MACKEY FUNCTORS AND BISETS

I. HAMBLETON, L. R. TAYLOR, AND E. B. WILLIAMS

Abstract. For any finite group G, we define a bifunctor from the Dress category of finite G-sets to the conjugation biset category, whose objects are subgroups of G, and whose morphisms are generated by certain bifree bisets. Any additive functor from the conjugation biset category to abelian groups yields a Mackey functor by composition. We characterize the Mackey functors which arise in this way.

1. Introduction

Let G be a finite group. A Mackey functor in the sense of Dress [5] is a bifunctor M: D(G) → Ab from the category of finite left G-sets and G-maps to abelian groups, satisfying a pull-back axiom (M1) and an additivity axiom (M2). These axioms express the classical Mackey properties from representation theory (see Section 2).

Many of the Mackey functors encountered in applications of the theory factor through the G-Burnside category A(G), whose objects are the subgroups H ⊂ G and whose morphisms are generated by H_{2} -H_{1} bisets with certain properties (compare [6, 1.A.4]). Let A_{•}(G) denote the additive completion of this category, defined in Section 3.

In Section 6 we define a subcategory B_{•}(G) ⊂ A_{•}(G), where the morphisms are restricted to be conjugation bisets, and construct a bifunctor j: D(G) → B_{•}(G) (see Definition 7.2). A Mackey functor M is said to be conjugation-invariant provided that the centralizer C_{G}(H) acts trivially on M(G/H) for all H ⊂ G, via the G-maps φ: G/H → G/H given by eH ↦ zH, for some z ∈ C_{G}(H) (see Definition 3.1). The main result (see Theorem 8.1) is a recognition principle for such Mackey functors.

Theorem A. Let G be a finite group. A Mackey functor M: D(G) → Ab factors through j: D(G) → B_{•}(G) if and only if M is conjugation invariant.

This result applies to classical Mackey functors such as the Swan ring and the Dress ring, and to algebraic K-theory and L-theory functors encountered in geometric topology and surgery theory. These applications, surveyed in [6] and [8], mostly depend on the following immediate consequence of our main result.

Corollary B. For any additive functor F: A_{•}(G) → Ab, the composition F ∘j: D(G) → Ab is a Mackey functor.

Remark 1.1. The idea of reformulating the bifunctor Mackey properties as a single functor out of an intermediate category has been carried out by several authors (see [11, Theorem 2], [9], [4, Chap. IV.8], [13]). The point of our biset version is to provide a useful method to check the Mackey properties, starting from input data that is often encountered in applications of the theory. The use of bisets as morphisms in a category has appeared in various settings in the literature (see, for example, [10], [1, §9, p. 454], [6, 1.A.4], [2, §2] and [3]).

Date: June 25, 2008.

Research partially supported by NSERC Discovery Grant A4000 and the NSF. The authors would also like to thank the SFB 478, Münster, for its hospitality and support in June 2008.

2. Mackey functors

We will first recall the basic definitions Dress used in his formulation of induction theory [5]. Let G be a finite group, and let D(G) denote the category whose objects are finite, left G-sets and whose morphisms are G-maps. A Mackey functor is a bifunctor M = (M_{∗}, M^{∗}): D(G) → Ab, where Ab denotes the category of abelian groups and groups homomorphisms, such that M_{∗}(S) = M^{∗}(S) for each object S ∈ D(G), and the following two properties hold:

(M1) For any pullback diagram of finite G-sets

\xymatrix{S \ar[r]^{Ψ} \ar[d]_{Φ} & S_{1} \ar[d]^{φ} \\ S_{2} \ar[r]^{ψ} & T}

the induced maps give an commutative diagram

\xymatrix{M(S) \ar[r]^{Ψ_{M}} & M(S_{1}) \\ M(S_{2}) \ar[r]^{ψ_{M}} \ar[u]^{Φ^{M}} & M(T) \ar[u]_{φ^{M}}}

Here we denote the covariant maps by ψ_{M} and the contravariant maps by φ^{M}.

(M2) The embeddings of S_{1} and S_{2} into the disjoint union S_{1} ⨆ S_{2} define an isomorphism M^{∗}(S_{1} ⨆ S_{2}) → M^{∗}(S_{1}) ⊕ M^{∗}(S_{2}). Let M(∅) = 0.

The property (M1) is the usual double coset formula, and (M2) gives additivity. We remark that for any bifunctor satisfying (M1), the composition M_{∗}(S_{1}) ⊕ M_{∗}(S_{2}) → M_{∗}(S_{1} ⨆ S_{2}) = M^{∗}(S_{1} ⨆ S_{2}) → M^{∗}(S_{1}) ⊕ M^{∗}(M_{2}) = M_{∗}(S_{1}) ⊕ M_{∗}(S_{2}) is just the identity matrix. It follows that any sub-bifunctor of a Mackey functor is Mackey.

Remark 2.1. We could replace the target category Ab of abelian groups throughout by the category R-Mod of R-modules, for any commutative ring R with unit.

3. The G-Burnside category A(G)

We will be interested in Mackey functors which factor through the G-Burnside category A(G), whose objects are subgroups H ⊂ G, and where Hom_{A(G)}(H_{1}, H_{2}) is the Grothendieck construction applied to the isomorphism classes of finite bifree H_{2} -H_{1} bisets (meaning both left and right actions are free). In contrast, the morphisms in the G-Burnside category of [1, §9, p. 454], and our category RG-Morita [6, 1.A.4] just have a one-sided isotropy assumption.

To make our recognition principle more precise, we will define a subcategory B(G) ⊂ A(G) by restricting its morphisms to conjugation bisets (see Definition 6.1).

Because of the Grothendieck construction, A(G) and B(G) are both Ab–categories: the morphism sets are abelian groups and the compositions are bilinear [12, I.8, p. 28]. Let u: A(G) → A_{•}(G) and u: B(G) → B_{•}(G) denote the associated universal free additive categories, and the universal inclusions (see [12, VII.2, problem 6, p. 194]).

It turns out that the Mackey functors which factor through B_{•}(G) have an additional property, called conjugation invariance, which can be expressed in terms of their restriction to the orbit category Or(G). Recall that the objects of Or(G) are the subgroups H ⊂ G, and the morphisms Hom_{Or(G)}(H_{1}, H_{2}) are the G-maps φ: G/H_{1} → G/H_{2}. Any such G-map is uniquely determined by eH_{1} ↦ gH_{2}, where g^{−1}H_{1}g ⊆ H_{2}. If H_{1} = H_{2} = H, then any element z ∈ C_{G}(H) in the centralizer of H gives a G-map φ_{z}: G/H → G/H.

Definition 3.1. A functor F: Or(G) → Ab is called conjugation invariant if the induced maps F (φ_{z}) = id: F (H) → F (H), whenever φ_{z}: G/H → G/H is given by eH ↦ zH, for some z ∈ C_{G}(H). A Mackey functor is said to be conjugation invariant if its restriction to Or(G) satisfies this condition.

We now relate this condition to the conjugation homomorphisms which will be used in the definition of B_{•}(G). If φ ∈ Hom_{Or(G)}(H_{1}, H_{2}) is represented by g ∈ G, we let c_{g}: H_{1} → H_{2} denote the associated conjugation homomorphism given by c_{g}(h_{1}) = g^{−1}h_{1}g ∈ H_{2}, for all h_{1} ∈ H_{1}. Since the G-map φ only depends on the coset gH_{2}, we may vary g ∼ gh_{2}, for any h_{2} ∈ H_{2}. The associated conjugation homomorphism c_{gh_{2}} = c_{h_{2}} ∘ c_{g} is thus well-defined (as a homomorphism) up to conjugation by elements of H_{2}. Let

c_{φ} := [c_{g}] ∈ Hom(H_{1}, H_{2})/(conjugation in H_{2})

denote the equivalence class of the associated conjugation homomorphism to a G-map φ(eH_{1}) = gH_{2}. Two different morphisms φ_{1}, φ_{2} ∈ Hom_{Or(G)}(H_{1}, H_{2}) yield the same equivalence class c_{φ_{1}} = c_{φ_{2}} if and only if g_{1}h_{2}g_{2}^{−1} centralizes H_{1}, for some h_{2} ∈ H_{2}.

Lemma 3.2. A functor F: Or(G) → Ab is conjugation invariant if and only if c_{φ_{1}} = c_{φ_{2}} implies F (φ_{1}) = F (φ_{2}) for all morphisms φ_{1}, φ_{2} ∈ Hom_{Or(G)}(H_{1}, H_{2}).

Proof. Suppose that φ_{1}, φ_{2} ∈ Hom_{Or(G)}(H_{1}, H_{2}) are given by φ_{i}(eH_{1}) = g_{i}H_{2}, for i = 1, 2. If there exists an element h_{2} ∈ H_{2} such that z := g_{1}h_{2}g_{2}^{−1} ∈ C_{G}(H_{1}), then we see that φ_{1} = φ_{2} ∘ φ_{z}. Therefore, if F is conjugation invariant and c_{φ_{1}} = c_{φ_{2}} we conclude that F (φ_{1}) = F (φ_{2}). Conversely, for any subgroup H ⊂ G, and any z ∈ C_{G}(H), the G-maps φ_{z}, id: G/H → G/H have the property c_{φ_{z}} = c_{id}. Therefore the given condition implies that F (φ_{z}) = F (id) = id, and hence F is conjugation invariant. □

The morphisms in A(G) are defined by the Grothendieck construction with addition operation the disjoint union of bisets. By convention, the empty biset ∅ represents the zero element. Composition comes from the balanced product:

H_{3}X_{H_{2}} ∘_{H_{2}}Y_{H_{1}} = (_{H_{3}}X_{H_{2}}) ×_{H_{2}} (_{H_{2}}Y_{H_{1}}).

The reader should check that this is well–defined on isomorphism classes of bisets and “bilinear” in that

(_{H_{3}}X_{H_{2}} ⨆_{H_{3}}Y_{H_{2}}) ∘_{H_{2}}Z_{H_{1}} ≅ (_{H_{3}}X_{H_{2}} ∘_{H_{2}}Z_{H_{1}}) ⨆ (_{H_{3}}Y_{H_{2}} ∘_{H_{2}}Z_{H_{1}})

with a similar formula for disjoint union on the right. The morphisms in A_{•}(G) are matrices of morphisms in A(G).

Definition 3.3. We define a contravariant involution τ: A(G) → A(G), by the identity on objects, and on morphisms it is the map induced on the Grothendieck construction by the function which takes the finite bifree H_{2} -H_{1} biset _{H_{2}}X_{H_{1}} to the finite bifree H_{1} -H_{2} biset _{H_{1}}X_{H_{2}} which is X as a set and h_{1} · x · h_{2} is defined to be h_{2}^{−1}xh_{1}^{−1}.

The reader needs to check that isomorphic bisets are isomorphic after reversing the order, and should also check that the transpose conjugate of a disjoint union is isomorphic to the disjoint union of the conjugate transposes of the pieces. This means that τ is a functor which induces a homomorphism of Hom–sets. It is clearly an involution, not just up to natural equivalence. Since τ is a homomorphism on Hom-sets, it induces an additive contravariant involution τ^{•}: A_{•}(G) → A_{•}(G), called conjugate transpose, which commutes with the functor u: A(G) → A_{•}(G). By definition, τ^{•} acts on a matrix of morphisms by applying τ to each entry, and then transposing the matrix.

4. Indecomposable H_{2} -H_{1} bisets

We will need some more detailed information about the bifree bisets used to define morphisms in A(G). An H_{2} -H_{1} biset is just a left (H_{2} × H_{1}^{op})-set and so any finite H_{2} - H_{1}-biset is a disjoint union of transitive (H_{2} × H_{1}^{op})-sets. Since there are three groups acting (H_{1}, H_{2} and H_{2} × H_{1}^{op}), the following definition should avoid confusion.

Definition 4.1. An H_{2} -H_{1} biset X is indecomposable if X is a transitive (H_{2} × H_{1}^{op})-set.

Since every H_{2} -H_{1} biset is a disjoint union of indecomposable H_{2} -H_{1} bisets, it follows that X_{1} ⨆ Y is isomorphic to X_{1} ⨆ Y if and only if X_{1} is isomorphic to X_{2}. One result of this remark is that the Grothendieck group of finite bifree H_{2} -H_{1} bisets is the free abelian group on the indecomposable bifree ones.

An indecomposable H_{2} -H_{1} biset X is isomorphic (via choice of base point) to a coset space (H_{2} × H_{1}^{op})/S, for some subgroup S ⊂ H_{2} × H_{1}^{op}. These models will be used extensively below, so we make some remarks and introduce some notation.

(i) To see (H_{2} × H_{1}^{op})/S as an H_{2} -H_{1} biset, define the H_{2} action to be left multipli-

cation in the first coordinate, but define (h_{2}, h_{1})S · g_{1} = (h_{2}, g_{1}^{−1}h_{1})S.

(ii) We introduce the following notation for points in (H_{2} × H_{1}^{op})/S: if h_{2} ∈ H_{2} and h_{1} ∈ H_{1}, write {h_{2}, h_{1}} = (h_{2}, h_{1}^{−1})S. In this notation, the H_{2} -H_{1} action is the

evident one: g_{2}{h_{2}, h_{1}}g_{1} = {g_{2}h_{2}, h_{1}g_{1}}.

(iii) The left isotropy group of (e, e) is just (H_{2} × {e}) ∩ S, and the right isotropy group of (e, e) is just ({e} × H_{1}^{op}) ∩ S, so the actions are free if and only if (H_{2} × {e}) ∩ S = {e × e} and ({e} × H_{1}^{op}) ∩ S = {e × e}.

(iv) Equivalently, the S-action is bifree if the compositions ι_{1} : S ⊂ H_{2} × H_{1}^{op} → H_{2} and S ⊂ H_{2} × H_{1}^{op} → H_{1}^{op} are injective. We will work instead with the composite ι_{2} : S → H_{2} × H_{1}^{op} → H_{1}^{op} ⟶^{≈} H_{1}, where the last homomorphism takes h_{1} to h_{1}^{−1}.

(v) Let L ⊂ H_{2} denote the image of the isotropy subgroup S under the injection ι_{1}, and let K ⊂ H_{1} denote the image of S under the injection ι_{2}. Define

γ: L ⟶^{ι_{1}^{−1}} S ⟶^{ι_{2}} K

and notice that this is an isomorphism.

(vi) Conversely, given subgroups L ⊂ H_{2}, K ⊂ H_{1}, and an isomorphism γ: L ⟶^{≈} K, let _{H_{2}}[L, γ, K]_{H_{1}} denote the indecomposable bifree biset (H_{2} × H_{1}^{op})/S, where S ⊂ H_{2} × H_{1}^{op} is the graph of

γ̄: L ⟶^{γ} K ⊂ H_{1} ⟶^{ι} H_{1}^{op},

and ι: H_{1} ⟶^{≈} H_{1}^{op} is the usual isomorphism which takes h to h^{−1}.

Given an indecomposable H_{2} -H_{1} biset X, a choice of base-point x ∈ X yields an isotropy subgroup S_{x} ⊂ H_{2} × H_{1}^{op} and a preferred biset isomorphism

(4.2)  Ψ_{x}: (H_{2} × H_{1}^{op})/S_{x} ⟶^{≅} X

defined by Ψ_{x}({h_{2}, h_{1}}) = h_{2}xh_{1}.

Definition 4.3 (The standard representation). Let X be an indecomposable bifree H_{2} - H_{1} biset, and x ∈ X a base-point. The standard representation for X at x is the data

(4.4)  Ψ_{x}: _{H_{2}}[L_{x}, γ_{X,x}, K_{x}]_{H_{1}} ⟶^{≅} X

given by the preferred biset isomorphism Ψ_{x}({h_{2}, h_{1}}) = h_{2}xh_{1}, where L_{x} ⊂ H_{2} is the image of the isotropy subgroup S_{x} under ι_{1}, K_{x} ⊂ H_{1} is the image of ι_{2}, and

γ_{X,x}: L_{x} ⟶^{ι_{1}^{−1}} S_{x} ⟶^{ι_{2}} K_{x}

is an isomorphism. We noted above that L_{x}, K_{x} and γ_{X,x} determine S_{x} as a (graph) subgroup of H_{2} × H_{1}^{op}. □

Remark 4.5. Any model coset space _{H_{2}}[L, γ, K]_{H_{1}} is the standard representation for some indecomposable bifree biset. We let X = (H_{2} × H_{1}^{op})/S, and choose x ∈ X as the coset of the identity, then L_{x} = L, K_{x} = K and γ_{X,x} = γ. □

Remark 4.6. Here is a second description of the standard representation. We present it so as to identify the component of any point x in an H_{2} -H_{1} bifree biset X, indecomposable or not. Let L_{x} = {h_{2} ∈ H_{2} | h_{2}x ∈ xH_{1} }. Check L_{x} is a subgroup of H_{2}. Since the H_{1} action is free, there exists a unique h_{1} ∈ H_{1} such that h_{2}x = xh_{1}. Define a function f: L_{x} → H_{1} by f(h_{2}) = h_{1}. Let K_{x} denote the image of f and let γ_{X,x}: L_{x} → K_{x} denote the restriction of f. Check that f is a homomorphism and hence γ_{X,x} is an isomorphism. Finally check that Ψ_{x}: _{H_{2}}[L_{x}, γ_{X,x}, K_{x}]_{H_{1}} → X defined by Formula 4.4 above is an injection of H_{2} -H_{1} bisets which is then automatically onto the component of X containing x. □

Hereafter we will call _{H_{2}}[L_{x}, γ_{X,x}, K_{x}]_{H_{1}} the standard representation of X at x even if X is not indecomposable. It follows that

⨆_{x∈X} _{H_{2}}[L_{x}, γ_{X,x}, K_{x}]_{H_{1}} \underrightarrow{⨆ Ψ_{x}} X

is a bijection of H_{2} -H_{1} bisets, where we take one x in each indecomposable component of X. Here are two useful properties of the standard representation for the reader to verify.

We note the effect of the involution τ: A(G) → A(G) on the standard representations.

Lemma 4.7. If [L_{x}, γ_{X,x}, K_{x}] is the standard representation for X at x, then [K_{x}, γ−1_{X,x}, L_{x}] is the standard representation for τ(X) at x.

Next we consider a change of base-point. If y is another point in the same indecomposable component as x, then y = h_{2}xh_{1} for some choice of h_{2} ∈ H_{2} and h_{1} ∈ H_{1}. For g ∈ G and any subgroup K ⊂ G, let K^{g} = g^{−1}Kg. If γ: L → K is a homomorphism between subgroups of G and if g_{1}, g_{2} ∈ G, define _{g_{2}}γ^{g_{1}}(h) = g_{1}^{−1}γ(g_{2}^{−1}hg_{2})g_{1} : L^{g_{2}^{−1}} → K^{g_{1}}.

Lemma 4.8. With notation as above, the standard representation for X at y = h_{2}xh_{1} ∈ X

is [L_{x}^{h_{2}^{−1}},_{h_{2}}γh_{1}_{X,x}, K_{x}^{h_{1}}]. In other words, L_{y} = L_{x}^{h_{2}^{−1}}, K_{y} = K_{x}^{h_{1}} and γ_{X,y} = _{h_{2}}γh_{1}_{X,x}.

It will be useful later to be able to identify X/H_{1} and H_{2}\X.

Lemma 4.9. If X is an indecomposable, bifree H_{2} -H_{1} biset, then the bijection of bisets Ψ_{x}: _{H_{2}}[L_{x}, γ_{X,x}, K_{x}]_{H_{1}} → X induces

(i) a bijection of left H_{2}-sets H_{2}/L_{x} → X/H_{1}
(ii) a bijection of right H_{1}-sets K_{x}\H_{1} → H_{2}\X.

Proof. The proof is immediate. □

5. Composition of bisets

Fix indecomposable, bifree bisets _{H_{2}}(X_{1})_{H_{1}} and _{H_{3}}(X_{2})_{H_{2}} and let us analyze X_{3} = X_{2} ×_{H_{2}} X_{1}. In general X_{3} will not be indecomposable, and the Mackey double coset formula enters the picture. We begin by analyzing the standard representation at a point.

Lemma 5.1. The standard representation for X_{3} = X_{2} ×_{H_{2}} X_{1} at [x_{2}, x_{1}] is given by

• L_{[x_{2},x_{1}]} = γ−1_{X_{2},x_{2}}(K_{x_{2}} ∩ L_{x_{1}}),

• K_{[x_{2},x_{1}]} = γ_{X_{3},[x_{2},x_{1}]}(L_{[x_{2},x_{1}]}) = γ_{X_{1},x_{1}}(K_{x_{2}} ∩ L_{x_{1}})

• γ_{X_{3},[x_{2},x_{1}]} = γ_{X_{1},x_{1}} ∘ γ_{X_{2},x_{2}}.

Proof. Since the projection X_{2} ×X_{1} → X_{2} ×_{H_{2}} X_{1} is onto, every point in X_{2} ×_{H_{2}} X_{1} is the image of a point in X_{2} × X_{1}. Given x_{1} ∈ X_{1} and x_{2} ∈ X_{2} write [x_{2}, x_{1}] ∈ X_{3} = X_{2} ×_{H_{2}} X_{1} for the image of x_{2} × x_{1}. In this notation, the H_{3} -H_{1} structure is the evident one: h_{3}[x_{2}, x_{1}]h_{1} = [h_{3}x_{2}, x_{1}h_{1}].

The point [x_{2}, x_{1}] ∈ X_{3} is a set of points {x_{2}h × h^{−1}x_{1}} ⊂ X_{2} × X_{1} for all h ∈ H_{2}. If h_{3}(x_{2} × x_{1})h_{1} = x_{2}h × h^{−1}x_{1} ∈ X_{2} × X_{1} for some h ∈ H_{2} then h_{3}x_{2} = x_{2}h and h^{−1}x_{1} = x_{1}h_{1}. Now if h_{3}x_{2} = x_{2}h then h = γ_{X_{2},x_{2}}(h_{3}) and h_{3} ∈ L_{x_{2}}. If h^{−1}x_{1} = x_{1}h_{1} then h_{1} = γ_{X_{1},x_{1}}(h^{−1}) and h^{−1} ∈ L_{x_{1}}. Since L_{x_{1}} is a group, h ∈ L_{x_{1}}. Since h = γ_{X_{2},x_{2}}(h_{3}),

h ∈ K_{x_{2}}. Therefore, h_{3} ∈ γ−1_{X_{2},x_{2}}(K_{x_{2}} ∩ L_{x_{1}}). Let L_{[x_{2},x_{1}]} = γ−1_{X_{2},x_{2}}(K_{x_{2}} ∩ L_{x_{1}}) and define

γ_{X_{3},[x_{2},x_{1}]} = γ_{X_{1},x_{1}} ∘ γ_{X_{2},x_{2}}. What we have seen so far is that if h_{3} [x_{2}, x_{1}] h_{1} = [x_{2}, x_{1}] in X_{3}, then h_{3} ∈ L_{[x_{2},x_{1}]} and h_{1} = γ_{X_{3},[x_{2},x_{1}]}(h_{3}).

If h_{3} ∈ L_{[x_{2},x_{1}]}, then h_{3}(x_{2} × x_{1})γ_{X_{3},[x_{2},x_{1}]}(h_{3}) = x_{2}γ_{X_{2},x_{2}}(h_{3}) × γ_{X_{2},x_{2}}(h_{3})^{−1}x_{1} and it further follows that for any h_{2} ∈ H_{2}, h_{3}(x_{2}h_{2} × h_{2}^{−1}x_{1})γ_{X_{3},[x_{2},x_{1}]}(h_{3}) = x_{2}γ_{X_{2},x_{2}}(h_{3})h_{2} × h_{2}^{−1}γ_{X_{2},x_{2}}(h_{3})^{−1}x_{1} = x_{2}(γ_{X_{2},x_{2}}(h_{3})h_{2}) × (γ_{X_{2},x_{2}}(h_{3})h_{2})^{−1}x_{1}. In other words, if L_{[x_{2},x_{1}]} and γ_{X_{3},[x_{2},x_{1}]} are defined as above, then for any h_{3} ∈ L_{[x_{2},x_{1}]}, h_{3} [x_{2}, x_{1}] γ_{X_{3},[x_{2},x_{1}]}(h_{3}) = [x_{2}, x_{1}]. If we let K_{[x_{2},x_{1}]} = γ_{X_{3},[x_{2},x_{1}]}(L_{[x_{2},x_{1}]}), then the proof is complete. □

Lemma 5.2. The double coset space K_{x_{2}}\H_{2}/L_{x_{1}} parametrizes the indecomposable components of X_{2} ×_{H_{2}} X_{1}, where x_{1} ∈ X_{1} and x_{2} ∈ X_{2} are base points.

Proof. To simplify the notation let X_{2} = _{H_{2}}[L_{3}, γ_{2}, K_{2}]_{H_{1}}, and X_{1} = _{H_{2}}[L_{2}, γ_{1}, K_{1}]_{H_{1}}. The set of indecomposable components is given by

H_{3}\(X_{2} ×_{H_{2}} X_{1})/H_{1} = (H_{3}\X_{2}) ×_{H_{2}} (X_{1}/H_{1}).

By Lemma 4.9, (H_{3}\X_{2}) ×_{H_{2}} (X_{1}/H_{1}) = (K_{2}\H_{2}) ×_{H_{2}} (H_{2}/L_{2}) = K_{2}\H_{2}/L_{2}. If {h_{2}} ∈ K_{2}\H_{2}/L_{2} is a set of double coset representatives with h_{2} ∈ H_{2}, then [x_{2}h_{2}, x_{1}] gives a set of points, one in each indecomposable component. □

Next we give an explicit formula for X_{3} = X_{2} ×_{H_{2}} X_{1} as a disjoint union of indecomposable bisets. We define a map

Φ_{h_{2}} : _{H_{3}}[L_{x_{3}}, γ_{X_{3},x_{3}}, K_{x_{3}}]_{H_{1}} → _{H_{3}}[L_{x_{2}}, γ_{X_{2},x_{2}}, K_{x_{2}}]_{H_{2}} ×_{H_{2}} _{H_{2}}[L_{x_{1}}, γ_{X_{1},x_{1}}, K_{x_{1}}]_{H_{1}}

by the formula Φ_{h_{2}}({h_{3}, h_{1}}) = [{h_{3}, h_{2}}, {e, h_{1}}], using the notation of Definition 4.3.

Theorem 5.3 (Mackey double coset formula). The H_{3}-H_{1} biset X_{3} = X_{2} ×_{H_{2}} X_{1} is given by the disjoint union of the left-hand vertical maps in the diagram

[figure: [γ−1_{X_{2},x_{2}}(K_{x_{2}} ∩ L_{x_{1}}^{h_{2}^{−1}}), γ_{X_{1},x_{1}}∘ c_{h_{2}} ∘ γ_{X_{2},x_{2}}, γ_{X_{1},x_{1}}(K_{x_{2}}^{h_{2}} ∩ L_{x_{1}})]; Φ_{h_{2}}; _{H_{3}}[L_{x_{3}}, γ_{X_{3},x_{3}}, K_{x_{3}}]_{H_{1}}; Ψ_{x_{3}}; _{H_{3}}[L_{x_{2}}, γ_{X_{2},x_{2}}, K_{x_{2}}]_{H_{2}} ×_{H_{2}} _{H_{2}}[L_{x_{1}}, γ_{X_{1},x_{1}}, K_{x_{1}}]_{H_{1}}; Ψ_{x_{2}}×Ψ_{x_{1}}; ≈; X_{2} ×_{H_{2}} X_{1}]

over the base-points x_{3} = [x_{2}h_{2}, x_{1}], for h_{2} ∈ K_{x_{2}}\H_{2}/L_{x_{1}}.

Proof. We first derive the explicit formula displayed in the top left corner of the diagram. Let [L_{x_{2}}, γ_{X_{2},x_{2}}, K_{x_{2}}] be the standard representation for X_{2} at x_{2}, and let [L_{x_{1}}, γ_{X_{1},x_{1}}, K_{x_{1}}] be the standard representation for X_{1} at x_{1}. Then the standard representation for X_{2} at x_{2}h_{2} is given by Lemma 4.8: [L_{x_{2}}, γ_{X_{2},x_{2}h_{2}}, K_{x_{2}}] = [L_{x_{2}}, γh_{2}_{X_{2},x_{2}}, K_{x_{2}}^{h_{2}}]. It further follows that the standard representation for X_{3} = X_{2} ×_{H_{2}} X_{1} at x_{3} = [x_{2}h_{2}, x_{1}] is given by

[L_{x_{3}}, γ_{X_{3},x_{3}}, K_{x_{3}}] = [L_{[x_{2}h_{2},x_{1}]}, γ_{X_{3},[x_{2}h_{2},x_{1}]}, K_{[x_{2}h_{2},x_{1}]}] .

By Lemmas 4.8, 5.1 and 5.2

• L_{[x_{2}h_{2},x_{1}]} = γ−1_{X_{2},x_{2}h_{2}}(K_{x_{2}h_{2}} ∩ L_{x_{1}}) = (γh_{2}_{X_{2},x_{2}})^{−1}(K_{x_{2}}^{h_{2}} ∩ L_{x_{1}}) = γ−1_{X_{2},x_{2}}(K_{x_{2}} ∩ L_{x_{1}}^{h_{2}^{−1}})

• K_{[x_{2}h_{2},x_{1}]} = γ_{X_{1},x_{1}}(K_{x_{2}h_{2}} ∩ L_{x_{1}}) = γ_{X_{1},x_{1}}(K_{x_{2}}^{h_{2}} ∩ L_{x_{1}})

• γ_{X_{3},[x_{2}h_{2},x_{1}]} = γ_{X_{1},x_{1}}∘ γ_{X_{2},x_{2}h_{2}} = γ_{X_{1},x_{1}}∘ γh_{2}_{X_{2},x_{2}} = γ_{X_{1},x_{1}}∘ c_{h_{2}} ∘ γ_{X_{2},x_{2}}

For h_{2} ∈ H_{2} the function defined by Φ_{h_{2}}({h_{3}, h_{1}}) = [{h_{3}, h_{2}}, {e, h_{1}}] makes the following diagram commute

\xymatrix{_{H_{3}}[L_{[x_{2}h_{2},x_{1}]}, γ_{X_{3},[x_{2}h_{2},x_{1}]}, K_{[x_{2}h_{2},x_{1}]}]_{H_{1}} \ar@{-->}[dr]_{Ψ_{[x_{2}h_{2},x_{1}]}} \ar[r]^{Φ_{h_{2}}} & _{H_{3}}[L_{x_{2}}, γ_{X_{2},x_{2}}, K_{x_{2}}]_{H_{2}} ×_{H_{2}} _{H_{2}}[L_{x_{1}}, γ_{X_{1},x_{1}}, K_{x_{1}}]_{H_{1}} \ar[d]^{Ψ_{x_{2}}×Ψ_{x_{1}}} \\  & _{H_{3}}(X_{2} ×_{H_{2}} X_{1})_{H_{1}}}

Since the Ψ maps are injective, it suffices to prove that the formula given for Φ_{h_{2}} makes the diagram commute. Over and then down takes {h_{3}, h_{1}} to [{h_{3}, h_{2}}, {e, h_{1}}] and then to [h_{3}x_{2}h_{2}, x_{1}h_{1}] whereas Ψ_{[x_{2}h_{2},x_{1}]}({h_{3}, h_{1}}) = h_{3}[x_{2}h_{2}, x_{1}]h_{1} = [h_{3}x_{2}h_{2}, x_{1}h_{1}].

The conclusion now follows immediately from the commutativity of the displayed diagram, since the disjoint union of the right-hand vertical maps Ψ_{x_{3}} is a bijection, and the map Ψ_{x_{2}} × Ψ_{x_{1}} is a bijection (see Definition 4.3). □

Remark 5.4. The map γ_{X_{3},[x_{2}h_{2},x_{1}]} can be displayed as the composition on the top row of the diagram

\xymatrix{L_{[x_{2}h_{2},x_{1}]} \ar[d] \ar[r]^{γ_{X_{2},x_{2}}} & K_{x_{2}} ∩ L_{x_{1}}^{h_{2}^{−1}} \ar[r]^{c_{h_{2}}} \ar[d] & K_{x_{2}}^{h_{2}} ∩ L_{x_{1}} \ar[d] \ar[rr]^{γ_{X_{1},x_{1}}} \ar[dr] &  & K_{[x_{2}h_{2},x_{1}]} \ar[d] \\ L_{x_{2}} \ar[r]^{γ_{X_{2},x_{2}}} & K_{x_{2}} \ar[r]^{c_{h_{2}}} & K_{x_{2}}^{h_{2}} & L_{x_{1}} \ar[r]^{γ_{X_{1},x_{1}}} & K_{x_{1}}}

where c_{g}(h) = g^{−1}hg is conjugation.

6. Conjugation bisets and B(G)

We will now define a subcategory B(G) ⊂ A(G), with the same objects (the subgroups of G), but with morphisms restricted to bifree conjugation bisets. As before, we perform the Grothendieck construction on the isomorphism classes of these bisets to get an Ab-category. The universal construction u: B(G) → B_{•}(G) then produces an additive subcategory (with involution) of A_{•}(G).

Definition 6.1 (Conjugation bisets). An indecomposable conjugation biset is an indecomposable bifree biset _{H_{2}}X_{H_{1}} so that the isomorphism in the standard representation of X at x ∈ X, γ_{X,x} = c_{g}, for some g ∈ G such that g^{−1}L_{x}g = K_{x}. A conjugation biset is a bifree biset, each of whose indecomposable subsets satisfies this condition.

Remark 6.2. The choice of g such that γ_{X,x} = c_{g} is not unique, but the conjugation biset only depends on γ_{X,x} as a homomorphism. This is the basic reason that functors out of B(G) are conjugation invariant. By definition, c_{g}(h_{2}) = g^{−1}h_{2}g ∈ K_{x}, for all h_{2} ∈ L_{x} ⊂ H_{2}, where g^{−1}L_{x}g = K_{x} ⊂ H_{1}. We have c_{g} = c_{g_{1}} : L_{x} → K_{x} if and only if g_{1}g^{−1} ∈ C_{G}(L_{x}), where C_{G}(L_{x}) denotes the centralizer of L_{x} in G.

The formula for the change of base-point in Lemma 4.8 shows that the definition of conjugation biset does not depend on the choice of point x used to compute γ_{X,x}: the element giving the conjugation may change, but not the fact that it is given by some conjugation. In particular, any biset which is isomorphic to a conjugation biset is itself a conjugation biset.

By Theorem 5.3, the composition of two conjugation bisets is again a conjugation biset. In addition, by Lemma 4.7, the involution τ restricts to give an involution τ: B(G) → B(G).

§6A. Induction and restriction. There are two extreme cases of indecomposable bifree bisets _{H_{2}}X_{H_{1}}.

(i) We say _{H_{2}}X_{H_{1}} is a restriction if the H_{1} action is transitive.

(ii) We say _{H_{2}}X_{H_{1}} is an induction if the H_{2} action is transitive.

We say that _{H_{2}}X_{H_{1}} is an isomorphism if it is both a restriction and an induction.

Proposition 6.3. Let _{H_{2}}X_{H_{1}} be an indecomposable, bifree biset. Pick a point x ∈ X and let [L_{x}, γ_{X,x}, K_{x}] be the standard representation.

(i) _{H_{2}}X_{H_{1}} is a restriction if and only if K_{x} = H_{1}.

(ii) _{H_{2}}X_{H_{1}} is an induction if and only if L_{x} = H_{2}.

Proof. Immediate from Lemma 4.9. □

Proposition 6.4. If _{H_{2}}X_{H_{1}} is an isomorphism, τ(_{H_{2}}X_{H_{1}}) is the inverse isomorphism. Conversely, if _{H_{2}}X_{H_{1}} has an inverse then it is an isomorphism. This justifies the terminology.

Proof. Let _{H_{1}}Y_{H_{2}} be the inverse for _{H_{2}}X_{H_{1}}. Then _{H_{2}}X_{H_{1}} ×_{H_{1}} _{H_{1}}Y_{H_{2}} is isomorphic to H_{2}H_{2}_{H_{2}}. Since (_{H_{2}}H_{2}_{H_{2}})/H_{2} is a point, so is (_{H_{2}}X_{H_{1}} ×_{H_{1}} _{H_{1}}Y_{H_{2}})/H_{2} = (X_{1}/H_{1})×(Y/H_{2}). It follows that Y is a restriction. The other equation for the inverse shows that Y is also an induction.

If _{H_{2}}X_{H_{1}} is both an induction and a restriction, it follows that the standard representation is H_{2}, H_{1} and some isomorphism γ_{X,x}. By Lemma 4.7 the standard representation for τ(X) is [H_{1}, γ−1_{X,x}, H_{2}]. By Theorem 5.3, τ(X) is the inverse for X. □

It further follows from Theorem 5.3 that the composition of two restrictions is a restriction and the composition of two inductions is an induction. More explicitly, we have the following.

Proposition 6.5. Let [L_{x_{1}}, γ_{X_{1},x_{1}}, K_{x_{1}}] be the standard representation for the biset _{H_{2}}X_{1}_{H_{1}} at x_{1} and let [L_{x_{2}}, γ_{X_{2},x_{2}}, K_{x_{2}}] be the standard representation for the biset _{H_{3}}X_{2}_{H_{2}} at x_{2}.

(i) If X_{1} and X_{2} are both restrictions then so is X_{2} ×_{H_{2}} X_{1}. (ii) If X_{1} and X_{2} are both inductions then so is X_{2} ×_{H_{2}} X_{1}. (iii) If X_{1} is an induction and if X_{2} is a restriction then [L_{x_{2}}, γ_{X_{1},x_{1}} ∘ γ_{X_{2},x_{2}}, K_{x_{1}}] is

the standard representation for the composition using the point [x_{2}, x_{1}]

Proof. This follows from Lemma 5.1 and some standard set theory. □

Definition 6.6. We define three subcategories of A(G), denoted Res_{A}(G), Ind_{A}(G) and Iso_{A}(G). The objects of any of these categories are all the objects of A(G). The morphisms in Res_{A}(G) are the set of all restrictions, the morphisms in Ind_{A}(G) are the set of all inductions and the morphisms in Iso_{A}(G) are the isomorphisms. □

Remark 6.7. Because Hom_{A(G)}(H_{1}, H_{2}) is the free abelian group on the indecomposable bifree H_{2} -H_{1} bisets, Hom_{Res(G)}(H_{1}, H_{2}) ⊂ Hom_{A(G)}(H_{1}, H_{2}), and indeed it is a summand. There is a similar statement for each of Ind(G) and Iso(G). Moreover, Iso(G) = Res(G) ∩ Ind(G).

Let _{H_{2}}X_{H_{1}} be an indecomposable bifree biset. After choosing a point x ∈ X, we can display X as a composition.

Definition 6.8. Let [L_{x}, γ_{X,x}, K_{x}] be the standard representation for X at x. We may regard H_{2} is an H_{2} -L_{x} biset, and H_{1} as a K_{x} -H_{1} biset, via group multiplication in G.

(i) We define Ind_{x} = _{H_{2}}(H_{2})_{L_{x}}, note that it is an induction and call it the standard induction for X at x.

(ii) We define Res_{x} = _{K_{x}}(H_{1})_{H_{1}}, note that it is a restriction and call it the standard

restriction for X at x. (iii) There are two evident isomorphisms. Make K_{x} into a left L_{x} set using γ_{X,x}, and

into a right K_{x} set via group multiplication. This makes K_{x} into an L_{x} -K_{x} biset. Similarly make L_{x} into an L_{x} -K_{x} biset using γ−1_{X,x}. Let _{L_{x}}(K_{x})_{K_{x}} be denoted ℜ_{x} and let _{L_{x}}(L_{x})_{K_{x}} be denoted 𝔏_{x}.

Note that 𝔏_{x} and ℜ_{x} are isomorphic as bisets via the bijection γ_{X,x}: L_{x} → K_{x}.

Proposition 6.9. As H_{2} -H_{1} bisets, an indecomposable bifree biset X is isomorphic to the composition Ind_{x} ∘ 𝔏_{x} ∘ Res_{x}.

(i) The standard restriction Res_{x} is always a conjugation restriction. (ii) The standard induction Ind_{x} is always a conjugation induction. (iii) The indecomposable bifree biset X is a conjugation biset if and only if 𝔏_{x} is a

conjugation isomorphism. (iv) Moreover, X is a restriction if and only if Ind_{x} is the identity; X is an induction

if and only if Res_{x} is the identity.

Proof. The standard representation for Ind_{x} is_{H_{2}}[L_{x}, id, L_{x}]_{L_{x}}, so it is clearly a conjugation biset. The case Res_{x} is similar. The standard representation for ℜ_{x} is _{L_{x}}[L_{x}, γ_{X,x}, K_{x}]_{K_{x}}, and so ℜ_{x} is a conjugation biset if and only if X is.

The function

Ind_{x} ×_{L_{x}} 𝔏_{x} ×_{K_{x}} Res_{x} → X

which sends the image of h_{2} × ℓ × h_{1} to h_{2}ℓxh_{1} can be checked to be a bijection of bisets. The remarks about X being a restriction or an induction are immediate. □

Remark 6.10. Of course the choice of x ∈ X is not unique, so let y = h_{2}xh_{1}. Let Res_{y}, Ind_{y} and ℜ_{y} be the corresponding bisets. Define an L_{y}-L_{x} biset V_{yx} to be L_{x} with right multiplication by L_{x} as the right action and use c_{h_{2}} : L_{y} → L_{x} to define the left action. Note that V_{yx} is an isomorphism and a conjugation biset. By Lemma 4.8, we have Ind_{x} ≅ Ind_{y} ×_{Ly} V_{yx}.

Similarly, we define a K_{y}-K_{x} biset W_{yx} to be K_{x} with right multiplication by K_{x} and left multiplication defined by c_{h_{1}} : K_{y} → K_{x}. We have Res_{x} ≅ W_{yx}^{−1} ×_{Ky} Res_{y} and 𝔏_{x} ≅ V_{yx}^{−1} ×_{Ly} 𝔏_{y} ×_{Ky} W_{yx}.

Definition 6.11. We define three subcategories of B_{•}(G), denoted Ind_{B}(G) ⊂ Ind_{A}(G), Res_{B}(G) ⊂ Res_{A}(G) and Iso_{B}(G) ⊂ Iso_{A}(G). These are the subcategories with the same objects as the bigger categories, but whose morphisms are all the morphisms which are conjugation bisets. □

7. Bifunctors into B_{•}(G)

In this section we will define the bifunctor j: D(G) → B_{•}(G) used in the statement of Theorem A. Let D_{∗}(G) denote the category whose objects are pairs (X, b), consisting of a finite G-space X and an ordered collection b = (b_{1}, ⋯, b_{n}) of base-points, one for each G-orbit of X. The morphisms are the G-maps (not necessarily base-point preserving). There is a functor

μ: D_{∗}(G) → D(G)

defined by forgetting the base-points. Since every object of D(G) is isomorphic to the image μ(X, b) of an object of D_{∗}(G), and μ induces a bijection on morphism sets, it follows that μ gives an equivalence between the categories D_{∗}(G) and D(G), with inverse functor μ′ [12, IV.4, Theorem 1, p. 91]. Moreover, the inverse functor μ′ can be chosen so that we have the additivity formula

μ′(X ⊔ Y) = μ′(X) ⊔ μ′(Y)

for any finite G-sets X and Y. This will be needed later to verify axiom (M2) for additive functors out of B_{•}(G).

We will now define the remaining functors in the following diagram:

(7.1)  \xymatrix{ & Or(G) \ar[r]^{i^{•}} \ar[r]_{i_{•}} \ar[d]^{𝔬} & B(G) \ar[d]^{u} \\ D(G) \ar[r]^{μ′} & D_{∗}(G) \ar[r]^{j^{•}} \ar[r]_{j_{•}} & B_{•}(G)}

The functor 𝔬: Or(G) → D_{∗}(G) sends H to the 1-tuple H and a G-set map G/H → G/K to the same G-set map. In fact Or(G) as defined is isomorphic to a full subcategory of D_{∗}(G).

The functor i_{•} is the identity on objects and sends a G-map f: G/H → G/K to the conjugation biset _{K}K_{g^{−1}Hg} where f(eH) = gK. This is well-defined, since a different choice gk, for k ∈ K, of representative yields an isomorphic biset.

Note that 1_{G/H}: G/H → G/H goes to _{H}H_{H} which is the identity. Check that if f_{1} : G/H_{1} → G/H_{2} and f_{2} : G/H_{2} → G/H_{3} are G-maps and if f_{1}(eH_{1}) = g_{1}H_{2} and f_{2}(eH_{2}) = g_{2}H_{3} then f_{2} ∘ f_{1}(eH_{1}) = (g_{1}g_{2})H_{3} and

H_{3}(H_{3})_{g_{2}^{−1}H_{2}g_{2}} ×_{H_{2}} _{H_{2}}(H_{2})_{g_{1}^{−1}H_{1}g_{1}}

is isomorphic to _{H_{3}}(H_{3})_{(g_{1}g_{2})^{−1}H_{1}(g_{1}g_{2})} by the map (h_{3}, h_{2}) ↦ h_{3}g_{2}^{−1}h_{2}g_{2}.

The functor i^{•} is also the identity on objects, but sends a G-map f: G/H → G/K to the conjugation biset _{g^{−1}Hg}K_{K} where f(eH) = gK. Rather than check identity and composition, just note that _{g^{−1}Hg}K_{K} is isomorphic to τ(_{K}K_{g^{−1}Hg}) by the function which sends k to k^{−1}, so i^{•} = τ ∘ i_{•} and hence i^{•} is a contravariant functor.

We define the functor j_{•} on objects by additivity: every object of D_{∗}(G) has the form

(X, b) = ⨆_{i=1}^{k} (X_{i}, b_{i}),

where (X_{i}, b_{i}) = G/H_{i} is an object of Or(G), and we send such an object to the ordered n-tuple (i_{•}(H_{1}), …, i_{•}(H_{k})). A morphism φ: (X, b) → (Y, c) in D_{∗}(G) is a collection (φ_{i}), 1 ⩽ i ⩽ k, of G-maps of the form φ_{i}: G/H_{i} → G/K_{f(i)}, where f: b → c is a function. Every morphism in B_{•}(G) is represented a finite matrix of bifree conjugation bisets, where ∅ = 0. We define j_{•}(φ) = α to be the morphism in B_{•}(G) represented by the matrix α = (α_{ij}), where α_{ij} = i_{•}(φ_{i}), if j = f(i), and α_{ij} = 0 otherwise.

The functor j^{•} is defined in a similar way. Notice that j_{•} factors through the subcategory Ind_{B}(G), and j^{•} factors through the subcategory Res_{B}(G).

Definition 7.2. We define the bifunctor

j: D(G) → B_{•}(G)

as the composition j = (j^{•}, j_{•}) ∘ μ′. □

8. The proof of Theorem A

We now state our main result, which is a more detailed version of Theorem A.

Theorem 8.1. If F: B_{•}(G) → Ab is an additive functor, then F ∘ j: D(G) → Ab is a conjugation invariant Mackey functor. Conversely, any conjugation invariant Mackey functor factors uniquely through an additive functor out of B_{•}(G).

Proof. Suppose that F: B_{•}(G) → Ab is an additive functor, and let M = F ∘ j: D(G) → Ab denote its composition with the bifunctor j. Then M is conjugation-invariant. The Mackey property (M2) is just additivity, so it remains to consider (M1). Let

(8.2)  \xymatrix{S \ar[r]^{Ψ} \ar[d]_{Φ} & S_{2} \ar[d]^{φ} \\ S_{1} \ar[r]^{ψ} & T}

be a pull-back diagram of finite G-sets. We first remark that a G-map u: S → S′ between finite G-sets is determined by its restriction to the disjoint G-orbits in S. In fact, each orbit in S is mapped by u into exactly one G-orbit of S′. In particular, if u: S → S′ is an isomorphism of finite G-sets, then by Proposition 6.4 the induced maps u_{M} and u^{M} are both isomorphisms with (u_{M})^{−1} = u^{M}.

This remark implies that property (M1) depends only on the isomorphism class of diagram (8.2). More precisely, let c: S_{1} ⟶^{≈} S′_{1} and d: S_{2} ⟶^{≈} S′_{2} be G-isomorphisms, and consider the commutative diagram of G-sets and G-maps:

\xymatrix{S′ \ar[drrr]^{l} \ar[dddr]_{n} \ar@{-->}[dr]_{v} \\  & S \ar[r]^{Ψ} \ar[d]^{Φ} \ar[ul]_{u} & S_{2} \ar[r]^{d} \ar[d]^{φ} & S′_{2} \ar[dl]^{j} \\  & S_{1} \ar[r]^{ψ} \ar[d]^{c} & T \\  & S′_{1} \ar[ur]_{i}}

where S′ is the pullback of i and j, with ψ = i ∘ c and φ = j ∘ d. There is an induced isomorphism u: S → S′ with inverse v, such that l ∘ u = d ∘ Ψ, and c ∘ Φ = n ∘ u. If we have the property (M1) for the pullback (S′, l, n), then the property (M1) holds for the pullback (S, Ψ, Φ). This is an easy diagram chase starting with the left-hand side of the required formula

(8.3)  φ^{M} ∘ ψ_{M} = Ψ_{M} ∘ Φ^{M}

and substituting in the expressions above for ψ and φ.

By the additivity property (M2), this formula follows from the basic cases where S_{1}, S_{2}, and T are transitive G-sets. Let S_{1} = G/H_{1}, S_{2} = G/H_{2} and T = G/K, and choose elements g_{1}, g_{2} ∈ G such that g_{i}^{−1}H_{i}g_{i} ⊆ K, for i = 1, 2. Since M is conjugation invariant, we may assume that ψ(eH_{1}) = g_{1}K and φ(eH_{2}) = g_{2}K (varying the choice of g_{1}, g_{2} will not affect the maps induced by M).

For i = 1 and i = 2 there are G-isomorphisms θ_{i}: G/H_{i} → G/H′_{i}, given by eH_{i} ↦ g_{i}H′_{i} where H′_{i} = g_{i}^{−1}H_{i}g_{i}, with the property that the pullback G/H′_{1} ⊂ G/K ⊃ G/H′_{2} is isomorphic to S = G/H_{1} ×_{G/K} G/H_{2}. It follows (by the remarks above) that it is enough to check formula (8.3) in the special case where H_{1} and H_{2} are actually subgroups of K. Hence, we may assume that g_{1} = g_{2} = e.

We will regard _{H_{2}}K_{H_{1}} as an H_{2}-H_{1} biset via the natural actions H_{2} ⊂ K and H_{1} ⊂ K. The pullback G-set

S = {(xH_{1}, yH_{2}) | xK = yK},

has the G-action defined by g · (xH_{1}, yH_{2}) = (gxH_{1}, gyH_{2}), for all g ∈ G. It follows that in each G-orbit there is always a representative of the form (xH_{1}, eH_{2}) with x ∈ K and (h_{2}xh_{1}H_{1}, eH_{2}) = (xH_{1}, eH_{2}) for all h_{2} ∈ H_{2} and all h_{1} ∈ H_{1}. In other words, the set of G-orbits in S is in bijection with the quotient H_{2}\K/H_{1} of the biset _{H_{2}}K_{H_{1}}.

The isotropy subgroup

G_{(xH_{1},eH_{2})} = {g ∈ G | gxH_{1} = xH_{1}, gH_{2} = eH_{2}} = H_{2} ∩ H_{1}^{x^{−1}}

where H_{1}^{x^{−1}} = xH_{1}x^{−1} and x ∈ K as above. Therefore, we have a bijection of G-sets

S = ⨆_{x∈H_{2}\K/H_{1}} G/H_{2} ∩ H_{1}^{x^{−1}} .

In terms of biset morphisms, the composition

(8.4) Ψ_{M} ∘ Φ^{M} = ∑ F (_{H_{2}}H_{2}_{H_{2}∩H_{1}^{x^{−1}}}) ∘ F (_{H_{2}∩H_{1}^{x^{−1}}}(H_{2} ∩ H_{1}^{x^{−1}})_{H_{1}∩H_{2}^{x}}) ∘ F (_{H_{1}∩H_{2}^{x}}H_{1}_{H_{1}})

where the middle biset is just the conjugation G-isomorphism c_{x^{−1}} : H_{1}∩H_{2}^{x} → H_{2}∩H_{1}^{x^{−1}}. On the other hand, the composition

φ^{M} ∘ ψ_{M} = F (_{H_{2}}K_{H_{1}}) .

To establish formula (8.3), we consider the H_{2} -H_{1} biset bijection

H_{2}K_{H_{1}} = ⨆_{x∈H_{2}\K/H_{1}} H_{2} x H_{1}

from the double coset decompostion. The standard representation of the component X = H_{2} x H_{1} expressed in the standard form is

H_{2} x H_{1} = _{H_{2}}[L_{x}, γ_{X,x}, K_{x}]_{H_{1}} = [H_{2} ∩ H_{1}^{x^{−1}}, c_{x}, H_{1} ∩ H_{2}^{x}] .

This follows from Remark 4.6: L_{x} = {h_{2} ∈ H_{2} | h_{2}x = xh_{1}, h_{1} ∈ H_{1}} = H_{2} ∩ H_{1}^{x^{−1}}, K_{x} = {h_{1} ∈ H_{1} | xh_{1} = h_{2}x, h_{2} ∈ H_{2}} = H_{1} ∩ H_{2}^{x} and γ_{X,x} = c_{x}.

Now by Proposition 6.9 we see that the composition

φ^{M} ∘ ψ_{M} = F (_{H_{2}}K_{H_{1}}) = ∑ F (Ind_{x}) ∘ F (𝔏_{x}) ∘ F (Res_{x}) .

But, by inspection, the right-hand side of this formula is just the expression for Ψ_{M} ∘ Φ^{M} in formula (8.4). This proves the property (M1).

Conversely, suppose that M: D(G) → Ab is a conjugation-invariant Mackey functor. We define the functor F on objects by setting F (H) = M(G/H), and extending additively. Since the bifunctor j: D(G) → B_{•}(G) is surjective on objects, this formula defines F uniquely on objects.

The morphisms in B_{•}(G) are finite matrices of bifree conjugation H_{2} -H_{1} bisets. Any such biset is a disjoint union of indecomposable conjugation bisets X, uniquely up to ordering, and after picking a base point x ∈ X we have the standard representation X ≅ _{H_{2}}[L_{x}, γ_{X,x}, K_{x}]_{H_{1}} and the factorization

H_{2}X_{H_{1}} ≅ _{H_{2}}[L_{x}, γ_{X,x}, K_{x}]_{H_{1}} ≅ Ind_{x} ∘ 𝔏_{x} ∘ Res_{x}

of Proposition 6.9. By Definition 6.8, it follows that the morphisms in B_{•}(G) are generated by compositions of the following three types of bifree conjugation bisets:

(i) Ind_{x} = _{H_{2}}(H_{2})_{L_{x}}, (ii) Res_{x} = _{K_{x}}(H_{1})_{H_{1}}, and (iii) 𝔏_{x} = _{L_{x}}(L_{x})_{K_{x}} .

Recall from Remark 4.6 that

L_{x} = {h_{2} ∈ H_{2} | h_{2}x = xh_{1}, h_{1} ∈ H_{1}} ⊂ H_{2}

and

K_{x} = {h_{1} ∈ H_{1} | xh_{1} = h_{2}x, h_{2} ∈ H_{2}} ⊂ H_{1} .

Since X is a conjugation biset, the map γ_{X,x}: L_{x} → K_{x} is given by c_{g_{x}}, where g_{x}^{−1}L_{x}g_{x} = K_{x}, so we have

γ_{X,x}(h_{2}) = c_{g_{x}}(h_{2}) = g_{x}^{−1}h_{2}g_{x} = h_{1} .

The element g_{x} need not be unique (see Remark 6.2). The isomorphism γ−1_{X,x} gives L_{x} the right K_{x}-action on L_{x} used to define the third biset.

We will define F on the basic morphisms of these three types, by associating to each of these bisets a G-map, and then applying the Mackey functor M:

• to Ind_{x} associate the G-map Ind_{L_{x}}^{H_{2}} : G/L_{x} → G/H_{2}, with Ind_{L_{x}}^{H_{2}}(eL_{x}) = eH_{2};

• to Res_{x} associate the G-map Res_{K_{x}}^{H_{1}} : G/K_{x} → G/H_{1}, with Res_{K_{x}}^{H_{1}}(eK_{x}) = eH_{1};

• to 𝔏_{x} associate the maps of G-sets 𝔏_{K_{x}}^{L_{x}}(g_{x}): G/K_{x} → G/L_{x}, with 𝔏_{K_{x}}^{L_{x}}(g_{x})(eK_{x}) =

g_{x}^{−1}L_{x} for the various choices of g_{x}.

Now define

(i) F (Ind_{x}) = M_{∗}(Ind_{L_{x}}^{H_{2}})  (ii) F (Res_{x}) = M^{∗}(Res_{K_{x}}^{H_{1}})  (iii) F (𝔏_{x}) = M_{∗}(𝔏_{K_{x}}^{L_{x}}(g_{x}))

Item (iii) is well-defined by Lemma 3.2, since M is conjugation invariant. We will write 𝔏_{K_{x}}^{L_{x}} for any of the 𝔏_{K_{x}}^{L_{x}}(g_{x}).

For any morphism X in B(G) we define F (X) by choosing a base point x ∈ X, writing X = Ind_{x} ∘𝔏_{x} ∘ Res_{x}, and defining F (X) = F (Ind_{x}) ∘ F (𝔏_{x}) ∘ F (Res_{x}). The formulas in Remark 6.10 show that the definition of F (X) is independent of the choice of base point.

The functor F is defined on all the morphisms in B_{•}(G) by the additive (matrix) extension of these formulas. Any relation in the Grothendieck group Hom_{B(G)}(H_{1}, H_{2}) leads to a isomorphism X ≅ Y of finite bifree H_{2} -H_{1} bisets. Since both sides are canonically (up to ordering) expressed as a disjoint union of (H_{2} × H_{1}^{op})-orbits, it is clear that F is well-defined.

Finally, we must check that F is a functor. Since F (_{H}H_{H}) = id, it remains to check that compositions are preserved. Suppose that X_{1} ∈ Hom_{B(G)}(H_{1}, H_{2}) and X_{2} ∈ Hom_{B(G)}(H_{2}, H_{3}). By additivity, we may assume that X_{1} and X_{2} are indecomposable. We must check that

F (X_{2} ∘ X_{1}) = F (X_{2}) ∘ F (X_{1}) .

Consider the left-hand side of the formula, where F (X_{2}∘X_{1}) = F (X_{2}×_{H_{2}}X_{1}) by definition. Pick a base point x_{1} ∈ X_{1}, and x_{2} ∈ X_{2}. By Lemma 5.2, the components of X_{3} = X_{2} ×_{H_{2}} X_{1} are indexed by elements h_{2} ∈ H_{2} representing the double cosets K_{x_{2}}\H_{2}/L_{x_{1}}. Each such component contributes a summand F ([L_{x_{3}}, γ_{X_{3},x_{3}}, K_{x_{3}}]), where x_{3} = [x_{2}h_{2}, x_{1}]. By Theorem 5.3, the standard representation at this base point is

(8.5) . [L_{x_{3}}, γ_{X_{3},x_{3}}, K_{x_{3}}] = [γ−1_{X_{2},x_{2}}(K_{x_{2}} ∩ L_{x_{1}}^{h_{2}^{−1}}), γ_{X_{1},x_{1}}∘ c_{h_{2}} ∘ γ_{X_{2},x_{2}}, γ_{X_{1},x_{1}}(K_{x_{2}}^{h_{2}} ∩ L_{x_{1}})]

The right-hand side of the formula is

F (X_{2}) ∘ F (X_{1}) = F (Ind_{x_{2}}) ∘ F (𝔏_{x_{2}}) ∘ F (Res_{x_{2}}) ∘ F (Ind_{x_{1}}) ∘ F (𝔏_{x_{1}}) ∘ F (Res_{x_{1}}) .

By the Mackey double coset formula for M, property (M1),

F (Res_{x_{2}}) ∘ F (Ind_{x_{1}}) = ∑_{h_{2}∈K_{x_{2}}\H_{2}/L_{x_{1}}} F (K_{x_{2}}) ∘ F (K_{x_{2}} ∩ L_{x_{1}}^{h_{2}^{−1}}) ∘ F (L_{x_{1}})

where

• K_{x_{2}} is a (K_{x_{2}}) - (K_{x_{2}} ∩ L_{x_{1}}^{h_{2}^{−1}}) biset via the two inclusions, and is a standard induction,

• K_{x_{2}} ∩ L_{x_{1}}^{h_{2}^{−1}} is a (K_{x_{2}} ∩ L_{x_{1}}^{h_{2}^{−1}}) - (L_{x_{1}} ∩ K_{x_{2}}^{h_{2}}) biset via the evident inclusion on the left and conjugation by h_{2}^{−1} on the right, and is a standard conjugation,

• L_{x_{1}} is an (L_{x_{1}} ∩K_{x_{2}}^{h_{2}}) - (L_{x_{1}}) biset via the two inclusions and is a standard restriction.

Hence F (X_{2}) ∘ F (X_{1}) = ∑_{h_{2}∈K_{x_{2}}\H_{2}/L_{x_{1}}} A[h_{2}] ∘ B[h_{2}] ∘ C[h_{2}]. where

A[h_{2}] = M_{∗} (Ind_{L_{x_{2}}}^{H_{3}}) ∘ M_{∗} (𝔏_{K_{x_{2}}}^{L_{x_{2}}}) ∘ M_{∗} (Ind_{K_{x_{2}}∩L_{x_{1}}^{h{−1 \atop 2}}}^{K_{x_{2}}})

B[h_{2}] = M_{∗} (𝔏_{L_{x_{1}}∩K_{x_{2}}^{h_{2}}}^{K_{x_{2}}∩L_{x_{1}}^{h{−1 \atop 2}}})

C[h_{2}] = M^{∗} (Res_{L_{x_{1}}∩K_{x_{2}}^{h_{2}}}^{L_{x_{1}}}) ∘ M_{∗} (𝔏_{K_{x_{1}}}^{L_{x_{1}}}) ∘ M^{∗} (Res_{K_{x_{1}}}^{H_{1}})

MACKEY FUNCTORS AND BISETS 17

To further analyze A[h_{2}] consider the commutative diagram

H_{3} ⊃ L_{x_{2}} ⟵^{γ_{X_{2},x_{2}}^{−1}} K_{x_{2}}
∪  ∪
γ−1_{X_{2},x_{2}}(K_{x_{2}} ∩ L_{x_{1}}^{h_{2}^{−1}}) ⟵^{γ_{X_{2},x_{2}}^{−1}} K_{x_{2}} ∩ L_{x_{1}}^{h_{2}^{−1}}

which implies

A[h_{2}] = M_{∗} (Ind_{γ_{X_{2},x_{2}}^{−1} (K_{x_{2}}∩L_{x_{1}}^{h{−1 \atop 2}})}^{H_{3}}) ∘ M_{∗} (𝔏_{K_{x_{2}}∩L_{x_{1}}^{h{−1 \atop 2}}}^{γ_{X_{2},x_{2}}^{−1} (K_{x_{2}}∩L_{x_{1}}^{h{−1 \atop 2}})}) .

To analyze C[h_{2}] first note that M_{∗} (𝔏_{K_{x_{1}}}^{L_{x_{1}}}) = M^{∗} (𝔏_{L_{x_{1}}}^{K_{x_{1}}}). Then consider the commutative diagram

L_{x_{1}} ⟶^{γ_{X_{1},x_{1}}} K_{x_{1}} ⊂ H_{1}
∪  ∪
L_{x_{1}} ∩ K_{x_{2}}^{h_{2}} ⟶^{γ_{X_{1},x_{1}}} γ_{X_{1},x_{1}}(L_{x_{1}} ∩ K_{x_{2}}^{h_{2}})

which implies

C[h_{2}] = M^{∗} (Res_{L_{x_{1}}∩K_{x_{2}}^{h_{2}}}^{L_{x_{1}}}) ∘ M^{∗} (𝔏_{L_{x_{1}}}^{K_{x_{1}}}) ∘ M^{∗} (Res_{K_{x_{1}}}^{H_{1}})
= M^{∗} (𝔏_{L_{x_{1}}∩K_{x_{2}}^{h_{2}}}^{γ_{X_{1},x_{1}} (L_{x_{1}}∩K_{x_{2}}^{h_{2}})}) ∘ M^{∗} (Res_{γ_{X_{1},x_{1}} (L_{x_{1}}∩K_{x_{2}}^{h_{2}})}^{H_{1}})

To analyze the conjugations, note that after substituting the new expressions for A[h_{2}] and C[h_{2}] we have three conjugations now occurring together in A[h_{2}] ∘ B[h_{2}] ∘ C[h_{2}]:

M_{∗} (𝔏_{K_{x_{2}}∩L_{x_{1}}^{h{−1 \atop 2}}}^{γ_{X_{2},x_{2}}^{−1} (K_{x_{2}}∩L_{x_{1}}^{h{−1 \atop 2}})}) ∘ M_{∗} (𝔏_{L_{x_{1}}∩K_{x_{2}}^{h_{2}}}^{K_{x_{2}}∩L_{x_{1}}^{h{−1 \atop 2}}}) ∘ M^{∗} (𝔏_{L_{x_{1}}∩K_{x_{2}}^{h_{2}}}^{γ_{X_{1},x_{1}} (L_{x_{1}}∩K_{x_{2}}^{h_{2}})}) =
M_{∗} (𝔏_{K_{x_{2}}∩L_{x_{1}}^{h{−1 \atop 2}}}^{γ_{X_{2},x_{2}}^{−1} (K_{x_{2}}∩L_{x_{1}}^{h{−1 \atop 2}})}) ∘ M_{∗} (𝔏_{L_{x_{1}}∩K_{x_{2}}^{h_{2}}}^{K_{x_{2}}∩L_{x_{1}}^{h{−1 \atop 2}}}) ∘ M_{∗} (𝔏_{γ_{X_{1},x_{1}} (L_{x_{1}}∩K_{x_{2}}^{h_{2}})}^{L_{x_{1}}∩K_{x_{2}}^{h_{2}}}) =
M_{∗} (𝔏_{γ_{X_{1},x_{1}} (L_{x_{1}}∩K_{x_{2}}^{h_{2}})}^{γ_{X_{2},x_{2}}^{−1} (K_{x_{2}}∩L_{x_{1}}^{h{−1 \atop 2}})})

Hence

\begin{matrix}A[h_{2}] ∘ B[h_{2}] ∘ C[h_{2}] = \\ M_{∗} (Ind_{γ_{X_{2},x_{2}}^{−1} (K_{x_{2}}∩L_{x_{1}}^{h{−1 \atop 2}})}^{H_{3}}) ∘ M_{∗} (𝔏_{γ_{X_{1},x_{1}} (L_{x_{1}}∩K_{x_{2}}^{h_{2}})}^{γ_{X_{2},x_{2}}^{−1} (K_{x_{2}}∩L_{x_{1}}^{h{−1 \atop 2}})}) ∘ M^{∗} (Res_{γ_{X_{1},x_{1}} (L_{x_{1}}∩K_{x_{2}}^{h_{2}})}^{H_{1}}) \end{matrix}

It now follows from (8.5) that A[h_{2}] ∘ B[h_{2}] ∘ C[h_{2}] = F (Ind_{x_{3}}) ∘ F (𝔏_{x_{3}}) ∘ F (Res_{x_{3}}) for each component x_{3} = [x_{2}h_{2}, x_{1}]. Therefore F (X_{2} ∘ X_{1}) = F (X_{2}) ∘ F (X_{1}). □

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Department of Mathematics & Statistics McMaster University Hamilton, ON L8S 4K1, Canada

  E-mail address: ian@math.mcmaster.ca

Department of Mathematics

University of Notre Dame

Notre Dame, IN 46556, USA

  E-mail address: taylor.2@nd.edu

Department of Mathematics

University of Notre Dame

Notre Dame, IN 46556, USA

  E-mail address: williams.4@nd.edu