Fused Mackey functors

Serge Bouc

A la mémoire de mon père, Charles Bouc (1925-2014).

Abstract: Let G be a finite group. In [6], 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 [6]. 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

1.1. This note is devoted to the frequently asked question of comparing Mackey functors for a single finite group G (cf. [9]) with biset functors (cf. [3]) defined only on subgroups of G and left-right free bisets as morphisms. More precisely, let F be a biset functor defined on the category D of all finite groups, where morphisms are given by Grothendieck groups of left-right free bisets. Let moreover G be a fixed finite group. Now:

• when H is a subgroup of G, set M(H) = F (H).

• when H ≤ K are subgroups of G, define a restriction map

r_{H}^{K} : M(K) → M(H)

by r_{H}^{K} = F (_{H}K_{K}), where _{H}K_{K} is the set K, viewed and an (H, K)- biset by left and right multiplication, hence also as a morphism from K to H in the category D.

• similarly, define a transfer map

t_{H}^{K} : M(H) → M(K)

by t_{H}^{K} = F (_{K}K_{H}), where _{K}K_{H} is the set K, viewed as a (K, H)-biset.

• finally, when x ∈ G, and H ≤ G, define a conjugation map

c_{x,H} : M(H) → M(^{x}H)

by c_{x,H} = F (_{^{x}H} xH_{H}), where _{^{x}H} xH_{H} is the coset xH, wiewed as an (^{x}H, H)-biset.

One checks easily that this yields a Mackey functor M for G. 1.2. The question is now to characterize those Mackey functors for G for which the restriction maps r_{H}^{K}, the transfer maps t_{H}^{K}, and the conjugation maps c_{x,H} only depend on the above bisets _{H}K_{K}, _{K}K_{H}, and _{^{x}H} xH_{H}, respectively. Equivalently, to characterize those Mackey functors for G which can be viewed as additive functors on the (non full) subcategory of D consisting of subgroups of G, where morphisms from H to K are linear combinations of isomorphism classes of (K, H)-bisets obtained by composition of the above three types of bisets.

These bisets have been called conjugation bisets by Hambleton, Taylor and Williams, who answered first the above question ([6]): the Mackey functors in question are the conjugation invariant Mackey functors, namely the Mackey functors M for G such that for any subgroup H of G, the centralizer C_{G}(H) acts trivially on M(H). However, the proof of this characterization given in [6] is rather computational and non canonical (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 ([4]) and Lindner definition ([7]) 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 [6]. 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) (cf. [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 [6] 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 preserves disjoint unions, and 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, where B(G) is the category introduced in Section 3 of [6]: its objects are the subgroups of G, and Hom_{B(G)}(H, K) = B_{K,H}^{G} for any subgroups H and K of G, the composition of morphisms being given by linear extension of the composition of bisets.

More precisely, the category B(G) 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 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_{B̂_{•}(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 ([7], see also [2]) that the additive functors on this category are precisely the Mackey functors for G. 2.10. With this equivalence of categories B̂_{•}(G) ≅ S(G), the main result of [6] can be viewed as a characterization of those Mackey functors for G, wiewed as additive functors on B̂_{•}(G), which factor through the functor Φ_{•} : B̂_{•}(G) → B_{•}(G).

This characterization amounts to a precise description of the identifications effected by Φ on morphisms: starting with f ∈ Hom_{S(G)}(X, Y), one can 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 a 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 morphisms 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 ↦ ^{g}c, 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 [6].

2.14. Remark : In view of Paragraph 1.1, one might be tempted to believe that such a conjugation invariant Mackey functor for G can always be obtained from a biset functor defined on all finite groups by the restriction procedure to subgroups of G described in Paragraph 1.1, but this is not true: for example, let G be an elementary abelian group of order 4, and let A, B, and C denote its subgroups of order 2. The simple Mackey functor M = S_{A,𝔽_{2}} for G over the field with 2 elements has value 𝔽_{2} at A, and {0} elsewhere (cf. [9] Lemma 15.1). The functor M is obviously a conjugation invariant Mackey functor, but if it were the restriction of a biset functor defined over all finite groups, then in particular its values at A, B, and C would be isomorphic to one other, as A, B and C are all isomorphic to C_{2}.

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. [5] 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. [10], [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 ([7]) 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 [6].

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}(G) (cf. [8], 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 ([9]).

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 ([9] 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] S. Bouc. Biset functors for finite groups, volume 1990 of Lecture Notes in Mathematics. Springer, 2010.

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

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

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

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

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

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

[10] 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/