A Note on Weak Algebraic Theories
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. A Note on Weak Algebraic Theories Daniel de Carvalho. N° 6643 septembre 2008. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--6643--FR+ENG. Thème SYM.
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(21) . C. T ◦T ◦T. µ ◦ idT. T ◦T. idT ◦ µ. (1). µ. ? T ◦T. µ. η ◦ idT. T. T ◦T. id. µ. T. -. T ◦η T ◦T . (2). ? T T (3). id. T. µ. ? - T. ? T. . v >¬° 7Y¬ v_¬ -$ C(C, C) ! ¬ v ±µ¦X¬5''¬)1=¬ '8Ûd´!1IM?>2S\]+>2PRw6sRt. IM?>2S]_>2PRwYsXt2PRws usKb?XPR`bH C Vªo®±R2° 1=¬ 2 ¯ v_¬ -$ Cat 1=¬
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(24) =¬ ¸ # %\¬ (T, µ, η) Vª 2ª¬®°¯±R!1 v¬ -$ C ! +/d'%/$ ! + 8 T Vª ª¬ °¯ !'>J? @_ C → C + 8 η Vª®' >M %K Md=ª ! ¸ +Û id ⇒ T + 8 ' µ Vª ' >b %K MYª ! ¸2 8Ûd T ◦ T ⇒ T C D ETF JH ILHLK F. v¬ X$ C !. s. . . .
(25). C. ÅkÅÍNOPO'Q R.
(26) z. 6Û¬%)R¬· % =. ª'> ' 1=5+! ® $#53¬' A !)7Y¬ v_¬ -$ C+ 7 Y¬®dÛ@GM®ª µT (A) - T ◦ T (A). T ◦ T ◦ T (A) T (µA ). µA. ? T ◦ T (A). T (A). (1). ? - T (A). µA. ηT (A) - T ◦ T (A) T( id. (2). µA. A). -. ? T (A). T (ηA ) T (A) T ◦ T (A) ) id A ( T. µA ? T (A). µA-. T ◦ T (A). (3). T (A) T (idA ). µ. A. bd®8>¬°H1=¬8 v¬ X$. C. (4). -. ? T (A). !. . ®¬Û('a6M7d8 Ûdª
(27) )! B+ Û@´GM± b¬bdP=M¬ ª)ª(#Y1=¬ ± 'M¸¬¸ªU#Y ¬'% 7+($·¬ %P! $ _! 1=¬%Û Md ª B+ 1± 1 ! Û Md ¬vYª 7Y¬·ªvª Û+ N7($ !E7Y¬D >% 8µ # %Ü v+Û µ Ogµ¬'a'2¬ -BG @ %P$ 7Y¬jª¬ Û@GMd ± ° b1=P¬%M¬ ª(R#Y¬('a±6 78Û_d ¤1= !!¬ +µ´±7RY ¬ !'¬'Û v ¬¸ªvMªd $ ª >BªP1>Y %/'d Û@ G Mq®ª%¬bd°L 11==¬ ¬'X¬5¬@'>Y b7 +Û%Ü7U¨$ !!%η ! 2 bdd 'µ#YR· ª¸7 S +Û#@> 5+ ! ±1=µ 8¬ ++µµ¬R dP Mµ±ª &¬ v#v¬ ¬2ªvª ¬ ¸>Y %P$ %= % ¬ T (id! ) ·=ªPid%P$B+µ¬O$X¬ Y¬ b¬±+ µ7¬ YM¬ >J7ÛM@¬8GM1=57Y ¬. bd®8>¬¸ª&OE7JVªd'¬RN¬¸ª® #"NÛ>Bª %P$!>T+ (id. b % %P$8)Y %\°21=¬8 v·ª¬®! ±R)! ª ¬¬®
(28) ?'Y ¬v¬+¬v 'ªo + v!Û18«=¬> %!2η+ ¯L#ª ¬ ¬%Ü- ®#Y v°¯+±ÛR¬v±µR#?J M N &V ª´!bªSVIp -! wY5 PdNK2°q±'¬v'ª ¸P k+°& M d% ! #Y! BQ +° >µo ¬®°Y«µ6µ M_Xp® ¬ °' _B @¬7YUY#?¬ %®M@¬ µ¬%¬vsd15QT¶XQ X¬ 7¬ 6Y±°0¬ M #?%MN+ Û! !b'7ª Y ¬ % ' +ÛQ > ) µ!¬ =% ±R¬ µ#?7 M7 1#%\¬ ±_q8 ± v #¸ >ME¬%8 1B>Vªp% +'# %Ü 8 vÛd +ÛOp µ! ¬ ! d % X5¬ ''¬o ¯ v ¬ X $µ =·ª¬ # ¯ 3¬& @8ªµ>%\ #v¬$ v¬ -¸Û¬ ª,+µY·ª¬oPMdµªµ>%\ #v¬pª¬ °¯ !'>J? @_vª d'µ2=·ª¬ 2 ¯LPMµª µ>%T #b¬' >M % MYª ! ¸ +Û=&ª ! *¸'X¬v¬b% + ' µ + 1=¬ÛX¬ 87 ! U$% d µ +1ª =¬¬ Y°¯ !'¸>J7 ?6N @ _ _ % C v→2 #=Dª¸7 v8Ûd6 ' #b¬ d < $ M ¬ " p + 7 5 Y · ª _ #b¬ Vªo±(F (id!'>J? @))_¸Û %d<$M¬ OD ! F Vª'/ (id!'>J ),+ 1=¬ id ◦ id 6= id !
(29) Y¬ ◦ ÃÄÅkÃ°Æ . . . . . . . . . . . A. . T (A). . A. . C. C∈C. F (C) C∈C. F. idC. F. .
(30) ¬®d¬vN¶ % - ¬ # Md
(31) Y¬b¸Û¬¸ª .
(32) !
(33) !! " Q$ b d=ª¸ÛR¬@¸ °M ¬5'Y7+ Û X d° +2µ¬q±¬ 1=®µ¬ v>%\ #v¬ #%\¬ R¬('a±¬ µY Vª 'X° ¯v _ ¬ - $°´d'7Y ¬ µ $ !R* 'X¬b¬v%+ d ¬´ ¯ b_¬ -$ ±d1#53¬' ! 1= ¬ 2 ¯v ¬ - 2$ C +a7 Vª %T¬b)7Y 7Y ¬) ¸ # %\¬ (C(C, C), ◦, id2 ) Vª ¸ C 2d'dÛR %A v¬ -C$ ! ¬' @ C !$7Y ¬ 2 ¯ v! ¬ X$ C Vª$1=¬Dªd2¬ 1B° ·
(34) ª = ¬ 2 ·2d¬' 2Û'Xv °H ¬°1X = ¬7 Y $ ªP¬ ¸ 2 ¯+ @v µ_ ¬¬ -R±¬ '$ ÛXC_ ¬ %#&b $_ 1¬ =- ¬ $ #43(C(C, C), ◦, id ) 7Y ¬ v ¬ X $µY ·ª¬!#53¬& @8 ª2b ¬2ª¬ °¯ C 1/2End(C) !'J> ? @ v ª + µ = ª ¬ P M d µ % ª b ! ¬ ' > M % M Yª ! ¸ + Û=ª,+µ= ª¬2v 2#=ªv7+ Û¦Vª 7Y ¬ ¬ P+ v %Kb dC→ #Y·ª¸C7+ Û • d' µ= ª ¬DÛX¬ 8 7U $®d%d!#53¬' F Vª (id ) !"2 ¬j % M ¬vR $ ªdµ 1= ◦ Vª' !'J> ? @_ & ! : ¬?v ¬B1= ¬B ¸ # %\¬ (1/2End(C), ◦, id ) Vª' qªP ¸ @j ±ÛX % 'v 8 ¬ ÛX dH $ 1= !RQ A>- a ¬ ± %ܬ b ·%ܪP76N@ ¬ ! ! ª !R1* = ¬ VªD'8 Ûd¬.´-Bd!o# ªP%\¬¸ @!g µ= ± µÛR¬) %Av v % %±_ 2¬ - ILKM `b$ STu ! KIM?>2S]_>2PRw=PXSTtYsdQYu5sKM?5RPX`MH+ C D ETF HJILHLK F ILKM`bSVu¸KgIM?>2S]_>2PRw=PXSTtYsdQÂusdKM?XPR`bHqVª ¸ # %T¬ (C, ⊗, I) v YªvVªP+ °M! ! 8 v ¬ X $ C S 8 2ª¬ °¯ !'J> ? @ ⊗ : C × C → C ªP> ' 1= = ! $« #43¬' @Û ª A + B ± C !H1= ¬Jv _ ¬ - $ C +µ¬H= ·¬ (A ⊗ B) ⊗ C = A ⊗ (B ⊗ C) = ± ! d<$ S PM µª f + g d' h ! 1= ¬Hv _ ¬ - $ C + µ¬ = %¬ (f ⊗ g) ⊗ h = f ⊗ (g ⊗ h) 8 ='%¬ d1#53¬' I !$1= ¬ v !_ ¬ - $ C ªP> 'B7Y ( +! <$ #43¬' @ A ! 1= ¬ b _ ¬ - $ C+µ¬ I ⊗A=A=A⊗I ¬1 = -M¬ v ¬@*¸B>#?ª' Y> 7 7Y Y %k¬ R / ¬(µ'aµ¬67M X8 Ûª,d +´ D ±ªP! ¬'¸ b IL ¬ KM @ªv`b± ªSTu ªK ¬ ¸ >%P$8°PX¯Y2w=%PRdSV't=¬ dsRidÛQ±R W=%?wY⊗b u¸Kbid_ PR¬ `%- ! = $ id(C, ⊗, !"I)
(35) V= ª ¬@jM ¬®ª' V¸ ª @' 21B d°' ÛX % 'v = d_ M¬ -- ¬ _$B + C D ETF HJILHLK F IeKb`MSVu¸Kj>PXw=PRSVt=sRQW=wYu KMPX` (F, , ) !'M dFªP ¸ / ª¬®°¯±ÛR % b _ ¬ - $ ªP ¸ ª¬®°¯±ÛX % v¬ X$ (C , ⊗ , I ) Vª" ¸ # %\¬ (F, , ) vd=ª¸VªP8° ! (C, ⊗, I) 8 !'J> ? @ F : C → C + 8 '¸ 2 %' >b %K M Yª ! ¸ 28 Ûd : ⊗ ◦ (F × F ) ⇒ F ◦ ⊗ + 8 ' PM dµ : I → F (I) ! C ªP> & 7Y ( +! ® d<$ #43¬' @Û ª A+ B ' C !)1= ¬8v ¬ X $ C+ 1= ¬Û b ⊗ id - F (A ⊗ B) ⊗ F (C) F (A) ⊗ F (B) ⊗ F (C) C. C. F (C) C∈C. C. . A. B. A⊗B. 0. 0. 0. 0. 0. 0. 0. idF (A) ⊗0. . 0. 0. 0. F (C). 0. . B,C. ? F (A) ⊗0 F (B ⊗ C). ÅkÅÍNOPO'Q R. A,B. . A⊗B,C. ? - F (A ⊗ B ⊗ C) A,B⊗C.
(36) . 6Û¬%)R¬· % =. bd®8>¬¸ªo°H1=¬8 v¬ - $ C ±B+ ! ® $#53¬' A !)7Y¬ v_¬ - $ C + 7Y¬ +µdÛ@ GM®ª 0. F (A). . ⊗0 idF (A) - F (I) ⊗0 F (A). id. I0. ⊗0. id. -. F( A). . ? F (A). . idF (A) ⊗0 F (A) ⊗0 F (I) . . I,A. F (A). 0. 0. A,I. ? F (A). ⊗. A). id I. ( id F. bd®8>¬°H1=¬8 v¬ X$ C ! C>D,EGFHJIJHLK F k¬@ (C, ⊗, I) + (C , ⊗ , I ) ' (C , ⊗ , I ) #v¬ 1 M¬v¬ªP ¸ Dª¬ °¯'dÛR % b _ ¬ - ¸ Û¬ ª&! #v¬ ¬@ (F,ª' ¸ @, ) #v±¬®ÛRªP % ¸ !' >J ±"!'ÛMXd % !'>J? @!'b _ (C, ⊗, I) _ (C!B,⊗¬,ªI¬ ) d' %\¬@ (F , , ) (C , ⊗ , I ) (C , ⊗ , I ) (F , , ) ◦ (F, , ) = (F ◦ F, , F ( ) ◦ ) , µ= ¬@M ¬ = ( ) µ77 = F ( ) ◦ ! *P Vª %\¬v 1= 1= ¬ ! % %\dµ°M X X¬5''¬ v ¬ X $ 1/2MonCat O 8 1= ¬#53¬' 8 ª M ¬ªP ¸ @ ª¬®°¯±ÛX % v _ ¬ - ¸ Û¬ ªTS 8 d´Pb µ C → C Vª ªP ¸ 2d'ÛX % !'J> ? @_ !'M C C S 8 1= ¬ ÛR¬ + 7( $dH1= ¬#53¬' (C, ⊗, I) Vª (id , (id ) , id ) S 8 v2#Y·ª¸78Ûd1Vª)1BVª R¬('a±¬v° ¬('aY7+Û
(37) ! µY ·ª¬#5! 3¬'
(38) = 8 ªD¬ ! b %¬j%\dª'µ ¸ ° M @ 2R¬(d'a'Y7+ÛXÛ ¬ % 7Y v - ¬ _A±¬ 1- ¬ = M ¬$ ¸ %ÜÛ7¬ 6N¬ ª¸¬¸ª)Vª1°ªP='+ ¬ %F%''1= #5¬8 3Û¬'d_ ¬@´ ¸ ® !E!1°±1>2= ¬%=PR!'w=>#5PX3% ST¬'%6t ªP) > ° #&! v 27Y_ ªP¬ ¬ - ¸ b _$ / ¬ -MonCat 2d $'d1/2MonCat ÛR %Av ¬ - $ ! C>D,EGFHJIJHLK F 2d'dÛ%° !ªP ¸ @ ª¬®°¯±ÛR % v ¬ X $ C Vª ¸ # %\¬ (A, µ, η) ªP> & 7Y 57Y ¬ b ¬¬.-XVªPÛ ª ªP ¸ @ 2d'dÛR %!'J> (F, , ) !'M d 1= ¬8 ¬@¸ ®°± %) #43¬' @ (1, ⊗, ∗) ! 7Y ¬ v ¬ X $ 1/2MonCat _ C ªP> ' 1= A = F (∗) + µ = ± η = ! 2d'
(39) ?Y ÛX¬ ! % v % %\dM µ¬ - °M $ #! M #Y·ª¸7+ Û R ·¬ ª! '= b G @_ ¬@¸ 76N+ Û¦ ! 7Y ¬2d'Ûª°£ ªP ¸ D ª¬®°¯ Vª/j2d'dÛg°gªP ¸ @' ª ¬°¯2d'ÛX %v ¬ X $ ¬ X ⊗, $ I) ! +aKd'' 2K %HJ ILVHLª %P$ K dF ! ´+ M APb
(40) VªDY µ ¬ ¸ # #5%\3¬ ¬'(A,. !Eµ,1=η)¬ v ¬ X $ C + µ VªDqPM µ A ⊗ A → A ! 1= ¬)v (C, ! 1= ¬8v ¬ - $ C ªP> ' 1= 1= ¬®Û@GM d ª C η I→A 0. 0. 0. 0. 00. 00. 00. 0. 0. 0. 0. A,B A,B∈. 0. 0. 0. A,B. (C). 0. 00. 00. 0. 0. 00. 0. A,B. F (A),F (B). 0. 0. C. A,B A,B∈. (C). I. ∗,∗. . A⊗A⊗A. µ ⊗ idA. idA ⊗ µ ? A⊗A. A⊗A µ. µ. ? - A. (1). ÃÄÅkðÆ.
(41) ¬®d¬vN¶ % - ¬ # Md
(42) Y¬b¸Û¬¸ª D η ⊗ idA. A. id. I. ⊗. A⊗A µ. id. - ? A. A. idA ⊗ η A⊗A µ. dA. ? A A⊗A. ⊗. A. id I. (3). i. idA ⊗ idA. A⊗A µ. µ. v >¬° 7Y¬ v_¬ -$. (2). (4). - ? A. !. C. °2# %T¬8 '=¬' v¶ ! Û@GM®ª + ± = %\ ! 2d'Û°1ªP ¸ a±ÛX % v¬ X$ ! Û Md >_d28 v % %/$JY %\ª µ=¬'¬&·¬@H1=¬ ªP ¸ ®ª¬ °¯2d'dÛR % v¬ X$ Vªq°±R¬v¬v¤ ªP ¸ @ '
(43) d=ÛR¬@M ¬ %A vVªo±¬ X1B°$ ! '=dM-¬ 1=¬ >BªP>Y %X¬5' Y7+Û´! >2PXw=PRSVt>2PX`M[=;YSTIM> ! C D ETF HJILHLK F ¬ ± #v¬ 8µ 2d'dÛ·ª1° ª' ¸ @®ª¬®°¯±ÛX %. v¬ X$ (C, ⊗, I) ! (A,>2µ,PXw=η)PRSVt>2PR`b(A[=;=0SV,ILµ> 0,!'ηM0) (A, µ, η) _ (A0, µ0, η0) VªDd PMdµ λ : A → ! 7Y¬ v¬ X$ C ªP> & 7Y 1=¬ +µÛ b®ª A0 µ A. A⊗A λ⊗λ. λ. ? A0 ⊗ A 0. µ0 η. I. ? - A0. - A. 0. η. λ. ÅkÅÍNOPO'Q R. -. v >¬° 7Y¬ v_¬ -$. C. !. ? A0.
(44) ,5~. 6Û¬%)R¬· % = 5+ ! ´<$ ª¬®°¯±ÛX %) b_¬ - $ C = (C, ⊗, I) + 7Y¬ ! % %\µ° ¨R _ R¬('a*P' ¬Vª v %\¬v_¬ - 1=$ Mon(C) O 8 1=¬#53¬' 8ª M¬ 2d'dÛ·Tª S 8 1=¬PMdµª M¬ 2d'dÛ U# BVª¸ Tª S 8 1=¬ ÛR¬ +7 ($d (A, µ, η) Vª 7Y¬ÛX¬ 87 U$ A S 1=8¬ b _¬ - $ C ! 8 v2 #Y·ª¸7 8Ûd1V)ª 1=¬ª¬®ª vd #Y·ª¸7 +Û´°H C>D,EGFHJIJHLK F N k@¬ C #v¬ v ¬ X $ ! ¬ WMon(C) = Mon(1/2End(C), ◦, id ) !H
(45) =¬ P#53M&¬d @µ8ªª® !! 11==¬8 ¬ b b __¬ ¬ -- $ $ WMon(C) bb¬ B¬ v v % %T% ¬b%T¬bq >2PR?NwYs sX%qt>2>2PRPRw6`bsR[=t=;=IoSVIMPX>2w1IB!Kb;=?usdKM?5RPR`bH C ± 1=¬ WMon(C)
(46) ?Y¬ ! % %\µ° #?M, #=ªv7 +Û R ·¬ ª 4 &Y MF @¬ ¸7 65 8Ûdx ! 7Y¬qµ¬vN¶ 'X·ª ¹H R ·¬. b_¬ -$ ! K K %HJILHLK F @¬ C #b¬® v ¬ X $ ! µ¬vN¶2'Xd1 v ¬ X $ C Vª ¸ # %\¬ (T, µ, η). bd=ª¸VªP8° ! 8 ª ¬°¯ !'>J? @_ T : C → C + 8 ' >M %K Md=ª ! ¸ +Û η : id ⇒ T + 8 d'' >M %K b=ª ! ¸2 8Ûd µ : T ◦ T ⇒ T ª'> ' 1= 5+! ® $ #53'¬ A !)7Y¬ v _¬ - $ C+ 7Y¬®dÛ@ GM®ª .
(47). C. . . C. T ◦ T ◦ T (A). µT (A) - T ◦ T (A) µA. T (µA ) ? T ◦ T (A). T (A). (1). ? - T (A). µA. ηT (A) - T ◦ T (A) id. T. (A ). (2). µA -. ? T (A). T (ηA ) T ◦ T (A) T (A) ) A d i T(. µA ? T (A) T ◦ T (A). (3). T (idT (A) ) - T ◦ T (A) µA. µA -. ? T (A). (4). ÃÄÅkðÆ.
(48) ¬®d¬vN¶ % -¬ # Md
(49) Y¬b¸Û¬¸ª. v >¬° 7Y¬ v_¬ -$ C ! °2# %T¬8 '=¬' v¶ !. ,G,. ! qª ¬ °Û@ °G7¯YM2¬%d R'¬(X'a Y!G7+2ÛÛ¬v£ Yb ª !®7ª Yª¬¬·°¯ªv2ªd ±'Û R+%N+ 7(# $ >D ¬b! 'dµ O1 =µ¬®Û¬ @±G'¬ M'> M 7 %7 (°$ ! 1!=¬ηÛ X! ¬5M' dY µ7+OÛµ ¬R 'a! >±2 d 'ÛR@+ G vMd d'% %P$ = %\·ª°12d'X O #v¬ vM¬ !'>% % +a7VªD± 1=¬®Û Md 7Yaµ>%\¬v21=¬'¸ %7($! µ ! ·ªvªK @¬ P8K Ûd%=HJIJª HLK FM¬¬ k>J @¬ %T(T,¬ µ,O η) #v¬ µ¬v5¶ ±R« 6 b_¬ -$ C !
(50) =¬ 7Y¬! % %\dµ°M 7Y¬ v ¬ X $ C S 8 7Y)¬ ¸ # %\¬ (T, µ, η) Vª ª ¬°¯2d'R!H 8 7Y¬' >b % MYª ! ¸ +Û µ : T ◦ T ⇒ T Vªo± ¸ % S 8 7Y¬oª¬ °¯ !'>J? @ T : C → C Vª !'>J? @ GS 1=8¬ v ¬ X $ C Vª 2d'R) ! 8 7Y¬µ¬v5¶'X (T, µ, η) H . . . . . . . . . . . . °2# %T¬8 '=¬' v¶ ! R¬ $ µ¬vN¶±R°'> v¬ ª 2ª¬ °¯'X ! C D ETF HJILHLK F
(51)
(52) ¬ C #v¬H b_¬ -$ ! ¬X¬ ±¬ #&$ S0 1=¬ !'>J? @8Ûd47Y _ ¬vG ' #43¬' @ ! 7Y¬ v_¬ -$ WMon(C) ·ªvª¸ dYª)7Y¬ ¸ # %\¬ (T, (idT ◦ idC) • µ, η) ! (T, µ, η) D
(53) k ¬ -$ ! k !$ #43 ¬' @ T ! 1=¬" v¬ X$ WMon(C) + S0(T) Vª #53 ¬' !) 1¬@= ¬8C v#v¬¬X $ v1/2Mon(C) ! °2# %T¬8 '=¬' v¶ ! C D ETF HJILHLK F
(54) MN ¬ . v¬qJ v¬ -ªbª¸$ R! =ª 2¬qR¬'_¬ #&$ S ! 1=¬ !'>J +Û 1= _ ¬vF & λ ∈ (id ◦ id ) • λ M k¬@ v¬ b _¬ - $ ! <$ λ ∈ WMon(C)(T, T ) +!µ¬ Y%¬ S (λ) ∈ ! #. C WMon(C)((T, µ, η), (T 0 , µ0 , η 0 )) # C 1/2Mon(C)(S0 (T), S0 (T0 )). D. 1. T0. 0. . ¬@ T = (T, µ, η) ±. T0 = (T 0 , µ0 , η 0 ). S1 (λ) • η. C. !B¬ Y%¬ &$R¬('a678Ûd´. = ((idT 0 ◦ idC ) • λ) • η # ! S1 = (idT 0 ◦ idC ) • (λ • η) = (idT 0 ◦ idC ) • η 0 . v¬' v>Bª¬ λ ∈ WMon(C)(T, T ) #. = η0 .. ÅkÅÍNOPO'Q R. 0 . 1.
(55) ,N} ¬ =%¬. 6Û¬%)R¬· % = S1 (λ) • µ = ((idT 0 ◦ idC ) • λ) • µ = (idT 0 ◦ idC ) • (λ • µ) = (idT 0 ◦ idC ) • (µ0 • (λ ◦ λ)) . v¬' b>Bª¬ λ ∈ WMon(C)(T, T ) #. 0 . = ((idT 0 ◦ idC ) • µ0 ) • (λ ◦ λ) = (µ0 • (idT 0 ◦T 0 ◦ idC )) • (λ ◦ λ) = (µ0 • (idT 0 ◦ idC ◦ idT0 ◦ idC )) • (λ ◦ λ) = µ0 • ((idT 0 ◦ idC ◦ idT0 ◦ idC ) • (λ ◦ λ)) = µ0 • (((idT 0 ◦ idC ) • λ) ◦ ((idT 0 ◦ idC ) • λ)) = µ0 • (S1 (λ) ◦ S1 (λ)) .. D . ¬ . v¬ v¬ X$ ! $ +aµ¬ =%¬. ' d<$. # C λ ∈ WMon(C)((T, µ, η), (T 0 , µ0 , η 0 )) 0 0 0 0 00 00 00 λ ∈ WMon(C)((T , µ , η ), (T , µ , η )) S1 (λ0 • λ) = S1 (λ0 ) • S1 (λ) . . . !. S1 (λ0 ) • S1 (λ). &$R¬('aY7+Û. . = ((idT 00 ◦ idC ) • λ0 ) • ((idT 0 ◦ idC ) • λ) # ! S1 = (idT 00 ◦ idC ) • (λ0 • (idT 0 ◦ idC )) • λ = (idT 00 ◦ idC ) • ((idT 00 ◦ idC ) • λ0 ) • λ = ((idT 00 ◦ idC ) • (idT 00 ◦ idC )) • (λ0 • λ) = ((idT 00 • idT 00 ) ◦ (idC ◦ idC )) • (λ0 • λ) = (idT 00 • idT 00 ) • (λ0 • λ) = S1 (λ0 • λ) .. k¬®·ª B + ¤ d' ª =dµ 7Y(+ ! <$ v_¬ -$ C +®µ¬ v R¬('a'¬1 !'>J S = µ71 S0 R¬('a±¬v ·ª° ®¬('a678Ûd 1d' µ77 S1 (S R¬('a0,'S¬v1) ·:ªWMon(C) ° ¬5'Y7+Û→ 1/2Mon(C) !.
(56) !
(57) !" 2 !"k k¬@ C #v¬ v¬ X$ ! *P Vª µ¬ % %ܯ8¶d'µ7Y 7Y¬q±+Û¹!¦>2PXwYsRt 7Y¬ v_¬ -$ Vª ¬ @>J %\¬ 8_J7Y¬ ! % %\dµ°M ' 8Ûd 8ª¬v"¬ +! o°=ªP? v¬"+ = ,A 1= v % %Ϊo7 sdQTR?5O=`bsRSTu KM;=?5PRC`bHS\w? ºBKM?5wYILSTPRwPR`b> ! C>D,EGFHJIJHLK F >PXwYsRtS\w®? ºBKM?5wYILSTPRwPR`b>" 1=¬ v ¬ - $ C Vª ) ¸ # %\¬ (T, η, − )+kµ Y¬ b¬ 8 T Vª !'>J +Û´ # (C) → # (C) S 8 η Vª !'>J +Û 7Y ¬vF &1 #53'¬ A ! C ªvªv R=ª®d1 Pbµ η : A → T (A) ° C S 8 − Vª !'>J? @8Ûd 1= _o¬vG ' f ∈ C(A, T (B)) ·ªvª¸ dYª¬ %T¬ ¬ f ! C(T (A), T (B)) 'ª > ' 1= ÃÄÅkÃ°Æ . . . .
(58) . ∗. A. ∗. ∗.
(59) ¬®d¬vN¶ % -¬ # Md
(60) Y¬b¸Û¬¸ª ,5| + ! $ f ∈ C(A, T (B))+µ¬$Y%¬ f ◦ η = f S _° ! $ f ∈ C(A, T (B)) d' ! d<$ g ∈ C(B, T (C))+µ¬ =%¬ g ◦ f = (g ◦ f ) S _°° ! $#53¬' A ! 7Y ¬ v _ ¬ - $ C+ η = id !
(61) = ¬d° ! 7J Vªpª¬' + Û VªE ª = µ 1= 1=¬D±+Ûq !%/?5sG%®>2PXwYsRt Vªo¬ >J %T¬ _ 1=¬ ! % %Tµ°M! ±+ Û 8ª¬v¬ M #Y·ª¸7+ Û ! /?5sG% >2PRwYsXt STw? ºBKb?wYIMS\PXw PX`M> dB1= ¬$v ¬ - $ C Vªj ¸ # %T¬ (T, η, − )+ µC = D ¬@ETM ¬F HJILHLK F 8 T Vª !'J> ? @8 Ûd1# (C) → # (C) S #53¬' A ! C ·ªvª¸ d Yª ´PM µ η : A → T (A) ° C S 8 η Vª® !'J> ? @8 Ûd27Y _ ¬vG 'q 8 − Vª!'J> + Û87Y j¬vG ' f ∈ C(A, T (B)) ·ªvª¸ d Yª/¬ %\¬ ¬ f ! C(T (A), T (B)) ªP> & 7Y + ! $ f ∈ C(A, T (B))+µ¬$Y %¬ f ◦ η = f S _° ! $ f ∈ C(A, T (B)) d' ! d<$ g ∈ C(B, T (C))+µ¬ = %¬ g ◦ f = (g ◦ f ) ! ª¸C¬vÛGD. ' ET ! F HJ#4¸ 3ILHL¬'K @F J1 = !)¬ 1= ¬ v ¬8_ Tv ¬ -= ¬ X $ (T,C $ η,! −>2ªv)ªvPR dR wY'=sXª tqTd>2´PR=¬ `b%\[=¬ (T;=SVIL¬,> η !', b − )! T#v¬8_ 8 µTµVª2¬v5 ¶!'J> ? @±+ªPRÛ> & ª® λ°7 Y 1¬.= -F ¬ 6_ ¯ A C λ C(T (A), T (A)) + ! $#53¬' A ! 7Y ¬ v _ ¬ - $ C+aµ¬ = %¬ λ ◦ η = η S _° ! $ f ∈ C(A, T (B))+µ¬$Y %¬ λ ◦ f = (λ ◦ f ) ◦ λ ! C D ETF HJILHLK F ¬ T = (T, η, − ) + T = (T , η , − ) d' T = (T , η , − ) #v¬B1 M ¬v¬ 1µ= ¬v5T ¶ ±¬v 'FX &%T¬ ·¢ ªλ°´#53#v¬'¬ ¬2 -F _ ¬=2ª¸Ûd d!H' R7! Y ¬ ¸ v U _d# ¬ HB- Vª¸1= $ ¬8!'b M _ ¬ -ªbª¸T R $ = CªH ! 1T= ¬ ¬ ! ¬ 2%\λ¬2¬®#b¬¬X ¬ ± ¬±#&R$ λ2•U λ#J 1=Vªv¬ !'J> !'M?d @ + ÛT ! A C (λ • λ) = λ ◦ λ . . ∗. A. . . ∗. . A. . ∗. ∗. ∗. ∗. T (A). .
(62) . ∗. A. ∗. ∗. . . ∗. A. . ∗.
(63) . ∗. 0. 0. ∗. ∗. ∗. ∗0. 0. 0. 0. A. . A. . . ∗. B.
(64). 0. ∗. 0. B. 0. 0. 0. 0. ∗0. 0 A. A. ∗0. A. 00. 00. 00. 00. ∗00. 0. 0 A A A ! C(T (A), T (A)) ±D'X%\¬@· ª °´ #v¬ ¬ kF- ¬@_¬ 2=Tª¸dÛ=d' R(T,! 2¸η, U−# J∗V)ªv7+ YT¬ !'0M v=d¬ (TX0,$ _η C0, −! ∗!8¬@)
(65) dλY'¬ #b ¬®T00 =±(TVRª002,2η00dU,'# −XB∗2Vª¸)2#v!'¬UM#1J Vbªv¬bT ¬_!'µ M¬bdTN ¶ 0 λ0 T0 T00 λ0 • λ 00 ! T T 0. 00. 0. 8 k <$ #43¬' @. A. 00. !)1=¬ v¬ -$ Ca+ µ¬ = ·¬. (λ0 • λ)A ◦ ηA. = (λ0A ◦ λA ) ◦ ηA = λ0A ◦ (λA ◦ ηA ). v¬& v>Bª ¬ λ Vª®2d'XU#JVª¸ !'b T _ T η #v¬& v>Bª ¬ Vª ±R2U#BVª¸ !'M d T T λ. = λ0A ◦ η 0 A. #. . =. ÅkÅÍNOPO'Q R. 0. . 00. A. 0. 0. 00 . ..
(66) ,{ 8. d<$ . f ∈ C(A, T (B)) (λ0 • λ)B ◦ f ∗. 6Û¬%)R¬· % =. +µ¬$Y%¬. = (λ0B ◦ λB ) ◦ f ∗ = λ0B ◦ (λB ◦ f ∗ ). v¬' b>Bª¬ Vª 'XU# BVª¸ '! M T 0. = λ0B ◦ ((λB ◦ f )∗ ◦ λA ) # λ. T0. . 0. = (λ0B ◦ (λB ◦ f )∗ ) ◦ λA 00. = ((λ0B ◦ (λB ◦ f ))∗ ◦ λ0A ) ◦ λA 00. = ((λ0B ◦ λB ) ◦ f )∗ ◦ (λ0A ◦ λA ) 00. = ((λ0 • λ)B ◦ f )∗ ◦ (λ0 • λ)A .. C>D,EGFHJIJHLK F
(67) k ¬@ T = (T, η, −∗) #v¬!µ¬vN¶%'X%° ¬ -G¬=ª¸Û ! ¸ J1=¬B b_¬ -$ !"¬DR¬'_¬ #&$ idT 1=¬!'>J? @+Û 1= ¬vG ' #43¬' @ A !51=¬$ v¬ X$ C ªvªv R=ª id ∈ C T (A) ! C(T (A), T (A)) D ¬ #v¬%1µ¬v5¶1'X1°«¬ -F_¬=ª¸Û ! ¸ dL1=¬ v¬ X$ ! C
(68) ?Y¬ idT Vª T'=X(T,η,U# −BV∗ª¸) !'M T T !
(69) ¸ N Û %7! D . ¸ 7Y¬ v¬ ¬ TX=$ (T,C η,±− %\¬@) λ±#v¬ T2=d(T'R, η,−U#J)Vª¸#v¬ !'+bµ µT¬v5¶ T ! ±
(70) R·=ª¬1°«µ¬.¬ -F=¬ Y·ª¸¬ Û _ id • λ = λ S _° λ • id = λ ! !. ∗. 0. 0. 0. ∗0. 0. . T0. . . T.
(71) ¸ N Û %7! µY·ª¬#53¬' 8ª M¬2_ kµ¬¬v®Na+ ¶!µ2·Yª d· 'ª¬+ X·! vª®2°' #=¬ ªvª-F7_+¬ÛY=´ª¸µÛdVª 1=! ¸±µ¬+ µµ vYY··ªª¬®R¬¬(Û'aR±P¬M¬ +µ17=(ª$¬ Td v→¬ XTVª 0 $ eWMon(C) M¬2±R!U# BVª¸ ª !'Md ! T T0 • T idT C>D,EGFHJIJHLK F
(72) ¬R¬'¬ #"$ M0 7Y¬ !'>J +Û 7Y ¬vF & #53¬& @ T = (T0, η, −∗) ! 7Y¬. b_¬ -$ eWMon(C) ·ªvª¸ dYª 1=¬ ¸ # %\¬ (T, µ, η) a+ µY¬ b¬ 8 T = (T0, T1) + µ=¬@M¬ T1 Vª !'>J +Û21= ¬vF & f ∈ C(A, B) ·ªvª¸ dYª (ηB ◦ f )∗ ! 8 µ Vª !'>J? @+Û 1= ¬vG '¨ #43 ¬' @ A ! 7Y¬ v¬ X$ WMon(C) ªvªv R=ª µA = ! (idT (A) )∗ D 1d<$ #43 ¬' @ ! 1=¬ b_¬ -$ eWMon(C) + M0(T) Vª #53 ¬' ! 1=¬. b_¬ -$ WMon(C) ! T. ÃÄÅkðÆ.
(73) ¬®d¬vN¶ % - ¬ # Md
(74) Y¬b¸Û¬¸ª . ¬ . ! ¬@. ,I. v¬d #53¬' ±!$ 7Y¬ v¬ -$ eWMon(C) ! ¬ M (T) = (T, µ, η) µ71. ! $ ¬ = · ¬ g ∈ C(B, C) T (g ◦ f ) = (η ◦ (g ◦ f )) #&$R(¬ 'a678Ûd´! T #. . T = (T0 , η, −∗ ) T = (T0 , T1 ) f ∈ C(A, B). 0. 1. . ∗. C. . 1. = ((ηC ◦ g) ◦ f )∗ = (((ηC ◦ g)∗ ◦ ηB ) ◦ f )∗ = ((ηC ◦ g)∗ ◦ (ηB ◦ f ))∗ = (ηC ◦ g)∗ ◦ (ηB ◦ f )∗. : ¬? v¬. Vªª¬ °¯ !'>J? @_+ µC¬$Y→%C¬ ! T $. = T1 (g) ◦ T1 (f ) .. f ∈ C(A, B). µB ◦ (T ◦ T )(f ) = id∗T (B) ◦ (ηT (B) ◦ (ηB ◦ f )∗ )∗ = (id∗T (B) ◦ (ηT (B) ◦ (ηB ◦ f )∗ ))∗ = ((id∗T (B) ◦ ηT (B) ) ◦ (ηB ◦ f )∗ )∗ = (idT (B) ◦ (ηB ◦ f )∗ )∗ ∗. = (ηB ◦ f )∗ = ((ηB ◦ f )∗ ◦ idT (A) )∗ = (ηB ◦ f )∗ ◦ id∗T (A) = T (f ) ◦ µA . : ¬? v¬. >M % MYª ! ¸28Ûd µ Vª$ f '∈C(A, +µ¬$Y%¬ B). T ◦T ⇒T. . T (f ) ◦ ηA. : ¬? v¬. X¬5'Y7+Û ! T d'. #"$. !. R¬('aY7+Û ! T. = (ηB ◦ f )∗ ◦ ηA . #&$. . µ .. . = η ◦f . V . ª ' > M % b=ª ! ¸28Ûd η $#53¬' A !)7Y¬ v¬ X$ Cid+aµ⇒¬$YT%!¬ µ ◦ T (µ ) = id ◦ (η ◦ id #&$R(¬ 'a678Ûd´ ! ' ) µ T B. C. A. A. . ∗ T (A). T (A). ∗ ∗ T (A). . = (id∗T (A) ◦ (ηT (A) ◦ id∗T (A) ))∗ = ((id∗T (A) ◦ ηT (A) ) ◦ id∗T (A) )∗ = (idT (A) ◦ id∗T (A) )∗ = id∗T (A) ∗ = (id∗T (A) ◦ idT (T (A)) )∗ = id∗T (A) ◦ id∗T (T (A)) = µA ◦ µT (A) .. $#53¬' A !)7Y¬ v¬ X$ Ca+ µ¬$Y%¬ µ ◦η = id ◦η #"$X¬5'Y7+Û´! µ ÅkÅÍ NOPO'Q R . A. T (A). . ∗ T (A). T (A). A.
(75) ,5z. 6Û¬%)R¬· % = k <$ #43¬' @ A !)1=¬8 v¬ X$ C=+aµ¬id= ·¬ . µ ◦ T (η ) = id ◦ (η ◦η ) #&$R(¬ 'aY7+Û ! d' µ T (A). . A. A. . ∗ T (A). A. T (A). ∗. T. . = (id∗T (A) ◦ (ηT (A) ◦ ηA ))∗ = ((id∗T (A) ◦ ηT (A) ) ◦ ηA )∗ = (idT (A) ◦ ηA )∗ = (ηA ◦ idA )∗ = T (idA ) .. k <$ #43¬' @ A !)1=¬8 v¬ X$ Ca+ µ¬ = ·¬ ◦ (η ◦ id ) µ ◦ T (id ) = id #&$R¬('aY7+Û ! d' T µ . A. T (A). . ∗ T (A). T (A). T (A). ∗. . = id∗T (A) ◦ ηT∗ (A) = (id∗T (A) ◦ ηT (A) )∗ = id∗T (A) = µA .. D ¬ 2U# BVª¸ !'Md . . T T. d' T!"
(76) #b¬= ¬+ µµ¬ #4=3%¬' @Û¬ ª2 !)1=¬ v¬ X$ eWMon(C) ! k¬@! λ #v¬22d'R T λ ∈ WMon(C)(M (T), M (T )) 0. 0. 0. ¬@ T = (T , η, − )+ T = (T , η , − )+ M (T) = (T, µ, η) ' 8 d< $ f ∈ C(A, B) +µ¬ =% ¬ 0. ∗. 0. T 0 (f ) ◦ λA. 0 0. ∗0. 0. 0. 0. 0. M0 (T0 ) = (T 0 , µ0 , η 0 ). !. 0. 0 = (ηB ◦ f ) ∗ ◦ λA. v¬& v>Bª ¬ Vª2d'XU#JVª¸ !'M T _ T = Λ ◦ (η ◦ f ) e@-° #v¬& v>Bª ¬ λ Vª2d'XU#JVª¸ !'b T _ = λ ◦ T (f ) . d<$#53¬' !)1=8¬ b _¬ - $ C+aµ¬ =%¬ A 0. = (λB ◦ ηB ◦ f )∗ ◦ λA # λ. 8. . B. 0. ∗. B. T0. . B. (λ • µ)A. = λA ◦ µA = λA ◦ idT (A) ∗. v¬' b>Bª¬ λ Vª 'XU# BVª¸ '! M T 0. = (λA ◦ idT (A) )∗ ◦ λT (A) . #. 0. T0. . 0. = (idT 0 (A) ∗ ◦ (ηT0 0 (A) ◦ λA )∗ ) ◦ λT (A) 0. = (idT 0 (A) ∗ ◦ T 0 (λA )) ◦ λT (A) 0. = idT 0 (A) ∗ ◦ (T 0 (λA ) ◦ λT (A) ) = µ0A ◦ (λ ◦ λ)A = (µ0 • (λ ◦ λ))A .. ÃÄÅkðÆ.
(77) ¬®d¬vN¶ % -¬ # Md
(78) Y¬b¸Û¬¸ª 8 k <$ #43¬' @ A !)1=¬ v¬ -$ C a+ µ¬ = ·¬ (λ • η)A. ,·. = λA ◦ η A. v¬' v>Bª¬ λ Vª ±RU# BVª¸ !'Md T T . µ! 71 M¬ 2=(Mª 2,dM')ª+µ=+ =d1¬@µ=M¬¬ 7MYVµª¬8R¬( b'ad±¬vX¬5'·ª'°¬ ¬!'5>J'Y 7+Û M: eWMon(C) ' M ! VªR¬('a→±¬vWMon(C) ·ª ! % %\µª&O λ ∈ eWMon(T, T ) M (λ) = λ ∈ WMon(M (T), M (T )) C D ETF HJILHLK F ¬X¬ ± ¬ #&$ e 1=¬ !'>J +ÛJ 1= _ ¬vF & #43'¬ @ T = ((T , T ), µ, η) ! 1=¬8 b_¬ - $ WMon(C) ·ªvª¸ R=)ª 7Y¬ ¸ # %T¬ (T , η, − ) +µ =@¬ M¬ 8 − Vª !'>J +Û 7Y _¬vF & f ∈ C(A, T (B)) ªvªv R=ª µ ◦ T (f ) ! D k d< $D #43'¬ @ T ! 7Y¬ v ¬ - $ WMon(C)+ e (T) Vª #53'¬ ! 7Y¬ b _¬ - $ ! 0 = ηA. . #. 0. . 0. 1. 0. 0. 1. 1. 0. 0. 0.
(79). 0. 0. 1. ∗. 0. ∗. 1. B. 0. eWMon(C). . ¬ . $. v¬®d´ #43¬' @+aµ!)¬ =7Y ¬ ·¬ v_¬ -$ #. T = (T, µ, η) f ∈ C(A, T (B)). f ∗ ◦ ηA. WMon(C). 2¬ª¬@ e (T) = (T , η, − ) ! !. 0. R¬('aY7+Û ! −. #&$. = (µB ◦ T (f )) ◦ ηA . 0. ∗. ∗. = µB ◦ (T (f ) ◦ ηA ) = µB ◦ (ηB ◦ f ) = (µB ◦ ηB ) ◦ f. $ . = idT (B) ◦ f = f .. f ∈ C(A, T (B)) g∗ ◦ f ∗. d' ! ®d<$ g ∈ C(B, T (C))+aµ¬$Y%¬ = (µ ◦ T (g)) ◦ (µ ◦ T (f )) #&$R¬('a678Ûd ! − . C. B. ∗. = µC ◦ (T (g) ◦ µB ) ◦ T (f ) = µC ◦ (µT (C) ◦ (T ◦ T )(g)) ◦ T (f ) = ((µC ◦ µT (C) ) ◦ (T ◦ T )(g)) ◦ T (f ) = ((µC ◦ T (µC )) ◦ T (T (g))) ◦ T (f ) = µC ◦ ((T (µC ) ◦ T (T (g))) ◦ T (f )) = µC ◦ T (µC ◦ T (g)) ◦ T (f ) = µC ◦ (T (g ∗ ) ◦ T (f )) = µC ◦ T (g ∗ ◦ f ) = (g ∗ ◦ f )∗ .. 1=¬8 b_¬ -$ WMon(C) !2 ¬@ λ #v¬®'X DU#JVª¸ !'b ¬@ TT_ d'T ! T
(80) =#b¬¬ 1+µµ¬ =#4 3·¬'¬ @Û몮∈! eWMon(C)(e ! (T), e (T )) ÅkÅÍNOPO'Q R
(81). 0. 0. 0. 0. 0.
(82) ,5. 6Û¬%)R¬· % =. . ¬@ kT= (T,<$µ, #43η)¬' @+ T = !)(T1=¬8, µ v,η¬ X)+ e$ (T)+aµ=¬ (T= ·,¬ η, − ) ±. . 0. 0. 0. 0. 0. A. λA ◦ η A. ∗. 0. 0. e0 (T0 ) = (T00 , η 0 , −∗ ). C. = (λ • η)A = λ0A. v¬& v>Bª ¬ λ Vª2d'XU#JVª¸ '! b T _ k <$ +µ¬ = ·¬ f ∈ C(A, T (B)) . λB ◦ f ∗. !. #. . T0 .. = λB ◦ (µB ◦ T (f )) = (λB ◦ µB ) ◦ T (f ) = (λ • µ)B ◦ T (f ). v¬& v>Bª ¬ Vª2d'XU# BVª¸ '! M T _. = (µ0 • (λ ◦ λ))B ◦ T (f ) # λ. T0. . = (µ0B ◦ (λ ◦ λ)B ) ◦ T (f ) = µ0B ◦ ((λ ◦ λ)B ◦ T (f )) = µ0B ◦ ((T 0 (λB ) ◦ λT (B) ) ◦ T (f )) = µ0B ◦ (T 0 (λB ) ◦ (λT (B) ◦ T (f ))) = µ0B ◦ (T 0 (λB ) ◦ (T 0 (f ) ◦ λA )) = µ0B ◦ ((T 0 (λB ) ◦ T 0 (f )) ◦ λA ) = µ0B ◦ (T 0 (λB ◦ f ) ◦ λA ) = (µ0B ◦ T 0 (λB ◦ f )) ◦ λA 0. = (λB ◦ f )∗ ◦ λA .. k¬®·ª ?. d77' 1ªR¬('aY±µ ¬b 7Y·ª2 ° µ¬ ®¬ v('a¨678RÛ(¬ d'a ±¬%?´ !''>J ? @µ_77 e =X(e¬5'',¬veq)·ª : ! WMon(C) µ % %Tµª&O ! → eWMon(C) e e λ∈ + 7Y¬ e (λ) = λ ∈ eWMon(e (T), e (T )) ! WMon(T, T ) K K %HJILHLK F
(83) Y¬ !'>J vª M ± e M¬ 8µVªU#JVª¸ ª! v¬ -¸Û¬ ª&! . 0. . 0. 0. . 1. 1. 1. 0. 0. 0. . . k¬@ T = ((T0, T1), µ, η) #v¬#53¬' k! ! ((T00 , T10 ), µ0 , η) k <$ +µ¬ = ·¬ f ∈ C(A, B) T10 (f ). WMon(C). ! . ¬@ e(T) = (T , η, − ) d' M (e (T)) = 0. ∗. 0. 0. = (ηB ◦ f )∗ = µB ◦ T1 (ηB ◦ f ) = µB ◦ (T1 (ηB ) ◦ T1 (f )) = (µB ◦ T1 (ηB )) ◦ T1 (f ) = T1 (idB ) ◦ T1 (f ) . v¬' v>Bª¬ ((T , T ), µ, η) Vª d1#53¬' p ! WMon(C) #. 0. . 1. = T1 (idB ◦ f ) = T1 (f ) .. ÃÄÅkðÆ.
(84) ¬®d¬vN¶ % -¬ # Md
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