맥키 퍼터의 탬바라이제이션과 위트‑번사이드 구성의 응용

T AMBARIZA TION OF A MA CKEY FUNCTOR AND ITS APPLICA TION TO THE WITT-BURNSIDE CO N STR UCTION HIRO YUKI NAKAOKA Abstract. F or an arbitrary group G , a ( semi -) Mackey functor is a pair of co v ariant and con tra v ariant functors fr om the category of G -sets, and is r e- garded as a G -biv ari an t analog of a commutat ive (semi- )group. In this view, a G -biv ar ian t analog of a (semi-)ri ng should b e a (semi- )T am bara f unctor. A T am bara functor is firstly defined by T am bara, which he called a TN R-functor, when G is finite. As shown by Brun, a T amb ara functor plays a natural role in the Witt-Burnside construction. It wil l b e a natural question if there exist sufficiently many examples of T am bara functors, compared to the wide range of Mack ey functors. In the first part of this article, we give a general construction of a T am bara functor from an y Mac ke y f unctor, on an ar bi trary group G . In fact, w e construct a functor from the category of semi-Mack ey functors to the cate gory o f T am bara functors. This functor giv es a left adjoint to the forgetful functor, and can be regarded as a G -biv ariant analog o f the monoid-ring functor. In the latter part, when G is finite, we invsetigate relations wi th other Mac ke y-functorial constructions — crossed Burnsi de ring, Elliott’s r ing of G - strings, Jacobson’s F -Burnside ring— all the se lead to the study of the Witt - Burnside construction. 1. Introduction and Preliminaries F or a n arbitra ry g roup G , a (semi-)Mack ey functor is a pair of cov ariant and contra v ar iant functors from the ca tegory of ‘ G -sets’. More precis ely , as defined in [1], if we a re given a Mackey system ( C , O ) on G , then subca teg ories o f the category of G -sets G Set C and G Set C , O are ass o ciated, and a Mackey functor is a pair M = ( M ∗ , M ∗ ) of a cov ariant functor M ∗ : G Set C , O → Ab and an a dditive contra v ar iant functor M ∗ : G Set C → Ab . If G is finite, we obtain an ordinary (semi-)Mack ey functor on G as in Remark 1.6. As in [15], for any finite group G , a (semi-)Mack ey functor can b e regarded a s a G -biv aria nt analo g of a commutativ e (semi-)gr o up. In fact if G is triv ia l, the category of (semi-)Mack ey functors are ca no nically eq uiv alent to the category of commutativ e (semi-)groups . In this view, a G -biv ariant analog of a (semi-)ring will be a (semi-)T am bar a functor. A T am bara functor is firstly defined by T a mb ara in [13], which he ca lled a TNR-functor, for any finite group G . As shown by Br un, a T ambara functor plays a na tural role in the Witt-Burnside construction ([4]). (See F act 3.3 in this article.) The author wishes to thank Professor T oshiyuki Katsura f or his enco uragemen t. The author wis hes to thank Professor F umihito Oda for stimulat ing arguments and useful commen ts, Professor Daisuke T am bara and Professor T omo yuki Y oshida for their comment s and advices. Supported by JSPS Grant - in-Aid for Y oung Scien tists (B) 2274000 5. 1 2 HIR OYUKI NAKAOKA It will be a natural question if there exist sufficiently many examples o f T ambara functors, c o mpared to the wide rang e of Mack ey functor s. In this a r ticle, we give a general construction of a T ambara functor from any Mack ey functor, on an arbitr a ry group G . In f act, we cons tr uct a functor from the category o f semi-Mac key functor s to the category of semi-T am bar a functor s. More precisely , in Theo rem 2.12, for an y T ambara system (Definition 2.1 ) ( C , O C , O • ) on G , we cons truct a functor S : SMack ( C , O • ) → ST am ( C , O • ) , where S Mack ( C , O • ) and ST am ( C , O • ) are appro priately defined catego ries of semi- Mack ey and semi-T a mbara functors, resp ectively: Theorem 2.12 . Let ( C , O C , O • ) b e a T am ba ra sys tem on G . The construction of S M in P rop osition 2.1 1 gives a functor S : SMack ( C , O • ) → ST am ( C , O • ) . Comp osing S with other k nown functors , we obtain functors T : SMack ( C , O • ) → T am ( C , O • ) , Mack ( C , O • ) → ST am ( C , O • ) , Mack ( C , O • ) → T am ( C , O • ) , where Mack ( C , O • ) and T am ( C , O • ) are the catego ries of Ma ck ey and T am bara func- tors, resp ectively . Moreov er in Theorem 2.15 , we show S is left adjoint to the forgetful functor ST am ( C , O • ) → SMack ( C , O • ) . Theorem 2.15 . Let ( C , O C , O • ) be any T am bar a system on G . The functor constructed in Theore m 2 .12 S : SMack ( C , O • ) → ST am ( C , O • ) is left adjoint to the forgetful functor µ : ST am ( C , O • ) → SMack ( C , O • ) . As a c o rollar y , comp os ition T = γ ◦ S : SMack ( C , O • ) → T am ( C , O • ) is le ft adjoint to the forgetful functor µ ◦ U : T am ( C , O • ) → SMack ( C , O • ) (Corollar y 2.17). Thus Theorem 2.1 5 means that T can b e r e garded as a G -biv ariant analog of the monoid-ring functor . In the latter part of this ar ticle, we res trict ourselves to the finite gro up ca se, and inv estigate the connection to other constr uctions, esp ecially in relation with the Witt-Burns ide construction. Firstly in section 3.1, we show the rela tion with the cross ed Burnside ring. In fact, if we are given a G -mono id Q , then by applying Corollar y 2 .17 to the fixed po int f unctor P Q asso ciated to Q , we obtain a T ambara functor T P Q . This general- izes the crossed B urnside r ing functor in [1 1], [12]. Indeed, if Q is a finite G -monoid, we hav e an isomorphism of T ambara functors b etw een T P Q and the crossed Burnside ring functor Ω Q : T P Q ∼ = Ω Q Secondly in section 3.2 , we sho w the relationship with the Witt-Burnside co n- struction. Historically , T am bar a functor s w er e applied to the Witt-Burnside con- struction fir s tly by Brun: T AMBARIZA T ION OF A M AC KEY FUNCTOR 3 F act 3. 3 (Theorem B, Theorem 1 5 in [4]) . F or a n y finite gr oup G , the ev alua tion at G/e T am ( G ) → G - Ring ; T 7→ T ( G/e ) has a r ig ht adjoin t functor L G . Here , G - Ring denotes the categ ory of G -rings . If G acts trivia lly on a ring R , then for any subg roup H ≤ G , there is an iso morphism W H ( R ) ∼ = L G ( R )( G/H ) . Here W H ( R ) is the Witt-Burnside ring as so ciated to H and R . Combining this with Theorem 2.15, we obtain a n isomorphism of functors, which g ives a description of the Witt-Burns ide ring o f a monoid-ring : Theorem 3.9 . F or any finite g roup G , there is an iso morphism o f functors W ◦ Z [ − ] ∼ = T ◦ L . Mon SMack ( G ) Ring T am ( G ) L / / Z [ − ]   T   W / /  Here, L is the functor defined in Claim 3.8. Theorem 3.9 can be also rega rded as an enhance men t of the Elliott’s isomo rphism of rings (Theorem 1.7 in [6]) to a functorial level. Thirdly in section 3.3, r e lating our construction to Jacobs on’s F -Burns ide ring functor, we investigate the underly ing Green functor structure o f T M . Jacobso n functorially ass o ciated a Green functor A F to any additive co ntrav ariant functor F ([8]). In fact, this functor g ives the underlying Green functor s tructure of T M . Namely , we hav e an isomorphis m of Green functors T M ∼ = A M ∗ . In this view, Theorem 2.12 can b e restated as follows: Theorem 2.12 ′ . Let G be a finite group and F b e an ob ject in Madd ( G ). If F is moreov er a semi-Mack ey functor, i.e., if there exists a semi-Mack e y functor with the co n trav ariant part F , then A F has a structure of a T ambara functor. In turn, as an application of Theorem 2.12 ′ , we can endow a T am bar a functor structure o n the Brauer ring functor defined by Jaco bson (Co r ollary 3 .20). Moreov er, combining this with Theorem 3.13 in [9], the underlying Green functor structure is also written by using Boltje’s ( − ) + -construction. Namely , w e have an isomorphism of Green functors T M ∼ = ( R Z [ M ∗ ] ) + (Corollar y 3.24). Throughout this article, we fix an arbitra ry group G . Its unit elemen t is denoted by e . H ≤ G means H is a s ubgroup of G . G Set denotes the ca teg ory of (no t necessarily finite) G -sets and G -equiv ar iant maps. The catego ry of finite G -sets is denoted b y G set . F or a G -set X and a point x ∈ X , we denote the stabilizer of x b y G x . F or a ny H ≤ G and g ∈ G , g H and H g denotes the conjugation g H = g H g − 1 , H g = g − 1 H g . Monoids are a ssumed to b e commutativ e and hav e multiplicativ e units 1. (Semi-)ring s are assumed to b e co mm utative, and ha ve additive units 0 a nd m ultiplicative units 1. W e denote the ca tegory of monoids b y Mon , the category of (resp. semi-)r ings by Ring (resp. SRing ), and the ca tegory o f ab elian gro ups by A b . A monoid homomo rphism pre serves 1 , and a (se mi-)ring homomor phis m preserves 0 and 1. 4 HIR OYUKI NAKAOKA F or any category K a nd any pair of ob jects X and Y in K , the set o f morphis ms from X to Y in K is denoted b y K ( X , Y ). F or each X ∈ Ob( K ), the comma category of K ov er X is denoted b y K / X . Definition 1.1 ([1]) . A Mackey system on G is a pair ( C , O ) of (a) a set C o f subgroups of G , closed under conjugation and finite in tersections , (b) a family O = {O ( H ) } H ∈C of s ubsets O ( H ) ⊆ C ( H ) , where C ( H ) is defined by C ( H ) = { U ∈ C | U ≤ H } for each H ∈ C , which satis fie s (i) [ H : U ] < ∞ (ii) O ( U ) ⊆ O ( H ) (iii) O ( g H g − 1 ) = g O ( H ) g − 1 (iv) U ∩ V ∈ O ( V ) for all H ∈ C , U ∈ O ( H ), V ∈ C ( H ) and g ∈ G . Example 1.2. Let C b e a set o f subgro ups as in (a) in Definition 1.1. If we define O C = {O C ( H ) } H ∈C by O C ( H ) = { U ∈ C ( H ) | [ H : U ] < ∞} for ea ch H ∈ C , then ( C , O C ) is a Mack ey system. In particular, if G is a topologic al gro up and C is the set of all close d (resp. o pen) subgroups o f G , we call ( C , O C ) the natu r al (resp. op en-natur al ) Mack ey s ystem on G (Definition 2.4 in [10]). Definition 1.3. F or a Mack ey system ( C , O ) on G , define subcateg ories G Set C , O ⊆ G Set C of G Set as follows. (1) G Set C is a full sub catego ry of G Set , who se ob jects are those X ∈ Ob( G Set ) satisfying G x ∈ C for a ny x ∈ X . (2) G Set C , O is a catego ry with the same ob jects as G Set C , who se morphisms from X to Y in G Set C , O are those f ∈ G Set C ( X, Y ) satisfying the follo wing . (i) f has finite fiber s, i.e., f − 1 ( y ) is a finite s et for each y ∈ Y . (ii) G x ∈ O ( G f ( x ) ) for each x ∈ X . R emark 1.4 . It c a n b e easily seen that morphisms in G Set C , O are stable under pull-backs in G Set C . Namely , for any pull- ba ck diagr am in G Set C X ′ Y ′ X Y ,  g ′ / / f ′   f   g / / g ∈ G Set C , O ( X, Y ) implies g ′ ∈ G Set C , O ( X ′ , Y ′ ). Definition 1.5. A s emi-Mackey functor M is a pair ( M ∗ , M ∗ ) of (a) contrav a riant functor M ∗ : G Set C → Mon , (b) cov ariant functor M ∗ : G Set C , O → Mon , satisfying the following conditions. (M0) M ∗ ( X ) = M ∗ ( X ) for ea ch X ∈ O b( G Set C ). W e put M ( X ) = M ∗ ( X ) = M ∗ ( X ). T AMBARIZA T ION OF A M AC KEY FUNCTOR 5 (M1) (Mack ey condition) If X ′ Y ′ X Y  g ′ / / f ′   f   g / / is a pull-back diagra m in G Set C with g ∈ G Set C , O ( X, Y ), then M ∗ ( f ) ◦ M ∗ ( g ) = M ∗ ( g ′ ) ◦ M ∗ ( f ′ ) . (M2) (Additivit y) F or an y dir e ct sum decompos ition X = a λ ∈ Λ X λ in G Set C , the natural ma p ( M ∗ ( i λ )) λ ∈ Λ : M ( X ) → Y λ ∈ Λ M ( X λ ) is an isomorphism, where i λ : X λ ֒ → X ( λ ∈ Λ) are inclusions. In particula r, M ( ∅ ) is triv ial. A semi-Mackey functor M is a Macke y functor if M ( X ) is an a b e lian gr oup for ea ch X ∈ Ob( G Set C ), na mely if M ∗ and M ∗ are functors to A b . A morphism of ( semi -) Mackey functors from M to N is a family of monoid homomo r phisms ϕ = { ϕ X : M ( X ) → N ( X ) } X ∈ Ob( G Set C ) , which is na tural with re sp e ct to b oth the contra v ar iant part and the cov ariant par t. The category of semi- Mack ey functor s (resp. Mack ey functors) on ( C , O ) is denoted by SMack ( C , O ) (resp. Mack ( C , O ) ). R emark 1.6 . Origina lly in [1], when G is a profinite group, the op en-natural Mack ey system is ca lle d the finite natur al Mack ey s ystem. A Mack ey functor on a finite natural Mack ey system is also called a G - mo dulation . In pa r ticular when G is a finite group, reg arded as a discre te top ologic a l group, then the natural and the finite natural Mac key sys tems coincide, and we abbreviate the categor y of se mi-Mack ey functors on the (finite) natural Mack ey system o n G to SMack ( G ). In the or dina ry definition o f a Mack ey (or a T am bar a) functor, we work on the category o f finite G -sets, instead of G Set . How ever, b y the additivit y of a Mack ey (or a T ambara) functor, the resulting categor ies b ecome equiv alent. Of co urse, if we res trict ourselves to finite groups, then the same a rguments as b elow are also po ssible using the ca tegory o f finite G -sets. Also remark that if G = { e } is trivial, the ev alua tio n a t the trivial one-p oint G -set { e } gives a n equiv alence of ca tegories SMack ( { e } ) ≃ − → Mon ; M 7→ M ( { e } ) . Similarly , Mack ( { e } ) is equiv alent to Ab . Example 1.7 . Let ( C , O ) be an y Mackey system o n G . F or a ny G -mono id Q , ther e exists a naturally asso ciated semi-Mack e y functor P Q ∈ O b( Mac k ( C , O ) ), which we call the fixe d p oint functor v alued in Q , as follows. (a) F o r ea ch X ∈ Ob( G Set C ), define P Q by P Q ( X ) = G Set ( X , Q ) . 6 HIR OYUKI NAKAOKA (b) F or each morphism f ∈ G Set C ( Y , X ), P ∗ Q ( f ) : G Set ( X , Q ) → G Set ( Y , Q ) is de fined as the co mpo sition by f . (b) F or each morphism g ∈ G Set C , O ( Y , X ), P Q ∗ ( g ) : G Set ( X , Q ) → G Set ( Y , Q ) is de fined by ( P Q ∗ ( g )( α ))( y ) = Y x ∈ g − 1 ( y ) α ( x ) , for any α ∈ G Set ( X , Q ) and y ∈ Y . 2. T ambariza tion of a Mackey functor 2.1. Review on T am bara functors. A T a m bara sy stem ( C , O + , O • ) on G con- sists of tw o Mack ey systems ( C , O + ) and ( C , O • ) satisfying some compatibility con- ditions for exp onential diag rams (see [10]). In this a rticle, we only consider the following sp ecial case (Pro po sition 4.5 in [10]). As in the pr evious section, if one only consider the finite group case, then one ma y work ov er the c a tegory of finite G -sets, using the o riginal definition by T a m bara [13]. Definition 2. 1 . Let e C be a set of subsets of G , clos ed under left and rig h t trans- lation, finite intersections and finite unions. P ut C = { H ∈ e C | H ≤ G } . As in Example 1.2, ( C , O C ) is a Mack ey sy s tem. W e s ay ( C , O C , O • ) is a T amb ar a system on G if it satisfies the following. (i) ( C , O • ) is a Mack ey system on G . (ii) O • satisfies H ∈ O • ( H ) for each H ∈ C . R emark 2.2 . F o r any X , Y ∈ Ob( G Set C ), we hav e G Set C , O C ( X, Y ) = { f ∈ G Set C ( X, Y ) | f has finite fib e r s } . Thu s, for any c o mmu tative dia gram in G Set C , A A ′ X f / / p   4 4 4 4 4 4 p ′          p ∈ G Set C , O C ( A, X ) implies f ∈ G Set C , O C ( A, A ′ ). By the ab ov e remark, categ ory S C , O | X used in [10] agrees with the comma cat- egory G Set C , O C /X , a nd w e hav e the following (Definition 4.1 and P rop osition 4.5 in [10]). R emark 2.3 . Let ( C , O C , O • ) be a T ambara system. Le t η ∈ G Set C , O • ( X, Y ) b e any morphism. F or an y ( A p → X ) ∈ Ob( G Set C , O C /X ), we define (Π η ( A ) π = π ( p ) − → Y ) in Ob( G Set C , O C / Y ) by Π η ( A ) =    ( y , σ )       y ∈ Y , σ : η − 1 ( y ) → A is a map of sets , p ◦ σ = id η − 1 ( y )    , π ( y , σ ) = y . T AMBARIZA T ION OF A M AC KEY FUNCTOR 7 W e abbreviately write Π η ( A p → X ) = (Π η ( A ) π → Y ). F o r any morphism a : ( A p → X ) → ( A ′ p ′ → X ) in G Set C , O C /X , define Π η ( a ) : Π η ( A p → X ) → Π η ′ ( A ′ p ′ → X ) by Π η ( a )( y , σ ) = ( y , a ◦ σ ). Then we obtain a functor Π η : G Set C , O C /X → G Set C , O C / Y , which is right a djo int to the pull-ba ck functor defined by η X × Y − : G Set C , O C / Y → G Set C , O C /X . By this adjoint pr op erty , for any p ∈ G Set C , O C ( A, X ), we have a commutativ e diagram (2.1) X Y A X × Y Π η ( A ) Π η ( A ) η   p o o λ o o ρ   π = π ( p ) o o  where ρ is the pull-back of η by π , and λ is the morphism corres po nding to id Π η ( A ) , and p ◦ λ b ecomes the pull-ba ck of π by η . Any commutativ e diagr am in G Set C (2.2) X Y Z X ′ Y ′ η   ξ o o ζ o o η ′   υ o o  isomorphic to (2 . 1) for some p ∈ G Set C , O C ( A, X ), is called an exp onential diagr am . Definition 2.4 (Definition 4.1 0 , Remark 4.11 in [10]) . L e t ( C , O C , O • ) be a T am- bara system on G . A semi-T amb ar a functor S o n ( C , O C , O • ) is a triplet ( S ∗ , S + , S • ) consisting of (a) a semi-Mack ey functor ( S ∗ , S + ) on ( C , O C ) (b) a semi-Mackey functor ( S ∗ , S • ) on ( C , O • ) which satis fie s the fo llowing conditions. (i) S ( X )(= S ∗ ( X ) = S + ( X ) = S • ( X )) is a semi-ring fo r each X ∈ Ob( G Set C ), with the addition and the multiplication induced from the monoid struc- tures of S ∗ ( X ) = S + ( X ) and S ∗ ( X ) = S • ( X ), r esp ectively . (ii) F or any exp o nential diag ram (2 . 2), we have S + ( υ ) ◦ S • ( η ′ ) ◦ S ∗ ( ζ ) = S • ( η ) ◦ S + ( ξ ) . F or any morphism f in G Set C (resp. G Set C , O C , G Set C , O • ), we abbreviate S ∗ ( f ) (resp. S + ( f ), S • ( f )) to f ∗ (resp. f + , f • ). These are ca lled the structu r e morphisms of semi-T ambara functor S . f + (resp. f • ) is called an additive (resp. mu ltiplic ative ) tr ansfer . If S ( X ) is moreov e r a ring for each X ∈ Ob( G Set C ), then S = ( S ∗ , S + , S • ) is called a T a mb ar a functor . If S a nd T are (semi-)T ambara functors, a morphism ψ from S to T is defined to b e a family ψ = { ψ X } X ∈ Ob( G Set C ) of semi-r ing homomorphis ms ψ X : S ( X ) → T ( X ), co mpatible with structure morphisms. In other w ords, ψ is a morphis m of 8 HIR OYUKI NAKAOKA (semi-)T a mbara functors if and o nly if it gives morphisms of additive and multi- plicative s emi-Mack ey functors ψ : ( S ∗ , S + ) → ( T ∗ , T + ) , ψ : ( S ∗ , S • ) → ( T ∗ , T • ) . W e denote the categ ory of semi-T a mbara functors on ( C , O C , O • ) b y ST am ( C , O • ) . (W e omitted O C , sinc e it is determined by C .) R emark 2.5 (Theorem 5.16 and P rop osition 5.17 in [10]) . By definit ion, if w e denote the multiplicative part by S µ = ( S ∗ , S • ) fo r each S ∈ Ob( ST am ( C , O • ) ), this g ives a forgetful functor µ : ST am ( C , O • ) → SMack ( C , O • ) ( ψ : S → T ) 7→ ( ψ : S µ → T µ ) . The categ ory of T am bara functors is denoted by T am ( C , O • ) . This is a full sub- category in ST am ( C , O • ) , whose inclusion w e denote b y U : T am ( C , O • ) → ST am ( C , O • ) . Conv ers e ly , if we are given a n ob ject S in ST a m ( C , O • ) , then the corresp ondence γ S : Ob( G Set C ) ∋ X 7→ K 0 ( S ( X )) ∈ Ob( R ing ) bec omes a T ambara functor, with a ppropriately defined structure morphis ms . Her e , K 0 : SRing → Ri ng is the ring- completion functor. Moreov er, for any ψ ∈ ST am ( C , O • ) ( S, T ), γ ψ := { K 0 ( ψ X ) } X ∈ Ob( G Set C ) gives a morphism o f T ambara functors γ ψ : γ S → γ T , and we obtain a functor γ : S T am ( C , O • ) → T am ( C , O • ) . In fact, γ is the left adjoint functor of U . (F or the finite gr oup case , see [1 3].) R emark 2.6 . As in Remar k 1.6, s imila rly to the categor y of Mack ey functors, when G is finite a nd ( C , O • ) is the natural Mack ey system, then T am ( C , O • ) is denoted by T am ( G ). If G = { e } is trivial, the ev aluation at the tr iv ial one-p oint G -s et { e } gives an equiv alence o f categories T am ( { e } ) ≃ − → Ri ng ; T 7→ T ( { e } ) . Similarly , ST am ( { e } ) is equiv alent to SRing . 2.2. Diagram Lemmas. In this se c tion, we in tro duce s o me diagra m lemmas con- cerning exponential diagr a ms, which will b e needed later. Their proofs are stra ight- forward, a nd we omit them. F or the finite g roup case, see [13]. Lemma 2.7. L et η ∈ G Set C , O • ( X, Y ) b e any morphism, and let ( A 1 p 1 → X , m A 1 ) , ( A 2 p 2 → X , m A 2 ) b e two obje cts in G Set C , O C /X . As s u me X Y A i X × Y Π η ( A i ) Π η ( A i ) exp η   p i o o λ i o o ρ i   π i o o T AMBARIZA T ION OF A M AC KEY FUNCTOR 9 is the exp onential diagr am for e ach i = 1 , 2 . If we let A b e the fib er e d pr o duct of A 1 and A 2 over X A = A 1 × X A 2 A 2 A 1 Y ,   2 / /  1   p 2   p 1 / / then Π η ( A ) Π η ( A 2 ) Π η ( A 1 ) Y  Π η (  2 ) / / Π η (  1 )   π 2   π 1 / / is also a pul l-b ack diagr am. If we put p = p 1 ◦  1 = p 2 ◦  2 and π = π 1 ◦ Π η (  1 ) = π 2 ◦ Π η (  2 ) , then we have an exp onential diagr am X Y A X × Y Π η ( A ) Π η ( A ) . exp η   p o o λ o o ρ   π o o Lemma 2.8. L et X Y A Z ∼ = X × Y Π η ( A ) Π η ( A ) exp η   p o o λ o o ρ   π o o b e an exp onential diagr am. If we pul l it b ack by a morphism ζ ∈ G Set C ( Y ′ , Y ) , t hen the obtaine d diagr am X ′ Y ′ A ′ Z ′ (Π η ( A )) ′ exp η ′   p ′ o o λ ′ o o ρ ′   π ′ o o is also an exp onential diagr am. Her e, ( − ) ′ is the abbr eviation of − × Y Y ′ . Lemma 2.9. L et (2 . 2 ) b e an exp onent ial diagr am, and let A A ′ Z X ′  ζ ′ o o p   p ′   ζ o o b e a pul l-b ack dia gr am with p ∈ G Set C , O C ( A, Z ) . If we let X ′ Y ′ A ′ X ′ × Y ′ Π η ′ ( A ′ ) Π η ′ ( A ′ ) exp η ′   p ′ o o λ ′ o o ρ ′   π ′ o o 10 HIR OYUKI NAKAOKA b e the exp onent ial diagr am, then X Y A X ′ × Y ′ Π η ′ ( A ′ ) Π η ′ ( A ′ ) exp η   ξ ◦ p o o ζ ′ ◦ λ ′ o o ρ ′   υ ◦ π ′ o o is also an exp onential diagr am. 2.3. T am barization of a Mac key functor. Fix a T am bara system ( C , O C , O • ). In this s ection, we a sso ciate a T am bara functor to any Mack ey functor. In fact, we construct a functor S from SMack ( C , O • ) to S T a m ( C , O • ) in Theo rem 2.1 2. As a consequence, we obta in the following functor s. γ ◦ S : SMack ( C , O • ) → T am ( C , O • ) Mack ( C , O • ) → ST am ( C , O • ) Mack ( C , O • ) → T am ( C , O • ) Definition 2.10. Let M be a semi-Mack ey functor o n ( C , O • ). F or any X ∈ Ob( G Set C ), define catego ry M - G Set C , O C /X by the following. (a) An ob ject in M - G Set C , O C /X is a pair ( A p → X, m A ) of p ∈ G Set C , O C ( A, X ) and m A ∈ M ( A ). (b) A mo rphism from ( A 1 p 1 → X, m A 1 ) to ( A 2 p 2 → X, m A 2 ) is a morphism f ∈ G Set C ( A 1 , A 2 ), such that p 2 ◦ f = p 1 and M ∗ ( f )( m A 2 ) = m A 1 . F or any tw o ob jects ( A 1 p 1 → X , m A 1 ) and ( A 2 p 2 → X , m A 2 ) in M - G Set C , O C /X , define their sum and pro duct as follows. (i) ( A 1 p 1 → X, m A 1 ) ∐ ( A 2 p 2 → X, m A 2 ) = ( A 1 ∐ A 2 p 1 ∪ p 2 − → X, m A 1 ∐ m A 2 ), (ii) ( A 1 p 1 → X, m A 1 ) × ( A 2 p 2 → X, m A 2 ) = ( A 1 × X A 2 p → X, m A 1 ⋆ m A 2 ). Here, m A 1 ∐ m A 2 ∈ M ( A 1 ∐ A 2 ) is the pull-back of ( m A 1 , m A 2 ) ∈ M ( A 1 ) × M ( A 2 ) by the isomorphis m ( M ∗ ( ι 1 ) , M ∗ ( ι 2 )) : M ( A 1 ∐ A 2 ) ∼ = → M ( A 1 ) × M ( A 2 ) , where ι 1 : A 1 ֒ → A 1 ∐ A 2 and ι 2 : A 2 ֒ → A 1 ∐ A 2 are the inclusio ns . m A 1 ⋆ m A 2 is the product of M ∗ (  1 )( m A 1 ) and M ∗ (  2 )( m A 2 ) in M ( A 1 × X A 2 ), where  1 and  2 are the pro jections in the following pull-ba ck dia gram, and p = p 1 ◦  1 = p 2 ◦  2 . A 1 × X A 2 A 2 A 1 X   2 / /  1   p 2   p 1 / / p = p 1 ◦  1 = p 2 ◦  2 m A 1 ⋆ m A 2 = M ∗ (  1 )( m A 1 ) · M ∗ (  2 )( m A 2 ) The iso morphism cla sses of ob jects in M - G Set C , O C /X forms a semi-ring with these sums and pr o ducts, which we deno te by S M ( X ). Prop ositio n 2.1 1. S M c arries a natur al structure of a semi-T amb ar a functor. Pr o of. Fir s t we co nstruct str ucture morphisms for S M . Let X , Y be any pair of ob jects in G Set C , a nd let ( A p → X, m A ) ∈ S M ( X ) b e any element. T AMBARIZA T ION OF A M AC KEY FUNCTOR 11 (1) F o r any ζ ∈ G Set C ( Y , X ), define ζ ∗ : S M ( X ) → S M ( Y ) by ζ ∗ ( A p → X , m A ) = ( A ′ p ′ → Y , M ∗ ( ζ ′ )( m A )) , where (2.3) A ′ Y A X  p ′ / / ζ ′   ζ   p / / is a pull-back diagra m. (2) F o r any ξ ∈ G Set C , O C ( X, Y ), define ξ + : S M ( X ) → S M ( Y ) by ξ + ( A p → X , m A ) = ( A ξ ◦ p − → Y , m A ) . (3) F o r any η ∈ G Set C , O • ( X, Y ), define η • : S M ( X ) → S M ( Y ) by η • ( A p → X, m A ) = ( Y ′ υ − → Y , M ∗ ( ρ ) M ∗ ( λ )( m A )) , where (2.4) X Y A X ′ Y ′ exp η   p o o λ o o ρ   υ o o is a n exp onential diagra m. W e only demo nstrate the following. The other conditions ca n be shown easily . (i) ξ + is a n additive homomorphism for any ξ ∈ G Set C , O C ( X, Y ). (ii) η • is a multiplicativ e homomor phism for any η ∈ G Set C , O • ( X, Y ). (iii) ζ ∗ is a semi-ring ho momorphism for any ζ ∈ G Set C ( Y , X ). (iv) S ∗ M is additive, namely , for a ny X , Y ∈ Ob( G Set C ), ( ι ∗ X , ι ∗ Y ) : S M ( X ∐ Y ) → S M ( X ) × S M ( Y ) is a n isomor phism, where ι X , ι Y are the inclusions. (v) F or any pull-back diagram in G Set C X ′ X Y ′ Y  ζ ′ / / ξ ′   ξ   ζ / / with ξ ∈ G Set C , O C ( X, Y ), we have ζ ∗ ◦ ξ + = ξ ′ + ◦ ζ ′∗ . (vi) F or any pull-back diagram in G Set C X ′ X Y ′ Y  ζ ′ / / η ′   η   ζ / / with η ∈ G Set C , O • ( X, Y ), we have ζ ∗ ◦ η • = η ′ • ◦ ζ ′∗ . 12 HIR OYUKI NAKAOKA (vii) F or any exp onential dia gram (2 . 2), we hav e υ + ◦ η ′ • ◦ ζ ∗ = η • ◦ ξ + . Confirmation of (i), (ii), (iii) Let ( A 1 p 1 → X , m A 1 ) and ( A 2 p 2 → X , m A 2 ) be an y pair of elements in S M ( X ). (i) is trivia lly satisfied, s ince ξ + (( A 1 p 1 → X, m A 1 ) ∐ ( A 2 p 2 → X, m A 2 )) = ( A 1 ∐ A 2 ξ ◦ ( p 1 ∪ p 2 ) − → X , m A 1 ∐ m A 2 ) = ξ + ( A 1 p 1 → X, m A 1 ) ∐ ξ + ( A 2 p 2 → X, m A 2 ) . T o show (iii), take the fo llowing pull-backs. A ′ i A i Y X  ζ i / / p ′ i   p i   ζ / / A A 2 A 1 X   2 / /  1   p 2   p 1 / / A ′ A Y X  ζ ′ / / p ′   p   ζ / / ( i = 1 , 2) p = p 1 ◦  1 = p 2 ◦  2 F ro m these, we obta in the following pull-back diagr ams. A ′ A ′ i A A i   ′ i / / ζ ′   ζ i    i / / ( i = 1 , 2) , A ′ A ′ 2 A ′ 1 Y   ′ 2 / /  ′ 1   p ′ 1   p ′ 2 / / Thu s w e hav e ζ ∗ (( A 1 p 1 → X, m A 1 ) × ( A 2 p 2 → X , m A 2 )) = ζ ∗ ( A p → X, M ∗ (  1 )( m A 1 ) · M ∗ (  2 )( m A 2 )) = ( A ′ p ′ → Y , M ∗ ( ζ ′ ) M ∗ (  1 )( m A 1 ) · M ∗ ( ζ ′ ) M ∗ (  2 )( m A 2 )) = ( A ′ p ′ → Y , M ∗ (  ′ 1 ) M ∗ ( ζ 1 )( m A 1 ) · M ∗ (  ′ 2 ) M ∗ ( ζ 2 )( m A 2 ) = ζ ∗ ( A 1 p 1 → X, m A 1 ) × ζ ∗ ( A 2 p 2 → X, m A 2 ) . Moreov er, s ince A ′ 1 ∐ A ′ 2 Y A 1 ∐ A 2 X  p ′ 1 ∪ p ′ 2 / / ζ 1 ∐ ζ 2   ζ   p 1 ∪ p 2 / / is a pull-back dia gram, we hav e ζ ∗ (( A 1 p 1 → X, m A 1 ) ∐ ( A 2 p 2 → X, m A 2 )) = ( A ′ 1 ∐ A ′ 2 p ′ 1 ∪ p ′ 2 − → Y , M ∗ ( ζ 1 ∐ ζ 2 )( m A 1 ∐ m A 2 )) = ( A ′ 1 ∐ A ′ 2 p ′ 1 ∪ p ′ 2 − → Y , M ∗ ( ζ 1 )( m A 1 ) ∐ M ∗ ( ζ 2 )( m A 2 )) = ζ ∗ ( A 1 p 1 → X, m A 1 ) ∐ ζ ∗ ( A 2 p 2 → X, m A 2 ) . T AMBARIZA T ION OF A M AC KEY FUNCTOR 13 T o show (ii), we use the nota tio n in Lemma 2.7 . F or any pa ir of e lement s ( A 1 p 1 → X , m A 1 ) and ( A 2 p 2 → X, m A 2 ) in S M ( X ), we hav e η • (( A 1 p 1 → X, m A 1 ) · ( A 2 p 2 → X , m A 2 )) = η • ( A p → X, M ∗ (  1 )( m A 1 ) · M ∗ (  2 )( m A 2 )) = (Π η ( A ) π → Y , M ∗ ( ρ ) M ∗ ( λ )( M ∗ (  1 )( m A 1 ) · M ∗ (  2 )( m A 2 ))) , η • ( A 1 p 1 → X, m A 1 ) · η • ( A 2 p 2 → X, m A 2 ) = (Π η ( A 1 ) π 1 → Y , M ∗ ( ρ 1 ) M ∗ ( λ 1 )( m A 1 )) · (Π η ( A 2 ) π 2 → Y , M ∗ ( ρ 2 ) M ∗ ( λ 2 )( m A 2 )) . Since we hav e M ∗ ( ρ ) M ∗ ( λ ) M ∗ (  i ) = M ∗ (Π η (  i )) M ∗ ( ρ i ) M ∗ ( λ i ) A 1 × X A 2 X × Y Π η ( A ) Π η ( A ) A i X × Y Π η ( A i ) Π η ( A i )  λ o o ρ / /  i   1 X × Y Π η (  i )   Π η (  i )   λ i o o ρ i / /  for i = 1 , 2, we obtain η • (( A 1 p 1 → X, m A 1 ) · ( A 2 p 2 → X, m A 2 )) = η • ( A 1 p 1 → X, m A 1 ) · η • ( A 2 p 2 → X, m A 2 ) . Confirmation of (iv) T o each ( A p → X, m A ) ∈ S M ( X ) a nd ( B q → Y , m B ) ∈ S M ( Y ), asso cia te ( A ∐ B p ∐ q − → X ∐ Y , M ∗ ( ι A )( m A ) · M ∗ ( ι B )( m B )) ∈ S M ( X ∐ Y ) , where ι A : A ֒ → A ∐ B and ι B : B ֒ → A ∐ B a re t he inclusions. This giv es the inv erse of ( ι ∗ X , ι ∗ Y ). Confirmation of (v) Let ( A p → X, m A ) b e any element in S M ( X ). If we take pull-backs A ′ X ′ Y ′ A X Y ,   p ′ / / ξ ′ / / ζ ′′   ζ ′   ζ   p / / ξ / / then we hav e ζ ∗ ξ + ( A p → X , m A ) = ( A ′ ξ ′ ◦ p ′ − → Y ′ , M ∗ ( ζ ′′ )( m A )) = ξ ′ + ζ ′∗ ( A p → X, m A ) . Confirmation of (vi) W e use the nota tion in Lemma 2.8. Let ζ ′′ ∈ G Set C ( A ′ , A ) b e the pull-back of ζ ′ (or ζ ). A ′ X ′ Y ′ A X Y   p ′ / / η ′ / / ζ ′′   ζ ′   ζ   p / / η / / 14 HIR OYUKI NAKAOKA If we let pr Z and pr Π be the canonical pro jections induced from ζ Z ′ (Π η ( A )) ′ Y ′ Z Π η ( A ) Y   ρ ′ / / π ′ / / pr Z   pr Π   ζ   ρ / / η / / then Z ′ A ′ Z A λ ′ / / pr Z   ζ ′′   λ / /  is commutativ e . By Lemma 2.8, we hav e η ′ • ζ ′∗ ( A p → X, m A ) = ((Π η ( A )) ′ π ′ → Y ′ , M ∗ ( ρ ′ ) M ∗ ( λ ′ ) M ∗ ( ζ ′′ )( m A )) , ζ ∗ η • ( A p → X, m A ) = ((Π η ( A )) ′ π ′ → Y ′ , M ∗ (pr Π ) M ∗ ( ρ ) M ∗ ( λ )( m A )) . Since M ∗ ( ρ ′ ) M ∗ ( λ ′ ) M ∗ ( ζ ′′ ) = M ∗ ( ρ ′ ) M ∗ (pr Z ) M ∗ ( λ ) = M ∗ (pr Π ) M ∗ ( ρ ) M ∗ ( λ ) , we obta in η ′ • ◦ ζ ′∗ = ζ ∗ ◦ η • . Confirmation of (vii) In the notation of Lemma 2.9, fo r any ( A p → Z, m A ) ∈ S M ( Z ), we have η • ξ + ( A p → Z, m A ) = η • ( A ξ ◦ p − → X, m A ) = (Π η ′ ( A ′ ) υ ◦ π ′ − → Y , M ∗ ( ρ ′ ) M ∗ ( ζ ′ ◦ λ ′ )( m A )) = υ + (Π η ′ ( A ′ ) π ′ − → Y ′ , M ∗ ( ρ ′ ) M ∗ ( λ ′ ) M ∗ ( ζ ′ )( m A )) = υ + η ′ • ( A ′ p ′ → X ′ , M ∗ ( ζ ′ )( m A )) = υ + η ′ • ζ ∗ ( A p → Z, m A ) .  Theorem 2. 12. L et ( C , O C , O • ) b e a T amb ar a system on G . The c onstru ction of S M in Pr op osition 2.11 gives a functor S : SMack ( C , O • ) → ST am ( C , O • ) . Pr o of. By Pr op osition 2.11 , S M bec omes an o b ject in ST am ( C , O • ) for any M ∈ Ob( SMack ( C , O • ) ). It suffices to constr uct a morphism S ϕ ∈ ST am ( C , O • ) ( S M , S N ) fo r each ϕ ∈ SMack ( C , O • ) ( M , N ) where M , N ∈ Ob( SMack ( C , O • ) ), in a functor ial wa y . F or ea ch X ∈ Ob( G Set C ), we ha ve a functor from M - G Set C , O C /X to N - G Set C , O C /X defined by M - G Set C , O C /X → N - G Set C , O C /X ( A p → X, m A ) 7→ ( A p → X, ϕ A ( m A )) , T AMBARIZA T ION OF A M AC KEY FUNCTOR 15 where ϕ A is the comp onent o f ϕ at A (monoid homo mo rphism from M ( A ) to N ( A )) . This functor preser ves sums and pro ducts, and th us we o btain a s emi-ring homomorphism S ϕ ( X ) : S M ( X ) → S N ( X ) . Claim 2.13. {S ϕ ( X ) | X ∈ Ob( G Set C ) } is c omp atible with structu re m orphisms of S M and S N . If this c laim is shown, it means S ϕ ∈ ST am ( C , O • ) ( S M , S N ). Moreov er, it ca n be easily chec ked that the corres p o ndence ϕ 7→ S ϕ preserves the iden tities and comp ositions, s o we obtain a functor S : SMack ( C , O • ) → ST am ( C , O • ) . Thu s it suffices to show Cla im 2 .1 3. Pr o of of Claim 2.13. Let ( A p → X, m A ) b e any element in S M ( X ). Compatibility with ξ + Let ξ ∈ G Set C , O C ( X, Y ) be any morphism. Obviously we hav e S ϕ ( Y ) ◦ ξ + ( A p → X, m A ) = ( A ξ ◦ p − → Y , ϕ A ( m A )) = ξ + ◦ S ϕ ( X )( A p → X, m A ) . S M ( X ) S N ( X ) S M ( Y ) S N ( Y ) S ϕ ( X ) / / ξ +   ξ +   S ϕ ( Y ) / /  Compatibility with η • Let η ∈ G Set C , O • ( X, Y ) be any morphism, and let (2 . 1) be the exp onential diagram. Then we hav e S ϕ ( Y ) ◦ η • ( A p → X, m A ) = (Π η ( A ) π → Y , ϕ Π η ( A ) M ∗ ( ρ ) M ∗ ( λ )( m A )) = (Π η ( A ) π → Y , N ∗ ( ρ ) N ∗ ( λ ) ϕ A ( m A )) = η • ◦ S ϕ ( X )( A p → X, m A ) . S M ( X ) S N ( X ) S M ( Y ) S N ( Y ) S ϕ ( X ) / / η •   η •   S ϕ ( Y ) / /  Compatibility with ζ ∗ Let ζ ∈ G Set C ( Y , X ) b e any mor phism, and let (2 . 3) b e the pull-back diagra m. Then we hav e S ϕ ( Y ) ◦ ζ ∗ ( A p → X , m A ) = ( A ′ p ′ → Y , ϕ A ′ M ∗ ( ζ ′ )( m A )) = ( A ′ p ′ → Y , N ∗ ( ζ ′ ) ϕ A ( m A )) = ζ ∗ S ϕ ( X )( A p → X, m A ) . 16 HIR OYUKI NAKAOKA S M ( X ) S N ( X ) S M ( Y ) S N ( Y ) S ϕ ( X ) / / ζ ∗   ζ ∗   S ϕ ( Y ) / /    Corollary 2.14. Comp osing S with γ : ST a m ( C , O • ) → T am ( C , O • ) , we obtain a functor T = γ ◦ S : SMack ( C , O • ) → T am ( C , O • ) . Mor e over, sinc e Mack ( C , O • ) is a ful l su b c ate gory of SMack ( C , O • ) , we also obtai n functors Mack ( C , O • ) → ST am ( C , O • ) , Mack ( C , O • ) → T am ( C , O • ) . 2.4. Adjoint prop e rt y. Theorem 2.15. L et ( C , O C , O • ) b e any T amb ar a system on G . The funct or c on- structe d in The or em 2.12 S : SMack ( C , O • ) → ST am ( C , O • ) is left adjoint to the for getful functor µ : ST am ( C , O • ) → SMack ( C , O • ) . Pr o of. W e construct natural ma ps Φ : ST am ( C , O • ) ( S M , T ) → SMack ( C , O • ) ( M , T µ ) Ψ : SMack ( C , O • ) ( M , T µ ) → ST am ( C , O • ) ( S M , T ) for ea ch M ∈ Ob( SMack ( C , O • ) ), T ∈ Ob( ST am ( C , O • ) ), and show Ψ and Φ ar e m utually inv er se. Let ψ ∈ ST am ( C , O • ) ( S M , T ) b e any morphism. F or each X ∈ Ob( G Set C ), define Φ( ψ ) X : M ( X ) → T µ ( X ) by Φ( ψ ) X ( m ) = ψ X ( X id X → X , m ) ( ∀ m ∈ M ( X )) . On the co n trary , for each ϕ ∈ SMack ( C , O • ) ( M , T µ ) and X ∈ Ob( G Set C ), define Ψ( ϕ ) X : S M ( X ) → T ( X ) by Ψ( ϕ ) X ( A p → X, m A ) = T + ( p ) ◦ ϕ A ( m A ) ( ∀ ( A p → X, m A ) ∈ S M ( X )) . Claim 2.16 . F or any ϕ ∈ SMack ( C , O • ) ( M , T µ ) and ψ ∈ ST am ( C , O • ) ( S M , T ) , the fol lowing ar e satisfie d. (i) Φ( ψ ) := { Φ( ψ ) X } X ∈ Ob( G Set C ) b elongs to SMack ( C , O • ) ( M , T µ ) . (ii) Ψ( ϕ ) := { Ψ( ϕ ) X } X ∈ Ob( G Set C ) b elongs t o ST a m ( C , O • ) ( S M , T ) . (iii) Ψ ◦ Φ( ψ ) = ψ , Φ ◦ Ψ( ϕ ) = ϕ . T AMBARIZA T ION OF A M AC KEY FUNCTOR 17 Pr o of of Claim 2.16. (i) It suffices to show the commutativit y of (ia) M ( X ) T ( X ) M ( Y ) T ( Y ) , Φ( ψ ) X / / M ∗ ( η )   T • ( η )   Φ( ψ ) Y / / (ib) M ( X ) T ( X ) M ( Y ) T ( Y ) . Φ( ψ ) X / / M ∗ ( ζ )   T ∗ ( ζ )   Φ( ψ ) Y / / for arbitr ary morphisms η ∈ G Set C , O • ( X, Y ) and ζ ∈ G Set C ( Y , X ). Le t m ∈ M ( X ) be any elemen t. Commutativit y o f (ia) follows fro m T • ( η )Φ( ψ ) X ( m ) = T • ( η ) ψ X ( X id X → X , m ) = ψ Y η • ( X id X → X , m ) = ψ Y ( Y id Y → Y , M ∗ ( η )( m )) = Φ( ψ ) Y M ∗ ( η )( m ) , since X Y X X Y exp η   id X o o id X o o η   id Y o o is an expo nent ial diag ram. Commutativit y o f (ib) follows from T ∗ ( ζ )Φ( ψ ) X ( m ) = T ∗ ( ζ ) ψ X ( X id X → X , m ) = ψ Y ζ ∗ ( X id X → X , m ) = ψ Y ( Y id Y → Y , M ∗ ( ζ )( m )) = Φ( ψ ) Y M ∗ ( ζ )( m ) . (ii) It suffices to show the commutativit y of (iia) S M ( X ) T ( X ) S M ( Y ) T ( Y ) , Ψ( ϕ ) X / / ξ +   T + ( ξ )   Ψ( ϕ ) Y / / (iib) S M ( X ) T ( X ) S M ( Y ) T ( Y ) , Ψ( ϕ ) X / / η •   T • ( η )   Ψ( ϕ ) Y / / (iic) S M ( X ) T ( X ) S M ( Y ) T ( Y ) . Ψ( ϕ ) X / / ζ ∗   T ∗ ( ζ )   Ψ( ϕ ) Y / / for ar bitrary ξ ∈ G Set C , O C ( X, Y ), η ∈ G Set C , O • ( X, Y ) and ζ ∈ G Set C ( Y , X ). 18 HIR OYUKI NAKAOKA Let ( A p → X, m A ) b e any element in S M ( X ). The co mmutativit y of (iia) follows from T + ( ξ )Ψ( ϕ ) X ( A p → X, m A ) = T + ( ξ ) T + ( p ) ϕ A ( m A ) = Ψ( ϕ ) Y ( A ξ ◦ p − → Y , m A ) = Ψ( ϕ ) Y ξ + ( A p → X, m A ) . In the notation of exp o nen tial diag r am (2 . 4), the co mm utativity of (iib) follows from T • ( η )Ψ( ϕ ) X ( A p → X, m A ) = T • ( η ) T + ( p ) ϕ A ( m A ) = T + ( υ ) T • ( ρ ) T ∗ ( λ ) ϕ A ( m A ) = T + ( υ ) ϕ Y ′ M ∗ ( ρ ) M ∗ ( λ )( m A ) = Ψ( ϕ ) Y ( Y ′ υ → Y , M ∗ ( ρ ) M ∗ ( λ )( m A )) = Ψ( ϕ ) Y η • ( A p → X, m A ) . In the notation of (2 . 3), the commutativit y of (iic) follows from T ∗ ( ζ )Ψ( ϕ ) X ( A p → X, m A ) = T ∗ ( ζ ) T + ( p ) ϕ A ( m A ) = T + ( p ′ ) T ∗ ( ζ ′ ) ϕ A ( m A ) = T + ( p ′ ) ϕ Y ′ M ∗ ( ζ ′ )( m A ) = Ψ( ϕ ) Y ( A ′ p ′ → Y , M ∗ ( ζ ′ )( m A )) = Ψ( ϕ ) Y ζ ∗ ( A p → X, m A ) . (iii) Let X b e a ny ob ject in G Set C . F or any m ∈ M ( X ), we have (Φ ◦ Ψ( ϕ )) X ( m ) = Ψ( ϕ ) X ( X id X → X , m ) = T + (id X ) ϕ X ( m ) = ϕ X ( m ) . Conv ers e ly , for any ( A p → X, m A ) ∈ S M ( X ), we hav e (Ψ ◦ Φ( ψ )) X ( A p → X, m A ) = T + ( p )Φ( ψ ) A ( m A ) = T + ( p ) ψ A ( A id A → A, m A ) = ψ X p + ( A id A → A, m A ) = ψ X ( A p → X, m A ) .   Corollary 2. 1 7. Comp ositio n T = γ ◦ S : SMack ( C , O • ) → T am ( C , O • ) is left adjoint to the for getful functor µ ◦ U : T a m ( C , O • ) → SMack ( C , O • ) . Pr o of. This immediately follows fr o m Remark 2.5 and Theorem 2.15.  T AMBARIZA T ION OF A M AC KEY FUNCTOR 19 With this cor ollary , T can be r egarded as a G -biv ar iant ana log of the functor taking mo noid-rings. Indeed if G is trivial, this is equiv alent to the mono id-ring functor: SMack ( { e } ) Mon T am ( { e } ) Ring ≃ / / T   Z [ − ]   ≃ / /  3. Rela tion with other constructions In the rest, G is assumed to b e finite, and we o nly consider the natural Ma ckey system on G , as in Rema rk 1.6 and Remark 2.6. In this case, we hav e G Set = G Set C . As in Remark 1.6, we may w or k over the catego ry G set of finite G -sets. F or ex ample, M - G Set C , O C /X in Definition 2.10 is replaced b y M - G set /X . 3.1. Relatio n wi th the crosse d Burnsi de ring. Le t Q b e a (not necess arily finite) G -mo no id, and P Q be the fixed point functor asso ciated to Q (Example 1.7). By T he o rem 2.12 , we obtain a T am ba ra functor T P Q . Reca ll that we hav e a sequence o f functors G - Mon P − → SMack ( G ) T − → T am ( G ) , where G - Mon is the ca tegory of G -monoids. W e show T P Q generalizes the crossed Burnside ring functor in [1 1], [12]. Indeed, if Q is finite, we co nstruct an isomo rphism of T am ba ra functor s b etw een T P Q and the cr ossed Burnside ring functor , in P rop osition 3.2 . First w e reca ll the definition of the cr ossed Burnside ring. Definition 3 .1. ( § 4 .3 in [11]) Fix a finite G -mono id Q . The categor y of cros sed G -sets G - xs et /QX is defined a s follows. (a) An ob ject in G - xset /QX is a triplet ( A p → X , m A ) of a finite G -set A , G -maps f ∈ G set ( A, X ) and m A ∈ G set ( A, Q ). (b) A mo rphism from ( A 1 p 1 → X, m A 1 ) to ( A 2 p 2 → X , m A 2 ) in G - xset / QX is a G -map f ∈ G set ( A 1 , A 2 ) sa tisfying p 2 ◦ f = p 1 and m A 2 ◦ f = m A 1 . A 1 A 2 X f / / p 1   4 4 4 4 4 p 2         A 1 A 2 Q f / / m A 1   4 4 4 4 4 m A 2         In G - x s et /QX , for a ny pair of o b jects ( A i p i → X, m A i ) ( i = 1 , 2), their s um (= c opr o duct in [11]) and pro duct (= ten s or pr o duct in [11]) are ( A 1 p 1 → X , m A 1 ) + ( A 2 p 2 → X, m A 2 ) = ( A 1 ∐ A 2 p 1 ∪ p 2 − → X, m A 1 ∪ m A 2 ) , ( A 1 p 1 → X, m A 1 ) · ( A 2 p 2 → X, m A 2 ) = ( A 1 × X A 2 p → X, m A 1 ∗ m A 2 ) , where A 1 × X A 2 A 2 A 1 Y   2 / /  1   p 2   p 1 / / 20 HIR OYUKI NAKAOKA is a pull-back dia gram, p = p 1 ◦  1 = p 2 ◦  2 , a nd m A 1 ∗ m A 2 is de fined by m A 1 ∗ m A 2 ( a 1 , a 2 ) = m A 1 ( a 1 ) m A 2 ( a 2 ) ( ∀ ( a 1 , a 2 ) ∈ A 1 × X A 2 ) . The c rossed Burnside ring Ω Q ( X ) is defined to b e the Grothendieck ring o f this category : Ω Q ( X ) = K 0 ( G - xset / QX ) Remark that if Q is tr ivial, then Ω Q ( X ) is nothing other than the ordinar y Burnside ring Ω( X ). As s hown in [11] a nd [12], Ω Q has a structure of a T am ba ra functor. F or any f ∈ G set ( X , Y ), the structur e morphisms f ∗ , f + , f • are defined in the following w ay , which generalizes those for the ordina ry Burnside ring functor Ω. (i) f ∗ : Ω Q ( Y ) → Ω Q ( X ) is the r ing homomo rphism induced from f ∗ ( B q → Y , m B ) = ( A p → X, m B ◦ f ′ ) ( ∀ ( B q → Y , m B ) ∈ Ob( G - x s et /QY )) , where A B X Y  f ′ / / p   q   f / / is the pull-back. (ii) f + : Ω Q ( X ) → Ω Q ( Y ) is the additive homomorphism induced from f + ( A p → X, m A ) = ( A f ◦ p − → Y , m A ) ( ∀ ( A p → X, m A ) ∈ Ob( G - x s et /QX )) . (iii) T o define f • : Ω Q ( X ) → Ω Q ( Y ), remark that ther e exists a ring is omor- phism Ω Q ( X ) ∼ = − → Ω( X × Q ) , which takes ( A p → X , m A ) to ( A ( p,m A ) − → X × Q ). F rom this, we can imp ort the m ultiplicative tra nsfers in to Ω Q from those for the o rdinary Burnside ring functor Ω. Let X Y X × Q X × Y Π f ( X × Q ) Π f ( X × Q ) exp f   p X o o e o o f ′   π o o be an exp onential diagra m, a nd let µ f : Π f ( X × Q ) → Y × Q be a morphism defined by µ f ( y , σ ) = ( y , Y x ∈ f − 1 ( y ) σ ( x )) for any ( y , σ ) ∈ Π f ( X × Q ). Remark that Π f ( X × Q ) a nd e is Π f ( X × Q ) =    ( y , σ )       y ∈ Y , σ : f − 1 ( y ) → X × Q is a map o f sets , p X ◦ σ = id f − 1 ( y )    , and e is defined by e ( x, ( y , σ )) = σ ( x ) for any ( x, ( y , σ )) ∈ X × Y Π f ( X × Q ). T AMBARIZA T ION OF A M AC KEY FUNCTOR 21 Using these, define f • : Ω Q ( X ) → Ω Q ( Y ) by f • = (Ω Q ( X ) ∼ = Ω( X × Q ) e ∗ − → Ω( X × Y Π f ( X × Q )) f ′ • − → Ω(Π f ( X × Q )) ( µ f ) + − → Ω( Y × Q ) ∼ = Ω Q ( Y )) , where f ′ • is the m ultiplica tive tra nsfer for the ordinary B ur nside ring functor defined in [1 3]. F or any S ∈ Ob( G set / ( X × Y Π f ( X × Q ))), its image f ′ • ( S ) is de fined to be Π f ′ ( S ), by the expo nential dia g ram: X × Y Π f ( X × Q ) Π f ( X × Q ) f ′   S T Π f ′ ( S ) exp o o o o   o o Prop ositio n 3.2. F or any finite G -monoid Q , ther e is a natu ra l isomorphi sm of T amb ar a functors ϕ : T P Q ∼ = − → Ω Q . Pr o of. F or each X ∈ Ob( G set ), categ ories G - x set /QX a nd P Q - G set /X in Defini- tion 2.10 are obviously equiv alent, throug h the na tur al functor G - xset /Q X → P Q - G set /X ob ject : ( A p → X, m A ) 7→ ( A p → X, m A ) , morphism : f 7→ f . Since this functor is co mpatible with sums and pro ducts , it yields a ring isomo r- phism ϕ X : T P Q ( X ) ∼ = − → Ω Q ( X ). So it remains to show ϕ = { ϕ X } X ∈ Ob( G set ) is compatible with the s tructure mor phisms of T P Q and Ω Q . Let f ∈ G set ( X , Y ) b e any mor phism. Obviously ϕ is co mpa tible with f + and f ∗ . Th us it suffices to sho w the compatibility with the multiplicativ e transfer f • . By the construction of f • = ( T P Q ) • ( f ) = ( γ S P Q ) • ( f ), it is the only alg ebraic ma p (see [5] for the definition) which makes the following diag ram commutativ e. S P Q ( X ) T P Q ( X ) S P Q ( Y ) T P Q ( Y )   / / ( S P Q ) • ( f )     / / ( T P Q ) • ( f )    Since f • : Ω Q ( X ) → Ω Q ( Y ) is also an algebra ic map as is a m ultiplicative trans- fer, it suffices to show the c o mmu tativity o f the fo llowing dia gram. S P Q ( X ) T P Q ( X ) S P Q ( Y ) T P Q ( Y ) Ω Q ( X ) Ω Q ( Y )   / / ( S P Q ) • ( f )     / / ϕ X ∼ = / / ϕ Y ∼ = / / f •   22 HIR OYUKI NAKAOKA T ake any element ( A p → X, m A ) ∈ S P Q ( X ). W e use the notation in Definition 3.1. If we let X Y X × Q X × Y Π f ( X × Q ) Π f ( X × Q ) exp f   p X o o e o o f ′   π o o A ′ Z Π exp q o o λ o o ρ    o o be tw o ex po nent ial diagrams, where X × Q A X × Y Π f ( X × Q ) A ′  ( p,m A )   e o o q   e ′ o o is a pull-back, then by Lemma 2 .9, the following diagra m b ecomes an e xp o nential diagram. X Y A Z Π exp f   p o o e ′ ◦ λ o o ρ   π ◦  o o Thu s, comp os ing appropr iate isomorphis ms , w e may assume Π = Π f ( A ) =    ( y , σ )       y ∈ Y , σ : f − 1 ( y ) → A is a map of sets , p ◦ σ = id f − 1 ( y )    , Z = X × Y Π f ( A ) . Let p 1 : X × Q → Q and p 2 : Y × Q → Q b e the pro jections onto Q . By the definition o f f • : Ω Q ( X ) → Ω Q ( Y ), we have f • ◦ ϕ X ( A p → X , m A ) = ( µ f ) + f ′ • e ∗ ( A ( p,m A ) − → X × Q ) = ( µ f ) + f ′ • ( A ′ q → X × Π f ( X × Q )) = ( µ f ) + (Π  → Π f ( X × Q )) = (Π µ f ◦  − → Y × Q ) ∈ Ω( Y × Q ) ( ∼ = Ω Q ( Y )) . On the other ha nd, ( ϕ Y ◦ ( S P Q ) • )( f )( A p → X, m A ) = (Π π ◦  − → Y , ρ ∗ ( e ′ ◦ λ ) ∗ ( m A )) = (Π π ◦  − → Y , ρ ∗ ( m A ◦ e ′ ◦ λ )) = (Π π ◦  − → Y , ρ ∗ ( p 1 ◦ e ◦ ( X × Y  ))) ∈ Ω Q ( Y ) . Thu s it rema ins to show these t wo elements co incide, under the isomorphism Ω( Y × Q ) ∼ = Ω Q ( Y ). Since pr Y ◦ µ Y ◦  = π ◦  , it suffices to show (3.1) p 2 ◦ µ f ◦  = ρ ∗ ( p 1 ◦ e ◦ ( X × Y  )) . Let ( y , σ ) ∈ Π f ( A ) = Π be a ny elemen t, where σ : f − 1 ( y ) → A is a map of s ets. Its image under  ca n b e written a s  ( y , σ ) = ( y , τ ) , T AMBARIZA T ION OF A M AC KEY FUNCTOR 23 with so me map o f sets τ : f − 1 ( y ) → X × Q . Then we hav e p 2 ◦ µ f ◦  ( y , σ ) = p 2 ◦ µ f ( y , τ ) = Y x ∈ f − 1 ( y ) ( p 1 ( τ ( x ))) , ρ ∗ ( p 1 ◦ e ◦ ( X × Y  ))( y , σ ) = Y x ∈ f − 1 ( y ) (( p 1 ◦ e ◦ ( X × Y  ))( x, ( y, σ ))) = Y x ∈ f − 1 ( y ) (( p 1 ◦ e )( x, ( y , τ )) = Y x ∈ f − 1 ( y ) ( p 1 ( τ ( x ))) , and (3 . 1) is s a tisfied.  3.2. Relatio n with the Witt-Burnsi de construction. Let W G ( R ) b e the Witt- Burnside ring a sso ciated to a ring R and a profinite group G . F or the definition, see [5]. If G is finite, as we assume in this section, then the Witt-Bur nside rings are rela ted to T am bara functors as follows. F act 3.3 (Theorem B , Theo rem 15 in [4]) . F or any finite gro up G , the ev aluatio n at G/e T am ( G ) → G - Ring ; T 7→ T ( G/e ) has a r ig ht adjoin t functor L G . Here , G - Ring denotes the categ ory of G -rings . If G acts trivia lly on a ring R , then for any subg roup H ≤ G , there is an iso morphism W H ( R ) ∼ = L G ( R )( G/H ) . R emark 3 .4 . If T is a T am bara functor on G , then T ( G/e ) carries a na tural G -ring structure induced from T ∗ of translatio ns . W e denote its G -fixed part by T ( G/e ) G . This gives a functor to Ring ev G : T am ( G ) → Ring ; T 7→ T ( G/e ) G , which we call the ‘ G -inv ariant ev aluation’ functor. Recall that the functor ta k ing the G -fixed par t G - Ring → Ri ng ; R 7→ R G is left adjoint to the natural functor Ring → G - Ring ; R 7→ R triv , which endows a r ing with the trivial G -action. Here, R triv denotes a ring R with the triv ial G -actio n. Thus by Brun’s theor em, we obtain: Corollary 3.5. G -invariant evaluation functor ev G : T am ( G ) → Ring has a left adjoint functor, which we denote by W : Ring → T am ( G ) , satisfying ( W ( R ))( G/H ) ∼ = W H ( R ) for any ring R and for e ach H ≤ G . Remark that the catego ry of s e mi- Mack ey functors a lso admits the ‘ G -in v ariant ev aluation’ functor ev G : S Mack ( G ) → Mon ; M 7→ M ( G/e ) G , 24 HIR OYUKI NAKAOKA compatible with that on T am ( G ): (3.2) T = ( T ∗ , T + , T • ) ( T ∗ , T • ) T am ( G ) Ring SMack ( G ) Mon ev G / / µ ◦U   multiplicative   part   ev G / /  _   W e construct the left adjoint functor o f ev G : SMack ( G ) → M on in the following. Remark that (cf. [3]), to give a semi-Mack ey functor M on G is equiv alent to g ive a data M ( G/H ) ∈ Ob( Mon ) r H K ∈ Mon ( M ( G/H ) , M ( G/K )) (restriction) t H K ∈ Mon ( M ( G/K ) , M ( G/H )) (transfer) c g,H ∈ Mon ( M ( G/H ) , M ( G/ g H )) (conjuga tio n map) for ea ch K ≤ H ≤ G and g ∈ G , sa tisfying t H K ◦ t K L = t H L , r K L ◦ r H K = r H L ( ∀ L ≤ ∀ K ≤ ∀ H ≤ G ) c g 2 , g 1 H ◦ c g 1 ,H = c g 2 g 1 ,H ( ∀ g 1 , g 2 ∈ G, ∀ H ≤ G ) c g,H ◦ t H K = t g H g K ◦ c g,K ( ∀ g ∈ G, ∀ K ≤ ∀ H ≤ G ) c g,K ◦ r H K = r g H g K ◦ c g,H ( ∀ g ∈ G, ∀ K ≤ ∀ H ≤ G ) and the Mack ey co nditio n (3.3) r H L ◦ t H K = X h ∈ L \ H/K t L L ∩ h K ◦ c h,L h ∩ K ◦ r K L h ∩ K ( ∀ K ≤ ∀ H ≤ G ) . In this descriptio n, a morphism ϕ : M → N b et ween semi- Mack ey functors M and N corr esp onds to a family ϕ = { ϕ H } H ≤ G of mo noid homomo r phisms ϕ H : M ( G/H ) → N ( G/H ) compatible with conjugations, restrictions a nd transfers . Definition 3.6. Let Q b e a monoid. Define a Mack ey functor L Q by L Q ( G/H ) = Q ( ∀ H ≤ G ) r H K = [ H : K ] : Q → Q ( ∀ K ≤ ∀ H ≤ G ) t H K = id Q : Q → Q ( ∀ K ≤ ∀ H ≤ G ) c g,H = id Q : Q → Q ( ∀ g ∈ G, ∀ H ≤ G ) where [ H : K ] denotes the multiplication by the index [ H : K ]. R emark 3.7 . Let P Q be the fixed point functor, where Q is re garded as a G -mono id with the trivial G -actio n. Then L Q ( G/H ) = Q = P Q ( G/H ) for each H ≤ G , and L Q is the Mack ey functor whose restrictions a nd tra nsfers are reversed from P Q . Claim 3.8. The c orr esp ondenc e Q 7→ L Q forms a functor L : Mon → SMack ( G ) , which is left adj oint to ev G . If Claim 3.8 is proven, we o btain the following. T AMBARIZA T ION OF A M AC KEY FUNCTOR 25 Theorem 3.9. F or any finite gr oup G , ther e is an isomorphism of fun ctors (3.4) W ◦ Z [ − ] ∼ = T ◦ L . Mon SMack ( G ) Ring T am ( G ) L / / Z [ − ]   T   W / /  Pr o of. Supp ose Claim 3 .8 is shown. Then, each functor in (3 . 4) is left adjoint to the corres p o nding functor in (3 . 2). Thus Theorem 3.9 follows from the co mm uta tivit y of (3 . 2) and the uniqueness of the left a djoint functor.  Pr o of of Claim 3.8. Let Q 1 , Q 2 be t wo monoids. T o each mono id homomorphism θ : Q 1 → Q 2 , w e can naturally asso ciate a mor phism of semi-Mackey functors L θ : L Q 1 → L Q 2 by L θ ( G/H ) = θ : L Q 1 ( G/H ) → L Q 2 ( G/H ) ( ∀ H ≤ G ) , and obtain a functor L : Mon → SMack ( G ). T o show the adjoin tness, let Q ∈ Ob( Mon ) and M ∈ Ob( SMac k ( G )) be a ny pair of ob jects. It suffices to construct natural ma ps Φ : Mon ( Q , M ( G/e ) G ) → SMa ck ( G )( L Q , M ) Θ : SMack ( G )( L Q , M ) → Mon ( Q, M ( G/e ) G ) which ar e inv er ses to each other. First we define Θ. Let ϕ ∈ S Mack ( G )( L Q , M ) b e any mor phism. Since G acts trivially on L Q ( G/H ) = Q , the comp onent o f ϕ at G/e ϕ G/e : L Q ( G/e ) → M ( G/e ) factors throug h M ( G/e ) G . This gives a monoid ho momorphism Θ( ϕ ) : Q → M ( G/e ) G . Second, w e define Φ. Let θ ∈ Mon ( Q, M ( G/e ) G ) b e any morphis m. W e define Φ( θ ) = { Φ( θ ) H } H ≤ G by the comp osition Φ( θ ) H = ( L Q ( G/H ) = Q θ → M ( G/e ) G ֒ → M ( G/e ) t H e → M ( G/H )) , for ea ch H ≤ G . Claim 3.10. Φ( θ ) = { Φ( θ ) H } H ≤ G ∈ SMack ( G )( L Q , M ) . If Claim 3.10 is shown, then we ca n easily show Φ ◦ Θ( ϕ ) = ϕ ( ∀ ϕ ∈ SMack ( G )( L Q , M )) , Θ ◦ Φ( θ ) = θ ( ∀ θ ∈ Mon ( Q, M ( G/e ) G )) , and thus obtain the desire d adjoint is omorphism Mon ( Q , M ( G/e ) G ) ∼ = SMack ( G ) ( L Q , M ) . Thu s it remains to show Claim 3 .10, namely , to sho w the compatibility of { Φ( θ ) H } H ≤ G with the structure morphisms o f semi-Mack ey functor s L Q and M . 26 HIR OYUKI NAKAOKA (i) compa tibility with conjugation maps F or a ny g ∈ G and H ≤ G , we hav e c g,H ◦ Φ( θ ) H = Φ ( θ ) ( g H ) ◦ c g,H . This follo ws from the following commutativ e dia gram. Rema rk c g,e is the ident it y on M ( G/e ) G . L Q ( G/H ) Q M ( G/e ) G M ( G/e ) M ( G/H ) L Q ( G/ g H ) Q M ( G/e ) G M ( G/e ) M ( G/ g H ) θ / /   / / t H e / / θ / /   / / t g H e / / c g,H   id   id   c g,e   c g,H       (ii) compa tibility with inductions F or a ny sequence of subgr oups K ≤ H ≤ G , we have t H K ◦ Φ( θ ) K = Φ( θ ) H ◦ t H K . This immediately follows fr o m the commut ativity of the fol- lowing diagram. L Q ( G/K ) Q M ( G/e ) G M ( G/e ) M ( G/K ) L Q ( G/H ) Q M ( G/e ) G M ( G/e ) M ( G/H ) θ / /   / / t K e / / θ / /   / / t H e / / t H K   id   id   t H K      (iii) compa tibility with restrictio ns F or any s equence of subgroups K ≤ H ≤ G , we hav e r H K ◦ Φ( θ ) H = Φ( θ ) K ◦ r H K . This follows from the commutativit y of the following diagr am. L Q ( G/H ) Q M ( G/e ) G M ( G/e ) M ( G/H ) L Q ( G/K ) Q M ( G/e ) G M ( G/e ) M ( G/K ) θ / /   / / t H e / / θ / /   / / t K e / / r H K   [ H : K ]   [ H : K ]   r H K      (Remark that, from the Mack ey condition, w e hav e r H K ◦ t H e ( x ) = X h ∈ K \ H t K e ◦ c h,e ( x ) = X h ∈ K \ H t K e ( x ) = [ H : K ] t K e ( x ) for any x ∈ M ( G/e ) G .)  Theorem 3.9 also gives a functorial enhancemen t o f E lliott’s is o morphism. F or any monoid Q and profinite g roup G , Ellio tt defined B Q ( G ) to be the Grothendieck ring of the ca tegory of G - strings over Q . The category of G -string s over Q is defined as follows. Definition 3.11 ( §§ 2.2 in [6 ]) . Let G b e a profinite gr oup, and let Q b e an ob ject in Mon . A G - s tring over Q is a pair ( A, m A ) of (a) an almost finite G -set A , (b) a map o f sets m A : A → Q , w hich is constant on G -or bits in A . T AMBARIZA T ION OF A M AC KEY FUNCTOR 27 The categor y of G -s trings is denoted by G - String / Q . F or the genera l definition of an almost finit e G -set, see [6]. Since w e are now assuming G is finite, it is nothing other than a finite G -set. F or eac h pair of ob jects ( A, m A ) and ( B , m B ), their sums and pro ducts are defined by ( A, m A ) ∐ ( B , m B ) = ( A ∐ B , m A ∪ m B ) , ( A, m A ) × ( B , m B ) = ( A × B , m A ∗ m B ) , where m A ∗ m B is defined by ( m A ∗ m B )( a, b ) = m A ( a ) · m B ( b ) for an y ( a, b ) ∈ A × B . In the context of the Witt-Burnside construction, E lliott s how ed the following. F act 3.12 (Theore m 1.7 in [6]) . F or any profinite group G and any monoid Q , there is an is omorphism of rings B Q ( G ) ∼ = W G ( Z [ Q ]). When G is finite, o ur construction of T M generalizes the Elliott’s ring. In fa c t, B Q ( G ) is obviously isomorphic to the ev aluation of T of M = L Q as follows: Prop ositio n 3.1 3. L et G b e a finite gr oup. F or any monoid Q and any sub gr oup H ≤ G , ther e is a ring isomorp hism T L Q ( G/H ) ∼ = B Q ( H ) . Pr o of. Since there is an equiv alence o f categor ies L Q - G set / ( G/H ) → H - String /Q ( A p → G/H, m A ) 7→ ( p − 1 ( eH ) , m A | p − 1 ( eH ) ) preserving sums and pr o ducts, the corres po nding Gro thendieck rings beco me iso- morphic: T L Q ( G/H ) = K 0 ( L Q - G set / ( G/H )) ∼ = K 0 ( H - String /Q ) = B Q ( H ) .  Since T L Q is a T ambara functor by Theorem 2.12, this isomorphism gives a T ambara functor structure on B Q . Thus, apply ing Prop osition 3.13 to the natural isomorphism T ◦ L ∼ = W ◦ Z [ − ] in Theorem 3.9, we obtain an iso morphism of T ambara functors B Q ∼ = T L Q ∼ = W ◦ Z [ Q ] . This functoria l isomorphis m gives bac k the Elliott’s r ing isomor phism in F act 3.12, by the ev aluation at G/H for each H ≤ G . 3.3. Underlying Green functor structure. Definition 3. 14. Let Madd ( G ) b e the categ ory o f additive contra v ar iant functors F from G set to Mon . As in Definition 1.5, ‘additive’ means that F takes direct sums to direct pr o ducts. R emark 3.15 . By Remark 1 .6, Madd ( G ) ca n be identified with the catego ry of additive contra v ar iant functor s from G Set C to Mo n , and w e hav e a natural forgetful functor SMack ( G ) → Madd ( G ) ; ( M ∗ , M ∗ ) 7→ M ∗ . 28 HIR OYUKI NAKAOKA R emark 3.1 6 . By definition, a Mackey functor ( M ∗ , M ∗ ) is a Gr e en funct or if it is equipp e d with a bifunctor ia l cro ss pro duct, namely , if a bilinear map cr : M ( X ) × M ( Y ) → M ( X × Y ) is given for ea ch X , Y ∈ Ob( G Set C ), naturally in X , Y with resp ect to M ∗ and M ∗ ([3]). F or a T am bara functor ( T ∗ , T + , T • ), the bilinear maps cr : T ( X ) × T ( Y ) → T ( X × Y ) ; ( x, y ) 7→ T ∗ ( p X )( x ) · T ∗ ( p Y )( y ) , where p X : X × Y → X a nd p Y : X × Y → Y are the pro jections, are natura l with resp ect to T ∗ and T + . Thus ( T ∗ , T + ) bec o mes a Green functor. T his defines a forgetful functor T am ( G ) → Gr e en ( G ), and we hav e a commutativ e diagr am of forgetful functors . SMack ( G ) Madd ( G ) T am ( G ) Gr e en ( G ) ( M ∗ ,M ∗ ) 7→ M ∗ / / ( T ∗ ,T + ,T • ) 7→ ( T ∗ ,T • )   ( M ∗ ,M ∗ ) 7→ M ∗   ( T ∗ ,T + ,T • ) 7→ ( T ∗ ,T + ) / /  Pr o of. The pro o f will be als o found in [14]. W e briefly de mo nstrate the naturality of cr with res pec t to the c ov ariant part. T ake an y f ∈ G set ( X , X ′ ) and g ∈ G set ( Y , Y ′ ). If we let X p X ← X × Y p Y → Y , X ′ p X ′ ← X ′ × Y ′ p Y ′ → Y ′ , X ′ π X ′ ← X ′ × Y π Y → Y be the pro ducts, then X X × Y X ′ X ′ × Y  p X o o f   f × 1 Y   π X ′ o o , X ′ × Y Y X ′ × Y ′ Y ′  π Y / / 1 X ′ × g   g   p Y ′ / / are pull-backs, and p X ′ ◦ (1 X ′ × g ) = π X ′ , π Y ◦ ( f × 1 Y ) = p Y . By the pro jection for mula ([13]), for any x ∈ T ( X ) a nd y ∈ T ( Y ), we hav e ( f × g ) + (( p ∗ X x ) · ( p ∗ Y y )) = (1 X ′ × g ) + ( f × 1 Y ) + (( p ∗ X x ) · (( f × 1 Y ) ∗ π ∗ Y y ))) = (1 X ′ × g ) + ((( f × 1 Y ) + p ∗ X x ) · ( π ∗ Y y )) = (1 X ′ × g ) + (( π ∗ X ′ f ∗ x ) · ( π ∗ Y y )) = (1 X ′ × g ) + (((1 X ′ × g ) ∗ p ∗ X ′ f + x ) · ( π ∗ Y y )) = ( p ∗ X ′ f + x ) · ((1 X ′ × g ) + π ∗ Y y )) = ( p ∗ X ′ f + x ) · ( p ∗ Y ′ g + y ) . T ( X ) × T ( Y ) T ( X × Y ) T ( X ′ ) × T ( Y ′ ) T ( X ′ × Y ′ ) cr / / f + × g +   ( f × g ) +   cr / /   T o ea ch F ∈ Ob( Madd ( G ) ), Jaco bson a sso ciated a Green functor A F , which we call the F - Burn side ring functor , in the following wa y . T AMBARIZA T ION OF A M AC KEY FUNCTOR 29 Definition 3 .17 ([8], [9]) . Let G b e a finite gro up and let F b e any ob ject in Madd ( G ). F or each finite G - set X , define a category ( G, X , F ) by the following. (a) An ob ject in ( G, X, F ) is a pair ( A p → X , m A ) o f a morphism p be t ween finite G -sets A and X , and a n element m A ∈ F ( A ). (b) A morphism in ( G, X , F ) from ( A p → X , m A ) to ( A ′ p ′ → X , m A ′ ) is a mor- phism o f G -sets f : A → A ′ , such that p ′ ◦ f = p and F ( f )( m A ′ ) = m A . F act 3.18 (Theorem 5.2 in [8]) . The co rresp ondence F 7→ A F gives a functor A : Madd ( G ) → Gr e en ( G ) . Caution 3.19 . I n J acobson’s paper [8], the contra v a riant part of a (se mi- )Mack ey functor is denoted by M ∗ , and the cov a riant pa rt is denoted by M ∗ . Thus, M ∗ in [8] is our M ∗ , a nd M ∗ in [8] is our M ∗ .) Since w e are assuming Mon is the ca tegory of co mmu tative mono ids, the resulting Green functor A F bec omes commutative. Obviously , if M = ( M ∗ , M ∗ ) is a semi-Ma ck ey functor and if X is a finite G -set, then our categ o ry M - G set /X a grees with ( G, X , M ∗ ). Thus, under the iden tifica- tion in Remark 1 .6, w e can show that ther e exis ts an is omorphism o f Green functor s (( T M ) ∗ , ( T M ) ∗ ) ∼ = A ( M ∗ ) . In fact, this g ives a natural iso morphism of functors: SMack ( G ) Madd ( G ) T am ( G ) Gr e en ( G ) ( M ∗ ,M ∗ ) 7→ M ∗ / / T   A   ( T ∗ ,T + ,T • ) 7→ ( T ∗ ,T + ) / /  In this view, we c a n resta te Theorem 2.1 2 as follows. Theorem 2.12 ′ . Let G be a finite group and F b e an ob ject in Madd ( G ). If F is mor eov er a s e mi-Mack ey functor , namely , if there exists a semi-Mack ey functor with the contra v ariant part F , then A F has a structure o f a T a m bara functor. As an example, Theo rem 2 .12 ′ gives a T amb ara functor structure o n the Br auer ring functor in [8]. Remark that the Br auer ring functor is de fined as A f Br , where f Br is the additive co n trav ariant functor defined by the comp osition f Br = ( G set P E → Ring Br → A b ֒ → Mon ) , where E / D is a finite Galois ex tension of fields with Galois group G . Corollary 3.20. L et E /D b e a fi n ite Galois extension of fields with Galois gr oup G . Ther e is a T amb ar a functor T f Br on G , whose un derlying Gr e en funct or agr e es with A f Br . Pr o of. This immediately follows fr o m that f Br is in fact a Mack ey functor .  R emark 3.21 . More gener ally , if E /D is a finite Galois extension o f rings with Galois gro up G , then f Br ∈ Ob( Madd ( G ) ) ca n b e defined in a similar way , a nd the same re sult a s Cor ollary 3.20 holds. This follows from the fact that f Br b ecomes a Mack ey functor by Prop ositio n 1 in [7]. 30 HIR OYUKI NAKAOKA Combining Theorem 2.12 ′ with Theorem 3.13 in [9], the underlying Green functor structure is a lso w r itten by us ing Boltje’s ( − + )-construction. In [2], Bo ltje defined the functor ( − ) + from the categ ory Re s alg ( G ) of algebr a r estriction functors to the category of Green functor s on G ( − ) + : R es alg ( G ) → Gr e en ( G ) . In [9], we co nstructed a functor R : R add ( G ) → Res alg ( G ) ; F 7→ R F which satis fie s R F ( H ) = F ( G/H ) for any F ∈ Ob( R add ( G )) and H ≤ G . Let R add ( G ) b e the categ ory of additive co n trav ariant functors from G set to Ring . The pair of adjoint functors Ring → Mon , forg etful (mult iplicative part) Z [ − ] : Mon → Ring yields ano ther adjoint pair R add ( G ) → Madd ( G ) , forgetful Z [ − ] : Madd ( G ) → R add ( G ) by the comp osition ont o the co domains. Caution 3.2 2 . Although monoids a nd r ings in [9] were not assumed to b e commu- tative, the ab ov e a djoint prop er t y still holds. R emark 3.23 . In [9], R add ( G ) is v alued in the catego ry of not ne c essarily c ommu- tative rings, and therefo re Ra dd ( G ) gav e an equiv alence of categories in that case (Prop osition 3.12 in [9]). A t any r ate, re stricting Prop ositio n 3 .12 in [9 ] to our commutativ e case, we obta in a n isomorphism of functors ( R ◦ Z [ − ]) + ∼ = A . Madd ( G ) Gr e en ( G ) R add ( G ) R es alg ( G ) A / / Z [ − ]   ( − ) + O O R / /  Namely , for eac h F ∈ O b( Madd ( G )), we hav e a natural isomorphism of Green functors ( R Z [ F ] ) + ∼ = A F . 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