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Tambara Functors on Profinite Groups and GeneralizedBurnside FunctorsHiroyuki Nakaoka aa Graduate School of Mathematical Sciences , The University of Tokyo, Komaba , Meguro,Tokyo, JapanPublished online: 22 Sep 2009.

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Communications in Algebra®, 37: 3095–3151, 2009Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927870902747605

TAMBARA FUNCTORS ON PROFINITE GROUPSAND GENERALIZED BURNSIDE FUNCTORS

Hiroyuki NakaokaGraduate School of Mathematical Sciences, The University of Tokyo,Komaba, Meguro, Tokyo, Japan

The Tambara functor was defined by Tambara in the name of TNR-functor, to treatcertain ring-valued Mackey functors on a finite group. Recently Brun revealed theimportance of Tambara functors in the Witt–Burnside construction. In this article,we define the Tambara functor on the Mackey system of Bley and Boltje. Yoshida’sgeneralized Burnside ring functor is the first example. Consequently, we can considera Tambara functor on any profinite group. In relation with the Witt–Burnsideconstruction, we can give a Tambara-functor structure on Elliott’s functor VM , whichgeneralizes the completed Burnside ring functor of Dress and Siebeneicher.

Key Words: Mackey functor; Tambara functor; Witt–Burnside ring.

2000 Mathematics Subject Classification: Primary 19A22; Secondary 18B40, 20J05.

1. INTRODUCTION

The Tambara functor was defined by Tambara in [7] for any finite group G, inthe name of TNR-functors. Roughly speaking, a Tambara functor on G is a ring-valued Mackey functor with multiplicative transfers, satisfying certain compatibilityconditions for exponential diagrams. Recently, Brun revealed that Tambara functorsplay an important role in the Witt–Burnside construction [2].

As Mackey functors admit a Lindner-type description (see [5]), the categoryof Tambara functors is equivalent to the category of product-preserving functors�U� �Set��0 from a certain category U to the category of sets [7]. This enables us amore functorial treatment of fixed point functors, cohomology ring functors, andBurnside ring functors, as examples of Tambara functors.

On the other hand, to consider Mackey functors on a possibly infinite groupG, Bley and Boltje defined in [1] general Mackey systems for arbitrary groups, onwhich Mackey functors are defined. This general class of functors include ordinaryMackey functors on finite groups, Mackey functors on profinite groups (so-called

Received October 15, 2007; Revised June 17, 2008. Communicated by D. K. Nakano.Address correspondence to Hiroyuki Nakaoka, Graduate School of Mathematical Sciences, The

University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan; Fax: +81-3-3422-4371. E-mail:[email protected]

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G-modulations), and has several applications in number theory as shown in [1].A Mackey system �� for an arbitrary group G is a pair of family of subgroupsin G with certain conditions, each of them is closed under conjugation and finiteintersections.

Independently, for any finite group G and any conjugation-closed family � ofsubgroups in G, Yoshida has defined in [8] the generalized Burnside ring ��G��,which has several properties similar to the ordinary Burnside ring ��G�. It is shownin [8] that if � is, moreover, closed under (necessarily finite) intersections, ��G��is equal to the Grothendieck ring of a category associated to �G��, and becomesa subring of ��G�. We only consider the case where � �= � is also closed under(finite) intersections, and in this article we generalize ��G�� to a Mackey functor�� on any Mackey system �� for an arbitrary group G.

In this article, we consider a generalization of Tambara functors, namely,we define a Tambara functor on any Mackey system with certain conditions.As a consequence, we can consider a Tambara functor on a profinite group.Our main theorem (Theorem 5.16) enables us to construct Tambara functors, forexample, we make the above Burnside functor �� into a Tambara functor��•�. In relation with the Witt–Burnside construction, on any profinite groupG, we give a Tambara-functor structure to Elliott’s functor VM , where M is anarbitrary multiplicative monoid. This functor is closely related to the Witt–Burnsideconstruction as shown in [4], which generalizes the completed Burnside ring functorconsidered by Dress and Siebeneicher in [3].

In Section 2, after fixing our notation, we introduce some known resultsand preparative properties concerning Mackey functors on Mackey systems. InSection 3, we show any Mackey functor on a Mackey system admit a Lindner-typedefinition. In this context, the above Burnside functor �� can be easily regardedas a Mackey functor. In Section 4, we define Tambara systems and (semi-)Tambarafunctors on them, generalizing the case of finite groups. In Section 5, we show howa semi-Tambara functor gives rise to a Tambara functor in Theorem 5.16. Thetheorem is as follows, and proven in a similar way as the finite-group case.

Theorem 5.16. Let S be a semi-Tambara functor on ��•�. With ��S��X� =K0S�X� and +, •, �∗ appropriately defined, �S becomes a Tambara functor.

By virtue of this theorem, we can show that the Burnside functor ��

becomes a Tambara functor. As a further example concerning the Witt–Burnsideconstruction, we make VM into a Tambara functor on a profinite group G.

2. PRELIMINARIES

First we fix a notation. For any group G, H ≤ G means that H is asubgroup of G. For any subgroup H ≤ G and any g ∈ G, define gH �= gHg−1 andHg �= g−1Hg. GSet denotes the category of G-sets and equivariant maps, and Gsetdenotes the category of finite G-sets, which is a full subcategory of GSet. If X is aG-set and x ∈ X, let Gx denote the stabilizer group of x in X. In this article, monoidsare assumed to be commutative and have an additive unit 0. A homomorphismof monoids preserves 0. Semi-rings are assumed to be commutative both for theaddition and the multiplication, and have an additive unit 0 and a multiplicative

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TAMBARA FUNCTORS ON PROFINITE GROUPS 3097

unit 1. A homomorphism of semi-rings preserves 0 and 1. For any category � andany objects X, Y ∈ Ob��, the set of morphisms from X to Y in � is denoted by��X� Y�.

The following definitions are based on [1]. When we consider a Mackeyfunctor, we will only treat the case of a �-Mackey functor and call it simply aMackey functor.

Defintion 2.1 (Definition 2.1 in [1]). Let G be an arbitrary group. A Mackeysystem for G is a pair �� with the following property:

a) � is a set of subgroups of G, closed under conjugation and finite intersections,b) � = ��H� H∈� is a family of subsets ��H� ⊆ ��H� �= �U ∈ � � U ≤ H ,

which satisfies:

(i) �H � U� < �;(ii) ��U� ⊆ ��H�;(iii) ��gHg−1� = g��H�g−1;(iv) U ∩ V ∈ ��V�,

for all H ∈ �, U ∈ ��H�, V ∈ ��H�, and g ∈ G.

Example 2.2. Let � be a set of subgroups of G, closed under conjugation andfinite intersections.

(1) If we define �� by

��H� �= �U ∈ ��H� � �H � U� < � �∀H ∈ ��

then �� becomes a Mackey system for G.(2) If we define �d by

�d�H� = �H �∀H ∈ ��

then ��d� becomes a Mackey system for G.

Remark 2.3. Both �� and �d satisfy H ∈ ��H� and H ∈ �d�H� for any H ∈ �.In the following, we often impose the condition

H ∈ ��H� �∀H ∈ �� (2.1)

to a Mackey system ��. If we fix �, then �� (resp., ��d�) is thelargest (resp., smallest) Mackey system, among all the Mackey systems ��satisfying (2.1).

Definition 2.4. In (1) in Example 2.2, if in particular G is a topological groupand � is the set of all closed (resp., open) subgroups of G, we call �� thenatural (resp., open-natural) Mackey system for G. If G is a profinite group, thenthis definition of the (resp., open-) natural Mackey system agrees with the definitionof the (resp., finite) natural Mackey system in [1].

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Definition 2.5 (cf. Definition 2.3 in [1]). Let �� be a Mackey system for anarbitrary group G. A semi-Mackey functor M on �� is a function which assigns:

a) a monoid M�H� to each H ∈ �;b) a homomorphism of monoids cgH � M�H�→ M�gH� to each H ∈ � and eachg ∈ G;

c) a homomorphism of monoids rHI � M�H�→ M�I� to each pair I ≤ H in �;d) a homomorphism of monoids tHI � M�I�→ M�H� to each H ∈ � and eachI ∈ ��H�,

in a compatible way as in [1]. If all the M�H� are abelian groups, then M is calleda Mackey functor. The maps cgH , r

HI , t

HI are called conjugations, restrictions, and

transfers, respectively.A morphism of (semi-)Mackey functors f � M → N is a set of monoid

homomorphisms f = �fH � M�H�→ N�H� H∈�, which are compatible with theconjugations, restrictions, and transfers in the obvious sense. We write the categoryof semi-Mackey functors (resp., Mackey functors) as �Mack�� (resp., Mack��).Note that Mack�� is a full subcategory of �Mack��.

Remark 2.6. For a finite group G, if we regard G as a discrete topological group,both the natural and open-natural Mackey systems are

� = �∀ subgroup of G

��H� = ��H� = �∀ subgroup of H �∀H ≤ G��

A (resp., semi-) Mackey functor on this Mackey system is nothing other than a(resp., semi-) Mackey functor on G. Thus the Mackey functor theory on finitegroups is contained in that on Mackey systems.

Definition 2.7 (Definition 2.6 in [1]). Let �� be a Mackey system for G.

(1) GSet� is defined to be a full subcategory of GSet, whose objects are thoseX ∈ Ob�GSet� which satisfy

Gx ∈ � for any x ∈ X�

(2) GSet�� is defined to be a category with the same objects as GSet�, whosemorphisms from X to Y are those f ∈ GSet��X� Y� satisfying the followingproperties:

(i) f has finite fibers (i.e., f−1�y� is a finite set for any y ∈ Y );(ii) Gx ∈ ��Gf�x�� for any x ∈ X�

Remark 2.8. Let � be a set of subgroups of G closed under conjugationand finite intersections, and consider the Mackey system ��. Then for anyf ∈ GSet��X� Y�, we have

f ∈ GSet�� X� Y�⇔ f has finite fibers.

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Proof. The necessity is trivial. Conversely, assume f has finite fibers. For anyx ∈ X, let Xx denote the orbit through x in X. Since Gf�x�/Gx � Xx ⊆ f−1�f�x�� andf−1�f�x�� is finite by the assumption, we have �Gf�x� � Gx� = �Xx < �, i.e., Gx ∈��Gf�x��. �

Remark 2.9. If the Mackey system �� satisfies �2�1�, then for any two objectsX� Y in GSet�, any injective map � from X to Y in GSet

� � X → Y

belongs to GSet�� .In particular, isomorphisms, inclusions are morphisms in GSet�� .Moreover, the folding maps

� � X � X → X

are morphisms in GSet�� .

Remark 2.10. Let �� be an arbitrary Mackey system. For any pullbackdiagram in GSet�

if g belongs to GSet�� , then g′ also belongs to GSet�� .

Definition 2.11 (cf. Definition 2.6 in [1]). Category �Bif�� (resp., Bif��) isdefined as follows.

An object M in �Bif�� (resp., Bif��) is a function which assigns:

a) a monoid (resp., abelian group) M�X� to each X ∈ Ob�GSet��;b) a monoid morphism f ∗ � M�Y�→ M�X� to each f ∈ GSet��X� Y�;c) a monoid morphism g∗ � M�X�→ M�Y� to each g ∈ GSet��X� Y�,

in such a way that the following conditions are satisfied:

(i) We have

�g′ � g�∗ = g′∗ � g∗� �f ′ � f�∗ = f ∗ � f ′∗

for all composable pairs in GSet� and GSet�� respectively, and

id∗ = id∗ = id�

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(ii) (Mackey axiom) If

is a pullback diagram in GSet� where g ∈ GSet��X� Y�, then

f ∗ � g∗ = g′∗ � f ′∗�

(iii) For any direct sum decomposition X = ∐�∈�X� in GSet�, the natural map

�i∗��∈� � M�X�→∏�∈�M�X��

is an isomorphism of sets, where i� � X� ↪→ X �� ∈ �� are the inclusions.M�∅� consists of a single element.For any M�N ∈ Ob��Bif�� (or Bif��), a morphism � from M to N is a

collection of monoid homomorphisms �X � M�X�→ N�X� �X ∈ GSet��, compatiblewith all g∗ and f ∗. Remark that Bif�� is a full subcategory of �Bif��.

Remark 2.12. An object in �Bif�� (resp., Bif��) is nothing other than apair of functors M = �M∗�M∗� which satisfies M∗�X� = M∗�X� �= M�X�� ∀X ∈Ob�GSet��, where:

a) M∗ � GSet� → �Mon� (resp., GSet� → �Ab�) is a contravariant functor withM∗�f� = f ∗;

b) M∗ � GSet�� → �Mon� (resp., GSet�� → �Ab�) is a covariant functor withM∗�g� = g∗,

which satisfies the above condition (ii) and (iii).In this view, a collection of monoid homomorphisms ��X � M�X�→

N�X��X∈Ob�GSet�� is a morphism in �Bif�� if and only if it is a naturaltransformation with respect to each of the covariant and the contravariant part.

Remark 2.13. Let �� be a Mackey system satisfying �2�1�, and M be an objectin �Bif��. Let X = ∐

1≤i≤n Xi be a finite direct sum of objects in GSet�, and let�i � Xi ↪→ X be the inclusion (1 ≤ i ≤ n). The inverse of the isomorphism

�= ��∗i �1≤i≤n � M�X��−→ ∏

1≤i≤nM�Xi� =

⊕1≤i≤n

M�Xi�

is ∑1≤i≤n

�i∗ �⊕

1≤i≤nM�Xi�→ M�X��

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Proof. Since

are pullback diagrams, we have

�∗i � �i∗ = id

�∗i � �j∗ = 0 �i �= j��

So we have

� ∑1≤i≤n

�i∗ = id�

Since is an isomorphism, this means −1 = ∑1≤i≤n �i∗. �

Corollary 2.14. Let �� be a Mackey system satisfying (2.1), and M be an objectin �Bif��. If the pullback of f ∈ GSet��X� Z� and g ∈ GSet��Y� Z� is written as

where ki ∈ GSet��Wi� X� and hi ∈ GSet��Wi� Y�, then we have

f ∗ � g∗ =∑

1≤i≤nki∗ � h∗i �

Proof. Put

W �= ∐1≤i≤n

Wi�

k �= ⋃1≤i≤n

ki�

h �= ⋃1≤i≤n

hi�

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and let �i � Wi ↪→ W be the inclusion (1 ≤ i ≤ n). If we put �= ��∗i �1≤i≤n � M�W��→∏

1≤i≤n M�Wi�, then by Remark 2.13, we have

f ∗ � g∗ = k∗ � h∗ = k∗ � −1 � � h∗= k∗ �

∑1≤i≤n

�i∗ � ��∗i �1≤i≤n� � h∗ =∑

1≤i≤nki∗ � h∗i �

�

Remark 2.15. Let K0 � �Mon�→ �Ab� be the group completion functor. Forany M = �M∗�M∗� ∈ Ob��Bif��, we define �M ∈ Ob�Bif�� by �M �=�K0 �M∗� K0 �M∗�. For any morphism � = ��X � M�X�→ N�X��X∈Ob�GSet�� ∈�Bif��M�N�, we define �� ∈ Bif��M� �N� by �� = �K0��X� � K0M�X�→K0N�X��X∈Ob�GSet��. Thus, we obtain a functor � � �Bif�� → Bif��.

Proposition 2.16. Let M = �M∗�M∗� be an object in �Bif��. For anyisomorphism v � V → V ′ in GSet�� , we have

M∗�v�M∗�v� = idM�V ′��

Proof. From the pullback diagram

we obtain

M∗�v�M∗�v� = M∗�id�M∗�id� = id

by the Mackey axiom. Since M∗�v� is an isomorphism, this means M∗�v� = M∗�v�−1.�

As Theorem 2.7 in [1], the following theorem can be shown.

Theorem 2.17. Let �� be a Mackey system for G. Then:

(i) �Mack�� is equivalent to �Bif��;(ii) Mack�� is equivalent to Bif��.

By this equivalence, we often identify (semi-)Mackey functors with objectsin �� Bif��.

3. A LINDNER-TYPE DEFINITION OF MACKEY FUNCTORS

First, we introduce a well known fact (Fact 3.1), essentially due to Lindner [5].For any group G, let Gset denote the category of finite G-sets and G-maps, andlet Sp�Gset� denote the span category of Gset (i.e., the classifying category of the

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bicategory of spans in Gset (cf. [6])). By definition, Ob�Gset� = Ob�Sp�Gset�� and

Sp�Gset��X� Y� = �span from X to Y in Gset / ∼equivalence

= ��Xf← V

g→ Y� � f� g are morphisms in Gset / ∼equiv.

for any X� Y ∈ Ob�Sp�Gset��, where �f� V� g� �= �Xf← V

g→ Y� and �f ′� V ′� g′� areequivalent if and only if there exists an isomorphism v ∈ Gset�V� V

′� such thatf = f ′ � v, g = g′ � v.

If we want to indicate v, we write as

�f� V� g�∼−→v�f ′� V ′� g′�

instead of �f� V� g� ∼ �f ′� V ′� g′�. We write the equivalence class of �f� V� g� =�X

f← Vg→ Y� as �f� V� g� = �X

f← Vg→ Y�. For any X� Y ∈ Ob�Sp�Gset��, the set of

morphisms Sp�Gset��X� Y� has a natural monoid structure. (This is an example ofProposition 3.5(ii).)

For any category �, let �� set��0 denote the category of covariant functorsfrom � to the category �Set� of sets, preserving arbitrary product (whenever theproduct exists in �). Here by the term arbitrary product, we mean a product

∏�∈� X�

of any objects X� ∈ Ob��, indexed by an arbitrary set �. If � = Sp�Gset�, we canview �� Set��0 also as the category of certain contravariant functors by the self-dualnature of Sp�Gset�. But we use this covariant way, in view of analogy with Tambarafunctors in the later sections.

Fact 3.1 (cf. Theorem 4 in [5]). Let G be a finite group.

(1) �Mack�G� is equivalent to �Sp�Gset�� set��0.(2) There exists a unique category ��Gset� with arbitrary finite products and a

functor � � Sp�Gset�→ ��Gset� with the following properties:

(a) Ob��Gset�� = Ob�Sp�Gset��;(b) � preserves arbitrary products;(c) ��Gset��X� Y� = K0�Sp�Gset��X� Y�� for any X� Y in Ob��Gset��, where K0

denotes the group completion, and the maps of �

�X�Y � Sp�Gset��X� Y�→ ��Gset��X� Y�

are the completion maps.

(3) Mack�G� is equivalent to ��Gset�� Set��0.

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Remark 3.2. For any set of objects �X��∈� in Sp�Gset� (or ��Gset�), their productis of the form

∐�∈� X� (cf. Proposition 3.5(1)). Since Ob�Sp�Gset�� = Ob��Gset�� =

Ob�Gset� consists of finite G-sets, when we consider the product of �X��∈�, thenX� must be equal to ∅ except finite � ∈ �. So, a functor F � Sp�Gset�→ �set� (or��Gset�→ �Set�) preserves arbitrary products if and only if F preserves finiteproducts.

Definition 3.3. Let �� be a Mackey system for an arbitrary group G whichsatisfies (2.1). We define a category � = Sp�� as follows:

Ob�� = Ob�GSet��

� �X� Y� = ��f� V� g� �V ∈ Ob�GSet�� f ∈ GSet��V�X�� g ∈ GSet��V� Y� / ∼equiv.

for any X� Y ∈ Ob�� , where �f� V� g� = �Xf← V

g→ Y� and �f ′� V ′� g′� areequivalent if and only if there exists an isomorphism v such that f = f ′ � v,g = g′ � v.

Let �f� V� g� = �Xf← V

g→ Y� denote the equivalence class of �f� V� g�.Composition in � is defined by

for any �f� V� g� ∈ � �X� Y� and �h�W� k� ∈ � �Y� Z�, where

is the pullback diagram.

To distinguish, we write the morphism in � by an arrow X ⇀ Y .For any f ∈ GSet��X� Y� and g ∈ GSet��X� Y�, we use the notation

Rf �= �f� X� idX� � Y ⇀ X

Tg �= �idX� X� g� � X ⇀ Y�

Any morphism �f� V� g� in � �X� Y� has a decomposition �f� V� g� = Tg � Rf .

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Remark 3.4.

(i) We have

T�g′�g� = Tg′ � Tg� R�f ′�f� = Rf � Rf ′

for all composable pairs in GSet� and GSet�� respectively, and

Tid = id� Rid = id�

(ii) If

is a pullback diagram in GSet� where g ∈ GSet��X� Y�, then we have

Rf � Tg = Tg′ � Rf ′ �

Proof. These can be easily checked directly from the definition of thecomposition law. �

In the rest of this section, let G be an arbitrary group, and let �� denote aMackey system satisfying (2.1), unless otherwise specified.

Proposition 3.5.

(i) For any X� ∈ Ob�� ∈ ��, if we put X �= ∐�∈� X�, then

�Ri� � X ⇀ X��∈�

is their product in � , where i� � X� ↪→ X is the inclusion (∀� ∈ �). ∅ is the terminalobject in � .

(ii) For any A�X ∈ Ob�� , � �A�X� has a monoid structure defined by

�f1� V1� g1�+ �f2� V2� g2� = �f1 ∪ f2� V1 � V2� g1 ∪ g2�

for any �f1� V1� g1�, �f2� V2� g2� ∈ � �A�X�. �A← ∅ → X� is the zero in this monoid,and we abbreviate this morphism to 0. Moreover, with this monoid structure, thefunctor

� �−� X� � � → �Set�

factors through the subcategory �Mon� of �Set�.

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Proof. (i) It suffices to show that for any Y ∈ Ob�� and any set of morphisms�f�� V�� g�� Y ⇀ X� �� ∈ ��, there exists a unique morphism �f� V� g� � Y ⇀ X

such that

Ri� � �f� V� g� = �f�� V�� g�� ∀� ∈ ��

As for the existence, we can easily see that the morphism

�f� V� g� �=[ ⋃�∈�

f��∐�∈�

V��∐�∈�

g�

]

satisfies the above commutativity condition. To prove the uniqueness, let �f ′� V ′� g′�be a morphism satisfying

Ri� � �f ′� V ′� g′� = �f�� V�� g�� ∀� ∈ �� (3.1)

Put

V ′� �= g′−1�X��

g′� �= g′�V ′��

i′� �= V ′� ↪→ V ′ inclusion�

Then since Ri� � �f ′� V ′� g′� = �f ′ � i′�� V ′�� g

′��, condition (3.1) is equal to the fact

that there exists an isomorphism v� � V��−→ V ′

� for each � ∈ �, such that g� = g′� �v�� f� = f ′ � i′� � v�.

By taking the direct sum, we obtain an isomorphism

∐�∈�

v� �∐�∈�

V��−→ ∐

�∈�V ′��

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which makes the following diagram commutative:

But since we have∐�∈�

V ′� = V ′�

⋃�∈��f ′ � i′�� = f ′�

∐�∈�

g′� = g′�

this means �f� V� g�∼−→∐�∈� v�

�f ′� V ′� g′�.

(ii) To show that � �X� Y� is in fact a monoid is easy. The latter half followsfrom the fact that for any morphism �k�W� h� � B ⇀ A in � , we have

where �f1 ∪ f2�′� f ′1� f

′2 and prV1�V2� prV1� prV2 are the appropriate pullbacks. �

Generally, let � be a category with finite products (hence has a terminal object∅). � is regarded as a symmetric monoidal category via the cartesian product. Anobject X ∈ Ob�� is called a monoid object if it is equipped with a pair of morphisms

mX � X × X → X

eX � ∅ → X

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which satisfies associativity, commutativity, and the unit law. Namely, it makes thefollowing diagrams commutative:

Here, tw � X × X → X × X is the twisting map. (Recall that monoids are assumedto be commutative in our notation.)

Proposition 3.6. Any object X in � is a monoid object with the structure

mX �= T�XeX �= T�∅ �

Proof. This follows from Proposition 3.5(i), Remark 3.4(i), and the commutativityof the following diagrams:

Here, tw � X � X → X � X is the twisting map and satisfies

Ttw = tw � X × X → X × X�

and �i � X ↪→ X � X is the inclusion into the ith component (i = 1� 2). �

Remark 3.7. By Yoneda’s lemma, an object X ∈ Ob�� is a monoid object if andonly if the functor

��−� X� � � → �Set�

factors through the subcategory �Mon� of �Set�.

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By Proposition 3.6, every object X ∈ Ob�� is a monoid object in � . Thecorresponding monoid structure on � �A�X� (∀A ∈ Ob�� ) is nothing other than (ii)in Proposition 3.5.

An arrow f ∈ ��X� Y� between monoid objects is called a morphism of monoidobjects if it makes the following diagrams commutative:

Again by Yoneda’s lemma, this is equivalent to that

f � − � ��−� X�→ ��−� Y�

is a natural transformation of functors � → �Mon�.

Proposition 3.8.

(i) For any f ∈ GSet��X� Y�,

Rf � Y ⇀ X

is a morphism of monoid objects.(ii) For any g ∈ GSet��X� Y�,

Tg � X ⇀ Y

is a morphism of monoid objects.

Proof. Since �f� V� g� = Tg � Rf , to show (i) and (ii) is equivalent to show that anymorphism in � is a morphism of monoid objects. But this can be shown in the sameway as in the proof of (ii) in Proposition 3.5. �

Definition 3.9. Let F � � → �Set� be a functor. We say F preserves finite productsif, for any finite product �pk � X → Xk�1≤k≤n, the induced map

�F�pk��1≤k≤n � F�X�→∏

1≤k≤nF�Xk�

is an isomorphism.

We can easily see the following proposition.

Proposition 3.10. Let F � � → �Set� be a functor which preserves finite products.

(i) If X ∈ Ob�� is a monoid object, then F�X� becomes a monoid.

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(ii) If f ∈ ��X� Y� is a morphism of monoid objects, then F�f� � F�X�→ F�Y� becomesa monoid homomorphism.

(iii) If Y is a monoid object in �, then for any X ∈ Ob��,FX�Y � ��X� Y�→ �Set��F�X�� F�Y��

is a monoid homomorphism, where the monoid structure of the right-hand side isthe one induced from that of F�Y�.

Remark 3.11. For any category � with finite products, we can define commutativegroup objects, semi-ring objects, and ring objects in �, in the same way. They satisfythe analogous statement of Remark 3.7, and Proposition 3.10 for any finite-product-preserving functor F .

Theorem 3.12. �Mack�� is equivalent to �Sp�� Set��0.

Proof. In view of Theorem 2.17, we show that there exists an isomorphism ofcategories

�Bif��−→ �� Set��0�

In one direction, for any M = �M∗�M∗� ∈ Ob��Bif��, we associate a covariantfunctor E � � → �Set� by

E�X� �= M�X� �∀X ∈ Ob�� E��f� V� g�� = M∗�g�M

∗�f� �∀�f� V� g� ∈ � �X� Y��

This definition of E��f� V� g�� is well defined. Indeed, for any

we have

M∗�g�M∗�f� = M∗�g

′ � v�M∗�f ′ � v�= M∗�g

′�M∗�v�M∗�v�M∗�f ′�

=Prop. 2.16

M∗�g′�M∗�f ′��

In other words, E is defined by

E�Rf � = M∗�f� �∀f ∈ GSet��X� Y��

E�Tg� = M∗�g� �∀g ∈ GSet��X� Y��

on morphisms. The associativity and unit law for morphisms can be checked easily.

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For any set of objects �X��∈� in GSet�, since M preserves arbitrary products,we have an isomorphism

�M∗�i��∈� � M�X��−→ ∏

�∈�M�X��

where X �= ∐�∈� X� and i� � X� ↪→ X is the inclusion. But this is nothing other than

the isomorphism

�E�Ri��∈� � E�X��−→ ∏

�∈�E�X��

which means that E preserves arbitrary products. Thus E becomes an objectof �� Set��0. If E� F ∈ Ob�� Set��0� are the objects corresponding to M�N ∈Ob��Bif�� as above, then for any morphism � � M → N in �Bif��, naturally,the same �

��X � E�X�→ F�X��X∈Ob��

gives a morphism � � E → F in �� Set��0.In the other direction, if we are given an object E ∈ Ob�� Set��0�, then we

define M = �M∗�M∗� by

M�X� �= E�X� �∀X ∈ Ob�GSet��M∗�f� �= E�Rf � �∀f ∈ GSet��X� Y��

M∗�g� �= E�Tg� �∀g ∈ GSet��X� Y��

By Remark 3.4(i) and Proposition 3.10, we can see easily that M∗ is a contravariantfunctor from GSet� to �Mon�, and M∗ is a covariant functor from GSet�� to �Mon�.Condition (ii) in Definition 2.11 follows from the property (ii) in Remark 3.4.By the same argument as before, we can see that M preserves arbitrary productssince E does. If M�N ∈ Ob��Bif�� are the objects corresponding to E� F ∈Ob�� Set��0� and � � E → F is a morphism in �� Set��0, then �X is a monoidhomomorphism for each X ∈ Ob�� . In fact, for each X ∈ Ob�� , the diagram

is commutative where �j � X ↪→ X � X is the inclusion into jth component (j = 1� 2).From this, we can show that the diagram

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is commutative, which means that �X is a monoid homomorphism. Thus, for anymorphism � � E → F in �� Set��0, naturally, the same �

��X � M�X�→ N�X��X∈Ob�GSet��

gives a morphism in �Bif�� . It can be seen easily that these correspondencesM ↔ E gives an isomorphism of categories

�Bif��−→ �Sp�� Set��0�

�

Definition 3.13. We define a category � = �� as follows:

a) Ob�� = Ob�� ,b) ��X� Y� = K0�� X� Y�� for any X� Y ∈ Ob��.

So, any morphism in ��X� Y� can be written as the difference of the images��1�, ��2� of two morphisms �1 and �2 in � �X� Y�. � actually becomes a categorywith the composition law defined by

��1� − ��2�� 1� − ��2�� = ��1 � �1 + �2 � �2� − ��1 � �2 + �2 � �1��where �i ∈ � �X� Y�, �i ∈ � �Y� Z� �i = 1� 2�. The well definedness of this compositioncan be shown easily, via the fact that the composition in �

� � � �X� Y�×� �Y� Z�→ � �X� Z�

is bi-additive. By this definition, the composition in �

� � ��X� Y�×��Y� Z�→ ��X� Z�

also becomes bi-additive.There is a functor � � � → � defined by ��F � X ⇀ Y� �= ��F� � X ⇀ Y�. For

any X� ∈ Ob�� ∈ ��, since �Ri� � X ⇀ X��∈� is their product in � , we have anisomorphism of sets

�− � Ri��∈� � � �Y� X��−→ ∏

�∈�� Y� X��

for each object Y ∈ Ob�� . By taking K0 of this isomorphism, we obtain anisomorphism for each Y ∈ Ob�� = Ob��

�− � �Ri��∈� � ��Y� X��−→ K0

( ∏�∈�

� �Y� X��)

= ∏�∈�

��Y� X��

which means that

��Ri�� X ⇀ X��∈�

is the product of �X��∈�. Thus � preserves arbitrary products.

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Theorem 3.14. Mack�� is equivalent to �� Set��0.

Proof. As before, we show the equivalence of Bif�� with �� Set��0. Let � �

�� Set��0�−→ �Bif�� denote the isomorphism constructed in Theorem 3.12. Let

�� Set��0 → �� Set��0 be the composition by �.

Let E be any object in Ob�� Set��0�. Since X ∈ � is a commutative groupobject by Remark 3.11, ��E��X� = E � ��X� becomes an abelian group. So we have��Ob�� Set��0�� ⊂ Ob�Bif��. Thus we obtain a functor

� � �� Set��0 → Bif��


Let E be any element in �� Set��0. Then ��E� belongs to Bif�� if andonly if E�X� is an abelian group for any X ∈ Ob�� . In this case, if we defineE ∈ Ob�� Set��0� by

E�X� �= E�X� �∀X ∈ Ob��E��1� − ��2�� = E��1��− E��2�� ∀�1� �2 ∈ � �X� Y��

then we have E = ��E�. Thus � is essentially surjective.To show � is fully faithful, it suffices to show that �� is fully faithful. Let

T1� T2 ∈ Ob�� Set��0� be any elements, and consider the map

We show this map is bijective. Injectivity is trivial, since � is identity on objects.To show surjectivity, let � = ��X � �

�T1�X�→ ��T2�X��X∈Ob�� be any element in�� Set��0��

�T1� ��T2�. By the naturality of �, we have

��T2�� X = �Y � ��T1��

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for any � ∈ � �X� Y�. Remark that ��Tj�� = Tj�� (j = 1� 2).

It suffices to show � is also natural with respect to any element ��1� − ��2� ∈��X� Y� (∀�1� �2 ∈ � �X� Y�). By (the proof of) Theorem 3.12, �X is a monoidhomomorphism (∀X ∈ Ob�� = Ob��. Moreover,

�Tj�X�Y � ��X� Y� −→ �Ab��Tj�X�� Tj�Y��

is a group homomorphism (cf. Proposition 3.10(ii)) (∀j = 1� 2). From these, we canshow easily

T2��1� − ��2�� X = �Y � T1��1� − ��2��

Example 3.15. Let �� be a Mackey system for G which satisfies (2.1) andG ∈ �. In this case, one-point set �∗ = G/G is the terminal object in GSet�. ByTheorem 3.14, we obtain a Mackey functor �� corresponding to the Hom functor��∗ �−� ∈ �� Set��0. For each X ∈ Ob�� , since the monoid

� ��∗ � X� = {��∗ ← V

g→ X� �V ∈ Ob�GSet�� g ∈ GSet��V�X�}/ ∼

equiv.

= {�V

g→ X� �V ∈ Ob�GSet�� g ∈ GSet��V�X�}/ ∼

equiv.

is the set of isomorphism classes of the comma category GSet��/X, we have

��∗ � X� = K0�GSet��/X��

where K0 denotes the Grothendieck group of GSet��/X. Thus �� satisfies

��X� = K0�GSet��/X�

for each X ∈ Ob�GSet��.

Definition 3.16. We call �� the Burnside functor on the Mackey system ��.

Remark 3.17. ��X� is in fact a ring, with a multiplication induced fromthe fiber product over X in GSet�� . In particular, if G is finite and � = ��,then ��∗ � = ��G/G� agrees with the generalized Burnside ring ��G��defined in [8].

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4. TAMBARA FUNCTORS ON MACKEY SYSTEMS

Let G be an arbitrary group. We consider a triplet ��+��•�, where ��+�and ��•� are Mackey systems for G. For X ∈ Ob�GSet��, we define a category��+ �X as follows:

a) Ob��+ �X� = Ob�GSet��+/X� = ��A→ X� �A ∈ Ob�GSet��+� = Ob�GSet�� ∈

GSet��+�A�X� ;

b) ��+ �X��A→ X�� A′ ′→ X�� = �� ∈ GSet��A�A

′� = GSet�A�A′� � ′ � � = for

any �A→ X�, �A′ ′→ X� in Ob��+ �X�.

Remark that for any X ∈ Ob�GSet��, ��+ �X is a full subcategory of GSet/X.

Definition 4.1. We call ��+��•� a Tambara system if it satisfies the followingcondition:

For any ∈ GSet��•�X� Y�, the pullback functor defined by

X ×Y− � ��+ � Y → ��+ �X

has a right adjoint

� � ��+ �X → ��+ � Y �

If � exists, we write the object ��A→ X�� ∈ ��+ � Y abbreviately as ��A�

��→ Y�.

As in the case where G is finite [7], it can be easily seen for any (possiblyinfinite) group G that the pullback functor defined by f ∈ GSet�X� Y�

X ×Y− � GSet/Y → GSet/X

always has a right adjoint. The construction is as follows:

a) For any �Ap→ X� ∈ Ob�GSet/X�, we define

�f�A� �= ��y� �� y ∈ Y� � � f−1�y�→ A a map� p � � = idf−1�y� �

q � �f �A� → Y� q�y� �� = y�

�f �A� is a G-set by g · �y� �� = �gy� g��, where g� is defined by �g��x� =g��g−1x� for any x ∈ f−1�gy�. We sometimes write this q as ��p� or �f �p�. Wedefine abbreviately

�f

((A

p→ X))�= (

�f�A�q→ Y

)�∈ Ob�GSet/Y��

b) For any morphism a � �Ap→ X�→ �A′ p′→ X� in GSet/X, we define a morphism

�f�a� � �f

((A

p→ X)) → �f

((A′ p′→ X′))

in GSet/Y by �f�a��y� �� = �y� a � ��.

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Then, this functor

�f � GSet/X → GSet/Y

is right adjoint to X ×Y−.

Lemma 4.2. Let ��+��•� be a triplet where ��+� and ��•� are Mackeysystems. This triplet ��+��•� becomes a Tambara system if it satisfies the followingcondition for any X� Y ∈ Ob�GSet�� and any ∈ GSet��•�X� Y�:

For any �A→ X� in Ob��+ �X�,

��A� ∈ Ob�GSet�� ∈ GSet��+��A�� Y�� (4.1)

Proof. Since ��+ �X (resp., ��+ � Y ) is a full subcategory of GSet/X (resp., GSet/Y ),the triplet ��+��•� becomes a Tambara system if for any ∈ GSet��•�X� Y� and

any �A→ X� ∈ Ob��+ �X�, the object ��A�

��→ Y� �∈ Ob�GSet/Y�� belongs toOb��+ � Y �. This is equivalent to condition (4.1). �

Thus if condition (4.1) is satisfied, then � can be taken as �.

Definition 4.3. Let � be a set of subsets of G, which is closed under left and righttranslation, finite intersections and finite unions (hence ∅ ∈ �). To �, we associatea set � of subgroups of G by

� �= �H ∈ � �H ≤ G �

Then � is closed under conjugation and finite intersections. In this case, we say“� arises from �.” Whenever we say “� arises from �,” we always assume that �satisfies the above condition.

For a given �, we often consider a Mackey system ��, where � arisesfrom � as above.

Example 4.4. Let G be a topological group. If � = �∀ closed subset of G (resp.�∀ open subset of G ), then the above �� is the natural (resp., open-natural)Mackey system for G.

From a Mackey system satisfying (2.1), we can construct a Tambara system.

Proposition 4.5. Let ��•� be a Mackey system satisfying (2.1), and assume �arises from some �, then ��•� becomes a Tambara system.

Proof. By Lemma 4.2, it suffices to show

(1) ��A� ∈ Ob�GSet��(2) �� ∈ GSet�� A�� Y�

for any ∈ GSet��•�X� Y� and any �A→ X� in Ob�� X�.

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Proof of (1). This is equivalent to

G�y�� ∈ � �∀�y� �� ∈ ��A��

We have

G�y�� = �g ∈ G � g · �y� �� = �y� ��

= ⋂x∈−1�y�

�g ∈ Gy � g��x� = ��gx�

= ⋂x∈−1�y�

( ⋃a∈−1�−1�y��

(�g ∈ Gy � g��x� = a ∩ �g ∈ Gy � ��gx� = a

))

= ⋂x∈−1�y�

( ⋃a∈−1�−1�y��

�Lx�a ∩ Rx�a�)�

where

Lx�a �= �g ∈ Gy � g��x� = a �

Rx�a �= �g ∈ Gy � ��gx� = a �

So, it suffices to show

Lx�a� Rx�a ∈ � �∀x ∈ −1�y��∀a ∈ −1�−1�y��

If Lx�a = ∅, then Lx�a ∈ �. Otherwise, we can take an element g0 ∈ Lx�a, and

Lx�a = �g ∈ Gy � g��x� = g0��x�

= Gy ∩ g0 ·G��x� ∈ ��

If �−1�a� = ∅, then Rx�a = ∅ ∈ �. Otherwise, since � is injective, there exists aunique element x0 ∈ −1�y� such that ��x0� = a. So we have

Rx�a = �g ∈ Gy � gx = x0 �

By a similar argument for Lx�a, we obtain Rx�a ∈ �.

Proof of (2). By Remark 2.8, it suffices to show that �� has finite fibers. Butsince

��−1�y� = �� −1�y�→ A a map� � � = id−1�y�

for any y ∈ Y , this follows from the fact that has finite fibers. �

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Tambara systems we mainly consider are of this type.

Example 4.6. If G is a topological group and ��•� is the natural (resp.,open-natural) Mackey system, then ��•� is a Tambara system. We call this��•� the natural (resp., open-natural) Tambara system.

Let ��+��•� be a Tambara system, and let ∈ GSet��+�A�X� and ∈GSet��•�X� Y� be any elements. We construct a certain commutative diagram from

and , called an exponential diagram. Take � of the object �A→ X� in ��+ �X ,

and take the fiber product of and ��:

By the adjoint isomorphism

��+ �X((X ×

Y��A�

��′→ X)� �A

→ X�) � ��+ � Y

((��A�

��→ Y)� ��A�

��→ Y�)�

there exists a morphism � � X ×Y��A�→ A corresponding to id��A�. Thus we obtain

a commutative diagram in GSet�

(4.2)

where ′ is the pullback of by ��.

Definition 4.7. A commutative diagram

(4.3)

in GSet� where �� are morphisms in GSet��+ and � ′ are in GSet��• is calledan exponential diagram if it is isomorphic to one of the diagrams (4.2) constructed

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above. We write as

to indicate that it is an exponential diagram.

Example 4.8. Let ��+��•� be a Tambara system where ��+� and ��•�satisfy (2.1). The following commutative diagrams are exponential diagrams.

(i)

(ii)

(iii) Let ��•� be a Tambara system as in Proposition 4.5. The exponentialdiagram constructed from � ∈ GSet�� X � X�X� and ∈ GSet��•�X� Y� is

where

V = ��y� C� � y ∈ Y�C ⊂ −1�y� �

U = ��x� C� � x ∈ X�C ⊂ −1��x�� x ∈ C �U ′ = ��x� C� � x ∈ X�C ⊂ −1��x�� x �∈ C �� U → X� �x� C� �→ x�

�′ � U ′ → X� �x� C� �→ x�

� � U → V� �x� C� �→ ��x�� C��

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�′ � U ′ → V� �x� C� �→ ��x�� C��

� � V → Y� �y� C� �→ y�

These are in GSet�. Moreover, it can be easily checked that �� ′ are morphismsin GSet��• , and � is a morphism in GSet�� .

Definition 4.9. In the notation of (iii) in Remark 4.8, the commutative diagrams

are called T -diagram and F -diagram of , respectively.

Now we define a (semi-)Tambara functor on a Mackey system. For theoriginal definition of a Tambara functor on a finite group, see [7].

Definition 4.10. Let ��+��•� be a Tambara system. A semi-Tambara functor Son ��+��•� is a function which assigns:

a) a semi-ring S�X� to each X ∈ Ob�GSet��;b) a homomorphism of additive monoids + � S�X�→ S�Y� to each ∈

GSet��+�X� Y�;c) a homomorphism of multiplicative monoids • � S�X�→ S�Y� to each ∈

GSet��•�X� Y�;d) a semi-ring homomorphism �∗ � S�Y�→ S�X� to each � ∈ GSet��X� Y�,

in such a way that the following conditions are satisfied:

(i) For any direct sum decomposition X = ∐�∈�X� in GSet�, the natural map

�i∗��∈� � S�X�→∏�∈�S�X��

is an isomorphism of sets, where i� � X� → X �� ∈ �� are the inclusions. S�∅�consists of a single element.

(ii)

�′ � �+ = ′+ � +� �′ � �• = ′• � •� ��′ � ��∗ = �∗ � �′∗

for all composable pairs in GSet��+ , GSet��• , GSet�, respectively, and

id+ = id• = id∗ = id�

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(iii) If

is a pullback diagram in GSet� where ∈ GSet��+�X� Y�, then �∗ � + = ′+ � �′∗.

(iv) If

is a pullback diagram in GSet� where ∈ GSet��•�X� Y�, then �∗ � • = ′• � �′∗.

(v) For any exponential diagram (4.3), • � + = �+ � ′• � �∗.If all S�X� are rings, S is called a Tambara functor.

Remark 4.11. The conditions from (i) to (iv) can be written in terms of Mackeyfunctors as follows:

(I) The morphisms +, •, �∗ yield functors:

a) a covariant functor S+ � GSet��+ → �Set�;b) a covariant functor S• � GSet��• → �Set�;c) a contravariant functor S∗ � GSet� → �Set�.

(II) The pairs �S∗� S+� and �S∗� S•� are Mackey functors on ��+� and ��•�,with respect to the additive and multiplicative structures of S�X� �X ∈Ob�GSet��, respectively.

Proposition 4.12. Let ��•� be a Tambara system as in Proposition 4.5, andlet S be a semi-Tambara functor on ��•�. For any ∈ GSet��•�X� Y�, considerthe T - and F -diagrams as in Definition 4.9:

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Then we have

•�x + y� = �+��• � �∗�x�� · ��′• � �′∗�y�� ∀x� y ∈ S�X��

Proof. Let �i � X ↪→ X � X and �′i � V ↪→ V � V be the inclusions into the ithcomponents (i = 1� 2). Since

is an exponential diagram, we have

• � �+ = �+ � �� ∪ �′�• � �� ′�∗ = �+ � �• � �� ′�• � �� ′�∗�

By Remark 2.13, for any x� y ∈ S�X� we have

��∗1� �∗2�

−1�x� y� = �1+�x�+ �2+�y� = �1•�x� · �2•�y��

So we have

• � �+��∗1� �

∗2�

−1�x� y�� = • � �+��1+�x�+ �2+�y�� = •�x + y��

and

�• � �� ′�• � �� ′�∗��∗1� �∗2�

−1�x� y�� = �• � �� ′�• � �� ′�∗��1•�x� · �2•�y��= �•��

′1•��• � �∗�x�� · �′2•��′• � �′∗�y��

= ��• � �∗�x�� · ��′• � �′∗�y��

Thus we obtain

•�x + y� = �+��• � �∗�x�� · ��′• � �′∗�y��

Definition 4.13. Let ��+��•� be a Tambara system, and S� T be two Tambarafunctors (resp., semi-Tambara functors). A morphism � from S to T is a collectionof semi-ring morphisms �X � S�X�→ T�X� �X ∈ Ob�GSet��, which commute withall +, •, �∗.

Tam��+��•� (resp., �Tam��+��•�) denotes the category of Tambara (resp.,semi-Tambara) functors and their morphisms. Tam��+��•� is a full subcategory of�Tam��+��•�.

Definition 4.14. Let ��+��•� be a Tambara system, where ��+� and ��•�are Mackey systems satisfying (2.1). We define a category � = ��+��•� as follows:

a) Ob�� = Ob�GSet��

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b) For any X� Y ∈ Ob��, the set of morphisms ��X� Y� is{(X

�← A→ B

→ Y) ∣∣∣∣ A�B ∈ Ob�GSet�� ∈G Set��+�B� Y�

∈ GSet��•�A� B�� ∈ GSet��A�X�

}/∼

equiv.�

where �X�← A

→ B→ Y� and �X

�′← A′ ′→ B′ ′→ Y� are equivalent if and only ifthere exists a pair of isomorphisms a � A→ A′ and b � B→ B′ such that = ′ � b,b � = ′ � a, � = �′ � a.

Let �X�← A

→ B→ Y� denote the equivalence class of �X

�← A→ B

→ Y�.Composition law is defined by �Y ← C → D→ Z� � �X ← A→ B→ Y� =

�X ← A′′ → D→ Y�, with the morphisms appearing in the following diagram:

Then � becomes in fact a category. This can be shown in the same way as in thecase of finite groups [7]. We write X ⇀ Y to indicate the morphism in �.

For any X, Y ∈ Ob��, we use the notation:

a) T �= �Xid← X

id→ X→ Y� for any ∈ GSet��+�X� Y�;

b) N �= �Xid← X

→ Yid→ Y� for any ∈ GSet��•�X� Y�;

c) R� �= �X�← Y

id→ Yid→ Y� for any � ∈ GSet��Y� X�.

As follows, � has analogous properties as in the case of a finite group [7].

Remark 4.15 (cf. Proposition 7.2. in [7]).

(i) Any morphism �X�← A

→ B→ Y� in ��X� Y� admits a decomposition �X

�←A

→ B→ Y� = T � N � R�.

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(ii) �→ T defines a covariant functor T � GSet��+ → �. Similarly, we have acovariant functor N � GSet��• → � and a contravariant functor R � GSet� → �.

(iii) If

is a pullback diagram in GSet�, where is a morphism in GSet��+ , then R� �T = T′ � R�′ .If

is a pullback diagram in GSet� where is a morphism in GSet��• , then R� �N = N′ � R�′ .

(iv) If

is an exponential diagram, then N � T = T� � N′ � R�.

Proof. These can be easily checked directly from the definition of thecomposition law. �

Proposition 4.16 (cf. Proposition 7.5. in [7]).

(i) For any set of objects �X��∈� in �,

�Ri� � X ⇀ X��∈�

is their product in �, where X �= ∐�∈� X� and i� � X� ↪→ X is the inclusion. ∅ is

the terminal object in �. Thus, arbitrary products exist in �.(ii) Any X ∈ Ob�� has a structure of a semi-ring object of � with addition T� ,

additive unit T�∅ , multiplication N� , multiplicative unit N�∅ .

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(iii) For any ∈ GSet��+�X� Y�, T preserves the additive structure on X and Y .For any ∈ GSet��•�X� Y�, N preserves the multiplicative structure on X and Y .For any � ∈ GSet��X� Y�, R� preserves the semi-ring structure on X and Y .

Proof of (i). This is proved essentially in the same way as in the proof ofProposition 3.5(i). For any Y ∈ Ob�� and any set of morphisms

f� =[Y

��← A��→ B�

�→ X�]� Y ⇀ X� �� ∈ ��

we define f � Y ⇀ X by

f �=[Y

�← ∐�∈�

A�→ ∐

�∈�B�

→ X

]�

where � = ⋃�∈� ��, =

∐�∈� �, =

∐�∈� �. Then, by virtue of Example 4.8(i),

we have

Ri� � f = f� �∀� ∈ ��

For any morphism f ′ = �Y�′← A′ ′→ B′ ′→ X� which satisfies

Ri� � f ′ = f� �∀� ∈ �� (4.4)

put

A′� �= �′ � ′�−1�X��

B′� �= ′−1�X��

�′� �= �′ �A′�� A′

� → Y

′� �= ′ �A′�� A′

� → B′�

′� �= ′ �B′�� B′

� → X��

Then we have Ri� � f ′ = �Y�′�← A′

�

′�→ B′�

′�→ X�, and by (4.4), there exists a pair ofisomorphisms �a�� b�� such that

By taking the direct sum, we obtain isomorphisms

a = ∐�∈�

a� � A→ A′

b = ∐�∈�

b� � B→ B′�

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This means f = f ′.

Proof of (ii). By Proposition 3.6, it suffices to show the distributive law. Butthis follows from Remark 4.15(iv), applied to the exponential diagram (ii) inExample 4.8.

Proof of (iii). This immediately follows from Proposition 3.8. �

Theorem 4.17 (cf. Proposition 7.7. in [7]). There is an isomorphism of categories

�Tam��+��•��→ �� Set��0�

Proof. Objects S ∈ Ob��Tam��+��•�� and ∈ Ob�� Set��0� correspond to eachother by

S�X� = �X� �∀X ∈ Ob�GSet�� = Ob��

+ = �T� �∀ ∈ GSet��+�X� Y��

• = �N� �∀ ∈ GSet��•�X� Y��

�∗ = �R�� ∀� ∈ GSet��X� Y��

For any semi-Tambara functors S1, S2 and corresponding object 1, 2 in�� Set��0, their morphisms � = ��X � S1�X�→ S2�X��X∈Ob�GSet�� in �Tam��+��•�and � = ��X � 1�X�→ 2�X��X∈Ob�� in �� Set��0 correspond in the obvious way.Details can be checked by using (the proof of) Theorem 3.12. �

By this theorem, we identify semi-Tambara functors and their morphisms withthe objects and morphisms in �� Set��0.

5. FROM SEMI-TAMBARA FUNCTORS TO TAMBARA FUNCTORS

Throughout this section, let ��•� be a Tambara system as inProposition 4.5. As in the case of a finite group [7], we construct a Tambara functor�S from a semi-Tambara functor.

We first recall the definition and some properties of algebraic maps from [3].

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Definition 5.1 (Section 5.6. in [3]). Let A be an additive monoid, M be an abeliangroup, and � � A→ M be a map. For any n elements a1� a2� � � � � an ∈ A, a mapD�a1�a2��an�

� � A→ M is defined by

D�a1��an��x� =

n∑k=0

( ∑1≤i1<···<ik≤n

�−1�n−k��x + ai1 + · · · + aik�

)�

When n = 1, we abbreviate D�a�� to Da�. Remark that Da��x� = ��x + a�− ��x�.The map � is said to be algebraic if either � ≡ 0, or there exists a positive integer nsuch that

D�a1��an�� ≡ 0 �∀a1� � � � � an ∈ A��

For any algebraic map � � A→ M , its degree is defined by

deg� �= max�n ∈ ≥0 � ∃a1� � � � � an ∈ A such that D�a1��an�� ≡ 0

if � �≡ 0, and deg� �= −1 if � ≡ 0.

By this definition, � is algebraic of degree ≤ n if and only if Da� is algebraicof degree ≤ n− 1 for all a ∈ A.

Remark 5.2.

(i) D�a1��an�� does not depend on the order of a1� � � � � an.

(ii) � is algebraic of degree ≤ 0 if and only if � is constant.(iii) � is algebraic of degree ≤ 1 if and only if � is a sum of a constant map and an

additive homomorphism.

Especially, an additive homomorphism is an algebraic map of degree ≤ 1.

Proposition 5.3 (Lemma 5.6.15 in [3]). Let � � A→ M be an algebraic map froman additive monoid A to an abelian group M . Let �A � A→ K0A be the groupcompletion of A. Then there exists a unique extension � � K0A→ M of � as analgebraic map, i.e., a unique algebraic map � such that � � �A = �.

Moreover, this � satisfies deg � = deg�.

From this proposition and Lemma 5.6.13 in [3], we have the following remark.

Remark 5.4. Let A be an additive monoid, andM�N be abelian groups. If � � A→M and ! � M → N are algebraic maps of degree m and n, respectively, then ! � � �A→ N becomes algebraic of degree ≤ mn.

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For the later use, we want to use the word “algebraic map” also in the casewhere the codomain is an additive monoid. So, we make the following definition.

Definition 5.5. Let � � A→ B be a map between additive monoids A and B. Wesay � is algebraic of degree n if �B � � � A→ K0B is algebraic of degree n, where�B � B→ K0B is the group completion map.

Remark 5.6. Let A be a semi-ring. For any element a ∈ A, the multiplicationmap by a

is algebraic of degree ≤ 1, with respect to the additive structure of A.

Proof. This follows from Remark 5.2(iii). �

Proposition 5.7. Let � � A→ B be an algebraic map between additive monoids Aand B. Then there exists a unique extension � � K0A→ K0B of � as an algebraic map,i.e., a unique algebraic map � such that � � �A = �B � �.

Moreover, this � satisfies deg � = deg�.

Proof. By Proposition 5.3, there exists a unique algebraic map ˜��B � �� K0A→K0B such that ˜��B � �� A = �B � �.

If we abbreviate ˜��B � �� to �, then this � satisfies the desired conditions. �

Remark 5.8. Let � be a semi-Tambara functor on ��•�.

(i) For any ∈ GSet��+�X� Y�, + � S�X�→ S�Y� has an extension

+ � K0S�X�→ K0S�Y�

as an algebraic map, which is also an additive homomorphism.

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(ii) For any � ∈ GSet��X� Y�, �∗ � S�Y�→ S�X� has an extension

�∗ � K0S�Y�→ K0S�X�

as an algebraic map, which is also a semi-ring homomorphism.

Proof. Indeed, the natural extensions + = K0�+� and �+ = K0��∗� give the

desired maps, where K0 denotes the functor from the category of semi-rings to thecategory of commutative rings, defined by the completion of semi-rings. �

We want to extend the maps • ( ∈ GSet��•�X� Y�).

Remark 5.9. Let A�B�C be additive monoids. If � � A→ B and ! � B→ C arealgebraic maps of degree m and n respectively, then ! � � � A→ C becomesalgebraic of degree ≤ mn.

Proof. Consider the diagram

where ! � K0B→ K0C is the unique extension of ! as an algebraic map andsatisfies deg ! = deg! = n. Since by definition �B � � is algebraic of degree ≤ m, thecomposition ! � ��B � �� becomes algebraic of degree ≤ mn by Remark 5.4. Thus�C � �! � �� = ! � �B � � is algebraic of degree ≤ mn, i.e. ! � � � A→ C is algebraicof degree ≤ mn. �

Proposition 5.10. Let A�B be semi-rings, let � � A→ B be an algebraic map withrespect to the additive structures of A and B, and let � � K0A→ K0B denote the uniqueextension of � as an algebraic map. If � is moreover a multiplicative homomorphism,then � also becomes multiplicative.

Proof. For any � ∈ K0A, consider the maps

and

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These are algebraic by Remark 5.6 and Remark 5.9. If � = a ∈ A, then each of �aand �a is an extension as an algebraic map of

which is also algebraic. By the uniqueness of the extension, we obtain �a = �a, i.e.,

��ax� = ��a��x� �∀a ∈ A� x ∈ K0A�� (5.1)

Then for any � ∈ K0A, each of the maps

�� K0A→ K0B

is an extension as an algebraic map of

which is also algebraic. Thus by the uniqueness, we obtain �� = ��, i.e.,

��x� = ��x� �∀�� x ∈ K0A��

This means � is multiplicative. �

Let f ∈ GSet��X� Y� be a morphism with finite fibers and assume Y consists ofone orbit. We define the degree of f by deg f = �f−1�y�. Here y is a point in Y , anddeg f does not depend on the choice of y ∈ Y .

Proposition 5.11. Let S be a Tambara functor on ��•�, and let X� Y ∈Ob�GSet��, ∈G Set��•�X� Y�. If Y consists of one orbit, then • � S�X�→ S�Y� isalgebraic, and satisfies deg�•� ≤ deg��.

Proof. Let

be the T-diagram of as in Definition 4.9. Put Vk �= ��y� C� ∈ V � �C = k �k ∈ ≥0�.If we put deg = n, we have Vk = ∅ (k > n), and

V = ∐0≤k≤n

Vk�

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Thus if �k � Vk ↪→ V denotes the inclusion, we have an isomorphism

��∗k�0≤k≤n � S�V��→ ⊕

0≤k≤nS�Vk�� (5.2)

Similarly, putUk �= ��x� C� � �C = k and let �′k � Uk ↪→ U be the inclusion �0 ≤ k ≤ n�.� � �n � Vn → Y is an isomorphism, and the inverse is

Put " �= �n � "n � Y → V and # �= �• � �∗ � S�X�→ S�V�. Let #k denote the kthcomponent of #:

#k �= �∗k � # � S�X�→ S�Vk� �0 ≤ k ≤ n��

Lemma 5.12. • = "∗ � # = "∗n � #n.

Proof of Lemma 5.12. Put �n �= ��Un � Un → Vn. Since

are the pullback diagrams, we have

• = �� n�∗−1�n•�� ′n�∗ = "∗n�n•�� ′n�∗= "∗n�

∗n�•�

∗ = "∗n�∗n#

= "∗n#n = "∗#��

By this lemma, to show that • is algebraic of degree ≤ n, it suffices to show#n is algebraic of degree ≤ n. (Remark that "∗n is algebraic of degree ≤ 1.) In fact,we show the following lemma by induction.

Lemma 5.13. #k is algebraic of degree ≤ k (0 ≤ ∀k ≤ n).

Since

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is a pullback diagram, we have

#0�x� = �∗0�•�∗�x� = 1 �∀x ∈ S�X�� (5.3)

Assume #s is algebraic of degree ≤ s (∀s ≤ k− 1). The T - and F -diagrams of � arenaturally isomorphic to

respectively, where

V �2� �= ��y� C1� C2� � y ∈ Y�C1� C2 ⊂ −1�y�� C1 ∩ C2 = ∅ U�2�1 �= ��x� C1� C2� � ��x�� C1� C2� ∈ V �2�� x ∈ C1

U�2�2 �= ��x� C1� C2� � ��x�� C1� C2� ∈ V �2�� x ∈ C2

$i�x� C1� C2� �= ��x�� C1� C2� �i = 1� 2�

%i�x� C1� C2� �= �x� C1 � C2� �i = 1� 2�

&�y� C1� C2� �= �y� C1 � C2��

By Proposition 4.12, we have

�•�u1 + u2� = &+�$1•%∗1�u1� · $2•%∗2�u2�� ∀�u1� u2 ∈ S�U��

If we put

�i � V�2� → V� �i�y� C1� C2� = �y� Ci� �i = 1� 2�

�i � U�2�i → U� �i�x� C1� C2� = �x� Ci� �i = 1� 2��

then

is a pullback diagram for each i = 1� 2. So for any x ∈ S�X�, we have

$i•%∗i ��

∗�x�� = $i•�∗i ��

∗�x�� = �∗i �•�∗�x� = �∗i #�x� �i = 1� 2��

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So, we obtain

#�x + a� = �•��∗�x�+ �∗�a�� = &+��

∗1#�x� · �∗2#�a�� ∀x� a ∈ S�X��

For any integers 0 ≤ s� t ≤ k with s + t = k, put

V�2�s�t �= ��y� C1� C2� ∈ V �2� � �C1 = s� �C2 = t

&s�t � V�2�s�t → Vk� �y� C1� C2� �→ �y� C1 � C2�

��2�s�t � V

�2�s�t ↪→ V �2�� inclusion�

Then since

is a pullback diagram, we have

�∗k&+��∗1#�x� · �∗2#�a�� =

∑s+t=k

�&s�t�+��2�s�t �

∗��∗1#�x� · �∗2#�a��

= ∑s+t=k

�&s�t�+��1�s�t�∗#s�x� · ��2�s�t�∗#t�a��

Here �1�s�t and �2�s�t are defined by

�1�s�t � V�2�s�t → Vs� �y� C1� C2� �→ �y� C1��

�2�s�t � V�2�s�t → Vt� �y� C1� C2� �→ �y� C2��

Thus we obtain

#k�x + a� = ∑s+t=k

�&s�t�+��1�s�t�∗#s�x� · ��2�s�t�∗#t�a��

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Since �1�k�0 = &k�0 is an isomorphism, we have

�&k�0�+��1�k�0�∗#k�x� · ��2�k�0�∗#0�a�� =

�5�3��&k�0�+��&k�0�

∗#k�x� · 1�= �&k�0�+�&k�0�

∗#k�x� =Prop 2.16

#k�x��

So for any a ∈ S�X� we have

Da#k�x� = #k�x + a�− #k�x�

=k−1∑s=0

�&s�k−s�+��1�s�k−s�∗#s�x� · Ak�s��

where

Ak�s �= ��2�s�k−s�∗#k−s�a� ∈ S�V �2�s�k−s�

is a constant independent of x for each 0 ≤ s ≤ k. By assumption #s is algebraic ofdegree ≤ s (0 ≤ ∀s ≤ k− 1). So Da#k becomes algebraic of degree ≤ k− 1, i.e., #k isalgebraic of degree ≤ k. �

Corollary 5.14. For any X� Y ∈ Ob�GSet�� and any ∈G Set��•�X� Y�, there existsa unique extension • � K0S�X�→ K0S�Y� of • such that:

(i) for any orbit Y� in Y , j∗� � • is the extension of j∗� � • as an algebraic map, where

j� � Y� ↪→ Y is the inclusion.

K0�S�X��•→ K0�S�Y��

j∗�→ K0�S�Y��

(ii) • is a homomorphism of multiplicative monoids.

Proof. Let Y = ∐�∈� Y� be the orbit decomposition and j� � Y� ↪→ Y �� ∈ �� be

the inclusions. For any � ∈ �, since •�� = j∗� � • is algebraic by Proposition 5.11(via the Mackey axiom), there exists a unique extension •�� K0S�X�→ K0S�Y�� of•�� as an algebraic map. Moreover, since •�� is a homomorphism of multiplicativemonoids, so is •��.

By the isomorphism

�j∗� ��∈� � K0S�Y��→ ∏

�∈�K0S�Y�� = K0�

∏�∈�

S�Y��

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we define • �= �j∗� �−1�∈� � �•��∈�.

This satisfies (i) and (ii), and • is unique by the uniqueness of •�� ∈ ��. �

Remark that for any finite subset �′ ⊂ �, if we put Y ′ �= ∐�∈�′ Y� ⊂ Y , then

the homomorphism of multiplicative monoids j′∗ � • � S�X�→ S�Y� is algebraic,where j′ � Y ′ ↪→ Y is the inclusion. This follows from the facts:

(1) �j′∗� ��∈� � S�Y′�

�→ ∏�∈�′ S�Y��, (here, j

′� � Y� ↪→ Y ′);

(2) j′∗� � j′∗ � • = j∗� � • is algebraic for any � ∈ �′; and(3) Finite product of algebraic maps is algebraic.

By the same reason, j′∗ � • � K0S�X�→ K0S�Y′� is also algebraic, and thus we

have the following lemma.

Lemma 5.15. Let • be the map constructed in Corollary 5.14. For any Y ′ ↪→ Y asabove, j′∗ � • � K0S�X�→ K0S�Y

′� is the unique extension of j′∗ � • as an algebraicmap.

Theorem 5.16. Let S be a semi-Tambara functor on ��•�. With ��S��X� =K0S�X� and +, •, �∗ constructed above, �S becomes a Tambara functor.

Proof. Conditions (i)–(iii) in Definition 4.10 concerning +, • are satisfied byRemark 2.15. So it suffices to show the following:

(A) For any ∈ GSet��•�X� Y� and ' ∈ GSet��•�Y� Z�, we have

˜�' � �• = '• � • and ˜�idX�• = idK0�S�X��

(B) If

is a pullback diagram in GSet�, where is a morphism in GSet��• , then �∗ � • =

′• � �′∗.

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(C) If

is an exponential diagram, then • � + = �+ � ′• � �∗.Proof of (A). By the direct product decomposition of S�Z�, it suffices to show forany orbit % � Z′ ↪→ Z�

%∗ � ˜�' � �• = %∗ � �'• � •��

Remark that %∗ � �' � �• = %∗ � �'• � •� is algebraic by Proposition 5.11 (via theMackey axiom). If we take the pullback of ' and %

then (since ' ∈ GSet��•�Y� Z�, ) Y′ = '−1�Z′� consists of finite number of orbits.

By the definition of '•, we have

%∗ � '• = '′• � %′∗� (5.4)

By Lemma 5.15, %∗ � ˜�' � �• is the extension of %∗ � �' � �• as an algebraicmap. On the other hand, by Lemma 5.15 (and Proposition 5.11), %′∗ � • and '′• arethe extensions of %′∗ � • and '′

• as algebraic maps, and thus '′• � �%′∗ � •� =�5�4�

%∗ �'• � • is the extension of %∗ � '• � • as an algebraic map. So, by the uniquenessof the extension, we obtain

%∗ � ˜�' � �• = %∗ � �'• � •��

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˜�idX�• = idK0�S�X��can be shown in the same way, or directly from the definition

of ˜�idX�•.Proof of (B). It suffices to show for any orbit j′0 � Y

′0 ↪→ Y ′,

j′∗0 � ��∗ � •� = j′∗0 � �′• � �′∗��

Remark that j′∗0 � ��∗ � •� = j′∗0 � �′• � �′∗� is algebraic by Proposition 5.11. There isan orbit j0 � Y0 ↪→ Y containing the image of Y ′

0 by �. We put �0 �= � �Y ′0� Y ′

0 → Y0and take the fiber product of ′ and j′0:

Then j′∗0 � ′• = ′0• � i′∗0 by (5.4). Since Y ′0 consists of one orbit, the map ′0• is

algebraic. Thus j′∗0 � ′• � �′∗ = ′0• � i′∗0 � �′∗ is algebraic. On the other hand, sincej∗0 � • is algebraic by Lemma 5.15, the map j′∗0 � �∗ � • = �∗0 � j∗0 � • is algebraic.

Thus, each of j′∗0 � �∗ � • and j′∗0 � ′• � �′∗ is the extension of j′∗0 � �∗ � • = j′∗0 �′• � �′∗ as an algebraic map, and we obtain j′∗0 � ��∗ � •� = j′∗0 � �′• � �′∗� by theuniqueness of the extension.

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Proof of (C). This can be shown in the same way as (A) and (B). For any orbitj � Y0 ↪→ Y , if we pull it back by �

then Y ′0 consists of finite number of orbits, and j∗ � �+ � ′• � �∗ = �0+ � j′∗ � ′• � �∗

becomes algebraic since j′∗ � ′• is algebraic by Lemma 5.15. Thus j∗ � • � + andj∗ � �+ � ′• � �∗ are extensions of j∗ � • � + = j∗ � �+ � ′• � �∗ as algebraic maps,and must agree by the uniqueness of the extension. �

Thus we obtained a Tambara functor �S. By the construction of �S, theset of completion maps � = ��X � S�X�→ K0S�X��X∈Ob�GSet�� gives a morphism ofsemi-Tambara functors � � S → �S.

Proposition 5.17. Let S be a semi-Tambara functor and T be a Tambara functor. Forany morphism � � S → T of semi-Tambara functors, there exists a unique morphism ofTambara functors � � �S → T such that � � � = �.

Proof. � is obviously unique, since the morphism of rings �X � K0S�X�→ T�X�satisfying �X � �X = �X is unique for each X ∈ Ob�GSet��, i.e., �X = K0��X�.So it suffices to show that ��X�X∈Ob�GSet�� is compatible with all +� •� �∗. Since��X�X∈Ob�GSet�� is indeed compatible with + = K0�+� and �∗ = K0��

∗�, it remains toshow the compatibility with •. But this immediately follows from the fact that foreach orbit j � Y ′ ↪→ Y , the following diagram is commutative:

�

By virtue of Theorem 5.16 and Proposition 5.17, with the same proof as inthe case of a finite group [7], we can show the following theorem. By Theorem 4.17,

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we identify �Tam��+��•� with �� Set��0. For example, for any X ∈ Ob��,��X�−� is regarded as a semi-Tambara functor. Remark also that if ∈Ob�� Set��0� is ring-valued, can be regarded as a Tambara functor.

Theorem 5.18. There exists a unique pair � � �� of a category with arbitraryproducts and a functor � � � → such that the following conditions are satisfied:

(i) Ob�� = Ob� �;(ii) �X� Y� = K0��X� Y�, where K0 denotes the Grothendieck ring of the semi-ring

��X� Y�, and any object in is a ring object;(iii) � is identity on objects, and for any X� Y ∈ Ob��, the component of �

�X�Y � ��X� Y�→ �X� Y�

is the completion map.(iv) � preserves arbitrary products.

Proof. Put Ob� � �= Ob��. For any X ∈ Ob� �, define a Tambara functorTX by

TX �= ��X�−��(Remark that the Hom-functor ��X�−� is a semi-Tambara functor.) As alreadyshown, there exists a morphism of semi-Tambara functors

�X � ��X�−�→ TX�

which satisfies the universality of Proposition 5.17. We define the morphism of by

�X� Y� �= TX�Y� = K0��X� Y�

for any X� Y ∈ Ob� �. For each � ∈ �X� Y�, by Yoneda’s lemma

TX�Y��−→ �Tam��Y�−�� TX��

there exists a corresponding morphism �( ∈ �Tam��Y�−�� TX�. This �( is theunique morphism satisfying

�(Y �idY � = �� (5.5)

By Proposition 5.17, there exists a unique morphism of Tambara functors�( � T Y → TX such that �( � �Y = �(.

(5.6)

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We define the composition law

�X� Y�× �Y� Z� −→ �X� Z�

∈ ∈

�� −→ � � �

by � � � �= �(Z��. The identity morphism in �X�X� is �XX�idX�.

Associativity of the Composition. For any � ∈ �X� Y� and � ∈ �Y� Z�, thereexists a commutative diagram

where the vertical arrows are Yoneda isomorphisms. So we have

�( � �( = (�(Z��

)(� ��Z�−�→ TX�

Thus, for any � ∈ �Z�W�,

� � �� = ˜(� � �)(

W�� = ˜(

�(Z��)(W�� = ˜(

��( � �()W��

while

�� = �(W �� = �(W ��(W ��

Since �( � �( makes the following diagram commutative:

it must agree with˜

��( � �(� by the uniqueness. Thus we obtain

� � �� = ˜(�( � �()

W�� = �(W

(�(W ��

) = �� Unit Law. For any � ∈ �X� Y�, we have

�YY �idY � � � = �(Y(�YY �idY �

) =�5�6�

�(Y �idY � =�5�5�

��

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On the other hand, by the uniqueness of the morphism satisfying (5.5), we have��XX�idX��

( = �X . Since �X = idTX , it follows that

� � �XX�idX� = ˜�XX�idX�(�� = ��

Thus becomes in fact a category.Functor � � � → is defined by

��X� = X

�X�Y �= �XY � ��X� Y�→ �X� Y� = TX�Y� �∀X� Y ∈ Ob��

Remark that �X�Y is the completion map of the semi-ring ��X� Y�. We show that �is in fact a functor. Obviously, � preserves identities. So, it remains to show

�� = �� ∀� ∈ ��X� Y��∀� ∈ ��Y� Z��

Since �X � ��X�−�→ TX is a natural transformation, we have

By the naturality of the Yoneda isomorphisms, we have

By the definition of ˜��XY ��(, we have

So, we have

�� = �YZ�� XY �� = ˜��XY ��(Z��

YZ��

= ��XY ��(Z�� = ��XY ��

(Z�� idY �

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= TX�� XY ��(Y �idY � =�5�5�

TX��XY ��

= �XZ�� = ��

Thus � � � → is in fact a functor. Remark that this � � �� satisfiesconditions (i)–(iii) in the statement of the theorem. To show (iv), let

�p� � X ⇀ X��∈�

be a product in �. This is equivalent to the fact that

�p� � −��∈� � ��A�X�→∏�∈�

��A�X��

is an isomorphism for each A ∈ Ob��. Since we have a commutative diagram

for each � ∈ �, thus we have

Thus

��p�� −��∈� � �A�X�→∏�∈�

�A�X��

is an isomorphism, which means that

��p�� X ⇀ X��∈�

is a product in . Thus � � �� satisfies all of the conditions in the statement of thetheorem. If any other � ′� �′� satisfies these conditions, then by Proposition 5.17,

for each X ∈ Ob� � there exists a unique isomorphism �′X � TX�−→ ′�X� �′�−��

which satisfies

�′X � �X = �′X

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By using (the uniqueness of) the universality, we can show that the followingdiagram is commutative for any � ∈ �X′� X� and Y ∈ Ob� �:

(See also the proof of Claim 5.21.) From this, we can see that �′X�Y �X� Y ∈ Ob� ��constitute an isomorphism �′ �

�→ ′, compatible with � and �′. �

In the proof of Theorem 5.18, the following remark was shown.

Remark 5.19. For any set of objects �X��∈� in , their product in can bewritten in the form

��p�� X ⇀ X��∈��

where

�p� � X ⇀ X��∈�

is the product of �X��∈� in �.

Theorem 5.20. Let ��•� be a Tambara system as in Proposition 4.5. There isan equivalence of categories

Tam��•��−→ � � �Set��0�

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compatible with the isomorphism of Theorem 4.17, i.e., the following diagram iscommutative:

where �� denotes the pullback by �.

Proof. Let � � �� Set��0�−→ �Tam��•� be the isomorphism in Theorem 4.17.

As in Theorem 3.14, there exists a functor � � � � �Set��0 → Tam��•�, whichmakes the following diagram commutative:

In the following, we show � is an equivalence. We use the notation in the proof ofTheorem 5.18.

To show � is fully faithful, it suffices to show �� is fully faithful. So for each 1� 2 ∈ Ob�� Set��0�, we show the map

� � �Set��0� 1� 2� −→ �� Set��0�� 1� �

� 2�

∈ ∈

� = ��X�X∈Ob� � �−→ � = ��X��X∈Ob��

(5.7)

is bijective. Injectivity is trivial. To show the surjectivity, take an element! ∈ �� Set��0�� 1� �

� 2�. Remark that

iX�Y � �X� Y�→ �Set�� i�X�� i�Y�� i = 1� 2�

is a ring homomorphism (cf. Proposition 3.10).For any � ∈ �X� Y�, since �X� Y� = K0��X� Y�, there are a� b ∈ ��X� Y�

such that � = �XY �a�− �XY �b�. Thus we have

2�� !X = ( 2

(�XY �a�

)− 2

(�XY �b�

)) � !X= �� 2�a� � !X − �� 2�b� � !X= !Y � �� 1�a�− !Y � �� 1�b�

= !Y �( 1��

)�

which means ! ∈ � � �Set��0� 1� 2�. Thus (5.7) is bijective, and �� becomes fullyfaithful.

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To show that � is essentially surjective, for any T ∈ Ob�Tam��•��, take thecorresponding object ∈ Ob�� Set��0� such that T = �� . It suffices to show theexistence of ∈ Ob�� Set��0� such that �� = . For each X ∈ Ob��, we havea morphism of semi-Tambara functors

X � ��X�−�→ �Set�� X�� −��

induced from . Remark that �Set�� X�� −�� is a Tambara functor (under theidentification by �). So by Proposition 5.17, we obtain a unique morphism ofTambara functors

X � TX → �Set�� X�� −��

such that

(5.8)

Claim 5.21. For any � ∈ �X� Y� = TX�Y�,

(5.9)

is commutative.

Proof. Since

��Y�−� �(−→ TX X−→ �Set�� X�� −��

are morphisms in �� Set��0, for any a ∈ ��Y� Z� we have a commutative diagram

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In particular, we have

�a� � XY �� = �a� � XY(�(Y �idY �

)= XZ

(�(Z�a�

)= XZ

(�(Z

(�YZ�a�

))�

Thus the following diagram in �� Set��0 is commutative:

Since Y � �Y = Y , we obtain the commutativity of (5.9) by the universality of �Y .�

By Claim 5.21, we can easily show

Y Z�� XY �� = XZ�� ∀� ∈ �X� Y��∀� ∈ �Y� Z��

�and XX��XX�idX�� = id by definition��

Thus if we define by

�X� �= �X� �∀X ∈ Ob� �� = XY �� ∀� ∈ �X� Y��

then becomes a functor from to �Set�. By (5.8), satisfies

� � = �

In particular, by Remark 5.19 (and the fact that preserves arbitraryproducts), it follows that also preserves arbitrary products. Thus we obtain ∈ Ob�� Set��0� satisfying �� = . This is what we wanted to show. �

As a corollary of Theorem 5.20, there exists a Tambara functor ��•� onany Tambara system ��•� as in Proposition 4.5, corresponding to the Hom-functor

�∅�−� ∈ � � �Set��0�

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For any X ∈ Ob� � = Ob�GSet��, we have

��•��X� = �∅� X� = K0��∅� X��

Since

��∅� X� = {(∅ ← ∅ �∅→ B→ X

) �B ∈ Ob�GSet�� ∈ GSet�� B�X�}/ ∼

equiv.

� c%�GSet��/X��

we obtain ��•��X� = K0�GSet��/X�.

Definition 5.22. We call ��•� the Burnside Tambara functor on the Tambarasystem ��•�.

Remark 5.23. Definition 5.22 gives a Tambara functor structure on Burnsidefunctor ��

defined in Definition 3.16.

For the rest of this section, we show that Elliott’s functor VM becomes aTambara functor on a profinite group G. According to Elliott, an almost finite G-setis a discrete G-space which satisfies

�U�X� < �

for any open subgroup U ≤ G, where �U�X� is defined by �U�X� = ��XU� = ��a ∈X � u · a = a �∀u ∈ U� . For any multiplicative monoid M , an M-valued G-set isan almost finite G-set with a map #X � X → M . M-valued G-sets form a categoryM-AlmG with finite products and finite sums as defined in [4], whose Grothendieckcategory is denoted by VM�G�. If M = 0, then V0�G� agrees with the completedBurnside ring defined in [3]. Since these rings have Mackey-functorial properties,we can expect them to be Tambara functors. Indeed, we give a Tambara functorstructure to VM in the following.

Let ��•� be the open-natural Tambara system on G (see Example 4.6).Remark that in this case, each of ��H��H��•�H� is simply the set of all opensubgroups of H , for each H ∈ �. For any p ∈ GSet��A�X�, we say p has almostfinite fibers when p satisfies

�U�p−1�x�� < � �∀x ∈ X�∀U ∈ ��Gx��

Morphisms Ap→ X with almost finite fibers form a full subcategory Alm/X of

GSet�/X. It can be easily seen that, for each X, Alm/X has finite products (i.e. fiberproducts over X) and finite sums.

For any multiplicative monoid M , a category M-Alm/X is defined as follows:

a) An object of M-Alm/X is a pair �Ap→ X� #A�, where �A

p→ X� is an object inOb�Alm/X� and #A � A→ M is a map of sets.

b) A morphism from �Ap→ X� #A� to �B

q→ X� #B� is a morphism f � �Ap→ X�→

�Bq→ X� in Alm/X which satisfies #B � f = #A.

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Then M-Alm/X also has finite products and finite sums, defined by(A

p→ X� #A)× (

Bq→ X� #B

)�= (

A×X B→ X� #A · #B)�(

Ap→ X� #A

)∐(B

q→ X� #B)�= (

A∐B→ X� #A ∪ #B

)�

where �#A · #B��a� b� �= #�a� · #�b� for any �a� b� ∈ A×X B.Remark that when X is a transitive G-set X = G/H (H ∈ �), then M-Alm/X

is equivalent to the category M-AlmH of M-valued H-sets. Thus we have VM�H� �K0�M-Alm/�G/H��. So, we define abbreviately

VM�X� �= K0�M-Alm/X�

for any X ∈ Ob�GSet��.

Theorem 5.24. VM is a Tambara functor on ��•�.

Proof. By Theorem 5.16, it suffices to show that the correspondence X �→ S�X� �=c%�M-Alm/X� is a semi-Tambara functor.

First, we describe the structure maps of S. Let �Ap→ X� #A� be an object in

M-Alm/X, and �Bq→ Y� #B� be an object in M-Alm/Y .

a) For any ∈ GSet�� X� Y�, define + � M-Alm/X → M-Alm/Y by +�Ap→

X� #A� = �A�p→ Y� #A�.

b) For any ∈ GSet��•�X� Y�, define • � M-Alm/X → M-Alm/Y by •�Ap→

X� #A� = ��A��=��p�−→ Y� #�A��, where #�A�� is defined by

#�A��y� �� =∏

x∈−1�y�

#A��x��

c) For any � ∈ GSet��X� Y�, define �∗ � M-Alm/Y → M-Alm/X by �∗�B

q→ Y� #B� =�B′ q′→ X� #B � �′�, where

is a pullback diagram.

These are well defined, i.e., we have

(A

�p→ Y� #A) ∈ Ob�M-Alm/Y��

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(��A�

�→ Y� #�A��) ∈ Ob�M-Alm/Y��(

B′ q′→ X� #B � �′) ∈ Ob�M-Alm/X��

for each � � � as above.We only show ��A�

�→ Y� ∈ Ob�Alm/Y�. The rest can be shown easily. Forany y ∈ Y , put −1�y� = �x1� � � � � xn . Then, �

−1�y� is bijective to∏

1≤i≤n p−1�xi� bythe map

By this bijection, we can give a Gy-action on∏

1≤i≤n p−1�xi�. When g ∈ Gxi�≤ Gy�,

this action satisfies

�g · �a1� � � � � an��i = g · aifor any �a1� � � � � an� ∈

∏1≤i≤n p−1�xi�, where the left-hand side denotes the ith

component of g · �a1� � � � � an�. Thus, under this bijection, we have

�−1�y�U ⊆ p−1�x1�× · · · × p−1�xi−1�× p−1�xi�U × p−1�xi+1�× · · · × p−1�xn�

for any U ∈ ��Gxi�.

Thus for any V ∈ ��Gy�, we have

�−1�y�V ⊆ ⋂1≤i≤n

p−1�x1�× · · · × p−1�xi�V∩Gxi × · · · × p−1�xn�

= ∏1≤i≤n

p−1�xi�V∩Gxi �

since V ∩Gxi∈ ��Gxi

�. So we obtain

�V ��−1�y�� = ��−1�y�V �

≤ ∏1≤i≤n

��p−1�xi�V∩Gxi �

= ∏1≤i≤n

�V∩Gxi �p−1�xi��

< ��

since �V∩Gxi �p−1�xi�� < � for each i.

We confirm the conditions in Definition 4.10. By definition, S�X� is a semi-ring. Since + preserves finite sums, the induced map + � S�X�→ S�Y� becomesan additive homomorphism. Similarly �∗ becomes a semi-ring homomorphism. Wedemonstrate how • becomes a multiplicative homomorphism. Let �A

p→ X� #A��

�Bq→ X� #B� be two elements in M-Alm/X. We show that ��A�×Y ��B�→

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3150 NAKAOKA

Y� #�A�� · #�B�� is isomorphic to ��A×X B�→ Y� #�A×XB��. Remark that#A×XB = #A · #B. Since there is an isomorphism

in Alm/Y , it suffices to show #�A×XB�� f = #�A�� · #�B��. This is satisfied, since

#�A×XB�� f��y� �� y� �� =∏

x∈−1�y�

�#A · #B�� x��

= ∏x∈−1�y�

#A��x��#B��x��

= ∏x∈−1�y�

#A��x��∏

x∈−1�y�

#B��x��

= #�A��y� �� · #�B��y� ��= #�A�� · #�B��y� �� y� ��

Since conditions (i)–(iv) in Definition 4.10 are shown in almost the same way,we only show condition (v).

Let

be an exponential diagram. For any object �Ap→ Z� #A� in M-Alm/Z, we have

• � +��Ap→ Z� #A�� =

(��A�

��p�−→ Y� #�A��)�

��+ � ′• � �∗��Ap→ Z� #A�� = ��+ � ′•

((A′ p′→ X′� #A′ = #A � �′

))= (

�′�A′��′ �p′�−→ Y� #�A′�′�

)�

where

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is a pullback diagram. So it suffices to construct an isomorphism between

��A��p�−→ Y� #�A�� and ��′�A

′��′ �p′�−→ Y� #�A′�′��.

Let �y′� �′� be an element of �′�A′�, where y′ ∈ Y ′ and �′ is a map from

′−1�y′� to A′. If we put y �= ��y′� then ′−1�y′� = −1�y�× �y′ , and we define

� ∈ Map�−1�y�� A� by

��x� = �′ � �′��x� y′�� ∀x ∈ −1�y��

In this notation, we define f � �′�A′�→ ��A� by f��y′� �′�� = �y� ��. It is easily

seen that this becomes an isomorphism in Alm/Y . It remains to show

#�A�� f = #�A′�′�� (5.10)

For any �y′� �′� ∈ �′�A′�, the left-hand side can be calculated as

#�A�� f�y′� �′� = #�A��y� �� =∏

x∈−1�y�

#A��x��

= ∏x∈−1�y�

#A��′ � �′�x� y′��

while the right-hand side is

#�A′�′��y′� �′� = ∏

x′∈′−1�y′�#A′��′�x′�� = ∏

x∈−1�y�

#A′��′�x� y′��

= ∏x∈−1�y�

#A � �′��′�x� y′��

Thus �5�10� is satisfied, and VM becomes a Tambara functor on ��•�. �

ACKNOWLEDGMENTS

The author wishes to thank Professor Toshiyuki Katsura for hisencouragement. The author is supported by JSPS.

REFERENCES

[1] Bley, W., Boltje, R. (2004). Cohomological Mackey functors in number theory.J. Number Theory 105:1–37.

[2] Brun, M. (2005). Witt vectors and Tambara functors. Adv. Math. 193:233–256.[3] Dress, A. W. M., Siebeneicher, C. (1988). The Burnside ring of profinite groups and

the Witt vector construction. Adv. Math. 70:87–132.[4] Elliott, J. (2006). Constructing Witt–Burnside rings. Adv. Math. 203:319–363.[5] Lindner, H. (1976). A remark on Mackey-functors. Manuscripta Math. 18:273–278.[6] Mac Lane, S. (1998). Categories for the Working Mathematician. 2nd ed. Graduate Texts

in Mathematics, 5. New York: Springer.[7] Tambara, D. (1993). On multiplicative transfer. Comm. Algebra 21(4):1393–1420.[8] Yoshida, T. (1990). The generalized Burnside ring of a finite group. Hokkaido Math. J.

19:509–574.

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