CERTAIN WELLFACTORED 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. Clark, 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 typewriter
.i,
TABLE 0F CO:"' ITS
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ABSTRACT ... o.
INTRODUCTION ...
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DIAGRAMI ( I)
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DIAGRAM (8A)
DIAGRAlI (83B)
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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 WELLFACTORED CATEGORIES
By
Stephen Jackson Maixwell
August, 1970
Chairman: Professor W. E. Clark
Major Department: Mathematics
A new kind of category, called a wellfactored 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 wellfactored
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 wellfactored 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
ERr.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 rins 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 compostion of two mo.rphiss belonging
to corr'esponlinS clrses of orphlsis.s. ,Roughly speaking,
for V1, : ha ve "odule" rmoro.piss of the form (.1,) and
''ring'" nmorphi sms of the form (f,1). These e::eplos and
the gen;ralizing concept of a wellfactored 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 wellfactored
cateCCr y b ..d .. a. very nice way into the "oduct"
of three categories. Then we show that a ;ellfactored
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, failiari~ 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 wollfactored
categories, can ,'tatc some knom results which are not
assured to be background knowledge for the readero
(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 Prmorphisms
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.onorrphic.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:A3 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 Nmorphisms
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 ... n1, 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 Gkl 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 Pimorphism implies f~ is an
identity, where i j, i = 1,2, and j = 1,2.
(2) If fl is grid connected to g, by Pr2morphisms
an1 f2 is rid colneted to G2 by Psnorphisms, 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 rFmorphism and
1. 
h is a _2morpphism.
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 rellfactored
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 catioryi .' 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). PToton. 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 C2morphisn and 1 = 10c. Let i 1(Cl1) Cl ObC
Define P2 ana:ogously; i.e., define by restricting the
second coordinate. 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 =ellfao
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 P2morphisns and
(f2,1) and (g2,1) ere grid connected by Frmorphsiss. 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,...,n1. 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 P2c 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,2morphi.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_ cardiality. Lot 2 have the standard
wellfastori o ion6 Let inc :1}~{ ,2L be the inclusion
function. Thsen (i1ncinc);:( 1 ,{1})>(1,2>,1, 2>) is a
Cmorphism. 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 vlellfatored 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 wellfatored.
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 csegory 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 Mi>
is a function so that (R,M,p) is a left Pmodule. A morphism
in R taking (R,!,p) into (R',iIQ,p') is a pair (f,g) so
that f is a ring 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 Emodules 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 morhisn
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
Proof Since P, and Pf are the standard wellfactori
zation, .: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~ PP 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 Rmodule, (M,S,p2)
is a right Smodale, 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 P2subcategories in 1.12
by holding 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 ellfactorization of Rp given in 1.13 induces a well
factorization of .
Proof. This is a straightforward 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 wiellfactcorod, 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:SX6 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:XX 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 XCX so that p is continuous
and andnd X are Hausdorff spaces. Then C is wellfactored
by the standard wellfctorization.
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 wellfactored
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).
Gspaoe is a
The image of
(1) For
(2) For
i ellfactorization 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 Gspace is called cffective provided xs = x implies
s = 1. Let X = 'xxs)EXXX x s in X and s is in C>,
where X is an effective Gspace. There is a function t:
X">3 defined by xt(x,x') = x' and hich is called the
transl.tion functio0. A Gspace X is called oril.cipal
provided X is an effective Gspace with a continuous trans
lation function. A rJirrr. Gbundle 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 Gbundles is a prnpQ p~al
no irphis o provided u:X is a GModule 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 wellfactori zation of Dan given in 1.11,
it suffices to prove that condition (3) of 1.4 holds.
Suppose (u,f):(X,p,EB)(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 GsTace X are
said to be Gequivalent 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 Gbundle and let F be
a left Gspa~ce. The equation (x,y)s = (xssly) defines
a right Gspace structure on XxF. Let Xp denote the quotient
space (Xe?)/G and let pF:XpB be the factoriza.tion of
XXF Pr'lX 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 fibre F (viewed
as a Gspace) ,sn as ccia tid :1 ,i2 nc3Dgl b'Idle dc A fibre
sorohiam 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 Gorphism 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/CGpp, ,)(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 tha (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 Gjmorphisn, 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" arrowrs 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 Afi>,2> Cr*... >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 AlB (
BE0 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
cateo;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 contiable
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 ? LatLle equivalence relation, Some sources refer
to a ca teorical, 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 .as C ff).
CHAP7_, .II
TWO E!': DF: TiHOREMS
In this chapter we show; the equivalence of the general
concept of w;ellfactored category to two special kinds of
wellfactored 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 14 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 idntity 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, and 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 the 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 orp' ...:' 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,SMM) 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:ST is a morphism
in Sgp(N) a 1nd g .:> is a morphism in M (JT = (,s,
the following diaraa comimutes
s0:,T i 1;
f03 g
)
The com~ ~sition is jnducod from I1. It is clear that Lact(i)
is inlcee a cateor y.
(2.7). .iflitio. ei't '1 = (L,0,s) be a multiplicative
category .:ithout. 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; 1g). The follouing diagram coammutes:
C
1 i
(KoK)oC
1 01, 1
C
Kv I
Ko C c9 .
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:Cl>ct(^) is clearly a functor and is onetoone
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 uellfactored categories into categories of left actions
so that the enb.ddin preservs 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 wellfactored 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 :ellfactorization. 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 :ellfactored 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. Howver, P
and 2 are not the standard wellfactorlzation. It is
clcar that by letting CJ = C~ C' = C and C = C that
.a . z2 3 3
A is then r:llfactored in the standard way by "1 and P
as a .u.boatese_,y of" .f.C ~ By 2,10, given aany cate
gory A wit h any ,ellfac.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 cxist categories
C~ .2, a nd C. such "tht we have an eLbdd ing F:S>
C1x C2xc^ th ch preserves Frmorphisss. Thus, F(s2) is
a r2morphis s r. F(s1) is a r1mIorphi~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 onetoone o0 norphisms so s2o1 sl 2. We
know, H is w.elldefined 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 onetocne.
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 subnonoids 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 1Y 1
is a normal subgroup of G. Thus, by uniqueness of 12F
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:A3, 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 catsorical 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 rinorphisms
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 Froroi.. '." grid connected by
P2morphi. 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 P1identlty) alor i"ae .gl' g n nsd a yrelated path
category 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 Porphiss 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
ifor 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 reans 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 P2categcry.
(b) Si = (Vp,*.. sk+1 Vk"ki, ..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
[2norphisms oa iad b so that asovk = o.b,
T: (a) corresponds to (1) in the definition onf,
(b) correspols 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 Pnorphiss grid connected by P2.orThias.
(2oin9).. _:'oCo to hon. Let h and 0h be .. 1 "phsns
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 Psorpl'isms grid connected by
Proof. The proof is th:e S aeo as for 2.18, :ilth 1 and 2
intercharsed.
(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:AB is a norophisn in C. Thon there
xist a uique facto'iza.tion f = hog .her g is a r rorphism
and h is a P2 orphion. Then define F(f) = (Ch7[gC3):
(1A2A))(B B 2,B). It is clear that F is .elldefineSd
Now ie must show that F is a functor. Since A1 1 1l
A A 6
and since 1A is both a 1or:phi2s an' a [2orph; s, P (1)
(CL. ) (A., A) F (A) Suppos A 
Then we have r1c 2 factorizations s indicated by the fol
loting co~utative 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 clsar that F is onctoone on objects. Suppose
F() (f) where f and f' are moiphisms 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 rmorphisms and g and g' are Fmorphisrms
grid connected by orp.his ras By definition of a well
factored category, wie then have f = hog = h'oge =: f.
Thus, F is onetoone 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 wellfactored 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 P2norphisms. This completes the proof.
(2.21). Re l~ak. NowT that we have gone through the
construction of C' C2 and C3 we may ask what Ci, C.,
1. 3 L1 2
and C are for some familiar cateSories.
3
Suppose S is a monoid willfactored 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 Smorphism. Hence,
C1 2' Similarlly, C. 0 and card(C) = card(Ob S) = 1.
It is easy to see that F:SC 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 (g1)_(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 rr' = 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). oiqoosijtion. LGo t N be as in 2.22. Let C =
Sgp(N) and C2 i Let C, be the discrete category with
objects z:SOSlA so that z is a rorph'.I.n in 1. Thcn Lct (l.)
is .:llf].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(_).
Froof. 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
SA >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 Porphism 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 iP2 factorization is unique so Lact(fN)
is wel). .lfactored in the standard mnsinzr as a subcategory of
C x C x C.3
(2.2!43). "Th \r. Let C be a :llfactored 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 ,ellfactored subsoatcory
of iact(JS ) so that the .'illfactorization of Lact (N) induces
the elfcton 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 nor 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 ,sellfactored 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 lfactored 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(!) ;ellfaotored 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 corespcnd 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 ntuples (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:UiVj(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 morphis 1 .)o For i =I ,.,.ck1,
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
l01,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,,..,t1, 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 rmorphism. 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 Le 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 folloini 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, f4cb2
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, colitions
(1) and (2) are as follos:
(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) where ,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 @:xi2.: On objects, 0 is the saS
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 kp 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 5ite 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 codition (2).
We need to sh r Q is alldefined 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; coi 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]. However, 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) will 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]][[u13
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 Cimorphism, =
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 welldefincd, 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 morphism 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 onetoone on morphisms. It is clear that each
Pimorphism in C maps to a Pimorphism in Lact(N), =
1,2. This completes the proof of 2.24.
CHAPTE'0. T7T
A SUFFICI:".'? COCDITION T : ) A ELLFACOrED.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" caL ories of La,.rvere and Linton.
(3.1). Rrik. We recall the folloiing definitions.
(A,U) is called a cop'~' cat ... 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 reation 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 epinorphi"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
er:\ 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. eserves and ref3>lcts "reu lr epi opiss.
(AsU) is said to be a vaijtais.t catepory 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 Rnodules 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 wellfacto.'
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 xegulr 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 dira:'i in M
commutes
a0b u@v
ABt AB A"OB"
A PC03cA> 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 corroduct 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). Pooit'.o. The transformaations d. and d are
natural.
1'rPoof. The proof of this proposition follo.s in a
straightforward manc'r from th3 definition of a coproduct.
(3.9), TPo~.s o ._o. Tne following ito diarams 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 eqation, do ( J. )o" = do (u u '
C(161)0eUl C = (1C'1 )) =
L o * ty
= 0 (10.d)od uA@(O3). Thus, by definition of coproduct,
(1) comiutes. The proof that diagr~a (2) comubtes 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
Dco(i(eouu E )) = uc (0 u i)) = ((l0u,)l)<
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@(RO))
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
shown 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 subtdiagSam (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) cor.u~;es by the definition of d and diagram
(4) col"luuli'e since d is natural.
Diagram (5) is redra,m as diaram~ (5) and is decom
posed into su" 2lig s 5.15.o2, and 5.3. Diagram 5.1 coi
mutes by conlition (d) in def.in. on 2..1, Digrac 5.2
coriutcs 5nri? c< is natural. Di c 1 5,3 C a ;ctCes since
(Rp) is an object inl Sp(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
5b 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
51~
C11*< ~ 
r1 *i == 0
zi32 i=
rd rl
z1.
' 121 >
0 0
p.; t\KCi r
O~~
\ \ 
r4
\ \ o
clas ci
'I"'
\ l\ rl
S c
rr7
^~ \ \^^ :
vIY L.
\ 00
5 > 121
\M arz v L .
4 \j w, \
r \ 0I \r
48
1
r:S1
SID
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(rn (6) corrnutes since .d is natural.
For diagram (7) to co~ r.auo need (1 p)Ol]oc = (1,p31).
By the definition of coroduct, 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)ou0 = 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 diagram 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 O1
~01 ** 1
C
(X! \
\ Mrl r^ 
V: 611
rI
0 t I
'I 0*
\ni 
0 0 'I '1
* ~ * 0
nicl 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 ., II
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 subdiagarams
(a) and (b) conniuteo
{ (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`Ledu 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 noea 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:GpG 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:AT(B), then there
exist unique f and f so that f is a Pmorphism, 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:
ATS(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:AT(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 r2orphism, 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 folloring 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 rLorphism,
so that h = h2oh1. Bat v is a Gpmorphism 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 nonidentity
morphisms a:AB and al:B>A with the indicated compo
sition. Let C = C' = AxA have the standard wellfactori
zation, whert P> (C) denoos 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 follo7s: 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,p1))
SB(ii) = I(B~) = (1,~l)o(1p) = SB(l,p)oSB(19p1).
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(BA) 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_)x79Lact ()
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
r1morphisrm, 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 [Pmorphism
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 PEmorphism in Lact(N) It is clear that the
components of pl and of p2 are uniquely detc rinedo 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 comiutativity 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 nonenpty.
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:CxEC
and p2:CCEE 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 ic'_'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 0U 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.
Proof. Since a rc_ i.ar epimorphismr is an extremal
epimorphism, we 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 :AOBB3 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
mAOBtS n
Then (1) commutes by the naturality of o. Also, (2), (3),
and ) commute since voz z No(uiv). Finally, (5) cOe I
mutes since u is a morphism in Sp(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 irplies 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. Hoever,
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 epiD
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 welldefined, 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 welldefined. It is clear that
t* is a morphism in Ab an it is left to the reader to verify
that (k*,t*) is a morphisi 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 onetoone.
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 elldefined, 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 Fnrthr 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 knor 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). Vaietj. 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 morphisis 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:
CinsxEns 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 BLOGRAPHY
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, University 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 Ct ,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 Universitly 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 Thoory at Eo;:doin Colle e in iMaine. He is a monbr
of the ir '"i. n Iaths',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 Collee of Arts and Sciences
and to the Graduate Council: and w;as approved as partial
fulf ilrlmnt of the rqu.rnents 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
