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- Permanent Link:
- https://ufdc.ufl.edu/UF00098432/00001
## Material Information- Title:
- Certain well-factored categories
- Creator:
- Maxwell, Stephen Jackson, 1945-
- Place of Publication:
- Gainesville FL
- Publisher:
- [s.n.]
- Publication Date:
- 1970
- Copyright Date:
- 1970
- Language:
- English
- Physical Description:
- vi, 68 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Adjoints ( jstor )
Algebra ( jstor ) Equivalence relation ( jstor ) Factorization ( jstor ) Functors ( jstor ) Hogs ( jstor ) Monoids ( jstor ) Morphisms ( jstor ) Semigroups ( jstor ) Uniqueness ( jstor ) Algebra, Homological ( lcsh ) Categories (Mathematics) ( lcsh ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Thesis:
- Thesis--University of Florida.
- Bibliography:
- Bibliography: leaf 67.
- General Note:
- Manuscript copy.
- General Note:
- Vita.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 022441283 ( AlephBibNum )
ACZ7662 ( NOTIS ) 13719940 ( OCLC )
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CERTAIN WELL-FACTORED CATEGORIES By STEPHEN JACKSON MAXWELL A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1970 To ry rnoth rand to the imoory of rdy father ACLUTO !LED'37 T I The author wouldd lik. to express his sincere gratitude to his director Profossor U. E. Clar-k, for his mathematical assistance his prince; ni, his Cen3riosity in giving aid on numerous cccasions. The author would also like to express his appreciation to his roommate, IMr. Burrow Brooks, for the loan of his typewr-iter .i, TABLE 0F CO:"' ITS AC O:'O-LEDG... '7 - LIST OF FIGL ... ABSTRACT ... o. INTRODUCTION ... Chaptc ' ...e....ecee;.0..*00*O*****0**0* ,....oeeo.oeo..OCO*,*eECo0.O0.0f *eeeOeoooe.o.*Oo*et@0e00OOOoo*O . .. .*.a . 0 oe0Q aeoe* **e a c ....e.oe ......a o*. .aea.. a oa6a 6 I. PRELIMIN~L RIES ........ c .e............. II. TWO EiM-BDD:I: TH O' -?! ..... .... .. . III. A SUFFICIENT CONDITIO:l FOR A WETLL- FACTOPED CATEGO- : TO - VARTI.TA, ..... BIBLIOGRAPHY .. .. o .. .. .. ... ... .... .... . BIOGRAPHiCAL .'CH ....... ...... c r c c .... o Page iii v Vi 1 3 17 40 6? 68 I ST OF FIGURES DIAGRAM (0 ') DIAGRAMI ( I) DIAGRAMi. (5') DIAGRAM (8A) DIAGRAlI (83B) Page 46 47 48 50 51 oLOJO*leeOe~eooee+ eo0e GcGY O0++4+OOOOI0100000OLl0e ..9 ......&o&aoe.a0 C L& .c .a a a 0 a 0 o..o0 a 0 e0o 6 & 0 0 6 a a 0 . . . ..C.. a c6& a60 &000a ca .. ..0. . . . . . 0 . 6 0....0 0c 0 ... 0 .04 ... & 00 0 0 0c c 0 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CERTAIN WELL-FACTORED CATEGORIES By Stephen Jackson Maixwell August, 1970 Chairman: Professor W. E. Clark Major Department: Mathematics A new kind of category, called a well-factored category, is defined. This is a generalization of the category of all modules over all rings, the category of all fibre bundles with fixed fibre, and the category of all topological acts. Two structure theorems are proved for well-factored categories. One is an embedding into a "product" category and the other is an embedding into the category of all left actions in a multiplicative category without unit. These embeddings preserve the factorization of morphisms given in the definition of a well-factored category. A sufficient condition for a category of left actions to be a variety, as defined by Herrlich, is given. As a corollary, it is noted that the category of all modules over all rings is a variety. INTROiCT 10N Let P be a 'rin3 and r be the category of all left ER-r.oul Gs. Th. study of the category .. is already well develop o and farilier. Thi. author set out to study the casteory of all noDiules over all rineg, denoted 71. Thus, each Ti is a subcatecory of 1. and morphisms in R are R, PI R_ semilinoir transfozrations (f,g); i.e., f is a ring homo- norophisr, g ris a group homoLorphism~ and g(rm) = f(r)g(m), for each r and u. The author then noted that numerous cat:eories sach a- the category of all topological acts, the category of all compact acts, the category of all fibre bundles vith fi'sd fibre' the csatgory of all ncnoi ds eating on sets, and the cateory of all blodules over all pairs of rin-s posses prop'ioLiC similar to those of RM Mos R'- imIpcrntly, iLn al.l those- ccatgcories, -e nay uniquely factor each morph srI as the compo-stion of two mo.rphiss belonging to corr'esponlinS clrses of orphlsis.s. ,Roughly speaking, for V1, -: ha v-e "odu-le" rmoro.piss of the form (.1,) and ''ring'" nmorphi sms of the for-m (f,1). These e::e-plos and the gen;ralizing concept of a w-ell-factored category are present Ln Chapter To In C.L ":tcxr IIl ; o prove' tio, structure theorems for l -factoreI cct;ories W show that a well-factored cateCCr y b .-.d .. a. very nice way into the "oduct" of three categories. Then we show that a ;ell-factored category may be embedded in a very nice ;.ay as a subcategory of the category of all left action in a rultipiicotive category -without unit, Herrlich i introJuceJ in [1] an aniciaatically defined varietal cateo ory whichh generalized the varieties of Lawvere and Linton. We give in Chapter III a sufficient condition for the category of all left actions in a multiplicative category without unit to be a varietal category in the sense of Herrlich. A corollary to this result is that the cate- gory of all nodules over all rings is a variety. le have a generalization of this condition and list some corollaries to the generlt'Trion but, due to the pressures of tinme we onit the proof. We hope to publish the complete results later. An important part of the proof for our condition given in Chapter III is that the forgetful functor U:Lact(T)--- EnsxEns has a left adjoint. We were partially motivated by the construction of a left adjoint for U when N is Ens with the product functor and the usual associativity trans- fornation. This construction was given in C2]. CHAPTER I PRELIT 1A TES We will use the 2notaLion of [-3] throughout, unless stated otherwise. Also, fa-iliari~ y on the part of the reader )ith Chapters I. II, and IV of [3] will be assumed In this chapter, wie will state the definition of a well- factored category, give some examples of woll-factored categories, can ,'tatc some kno-m results which are not assured to be background knowledge for the r-eadero (1.1), Rrji o For the following definitions, let C be a category and let Pr and P2 be two :classes" of sub- catego:.ies of C such that each object C in C is an object in ecly on b of and in exactly one nrber of P2 Let f be a morphism in C. Then f is said to be a P,-r0orphism if an6 only if f is a norphisia in some category i.n P~ i=!~,2 Note then that the composition of Pr-morphisms is a P. -oorphisr i1;,2 (1.2). Notaticon. Ab wrill denote the category of all abslian groups. R uill cdeoote the category of all rings which do not paces arly have an identity, R' wil. denote the category of all rings having an identity and all identity- preservi.-g rig ho.onorr-phic.s. Enl. is the category of all sets, (1.3). Din;iTti~c, .t : =-U:IorA and U i:orA or AE AcP 1 2 vice versa. Lot f and g be in N, say f:A-3 and g:A'-^BA. We say 'f and reS 'd f' l!i by M.-morphisms if and only if there exist sequen e Si, Sn of N-morphisms sach that each S. is not the empty sequel nce the elements of each Si need not be coach :able, S. = (f) S = (g), and, for each i = 1 ... n-1, one of the following (or one of the following with S. and S, interxchaiged) occurs: (1) S = (g 9 ., 1' -il kk- ,. Si+1 = n "* g:-1 kd-:' kl* *0$1 where 'g indicates gk has been deleted from the sequence. Also, ck is an identity such that if N is the collection of all P. morphi~ then cod Gk-l dom gk, and dom ongk are all objects in the some f -subcategory of C, i=1,2 and i $ j If k = n, then we have the above conditions for k and k, If k :- 1, then we have the above conditions for k* and k-+1 (2) SS i * k+1 * Si+1 i (, k+1 k,. *- 1 where fofl is defird a i equal to g (3) S = (k"i 9"'"f +1 r k"*** S ;, :': ( 9 k fi'91 9Sk '- ) and there cxist -. C';;lphiss i and 1' so that moo = gk+lm; ieoe, the follow .~-ri dinZ':caj! co prLItos : f E--->,l E '-+W t (-i4). .fi.n 1on, We say th.at C is c. factored by sn P2 if oi I the follow, th'- conditions hold. 5 (1) If fl and f2 are ~-,-orphi ss as indicated by 4- subscript, then flof2 is a Pi-morphism implies f~ is an identity, where i j, i = 1,2, and j =- 1,2. (2) If fl is grid connected to g, by Pr2-morphisms an1 f2 is rid colneted to G2 by Ps-norphisms, where A ,---J.- and A- -2-, then f2o 1 = g2og"1 (3) For each orphlism f in C, there is a unique factorization f = hog such that g is a rF-morphism and 1. - h is a _2-morpphism. We will denote this situation by C= P X f . (1.5). Examonpe. Let S be a monoid. We may consider S as a category in the usual manner. If S is rell-factored by P1 and P2, then the cardlnality of Fi is one. i = 1,2. Also, S is isomorphic to P ~ where 1 is identified with its menlir, for i = 1,2, and X denotes uonoid product. Proof, We must put off the proof that this example is true until the next chpter. (1.6). ,voile. Lot G tbe a group. Then G may be con- sidered, in the v -'3.1 manner, as a category. Suppose G is well factored by FP and ,. Then the cardinality of Pi is one, for i -.= 1,2 Identifying Pi with its member, for i = 1,2, :we have G = P1Xr where X denotes the direct product of groups. Pr.cf. We rmst put' off the proof that this example is true until the next chantr;i (1.7). r 1i t:.t'o." Lt.a C C and C be categories. Define the cati-oryi .' 2 '' as foll.owsc the object class of C X C x C is equal to the object class, of -1 -2 3 C 22 x C 3 Also, a "orphlsn in C x C x C is of the form (fGg)(C ,,C2,C )---(C C. ,), where f:C --> is a morphlsm in C, and 0:C---->3 is a morphism in C. It i.s -- 2 2 clear -that C1 x 2C x C is indeed a cabe oryo 1 -) 2 -,- (1.8). PT-o-ton. Let C C and C be categories. Let CI be an object in C1 Let P ( CC) be the subcategory of C x C.2 C given by: Obq(C1) = {(C1,C2C ) Ci 6 Ob, i = 2,3i (1,G) c c and Morf1(CI) = (0c,&C2 C3): >(C1,CC3) g.:C-- >C' is a C2-morphisn and 1 = 10c. Let i 1(Cl1) Cl O-bC- Define P2 ana:ogously; i.e., define by restricting the second coordin-ate. Then any suboategoryAof x x 3 in which (3) of 1.4 holds and such that (1,;1)(AP A2,A )- (A1A.A2,A3) is an A-.morphism, implies A 3 is =ell-fao- tored by the restrictions of ar and t to A. We call this the standard ',ell- factorization of A, Proof. Let Abes such a category. It is clear that each object in A is an object in exactly one member of P and in exactly one member of rP. First, we prove that (1) of 1.4 holds in A. Lret (1p<) : (B3.B2, E) --(B ,A2,) and ( ,1)'(C1,B2,C3)--~(Bi B2,Bt ) Then (1,) o(P,1) = (~-; ) implies c = I, Then by hypothesis B = A. Thus, (1,<<) is an identity. Similarly, (1,0)() (l,,) implies (p,i ) is an identity. Thus, (1) of 1.4 holds. Suppose A(:t -) (2f ,l) (AA, A,)------- ->( AI, E ,B (CI9BD2C') aad (1, ) (g ,1) (A,,A )) >(AD2,B')- 2 >(C1,B2,C3) where (l,f1) and (1,G1) a:.' grid connected by P2-morphisns and (f2,1) and (g2,1) ere grid connected by Fr-morphsiss. It suffices to prove If = g- and f g2o e will. prove f = g since the proof that f2 g2 is analogous. We use the sequences of P- nohis. front 1.3, here S1 (1) and Sn = (gl). wHe ill prove the following inductively: if Si = (Sni.0. ,! then gni = (1,hni )" ^1 = )(1hl) hn is_ "_is defined and if S1 (g ,.,g gn+ = ni +1 1 ni+1 (~,hn,.i) '' 1 = (,hlh),9 then hnio **oh1 = he "'..oh1, for i = 1,...,n-1. Suppose i = 1. By 1.3, we have three possible cases. (1) S1 = ((1~1)) and S2 = ((1,1),(,f)) or S2 ((1,f2)(1,1)). Then cod(1,fl) and dom(l,l) are in the same P2-c at gory implies cod(!,fl) dom(1,1) = cod(1,1). Thus, the inductive statcnent is true here for i = 1. (2) S1 ((1, f)) e S = ((1,f), (.1f)), where (1if)o(1,) = (1. f). -Ti-, the inductive statement for i = 1 is obviously true here. (3) S ((1f )) and S2 = ((l,h)) where there exist P,2-morphi.sms (m, ) arnd (i,1I) such that (m' ,1)o(l,h) = (l,fi)o(mn1). Thus, h = fl, so the inductive statement for i = 1 is true here. Going from Si to Si+1 is quite similar to the above. Thus, fl = g1" Similarly, f2 g Hence, (f2,1)o(,f) = (2)(1G).Thu, is well-,factored. (1.9), -: ,I 1-. ne ,7 eh cibit a product category such that condition (3) of 1,4 does not hold in all of its subcategories, Let C be the full subcategory of Ens x E ns Ens x Enis 0 whose objects are pairs (X,Y) of sets X and Y which have the saz_ cardi-ality. Lot 2 have the standard well-fastori o ion6 Let inc :1}--~{ ,2L be the inclusion function. Thse-n (i1ncinc);:( 1 ,{1})-->(1,2>,1, 2>) is a C-morphism. -o;erer, it is easy to see that this has no P1- -2 factorization, (1.10). xiU:;pe We now, exhibit a category C x C x C -1 2 x3 which is not vlell-fatored in the standard way. Let C = C = C 3 Ens. Then it is easy to see that the morphism (1 ,1) (1, {1~3 3 1})-> 1(1, ( ,1i 2>) has infinitely many P~ faotorizations. Note also that the condition (1,1): (C1PC2,C3)-->(C,C2pC ) is a morphism implies C C fails. (1.11). 1ole. Let A be a category. Let ) be the diagram schere given by 1-->2. h-,[.. ,A] is the category of all diagr is in A over. For convince of notation, we will denote t:A1--->A2 as (A1,A2t), Let C = C A and let C be the discrete category such that ObC = HIorA. 3 3 = Then[C ,A] is a subcategory of C x C x C Suppose (1,1): (A~A2 ,t)--->( ,A2 ). Then lot = to1 so t = t'. Suppose (fg):(A1,A2 --- t) Then (1 g):(A,A2,t) > (AAtof) and (fAA t) a nd (f,g) = (f,) o(i,g). Furthermore, this is the uniouc F factor- ization of (f,g). Thus, by 1.8,5C ,A is well-fa-tored. Special cases of interest are: (1) If =-R, then l[.,A] is denoted R nd is well- factored. (2) If A = 1, then L[ -l is denoted by 1R1. (3) If A = Top, the cs-egory of all topological spaces, then iEA is denoted as ?-n, the category of all bundles. (1.12). Exgagle, Let RI be the category of all modules over all rings. That is, an object in M is a triple (R,-,p) where R is a rin, i is an abelian group, and p:R x M-i-> is a function so that (R,M,p) is a left P-module. A morphism in R taking (R,!,p) into (R',iIQ,p') is a pair (f,g) so that f is a rin-g morphisn taiRing R into B', g is a group morphism taking I into ,'I and, for each r in R and m in i, we have that g(rm) = f(r)g(m). Lot be the subcategory of RJ consisting of all E-modules and so that a morphisr (f,g) in Ri is a morphism in Ri if and only if f = 1R. Whenever possible, (R,I,p) is denoted by (R3,.;). Let R. be the subcategory of RL consisting of all modules having carrier equal to 1. A morphism (fzg) in 71 is a mor-hisn in 1:! ilf ani only if g = 1I. Let P1 be thB collection of all H such that E is a ring. Let P be the collection of all such that N is on anablian group. Then Ni = 0 x Po Pr-oof Since P, and Pf are the standard well-factori- zat-ion, -.:e may apply 1.8. Suppose (1,1): (;I,p)--(RM;p'). In the dcain; rm = p(r,n). In the codomain: rt'' = p'(r,m). Thus, p'(rmn) = rm = (r)*1(m) = (rm) = l(p(r,n)) = p(r,n). This says p' = p. Suppose (fg): (R,,;p)--( ,.(p') Then (1,g) : (,I:,p)--(RI ,p (fx1)) ai (f,1):r, (fxl)) -->(R,H' p') and (f,1)o(ig) = (f,g). Moreover, this is the un'i.q~ P-P fctorizaic Then by 1.8 -we have that 2 R- 1 2 (1.13). .lo. Let be the category of all bi- modules over all rings. Am. object (R,i,S,p19p2) in i such that R and i are rings, II is an abelian group, pl: Rxn--)I9, p2:.Xs-->::, (R;,MIl) is a left R-module, (M,S,p2) is a right S-modale, and, for each r in R and m in MI1 and s in S, we have (rm)s = r(ms). A morphism in R_ taking (R,I,S) into (R1Il,S') is a triple (f,g,h) so that f:_R-->P is a ring morphism, h:S-- 3 is a ring m norphism, arnd g: I.-->iM is a group morphisn so that, for each r in R and m in M and s in S, g(rm) = f(r)g(m) and g(ms) = g(m)h(s). Whenever possible, we denote (R,',S,pl,p2) by (R,MS). We define P.-subcategories analogously to P -subcate- gories in 1.12 by holding the rings constant. We define P2-_subcategories analogously to P2-subcategories in 1.12 by holdi-ng the abelian group constant. Then 1, = P x 2. Proof. This proof is analogous to the proof of 1.12. We chance. notation from (RllS,pi,p2) to ((RS),iG,(pip2)) and from (f,g,h) to ((fh),g). (1.14). F amr.ple Let rm be the full subcategory of P' determined by the objects of the form (R,;, R). Then the ell-factorization of Rp given in 1.13 induces a well- factorization of . Proof. This is a straightforw-ard application of 1.8. (1.15). .--"--~1es. We define the categories IM R M 1' and I, -, = and R1~! analogously to respectively I by replacing R with R1. We also obtain :i 1 -factorizations of these categories in an analogous fashion. Finally, Sand I are wiell-factcorod, where no:. ;e have right nodules rather than left modules. Proof. It is clear that these examples are valid. (1.16). E,:;-e. Lt C bt the category of all topo- logical acts. An object in C is a triple (S,Xp:S-X--6 X) such that S is a Hausdorff topological senigroup, X is a Hausdorff space, p is continuous and gives the multipli- cation, and, s and t in S and x in X, (sb)x = s(tx). We denote (S,X,p) by (S,X) Thenever possible. A morphism in C taking (S,X) into (S',X') is a pair (f,g) such that f:S-->S is continuous and preserves the semicroup multiplication. g:X---X is coontinuou.s, and., for each s in S and x in X, g(sx) = f(s)g(x). C is a subco.t : of C x C2 x C There C is the -1 -2 3 C1 category of all topological senigroups, C2 is the category of all Eausdorff spaces, an.d C3 is the discrete category having as objects al:l p:S X--CX so that p is continuous and andnd X are Hausdorff spaces. Then C is well-factored by the standard well-fctorization. Proof. This is an easy application of 1,8. (I.7?). ',, plc Let Co:np C be the category of all compact topological acts. That is, Comp C is the full subcategory of C deteriincd by the objects (SX) in C such that S and. X are compact. Then Comp C is well-factored by the factcri ation induced from C. Proof. ',We not tht the i the nduced factor ization is just the standard the proof is (1.18). G-spaoe is a The image of (1) For (2) For i ell-factorization of Corp C. With this fact, an easy al?.cation of 1.8. EBan2DJ. If C is a topological group, a right topolloical space X along with a map X>3-->X. (x,s) is denoted xs. We require the following: each x in X and' s and sl in G, x(ss') = (xs)se. each x in X; .71 = x, .where 1 is the identity in G. A G-space is called cffecti-ve provided xs = x implies s = 1. Let X = 'xxs)EXXX x s in X and s is in C->, where X is an effective G-space. There is a function t: X"-->3 defined by xt(x,x') = x' and -hich is called the transl.tion functio0.- A G-space X is called -oril.cipal provided X is an effective G-space with a continuous trans- lation function. A rJirr-r-. G-bund-le is a bundle (X,p,B), where X is a principal G spa e. A morphism (u,f):(Xp,3)--> (X',p',BA) between two principal G-bundles is a prnpQ p~al no irphi-s o provided u:X--- is a G-Module homomorpnhisn and is continuous. Let PDBn be the category of principal bundles and principal rorphisSs. Let Pauni be the subcategory of PBan obtained by holding B in (X~p,B) constant and PEunX be the subcategory obtained by holding X in (Xp,B) constant. Then morphisns have identities in the "constant" coordinate. Let P. be the classs of all P:BnX sand let P2 be the "class" of all PEn Then PEun = 1 x F2. ?Proof Since 1i and r are the factorization induced on PEnn 1 the w-ell--factori zation of Dan given in 1.11, it suffices to prove that condition (3) of 1.4 holds. Suppose (u,f):(X,p,E--B)--(X' p',). Then (1,f):(X,p,B)-- (X,fop,B') and (u,1) :(: fo p,C )-- (X',p',B') and (X, fp, B) is an object in P-.n,, The rcest is clear. (1. 19). :---rl. For fu'rter details about the cate-' gory of fibre handles ai, about 1.18, the reader should consult [43 To elements x and x' in a G-sTace X are said to be G-equivalent if and only if there exists s in G such that xs = x'. This is an equivalence relation, so we may foin the quotient space, denoted X/G, Let c = (X.p,B) be a principal G-bundle and let F be a left G-spa~ce. The equation (x,y)s = (xss-ly) defines a r-ight G-space structure on XxF. Let Xp denote the quotient space (Xe?)/G and let pF:Xp---B be the factoriza.tion of XXF Pr'l-X by the canonical map X;F-->X. Thus, for (xy) in X?, pp((xy)G) = p(x). Then (Xpp",B), de- notbed d[FC, is the fib .e inldle over B with fib-re F (vie-wed as a G-space) ,sn -as ccia tid :1 ,i2 nc3Dgl b'-Idle dc A fibre s-orohiam from dWC] to d'[F] is a bundle morYphism of the form (uFpf):d[.]--'d'[F3, where (uif):d- is a principal bundle norphisi an d up is obtained as follows: u:X--)>X induces a G--orphism u;:I:XX;--;X; F aend u? is the induced quotient !ap from (WXP)/G to (X'.)/G. Let iun[F] denote the category of all fibre bundles ,ith fibre F. Let Bun[P]X be the subcategory obtained by hodi.ng X constant in d and .ul~l b the subcategory obtained by holding B con- stant in dc Let Pi be the "class" of all Pan[rpJ a:d let P2 be the "clasr" of all un[F1 g. Then Ean[F] "= Pf X ,. Proof. As in 1.18, it suffices to prove (3) of 1.4. Suppose (uF,f): (X F/CG-pp, ,)---(X F/G,p'F, B'). Then we have a unique Pf- factLtorization (Xp, pp, 3)- -----~" __-_ (X;, 3pB') (,f ) (up, I) (X7p, ou,,B') as soon as wee pove t-ha (XplouF,B") is a fibre bundle. However, this is a simple consequence of the fact that we are working rith quotient n>,i .e d the details are left to the reader. (1.20). Path Ct -. C Recall that a graph is a class of objects, a class of morphisLs, a domain function and C. codonain function; i.e., a graph is a category without composition. Thus, there is an obvious forgetful functor from the c.--'- y of all categories to the category of all graphs. ( e should point out that a morphis. of graphs consists of a pair of functions (F' F2), so that for graphs G1 and G2s F1:Ob',-- bC- 2 and F2:Kor G1---:or G2 so that if g is a Gj-morphisn, then fl(dom g) = dom F2(g) and F(cod g) = cod F2(g).) This forgetful functor has a left adjoint; i.e., free categories (over graphs) exist. (These are also called oth catofories. ) This construction is done by using finite sequences of "composable" arro-wrs in the graph as morphl ns in the free category and juxtaposition as composlltio:, Given an object A in the graph, ( )A (the enpty sequence from A to A) is the identity morphism on A. The path category on a graph G is denoted Pa C. Two systems of notation -:ill 'e used in path categories. We may denote morphisss pictorially as A-fi->,-2-> C--r*..-. ->D or as sequences as (g;,..'~* 3 2:g ). Note that -hen writ- ten as a sequence, the morphisms are written in their Vcomposable?' order. The identity morphism on A may be de- noted as A-)L or just by ( )A. We will riite A-l-B ( B--E0 as (g,f). (1.21). Cateyorical Relations. Let C be a category and let E be a relation on MIor C, Then E is called a cate-o;ical rlation if and only if f E g implies dom f = don g and cod f = cod g. Also, E is said to be co-ntiable if and only if (1) f E gs hog defined, and hof defined implies her E hog and. (2) fE g fch defined, and g'h defined implies foh E goh. It is clear that U(hoir,(A,B) x hoi(BA)) is a categor- AB e ObC ical, compatible equivalence relation. Since the inter- section of any family of categorical,compatible equivalence relations is a categorical, compatible equivalence relation, any subset of U (hon(A,3) x hom(BA)) generates a categor- A, BObC ical, co ? LatL-le equivalence relation, Some sources refer to a ca te-orical, compatible equivalence relation as a congruence relation., Finally, if 2 is a categorical, compatible equivalence relation, thc-- ::e nmay form the quotient category of C by E, denoted by C/0, as foll s: Ob C/3 Ob C and Mor(C/J) = 16 (NIor C)/E, dao2aLrU- domr f an l coi? ~ f] od f, aa [fjo[ge] = fog, whore 7 :'Trite w zirIn iYi E 9 -ith repJreoent alt ve f in .a-s C ff). CHAP7-_, .II TWO E!': --DF: TiHOREMS In this chapter we show; the equivalence of the general concept of w;ell-factored category to two special kinds of well-factored categories. (2.1). Definition. A rultipl.ca tive ca~teory without unit is a triple N = (i,0,<) such that-, (a) M is a category, (b) Q:i. x 1---4 s a bi functor. (c) For all objects L,I,NI in K, rLNN:Lc ,.)---(LC )~ i is an isomorphism and is natural in LI, and N. We will usually vrite L,. ,N as just c<. (d) For all objects L,i,N, and P of ~, the fojlorins diagram coirutes: L( 0( N ( )) ('0 (L(I)N)PP L ((I)0? )--------- >(LO(I;N) ) (2.2). Re .r. Definition 2.1 and the definitions of a semigroup and of an act are motivated by si._ilar defi- nitions in 5] Conditions abh; a^n3. c do not imply con-- dition d. For example, let N i ,I. where R is a coinutative R- ring viith id-ntity Let 0 b3 the usual tensor product of raodules over R. Let < be defined by: 1(m@n)O---(-10ln)n. Then conditions a, b, a-nd c hold but cd fails, if char R 2 2. (2.3). DPfin.tion. A sei'!:r7oup in N is a pair (Su) such that t-he follo.;ing holld: (a) S is an object in 11 where =- (e,,NT ). (b) SxS-~-S is a morphism in L, (c) The following diasgram commutes: 10u S( S.3).----------....--- so I (S 3 )S ul1 u S@ S3--- S Whenever: possible, re denote (S,u) as S. (2.4). ^ A.-it.o. If N is a maltiplicativo category without unit, then Sgp(N) is the category of scmigroups in 1. A or-p' ...:' i in Sgp(N) from (S,u) to (T,v) js a mior- phiswm f:S->T in so that the following diara c. ;-es: u SOS >S TOS'------------- T f~f f T0I1 -11-------------> Q1 Composltlon is ind3ued from I.. It is clear that Sgp(-) is indeed a category. (2.5). Definition._ Let (S,u) be a semiigroup in.N = (M,<,<) and let M be an object in I,. Then a left c.t.ion of (S,u) on K in N is a triple ((Su).:s, -l---) :.hre z is a morphisi in a!- i the- ll...i diagra coit 10< S (S ) .: -------------s- com (03)0:: z Si ______________________ o Cy. i . z TWhenever possible, r> :;ill denote ((Su),i,SM--M) by (S,MI, z). (2.6). DeLiition. Lact(N) is the category of all left actions in N. A morphism in Lact(N) taking (S,M,z) into (T,N,z') is a pair (f,g) where f:S--T is a morphism in Sgp(N) a 1nd g .:--> is a morphism in M (JT = (,s-, the following di-araa comimutes s0:,T i 1; f03 g ) The com~ ~sition is jnducod from I1. It is clear that Lact(i) is inlcee a cate-or y. (2.7). .iflitio-. ei't '1 = (L,0,s) be a multiplicative category .:itho-ut. unit. is said to be e::ct if and only if c< is al:zays a identity morphism. (2.8). P in o t.,ion. Let C be a category. Then there exists an e:.;t ultipll.cative category without unit, such that C is ioo .tc to a subcatc cry of lect(0,). ionf01 ah :- t" the 3 c-~t ory of all e.njofunctors of C. Then N = (2,composition, ~euality), is an exact multiplicative category ith ct unit since t in-:techane la,.r says compo- sition is a bifunctor takin x ~xM into E. Let C be an object in C. T'-.: i et C.C-->C be defined by ~ (A) = C, for each object A in C, and C(:) -- IC for each morphism f in C. The;] CU is an endofunctor of C. Let K be the identity functor on C. Then it is easy to see that (K.1 ) is a semigroup in N. Define F:C-->Lact()) by F(C) = ((KiKI),C; 1-g). The follo-uing diagram coammutes: C 1 i (KoK)oC 1 01, 1- C Kv I Ko C-- c----------9 . This sho;,s that F(C) is an object in Lact(N). If f is a morphirn in C, then F(f) = (1 ,) here = t'AA b and, for each object A in C, tA = f. lt is clear that f is a morphism in 'N and, that (1,7) is a morphisn in Lact(N). Thus, F:C--l>ct(^) is clearly a functor and is one--to-one on objects and on norz-' , Now ,e sho:- that ia(.') is a subcategory of Lact(N). Suppose F(f)o?(g) is defined. Then cod F(g) = (K.,,1 ) and dom F(f) = (K,,1-) and cod F(S) = don F(f), Thus, S= C so B = C. ut then cod g = B = C = do f so fog is defined. Hence, F(f)oF(g) = F(fog) so ira(F) is closed under composition. Therefore, ir(F) is a subcategory of Lact(`) so C is iJ o ;orphic to ii(P). (2.9). q RA .i. Although proposition 2.8 is a very general embedding theoren, it seems to have very little practical use. We wil prove later an embedding theoren of uell-factored categories into categories of left actions so that the enb.ddin preser-vs F'-factorizations. This is a nuch more useful enscr din, Hoever, we will first prove an embedding theore of a somewhat different nature. (2.10). Tierorem. Iet C be a well-factored category. Then there exist categories C C; and C such that C e- there e1 -t - is isomorphic to a subcategory of C x C x C which is well 1 -2 -3 factored by the standard :ell-factorization. Moreover, each iP.-orphism in C maps to a P.-morphism in the subcate- gory of C' x Cx C for i = 1,2. (2.11). PyRr'r. Before proving 2,10, :,e apply it to prove 1,5 and 1,6. Then several preliminary propositions leadinc- to the poof of 2.10 rill be given. (2.12). Helri, ote that if A is a subcategory of C x C x C, so that A is :ell-factored in the standard way by P, and F, then letting P = P2 and P = [, we have that A = F rx P provided (3) of 1.4 holds. How-ver, P and 2 are not the standard -well-factorlzation. It is clcar that by letting CJ = C~ C' = C and C = C that .a . z-2 ---3 -3 A is then r:ll-factored in the standard way by "1 and P as a .u.boates-e_,y of" .f.C ~ By 2,10, given aany cate- gory A wit h any ,ell-fac.torization, we can find an isomor-. phic copy of A so t'hat the isomorphic copy is then well- factord i. th stad.ird ni c- by the corresponding :1ll- facto: a on, (2.13). P~oof f 1.5 Definc H :S-->P >x by H(s) = (s1is2), 'hee s = s2s1. (He are usin, subscript to indicate .. -zlorphisri ) By 2.10, there cx-ist categories C~ .2, a nd C. such "tht we have an eLbdd ing F:S----> C1x C2xc^ th ch preserves Fr-morphisss. Thus, F(s2) is a r2morphis s r. F(s1) is a r1-mIorphi~s i.e., F(s2) (a,1) and P(si) = (l,b), for some a and b. The we have F(s2)oF(s1) = (a,l)o(l,b) (a,b) = (1,b)o(a,l) = F(sj)o(s2). Dat F is one-to--one o0 norphisms so s2o1 sl 2. We know, H is w.ell-defined since the -P2 f actorization of a norphisn is unique. Let s and s' be in S. T:. sos = (s2o0s1)(s2os.) = s o (s s')os' = s o(sos )os' = 2 2 1 2 1 2 2 1 1 (s2s00o s,! Thus, H(sos') = (s1si s2os') = (si,s2)o(s),s) = H(s)o H(s'), so is a functor and a monoid +orphisH. H(s) = H(s') implies s = s20s = so s = s'. Thus, H is one-to-cne. Let (s1s2) be ai elenmnt of Jx P. Then sO is in S and H(s2os1) = (s1,s2). T',-;, H is onto. Since F[ and P2 are subcategories of S, 1 arind are subn-onoids of S. Hence, H i.s an isomorphism and so S is iso hic to F x U. (2.14). o- of of 1.6. e w:ill first prove that P2 is a normal subgroup of G. Since P2 is a no'--_ty subcategory of G, 1 is in r2 and f2 is closed under multiplication. Let g2 be in f2, Then 2 = h2hl, whe:r h2 and h are unique elements of P2 and respective y. Theni = g,2o1 i20h2oh) = ( 2h2)oh11 = iol 7 P- uni quen of rP 2 factoiza ion, i 1 so g = h nd h i in - aco h = 1. so g2- = h2 and h is i F2 is a subgroup of G. Let g be in V and 2 be in P. Then by an argurlnt similar to the one in 2.13, w e have that glog2 2 o2 Thus, for each h in F g, gohog1 = o ho g gich I s in 1 2oglOhOg og 1 = 21 lP g2 2hog 1whiccl is in f2. Thus, i2 is a normal subgroup of G. Similarly, Il 2 1-Y 1 is a normal subgroup of G. Thus, by uniqueness of 12-F factorization, G = rx where x now denotes direct prod- uct. (2.15). DfJinition. Let C be a category wThich is well- factored by l ancd i r Then a new category C. may be formed j2. as follows. Form a graph denoted as G1 by taking Ob G1 . For each P -morphism g:A---3, there exist unique A and B in P such that A is an object in A and B is an object in 1 B. Let there correspond a unique arrow g:A--) in the graph G1. Define o on Pa(G1) by: (1) (A A >A) y A A( ) A) for each object A in A. (2) (W-- - ->U) y (-g- -?U) whenever gof is defined. (3) (--- ) () povided there exist Fi- morphisms a and b so that the follo:Ting diagram comu:Ttes: f a b Sg W----->----------> . Now u is a cat-sorical relation on Pa(G), Let v be the catcgor.' cnly cop 'tible equivalent e relt: io:'n ;:.nated by r. Then let Cj~ (Pa GK)/y. (2.16). PF-.?-. TLt C be a category which is well- factored by ai and 2" :r, e may form a category C2 by inter- changin 1 and 2 throughout definition 2.15. (2.17). o;oqiitop. Let Y be a categorical relation. Let v be the categorical, compatible equivalence relation generated by y. ihen consists precisely of those pairs (gg') of morphisms such that don g = dom g' and cod g = cod g' and the following holds: there exist morphisms g19 o. such that for each i = l,...,n, dom gi = dom g and cod gi = cod g and, for each i = 0,...,n, (letting S= g and gr+1 = g') there exist g and' such that ii+1 ii+1 gi = o ***ogi- and gi+1 = i oLi1 "og1ii andc, for each j 1= 1,...,ki, E gg where E is =, or 1 Proof. The proof of this proposition is routine and will be left to the reader. (2.18). Fropostito;. Let h and h' be ri--norphisms in C where C = r x y. Lot Pa(G2) be the free category formed in 2.16 and let h s.nd hC denote the morlphisms in Pa(G2) correspon:.ing respectively to h and h'. If hh', thenl in C, h and h' are F-roroi.. '." grid connected by P2-morphi. sms Eroof. In Corder to prove this proposition, it is nec- essarOy to m2. a careful analysis of what h ? h' means. Namely, we he have morphisms gi'" as in 2.17. Note that we may avoid the possibility that sone Ci is the cmpty morphism by, if so, pu'iting -an ienti.ty (corresponding to a P1-identlty) alo-r i"ae .gl' g n nsd a y-related path categor-y idezntity alongside h and. h'. Nowu e write dowun equivalent cc itions in C to describe the process in 2.17. We have that there e-:ist seuences of P-orphiss SO, Sn+ such that S = (h)1 9n+1 = (h). no Si is empty and (since *einay i thout loss of generality assu-. that gi - 11 i-for each j except sonm one value of j) for S. and Si+1 w;e have (or with S. and Si+1 interchanged) one of the follo-.ing occurs: (a) 1A ( )A r-eans that A is an object in A so -e have Si (Vp ... ,k+v,vk+k,7 l. v) and Si+1 = (p * *;k+.-s'LkVk, ..*v1) here ^k vk has been deleted from the sequence, vk is an identity, and cod vk_1, dora k+, and co = do! vk. are all in the same P2-categcry. (b) Si = (Vp,*.. sk+1 Vk"k-i, ..1) and ;-1 (vp . .,Vk .,stvk- ,. 1 ) where sot = vk. (c) Si = (vp9 ...,Vk+.vv . vi ) lnd Siv-"1 **. ,t1' 1.-1 . .v ,) ;:h e thore exist [2-norphisms oa iad b so that asovk = o.b, T: (a) corresponds to (1) in the definition onf, (b) correspo-ls to (2) in the definition of (t and (c) corres onds to (3) in the definition of Ty. ,Thus, by 1.3, h and h' are P--nor-phiss grid connected by P2-.orThias. (2oin9).. _-:'oCo to hon. Let h and 0h be --.. 1 "p-h-sns in C T.here C x L et Pa( Gi) be the path category fo'.-- in 2.15 and let -h E- h' denote the no:; -* :. In Pa(G1) corres'-i: r ti to h n h, If h then in C, h ani h' are P-sorpl'isms grid conne-cted by Proof. The proof is th:e S aeo as for 2.18, :ilth 1 and 2 interchar-s-ed. (2.20). Pr;f' c-" 2.10. Lt an be the cato formed in 2.15 snd 2o16 respc _tivevly. Let C be the dis- crete category for.7ed by the objects of C. Let A be an object in C. Then there exist unique A. in [. and unique A2 in r2 such that A is ar objot-. in Ai a.nJ A is an object in A,. Define F:-C1-. x C2x C on objects by F(A) = (A1 A). Suppose f:A---B is a norophisn in C. Thon there xist a uique facto-'iza.tion f = hog -.he-r g is a r r--orphism and h is a P2 -orphion. Then define F(f) = (Ch7[gC3): (1A2A))----(B B 2,B). It is clear that F is .ell-defineSd Now ie must show that F is a functor. Since A1 1 1l A A 6 and since 1A is both a 1-or:phi2s an' a [2-orph; s, P (1) (CL. ) (A., A) F (A) Suppos A-- - Then we have r1c 2 factorizations s indicated by the fol- loting co~uta-tive dia --r: f f D 6' h' I By definition of v in 2 15 and in 2.16, h=p' in Pa(Gi) and pyFg in Pa(C). : [ h = [p ] l P: [J. (f)o(f) = ([h' ,[Cs'])o(h],rs) = ([hoh][o, go]) = ([hrl [p [p]o[]) -= ( [h' p'C pog]) F(fo f). Conse- quenbly, F is a functor. It is cl-sar that F is onc-to-one on objects. Suppose F() (f) where f and f' are moi-phisms in C. Then dom f don f annd ccd f =- cod f. Also, f and f0 have fac- torizations as indicated by the following diagram where the top and bottom triangles are cormmutative: r9 h _g 2_2 Thus, F(f) (ChJ,r]) aind F(f') ([h'],7 gl]) so [h1] =[htl and [zj = e ; i.e., as Pa(GI) ond Pa(G2) morphismis, hh' and gCg'. By 2.18 and 2.19, h and h are F2-.:orphisms grid connect by r-morphisms and g and g' are F--morphisrms grid connected by -orp.his ras By definition of a well- factored category, wie then have f = hog = h'oge =: f. Thus, F is one-to-one on morphisms, We next show that in(F) i.s a suboategoory of Cx Cx C. It suffices to sho:e J.m(F) is dco under comooosition. Suppose F(f)oF(g) is defied. Then doi F(f) (A1 A2,A) and cod F(s) = i'B,2, B) so A = B, But by definition of F, A = do:s f as, B = cod g so fog is defined. Hence, F(f)o'F(c) = -(u-) so ij(F) is cloCed- u nicr counos.tion. Since r: ha that C is i sor.c.chi to ir(F). :,; nowr show; im(F) is well-factored in the standard manner. It suffices to sho-: that each P.--noro'hism in C nans to a i- morph is in 9C x C.x C, i = 1,2. If f is a r;-morphisn, then f = lof is the fP- factorization of f so F(f) = ([f),[13) anid [1 is al -.-s an identity in Ci Hence, F(f) is a r -morphi:-, Similarly, we can sho; that F pre- serves P2-norphisms. This completes the pr-oof. (2.21). Re l~ak. Now-T that we have gone through the construction of C' C2 and C3 we may ask what Ci, C., 1.- -3 L-1 2 and C are for some familiar cateSories. -3 Suppose S is a monoid wi-ll-factored by and r 1 2 Then as noted. earlier, discounting empty subcategories, card( 1) = card( ) = 1 and, identifying 1 with its member 2 2 and P? with its Ynaembeis S x r as a monoid. Put by 2.10, S is isomorphic to a subcategory of Cx C2x C . j.1 -2 3 We ask, "Ihat :are ~., C, and C'?" According to the con- struction, C = (Pa G.)/Y whire GC is the graph of i -mor- phisms. Then G1 is P2 considered as a gropho Bt we identify Ip2 i7ith ( )GI and (f,g) with (fog), Finally, we say ftg if theeare e -morphi isIs a and b so that bof = goa. But as noted in 2.13, goa = aog, so bof aog. We then have two 7 -F fractorizations of an S-morphism. Hence, C1 2' Similarlly, C. 0 and card(C) = card(Ob S) = 1. It is easy to see that F:S--C x C x C from 2.10 is onto. Similarly, if G is a group and is ;well.-factored by 1 and P then card(P) card( ) 1 and, making the identificti of above C., C 0 ,, -.0 crd(C3) 1 -. c 2 -. 3 No;.w 'we consider the category of all modules over all rings. Ob(G1) ,jM' I is a ring- Arrows in G1 are _R -- gso that (5,l):(Ri:, )-- :(ER,;) for some M. Suppose (g-1)_(RI ,)--,-''E75 I'),. The following diagram coL.mUli.es (g,1) (, ,.)---- --- ---------( ', -) (1,0) (g,) (1,0) (Rs1, v )-1 ->(R Mi', .: ) Consequently, (G,iT)(S'(gj ). Given a ring morphlism g: R-->R'- (g,i):(ER R.)--,(RHS~^,f,) where r-r' = g(r)-:ra (6,1) (g',1) Also, given ------- we have (g,1): (, (.)-->(R ,i.S) and (gr ): (R 9 1,.7')-----R ?, ,9 ). The following die.g-,. comrnutes ;-here r. r'm = g(r)Vsnm' (1,0) (1,0) '(R g)--- 1,-:R' ) .so - Thus, ((g',i' ), (g ~l) )F(g'og, 1 ) so C, R. G2 has as objects all RK such thab Ii is an abelian group. Ar2 n G2 are RM--- : so that for some R, (RM,)---- (RI,',). Then- the following diagra commnutes: (00M N, )---------- (, . (0,1) (0,1) (. ^, )---_--- c-_,>(,:i: 1 ..:.) iiv Go(1p Ig)phisu Conseuetly, (1R<) (1~0,c) G.ivn &niy group mc:;hisu g:---:-I', where and '1n are abalian groups, ( (,g):(0,Mo) --) *()) so (1),o .: -u'. Given ) v g we note that ((1R: s'), ,))(1E Thus, we have couposability of morphions. Consequently, C = (Pa G2)/P is isomomorophic to Ab. Our:- results then are that for RM, C1 C2 Ab, C is discrete, and card(C3) = card(R ). (2.22) Propos..trion, Let N = (I,,,<) be a multiplicative category without unit. Let (S,u) be an object in Sgp(N), A be an object in ,1, ((T,v),B,z) be an object in Lact(N). Suppose ft,(S,u)-->(Tsv) is a morphism in Sgp(N) and that g:A-->3 is a uorphism in I. Then ((S,u),B,zo(fOl)) is an object in Lact(Hi). Proof. This is equivalent to showing that the following diagra n co:-iutes: 10(f0) 10z so(S)3) >S(>(T( ) >SO3 C< fl (SS )@B TOB .u01 ;u3z S03-- T3B T>3 . ful z The proof that the above dlagrrai7 commutes is routine and is left to the reader. 1o use that f is a semigroup mor- phism, o is natural,& is a bifunctor, i.nd (T,Bz) is an object In Lrct(N). (2.23). oiqo-osijtion. LGo t N be as in 2.22. Let C = Sgp(N) and C2 i Let C, be the discrete category with objects z:SOSl---A so that z is a rorph'.I.n in 1. Thcn Lct (l.) is .:ll-f].ot. d. in the standard Iannor as a subcanegory of x C x C2 ':3. 2o will refer to this as the standard well- factorization of Lact(_). F-roof. By 1.8, it suffices to sho.; that (3) of 1.4 holds and that if (1,1):(SA,z)--->(S,Az') is a morphism in Iact(), thein z =Z'. Given the morphism (1,1), the following diagram. co:imutes: z S--A ---->A S 0A-:%A . 101 1 Thus, z = z'. Suppose (f,s):((S,u),AzA)-->((T.v),BzB) is a morphisn in LBct(N). By 2.22, ((S,u),B,zBo(fi)) is an object in Lact(N). Aso, gozA = zBo(f), since (f,g) is a morphism in Lct(I). Hence, gozA = ZB (fi1) (1i3) so (.i:):(S,~ A)----:S BB;zB (f0i)) is a morphism in Lact (). It is cl].er that (f,1):(S,,zBO(z fi))----(TB,zB) is a mor- phisn in Lnct(!_). Also, (1,g) is a P---orphism end (f;1) is a P-. orphls" and (fsg) = (f,1)o(1~, ). To see that the factoriztion is unique, it suffices to prova (S,B,zgB(f@l)) is unique. If (f"g) = (f',1)o(1,lg), then f' = f, g' -- , and dom(fC1) = (S,Bp). Also, lop = z B(fol) so p zBO (f01). Thus: the i-P2 factorization is unique so Lact(fN) is wel). .l-factored in the standard mnsinzr as a subcategory of C x C x C.3 (2.2!43). "Th \r. Let C be a :ll-factored in the stand- ard rcJ.mner subost :- of C x Cx C Then there exists an exact u.1tip'l te ,-c T -~ (;tc<) Tthout' unit such that C is isoorphiLc to a ,ell-factored subsoatcory of iact(JS ) so that the .'ill-factorization of Lact (N) induces the el-fcton of the subcategory. Also, each f- lorphism of C maps to a s-:aorphisnm n Lact(N), i = 192. (2.25). e--' EBsfore proving 2.24;, we obtain a pre- liminary result. In fac'i this no-r summarizes 2.1.0 and 2.24. (2.26). eot. Lot C be. a category and let PF and 2, be 'classes" of subcategories of C. Then the following are equivalent: (1) C is ,sell-factored by F and r. (2) Thero exist categories C C, an. C3 such that C is isonorohic to a subcategory of C, x C2 C3 rhich is vwel l-factored in the standard manner and such that each r -norphi~si in C naps to a P.-noropis n x -2x 3. (3) T 'e exists an exact nu.ltipJlicative category S= (i,3,c<) .without unit such that C is isomo:p:hic to a suboategor.y of I.Ct(!) ;ell-faotored by the factorization induced b; the nl_-factorioation of Lact(N) and such that each r._0.rphis in C 1i ps to a a -orphisR in Lact(i), for i = 1,2. Proof~ It follows from 2.10 that (1) implies (2). It follow s from 2.24 that (2) implies (3). It is clear that (3) inplics (1). (2.27)o. Y'cof of 2.24. Disjointify the catoories C. and C2. Form a gri:',l as follows: objects in P consist of pairs (A1.A2) where Ai is an object in C, for i = 1,2, or objects in P r T: th -selv.s bo objects in C.,. For each object (A~,A2jA-) in C, Ie let there cor--espcnd a unique 1 2 - morphism front (A1,A2) to A2 whichh we will denote by A, Now we form another graph G as follc;3: Objects in G are n-tuples (UlJ,...,U ) .There each Ui is an object in 01 or C2 and n is greater- than or equal ons. A morphism f from (U1,...,Un) to (V1' ..,Vm) in G is a set of morphisms (in Ci, C2, orP) denoted fi or fi +, here fi:Ui--Vj(i) is a morphism in C1 or in C2 lnd fi, s (U. )---T (i) is a norphism in i. Also, we require n>m and that, for each i = l,.. r n, i appears c:actly co anco a subscript of sonm fi or fi i+1 .There fi+1 is covtntedr as having two subscripts. Given an object (X!,,..,X) in Pa(G), denote Q : reU tp-.e f and g (X1. .,qX )-->{ ...,X ) as 1( ')e. Suppose f and g are two morphisms in Pa(G) going front (U ,..Un) to (V1,...V ) We obtain from f a ne:; n i! i. ,F and from g a new morphis:u G = (Gs,,., G1). Suppose f (U, Un ) Then 3.et F = ( (U1 .. U)). Otherwise, f = (fk f1) Insert bfore f1 the norphisna S(Ul ,, ..Un) and after fk the morph-is 1 .)o For i =I ,.,.ck-1, let cod f (Uiu ... -Ui)). Insert bset'ecn fi and fi'+ the mlorphism 1(U .U Cslling this ncw sequence (Ft ,..,P1) -e see it is a path in Pa(G) and we let F = (Ft~ ~oeF1)) G is obtained from g in a similar manner. Let iO be such that liCr< L. :IU F> ----> ; in fact~ ', i0 "O F 1i Lt I-, 0 Exactly one of the following 10 J0 = i 0 L0 possibilities occv.rs 2 _ F i:U ---.- : (UiU )--"U I or, finally, 0 0O o0 l0-1,i 0 i(U 01 )--1 p Whichever of these occurs, rwe call it F~U- and P = Let F F 0 P P P p p p o There are three possibilities for W which are analogous to the three po.si. iti for U1 hichever of these occurs rwe call F.'. Continuing in this : w ;e obtain the sequence (t,,..,F 1). For e = 1,,..,t-1, either 0e 0 +o e O ot er F0 e 0 is (defined in C. or in C or one of F~' and F0e+ 0 0 0 0 is a r-morphism. By compo3ing ;hatevor adjacent norphisms are col: ,sable, we obtain a now sequence (F. i0, 71) -0 0 so that for l6cT -., exactly one of F? and Fe+1 is 0 "-0 0 orphisr. Similarly, from G we obtain (G 0 o,,G ). O 0 If, for 0 = ,n, (FTiO (G0 ...G ), then L-e say fyg. It is easy and is left to the readr to see that Y is a categorical, compatible equivalence relation. L~t A = (Pa G)/y. Lot h !/j where fp is the categorcal. comatible equivalence- relation generated byp .: catcocal la- tionp consists of all pairs (aa') such that aa (U,"c..,U)-r-,.(Vi',* ) and there exist ..- native b and bh in Lor Pa G so that a = b sid a' = b' and either a = a' or at least one of thce follo-ini t-- o condi- tions holds or with a and a' interchan ~d at least one of the following toco conditions holds: (1) b = uobbodt, blb2 C::or G, ffrbj f1:X--- , f2"i"'* fb2 Cf (;bj C4 11 "---,v b0 ub!bod b:b;oar G, foEb q (X.-g- f ^2 2f -t .2 -0 J o tha G (= b 1 1 1 2 ,- ik 2 )~ (2) b uob2obldc, b bE..:or- G5 fIl,f2 i3b1 f1:Xi >Tj5 f2 'X-- j t fl 1' 3 ara irdntltics, f4c-b2 J f3 1.++' 12f 329 and f4:( )-->Y b' = uT 'o bb- ', ,fbI, f X --, is an identity, f(X, ) and fb so that f:('! T )-- Also, (X 1,X -1 X ) n Ykj. = Y .... "k = 'b Pictorial.ly, co-litions (1) and (2) are as follo-s: (1) (U1.66 ee U1) (X. See I (Y, .....0 (V 9...... (V .. +1, Co. ,x) f2 3 .... .*,V ) here (XiX~ ) (X (2) ( U10 Sd o \^ ( I . -c.6 (VY . . . . I3lae~ 3 0 0.e. S..o.o.. (U .. to . .... o o Un) (x{ c 0 x a ) d9 1 5 9 T. 9' 0 0 a I , f 1 2 (Y 1 ... 'Q C Y- ,) ut (V1 0.. .c ........cV0) qUn) (U1) ................. ,,.Un) d' Xx) (Xi' X X- 1 __-X.. +1X7 . 1 I2 -(-w c T. F a-. ,") (Vw..i ,", .) u1 VI C 9 o c c o ( Vii -i V; where fcf:-".. and "i are identioc, (. ..,X, ) = ^ <1 1 .,- A J. Y i+ + 'i Ll . Let (UI"..,U ) and (Vl',.." V ) be objects in G. Let (U1,i... .U ,1) = (U ,.. L,UV),, V ); ie., let it be ( 0...,~,ln+m) rhere, for i = l,...,n, W J = Ui and, for i = n+1,...,--, Vi Similarly, if f:(U1,...,U) ->(V1 I. f:(U ..,U)-0 U(V ...,V ,), f =f U f1,1+- an~ f = f ] then f f :(U ,...,Un)9 (U ... ,U )--(1,.., V)(V ,...,V s) w-here ,e let f0fC = where, for = 1,..,n, g = f and g .i+l and, Or i fgi for i = n+1,...,n, g = f and g = f . Nou :e e fie @:xi---2.: On objects, 0 is the sa-S as for G. Given [[Cfl and C[]s th3 n f -- (fk 0f and g = (gP,..,, -i). Assume that kp. By definition of y , we see that if (X1..,. : ) is cod g, then g'(,...,l,gP,...,gl where we ta: e k-p 1(XI ,s a,d wher (X i...oX ) - cod GP = cod. g, He define [fJ.13@[[Cs] = [[hC] where h = (f 01.,- C, 0'1 z- '-..f 1). If p>,k, then us lengthen f similarly. For noto tional conyveience, ;e vill 5-ite a (1) b to indicate that a and b satii y condition (1) in the defi- nition of Ue use a sil :.'l- n ot.ation for co-dition (2). We need to sh- r Q is all-defined on corphisrs. S 1)ose [Cv33 = [[v] c '* [[u]J =[Cu'3 We r.ee:. to sho:r [ Cv3]0 [[ul] I= Cv][C[u'l. Thus; it suffice to showu CLv] [[u1] ECCv lJ@[[u]] and [Cv'33CCuj = C[[-,, ([uC3 Since the proofs are similar, we shoC 0 only thCt CCv]3[iCu]3 = CC[v]][[ul1 In A, we have C -v]-vT'3. Since is a c.t3 ;oricl; co-i pat- ibleS reflexiv, sy. stric relati- [ v J]v'3 if fnd I Sy2_, v -eJ;; j nd only if there cxist VlI ..vn so that C[vfP7v)f[v2jo.. [7'1 By the definition of p, ECv]r[v means that there exist representatives e and el so that [e] = [v] and [ell = [v1] and condition (1) or (2) in the definition of p holds be- tween e and el or that [v] =[vl]. How-ever, we may assume without loss of generality that e = v and el = v1. However, for [vl-]p[v2] since we have already used vi in conjunction with v, we must use a new representative for vl say v1. Thu.s, we have v = or (1) or (2) v1lV1 = or (1) or (2) v2 v2 *'"n = o2r (1) or (2) v'. Thus, it suffices to show: b = or (1) or (2) r implies [[b]3][Eu]3 = CCw]]3 Lul1 andi that b~w.: implies [[b]~3[[ulJ = CC3[w3 [[u]]. To prove this, suppose b (1) or (2) U. Then b = (b k,.osb ) and w = (,..u = () Assume k.,m and p>m. Then [[b]]J@[[u3L = [[(bk 1.. b1 lbm um...eb u and [[wlO[[u]] = p[[(wp 1 o,,-: 1,w u1, ,..l U 1 m. EBt then (1) or (2) w-ill hold between the representatives given above for [Cb3]0[[u]J and C[[w3][Lu] so [[b]]3[[uu3 = C[[l[[u-]] 1;'-r! other possibilities for k and p with m lead to differences. only in notation. Now suppose byrw Then, recalling the definition of we see that composi- tion of morphisns of the form 1 1. vith either b or w preserves the relation y, so [[bJ] [[ul3 = [[C]][[u-13 Thus, 0 is Tell defined. It is easy to see that 0 is in fact a, funictor, Letting c< 'c equality, N = (MT,,c) is clearly an exact mult 1prlitctive ca ;ocry l thcut unit. D"fine, F:C--->Lct () as follo ::. If (A 1A2,A.) is an object In C, let F(AuAsAq ) - ((A,(A1,A1) uC1 ,i ->A ~) ,A2, (A, 1,2 ) -3 3 Note that we now identify Ai wiith (Aij), i = 1,2. Also, we identify f. with (f ) where fi is a Ci-morphism, = 1,2. We do this since this identification embeds Ci into , i = 1,2. Given (frf2 ):(A1,A2,9A)----(A{,A ,A2) in C, define F(f.lf2) = ([[f3 [[f2] ). That (Aj,(AA.) [[1,1] is an object in Sgp(L) follows immediately from the defi- nition of y That F(A1,A2,A3) is an object in Lact(N) follows from condition (2) in the definition of, That [[f]] is a morphism n Sgp(N) follows from the definition of That ([[Cf l [[fJ2] ) is a morphism in Lact(N) follows from condition (1) in the definition off. It is now clear that F is a functor. Suppose F(f)Fo(g) is clefined. Then don f = (A1A,A2 A) and cod g = (A.A2,A 3) for some (A. AnA ) in C. Thus, fog is defined so F(f)oF(g) = F(fog). Hence, the i.ase of F is a subcatesory of "lct(o). Suppose F(f) = F(s). Then f = (flif2) and g = (g1g2) andnd [[f2f C[- 2] in i. Thus, [fj] f=C, Thus, as noted whenn proving 0 to b3e well-defincd, there exist hi~,1,h' 2h2,...hnh so that fl or (1) or (2) hi S=or (1) or (2) h Zrh2 ".hn = or (1) or (2) gl. Since doi fl has length one, (1) and (2) cannot' hold, so we have f h = hi. As in the definition of y we obtain 1 1 from h1 and. the o lrph1..H) en (from 1) the morphism IH (Hi, ..a ,)" Then since th~e .en-"th of dom h1 = 1 and. dom = dor Hi. = don H. = dora 'i ^ 1 1 .. 1 39 have H o.co = Ho ... 1. Since fl is a morp-hism in C and fl = h!, than H1 = (i,h91) so h = HtO ** Cortinuin, in his nanrr, :we obtain f = h = H o*o H 1 1 1 1 H .". .oi . = .2= g1. Thus, = g. Simi- larly, f = g ence f = g. Thus, F is an embedding since F is one-to-one on morphisms. It is clear that each Pi-morphism in C maps to a Pi-morphism in Lact(N), = 1,2. This completes the proof of 2.24. CHAPTE'0. T7T A SUFFICI:".'? COCDI-TION T : ) A ELL-FACOrED.D CA'EGORY TO E3 In this chapter, we will be concerned with a type of varietal category introduced by Herrlich in [1] which gen- eralizes the 'varietal" ca-L ories of La,.rvere and Linton. (3.1). Rrik. We recall the follo-iing definitions. (A,U) is called a cop-'-~' c-at ..--. if and only if U:A->ns and U is faithful. If p,q, and f are i or.:i.-': ,3 in A A any category, then (pq) is called the c-- -.1- i r-eation of f if and only if (pq) is the pullback of f with itself. If A is any category. f is said to be an E:^r .l..-.-. a-: -' if a6n only if f is an epinor-phi"A: S- Il f = Log is any factorization of f so that n is a ronororphism implies n is an isonorphism. A norphism f is said to be a r.. 1- alr e-r:\ if an1 only i f is the coeoualizer of some two m orphisms. It is easy to se that if f is a regular epinor~cr .... ', thce f is an: eztr;~ 1 epinor' c (3.2). Dn fiJito (-errlih ). Let (A,U) be a concrete category. (AU) is said to be an '- c t-. -- if and only if it satisfies the follo:.in three con .itio (Al) A has congruonce relations arrd oeo cr.izers, (A2) U has a left r. joint. (A3) U .-l. es-erves and ref3>lct-s "reu lr epi opiss. (AsU) is said to be a vaijtais.t catepo-ry if and only if it is ale~braic and satisfies the following condition: (V) U reflects congruence relations. (3.3). ES:v2le (HErrlich). (1) The following 3ar varietal categories: Ens, cat- egory of pointQe sets, category of all groups, category of all semigroups, category of all monoids, R, for any ring R, category of all unital R-nodules where R is any ring with identity, category of all lattices, category of all Boolean algebras, category of all compact; Hauslorff spaces, category of all compact, Hausdoorff groups, and Abe. (2) The category of all torsion free ab.elin groups is an algebraic category uhich is not varietal. (3.4)t emai'rk. We now prove a sequence of propositions leading to our sufficient condition for a well-facto.-' category to be algebraic or varietal. We state this suf- ficient condition in 3,5. T first fe- propositions will prove 3.63 *which is an important part of the proof of 3.5. It may be helpful to tl: reader to verify the folloTingr propositions only for the category of all modules over all rings or for the category of ronoids acting on pointed sets. (3.5)* 1"-. '.2 Let N = (nI,,Qo) b:C a multiplicativo category .nithoui unit such that. 5 has finite coproduc'ts. Also_, assure that for eachI object I in I! that i;_. and 0:.! preserve fiite coproduc.ts. Let U:Laot()x be the obvious fo_ .tful functor. Ass;ie the follo-?.ng to be true (1) If f is a x-egulr cepinorphinm in Sgp(I) and g is a regular epimorphizm in 1, then f5; is an epinorphisn in H. (2) (Sgp(1!)sU2) and (W,U3) are algebraic categories so that neither U2 nor U- assumes the e;pty set as a value. (3) lact(2) has c ."-'.sce relations and coequalizers. (4) If a and b are regular epimor;phisms respectively in Sgp(W) and in [> and v is a nonomorphism and the dia- gramL belo:7 consisting of the solid arror:s commutes, then there exists ZA B:AB-->B so that the entire di--ra:'i in M commutes a0b u@v A-Bt AB A"OB" A PC03c---------A> ---------------A "03 "-- zA',B: zAB zA' B" b v B B- -" o Jpet U = T[o(U2xU )oU :Lact( N--->iis, ,rhere 7T 3Enxs--,EnsTT is the product functor. Then (I...ct(),U) is an algebraic category. If, in addition, Twe assume that (Sgp(N),U2 and (i U ) are varietal categories, then (Lact(:),U) is a varieta! ca'.. ory (3,6), PrcnosoJi.ton Iet N and U, be as in 3.5. Then U1 has a left adjoint. (3.7). Not~ ion, We wrill let Ni U1, U2, Uj and T be as in 3.5 for the rest of this chapter. I t uE denote an injection into a co-rroduct involving E. Define dA.B C (A3B)l(AC)-->AO( OL!) to be the unique morphism d such that douA03 = AuB- aeid dOuA@C = A*C. henIevcr possible, we will denote dABC as d. Sim ilarly, we define cB, C, A (EBA)JL(CCA)--4(BC)A, Whenvor possible, Ce denote C as d. The assumption that AO_ and 2@A reserve finite coproducts s..1': that d an d 6 are iso:norphisms (3.8). Po-oit'.o. The transformaations d. and d are natural. 1'rPoof. The proof of thi-s proposition follo-.s in a straightforward manc'-r from th3 definition of a coproduct. (3.9), TPo~.s o ._o. Tne following ito dia-rams commute: (1) A C( F^)$L(B D)3 A B C9B D 'dB, CD A(EC (BD AO( EB CiD)) d.A B,C sD [(AOB)9 ] AA3)D]aA B (D3)0( CAD) (2) dA B,B CD, aB,CG 1 [(A B).] E Dc D ( (AiB) 0) J I< Il c< ^ [A(C(D)] c-.)] aABC D --- >(AjIE)(C.) P oo To see that diagram (1) co -os, it suffices to show; that (1) comrautes when composed wiith the coproduct injection into the coproduct of A&(5W,) and A( -)., We 1ill prove it for uA.O- l ') only. We calculate the fol- louj eq--ation, do ( J. )o" = do (u u ' C(161)0eUl C = (1C'1 )) = = 0 (10.d)od uA@(O3). Thus, by definition of coproduct, (1) comi-utes. The proof that diagr~a (2) com-ubtes is anal- ogous. (3.10). ?-.roc^Iticn The following diagram comlutesS EA( BD)] 1J [A( C.-J)] --------- >[[(A 3)D] (A)C. d A,B DiC D {A B,A C,D A [(BD )(COD) [(A)3 )(A3)] C D I0dBtC,D dA BA C@1 AO [(BJ0C)0D- 0[AO( BJiC) D PEoof. As in 3.9, it suffices to prove comnutativity when composed -.ith the injections into the coproduct of A(Pf.) and A(C' )). Since the procnedures are the same, we will consider only uAO( )" Then we have the following qualities: o iD0 iB B 'B (d0l)(u 01)oc< (al)oUc(A)~* = (Cl)'o (I)u Thus; the diagram conmz.uts. (3.11) roposio~:i 0. Let R (RsP) be an object in Sgp(N). Let I,- be an object in M. Define F( I,) to be -1 (R,K'II(~i::),z) where z is given by R(MI(?.^D:))-- d(RB;I)JL(R@(R-O)) -1 >(R:.i)lL( (RI) ( p01) : :(ROI ), i hore UROI is the coprocduct injection. Then F(R:M) is an object in Laot(i). jroof- It suffices to sho tat t ht e following diagran (0) co=ut.es. Diagrai (0) is reraw~ in ~dig r (Ot) a;:ith the values of .z indicated, Also, in (0'), die". (0) is sub- divided into cight subdia'3r- s,~ each of uhich is subsequently sho-wn to co:'iube. This conc-",ibtucbes the proof that (0) com utes, (0) Re[(MO.(RE:) )] p1O z l 0z O(H0 H(M ,'::) )- --->-M:,I (W;(._(Ai) Subdiagrsm 1 in (0) is redrain in (16) and decomposed into diagrams 1.1,1.2,1.3,1.4, and 1 5. Diagrams 1.1 and 1.5 comcute by 3.10, 1.2 conxutes since c is natural, and 1,3 and 1.4 co:iute by simple computations. For subtdiagS-am (2) to co ': :, it suffices to sho; [(l,p)lo = (lp0i). It suffices here to show equality when we com- .-; on both sidos with the injections into the coproduct of ROI and (ROR)ON. Since the p.ro.eclures are similar, ow consider only UEp -. We have the follo-::ins: [(1,p)&if'o .i = [(1,p)l]o(u 1) = C(l,p)aou l = i1 1 =(1,p@1l)ou. Thus, (2) commutes Diegram. (3) co-r.u~;es by the definition of d and diagram (4) col"luuli-'e since d is natural. Diagram (5) is redra-,m as dia-ram~ (5-) and is decom- posed into su" 2li--g s 5.15.o2, and 5.3. Diagram 5.1 coi- mutes by con-lition (d) in def.in. on 2..1, Digrac 5.2 cori--utcs 5nri? c< is natural. Di c 1 5,3 C a ;ctCes since (Rp) is an object inl S-p(Ni), 46 _. _-_-________- --- 1_ `'-__ ------ --- o co RI I =1 'j a: :jti / i T o "7M / ,LT~ v-: " 5 l.5 / / II RI 1t V/ -- ._i ./ '(.\ -. / / /C-. - / i W g M5 91 ; -. l5 -^^' ^ c- .5? ^ ^ , \f r RI Q) g =={ f- = 5- I 0 5 RI~ RI RI J N.YL, -^lJ^^ g ^ RI I II (1@ ~1 01 'H -' ,/-'~ ~ _ FlL 01 cr 1 0 cr1~~ RIR R r 5-b LJJ '- RI / r r" /i /iw 5--- C3 5--' I .r I / r C^ ^ "- r!/( r RI, R^iI. / - RI~~o ~ -: = a i -! 'H~ -I l-: 0 j -- | _i _J j LI RI S: C7 RI R Ig p"1 / cr! t 47 .9cc r- rr^ 0 - rr- 0 0 - I CO rd d -1 I Y II te: pL 5-1~ C11-*< ~- - r-1 *-i == 0 zi32 -i= rd rl z1. '- 121 > 0 0 p.; t\KCi r O~~ \ \ - r-4 \ \ o clas ci 'I"' \ l\ rl --S c rr7 ^~ \ \^^ : v-IY L-. \ 00 -5- -----> 121 \M arz v- L . 4 \j w, \ r- \- 0I \r 48 --1 r:S1 S-I-D 11 / "a a cc c^ (-- < -i ~4-- 01 04 01 \1 1 I g i - I/ /L r; ^-~V_ S ? &"i iI We note that dia(r-n (6) corrnutes since .d is natural. For diagram (7) to co-~ r.-auo need (1 p)Ol]oc = (1,p31). By the definition of co-roduct, it suffices to prove the above equation with u ,. or -.ith u -),. composed with both sides. o .e obtain the equations: C(1,P)i ] ouR. ( (lp)l)o (u fl)) [(1,p)ou-0 = 101= 1 = (1,pol)oupO1. The proof is just a.s easy for u( ) ., and is left for the reader, Thus, diesram (7) co=mutes. To prove that diagram (8) commutes, we adjoin uRv. 1 and u .o)01 to the diacrr''. to forn respectively diagrams 'RO(Ri) (8A) and (8'). It is clear by inspection that (8A) and (8B) commute. Thus, diagrams (1) throuSh (8) commute so our original diasram (0) conautes. This completes the proof of 3.11. (3.12). Pr (^osititlo- G iven that (1,g):(R,~ )--)U1 (ENz) is a morphisr: in Sgp(N)xi, then there exists a unique Lior.- phism (Ih) i- JL- (ct(Q) so that the followiL di. cgrcam coc utes: (1 ,v. ) U (RNz) . Furthrmoor:e, h (g z o( 10)). Proof. Co ~rutativity of the above diagr-am is clear. We u.st sfho (1, (g,zo(1'5)))) is a morph.is2: in Lact(N). It suffices to shot: the ditc. --* below coriautes: -d 1 ( '(1, p i) R^__ -___- -____i^ iaz^ frr '-a I- I.-) =0 " 0i 0 * vii @,, 8 . p.; ---I \ / 1 - C O-1 ~01 ** 1 C (X! \ \ Mrl r^ --- V: 611 r--I 0 t -I- '-I 0* \n-i - 0 0 '-I '1 * -~ -* 0 n-icl 0 vi '-I -- --- 0- P1 */ PPl vi Ir, e- x --^'- `- -I .^---^ c: = LJ %- 14- ^ . / ^----'^^^ / ; ) C-. /'*-'----*-----^Pl 1- S /c.- ^ / ci V I= Ph% Ph Ph P4 Ph 04 oh ----- --- L~J ., I-I As usual, 1.7 provo cori'unativity by adjoining first u,,., and then R Wc obbtif thI equations: z -(plz o( ~g) V We o U nthL [0g o de -a z El [ 0 (qz 0(1cD))1 a(iC>,[Q) 0 um zo = izo(1&c~ I = Z -(101)) o (l I p&l)-UROu 1o (z l))u (Note that -,e sec fro-- this ay?.: th-, :c ',iuad~ come mtaativity that h is unique.) z 316,zo(1Oc ))]odou,, () = zFlo :7:zo(12:)0 1( 10&z) 0 C,= zU[(z (1Qz))j = Zo(1OZ)o[1(1(lOg3D Bult. ( G)ROM) = (zo(1g)) a (IpBO-)-VB(n)T)O OC=zo(g1)o o = zp(p~-)oo. Thu s it sufj ces to Drove tho diagram belouy Ith sub-diagarams (a) and (b) conn-iuteo- { (a)->R IT RON, (b) z Diagx~z (a) is cuxpmi&ed b~lol amr clearly commrites: I p01 Diagrei- (b) ccl `;es sirce (R,i\,z) is an object in Lact(. Sinc -,Tc- have ino`L-ed-u that h i'Un? it- is easY to see that (l,h) is unique. This completes the proof of 3.12. (3.13) P"roap--''it. Le (t (i(R) denote the standard r.-subcategory of Lact(N) determined by the semigroup R. Let Up = Ul ( Then Up: F (R)--> R&ix has a left adjoint F. F(R~M) = (RMiU(Rc:,),z) is the same as in 3.11. F is defined on morphisns by: (R,) (--) ----------- R, I (RM) zy) (1,g) F(1,g) = N (1,(uNug,zN(10(uNo g)))) (R,N) I ->(R, N(RON) ) Proof. This proposition follows inr.ediately from 3.12. (3.14). rE~ k. We no-ea pause to prove some propositions of a different nature and which will be needed for the proof of 3.5. (3.15). .i'"* --'- ;o'--. Let C and C be categories wel1- factorced respcti- ly by 1 2 and 1 P Let P be a class which indexes both rand f. Let T:C--)C be such that for each p in P, l:Gp-- p Let TG = T Assume that K p P for each p in P, there exists Sp:Gp---G so that S is left adjoint to T Also, T(C) is an object in Gp implies C is an object in GI, Finelly, if f:A--T(B), then there exist unique f and f so that f is a P-morphism, f2 is a P.-nmorphism, and T(f2 )f = f. Then there exists S:C~-)' so that S is left adjoint to T and, for each p in P, SGp = Sp. Pro ?f. Let A be <,n object in C, Then there exists a unique Gp in P, so that A is an object in Gp. Define Ip Dein S(A) = Sp(A) Thus, TS(A) = T S (A), so there exists v: A----TS(A), a ncrphi.u n Gp, so that if f:A-->T p(B) is in Gp, then there exists a unique g:S(A)-- B in GP so that T(g)ov = f. Suppose f:A--T(B) is in C a9nd that B is an arbitrary object of C'. By hypotLli s, there exists unique f,' a G p orphiSm, and f2, a r2-orphism, so that the following diagram commutes: A-- T------(C) f T(f2 T(B) Then T(C) is in G so C is in G1. Thus, T(C) = T (C) and we have that there exists a unique g in GI so that T(g)ov = f" Thus, the follo-ring cliagra cc utes: V A>TS (A) Sfl f T(g) f T(C) f/ ( T(f2 T(B) . We now prove that f2og is unique. Suppose T(h)ov = f. Then there exist h1, aP orphi, nd h, La r-Lorphism, so that h = h2oh1. Bat v is a Gp-morphism and so is T(hl), so f = T(h2)o(T(hl)ov) implies h2 = f2 and T(hl)ov = fit Then by uniqueness of g, hi = g so h = h2 hi = f2og. Thus, by the front adjunction thoorem, S is the left adjoint of T. Also, S is defined on norphi:.. via the front ad- junction diE.agrm. Since Sp is the left adjoint of T , S l = SP, for each p in P. (3.16). .gAviopr of S. The functor S is not necessarily well behaved cn P2 ctegories. Tn fact, it tay be that S(G), ;here G is any P2 subcategory, is not contained in any P2 or .' subc.atcgory. ec no- give an example of this a 2 i situation. Let A hate objects A and B and non-identity morphisms a:A---B and a-l:B-->A with the indicated compo- sition. Let C = C' = AxA hav-e the standard well-factori- zation, whert P> (C) deno--os the i subcategory of A/A deter- mined. by C, an object in A, Let SA: P(A)---P (A) = p1(A)* Define. SB': ()--- ^(B) as follo-7s: S(B(BA) (BB), SB(B,B) = (B,A), SB(I(BA)) 1(BB), SB(i(B,B)) 1(BA)' SB(1,p) = (l,;pI-), and S (1,p"l) = (1,p). To see that SB is a func- tor, :o compute the folloA.ins equations. SB((1,p) (1,p-1)) SB(ii) = I(B~) = (1,~l)o(1p) = SB(l,p)oSB(19p-1). Siilarl.-y, SB((1,pi)o (1,p)) = SB(lp--)oS B(Jp). Thus, S is a functor. Let T = IAxA .It is clear that all the hypotheses in3.15 are satisfied, -iith the exception that SB is the left adjoint of T 1 (B) How:ever, this is a consequence of the front adjunction theorem and of some easy calculations and the details uill be left to the reader. Thus, by 3.15 there exist S:C-~ ~0 so that S is a left adjoint of T and S q(f ) S and S -(B) = SB. it S(B;A) = SB(BA) (B,B). Thus, (B.A) is an object in P (A) but S(B,A) is not an object in [!(A), _n foct, S(B-A) is an object in P (3). 2 Thus, S( 2(A)) is not contained in (A) or in P (). (3.17). .?'oi ..il, p ..'re exists F :S3p(N_)x7--9Lact () such that F1 is a left adjoint of UI andl F ,I is the same as F in 3.13. Proof. By 3.15, it suffices to siho- that for (f,g):((R,M) -->Ui(S,.Nz), there exist unique pl:(RMI)---3UI(C), p1 a r1-morphisrm, and p2:C-->(S,N,z), p2 a -morphism, so that (f,g) = UI( 2)op It suffices to show-. this since by definition of Ui and by 3.13, the remainder of the hypoth- ses in 3.15 are satisfied. Let C = (R,Nzo(f01)). By 2.22, C is an object in Lact(N), Lct pl = (1,g). Clearly p, is a [P-morphism in Sgp(H)x.I Let p2 = (f1l). To see that p2 is an act morphism, it suffices to note that the folloT.-ng diagram comnlutes: RON f --------- -;> S:OzN---- A f01 1 SN- .--. Thus, P2 is a PE-morphism in Lact(N) It is clear that the components of pl and of p2 are uniquely detc r-inedo All that is left to showit is the uniqueness of C. In particular, we must show that zo[f0l] is unique. However, this follows from the fact that (f,l) must be an act rmor:phism and from, consequently, the co-miutativity of the above C.icer'?. Thus, the factorization is unioque. This completes the proof of this proposition. (3.18). -rr Ute see that 3.17 is just a restat ement of 3.6, so 3.6 is proved. (3.19). Po.c'oooit o.p.- The product functor T:EnsxEnIs--> Ens has a left adjoint F4, where F4(A) = (A,A) and F4(f) = (f,f). PEoof. We will prove this using the front adjunction theorem. For the remainder of this proof, we denote F4 as F. Define DA:A-->AxA by DA(a) = (aa),.if A is non-enpty. If A is the empty set, then let DA -- A. If f:A-->B, then we have the following: [(fxf)DA (a) = (f1rf)(aa) (f(a),f(a)) = (DBof)(a). Thus, D:I---- F is a natural transformation Let f:A---(C(E) (=C E). Let pS:CxE---C and p2:CCE---E be the usual projection functions. Then (pofsP2of): (A:A)--->(C,E) is a morphism in EnsxEns. 1 T(pl1 fp2_f) = (p10f)x(p2f). L((P- fx(p of))DA ](a) = ((piof) x(p2of))(aa) = (p (f(a)),p2(f(a))) = f(a,). Thus, 1 f1 '2 f)DA = fI Suppose (uv)oDA = f. Then if a is in A; (uxv)(a a) = f(a)s so f(a) = (u(a) v(a)). Thus; (Plof)(a) = u(a) and (P2.f)(a) = v(a), so p of = u and P2of = v. Hence, (plf,p2 f) is uir.iue, By the front adjunction theorem, -.w then have that F is the left adjoint of T. (3.20). ProDosition. The functor U -= To(U2% U )U1 of 3.5 has as left adjoi nt the functor F -,-o (F2xF )oF L 2 3 4 where FI, F2, F 9 and F4 are respectively the left adjoints of UL, U2, U3 eundT. Proof. The left adjoints exist by 3.5 and by 3.20, and by 3,6. W'e refer only to the hypotheses of 3.5. The rest is straightforward and is left to the reader. (3.21). ,'?os,:to.~_ Let U_ be as in 3.5. Suppose f is a regular epimrorphism in Sgp(~ ) and g is a regular epimorphism in i implies fog is an epimrc:phism in M. Then U reflects regular epimorphisci'C Proof. .Lt (f'g) (Ri~,z)-->(SNzN) be a morphism in Lact(,). Suppose U(fg) is a regular epinorphisn in Sgp(N)x!. say U(f,g) is coeq((a,b),(a',b')) -where (a,b), (abt):(TL)-->(RiII). By proposition 3.12, (1, (1,z o(11))): F(R,~M)-->(B,Fz,) is a norphism in Lact(N). By 3.1i7 F(a;b): F(T,L)--,P(R, ) is also a morphism in L.act(N\). By 3.13 and by the proofs of 3.15 and 3.17, we have F(a,b) = (ah) where h = (u ob zo(al )o(10)(uMob))), where z is as described in 3l11, Thus, (1, (,z ))o(ab):'(TL)-- -RM,). (1, (19il))oI(ab) = (1, (i,z. )o(ah) = (a, (i,M)h). We wish to express (.,z )oh in a Cimpler form. (1,z-,)hou (1z )ou ob = lb = b. (Iz2)h)ohouT = (1, z)ozo(a1) (10(ui ob)) = (l,z )ozo(a,(u MOb)) = (,z ))oZo(@u u)- (ab) I M M M = (1, ,i)ou (1., 1,p l) (1Lo).d"-o1 (10u-) o(a@b.b) = z.o(1,p01) o(rlc<). u- (ab) = z, (1. l.p1)u o .. (aob) = z-..- (aSb) - z-(aeb). Thus, by the definition of coprocduct, we have (l,zH )h = (b,z Io(a@b)). Hence, (a, (b,z, (aob))):F(T,L)--> (P.BIIzi) is a i-c-'_'Il:: in Lct(N). Similarly, (a', (b',z o (a' ') ) ) :F(T,L.)--() ,MT z ) is a mor phi n in Lact(H). In tho above, p is the muLltiplication on R. foa = foa'. g'e(b,zN(ago))u = gob = gob = g(b',z .(a'QbP))-ur. g.(b,z,..(ab))tu--. = gzi.o (aGo) = ll L u TL, Mi zN (f-)o(ab) = zN((fa)o('f b)) ,oz ((foa) C)()-b')) ziN*(f03) (a ? ) gz o(a"'bI ) = G -(b o~, '(aROb' ) )ucoiL Thus, from the definition of coproduct, we have go(bz.o(aob)) Sgo(bez (a~@b~)). Tais implies (fg) (a, (bz i(b))) (f, )(a'(bOz o(a19b'))). Suppose (uIv):(RNz) --- (W,K,?K) in lact() so that v (u;v)(a (b,z (aOb))) = (uv) o(a', (b'M o(a'@b))). Then, in the same manner as for (fg). wes obtain uoa = uoa' and vcb = vrbl. Then there exists a unique morphism (u*,vi) in Sgp(T)xN so that (,v) = (u'v'0)o(fg). Since U1 is faithful, uniqueness in Sgp(N)YM implies uniqueness in Lact(N). Thus, it suffices to prove (u',v:) is a morphism in Lact(N) in order to prove that (fg) is the coequalizer of (a,(bz,, (aBb))) and of the morphism (a, (bqzo(a"b'))) which proves that U1 reflects regular epinorphisms. In the following iag--.-* portions (1), (2), and (3) and the outer rectangle coin:ute: BZ __I___-------------___ N. zMM, u (2) v fes z (1) WOK--- (3) uS 0--U W V ,- sNZN zN ThusE we have zzo(u'^s)o(,.-j) = v'ozNo (f@g). Our hypothesis then implics that f is an epimorphiim so z (u'-ov) = v..ozNo Consequently, (u, ,v-) is a morphism in Lact (N), This completes the proof. (3o22). Proposition. Suppose H and Sgp(N) have regular epinorphism, monomorpohism factorization of morphirrZas and that (4) in the hypotheses of 3.5 holds. Then UI of 3.5 preserves regular epi~orphisms. In fact, U1 raps cxtremal .... c~ U1 C c o x rema3. epimorphisms onto regular epimorphisms. Pr-oof. Since a rc_ i.ar epimorphismr is an extremal epimorphism, w-e mny prove only that U1 maps extremal epi- morphisms onto regular epi]orphisms. Let (fg):(RFM,zq ) -- SN,zN) be an extremal epinorphism in Lact(N). Then we have vob = g and uoa = f, where a and b are regular epimorphisms in respectively S3p(N) and M and u and v are monomorphisms respectively in Sgp(N) and M. Let dom v = B and dom u = A. Then (fg) is a morphisn in Lact(N) implies goZe = z N(fOg) so vobozg = zNO(uv) (adb). Theng by hy- pothesis, there exists z :AOB--B3 so that bz Z = z B(ab) and vzg = zN (uv). Let iu be the seiAgroup multiplication on A. We have the follo;-.ing diagravn: S (S)-----------------> SN t du (uitv) (2) C4A@C(A@3) 10ZB (ss ()u_, )( (1)) SNN S(4) o(5)te sc v (uv. ially ) cor(AA) z B mAOB------t--S n Then (1) commutes by the naturality of o. Also, (2), (3), and ) commute since voz z No(uiv). Finally, (5) cO-e I mutes since u is a morphism in S-p(N), T')n -e have that vozBo(ul0l)o V= vZB (1B) Since v is a monomorphismi this implies zB (:!) c = zBO(0zB). Thus, (A,B,zB) is an object in Lact(I)o Then we have (a;b) is a norphism in Lact(N). Also, (u,v) is a monomorphisr in Lact(Ni), But sinse (f, ) is an extreal eplmorphism, (fg) = (u,v) o (a,b) implies (u,v) is an isomorphism in Lact(N) so Ul(uv) is an isonorphica in Sgp(T)xRM, iut Ul(f,s) = U1(uv)oU,(a,b) and Ul(a,b) is a regular epimorphism in Sgp())xMl since a and b are regular epimorphisms respectively in Sgp(N) and ... Consequently9 Ul(fg) is a regular cpiiorphism in Sgp(N)x;i so U1 preserves regular epimorphisms. (3.23). ..roositio0 U1 of 3.5 reflects congruence relations. Proof, Suppose the following is a diagram in Lact (N): (PQz ) P 2 (R;z (P292) (f)s) (,I, (------S, N zN) Denote U1(P,Q,zQ) as (PQ) and U (p1 q1) as (pi, 1), and so forth. Suppose the folloulng is a pullback square in Sgp ()x .: (PliqJ (P, 9Q)-- >( 1M) (P25q2) (f(g) ( R )------ ->(s, N) . Suppose that the foll.ouus C-.rra.mi L con mu.bes in -eact(N): (P(P) _, ,ZQ.s)--- ------- ------- ,(RH,!i, zi~i) (P() (f,g)) 1 2 (R,, Ii,z ) Then we also have (fsg) = (p.) " Hence, there exists a unique (p,q): (P,Q' -->(P,Q) such that, in Sgp()x, (pi,q)o(p,q) = (p,), for i = 1,2. Since U1 is faithful, unimquness in S"p(")xi ir-plies unique-. ness in Lact(N). Thus, it suffices to show that (p,q): (P'eQOsZq )---(PQsz ) is a morphisn in Lact(N). Since (ql'q2) is the pullback in 1- of g -:lth itself and since goq : ZQ: = goqCozQ ,e have that there exists a unique r:PF~Q-->Q so that qo r = O!ozQ for i = 1,2. Ho-ever, qioqZQ. = qO! ,, i 1= 12. Also, q oz (pq) = z (p o(pq.) = z( o(peq) = qoz ,. Thus, qoz2, = r = zQ (p^). Consequ- --'-. (pi,):(', Q ,Z: )-- (PQoz ) is a morphism in Lact(). ".s, U reflects congruence relations. (3.24). P oz.-siion. Let (A,U) and (B,V) be algebraic categories so that neither U nor V assumes the empty set as a value. Then (;B, TTo(UxV)) is an algebraic category, where VT is as in 3.19. Proof. If F and G are respectively the left adjoints of U and V and if H denotes the left ad. joint of T (see 3.19), then it is easy to see that (FxG)oH is the left adjoint of I-(UxV). The r, -'inug properties are easily verified (3.25). 4P : Proposition 3.24 holds ,with "el. "bz'ric" replac by varietyt al. " ?roof. The proof is straightfor.a:'d and is left to the readers (3.26). Proof of 3.5. This is immediate from propo- sitions 3.6 and 3.21 through 3.25 and from the fact that since Ssp(N) ead .I are algebraic, they have regular epi-D morpnism, nononorphism factorization of morphisms (see [Il). (3.27). Cooll3r5 to 3.5. H, the category of all modules over all rings, is a varietal category. Proof. Let N = (Ab69,) wThere denotes the tensor product over the integers and o is the usual associativity transformation. Then it is well known that N is a multi- plicative category without unit. Also, it is well kno:.m that for any abelian group E, OE and EO_ preserve finite (in fact, arbitrary) coproducts. ( For details about the tensor product, the reader may sec C6].) It -s easy to see that Sgp(N) is isomorphic to R and tht. Lact(!) is isomorphic to Ri. Since isomorphisms pre- serve all the properties concerned, we will check in LM R- the hypotheses to 3.5. (1) Let f be a regular epimorphism in R and g be a regular epimorphism in Ab. Then f and g are onto functions so f6g is an onto function, Hence, f0g is an epimorphism, ( For background ma.terial on R, see [7] .) (2) It is well knon.m that Ab and R. with their forget- ful functors are varietal categories and that their forget- ful functors never assume the empty set as a value, (3) Let (ftg):(R,K,. )--(S,N,-:,-) be a morphism in RM. It is easy to short that the following is a congruence relation wherc pi and qi are the natural. projections, i = 1,2, and where wee use the induced module multiplication on the group (P19qI) (mm'i (m) = n m= mR)}.m ((O.Tr')lf(r) = f(r')}, (mn') g(m) = gda')},* IRM,*) (P2 q2) (fg) (R, ) ---(S N, ) The details are left to the reader. Now we construct coequalizerso Let (f,g) and (f',g) taking (RiM,*) into (S,N,-) be two mcrphisms in i Let N' = g g(m)-ge (m)I mleGk. Let I be the ideal of S generated by &f(r)-f'I(r) rc}. It is easy to see that p:S-->S/I is the coequalizer of f and fY in RH where p is the natural projection. There is an induced module multiplication for (S/i 5/(N',+IT+SN' )), Let q:,N-->N/(1 +IN+SN' ) be the natural L!.p. By def nition., of the .induced multiplication, (psq) is a morphism in R. It is clear that (pq)o(fg) (pvq)o(fSg ). Suppose (k,t).(f,.) 1= (kt)O("P,), where (k,t): (S,N,-)---(T,L,). Since p = coeq(f,'), there exists a unique ring morphism k* so thai k"op = k. Define t*(q(n)) = t(n). To see that t-: is well-defined, suppose x is in Nc, IN, or SN'. If x is in N', then x g(m)-g' (m) for some m, so t(x) = t(g(m)-g' (m)) = t(g(m))-t(gr (m)) = 0. For each r in R, t((f(r).-f(r)n) = k(f(r)-.f(r))t(n) O't(n) = 0. Since I is generated by (f(r)-f' (r) rE , t(x) = 0, for each x in IN. t(s(g()-g(n'))) = k(s)t(g(m)-g()) = k(s).0 =0. Thus. for each x in SN', t(x) = 0. Hence, t* is well-defined. It is clear that t* is a morphism in Ab an it is left to the reader to verify that (k*,t*) is a morphis-i 1l. Since p and q are onto, (k1:t*) is unique Thus, (p,q) is the coequalizer of (f,g) and (f-,g'). (4) Suppose' a and b are regular epimorphisms respec- tively in R and Abo Also, assume that v is a monomorphism in Ab. Then a b, b, nd ab are onto and v is one-to-one. Suppose the following diagram in Ab commutes: a~b UBV A'B' a.b M03 Au3v A" B" z B, z B b v Define zB :AO-->3 as follows. For x in AB, asb is onto implies that there exists x' in A'OB1 so that (a@b)(x') = x. Define zB(x) = (bozB,)(x'), It is left to the reader to show that zB is -ell-defined, is a morphism in Ab, and! that the required commutativity holds. Thus, all the hypotheses of 3.5 are satisfied. Hence, by 3.5;, is a varietal category. (3.28). G- erali.io's and Fnrth-r Results. We have obtained a generalization of 3.5 which allows us to deduce that certain subcategories of Lact(N) are variety] categories. Due to time pressures, we are unable to include the details. However, we would like to list a number of well kno-r cat- egories t:hich are varietal. The proofs of these assertations are along the same lines as the proof that RIM is varietal. (3.29). Va-ietj. C .- copes, The following categories are vartital: (a) the category of all not necessarily unital modules over all rings having an identity, ere ring morphiss preserve identies and. morphisms of the category are analo- gous to mor-phisis in (b) the category of all monoids acting on pointed sets, not nece:- --ily in a unital fashion but so that if (M,(X,x),-) is an object in the category, then 1.x = x, where morphiism in the category are analogous to morphisms in RI. anY (c) the category of all compact, Hausdorff monoids acting on compact, Hausdorff spaces analogously as in (b). It is our hope to publish at a later date a complete proof of 3.29. (3.30) Ca .'--.: Whch A- Ar:ost Varietal. Let C be either the category of all co pact acts (see 1.17) or the cp-'- cf all s- groups acting on sets. Let U: C---insxEns be the forgetful functor. Then U satisfies properties A2 A3, and V in definition 3,2. Furthermore ~ C has congruence relations and coequalizers. Proof U has a l.eot adjoint by 3.6, We hope that the recan.iier of this proof will be published at a later date. DIi to time pressures, it is not given now. BI BLOGRAP-HY 1. H. Herrl.lch, -.-tr.n ca- zoeies:;i- an a orati gpor8'D.,c to appear. 2. E. M. Norris, S, structure thlcrns for to olozical machimnjs, Univer-sity of Florida, 1969. 3. B. litche!lls Theory oi cf a- ores, Acad eic Press 1965. 4. De Husscoller-,; Fib:re buindies, MIcGraT:-ill Boo Company, 1966. 5. S. IMacL.~cnec C-t ,Ical al c 7-., ba, -NSF.L Advanced. Science Seminar Lectures at EBo,.oin College, 1969. 6. C. W. Curtis and I. Reiner, RLo''ratJ.son tteo" rr. of fiite rmo-. -o. I' `r '- R S Inters ci'ce P.blishiers 1962. 7< K, Koss 033, i '' ." 1 O Urnivorsity of Florida, 1969. BIOGRAHI CAL S:' TCH Stephe.n Jacikson :~ -caTel was born September 21, 1945, in Plant City, Flori.da. In June, 1963, he graduated from Plant City Senior High School. In June, 1966, he received the Bachelor of aArts in mathematics from the University of South Florida0 In August, 1967, he received the Master o Arts in mnthenatics from the University of South Florida. From Septemibera 1967, until the present time, he has studied at the Universit-ly of Florida toward the C.l :' c- of Doctor of Philosophy. He has held throughout his stay at the University of Florida a NASA Tralneeship, During the surer of 1969, he attended an I". Advanced Science Seminar ian Category Thoo-ry at Eo;-:doin Colle e in iMaine. He is a monb-r of the ir '"i. n I-aths',atical Society. This disse:rtation ivras prepared under the direction of the cha rnai. of. *th cr-.d.idate s supervisory conmaitte and has been approved by all inebers of that comnuittes, It was submitted to the Dean of the Colle-e of Arts and Sciences and to the Graduate Council: and w;as approved as partial fulf ilrlmnt of the rqu.r-nents for the degree of Doctor of Philosophy. Augiust, 1970 D Collee oT'ts and Sciences an. a z ,1;.t e cSchool SupC..vi a. c:y Co&ii:';;lee: Cha5. iiL -1c;- /j1 |

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PAGE 1 CERTAIN WELL-FACTORED CATEGORIES By STEPHEN JACKSON MAXWELL A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTUL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1970 PAGE 2 UNIVERSITY OF FLORIDA 3 1262 08552 3073 PAGE 3 To my mother and to the raamory of my father PAGE 4 ACKNOV.'LEDGI'ESNTS The author would like to express his sincere gratitude to his director, Professor V/. E, Clark, for his mathematical assistance, his patience, and his generosity in giving aid on numerous occasions. The author would also like to express his appreciation to his roommate, I'lr. Burrox-j Brooks, for the loan of his typetrrlter. iii PAGE 5 TABLE 0? CONTENTS Page ACKNOl'JLEDGMSNTS .<,Â».. ..*<>< <. ..oc.o. iil LIST OP FIGUR3S , . . c e v INTRODUCi lOri Â•Â•>cat9Â«Â«c*Â«eÂ«*o>Â«<*ee*oÂ«Â«Â»Â«et'C*Â«Â»Â» 1 Chapter II. TWO EM.B3DDING THJ^ORKi'IS ..,Â«..Â„*,. 1? III. A SUPFICIEWT CONDITION FOR A VJELLFACTOEED CATEGORY TO BS VARIETiil. ..... ^0 iv PAGE 6 LIST OF PIGUR2S Page DI AGR AI'l (0') ,Â»,,Â«,cÂ»...Â»Â».Â»Â»Â»eoÂ«Â«Â«Â»Â»Â»Â«*Â»<=Â«*Â»''*Â« ^^ DI AGRAIi (1*) ,.Â«eÂ»eÂ«..Â«ee.Â».Â«.Â»Â«<>Â»Â»Â»Â»Â»Â»Â»eÂ»eÂ«.Â«Â«Â»Â» '^7 DI AGRAI'I (5*) Â•Â»Â»<><Â•. i>Â»Â»Â«ecÂ»eÂ»Â»cÂ«cÂ«Â«Â»i>8Â«cÂ«s*Â»Â«Â«9cee ^I'O DIAGHAI'I ( 8 a) .c.Â«Â»Â«oÂ»Â«c..Â»ii .Â«.o9oÂ« 30 DIAGRAM (8B) .Â«i:.6<.ee.eoÂ»ceÂ» 9.,.oÂ»veÂ«>eet>.Â»Â»Â«e 3- PAGE 7 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CERTAIN WELL-FACTORED CATEGORIES By Stephen Jackson Man-fell August, 1970 Chairman: Professor W. E. Clark Major Department: Mathematics A nev7 kind of category, called a vjell-factored category, is defined. This is a generalization of the category of all modules over all rings, the category of all fibre bundles with fixed fibre, and the category of all topological acts.. Two structure theorems are proved for vrell-factored categories. One is an embedding into a "product" category and the other is an embedding into the category of all left actions in a multiplicative category without unit. These embeddings preserve the factorization of morphisms given in the definition of a vrell-factored category. A sufficient condition for a category of left actions to be a variety, as defined by Herrlich, is given. As a corollary, it is noted that the category of all modules over all rings iy a variety. vi PAGE 8 INTROrUCTION Let R 02 a ring and K be the category of all loft R-modules. Th3 study of th3 catcjgory Â„M is already vrell developed and femilj.ei-. The author sot out to study the category of all nodules over all rings, denoted _,1'I. Thus, eaoh jjM is a subcategory of H and Diorphisms in _M are semilinear transformations (f,g); i.e., f is a ring honio-. morphisuij g is a group homomorphism, and g(rm) = f(r)g{m), for each r and m. The author then noted that numerous categories such as the category of all topological acts, the category of all compact acts, the category of all fibre bundles vdth fixed fibre, the category of all ncnoids acting on sets, and the category of all biinodules over all pairs of rings possess properties similar to those of Â„Mo Most importantly, in all these categories, ve may unic/uely factor each laorphism as the composition of tvro inorphisas belonging to corresponding classes of morphisms. Roughly speaking, for pK, v:q have "module" morphlsns of the form (l,g) and "ring" morphisES of the foi-in (f,l). These examples and the generalizing concept of a X'Tell-factored category are presented in Chapter I, In Chapter II, we prove tvro structure theorems for v:ell-'factored categories. V/e show that a TJsll-factored category may be embedded in a very nice way into the "product' PAGE 9 2 of three cateeories. Then we shovr that a ^rsll-factored category may be embedded in a very nice \-ia.y as a subcategory of the category of all left actions in a EOiltiplicatlYe category without unit. Herrlich introduced in [jQ an axicmatically defined varietal category which generalized the "varietes" of LaTivere and Linton. v;e gi'v^e in Chapter III a sufficient condition for the category of all left actions in a multiplicative category without unit to be a varietal category in the sense of Herrlich. A corollary to this result is that the category of all modules over all rings is a variety. Vie have a generalisation of this condition and list some corollaries to the generalization but, due to the pressures of time, we onit the proof. We hope to publish the complete results later. An iraportant part of the proof for our condition given in Chapter III is that the forgetful functor U:Lact(N) > EnsxEns has a left adjoint. We vrere partially motivated by the construction of a left adjoint for U when N is Ens with the product fu.nctor and the usual asGociativity transformation. This construction v:as given in Q2I]. PAGE 10 CHAPTER I PHSLIMINAHIES We will use the notation of CJJ throughout, unless stated othervrise. Also, familiarity on the part of the reader vrith Chapters I, 11, and IV of C3 ^ will be assumed. In this chapter, vre vrill state the definition of a vjellfactored category, give some examples of well-factored categories, and state some knoT-m results which are not assuiiiod to be background knovjledge for the reader. (i.l), Rgiaai^, For the follovring definitions j let C be a category and let P. send P^ be tiro ''"classes" of subcategories of C such that each object C in C is Bjn object in exactly one cieiaber of P. and in exactly one nainbsr of P^. Let f be a morphism in C. Then f is said to be a P -laorphism if and only if f is a morphism in some category in P^, i=ls2. Note then that the composition of P, -morphisms is a P, -morphisn, i=l,2. (1.2)c Notation. Ab will denote the category of all abslian groups, R will denote the category of all rings which do not necessarily have an identity, R wilD. denote the category of all rings having an identity and all identitypreserving ring homomorphi ams . Ens is the category of all setSc (1.3). Definition. I^t N =L/IIorA and M =^ U KorA or ~" "" A e P^ " " A e P^ ~ PAGE 11 vice -versa. Let f and. g be inN, say f:AÂ— >3 and gsA*-^BÂ». We say f and g are srid, connected by M-riorphlsms if and only if there exist sequences S.jc,.,S of N~morphisms sach that each S. is not the empty sequence, the elenents of each S, need not be composable, S., = (f)^ ^-j^ = (Â£)s and, for each i Ij.e.-n-l, one of the folloT\ring (or one of the follovjing with S. and S, ,. interchanged) occurs: 1 I'. 1 (1) Sj^ = (Sj^,jÂ«Â»Â»s St,.|.< j Sjj j S]^-'1 9 Â• ' * j S-j_ J where 'g,, indicates gj^ has been deleted from the sequence. Also, g, is an identity such that if N is the collection of all n, -raorphisms, then cod Sj^^^j ^om g^, and dom g]^.uj_ are all objects in the saroe P^ -subcategory of C, i-1,2 and i 7^ j. If k := n. 5 then we have the above conditions for k and k"l. If k ~ Ij then we have the above conditions for k and k-i-l. ( 2 ) S^ (g^. 5 . c J g].--i.1 ) SkÂ» S],-;Â„l J Â• Â» Â• 5 C^ ) '-'j.j.;]^ ~ ^gp. S Â» Â• ' Sot, r ^ JÂ« -'Â• sg],-^3^S Â• Â• Â« 5o^ / Where f^f * is defined and is equal to g-|,_. i -M ~ ^ ^hi-] s " Â» Â» s ^k+1 ' * ^kJ s Â» o ? S'l / and there exist W">inorphisi>is iv. and lo.' so that la'of = g-^^j^m^ i^ees the following diagram comicutesj PAGE 12 (1) If f^ and f2 are p. -norphlsms as indicated bysubscript, then fi''f2 ^^" ^ Pj_-morphisin implies f j is sii identity, v;here i r' jÂ» i Ij?-* and j =Â• 1,2. (2) If f^ is grid connected to g^ by P^-norphisms and f 2 ts -rid comiected to Â£2 ^^ P^-aorphisns, i-rhere A-4ii-^S and A-^^L^S, then f2Â°"l ^ S2''Gl' (3) For each morphlsm f in C, there is a unique factorisation f h^s such that g is a P. -morphlsm and h is a Pp-morphism. We villi denote this situation by C =^H, X' Pg* (1.5), Exarnp]^. Let S b3 a monoid. VJe may consider S as a category in the usual manner. If S is -irell-factored by p. and P^j then the cardinality of P is one, 1 1,2. Also, S is isomorphic to P^ X P^, where P. is identified V7ith its member, for 1 1,2^ and X denotes monoid product. Proof. We must put off the proof that this example is true until the next chapter. (1.6). ExaraÂ£lÂ£. LÂ£!t G be a group. Then G may be considered, in the usual manner, as a category. Suppose G is v;ell factored by P and P^. Then the cardinality of p. is one, for i 1,2. Identifying V^ with its member, for i = 1,2, we have G = H^^* '^'^hex-e X denotes the direct product of groups. ^qcf . We must put off the proof that this example is true until the next chapter, ^^'?)' Definitioii. Let Â£ , C ^, andC^, be categories. Define the category "^X^'cY^ifC'^ as follows.the object PAGE 13 class of C X C X C is equal to the object class, of X tL J C, X C,, X C^. Also, a morphlsm in C. ;< C ,. x^ o is of the form (f5s)s(G3_,C2,C3) >(C5 ,C^,C.^) , where fjC^^ >CJ^ is a morphissi in C. Sjicl gsC->C* is a morphism in C^. It is clear that "C . x C ^ x G ^ is indeed a catesor,],'-, L w _y (1,8). PiÂ£oso.?AtviÂ£?iÂ» I'2t C^, C2Â» ^-'"^ ^-^ ^-"^ categories. Let C^ fcs an object in C^, Let P^ ( Q ) be the subcategory of C ^ x~l' 2^2 o Si"cen by s Obq(C^) = {^(G;^,C2sC3)| C\ 6 OtÂ£^, i = 2,3}= and Morq(Ci) = {(C^jC^sC^) :Â• -MC^CJ^O* ) | g:C^ Â— ^Cg is a C2-norphisin and 1 IcJV. L^^t P-^ "(^P^^CC^^) | C^e OcG^V. Define P2 analogously! i.sÂ«} define P2 by restricting the second coordinate. Then any subcategoryA of C^ x ^2 ^ ^3Â» in vj-hich (3) of 1,^holds and such that (Ijl) ? (A|j_5A2,Ao)Â— > t (A-|^sA25A-3) is an A^iiiorphism, implies A3 = Ao is vrell-factorod by the restrictions of P^ and (2 to A. \le call this the standard wellfactorization of A, ^;oof_. Let Abe such a category. It is clear that each object in A is an object in exactly one member of P-^ and in exactly one member of P2. B'irst, vie prove that (1) of 1 . ''-Iholds in A , Let ( 1 ;^) 5 ( B^ , B2 -, E-.. y Â— >( Bj_ , A2 ; A-, ) and (jB,l) J (0^,32,03) 5(3^,B2,B PAGE 14 (A^.Ag.A ) -Â± >(A^,E^,B') 2 Xc^jB^jC^) where (Ijf^) and (1,0-j) are grid connected by fl-siorphl sns and (fgjl) and (gÂ„s1) are grid connected by n-raorphls^is. It suffices to prove f^ = g^ and f^ = gp^ We will prove f^ = g^ since the proof that f^ ~ gp is analogous. VJe use the sequences of f^-Eiorphisms from 1,3, where S. = (f.) and ^n ~ (S;j^). We vLlll prove the following inductively: if ^i " (S^^,...,g^), then g^^ = (i,hj.^. ) , . . . ,g^ = (l.,h^), V ""^'1 ^^ defined and if S^^^.^ =. ^^^.. ^^ Â° Â« 'S^) , S^^^^ = ^-s^nÂ« ,.)'Â•Â•Â• jS-i '-'' (Ijh.), then h^ => Â•"oh. h^^ * " Â°h. , for i = l,..o,n"l. Suppose i i. By 1.3Â» we have three possible cases. (1) S^ = (dsf^)) and Sg = (d^Dsd^f^)) or S^ r. ((l,f^), (1,1)). Then cod(l,f^) and doa(l,l) are in the same P^-category implies coddjf ) domd,!) = codd,l). 1 Thusj the inductive statement is true here for i = 1. (2) S^ =-(dy f^)) and S^ = ( d,f ) , d,f Â» ) ) , where . d,f)od5fÂ«) -.-. d^f^). Then the inductive statement for i = 1 is obviously true here. (3) S^ :(djf^)) and S^ ^(d,h)) where there exist Pg-morphisBs (nijl) and (m%i.) such that (E.',l)Â«d,h) = djf ^) oC'^'jl) , Thus, h f., so the inductive statement fo2.^ i 1 is true here. Going from S^ to S^^.,.^ is quite similar to the above. Thus, f = g. . Similarly, i' g^. Hence, (f ,1)0(1, f.) --= (S2s 1 ) " d i S^ ) . Thus , A is vrell~f actored. (1.9). E:caiao],e^. V/e now exhibit a product category PAGE 15 8 such that conlition (3) of lA does not hold in all of its subcategories e Let C be the full DUbcategory of Ens x Ens = Ens X Ens X irhose objects are pairs (X,!) of sets X and Y which have the cane cardinality. Let 9. have the standard well-factorisation. Let inc;<{i}Â— ^ilt^y be the inclusion function. Then ilyiOylna) %{il},^iy) ^>('{ls2>, Â«(l,2>) is a C-morphism. Eoiiever, it is easy to see that this has no P. " rÂ„ factorization. (1.10). Example. He now exhibit a category C x pTx~^o V7hich is not Trell-factored in the standard way. Let Â£^ ~ Co = C^ Ens, Then it is easy to see that the morphism (i,l)sh^l>. PAGE 16 (2) If A = R^, then D:, a] is denoted by ^R^. (3) If A = Top, the cateGorjr of all topological spaces, then ^,4] is denoted as Rm, the category of all bundles. (1.12), Exe22iÂ§.' ^'^^ ^K be the category of all modules over all rings. That is, an object in M is a triple (R,M,p) R"~ where R is a ring, 14 is an abelian group, and p:R y l PAGE 17 10 (1.13). Exanrole . Let pMp^ be the category of all bimodules over all rings. An object (R,I-ijS,pji jp ) in ^K^ ^^ such that R and H are rings, K is an abelian group, p^: RxI'I Â— >II, pgiM^S Â— ^H, (R5ll,p]_) is a left R-module, (M,S,P2) is a right S-nodule, and, for each r in R and m in M and s in S, \:e have (rrri).? = r(ins). A morphiam in ^K-^ taking (H,K,S) into (R',M',SÂ«) is a triple (f,Ssh) so that fsR Â— ?RÂ» is a ring morphisni, hsS Â— >>3' is a ring morphism, and gsM Â— >i'i' is a group morphism so that, for eac?i r in R and n in M and s in S, g(rn) = f(r)g(m) sjid g(Lis) g(m)h(s), VJlienever possible, we denote (R,MjS,p^,p2) by (RjKjS). VJe define H, -subcategories analogously to H. -subcategories in 1.3.2 by holding the rings constajit. We define r2-subcategories analogously to P -subcategories in 1.12 by holding the abelian group constant. Then jdH-j = V-^ x V2* j^q of . This proof is analogous to the proof of 1.12. V/e change notation from (RjHjSjp^jp^) to ( {R5S) ,H, (pj_,P2) ) and froL(f,g,h) to ({f,h),g). (1.1^). Example. Let ^M-o be the full subcategory of jjMjj determined by the objects of the forM (R,M,R). Then the vrell-factorlzation of ^Kjj given in I.13 induces a wellfactorization of ^Ijj. ^oof. This is a straightforward application of 1.8. (1.15). Examples. We define the categories ^-.M, ^K 1, ^^^ rI^rI analogously to respectively ^M, ^^, ^K^ by replacing R with R^. V/e also obtain well-factorizations PAGE 18 11 of these catesories in aii analogous fashion. Finally, K^ and, K^l ai^e ;;ell~factored, where no;r v/e have right nodules rather than left modules. &2Â£ilÂ» It is clear that these examples are valid. (1.16), gxampie. Let C te the category of all topological acts. An object in C is a triple (S,Xsp:Sj^'X Â— ->X) such that S is a Hausdorff topological senigroup, X is a Hausdorff space, p is continuous and gives the multiplication, and, s and t in S and x in X, {st)x ^-s(tx). VJe denote (S5X5P) by (S,X) irhenever possible. A morphlsn in C taking (S,X) into (SÂ»,XÂ») is a pair (f,g) such that f;SÂ— >3Â» is continuous and preserves the semigroup multiplication, S'X ^^' Is continuous, and, for each s in S and x in X, g(sx) = f(s)g(x). C is a subcategory of C -j^ x Â£2 x C^ where C^ is the category of all topological semigroups, C^ is the category of all Hausdorff spaces, and C^ is the discrete category having as objects all pjS-^^X Â— ->X so that p is continuous and S and X are Hausdorff spaces. Then C is well-factored by the standard ifell-factorization. ^.f^^t' This is an easy application of 1.8. (1.1?). Kxs^mie_, Let Comp C be the category of all compact topological acts. That is, Comp C is the full subcategory of C determined by the objects (S,X) in C such that S and X are compact. Then Comp C is well-factored by the factorization induced from C. ^2Â£Â£.Â« We note that the induced factorization is just PAGE 19 12 the standard T-rell-factorl nation of Coap C. V/ith this fact, the proof is an easy application of 1.8. ^-Â•i^')Â« ^^^^2lS.Â« If G is a topological group, a right G-s^aoe is a topological space X along with a map XxQ >X, The image of (x,s) is denoted xs. We require the follovring: (1) For each x in X and s and s' in G, x(ss') = (xs)sÂ». (2) For each x in X^ xl = x, where 1 is the identity in G. A G-space is called efXÂ§CiJAze provided xs = x implies s 1. Let X-= {"(x,xs)eX> PAGE 20 13 it suffices to prove that condition (3) of 1.^:holds. Suppose (u,f)j(X5p,E) ->(Z',pSB'). Then (l^f ) s (X^p.B) y (X%fop,3') and (u,l):(:{,fop,E') >(X',pÂ»,B') and (X,fcp,3') is an object in PBan. The rest is clear. (1.19). SxaEple. For further details about the category of fibre oondles and about 1.18, tlia reader should consult [t] . iv.To elements x and xÂ» in a G-cpace X are said to be G-equivalent if and only if there exists s in G such that xs = x'. This is an equivalence relation, so we may form the quotient space, denoted X/G. Let d =(X.p,B) be a principal G-bundle and let F be a left G~space. The equation (x5y)s = (xSjS"^^) defines a right G-space structure on Xx?. Let Xp denote the quotient space (X.-?)/G and let pptXp >B be the factorization of XX? Â— 2Â£.>C_vX Â— P_>3 by the canonical map X^^F 5-Xp. Thus, for (x,y) in X,\'?, Pp( (x;y)G) = p(x). Then (Xp,pp,E), denoted d[FL)s i^' the fibre bundle over B with fibre F (vievred as a GÂ»space) and associated principal bundle d, A fibre EiL^.^lism from d[?3 to d'[P3 is a bundle morphism of the form (up,f):d[?0 ->-'[fJ, where (usf)jd Â— >dÂ» is a principal bundle morphism Â©jid Up is obtained as follows s u:X Â— '>XÂ» induces a G-morphism u%l:X>c? Â— >XÂ»xP end Up is the induced quotient map from (XkF)/G to (X'>c.v)/G, Let BanLF] denote the category of all fibre bundles with fibre F. Let Bin[Fj^ be the subcategory obtained by holding X constant in d and BonLFJg be the s-ubcategory obtained by holding B constant in d. Lot Pj be the "class" of all Fan[?J-^ and let PAGE 21 Pg te the "class" of all EonCFOg. Then Bin[P] P^ X H,. ^Â£Â£PXÂ« ^^^ 3.n I.I85 it suffices to prove (3) of Uk Suppose (up,f):(XXF/G,pp,B) (X^x?/G,pÂ»p, BÂ» ) . Then we have a unique [\f^ factorization (Up,f) (UpsD r i' as soon as vre pr-ove that (Xpfp'ou,,, B' ) is a fibre bundle. Hovrever, this is a siEple consequence of the fact that vre Bxe working with quotient Raps and the details are left to the reader. (1.20), Path Categ^ories^. Recall that a graph is a class of objects J a class of morphisms, a domain functions and a codonain function; i.e.j a graph is a category tjithout composition. Thus, there is an obvious forgetful functor from the category of all categories to the category of all graphs. ( VJe should point out that a morphism of graphs consists of a pair of fimctions (F^^P^), so that for graphs Gj and G^, P^:ObGj Â— ^ObG^ and FgtHor G^ Â— ^Hor G., so that if g is a G^-norphisn, then F^^Cdom g) = dom FgCs) and P-(cod g) = cod F2(s).) This forgetful functor has a left adjoint; i.e., free categories (over graphs) exist. (These are also called Â£ath categories. ) This construction is done by using finite sequences of "composable'arrows in the graph as morphisns in the free category and juxtaposition as composition. Given an object A in the graph, ( )j^ (the PAGE 22 15 ecipty sequence from A to A) is the identity morphism on A. The path cateÂ£ory on a graph G is denoted Pa G. Tvfo systems of notation v:ill be used in path categories. We may denote norphisns pictorially as AÂ— 5l->3-^-2->C-^i3. . .-^-pD or as sequences as is^^, . . . ,Sj,s^,s^) , Note thsit vjhen raitten as a sequence, the morphisms are irritten in their *'composable" order. The identity norphism on A may be denoted as A-1-1a^A or just by ( ),. We will VTrite A-^B-LK B-^C as (s,f). (1Â»21), Categorical Relations. Let C be a category and let E be a relation on Mor C. Then E is called a 23^MS23L\^Â§1. Â£Â£lsMÂ£^ if and only if f E g implies dom f = dom g and cod f = cod g. Also, E is said to be corapatible if and only if (1) f S g; hog defined, and hof defined implies hof E hog and (2) f S g, fÂ»h defined, and g-h defined implies foh E goh. It is clesTthat U(ho!n(A,B) x hom(B,A)) is a categorA, B e ObC ical, compatible equivalence relation. Since the intersection of any family of categorical, compatible equivalence relations is a categorical, compatible equivalence relation, any subset of (J (hom(A,3) xhoiTi(B,A)) generates a categorAjBeObC ical, compatible equivalence relation. Some sources refer to a categorical, com.patible equivalence relation as a congruence relation. Finally, if E is a categorical, compatible equivalence relation, then we may form the quotient category of C by E, denoted by C/E, as follous; Ob C/E = Ob C and Kor(C/E) - PAGE 23 16 (Mor C)/E, dou[f] = don f and cod[f] = cod f, ana [f] o [g] = fog, vrhere x:e vrrlte a morphisni in C/S Trith representative f in C as ffj. PAGE 24 CHAPTER II TWO EHBSDDING TIIHORSMS In this chapter we show the equivalence of the general concept of well-factored category to tvro special kinds of well-factored categories. (2.1). D|finitlÂ£n, A nultiplicative catep^ory without unit is a triple N = (MjÂ©,^) such that! (a) M is a category, (b) Â®sK X K Â— ->I1 is a blfunctor. (c) For all objects L,M,N in K, '<^ j,^ j^:LÂ® PAGE 25 IS ring with identity. Let g) he the usual tensor product of modules o-ver R. Let Â« be defined by: lS>i:a0n)\ Â— ->(-10u)0n. Then conditions a, b, and c hold but d fails, if char R j-' 2, ^2.3). PjIinU^iSS.' A send^roujD in N Is a pair (S,u) such that the folloulns hold: (a) S is an object in M where N = (KjfS't'x), (b) SxS-^.S is a morphism in K. (c) The following diagram commutes: lÂ®u S0{S0S)-&S0S 5(y.:; )(g)S uÂ©l S@3u u V/henever possible, we denote (S,u) as S. ^2.^). iMiS^tioii* If M is a multiplicative category without unit, then Ssp(N) is the category of semigroups in N. A morphism in Sgp(N) from (S,u) to (T,v) is a morphism f :S Â— >T in K so that the following diagram commutes: u S03f0f -?S \^ V TÂ®T-^T Composition is induced from M. It is clear that Sgp(N) is indeed a category. (2.5). EsCi^iiiciS' Let (S,u) be a semigrou-o in N = (MsÂ®,^0 and let K be an object in H. Then a lej;t aotdon of (S,u) on ^'^ in N is a triple ( (S,u) ,M,SÂ®M-^->H) where z PAGE 26 19 is a morphism in H and the follo-.)-ing diasram commutes: SÂ®(S0M)^SÂ®M Â« {203)0^1 U01 S^l'r Â— __ J2,j 2 V/henever possible, ire ttIII denote ( (S,u) jHjSÂ®::-^^) by (S,M,z). (2.6). Definition. Lact(N) is the category of all left actions in N. A morphism in Lact(N) taking (S,M,z) into (T,N,z*) is a pair (f,g) where f:SÂ— ^T is a morphism in Sgp(N) and gsM Â— J-N is a morphism in H (N = (K5Â®,c<) ) s^d the follo'/iing diagram cojonutes: S0I f^g S line composition is induced from H. It is clear that Lact(N) is indeed a category, (2.7). 2ltiSijyi2a' I^t M = (iiiÂ©5'=<) be a multiplicative category vrithout unit. N is said to be exact if and only if o< is alirays an identity morphism. (2.8). St;:op^sitiOTL, Let C be a category. Then there exists N, an exact multiplicative category without ujait, such that C is isomorphic to a subcategory of I^ct(N). ^2Â£tÂ« I'-'t li cs the category of all endofunctors of C. PAGE 27 20 Then N = (M,conposition, equality) is an exact multiplicative category without unit since the interchange law says composition is a bifunctor taking UxE into M. Let C be an object in C. Then let CjC Â— >C be defined by C(A) = C, for each object A in C, and C(f ) = 1^, for each morphism f in C, Then C is an endofunctor of C. Let K be the identity functor on C. Then it is easy to see tliat (K,!^) is a semigroup in N. Define F:C Â— >Lact(N) by F(C) ((K,l..),Cs 1-j). The following diagram coramutesj Ko(KoC) ^ ^ 1 ->K*C ^0 ->C This shows that P(C) is an object in Lact{N). If f is a morphism in C, then F(f ) =. (if) ,,here f --= {t > and ^ ^ A AeOb C ' foreach object A in C, t^ = f . it is clear that f il a morphism in M and that {l^J) is a morphism in lact(N). Thus, F:CÂ— >Lact(N) is clearly a functor and is one-to-one on objects and on aorphisms. Now we show that im(P) is a subcategory of Lact(N). Suppose P(f)oF(g) is defined. Then cod F(g) = (K,B,1-) and dom P(f) =. (K,C,1~) and cod F(g) = don, F(f), Thus, B = C so B = C. Eut then cod g = B = C = dom f so fog is defined. Hence, F(f)oP(g) = F(f,g) g^ im(F) is closed under composition. Therefore, im(F) is a subcategory of Lact(N) so C is isouorphic to im{F). PAGE 28 21 (2.9). Remark . Although proposition 2.8 is a very general embedding theorem, it seems to have very little practical use. We will prove later an embedding theorem of ivell-factored categories into categories of left actions so that the embedding preserves TL -factorizations. This is a much more useful embedding. However, we will first prove an embedding theorem of a somewhat different nature, (2.10). Theorem. Let C be a well-factored category. Then there exist categories C. $ C,^? Â£-'i C such that C 1
PAGE 29 22 (2.13). Proof of 1.5. Define H:S Â— >P^ >c f^ by H(s) = (s^jS^), tThere s = s^^s^. (v/e are using subscript to indicate r^^-norphism, ) By 2.10, there exist categories C^s C2, and CÂ„ such that ire have an embedding .F:S -> ~l^-2^% ^"^-^^ presei-ves HL-morphisms. Thus, F{s^) is a r^-morphissi and F(s^) is a ri-morphism; i.e., FCs^) = (a,l) and P(s-^) = (l,b), for some a and b. The we have F(s2)Â°F(s^) --(a,l)o(i,b) = (a,b) = (l,b)o(a,l) = F(cJop(s ). But F is one-to-one on morphisms , so Sp^s^ = s-^s . \Is know H is well-defined since the r^~ fl factorization of a morphism is unique, I/st s and s* be In S. Then se>sÂ« _ (Sp^'S. )o(s*Â«s' ) = S o (3 os')<Â»S* = S o(sÂ»os )<=S*= ^ ^i 212i 22 11 (s^-spoCs^osp. Thus, H(sosM = (s^os|,S2Â«'s') = (33^,82)0 (sÂ»,sp = H(s)^K(sÂ»), so PI is a functor and a monoid mor-phis:ii. H(s) = H(s') implies s = 82*3 = sÂ» so s = s*. Thus, H is one-to-one. Let (s.jSp) be a>i element of f] ^' To. Then So^s. "is ^ J. 2 iC 1 ~ in S and HCsg^s^) = (5^,82). Thus, PI is onto. Since Pi and p2 SJTS subcategories of S, P^ and P2 are submonoids of S. Hence, H is an isomorphism and so S is isomorphic to Px H. ^2.1^-). Sioof of 1.6. We will first prove that P^^ is a normal subgroup of G. Since P^ is a nonempty subcategory of G, 1 is in P^ and P2 is closed under multiplication. Let g2 t)e in P^, Then g'^ = ^a"^'^!' ^''"^^Â®--Â® ^2 ^"^ ^1 ^^^Â® unique elements of P^ and P^ respectively. Then 1 = g2''S:^ = g2Â°(h2''^l) = (S2*'J'i2)Â°^i = lol. By uniqueness of P^.>P factorization, h^ 1 so g^^ = hg and h^ is in T^. Thus, PAGE 30 23 P is a subjro-ap of G. Let g be in P and g bs in P . Then by an argument similar to the one in 2.13, ^'^e have that Sj^^gg S2''Sj. Thus, for each h in P^* SÂ°h''S"-^ = -1 -1 -1 S2Â°^-\''^-] "^'^"Sp ~ ^^/^''''Sp ^^i^iich is in g2Â°SiÂ°h='gT-'-o Â£"'Â• ^2 P2. Thus, P^ is a normal subgroup of G. Similarly, P is a normal subgroup of Go Thus, by uniqueness of ^'.-1! factorization, G = '1 ^ ^Â» vrhere x now denotes direct product. (2.15). l^^fXlRlMPJL' Let C be a category which is wsllfactored by f^ and P^. Then a neu category Cmay be formed as follows. Form a graph denoted as G^ by taking Ob G^ P.. For each Pg-morphism g:A S, there exist unique A and B in P^ such that A is an object in A and B is a^i object in B. IÂ«t there correspond a unique arrow gjA >3 in the graph G^c Define Â•&on Pa(G^) byi (1) (A ^A yk) js(A iÂ— >A), for each object A in A. (2) Vj~~^lÂ—^Â—>2) V (W ^Â°^ > U) whenever gof is defined. X^T (3) (liÂ— ^-^V) r (WÂ— ^Â— >V) provided there exist P^morphisms a and b so that the following diagram coinautes: f Now r is a categorical relation on Pa(G^), Lst ? be the categorical, conpatible equivalence relation generated by y . T?ien let C^ rr (?a Gj^)/^. PAGE 31 2^ (2.16). Hem.^lÂ£. Let C be a category Tvhich is wellfactored by r^ ai'id P^, l/e nay fern a category C^ by interchanging 1 and 2 throughout definition 2.15. (2'17). .Â£Â£Â£2!ial^.HiÂ« ^"f^ y be a categorical relation. Let ? be the categorical, compatible equivalence relation generated by jr. Then ? consists precisely of those pairs (sÂ»S*) of sorphisms such that don g = dom gÂ» and cod g = cod g' and the following holds: there exist morphisms Cl5...Â»G^, such that for each i 1, . . . ,nj dom g^ = dom g and cod g^ = cod g and, for each i = 0,..o,n, (letting go = g and Sy:^.'ri = g') there exist g^^ and g.^*^ such that g. = g:^'o ...ogl.i and g.^^ = g^i^^. -"-e^;^ -nd, for each = l,..,,v^, S^^i E g^;^ V7here E is -, y , or y^l. SZÂ£2i^ The proof of this proposition is routine and vrill be left to the reader. . (2.18). Propj)sition. Let h and hÂ« be Pj: -morphisms in C where C = P^x P^. Lot Pa(C-2) be the free category formed in 2.16 and let h and hÂ« denote the morphisms in ^a.iG^) corresponding respectively to h and h'. If h?h', then in C, h and h* are T^ -morphisms grid connected by r^-morphisms, feioof . In order to prove this proposition, it is necessary to make a careful analysis of what hFh' means. Namely, we have morphisms g^,...,g^ as in 2.1?. Note that we may avoid the possibility that some g^ is the empty morphism by, if so, putting an identity (corresponding to a P^^identlty) alongside gi,...,g^ and a r -related path PAGE 32 25 categoi-y identity alongoide h o>ia h'. Now we write do^rn equivalent conditions in C to descrioe the process in 2.1?. We have that there esist sequences of Q-Eorphisins S , ..., ^n+l such that Sq = (h), S,^..i = (h'), no S^ is eapty and (since vis ns.y '-.-ithout loss of generality assume that g^'^ = j,i -^ gjL-i-i* ^^^' e^^^i 5 except sone one value of j) for S. and Si+1 we have (or with S. and S^^^^ interchanged) one of the following occurs: (a) l^y ( )^ means that A is an object in A so we have Si = (^pÂ»---.Vj,+i,Vj^,Vj^^^,...,v^) and ^1-rl = ^"'p* Â• Â• Â• s'^]r:-hl>'^li5Vj,_^^, . . . jV^) where v-^ means Vj. has been deleted from the sequence, vj. is an identity, and cod y^^i, dom Vj^^^, and cod r-^ = dom Vj^ are all in the same ^g'^^^'^^sory, (b) S^ :^^p""Â»^ka.iÂ»Vj^jVj^Â„lÂ»...jV^) and Si +1 = (Vp 5 . . . , Vj.^ J , s s t , Vj^_^ , . . . J Y^ ) vrhere s ot = v,, ( c ) S^ -(vp s . . . , \\j^i , '^ir Â» Vj^Â„ J , , . , 5 V J ) aiod Sj_^.i ('^p* Â• Â• Â• Â»"^"k-MÂ»^"^Â»'^k-lÂ»' Â• Â• Â»"^i) where there exist l^-Eiorphisns a ajid b so that aÂ«Vj,= wob. Then (a) corresponds to (1) in the definition of y, (b) corresponds to (2) in the definition of i-, and (c) corresponds to (3) in the definition of y. Thus, by I.3, h and hÂ» ar-e /J^-sorphisns grid connected by l^-morphisms. (2,19). I;Â£o.S03itj,on. Let h and hÂ» be Pp-norphisms in C where C q x f^. let PaCC-^) be the path category formed in 2.15 and let h and hÂ« denote the norphisms in Pa(G^) corresponding respectively to h and h*. If hrh', PAGE 33 26 then in C, h aiid h' are P^-norphisms grid connected by I -.-iiiorphlsns. Proof. The proof is the sacie as for 2.18, with 1 and 2 interchaiiged. (2.20). ^C2.f of. 2.10. Let C^ and C^ be the categories forced in 2.15 Â£nd 2. 16 rospsotlvely. Let C be the discrete category forced by the objects of C. Let A be aji object in C. Then there exist unique A^ in H, and unique A^ in r^ such that A is an object in A^ ajid A is sn object in Ag. refine T t CÂ—^-Q^TJ^T^^ on objects by P(A) = ^A-i^>A^>^)' Suppose f:AÂ— >B is a morphism in C. Then there exist a unique factorization f = hog ,ihere g is a P^ -morphism and h is a r^-aorphisn. Then define P(f) --. ([hl^Cs])? (^;^jA2Â»-^) ^(B^sB^jB). It is clear that P is vxell-defined. Now ^le must show that F is a functor. Since 1. = i o i A A -"a and since 1^ is both a q^norphisn and a P^-morphisia, P(l ) = (Cl^;).[l^3) i(^^^^^^^^j = l^^^y Suppose A-il-^.3-^-!_>D. Then we have Pi-Pg factorizations as indicated by the fol~ lowing cormutative diagrams f By definition of ^ m 2.15 and in 2.16, h?pÂ» in Pa(G^) and P?g' in PaCc-^). Thus, [ h] = [p'J and CpD Cs'J. PAGE 34 2? F(f )oF(f) = (Ch'J,CGÂ»3)o(Chl,rG3) ([:i!]Â°M,i;s']-Cs]) = (D^-'J^pG^H-H) ([h'-P'l^CP-s]) F(fof). Conssquently, P is a functor. It is clear that F is one-to-one on objects. Suppose F(f) = F(f*) where f and fÂ» are morphisiis in C. Then dom f dom f and cc:l f ^cod fÂ» . Also, f and fÂ» have ^^-V^ factorizations as indicated by the follouing diagram where the top ajid bottom tricoigles are commutative: Thxi.s, F(f) = (Chl^rsl) and F(f') ([h'Jj^g']) so [h] -[h'] and Â£^ = Ls'3 Â» i'G., as Pa(G^) and PaCGg) Horphises, h?h* and gyg*. B-j 2.18 and 2.19j h s.nd h* are fl-Korphisms grid connected by f^-morphisms and g and g' are ri-niorphisias grid connected by R-morphisms. ^ definition of a wellfactored category, we then have f = h^g = h*Â«g* ~ f. Thus, P is one-to-one on morphisms. . We next show that im(F) is a subcategory of C x C x C . It suffices to show im(F) is closed under composition. Suppose P(f)oF(g) is defined. Then dom F(f) ^ (A^,Ap,A) and cod F(g) =-iB-jB^jB) so A = B. Eu.t by definition of F, A do:n f and B ~ cod g so fog is defined. Hence, F(f)''F(Â£) = P(fcg) so im{P) is closed under composition. Since we have that C is isomorphic to im(F), we now PAGE 35 28 show im(F) Is well-factored in the standard maimer. It suffices to she-.; that each P^-norphism in C maps to a Pmorphis.^,, in C^xC^xC^, i = 1^2, If f is a P^-morphisia, then f lof is the P^/^factorization of f so P(f) = (CfJ^UJ) and Ci3 is always an identity in C,. Hence, F(f ) is a r^-morphism. Similarly, ire can show that P preserves P^^norphisrris. This conpletes the proof. (2.21). Remark. Now that we have gone through the construction of C^, c^, and C3, we may ask what C^, c,, and C^ are for sone familiar categories. Suppose S is a monoid well-factored by P and P . Then as noted earlier, discounting empty subcategories, card(r^) = caixKP^) = 1 and, identifying P^ with its member and r^ with its member, S ^ P x P as a monoid. But by 2.10, S is isomorphic to a subcategory of IT' x C x ""C . We ask, "ifhat are C^, C^, and C^?''" According to the construction, C^ . (Pa G^)/? where G^ is the graph of P^-morPhisms. Then G^ 3s P^ considered as a graph. Bat we identify 1^ ,,ith ( )^^ and (f,g) nlth (fog). Finally, >:e say fFg if there are ^^morphisms a and b so that bof = g-a. Bat as noted in 2.I3, gca = a.g, so bof =. a^g. We then have two P^-^p^ factorizations of an S^.morphism. Hence, % Pg. Similarly, C^. ^ P^ and card(C3) " ^^3.'Â«d(0b S) =. 1. It is easy to see that F-s-^c^irc^-Tc. fro::. 2.10 is onto. Similarly, if g is a group and is -.-ell^factored by Pi and P then card(P) ^car^cK P \ . . ^ ^ J.; ccrcu^) 1 and, making the identifications o-f" abovp r 'y P r^ ^ H -, , v aoovc, Cj ~ \ 2^ 2.2'-= 'is end card(C3) ^' ^' PAGE 36 29 Now we consider ^^H, the category of all modules over all rings. Ob(G^) =^^JIJH is a ring}'. ib.TovTs in G^ are gH Â— iKilL__^H so that (s,l):(R,I-l,.) j(RÂ»,I'I,-^) for some H. Suppose (g,l)E(R5HJ.M ->(H',MÂ»-x-Â»). Tne folloirins diasram commutes: (s,i) (R,II,.)~(1,0) (rJi^iS-*)(s,i) ^(HSlij-'O (1,0) -$(RSM',.::-M . Consequently, (g, 1^4)^(5, lj,j, ) . Given a ring morphism gs R Â— >RS (s,l)i(H,R%.) ->(RÂ«,RS-:0 where r-rÂ« g(r)--T^ Aa.so, given Â„M (s.i) (sM) ti-lr ->M, we have (g,!) s (R,M, . ) ^(R' ,1.1, -^) aaid (gM) : (R* ,M' , -;:-Â« ) $(RÂ« * , IV , -Â•;-' Â« ) . The following diagram cornrautes where rÂ« *m' = g(r)':;-'m*: (S,l) (H,K,.)(1Â»0) (R,MS-Â»)(g,l) (1,0) -^(RMiS-x-') . Thus, ((sM^. ),(g,l-_jj)y(gÂ«cg,l^.,), so C^ ^ R. Gg has as objects all jjM such that M is an abelian group. Arrows in C^ ^^^ jj^'^ U^^MÂ» so that for some R, (RjH,') LLiÂ£.LÂ„.(h 5 K Â«,,>:Â•). Then the following diagrsin commutes J (1,S) _>ro , n^ , . ) (0,M,O(0,1) (r,m;o(i,s) (0,1) >(R,HS:0 Consequently, (lj^,g) y (lo,s). Given any group morphism PAGE 37 30 gsl^l ->II*s where lA and H* are abalian groups, (Ijs) s (0,1'I, Â« ) K 0,11',') BO ( 1 , s ) s H^'^ i^ * Â• ^^"^^^ R^'^ ^"^"^ Â— gAlÂ—Ll^Ell^A!; 1'js note that ( (l^jsS* ) Â» (1j.jS) )y (IqjS'" s) . Thus, ire have composability of morphisms. Consequently , Co = (Pa G2)/y . is isoffiomorphic to Ab, Our results then are that for j^M, C, = R, Co = Ab, Co is discrete, and card (Co) = card(pM). (2.22) ^:Â£EQSltion. Let N = (K,Â®, ^) be a multiplicative category without unit. Let (S,u) be an object in Sgp(N), A be an object in M, ((T,v),B,z) be an object in Lact (N) . Suppose fs;(S,u) ->(Tsv) is a morphism in Sgp(N) and that gsA Â— >B is a morphism in M. Then ( (S,u) ,B,zo(fÂ®l) ) is an object in Lact(N), Proof, This is equivalent to shoi-ring that the following diagrarxi coiimiutes: S0(S03)Â— {SÂ®S)Â®B S&>3 10(f@l) 1Â®Z ->S(^(TÂ®3)->S(giB ^T^Bf01 T0B ->3 fÂ©i z The proof that the above diagram commutes is routine and is left to the reader. We use that f is a semigroup morphism, o^ is natural, iS> is a bifunctor, S2id (T,B,s) is an object in Lact(N), (2.23). SLPiSS^Aioil* ^-^^ N be as in 2.22. Let Cj_ Sgv)(N) and C^ M. Let Co be the discrete category V7ith objects zsS(S>^l ->A so that z is a morphism in M, Then lact (N) is well-factored in the standard manner as a subcategory PAGE 38 of* Â£ix C^x C' . '^'^ xjill refer to this as the standard well' factorization of Laot(N). ^loo,f. Ey 1.8, it suffices to show that (3) of 1.^ holds and that if ( i , 1) : ( S , A, z ) >{ S , A , z ' ) is a norphi sn in I.act(N), then 2 =2'. Given the raorphism (1,1), the follovjing diagram commutes: z SÂ®A 101 SÂ®AThus, z = zÂ». Suppose (f ,s) : ( (S,u) ,A,2^) ?' (T,v) , B^Zg) is a morphisn in Lact(N). B-/ 2.22, ( (S,u),B,ZgÂ° (fÂ®l) ) is an object in Lact(N). Also, goz^^ = z-QÂ°{T0s)i sines (f,s) is a morphism in Dact(N). Hence, S^z^ = z^Â° ifÂ®!)" iW^) so (1jg)c (S,A,z^J -^(SjBjZgC (f^D) is a morphism in Lact(N). It is clear that (f ,1) j (S,B;Z_,o (f01) ) ->(T,B,Zp) is amorphism in Lact(N). Also, (Ijg) is a f! -morphism and (f,l) is a r^-morphism and {ffs) = (f,l)Â°(lss). To see that the factorization is uniqi^e, it suffices to prove (S,B,ZÂ£Â°(f^l) ) is unique. If (f,g) = (fM)r(l,s*), then f = f, gÂ» = g, and dorn(fSi) ^(SjBsp). Also, l-p = z^o (f(^l) so p = ZgÂ»(f@i), Thus, the ^^-^2 factorization is unique so Lact(N) is well"factored in the standard manner as a subcategory of C^x 0^x03. (2.2^-). Theoi^ii. Let C be a well-factored in the standard manner subcategory of C^ x C, x"C_. Then there exists an exact multiplicative category N (MjÂ®,o<) without unit PAGE 39 32 such that C is isosorphic to a uell-factored subcatesory oi" iKact(IT) so that ths v^ell-factorization of Lact{N) induces the weil-factorization of the subcategory. Also, each flmorphisn of C maps to a l^-norphisni in lact(N), i = 1,2. (2.25). He5.Si:k. Bsfore proving 2.2^, we obtain a preliminary result. In fact, this noi-r summarizes 2.10 and 2.2^. (2.26). ThB02:&^, Let C be a category and let P^ and P^ b-3 "classes" of subcategories of C. Then the follovring are equivalent: (i) C is well-factored by Pand P. (2) There exist categories C^, C^, and C^ such that C is isomorphic to a subcategory of C. x C^ v c^ vhlch is well-factored in the standard maimer and such that each Pj^-morphism in C maps to a P^-morphism in C.x C x C . (3) There esists an exact multiplicative category N = (n,05c<) without unit such that C is isomorphic to a subcategory of Lact (N) xjell=factored by the factorization induced by the wall-factorization of Lact(N) and such that each P^-morphisri in C maps to a P^-morphism in Lact(N), for i = 1,2c ^22t' 3:t follows from 2.10 that (1) implies (2). It follows from 2.2^ that {2) imp.lies (3). it is clear that (3) implies (i). (2.27). ^rcof of 2.2^. Disjointify the categories % and C^o Form a graph Pas follows: objects in P consist of pairs (A^jA^) where A^ is an object in C^, for i 1,2, or objects in P may themselves be objects in C^. For each object (Aj,,A2,A-) in C, we let there correspond a unique PAGE 40 33 morphisn fron (A. ,A^) to Ap Khich we vrill denote by A^o Now v:e form another graph G as follc^rss Objects in G are n-tuplos (U-j^, , . . ,11^^) where each Uj^ is an object in C^ or Â£2 and n is greater than or equal one, A morphism f from (U^, . . . jUj.^) to (V^s...,Vj^) in G is a set of morphisms (in C^, C^, orP) denoted f^^ or f^ ^.j.^ where f^ :Uj_ ^'^1(1) is a morphisn in C^^ or in C^ and fi.i-M s (U. jU^^^. ) ^'^%(i) is a norphism in P. Al.so, v:e require n^iti and that, for each i = l5,..jnj i appears exactly onoe as a subscript of some f j_ or f .^ ^^^ where f . . is counted as having two subscripts. Given an object (X. ,. .ejX^, ) in Pa(G)3 denote {i^l ^ : ^4 X^ irzl (X^,,..,X ) >(X.j5,..sX^) as 1, V. Suppose f and g ^ i are two laorphisas in Pa(G) going from (lJ^,,..Uj^) to (V^j.e.jV ). \]q obtain from f a new norphisn P = (P", .. Â» . ,F^) and from g a nevr morphism G = (G^j . . . jG^). Suppose f = ( ) /IT, u )' '^^^'^ le'i^ P = (1/n TT ^)Â• Otherwise, f = (f-^sc.sf ). Insert before f-'the morphism 1, , and after f the morphism 1,,^ ^^ w For i ^ ls...ck-l, let cod fr. (U-^s.e.sir ^j)Â„ Insert between f^ and f^"*"^ the morphism 1, 5. Â± .. Calling this nevr sequence vuj , , . . ,Uii(i) ; (F^,..,,P^), T-e see it is a path in Pa(G) and v;e let F = / t 1 (P ,,..5 ?), G is obtained from g in a similar manner. Let Iq bo such that l^i^^n. Then t} :U^ >U. ; in fact, Ff = 1 . L-et f;^ = pj. Exactly oiie of the following possibilities occu^rs! Pi :Ut y:l PJ j(Ui ,U . ) ->'.7 , or, finally. PAGE 41 2 ^iQ-l,ij^^* ^'^^i ~lÂ»^-^'i^ ''''p' Vfnichever cf these occurs, vre call it F--. F3..v;^-.Wp and p3 = i^ . Let f3 = fJ[3. Kiere are three possibilities for V/^^hich sxe analogous to the three possibilities for U^ . Whichever of these occurs we call F|^ Continuir.s in this manner, vre obtain the sequence (PÂ£t, . . . ^p,l) . p^^. ^ ^ i,,..,t^l, either ^i^ " ^1^^ is defined in C^ or in C^ or one of F./and F'^*^! is a r-morphism. By composins whatever adjacent norphisms are conposable, vre obtain a no'7 sequence (pJ^^O,.,, pl^ 3-rt i' so that for l. PAGE 42 35 (2) b = uob^'-'b^^d, b^jb^el'Ior G, f 3^5f2jf3Â€b3^, ^I'-^ i-l ^ ''0* ^2''"^! Â— ^''jÂ« ^"3*^^1+1 ^Â•j+iÂ» ^1Â»^2Â»^3 ^""^ idontitics, f/^c-bg, and fÂ£^:(VJj,W^.j.^) Â— 5>Yj^. b' = uÂ»ob^ob5o dÂ» , f{,f^eb|, fj: ^r-l Â— ^-'1 is an identity, ^2* ^^^'^4i^ Â— ^-^s+l' ^^'^'^ ^3*^-^^ so that f^!(W|,W^^i) ^Y^, Also, (X^^^jX^jX^ ...^) = (x;,.i,X^,z;,^l) and Yj^ = Y?. Pictorially, coiolltions (1) PJid (2) ere as f ollovrs : (1) d V V Xj^ ) e Â• . J Xj^ , X^ .l.^ y e Â• Â• ) X ) dÂ» V XÂ£ J t Â• Â• s X^, J Xj^.i--] J Â• Â• 8 5 X^ J ) I -> ^1 ^2 V V -1 s Â» Â• e Â« U lere (X^^.X^^^.^) (X^,X^^.) and Y^^ = Y^ (2) tm Â« Â• e f Â»oceÂ«Â«Â«c}V^./ 1 'X^ J Â• Â• Â• s-^x-.lÂ»-^j jX^.i.^* Â• Â• Â• jX^) (X , c.ojX' ^ jXpjXp^^, . . . jX^.j ) fÂ« Â«Â»Â«Â•Â•< t^l' v ^ /'rÂ» T.ri m9 t.tÂ« \ ^l:. ^ v Â• Â• > 1 1 u .Y ) fv* Vi ^3 jY' ) UÂ» ^'l***'**'** ^e Â»e.Â«Â»Â«eoÂ«.j V, j VV,-5,Â«Â«,,eci9CÂ»5Â«escÂ«Â»l>6s M,-, } J "liiWhere f ^^f^jf^, and f J are identies, ('\Â»ijX.^Â»X^^^) (x;,Â„^,X^,X^..l), ejid Yj^ = Yl, PAGE 43 36 IÂ«t (U^,....Uj^) and (V^,...5V ) be objects in G. Let (Ui,...sU^)Â®(V..,...,V^) = (Ui,..,,U^.,V^,...,Vj^); 1.9., let it be (W^, . . . ,Wj^^j^) where, for i = i,..,,!!, Wj_ = U^ and, for i = n-M,...,n-:-Li3 W^ = v^^^^. Sinilarly, if f : (U^, . , . .u^) Â— >(Vis...,\), f';(UJ,...,U.^,) HV{...,,V^,), f=if{L[j (U| 5 . . . , Uj^^ , ) ^( v^ 9 . . . , V^ )Â®( V| , . . . 5 V^ 5 ) where \:e let f 0f Â« = Â°C^i^i^ui-riU<}-l}Vi^l^^i I i.e., fÂ®fÂ» =fe>Ufe,i-M> where, for i 1, . . . ,n, g^ = f^ and 2^^^^^ = f^ ^^^ and, for i n+i,...,ri, g. = fÂ» Â„ a-'id a = f Â« , Â» Â» > fc>i i-n ^"^ ^i,i-hi H'n,i"(n-1)* Noir we define @:KxK Â— ^, On objects, is the same as for G. Given Zlf}J and CCslD s then f = (f^',...,fl) and g = (sPj...,s-^). Assume that k>p. By definition of y , we see that if (X^,...,}^^) is cod g, then gjr(l, . . . ,l,gP, . . . ,gi) where we take k-P 1^.. ^ /s and where (X,,..e,X ) = cod gP = cod g. We define rCf.lJÂ®Cr533 = CChDD where h = (f^^0l,.e.,fP"''^-01,fPÂ®gP,..,,f^Â©gl). If p>,i,, then we lengthen f similarly. For notational convenience, we will write a (i) b to indicate that a and b satisfy condition (1) in the definition of yi . Uq use a similar notation for condition (2), We need to show is well-defined on norphisms. Suppose CCv:3 CCv-J3 and [ru'JT-CCu'33 . VJe r.-e-ed to show rCv3]Â®rCu3: = CCv^j](g)[[u-33. Thusj it suffices to show Ct.vJD^CCull = CCv!3DÂ®C[u33 e-nd rCVllÂ®CCu]T = CCvO] 0[CuÂ«JJ . Since the proofs are sirrdlar, vre shc;i only that rCva3S[Cul} CCv^J]Â® [[lO]. In A, V7e have Cv]/[vÂ«j. Since _/> is a categorical, coiapatible, reflexive, and symmetric relatio:i, CvD/CVJ if a^id PAGE 44 37 only if there exist "^ jl * * ' * * "^"^n ^Â° *^^^ Mf[.'7Â£if[y^f"'fl'^*}Â» By the definition of /> , Â£'/]/>["-;] naajis that there exist representatives e and e^ so that [e] [v] and [e^l = [v-l and condition (1) or (2) in the definition of /> holds "between e and e. or that [v] "[v^] , HoT'^ever, ue may assu.me without loss of generality that e = v and ^i == "^-i Â• However, for [v^3j=[.Vpl5 since we have already used r^^ in conjunction with v, we Eust use a new representative for v-^ , sa3'v^. Thus J W3 have v or (1) or (2) v-yvj^ = or (1) or (2) Vp yv ... Vj^ = or (1) or (2) v*. Tnus, it suffices to show: b = or (1) or (2) w implies CCb]]Â®aul3 = CCwD'J^LLu]! and that bjfw implies [Lb3]Â®[[u33 =r [Cw3]^[[ul] . To prove this, suppose b (1) or (2) w. Then b = (b j...jb ) and v; = (w-^5..,sW ) and u = (u^^, . . . sti ) . Assume k^m and p>m. Then [Cb]]@[[ul3 = CC(V^ 1, . . . sb"""'"^ l,b^ u^,...,bi u>)]3 and CD0]Â®[M3 [[(w^^ l,...,ir^"^ l,i;^ u^, . . . ,w^ u^D]. But then (1) or (2) will hold between the representatives given above for [Co3]0[lvJ] and CDtJ]0[Cu]] so [[b]lÂ®[[u]l = CDO]Â®rMl . The other possibilities for k and p with m lead to differences, only in notation. .Now suppose byw. Then, recalling the definition of y , we see that composition of morphisms of the form I.t with either b or w preserves the relation y, so [Cbj3.g)[[u]j ~ ["D-'DD^CCi-Ol <> Thus, Â© is well defined. It is easy to see that (S" is in fact a functor. Letting o< bo equality, N = (M,Â®,o!') is clearly an exact multiplicative category ^-rithout unit. Define F':C Â— >Lact(N) as follows. If (A^jAgjA.,) is an object in Cj let P(A^5A2,A3) PAGE 45 38 Note that vre no^T identify A^ Kith (A^), i = 1,?. Also, V7e identify f^ with (f^) vrhere f^ is a C^-aorphism, i = 1,2, We do this since this identification embeds C, into M, i 1,2. Givon (f^,f2)j(A^,A2,A^) >(A{,A*,Aj) in C, define FCf^^f^) ^ ( [[f ^H , [[f^"]] )^ That (A^, (A^,A.J -10^31^^^) is an object in Sgp(N) folloTjs imjuediately froii the definition of y. That P{A^,A2,A ) is an object In Lact(N) follovrs from condition (2) in the definition of ^^ . That CCf J] is a morphism in Sgp(N) follows from the definition of y. That ([Cf.]] jCCf^l] ) is a morphism in Lact(N) follows from condition (i) in the definition of/>. It is now clear that F is a functor. Suppose F{f)oF(g) is defined. Then dom f ~^ {k^^k ,A^) and cod g (A.,5A2,A^) for soiae (A^^A^jA ) in C, Thus, f-S is defined so F(f)-P(s) = F(fog), Hence, the image of P is a subcategory of Lact(N), Suppose F(f) = P(s). Then f (f^^f^) and g = (g^.g^) and rCf^l] = CCcpi and {X-f^-} r-. [[g^:] i^ M. Thus, [fi3/[g^l. Thus, as noted when proving to be uell-Â»definsd, there exist h^jh^.h^^hgj.e.jh^jh^ so that f^ -or (1) or (2) h JtEj = or (1) or (2) V^2Â°*"V^n " Â°^^-^ ^'2(?-) ^r Since dof^ has length one, (1) and (2) cannot hold, so we have f^ =. h^. An in the definition of zr , we obtain from h^ and E^ the morphisms H^ =: (K^'l,...,Hh and Cfrom h^) the morphism \ = (H^l, . . . ^hJ) . Then since the length of dom h^ = 1 and dom h^ = dom H^ = don H^ = dom h. , T/e PAGE 46 39 .tl ^1 _. ^ri ^ =1 have H^-'-o Â•Â• coHjh'^-o . .,oH^, Since f is a morphism in C^ and f^ = h^, th3ii H^ = (l,h^5l) so h^ = H*^c= Â°hJ. Continuirig in this manner, t:e obtain f . = h = E^'Io^.ooh = Jti -1 t2 1 111 1 H^ ^.-.oH^ = E^'i Â»**-E^ = .Â•.== g^. Thus, f^ = g^. Similarly, f^ = g^. Hence, f = g. Thus, P is an embedding since F is one-to-one on morphisms. It is clear that each r^ -morphism in C maps to a P, -morphi sm in Lact(N), i = 1,2. This completes the proof of 2.24. PAGE 47 CHAFTEH III A SUFFICIENT COIJDITION FOR A VJEXL-FACTORED CATSGOHY TO BE VARIETAL In this chapter, vie will be concerned with a. type of varietal category introduced by Herrlich in ClD which generalizes the 'Varietal" categories of Lavrvere and Linton. (3.1). RsMiTii. We recall the follouing definitions. (A,U) is called a concrete cate.g;ory if and only if U:A Â— ^Ens and U is faithful. If pjq, and f are norphisms in A, A any category, then (pjo) is called the congruence relation of f if and only if (p,q) is the pullback of f with itself. If A is any category, f is said to be an extremal eplmorEhism if and only if f is an epimorphism and f nog is any factorization of f so that m is a monomorphism implies m is an isonorphism, A norphisa f is said to be a regular? fi;iii2Â£S12il^iSii if and only if f is the coequalizer of some two norphisms. It is easy to see that if f is a regular epimorphism, then f is an eztremal epinorphisa. (3.2). Definition (Kerrlich). Let (A,U) be a concrete category. (A,U) is said to be &:a algeb raic cate^or^ if and only if it satisfies the following three conditions: (Al) A has congruence relations and coequalizers, (A2) U has a left adjoint. (A3) U preserves and reflects regular epinorphisns. 40 PAGE 48 4.1. (A,U) is said to be a yg.yietg.l category if and only if it is algebraic and satisfies the follouing condition; (V) U reflects congruence relations, (3.3). ExanrOes (Ksrrlich). (1) The follo-.rlns at-e varietal categories,* Ens, category of pointed sets, category of all groups, category of all semigroups, category oj? all monoids, R, R-^, M for any ring R, category of all unital R-nodules ;-;here R is any ring vrith identity, category of all lattices, category of all Boolean algebras, category of all compact, Hausdorff spaces, category of all compact, Hausdorff groups, and Abo (2) The category of all torsion free abelian groups is an algebraic category which is not varietal. (3.4) Â«. Remark. We now prove a sequence of propositions leading to our sufficient condition for a xrell-factored category to be algebraic or varietal. Vfe state this suf~ ficient condition in 3,5, The first few propositions will prove 3,63 which is an important pai't of the proof of 3.5. It may be helpful to the reader to verify the following propositions only for the category of all modules over all rings or for the category of monoids acting on pointed sets, (3.5). Theorem, 1st N = (HpQ.cK) bD a multiplicative category without unit such that H has finite coproducts. Also, assume that for each object M in H that K^_ and _ 0M preserve finite coproducts. Let U^sLact(N) ?Ss-p(N)xM be the obvious forgetful functor. Assume the following to be true; PAGE 49 (i) If f is a resular epimorphism in Ssp(N) and g is a regular epimorphism in H, thsn f@5 is an epimorphism in M. (2) (Sgp(N)jU2) and (H5U-) are algebraic categories so that neither Up nor U^ assumes the empty set as a value. (3) Lact(N) has congruence relations and coequalizers, {k) If a and b are regular epimorphisms respectively in Sgp(2T) and in I;I and v is a nononorphism and the diagram below consisting of the solid arrows commutes, then there exists z^^ g;A03 Â— >B so that the entire diagram in M commutes: aÂ©b u^v AÂ»<^3Â«km A"Â®B" Let U = TrÂ»(U2>^U )oU.:Lact(N) ->Ens, where TTs Ens x Ens Â— ^Ens is the product functor. Then (Lact(N),U) is an algebraic category. Ifj, in addition, we assume that (Sgp(N)5U2) and (M5U0) are varietal categories, then (Lact(N),U) is a varietal category. (3c 6), ProT2os3.tion, Let N and U^ be as in 3.5. Then Ui has a left adjoint. (3o7). Notationc V.'e i-rill let N, U^^ U^, U r and TT be as in 3.5 for the rest of this chapter. let u-r, denote an injection into a coproduct involving E, Define d. _ _s A, Sj C (AÂ®3)iI(AÂ£0) >A0(3IiO) to be the unique morphism d such that dÂ°u^^|gj2 = -^^'-^B ^^^ '^""^AÂ®^ ~ AÂ®a . VJiienevcr possible, we will denote ^^ ^ ^ ^s d. Similarly, we define "K^ ^ ^: (BÂ®A)il(caA.) >(BJ1G)Â©A. vrnenever possible, ^re denote dg c,A PAGE 50 ^3 as d. The assumption that AÂ®__ and __Â®A preserve finite coproducts says that d and d are isomorphisms. (3.8), ^o20Â£itionÂ« The trsjis format ions d and d are natural . ^J>.2L' ^'i^-e proof of this proposition follovis in a straight forvrard manner from the definition of a coproduct, (3Â»9)Â« Pi^oposltion. The following tvro diagrams commute: (1) ^A B CsB D (AÂ®( Eg^C )] Ji|;iÂ®'v BSD )| (2) ^A BjB C,D [(aÂ©c)Â®d] II |(bÂ®c)Â®d] A0C(P^Â©j)iUBÂ®D)l *^A BjC.D -^(AÂ®3)@(C1.D) [(AÂ®C)1[(E0G)]Â®D -^,B,CÂ®1 A ^A,B.C D ({A1(3)Â®G}(S)D -^(AJiB)Â®(CÂ®D) ^oof. To see that diagram (1) commutes, it suffices to sho-j that (1) commutes -vihon composed ifith the coproduct injection into the coproduct of AÂ®(B<2^G) and AÂ®(E2)D). We vrill prove it for ^AQfEfy^) only. We calculate the fol~ lovring e-iuation. PAGE 51 kk = "^^ (i^d)" G.Â°u.^,^_^,,,, , Thus, by definition of coproductj (1) coiriuiutesÂ» The proof that diagram (2) comznutes is analogous. (3 Â•10). P2-'opo3ition. Th9 follo'/dng diagram conmiutesj lA PAGE 52 ^5 divided into eight suMiagrams, each of which is subsequently shovjn to coiamute. This constitutes the proof that (0) comniites. (0) (BÂ®H)0{iii/(RÂ®::))RÂ©[HÂ®(I'IIi(HÂ®M))] 10Z R0{KJi(RÂ®M) ) p2)j ->30(Ki[(R(S:'i)) -^Mil(H@H) SuMiasran 1 in (0Â») is redra^m in (1^ and decomposed into diasrans 1. IsLa,! .3sl.-i , and lo5. Diasrams 1.1 and 1.5 commute by 3,105 1.2 coiniautes since d is natural, and 1.3 and 1.^ commute by simple computations. For sutdiagr-am (2) to conmute, it suffices to shoir L(ljP)Â©l]Â°d (ljp01). It suffices hers to show equality when VJ9 compose on both sides with the injections into the coproduct of RÂ©I'I and (R0H)Â®H. Since the procedures are similar, we consider only uÂ„^,,.. We have the following! [{l,p)S;i]Â°doUp^^,^j = [(l5P)0i]''(up@l) = lil,v)ov.^m 1Â©1 = 1 = (l5p(51)Â»Upg,^j. Thus, (2) commutes. Diagram. {3) coDJiiutes by the definition of d and diagram (A'-) coiiMutes since d is natural. Diagram (5) is redraim as diagram (5*) and is decomposed into sul>diagrans 5.1j5o2, and 5.3. Diagrasi 5,1 commutes by condition (d) in definition 2.1. Diagram 5.2 commutes since ci iy natural. Diagrajn 5.3 oom:uiutes since (RsP) is an object in Sgp(N). PAGE 53 kG o d U PAGE 54 47 to O PAGE 55 ^8 v^ U CD PAGE 56 ^9 We note that dia^ratQ (6) coronutes since d is natural. For diagram (7) to cominits, vre need [(l5p)Â©i]od = (l,p5l). By the definition of coproduct, it suffices to prove the above equation irith u^-,,. or vilth u,___v^,. composed \ilth both sides, VJe obtain the eqviations: Ql9p)01]Â° douj^gi^r = The proof is J^ist as easy for u,Â„-^v^., and is left for the reader. Thusj diagram (7) commutes. To prove that dia.gram (8) commutes, we adjoin u_ 01 and u , JQl to t'ne diagram to form respectively diagrams (8A) and (SB). It is clear by inspection that (8A) and (8B) commute. Thus, diagrams (1) through (8) commute so our original diagram (0) commutes. This completes the proof of 3.11. (3.1?,). Proposition. Given that (l,g) : (R,H) ^>U^(R5N,2:) is a morphism in Sgp{N)xH5 then there exists a unique morphism (Ijh) in LÂ£ict(N) so that the following diagram commutes s (1,1%.) (R,K) ^i ->U_^?(R,!.i) = (R,Mli(RÂ®i:)) {1,S) ^"^^.^ /^ U^djh) Uf(R,N,z) . Furthermore , h ~ (g , z Â«> ( 10 z ) ) . ^oof, Comi-iutatlvity of the above diagram is clear. We must sho-.T (1, (ssZ^dOo) ) ) is a morphism in Lact(N). It suffices to sho^-r the diagram below comisutes: d"'^ (illw) (1 PÂ®1) RÂ® ( K li( RÂ®I'! ) )Â— ^C ?.&:! )i[H Â© ( R0ri )3~ > ( RQH ) JI [ ( R5H ) Â®: : 1 Â— ~ Â— >R<^M. Â•I; I 1 (s,?:-(l PAGE 57 50 CO hi to d ft PAGE 58 51 9i PAGE 59 52 As usual, we prove commutativity by adjoining first Up^^^^.j and tlien ^'oa^/u^v') Â• ^'Â® obtain the equations; 2 =Â• [lÂ®(g,2o( 10^^) )J Â°^'^^HÂ®I'I ^ 2Â° &Â®(sÂ»zÂ°(l03))] Â°(lÂ®Uj,i) = Z-. [l@(s5.2Â°(lÂ®s))Â°ujj = (gsz od^j) )Â°Uj^0.,.Â°(lspgil)Â° (:lJio<)''Uj^g5,j, (Note that we see from this and th3 required conimutativity that h is unique.) 2o[l0Cs,zÂ°(lÂ®G))]Â°dou^g,(p^g-,j = z4l0(G,2Â°(lÂ®i-))] Â°(l^aj^0^^) = z4lÂ®[(SvZ^(lÂ©5))Â°U-H0I.i] ) = 2o[ig)(znlÂ®s))] = z<'(l<9z)Â»[l0(l^s)]. aits (g,z-(lÂ®s))Â«uj^g2,j''(l,p01)Â»(lliÂ«)-uj^g,(j^^j,j) = (z'(10s)) (l,p^j.)"U^g^jjjg,_,,Â°o( = zÂ»(l%)o(p01)'>^ = z.(p%)Â«cx. Thus, it suffices to prove the diagram below with subdlagrams (a) and (b) comiautesj EÂ®( Rm ) lÂ®{Wf^) ->RÂ® ( RÂ®N ) (RÂ®R)Â®M RÂ®N B. PAGE 60 53 (l,h) is unique. This completes ths proof of 3.12. (3Â»13)Â» Rcoposition. Let P.iB.) donote the standard n. "Subcategory of Lact(W) determined hy the semigroup R. Let Up = uJ_ . Then Up: C (R) Â— ->fn}xM has a left adjoint F. F(RsI'l) = (RsHlKRGlOsz) is the same as in 3.1i. F is defined on morphisms by: (1. "M> R,Mi((RÂ®M),Zj,i) F(l,g) = (Is (Ui^'-gjZji'' (lÂ®(Uj^os) ) ) ) ->(R,Ni((RÂ®N),Zj^.) . grppf . This proposition folloirs inraediately from 3Â»12. (3.1^J')Â» Rg-^ark. We now pause to prove sosie propositions of a different nature and iThich will be needed for the proof of 3.5. (3Â»15)Â» Proposition. Let C and C* be categories wellfactored respectively by ^< t T^ and PUpl* Let P be a class -'^ 12 which indexes both P. and P*. Let T:C'~ Â— >0 be such that 1 ' 1 Â». ~ "for each p in P, alc:G.^ >GÂ„. Let TLf = TÂ„. Assume that *^p ^ ^ Gp P for each p in P, there exists S sGp Â— )G' so that S is left adjoint to T , Also, T(C) is an object in G implies C is an object in G'. Finally, if f:A Â— ->T(B), then there exist unique f. and f so that f is a T^-iaorphisn, f^ is a P^~morphism, and T(f2)Â°f. " f. Then there exists S:C Â— ->C* so that S is left adjoint to T and, for each p in P, S|^^ = Sp. Prc)of . Lit A be an object in C. Then there exists a unique Gp in V^ so that A is an object in Gp. Define PAGE 61 5^ S(A) = S (A). Thus, TS(A) = T S (A) j so there exists v: A Â— ^T3(A)5 a ncrphlsm in G , so that if f :A Â— >-T (3) is in x^ x^ G s then there exists a unique g:S(A) >B in G* so that T(g)cv = f. Suppose f !A Â— ^T(B) is in C arid that B is an arbitrary object of C*. By hypothesis, there exists unique f^, a G "inorphiEm, and f^j a r'-morphism, so tliat the following diagram conimutes: A^T(C) *T(B) . " Then T(C) is in G so C is in GÂ». Thus, T(C) =^ T (C) and we have that there exists a unique g in G' so that T(s)Â°v Thus, the following diagram commutes s V A: > To ( A ) T(Â£) = fT(B) We now prove that f^^S is unique. Suppose T(h)ov = f. Then there exist h., a l '-raorphisEi, and h^s a P'-niorphism, so that h = ^2Â°^"'!* ^'''*' "^^ ^^ ^ G -Eiorphism and so is T(hj), so f =-T(h^)o (T(h )ov) implies h^ = f^ ^-^^^ T(h^)ov = f^. Then by uniqueness of g, h^ = g so h = hpÂ° h. f2''S. Thus, by the front adjunction theorem, S is the left adjoint of T. Also, S is defined on norphisms via the front ad- PAGE 62 55 Junction dias'^-am, Sinoe S is the left adjoint of T , S\ = S , foieach p in P. (3.16), _B3havi..or of S. The functor S is not necessarily vrell behaved ci\ Pp categories. In fact, it ii:ay be that S(G)j TThore G is any P^ subcategory, is not contained in any Vl or Pj subcategory. Vic now give an example of this situation. Let A ha^e objects A and B and non-identity morphisins ajAÂ— >3 and a~^5B Â— >A with the indicated composition. Let C = C* = AxA have the standard vrell-factorization, uhere T^CC) denotes the V. subcategory of A/A determined by C, an object in A, Let S^xV^{k) >r|(A) = Ip^(A). Define S^tP^Cs) Â— >PHB) as follows! 3^(3, A) = (BjB), S (B5B) = (B,A), Sgd^g^^j) = l(B^B)^ ^B^'^(B,B)^ " '^B,A)' ^B^^"^^^^ = (l,p"^), and Sg(ljp~i) = (Ijp), To see that S^ is a functor, wo compute the following equations, S-^( (i,p) " (l,p"l) ) ~ Sg(i,i) = i(B^^^) (l,P"^)<^(lsP) = S3(l,p)oSg(l5p"l). Similarly, SgC (l,p"l) c (i,p) ) = Sg(l,p-1) oSg(l,p). Thus, S-, is a functor. Lot T ^^xAÂ» ^"^ is clear that all the hypotheses in3,15 are satisfied, i^ith the exception that Sg is the left adjoint of T' =Ip , ,. Ho^rever, this is a consequence of the front adjunction theorem and of some easy calculations and the details will be left to the reader. Thus, by 3,15s there exists SsC Â— >C* so that S Is a left adjoint of T ^-^'^lr(A) '-" ^A ^^^^ir(B) " ^B* -^^ S(B,A) = Sg(B,A) = (B,B). Thus, (B;A) is exi object in P (A) but S(B,A) is not an object in P'(A). In fact, S(B,A) is an object in Pg^B). PAGE 63 56 Thus, S(P2(A)) is not contairxed in pMA) or in PM3). (3.1?). .Proposition. There exists Fi:Ssp(N)>!;: Â— ^Lact (N) such that Fj_ is a left adjoint of U^ and F^ I ^ is the same as F in 3. 13. "^ ^:Â£2.t. By 3ol5, it suffices to shov/ that for (f,g)s(H,H) >U-j_(S,I\i,2)5 th';rs exist unique p rCRjM) ?U. (C), p a r\-morphis!n, and P2:C Â— >{o^^7.)^ p^ a P^-norphism, so that (f,s) = U^(P2)Â°Pj^. It suffices to sho/r this since , by definition of U^ and by 3.13* the remainder of the hypothses in 3.15 ^x-e. satisfied. Let C = (H,N52o(f01)). By 2.22, C is an object in Lact(N). Let p^ = (l.g). caearly^ p^ is a P.-morphism in Ssp(N)xri. Let p^ = (f,l). To ses that p^ is sji act morphisn, it suffices to note that the follo^ring diagram conmiutes: ^^ Â— Â£Â®1_ ^ sÂ®N ^ __^ mi >u z SÂ®iJ ^ Thus, pg is a P^-morphism in Lact(N). It is clear that the components of p^ and of p^ are uniquely determined. All that is left to sho'^T is the uniqueness of C. In particulaT, XTe must show that zÂ«[fÂ®f) is unique. However, this follows from the fact that (f,l) must be aji act morphism and from, consequently, the commutativity of the above diagram. Thus, the factorization is unique. This completes the proof of this proposition. (3.13). Remark. V7e see that 3.i? is just a restatement PAGE 64 51 of 3Â»6, so 3.6 is proved. (3Â»19). ^.o;^23iJtion, The product functor IT: Snss^Eias > Ens has a left adjoint P^j,, where F^_(A) = (A, A) and F/^Cf) = (f,f). Pi' oof . We will pro-/e this usin^ the front adjunction theorenie For the remainder of this proof, we denote Fjij, as P. Define D^JA ^A^A by '^A^) ~ (a, a),, if A is non-empty. If A is the empty set, then let D. = A. If f :A ->B, then we have the following j [(fxf )Â»D "] (a) -(f^f)(a5a) (f{a),f(a)) = (Dg-f)(a). Thus, D;l^^-^ -^F is a natural transformation. Let f:A ^(C,S) (-C E). Let p.sCxE Â— ^C and PgtC^E Â— ^S be the usual projection functions, Tlien (P;j_'' I'sPo'i") 5 (A5A) ^(CjS) is a morphism in EnsxEns. Tr(Pi-f,P2-f) = (p^Â°f)x(p2of). L((PiÂ°i")^(p2'>f))Â°I^^] (pO = ((Plof)^(P2Â°f))(a,a) = (p^(f(a)),p2(f{a))) f(a). Thus, ITCp^ fsPg f)Â°OjN^ = f. Suppose (u,v)-D^ = f. Then if a is in Aj (uxv)(aja) == f(a)s so f(a) = (u(a),v(a)). ThuSj (p^Â°f)(a) = u(a) and (P2Â°f)(a) = v(a), so p., ^f u and P2Â°f = V. HencGj (p Â°f,p Â°f) is unique, By the front adjunction theorem, X'xe then have that F is the left adjoint of IT. (3.20). ^PjDosltijon. The functor U ^^ FÂ°(Up^U^)^U. of 3.5 has as left adjoint the functor F =-F^o(F xP_^)oF where P^, F^, P^, and F^^ are respectively the left ad joints of Uj, Ug, U , aridTT. ^loof. The left ad joints exist by 3.5 end by 3.20. and by 3.6. Ue refer only to the hypotheses of 3.5. The rest is straight forv^ai'd and is left to the reader. PAGE 65 58 (3.21). Proposition, Let U., be as in 3.5, Suppose f iG a resular epimorphism in SÂ£p(n) and g is a regular epiinorphism in M implies fÂ®s is an epimorphism in M. Then Vl reflects regular epimorphismse iroof, let (f,g).-(R,M,Zj,j) >(S,N,z^) be a norphism in Lact(N). Suppose V{f,g) is a regiilar epimorphism in Ssp(N)xM. say U(f,g) is cosq( (a,b) , (a' ,b' ) ) vrhare (a,b), (a',bÂ«)s(T,L) ->(R,M). By proposition 3.I25 (1 j (1,2^.,Â° (1Â®!) ) ) F(RjI-I) ->(E,K5Z_^) is a norphism in Lact (N) . By 3.I?, F(a,b)! P(T,L) >?(HsK) is also a norphism in Lact(N). By 3.I3 and by the proofs of 3. 15 and 3.17, we have F(a,b) = (a,h) Tfhere h = (u^^.^ob^zo (aÂ®i) Â» (i^^^ob) ) ) , where z is as describ-d in 3cli, Thus, (l,(l,Zj^j))oF(a,b).-P(T,L) ?(R,M,z^,.). (ls(lsZ-^j))oP(a,b) = (l,(l,2^^)c,(a,h) (a, (l5Zj,j)oh)o We wish to express (l5Zj^>h in a simpler form. {l,z-_,)^h<Â»u = ^^'^M^'V^ lÂ°b b, (i,z.^)ohou^^^ = (I,z-.)o2o(a0i) ^(iÂ®(u,,ob;) =: (i,z.,poz.{aÂ©(u .b)) (i,J2^,)o2o(iÂ®u, )'.(aÂ®b) = (l,z.,j)oup^^-,= (i,pÂ®l).(ll(.c<).d-lo (10u,^) = (a^b) := Zj^jo(i,p01) Zj,jÂ»(a&b). Thus, by the definition of coproduct, \je have (l5Z.,)Â»h (b,z^,jc(p.^b)). Hence, (a, (b,z,^jÂ« (aÂ©b) ) ) jP(T,L) > (R,M,Zj,0 is a norphism in Lact(N). Similarly, (aS(bSz,,.o(a'Â©bÂ«))):F(T,L) >(R,M,z^.) is a morphism in Lact(N). In the above, p is the multiplication on R, fÂ»a = foaÂ«. |r6>(b,z . (a<^D))-u = gob ^ g^b* == g-(bSz^,j.(a'Â©b^))eu^. SÂ«(b,Zj^.o(a^b)).u^^^^ = g-Zj^j^(a PAGE 66 59 Thus, from the definition of coproduct, xre have go(b,2;j,;c(aÂ®b) ) = go{bÂ»,Zj,.Â°(aÂ»Â®b'))o Tnis implies (f ,g)o (a, (b^z.^^o (aÂ®b) ) ) = (f ,g) ^ (a* , (bÂ« , Zj^o (aÂ«Â®bÂ« ) ) ) . Suppose (Ujv) : (RjMsZ-^j) > (W,K,z^) in lact(N) so that (u,v) Â» (a, (bjZ <> (a6)b) ) ) = (UjV) o(a', (b* jz^o (a-Sb' ) ) ), Then, in the ssj^e manner as for (fÂ»g)Â» "t'-s obtain uÂ°a = u^a* and v^b = v^b*. Then there exists a unique morphisn (u",v*) in Ssp(N)xM so that (u,v) = (u",v-'-)Â°(f,g). Since U^ is faithfulj uniqueness in Ssp(N)xM implies uniqueness in Lact(N), Thusj it suffices to prove (u-'%v-'') is a norphisn in Laot (N) in order to prove that (f,s) is the coequalizer of (a, (bjZ,,o (a^b) ) ) and of the morphisn (a' , (b' 5Z,_^= (a*Â®bÂ» ) ) ) which proves that U. reflects regular epimorphisms. In the folloTring diagram, portions (1), (2) J and (3) and the outer rectangle commute j z. Thus, V7e have Zjro(u"-@v*)'Â» (fÂ®s) = v"-oz " (f PAGE 67 60 eplmorphisms onto regular eplnorphisns. Proof, Since a regular epimorphiGin is eji eztremal epiraorphism, ire nay prove only that U. maps extremal eplmorphisms onto regular epimorphlsmso Let (f jg) ? (R,M,Zj^j) ^s ,1^2^^ ) bs an extreiaal epinorphism in Lact(N). Then vre have Vob = g and u^a = f , where a and b are regular eplmorphisms in respectively Ssp(N) and M and u and v are monomorphisms respectively in Sgp(N) and M. Let dom v B and dom u = A, Then (f,s) is a morphisn in Lact (N) implies SÂ°2j.j = z.^^Â°{f0Q) GO vobÂ°Z|,j = Zj^<'(uÂ®v)o (aÂ®b). Then, by hypothesisj there exists z^:A(S)B >3 so that b-z,. = z-,''(aÂ®b) and v-Zg = z^Â« (uÂ®v). Let u be the semigroup multiplication on A, Ue have the follovrlng diagram: lÂ®z. Sl^(SÂ®N) N uc5)(i;.Â€>v) (2) -^sm UÂ®'A 1Â®Z B jA^B(SÂ®S)Â®N (u(Â»u)Â®v (A0A)Â®B UÂ®1 uÂ®i V z -AÂ®BB (3) z B (^) Â— ^h N uÂ®v SfgiN^ 'n Then (1) commutes by the naturality of ^. Also, (2), (3), and (/|-) commute since voz^ = z^^Â°(uÂ®v). Finally, (5) commutes since u is a morphism in Sgp(N). Then we have that PAGE 68 61 VoZt5o(u' PAGE 69 62 (P%QÂ»,2:^,) li--^. ^(R,II,2j,j) (f,s) Then WG also have (f ,g} o (p] ,0 * ) = (f,s) Â» (p^,q Â» ) in Sgp(N)xK. Hence, there exists a unique (p,q) : (PÂ»,Q' ) ^(P,Q) such that, in Ssp(N)xK, (p^,q^ )o (p,q) = (p.Sq^), for i = 1,2. Sinc9 U^ is feithful, uniqueness in SÂ£p(N)xH implies uniqueness in lact(IJ). Thus, it suffices to shov: that (p,q): (PSQ*,Zqs) -)(P,Q5Z ) is a norphisni in Lact(N). Since (qiÂ»q2) is the pullbs.ck in K of g with itself and since eÂ°q|'>ZQS = goq^-'Z J, lie have that there exists a unique r!P'Â®QÂ« Â— >Q so that q^or = ^-i^^Qcs fo^" i = 1,2. HoT:sver, ^iÂ°^"^Q' " Â°-i"^QeÂ» ^ ^ ^Â»2. Also, q^^Z "(perj = z^.-Cp^q^) o(pÂ®q) z-.j-(p|^q^) = qi'ZQg. Thus, qoz^^, = r = z^oCp^q). Consequentlj^ (pjq) 5 (?' jQS^^t ) >(P,QsZ^) is a morphisa in Lact(N). Thus, U^ reflects consru.ence relations. 0.2^). Siososition, Let (A,U) and (3,7) be algebraic categories so that neither U nor V assumes the empty set as a value. Then (AxB, Tr<'(UxV) ) is an algebraic category, where ff is as in 3,19. i^92L* If F and G are respectively the left ad joints of U and V and if H denotes the left adjoint of F (see 3.19), then it is easy to see that (FxG)cH is the left adjoint of JT-dlxV). The remaining properties are easily verified, (3.25). I^pposition. I^oposition 3.2^4holds with "algebraic" replaced by "varietal." PAGE 70 63 Proof. The proof is straightfor-i'rard and is left to the reader , (3.26). Proof of 3.5, This is ir^mediate from propositions 3.6 and 3.21 through 3*25 and from the fact that since Ssp(N) and M are algebraicj they have regular epimorpnism, nonomorphisra factorization of morphisms (see Hll)* (3.27). Ccrollarv to 3.5. ^H, the category of all modules over all rings, is a varietal category. ^voof . Let N = (Ab,Â®sO^) where Â® denotes the tensor product over the integers and ex is the usual associativity transformation. Then it is vxell kno^ai that N is a multiplicative category iJithout unit. Also, it is vrall knovm that for any abelian group E, _Â®E and W_ preserve finite (in fact J arbitrary) coproducts. ( For details about the tensor product, the reader may see C6}^) It is easy to see that Sgp(N) is isomorphic to R and that Lact(N) is isomorphic to _M. Since isomorphisms preserve all the properties concerned, we will check in Â„H the hypotheses to 3.5. (i) Let f be a regular epimorphism in H and g be a regular epimorphism in Ab. Then f and g are onto functions so fÂ©g is an onto function. Hencef^g is an epimorphism, ( For background material on R, see C7J . ) (2) It is well known that Ab and R with their forgetful functors are varietal categories and that their forgetful functors never assume the empty set as a value. (3) Let (f,g):(R,M,. )Â• ->(S,N,,"0 be a morphism in M. R It is easy to show that the following is a congruence relation PAGE 71 64 vjhere p^ and q^ are the natural proj-ections, i = l,2j and where vje use the induced nodule multiplication on the group Tf(m,m')[s(ra) = s(nÂ»)}; ({(r,rMlf(r) ^ f (r' )} , {(in,n' )| g(m) = s(inÂ» )}, . )-^^^M(R,M, Â« ) (Pg.Qg) (R,I-I,.)(f,s) The details are left to the reader. Now vre construct coequalizers. Let (f,g) sr.d (f',g*) taking (R,M,') into (SjNs-^) be tvro morphisms in M . Let NÂ» = fg(2n)"SÂ«(m)|nieI.i}c Let I be the ideal of S generated by {f(r)Â»fÂ«(r)|rc-R}. It is easy to see that psSÂ— >S/I is the coequalizer of f and f * in R, where p is the natural projection. There is an induced module multiplication for (S/I,N/(NÂ»-MN+SN')). Let q.'NÂ— >xV(N'+IN-}-SNÂ» ) be the natural map. By definition of the induced multiplication, (Pso) is a morphism in j^M. It is clear that (p,q)Â°(f5G) = (P.q)Â«(fSg'). Suppose (k,t)o(f,s) = (k,t) => (f * ,sÂ» ) , where (kjt)s(S,N,-^) >(T,L5.). Since p = coeq(f,fÂ»), there exists a unique ring norphism k"so that k-^^^p ^ k. Define t^*(q(n)) = t(n). To see that t'"is vzell-defined, suppose x is in NÂ», IN, or SN'. If X is in N', then x g(Li)-^sÂ» (n) , for some m, so t(x) = t (g(ra)-s' (m) ) t (g(m) )-t (gÂ« (m) ) = 0. For each r in R, t ( (f (r).^f Â• (r)n) = k(f (r)~f Â« (r) )t (n) = 0Â»t(n) --. 0. Since I is generated by {"f(r)-fÂ»(r) reR}, t(x) = 0, for each x in IN. t (s (g(]n)-'-g(m' ) ) ) = k(s)t(g(ni)-g(mÂ»)) = k(s).0 =0. Thus, for each x in SNÂ», PAGE 72 (>5 t{:<:) =Â• 0. Hence, t* is vrell-definGd, It is clear that t* is a morphism in Ab and it is left to the reader to verifythat (k-,t'"0 is a raorphism II, Since p and q are onto, (k-"-,t--0 is uniquBo Tnus, (Pjq) is the coequalizer of (f,g) and (f%Â£"). (4) Suppose a and b are regular epimorphisms respectively in R and Ab. Also, assume that v is a monomorphism in Ab. Tlien a, b, and aÂ®b are onto and v is one-to-one. Suppose the following diagram in Ab commutes: A.Â® 3*^^ ->AÂ®3 "^^ >A"Â€)B" Define z sAÂ®3 >3 as follovxs. For x in A PAGE 73 66 are varietal j (a) the category of all not necessarily unital modules over all rings having an identity, where ring laorphisms preserve identies and morphisms of the category are analogous to norphisjiis in K, (b) the category of all monoids acting on pointed sets, not necessarily in a unital fashion but so that if (M, (Xjx), Â• ) is an object in the category, then lÂ«x = x, where morphisms in the category are analogous to morphisms in LI, and (c) the category of all compact, Hausdorff monoids acting on compact, Hausdorff spaces analogously as in (b). It is our hope to publish at a later date a complete proof of 3.29. (3.30). Categories 1-Jhich ^.flre Almo_s_t yarietalÂ« Let C be either the category of all compact acts (see 1.1?) or the category of all semigroups acting on sets. Let Us C Â— ?SnsxKns be the forgetful functor. Then U satisfies properties A2j A3, and V in definition 3.2. Furthermore, C has congruence relations and coequalizers. ^CÂ£Sl.LU has a left adjoint by 3,6. V7e hope that the remainder of this proof will bs published at a later date. Dj.e to time pressures, it is not given noi;. PAGE 74 BIBLIOGRAPHY 1. K. Herrlich, A!?._ge>)r..aj.,c cats goriest an axiomatic approac h, to appear, 2. E, M. No2.TiS5 Soi:^ structure theorems for topolo^ical machin 5sÂ« University of Florida, I969. 3. B. Mitchell, Theory of categories. Academic Press, 1965. 4. De Huseiaoller, Fibre bundles, McGran-Kill Bool^ Company J I 96 6. 5Â» S, MacLane, Categ:orical algebra, NSF Advanced Science Seminar Lectures at Eo-Tdoin College, I969. 6. C. \'l . Curtis and I. Reiner, Representation theoi'^ of finite p:rouP3 and asso cjl s^t ivÂ£ algebras , Inter science Publishers, 1962, 7. K, I'loss, The. cate?;ory of rings, Uni'v^ersity of Florida, 1969. 67 PAGE 75 BIOGRAPHICAL SKKTCH Stephen Jaclcson Maja-rell was born September 21, 19^59 in Plant City, Florida. In June, I963, he graduated from Plant City Senior High School. In June, I966, he received the Bachelor of Arts in mathematics from the University of South Florida. In August, 196? $ he received the Master of Arts in mathematics from the University of South Florida. From September, 196? , until the present tims, he has studied at the University of Florida toward the degree of Doctor of Philosophy. He has held throughout his stay at the University of Florida a NASA Tralneeship, Iv.ring the sumjner of 1969, he attended an NSF Advanced Science Seminar in Category Theory at Bo-;rdoin College in Maine. He is a member of the American Mathematical Society. 68 PAGE 76 This dissertation was prepared under tho direction of the chairmairi of the candidate's supervisory committee and has been approved by all meubers of that committso. It was submitted to thG Dean of the College of Ai'ts and Sciences and to th3 Graduate Counciland T;as approved as partial fulfillEiant of the roquirements for the degree of Doctor of Philosophy. August, 1970 ...^Â„slL^,A^.^B:'Â± Diian, College of &ft3 and Sciences Supervisory Committee; Dean, Graduate School Chairman PAGE 78 \ 7946 B i xml version 1.0 encoding UTF-8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID EEKRD9N9K_A9MG6U INGEST_TIME 2017-07-17T20:27:33Z PACKAGE UF00098432_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES |