since they depend only on self-dual properties of S and O. In many of our examples, the question of a preferred variance is moot, either because O and O^{op} are isomorphic or because functors of both variances are of interest. Nevertheless, in a few key cases, variance is very significant. The preferred variance in these cases is perversely split about evenly. Since there is no clearly preferred variance and the choice of variance is often irrelevant, 𝔐_{O} is a deliberately ambiguous symbol denoting either of the two categories of additive functors from O to Ab. In the sections where functors of both variance are considered, this notation is used only in remarks applicable to both categories. When variance really matters, the symbols 𝔐O^{cov} and 𝔐O^{cont} are used to denote the categories of covariant and contravariant additive functors from O into Ab, respectively.

The case of Day’s approach considered here is quite similar to that described in section 4 of [21]. However, two differences are worth noting. In [21], it is assumed that O contains the unit for the tensor product on S. Here, that constraint is replaced by a weaker condition on O which fits more naturally into our discussion of the connection between the properties of O and our axioms. The second difference is that all the functors in [21] are contravariant. Since the contravariant case is discussed in detail in [21], the introductory discussion here is focused on the covariant case. Readers interested in the contravariant case may either make the necessary notational adjustments for themselves or look them up in [21].

The definition below provides the basic components of Day’s symmetric monoidal closed structure on the category 𝔐O^{cov}. In this definition, and throughout the rest of the article, objects of O are denoted by A, B, C, etc., and the objects of S that need not be in O are denoted by Z, Y, X, etc. The unit for the tensor product of S is denoted • (or •_{S}).

2.1. Definition.

(a) For X in S, let HX^{cov} : O [figure]Ab and HX^{cont} : O^{op} [figure]Ab be the functors given by HX^{cov}(A) = S(X, A) and HX^{cont}(A) = S(A, X). Note that HX^{cov} is in 𝔐O^{cov} and HX^{cont} is in 𝔐O^{cont}. In remarks applicable to the functor category of either variance and in contexts where the desired variance should be obvious, the notation H_{X} is used for the functor of the appropriate variance associated to X.

(b) If M and N are objects of 𝔐O^{cov}, then the functor M □ N in 𝔐O^{cov} is given on an object A of O by

(M □ N)(A) = ∫ ^{B,C∈O} M(B) ⊗ N(C) ⊗ S(B ∧ C, A).

The naturality of coends provides the definition of M □ N on the morphisms of O.

(c) If M and N are objects of 𝔐O^{cov}, then the functor ⟨M, N⟩ in 𝔐O^{cov} is given on an object A of O by

⟨M, N⟩(A) = ∫_{B,C∈O} hom (M(B) ⊗ S(A ∧ B, C), N(C)).

The naturality of ends provides the definition of ⟨M, N⟩ on the morphisms of O.

(d) Let A, C be in O and X, Y be in S. Then the evaluation map

S(X, C) ⊗ S(Y ∧ C, A) [figure]S(Y ∧ X, A)

induces a homomorphism

ρ^{cov}(X, Y; A) : ∫ ^{C} S(X, C) ⊗ S(Y ∧ C, A) [figure]S(Y ∧ X, A)

of abelian groups. Note that, by Lemma 4.3(b) of [21], this map is an isomorphism if X is in O. There is an an analogous map

ρ^{cont}(X, Y; A) : ∫ ^{C} S(C, X) ⊗ S(A, Y ∧ C) [figure]S(A, Y ∧ X)

which must be used instead of ρ^{cov}(X, Y; A) in the context of categories of contravariant functors. This map is denoted θ in [21] and is discussed there in the proof of Proposition 5.2. In remarks applicable to functors of either variance, ρ(X, Y; A) is used to denote the appropriate one of these two maps.

It is easy to see that the □ operation is symmetric, and a simple end calculation gives the desired adjunction relating □ and ⟨ , ⟩. Thus, to show that these constructions provide 𝔐_{O} with a symmetric monoidal closed structure, it suffices to exhibit a unit for □, construct an associativity isomorphism, and prove that the appropriate diagrams commute. The unit for 𝔐_{O} should be H_{•_{S}}. The associativity isomorphism for 𝔐_{O} should be easily derived from the associativity isomorphism for S, and the commutativity of the required diagrams for 𝔐_{O} should follow from the commutativity of the analogous diagrams in S. Nevertheless, these last three pieces of the monoidal structure need not fit properly into place unless the map ρ(X, Y; A) is an isomorphism under the appropriate conditions.

2.2. Theorem. Let S be an Ab-category with a symmetric monoidal structure derived from a bilinear tensor product ∧, and let O be a skeletally small full subcategory of S. If the map ρ(X, Y; A) is an isomorphism whenever X = •_{S} and Y ∈ O and whenever X and Y are both finite ∧-products of objects in O, then 𝔐_{O} is a symmetric monoidal closed category.

Proof. We prove this result for 𝔐O^{cov}; the proof for 𝔐O^{cont} is analogous. In this and several other proofs, we make use of a variety of folklore results about ends and coends. The contravariant analogs of these results are discussed in section 4 of [21]. The proofs of these results are formal, and obviously translate to the covariant context. Note, however, that the results in section 5 of [21] typically apply only in the contravariant case since their proofs make use of more geometric arguments. Let M, N, and P be in 𝔐O^{cov}. The unit isomorphism M □ H•_{S}^{cov} ≅ M of 𝔐O^{cov} is given at A ∈ O by the composite

(M □ H_{•_{S}})(A)  =  ∫ ^{B,C∈O} M(B) ⊗ S(•_{S}, C) ⊗ S(B ∧ C, A)
[figure: ∫^{B} ρ] ∫ ^{B∈O} M(B) ⊗ S(B ∧ •_{S}, A)
≅  ∫ ^{B∈O} M(B) ⊗ S(B, A)
≅  M(A),

in which the last isomorphism is given by Lemma 4.3(c) of [21]. The associativity isomorphism for 𝔐O^{cov} is obtained by using the maps ρ^{cov} to identify both ((M □ N) □ P)(A) and (M □ (N □ P))(A) with

∫ ^{B,C,D∈O} M(B) ⊗ N(C) ⊗ P (D) ⊗ S(B ∧ C ∧ D, A)

for each A ∈ O. The maps ρ^{cov}(X, Y; A) are used in much the same fashion to reduce the question of the commutativity of each of the necessary diagrams in 𝔐O^{cov} to the commutativity of a corresponding diagram in S.

2.3. Scholium. The requirement in this theorem that ρ(X, Y; A) be an isomorphism whenever both X and Y are finite ∧-products of objects in O was unfortunately overlooked in [21]. However, the map θ introduced in the proof of Proposition 5.2 of [21] is our map ρ^{cont}(X, Y; A). In the proof of that proposition, θ is shown to be an isomorphism under far broader conditions than those needed to ensure that the categories considered in [21] are symmetric monoidal.

2.4. Remark.

(a) If •_{S} is in O, then the map ρ(•_{S}, Y; A) is an isomorphism by Lemma 4.3(b) of [21]. Thus, the restriction imposed on •_{S} in the theorem above is weaker than that imposed in [21].

(b) If O contains •_{S} and is closed under finite ∧-products, so that it is a symmetric monoidal subcategory of S, then both conditions on the maps ρ(X, Y; A) are satisfied, again by Lemma 4.3(b) of [21].

We conclude this section with a list of examples of pairs (S, O) satisfying the hypotheses of Theorem 2.2. These examples serve a three-fold purpose. The next section is devoted to positive results giving conditions on a pair (S, O) which ensure that the associated functor category 𝔐_{O} satisfies one or more of our axioms. The power of those positive results is illustrated by the fact that they apply to several of the most important special cases of the examples below, and thus assure us that the associated functor categories are well-behaved. The limitation of our positive results is that they give sufficient, but not necessary, conditions for the good behavior of 𝔐_{O}. Some special cases of the examples below are used to provide a measure of this lack of necessity. Section 6, and most of the sections following it, describe functor categories which fail to satisfy our compatibility axioms. These badly behaved categories are also special cases of the examples below.

2.5. Example.

(a) Let R be a commutative ring, S be the category R-Mod of R-modules, and O be the full subcategory of S containing R as its only object. In this case, S is a symmetric monoidal closed category, and O is a symmetric monoidal closed subcategory. Of course, 𝔐_{O} is just R-Mod, and Day’s symmetric monoidal closed structure is identical to the standard one. In some sense, this example is frivolous in that we have applied a vast machine to R-Mod with its symmetric monoidal closed structure only to recover that category with the same structure. However, in our discussion of the connections between the structure of O and our compatibility axioms, this example nicely illustrates the benefits of a very well-behaved category O.

(b) Let R be a commutative ring, G be a finite group, and S be the category R[G]-Mod of R[G]-modules. In this context, there are two reasonable choices for O. The smallest, which we denote O_{1}, is the full subcategory of S containing R[G] as its only object. The other category, O_{2}, is the full subcategory of S containing the n-fold direct sum of copies of R[G] for all n ≥ 1. Note that neither of these subcategories contains the unit for the tensor product on S, which is R with trivial G-action. The advantage of O_{2} over O_{1} is that O_{2} is closed under both the tensor product and the internal hom operations on S. Of course, 𝔐_{O_{1}} and 𝔐_{O_{2}} are just R[G]-Mod with its usual symmetric monoidal closed structure, so this example is just as frivolous as the previous one. However, it too serves to illustrate the relation between the structure of O and the compatibility axioms satisfied by 𝔐_{O}.

(c) Let G be a compact Lie group, U be a G-universe, and S be the equivariant stable category h̄GSU of G-spectra indexed on U (see chapter I of [23]). The obvious choice for the associated category O is the stable orbit category O_{G}(U). This is the full subcategory whose objects are suspension spectra Σ_{U}^{∞}G/H_{+} associated to the orbits G/H derived from the closed subgroups H of G. The category O_{G}(U) contains the unit for S. However, if G is nontrivial, O_{G}(U) is closed under neither the tensor product nor the internal hom on S. A contravariant functor out of O_{G}(U) is called a (G, U)-Mackey functor, and the category of such functors is denoted 𝔐_{G}(U). If U is a complete G-universe, then O_{G}(U) and 𝔐_{G}(U) are abbreviated to O_{G} and 𝔐_{G}, respectively. If G is finite and U is complete, then (G, U)- Mackey functors are the classical Mackey functors introduced by representation theorists for the study of induction theorems (see [8, 11, 16, 22] and Proposition V.9.9 of [23]). The case in which U is incomplete plays a role in the equivariant Hurewicz and suspension theorems [18, 19] and in the study of change of universe functors in equivariant stable homotopy theory [20]. The category 𝔐_{G}(U) is discussed in greater detail in sections 4, 6, 7, 8, 9, and 10.

(d) If the group G in the previous example is finite, then the Burnside category B_{G}(U) is another choice for O. This is the full subcategory of S containing the suspension spectra Σ_{U}^{∞}X_{+} of the finite G-sets X. Note that O_{G}(U) is a subcategory of B_{G}(U). The advantage of B_{G}(U) over O_{G}(U) is that it is closed under the tensor product operation on S. If the universe U is complete, then B_{G}(U) is a symmetric monoidal closed subcategory of S. Its closed structure is even nicer than that of the category of finite dimensional vector spaces over a field in that objects in B_{G}(U) are canonically self-dual. However, if U is incomplete, then B_{G}(U) is not closed under the internal hom operation on S. As with O_{G}(U), one is usually interested in contravariant functors out of B_{G}(U). For either variance, the categories of functors out of O_{G}(U) and B_{G}(U) are equivalent via the restriction functor induced by the inclusion of O_{G}(U) into B_{G}(U). Thus, we abuse notation and employ 𝔐_{G}(U) to denote the category of contravariant functors from B_{G}(U) to Ab.

(e) The objects of the global Burnside category B_{∗} are the finite groups. If G and H are finite groups, then the set of morphisms from G to H in B_{∗} is the Grothendieck group of isomorphism classes of finite (G×H)-sets. The composition of a (G×H)-set X, regarded as a morphism from G to H, with an (H × K)-set Y, regarded as a morphism from H to a finite group K, is (X × Y)/H, where the passage to orbits is over the diagonal action of H on X × Y. The cartesian product of groups makes B_{∗} into a symmetric monoidal Ab-category. There is an obvious duality functor 𝔇 : B_{∗} [figure]B_{∗}^{op} which is the identity on objects and which sends the (G × H)-set X to itself regarded as an (H × G)-set. This duality functor provides B_{∗} with a closed structure like that for finite dimensional vector spaces; the internal hom object associated to groups G and H is 𝔇(G) × H. Covariant additive functors from B_{∗} to Ab are the most structured kind of globally defined Mackey functors. The category of such functors is denoted 𝔐_{∗}.

Most globally defined Mackey functors carry a much less rich structure than that carried by the functors in 𝔐_{∗}. These less structured Mackey functors are additive functors from some subcategory of B_{∗} into Ab. The subcategories of B_{∗} of interest to us here can be described in terms of pairs (𝔓, 𝔔) of sets of integer primes. A (G × H)-set X is said to be a (𝔓, 𝔔)-set if, for each x ∈ X, the G-isotropy subgroup G_{x} of x has order divisible only by the primes in 𝔓 and the H-isotropy subgroup H_{x} of x has order divisible only by the primes in 𝔔. The subcategory B_{∗}(𝔓, 𝔔) of B_{∗} has the same objects as B_{∗}, but the set of morphisms from G to H in B_{∗}(𝔓, 𝔔) is the Grothendieck group of isomorphism classes of finite (G × H)-sets which are also (𝔓, 𝔔)-sets. Observe that B_{∗}(𝔓, 𝔔)^{op} is just B_{∗}(𝔔, 𝔓). The category B_{∗}(𝔓, 𝔔) inherits a symmetric monoidal structure from B_{∗}, but the internal hom on B_{∗} does not restrict to give B_{∗}(𝔓, 𝔔) a closed structure. In fact, B_{∗}(𝔓, 𝔔) is typically not a closed category. Covariant additive functors from B_{∗}(𝔓, 𝔔) to abelian groups are called global (𝔓,𝔔)-Mackey functors. The category of such is denoted 𝔐_{∗}(𝔓, 𝔔).

The category B_{∗}, and each of its subcategories B_{∗}(𝔓, 𝔔), can serve as both S and O in a pair (S, O) satisfying the hypotheses of Theorem 2.2. Note that this is our only example in which S need not be a closed category. The categories B_{∗}(𝔓, 𝔔) appear, under various names and in various guises, in [1, 2, 4, 5, 7, 9, 10, 12–14, 24–26, 28, 29], and are discussed in greater detail in sections 5, 6 and 11.

2.6. Remark. The pairs (S, O) from Examples 2.5(a), 2.5(d), and 2.5(e) satisfy the hypotheses of Theorem 2.2 by Remark 2.4 since, in each of these cases, O is a symmetric monoidal subcategory of S. Simple direct computations indicate that the pairs (S, O) from Example 2.5(b) satisfy the hypotheses of Theorem 2.2. If the group G is finite in Example 2.5(c), then the pairs introduced in that example must satisfy the hypotheses of Theorem 2.2 since the resulting functor categories 𝔐_{G}(U) can be identified with the functor categories introduced in Example 2.5(d). If G is a nonfinite compact Lie group, then the argument needed to show that the pairs (S, O) of Example 2.5(c) satisfy the hypotheses of Theorem 2.2 is described in Scholium 2.3.

3. Positive results on functor categories

Throughout this section, (S, O) is assumed to be a pair of categories satisfying the hypotheses of Theorem 2.2. The focus of most of this article is on functor categories which do not satisfy our compatibility axioms. However, this section is devoted to positive results giving conditions on a pair (S, O) which ensure that the associated functor category 𝔐_{O} satisfies our various compatibility axioms. Our compatibility axioms naturally fall into three pairs. Associated to each of these pairs of axioms there is a fairly natural closure property on the subcategory O which ensures that the associated functor category 𝔐_{O} satisfies that pair of axioms. Unfortunately, as various pairs drawn from Example 2.5 illustrate, these sufficient conditions are far from necessary.

3.1. Proposition. Let S be an Ab-category with a symmetric monoidal structure derived from a bilinear tensor product ∧, and let O be a skeletally small full subcategory of S such that the pair (S, O) satisfies the hypotheses of Theorem 2.2. If the unit •_{S} of S is in O, then the unit of 𝔐_{O} is projective, and 𝔐_{O} satisfies the axioms IPiP and IIiI.

Proof. The unit for 𝔐_{O} is the functor H_{•_{S}}. Since •_{S} is in O, this functor is representable and therefore projective. The rest of the proposition follows immediately from Lemma 1.2.

In all the pairs (S, O) from Example 2.5 except those from Example 2.5(b), O contains the unit of S so that the proposition above applies. However, in the two pairs introduced in Example 2.5(b), the unit of S is definitely not in O. In this case, both S and 𝔐_{O} are the category R[G]-Mod of modules over the group ring R[G] of a finite group G. The unit for this category is the ring R with trivial G-action, which is typically neither projective nor injective in R[G]-Mod. However, being the unit, R is necessarily internally projective in R[G]-Mod. Moreover, if R is a field, then, regarded as a trivial R[G]-module, it is internally injective in R[G]-Mod. Thus, R[G]-Mod illustrates how badly behaved the category 𝔐_{O} can be when the hypotheses of the proposition don’t hold. On the other hand, if R is a field of characteristic prime to the order of the group G, then R with trivial G-action is projective in R[G]-Mod so that R[G]-Mod satisfies the axioms IPiP and IIiI. Thus, the sufficient condition in the proposition is far from necessary.

3.2. Proposition. Let S be an Ab-category with a symmetric monoidal structure derived from a bilinear tensor product ∧, and let O be a skeletally small full subcategory of S such that the pair (S, O) satisfies the hypotheses of Theorem 2.2. If O is closed under the tensor product operation on S, then the category 𝔐_{O} satisfies the axioms TPPP and PiIP.

Proof. The representable functors form a set of projective generators for 𝔐_{O}. Thus, to show that 𝔐_{O} satisfies TPPP, it suffices to show that, if A and B are in O, then H_{A}□H_{B} is projective. Lemma 4.4(b) of [21] gives that H_{A} □ H_{B} ≅ H_{A∧B}. Since A ∧ B is in O, H_{A∧B} is a representable functor in 𝔐_{O} and so projective. The rest of the proposition follows from Proposition 1.3(a).

The sufficient condition given by this proposition is not strictly necessary since the categories R[G]-Mod and 𝔐_{G}(U) associated to a finite group G in Examples 2.5(b) and 2.5(c) both satisfy TPPP and PiIP, but are of the form 𝔐_{O} for a category O that is not closed under the tensor product operation on S. However, these examples are misleading in the sense that, in both cases, O is a proper subcategory of a larger subcategory O′ of S which is closed under the tensor product and whose associated functor category 𝔐_{O′} is equivalent to 𝔐_{O} under the obvious restriction functor. Theorem 3.8 below provides a more convincing example of non-necessity. 3.3. Proposition. Let S be a symmetric monoidal closed Ab-category, and let O be a skeletally small full subcategory of S such that the pair (S, O) satisfies the hypotheses of Theorem 2.2. If O is closed under the internal hom operation on S, then the category 𝔐O^{cov} satisfies the axioms PiF and IiII.

As with the previous proposition, the sufficient condition given by this proposition is not strictly necessary since the category R[G]-Mod of Examples 2.5(b) satisfies PiF and IiII, but is a category of the form 𝔐O^{cov} for a category O that is not closed under the internal hom operation on S. Another such example is provided by the category O_{G}(U) of Example 2.5(c) in the special case where G is finite and U is complete. Again, however, these examples are misleading in the sense that, in each of them, O is a proper subcategory of a larger subcategory O′ of S which is closed under the internal hom operation and whose associated functor category 𝔐_{O′} is equivalent to 𝔐_{O} under the obvious restriction functor. Theorem 3.8 gives a more convincing example of non-necessity. 3.4. Remark. In all of our examples derived from the equivariant stable category h̄GSU of G-spectra indexed on a G-universe U, the category S, which was assumed to be h̄GSU, can be replaced with its full subcategory S′ consisting of the G-spectra with the Ghomotopy type of finite G-CW spectra. Equivariant Spanier-Whitehead duality provides a functor 𝔇 : S′ [figure](S′)^{op} which is an equivalence of symmetric monoidal closed categories between S′ and its opposite category. The internal hom in S′ associated to objects X and Y is just 𝔇(X) ∧ Y. The category B_{∗} of Example 2.5(e) has a similar closed structure. Categories with this sort of symmetric monoidal closed structure are sometimes called *-autonomous categories (see, for example, [3]). Whenever S, or some full subcategory S′ of S containing O, has a closed structure of this sort, the category O is closed under the internal hom operation on S if and only if O^{op} is analogously closed. Thus, the restriction of Proposition 3.3 to covariant functors is unnecessary in this special case.

To prove the proposition above, we need to introduce an important endofunctor on

𝔐O^{cov}.

3.5. Definition. Let S be a symmetric monoidal closed Ab-category, and let O be a skeletally small full subcategory of S such that the pair (S, O) satisfies the hypotheses of Theorem 2.2. Assume also that O is closed under the internal hom operation on S. If X and Y are in S, then denote the internal hom of this pair in S by ⟨⟨X, Y ⟩⟩. For N in 𝔐O^{cov} and D in O, let N^{D} be the functor in 𝔐O^{cov} given by N^{D}(A) = N(⟨⟨D, A⟩⟩). 3.6. Lemma. Let D be in O.

(a) The assignment of N^{D} to N ∈ 𝔐O^{cov} is an exact additive functor on 𝔐O^{cov}.

(b) There is an isomorphism

HD^{cov} □ N ≅ N^{D}

which is natural in both N ∈ 𝔐O^{cov} and D ∈ O.

Proof. Clearly the assignment of N^{D} to N and D is functorial and is additive in each of N and D separately. It is exact in N because the exactness of sequences in the functor category 𝔐O^{cov} is determined pointwise. On an object A of O, the isomorphism between H_{D} □ N and N^{D} is given by the composite

(H_{D} □ N)(A) = ∫ ^{B,C∈O} H_{D}(B) ⊗ N(C) ⊗ S(B ∧ C, A)
= ∫ ^{B,C∈O} S(D, B) ⊗ N(C) ⊗ S(B ∧ C, A)
≅ ∫ ^{C∈O} N(C) ⊗ S(D ∧ C, A)
≅ ∫ ^{C∈O} N(C) ⊗ S(C, ⟨⟨D, A⟩⟩)
≅ N(⟨⟨D, A⟩⟩) = N^{D}(A).

Here, the first and third isomorphisms are given by Lemmas 4.3(b) and 4.3(c) of [21], respectively. The second isomorphism comes from the adjunction isomorphism making S a closed category. Clearly the composite above is natural in A, D, and N.

3.7. Remark. The construction N^{D} is very closely related to the construction N_{D} introduced in the context of Example 2.5(d) by Dress [8]. In fact, if the G-universe U of that example is complete and O is taken to be B_{G}(U), then the very simple nature of the symmetric monoidal closed structure carried by B_{G}(U) implies that the two constructions are isomorphic.

Proof of Proposition 3.3. Note first that the adjunction making S a closed category forces the tensor product on S to be bilinear. Since the representable functors form a set of projective generators for 𝔐O^{cov}, showing that 𝔐O^{cov} satisfies PiF is easily reduced to showing that the functor HD^{cov}□? is exact for any D ∈ O. This is established in Lemma 3.6. Since 𝔐O^{cov} has enough projectives, it must also satisfy IiII by Proposition 1.3(b).

The following positive result about the category of Mackey functors for the circle group S^{1} stands in sharp contrast to the host of negative results contained in section 6. It also provides clear evidence that the rather formal sufficient conditions contained in Propositions 3.2 and 3.3 above are very far from necessary. This result is proven in section 10.

3.8. Theorem.

(a) The category 𝔐_{S^{1}} of S^{1}-Mackey functors satisfies the six axioms PiF, PiIP, IiII, IPiP, IIiI, and TPPP.

(b) Let O′ be a full subcategory of the complete S^{1}-stable category which contains the stable orbit category O_{S^{1}}. If O′ is closed under either ∧-products or function objects, then the restriction functor

𝔐O′^{cont} [figure]𝔐O_{S^{1}}^{cont} = 𝔐_{S^{1}}

is not an equivalence of categories.

4. An introduction to (G, U)-Mackey functors

Let G be a compact Lie group, and U be a possibly incomplete G-universe. This section is intended to provide a basic introduction to the categories O_{G}(U) and 𝔐_{G}(U). The results presented here provide both some sense of why these categories have something to do with the classical notion of a Mackey functor, and some intuition about why, for various choices of G and U, the category 𝔐_{G}(U) fails to satisfy our various axioms. These results also form the foundation for the proofs, given in later sections, of Theorems 3.8, 6.1, 6.5, and 6.9.

The structure of the morphism sets of the stable orbit category O_{G}(U) of Example 2.5(c) is described in Corollary 5.3(b) of [15] and Corollary 3.2 of [21]. An object of O_{G}(U) is the suspension spectrum Σ_{U}^{∞}G/H_{+} of an orbit G/H of G; however, to avoid unnecessary notational complexity, we hereafter denote this object by G/H. The set of morphisms in O_{G}(U) from G/H to G/K is a free abelian group whose generators are certain allowed equivalence classes of diagrams of the form

G/H [figure: α] G/J [figure: β]G/K,

in which α : G/J [figure]G/H and β : G/J [figure]G/K are space-level G-maps. Two such diagrams are equivalent if there is a G-homeomorphism γ : G/J [figure]G/J′ making the space-level diagram

\xymatrix{ & G/J \ar@{-->}[dl]_{α} \ar[dd]_{γ} \ar@{-->}[dr]^{β} \\ G/H &  & G/K \\  & G/J′ \ar@{-->}[ul]^{α′} \ar@{-->}[ur]_{β′}}

commute up to G-homotopy.

The morphism in O_{G}(U) represented by the diagram

G/H [figure: α] G/J [figure: β]G/K

is the composite of the map, denoted τ(α), which is represented by the diagram

G/H [figure: α] G/J [figure: 1_{G/J}]G/J

and the map, denoted ρ(β), which is represented by the diagram

G/J [figure: 1_{G/J}] G/J [figure: β]G/K.

If J is a subgroup of H, hereafter denoted J ≤ H, then there is a canonical G-map π_{H}^{J} : G/J [figure]G/H which takes the identity coset eJ of G/J to the identity coset eH of G/H. The associated maps τ(π_{H}^{J}) : G/H [figure]G/J and ρ(π_{H}^{J}) : G/J [figure]G/H in O_{G}(U) are denoted τ_{H}^{J} and ρ_{H}^{J}, respectively.

The connection between the category O_{G}(U) and the classical notion of a Mackey functor introduced by representation theorists can be seen from this sketch of the structure of the morphism sets of O_{G}(U). A contravariant functor M from O_{G}(U) to Ab assigns an abelian group M(G/H) to each orbit G/H of G. This abelian group is the value of the Mackey functor M at the subgroup H of G and is often denoted M(H) rather than M(G/H). The map τ(α) associated to a G-map α : G/J [figure]G/H induces an induction (or transfer) map from M(G/J) to M(G/H), and the map ρ(α) induces a restriction map from M(G/H) to M(G/J).

Not every equivalence class of diagrams represents a generator of a morphism group in O_{G}(U). Each equivalence class of diagrams contains at least one diagram of the form

G/H [figure: π_{H}^{J}] G/J [figure: β]G/K.

The allowed equivalence classes are those having a representative of the above special type in which the subgroup J of H satisfies both the condition that H/J embeds in the universe U as an H-space and the condition that the index of J in its H-normalizer N_{H}J is finite. For our purposes, the essential property of a complete G-universe U is that, for such a universe, the embedding condition on H/J is always satisfied.

If the universe U is contained in a larger universe U′, then O_{G}(U) can be identified with a subcategory of O_{G}(U′). In particular, since any G-universe is isomorphic to a subuniverse of a complete G-universe, O_{G}(U) is always a subcategory of O_{G}. Viewing O_{G}(U) as a subcategory of O_{G} reveals a difference between the two restrictions imposed on the equivalence classes which index the generators of the morphism sets of O_{G}(U). If H/J does not embed in U, but the finiteness condition holds, then the diagram

G/H [figure: π_{H}^{J}] G/J [figure: β]G/K,

represents a morphism of O_{G} which has been omitted from O_{G}(U) by our choice of U. However, if the index of J in N_{H}J is not finite, then this diagram does not represent a generator in O_{G}(U) for any U.

If the index of J in N_{H}J is not finite, then for certain geometric reasons it is best to think of the diagram

G/H [figure: π_{H}^{J}] G/J [figure: β]G/K,

as representing a morphism in O_{G}(U) which, in some vague sense, should have been a generator, but has instead been identified with the zero map. This distinction between a morphism that has been omitted from O_{G}(U) and one that has been identified with zero is important for understanding composition of morphisms in O_{G}(U). It is also important for understanding the homological anomalies described in this paper. If the group G is finite and the G-universe U is incomplete, then the morphisms missing from O_{G}(U) may cause the axioms PiF and IiII to fail in the functor category 𝔐_{G}(U). If the group G is a nonfinite compact Lie group, then the “unexpected” zero morphisms in O_{G} may cause the axioms TPPP, PiIP, TPPP, and IiII to fail in the functor category 𝔐_{G}. The arguments establishing these two types of failures are rather different since the failures happen for quite different reasons.

5. An introduction to globally defined Mackey functors

Here, we examine the structure of the category B_{∗}(𝔓, 𝔔) associated to the category of global (𝔓, 𝔔)-Mackey functors. Thus, throughout this section, all groups are assumed to be finite, and 𝔓, 𝔔, and ℜ are assumed to be sets of primes. The observations presented here should provide both a sense of why the functors out of B_{∗}(𝔓, 𝔔) should be regarded as globally defined Mackey functors and an understanding of the way in which the choice of the sets 𝔓 and 𝔔 controls the level of structure carried by those functors. These remarks also lay the foundation for the proof of Theorem 6.10 in section 11.

Recall from Example 2.5(e) that the set of morphisms from G to H in B_{∗}(𝔓, 𝔔) is the Grothendieck group of isomorphism classes of (G × H)-sets which are also (𝔓, 𝔔)-sets. This is a free abelian group whose generators are the isomorphism classes of (G × H)- orbits which are (𝔓, 𝔔)-sets. For any (G × H)-orbit (G × H)/J, the inclusion of J into G × H can be composed with the projections from G × H to G and H to produce group homomorphisms α : J [figure]G and β : J [figure]H. If M is a covariant functor out of B_{∗}(𝔓, 𝔔), then the appropriate intuitive understanding of the map from M(G) to M(H) induced by the (G × H)-orbit (G × H)/J is that this map is the composite of a restriction map from M(G) to M(J) associated to the homomorphism α and an induction map from M(J) to M(H) associated to the homomorphism β. The imposed constraint that the (G × H)-set (G × H)/J must be a (𝔓, 𝔔)-set translates easily into the restriction that the kernel of α must be a 𝔔-group (that is, have order divisible only by the primes in 𝔔) and the kernel of β must be a 𝔓-group. Hereafter, we refer to a homomorphism whose kernel is a 𝔓-group as a homomorphism with 𝔓-kernel.

The most common choices for 𝔓 and 𝔔 are the empty set ∅ of primes and the set of all primes, which we denote by ∞. If 𝔓 = ∅, then the trivial group is the only 𝔓-group. On the other hand, if 𝔓 = ∞, then all finite groups are 𝔓-groups. Thus, a homomorphism with ∅-kernel is a monomorphism, and every group homomorphism has ∞-kernel. At the level of global Mackey functors, this means, for example, that a global (∅, ∞)-Mackey functor has induction maps only for injective group homomorphisms, but restriction maps for all homomorphisms. Note that B_{∗}(∞, ∞) = B_{∗}.

In order to justify the intuitive description of the maps in B_{∗}(𝔓, 𝔔) presented above, we must first identify the morphisms in B_{∗}(𝔓, 𝔔) which are derived directly from ordinary group homomorphisms. 5.1. Definition. Let α : H [figure]G be a group homomorphism. Then τ(α) is the set G considered as a (H × G)-set with action given by

(h, g)x = α(h)xg^{−1},

for (h, g) ∈ H × G and x ∈ G. Also, ρ(α) is the set G considered as an (G × H)-set with action given by

(g, h)x = gxα(h^{−1}),

for (g, h) ∈ G × H and x ∈ G. Note that, if α has ℜ-kernel, then τ(α) is an (ℜ, ∅)-set, and ρ(α) is an (∅, ℜ)-set. If ℜ ⊂ 𝔓, then τ(α) is a generator of the free abelian group of morphisms from H to G in B_{∗}(𝔓, 𝔔). It should be thought of as the induction, or transfer, map associated to α. Similarly, if ℜ ⊂ 𝔔, then ρ(α) is a generator of the free abelian group of morphisms from G to H in B_{∗}(𝔓, 𝔔). It should be thought of as the restriction map associated to α. If α is the inclusion of a subgroup H into the group G, then τ(α) and ρ(α) are denoted τ_{G}^{H} and ρ_{G}^{H}, respectively. If G = H and α is the identity map, then τ(α) and ρ(α) are equal and serve as the identity map 1_{G} of G in B_{∗}(𝔓, 𝔔).

If α : J [figure]G and β : J [figure]H are the group homomorphisms associated in our introductory remarks to the (G × H)-orbit (G × H)/J, then it is fairly easy to see that, in B_{∗}, (G × H)/J is the composite τ(β) ∘ ρ(α). Further, the orbit (G × H)/J is a (𝔓, 𝔔)- set precisely when α has 𝔔-kernel and β has 𝔓-kernel. These observations allows us to think of the generators of the free abelian group B_{∗}(𝔓, 𝔔)(G, H) as equivalence classes of diagrams of the form

G [figure: α] J [figure: β]H,

in which α is a group homomorphism with 𝔔-kernel, β is a group homomorphism with 𝔓-kernel, and the induced homomorphism (α, β) : J [figure]G × H is a monomorphism.

This description of the morphism sets of B_{∗}(𝔓, 𝔔) plays a central role in the proof of Theorem 6.10. Three notes of caution are, however, necessary. First, since 𝔐_{∗}(𝔓, 𝔔) is the category of covariant functors out of B_{∗}(𝔓, 𝔔), the Mackey functor interpretation of this diagram is the reverse of the interpretation associated to the analogous diagrams for O_{G}(U). Here, α is plays the role of the restriction map, and β plays the role of the induction map. Second, the equivalence relation which must be imposed on these diagrams cannot be described as neatly as the corresponding relation for O_{G}(U). For our purposes, it suffices to say that the diagram

G [figure: α′] J′ [figure: β′]H,

is equivalent to the unprimed diagram above if and only if the two (G×H)-sets τ(β)∘ρ(α) and τ(β′) ∘ ρ(α′) are isomorphic. The third note is that composition in B_{∗}(𝔓, 𝔔) cannot be computed using pullback diagrams like those used to compute composition in O_{G}.

5.2. Remark. There is an alternative approach to globally defined Mackey functors in which the category B_{∗}(𝔓, 𝔔) is replaced by a somewhat larger category B′_{∗}(𝔓, 𝔔) whose objects are finite groupoids rather than finite groups. In many ways, the relation between B_{∗}(𝔓, 𝔔) and B′_{∗}(𝔓, 𝔔) is similar to that between O_{G}(U) and B_{G}(U). In particular, the categories B_{∗}(𝔓, 𝔔) and B′_{∗}(𝔓, 𝔔) are similar enough that the associated categories of covariant functors into Ab are equivalent under the restriction functor derived from the inclusion of B_{∗}(𝔓, 𝔔) into B′_{∗}(𝔓, 𝔔). The larger category B′_{∗}(𝔓, 𝔔) is somewhat more difficult to define because one has to deal with functors from finite groupoids into finite sets rather than sets carrying actions by finite groups. However, it has the advantages of an easily described equivalence relation on the generators of its morphism sets and a simple pullback formula for its composition. Further advantages of B′_{∗}(𝔓, 𝔔) over B_{∗}(𝔓, 𝔔) are noted at the end of this section and in Remark 6.11(b).

One technical result describing the behavior of composition in B_{∗}(𝔓, 𝔔) in the context of cartesian products of groups is needed for the proof of Theorem 6.10. The information this result provides about B_{∗}(𝔓, 𝔔) is similar to that provided about O_{G}(U) by Lemma 3.3 of [21].

5.3. Lemma. Let ψ : J [figure]P, ξ : J [figure]Q and ζ : Q′ [figure]Q be homomorphisms between finite groups such that ξ has 𝔔-kernel and ζ has 𝔓-kernel. Assume that the integers {n_{i}}_{1≤i≤m} and the diagrams

Q′ [figure: α_{i}] K_{i} [figure: β_{i}]J,

for 1 ≤ i ≤ m, are chosen so that the composite ρ(ξ) ∘ τ(ζ) in B_{∗}(𝔓, 𝔔) is the sum over i of n_{i} times the generator of B_{∗}(𝔓, 𝔔)(Q′, J) represented by the i^{th} diagram. Then the composite ρ((ψ, ξ)) ∘ τ(1 × ζ) of the restriction map associated to the homomorphism (ψ, ξ) : J [figure]P × Q and the transfer map associated to 1 × ζ : P × Q′ [figure]P × Q is the sum over i of n_{i} times the generator of B_{∗}(𝔓, 𝔔)(P × Q′, J) represented by the diagram

P × Q′ [figure: (ψ∘β_{i},α_{i})] K_{i} [figure: β_{i}]J.

This result can be proven by brute force computations with the obvious finite sets carrying the appropriate group actions. A further advantage of the category B′_{∗}(𝔓, 𝔔) described in Remark 5.2 is that the simple pullback formula for composition in B′_{∗}(𝔓, 𝔔) trivializes the proof of this lemma.

6. A bestiary of symmetric monoidal closed abelian categories

This section contains a catalog of functor categories which fail to satisfy the hypotheses of Propositions 3.2 and 3.3 and also fail to satisfy various of our compatibility axioms. These badly behaved categories come from the families of categories 𝔐_{G}(U) of Example 2.5(c) and 𝔐_{∗}(𝔓, 𝔔) of Example 2.5(e). The results stated in this section about the misbehavior of our functor categories are proven in the subsequent sections.

Let G be a compact Lie group, and U be a possibly incomplete G-universe. The hypotheses of our theorems about the homological misbehavior of the category 𝔐_{G}(U) of (G, U)-Mackey functors are certainly not as weak as they could be. Nevertheless, they are weak enough to produce an almost overwhelming supply of badly behaved categories.

6.1. Theorem. Let G be a finite group, U be a G-universe, and C ≤ D ≤ H be subgroups of G such that

(i) C is normal in D and D/C ≅ ℤ/p for some prime p

(ii) H/C embeds as an H-space in U

(iii) H/D does not embed as an H-space in U.

Then H_{G/D} is not flat in 𝔐_{G}(U), and 𝔐_{G}(U) satisfies neither PiF nor IiII.

6.2. Remark. One way of understanding the misbehavior of 𝔐_{G}(U) implied by Theorem 6.1 is that, if the subgroups C ≤ D ≤ H satisfy the hypotheses of the theorem, then the induction maps τ_{D}^{C} and τ_{H}^{C} are contained in the subcategory O_{G}(U) of O_{G}, but the induction map τ_{H}^{D} is not. Thus, even though the map τ_{H}^{C} can be factored as the composite τ_{D}^{C} ∘ τ_{H}^{D} in O_{G}, it cannot be so factored in O_{G}(U). This failure of τ_{H}^{C} to factor properly in O_{G}(U) seems to be the source of the homological misbehavior of the category 𝔐_{G}(U).

6.3. Example. Let p be a prime, G = H = ℤ/p^{2}, D = ℤ/p ⊂ ℤ/p^{2}, and C be the trivial subgroup of G. Let U be a G-universe whose only irreducible summands are the trivial irreducible G-representation and a free irreducible G-representation. Then G/C embeds in U as a G-space, but G/D does not. By the theorem, the projective Mackey functor H_{G/D} is not flat in 𝔐_{G}(U). Beyond the context of this paper, the special significance of this universe is that it has natural connections to the study of semi-free actions of G.

6.4. Corollary. Let p be a prime, G be a finite p-group, and U be a G-universe. Then the category 𝔐_{G}(U) cannot satisfy either PiF or IiII unless, for each subgroup H of G, the set

{K : K ≤ H and H/K does not embed as an H-space in U}

is closed under passage to subgroups. In particular, if the free orbit G/e embeds in U as a G-space, then 𝔐_{G}(U) cannot satisfy either PiF or IiII unless every G-orbit G/K embeds in U.

The statement of our second main theorem requires a bit of additional notation. The Weyl group N_{G}H/H of a subgroup H of G is denoted W_{G}H, and the set of G-conjugates of H ≤ G is denoted (H)_{G}. If H and K are subgroups of G, then the notation (K)_{G} ≤ (H)_{G} indicates that K is subconjugate to H in G.

6.5. Theorem. Let G be a compact Lie group, and U be a complete G-universe. Also, let C ≤ D ≤ H be subgroups of G such that

(i) C is normal in D and D/C is a finite p-group for some prime p

(ii) W_{H}D is finite

(iii) for every K < D such that (C)_{G} ≤ (K)_{G}, W_{H}K is not finite.

Then H_{G/D} is not flat in 𝔐_{G}, and 𝔐_{G} satisfies neither PiF nor IiII.

6.6. Remark. One way of understanding the misbehavior of 𝔐_{G} implied by Theorem 6.5 is that, if the subgroups C ≤ D ≤ H satisfy the hypotheses of the theorem, then the induction maps τ_{D}^{C} and τ_{H}^{D} are generators of the morphism sets of O_{G}. Their composite is the induction map τ_{H}^{C} associated to the containment C ≤ H. This map τ_{H}^{C} should also be a generator of the appropriate morphism set. However, since W_{H}C is not finite, τ_{H}^{C} is actually the zero map. This vanishing of the composite of two generators of the morphism sets of O_{G} seems to be the source of the homological misbehavior of the category 𝔐_{G}(U).

6.7. Example.

(a) Let G and H both be the orthogonal group O(2), D be the dihedral group of order 2n (for n ≥ 3) regarded as a subgroup of G, and C be the cyclic group of order n regarded as a subgroup of D. Then C is normal in G, and D/C ≅ ℤ/2. Moreover, N_{G}D is the dihedral group of order 4n, so W_{G}D is finite. If K is a proper subgroup of D such that (C)_{G} ≤ (K)_{G}, then K = C, and W_{G}K is G/C, which is not finite. Thus, the projective Mackey functor H_{G/D} is not flat in 𝔐_{G}, and 𝔐_{G} satisfies neither PiF nor IiII.

(b) If C ≤ D ≤ H are subgroups of G which satisfy the hypotheses of the theorem and ϵ : G′ [figure]G is a surjective group homomorphism, then the subgroups C′ = ϵ^{−1}(C), D′ = ϵ^{−1}(D), and H′ = ϵ^{−1}(H) are subgroups of G′ which also satisfy the hypotheses of the theorem. Therefore, 𝔐_{G′} does not satisfy the axioms PiF and IiII.

(c) We would like to argue that, if C ≤ D ≤ H are subgroups of G which satisfy the hypotheses of the theorem and G ≤ G′′, then C, D, and H, regarded as subgroups of G′′, satisfy the hypotheses of the theorem. However, condition (iii) in the hypotheses of the theorem might fail since there might be a subgroup K of D such that C was G′′- subconjugate to K, but not G-subconjugate to K. Nevertheless, if D/C ≅ ℤ/p and C is the unique subgroup of D of index p, as in part (a) of this example, then the triple C ≤ D ≤ H, regarded as subgroups of G′′, must still satisfy the hypotheses of the theorem. In fact, it suffices to assume that D/C ≅ ℤ/p and that every subgroup C′ of index p in D is H-conjugate to C. Combining this observation with part (a) of this example, we see that, if a compact Lie group G contains the orthogonal group O(2), then 𝔐_{G} satisfies neither PiF nor IiII.

Combining the various parts of the example above yields the following corollary of Theorem 6.5.

6.8. Corollary. If G is any one of

O(m), SU(m), U(m), for m ≥ 2;
SO(n), Spin(n), for n ≥ 3;
Sp(q), for q ≥ 1;

then 𝔐_{G} satisfies neither PiF nor IiII.

The family of categories 𝔐_{G}, for G a compact Lie group, also provides us with some examples of symmetric monoidal closed categories which fail to satisfy TPPP and PiIP.

6.9. Theorem. Let G be the orthogonal group O(m), for m ≥ 2, or the special orthogonal group SO(n), for n ≥ 3. Then 𝔐_{G} satisfies neither TPPP nor PiIP.

Now we turn to the context of global Mackey functors. Recall that, in Example 2.5(e), we associated a category 𝔐_{∗}(𝔓, 𝔔) of global (𝔓, 𝔔)-Mackey functors to each pair (𝔓, 𝔔) of sets of integer primes.

6.10. Theorem. Let 𝔓 and 𝔔 be sets of integer primes. If the prime p is not in 𝔓, then the representable functor in 𝔐_{∗}(𝔓, 𝔔) associated to the cyclic group ℤ/p is not flat in 𝔐_{∗}(𝔓, 𝔔). Thus, 𝔐_{∗}(𝔓, 𝔔) satisfies neither PiF nor IiII.

6.11. Remark.

(a) Recall that the empty set of primes and the set of all primes are denoted ∅ and ∞, respectively. The three types of global Mackey functors that appear most often in the literature seem to be (∅, ∅)-, (∅, ∞)- and (∞, ∅)-Mackey functors (see, for example, [1, 2, 4, 5, 7, 9, 10, 12–14, 24–26, 28, 29]). Theorem 6.10 indicates that the first two of these categories of global Mackey functors are badly behaved.

(b) As noted in Remark 5.2, the category B_{∗}(𝔓, 𝔔) used to define global (𝔓, 𝔔)- Mackey functors can be replaced by a somewhat larger category B′_{∗}(𝔓, 𝔔) whose objects are finite groupoids. One advantage of this replacement is that, for any set of primes 𝔔, the category B′_{∗}(∞, 𝔔) is a symmetric monoidal closed category. Proposition 3.3 therefore implies that 𝔐_{∗}(∞, 𝔔) satisfies PiF and IiII. Thus, the theorem above gives sharp conditions under which 𝔐_{∗}(𝔓, 𝔔) fails to satisfy PiF and IiII.

7. Mackey functors for incomplete universes and the proof of Theorem 6.1

Each of Theorems 6.1, 6.5, and 6.10 asserts that a projective object P in a certain functor category 𝔐_{O} is not flat. The proofs of these results follow the same pattern. It suffices to show that, for some object A in O, the functor sending M in 𝔐_{O} to (P □M)(A) does not preserve monomorphisms. To show this, we restrict the domain of this functor to a nice full subcategory 𝔐′ of 𝔐_{O}. We then identify a natural direct summand ℨ : 𝔐′ [figure]Ab of the restricted functor, and show that the summand fails to preserve monomorphisms. In this section, we prove Theorem 6.1 by carrying out the appropriate special case of this general program. Thus, throughout the section, G is a finite group, U is a G-universe, and C ≤ D ≤ H are subgroups of G. The appropriate projective object P in this context is the representable functor H_{G/D} in the category 𝔐_{G}(U), and the appropriate object A of O = O_{G}(U) is the orbit G/H.

We begin the proof of Theorem 6.1 by introducing the appropriate subcategory 𝔐′ of 𝔐_{G}(U), and the appropriate direct summand functor ℨ.

7.1. Definition.

(a) A Mackey functor M is concentrated over C if, for K ≤ G, M(G/K) = 0 unless (C)_{G} ≤ (K)_{G}. The subcategory 𝔐′ appropriate for the proof of Theorem 6.1 is the full subcategory of 𝔐_{G}(U) consisting of Mackey functors which are concentrated over C; this subcategory is denoted M_{G}^{C}(U).

(b) The Weyl group W_{G}C is contained in the morphism set O_{G}(U)(G/C, G/C) as the set of maps ρ(β) associated to the endomorphisms β of the G-set G/C. Thus, the group W_{G}C acts on the value M(G/C) of a Mackey functor M at G/C. Let ℨ_{C} : 𝔐_{G}(U) [figure]Ab be the functor sending M in 𝔐_{G}(U) to the quotient group M(G/C)/W_{D}C of M(G/C).

The technical foundation of the proof of Theorem 6.1 is the following result:

7.2. Proposition. Let G be a finite group, U be a G-universe, and C ≤ D ≤ H be subgroups of G such that

(i) C is normal in D, and D/C ≅ ℤ/p for some prime p

(ii) H/C embeds as an H-space in U

(iii) H/D does not embed as an H-space in U.

Then, for M in M_{G}^{C}(U), ℨ_{C}(M) splits off from (H_{G/D} □ M)(G/H) as a natural direct summand.

Before proving this result, we show how it can be used to complete the proof of our theorem.

Proof of Theorem 6.1. Assume that C is normal in D, and that D/C ≅ ℤ/p for some prime p. Also assume that H/C embeds as an H-space in U, but that H/D does not so embed. We must construct a monomorphism ι : A [figure]B which is not preserved by the functor ℨ_{C}. Define the object B of 𝔐_{G}(U) by the exact sequence

⨁_{K<C} H_{G/K} ⊗ ℤ/p [figure: ⊕ ρ̃_{C}^{K}⊗1]H_{G/C} ⊗ ℤ/p [figure]B [figure]0.

Here, the map ρ̃_{C}^{K} : H_{G/K} [figure]H_{G/C} is derived from the restriction map ρ_{C}^{K} in O_{G}(U). By construction, B is concentrated over C. Let A in 𝔐_{G}(U) be the image of the composite map

H_{G/D} ⊗ ℤ/p [figure: τ̃_{D}^{C}⊗1]H_{G/C} ⊗ ℤ/p [figure]B,

in which τ̃_{D}^{C} is the map derived from the induction map τ_{D}^{C} in O_{G}(U). Clearly, this gives a monomorphism ι : A [figure]B and an epimorphism μ : H_{G/D} ⊗ ℤ/p [figure]A. Since A embeds in B, it must also be concentrated over C.

The composition of morphisms in O_{G}(U) is discussed in section 3 of [21]. Using the results presented there, it is easy to see that B(G/C) is just the group ring ℤ/p[W_{G}C]. Let t ∈ ℤ/p[W_{G}C] be the sum ∑_{d∈W_{D}C} d. A straightforward double coset computation indicates that A(G/C) is the image of the endomorphism of ℤ/p[W_{G}C] given by multiplication on the right by t. Thus, A(G/C) can be identified with the ℤ/p vector space ℤ/p[W_{G}C/W_{D}C] whose basis set is the orbit W_{G}C/W_{D}C. Under these identifications, the actions of W_{G}C on A(G/C) and B(G/C) are the obvious ones coming from the left action of W_{G}C on itself. Moreover, the map

ι(G/C) : A(G/C) [figure]B(G/C)

is just the trace map

\xymatrix{ℤ/p[W_{G}C/W_{D}C] \ar[r] & ℤ/p[W_{G}C]}

which sends a coset in W_{G}C/W_{D}C, regarded as a basis element of ℤ/p[W_{G}C/W_{D}C], to the sum of its elements in ℤ/p[W_{G}C]. These identifications allow us to analyse the map

ℨ_{C}(ι) : ℨ_{C}(A) [figure]ℨ_{C}(B).

The objects ℨ_{C}(A) and ℨ_{C}(B) are the ℤ/p vector spaces ℤ/p[W_{D}C\W_{G}C/W_{D}C] and ℤ/p[W_{D}C\W_{G}C] whose bases are the sets W_{D}C\W_{G}C/W_{D}C and W_{D}C\W_{G}C, respectively. The map ℨ_{C}(ι) is the obvious one derived from the trace map ι(G/C). It follows that the double coset W_{D}CeW_{D}C is a basis element in ℨ_{C}(A), but maps to zero in ℨ_{C}(B). Thus, the map ℨ_{C}(ι) is not a monomorphism.

Note that Corollary 6.4 follows directly from Theorem 6.1 and the fact that p-groups are solvable. The remainder of this section is devoted to the postponed proof of our splitting result.

Proof of Proposition 7.2. By Lemma 4.4(a) of [21],

(H_{G/D} □ M)(G/H) = ∫ ^{G/Q} M(G/Q) ⊗ [G/H, G/D × G/Q].

Here, [?,?] is used to denote the morphism set in S = h̄GSU, and G/D × G/Q is used to denote the spectrum Σ_{U}^{∞}G/D_{+} ∧ Σ_{U}^{∞}G/Q_{+} = Σ_{U}^{∞}(G/D × G/Q)_{+}. By definition, this coend is a quotient of the abelian group ⨁_{G/Q} M(G/Q) ⊗ [G/H, G/D × G/Q]. We prove the proposition by constructing a natural map

λ : (H_{G/D} □ M)(G/H) [figure]ℨ_{C}(M),

and then showing that λ is a naturally split epimorphism. This map λ is derived from a map

λ̃ : ⨁_{G/Q} M(G/Q) ⊗ [G/H, G/D × G/Q] [figure]ℨ_{C}(M)

by factoring λ̃ through the quotient group (H_{G/D} □ M)(G/H).

The map λ̃ may be defined by specifying its restriction λ̃_{Q} to each of the summands M(G/Q) ⊗ [G/H, G/D × G/Q] of its domain. Proposition 3.1 of [21] indicates that [G/H, G/D × G/Q] is a free abelian group. Thus, M(G/Q) ⊗ [G/H, G/D × G/Q] is a direct sum of one copy of M(G/Q) for each generator of [G/H, G/D × G/Q]. The generators of [G/H, G/D × G/Q] are certain equivalence classes of diagrams of the form

G/H [figure: δ] G/J [figure: (α,β)]G/D × G/Q,

in which α : G/J [figure]G/D, β : G/J [figure]G/Q, and δ : G/J [figure]G/H are maps of Gspaces. Denote the equivalence class of this diagram by (δ; α, β). The map λ̃_{Q} may be defined by specifying its restriction λ̃_{(δ;α,β)} to the copy of M(G/Q) associated to each equivalence class (δ; α, β).

If the equivalence class (δ; α, β) does not contain a diagram of the form

G/H [figure: π_{H}^{C}] G/C [figure: (π_{D}^{C},β)]G/D × G/Q,

then λ̃_{(δ;α,β)} is defined to be the zero map. On the other hand, the map λ̃_{(π_{H}^{C};π_{D}^{C},β)} is defined to be the composite

M(G/Q) [figure: M(ρ(β))]M(G/C) [figure]M(G/C)/W_{D}C = ℨ_{C}(M).

There is, in fact, more than one choice for the map β in the diagram chosen to represent the equivalence class (π_{H}^{C}; π_{D}^{C}, β). However, the passage to orbits in the codomain of λ̃_{(π_{H}^{C};π_{D}^{C},β)} kills off the uncertainity that might have arisen in the definition of λ̃_{(π_{H}^{C};π_{D}^{C},β)} from the possible choices for β.

To show that the map λ̃ induces the desired map λ, it suffices to show that, for each morphism f : G/Q [figure]G/Q′ in O_{G}(U), the diagram

\xymatrix{M(G/Q′) ⊗ [G/H, G/D × G/Q] \ar[r]^{M(f)⊗1} \ar[d]_{1⊗(1×f)_{∗}} & M(G/Q) ⊗ [G/H, G/D × G/Q] \ar[d]^{λ̃_{Q}} \\ M(G/Q′) ⊗ [G/H, G/D × G/Q′] \ar[r]^{λ̃_{Q′}} & ℨ_{C}(M)}

commutes. In fact, the commutativity of this diagram needs to verified only for the morphisms f which are generators of the free abelian group O_{G}(U)(G/Q, G/Q′). Moreover, since each morphism in O_{G}(U) is the composite of a restriction map and an induction map, it suffices to check the commutativity of the diagram for the cases in which f is either the restriction map ρ(β) associated to a G-map β : G/J [figure]G/K or the induction map τ_{Q}^{Q′} associated to a containment Q′ ≤ Q.

Lemma 3.3 of [21] is the key to verifying the commutativity of the required diagram for both of these types of maps. Verifying the commutativity of the diagram for a restriction map ρ(β) is entirely elementary, and uses none of the conditions imposed on C, D, and H other than the containments C ≤ D ≤ H. Thus, we can assume that f is the induction map τ_{Q}^{Q′} associated to the containment Q′ ≤ Q.

Note that M(G/Q′) ⊗ [G/H, G/D × G/Q] is a direct sum of one copy of M(G/Q′) for each generator of [G/H, G/D × G/Q]. The commutativity of the diagram can therefore be verified by checking it on each of the copies of M(G/Q′). If the diagram

G/H [figure: δ] G/J [figure: (α,β)]G/D × G/Q,

represents a generator of [G/H, G/D × G/Q], then (J)_{G} ≤ (D)_{G}. Using the equivalence relation, we may adjust the diagram so that J ≤ D and α = π_{D}^{J}. It is easy to verify that, if (C)_{G} ≰ (J)_{G}, then, on the copy of M(G/Q′) indexed on the equivalence class (δ; π_{D}^{J}, β), both of the composites in the diagram are zero. Thus, we may assume that (C)_{G} ≤ (J)_{G}. Since D/C is ℤ/p, our assumptions now give that either J = D or (J)_{G} = (C)_{G}.

First, assume that J = D so that the generator of interest in [G/H, G/D × G/Q] is that associated to the equivalence class (δ; 1_{G/D}, β). We wish to show that both of the composites in the diagram are zero on the copy of M(G/Q′) indexed on this class. Clearly, the composite along the top and right of the diagram must be zero on this copy by the definition of λ̃_{Q}. Assume that δ takes the identity coset eD of G/D to the coset gH of G/H. Then g ∉ H since H/D doesn’t embed in U and the equivalence class (δ; 1_{G/D}, β) is assumed to give a generator of [G/H, G/D × G/Q]. The image of this generator of [G/H, G/D × G/Q] under the map

(1 × f)_{∗} : [G/H, G/D × G/Q] [figure][G/H, G/D × G/Q′]

is given by the double coset formula, and is a sum of generators represented by diagrams of the form

G/H [figure: δ∘π_{H}^{J}] G/J [figure: (π_{H}^{J},β)]G/D × G/Q′,

in which J is a subgroup of D. The composite δ ∘π_{H}^{J} appearing in this diagram cannot be π_{H}^{J} since it must take eJ to gH. But then λ̃_{Q′} kills the copies of M(G/Q′) in its domain indexed on these generators of [G/H, G/D × G/Q′]. It follows that, on the (δ; 1_{G/D}, β)- indexed copy of M(G/Q′), the composite along the left and bottom of the diagram also vanishes.

Now let us assume that (J)_{G} = (C)_{G}. Since J ≤ D and C is normal in D, either J = C or (J)_{D} ≠ (C)_{D}. If (J)_{D} ≠ (C)_{D}, then the definitions of λ̃_{Q} and λ̃_{Q′} easily imply that both of the composites in the diagram vanish on the (δ; π_{D}^{J}, β)-indexed copy of M(G/Q′). Thus, we may assume that J = C. If δ ≠ π_{D}^{C}, then the definitions of λ̃_{Q} and λ̃_{Q′} again imply that both of the composites in the diagram vanish on the copy of M(G/Q′) in question. Thus, we need only consider the copy of M(G/Q′) indexed on the generator (π_{H}^{C}; π_{D}^{C}, β). On this copy, simple double coset arguments indicate that the two composites in the diagram are equal. In these arguments, the fact that M is concentrated over C is used to ensure the vanishing of certain contributions to the composite along the top and right which might otherwise not vanish.

We have now constructed the map λ : (H_{G/D} □ M)(G/H) [figure]ℨ_{C}(M) for an arbitrary M in M_{G}^{C}(U). It is easy to see that λ must be natural in M. We must still show that λ is naturally split. There is an obvious map σ̃ of M(G/C) into (H_{G/D} □ M)(G/H) coming from the inclusion

M(G/C) ⊂ ⨁_{G/Q} M(G/Q) ⊗ [G/H, G/D × G/Q]

associated to the copy of M(G/C) in M(G/C) ⊗ [G/H, G/D × G/C] indexed on the generator represented by the diagram

G/H [figure: π_{H}^{C}] G/C [figure: (π_{D}^{C},1_{G/C})]G/D × G/C.

It follows directly from the definition of λ that the composite λ ∘ σ̃ is just the projection M(G/C) [figure]M(G/C)/W_{D}C = ℨ_{C}(M). Moreover, the map σ̃ is obviously natural in M. Thus, it suffices to show that the map σ̃ factors through the projection M(G/C) [figure]M(G/C)/W_{D}C = ℨ_{C}(M) to give a map

σ : ℨ_{C}(M) [figure](H_{G/D} □ M)(G/H).

For each Q ≤ G, let

ν_{Q} : M(G/Q) ⊗ [G/H, G/D × G/Q] [figure](H_{G/D} □ M)(G/H)

be the map obtained from our description of (H_{G/D} □ M)(G/H) as a quotient group of ⨁_{G/Q} M(G/Q) ⊗ [G/H, G/D × G/Q]. Then, for any morphism f : G/Q [figure]G/Q′ in O_{G}(U), the diagram

\xymatrix{M(G/Q′) ⊗ [G/H, G/D × G/Q] \ar[r]^{M(f)⊗1} \ar[d]_{1⊗(1×f)_{∗}} & M(G/Q) ⊗ [G/H, G/D × G/Q] \ar[d]^{ν_{Q}} \\ M(G/Q′) ⊗ [G/H, G/D × G/Q′] \ar[r]^{ν_{Q′}} & (H_{G/D} □ M)(G/H)}

commutes. Take Q = Q′ = C in this diagram, and consider the generators f in O_{G}(U)(G/C, G/C) which give the action of W_{D}C on M(G/C). It follows immediately that σ̃ does factor through the indicated projection to give the desired splitting map σ for λ.

8. Mackey functors for compact Lie groups and the proof of Theorem 6.5

In this section, G is a compact Lie group, U is a complete G-universe, and C ≤ D ≤ H are subgroups of G satisfying the conditions of Theorem 6.5. Following the general program outlined in section 7, we show that H_{G/D} is not flat in 𝔐_{G} by showing that the functor sending M in 𝔐_{G} to (H_{G/D} □M)(G/H) does not preserve monomorphisms. In the proof, the domain of this functor is restricted to the full subcategory M_{G}^{C} of 𝔐_{G} consisting of the objects which are concentrated over C. This restricted functor has a natural direct summand which is easily computed and can be shown not to preserve monomorphisms.

Recall from Definition 2.1(b) of [21] that the Brauer quotient br_{D}M of an object M in 𝔐_{G} at D is defined by the exact sequence

⨁_{K} M(G/K) [figure: ⊕M(τ_{D}^{K})]M(G/D) [figure]br_{D}M [figure]0.

Here, the direct sum is indexed on the proper subgroups K of D for which W_{D}K is finite. This Brauer quotient is the natural direct summand that is to be split off from our functor.

8.1. Proposition. Let G be a compact Lie group, U be a complete G-universe, and C ≤ D ≤ H be subgroups of G such that

(i) W_{D}C and W_{H}D are finite

(ii) for every K < D such that (C)_{G} ≤ (K)_{G}, W_{H}K is not finite.

Then, for M in M_{G}^{C}, br_{D}M splits off from (H_{G/D}□M)(G/H) as a natural direct summand.

Before proving this result, we show how it can be used to complete the proof of our theorem.

Proof of Theorem 6.5. Assume that W_{H}D is finite, C is normal in D, and D/C is a finite p-group for some prime p. Also assume that, for every K < D such that (C)_{G} ≤ (K)_{G}, W_{H}K is not finite. We must construct a monomorphism ι : A [figure]B which is not preserved by the functor br_{D}. Define the object B of 𝔐_{G} by the exact sequence

⨁_{K<C} H_{G/K} [figure: ⊕ρ̃_{C}^{K}]H_{G/C} [figure: ϕ]B [figure]0.

Here, the map ρ̃_{C}^{K} : H_{G/K} [figure]H_{G/C} is derived from the restriction map ρ_{C}^{K} in O_{G}. By construction, B is concentrated over C. Let A in 𝔐_{G} be the image of the composite map

H_{G/D} [figure: τ̃_{D}^{C}]H_{G/C} [figure]B,

in which τ̃_{D}^{C} is the map derived from the induction map τ_{D}^{C} in O_{G}. Note that there are an obvious monomorphism ι : A [figure]B and an obvious epimorphism μ : H_{G/D} [figure]A. Since it is a subobject of B, A must be concentrated over C.

Using the results in section 3 of [21] about the composition in O_{G}, it is easy to see that B(G/D) is a free abelian group whose generators are the images under ϕ : H_{G/C} [figure]B of the induction maps τ(α) in H_{G/C}(G/D) associated to G-maps α : G/C [figure]G/D. It follows that br_{D}B is zero. Thus, to show that the functor br_{D} does not preserve monomorphisms, it suffices to show that br_{D}A is nonzero.

Let x ∈ A(G/D) be the image under μ : H_{G/D} [figure]A of the identity map 1_{G/D} in H_{G/D}(G/D) = O_{G}(G/D, G/D), and let y ∈ B(G/D) be ι(G/D)(x). It is easy to see that y is the generator of B(G/D) derived from the transfer map τ_{D}^{C} in H_{G/C}(G/D). To show that the image of x in br_{D}A is nonzero, it suffices to show that x is not in the image of the map

⨁_{K} A(G/K) [figure: ⊕A(τ_{D}^{K})]A(G/D)

whose cokernel is br_{D}A. In fact, px is the smallest nonzero multiple of x which can appear in this image. This can be proven by showing that py is the smallest nonzero multiple of y which can appear in the image of the composite

⨁_{K} A(G/K) [figure: ⊕A(τ_{D}^{K})]A(G/D) [figure: ι(G/D)]B(G/D).

This claim about y can be proven using the commuting diagram

\xymatrix{H_{G/D}(G/K) \ar[r]^{H_{G/D}(τ_{D}^{K})} \ar@{->>}[d]_{μ(G/K)} & H_{G/D}(G/D) \ar[r]^{τ̃_{D}^{C}} \ar@{->>}[d]^{μ(G/D)} & H_{G/C}(G/D) \ar@{->>}[d]^{ϕ(G/D)} \\ A(G/K) \ar[r]^{A(τ_{D}^{K})} & A(G/D) \ar@{^{(}->}[r]^{ι(G/D)} & B(G/D).}

Let w be a generator of H_{G/D}(G/K) = O_{G}(G/K, G/D), and z be its image in B(G/D) under either composite in the diagram above. It suffices to show that, if z is written in terms of our standard basis for B(G/D), then the coefficient of y is a multiple of p. We analyze z by first computing the element τ̃_{D}^{C}(H_{G/D}(τ_{D}^{K})(w)) = τ_{D}^{C} ∘ w ∘ τ_{D}^{K} of H_{G/C}(G/D) = O_{G}(G/D, G/C). The generator w is represented by a diagram of the form

G/K [figure: α] G/J [figure: β]G/D,

and we can select this diagram so that J ≤ D and β = π_{D}^{J}. The composite w ∘ τ_{D}^{K} is then represented by the diagram

G/D [figure: π_{D}^{K}∘α] G/J [figure: π_{D}^{J}]G/D.

Since G is a nonfinite compact Lie group, computing the composite of this map with τ_{D}^{C} would normally be rather difficult since it requires an analysis of the pullback diagram

\xymatrix{P \ar[r] \ar[d] & G/C \ar[d]^{π_{D}^{C}} \\ G/J \ar[r]^{π_{D}^{J}} & G/D}

in the category of G-spaces. However, it is easy to see that, if the composite π_{D}^{K} ∘ α is not π_{D}^{J}, then the coefficient of y in the expression for z in terms of our basis is zero. Further, if J does not contain C, then the coefficient of y also has to be zero. Thus, we may assume that π_{D}^{K} ∘ α = π_{D}^{J} and C ≤ J. Now we can use the fact that D/C is a finite group to complete the computation. The D-orbits D/C, D/J, and D/D may be regarded as orbits of the finite group D/C. The diagram

\xymatrix{D/J × D/C \ar[r]^{π_{2}} \ar[d]_{π_{1}} & D/C \ar[d]^{π̃_{D}^{C}} \\ D/J \ar[r]^{π̃_{D}^{J}} & D/D}

is obviously a pullback in the category of D/C-sets. The functor G×_{D}? preserves pullbacks, and so takes this pullback diagram into the pullback diagram we must analyze in the category of G-spaces. Since D/J × D/C decomposes into a disjoint union of finitely many D/C-orbits, the pullback P in our diagram of G-spaces is just a disjoint union of the analogous G-orbits. The free D/C-orbit D/C appears |D/J| times in D/J × D/C, and |D/J| = p^{n} for some n ≥ 1. From this, it follows that, if τ̃_{D}^{C}(H_{G/D}(τ_{D}^{K})(w)) is written in terms of the standard generators of H_{G/C}(G/D), then the coefficient of the generator τ_{D}^{C} is |D/J| = p^{n}. The coefficient of y in the expression for z in terms of the standard basis must then also be |D/J| = p^{n}.

Corollary 6.8 follows from Theorem 6.5 and the remarks in Example 6.7 by observing that the listed groups either contain O(2) or map onto a group containing O(2). In particular, the groups O(m), U(m), and Sp(m) contain O(2) for m ≥ 2. For n ≥ 3, the groups SO(n) and SU(n) also contain O(2). The groups Spin(n) map onto the groups SO(n). The special cases SU(2) and Sp(1) are handled by their surjections onto SO(3).

The remainder of this section is occupied by the postponed proof of our splitting result.

Proof of Proposition 8.1. Let M be in M_{G}^{C}. As in the proof of Proposition 7.2, we first construct a natural map

λ : (H_{G/D} □ M)(G/H) [figure]br_{D}M,

and then show that it is a split epimorphism. The map λ is, as in the previous section, derived from an appropriately behaved map

λ̃ : ⨁_{G/Q} M(G/Q) ⊗ [G/H, G/D × G/Q] [figure]br_{D}M.

As before, λ̃ is defined by specifying its restriction λ̃_{Q} to the summand associated to each orbit G/Q. Moreover, λ̃_{Q} can be defined by giving its restriction to the copy of M(G/Q) in M(G/Q) ⊗ [G/H, G/D × G/Q] associated to each generator of [G/H, G/D × G/Q]. These generators correspond to equivalence classes of diagrams of the form

G/H [figure: δ] G/J [figure: (α,β)]G/D × G/Q.

If the equivalence class (δ; α, β) of such a diagram does not contain a diagram of the form

G/H [figure: π_{H}^{D}] G/D [figure: (1_{G/D},β)]G/D × G/Q,

then the restriction λ̃_{(δ;α,β)} to the associated copy of M(G/Q) is the zero map. On the other hand, the map λ̃_{(π_{H}^{D};1_{G/D},β)} is defined to be the composite

M(G/Q) [figure: M(ρ(β))]M(G/D) [figure]br_{D}M.

To show that λ can be derived from λ̃, it suffices to show that, for each morphism f : G/Q [figure]G/Q′ in O_{G}(U), the diagram

\xymatrix{M(G/Q′) ⊗ [G/H, G/D × G/Q] \ar[r]^{M(f)⊗1} \ar[d]_{1⊗(1×f)_{∗}} & M(G/Q) ⊗ [G/H, G/D × G/Q] \ar[d]^{λ̃_{Q}} \\ M(G/Q′) ⊗ [G/H, G/D × G/Q′] \ar[r]^{λ̃_{Q′}} & br_{D}M}

commutes. Again, we need only check the cases in which f is either a restriction map ρ(β) or an induction map τ_{Q}^{Q′}.

As in the previous proof, the commutativity of the diagram is purely formal if f is a restriction map ρ(β). If f is an induction map τ_{Q}^{Q′}, then the required commutativity can be verified by checking it on each summand M(G/Q′) of M(G/Q′) ⊗ [G/H, G/D × G/Q]. On the summand associated to an equivalence class that does not contain a diagram of the form

G/H [figure: π_{H}^{D}] G/D [figure: (1_{G/D},β)]G/D × G/Q,

it is easy to see that both composites in the diagram are zero. Thus, we restrict our attention to the summands indexed on equivalence classes of the form (π_{H}^{D}; 1_{G/D}, β).

To compute the composite along the top and right side of the rectangle on such a summand, it is necessary to commute the composite in O_{G} of f and the restriction map ρ(β) : G/D [figure]G/Q. This composite is a sum of generators of O_{G}(G/D, G/Q′) represented by diagrams of the form

G/D [figure: π_{D}^{K_{i}}] G/K_{i} [figure: ζ_{i}]G/Q′,

in which K_{i} ≤ D. Because the target of the composite along the top and right side is br_{D}M, the generators for which K_{i} ≠ D can be ignored. The implications of the remaining terms for the composite along the top and right are easily understood.

To understand the composite along the left side and bottom of the rectangle, it is necessary to compute the image of the generator associcated to the equivalence class (π_{H}^{D}; 1_{G/D}, β) under the map

(1 × f)_{∗} : [G/H, G/D × G/Q] [figure][G/H, G/D × G/Q′].

By Lemma 3.3 of [21], this image is the sum of generators represented by diagrams of the form

G/H [figure: π_{H}^{K_{i}}] G/K_{i} [figure: (π_{D}^{K_{i}},ζ_{i})]G/D × G/Q′,

in which exactly the same subgroups K_{i} and the same morphisms ζ_{i} appear as those encountered in our examination of the other composite. By the definition of the map λ̃_{Q′}, the generators for which K_{i} ≠ D can be ignored. The implications of the remaining terms for the composite along the right and bottom are easily understood, and it follows that two composites in the diagram are equal. Thus, the desired map λ can be obtained from the defined map λ̃. Clearly, λ is natural in M ∈ M_{G}^{C}.

It is still necessary to show that λ is naturally split. There is an obvious map

σ̃ : M(G/D) [figure](H_{G/D} □ M)(G/H)

derived from the inclusion

M(G/D) ⊂ ⨁_{G/Q} M(G/Q) ⊗ [G/H, G/D × G/Q]

associated to the copy of M(G/D) in M(G/D) ⊗ [G/H, G/D × G/D] indexed on the equivalence class (π_{H}^{D}; 1_{G/D}, 1_{G/D}). Clearly, the composite λ ∘ σ̃ is just the canonical projection M(G/D) [figure]br_{D}M. Thus, to show that λ is a split epimorphism, it suffices to show that the map σ̃ factors through the projection M(G/D) [figure]br_{D}M. This is equivalent to showing that, for each K < D such that W_{D}K is finite, the composite

M(G/K) [figure: M(τ_{D}^{K})]M(G/D) [figure: σ̃](H_{G/D} □ M)(G/H)

is zero. Unless (C)_{G} ≤ (K)_{G}, M(G/K) is zero, and the composite is trivially zero. Thus, we assume that (C)_{G} ≤ (K)_{G}.

For each Q ≤ G, let

ν_{Q} : M(G/Q) ⊗ [G/H, G/D × G/Q] [figure](H_{G/D} □ M)(G/H)

be the map obtained from our description of (H_{G/D} □ M)(G/H) as a quotient group of ⨁_{G/Q} M(G/Q) ⊗ [G/H, G/D × G/Q]. The diagram

\xymatrix{M(G/K) ⊗ [G/H, G/D × G/D] \ar[r]^{M(τ_{D}^{K})⊗1} \ar[d]_{1⊗(1×τ_{D}^{K})_{∗}} & M(G/D) ⊗ [G/H, G/D × G/D] \ar[d]^{ν_{D}} \\ M(G/K) ⊗ [G/H, G/D × G/K] \ar[r]^{ν_{K}} & (H_{G/D} □ M)(G/H)}

commutes by the definition of (H_{G/D} □ M)(G/H). The composite σ̃ ∘ M(τ_{D}^{K}) can be obtained by restricting the composite along the top and right hand side of the rectangle to the copy of M(G/K) indexed on the generator of [G/H, G/D × G/D] associated to the equivalence class (π_{H}^{D}; 1_{G/D}, 1_{G/D}). The image of this generator under the map

(1 × f)_{∗} : [G/H, G/D × G/D] [figure][G/H, G/D × G/K].

is the element of [G/H, G/D × G/K] that should be represented by the diagram

G/H [figure: π_{H}^{K}] G/K [figure: (π_{D}^{K},1_{G/K})]G/D × G/K.

However, by assumption, (C)_{G} ≤ (K)_{G} and so W_{H}K is not finite. Thus, this diagram represents not a generator of [G/H, G/D × G/K], but the zero element. The composite along the left and bottom of the rectangle must then restrict to zero on the appropriate copy of M(G/K). Thus, the composite σ̃ ∘ M(τ_{D}^{K}) is zero. It follows that the map σ̃ : M(G/D) [figure](H_{G/D} □ M)(G/H) induces a map

σ : br_{D}M [figure](H_{G/D} □ M)(G/H)

which splits λ.

9. Mackey functors for compact Lie groups and the proof of Theorem 6.9

Throughout this section, G is a compact Lie group, and U is a G-universe. Here we convert the question of whether 𝔐_{G}(U) satisfies TPPP and PiIP into a question in equivariant stable homotopy theory. This conversion is used here to prove Theorem 6.9 by showing that 𝔐_{G} fails to satisfy TPPP and PiIP if G is either the orthogonal group O(m), for m ≥ 2, or the special orthogonal group SO(n), for n ≥ 3. It is used in the next section to show that, if G = S^{1}, then 𝔐_{G} does satisfy TPPP and PiIP.

The category 𝔐_{G}(U) satisfies TPPP and PiIP if and only if, for every subgroup H of G, the functor ⟨H_{G/H},?⟩ : 𝔐_{G}(U) [figure]𝔐_{G}(U) preserve epimorphisms. Associated to any short exact sequence

0 [figure]M′ [figure: ι]M [figure: ϵ]M′′ [figure]0

in 𝔐_{G}(U), there is a fibre sequence

K(M′, 0) [figure: ι̃]K(M, 0) [figure: ϵ̃]K(M′′, 0)

of zero-dimensional equivariant Eilenberg-Mac Lane spectra in the stable category of Gspectra indexed on U. Proposition 5.2 of [21] allows us to use the long exact homotopy sequence derived from this fibre sequence to investigate the exactness of the functor ⟨H_{G/H},?⟩.

That proposition gives an isomorphism

⟨H_{G/H}, M⟩(G/J) ≅ [G/J_{+} ∧ G/H_{+}, K(M, 0)]_{G}

which is natural in M ∈ 𝔐_{G}(U) and G/H, G/J ∈ O_{G}(U). Here, [ , ]_{G} is used to denote the morphism sets in the G-stable homotopy category. This isomorphism identifies the map

⟨H_{G/H}, M⟩(G/J) [figure: ϵ_{∗}(G/J)]⟨H_{G/H}, M′′⟩(G/J),

which must be an epimorphism if 𝔐_{G} is to satisfy TPPP and PiIP, with the map

[G/J_{+} ∧ G/H_{+}, K(M, 0)]_{G} [figure: ϵ̃_{∗}][G/J_{+} ∧ G/H_{+}, K(M′′, 0)]_{G}.

From the long exact homotopy sequence, it follows that ϵ̃_{∗} is an epimorphism if and only if the map

[G/J_{+} ∧ G/H_{+}, ΣK(M′, 0)]_{G} [figure: ι̃_{G}][G/J_{+} ∧ G/H_{+}, ΣK(M, 0)]_{G}

is a monomorphism. The change of group isomorphisms given in section II.4 of [23] identify this map with the map

[G/H_{+}, ΣK(M′, 0)]_{J} [figure: ι̃_{J}][G/H_{+}, ΣK(M, 0)]_{J}.

We have now reformulated the question of whether 𝔐_{G}(U) satisfies TPPP and PiIP into the form needed for our discussion of S^{1} in section 10.

9.1. Proposition. Let G be a compact Lie group, and U be a G-universe. Then the category 𝔐_{G}(U) satisfies TPPP and PiIP if and only if, for every monomorphism ι : M′ [figure]M in 𝔐_{G}(U) and every pair H, J of subgroups of G, the map

[G/H_{+}, ΣK(M′, 0)]_{J} [figure: ι̃_{J}][G/H_{+}, ΣK(M, 0)]_{J}.

induced by ι is a monomorphism.

For the proof of Theorem 6.9, it suffices to exhibit a monomorphism ι : M′ [figure]M in 𝔐_{G} and two subgroups H and J of G for which the map ι̃_{J} is not a monomorphism. It is possible to select these subgroups H and J so that H ≥ J. For such a pair of subgroups, the identity coset eH of G/H is J-invariant and so provides a basepoint for G/H regarded as a J-space. The existence of this basepoint implies that G/H_{+} is equivalent to G/H ∨S^{0} in the J-stable category. Thus,

[G/H_{+}, ΣK(M′, 0)]_{J} ≅ [G/H ∨ S^{0}, ΣK(M′, 0)]_{J}
≅ [G/H, ΣK(M′, 0)]_{J} ⊕ [S^{0}, ΣK(M′, 0)]_{J}.

Similar observations apply to [G/H_{+}, ΣK(M, 0)]_{J}. The dimension axiom implies that [S^{0}, ΣK(M′, 0)]_{J} and [S^{0}, ΣK(M, 0)]_{J} are both zero. The map which we wish to show is not a monomorphism can therefore be identified with the map

[G/H, ΣK(M′, 0)]_{J} [figure: ι̂_{J}][G/H, ΣK(M, 0)]_{J}.

The task of showing that 𝔐_{G} fails to satisfy TPPP and PiIP is now reduced to the two problems of analyzing the G-orbit G/H as a J-space so that the map ι̂_{J} can be understood and of selecting a short exact sequence in 𝔐_{G} for which this map is not a monomorphism. We address these two problems only for the case in which G is either O(n) or SO(n), and the universe U is complete.

If G is either O(n) or SO(n), then the subgroup J in the analysis above can be taken to be H. For either choice of G, our argument employs an appropriately chosen proper subgroup K of H. If G = O(n), then H = O(n−1) and K = O(n−2). If G = SO(n), then H = SO(n−1) and K = SO(n−2). Here, O(1) = ℤ/2, and O(0) = SO(1) = e. For both O(n) and SO(n), standard geometry gives nonequivariant homeomorphisms G/H ≅ S^{n−1} and H/K ≅ S^{n−2}. Moreover, for either choice of G, the G-orbit G/H, considered as an H-space, may be identified with the unreduced suspension S(H/K) of the H-orbit H/K. Under this identification, the map ι̂_{J} becomes the map

[S(H/K), ΣK(M′, 0)]_{H} [figure: ι̂][S(H/K), ΣK(M, 0)]_{H}.

The unreduced suspension S(H/K) fits into a cofibre sequence

S^{0}  [figure]S(H/K) [figure]ΣH/K_{+}  [figure]S^{1}

of H-spaces. From this cofibre sequence, we obtain the diagram

[figure: [S^{1}, ΣK(M′, 0)]_{H}; ι′′; [S^{1}, ΣK(M, 0)]_{H}; θ; [ΣH/K_{+}, ΣK(M′, 0)]_{H}; ι′; [ΣH/K_{+}, ΣK(M, 0)]_{H}; [S(H/K), ΣK(M′, 0)]_{H}; ι̂; [S(H/K), ΣK(M, 0)]_{H}; [S^{0}, ΣK(M′, 0)]_{H}; ∥; [S^{0}, ΣK(M, 0)]_{H}; ∥; 0; 0]

in which the columns are exact sequences derived from our cofibre sequence. Consider the pullback diagram

\xymatrix{P \ar[r] \ar[d] & [S^{1}, ΣK(M, 0)]_{H} \ar[d]^{θ} \\ [ΣH/K_{+}, ΣK(M′, 0)]_{H} \ar[r]^{ι′} & [ΣH/K_{+}, ΣK(M, 0)]_{H}}

associated with the top rectangle of the larger diagram. The commutativity of the top rectangle in the larger diagram implies the existence of a map

ϕ : [S^{1}, ΣK(M′, 0)]_{H} [figure]P.

Assume that the map θ in these two diagrams is a monomorphism. A simple diagram chase then gives that the map ι̂ of interest to us fails to be a monomorphism if and only if the map ϕ fails to be an epimorphism.

The change of group isomorphisms from section II.4 of [23] allow us to identify the top rectangle of our larger diagram with the diagram

\xymatrix{M′(G/H) \ar[r]^{ι(G/H)} \ar[d]_{M′(ρ_{H}^{K})} & M(G/H) \ar[d]^{M(ρ_{H}^{K})} \\ M′(G/K) \ar[r]^{ι(G/K)} & M(G/K).}

We can think of P as the pullback associated with this rectangle and of ϕ as a map from M′(G/H) into P. Our geometric argument has therefore reduced the question of whether the functor ⟨H_{G/H},?⟩ preserves epimorphisms to the question of whether there exists a monomorphism M′ [figure]M in 𝔐_{G} for which the map M(ρ_{H}^{K}) : M(G/H) [figure]M(G/K) is a monomorphism and the map ϕ : M′(G/H) [figure]P is not an epimorphism.

Let M be the Mackey functor \underline{Z} given by Proposition V.9.10 of [23]. For any subgroup J of G, M(G/J) = ℤ. Also, for L ≤ J, the restriction map M(ρ_{J}^{L}) : M(G/J) [figure]M(G/L) is the identity map, and the induction map M(τ_{J}^{L}) : M(G/L) [figure]M(G/J) is multiplication by the Euler characteristic χ(J/L). Let M′ be the image of the canonical map H_{G/K} □ M [figure]M. By Proposition 5.5 of [21], the map (H_{G/K} □ M)(G/K) [figure]M(G/K) is a split epimorphism. Therefore, M′(G/K) = M(G/K), and the map ι(G/K) is the identity map. The image of the map (H_{G/K} □ M)(G/H) [figure]M(G/H) is contained in the image of the transfer map M(τ_{H}^{K}) : M(G/K) [figure]M(G/H) by Proposition 5.7 of [21]. This transfer map is multiplication by the Euler characteristic of H/K ≅ S^{n−2}, which is either 2 or 0. Thus, M′(G/H) is either ℤ or 0. If M′(G/H) = ℤ, then the map ι(G/H) : M′(G/H) [figure]M(G/H) is multiplication by 2. In either case, the map ϕ : M′(G/H) [figure]P is not an epimorphism. Thus, the functor ⟨H_{G/H},?⟩ : 𝔐 [figure]𝔐 does not preserve epimorphisms, and the category 𝔐_{G} satisfies neither TPPP nor PiIP.

10. S^{1}-Mackey functors and the proof of Theorem 3.8

To prove Theorem 3.8, we must show that 𝔐_{S^{1}} satisfies all six of the axioms PiF, PiIP, IiII, IPiP, IIiI, and TPPP. It follows immediately from Proposition 3.1 that it satisfies IPiP and IIiI. Tools from equivariant stable homotopy theory are used to show that it satisfies the other four axioms. Proposition 9.1 provides the means of applying these tools to the question of whether 𝔐_{S^{1}} satisfies TPPP and PiIP. This section begins with an analogous proposition providing a method for applying these tools to the question of whether the category 𝔐_{G}(U) associated to a compact Lie group G and a G-universe U satisfies PiF and IiII. After proving this proposition, we specialize to the case in which G = S^{1} and U is a complete G-universe, and prove Theorem 3.8.

As in section 9, we assume here that ι : M′ [figure]M is a monomorphism in 𝔐_{G}(U), and ι̃: K(M′, 0) [figure]K(M, 0) is the induced map between the equivariant Eilenberg-Mac Lane spectra associated to M′ and M.

10.1. Proposition. Let G be a compact Lie group, and U be a G-universe. Then the category 𝔐_{G}(U) satisfies PiF and IiII if and only if, for every monomorphism ι : M′ [figure]M in 𝔐_{G}(U) and every pair H, J of subgroups of G, the map

[S^{0}, G/H_{+} ∧ K(M′, 0)]_{J} [figure: ῐ_{J}][S^{0}, G/H_{+} ∧ K(M, 0)]_{J}

induced by ι is a monomorphism.

Proof. To prove that 𝔐_{G}(U) satisfies PiF and IiII, it suffices to show that, for every monomorphism ι : M′ [figure]M in 𝔐_{G}(U) and every pair H and J of subgroups of G, the map

(1 □ ι)(G/J) : (H_{G/H} □ M′)(G/J) [figure](H_{G/H} □ M)(G/J)

is a monomorphism. Proposition 5.2 of [21] provides an isomorphism

(H_{G/H} □ M)(G/J) ≅ [G/J_{+}, G/H_{+} ∧ K(M, 0)]_{G}

which is natural in M ∈ 𝔐_{G}(U) and G/H, G/J ∈ O_{G}(U). Under this isomorphism, the map (1 □ ι)(G/J) is identified with the map

[G/J_{+}, G/H_{+} ∧ K(M′, 0)]_{G} [figure: ῐ_{G}][G/J_{+}, G/H_{+} ∧ K(M, 0)]_{G}

induced by the map ι̃: K(M′, 0) [figure]K(M, 0). The change of group isomorphisms given in section II.4 of [23] allow us to identify the map ῐ_{G} with the map ῐ_{J} of the proposition.

Henceforth, we assume that G = S^{1} and that the universe U is complete. The remainder of this section is devoted to the proofs of the two parts of Theorem 3.8.

Proof of Theorem 3.8(a). Propositions 3.1, 9.1 and 10.1 reduce the proof of this part of the theorem to showing that, for every monomorphism ι : M′ [figure]M in 𝔐_{G} and every pair H and J of subgroups of G = S^{1}, the two maps

[G/H_{+}, ΣK(M′, 0)]_{J} [figure: ι̃_{J}][G/H_{+}, ΣK(M, 0)]_{J}.

and

[S^{0}, G/H_{+} ∧ K(M′, 0)]_{J} [figure: ῐ_{J}][S^{0}, G/H_{+} ∧ K(M, 0)]_{J}

are monomorphisms. For H = G, the dimension axiom indicates that the morphism set [G/G_{+}, ΣK(M′, 0)]_{J}, and the analogous set for M, are both zero. Thus, ι̃_{J} is trivially a monomorphism. Further, the isomorphism

[S^{0}, G/G_{+} ∧ K(M′, 0)]_{J} ≅ M′(G/J),

and the analogous isomorphism for M, identify the map ῐ_{J} with the map

ι(G/J) : M′(G/J) [figure]M(G/J),

which is assumed to be a monomorphism. Thus, we may assume that H ≠ G.

If J = G and H ≠ G, then both ι̃_{J} and ῐ_{J} have vanishing range and domain, and so are monomorphisms. For ι̃_{J}, this follows directly from the dimension axiom. To see this for ῐ_{J}, note that in the equivariant stable category G/H_{+} ∧K(M, 0) can be identified with a G-CW spectrum whose zero skeleton is a wedge ∨_{i}G/K_{+}^{i} of orbits G/K^{i} associated to certain subgroups K^{i} of H. A simple connectivity argument gives that the map

⨁_{i}[S^{0}, G/K_{+}^{i}]_{G} ≅ [S^{0}, ∨_{i}G/K_{+}^{i}]_{G} [figure][S^{0}, G/H_{+} ∧ K(M, 0)]_{G}

induced by the inclusion of this zero skeleton is an epimorphism. Thus, to show that [S^{0}, G/H_{+} ∧ K(M, 0)]_{G} is zero, it suffices to argue that [S^{0}, G/K_{+}^{i}]_{G} is zero. Corollary 3.2 of [21] gives that [S^{0}, G/K_{+}^{i}]_{G} = O_{G}(G/G, G/K^{i}) is a free abelian group whose generators are equivalence classes of diagrams of the form

G/G [figure: α] G/L [figure: β]G/K^{i}

in which W_{G}L is finite. However, in any such diagram, L ≤ K^{i} ≤ H. Since H ≠ G, L must be a finite cyclic group, and so W_{G}L cannot be finite. Thus, there are no generators, and O_{G}(G/G, G/K^{i}) = 0. Analogously, the domain of the map ῐ_{J} is zero.

We can now assume that neither H nor J is equal to G. If J ≤ H, then the identity coset eH of G/H is J-invariant, and so provides G/H, considered as a J-space, with a basepoint. Thus, in the J-stable category, G/H_{+} ≅ G/H ∨ S^{0} = S^{1} ∨ S^{0}, where S^{1} is assumed to have trivial J-action. This decomposition of G/H_{+} and the dimension axiom provide isomorphisms

[G/H_{+}, ΣK(M, 0)]_{J} ≅ [S^{1}, ΣK(M, 0)]_{J} = M(G/J)

and

[S^{0}, G/H_{+} ∧ K(M, 0)]_{J} ≅ [S^{0}, S^{0} ∧ K(M, 0)]_{J} = M(G/J).

These isomorphisms, and the analogous ones for M′, identify the maps ι̃_{J} and ῐ_{J} with the map ι(G/J) : M′(G/J) [figure]M(G/J), which is assumed to be a monomorphism.

We can now assume that H, J ≠ G and J ≰ H. Thus, J ∩ H is a proper subgroup of J. In this case, G/H, considered as a J-space, can be identified with the unit sphere Sξ of an irreducible complex representation ξ of J whose kernel is J ∩ H. The inclusion of any J-orbit into G/H = Sξ yields a cofibre sequence

J/(J ∩ H)_{+}  [figure]G/H_{+}  [figure]ΣJ/(J ∩ H)_{+}.

The next map ΣJ/(J ∩ H)_{+} [figure]ΣJ/(J ∩ H)_{+} in this sequence is the difference of the identity map of ΣJ/(J ∩ H)_{+} and the map j̄ : ΣJ/(J ∩ H)_{+} [figure]ΣJ/(J ∩ H)_{+} given by multiplication by an appropriate generator j of J (see, for example, the proof of Lemma A.1 in the appendix of [17]). To understand the map ι̃_{J} in this context, consider the exact sequence

[ΣJ/(J ∩ H)_{+}, ΣK(M, 0)]_{J}  [figure: (1−j̄)^{∗}] [ΣJ/(J ∩ H)_{+}, ΣK(M, 0)]_{J}
[figure: [G/H_{+}, ΣK(M, 0)]_{J}]
[figure: [J/(J ∩ H)_{+}, ΣK(M, 0)]_{J} = 0]

derived from our cofibre sequence. The last group in this sequence is zero by the dimension axiom. The first map in this sequence can be identified with the map

M(G/(J ∩ H)) [figure: 1−j̃]M(G/(J ∩ H)),

in which the map j̃ is given by the action of j, considered as an element of the Weyl group W_{G}(J ∩ H), on M(G/(J ∩ H)). Since W_{G}(J ∩ H) = S^{1} is connected, it acts trivially on M(G/(J ∩ H)). Thus, the map (1 − j̄)^{∗} in the exact sequence above is trivial, and there is an isomorphism

[G/H_{+}, ΣK(M, 0)]_{J} ≅ [ΣJ/(J ∩ H)_{+}, ΣK(M, 0)]_{J} = M(G/(J ∩ H)).

This isomorphism, and the analogous isomorphism for M′, identify the map ι̃_{J} with the map ι(G/(J ∩ H)) : M′(G/(J ∩ H)) [figure]M(G/(J ∩ H)), which is assumed to be a monomorphism.

The cofibre sequence of the inclusion J/(J ∩ H)_{+} [figure]G/H_{+} can also be used to analyze the map ῐ_{J} whenever H, J ≠ G and J ≰ H. For this analysis, consider the exact sequence

[S^{0}, J/(J ∩ H)_{+} ∧ K(M, 0)]_{J}  [figure: ((1−j̄)∧1)_{∗}] [S^{0}, J/(J ∩ H)_{+} ∧ K(M, 0)]_{J}
[figure: [S^{0}, G/H_{+} ∧ K(M, 0)]_{J}]
[figure: [S^{0}, ΣJ/(J ∩ H)_{+} ∧ K(M, 0)]_{J} = 0,]

in which the last term is zero by the dimension axiom. Since J is finite, equivariant Spanier-Whitehead duality provides an identification of [S^{0}, J/(J ∩ H)_{+} ∧ K(M, 0)]_{J} with [J/(J ∩ H)_{+}, K(M, 0)]_{J}. Using this identification, the first map in our exact sequence can be identified, as in the previous exact sequence, with the map

M(G/(J ∩ H)) [figure: 1−j̃]M(G/(J ∩ H)),

which is known to be zero. Thus, there is an isomorphism

[S^{0}, G/H_{+} ∧ K(M, 0)]_{J} ≅ [J/(J ∩ H)_{+}, K(M, 0)]_{J} = M(G/(J ∩ H)).

This isomorphism, and the analogous isomorphism for M′, identify the map ῐ_{J} with the map ι(G/(J ∩ H)) : M′(G/(J ∩ H)) [figure]M(G/(J ∩ H)), which is assumed to be a monomorphism.

Proof of Theorem 3.8(b). Recall that G = S^{1}. In this proof, we make use of the ideas presented in Remark 3.4. The stable orbit category O_{G} is a full subcategory of the category S′ of G-spectra which have the G-homotopy type of finite G-CW complexes. The category S′ is a symmetric monoidal closed category whose internal hom functor ⟨⟨?,?⟩⟩ is derived from a duality functor. Thus, for objects X and Y in S′,

X ∧ Y ≅ 𝔇(𝔇(X)) ∧ Y = ⟨⟨𝔇(X), Y ⟩⟩ ≅ ⟨⟨⟨⟨X, •_{S′}⟩⟩, Y ⟩⟩.

It follows that, if O is a subcategory of S′ which is closed under the internal hom operation on S′, then it must also be closed under the ∧-product operation on S′.

To prove Theorem 3.8(b), it therefore suffices to show that, if O′ is a full subcategory of the G-stable category which contains O_{G} and which is closed under smash products, then the restriction functor

𝔐O′^{cont} [figure]𝔐O_{G}^{cont} = 𝔐_{G}

is not an equivalence of categories. We can regard the functors H_{G/e} and H_{G/e×G/e} as contravariant functors out of O′. The diagonal map Δ : G/e [figure]G/e × G/e induces a map Δ̃ : H_{G/e} [figure]H_{G/e×G/e}. Since G/e and G/e × G/e are path connected, Proposition 3.1 of [21] implies that the map

Δ̃(G/H) : H_{G/e}(G/H) [figure]H_{G/e×G/e}(G/H)

is an isomorphism for H ≤ G. However, by the change of group isomorphisms in section II.4 of [23],

H_{G/e}(G/e × G/e) = [G/e_{+} ∧ G/e_{+}, G/e_{+}]_{G}
≅ [G/e_{+}, G/e_{+}]_{e}
≅ [S^{1} ∨ S^{0}, S^{1} ∨ S^{0}]_{e}
≅ ℤ ⊕ ℤ ⊕ ℤ/2

and

H_{G/e×G/e}(G/e × G/e) = [G/e_{+} ∧ G/e_{+}, G/e_{+} ∧ G/e_{+}]_{G}
≅ [G/e_{+}, G/e_{+} ∧ G/e_{+}]_{e}
≅ [S^{1} ∨ S^{0}, (S^{1} ∨ S^{0}) ∧ (S^{1} ∨ S^{0})]_{e}
≅ ℤ ⊕ ℤ ⊕ ℤ ⊕ ℤ/2.

Thus, Δ̃ is not an isomorphism in 𝔐O′^{cont}, but its restriction to 𝔐O_{G}^{cont} = 𝔐_{G} is an isomorphism.

11. Globally defined Mackey functors and the proof of Theorem 6.10

In this section, all groups are assumed to be finite, and 𝔓 and 𝔔 are sets of primes. In proving Theorem 6.10, we follow the general pattern described in section 7. First, we identify a natural direct summand ℨ_{p}(M) of the functor sending a (𝔓, 𝔔)-Mackey functor M to the abelian group (H_{ℤ/p} □ M)(ℤ/p). Then we show that the functor ℨ_{p} does not preserve monomorphisms. The argument here is somewhat simpler than the previous ones in that no restriction on M is needed to split off ℨ_{p}(M).

11.1. Definition. Assume that the prime p is not in the set 𝔓 of primes, and that M is in 𝔐_{∗}(𝔓, 𝔔). Denote the trivial group by e, and the direct sum of p copies of an abelian group A by A^{p}. Then ℨ_{p}(M) is the pushout in the diagram

\xymatrix{M(e)^{p} \ar[r]^{M(τ_{ℤ/p}^{e})^{p}} \ar[d]_{∇} & M(ℤ/p)^{p} \ar[d]^{κ_{M}} \\ M(e) \ar[r]^{η_{e}} & ℨ_{p}(M).}

Here, ∇ is the folding map. Regard M(ℤ/p)^{p} as being the direct sum of a collection of copies of M(ℤ/p) indexed on the set of group endomorphisms of ℤ/p. Given such an endomorphism γ : ℤ/p [figure]ℤ/p, let η_{γ} : M(ℤ/p) [figure]ℨ_{p}(M) be the composite

M(ℤ/p) ⊂ M(ℤ/p)^{p} [figure: κ_{M}]ℨ_{p}(M),

in which the first map is the inclusion of M(ℤ/p) into the direct sum as the copy indexed on γ. Analogously, the map η_{e} in the pushout diagram above should be thought of as the map from M(e) into ℨ_{p}(M) associated to the unique group homomorphism e [figure]ℤ/p.

11.2. Proposition. Let p be a prime and 𝔓 and 𝔔 be sets of primes such that p ∉ 𝔓. Then, for all M in 𝔐_{∗}(𝔓, 𝔔), ℨ_{p}(M) splits off from (H_{ℤ/p} □ M)(ℤ/p) as a natural direct summand.

As in the previous sections, we postpone the proof of this result until after we have shown how it can be used to complete the proof of our theorem Proof of Theorem 6.10. Let p be a prime, and let 𝔓 and 𝔔 be sets of primes such that p ∉ 𝔓. We must construct a monomorphism ι : A [figure]B which is not preserved by the functor ℨ_{p}. Let B be the representable Mackey functor H_{e}, and let A be the image of the map

τ̃_{ℤ/p}^{e} : H_{ℤ/p} [figure]H_{e} = B.

induced by the map τ_{ℤ/p}^{e} in B_{∗}(𝔓, 𝔔). Note that there are an obvious monomorphism ι : A [figure]B and an obvious epimorphism μ : H_{ℤ/p} [figure]A.

Consider the diagram

\xymatrix{A(e) \ar@{-->}[dr]_{θ} \ar[drr]^{ι(e)} \ar[ddr]_{A(τ_{ℤ/p}^{e})} \\  & P \ar[r]_{ι′} \ar[d]^{τ′} & B(e) \ar[d]^{B(τ_{ℤ/p}^{e})} \\  & A(ℤ/p) \ar[r]^{ι(ℤ/p)} & B(ℤ/p),}

in which the rectangle is a pullback, and θ is the induced map into the pullback. We show first that the map θ is not surjective, and then use this to show that the map ℨ_{p}(ι) : ℨ_{p}(A) [figure]ℨ_{p}(B) is not a monomorphism.

To show that θ is not surjective, it is necessary to compute several values of the representable functors used to define A and B. It is easy to see that the morphism sets [ℤ/p, e] = H_{ℤ/p}(e) and [e, e] = H_{e}(e) of B_{∗}(𝔓, 𝔔) are infinite cyclic groups generated by the morphisms ρ_{ℤ/p}^{e} and 1_{e}, respectively. The map τ̃_{ℤ/p}^{e}(e) takes ρ_{ℤ/p}^{e} to p · 1_{e}.

The morphism set [e, ℤ/p] = H_{e}(ℤ/p) has either one or two generators, depending on whether or not p is an element of 𝔔. The morphism τ_{ℤ/p}^{e} is always a generator. If p ∈ 𝔔, then there is a second generator represented by the diagram

e [figure] ℤ/p [figure: 1_{ℤ/p}]ℤ/p.

The morphism set [ℤ/p, ℤ/p] = H_{ℤ/p}(ℤ/p) is generated by the composite τ_{ℤ/p}^{e} ∘ ρ_{ℤ/p}^{e} and by the morphisms in B_{∗}(𝔓, 𝔔) of the form ρ(α), where α is an endomorphism of ℤ/p. If p ∉ 𝔔, then α must be a nonzero endomorphism; otherwise, it can be any endomorphism. The map τ̃_{ℤ/p}^{e}(ℤ/p) takes τ_{ℤ/p}^{e} ∘ ρ_{ℤ/p}^{e} to p · τ_{ℤ/p}^{e} and ρ(α) to τ_{ℤ/p}^{e}.

From these computations, it follows immediately that the diagram above has the form

\xymatrix{ℤ \ar@{-->}[dr]_{p} \ar[drr]^{p} \ar[ddr]_{p} \\  & ℤ \ar[r]_{1_{ℤ}} \ar[d]^{1_{ℤ}} & ℤ \ar[d]^{i_{1}} \\  & ℤ \ar[r]^{i_{1}} & ℤ ⊕ C.}

Here, C is ℤ if p ∈ 𝔔; otherwise, C = 0. Also, the map i_{1} is the inclusion of the first summand of a direct sum, and p denotes the map given by multiplication by p. Clearly, θ is not surjective.

Pick x ∈ A(ℤ/p) and y ∈ B(e) illustrating the fact that θ isn’t onto; that is, such that ι(ℤ/p)(x) = B(τ_{ℤ/p}^{e})(y) = z ∈ B(ℤ/p) but there is no element of A(e) hitting both x and y under the obvious maps. The choices x = μ(1_{ℤ/p}) and y = 1_{e} are, for example, acceptable. Consider the elements

(x, −x, 0, …, 0) ∈ A(ℤ/p)^{p}
(y, −y, 0, …, 0) ∈ B(e)^{p} (z, −z, 0, …, 0) ∈ B(ℤ/p)^{p}

in the context of the pushout diagrams defining ℨ_{p}(A) and ℨ_{p}(B). From the equations

∇(y, −y, 0, …, 0) = 0

and

B(τ_{ℤ/p}^{e})^{p}(y, −y, 0, …, 0) = (z, −z, 0, …, 0),

it follows that

κ_{B}(z, −z, 0, …, 0) = 0 ∈ ℨ_{p}(B).

However, κ_{A}(x, −x, 0, …, 0) must be a nonzero element of ℨ_{p}(A) since x and y illustrate the fact that θ isn’t surjective. Since

ℨ_{p}(ι)(κ_{A}(x, −x, 0, …, 0)) = κ_{B}(z, −z, 0, …, 0) = 0,

it follows that ℨ_{p} does not preserve monomorphisms.

The rest of this section is devoted to the postponed proof of our splitting result.

Proof of Proposition 11.2. Let M be in 𝔐_{∗}(𝔓, 𝔔). As in the proof of Proposition 7.2, we first construct a natural map

λ : (H_{ℤ/p} □ M)(ℤ/p) [figure]ℨ_{p}(M),

and then show that it is a split epimorphism. The map λ is, as before, derived from an appropriately behaved map

λ̃ : ⨁_{Q} M(Q) ⊗ [ℤ/p × Q, ℤ/p] [figure]ℨ_{p}(M).

Again it suffices to specify the restriction λ̃_{Q} of λ̃ to the summand indexed on each group Q. That summand is itself a direct sum of copies of M(Q) indexed on the generators of [ℤ/p × Q, ℤ/p]. These generators correspond to equivalence classes (α, β; δ) of diagrams of the form

ℤ/p × Q [figure: (α,β)] J [figure: δ]ℤ/p,

in which the induced map (α, β, δ) : J [figure]ℤ/p × Q × ℤ/p is a monomorphism. Denote the restriction of λ̃_{Q} to the copy of M(Q) indexed on the generator associated to this diagram by λ̃_{(α,β;δ)}.

Some manipulation of the diagram above must be done to define λ̃_{(α,β;δ)}. Let L = δ(J) so that L is either e or ℤ/p, and regard δ as a map into L. Since the order of the kernel of δ is not divisible by p, there is a unique map γ : L [figure]ℤ/p making the diagram

\xymatrix{ & J \ar[dl]^{α} \ar[dr]^{δ} \\ ℤ/p &  & L \ar[ll]^{γ}}

commute. This map is perhaps best understood by looking at a p-Sylow subgroup P of J. The map δ must restrict to an isomorphism from P to L. The map γ is the composite of the inverse of this isomorphism and the restriction of α to P.

If the kernel of β is not a 𝔔-group, then define λ̃_{(α,β;δ)} to be zero. Otherwise, define it to be the composite

M(Q) [figure: M(ρ(β))]M(J) [figure: M(τ(δ))]M(L) [figure: η_{γ}]ℨ_{p}(M),

where η_{γ} is the map from Definition 11.1 associated to a homomorphism γ from either e or ℤ/p into ℤ/p.

To show that λ can be derived from λ̃, it suffices to show that, for each morphism f : Q′ [figure]Q in B_{∗}(𝔓, 𝔔), the diagram

\xymatrix{M(Q′) ⊗ [ℤ/p × Q, ℤ/p] \ar[r]^{M(f)⊗1} \ar[d]_{1⊗(1×f)^{∗}} & M(Q) ⊗ [ℤ/p × Q, ℤ/p] \ar[d]^{λ̃_{Q}} \\ M(Q′) ⊗ [ℤ/p × Q′, ℤ/p] \ar[r]^{λ̃_{Q′}} & ℨ_{p}(M)}

commutes. Further, we need only check the cases in which f is a generator of the form ρ(ξ) associated to a homomorphism ξ : Q [figure]Q′ with 𝔔-kernel, or of the form τ(ζ) associated to a homomorphism ζ : Q′ [figure]Q with 𝔓-kernel. As in the previous proof, the commutativity of the diagram is purely formal if f = ρ(ξ) for some homomorphism ξ. Thus, we assume that f = τ(ζ) for some homomorphism ζ : Q′ [figure]Q with 𝔓-kernel.

We can verify the commutativity of the diagram by checking it on each summand M(Q′) of M(Q′) ⊗ [ℤ/p × Q, ℤ/p]. On the summand indexed by a diagram

ℤ/p × Q [figure: (α,β)] J [figure: δ]ℤ/p,

in which the map β : J [figure]Q does not have 𝔔-kernel, it is easy to see that the two composites in the diagram are both zero. Thus, we restrict our attention to the summands indexed on diagrams in which β does have 𝔔-kernel.

For this case, we must use Lemma 5.3 in much the same way that Lemma 3.3 of [21] is used in the proofs of Propositions 7.2 and 8.1. We also need to know that, for any group endomorphism γ : ℤ/p [figure]ℤ/p, the diagram

\xymatrix{M(e) \ar[r]^{M(τ_{ℤ/p}^{E})} \ar[dr]_{η_{e}} & M(ℤ/p) \ar[d]^{η_{γ}} \\  & ℨ_{p}(M)}

commutes. Ensuring the commutativity of this diagram is, however, the whole point of the pushout diagram used to define ℨ_{p}(M). The commutativity of our naturality diagram on the appropriate summand M(Q′) of M(Q′) ⊗ [ℤ/p × Q, ℤ/p] follows from these observations by an easy diagram chase. Thus, the desired map λ can be obtained from the map λ̃. Clearly, λ is natural in M.

We still must show that λ splits naturally. Let D be the summand of [ℤ/p × ℤ/p, ℤ/p] whose generators are represented by the diagrams

ℤ/p × ℤ/p [figure: (γ,1_{ℤ/p})] ℤ/p [figure: 1_{ℤ/p}]ℤ/p,

in which γ is an endomorphism of ℤ/p. Then D ≅ ℤ^{p}, and M(ℤ/p) ⊗ D ≅ M(ℤ/p)^{p}. Let σ̃ : M(ℤ/p)^{p} [figure](H_{ℤ/p} □ M)(ℤ/p) be the composite

M(ℤ/p)^{p} ≅ M(ℤ/p) ⊗ D ⊂ M(ℤ/p) ⊗ [ℤ/p × ℤ/p, ℤ/p]
⊂ ⨁_{Q} M(Q) ⊗ [ℤ/p × Q, ℤ/p]
[figure: (H_{ℤ/p} □ M)(ℤ/p),]

in which the last map is the standard projection. It is easy to see that the composite λ ∘ σ̃ : M(ℤ/p)^{p} [figure]ℨ_{p}(M) is just the projection κ_{M} : M(ℤ/p)^{p} [figure]ℨ_{p}(M) of Definition 11.1. Thus, to show that λ is a split epimorphism, it suffices to show that the map σ̃ factors through the projection κ_{M}.

For each Q, let ν_{Q} : M(Q) ⊗ [ℤ/p × Q, ℤ/p] [figure](H_{ℤ/p} □ M)(ℤ/p) be the map obtained from our description of (H_{ℤ/p}□M)(ℤ/p) as a quotient of ⨁_{Q} M(Q)⊗[ℤ/p×Q, ℤ/p]. The diagram

\xymatrix{M(e) ⊗ [ℤ/p × ℤ/p, ℤ/p] \ar[r]^{M(τ_{ℤ/p}^{e})⊗1} \ar[d]_{1⊗(1×τ_{ℤ/p}^{e})^{∗}} & M(ℤ/p) ⊗ [ℤ/p × ℤ/p, ℤ/p] \ar[d]^{ν_{ℤ/p}} \\ M(e) ⊗ [ℤ/p × e, ℤ/p] \ar[r]^{ν_{e}} & (H_{ℤ/p} □ M)(ℤ/p)}

commutes by the definition of (H_{ℤ/p} □ M)(ℤ/p). For any endomorphism γ of ℤ/p, the image under (1 × τ_{ℤ/p}^{e})^{∗} of the generator of [ℤ/p × ℤ/p, ℤ/p] represented by the diagram

ℤ/p × ℤ/p [figure: (γ,1_{ℤ/p})] ℤ/p [figure: 1_{ℤ/p}]ℤ/p

is just the morphism represented by the diagram

ℤ/p × e [figure] e [figure]ℤ/p

in which all the maps are the obvious ones. This observation, together with the commutativity of the above naturality diagram for the maps ν_{e} and ν_{ℤ/p}, easily implies that σ̃ factors through the projection κ_{M}. It follows that σ̃ induces a map σ which splits λ. The map σ is obviously natural.

Acknowledgments

During the research leave on which I completed this article, I was a visitor in the MIT Mathematics Department. The hospitality of that department, and especially of its algebraic topologists, is greatly appreciated. Comments of Peter May on an earlier draft of this article were quite helpful and are also deeply appreciated.

References

[1] J. F. Adams, Graeme Segal’s Burnside ring conjecture, Symposium on Algebraic Topology in honor of José Adem (Oaxtepec, 1981), Contemporary Mathematics, vol. 12, Amer. Math. Soc., Providence, R.I., 1982, pp. 9–18.

[2] J. F. Adams, J. H. Gunawardena, and H. Miller, The Segal conjecture for elementary abelian p-groups, Topology 24 (1985), no. 4, 435–460.

[3] Michael Barr (with an appendix by Po Hsiang Chu), ∗-autonomous categories, Lecture Notes in Mathematics, vol. 752, Springer, Berlin, 1979.

[4] Dave Benson, Stably splitting BG, Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 2, 189–198.

[5] Serge Bouc, Foncteurs d’ensembles munis d’une double action, J. Algebra 183 (1996), no. 3, 664–736.

[6] Brian Day, On closed categories of functors, Reports of the Midwest Category Seminar, IV, Lecture Notes in Mathematics, vol. 137, Springer, Berlin, 1970, pp. 1–38.

[7] Tammo tom Dieck, Transformation groups, de Gruyter Studies in Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1987.

[8] Andreas W. M. Dress, Contributions to the theory of induced representations, Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lectures Notes in Math., vol. 342, Springer, Berlin, 1973, pp. 183–240.

[9] , Induction and structure theorems for orthogonal representations of finite groups, Ann. of Math. (2) 102 (1975), no. 2, 291–325.

[10] W. G. Dwyer, E. M. Friedlander, and S. A. Mitchell, The generalized Burnside ring and the K-theory of a ring with roots of unity, K-Theory 6 (1992), no. 4, 285–300.

[11] J. A. Green, Axiomatic representation theory for finite groups, J. Pure Appl. Algebra 1 (1971), no. 1, 41–77.

[12] I. Hambleton, Laurence Taylor, and Bruce Williams, Detection theorems for K-theory and L-theory, J. Pure Appl. Algebra 63 (1990), no. 3, 247–299.

[13] John C. Harris, On certain stable wedge summands of B(z/p)_{+}^{n}, Canad. J. Math. 44 (1992), no. 1, 104–118.

[14] John C. Harris and Nicholas J. Kuhn, Stable decompositions of classifying spaces of finite abelian p-groups, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 3, 427–449.

[15] L. G. Lewis, Jr., Splitting theorems for certain equivariant spectra, to appear in the Memoirs of the Amer. Math. Soc.

[16] , The theory of Green functors, mimeographed notes, 1980.

[17] , The RO(G)-graded equivariant ordinary cohomology of complex projective spaces with linear ℤ/p actions, Algebraic topology and transformation groups, Proceedings, Göttingen 1987, Lecture Notes in Math., vol. 1361, Springer, Berlin, 1988, pp. 53–122.

[18] , Equivariant Eilenberg-Mac Lane spaces and the equivariant Seifert-van Kampen and suspension theorems, Topology Appl. 48 (1992), no. 1, 25–61.

[19] , The equivariant Hurewicz map, Trans. Amer. Math. Soc. 329 (1992), no. 2, 433–472.

[20] , Change of universe functors in equivariant stable homotopy theory, Fund. Math. 148 (1995), no. 2, 117–158.

[21] , The category of Mackey functors for a compact Lie group, Group representations: cohomology, group actions and topology (Seattle, WA, 1996), Proc. Sympos. Pure Math., vol. 63, Amer. Math. Soc., Providence, RI, 1998, pp. 301–354.

[22] L. G. Lewis, Jr., J. P. May, and J. E. McClure, Ordinary RO(G)-graded cohomology, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 208–212.

[23] L. G. Lewis, Jr., J. P. May, and M. Steinberger (with contributions by J. E. McClure), Equivariant stable homotopy theory, Lecture Notes in Math., vol. 1213, Springer-Verlag, Berlin, 1986.

[24] John Martino and Stewart Priddy, The complete stable splitting for the classifying space of a finite group, Topology 31 (1992), no. 1, 143–156.

[25] J. P. May, Stable maps between classifying spaces, Conference on algebraic topology in honor of Peter Hilton (Saint John’s, Nfld., 1983), Contemporary Mathematics, vol. 37, Amer. Math. Soc., Providence, R.I., 1985, pp. 121–129.

[26] Robert Oliver, Whitehead groups of finite groups, Cambridge University Press, Cambridge, 1988.

[27] Bo Stenström, Rings of quotients: An introduction to methods of ring theory, Die Grundlehren der Mathematischen Wissenschaften, vol. 217, Springer-Verlag, New York, 1975.

[28] Peter Webb, Two classifications of simple Mackey functors with applications to group cohomology and the decomposition of classifying spaces, J. Pure Appl. Algebra 88 (1993), no. 1-3, 265–304.

[29] C. Witten, Self-maps of classifying spaces of finite groups and classification of lowdimensional Poincaré duality spaces, Ph.D. thesis, Stanford University, 1978.

Department of Mathematics, Syracuse University, Syracuse NY 13244-1150

Email: lglewis@syr.edu

This article may be accessed via WWW at http://www.tac.mta.ca/tac/ or by anony-

mous ftp at ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1999/n9/n9.{dvi,ps}

THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods.

Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication.

The method of distribution of the journal is via the Internet tools WWW/ftp. The journal is archived electronically and in printed paper format.

Subscription information. Individual subscribers receive (by e-mail) abstracts of articles as they are published. Full text of published articles is available in .dvi and Postscript format. Details will be e-mailed to new subscribers and are available by WWW/ftp. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. For institutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh, rrosebrugh@mta.ca.

Information for authors. The typesetting language of the journal is TEX, and L^{A}TEX is the preferred flavour. TEX source of articles for publication should be submitted by e-mail directly to an appropriate Editor. They are listed below. Please obtain detailed information on submission format and style files from the journal’s WWW server at URL http://www.tac.mta.ca/tac/ or by anonymous ftp from ftp.tac.mta.ca in the directory pub/tac/info. You may also write to tac@mta.ca to receive details by e-mail.

Editorial board.

John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@barrs.org Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, Università dell Insubria: carboni@fis.unico.it P. T. Johnstone, University of Cambridge: ptj@pmms.cam.ac.uk G. Max Kelly, University of Sydney: maxk@maths.usyd.edu.au Anders Kock, University of Aarhus: kock@imf.au.dk F. William Lawvere, State University of New York at Buffalo: wlawvere@acsu.buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.ruu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mscs.dal.ca Andrew Pitts, University of Cambridge: ap@cl.cam.ac.uk Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz James Stasheff, University of North Carolina: jds@charlie.math.unc.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Sydney: walters b@maths.usyd.edu.au R. J. Wood, Dalhousie University: rjwood@mscs.dal.ca