MAYBE XEROXEO
ATHESIS
SUBMITTE DTO
ne
aM-lITT EEONGRADUATE STUDIES INPARTIAL FULFILLMENT OFTHE REQUIREMENTSFORTHEDEGREE OF MASTEROF ARTS
MfJ.DRI ALl.U VERSI TYOFNEWFOLflDLAN D ST. JOHN'S.NEWFOLflDLAND
AUGUST, 19 72
(i)
Thisthesishas beenexami n e d and approvedby:
Supervisor
In te rnalExaminer
Exte rna lExami ne r
Dat e--- --~- --- ---- ---
~
IACI'NOWLEDGEMENTS
Itis agreatpleasure toacknowledge th e inv alu able ass istan ceand sup e rvi s i ongi ve nsowillinglybyDr.Peter800thduringthe preparat i on ofthisthes is. Als oth anksare due toMrs . Rosali n d Wellsfor her patience andsk i llsh own intyPingIIYsometiae s illegi ble writing.
Finall y , Isayth anksto mywife,Jane, forher enc o ura g ement and fore- bear ance.
(iii)
ABSTRACf
'Le t us
conside~
the followingsYJlllletric monoida lclosedc~tegOries:
(i) 8M'thecategory of setsalder the actionof a commutative mono id M; in short ,aca t e go ry of M-sets;
(il) SG,thecategoryof u-sets , where G is an abelian group;
(iii) MK,th e categoryofmoduloi d s over a commutative semiring K (a moduloidis basicallyamono i d actedon by a semiring);
(iv) ModK,t h ecategoryof modulesover a commutativering K;
(v) "F,t h e categoryof vecto rspaces over a field F.
Let C be an arbitrary clos edcategory. We are concernedwit h the followi ngques tion:
Whatconditions have to be itllposedon (J to ensure thatit can be embedded(in some canonicalway) in toone or more of theabove categories?
Thebas i c categorythe oryne ed e d in th i sthe s i s is provided in chaptersI and II. Inchapter I weha ve provided the detailS of how, inII categorywith bd'products ,the set hom(A, B) can be given the structureof a coll\Jllutative monoid(underadd i tion). Chapter II gives a sUllllllary of the standardde f i n itions andresu l ts le adin gup toth e conceptof a symmetric monoidal closed category.
Sincetheproperties of categories (i) and (iii) are not sowell known, these categor iesare discussed in some detail in chapte rs III and IV. It is shOWnthateach ofthe categoriesisin fact a symmetricmonoidal closed category.
In chapter Vwe answer ouroriginal question by establishingfive embeddingtheorems.
Ea ch of thesetheoremsgives su f f i ci en tconditionsfor aclosed cat e g ory tobe embeddabl~in~~eof theebcve cate gories. F:rl;lyeleme"Rtary examplesare given to illustrateeach of the theorems.
In the appendixa detailedexample isgivento sh ow that theseembeddings ar e not.ingen e r al . fullembe ddi ngs.
(v)
ACkNOWLEDGEMENTS
ABSTRACT
Page (i)
(ii)
OfAPTERr. CATEGORIES ANDBIPRODUCTS 1. Categories 2. h~ cto :rs
3. NaturalTransformations 4. Zero Ob jects S. ProductsandSums 6. 8iprodu cts
OiAPTERII. CLOSED CATEGORIES 16.
1. Clos ed Catego ries withFait hful Bas i cFunctors 2. Mono lda! Categorie s
3. Mono i da]Closed categ o ries 4. S)'DIIOtrlc Monoid alClosedCat e gorie s
OiAPTERIII. M-SETS
1. Definitionsand Examples 2. Quotient x- s ee s 3. M-Setsof Morp his ms 4. M-Biaorph isllL'l S. Tens or Produ ctsof M-Sets 6. The Cate gory 8
M of x-s ee s
CHAPTER IV. loIlOULOIDS
1. Definitionsand EXaJDPl e s 2. SublllOduloi ds
3. Cong ruence s and QuotientNod uloids 4. Mo rph isas ofModul o i ds
29.
44.
s. FreeModuloids 6.·Biproducts. 7. Modwoids ofMorphism!>
8. Bilin e ar FtDl.ctions 9. TensorProduct ofK-Mod wo i ds 10. The Category ~ of K-Modu loids
otAYrERV. EMBEDDINGTHEOREMS 70.
APPENDIX BI BLIOGRAPHY
I. Embedd i n gTheo remsforM-Sets andG-Sets
2. Embe dding Theo r ems forModulo i ds ,Modul e sandVectorSpaces 3. Ex amples.
89 .
94.
1.
OiAPTER ONE
CATEGORIES AND BIPRODOC'TS
Itis well knownthat inan Abelian category 'theset hoa(A..B) of aorphis asfrolll A to B can be enriched withan Abeliangroupstructure.
In this chapterwewil lprovi de50 _ basiccategorytheory and show,by a fairlyst an d ard argUlllt!nt . that in acategory wi th bfprcductis,the set homeA.B) can be given thestr uct ureofaco ...utativeecnctd.
1. Cate gor i e s
DefI nI tion 1.1
Acateg ory C cons ists of (1) a classofobjects A,B.C•••
(ii) for eachpair (A,B) of objectsa set hOOl (", 8 ) theel emen tsof which are calle dIIIOrphisasfroa Ato B of C. witildolllain A and codomain B. (We write x:A- .B or A2.tS for each. x£hoa(A,B) (Ui) foreadl.trip le (A,B,C) of objectsa fUletioD
ho a ( A, B) xhoa (B. C)-+h Oll (A, C) called co~itioD of.arp hislllS ; the s edat a beiDa:s\tl j eettothetwo axiolll5
(1 ) If x~ho m( A. B) . y£hOIl(B,C). %EohOll (C. D) then :to(r0 x) • (z 0ylox
(2) For each obje ct A th e r e exi sts an ele ment lA~ hom(A , ;.) calledan ide ntityIIIOrphism such that i f xco:hom(A,B ) the n
ROllIark : The.arp hb. lA whos e existen ce isrequiredby(2) islD11quel y defined;be causeif.1" isasecon dIIlOrphis.,with the same propertythe n 1,\0 lA-
1 1 "
lADuringthe courseof thisthesiswewillfrequentlyre f er to the fo llowingcategorie s:
S. thecategoryof allsets ;
S ..
the catego ryofpoi nte dse ts;SM' the cate go ry ofse ts under the acti on ofa colQlJlutativc monoi d H;
SGOthe catego ryof sets undertheact ionof anAbeliangroup G;
Mod IC' the categoryofmodulesover acollllDutat ivering K.
"\::. thecatego ryof .aduloids over a co. .utative seairlng K (the tens llOduloi d andse llli ri ng w111 be:1efined in chapter4);
Vp " th e category of vectorsp a ces overa field F.
Becausethe abovecategories ha veobjectswithunderlyingsets . that is the reisa fai t hful functor C-+S. they ar e mor e speci fica lly referred to as conc retecatego ries. Howev er. it canbe easilysh ownthateve ry concretecategoryis acat e go ry.
A cate go ry C' isa subcateg oryof C Wider the followi ngcon ditions (1) ObD!CObC
(2) homCl (A, B)C homC(A ,B) for all (A,B)E:ClxC' (3) the compos i tion of anytwo morphisDlSin C' isthesame as
th ei r colllpOsitionin C
(4) IA isthe same in C' as in C forall A
e
C'If furthermore heIl.C,(A,B) •hca C(A, B) for all (A, B)
c
C' xC' ~3.
saythat C' is afullsubcategory of C. For exampl e . th ecauiOO'of
.Abe liangroupsisa fullsu,cst eg o ryof the cat ego ry of-allgroups.
De f i n iti on 1. 2
For every category C we definethe dual catego ry
e-
as follows:(i) DbC*= (ii) MorC* ..
{A*
I
Ae:
DbC}{x*
I
xE:.Mer C} where x*0y*• (y0x)· That istheobjects of C· are the same as the objectsof C and a aorph i s_ A---fB in C*is a.orp h i s lI 8~A.in C.Definition1.3
Foreach pair of categorie s C.C·. there exis ts aprod uct category ex C'. Anobjectof th is product isanorderedpair (A,A ') of objects of Cand C' respectively; a morphis m (A .A')~(B.B') wit hthe indicateddomainand codomainisanordered pair (f,f') of morphisms
£:A---"S.it:A'--4S'. Thecompos it e ofIIlOrphismsisdefine dterm-wi s e; Thus (f,f') as above and ase cond suchordered pair (g.g'):("B.B·) ~(D.D·) have the collpOsite (g,g')0 (f, f')..(g0 f," 0 i'): (A.A·) ...(D. O·).
Def i ni ti on1.4
A morphis.. x:A_ B isin vertible(isanisOllOrphi slI) in C iff ther e is a JDOrphisl'l x·:B_ A in C with both x'0x ..lA and x0x' t "e-
Afamilia r argwnent showsthat if sucha morphismexis ts,it isunique;
henceitis usua lly written x' .x-I . Two objects AandB areeq uiv alent (Le.isomorphic)in Cifthereisan invertiblemorphism x:A ~B.
2. Fun ct o rs Definition 1.S
A covarian t flnctor (orsimplyfutelar)fro a a category A toa cat e go ry B isa function F:A---tB whi chassignstoeveryobject A of A anobject. FCA) of B and toeve ry.arphis. x:"J.---+Az in A aIIOrphism F(x) :F(Al)~F(~) in B su ch that
(2) F(x0y) •F(x) 0Fey)
~ I fcondition(2) isreplaced by (2') F( x 0y) •Fey) 0F(x )
wespeakof a con t rava ri an t ftrlctor F:A---'B wh i chassignsto every .:>rphiu l x:Az---"'~ in A amorphisaF(x) :F(Al )-+F(~ )in I Expp le! (The Standard hOlD f\rlctors)
Let S bethecategoryofsets, A anarbitrarycateg o ryand A an object in A.
(1) The fmeter hOIlCA,.):A--+ S defi ned as fol lows: For BE:A hOIl(A.- ) ( B) ..hOIl {A,B) . For X:Bl~B2 £. A. hOIll(A,-) (x) isthe function hOll (A.Bl ) ~hOIll(A,Bz) definedby hOIl (A.X ) (y ) •:It0 Y whe re y:A--+81 ; Lscovariant .
(2) The functor home-,A) :A~Sdefined as foll ows : For B£A home-,Al( B)• hom(S,A). For x:Bl ~B2E A hOID ( - , A) (x) isthe functi on hom(B2,A)~om (Bl,A) definedby hOlD (x,A)(y )•y 0x where y:B~A.; is cont ravariant.
1
s.
Definition 1. 6
Given.• functor P:A~B bet weentwo categories-It and B. we write PAD:ho.(A,B)- . , .ho_ (PA, FB) fortheassocia t e dfunct i o nson the setsof 1IlOrphisru . The ftmctor F is calledfai thful ifeach FA8 is inj e c tive.
Thatisif F isa fai thfulfunctor, then themorphism f:A---J,B is compl e telydeterminedby Ff:FA ~FB since f.g:A~B with pi-Pg :FA--lFB impliesthat f"g.
~
If P:A--+ B isfaithful,then adia grlllll ofIIOrphisas in A co~es
iff F of itco_utes in B••
Proof (Easy)
Definition1.7
Afunct or e:AxB----+C on apro d uc t cate gory A x B toanother cat e go ry C is called a~on A and B to C.
Forexample.meMetor ho.:A*x A--+S I calledtheusualhOIa functo r tosetsJisa bifun cto r.
Defini t:ion 1.8
Let F:C---tS be afunctor (co varian t ) frolll a cat egory C toth e categoryofse t s S. Aunive rs a l element for F isapair (u .R) consis t ing of an obje ct Rof C and anelemen t u€ FeR) withthefollow i ng property.
To eve ry object Aof C andeve ry element se:F(A) thereis ex a ctly oneJDOrp hisll f:R- tA with P(f)( u) "" s ,
Remark: Auniversalelemen t (v. R) foracon t ravari an tfun c tor K:C~S con:sists ofan ob j ect Rof C and ane'lelDent vII.K( R).·-such that foreachelement aE.K(A) thereisexactly one. .rphis.
f:A...-fR with K(fl(v) '" a. Since X iscontr a va riant , KCf) is a ftml:tio n K(f ) : K( R)--+K(A) .
3. Nat ur al Transfo rmat ions Definition 1.9
If F.H :A - ;B arefWlCtors .a natural tr ansform atio n 6:P-...R fro m F to H is afuncti onwh ich assigns toeachob j e ct Aof A a Ilorphis ll SA:F{A)~H(A) of B insucha way that every morphism f:A~A' of Ayields acOJlllllutativediagrll/ll
' A
H I
lH(f)'A'
Anaturaltransfo1'1lla t!on 9:F--+B isalsocal leda".,rphis.of flrlcto rs".
Definition 1.1 0
Ifeach 8A isaniSOlllOrphi s lIlincate gory B. wecall e:F--...+B
•natural isomorphismoranaturalequivalence.
~ A generalizationofthenotionof ana t ur a ltransf o rma ti on has beengivenby EilenbergandKelly [6J. Rath er than pre s en t i ng a detailed acco untof thi s gene r a liza tion , we willgive the particular detail s for ea chsit uationinwhi chthis ge ne ralized notionisused .
7.
4. ZeroObj e cts Defini tion 1.11
Anobjec t 0 iscalled azero object in aca te go %)" C if for each ob je ct A in C, hom(A,O) and hom(D,A} containexa ct lyoneelemen t .
Proposit i on1.1
Nlytwozero objectsareisomorph i c.
Proof: Assune that 0 and O' are distinctzeroobjectsin C. Sinc e 0 Isa ze roobject wehav e ho m(O' ,0) and hom(D,O') eachcon t ai n i ng ex a c t lyoneel e ment. Thus , O·....L..,.O-J:::....O' impl ies that yox'"10" 5f_ita r ly, x0y • 10 andccns equerrtIy 0andO' areiso lllOrphi c.
Definition1.12
If C has azero objectthenthemap A....!..., B is call e d ze roifi t fact orsthrough thezero object.
Pro position 1.2
boa(A..B) contai ns exactlyonezero.ap.
Proof: Conside r A~0
....!....t
B where a.v0u. Since 0 isazero ob ject for C. uand v are unkque, Therefore a·v0U isum que, Wri te a·O.Proposition1. 3 xoO -() ox_o
1
~: Consider C""!'+A.~0~B
o0x . (v0u}ox.v 0(u-e-x ) ..0 since C---...,.B"facto rs
through O. Sillilarly x 0 0=O.
S. Produ cts andSlDS
Definition 1.13
Gi ve napair ofobjects A, B ofa cat egory C, we say thattheobj e ct P isa~of Aand B ifthereexi st morp hisllls P~A and P24B suchth a tforeverypair of morphisms X--iA and X-4B thereis a unique X---+P such that
~ fpl
' ~l
8I tcan be eas ily shown that p. th e produc t of A andB. is uniq ue up to iso.crphis.. P iswritt en as AxB.
I f
~r;,
' ~l"
"
Wedefinethe diagonal map
lI :A~ A
x A by IJ..[~].
Le.9.
Definiti on1.14
. Givenapairofobjects Aand_B, ·we say th atan obj ect S·isa SlD (or cop ro d uct ) of A and 8 i fthereexis t IIOrph isas 11:A--tS and 12:B~S su ch thatforeve ry pair of JIlorphislllS A---fX and B~X there is a tmique morphism 5~X such that
Itcanbe easil ysh own that S. the S\IIII of A and B istrIiqueup to isomorphis lll. 5 iswrit ten A+B
If
we write f· [fl' £2]
,. [£il'£12)
We definethefolding••p orcod iago na l _p V:A+It. - A by V .[1, 1 ]. t,e,
Remar k: Maps Al+~----+81X82 can be wri ttenasmatr i c e s a,B=1, 2
This canbese e nbyconsideri n g thefollowi ng diagram s
Tosh ow thisinvarian ceweuse
p~, [:~~
'" Pt/ where a-1or 2and [fit,fi 2]i6=fi
13 whe r e 6=1 or2.
,
11.
Take k..
"'d
Paki a"" Pa ([fi l'fiz]ia) "Pa ( fia) Therefore Pah i6 ;; Pakia for a,aIE.(l,2) Thus h "" k,
Definition1.15
If C is acate gorywith aze ro object then a biproductof Al and Arl.
isa diagram
su ch th at
(i )(11 '~) isaSimi (i i ) (PI' Pz ) isa product (iii) Plil..lAI; Pz i
z"l~ ;Pliz "0;Pzi1=0 Wewrite A=Al tt)A
z
Let C beacate gorywitha zero obje ct su ch that anytwoobject shave a biproduct.
Definition 1.16
If x,y4:hom(A. B) wedefine
x+ Ly:A--'l"B by
A ~A +A ~B
X +RY:~A~B
byA 1UB + B ~B
Proposition1.4
O+LX. X+LO .x
o
+ RY .. Y+R 0..y.
~ 0+Lx isbydefinitionthe colllpOsite A~A+ A.124B
Byth edefinitionofSUIII [O. x) i1..0 and [O,x] 12..x, But xP2ilc
x.u ..
0 and xP212.. x.l..x, Therefore , xP2'" [O, x ]. So [O, xJ4.. xP14 ..xl..x,Theothe rresults _aybeobtained sWlarly and by duality.
ProPosition 1.5
(1) Givena')rphiUIS A~B B~C then ux+L uy..u(x+Ly) (i i ) If z:C --+" isa morphisll. then xz+Ryz..(x+RY) z
Proof: (1) By defi nition u(x+L y) isth e compos ite A~A+A~B~C
Thatis
13.
Bydefinition ux. L uy is
Thus it issuf f i cient to prove that u[x. y]. (ux,ur]. If u[x.y]
replace s lux,uy) in the se co n d dlagrlUl.the triangle son theright si deof thi sdiagraJl willst il l cceeute• Thus bythe ooi q ue nes s property
u[x.Y] •lux, uy]
The other result is obtainedin asi mi l arway.
froposition1,6
(1) +L and +R arethe same (written+). (ii) (h o_CA, B).+
>
isaCOlIID.utative .ano i d . Proof: (1) Cons i d e r the four-orp h isllSA~A @ A -) B EEl B~ B
•• [w. r,:.1 Xl
v (e6.)
Reme"' '"that
G:~J ~. [&;~:?J
=IT;] [Q]
Therefo r e . ee =
IT;] [~J . ~]
>L[~]
V( [; ] >L
[~J)
V
~J
>L V[~J
In a similarway
(ve) l!... [w,xJA +R [r,z].6.
'" (w ..Lx) +R (Y·+Lz)
(1) Put X"Y=0
Then and wewrite + for +R and +L
(ii ) Put y"O. Then w+( x ..z)'" (w..x) .. z Put w=z'"O. Then y+x•x ..y Thus (hom(A.B). ") isa ccamut at.Lvemonoid.
Proposition1.7
Givena biproduct of Al and~ in C. then 11 Pl+izPz ,.lA
1(£lA 2
Proof: Considerthefollowi ngdiagram
Sin ce i1 •i2 isaSUII. thereexist saUliquemorphism
Clearly x· IAl @Az issuch a-arp his••
But (li PI+i2P2)i2 - il PI!1+12P2 il _i
l +0 ail
and (llPl+i2P2 )i2-IlP 1i2+i2P2i~
- 0+i2 oi, Therefore bytheuniquene s s property
is.
OtAPTER TWO
CLOSEDCATEGORIES
1. Closed Cat eg orieswithFai t hfulBasicFunctor Def in ition2.1
Aclosed category C. (C
o. F. H. K.1f,e.L) consis tsof thefollowing sev e n da t a~
(1) acategory Co; (ii) a futctor F:Co- JS ; (iii) afunctor H:~x
C o-+ c
i(iv) an object K of Co;
(v) a naturalisomorphislII "."A:A~H (K . A) in Co; (vi) a~tr an s fo l'll.aU on e· 9
A:K~H( A. A) in Co ; (vii) a na tural transfonn at i on
L..L: C:H(B.C)
---+
H(H(A,S),H(A,C» ) These dataare to satisfythe followingsix axioms~ceo. The fo llowing diagraJIIof functors co. .utes:
...9a:.
The followi ngdiagram comm utes:H( B , B) LA ) H(H{A, B) ,H( A, B) )
17.
CC2. The followingdiagTam commutes: LA
H{H(B,C) ,H( B.D»
1 "(I.L'1
Thefollowingdiagr~commutes:
H(C,D) L
H(HCA.C).H(A,D»
1
H(H(B,C),H(H (A. B) . H(A,D»)L"(A.B)
I
H(L ,AI)H(H(H CA, B) ,H(A, C)) , H(H(A, S).H(A,D»))
CC4. The followingdialramcolllllutes:
H(B, C) L ) H(H(l,B) .H( K, e))
~ 1 " ( · · 1)
H(B,H(K,C»
CCS. The lIap
P1rH(A,A}:RI(A. A)- - - }FH(K.H(A,A»
whi ch by COO may alsobe wri t ten F'rrH(A,A): hom( A, A)~ham eK,H CA, A»
sends lA€ hOlll(A. A) to 9A£hOIlCK. H CA,A»
,
~(1) For a closed category C.C0 iscalledtheUlderlyingcategory . f:Co-:-"'S the basi c: functorand H.t.he i nt e rnalHOIIl- f w eto r . TO- si mpli fy notation C willbe used for bot h aclosedcat egoryandits lDlderlyingcate gory
(ii) Theword natural isused in condition(vi)of aclose d category ina genera l hed sense. Ellenbergand hUy [6] have gi yena detailed accoun t of th i s generalizednotion of a natural tran s fonaat ion .whichrequiresth a t the followingdiagr8llco_utes :
H( -, B)
hoa(H(A.A} .H( A, B)~haa(K.HeA,B»
hoa( e A,I ) Eval uating this diagramat fE:hom(A.B) . thatthe followingdiagramco~utes:
8 .
1
H(',f]thereq ui rementis
H(B.B)
- - - - --t
H(A, B)H(f.B)
Note thatbyceo RiCA. B)..hOIllCA,S) and FH(f.g ) ..hOlll(f,g )•
Propo s i tion2.1
Inthe presenceof
ceo
andCCS.th e axiOllleel isequivalent to any of the followi ng:Ca> f~ ef)..H(l.f)'ChOD,CHeA.B).H(A~C)) for fEha.(8~C)
.,
19 .
(b) (FL:B)( l B)·IHeA,B)
ee)FL:C. H(A.-) :hOIl(B.C)~holll(H (A.B)."(A,C») Proof: {Se e(5} page 430).
Proposition2.2
If the badefunctor F isfaithful,the axi oms ee2,CC3and CC4 arecons eque ncesofceo,CCI and
ecs.
~ (See (S]page432 ).
Als o wi ththesilllpli fi ca tlonthat thebasicfmetor F isfaithful, itis not ne c e s saryto asS\PCthat L~C:H(B.C)- - ) H ( H( A.B ).H(A, C»
isna t ura l in B andC. Thiswillbe shownto be a ecnsequeneeofth e fai thfulne ssof F.
~
Given A"....!:-.A· ~A and B~B·...I:..,.au in any close d cat e go ry C widl faithful bas ic. functor F. Then
H(f0 f'.I'0a):H(A,B) - - J H (A".S") is equa lto H(ft.g ' ) o U(l,g ) .
Proof: PH(l0i',g' 0g) •hom(l0£ ", g'0i)
•hom(f',g')0he m( f , g )
•m(ll,g')0FH(f.g )
• FCH(f',g')0H(f,g » Si n ce F isfaithful,
HCf0 £',g' 0 g)'" H(f',g') o H(f,g)
Pro po s i ti on 2.4
LA is anatural'transformationin B anac,
Proof: Consider thefOllO;~Rgdiagram
FHeB,e) Be ) FH(H(A,B)'H(A, C»
FH(p,q) i iFH CHCl'Pl. H(l,q»
FH{M, N)
F~
) FH{H(A,M).H(A ,N ) where p:M-4B and q:C~ N.If fe. FH(B,C), thenbyProposition2. 1 (part (a»
FL: C (f)=H(l,£). Thereforethe above dia gramcommutes i f H(l,q0f op) ..H(l,q)0H(l.f) 0HCl.p).
Thisequalityfollowsimmediately fromLemma 2• .3. Since F is faithful,the naturalityof LA isassuredby LeMa1.1.
Propos! tion 2.5
Forany f:H(K,K)~X in a. the composite
K-;---J H(K,K)~X isthe illlage of 1€hOID(K,Ie) under the compos!te map
hom(K.K)~ FX~hom(K. X) •
Proof: Evaluate at 1£hom(K,K) the diagrllll.
Ff
homCK,H eK,K).)-'h-om""'(""I""','::-)
---»
homCK,X) which ccemute s bythenaturali t yof 71. Note that21.
fYH(K.K)..FHCI ,.) .. hOll(I }. Therefore hoa(l.f)(hOlll(l._ ){I» ..f0 ...
Corollary1
The followingdi a gr am commutesforea ch f£.hom(K.K)
- - - - - -»
H(K. K)· 1 1
H(l.nH(K,K)I- - - ->,H(K,K)
Proof: Inpropo sit ion3.6 replace f by H(l , £> and X by H(l,K)•
Then H(l,l)011" ..PlI'(FH(l.f) (I».whe re 1EhOD.(l , K)
..Pw(f 010 1) ..FIr(f) and H(f,l)0'II'.. FlI'(FH(f.l)(l))
IIFlI'(l0 10 f)
Thatis.H(l,f)0'" ..H(f ,l) 0....
.Coro llary2 For flIEhOJll(K.K )
H(l , f) ..H(f, l) :H(K .K)~H(K ,l )
Proof: The resul t followsbecause 11" isanisomorph i s m.
Corollary3.
The~noid ho .. (I::.K) ofen dollOrph isas of Kisco. .uta tive.
Proof: Applying F to corollary 2 gives
h01ll(l. f) - ho.( f . l ):hOlR(J:.K)~hom( K,K). Eval uatingat gE:.hom(K,K) nowgives fo g..Ii0f.
2. MoRoida!Categorie s Definition2. 2
A .moidalcategory C • (Co-&J: . r ..! .a) consists of the followina:
six data:
(1) acatf'gory Co;
(ii) afmeto r ~Cox Co~Co (writtenbet ween its arg Ulllentsand calle dthete ns orproductof C); (iii) an object K of Cf¥';
(iv) anaturaltsoeorphis. r- r
A:AI&> K--.A;
(v) •naturalisollOrp hism.t •LA:K~A~A;
(vi) anatural isOlllOrphislIl a· BABC: (A~B) 0C---+Al&l(8<&!C).
Thes edata are tosatis fythe followingfi ve extcesr
~ The following dia gram
.
co_utes:~(AThe@K) (ZlBfollowing di agram
.
eoeeuues e)A(!)(K(&Bli '·4
23.
~ The followingdiagr8ll.commutes:
«(A~ B)
0 C)C!>
D~(A@B) @(C~D)-~A(8)CBl»(C"DJ)
(A@ (B~CJ)~D
- - - ---"--- - --+ .
A@((B c!>CJ 19DJ~ The followi ngeliagr. .collUllutes: (A<8lB J<8lK 'A ® ( B ® ')
~(1) The above axioas are not independen t. It hasbe en shown (Kelly(l.l) th a t MCl.MC4 and MCS are consequencesof K:2 and He3.
(2) Naturalisa.orphism such as a.r..l. aresai dto be coherent i f, roughl y speaking,alldiag ralllSmad e bythe ir use alo ne (with thei r in'V':rses,1and@ suchas the diagramsof Mel -MeS.co.-ute.
It hasbeenshown (MacLane (L~ ) . that Mel ·MCSilllpl ytha t the isomorphisms _.r,t are cohere nt.
(3) In theterminolo gy ofS#oabo u[1]. amon o idal ca tegoryis a cate go ryavecllIultiplication•
3. Mono id.! Closed Categories Defio!ti m 2.3
AIIIOnoidalclo s edcategory (orequa llyclosedIIlOn oi d al catego ry) C.(llIe•p•CC) consistsof thefollowingth reedata:
(i) a.ono idal category ..c..(Co.@.I.r .l. a );
(ii) a clos e dcategory cC • (C
o.F.H,J:. 'II.e ,L) withthesame Co and t::asin lIle;
(iii) a naturalis omorphisa P" PARe:HCA@B,C)--}RCA,HCB,C» . Thesedata are to sat isfy thefollowingfo ur axioms:
~. 'r.tefollowi ng diagramcoeectee:
H(I::~A.B)
- - -=---- -'»
H(t::, H(A,B»H(t ,~ A', B)
H(A,B)
~ Thefollowingdi a gram coea uees:
H«Al8I B) ®C.D)~H(.\(8)B.H(C.D»~H(A.H(B.HCC.D»))
H( a,l)
1 1
H(l,p)H(A~B)'9C,O) -;;- --7) H( A. H(B(3lC .D»
.,
HC~
25.
MCC4. Thefoll owingdia gr. .cO_ut es:
HCA@ K.B j--..:...- - - tJH CA,H(K,B))
~" J
H(A,B)
ae.ark.(i) WeshalldenotetheDOna l d a!cate gory ·C and theclos ed cate go ry Cc bythe same syd>o l C as theIIOnoldalclosed cate gory.except. whenit isnece s s aryto dis ting uish between thethreest ru ct ures.
(li) Both the dataand theaxioms for a IIlOnoidal cl osed cate gory contai nredun d ancies. The interconn e ction shavebe enshown by Ellenbe rg and Kelly[5] pp.477- 489. These in terco nnections lead toaneconolllicalwayofgi vina:a monoidalclos e dcate go ry.
Consider theso called''basi csituat i on " ([S] pp. (77 ) in which we aregivenacategory Co' fmeto rs @eoxC~Coand H:C;x Co--'t'Co' anatural iSOIIOrph.lsa
r •PABC ;ho.(AQ1}B.C)~holll(A . H (B .C». and a functor F:C
o---+S satis fy ing ceo.
Si nc e P is a naturalisomorp hism,theYonedarep re s ent at i on theorem[5
J
showsthatthecommutativi ty of thediagru(2.1 )
hc.(A@K.B) - _ --"-_ _ --»ho_ CA.,H(K, 8)
' h~: ~ ~ . . )
hoaC A,8)
sets up a bijectionbetweennaturalisolllOrp hisllS r:A(3)K- ) A and natura l iSOIIIorphislIS ...:A--JH( K, A). Putting BaA and eval ua ti ngat 1 gives ~r"" '11".
In the sameway commutat ivi t yof thedia gr8111
home (A@8) @C,D)
-L....t
hom( A..8, H(C, D)~hom(A.H(B.H(C. O)) )(2 . 2)
1
hom(a .1 )r
hom(l, p )ho_(A(g)(8 @ C) .D) p ) ho_CA.H(B@C ,D»
setsupa bijectionbetwee n naturalisOlDOrphisllS a:(A @B)<5lC~A~(B @ C) and naturaliso-orphisllS P:H(8(g)C.D)~H(B.H( C.D)).
Proposi tion2.6
Given (i) a category Co • (ii) a functor 0:Cox Co---"Co • (iii) afun ctor H:C~x CO~CO •
(iv) afunct or F:Co---tS suc h that FH·hom. (v) anobject K of Co •
(vi) anat ural isOlllOrph ism
'II'•"'A:A~H(K . A)•
27.
and (vi i) a !1a t ur al iso-orph is•
.p.•P
ASC..H(A ®B. C )~H (A.H(B. C» .-
The n thesedatacan becompletedto give a monoida lclos edcategory i fandonlyifthe r and a dofinedbydia grams 2.1and 2.2,where p..Ftl. sa t i s fy MCC4and MCC2. Moreove r,,!! F isfaithful,the satisf actionofMCC4andM:C2 is autOllia ti c.
E=>f' ((51p.495)
Theexample s consideredinthispaper will besuchthat the basic functor F is fa!th ful. In thiscasei tissuf f i c i en t to est ab lis h the exist e nce of the seve n piecesof dataofProposition2.6to sh ow th a ta catego ry isclosed.anoi dal. Thisisso,si n ce thefaithfulnes s of F auto maticallylivesusthe"bas i csi tuat i on",whichin turn givesfroll diagrams 2.1 and 2.2thenatura l isolllorphisJlS randa;
the s e automaticallysatisfyi ng MCC4andMCC2.
4. S)'!!IlOtrl c Monoid.lCl o s ed categories
Defini ti on 2.4
A~for a -ano i d a l cate gory C consistsofanatural isomorph ism C '"CAB:A(3)B---+B@A in C 5at lsfyiorthefol lowin g twoaxioms:
MC6. CBACAB- 1:A®B~B~A.
MC7. Thefollowin gdi a gram COlDJlllJtes:
,
(A@ B)®C
...L.t
A@(S@ C)~(B @ C)@ Ac@ ll 1 -
('<!)A)&C~'@(A®C)~'®(C®A)
A.anoi dalclosedcate gory C togetherwith aS)'lIIIIle t ry C for
·c
is calledasywJletric monoidal clos e dcategory.
Rellark : A monoidal category. even a clos edone.may admit se vera l distinctsr -tries . fo r exampl e l Eilenb erg and Kelly{5 ] ha vesh own thatthe closed .:lno i d a l catego ry GK ofgr a ded K-.xiulesadmit s one sr-et ry forevery k<EIewith k2• l.
Howev er.ithas alsobeensh own(Eile nberg and Kell y [5])that if the basic func tor f isfaith f Ul,the nthemonoi da ! closed cat e gory C admi t sat mostone symmetry.
To show that a category C wit hfaith fulbasicfunctoris-asr - etrl c lIOTloid a lclosedca te go ry,weshall'havetoestab lis hthe sovendataof Proposition 2.6plus th e existenceof anatura liso..orph is.
C.CAB:AcgB---+B@A in C.sa tis fying MC6andMC7.
29.
otAPTER nfREE
M-SETS
In thischa pterwe will give afai r ly det ai led.CCOWlt ofthecat e gory ofsets under the actionofa co_utat ivelIOJ1oi dM.inshort.a category of M-sets. The objectiveisto show that any category of M-setsis a S)'IUIIetric mon o i d a l closed cate gory.
1. Definition sandEXUlples
Definition 3.1
A .anoi d Macts ona set X when the reis a give n function MxX----tX written (a,x)t--+mx andcalled the"action"of 11£ M on x£OX. suchthatfor all x~X and m€ M
Anypair (X.MxX- - - iX) consistingofa set X to getherwithan
"a ction" of MonX is calledan M-set. If M.. G.a group, then the pai r
ex.
GxX- - - .X} isthewellknown c-s ee,E.!!!!!P.!!!.
(i) Eve ry set X isanM-s etwhe re M.. {l} is th e trivi al IIlOnoid; the actionbeing de f ine d by lx ..x forall xE.X.
(ii) Every pointed,set
. x...
is anM-s et whe re M" {O.l} ;the action bei ng define dby 1x.. x lind OX"* forall x",X.(iii)Amonoid M isan M-s etwiththe ob vi o usaction.
(iv) If T is atransfol"llati ongroupconsisti ngof permutations t ofX. th e assignment (t.xH-- ---tt(x) definesan action of T onX.
(v) More genera lly. any represen tation h:G--+T of a group G givesby -(g.x)t--'I'h(g)(x) an actionof G on X.
Consider for a particular monoid M any two M-s ets X and Y.
Denni tion 3. 2
A morphi s m ofM-setsconsistsof a fmet ioR f:X. - jY.such tha t f(mx),.IIIf(x) for al l xE'.X.IIIE:M.
AIIOT1Jh i s. f:X---+Y £1"011.the M-set X to the M-set Y 1.5sai d tobe aIIOJlO.arphis_iffi tisin j e c t i ve; f is sai d to bean epi lllOrphisli iffi tis surjective. A bijective morp his mofM-setsis called aniso morphis m. Clear lycomposi tionofmorp hisllIS of M- s e t."is aIIIOrp hislllof M-sets. A.lso,theidentityftmctionis obviously a JIIOrp h i s. of M-sets.
Wenowre s t ri ct ourdis cus sion to M-s etswhere M isacollllllutative monoid andmoregenera lly to ca tego ri es ofM- sets.denot ed 8M for each fixed lDOnold M. Analternateformulationof Defini tion3.2 istha t a~rphis.of a-s ee s Xand Y isa fUlction f:X- - 'Y suchth a tthe follOWinjdi agraa co_ute s :
MxX- - - ....J X·
MxY ...., Y
Note that the cate gory S of setsisthe category of {l}-se es,whe reas thecategory S. of pointedsets isa fullsub c a tegory of thecate gory of {O. ll -s et.s, The la t te r..tatelDeT1t iseasil y verifiedsinc e the objectsof S. are pairs (X.MxX-.+X) where,as indicated inexa.ple (li)abo ve, the action M x X--+X is fixed for eachpointed set X."
31.
2.QuotientH-Sets.
Let·X be"anarbitraryM-s et
and
let E bean.
eq uival e n ce relationon X such that x e x· implies Il.XEJIlX' for all IIIE.N;tha t is, theequivalencerelationis colllpatiblewith theactionon X. Two obvious ex amples ofsuch aooll.pat i bleequivalence relationarethetri vi al relation X:It X (allelements of X are related)andth e equali ty relation Ix (twoelementsare relatedi fand onlyifthey arethe5811I0).
Considerthequotient set ~. iifedefinea functiononthissetas fo llows
em, [x])t - - +[.xl
ThisfUlctioniswell~definedsincefor all ..E:M. (xl"[x"] implie s that [mx].. lU i] bythe compatibilityof E wit h theacti onon X.
Clearlythisfutction isanactionon ~. and
ci-.
M xi
--+~) is an M-setcalleda quotientM-setof X.3.M.SetsofMorphislllS
If X and YaretwoM_sets.defineforall
s
c hOlll(X.Y) _ilL M Mxhom(X,Y)----i>hom(X,Y)such that (mf) (x) =f(lIX) forall xE.X.
Since M iscommutative,i t canbe readilysh ownth a t IIlE is a l10 rphi sll of M-sets;therefo re,th e aboveflAlctioniswelldefined . Also forall xEoX.
-I '
~E.M ande e
ha. (X. 'f)«11.1-2) f)(X) •£« a 1~)X)
• f« a;Z-1II1) X)
• f(~(.1
x »
•~f (Jllxl
- Ill)(117: f)(x) and (1f)(x)..f(lx) •f(x)
Thereforethese t hameX, Y) togetherwit h theabo ve function MxhOIl{X. Y)----JhOIll(X,Y) isanM-s etandiswritten h0!l)t(X.Y).
Nowlet f:X'~x and g:Y--+Y' denotearbit rari l ygiven -.orphisllSof M.s etsandconsi der thex-sees ho....CX.Y) and h0-w(X,,Y' ) . Def ineafun cti on
.:hOD){ ex.y)- - »honx (X'IYI)
bytaking +(h )-g0h0 f for every ae.ho'\!(X.Y). Itisa routi neexerciseto showthat f is a morphismof x-seee, Denote ~ by ho~(f.g).
Propos! ti on3.1
For anyset X actedon byaco.-utativfI.moid M . . . . .x:X- - th~(M.X ) defined by wex) .f
x suchthat {x C.) ..IIlX forall XlIE.X.III€.M isa na tural iso-.orp hislll.
Proof: "is well -defi n e dsince for n.m<Ii:M fxCnm)• (om)x-n(mx)•nf" em)
Also '/I' isa morp his mofM-setssi nce for all n.mE: M xo<X
11'(nx) ell ) • f nx(Ill) •a(nx)
.. (l:WI) x
=(n-)x
33.
'" n(mx) • ..R1r(X)(m)
To show that 'If is a monomorphism of x-sees ,consider xl. X2E:.X such that xl ' ~.
For 1 eM.f
XI(1)'"xl ;. X2'"f
X2(1); Le. f XI#f
Xl To show that .... is an epimorphism of M-sets. co ns i d e r f€.h0Il)j:(M.X). For all moEN
f(m)=f(mI)..mf(l)..mx' where x'..f(l) Therefore 1I'(x')..f.
A routine check showsthat the above isomorphism isnatural in X.
4. M-8illlOrphisms
Let X and Y denote arbitrarily given M-setswhere lotis a conuuutative monoid andconsid~rth e Cartesian product of the sets X and Y.
Afunction g:XxY-..+Z from XxY to anM-set Z is calledan M-bimorphislllif f
g(mx,y)..g(x,llIY).. mg(x.y) for all xe,X.y€. Y, m£ M.
Let Bimorph (X,Y:Z) denote the set ·of all M-bimorphislIS h:Xx Y-+Z.
If t:Z--)W isamorphism of M-sets,th e composite to h:X x Y---+W is an M-bimorphism. For fixed M-sets.X and Y. thefo1'lllulae F(Z)..Bimorph(X,Y;Z). F(t)(h) "t0h, h~F(Z) definea funct.or- F from the category 8
Mof sets acted on by a commutativemonoid M to the category 8 of sets.
A universal element h
o for this flDlctor F is ca lle d a "universal M-bimorphism"on X x Y. That is h:X xY~Z isa universal M-bimorphismiff for every M-bimorphism h:XxY~W there exists a unique morphi smof M-sets t :Z- . ,W such that thefollowing diagraJP
4. Tensor Product ofM- s et.s
For anyt.woM-s ets X and Y act.e d on by a commut a t i veaonoid M weshallconstruct a lmiversal M-bi lllOrphisa on XxY. That iswe const ructanew M-5et . XxY and anM-bil,orphisa Xx Y--foX&Ywhich isuniversal a..on g M-bi.arphis asfroa X x Y to an arbitraryM-5 et.
To definethe tensorproduct of M-set s X and Y welIus t tak ethe
"biggest pos sible"qtDtiont M-set X; Y so that X x Y- t~ is anM-b i morph i s li . To do this let R be the following relation on XxY:
(1IIX.y) R (x,my) forall xE. X. Y£ Y. III€, M.
Wewillnowconstructthe "fine st"equivalencerelation E on XxY whichcontains R. Thatisweconstructanequi val en ce relation E:JR su chth at anyother equivalencerelatio n E'=,R aust have EC E'.
Putting II- 1 clearlysh ows that R isreflexive. Let T_RUR- I• Then T isboth reflexive andsymmet ri c , and furthenlOre has the fol lowingproperty:
LellllDa3.2
If (x , y)T (x',y'). th e n (1) (lI1X'IlY ) T (o ',my ' ) , (ii) (lIlX,y )-T(x , a y )T (x'._y') and (iii) (ax,y)T (ax'.y')T(x'.-r' )
35.
forall IIIE:.M.
Proof: (x.y) T(x',y') illlpliesthat (x, y )R(x'or')or(x' ,y') R (x. y ) If (x,y)R (x' ,y') then forso_ n~M
x•nx' and y' •ny
Forall me M mx,.JII(nx ')• (1IlJl)x' • (nm)x' _n(mx') and Illy' .m(DY) .. (mI)y ..(na)y•n(ay ) . Us ingthe s e equa ti onsand the definitionof R. wehave
(ax.lIY)R (ax'.IIY') (ax.y) R(x,lIY) R(x',my') (lIIX,y) RCmx',y'}R(x'.II Y' ).
Since RCT theseabove st atemen ts gi ve (1) (11) and(iii ). If (x',r')R(x. y). then a shdlar argUJDentccepfe testheproof.
We arenow able todefinethe requiredrelat io n E.
Defini ti on3.3
For (x,y). (x',y') in X x Y
(x .y)E(x',yO) if fthere exis tsafinite seq uence (x,y)T(xl'Yl)..(xI'Yl ) T(x2' Y2).•• •• (xn'Yo)T (x',y ') for xi€.X,fi£ Y. i· 1.2, 3•••• •n,
Proposi t ion 3.3
Therel a t ion E defi ne daboveisthefine st equi valen ce relationon x xY cont aini ng R.
Proof : Th~reflexi,?-ty and s~et ryof T ens uresthat E isreflexive and sYJlllletri c. If (x.y) E(x',y') and (x',y' )E (x",y") then jwet~sitionofthetwo impli e d finitesequen ces gi vesanother
1
I I
finite sequencewh ichimplies that (x,y) E (x",r ") . Thus36.E is also transitiveand is therefore an eqdf vatenc erelationon XxY.Clearly RCE. To show tha t E is the finestsuchequivalence rel ad on , supp05e that F is any other equivalencerelat ion with RC F. Since T isessen t ial ly R with symmetry .
RC Te F.
If (x.y)E(x'.Y·)' thenthere exists a finite seq ue nce (x ,Y)T (x!' y! ). (x I,Yt)T(X2.f2) •
whi ch givesth e followingsequence
(x,Y)F (xl,rt), (xlOY1) F(x2'Y2) • ••• •(~'Yn)F (x',y') By th etransitivityof F wehave (x ,y) F(x'. y' ).
There f or e ECF.
We sh a ll now showthat theqoo t ient set
- x x .-
Y canbe give n the scructureofan M_s et.which we shallcall thetensor productof the M·s ets X andY.Def i ne MX X; Y ~ X;Y by
(.. ,((x,r») .[(ax,y» )• ((x. III)')]
whe re lCu, y )] . [(x,. y )] si nce RC E.
Pro positi on 3.4
M x X;Y~ X~Y is welldefined.
Proof: If (x,y) E (x' ,y'), then thereex is ts a finiteseq uen ce (x,y )T(x1, Y1J, (xl'Y l) T (x 2 ' Y2)' •.•• (xn 'Yn) T(x',y')
37.
ByLellUlla3.2
(lIlX.r)T (llXl'Yl). (IIlXt.rl)T(U2,f2) ••• • •(un,Yn) T(u ',y') for all III€.M.
Therefo re [(x,r)} . l(x.y')] impliesthat
[(mx ,r)] • [(mx', y')] forall DIEoN. Tha tis,theabove flUlet i o n is welldefined.
It follows easilythatthe set X; Y wi t h the above actionis anM-s et.
Defini t i on3.4
Thetens o r product XG)Y of the M-sets X andY is theM-set X; Y Notetha t writing[(x ,r)] as x£&)y weha ve
lIl(xG)Y)• (1IlX)(8)y. xQtl(lIlY) for all II€.M.
The fun c ti on 0:X xy~ X~Y isevi den tl y anM-bi~rphis.. Wesh all nowsh owtha t this functionisunive rsal&IIOngM-billlorphislllS froll x xY toany M-set.
Proposition3.5 (Univers alBi -orphis. Property)
Toea ch M-billOrph islD h:XxY---tZ there isexa ctlyone IIlOrphisli ofM-s ets t:X @Y---tZ suchthat t(x <XIY)""h(x , y).
Pro o f : In the followingdiagra.wearegi ve nthe so li darrow s
x xY- - _ ...,
y=
X@ yTosh owtha t t iswell defi ned.conside r (x.y)E(x',r')
r
lThen there exists a finitesequence (x,y)T(xt ,rt) .ll};l'Y.)T ('-Y2), '" • (Xn,Y
n) T (x' ,y')- for xi€.X fiE Y and REIJ.J
But (x,y)T (xp Y1) implies (x,y)R (xpYt)or (xt.rt)R (x, y ) Thus x.. axl and Yl"ay forsome aE.M.
orXl'"'bx and y..by} forsome
se
M.That is h(x,r ) .. h (ax l.r) ..ah(x1,r) ..h(xt·ay) ..h{x 1 . Yt )
hex, y )..h(x,bYt ) ..bh(x,Yt) ..h(bx,yt) ..h(xpYt)
Si ncea shular resu ltobviouslyho l ds forcad!.ofthe reaaini ng terJIIS of the above seque n ce .weha ve
h(x,r)..h(xl'Yt)..h(x2'Y2),. ... h( xn , yn) ..hex',y' ) The refore (x,y)E(x',y') impliesthat t(x/Zlr)"text <pyl) Forall xE: X.r
e:
Y.IIIE: Mt(lD{ x c&l Y» ..t(ax @ y ) ..h(III.X , r ) .. mh(x,Y) .. m(xC!>y) He n ce t isa aorphisll of N-scts whichis obviouslyunique for eachM-b i.arphisla h.
Po r M-s ets X.YandZ consider the setBi-orph (X.V;Z) ofall M.b inlo rp hislllS X Xy~Z.
Define MxBimorph (X, Y;Z )---+8imorph (X,Y;Z) by (m, f)1-1
- - - - - - - >
mfsuchthat (.f)(x.y).Il.f(x,y) forall x€.X.yE:. Y.Ill€.M. and f€..Bi llOqi't (X.Y;Z). To showthatthe above fmetion is welldefined we haveto show that af is an M_billO rp his. fro. Xx Y to Z. For all
~ i
39.
D£' M•
.(1If)(nx,r)•IIf(nx.r)... (n l(x,r»
.,(mn)f(x.r)
• (na } f ( x,r) - ft(lIl( x,Y»
•n(mf) (xy)
Silrl l arly (1If)(x,ny) •n(lIf)(x,y) and the above fWlctioniswelldef ined . It followsreadilythat Bimorph (X,Y.Z) wi th theabov eactionisan
"I-set.
Wenowshowthat
BlllOrph(X,V;Z) it h0"\.t (X.h0'\t('f.Z»
To do this. consider anM- billlOrp h is ll £:XxY- +Z.
Writ e l(x , y ).FxCY)' Therefore rx:y--t>Z isa "parti alfuncti on"for f. Since f isanM-b imo rphis m.i tfollowseasily that Fx isa morphislll ofx- s e es .
De fi n e11function F: X----+h~(Y.Z) bythe ass ignment x~Fx' Si nce for all x6. X.re Y, ID£.M
F(u:) (y )• faxCY) ,. f(ax. r)
•mf{x , y )
•IIFx CY)
•m(f(x» (y)J F isamorphism of M- seU.
Proposition3.6
q>,"~yz : Bimorp h(X,YiZ) - - 4 h0'\t:(X.hOUM(Y',Z»)
f
I )
Fis anatural isomorphismofx-sets.
Proof: The universalityofthetens orproductstates th at every N-bblorphisa h:XxV- - . Z has th eform h(x.y) ..lex@y ) fora tmique morphism of x-s ets £: X
®
y~Z. That is ft_~h is a bijection.~ Firstwe showth at the assignment :fiI-1,F is a-bijectionby roostruetto, an inverse. Gi ven any f:X---iohOll}(ex .hoa(Y,Z»
de fine f by l(x,r) ..Fx(y); then since F is a morphism of M-sets
also since Fx:Y~Z is amorphism ofx-sees l(x.my)..Fx{my) •mFxCY)..IIf(x,Y).
The re fore f isan M- bimo rp hisll.and
F+--tf
isthe desired inverse.The assignment f...F isa . ,rph is ll of M-sets (and hencean isomorphis m) because the"a cti on s"on bo th f and F aredefine d pointwise.
Naturalityfollowsby consideringaconfiguration ofthreesquares;
onefor each of X,Y,Z varying.
Proposition3.7 for M-sets X.Y,2
p..PXyz: hOI\i (Xa. Y.Z)- - +h0'\t (X.hoDM(V,Z )) definedby p(f)(x){y)" f(x @ y ) isa nat ural isomorph isli.
Therefore consider
c : hOl\t(X@Y .Z)- - t B h , o r p h(X.Y;Z)
def inedby a(f) . h. Fo r all xex.y£Y, Ill£. M
, I
41.
1I(1If)(x. y):>; (1If)(X@Y)
"-m..f(x @y) .. lII.h ex,r)
==lI.a(f)(x.r )
Hence a. isan is omorphis.oflot-sets. Againby cons i de ri ng a configurati on ofth ree squa res ;one forea ch ofX, Y and Z varyin g,}" ecansh ow that 11
isnatural.
Thisisomorphis mfollowedby thatof Proposition 3.6isthede si red isomorphis Pl p,
Proposition 3.8
Forlot-sets X3IId Y
Considerthe followingdiag ram
"
and t1•t2 are theprevio usly describeduni ve rsalM-bimorphisms.
Clear ly f isabije cti vefm etion and thereforehas an inverse
£- 1 suchtha t i- ICy ,x)..(x,y). The colllpositions t l0 £-1 isdefinedby l( x , r)" (Y.x)
£:XxY~YxI
"
wher e the function
and t20 f areeasilyveri f ied tobeanM- bborp hi slIlS. Therefore byth euni versali t y of t
1 the re extaesa unique IIOrpbislllof M-sets Cxy:Xl!> Y----+Y@X suchtha t
S-y(x@y).. (~of)(x,y ) '"t2(y.x)• y@x
I I
r
i Siailarly the reexis ts a uniqueIIlOrphis.o.:..~=~u_ . _ ._ . _~ _ _ - ~
Cn. :
y<z>x~Xl&)Y. .suchthatCvx(y@x)• (t
l0f-l)(y.x)..t1 (x .y ) •x(&:/y
. .
·Th e refo re C. Cxy:X@Y---+Y(!)X is an isollorphls.whichcanbeshown tobenaturalin X andY by a routinemet h od.
4. TheCategory 8 Mof !of-sets
Wearenow ina posi tloo toshowtha t for a fixedco_utativeIDOnoid M.
theca tego ry 8M ofM-se tsisasymmetric monoidalclos edcate gory.
To dothiswefi r stofallshowtha t 8
Misa closedIlIOno i d al category byshowi ng th atthe seven dat a of Proposition2.6are pro vide d.
(1)
v, :
8 M(i i ) Thetens orproductdefinedinsecti on 3iscle arly afunctor
(iv) Take F: SM~S to be the' forg e t f ul ftmctor' . Clearly F isfaithful.
(v) K.M, the co.utaUve.onoid.
(vi) Propositi on 3.1pro vi des thenat ur al isollOtphism 'Ir• 'll"X :X~ ho",, (M.X )
(vi i ) The natura liso llOrph ism
p - PXYZ:h~(X<!>Y.Z)----+ho~(X.h~(Y.z»
isprovi d edbyPropo s ition 3.7.
Since thebasi cfunctor F is faithful, it followsfrom Pro po siti on 2.6 th at 8
M is a.ono i d al closedcategory.
Notetha t for 8
M data (vi) and(vii) ofDefinition2.1are ex:M---+ho,\\(X,X ) de finedby eX·\om ~i~I) and
43.
definedby -L (f)(aJ=-f o g where f: Y-...+Z,g:X---.Y.
Since the basic fun ctor F isfaithful.thenatural isoaorp hisll C;y:XQ;lY~YOPX def inedinProposi t ion 3.8isa tni q ue S)'IIllIletry for 8M ifitsat isfiesMC6and MC7. ClearlyMC6issa t is fied. Itis arouti ne exercis eto show that if f:X~XI and g=V----tY ' are IIOrphisll5 of M-s ets.then f@g:X@Y--+X' @Y' . defined by (f ~g)(x(8)y)• f(x )@ g (y isamorphismof M_s ets. With thisdefinitionthe cOllllllutativityof the dia gr8la inMe7follows trivially forthe cat ego ry
Sw.
The refore 8M is a S)'llIIIIet rlc.onoid al closedcategory withatniq ue SyBDet ry.I
I I ~
I ~
I .
1
CHAPTERFOUR
MODULOIDS
Many bookson modem algebraprovide detailedaccount sofal ge b raic st ruct u res . called modules. In thischapter definitionsand basi cresults willbe given forsligh tly morege ne r alst r uct ures.whichwesha ll call modul oids .
1. Defio! tionsand Example s
Forth e purposes of this th e s is weformulat ethefollowingmodified definitionofa semi ring.
Definition4.1
Asyst em <R; +••) iscalledasemi ringif (1) (R•• ) is acollUllutativemonoid;
(ii)
<R •• )
isa monoid such that ~O..0.:..0 for all f(e R, where o isthe identity element for (R.+) ;(iii) .. isdistributive (on bothsi des ) over +.
Notetha t in termsof the usual definition of asemiring the above st ru ct ure is ase mi ring withamultiplicative identity 1 and an additive id e nti t y O. which isan absorbentelement. Theset of nonnegative inte gersisan obvious exampleofsuch a structure. Alsoeveryringis clearlyasemi ri ngof thetype definedabove if the definiti on ofarin g is , as given byMa cLan eandBirk o f £ [l7]. a"ringwith identity".
Acommut a t ivesemi ring Kisonein whichthe multipli cation is commutative.
Let <R;+••
>
beanysemi ri ng .45 .
Definition 4.2
AnR-moduloid A ~n~istsof (i) a coJJDllutativemonoid A; (ii) a f\.netioD RxA--+A suchthat
Cal I((a+b) .. lea+ eb forall a.b €,A. tr::E.R; (b) (I{+A)a. IC&+ >.a forall aE:A .K, ),€.R;
ee) (d)a.I( Aa) forall aE.. A. Ie,A€R; Cd) la . a for all a£ A; IE. R; (e ) Oa..Q. forall a€. A;Q.€.A. De:R.
Remark: (1) Thetel1lmoduloi d hasbeenusedwitha differentaeaningby some aut hors [esg,Ros enfeld[21))
(ii) No tethat becaus etherearenoaddi t i ve inversesinth e abo ve struct uresit isnec ess a rytoass ...ethat 0to;R isan abso rben t for the seJd ri ni R and to arlo-athe asiailar conditionfor R-lIlOduloi ds. (axiom (e ».
it.Q."~ for all ICE:R; Q€ A
f!:22!l IC.Q.. Ie(Oa) .. (11:;0).
• 08
• !l
axiOil(e) axiom (e) Defin itionofsemi r i ng axiOlll (e )
~
(1) .Take .R bethe-se.iting.Z· of all nc n a e g etdve inte§ers.···Any
colllllutati ve IlOnoid A can be consideredas a Z· -lllOduloid. Also eve ry Z·-moduloi d is a coDlJllutativeecnc f d.
(2) Eve ry R-modul e, where R Isaring,iscle ar l y an R-mod ul oid.
(3) Given asemiri ng R and a posi t ivein teger n·)th e set an isan R-modulo i d unde r th e termwiseoperationsdefinedby
(rl,rz• ••• • rn ) +(51.5 2,••• • so) • (r1+St . rz +52' ••••r n+so) and c(rt . r2• ..• •r
n) • (.::r 1.lCrZ' ••• •lCr n).
(4) Byasub seDd ri n g of the selliri n g X we _ananonetlpty slbset RofX whi Ch isits elfa sellinngunder the binary ope r ations definedin X.
Anyse miring X witha slil s elli. ring ReX isanR-modulo id. The operati o nsaretheaddi t ionin X, anda restri cti onofthe multiplicati on in X; namel yth e funct i on
whichtake sthe productof an el ement c inthe s\tlsellliringwith any a inth e wholeseJl.irln g X. The case X. R giv e sthe aoduloid R'• R. Thusitisa specialcaseof exampl e s three and fourthat eve ry seuringisa J:IOd ul o i d over itse lf.
(5) ExamplentDberthree lIIay begeneralizedinthe fo llowing wa y. If X is any set , the func tio nmoduloi d AX isthesetof allfunctions f:X~A fromthe set X tothe R-moduloid A wi t htheusua l
"poi ntwi se" modu lo i doperat ions . Cf+g)(x)'" f(x )+g(x)
(cOCx ) • cf (x ) for a l l x e Xee; R The lIlOduloi d axiOIlSfor theseoperat ions followaton ce.
47.
(6) A Z- modul oidis an abeliangro up . (Define -ae.(-I).)
(7) Ina51_ ilarway fora givenri ng R, an R-modulo i disan R-lIlOdule.
In pa rticular,if R.F.afield,thentheR- mod ul o i d is avecto r
space over F.
Mo reexp lic itlythemodulo idsdefinedand de scribedabove areleft aodulcdds, RightIIIOd u loi ds canbedefined in asi mil arway. If R.K, a co_utative sem.irin g . thenit followsthatrightandleft .odulo i ds areessenti al l ytheSaJIe.
2.SubllOd u loi ds
Let X be an arbit raryR- lIDdulo i d. By a sti>.aduloi dof X we_an anon empty sms et A of Xwhichisitself a IIlOduloidover R relat ive toaddition and sc alarlIu l tiplica tionofthe moduloi d X.
Aalongthes\iJ -odul oi ds of X are X itse lf andthe set to) consisting ofthe zeroelellen t alone . Anysoo llOd ul oidof X different from thesetwo issai dtobe a propersubmoduloid. Clearlyeverysubmodul e isasubllOduloid.
Let 5 beanarbitrarynon e mpt ysubsetofanR-.aduloid X. Then 5 is con tai ne dinat le astonestbmoduloidof X, namely X itself. It can be easilyshownthattheinte rsecti on A ofallstDlIOdulo i dsof X containing S isa stmllOduloid of X. In fact It. is thes-alles t stmllOd ul o i dof X th atcontai nsthegivensubset S. Thissub lllOd u lo i d of X is called thestD bIOduloi dgenerate d by S. In ease A· X. we sayth a t S isa set ofgenerators of X and that S is gen era t ed by S.
Anereeent; a ofan R-moduloid X is said to bea linearcombination ofelementsin asubs et Sof X iff thereexis ts afinite nUlAb er of elements
n
x1.x2'••. • so E.5 such that a-i!lAi Xi holdswithcoefficients
3. Congruencesand QuotientModuloids
Asalready indicated inthechapter on N-sets.a relation '" on anR- lIOdw o i d It. is said to be coapatiblewith the actionon It. i f a'\0b (a.bin A) impliestha t h ....Ab forevery ), in R. 51l1il arly. arel ation '" onan R-lIlOdul oi d A is said tobe compatib le withthe additionon A i f a'\0b (a.b inA) iap liesth a t a+x'"b+x for all x in A.
Note thatifthe relation '"on A isan equivalence relation, the n theabove condit ionfor compatib il itywith additionisequivalen ttothe following condi ti on:
a ... b and c'"d (a .b,c.dinA) implyth a t a+c""h+d
Definition 4.3
By.acongruence E onanR-lIOduloid It. we _an an equivalencerelation whi ch is compatib l e with bothscalarmUl tiplicat i onand addition. Thatis aE b (a. binA) impliestha t Aa+AC E xb+),c for all c inA andall AinR.
~
(1) The equivalence relations A x A and IA"{(a,a )
I
a€.A}con gruencesonany R-mod u loid A.
(2) If E is a congruence on an R-moduloid A and X is asub modulo i d of A, then the re striction of E to X isacong ruen c eon X.
Si n ce acon g ruenceis sb p ly asp e cial kind of equivalencerelation, it ismeaningf ul to say that onecongruenceis~thananot her. Rec a ll
,
49.
that E isfinerthan E' iff aEb illJlli e sthat a£' b. thatis .E
c;:. e' .
Cl e arly lA' the equa li ty relation , -isthe fine stcongruenceonany R-moduloid A.
Definition 4.4
Theordered set 5 isca lle da~ifeve:ry noneaptyfiniteswset of S has asup andanin fo 5 iscalleda co,llJ)let e latti ce ifevery subset of S has asupandan info
Note thata completelatticemustha ve aleastelement o . supII.inf5 and agre atest element 1..ini ... sup s.
Proposition4.2
Let I be any set,and S any setofstb s e ts of Xwh ichcontains X its e lf andis"c l osedalde r inte rs e ction"- that is , forall non ellJlt y TS;S weha ve
n
TY€ S. Then S. ord e red by ~. isaccepr eee lattic e.~ Forany nonempt y T.s;:.5 we have infT..
f}.
Y. Onthe other hand, le t T* be theset of upper boun dsof T in S -th at is ,the set of y*e 5 such that y~y* forall Y€ T. Then T*I- l' since XE:.X'" andn
T* y*..sup.T.Su chan 5 isca ll ed an I-lattice on l((21]). The slbgro ups ofa gro up. the sttlrings of a ringand the subs p ac esof avec torspaceare nontrivial ex ample sof r-reec t ees,
Proposition4.3 The set C
A of allcon g ruen ceson anR-llOduioid A,orde re d by in c lus ion , isacomple te latticewithleas t eteee nt I
A and grea testelellent A x A.
so.
Proof: Weneed on ly showth a t anyinte rse ctionofcong ruen ce s on th e R-moduloid A isacong ruence.·ConsidEtr T a non;mp ty se tof congruen ceson A. Wewi llshow that
0 2'
~€CA" It isa standa rd result that
n
r E isan equiva lencerelation.for (a ,b)E.
n2'
E weha ve (a. b)efl ' where Ef isth efine stcongruenceon A suchth a t ErE.T; clearl y IA~Elo The refo re . (a+x,b+x)€. Ef~
n r
E for all x in Awhich showsthatn~ iscompatible with addi tionon A.
Sif'11 1arly
n
r E iscompatiblewithsc a lar..ultiplicationandhence isa congruence forthe R-lIIOduloid A.Propo sition4.4
Let E be acongruen ce on theR_moduloid A. Thenthe a-crass contai ni ng 0 is asub llOduloi d of A which we Sh allcall ano raal stilltOdu loid of A.
~ Let Eo " {a
I
(a,O)€ E) where E~AxA.Eo isclosedunder additionsince a.bE:Eo ~
«.».
(b.O)£.E~( a+b.O )£.E
~a+bIEEo
Als o Eo isclo s e dunder sc ala r multi pli c ati onfrom R sinc e e
e.
Eo ~(a,O)EE~().a, O)€.E for all X€.R forall XE:R
Themodul o i daxiOlll5 areilllllledi at e.
,
51.
Clearlythe A x A-classcontaining0is the R-moduloid A itself.
and thelA- cl as s containing-O-is thetrivi alslblllOduloid of A.
Us ingthisconceptofcong r ue nc e we will nowcons t ruct aqtDt l e nt lIIoduloid of the R-Illoduloid A. Given acong r ue n ce Ii onthe R-moduloid A. wefirst show that A/
E wit han app rop ri ate binary operationisaco_utatiye.moid.
For aEA.[a].. {x
I
(x, a)E.E) is an elementof A/E" We define fa]
G
[b] ..[a+b]. Theoperation0
is welldefinedsincei fraJ..[a']and [b]"[b'], th e n (a,a')E.E and (b,b') E E implythat ea+b.a'+b')G E; that is [a+b]..ra'+b '].
It follows easily that (A/E_@) isa COlIlIIutative IIOno i d. De f i ne Rx
A lE
~A lE
by(l. [ a ))
t---+
[laJThi s scalar Ilul ti p li c a t ioniswell defi n edsince if raJ ..[a'], then a Ea'. whichilllplie s that la E},a ' for all A'"R. Thatis[h I " [Aa '}
Aroutine check showsthat <A/E'<9••) isan R-lIlOduloidwhich we shall call aquo tientmodu l o i dofA.
4. Morp hismofMo du l o i ds
GiventwoR-lIlOdul o ids X and Y, a IIOrphisaof R_modu loi dsis afuncti on f:X~Y su chth a t
f( a+b)• f( a )+feb) and, f(b ) : >.f(a)
a,bin X and all>. inR
Notethat condi tion(e) ofDefini tion4.2,coJDbinedwi th thesecondpart ofthe abov edefinitionens ure sthata IIlOrp his lll ofR-lIIOduloidspres erves
,
ad ditive identi ties.
It canbe.e asily :shownthatde collpOsiteof·twoDOrphiS iI.Sof-R-llOduloids.
whendefined,is a IIlOrphisa ofR- lIlOdulo i ds.
A IKIrphisJ:l ofR-lIIOduloids £:X- 'Yissai d to bea.anollOrphiSIl iff itisinjective; f issaidto be an epillO rphisai ffit issurj ecti v e. A bij ec tive IIlOrph ismof R-lIOduloi ds iscalled an isOlllOrphisli .
Definetheprojec t i on p:A- - ,A/E fromany R-lIOd uloid A to aqlDtient mod uloi d of A by p(a ).{a]. I tfollowsrea dil y from thedefinitions of additionandsc a l ar lIlUitipli c a t ionon AlE that p isamorphi s . of R-moduloids.
Proposition4.5 (Themi ve rs al prope rtyof p)
Let E bea congruen ceon theR-JIlOduloid A. Toea ch morp hisllof R- .oduloi ds t:A---+B such that al'~€Awi th (al '&z)~E i.-pli es teal ) •t( a
2) there is amiq ue -arphis.ofR-aoduloids s:A/E~Bwith sOP '"t
A ~ P )i :/E
t I
,
8
Proof: Define s:A/E---+B by s([a]) •tea). 5 iswelldefinedsi nceif [a] ..[al ], thatis (a,a l)£.E. then t(a) .tea '). Because t isa morphismofR_modu l o i ds ,i tfollowsthat s isalIo rp his lll of R-lIlOdulo ids. ThisIIlOrp his m s has the req ui redprope rty sinc e
(s 0 pHa)•s(p(a)) ..s([ a ])• tea) for all ainA.
,
53.
Moreover 5 isuniquelydetenrine d. For sup pos e sI:A/E~B issu ch that s' o'p=e, then
s([ a ])..tea )..(5'0pHa) •st ela]) fo r all {a]inAlE-
S. FreeModulolds
Let 5 beanarbi trari ly given se t. By afree~ove r R on th eset 5 we eean a lIIoduloid F over R togethe r wit hII.funct.Lcnf:S ---+F suchth a t foreve ry Elmction g: S~X fro_the set 5 into a moduloid X over R. there isa lniq ue morphis.ofaodulo i ds h:F---+Xsuch that the ca.lutati vi ty relation h0 f..g holdsin thefollowin gdiagram
- --=---> ,
F~l
Xh
Thefollowingtwo the orellScan be eas il y proved in th eusualway .
Theorem 4.6
IfanR-modul oi d F to gether withafl.Dlcti on f:S--+F isa free R-lD()du l oid onthese t 5,then f isinjectiveand itsillage itS) generate s F.
Theon. 4.7 (UniquenessTheorelll)
If (F.f) and (P',f') arefreeR-ooduloldson the same set S. then thereexis tsa unique isolllorphism j:P---+F' suchth at j 0f..£1
Wenow establishthe following th eo rem.
~ (Exis ten ceTheorell)
Foranyset S. therealways existsafre eR_modul oidon S.
,
~ Let R den oteth e given sem.i.r i ngandconsi der theset ofallfuncti ons . f:S--:-t.R sat.1sfy ing f( x ) . 0 forallexcept at-"lst a fini te
nUlllberofereeen ea xE.S. l'hisset is closed underpointwis e addi ti on andscal ar.u l dplicati o n. I tisa stbllOduloidofthe function~uloi d RS an disdenote d by R' S ).
Ne xt letus defineafUletio n E:S---+R{S) by assi gni ngtoeach eleme n t xE.S theftlnct1M
E
: S ~ R
defi nedby E(y) •[i
i f x· rx x ~ if x~y :l.r
e.
5If
r e
R(S) then f has nonzerovaluesfor at most n elements of S. sa y Ix ,x ••. .•xo} ' Then f isdeterminedbyits nandin deed f• i!lf(xi) E
Xi
That isthestbllO dul o id R'S ) is spannedbyaU the elements Ex . Let h:S---+A be anarb it rary functi on£I'0Ilthe set 5 to an R-lIOdu lo id A. Wenow shOW' thatthere is exac t ly one IlOrph i s. of llOduloids t: R(S )----:,A wit h to E .h as inthediagrBJll•
S E ) ReS)
~
h... : t
:ANow to E'"h statesth at t(Ex > '"h(x ) forallx; so any su ch morphism t must hav e n
t(£) . it f( xi )h (xi)
foreach f£.R(5) . Thissh owsthat t isuniquei fitexists ; conversely cne aay veri fyth atth efuncti on t:R(5)- +A define dby
55.
thisformulais indeeda morphismof R-moduloids .
Thus eve ryset 5 of elementsdetetwine san essentiallym.ique free R- llIOduloid . Sinc e the function
E:S- - tR{S)
isin j ective.we mayiden ti fy 5 with itsimag e 8(5 ) in R( S) . This havingbeen done. the set 5 becos e sasubset of R(S) which generates R(S). This R-lIOdwo id R(S) will be referred toas the freeR-lIIOdulo id generatedby the given set S.
6. Bipro du c ts
Considerth e cartesianproduct Ax B of the R-lIOduloids A and 8.
Undertheusualpair addition A x B is cle arlyaco_ut ad ve arnold si n c e Aand B arecollllu tativemonoi ds. Defin escalarmultip lication by A(a,b)• (A8.Ab ) for all ),IE R. a~A, be:B. It fo llows readilyth at the set A x 8 isan R-lIOduloidtnder the aboveoperations.
We now definethefolloWing ftnetions
PI P2
AI lA ),. S~B
11 12
by PI(a,b) • a; P2(a,b) •b; 11(a) • (a,O) and 12(b)• (O,b) forall ae-.A.
s
e.B. Clear l y Pl ,P2' 11andi2 aremorph isas of R-aoduloids, andthe following le_acan beestablishedinexa c tly the salle way as for modu les.
Pl il• IA' P2i2"lB; Pll2"0~ P2i
l• 0; and iiPI +izpz• lA x B
The IIOd u l o i d A x B willnow be shown tobe both theproductandSUllof th~.R-~ul~ids. A~dB.
Theo rea4.10
If C isanyR-modul oi d and f:C---+A. ,:C---+8 are two -orphisas of lIOduloids.the reisa unique IIOrph isJII ofmoduloids
t:C---+AxB
such that Pl Ot=f and Pz 0t Itg. Thatis, AxB isapro dut t;;object.
Proof: We mustshowthatthe foll owi ng di a gram
can befilled in withaU\iqueIlOrphis a t soasto be co_utati ve . Nowthisco. .utativi ty PItI tf and Pzt..& iJlp lies
ii i+12&..llPl t +i:zp:zt.. (llPt+i:zpz)t..t Hence c, i fit exi s t5~.ustbe t..i l l+121. Conversely,thi s
I}f+i21 isaIIOrphisJIIofR-lIOduloids C- - .Ax B such that Pl(itf+121) "Plil i. Pll2e"f+os>
e
and p:z(itf+i21)..p:zi1f+P2i:zg ..Of+..g..g Hence t=11f+~g is t:he morph isilrequired.
~
If C isany R-lllOdulo idand f:A--+C. g:B--tC aretwo~rphisll5 of R-lIOduloids.thereisa UliqueIIOrph isll s:A x B----JC ofIIKldu l o idssuch that S 0I}'"f
object.
and S0i2.. B. That is , AxB is aSlIIII(orcoproduc t)
57.
Proof: (Siai l ar totheproofofTheorelll4.10 )•
.SinCe A.xB is botha5U1ll and apro du ct. wewrite-it ASB. AG)B willtherefore be calledth e biproductofthe R-moduloids Aand B.
7. Moduloids of Morph isms
If A andB a r emoduloidsoverthecommutativese mi ringK.the set hom(A,B) ofIDOrphismsfrom A toB, underthe usua lpoi n t wiseaddi t ionof IlOrph isltS willbeassociative,have aniden t ity.vizthezero-orphislUand willbecOIIDutativesince 8 is co_utad ve. Howe ver,in general, there willbe noinverse eleJlleltts WIderpointwis eaddit ionin hora(A. B) since B isaIlOnoid. Thus hOla(A.B) canbegiventhe st ruct ureof aco-utative .canoidunderpoint wiseaddition.
Ne xt for any I(E K and any £E.hom(A, B}. conside r thefUl eticD ocf:A- - +B definedby (",fH a) . lC(f ( a» forevery aEA. Since K is a cOlMlut ati vesemiri ng.i tcanbeeasilyverifiedthat ICE isa morphism ofthe IIlOdu loid A intothe mod uloi d B. Theassignlllen t (IC.f)~ICf define sasca lar multipli c ati onin ho_(A. B) andlives hom (A.B) th e st ruct u re ofa K.llOdulo idcalledthelIOdulo idof aUIlOrph islllSofthe..cdu lo i d A intoth e.odu lo i d B. Whengiventh is extrastruct u re hoa(A.B ) is written as HOlI(A. B).
Nowlet f:A'~A and I:B~B' denotearbitrari ly liv e n IIOrphisms of K.modulo ids andconside rtheIIlOdul oids Hom(A. B) and Hom(A'.B' ).
De fin eafWlction
t:Hom(A. B)- - o J -HOII (At.B') bytaking
+(h). gohof
forevery h in tlol.(A.B). Cl e arly + is aIlOrphis. ofK-lDOdulo i d.s.
Denote + by Ho.(f.g) .
Proposition 4.12
Foranymod u19id A...overa':Ommut a t i ve semiri ng K 11''' :.'A: A- - - - ' tHOIIICK. A) define dby 'II'(a)" fa such that f. (k ) ..ka for all Be.A. k€ K, is anat u raliSOllOrp h is ll of K-lDDdulo ids.
iswellde f i n ed since
faCXk)'"(Ak)a'"'ACka).. Ha Ck)
Toprove th at ... is a IIIOrphis.of -oduloids consider for all ke.K T( al+Ilz)(t.)..fal +az(k)..t(al+8
2) ..k~+kllz ..f
.,
Ct)+f.,
(k).. (f +f )(k)
a1 &z ..(.Cal)+'ll'Caz )) (k)
Thus 1l'(~+82)" . (al)+1'( a2) for all &1,82in A.
'or all k.,\,tE..K
.... ().a)(k )'"f),a (k)..t(). a)
=-(kAla .. elk).
'"Atka )
..HaCk)
.. ),'II(a ) (k )
Thus .C).B) "' A.wCa) fo rall aE:A, Xc K To prove tha t ... isan ise-::trphi s . we firstshow that IIOROmorphis ... Consider &1'82to.A such that ~; S:z
,. is a
