Biset functors as module Mackey functors, and its relation to derivators

0.44 MB·

by  Hiroyuki Nakaoka

BISET FUNCTORS AS MODULE MACKEY FUNCTORS, AND
ITS RELATION TO DERIVATORS.
6
1
HIROYUKINAKAOKA
0
2
Abstract. Inthisarticle,wewillshowthatthecategoryofbisetfunctorscan
n
be regarded as a reflective monoidal subcategory of the category of Mackey
a
functors on the 2-category of finite groupoids. This reflective subcategory
J
is equivalent to the category of modules over the Burnside functor. As a
2
consequenceofthereflectivity,wecanassociateabisetfunctortoanyderivator
2
onthe2-categoryoffinitecategories.

]
T
C
1. Introduction and Preliminaries
.
h
t AMackeyfunctorisausefultooltodescribehowanalgebraicsystemassociated
a
to finite groups (such as the Burnside rings or the representation rings, or the
m
cohomologygroups,etc.) behavesunderthechangeofsubgroupsofafixedgroupG.
[
Recently,Bouc[5]hasdefinedthenotionofabiset functor,whichmoreoverenables

1 us to deal with the behavior of algebraic systems named as above, with respect
v to bisets among all finite groups. The category of biset functors Fk becomes a
B
3
symmetricclosedmonoidalcategory([5,II.8]),anditsmonoidobjectscalledGreen
9
biset functors, are also investigated in the literature [5],[19].
1
6 As shown in [16], a biset functor can be regarded as a special class of Mackey
0 functor on some 2-category S. This special Mackey functors are called deflative
. Mackey functors there. Indeed, the category of biset functors Fk has shown to be
1 B
0 equivalent to the categoryMackk (S) of deflative Mackeyfunctors on S. Relations
dfl
6 between(global)Mackeyfunctorsandbisetfunctorsarealsoinvestigatedbyseveral
1
researcherssuch as [4], [8], [10], [11].
:
v We can also show that the 2-category S becomes biequivalent to the 2-category
i of finite groupoids. It allows us to relate Mack(S) to derivators. In fact, to any
X
derivatoronthe2-categoryoffinitecategories,wecanassociateaMackeyfunctoron
r
a S. This gives a way to construct biset functors from derivators (since the inclusion
Fk ֒→ Mackk(S) is shown to have a left adjoint, as below). We remark also that
B
the theory of derivators is of recent interest by several researchers, as seen in [7],
[9].
Oneofthemotivationsofthisarticleistopursuethisinterpretation,todescribe
Green biset functors in terms of Mackk(S). By the result of Panchadcharam and
Street [18], the category Mackk(S) can be equipped with a symmetric monoidal
structure. WewillshowthatthismonoidalstructurerestrictstoMackk (S),andthe
dfl
equivalenceFk −≃→Mackk (S)constructedin[16]isinfactamonoidalequivalence.
B dfl
Theauthor wishestothankProfessorSergeBoucforhiscommentsandadvices.
The author wishes to thank Professor Ivo Dell’Ambrogio, for his interest and stimulating
discussions.
ThisworkissupportedbyJSPSKAKENHIGrantNumbers25800022, 24540085.
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2 HIROYUKINAKAOKA
As a consequence, their categories of monoids become equivalent. Namely, we
obtainanequivalencebetweenthecategoryofGreenbisetfunctorsandthecategory
ofdeflativeGreenfunctorsonS. Besides,weseethattheinclusionFk ֒→Mackk(S)
B
has a left adjoint, given by the tensor product with a monoid object.
This motivation comes from our ongoing work on Tambara biset functors. It is
expected to be a framework for biset functors equipped with multiplicative induc-
tions, such as the Burnside functor and the representation ring functor. This will
be an analog of the notion of Tambara functors in the ordinary Mackey functor
theory. In [17] and forthcoming works, we will formalize ‘Tambara’ properties for
biset functors, using our interpretation as Mackey functors on S. In the analogy
withthe ordinarycase,we expectthat the underlying Mackeyfunctor ofanyTam-
bara biset functor should naturally become a Green functor on S. In this article,
we develop what will be necessary for this purpose.
Throughout this article, k denotes a commutative ring with multiplicative unit.
Any group is assumed to be finite. The unit of a group is denoted by e. Abbre-
viately, trivial group is denoted by e. For a finite group G, let set denote the
G
category of finite G-sets, where morphisms are equivariant G-maps. The discrete
category with only one object is denoted by e. For any category K and any pair
of objects X and Y in K , the set of morphisms from X to Y in K is denoted by
K (X,Y). Let C(K ) denote the set of isomorphismclasses of objects in K . For a
functor F: K →K ′, let C(F): C(K )→C(K ′) denote the map induced from F.
For a strict 2-category K, we use the following notation.
(0) K0 denotes the class of 0-cells in K.
(1) For any pair of0-cells X,Y ∈K0, the setof 1-cellsfromX to Y is denoted
by K1(X,Y). Together with the 2-cells among them, they form a category
K(X,Y) satisfying Ob(K(X,Y))=K1(X,Y).
(2) For any X,Y ∈K0 and any f,g ∈K1(X,Y), the set of 2-cells from f to g
is denoted by K2(f,g). Namely, we put K2(f,g)=(K(X,Y))(f,g).
Horizontal composition and vertical composition are denoted by “◦” and “·” re-
spectively.
Let Cat denote the 2-category of small categories ([3, Example 7.1.4a]). Let
finCat⊆ Cat denote the full 2-subcategory of finite categories (this is an example
of category of diagrams ([9, Definition 1.12])), and let finGpd ⊆ finCat denote the
full 2-subcategory of finite groupoids.
Let us recall the definition and properties of S from [16]. For the generalities of
2-categries and bicategories, see [14],[3],[13].
Definition 1.1. ([16, Definition 2.2.12]) 2-category S is defined as follows. (See
[16, section 2.2].)
(0) A 0-cell is a pair of a finite group G and a finite G-set X. We denote this
pair by X.
G
(1) For any pair of 0-cells X and Y , a 1-cell α: X → Y is a pair of a map
G H θ G H
α: X →Y and a family of maps {θ : G→H} satisfying
x x∈X
(i) α(gx)=θ (g)α(x)
x
(ii) θx(gg′)=θg′x(g)θx(g′)
BISET FUNCTORS AS MODULE MACKEY FUNCTORS 3
for any x∈X and any g,g′ ∈G. We denote such a pair by α.
θ
Ifθsatisfiesθ =f (∀x∈X)forsomegrouphomomorphismf: G→H,
x
then the 1-cell α is called f-equivariant, and simply written as α. In
θ f
particular when G = H, a 1-cell α: X → Y is called G-equivariant or
θ G G
simply equivariant, if it is id -equivariant. In this case, we denote this
G
1-cell by α = α.
idG G
For any consecutive pair of 1-cells
X α Y β Z
−→θ −τ→ ,
G H K
we define their composition (β)◦(α)= β◦α by
τ θ τ◦θ
- β◦α: X →Z is the usual composition of maps of sets,
- τ ◦θ is defined by
(τ ◦θ) (g)=τ (θ (g)) (∀g ∈G),
x α(x) x
namely, (τ ◦θ) =τ ◦θ for any x∈X.
x α(x) x
The identity 1-cell for X is given by id = idX.
G X G
G
(2) For any pair of 1-cells α,α′: X → Y , a 2-cell ε: α ⇒ α′ is a family of
θ θ′ G H θ θ′
elements {ε ∈H} satisfying
x x∈X
(i) α′(x)=ε α(x),
x
(ii) ε θ (g)ε−1 =θ′(g)
gx x x x
for any x∈X and g ∈G.
If we are given a consecutive pair of 2-cells
α
θ
ε
(cid:11)(cid:19)
X α′ // Y
G θ′ >> H
(cid:11)(cid:19)ε′
α′′
θ′′
then their vertical composition ε′·ε: α ⇒ α′′ is defined by
θ θ′′
(ε′·ε) =ε′ε (∀x∈X).
x x x
The identity 2-cell id: idα ⇒ idα is given by idx = e (∀x ∈ X). With
θ θ
this definition, any 2-cell becomes invertible with respect to the vertical
composition.
α β
Horizontal compositions are given as follows. Let X −→θ Y −τ→ Z be a
G H K
sequence of 1-cells.
(a) For a 2-cell
α
θ
&&
X ε Y ,
G (cid:11)(cid:19) 88 H
α′
θ′
define (β)◦ε: (β)◦(α)⇒(β)◦(α′) by
τ τ θ τ θ′
β
(1.1) (( )◦ε) =τ (ε ) (∀x∈X).
τ x α(x) x
4 HIROYUKINAKAOKA
(b) For a 2-cell
β
τ
&&
Y ρ Z ,
H (cid:11)(cid:19) 88 K
β′
τ′
define ρ◦(α): (β)◦(α)⇒(β′)◦(α) by
θ τ θ τ′ θ
α
(1.2) (ρ◦( )) =ρ (∀x∈X).
θ x α(x)
Remark 1.2. In the following, a 1-cell α is often abbreviately written as α.
θ
Define a 2-functor E: S→finGpd in the following way (Definition 1.3, Proposi-
tions 1.4, 1.5, Corollary 1.6).
On 0-cells, E is defined as follows.
Definition 1.3. Forany0-cell X ∈S0, associatethe categoryofelements E(X)=
G G
eℓ (X) ([1, Example A.14]) defined by
G
- Ob(E(X))=X.
G
- For any x,x′ ∈X, define (E(X))(x,x′)={g ∈G|gx=x′}.
G
On 1-cells, E is given by the following.
Proposition 1.4. Let X, Y ∈ S0 be any pair of 0-cells. There is a bijective map
G H
on the set of 1-cells
E: S1(X, Y )−∼=→finGpd1(E(X),E(Y )),
G H G H
which is given explicitly in the proof.
Proof. Any functor F: E(XG)→E(HY ) shouldconsistofa pair (α,{Fx,x′}x,x′∈X) of
(i) a map on the set of objects α: X →Y,
(ii) a family of maps
X Y
(1.3) {Fx,x′: (E( ))(x,x′)→(E( ))(α(x),α(x′))}
G H
preserving the identities and the composition.
A family of maps (1.3) can be regarded as a map
Θ: X×G→∼= {(x,x′,g)∈X ×X×G|x′ =gx}→H
(x,g)7→(x,gx,g)7→F (g)=:θ (g).
x,gx x
Thisfamilypreservesthecompositionifandonlyif,foranysequenceofmorphisms
x→g′ g′x→g gg′x in E(X), it satisfies
G
θx(gg′)=θg′x(g)◦θx(g′).
This is nothing but the condition for a 1-cell α: X → Y in S. Preservation of the
θ G H
identitiesalsofollowsautomaticallyfromthisequation. ThusfunctorsF correspond
bijectively to 1-cells α by the equation
θ
F (g)=θ (g) (∀x∈X,g ∈G).
x,gx x
(cid:3)
BISET FUNCTORS AS MODULE MACKEY FUNCTORS 5
On 2-cells, E is given by the following.
Proposition1.5. Let α,β: X → Y beanypairof1-cellsinS. Thereisabijective
θ τ G H
map on the set of 2-cells
E: S2(α,β)−∼=→finGpd2(E(α),E(β)),
θ τ θ τ
which is given explicitly in the proof.
Proof. Remarkthateachelementinboth handsides is givenas afamily ε={ε ∈
x
H} satisfying some conditions.
x∈X
For the left hand side, the conditions are
(i) ε α(x)=β(x) (∀x∈X).
x
(ii) ε θ (g)=τ (g)ε (∀x∈X,g ∈G).
gx x x x
For the right hand side, the conditions are
(i) ε ∈(E(Y ))(α(x),β(x)) (∀x∈X).
x H
(ii) For any x∈X and any g ∈G (viewed as a morphism g: x→gx),
α(x) εx //β(x)
θx(g) (cid:8) τx(g)
(cid:15)(cid:15) (cid:15)(cid:15)
α(gx) // β(gx)
εgx
is commutative.
Obviously, these conditions are equivalent. (cid:3)
Corollary 1.6. E: S → finGpd is a strict 2-functor ([3, Definition 7.2.1]). This
induces isomorphism of categories E: S(X, Y ) → finGpd(E(X),E(Y )) for any
G H G H
X, Y ∈S0.
G H
Proof. It is straightforward to check E is in fact a 2-functor. The latter part also
follows from Propositions 1.4, 1.5 immediately. (cid:3)
Definition 1.7. ([16, Definition 3.1.1]) Let ι: H ֒→ G be a monomorphism of
groups. For any X ∈Ob( set), we define Ind X ∈Ob( set) by
H ι G
Ind X =(G×X)/∼,
ι
where the equivalence relation ∼ is defined by
- (ξ,x) and (ξ′,x′) in G×X are equivalent if there exists h∈H satisfying
x′ =hx, ξ =ξ′ι(h).
Wedenote theequivalenceclassof(ξ,x) by[ξ,x]∈Ind X. TheG-actiononInd X
ι ι
is defined by
g[ξ,x]=[gξ,x]
for any g ∈G and [ξ,x]∈Ind X.
ι
The following facts have been shown in [16].
Fact 1.8. ([16, Proposition 3.1.2]) Let ι: H ֒→ G be a monomorphism of groups.
For any X ∈Ob( set), if we define a map υ: X →Ind X by
H ι
υ(x)=[e,x] (∀x∈X),
6 HIROYUKINAKAOKA
then the 1-cell
υ X Ind X
ι
: →
ι H G
becomes an equivalence (cf. Remark 1.17).
Fact 1.9. ([16, Proposition 3.2.13]) Let G be any finite group. For any X,Y ∈
Ob( set), let
G
X ֒υ→X X ∐Y ←υY֓Y
be the coproduct in set. Then
G
X υX X ∐Y υY Y
(1.4) −G→ ←G−
G G G
gives a bicoproduct of X and Y in S.
G G
Remark 1.10. E sends (1.4) to the coproduct of categories
X X Y Y
E( )֒→E( )∐E( )←֓E( ).
G G H G
Fact 1.11. ([16, Proposition 3.2.15]) Let X and Y be any pair of 0-cells in S.
G H
Denote the monomorphisms
G→G×H ; g 7→(g,e)
H →G×H ; h7→(e,h)
by ι(G) and ι(H) respectively, and denote the natural maps
X →Ind X∐Ind Y ; x7→[e,x]∈Ind X
ι(G) ι(H) ι(G)
Y →Ind X ∐Ind Y ; y 7→[e,y]∈Ind Y
ι(G) ι(H) ι(H)
by υ and υ . Then
X Y
X υX Ind X ∐Ind Y υY Y
−ι(→G) ι(G) ι(H) ←ι(H−)
G G×H H
gives a bicoproduct X ∐ Y of X and Y in S.
G H G H
Remark 1.12. As a consequence, for any X, Y ∈S0, its image E(X ∐ Y ) becomes
G H G H
equivalent to the coproduct of E(X) and E(Y ) by
G H
X Y Ind X∐Ind Y
E( ∐ ) = E( ι(G) ι(H) )
G H G×H
Ind X Ind Y X Y
= E( ι(G) ) ∐ E( ι(H) ) ≃ E( )∐E( ).
G×H G×H G H
Fact 1.13. ([16, Proposition 3.2.17]) Let α: X → Z and β: Y → Z be any pair
θ G K τ H K
of 1-cells in S. Denote the natural projection homomorphisms by
pr(G): G×H →G, pr(H): G×H →H.
If we
- put F ={(x,y,k)∈X ×Y ×K |β(y)=kα(x)}, and put
℘ : F →X ; (x,y,k)7→x,
X
℘ : F →Y ; (x,y,k)7→y,
Y
BISET FUNCTORS AS MODULE MACKEY FUNCTORS 7
- equip F with a G×H-action
(g,h)(x,y,k)=(gx,hy,τ (h)kθ (g)−1)
y x
(∀(g,h)∈G×H, ∀(x,y,k)∈F),
- define a 2-cell κ: α◦℘ ⇒β◦℘ by
X Y
κ =k,
(x,y,k)
then the diagram
℘X
F pr(G) // X
G×H G
(1.5) pr℘(YH) (cid:15)(cid:15) {(cid:3)⑧⑧⑧⑧⑧⑧κ (cid:15)(cid:15) αθ
Y // Z
H β K
τ
gives a bipullback ([12, P.155]) in S.
℘X ℘Y
In particular if Z = 1, then X p←r(G−) F p−r(→H) Y gives a biproduct of X and
K e G G×H H G
Y . Here 1 denotes a set with one element, with the trivial group action.
H
Remark 1.14. Esends(1.5)tothecommasquare([3,Diagram1.12],[9,Proposition
1.26]) of categories
E(α)/E(β) //E(X)
θ τ G
(cid:15)(cid:15) {(cid:3)⑧⑧⑧⑧⑧⑧ (cid:15)(cid:15) E(αθ) .
E(Y ) //E(Z)
H E(β) K
τ
Proposition1.15. ForanyG ∈finGpd0, thereis X ∈S0 admittinganequivalence
G
E(X)≃ G. In particular, E: S→finGpd is a biequivalence as in [13, section 2.2],
G
by Corollary 1.6.
Proof. By Remark 1.12, we may assume G is connected. Take any x ∈ Ob(G)
and put Aut (x) = G(x,x). Then the 0-cell {x} ∈ S0 is sent by E to a full
G AutG(x)
subcategory of G.
For any x′ ∈ Ob(G), there is an isomorphism x → x′ since G is connected
groupoid. This means E( {x} ) is dense in (and hence equivalent to) G. (cid:3)
AutG(x)
Definition 1.16. Category C =S/2-cells is defined as follows.
(i) Ob(C)=S0, namely, an object in C is a 0-cell in S.
(ii) For any pair of objects X, Y ∈Ob(C), define as
G H
X Y X Y
C( , )=C(S( , )).
G H G H
Theisomorphismclassof α ∈Ob(S(X, Y ))=S1(X, Y )isdenotedby(α),
θ G H G H θ
or abbreviately by α.
Remark 1.17. For any 1-cell α: X → Y in S, the following are equivalent.
G H
(1) α is an isomorphism in C.
(2) α is an equivalence in S. Namely, there is a 1-cell β: Y → X and 2-cells
H G
ρ: β◦α⇒id, λ: α◦β ⇒id.
8 HIROYUKINAKAOKA
(As stated in Remark 1.2, 1-cells are abbreviated by α,β.)
Definition 1.18. While biproducts and bicoproducts in S yield products and co-
products in C, remark that the image of a bipullback in S becomes only a weak
pullback in C.
A weak pullback in C
δ
W // Y
L H
γ (cid:8) β
(cid:15)(cid:15) (cid:15)(cid:15)
X // Z
G α K
is called a natural weak pullback (of α and β) if it comes from some bipullback
W δ // Y
L H
γ (cid:15)(cid:15) {(cid:3)⑧⑧⑧⑧⑧⑧κ (cid:15)(cid:15)β
X // Z
G α K
in S. We write as
δ
W // Y
L H
(1.6) γ nwp β
(cid:15)(cid:15) (cid:15)(cid:15)
X // Z
G α K
to indicate it is a natural weak pullback. In this way, we can give C a class of
natural weak pullbacks.
Corollary 1.19. The category C has the following.
(1) Initial object ∅= ∅, and terminal object 1.
e e
(2) Any binary product and binary coproduct.
(3) A class of natural weak pullbacks.
In particular by (1),(2), C has any finite product and any finite coproduct.
Remark 1.20. Let K be any category. To give a functor F: C →K is equivalent
togiveastrict2-functorF: S→K ,whereK isregardedasa2-categoryequipped
only with identity 2-cells.
Remark 1.21. If one define a category C′ = finGpd/2-cells in the same manner,
then there is an equivalence C ֒→≃ C′ which preserves the class of natural weak
pullbacks, by Remark 1.14 and Proposition 1.15.
We define the notions of a (semi-)Mackey functor on S and on C, which are the
same by Remark 1.20.
Definition 1.22. Asemi-Mackey functor(respectively,ak-linear Mackey functor)
M = (M,M∗) on C is a pair of a contravariant functor M∗: C → Set (resp.
!
M∗: C → kMod) and a covariant functor M : C → Set (resp. M : C → kMod)
! !
which satisfies the following.
(Set denotes the category of sets, where morphisms are maps of sets. kMod
denotes the category of k-modules, where morphisms are k-linear maps.)
(0) M∗(X) = M(X) for any object X ∈ Ob(C). We denote this simply by
G ! G G
M(X).
G
BISET FUNCTORS AS MODULE MACKEY FUNCTORS 9
(1) [Additivity]Foranypairofobjects X and Y inC,ifwetaketheircoproduct
G H
X −υ→X X ∐ Y ←υY− Y
G G H H
in C, then the natural map
X Y X Y
(M∗(υ ),M∗(υ )): M( ∐ )→M( )×M( )
X Y
G H G H
is isomorphism. Also, M(∅) is trivial (i.e., the terminal object).
(2) [Mackey condition] For any natural weak pullback
δ
W // Y
L H
γ nwp β
(cid:15)(cid:15) (cid:15)(cid:15)
X // Z
G α K
in C, the following diagram in Set (resp. kMod) becomes commutative.
M∗(δ)
M(W)oo M(Y )
L H
M!(γ) (cid:8) M!(β)
(cid:15)(cid:15) (cid:15)(cid:15)
M(X)oo M(Z)
G M∗(α) K
We can alternatively define a semi-Mackey functor by using S. In the following,
whenwespeakofa2-functorfromStoSet ortokMod,weregarditasa2-category
equipped only with identity 2-cells. (See Remark 1.20.)
Asemi-Mackey functorM =(M,M∗)onSisapairofacontravariant2-functor
!
M∗: S→Set and a covariant 2-functor M : S→Set which satisfies the following.
!
(0) M∗(X)=M(X) for any 0-cell X ∈S0. We denote this simply by M(X).
G ! G G G
(1) [Additivity]Foranypairof0-cells X and Y inS,ifwetaketheirbicoprod-
G H
uct
X −υ→X X ∐ Y ←υY− Y
G G H H
in S, then the natural map
X Y X Y
(1.7) (M∗(υ ),M∗(υ )): M( ∐ )→M( )×M( )
X Y
G H G H
is isomorphism. Also, M(∅) is trivial.
(2) [Mackey condition] For any bipullback
W δ // Y
L H
γ {(cid:3)⑧⑧⑧⑧⑧⑧κ β
(cid:15)(cid:15) (cid:15)(cid:15)
X // Z
G α K
10 HIROYUKINAKAOKA
in S, the following diagram in Set (resp. kMod) becomes commutative.
M∗(δ)
M(W)oo M(Y )
L H
(1.8) M!(γ) (cid:8) M!(β)
(cid:15)(cid:15) (cid:15)(cid:15)
M(X)oo M(Z)
G M∗(α) K
This is just a paraphrase of the definition using C. With this view, for any mor-
phism α in C, we write M∗(α) and M(α) also as M∗(α) and M(α).
! !
Remark 1.23. In [16], we used the notation (M∗,M ) to denote a Mackey functor,
∗
where M∗ is contravariant, and M is covariant. In this article, we prefer to use
∗
(M,M∗), because of the following reason.
!
In analogy with the ordinary Mackey functor theory, a Tambara biset functor is
expected to be defined as a triplet of functors (M,M∗,M ), consisting of an addi-
! ∗
tiveMackeyfunctor(M,M∗)andamultiplicative semi-Mackeyfunctor(M∗,M ).
! ∗
The additive part (M,M∗) is expected to become a Green functor on S. The aim
!
of this article is to give a framework to deal with Green functors, which will serve
to the study of this additive part.
Remark 1.24. If M = (M,M∗) is a semi-Mackey functor on C, then each of M∗
!
and M becomes a functor to Mon, as shown in [16, Proposition 5.17]. Thus in
!
the definition of a semi-Mackey functor, we may assume M∗,M are functors to
!
Mon, from the beginning. (Mon denotes the category of commutative monoids
with units, where morphisms are monoid homomorphisms preserving units.)
Remark 1.25. Mackk(S) is a k-linear abelian category.
Example 1.26. ([16,Example5.4.1])Asemi-Mackeyfunctor A onC isdefined
big
as follows.
(1) Forany X ∈Ob(C),thesetA (X)isdefinedtobethesetofisomorphism
G big G
classes of the slice categoryC/X. Coproducts and natural weak pullbacks
G
in C give a commutative semi-ring structure on A (X). We denote the
big G
a a
isomorphism class of an object (A → X) by [A → X].
K G K G
(2) Let α∈C(X, Y ) be any morphism.
G H
(i) A (α) = α◦−: A (X) → A (Y ) is defined by the composition
big! big G big H
with α.
(ii) A∗ (α)= X ×−: A (Y )→A (X) is defined by the natural weak
big G big H big G
Y
H
pullback by α.
If we compose with the additive completion functor K : Mon → Ab, we obtain a
0
Z-Mackey functor
Ω =(Ω ,Ω∗ )=(K ◦A ,K ◦A∗ )∈Ob(MackZ(C))
big big! big 0 big! 0 big
which we call the bigger Burnside functor. Furthermore, by composing the coeffi-
cient change functor k⊗Z−: Ab →kMod, we obtain a k-linear Mackey functor
Ωkbig =(Ωkbig!,Ωkb∗ig)=((k⊗Z−)◦Ωbig!,(k⊗Z−)◦Ω∗big)∈Ob(Mackk(C)).