Mackey functors on compact closed categories

MA CKEY FUNCTORS ON COMP A CT CLOSED CA T EGORIES ELANGO P ANCHADCHARA M AND R OSS STREET De dic ated to the memory of Saund ers Mac L ane Abstract. W e dev elop and extend the the ory of Mack ey functors as an appli- cation of enriched category theo ry . W e define Mack ey f unctors on a lextensive category E and inv estigat e the properties o f the category of M ack ey func tors on E . W e sho w that it is a monoidal category and the m onoids are Green func- tors. M ac ke y functors are seen as providing a setting i n which mere n umerical equations o ccurring i n the theory of groups can be given a structural founda- tion. W e obtain an explicit description of the ob jects of the Cauch y completion of a monoidal functor and apply this to examine Morita equiv alence of Green functors. 1. Introduction Groups ar e used to ma thematically understa nd symmetry in nature and in math- ematics itself. Classica lly , gr oups were studied either directly or via their represen- tations. In the last 40 y ears, arising from the latter, gro ups have b een studied using Mack ey functors. Let k be a field. Let Rep ( G ) = R ep k ( G ) b e t he categor y of k -linea r repr esen- tations of the finite g roup G . W e w ill study the str ucture of a monoidal categ ory Mky ( G ) where the ob jects are calle d Mack ey functors . This provides an exten- sion of ordinary repr esentation theory . F or example, Rep ( G ) can b e re g arded a s a full reflective sub-ca tegory of Mky ( G ); the reflection is strong monoidal (= tenso r preserving ). Representations of G are equally representations o f the group algebra k G ; Mack ey functors ca n b e r egarded as r epresentations of the ” Ma ck ey alg ebra” constructed fro m G . While Rep ( G ) is compac t closed (= a utonomous monoida l), we a re only able to show that Mky ( G ) is star- autonomous in the sense of [Ba]. Mack ey functors and Green functors (which are monoids in Mky ( G )) have b een studied fairly ex tensively . They provide a setting in which mer e numerical equa tions o ccurring in gro up theory ca n b e g iven a structural foundatio n. One applicatio n has b e e n to provide r elations b etw een λ - and µ -inv ar iants in Iwasaw a theory and betw een Mordell-W eil groups, Sha fa revich-T ate gro ups, Selmer gro ups and z e ta functions of elliptic curves (see [BB]). Our pur p o se is t o g ive the theory of Mackey functor s a categorical simplification and generaliza tion. W e sp ea k of Mack ey functors o n a compact (= rigid = au- tonomous) closed categor y T . How ev er, when T is the catego r y Spn ( E ) of spans in a lextensive category E , we speak of Mackey functors on E . F urther, when E is the catego r y (t op os) o f finite G -sets, we spea k o f Mack ey functors on G . The authors are grateful for t he supp ort of the Australi an Rese arch Council Di sco ve ry Gran t DP0450767, and the first author f or the supp ort of an Australian In ternational Postgradua te Researc h Scho larship, and an In ternationa l Macqua rie Uni v ersity Researc h Sc holarship. 1 2 ELANGO P ANCHADCHARAM AND R OSS STREET Sections 2-4 set the stage for Lindner’s result [Li1] tha t Mack ey functors, a concept g oing back at lea st as far as [Gr ], [Dr] and [Di] in group representation theory , can b e regarded as functors o ut of the category of spans in a suitable category E . The impo rtant prope rty of the catego r y of spans is that it is compact closed. So, in Section 5 , we lo ok at the categ ory Mky o f additive functors from a gener al compac t clos ed catego ry T (with direct sums) to the c ategory o f k - mo dules. The c onv olution monoidal s tructure on M ky is describ ed; this general construction (due to Day [Da1]) agrees with the usua l tensor pro duct of Mack ey functors app earing , for example, in [Bo1]. In fact, again for genera l reasons, Mky is a closed category; the in ternal hom is describ ed in Section 6. V arious conv olution structures hav e b een studied by Lewis [Le] in the co ntext of Mack ey functors for compact Lie groups mainly to pr ovide coun ter examples to familiar b ehaviour. Green functors a re int ro duced in Sectio n 7 as the mo noids in Mky . An ea sy construction, due to Dress [Dr], which cr eates new Ma ck ey functors from a given one, is describ ed in Section 8. W e use the (lax) centre construction for mono idal categorie s to explain re s ults of [Bo2] and [Bo3] about when the Dress constructio n yields a Green functor. In Section 9 we apply the work of [Da4] to show that finite-dimensional Mack ey functors form a ∗ -a utonomous [Ba] full sub-monoida l category Mky fin of Mky . Section 11 is rather sp eculative about what the cor rect notion of Mack ey functor should b e for quantum groups. Our appr o ach to Morita theor y for Gre e n functors inv olves even mor e serious use of enr iched catego ry theory: esp ecially the theory of (tw o-sided) mo dules. So Section 12 r eviews this theory of mo dules and Section 13 adapts it to our con- text. Two Gr een functors a re Morita equiv alent when their Mky - enriched cate- gories of mo dules a re equiv alen t, and this happ ens, b y the genera l theory , when the Mky -enriched categories of Cauch y mo dules are equiv alen t. Section 14 provides a characterization of Cauc h y mo dules. 2. The comp act closed ca tegor y Spn ( E ) Let E b e a finitely complete category . Then the categ o ry Spn ( E ) can be defined as follows. The o b jects a re the o b jects o f the ca tegory E and mor phis ms U / / V are the iso mo rphism clas ses of sp ans fr o m U to V in the bicategor y of spans in E in the sense of [B´ e]. (Some prop er ties o f this bicategor y can be found in [CKS].) A sp an from U to V , in the sense of [B´ e], is a diagra m of tw o mo rphisms with a common domain S , as in ( s 1 , S, s 2 ) : S V . s 2   < < < < < < < U s 1          MACKEY FUNCTORS ON COMP A CT CLOSED CA TEGORIES 3 An isomorphism of tw o spa ns ( s 1 , S, s 2 ) : U / / V and ( s ′ 1 , S ′ , s ′ 2 ) : U / / V is an inv ertible arrow h : S / / S ′ such tha t s 1 = s ′ 1 ◦ h and s 2 = s ′ 2 ◦ h . S V s 2 # # F F F F F F F F F U s 1 { { x x x x x x x x x S ′ h ∼ =   s ′ 1 a a D D D D D D D D D s ′ 2 = = z z z z z z z z z The co mpo site of t wo spans ( s 1 , S, s 2 ) : U / / V and ( t 1 , T , t 2 ) : V / / W is defined to be ( s 1 ◦ pro j 1 , T ◦ S, t 2 ◦ pro j 2 ) : U / / W using the pull-back diagram as in S × V T = T ◦ S T pro j 2   9 9 9 9 9 9 W . t 2   9 9 9 9 9 9 S pro j 1         U s 1         V s 2   9 9 9 9 9 9 t 1         pb This is well defined since the pull-back is unique up to is omorphism. The identit y span (1 , U , 1 ) : U / / U is defined by U U 1   < < < < < < < U 1          since the co mpo site of it with a s pa n ( s 1 , S, s 2 ) : U / / V is given by the following diagram and is equa l to the span ( s 1 , S, s 2 ) : U / / V S S 1   9 9 9 9 9 9 V . s 2   9 9 9 9 9 9 U s 1         U 1         U 1   9 9 9 9 9 9 s 1         pb This defines the catego ry Spn ( E ). W e ca n write Spn ( E )( U, V ) ∼ = [ E / ( U × V )] where square brackets denote the is o morphism classe s of morphisms. Spn ( E ) b eco mes a monoidal categ o ry under the tensor pro duct Spn ( E ) × Spn ( E ) × / / Spn ( E ) defined by ( U, V )  / / U × V [ U S / / U ′ , V T / / V ′ ]  / / [ U × V S × T / / U ′ × V ′ ] . This us e s the cartesian pro duct in E yet is not the cartesian pro duct in Spn ( E ). It is also c ompact closed; in fact, w e ha v e t he following isomorphisms : Spn ( E )( U, V ) ∼ = 4 ELANGO P ANCHADCHARAM AND R OSS STREET Spn ( E )( V , U ) a nd Spn ( E )( U × V , W ) ∼ = Spn ( E )( U, V × W ). The second isomor- phism can b e shown b y the following diagram S W   < < < < < < < U × V          o o  / / S W   < < < < < < < U          V   o o  / / S V × W .   < < < < < < < U          3. Direct sums in Spn ( E ) Now w e assume E is lextensive. References for this notion are [Sc], [CL W ], and [CL]. A categor y E is called lextensive when it has finite limits and finite co pro ducts such tha t the functor E / A × E / B / / E / A + B ; X f   A , Y g   B  / / X + Y f + g   A + B is a n equiv alance of catego ries for all ob jects A and B . In a lextensive ca tegory , copro ducts ar e dis joint and univ ersal and 0 is strictly initial. Also we hav e that the canonical morphism ( A × B ) + ( A × C ) / / A × ( B + C ) is inv ertible. It follows that A × 0 ∼ = 0 . In Spn ( E ) the o b ject U + V is the dir ect sum of U and V . This can be shown as follows (where w e use lextensivity): Spn ( E )( U + V , W ) ∼ = [ E / (( U + V ) × W )] ∼ = [ E / (( U × W ) + ( V × W ))] ≃ [ E / ( U × W )] × [ E / ( V × W )] ∼ = Spn ( E )( U, W ) × Spn ( E )( V , W ); and so Spn ( E )( W , U + V ) ∼ = Spn ( E )( W, U ) × Spn ( E )( W, V ). Also in the catego ry Spn ( E ), 0 is the zero ob ject (b oth initial and terminal): Spn ( E )(0 , X ) ∼ = [ E / (0 × X )] ∼ = [ E / 0] ∼ = 1 and so Spn ( E )( X , 0) ∼ = 1. It follows that Spn ( E ) is a category with ho ms enriched in commu tative monoids. The addition of tw o spans ( s 1 , S, s 2 ) : U / / V and ( t 1 , T , t 2 ) : U / / V is given by ( ∇ ◦ ( s 1 + t 1 ) , S + T , ∇ ◦ ( s 2 + t 2 )) : U / / V as in S V s 2   < < < < < < < U s 1          + T V t 2   < < < < < < < U t 1          = S + T V + V s 2 + t 2   5 5 5 5 5 5 V . ∇   5 5 5 5 5 5 U + U s 1 + t 1         U ∇         [ s 1 ,t 1 ]   [ s 2 ,t 2 ]   Summarizing, Spn ( E ) is a monoidal commutativ e-monoid-enriched category . MACKEY FUNCTORS ON COMP A CT CLOSED CA TEGORIES 5 There ar e functors ( − ) ∗ : E / / Spn ( E ) a nd ( − ) ∗ : E op / / Spn ( E ) which are the iden tit y on ob jects and t ake f : U / / V to f ∗ = (1 U , U, f ) and f ∗ = ( f , U, 1 U ), resp ectively . F or an y tw o arrows U f / / V g / / W in E , w e ha v e ( g ◦ f ) ∗ ∼ = g ∗ ◦ f ∗ as w e see from the following diagram U V f   ; ; ; ; ; W . g   ; ; ; ; ; U 1        U 1        V f   ; ; ; ; ; 1        pb Similarly ( g ◦ f ) ∗ ∼ = f ∗ ◦ g ∗ . 4. Mackey functors on E A Mackey functor M fr om E to the categor y Mo d k of k - mo dules consists of t wo functor s M ∗ : E / / Mo d k , M ∗ : E op / / Mo d k such tha t: (1) M ∗ ( U ) = M ∗ ( U ) (= M ( U )) for all U in E (2) for all pullbacks P V q / / W s   U p   r / / in E , the square (which w e call a Mackey squar e ) M ( U ) M ( W ) M ∗ ( r ) / / M ( V ) M ∗ ( s ) O O M ( P ) M ∗ ( p ) O O M ∗ ( q ) / / commutes, and (3) for all copro duct diagr ams U i / / U + V V j o o in E , the diagra m M ( U ) M ∗ ( i ) / / M ( U + V ) M ∗ ( i ) o o M ∗ ( j ) / / M ( V ) M ∗ ( j ) o o is a dire c t sum situatio n in Mo d k . (This implies M ( U + V ) ∼ = M ( U ) ⊕ M ( V ).) 6 ELANGO P ANCHADCHARAM AND R OSS STREET A morphism θ : M / / N of Mack ey functors is a family of mor phisms θ U : M ( U ) / / N ( U ) for U in E which defines natural transforma tions θ ∗ : M ∗ / / N ∗ and θ ∗ : M ∗ / / N ∗ . Prop ositio n 4. 1. (Lindner [Li1] ) The c ate gory Mky ( E , Mo d k ) of Mackey fun c- tors, fr om a lextensive c ate go ry E t o the c ate gory Mo d k of k -m o du les, is e qu ivalent to [ Spn ( E ) , M o d k ] + , the c ate gory of c opr o duct-pr eserving functors. Pr o of. Let M be a Ma ck ey functor from E to Mo d k . Then we have a pa ir ( M ∗ , M ∗ ) such that M ∗ : E / / Mo d k , M ∗ : E op / / Mo d k and M ( U ) = M ∗ ( U ) = M ∗ ( U ). Now de fine a functor M : Spn ( E ) / / Mo d k by M ( U ) = M ∗ ( U ) = M ∗ ( U ) and M     S V s 2   9 9 9 9 9 9 U s 1             =  M ( U ) M ( S ) M ∗ ( s 1 ) / / M ( V ) M ∗ ( s 2 ) / /  . W e need to see that M is well-defined. If h : S / / S ′ is an isomorphism, then the following dia gram S ′ S ′ 1 / / S ′ 1   S h − 1   h / / is a pull ba ck diagr am. Ther efore M ∗ ( h − 1 ) = M ∗ ( h ) and M ∗ ( h − 1 ) = M ∗ ( h ). This gives, M ∗ ( h ) − 1 = M ∗ ( h ). So if h : ( s 1 , S, s 2 ) / / ( s ′ 1 , S ′ , s ′ 2 ) is an is omorphism of spans, we ha ve the following commutativ e diagram. M ( U ) M ∗ ( s 1 ) ? ?           M ∗ ( s ′ 1 )   ? ? ? ? ? ? ? ? ? ? M ( S ) M ∗ ( s 2 )   ? ? ? ? ? ? ? ? ? ? M ∗ ( h )   M ( S ′ ) M ∗ ( s ′ 2 ) ? ?           M ∗ ( h ) O O M ( V ) Therefore we get M ∗ ( s 2 ) M ∗ ( s 1 ) = M ∗ ( s ′ 2 ) M ∗ ( s ′ 1 ) . MACKEY FUNCTORS ON COMP A CT CLOSED CA TEGORIES 7 F rom this definition M b eco mes a functor, since M           P T p 2   < < < < < < < W t 2   < < < < < < < S p 1          U s 1          V s 2   < < < < < < < t 1          pb           = M ( U ) M ( P ) M ∗ ( p 1 s 1 ) / / M ( W ) M ∗ ( t 2 p 2 ) / / M ( S ) M ∗ ( s 1 )   9 9 9 9 9 9 9 M ( V ) M ∗ ( s 2 )   9 9 9 9 9 9 9 M ( T ) M ∗ ( t 1 ) B B        M ∗ ( t 2 ) E E       M ∗ ( p 1 ) C C       M ∗ ( p 2 )   < < < < < < < Mack ey = ( M ( U ) M ( V ) M ( s 1 ,S,s 2 ) / / M ( W ) M ( t 1 ,T ,t 2 ) / / ) , where P = S × V T and p 1 and p 2 are the pro jections 1 a nd 2 resp ectively , so that M (( t 1 , T , t 2 ) ◦ ( s 1 , S, s 2 )) = M ( t 1 , T , t 2 ) ◦ M ( s 1 , S, s 2 ) . The v alue of M a t the ident ity span (1 , U, 1) : U / / U is given b y M     U U 1   < < < < < < < U 1              = ( M ( U ) M ( U ) 1 / / M ( U ) 1 / / ) = (1 : M ( U ) M ( U ) / / ) . Condition (3) on the Mack ey functor clea rly is equiv alen t to the requirement that M : Spn ( E ) / / Mo d k should pres e rve copro ducts. Conv ersely , let M : S pn ( E ) / / Mo d k be a functor. Then we c a n define tw o functors M ∗ and M ∗ , referr ing to the diagr a m E Spn ( E ) ( − ) ∗ / / Mo d k , M / / E op ( − ) ∗ 7 7 n n n n n n n n by putting M ∗ = M ◦ ( − ) ∗ and M ∗ = M ◦ ( − ) ∗ . The Mack ey square is obtained by us ing the functoriality of M on the comp os ite spa n s ∗ ◦ r ∗ = ( p, P , q ) = q ∗ ◦ p ∗ . The remaining details ar e routine.  5. Tensor product of Mackey functors W e now work with a gener al compact closed category T with finite products. It follows (see [Ho]) that T has direc t sums and therefor e that T is enr iched in the monoidal category V of comm utativ e monoids. W e write ⊗ for the tensor pro duct in T , write I for the unit, and write ( − ) ∗ for the dual. The main exa mple we have in mind is Spn ( E ) as in the last s ection wher e ⊗ = × , I = 1 , and V ∗ = V . A Mack ey functor on T is a n additive f unctor M : T / / Mo d k . Let us review the mono idal s tructure on the categor y V of commutativ e monoids ; the binary op eratio n of the monoids will b e written additively . It is mono ida l clo sed. F or A, B ∈ V , the c o mm utative mo no id [ A, B ] = { f : A / / B | f is a co mm utative monoid morphism } , 8 ELANGO P ANCHADCHARAM AND R OSS STREET with p oint wise addition, provides the internal hom and there is a tensor pro duct A ⊗ B s atisfying V ( A ⊗ B , C ) ∼ = V ( A, [ B , C ]) . The construc tio n of the tensor pro duct is as follows. The free comm utativ e monoid F S o n a set S is F S = { u : S / / N | u ( s ) = 0 for all but a finite num ber of s ∈ S } ⊆ N S . F or any A, B ∈ V , A ⊗ B = F ( A × B ) / ( a + a ′ , b ) ∼ ( a, b ) + ( a ′ , b ) ( a, b + b ′ ) ∼ ( a, b ) + ( a, b ′ ) ! . W e r egard T and Mo d k as V -categories . E very V -functor T / / Mo d k pre- serves finite direct sums. So [ T , Mo d k ] + is the V -categ o ry of V -functors. F or each V ∈ V a nd X an ob ject of a V -categor y X , we w r ite V ⊗ X fo r the ob ject (when it exists) satisfying X ( V ⊗ X , Y ) ∼ = [ V , X ( X, Y )] V -naturally in Y . Also the co end we use is in the V -enriched se nse: for the functor T : C op ⊗ C / / X , we hav e a coequa lizer X V ,W C ( V , W ) ⊗ T ( W, V ) / / / / X V T ( V , V ) / / Z V T ( V , V ) when the copro ducts and tensors exist. The tensor pro duct of Mack ey functors can b e defined by convolution (in the sense of [Da1]) in [ T , Mo d k ] + since T is a monoida l categor y . F or Mack ey functors M and N , the tens o r pro duct M ∗ N can b e wr itten as follows: ( M ∗ N )( Z ) = Z X,Y T ( X ⊗ Y , Z ) ⊗ M ( X ) ⊗ k N ( Y ) ∼ = Z X,Y T ( Y , X ∗ ⊗ Z ) ⊗ M ( X ) ⊗ k N ( Y ) ∼ = Z X M ( X ) ⊗ k N ( X ∗ ⊗ Z ) ∼ = Z Y M ( Z ⊗ Y ∗ ) ⊗ k N ( Y ) . the last tw o isomorphisms are given b y the Y oneda lemma. The Burnside functor J is defined to b e the Mack ey functor on T taking an ob ject U of T to the free k -mo dule on T ( I , U ). The Burns ide functor is the unit for the tensor pro duct o f the categ ory Mky . This conv olution satisfies the necessary commutativ e and asso ciative conditions for a symmetric monoidal c ategory (see [Da1]). [ T , Mo d k ] + is also an ab elian category (see [F r]). When T and k are understo o d, we simply wr ite Mky for this ca teg ory [ T , Mo d k ] + . MACKEY FUNCTORS ON COMP A CT CLOSED CA TEGORIES 9 6. The Hom functor W e now make explicit the c losed str uc tur e on Mky . The Hom Mackey functor is defined by taking its v a lue at the Mackey functors M and N to b e Hom( M , N )( V ) = Mky ( M ( V ∗ ⊗ − ) , N ) , functorially in V . T o see that this hom ha s the usual universal pr o p erty with resp ect to tensor, notice that w e ha ve the natural bijections b elow (re presented by horizontal lines). ( L ∗ M )( U ) / / N ( U ) natura l in U L ( V ) ⊗ k M ( V ∗ ⊗ U ) / / N ( U ) natura l in U and dinatural in V L ( V ) / / Hom k ( M ( V ∗ ⊗ U ) , N ( U )) dinatura l in U a nd natural in V L ( V ) / / Z U Hom k ( M ( V ∗ ⊗ U ) , N ( U )) natura l in V L ( V ) / / Mky ( M ( V ∗ ⊗ − ) , N ) natural in V W e can obtain another expr ession for the hom using the isomor phism T ( V ⊗ U, W ) ∼ = T ( U, V ∗ ⊗ W ) which s hows that we ha ve adjoint functors T ⊥ V ⊗− + + T . V ∗ ⊗− j j Since M and N are Mack ey functors on T , we obtain a diagra m T ⊥ V ⊗− + + N   < < < < < < < T V ∗ ⊗− k k M          Mo d k and an equiv alence of natural transfor mations M = ⇒ N ( V ⊗ − ) M ( V ∗ ⊗ − ) = ⇒ N . Therefore, the Hom Ma ckey functor is also given b y Hom( M , N )( V ) = Mky ( M , N ( V ⊗ − )) . 7. Green functors A Gr e en functor A on T is a Mack ey functor (that is, a copro duct pr eserving functor A : T / / Mo d k ) equipped with a monoidal structure made up o f a natura l transformatio n µ : A ( U ) ⊗ k A ( V ) / / A ( U ⊗ V ) , for which we use the notation µ ( a ⊗ b ) = a.b fo r a ∈ A ( U ), b ∈ A ( V ), a nd a morphism η : k / / A (1) , 10 ELANGO P ANCHADCHARAM AND R OSS STREET whose v alue at 1 ∈ k we denote by 1. Gr een functors are the mono ids in Mky . If A, B : T / / Mo d k are Green functors then we have a n isomor phism Mky ( A ∗ B , C ) ∼ = Nat U,V ( A ( U ) ⊗ k B ( V ) , C ( U ⊗ V )) . Referring to the sq ua re T ⊗ T Mo d k ⊗ Mo d k A ⊗ B / / Mo d k , ⊗ k   T ⊗   C / / we wr ite this more prec is ely as Mky ( A ∗ B , C ) ∼ = [ T ⊗ T , Mo d k ]( ⊗ k ◦ ( A ⊗ B ) , C ◦ ⊗ ) . The Bur nside functor J and Hom( A, A ) (for any Mack ey functor A ) a re mono ids in Mky and so are Gree n functors. W e denote by Grn ( T , Mo d k ) the ca tegory o f Green functors o n T . When T and k a re unders to o d, we simply write this as Grn (= Mon ( Mky )) co nsisting of the monoids in Mky . 8. Dress construction The Dr e s s constr uction ([Bo2], [Bo3]) provides a family of endofunctors D ( Y , − ) of the category Mky , ind exed b y the ob jects Y o f T . The Mac key functor defined as the comp osite T −⊗ Y / / T M / / Mo d k is deno ted by M Y for M ∈ Mky ; so M Y ( U ) = M ( U ⊗ Y ). W e then define the Dr ess c onstruction D : T ⊗ Mky / / Mky by D ( Y , M ) = M Y . The V -category T ⊗ Mky is monoidal via the p o int wise structure: ( X, M ) ⊗ ( Y , N ) = ( X ⊗ Y , M ∗ N ) . Prop ositio n 8.1. The Dr ess c onstruction D : T ⊗ Mky / / Mky is a str ong monoidal V -functor. MACKEY FUNCTORS ON COMP A CT CLOSED CA TEGORIES 11 Pr o of. W e need to show that D (( X , M ) ⊗ ( Y , N )) ∼ = D ( X , M ) ∗ D ( Y , N ); tha t is, M X ∗ M Y ∼ = ( M ∗ N ) X ⊗ Y . F or this we hav e the calculation ( M X ∗ N Y )( Z ) ∼ = Z U M X ( U ) ⊗ k N Y ( U ∗ ⊗ Z ) = Z U M ( U ⊗ X ) ⊗ k N ( U ∗ ⊗ Z ⊗ Y ) ∼ = Z U,V T ( V , U ⊗ X ) ⊗ M ( V ) ⊗ k N ( U ∗ ⊗ Z ⊗ Y ) ∼ = Z U,V T ( V ⊗ X ∗ , U ) ⊗ M ( V ) ⊗ k N ( U ∗ ⊗ Z ⊗ Y ) ∼ = Z V M ( V ) ⊗ k N ( V ∗ ⊗ X ⊗ Z ⊗ Y ) ∼ = ( M ∗ N )( Z ⊗ X ⊗ Y ) ∼ = ( M ∗ N ) X ⊗ Y ( Z ) . Clearly we ha v e D ( I , J ) ∼ = J . The coherenc e conditions ar e readily chec k ed.  W e shall analyse this situation more fully in Remark 8.5 b elow. W e a re interested, after [Bo2], in when the Dress construction induces a family of endofunctors on t he category Grn of Green functors. That is to say , when is there a natural structure o f Green functor on A Y = D ( Y , A ) if A is a Green functor? Since A Y is the comp osite T −⊗ Y / / T A / / Mo d k with A monoidal, what we r equire is that − ⊗ Y sho uld b e monoidal (since mo noidal functors comp ose). F or this we use Theore m 3.7 of [DPS]: if Y is a monoid in the lax c entr e Z l ( T ) of T then − ⊗ Y : T / / T is c anonic al ly monoidal. Let C be a monoidal category . The lax ce ntre Z l ( C ) of C is defined to have ob jects the pairs ( A, u ) where A is an ob ject of C and u is a natural family of morphisms u B : A ⊗ B / / B ⊗ A such that the following t w o diagrams commute A ⊗ B ⊗ C B ⊗ C ⊗ A u B ⊗ C / / B ⊗ A ⊗ C u B ⊗ 1 C # # G G G G G G G G G G 1 B ⊗ u C ; ; w w w w w w w w w w A ⊗ I I ⊗ A u I / / A . ∼ = { { w w w w w w w w w w ∼ = # # G G G G G G G G G G Morphisms o f Z l ( C ) ar e mor phisms in C compatible with the u . The tensor pro duct is defined by ( A, u ) ⊗ ( B , v ) = ( A ⊗ B , w ) where w C = ( u C ⊗ 1 B ) ◦ (1 A ⊗ v C ). The centre Z ( C ) o f C consists o f the ob jects ( A, u ) o f Z l ( C ) with ea ch u B inv ertible. It is p ointed out in [DPS] that, when C is cartesian mono idal, an o b ject of Z l ( C ) is merely an o b ject A o f C together with a natura l family A × X / / X . Then we hav e the natural bijections b elow (represented b y hor izontal lines) for C cartesian 12 ELANGO P ANCHADCHARAM AND R OSS STREET closed: A × X / / X natural in X A / / [ X , X ] dinatural in X A / / Z X [ X , X ] in C . Therefore we obtain an equiv a lence Z l ( C ) ≃ C / R X [ X , X ] . The internal ho m in E , the catego ry of finite G -sets for the finite gro up G , is [ X , Y ] which is the set of functions r : X / / Y with ( g .r )( x ) = g r ( g − 1 x ). The G -set R X [ X , X ] is defined by Z X [ X , X ] =  r = ( r X : X − → X )    f ◦ r X = r Y ◦ f for all G -maps f : X − → Y  with ( g .r ) X ( x ) = g r X ( g − 1 x ). Lemma 8.2. The G -set Z X [ X , X ] is isomorphic to G c , which is t he set G made a G -set by c onjugatio n action. Pr o of. T ake r ∈ R X [ X , X ]. The n we ha v e the commutativ e square G G r G / / X ˆ x   X ˆ x   r X / / where ˆ x ( g ) = g x fo r x ∈ X . So we see that r X is determined by r G (1) and ( g .r ) G (1) = g r G ( g − 1 1) = g r G ( g − 1 ) = g r G (1) g − 1 .  As a consequence of this Lemma, we hav e Z l ( E ) ≃ E /G c where E /G c is the category o f cro ssed G -sets of F reyd-Y etter ([FY1], [FY2]) who showed that E / G c is a braided monoidal categ o ry . O b jects are pair s ( X , | | ) where X is a G -se t and | | : X / / G c is a G -set morphism (“ equiv a riant function”) meaning | gx | = g | x | g − 1 for g ∈ G , x ∈ X . The mo rphisms f : ( X, | | ) / / ( Y , | | ) are functions f such that the following diagram comm utes. X Y f / / G c | | } } z z z z z z z z z z | | ! ! D D D D D D D D D D That is, f ( g x ) = g f ( x ). T ensor pro duct is defined by ( X, | | ) ⊗ ( Y , | | ) = ( X × Y , k k ) , where k ( x, y ) k = | x || y | . MACKEY FUNCTORS ON COMP A CT CLOSED CA TEGORIES 13 Prop ositio n 8 .3. [DPS, Theo rem 4.5] T he c entr e Z ( E ) of the c ate gory E is e quiv- alent t o the c ate gory E /G c of cr osse d G -set s . Pr o of. W e ha ve a fully faithful f unctor Z ( E ) / / Z l ( E ) and so Z ( E ) / / E /G c . On the other hand, let | | : A / / G c be an ob ject of E / G c ; so | g a | g = g | a | . Then the corres p o nding o b ject of Z l ( E ) is ( A, u ) wher e u X : A × X / / X × A with u X ( a, x ) = ( | a | x, a ) . How ev er this u is inv ertible since w e see that u X − 1 ( x, a ) = ( a, | a | − 1 x ) . This prov es the prop osition.  Theorem 8.4 . [Bo3, Bo2] If Y is a monoid in E /G c and A is a Gr e en functor for E over k t hen A Y is a Gr e en functor for E over k , wher e A Y ( X ) = A ( X × Y ) . Pr o of. W e hav e Z ( E ) ≃ E /G c , so Y is a monoid in Z ( E ). This implies − × Y : E / / E is a mo noidal functor (se e Theorem 3.7 of [DPS]). It als o pr eserves pullbacks. So − × Y : Spn ( E ) / / Spn ( E ) is a mono idal functor . If A is a Green functor for E ov er k then A : Spn ( E ) / / Mo d k is monoidal. Then we get A Y = A ◦ ( − × Y ) : Spn ( E ) / / Mo d k is monoidal and A Y is indeed a Green functor for E ov er k .  R emark 8.5 . The r eader may hav e noted that Pro p o sition 8.1 implies that D tak es monoids to monoids. A monoid in T ⊗ M ky is a pair ( Y , A ) wher e Y is a monoid in T and A is a Green functor; so in this cas e , we hav e that A Y is a Green functor. A monoid Y in E is certainly a monoid in T . Since E is cartesia n monoidal (and so symmetric), each mo no id in E gives one in the cen tre. How ever, not every monoid in the centre ar ises in this way . The full r esult b ehind Pr op osition 8.1 and the centre situation is: the Dress constr uction D : Z ( T ) ⊗ Mky / / Mky is a strong monoidal V -functor; it is merely mono ida l when the cen tre is replaced by the lax centre. It follows that A Y is a Green functor whenever A is a Green functor and Y is a monoid in the lax c ent re of T . 9. Finite dimensional Mackey functors W e make the following further a ssumptions on the symmetr ic compact c losed category T with finite direct sums: • there is a finite set C of ob jects of T s uch that every ob ject X of T can be written a s a direct sum X ∼ = n M i =1 C i with C i in C ; and • each hom T ( X , Y ) is a finitely genera ted comm utative mo noid. 14 ELANGO P ANCHADCHARAM AND R OSS STREET Notice that these assumptions hold when T = Spn ( E ) where E is the categor y of finite G -sets for a finite group G . In this case we can take C to cons is t of a representative set o f connected (transitive) G -sets. Moreover, the spans S : X / / Y with S ∈ C g enerate the monoid T ( X, Y ). W e also fix k to b e a field and write V ect in place of M o d k . A Mack ey functor M : T / / V ect is called fin ite dimensional when eac h M ( X ) is a finit e-dimensiona l v ector spa ce. W rite Mky fin for the full sub categor y of M ky consisting of these. W e regard C as a full subcategor y o f T . The inclusion functor C / / T is dens e and the density colimit prese ntation is preser ved b y all additive M : T / / V ect . This is shown as follo ws: Z C T ( C, X ) ⊗ M ( C ) ∼ = Z C T ( C, n M i =1 C i ) ⊗ M ( C ) ∼ = n M i =1 Z C T ( C, C i ) ⊗ M ( C ) ∼ = n M i =1 Z C C ( C, C i ) ⊗ M ( C ) ∼ = n M i =1 M ( C i ) ∼ = M ( n M i =1 C i ) ∼ = M ( X ) . That is, M ∼ = Z C T ( C, − ) ⊗ M ( C ) . Prop ositio n 9. 1. The ten s or pr o duct of finite-dimensional Mackey functors is finite dimensional. Pr o of. Using the last isomor phism, w e hav e ( M ∗ N )( Z ) = Z X,Y T ( X ⊗ Y , Z ) ⊗ M ( X ) ⊗ k N ( Y ) ∼ = Z X,Y ,C,D T ( X ⊗ Y , Z ) ⊗ T ( C , X ) ⊗ T ( D, Y ) ⊗ M ( C ) ⊗ k N ( D ) ∼ = Z C,D T ( C ⊗ D , Z ) ⊗ M ( C ) ⊗ k N ( D ) . If M and N a re finite dimensional then so is the vector space T ( C ⊗ D, Z ) ⊗ M ( C ) ⊗ k N ( D ) (since T ( C ⊗ D , Z ) is finitely generated). Also the c o end is a quotient o f a finite direc t sum. So M ∗ N is finite dimens io nal.  It follows tha t Mky fin is a mono idal sub category of Mky (since the Burns ide functor J is certainly finite dimensiona l under our a ssumptions on T ). MACKEY FUNCTORS ON COMP A CT CLOSED CA TEGORIES 15 The promono ida l str ucture o n Mky fin represented by this mo noidal structure can b e express ed in man y wa ys: P ( M , N ; L ) = Mky fin ( M ∗ N , L ) ∼ = Nat X,Y ,Z ( T ( X ⊗ Y , Z ) ⊗ M ( X ) ⊗ k N ( Y ) , L ( Z )) ∼ = Nat X,Y ( M ( X ) ⊗ k N ( Y ) , L ( X ⊗ Y )) ∼ = Nat X,Z ( M ( X ) ⊗ k N ( X ∗ ⊗ Z ) , L ( Z )) ∼ = Nat Y ,Z ( M ( Z ⊗ Y ∗ ) ⊗ k N ( Y ) , L ( Z )) . F o llowing the terminology of [DS1], w e say that a promonoidal category M is ∗ - autonomous when it is equipped with a n equiv a lence S : M op / / M and a natur al isomorphism P ( M , N ; S ( L )) ∼ = P ( N , L ; S − 1 ( M )) . A monoidal ca tegory is ∗ -autonomo us when the as so ciated promonoida l catego ry is. As a n application of the work of Day [Da4] we obtain that Mky fin is ∗ -a utonomous. W e shall give the details. F o r M ∈ Mky fin , define S ( M )( X ) = M ( X ∗ ) ∗ so that S : Mky op fin / / Mky fin is its own inverse equiv alence. Theorem 9.2 . The monoidal c ate gory M ky fin of finite-dimensional Mackey func- tors on T is ∗ -autonomous. Pr o of. With S defined a s above, w e ha ve the ca lculation: P ( M , N ; S ( L )) ∼ = Nat X,Y ( M ( X ) ⊗ k N ( Y ) , L ( X ∗ ⊗ Y ∗ ) ∗ ) ∼ = Nat X,Y ( N ( Y ) ⊗ k L ( X ∗ ⊗ Y ∗ ) , M ( X ) ∗ ) ∼ = Nat Z,Y ( N ( Y ) ⊗ k L ( Z ⊗ Y ∗ ) , M ( Z ∗ ) ∗ ) ∼ = Nat Z,Y ( N ( Y ) ⊗ k L ( Z ⊗ Y ∗ ) , S ( M )( Z )) ∼ = P ( N , L ; S ( M )) .  10. Cohomological Mackey functors Let k b e a field and G b e a finite gro up. W e are interested in the r elationship betw een ordinar y k - linear representations of G and Mack ey functors on G . W rite E fo r the catego ry of finite G -s e ts a s usual. W rite R for the categor y Rep k ( G ) of finite -dimensio nal k -linear repre sentations o f G . Each G -set X de ter mines a k -linear r epresentation k X of G by extending the action of G linearly o n X . This gives a functor k : E / / R . W e extend this to a functor k ∗ : T op / / R , where T = Spn ( E ), as follows. O n ob jects X ∈ T , define k ∗ X = kX . 16 ELANGO P ANCHADCHARAM AND R OSS STREET F o r a span ( u, S, v ) : X / / Y in E , the linear function k ∗ ( S ) : k Y / / k X is defined by k ∗ ( S )( y ) = X v ( s )= y u ( s ) ; this preserves the G -actions since k ∗ ( S )( g y ) = X v ( s )= g y u ( s ) = X v ( g − 1 s )= y g u ( g − 1 s ) = g k ∗ ( S )( y ) . Clearly k ∗ preserves copro ducts. By the usual argument (g oing back to Ka n, and the g eometric realiza tion a nd singular functor adjunction), we obtain a functor e k ∗ : R / / Mky ( G ) fin defined by e k ∗ ( R ) = R ( k ∗ − , R ) which we shall write as R − : T / / V ect k . So R X = R ( k ∗ X , R ) ∼ = G - Set ( X , R ) with the effect on the span ( u, S, v ) : X / / Y transp orting to the linea r function G - Set ( X , R ) / / G - Set ( Y , R ) which ta kes τ : X / / R to τ S : Y / / R wher e τ S ( y ) = X v ( s )= y τ ( u ( s )) . The functor e k ∗ has a left a djoint c oli m ( − , k ∗ ) : Mky ( G ) fin / / R defined by c oli m ( M , k ∗ ) = Z C M ( C ) ⊗ k k ∗ C where C runs ov er a full sub categor y C of T consis ting o f a repres e n tative set of connected G -sets. Prop ositio n 10.1. The functor e k ∗ : Rep k ( G ) / / Mky ( G ) is ful ly faithful. Pr o of. F or R 1 , R 2 ∈ R , a morphism θ : R − 1 / / R − 2 in Mky ( G ) is a fa mily o f linear functions θ X such that the following squa re commutes for all spans ( u, S, v ) : X / / Y in E . G - Set ( X , R 1 ) G - Set ( X , R 2 ) θ X / / G - Set ( Y , R 2 ) ( − ) S   G - Set ( Y , R 1 ) ( − ) S   θ Y / / Since G (with m ultiplication action) forms a full dense sub category of G - Se t , it follows that we obtain a unique morphism f : R 1 / / R 2 in G - Set such that f ( r ) = θ G ( ˆ r )(1) MACKEY FUNCTORS ON COMP A CT CLOSED CA TEGORIES 17 (where ˆ r : G / / R is defined by ˆ r ( g ) = g r for r ∈ R ); this is a sp ecial case of Y oneda ’s Lemma. Clearly f is line a r since θ G is. By taking Y = G, S = G a nd v = 1 G : G / / G , commutativit y of the ab ov e square yields θ X ( τ )( x ) = f ( τ ( x )); that is, θ X = e k ∗ ( f ) X .  An imp ortant pro per ty o f Mackey functors in the image of e k ∗ is that they are c oh omolo gic al in the sens e o f [W e], [Bo4] and [TW]. First we rec a ll some clas sical terminology asso cia ted with a Mack ey f unctor M on a g r oup G . F o r subgroups K ≤ H of G , we hav e the cano nical G - s et morphism σ H K : G/K / / G/H defined on the connected G -sets o f left cos ets by σ H K ( g K ) = g H . The linear functions r H K = M ∗ ( σ H K ) : M ( G/H ) / / M ( G/K ) and t H K = M ∗ ( σ H K ) : M ( G/K ) / / M ( G/H ) are called r estriction and tr ansfer (or tr ac e or induction ). A Mack ey functor M on G is called c oho molo gic al when each composite t H K r H K : M ( G/H ) / / M ( G/H ) is equal to m ultiplication b y the index [ H : K ] of K in H . W e supply a pro of of the following kno wn example. Prop ositio n 10. 2. F or e ach k -line ar r epr esentation R of G , t he Mackey fun ctor e k ∗ ( R ) = R − is c oho molo gic al. Pr o of. With M = R − and σ = σ H K , notice that the function t H K r H K = M ∗ ( σ ) M ∗ ( σ ) = M ( σ , G/ K, 1) M (1 , G/K , σ ) = M ( σ, G/K, σ ) takes τ ∈ E ( G/H , R ) to τ G/K ∈ E ( G/H, R ) where τ G/K ( H ) = X σ ( s )= H τ ( σ ( s )) = X σ ( s )= H τ ( H ) = ( X σ ( s )= H 1) τ ( H ) and s runs ov er the distinct g K with σ ( s ) = g H = H ; the num ber o f distinct g K with g ∈ H is of course [ H : K ]. So τ G/K ( xH ) = [ H : K ] τ ( xH ).  Lemma 1 0.3. The functor k ∗ : T op / / R is s t r ong monoidal. Pr o of. Clea rly the canonical iso morphisms k ( X 1 × X 2 ) ∼ = k X 1 ⊗ k X 2 , k 1 ∼ = k show that k : E / / R is strong monoida l. All that r emains to be seen is that these isomorphisms are natura l with resp ect to s pans ( u 1 , S 1 , v 1 ) : X 1 / / Y 1 , ( u 2 , S 2 , v 2 ) : X 2 / / Y 2 . This comes down to the bilinear ity of tensor pro duct: X v 1 ( s 1 )= y 1 v 2 ( s 2 )= y 2 u 1 ( s 1 ) ⊗ u 2 ( s 2 ) = X v 1 ( s 1 )= y 1 u 1 ( y 1 ) ⊗ X v 2 ( s 2 )= y 2 u 2 ( y 2 ) .  W e can now see that the adjunction c oli m ( − , k ∗ ) e k ∗  18 ELANGO P ANCHADCHARAM AND R OSS STREET fits the situation o f Day’s Reflection Theor em [Da2] a nd [Da3, page s 24 and 25]. F o r this, r ecall that a fully faithful functor Φ : A / / X in to a closed categ ory X is s aid to b e close d u n der exp onentiation when, for a ll A in A and X in X , the int ernal hom [ X , Φ A ] is isomor phic to an ob ject of the form Φ B for some B in A . Theorem 1 0.4. The functor c oli m ( − , k ∗ ) : Mky ( G ) fin / / R is st ro ng monoidal. Conse qu ently, e k ∗ : R / / Mky ( G ) fin is monoidal and close d under exp onentiation. Pr o of. The firs t sentence follows quite for mally from Lemma 10.3 and the theory of Day c onv olution; the main calculation is: c oli m ( M ∗ N , k ∗ )( Z ) = Z C ( M ∗ N )( C ) ⊗ k k ∗ C = Z C,X,Y T ( X × Y , C ) ⊗ M ( X ) ⊗ k N ( Y ) ⊗ k k ∗ C ∼ = Z X,Y M ( X ) ⊗ k N ( Y ) ⊗ k k ∗ ( X × Y ) ∼ = Z X,Y M ( X ) ⊗ k N ( Y ) ⊗ k k ∗ X ⊗ k ∗ Y ∼ = c oli m ( M , k ∗ ) ⊗ c oli m ( N , k ∗ ) . The second sentence then follows fr o m [Da2, Reflection Theor em].  In fancier words, the adjunction c oli m ( − , k ∗ ) e k ∗  lives in the 2-categ ory o f mono idal catego ries, monoidal functors and mono idal natural transformatio ns (all enriched o v er V ). 11. Mackey functors for Hopf al gebras In this section we provide another exa mple of a compa ct closed categ ory T constructed fro m a Hopf algeba r a H (or quant um gr oup ). W e sp eculate that Mack ey functors on this T will prov e as useful for Hopf algebra s as usual Mac key functors hav e f or gr oups. Let H b e a br a ided (semis imple) Hopf algebra (ov er k ). Let R deno te the category o f left H -modules which ar e finite dimensional as vector spaces (ov er k ). This is a compa c t clo sed braided monoidal ca teg ory . W e write Com o d ( R ) for the ca tegory obtained from the bicatego ry of that name in [DMS] by tak ing isomorphisms classes of mor phisms. Explicitly , the ob jects are co monoids C in R . The morphisms are isomorphism classes of como dules S : C  / / D fro m C to D ; suc h an S is equipped with a coactio n δ : S / / C ⊗ S ⊗ D satisfying the coas so ciativity and counity conditions; we ca n break the tw o-sided coaction δ in to a left coa ction δ l : S / / C ⊗ S and a right coaction δ r : S / / S ⊗ D connected by the bicomo dule co ndition. Compositio n of co mo dules S : C  / / D and T : D  / / E is defined by the (coreflexive) equalizer S ⊗ D T / / S ⊗ T 1 ⊗ δ l / / δ r ⊗ 1 / / S ⊗ D ⊗ T . MACKEY FUNCTORS ON COMP A CT CLOSED CA TEGORIES 19 The identit y c omo dule of C is C : C  / / C . The categor y Como d ( R ) is compa ct closed: the tensor pro duct is just that for vector spaces equipp ed with the extra structure. Direct sums in Como d ( R ) are given by direct s um as vector spaces . Consequently , Como d ( R ) is e nr iched in the monoidal category V of co mm utative monoids: to add comodules S 1 : C  / / D and S 2 : C  / / D , w e tak e the dir ect sum S 1 ⊕ S 2 with coaction defined as the comp os ite S 1 ⊕ S 2 δ 1 ⊕ δ 2 / / C ⊗ S 1 ⊗ D ⊕ C ⊗ S 2 ⊗ D ∼ = C ⊗ ( S 1 ⊕ S 2 ) ⊗ D . W e can now apply our ea rlier theory to the example T = Como d ( R ). In pa r- ticular, w e call a V -enriched functor M : Como d ( R ) / / V ect k a Mackey functor on H . In the ca se where H is the gr o up algebr a k G (made Hopf by means of the diagonal k G / / k ( G × G ) ∼ = k G ⊗ k k G ), a Mackey functor on H is no t the same as a Mack ey functor on G . Howev er, ther e is a strong relationship that we shall now explain. As usual, let E denote the cartesian monoidal ca tegory o f finite G -sets. The functor k : E / / R is str ong mono idal and preserves coreflexive equalizers. There is a monoidal equiv alence Como d ( E ) ≃ Spn ( E ) , so k : E / / R induces a stro ng monoida l V -functor ˆ k : Spn ( E ) / / Como d ( R ) . With Mky ( G ) = [ Spn ( E ) , V ect ] + as usual and with Mky ( k G ) = [ Como d ( R ) , V ect ] + , we o btain a functor [ ˆ k , 1 ] : Mky ( k G ) / / Mky ( G ) defined by pre-compo sition with ˆ k . Pr op osition 1 of [DS2] a pplies to yield: Theorem 11.1. The functor [ ˆ k , 1 ] has a str ong monoidal left adjoint ∃ ˆ k : Mky ( G ) / / Mky ( k G ) . The adjunction is monoidal. The formula for ∃ ˆ k is ∃ ˆ k ( M )( R ) = Z X ∈ Spn ( E ) Como d ( R )( ˆ k X , R ) ⊗ M ( X ) . On the o ther hand, we alr eady have the compact clo sed catego r y R of finite- dimensional repres ent ations of G and the stro ng monoidal functor k ∗ : Spn ( E ) op / / R . Perhaps R op ( ≃ R ) s hould b e our candidate for T ra ther than the mo re complica ted Como d ( R ). The r e s ult of [DS2] applies also to k ∗ to yield a monoida l adjunction [ R op , V e ct ] ⊥ [ k ∗ , 1] / / Mky ( G ) . ∃ k ∗ o o Perhaps then, additive functors R op / / V ect would pro vide a suitable generaliza- tion o f Ma ck ey functors in the ca se o f a Hopf alg ebra H . These matter s r equire inv estigation at a later time. 20 ELANGO P ANCHADCHARAM AND R OSS STREET 12. Review of some enriched ca tegor y theor y The basic refer ences are [Ke], [La] and [St]. Let COCT V denote the 2 -categor y w ho se ob jects ar e co complete V -categ ories and whose morphisms are (w eigh ted-) colimit-preser ving V -functor s ; the 2-cells are V -natural transfor ma tions. Every small V -category C determines a n ob ject [ C , V ] of COCT V . Let Y : C op / / [ C , V ] denote the Y oneda embedding: Y U = C ( U, − ). F o r any ob ject X of CO CT V , we ha ve a n equiv a lence of catego ries COCT V ([ C , V ] , X ) ≃ [ C op , X ] defined by res triction alo ng Y . This is expressing the fact that [ C , V ] is the free co completion of C op . It follows that, for small V -c ategories C and D , we ha ve COCT V ([ C , V ] , [ D , V ]) ≃ [ C op , [ D , V ]] ≃ [ C op ⊗ D , V ] . The wa y this works is a s follows. Supp ose F : C op ⊗ D / / V is a ( V -) functor. W e obtain a colimit-pres erving functor b F : [ C , V ] / / [ D , V ] by the formula b F ( M ) V = Z U ∈ C F ( U, V ) ⊗ M U where M ∈ [ C , V ] and V ∈ D . Conv ersely , given G : [ C , V ] / / [ D , V ], define ∨ G : C op ⊗ D / / V by ∨ G ( U, V ) = G ( C ( U, − )) V . The main calculatio ns pro ving the equiv alence are as follows: ∨ b F ( U, V ) = b F ( C ( U, − )) V ∼ = Z U ′ F ( U ′ , V ) ⊗ C ( U, U ′ ) ∼ = F ( U, V ) b y Y oneda ; and, b ∨ G ( M ) V = Z U ∨ G ( U, V ) ⊗ M U ∼ = ( Z U G ( C ( U, − )) ⊗ M U ) V ∼ = G ( Z U C ( U, − ) ⊗ M U ) V since G preserves w eigh ted colimits ∼ = G ( M ) V by Y oneda aga in. MACKEY FUNCTORS ON COMP A CT CLOSED CA TEGORIES 21 Next we look how compo sition of G s is transp orted to the F s. T ake F 1 : C op ⊗ D / / V , F 2 : D op ⊗ E / / V so that c F 1 and c F 2 are comp osa ble: [ C , V ] [ D , V ] c F 1 8 8 p p p p p p p p p p p p [ E , V ] . c F 2 & & N N N N N N N N N N N N c F 2 ◦ c F 1 4 4 Notice that ( c F 2 ◦ c F 1 )( M ) = c F 2 ( c F 1 ( M )) = Z V ∈ D F 2 ( V , − ) ⊗ c F 1 ( M ) V ∼ = Z U,V F 2 ( V , − ) ⊗ F 1 ( U, V ) ⊗ M U ∼ = Z U ( Z V F 2 ( V , − ) ⊗ F 1 ( U, V )) ⊗ M U. So we define F 2 ◦ F 1 : C op ⊗ E / / V by (1) ( F 2 ◦ F 1 )( U, W ) = Z V F 2 ( V , W ) ⊗ F 1 ( U, V ); the last calculatio n then yields c F 2 ◦ c F 1 ∼ = \ F 2 ◦ F 1 . The identit y functor 1 [ C , V ] : [ C , V ] / / [ C , V ] corr esp onds to the hom functor o f C ; that is, ∨ 1 [ C , V ] ( U, V ) = C ( U, V ) . This gives us the bica tegory V - Mo d . The ob jects ar e (sma ll) V -catego ries C . A morphism F : C  / / D is a V - functor F : C op ⊗ D / / V ; we call this a mo dule fr om C to D (others call it a left D -, right C - bimo dule ). Comp ositio n of mo dules is defined by (1) above. W e can sum up now by s aying tha t c ( ) : V - Mo d / / COCT V is a pseudofunctor (= ho momorphism o f bicategor ies) taking C to [ C , V ], taking F : C  / / D to b F , and defined o n 2 -cells in the obious way; mor e over, this pseudofunctor is a lo cal eq uiv a le nc e (that is, it is an equiv alence on ho m-categor ies): c ( ) : V - Mo d ( C , D ) ≃ COCT V ([ C , V ] , [ D , V ]) . A monad T on a n ob ject C of V - Mo d is called a pr omo nad on C . It is the same as giving a co limit-preserving mo na d b T on the V -catego ry [ C , V ]. One way that promona ds aris e is fro m mono ids A for so me co nv olution monoidal structure on [ C , V ]; then b T ( M ) = A ∗ M . 22 ELANGO P ANCHADCHARAM AND R OSS STREET That is, C is a promonoidal V -catego ry [Da1]: P : C op ⊗ C op ⊗ C / / V J : C / / V so that b T ( M ) = A ∗ M = Z U,V P ( U, V ; − ) ⊗ AU ⊗ M V . This means that the mo dule T : C  / / C is defined by T ( U, V ) = b T ( C ( U, − )) V = Z U ′ ,V ′ P ( U ′ , V ′ ; V ) ⊗ AU ′ ⊗ C ( U, V ′ ) ∼ = Z U ′ P ( U ′ , U ; V ) ⊗ AU ′ . A promona d T o n C has a unit η : ∨ 1 / / T with co mpo nent s η U,V : C ( U , V ) / / T ( U, V ) and so is determined by η U,V (1 U ) : I / / T ( U, U ) , and has a multiplication µ : T ◦ T / / T with co mpo nent s µ U,W : Z V T ( V , W ) ⊗ T ( U, V ) / / T ( U, W ) and so is determined by a natural family µ ′ U,V ,W : T ( V , W ) ⊗ T ( U, V ) / / T ( U, W ) . The Kleisli c ate gory C T for the promonad T on C has the sa me ob jects as C and has homs defined by C T ( U, V ) = T ( U, V ); the identites are the η U,V (1 U ) and the comp osition is the µ ′ U,V ,W . Prop ositio n 12.1. [ C T , V ] ≃ [ C , V ] b T . That is, the functor c ate gory [ C T , V ] is e quiva lent to the c ate gory of Eilenb er g-Mo or e algebr as for the monad b T on [ C , V ] . Pr o of. (sketch) T o giv e a b T -algebra structure on M ∈ [ C , V ] is to give a morphism α : b T ( M ) / / M satisfying the tw o axioms for an ac tio n. This is to give a natural family of morphisms T ( U, V ) ⊗ M U / / M V ; but that is to g ive T ( U, V ) / / [ M U, M V ]; but that is to g ive (2) C T ( U, V ) / / V ( M U, M V ) . Thu s we can define a V -functor M : C T / / V which agrees with M on ob jects and is defined by (2) on homs; the action axioms are just what is needed for M to b e a functor . This pr o cess can b e reversed.  MACKEY FUNCTORS ON COMP A CT CLOSED CA TEGORIES 23 13. Modul es over a Green functor In this section, we present w ork inspired b y Chapter s 2 , 3 and 4 of [Bo1], casting it in a mor e categorical framework. Let E denote a lextens ive category a nd CMon denote the categor y of commu- tative monoids; this la tter is what we called V in earlier sections. The functor U : Mo d k / / CMon (which forgets the a c tion of k o n the k -mo dule a nd retains only the additive monoid structure) has a left adjoint K : CMon / / Mo d k which is strong monoidal for the ob vious tensor pro ducts on CMon and Mo d k . So each category A enriched in CMon determines a categor y K ∗ A enr iched in Mo d k : the ob jects of K ∗ A are those of A and the homs a re defined by ( K ∗ A )( A, B ) = K A ( A, B ) since A ( A, B ) is a co mm utative monoid. The p o int is that a Mo d k -functor K ∗ A / / B is the same as a CM on -functor A / / U ∗ B . W e know that Spn ( E ) is a CMon -categor y; so we o bta in a monoidal M o d k - category C = K ∗ Spn ( E ) . The Mo d k -categor y of Mackey functors on E is Mky k ( E ) = [ C , Mo d k ]; it bec omes monoidal using con volution with the monoidal structure on C (see Section 5). The Mo d k -categor y o f Gr e en functors on E is Grn k ( E ) = Mon [ C , Mo d k ] consisting of the mono ids in [ C , Mo d k ] for the co nv olution. Let A b e a Green functor. A m o du le M ov er the Green functor A , or A -mo dule means A acts on M via the conv olution ∗ . The monoidal actio n α M : A ∗ M / / M is defined by a family of morphisms ¯ α M U,V : A ( U ) ⊗ k M ( V ) / / M ( U × V ) , where w e put ¯ α M U,V ( a ⊗ m ) = a.m for a ∈ A ( U ), m ∈ M ( V ), satisfing the follo wing commutativ e diagrams for morphisms f : U / / U ′ and g : V / / V ′ in E . A ( U ) ⊗ k M ( V ) M ( U × V ) ¯ α M U,V / / M ( U ′ × V ′ ) M ∗ ( f × g )   A ( U ′ ) ⊗ k M ( V ′ ) A ∗ ( f ) ⊗ k M ∗ ( g )   ¯ α M U ′ ,V ′ / / M ( U ) A (1) ⊗ k M ( U ) η ⊗ 1 / / M (1 × U ) ¯ α M   ∼ = ' ' N N N N N N N N N N N N A ( U ) ⊗ k A ( V ) ⊗ k M ( W ) A ( U ) ⊗ k M ( V × W ) 1 ⊗ ¯ α M / / M ( U × V × W ) . ¯ α M   A ( U × V ) ⊗ k M ( W ) µ ⊗ 1   ¯ α M / / If M is an A -mo dule, then M is in pa rticular a Mackey functor. Lemma 1 3.1. L et A b e a Gr e en fun ct or and M b e an A - m o dule. Then M U is an A -mo dule for e ach U of E , wher e M U ( X ) = M ( X × U ) . Pr o of. Simply define ¯ α M U V ,W = ¯ α M V ,W × U .  24 ELANGO P ANCHADCHARAM AND R OSS STREET Let Mo d ( A ) denote the category of left A -modules for a Green functor A . The ob jects are A -mo dules and mor phisms ar e A -mo dule mo r phisms θ : M / / N (that is, morphisms of Ma ck ey f unctors) satisfying the following comm utative diag ram. A ( U ) ⊗ k M ( V ) M ( U × V ) ¯ α M U,V / / N ( U × V ) θ ( U × V )   A ( U ) ⊗ k N ( V ) 1 ⊗ k θ ( U )   ¯ α N U,V / / The category Mo d ( A ) is enriched in Mky . The homs ar e given b y the equalizer Mo d ( A )( M , N ) Hom( M , N ) / / Hom( A ∗ M , N ) Hom( α M , 1) / / Hom( A ∗ M , A ∗ N ) . ( A ∗− ) $ $ J J J J J J J J J J Hom(1 ,α N ) ; ; w w w w w w w w w w Then w e see that Mo d ( A )( M , N ) is the sub-Mac key functor of Hom( M , N ) defined by Mo d ( A )( M , N )( U ) = { θ ∈ Mky ( M ( − × U ) , N − ) | θ V × W ( a.m ) = a.θ W ( m ) for all V , W, and a ∈ A ( V ) , m ∈ M ( W × U ) } . In pa r ticular, if A = J (Burns ide functor) then Mo d ( A ) is the category o f Mac key functors and Mo d ( A )( M , N ) = Hom( M , N ). The Gre en functor A is itself an A -mo dule. Then by the Lemma 13.1, we s ee that A U is a n A -mo dule for each U in E . Define a catego ry C A consisting o f the ob jects of the form A U for ea ch U in C . This is a full subcategor y of Mo d ( A ) and we have the following equiv alences C A ( U, V ) ≃ Mo d ( A )( A U , A V ) ≃ A ( U × V ) . In other words, the categ ory Mo d ( A ) of left A -mo dules is the categor y o f Eilenberg-Mo ore algebra s for the monad T = A ∗ − o n [ C , Mo d k ]; it prese r ves colimits since it has a right adjoint (as usual with convolution tensor pro ducts). By the above, the Mo d k -categor y C A (tec hnically it is the Kleisli category C ∨ T for the promonad ∨ T on C ; see Pr op osition 12.1) satisfies an eq uiv alence [ C A , Mo d k ] ≃ Mo d ( A ) . Let C b e a Mo d k -categor y with finite dir e ct sums a nd Ω be a finite set of ob jects of C such that every ob ject of C is a dir e c t sum of ob jects from Ω. Let W be the algebr a of Ω × Ω - matrice s whose ( X , Y ) - entry is a morphism X / / Y in C . Then W = { ( f X Y ) X,Y ∈ Ω | f X Y ∈ C ( X , Y ) } is a vector space o ver k , and the pro duct is defined by ( g X Y ) X,Y ∈ Ω ( f X Y ) X,Y ∈ Ω =  X Y ∈ Ω g Y Z ◦ f X Y  X,Z ∈ Ω . Prop ositio n 13.2. [ C , Mo d k ] ≃ Mo d W k (= the c ate gory of left W -mo dules). MACKEY FUNCTORS ON COMP A CT CLOSED CA TEGORIES 25 Pr o of. P ut P = M X ∈ Ω C ( X , − ) . This is a small pro jective genera tor so Exercise F (page 106 ) o f [F r] applies and W is ident ified as E nd(P).  In pa r ticular; this applies to the category C A to o btain the Gr e en algebr a W A of a Green functor A : the p oint being that A and W A hav e the same modules. 14. Morit a equiv alence of Green functors In this se c tion, we lo ok at the Morita theo ry of Green functors ma k ing us e of adjoint tw o-sided mo dules ra ther than Morita c ontexts as in [Bo1]. As for a ny s ymmetric co co mplete clos ed mono idal categ ory W , we hav e the monoidal bica tegory Mo d ( W ) defined as follows, wher e we take W = Mky . Ob- jects a re mono ids A in W (that is, A : E / / Mo d k are Gr een functor s) a nd mor - phisms are mo dules M : A  / / B (that is, algebr a s fo r the mona d A ∗ − ∗ B on Mky ) with a tw o-sided action α M : A ∗ M ∗ B / / M ¯ α M U,V ,W : A ( U ) ⊗ k M ( V ) ⊗ k B ( W ) / / M ( U × V × W ) . Comp osition of morphisms M : A  / / B and N : B  / / C is M ∗ B N and it is defined via the co e qualizer M ∗ B ∗ N α M ∗ 1 N / / 1 M ∗ α N / / M ∗ N / / M ∗ B N = N ◦ M that is, ( M ∗ B N )( U ) = X X,Y Spn ( E )( X × Y , U ) ⊗ M ( X ) ⊗ k N ( Y ) / ∼ B . The iden tity morphism is given b y A : A  / / A. The 2-cells are na tural transfor ma tions θ : M / / M ′ which r esp ect the ac tio ns A ( U ) ⊗ k M ( V ) ⊗ k B ( W ) M ( U × V × W ) ¯ α M U,V,W / / M ′ ( U × V × W ) . θ U × V × W   A ( U ) ⊗ k M ′ ( V ) ⊗ k B ( W ) 1 ⊗ k θ V ⊗ k 1   ¯ α M ′ U,V,W / / The tenso r pro duct on Mo d ( W ) is the co n volution ∗ . The tensor pro duct o f the mo dules M : A  / / B a nd N : C  / / D is M ∗ N : A ∗ C  / / B ∗ D . Define Green functors A and B to b e Morita e quivalent when they are equiv alent in Mo d ( W ). Prop ositio n 14.1. If A and B ar e e quivalent in Mo d ( W ) then Mo d ( A ) ≃ Mo d ( B ) as c ate gories. Pr o of. M o d ( W )( − , J ) : M o d ( W ) op / / CA T is a pseudofunctor and so takes equiv a lences to equiv alences.  26 ELANGO P ANCHADCHARAM AND R OSS STREET Now we will loo k at the Cauc hy completion of a monoid A in a mono idal category W with the unit J . The W -category P A has underlying catego ry Mo d ( W )( J, A ) = Mo d ( A op ) wher e A op is the monoid A with comm uted multiplication. The ob jects are mo dules M : J  / / A ; that is, rig h t A - mo dules . The homs of P A are defined by ( P A )( M , N ) = Mo d ( A op )( M , N ) (see the equalizer o f Section 13). The Cauc hy completion Q A of A is the full sub- W -categor y of P A consisting of the mo dules M : J  / / A with r ig ht a djoin ts N : A  / / J . W e will ex a mine what the ob jects of Q A ar e in more explicit terms. F o r motiv a tion and prepar ation we will lo ok a t the monoida l catego ry W = [ C , S ] where ( C , ⊗ , I ) is a mo no idal ca tegory and S is the cartesian monoidal category o f sets. Then [ C , S ] b ecomes a mo noidal ca teg ory by conv olution. The tensor pro duct ∗ and the unit J are defined by ( M ∗ N )( U ) = Z X,Y C ( X ⊗ Y , U ) × M ( X ) × N ( Y ) J ( U ) = C ( I , U ) . W rite Mo d [ C , S ] fo r the bicategor y w ho se ob jects ar e mo no ids A in [ C , S ] and whose morphisms are mo dules M : A  / / B . These mo dules have t w o-sided action α M : A ∗ M ∗ B / / M ¯ α M X,Y ,Z : A ( X ) × M ( Y ) × B ( Z ) / / M ( X ⊗ Y ⊗ Z ) . Comp osition of mo rphisms M : A  / / B and N : B  / / C is g iven by the coeq ua lizer M ∗ B ∗ N α M ∗ 1 N / / 1 M ∗ α N / / M ∗ N / / M ∗ B N that is, ( M ∗ B N )( U ) = X X,Z C ( X ⊗ Z, U ) × M ( X ) × N ( Z ) / ∼ B where ( u, m ◦ b, n ) ∼ B ( u, m, b ◦ n ) ( t ◦ ( r ⊗ s ) , m, n ) ∼ B ( t, ( M r ) m, ( N s ) n ) for u : X ⊗ Y ⊗ Z / / U, m ∈ M ( X ) , b ∈ B ( Y ) , n ∈ N ( Z ) , t : X ′ ⊗ Z ′ / / U, r : X / / X ′ , s : Z / / Z ′ . F o r each K ∈ C , we o btain a mo dule A ( K ⊗ − ) : J  / / A . T he action A ( K ⊗ U ) ⊗ A ( V ) / / A ( K ⊗ U ⊗ V ) is defined by the monoid structure on A . Prop ositio n 14.2. Every obje ct of the Cauchy c ompletio n Q A of the monoid A in [ C , S ] is a r etr ac t of a mo dule of the form A ( K ⊗ − ) for some K ∈ C . Pr o of. T ake a mo dule M : J  / / A in Mo d [ C , S ]. Supp ose that M has a rig ht adjoint N : A  / / J . Then w e ha ve the f ollowing a ctions: A ( V ) × A ( W ) / / A ( V ⊗ W ), M ( V ) × A ( W ) / / M ( V ⊗ W ) , A ( V ) × N ( W ) / / N ( V ⊗ W ) since A is a monoid, M is a rig ht A -mo dule, and N is a left A -mo dule resp ectively . MACKEY FUNCTORS ON COMP A CT CLOSED CA TEGORIES 27 W e have a unit η : J / / M ∗ A N and a counit ǫ : N ∗ M / / A fo r the adjunction. The comp onent η U : C ( I , U ) / / ( M ∗ A N )( U ) of the unit η is deter mined by η ′ = η U (1 I ) ∈ X X,Z C ( X ⊗ Z, I ) × M ( X ) × N ( Z ) / ∼ A ; so there exis t u : H ⊗ K / / I , p ∈ M ( H ) , q ∈ N ( K ) s uch that η ′ = [ u, p, q ] A . Then η u ( f : I / / U ) = [ f u : H ⊗ K / / U, p, q ] A . W e also have ¯ ǫ Y ,Z : N Y × M Z / / A ( Y ⊗ Z ) co ming fro m ǫ . The c o mm utative diagram M ( U ) X X,Y ,Z C ( X ⊗ Y ⊗ Z, U ) × M ( X ) × N ( Y ) × M ( Z ) / ∼ η U ∗ 1 / / M ( U ) 1 ∗ ǫ U   1 * * T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T yields the equations m = (1 ∗ ǫ U )( η U ∗ 1)( m ) = (1 ∗ ǫ U )[ u ⊗ 1 U , p, q , m ] A = M ( u ⊗ 1 U )( p ¯ ǫ K,U ( q , m )) (3) for all m ∈ M ( U ). Define M ( U ) i U 1 1 A ( K ⊗ U ) r U r r by i U ( m ) = ¯ ǫ K,U ( q , m ) , r U ( a ) = M ( u ⊗ 1 U )( p.a ). These are ea sily seen to b e natural in U . Eq ua tion (3) sa ys that r ◦ i = 1 M . So M is a retract of A ( K ⊗ − ).  Now we will lo ok at what are the o b jects of Q A when W = Mky which is a symmetric monoidal clos ed, complete and co complete ca tegory . Theorem 14.3. Th e Cauchy c omple tion Q A of t he monoid A in Mky c onsists of al l the re tr acts of mo dules of t he form k M i =1 A ( Y i × − ) for some Y i ∈ Spn ( E ) , i = 1 , . . . , k . Pr o of. T ake a mo dule M : J  / / A in Mo d ( W ) and supp ose that M has a right adjoint N : A  / / J . F or the adjunction, we hav e a unit η : J / / M ∗ A N a nd a co unit ǫ : N ∗ M / / A . W e write η U : Spn ( E )(1 , U ) / / ( M ∗ A N )( U ) is the comp onent of the unit η and it is determined by η ′ = η 1 (1 1 ) ∈ k X i =1 Spn ( E )( X × Y , 1) ⊗ M ( X ) ⊗ N ( Y ) / ∼ A . 28 ELANGO P ANCHADCHARAM AND R OSS STREET Put η ′ = η 1 (1 1 ) = k X i =1 [( S i : X i × Y i / / 1) ⊗ m i ⊗ n i ] A where m i ∈ M ( X i ) and n i ∈ N ( Y i ). Then η U ( T : 1 / / U ) = k X i =1 [( S i × T ) ⊗ m i ⊗ n i ] A . W e also have ¯ ǫ Y ,Z : N Y ⊗ M Z / / A ( Y × Z ) co ming fro m ǫ . The c o mm utative diagram M ( U ) k X i =1 Spn ( E )( X i × Y i × U, U ) ⊗ M ( X i ) ⊗ N ( Y i ) ⊗ M ( U ) / ∼ A η U ∗ 1 / / M ( U ) 1 ∗ ǫ U   1 * * T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T yields m = k X i =1 [ M ( P i × U ) ⊗ m i ⊗ ǫ ( n i ⊗ m )] where m ∈ M ( U ) and P i : X i × Y i / / U . Define a na tur al retraction M ( U ) i U 0 0 k M i =1 A ( Y i × U ) r U r r by r U ( a i ) = M ( P i k × U )( m i .a i ) , i U ( m ) = k X i =1 ¯ ǫ Y i ,U ( n i ⊗ m ) . So M is a r e tract of k M i =1 A ( Y i × − ). It remains to c hec k that each mo dule A ( Y × − ) has a righ t adjoin t since retracts and direct sums of mo dules with right adjoin ts hav e right a djoint s. In C = Spn ( E ) each ob ject Y ha s a dual (in fact it is its own dual). This implies that the mo dule C ( Y , − ) : J  / / J has a right dual (in fact it is C ( Y , − ) itself ) since the Y oneda embedding C op / / [ C , Mo d k ] is a stro ng monoidal functor. Moreover, the unit η : J / / A induces a mo dule η ∗ = A : J  / / A with a right a djoin t η ∗ : A  / / J . Ther efore, the co mpo site J  C ( Y , − ) / / J  η ∗ / / A , which is A ( Y × − ), has a rig ht adjo int.  Theorem 14.4. Gr e en functors A and B ar e Morita e quivalent if and only if Q A ≃ Q B as W -c ate gories. Pr o of. 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