Fused Mackey functors

Serge Bouc

Abstract: Let G be a finite group. In [5], Hambleton, Taylor and Williams have considered the question of comparing Mackey functors for G and biset functors defined on subgroups of G and bifree bisets as morphisms.

This paper proposes a different approach to this problem, from the point of view of various categories of G-sets. In particular, the category G-\underline{set} of fused G-sets is introduced, as well as the category \underline{S}(G) of spans in G-\underline{set}. The fused Mackey functors for G over a commutative ring R are defined as R-linear functors from R \underline{S}(G) to R-modules. They form an abelian subcategory Mack_{R}^{f}(G) of the category of Mackey functors for G over R. The category Mack_{ℤ}^{f}(G) is equivalent to the category of conjugation Mackey functors of [5]. The category Mack_{R}^{f}(G) is also equivalent to the category of modules over the fused Mackey algebra μ_{R}^{f}(G), which is a quotient of the usual Mackey algebra μ_{R}(G) of G over R.

AMS Subject classification : 18A25, 19A22, 20J15.

Keywords : Mackey functor, biset functor, conjugation, fused.

1. Introduction

This note is devoted to the frequently asked question of comparing Mackey functors for a single finite group G with biset functors defined only on subgroups of G and left-right free bisets as morphisms. The answer to this question has already been given by Hambleton, Taylor and Williams ([5]), but in a rather computational and non canonical way (in particular, in Section 7, the definition of the functor j_{•} requires the choice of sets of representatives of orbits of any finite G-set).

The present paper makes a systematic use of Dress definition ([3]) and Lindner definition ([6]) of Mackey functors, to avoid these non canonical choices. This leads to the definition of the category of fused G-sets (Section 3), and the category of fused Mackey functors (Section 4) for a finite group G, which is equivalent to the category of “conjugation invariant Mackey functors” of [5]. This category is also equivalent to the category of modules over the fused Mackey algebra, introduced in Section 5.

2. Conjugation bisets revisited

2.1. First a notation : when G is a finite group, and X is a finite G-set, let G-set↓_{X} denote the category of (finite) G-sets over X: its objects are pairs (Y, b) consisting of a finite G-set Y, and a morphism of G-sets b : Y → X. A morphism f : (Y, b) → (Z, c) in G-set↓_{X} is a morphism of G-sets f : Y → Z such that c ∘ f = b.

There is an obvious notion of disjoint union in G-set↓_{X}, and the corresponding Grothendieck group is called the Burnside group over X. It will be denoted by B(_{G}X), or B(X) when G is clear from the context.

Similarly, when G and H are finite groups, and U is a (G, H)-biset, one can define the category (G, H)-biset↓_{U} of (G, H)-bisets over U, and the Burnside group B(_{G}U_{H}) of (G, H)-bisets over U. 2.2. When H is a subgroup of G, and Y is an H-set, induction from H-sets to G-sets is an equivalence of categories from H-set↓_{Y} to G-set↓_{Ind_{H}^{G}Y}. A quasiinverse equivalence is the functor sending the G-set (X, a) over Ind_{H}^{G}Y to the H-set a^{−1}(1×_{H}Y) (see [2] Lemma 2.4.1). In particular B(_{H}Y) ≅ B(_{G}Ind_{H}^{G}Y).

2.3. Now an observation: when H and K are subgroups of G, the conjugation (K, H)-bisets defined in Section 6 of [5] are exactly those over the biset _{K}G_{H} (the set G on which K and H act by multiplication), i.e. the (K, H)-bisets U for which there exists a biset morphism U → _{K}G_{H}.

Indeed, a conjugation (K, H)-biset U is a bifree (K, H)-biset isomorphic to a disjoint union of bisets of the form (K × H)/S, where S is a subgroup of K × H of the form

S_{g,A} = {(^{g}x, x) | x ∈ A}

where A is a subgroup of H, and g is an element of G such that ^{g}A ≤ K. For such a transitive biset (K × H)/S, the map

∀(k, h)S ∈ (K × H)/S, (k, h)S ↦ kgh^{−1}

is a morphism of (K, H)-bisets.

Conversely, let U be a (K, H)-biset for which there exists a biset morphism α : U → _{K}G_{H}. Then for any u ∈ U, the stabilizer S_{u} of u in K × H is the subgroup

S_{u} = {(k, h) ∈ K × H | k · u · h^{−1} = u}

of K × H. Then if (k, h) ∈ S_{u},

α(k · u) = kα(u) = α(u · h) = α(u)h .

Let A_{u} denote the projection of S_{u} into H, and set g_{u} = α(u). It follows that S_{u} ⊆ S_{g_{u},A_{u}}.

Conversely, if (k, h) ∈ S_{g_{u},A_{u}}, then k = ^{g_{u}}h, and there exists some x ∈ K such that (x, h) ∈ S_{u}, since h ∈ A_{u}. Thus x · u · h^{−1} = u, from which follows that

α(x · u) = xg_{u} = α(u · h) = g_{u}h ,

hence x = ^{g_{u}}h = k, and S_{u} = S_{g_{u},A_{u}}. Observation 2.3 follows. 2.4. In other words, conjugation (K, H)-bisets form a category Conj_{K,H}^{G}, and there is a forgetful functor Φ : (K, H)-biset↓_{_{K}G_{H}} → Conj_{K,H}^{G} sending (U, a) to U. This functor is full, preserves disjoint unions, and moreover it induces a surjection on the corresponding sets of isomorphism classes. This means that Φ induces a surjective group homomorphism (still denoted by Φ) from B(_{K}G_{H}) to the Grothendieck group B_{K,H}^{G} of conjugation (K, H)-bisets.

2.5. If H, K and L are subgroups of G, if (U, a) is a (K, H)-biset over _{K}G_{H} and (V, b) is an (L, K)-biset over _{L}G_{K}, the composition (V, b) ∘ (U, a) is the (L, H)-biset over _{L}G_{H} defined by the following diagram:

\xymatrix{V \ar[dd]^{b} &  & U \ar[dd]^{a} &  & V ×_{K} U \ar[dd]^{b×_{K}a} \\  & ∘ &  & = \\ _{L}G_{K} &  & _{K}G_{H} &  & G ×_{K} G \ar[d]^{μ} \\  &  &  &  & _{L}G_{H}}

where μ is multiplication in G. This composition is associative, and additive with respect to disjoint unions. Hence it induces a composition

ˆ∘: B(_{L}G_{K}) × B(_{K}G_{H}) → B(_{L}G_{H}) .

Hence, one can define a category B̂(G) whose objects are the subgroups of G, and such that Hom_{B̂(G)}(H, K) = B(_{K}G_{H}), for subgroups H and K of G. Composition is given by ˆ∘, and the identity morphism of the subgroup H of G in the category B̂(G) is the class of the biset (_{H}H_{H}, i_{H}), where i_{H} : _{H}H_{H} → _{H}G_{H} is the inclusion map from H to G.

Since the functor Φ maps the composition ˆ∘ to the composition of bisets, and the identity morphism of H in B̂(G) to the identity biset _{H}H_{H}, one can extend Φ to a functor B̂(G) → B(G), which is the identity on objects.

In other words, the category B(G) introduced in Section 3 of [5] is the quotient of the category B̂(G) obtained by identifying morphisms which have the same image by Φ. 2.6. By the above Remark 2.2, when H and K are subgroups of G, there is a group isomorphism

B(_{K}G_{H}) ≅ B(Ind_{K×H}^{G×G}(_{K}G_{H})) ,

(with the usual identification of (K, H)-bisets with (K × H)-sets). Now the biset _{K}G_{H} is actually the restriction to (K × H) of the (G, G)-biset G. By the Frobenius reciprocity, it follows that

Ind_{K×H}^{G×G}(_{K}G_{H}) ≅ Ind_{K×H}^{G×G}Res_{K×H}^{G×G}(_{G}G_{G}) ≅ (Ind_{K×H}^{G×G} • ) ×_{G}G_{G} ,

where • is a set of cardinality 1. Since Ind_{K×H}^{G×G}• ≅ (G/K)×(G/H), it follows (after switching G/H and G) that

Ind_{K×H}^{G×G}(_{K}G_{H}) ≅ (G/K) × G × (G/H) ,

where the (G, G)-biset structure of the right hand side is given by

∀(a, b, x, y, g) ∈ G^{5}, a · (xK, g, yH) · b = (axK, agb, b^{−1}yH) .

2.7. It should now be clear that the additive completion B̂_{•}(G) is equivalent to the category whose objects are finite G-sets, where for any two finite G-sets X and Y

Hom_{B_{•}(G)}(X, Y) = B(_{G}(Y × G × X)_{G}) ,

the (G, G)-biset structure on (Y × G × X) being given as above by

∀(a, b, g, x, y) ∈ G^{3} × X × Y, a · (y, g, x) · b = (ay, agb, b^{−1}x) .

Keeping track of the composition ˆ∘ along the above isomorphism shows that the composition in the category B̂_{•}(G) can be defined by linearity from the following: if X, Y, and Z are finite G-sets, if

\xymatrix{ & V \ar[ddl]_{f} \ar[dd]^{e} \ar[ddr]^{d} &  &  &  & U \ar[ddl]_{c} \ar[dd]^{b} \ar[ddr]^{a} \\  &  &  & and \\ Z & G & Y &  & Y & G & X}

are (G, G)-bisets over (Z × G × Y) and (Y × G × X), respectively, their composition is given by the following (G, G)-biset over (Z × G × X)

\xymatrix{ & (V ×_{d,c} U)/G \ar[dl]_{γ} \ar[d]^{β} \ar[dr]^{α} \\ Z & G & X}

where V ×_{d,c} U is the pullback of V and U over Y, i.e. the set of pairs (v, u) ∈ V × U with d(v) = c(u), and (V ×_{d,c} U)/G the set of orbits of G on it for the action given by (v, u) · g = (vg, g^{−1}u). This makes sense because d(v · g) = g^{−1}d(v) = g^{−1}c(u) = c(g^{−1} · u) if d(v) = c(u). The map (γ, β, α) is given by

(γ, β, α)((v, u)G) = (f(v), e(v)b(u), a(u)) .

2.8. The functor Φ : B̂(G) → B(G) extends uniquely to an additive functor Φ_{•} : B̂_{•}(G) → B_{•}(G), and the category B_{•}(G) is the quotient of B̂_{•}(G) obtained by identifying morphisms which have the same image by Φ_{•}. Clearly, two morphisms f, g ∈ Hom_{B̂_{•}(G)}(X, Y) are identified if and only if f − g is in the kernel of the group homomorphism

ϕ : B(_{G}(Y × G × X)_{G}) → B(_{G}(Y × X)_{G})

induced by the correspondence

\xymatrix{ & U \ar[ddl]_{c} \ar[dd]^{b} \ar[ddr]^{a} &  &  &  & U \ar[ddl]_{c} \ar[ddr]^{a} \\  &  &  & ↦ \\ Y & G & X &  & Y &  & X}

on bisets. In other words, a morphism f in B̂_{•}(G) gives the zero morphism in B_{•}(G) if and only if it belongs to Ker ϕ. 2.9. Now the (G, G)-biset _{G}G_{G} is isomorphic to Ind_{Δ(G)}^{G×G}•, where Δ(G) is the diagonal subgroup of G × G. It follows that there is an isomorphism of (G, G)-bisets

Y × G × X ≅ Ind_{Δ(G)}^{G×G}(Y × X) .

Hence, by Remark 2.2 again, since Δ(G) ≅ G,

B(_{G}(Y × G × X)_{G}) ≅ B(_{G}(Y × X)) ,

where _{G}(Y ×X) is the usual cartesian product with diagonal G-action. More precisely, this isomorphism is induced by the correspondence

\xymatrix{ & U \ar[ddl]_{c} \ar[dd]^{b} \ar[ddr]^{a} &  &  &  & b^{−1}(1) \ar[ddl]_{c} \ar[ddr]^{a} \\  &  &  & ↦ \\ Y & G & X &  & Y &  & X}

It is then easy to check that the composition of

\xymatrix{ & V \ar[ddl]_{f} \ar[dd]^{e} \ar[ddr]^{d} &  &  &  & U \ar[ddl]_{c} \ar[dd]^{b} \ar[ddr]^{a} \\  &  &  & and \\ Z & G & Y &  & Y & G & X}

corresponds to the usual pullback diagram

\xymatrix{ &  & e^{−1}(1) ×_{d,c} b^{−1}(1) \ar[dl] \ar[dr] \\  & e^{−1}(1) \ar[dl]_{f} \ar[dr]^{d} &  & b^{−1}(1) \ar[dl]_{c} \ar[dr]^{a} \\ Z &  & Y &  & X}

In other words, the category B̂_{•}(G) is equivalent to the category S(G) whose objects are the finite G-sets, where

Hom_{S(G)}(X, Y) = B(_{G}(Y × X)) ,

and composition is induced by pullback. It has been shown by Lindner ([6], see also [2]) that the additive functors on this category are precisely the Mackey functors for G. 2.10. It remains to keep track of identifications by Φ, i.e. to start with a morphism f ∈ Hom_{S(G)}(X, Y), to lift it to

f^{+} ∈ Hom_{B̂_{•}(G)}(X, Y) = B(_{G}(Y × G × X)_{G}) ,

and see when f^{+} lies in Ker ϕ. Now f is represented by a difference of two G-sets over _{G}(Y × X) of the form

\xymatrix{ & Z \ar[ddl]_{b} \ar[ddr]^{a} &  &  &  & Z′ \ar[ddl]_{b′} \ar[ddr]^{a′} \\  &  &  & − \\ Y &  & X &  & Y &  & X .}

By induction from Δ(G) to G × G, the G-set on the left hand side lifts to the following (G × G)-set over _{(G×G)}(Y × G × X)

\xymatrix{ & G × Z \ar[dl]_{γ} \ar[d]^{β} \ar[dr]^{α} \\ Y & G & X}

where the (G × G)-actions on G × Z and Y × G × X are given respectively by (s, t) · (g, z) = (sgt^{−1}, tz) and (s, t) · (y, g, x) = (sy, sgt^{−1}, tx), and where

(γ, β, α)(g, z) = (gb(z), g, a(z)) .

Similarly the G-set (Z′, (b′, a′)) lifts to (G × Z′, (γ′, β′, α′)).

Now f^{+} is in Ker ϕ if and only if there is an isomorphism

\xymatrix{ & G × Z \ar[ddl]_{γ} \ar[ddr]^{α} &  &  &  & G × Z′ \ar[ddl]_{γ′} \ar[ddr]^{α′} \\  &  &  & ⟶^{θ} \\ Y &  & X &  & Y &  & X .}

of (G × G)-sets over Y × X. Since (g, z) = g · (1, z) for any (g, z) ∈ G × Z, it follows that θ is a map from G × Z to G × Z′ of the form

(g, z) ↦ (gu(z), v(z)) ,

where u is a map from Z to G and v is a map from Z to Z′. Now for any (s, t) ∈ G × G, the equality

θ((s, t) · (g, z)) = (s, t) · θ((g, z))

gives

(sgt^{−1}u(tz), v(tz) = (sgu(z)t^{−1}, tv(z)) .

This is equivalent to

u(tz) = ^{t}u(z) and v(tz) = tv(z) .

This means that u is a morphism of G-sets from Z to G^{c}, which is the set G with G-action by conjugation, and v is a morphism of G-sets.

Moreover θ is a bijection if and only if v is.

Finally θ is an morphism of (G, G)-bisets over Y × X if and only if α′ ∘ θ = a and γ′ ∘ θ = γ, i.e. equivalently if

a′ ∘ v = a and gu(z) · b′ ∘ v(z) = g · b(z)

for any (g, z) ∈ G × Z. In other words

a = a′ ∘ v and b = u ∗ (b′ ∘ v) ,

where, for any map w : Z → Y, the map u ∗ w : Z → Y is defined by (u ∗ w)(z) = u(z) · w(z). The map u ∗ w is a map of G-sets if u : Z → G^{c} and w : Z → Y are. Note that w′ = u ∗ w if and only if w = ū ∗ w′, where ū : Z → G^{c} is defined by ū(z) = u(z)^{−1}.

It follows that f maps to the zero morphism in B(G) if and only if there exists u : Z → G^{c} and an isomorphism v : Z → Z′ such that

a′ ∘ v = a and b′ ∘ v = u ∗ b ,

But then v is an isomorphism

\xymatrix{ & Z \ar[ddl]_{b′∘v} \ar[ddr]^{a′∘v} &  &  &  & Z′ \ar[ddl]_{b′} \ar[ddr]^{a′} \\  &  &  & ⟶^{v} \\ Y &  & X &  & Y &  & X .}

of G-sets over Y × X, and f is also represented by the difference

\xymatrix{ & Z \ar[ddl]_{b} \ar[ddr]^{a} &  &  &  & Z \ar[ddl]_{u∗b} \ar[ddr]^{a} \\  &  &  & − \\ Y &  & X &  & Y &  & X ,}

since a′ ∘ v = a and b′ ∘ v = u ∗ b. These are the morphisms in the category S(G) that vanish in B_{•}(G). In other words:

2.11. Theorem : Let G be a finite group. Let \underline{S}(G) denote the quotient category of S(G) defined by setting, for any two finite G-sets Y and Y

Hom_{\underline{S}(G)}(X, Y) = B(_{G}(Y × X))/K(Y, X) ,

where K(Y, X) is the subgroup generated by the differences

(2.12)  \xymatrix{ & Z \ar[ddl]_{b} \ar[ddr]^{a} &  &  &  & Z \ar[ddl]_{u∗b} \ar[ddr]^{a} \\  &  &  & − \\ Y &  & X &  & Y &  & X ,}

where a : Z → X, b : Z → Y, and u : Z → G^{c} are morphisms of G-sets. Then the functor Φ_{•} induces an equivalence of categories \underline{S}(G) ≅ B_{•}(G).

Since the difference 2.12 factors as

[figure: Z; b; Id; ∘; Y; Z; (; Z; Id; Id; −; Z; Z; Z; u∗Id; Id; Z; Z; ); Z; Id; a; ∘; Z; X]

the morphism vanishing in \underline{S}(G) are generated in the category S(G) by the morphisms of the form

\xymatrix{ & Z \ar[ddl]_{Id} \ar[ddr]^{Id} &  &  &  & Z \ar[ddl]_{u∗Id} \ar[ddr]^{Id} \\  &  &  & − \\ Z &  & Z &  & Z &  & Z .}

2.13. It follows that the additive functors from \underline{S}(G) to the category of abelian groups are exactly those Mackey functors (in the sense of Dress) such that for any G-set Z and any u : Z → G^{c}, the morphism M_{∗}(u ∗ Id) is equal to the identity map of M(Z).

This condition is additive with respect to Z, since the map u ∗ Id_{Z} maps each G-orbit of Z to itself. Hence these functors are exactly the functors for which the map M_{∗}(u ∗ Id) is the identity map of M(G/H), for any subgroup H of G and any u : G/H → G^{c}. Such a map is of the form gH ↦ gcH, where c ∈ C_{G}(H). The map u ∗ Id : G/H → G/H is the map gH ↦ gcH.

Translated in terms of the usual definition of Mackey functors, this map expresses the action of c on M(H) = M(G/H). This shows that additive functors from \underline{S}(G) to abelian groups are exactly the Mackey functors for the group G such that, for any H ≤ G, the centralizer C_{G}(H) acts trivially on M(H). These are the “conjugation invariant Mackey functors” introduced in [5].

3. Fused G-sets

Let Z be any (finite) G-set. The multiplication (u, v) ↦ u ∗ v endows the set Hom_{G-set}(Z, G^{c}) with a group structure. Moreover, for any finite G-set X, this group acts on the left on the set Hom_{G-set}(Z, X), via (u, f) ↦ u ∗ f. This action is compatible with the composition of morphisms: if Y is a finite G-set, if u : Z → G^{c} and v : Y → G^{c} are morphisms of G-sets, then for any morphisms of G-sets f : Z → Y and g : Y → X, one checks easily that

(3.1)  (v ∗ g) ∘ (u ∗ f) = (u ∗ (v ∘ f)) ∗ (g ∘ f) .

3.2. Notation : Let G-\underline{set} denote the category of fused G-sets: its objects are finite G-sets, and for any finite G-sets Z and Y

Hom_{G-\underline{set}}(Z, Y) = Hom_{G-set}(Z, G^{c})\Hom_{G-set}(Z, Y) .

The composition of morphisms in G-\underline{set} is induced by the composition of morphisms in G-set.

3.3. Remark : For any G-set Y, set Y^{I} = Y × G^{c}. This notation is chosen to evoke a path object in homotopy theory (cf. [4] Section 4.12). There is a natural morphism p : Y^{I} → Y × Y, defined by p(y, g) = (y, gy), for y ∈ Y and g ∈ G, and a morphism i : Y → Y^{I} defined by i(y) = (y, 1), for y ∈ Y. The composition p ∘ i is equal to the diagonal map Y → Y × Y.

Two morphisms a, b : Z → Y in G-set are equal in the category G-\underline{set} if and only if the morphism (a, b) : Z → Y × Y factors as

\xymatrix{ & Y^{I} \ar[d]^{p} \\ Z \ar[ur]^{φ} \ar[r]_{(a,b)} & Y × Y}

for some morphism of G-sets φ : Z → Y^{I}.

3.4. Remark : It follows from 3.1 that the map u ↦ u ∗ Id_{Z} is a group antihomomorphism from Hom_{G-set}(Z, G^{c}) to the group of G-automorphisms of Z. Hence a morphism \underline{f} : Z → Y in the category G-\underline{set} is an isomorphism if and only if any of its representatives f : Z → Y in G-set is an isomorphism.

3.5. Weak pullbacks of fused G-sets. Disjoint union of G-sets is a coproduct in G-\underline{set}. There is also a weak version of pullback in G-\underline{set} : let

\xymatrix{ & T \ar[dl]_{\underline{c}} \ar[dr]^{\underline{d}} \\ X \ar[dr]_{\underline{a}} &  & Y \ar[dl]^{\underline{b}} \\  & Z}

be a commutative diagram in G-\underline{set}, where underlines denote the images in G-\underline{set} of morphisms in G-set. This means that \underline{a} ∘ \underline{c} = \underline{b} ∘ \underline{d}, i.e. that there exists u ∈ Hom_{G-set}(T, G^{c}) such that

b ∘ d = u ∗ (a ∘ c) .

But u ∗ (a ∘ c) = a ∘ (u ∗ c). It follows that there is a unique morphism e ∈ Hom_{G-set}(T, X ×_{a,b} Y) such that the diagram

\xymatrix{ & T \ar[ddl]_{u∗c} \ar@{-->}[d]_{e} \ar[ddr]^{d} \\  & X ×_{a,b} Y \ar[dl]^{p} \ar[dr]_{q} \\ X \ar[dr]_{a} &  & Y \ar[dl]^{b} \\  & Z}

is commutative in G-set, where p : X ×_{a,b} Y → X and q : X ×_{a,b} Y → Y are the canonical morphisms from the pullback X ×_{a,b} Y. In other words, the diagram

(3.6)  \xymatrix{ & T \ar[ddl]_{\underline{c}} \ar[d]_{\underline{e}} \ar[ddr]^{\underline{d}} \\  & X ×_{a,b} Y \ar[dl]^{\underline{p}} \ar[dr]_{\underline{q}} \\ X \ar[dr]_{\underline{a}} &  & Y \ar[dl]^{\underline{b}} \\  & Z}

is commutative in G-\underline{set}.

But still (X ×_{a,b} Y, \underline{p}, \underline{q}) need not be a pullback in G-\underline{set}, since the morphism \underline{e} making Diagram 3.6 commutative is generally not unique, as e itself depends on the choice of u. Moreover, the lifts a and b of \underline{a} and \underline{b} to G-set are not unique : it should be noted however that if a′ = v ∗ a and b′ = w ∗ b are other lifts of a and b, respectively, where v ∈ Hom_{G-set}(X, G^{c}) and w ∈ Hom_{G-set}(Y, G^{c}), then the map f : (x, y) ↦ (v(x)x, w(y)y) is an isomorphism of G-sets from X ×_{a′,b′} Y to X ×_{a,b} Y, such that the diagram

[figure: X ×_{a′,b′} Y; p′; q′; f; X; a′; v∗Id; Y; b′; w∗Id; X ×_{a,b} Y; p; q; Z; Id; X; a; Y; b; Z]

is commutative in G-set. Since \underline{a′} = \underline{a}, \underline{b′} = \underline{b}, \underline{v ∗ Id} = \underline{Id}, and \underline{w ∗ Id} = \underline{Id}, this yields a commutative diagram

[figure: X ×_{a′,b′} Y; \underline{f}; \underline{p′}; \underline{q′}; X ×_{a,b} Y; \underline{p}; \underline{q}; X; \underline{a}; Y; \underline{b}; Z]

in G-\underline{set}, and \underline{f} is an isomorphism. This shows that the weak pullback X ×_{a,b} Y only depends on \underline{a} and \underline{b} in the category G-\underline{set}. For this reason, it may be denoted by X ×_{\underline{a},\underline{b}} Y. 3.7. Spans of fused G-sets. Recall (cf. [9], [1] for the general definition) that if X and Y are finite G-sets, then a span Λ_{Z,\underline{a},\underline{b}} over X and Y in the category G-\underline{set} is a diagram of the form

\xymatrix{ & Z \ar[dl]_{\underline{a}} \ar[dr]^{\underline{b}} \\ X &  & Y}

where Z is a finite G-set and \underline{a}, \underline{b} are morphisms in the category G-\underline{set}. Two spans Λ_{Z,\underline{a},\underline{b}} and Λ_{Z′,\underline{a′},\underline{b′}} over X and Y are equivalent if there exists an isomorphism \underline{f} : Z → Z′ in G-\underline{set} such that the diagram

\xymatrix{ & Z \ar[dd]^{\underline{f}} \ar[dl]_{\underline{a}} \ar[dr]^{\underline{b}} \\ X &  & Y \\  & Z \ar@{-->}[ul]^{\underline{a′}} \ar@{-->}[ur]_{\underline{b′}}}

is commutative. The set of equivalence classes of spans of fused G-sets over X and Y is an additive monoid, where the addition is defined by disjoint union (i.e. Λ_{Z_{1},\underline{a}_{1},\underline{b}_{1}} + Λ_{Z_{2},\underline{a}_{2},\underline{b}_{2}} = Λ_{Z_{1}⊔Z_{2},\underline{a}_{1}⊔\underline{a}_{2},\underline{b}_{1}⊔\underline{b}_{2}}). The corresponding Grothendieck group is isomorphic to Hom_{\underline{S}(G)}(Y, X).

It should be noted that even if there is no pullback construction in the category G-\underline{set}, the isomorphism classes of spans in G-\underline{set} can still be composed by weak pullback, and this induces the composition of morphisms in \underline{S}(G).

4. Fused Mackey functors

4.1. Definition : Let R be a commutative ring. Let R S(G) (resp. R \underline{S}(G)) denote the R-linear extension of the category S(G) (resp. \underline{S}(G)), defined as follows:

• The objects of R S(G) and R \underline{S}(G) are finite G-sets.

• For finite G sets X and Y,

Hom_{R S(G)}(X, Y) = R ⊗_{ℤ} Hom_{S(G)}(X, Y) ,
Hom_{R \underline{S}(G)}(X, Y) = R ⊗_{ℤ} Hom_{\underline{S}(G)}(X, Y) .

• Composition of morphisms is induced by the pullback in G-set (resp. the weak pullback in G-\underline{set}).

A Mackey functor for G over R in the sense of Lindner ([6]) is an R-linear functor from R S(G) to the category R-Mod of R-modules.

Similarly, a fused Mackey functor for G over R is an R-linear functor from R \underline{S}(G) to R-Mod. A morphism of fused Mackey functors is a natural transformation of functors. Fused Mackey functors for G over R form a category denoted by Mack_{R}^{f}(G).

The following is an equivalent definition of fused Mackey functors, à la Dress:

4.2. Definition : Let R be a commutative ring. A fused Mackey functor for the group G over R is a bivariant R-linear functor M = (M^{∗}, M_{∗}) from G-\underline{set} to R-Mod such that:

1. For any finite G-sets X and Y, the maps

\xymatrix{M(X) ⊕ M(Y) \ar[r]^{(M_{∗}(\underline{i}_{X}),M_{∗}(\underline{i}_{Y} )} & M(X ⊔ Y) \ar[l]^{(M^{∗}(\underline{i}_{X}),M^{∗}(\underline{i}_{Y} )}}

induced by the canonical inclusions i_{X} : X → X⊔Y and i_{Y} : Y → X⊔Y are mutual inverse isomorphisms.

[figure: 2. If; X ×_{\underline{a},\underline{b}} Y; \underline{p}; \underline{q}; X; \underline{a}; Y; \underline{b}; Z]

is a weak pullback diagram in G-\underline{set}, then M^{∗}(\underline{a})M_{∗}(\underline{b}) = M_{∗}(\underline{p})M^{∗}(\underline{q}).

A morphism of fused Mackey functors is a natural transformation of bivariant functors.

The category Mack_{R}^{f}(G) can be viewed as a full subcategory of the category Mack_{R}(G) of Mackey functors for G over R. In the case R = ℤ, this category is equivalent to the category of conjugation invariant Mackey functors introduced in [5].

The inclusion functor Mack_{R}^{f}(G) ↪ Mack_{R}(G) has a left adjoint:

4.3. Definition : Let M be a Mackey functor for G over R, in the sense of Lindner, i.e. an R-linear functor R S(G) → R-Mod. When X is a finite G-set, set

M^{f}(X) = M(X)/ ∑_{Z,a,u} Im(M(Λ_{a,Id_{Z}}) − M(Λ_{u∗a,Id_{Z}})) ,

where the summation runs through triples (Z, a, u) consisting of a finite Gset Z, and morphisms of G-sets a : Z → X and u : Z → G^{c}, and Λ_{a,Id_{Z}} denotes the span

[figure: Z; a; _{Id_{Z}}; X; Z]

of G-sets.

4.4. Proposition : Let R be a commutative ring, and G be a finite group.

1. Let M be a Mackey functor for G over R. The correspondence

X ↦ M^{f}(X)

is a fused functor M^{f} for G over R.

2. The correspondence F : M ↦ M^{f} is a functor from Mack_{R}(G) to Mack_{R}^{f}(G), which is left adjoint to the inclusion functor

I : Mack_{R}^{f}(G) ↪ Mack_{R}(G) .

Moreover F ∘ I is isomorphic to the identity functor of Mack_{R}^{f}(G).

Proof : For Assertion 1, to prove that M^{f} is a Mackey functor, observe that if Λ_{Z,a,b} is a span of finite G-sets of the form

\xymatrix{ & Z \ar[dl]_{a} \ar[dr]^{b} \\ X &  & Y}

and u : Z → G^{c} is a morphism of G-sets, then

Λ_{Z,a,b} − Λ_{Z,u∗a,b} = (Λ_{Z,a,Id_{Z}} − Λ_{Z,u∗a,Id_{Z}}) ∘ Λ_{Z,Id_{Z},b} .

It follows that the R-module

∑_{Z,a,u} Im(M(Λ_{a,Id_{Z}}) − M(Λ_{u∗a,Id_{Z}}))

is equal to the sum

∑_{Z,a,b,u} Im(M(Λ_{a,b}) − M(Λ_{u∗a,b})) .

In other words, it is equal to the image by M of the R-submodule K_{R}(X, Y) of Hom_{R S(G)}(Y, X) generated by the morphisms Λ_{a,b} − Λ_{u∗a,b}, i.e. to the kernel of the quotient morphism

Hom_{R S(G)}(Y, X) → Hom_{R \underline{S}(G)}(Y, X) .

This shows that K_{R} is an ideal in the category R S(G). So if M is an R-linear functor R S(G) → R-Mod, the correspondence

X ↦ M^{f}(X) = M(X)/ ∑_{f∈K_{R}(X,Y)} ImM(f)

is an R-linear functor from the quotient category R \underline{S}(G) to R-Mod.

Assertion 2 is straightforward: first it is clear that F ∘ I is isomorphic to the identity functor, since N^{f} = N when N is a fused Mackey functor. This isomorphism F ∘ I ≅ Id_{Mack_{R}^{f}(G)} provides the counit of the adjunction. Next for any Mackey functor M, there is a projection morphism M → IF(M), and this yields the unit of the adjunction.

4.5. Remark : Assertion 2 shows that Mack_{R}^{f}(G) is a reflective subcategory of Mack_{R}^{f}(G) (cf. [7], Chapter IV, Section3).

4.6. Remark : If the Mackey functor M is given in the sense of Dress, then for any finite G-set X

M^{f}(X) = M(X)/ ∑_{{a:Z→X \atop u:Z→G^{c}}} Im(M_{∗}(a) − M_{∗}(u ∗ a)) ,

where Z is a finite G-set, and a, u are morphisms of G-sets.

4.7. Corollary :

1. If P is a projective Mackey functor, then P^{f} is projective in the category

Mack_{R}^{f}(G).

2. The category Mack_{R}^{f}(G) has enough projective objects. More precisely, if N is a fused Mackey functor, and θ : P → I(N) is an epimorphism in Mack_{R}(G) from a projective Mackey functor P, then F(θ) : P^{f} → N is an epimorphism in Mack_{R}^{f}(G).

Proof : Assertion 1 follows from the fact that F is left adjoint to the exact functor I. Assertion 2 is then straightforward.

5. The fused Mackey algebra

When G is a finite group, set Ω_{G} = ⊔_{H≤G} G/H, and let RB_{Ω_{G}} denote the Dress

construction for the Burnside functor RB over the ring R. Recall that RB_{Ω_{G}}, as a Mackey functor in the sense of Dress, is obtained by precomposition of RB with the endofunctor X ↦ X × Ω_{G} of G-set.

Also recall (cf. [2] Lemma 7.3.2 and Proposition 4.5.1) that the functor RB_{Ω_{G}} is a progenerator of the category Mack_{R}(G), and that the algebra End_{Mack_{R}(G)}(B_{Ω_{G}}) ≅ B(Ω_{G}^{2}) is isomorphic to the Mackey algebra μ_{R}(G) of G over R, introduced by Thévenaz and Webb ([8]).

It follows from Corollary 4.7 that the functor (RB_{Ω_{G}})^{f} is a progenerator in the category Mack_{R}^{f}(G). Hence this category is equivalent to the category of modules over the algebra End_{Mack_{R}^{f}(G)}((RB_{Ω_{G}})^{f}).

5.1. Definition : The fused Mackey algebra of G over R is the algebra

μ_{R}^{f}(G) = End_{Mack_{R}^{f}(G)}((RB_{Ω_{G}})^{f}) .

5.2. Lemma : Let X be a finite G-set. Then (RB_{X})^{f} is isomorphic to the Yoneda functor Hom_{R \underline{S}(G)}(X, −).

Proof : Denote by Y_{X} the Yoneda functor Hom_{R \underline{S}(G)}(X, −). For any fused Mackey functor N for G over R

Hom_{Mack_{R}^{f}(G)}((RB_{X})^{f}, N) ≅ Hom_{Mack_{R}(G)}(RB_{X}, I(N))
≅ I(N)(X) ≅ N(X)
≅ Hom_{Mack_{R}^{f}(G)}(Y_{X}, N) .

The lemma follows, since all these isomorphisms are natural.

5.3. Theorem : The fused Mackey algebra μ_{R}^{f}(G) is isomorphic to the quotient of the algebra RB(Ω_{G}^{2}) ≅ μ_{R}(G) by the R-module generated by differences of the form

\xymatrix{ & Z \ar[ddl]_{b} \ar[ddr]^{a} &  &  &  & Z \ar[ddl]_{u∗b} \ar[ddr]^{a} \\  &  &  & − \\ Ω_{G} &  & Ω_{G} &  & Ω_{G} &  & Ω_{G} ,}

where a, b : Z → Ω_{G} and u : Z → G^{c} are morphisms of G-sets.

Proof : This follows from Lemma 5.2, since the quotient in the theorem is precisely End_{R \underline{S}(G)}(Ω_{G}).

5.4. Remark : One can deduce from this theorem that the fused Mackey algebra μ_{R}^{f}(G) is always free of finite rank as an R-module, and this rank does not depend on the commutative ring R. More precisely, Thévenaz and Webb have shown ([8] Proposition 3.2) that the Mackey algebra μ_{R}(G) has an R-basis consisting of elements of the form

t_{K}^{H} c_{g,K} r_{K^{g}}^{L} ,

where (H, L, g, K) runs through a set of representatives of 4-tuples consisting of two subgroups H and L of G, and element g of G, and a subgroup K of H ∩^{g}L, for the equivalence relation ≡ given by

(H, L, g, K) ≡ (H′, L′, g′, K′) ⇔ \{ \begin{matrix}H = H′, L = L′, \\ and \\ ∃h ∈ H, ∃l ∈ L, g′ = hgl, K′ = ^{h}K . \end{matrix}

Similarly, the quotient algebra μ_{R}^{f}(G) of μ_{R}(G) has a basis consisting of the images of the elements t_{K}^{H} c_{g,K} r_{K^{g}}^{L}, where (H, L, g, K) runs through a set of representatives of 4-tuples as above, modulo the relation ≡^{f} defined by

(H, L, g, K) ≡^{f} (H′, L′, g′, K′) ⇔ \{ \begin{matrix}H = H′, L = L′, \\ and \\ ∃h ∈ H, ∃l ∈ L, ∃x ∈ C_{G}(K), \\ g′ = hxgl, K′ = ^{h}K . \end{matrix}

References

[1] J. Bénabou. Introduction to bicategories, volume 47 of Lecture Notes in Mathematics, pages 1–77. Springer, Berlin, 1967.

[2] S. Bouc. Green-functors and G-sets, volume 1671 of Lecture Notes in Mathematics. Springer, October 1997.

[3] A. Dress. Contributions to the theory of induced representations, volume 342 of Lecture Notes in Mathematics, pages 183–240. Springer-Verlag, 1973.

[4] W. Dwyer and J. Spalinski. Homotopy theories and model categories, pages 73–126. North-Holland (Amsterdam), 1995.

[5] I. Hambleton, L. R. Taylor, and E. B. Williams. Mackey functors and bisets. Geom. Dedicata, 148:157–174, 2010.

[6] H. Lindner. A remark on Mackey functors. Manuscripta Math., 18:273– 278, 1976.

[7] S. Mac Lane. Categories for the working mathematician, volume 5 of Graduate texts in Mathematics. Springer, 1971.

[8] J. Thévenaz and P. Webb. The structure of Mackey functors. Trans. Amer. Math. Soc., 347(6):1865–1961, June 1995.

[9] N. Yoneda. On Ext and exact sequences. J. Fac. Sci. Univ. Tokyo, I, 7:193–227, 1954.

Serge Bouc - CNRS-LAMFA, Université de Picardie, 33 rue St Leu, 80039, Amiens Cedex 01 - France.

email :  serge.bouc@u-picardie.fr
web   :  http://www.lamfa.u-picardie.fr/bouc/