 Title Page 
 Acknowledgement 
 Table of Contents 
 Abstract 
 Introduction 
 Preliminaries 
 Generalities 
 Categorical congruences 
 Categorical equivalence relations... 
 Images 
 Unions 
 Rectangular relations 
 Bibliography 
 Biographical sketch 

(RxZ) (XxS)   XxYxZ  > XxZ
cl.T
RoS
It is evident that T RRoS whence clT C RoS. But (RoS,j') is the
intersection of all closed subsets of XxZ through which Y factors.
Thus (RoS,j') C (clT,j). Hence (RoS,j') (clT,j).
1.35. Example. With the hypothesis of Example 1.34, T and clT do not
necessarily coincide.
Proof. Let X = Z be the closed unit interval with the usual subspace
topology induced from the real line. Let Y be the closed unit interval
with the discrete topology. Let R = {(x,y): y = x} considered as a
subspace of XxY. Let S = {(y,z): 0 < y < } considered as a subspace
of YxZ. It is easy to see that both R and S are closed in XxY and YxZ
respectively.
Clearly T = {(x,z): 0 < x < } and c3T = {(x,z): 0 < x < }
whence T clT.
1.36. Example. In the category Top, the composition of relations is
2
not necessarily associative.
Proof. Let X = Z be the closed unit interval with the usual subspace
topology induced from the real line. Let Y be the closed unit interval
with the discrete topology. Let R = {(,)} considered as a subspace
of XxX. Let Sbe{(x,y): y = x} considered as a subspace of XxY and let
Tbe{(y,z): 0 < y < } where T is considered to be a subspace of YxZ.
Hence, each together with its inclusion map is a relation since each
of R, S, and T is a closed subspace of XxX, XxY, and YxZ respectively.
It follows that RoS = {(,)} and that (RoS)oT = 0. But
SoT = {(y,z): 0 < y < } and from this it follows that
Ro(SoT) = {(,z): zeZ}. Hence Ro(SoT) # (RoS)oT.
1.37. Remark. At first glance, the results of Examples 1.34, 1.35 and
1.36 seem to be pathological, thereby casting doubt on the usefulness
of the categorical definition of composition of relations (1.26). How
ever, this should cause no more anxiety that1 does the fact that the
set theoretic union of two subgroups of a group is seldom a subgroup.
Furthermore, the results 1.31, 1.38, 1.39, 2.4, 3.1, 3.6, 3.9,
3.10, 3.12, 4.22, 5.20, 5.23, 5.25, 5.26, 5.27, 5.34, 6.13 and 6.27
seem to indicate that this definition (1.26) yields nice theorems which
reenforces its appropriateness.
1.38. Theorem. Let (R,j) be a relation from X to Y and let (S,k) be a
relation from Y to Z. Then (RoS)1 and SloRI are isomorphic relations
from Z to X.
Proof. The following products shall be used:
(Xxy,T1~ 2), (YxZ,1, 2), (RxZ,pl*,o2*), (XxZ,Pl,p2), (XxS,1 ,51 2)
(Xx(YxZ),lT '2*), (XxyxZl, 1'2,iT3), and ((XxY)xZ,pl,p2).
The notation " over a projection morphism shall denote the
projection morphism of that product object where the product is taken
in reverse order; i.e., the projections of YxX are r1 and 72 and the
projections of ZxY are TfI and T2.
Consider the following diagram. It will be shown to be commutative.
ZxR1
zP 2,' 1 >
RxZ *y (XXY)xZ '>  "P ZX (Yxx)/
V01 IV )
(RxZ)CI(xxS) X. : xyxZ >~  Zxy>rX
X2 02 02
x XS 11') [X(YXyZ) t~s (Zxy) cX
k*xl1
"X,
37
01 = < `Ip2',2P2> and 02 = <171 T*, 2 27*,'7 2
11 = P1P> = P2"
^1 <3'2 1 = 3l = P2
2 1 p 1> 71P2 = I <7r2,17T>p1 = 72PP.
^72
1 30]p]> = '2P2pl> = 7T2<'r2,7I>PI = n1PI.
I 1 1 2' < 2I' 311 2 1 >(D 1 7 12 1 2 2T12*
" 3<13'T2,1> = 31 2 = 2 22.
i 02<2 1,T1 *> 2= 2 1 T" 2, 1> T<, T l*> f2< 2,']>7n2* = 1712*.
2 <7T 3 ,1T r> = 12 1 2 2 = 2
3 0222" ,7 i*> = 2T1*T<2',12f>72*,71*> 71= '
3n22 = 212 = 2
Thus 01p1 > = 1 and 02 = I2 <2, >>i7r*,71*> .
0lp >(jxl) = P2(jx1) = p *.
P(ij*X p * = *.
1 2 1 1 2 12
2 2 1 2 2 1 21p
p (lxj*) = J*p2* = JPI*".
Thus >(jx]) = (Ixj:)
1<2' 2 I>*>(1xk) = < 2' > ,2*(1Xk) kp2 = k*T
Il(k*xl) ,1> = k*pl = k*T2.
2 2<2 T *>(lxk) = 1*(1xk) = PI.
2(k*xl) = 2 = ~1
Thus <.T2*1*>(lxk) = (k*xi).
Hence the diagram is commutative.
Consequently by the definition of intersection, there exists a
unique morphism & such that the following diagram commutes.
Y
It is easy to see that the following diagram commutes.
Y X
(RxZ) r (XxS) > XxxZx
(SlxX)( (ZxR1) < ~I'3 Z xX
S1oR1
Since (RoS,6) is the intersection of all extremal subobjects
through which y factors, then (RoS, ) < (S oR ,a ).
1 
Hence there is some morphism p such that a'p = B. Consequently,
ip=
S = ts' = t t and the following diagram commutes.
S oR1
2 i \ 2.'1> a2
SB
RoS XxZ  ZX
(RoS)1 Y'
Since ((RoS)1,B*) is the intersection of all extremal subobjects
through which B factors then ((RoS)1,B*) < (SloR1,a').
Now applying the above result to (S1,k*) and (R1,j*), it fellows
that ((SloR1>)1,a') < ((R1)o(S1)1,B#) E (RoS,Z) (1.11) whence
(SloR1,a') < ((RoS)1,6*) (1.12), so that
1.39. Corollary. Let (R,j) be a relation from X to Y. Then (RoR1,j#) is
a symmetric relation on X and (RIoR,j') is symmetric on Y.
Proof. ((RoR1)1,j#*) E ((R1)1oR1,j) (RoR ,j#) and
((RloR)1,j*) = (R1o(Rl)I,j) = (RIoR,j") (1.38 and 1.11).
1.40. Proposition. Let (R,j) be a relation from X to Y and leL (S,k) and
(T,m) be relations from Y to Z. Then
(Ro(S(T),e) < ((RoS)(1(RoT),6).
Proof. By Proposition 1.5 there exist canonical isomorphisms:
i: (RxZ) (Xx (S (' T)  (RxZ) .( (XxS) 0 (XxT)
y: Xy(S AT) r > (XxS) 1(XxT).
Consider the following commjutative diagrams.
X2
S TT  S
A1 O^ k
T  yz
m
ix1
RxZ (XxY)Z
'B 01
xx (sr (T) >S R (Xxs) xT) (X T)   (yxZ)
Let (T,B) be the epiextremal mono factorization of <, ,w3>y. Thus
the codomain of T (domain of S) is Ro(S{ T). Since this is the intersec
tion of all extremal subobjects through which yI factors it follows
that (Ro(SfT),6) (RoS,61) and (Ro(SX.T).) < (RoT,62). Thus
(Ro(SAT),8) < ((oS)A)(RT),C) (1xa19).
SECTION 2. CATEGORICAL CONGRUENCES
2.0. Remark. Lambek [16,pg 93 presents the following definitions for
dealing with rings which have identities.
More general than homomorphism is the concept of homo
morphic relation. Thus let 6 be a binary relation between rings
R and S, that is essentially a subset of the Cartesian product
RxS, then 6 is called homomorphic if 000, 191, and rl0s1, r20s2
imply (rl)6(sl), (rl+r2)6(sl+s2), (r r2)6(s1s2). Of course a
similar definition can be made for any equationally defined
class of algebraic systems.
lHe goes on to add:
A homomorphic relation on R (that is, between R and itself)
is called a congruence relation if it is an equivalence rela
tion, that is reflexive, symmetric, and transitive.
Lajibek notes that a symmetric transitive relation is not neces
sarily reflexive, but is a congruence on a subring. He also notes that
a reflexive homomorphic relation is a congruence. This latter result
is due to the fact that all homomorphic relations are difunctional
(see 5.22).
We will generalize all of these results. However it must be noted
that in the category Rng a congruence is an equivalence relation and
conversely. Thus we shall obtain the result that if (R,j) is a symme
tric transitive relation on an object X then R is an equivalence rela
tion on an extremal subobject of X. However, this result must be post
poned until Section 3 (see 3.4 and 3.10).
Also the result that the reflexive diiunctional relations are pre
cisely the equivalence relations must be postponed until Section 5.
In this ?ecticn it will be shcwn that a (categorical) congruence
is a (categorical) equivalence relation and that congruences (when '
has coproducts) are determined by (categorical) quotients (2.12).
If f is a set function from a set X to a set Y then the set
{(x1,x2)EXxX: f(x1) = F(x2)}
is called the congruence (sometimes kernel) determined by f. It will
be shown that (categorical) congruences have behavior similar to that
of the above set (2.8, 2.10, 2.11, and 2.12).
2.1. Definition. If (R,j) is a subobject of XxX then (R,j) is called a
congruence if and only if there exists a morphism f with domain X such
that (R,j) is the equalizer of fnl and fr2.
j 1 f
R  XxX .X > Y
T2
If g is a morphism with domain X then the equalizer of girl and gn2
denoted by (cong(g),i ) is called the congruence generated by g.
2.2. Remark. If X is a object then (Ax,i ) is the congruence gener
ated by 1X'
2.3. Remark. It is easy to see that (R,j) is a congruence on X if and
1
only if (R ,j*) is a congruence on X.
2.4. Theorem. If (R,j) is a congruence on X then (R,j) is an equivalence
relation on X.
Proof. Since (R,j) is a congruence on X there exists a miorphism f with
domain X such that (R,j) is the equalizer of fir, and firi. Recall that
(AX,i ) is the congruence generated by I whence iL Xir = 1XT2iX. Thus
fTi = fn2Si so by the definition of equalizer there exists a morphism
X Y
A from AX to R for which jX = iX. This implies that (AXiX) < (R,j) so
that (R,j) is reflexive.
To see that (R,j) is symmetric, observe that
fTrlj fr2 = f1r1jT^J Tr2j. Thus flrj*T f72j*T so that
since T is an epimorphism it follows that flj* = fj*. Hence, from the
definition of equalizer, there exists a morphisnm from RI to R for
which jn = j*. This implies that (R",j*) < (R,j) so that (R,j) is sym
metric.
Consider the following products: (XxX,Tl,Tr2), (XXXxX,712,T 23),
((XxX)xX,plp2), (Xx(XxX),P1,p2), (RxX,1 *,T2*) and (XxR,I,i2). To see
that (R,j) is transitive, consider the following commutative diagram.
jx1X
RxX >> (XxX)xX
Y
(RxX) ri (XxR)   XxxxX
XxR  Xx(XxX)
1Xxj
Let (T#,j#) be the epiextremal mono factorization of y.
Recall that the codomain of T# (domain of j#) is RoR.
Next, it will be shown that fiy = f2y = fY.
y = f 302(]xj)X2 = fT22(1Xxj).2 = fT2ji2,2 = fljt2A2 =
fl 2(PX:<)X)2 = f 022(lXxj)'2 = f 2y.
fly f= 10i(jxlx)A f1pI1(jxlX)i f1 jTl*'/ = f2J 1i*l
fr2p1(jxly)XA = f 12il(jx1X)X1 = f 2y.
Thus fT,;~i 'i, 3>"' = fy = fTy fT 2<,3>y; so fTrj/T# fi#2
Again, since T# is an epimorphism, it follows that firT3j = frr2j#. By
the definition of equalizer there exists a morphisrm k from RoR to R for
which jk = j#. This implies that (RoR,j#) < (R,j) so that (R,j) is tran
sitive.
2.5. Theorem. The intersection of any finite family of congruences on
any object is a congruence.
Proof. Let {(Ei,ei): iI)} be a finite family of congruences on X. Then
there exist morphisms fi with domain X such that (Ei,ei) is the equal
izer of fini and fi72 (2.1). Let the codomain of each fi be denoted Yi.
Consider the morphism from X to TTYi and consider
iel isI
the intersection ( ( Ei,e).
ieI
It will be shown that (f1 Eie) is the equalizer of 7rl and
iel iel
T .2'
iel
e 71 f i
SEi XxX ~~_. X  " Y
Xi ei fi V Pi
Ei Yi
ie
First observe that: pjlne = fj e = f 72e = p r2e for
ici isI
each jIe. Thus 7Te 7 e.
iel iel
Now if g is a morphism from W to XxX such that rr2g
irI ieI
then fj7Tg = p 1g = PjTr2g = fj2g.Thus by che definition of
iel iel
equalizer there exist morphisms ki from W to Ei so that eiki = g for
each ilI. Thus by the definition of intersection there exists a morphisn
k from W to (\ Ei such that ek = g. This implies that (f Ei,e) is the
ieI icI
equalizer of 72.
ieI ieI
2.6. Proposition. If is complete then the intersection of any fam
ily of congruences on any object is a congruence.
Proof. Repeat the proof of 2.7 assuming I to be infinite.
2.7. Proposition. Let Q be the family of all congruences on X and let
(fQ,p) be the intersection of this family. Then ( O and Ax are isomor
phic relations on X.
Proof. If (E,e)eQ then (Ee) is an equivalence relation and hence is
reflexive (2.4). Thus (xX,ix) < (E,e). Hence (Ax,iX) < (r(Q,p). But
(Ax,ix) is a congruence; hence (i'P,o) < (AX,iX).
2.8. Proposition. Let f be a morphis from X to Y. Then f is a
monomorphism if and only if AX and cong(f) are isomorphic relations on X.
Proof. Since (cong(f),if) is an equivalence relation (2.4) it is reflex
ive and hence (Ax,iX) < (cong(f),if). If f is a monomorphism then
fTlif = fn2if implies that "lif = T2if. Hence there exists a morphism k
for which iXk = if and consequently (cong(f),if) < (Ax,iX).
Conversely, suppose that (cong(f),if) (Ax,i') a and a and are
morphisms having domain Z and codomain X such that fa = f. Consider
the morphism from Z to XxX. fT = fa = f, = fa2 so that
there exists a morphism X from Z to AX for which iX = . Thus
a = 1:1 = nlijXX = 2 = B. Consequently a = 8 so that
f is a monomorphism.
2.9. Definition. A morphism f from X to Y is said to be constant if
ad only if for all pairs of mrphiss a
aud only if for all pairs of morphisms Z TLT2ET^. X, fa = f6.
2.10. Proposition. Let f be a morphism from X to Y. Then f is constant
if and only if (cong(f),if) F (XxX,1X).
Proof. If f is constant then fir = f2r so that fIrlx = f2X 1 Thus
2X 1 AA 2 XXX
there exists a unique morphism k from XxX to corig(f) for which ifk = 1XxX
whence if is a retraction. But since if is an equalizer, it must be an
isomorphism (0. 4 ) so that (cong(f),if) and (XxX, 1XxX) are isomorphic
relations on X.
Conversely, suppose that (XxX,1XxX) (cong(f),if) and that a and
6 ere morphisms with common domain,and codomain X. Consider from
Z to XxX where Z is the common domain of a and B. Since
falXx, = f21X, it follows that fir = fT2 so that
fa = fT1 = fi2 = fB. Thus f is a constant morphism.
2.11. Proposition. If f from X to Y, g from Z to Y, and h from X to Z
are ( morphisms such that f = gh then (cong(h),ih) < (cong(f),ir).
Furthermore if g is a monomorphism then (cong(h),ih) (cong(f),if).
Proof. Since hrlih = h 2ih it follows that ghrlih = ghT2ih so that
fTlih = fT2ih. Thus there exists a morphism k from cong(h) to cong(f)
for which ifk = ih. Whence (cong(h),ih) < (cong(f),if).
If g is a monomorphism then f'lif = fn2if = ghl1if = ghr2if
implies that hTlif = hi 2if. Thus there exists a morphism k* from cong(f)
to cong(h) for which ihk* = i,, whence (cong(f),if) < (cong(h),ih). Con
sequently (cong(f),if) = (cong(h),ih).
2.12. Proposition. If ( has coequalizers and f is a I morphism from
X to Y and if (f*,Z) is the coequalizer of Tiif and r2if then cong(f)
and cong(f*) are isomorphic relations on X.
Proof. Since fnif fiT2if then by the definition of coequalizer there
exists a morphism k* from Z to Y for which k'f* = f. Since
f*li = f2if it follows that flif = k*f*rri = k*f*if =
fr 2i ,. Thus there exists a morphism k from cong(f*) to cong(f) for
which i k = i Consequently (cong(f*),if,) : (cong(f)i ).
f f C f
Now since f* is the coequalizer of r if and i2 i then
f*7lr i f2r i Hence there exists a morphism k' from cong(f) to
cong(f*) for which i fk' = i Consequently (cong(f),if) <
(cong(f*),if,).
2.13. Proposition. If is complete and 0 is a family of congruences
on X generated by morphisms f: X >Y and if (,r1,p) (A ,iX) then
the unique morphism 0 from X to TT Y such that rfO = f, is a monomor
phism.
Proof. Observe that for each f, fir i = T~f, r i = 2i = f i 0. Thus
 10 r 2f 10f
f
it follows that (cong(),i ) < (cong(f),if) for all X f Hence
(cong(0),i ) < ((n,p) (1.19). Since ((),p) (A,iX) < (cong(0),i)
(2.4) it follows that (f ,p) (A ,iX) = (cong(O),i). Thus 0 is a
monomorphism (2.8).
2.14. Corollary. If is complete and 0 is a family of congruences
on X generated by morphisms f: X Y and for some g: X  Yg,
f 0
g is a monomorphism, then the unique morphism 0 from X to TT Yf such
that f 0 = f is a monomorphism.
Proof. Since g is a monomorphism then (cong(g),i ) = (A ,iX) (2.8). Thus
g X X
(C0I,p) < (A,i X) by the definition of intersection. But
(AX,iX) < (f[I,p) (2.4 and 2.6). Consequently (6X.iX) = ((In,p) and the
result follows from Proposition 2.13.
SECTION 3. CATEGORICAL EQUIVALENCE
RELATIONS AND QUASIEQUIVALENCE RELATIONS
3.1. Theorem. If {(E.,4 ): icl} is a family of equivalence relations on
a object X then their intersection (rf E.,4) is an equivalence rela
iel
tion on X.
Proof. Since (AX,i ) < (E., .) for each iel it follows that
X k 1 1
(A ,i ) < ( Ei.4) (1.19). Hence (\n E.,<) is reflexive.
iel iel
Since (fl E.,4) < (E.,V.) for each iel and since each (E.,i.) is
isl
symmetric it follows that (( \ Ei)l,*) < (E 1, i*) < (E.,,.) for each
I 1 1 1
iEI
isl (1.12). Thus ((ft E.)1,4*) < (r\ Ei,4) and hence (r Ei,r) is sym
ilI iel iel
metric.
Since ((~l Ei,) < (Ei,4i) for each ilI then
iEI
((C E.)o(R E.),#/) < (E.oE'.,4#) for each ilI (1.30). And since
idI icI
(EioEi',i#) < (Ei,4i) for each icI it follows that
((f, Ei)o( ( Ei),#) < (fn E. ,) (1.19) whence (f Ei, ) is transitive.
ieI iel iIe iil
Thus it is an equivalence relation.
3.2. Definition. A quasiequivalence (R,j) ou X is a relation on X which
is both symmetric and transitive.
This term is due to Riguet [221 ; however, Lambek [13] calls
symmetric transitive relations subcongruences. Wiile this term sub
congruence is appropriate in the categories Grp and Ab, it does not seen
co be appropriate in more general categories. MacTane I calls such
relation .s sy1Jcric idemporents.
3.3. Proposition. If (A,a) is an extremal subobject of X then (AxA,axa)
is a quasiequivalence on X.
Proof. Consider the products (AxA,pl,p2) and (XxX,Tr', T2). Since a is an
extremal monomorphism then axa is an extremal monomorphism (0.20) and
hence (AxA,axa) is a relation on X.
Consider the following commutative diagram.
axa < il ,Il>
AxA >  XxX > XxX
T (axa)*
(AxA)1
Since 7l <;T2',~>(axa) = 2(axa) = ap2;
ir (axa) = ap1 = ap2;
7r,2 <2'l>(axa) = l (axa) apl; and
,2(axa) = ap2 = apl
then it follows that (axa) = (axa). But is an iso
morphism hence an epimorphisn and axa is an extremal monomorphism;thus by
the uniqueness of the epiextremal mono factorization of <72',71>(axa)
(0.13), ((AxA) ,(axa)*) = (AxA,axa). Thus (AxA,axa) is symmetric.
To see that (AxA,axa) is transitive, first consider
(((AxA)xX)tA(Xx(AxA)),y) where y is the unique extremal monomorphism
induced by the indicated intersection. It will next be shown that
(AxAxA,axaxa) and (((AxA)xX),1 (Xx(A';A)),y) are isomorphic as extremal sub
objects of XxxxX. To show this it will be shown that (AxAxA,axaxa) is
precisely tha intersection of ((AxA)xX,0 ((axa)xlX)) and
(X:(A,A),02 (l x (ax;a)).
Consider the products: (XxXxX, 1, 2, 3), (Xx(AxA),plp2),
((AxA)xX,I, 2), (Xx(XxX),l '*r *), ((XxX)xX, 11 2*) and
(AxAxA,p 1,p2'p3
Observe the following qualities.
S1] ((axa)x) 1,> = ((axa)xl))< ,a3 > =
1T(axa)1P'P2> ,ap3> = l (axa) 2
ap1 = ap1 = (axaxa).
T2 1((axa)xl x)< ,ap3 >
3, 1((axa)xl x ,ap3>
102(lx (axa))>
T202 (2Xx(axa))>
r302 (x( (axa))>
= 2TrI((axa)xl )< ,a 3>
T2(aa)< P'P2> = aP2
ap2 = T2(axaxa).
2 *((axa)xl)1',p2>,ap3> =
1XP2<'ap3> = lXap3 = ap3 = '3(axaxa).
= Ir*(1Xx(axa))
1Xp = 1 ap1 = aP1 = l(a>aaxa).
SiTr2*(1Xx(axa))
7l(axa)p2> = l(axa) =
ap1 = ap2 = T2(axaxa).
= T2'r2*(lx(axa))> =
2(axa) = ao2 = ap3
77,(axaxa).
.3
Thus by the definition of product the following diagram conmlutes.
(axa)xlX
(AxA) xX  (XxX) xX
<<'1 2>,ay axaxa
AxAxA ,  XxXxX
,P23 Xx (AxA)  Xx(XxX)
lxx(axa)
Now, if (W,6) is a subobject of XxXxX so that there exist mor
phisms y and y such that 0 ((axa)xl )y = 6 = (1 Xx(axa))y then con
1 2 1 X i 2 X 2
side the morphism = 2 from W to AxAxA. It will
1 1 1 2 1 1 2 2 2
be shown that << ,p >,ap >E = y and > = y
1 2 3 1 1 2 3 2
Since p1< ,1'p2> = 1T = pi1Y1 and p2< 1'P2>i = 2 = p2P1Y1 it
follows that P1<,a3 >E = S P= "
Now since G((axa)xl1X)Y = 02(1Xx(axa))y2 it follows that
P2y = f2*((axa)xl )y = 30 1((axa)xlX)y, = ,3 02(l x(axa))y2
S22*(I (axa))y2 = T2(axa) 2y2 = ap2a2y2.
ILhence p2 1 p2<<1p,p2>,ap3> = ap3& = aP2P2Y2. Thus
< ,ap >E = y .
1 2 3 1
Again since Q ((axa):.l ) = 0 (1 x(axa))y it follows that
X 2 X 2
P1 2 1= *(Xx(axa))y2 = 2 0 (1 X(axa))y2 1 11((axa)x1 )yX =
T Ift 1*((axa)xl )y = Ir (axa)pl = ap
Hence pl> = ap a = ap 1 Y= 1
1 3> 1 111 1 2
Since p 2 P = p2 = P ahd p2P2 = = P2Y 2 2 it
follows that F p y2. Hence p > = = Y
2 3 2 2 2 3 2 3" 2
Consequently
1 2 3 2
From this it follows that (axaxa) = 6. Since (axaxa) is a mono
morphism the morphism & is unique. Thus it has been shown that
(AxAxA,axaxa) is the intersection of (Xx(AxA) ,2 (1 x(axa))) and
((AxA)xX,01((axa)x]X)). It next will be shown that the following diagram
commutes.
axaxa < Tr1,3
AxAxA  XxXxX  XxX

"'AxAA A
(Ti,j#) is the epiextremal mono factorization of (axaxa).
Now r1(axa) = ap1 api = 71(axaxa) and
[2(axa) = ap2< i,P3> = ap3 = 72<1 T, 3>(axaxa). Thus the above
diagram commutes..
Since ((AxA)o(AxA),j#) is the intersection of all extremal sub
objects through which 3>(axaxa) factors then
((AxA)o(AxA),j#) < (AxA,axa)
whence transitivity is obtained.
3.4. Canonical Embedding. Let (R,j) be a relation on X. Let (Tl,jIl
(c2,J2), and (3,j3) be the epiextremal mono factorizations of T1j,
21j, and T2j* respectively. Let RX, XR. and XR1 denote the domains of
ji, j2, and j3 (codomains of TIp2 and T3) respectively.
R  > xX  X
TI
TTr
j' "X2 3
R  c XxX  X
J 12
R  XXX X
SXR
In the categories Set, FGp, Grp, Ab, (RX,j ) may be taken to be
the set {xcX: there exists ycX such that (x,y)ER} together with the in
clusion map. Similarly, in these same categories, (XR,j,) may be taken
to be the set {ycX: there exists xcX such that (x,y)eR} together with
the inclusion map and (XR,j 3) may be taken to be the set
{xcX: there exists yCX such that (y,x)cR1} together with the inclusion
map.
In the categories Top1, and Top2 the extremal subobjects (RX,j ),
(XR,j2), and (XR,j 3) of X have precisely the same underlying sets as
above endowed with the subspace topology induced by the topology of X.
See Section 4 (4.1, 4.2, and 4.3) for a more detailed discussion.
It is easy to see that in the category Set, a symmetric, transi
tive relation on a set X is an equivalence relation on a subset of X.
Recall the discussion in Section 2 (2.0) of the remarks of Lambek who
obtains the similar result for homomorphic relations on rings with
identity. This result we wish to generalize. In order to do thils we must
first be able to pick out the subobject.of X on which the relation is an
equivalence relation.
Referring to the above diagrams, since T is an isomorphism (1.9)
and sialce the euiextremal mono factorizaticr is unique (0.18) it is
clear that (RX,j1) = (XR1,j3). That is, there exists an isomorphism k
from RX to XR1 such that j3k = jl (se, 4.4 and 4.5).
Consider the product (RXxXR,l,' 2). Also, consider the morphism
(jlxj2)< r1'2> from R to XxX. Since
7i(jlxj2)<9lT2> JlP1 = JiTI = 2Tij and
^T2( 1xj2) = .222< 1,T2> = j2T2 = 72j it follows from the defini
tion of product that (jlxj2) = j. Note, is an extremal
monomrorphism since j is an extremal monomorphisr (0.16).
Now suppose that (R,j) is symmetric on X. Then it follows that
there exists an isomorphism a so that ja = j* (1.13). Thus 72ja = 7T2J*
and this together with the fact that a is an isomorphism and the unique
ness of the epiextremal mono factorization implies that
(XR,j2) (XRl,j3). That is, there exists an isomorphism B so that
J2 J3. Thus it is routine to see that the following diagram commutes.
T1
a XXX X
R 1. XR
2 12
SXxX  X
R 1  R
R i
Consider the following products: (XRxXR,pI,p2) and (XxX,~1I ,2).
Letting 4 = (kkxlxR) then (2j2x2) = j, since
11(j2xj2)p = J2P1 = j2BklP = j2BkTl = J1lT = IrTj and
2(J2xj2) = J2o02 j21XRP2 = 2T2 = 2J. Thus the following
diagram commutes and the relation (R,i) on XR shall be called the
canonical embedding of R into XRxXR.
R *RXxXR
Skx1XR
XRxXR
I j2xj2
XxX
3.5. Lemma. Let (R,j) be a symmetric relation on X. Then (R,i) is a
symmetric relation on XR.
Proof. Suppose that (R,j) is symmetric on X and let (R,4*) be the inverse
of (R.~) on XR. Then Y = i*T*. It is easy to verify that
p. Thus since (Rl,j*) is the intersection of
all extremal subobjects through which <72,r1>j facturs there exists a
morphimn X from R1 to R so that (j2xj2)W*X = j*. But j*'T = <2,7T1.>j
whence (j2xj22)T* = (j2xj2) = <2,T>j = j*T. So
j*'T = (1(j2xj2) ) ((2xj2)) *)T*. Since (j2xj2)i* is a mDonomorphism
it follows that XT = T*. Recall that T and T* both are isomorphisms (1.9).
Rence .\ is an isomorphism.
Recall that by the definition of symmetry (1.10), there exists a
morphism c from R1 to R so that ja = j*. Thus
(j2xj2),"1 = j 1 = j j= (j2xj2)~. But since (j2xj2) is a monomor
phism this implies that *Xca = Thus since X and a are isomorphisms,
we have (R,4*) E (R,4). Hence (R,i) is symmetric on XR.
3.6. Lemma. If (R,j) is a quasiequivalence on X then (R,i) is a
quasiequivalence on XR.
Proof. In view of Lemma 3.5 it need only be shown that (R,p) is transi
tive on XR. To that end first consider the following diagram. It will
be shown that there exists a morphism X such that the diagram commutes.
Ox1XR
RxXR  (XRxXR)xXR
xP 01 
II6 "
S(RxXR) (XRxR) XRXPRxXR  XRxXR
2 } x2
XRxR , >XRx (XRXR) j 2xj2
X l J2xj2xj2
RxX : X  (XXX) xx
SI
S(RxX) t, (Xx
XxR   Xx (XxX)
1xj
... XxX
Clearly <, 3>(j2xj2xj2) = (j2xj2)<1 ,P3>. Also,
01 (jxX) (1Rxj2)1 01 = j01 ((j2xj2) jxj2)X =1
O ((j2xj2)xj2)(x1XRlX) = (j2xJ2xj2)61(x1XR)Xl and
02(1Xxj)(j 2x1) )2 = 2(j2xj)X2 = 02(j2x(j2x, ) )2 =
O2(j2x(j2xj2))(IXRpx)X2 (j2xj2xj2)02(lXRX1)X2
as can be verified in a straightforward manner. Thus the diagram above
is commutative, and in particular,
ol(jx1x)(1Rxj2)l = (j2xj2xj2)' = 92(1Xxj)(j2x1R)x2. Hence, by the
definition of intersection there exists a unique morphism X such that
yX = (j2xj2xj2)6.
Let (RoR,i#) denote the composition of (R,p) with (R,u) on XR. Let
(RoR,j') be the composition of (R,j) with (R,j) on X. Then
6 !T# where T# is an epimorphism and <1l, 3>y = j'T' where r'
is an epimorphism. But since yX = (j2xj2xj2)6, it follows that
<'1f,3>yX = (2xj2)6 so that (j2xj2)0#T# = j'T'A. Hence the fol
lowing diagram commutes.
(RxXR) C (XRxR) > RoR
(RxX) n (XxR) XRxXR
T' j2Xj2
RoR '> XxX
1
Since T# is an epimorphism and j' is an extremal monomorphism, by
the diagonalizing property (0.19) there exists a unique morphism & such
that j'S = (j2xji2)# and CT# = T'X. But this says that
(RoR,(j2xj2)i#) < (RoR,j'). Since (R,j) is transitive (RoR,j') < (R,j)
hence (RoR,(j2xj 2)#) < (R,j) =: (R,(j2xj2)i). Hence there exists a mor
phism o from RoR to R such that (j2xj2)io = (j2xj2)y1#. Again, j2xj2 is a
monomorphism so that ipo = p# which says that (RoR,4#I) < (R,y) hence
(R,y) is transitive.
3.7. Theorem. If (R,j) is a relation from X to Y and 7rlj is an epi
morphism then (RoRl,j#) is reflexive on X.
Proof. It will first be shown that the following diagram commutes.
RxX  > (XxY) xX
<1RKlj> 01
R   > (RxX) r (XxRl) 1 ; XXYxX
< 2
XxR  Xx(YxX)
Consider the following products: (XxY,7Tl,r2), (XxYxX,T~1,7r2, 73),
(RxX,p1,o2), (XxRl,pl*,p2*), ((XxY)xX,l*,'2*), (Xx(YxX), 1i 2.),
(YxX, 1, 12), and (XxX,T1, 2).
Now,
1 (jxlX)= il1*(jxlx) r ljPl
2O1(jxl) = T27Tl*'(jx = 2" 
730 (jxl)1R, 1R ,3IJ> = T2*(jxlx)<1R,'1>J = ; 2< R iP> = Tr13
102(iXxj) = iI(Ixxj*)> = P1*Tj,T> = Tlj.
72 2(1XxjT)<1]jT> = (T 2(IXXj*) = ,IJ*j * =
^Tfj*' = 11j = r2J
B302(1Xxj*)<3)j.T> = 722(iXxj*)<' .,T> = 22j*T = 2<2,lI>j = 1J"
Thus by the definition of pr'uct the diagram commutes. Hence
there exists a morphism E so that X1E = <1R,'Ij> and X2E = .
From the above it is easy to see that
1l< ,1T3>YE = TlY = 1Tlj = T = 3Y 2YE. Recall that (Ax,iX) is
the equalizer of T1 and T2 hence there exists a morphism such that
ix = <11l,73>YE.
Let (RoR ,j#) be the indicated composition of relations and let
T# denote that epimorphism for which j#T#/ = y. Thus, combining
the above results, <7Tij,>Tj> = YE = j#T#E = iX.
Since (Ax,ix) and (X,) are isomorphic as extremal subobjects
of XxX (1.21), there exists an isomorphism X such that <1,1x>X = iX.
Consequently, Xt = iX = Y = .
Now 7ITX = l XX = X} = Xrl<7T J,7 rj> = 7rlj and by hypothesis
Trlj is an epimorphism; thus, since is an isomorphism,it follows that
( must be an epimorphism.
Thus has ((,iX) as its epiextremal mono factcrization.
But this means that (Ax,ix) is the intersection of all extremal subob
j.cts of XxX through which factors (0.21). Recall that
= j#t#/E, thus (A ,iv) < (RoR ,ji#) which was to be proved.
3.8. Corollary. If (R,j) is symmetric on X then (RoR,ipV), the composi
tion of (R,i) ,ith (R,.) on XR, is reflexive on XR.
Proof. Since (R,j) is symmetric on X then (Ri) is symmetric on XR (3.5)
hence (O,,) 1 (R1. :*) (1.13). Referring to the diagram in (3.41) fol
lowing the c0 'inition of the canonical embedding it is immediate that
pl is an e; rphism since plI = Bkt and each of B, k, and T1 is an
1 ,.
epimorphisr. Thus (RoR ,1) (RoR,/#) is reflexive on XR (3.7).
3.9. Corollary. If (R,j) is a quasiequivalence on X then (R,j) is an
equivalence relation if and only if lrlj is an epimorphism (respectively
if and only if r2j is an epimorphism).
Proof. If (R,j) is an equivalence relation then (R,j) is reflexive and a
quasiequivalence. Thus by Proposition 1.24, 7rj and T2j are retractions
hence epimorphisms.
Conversely, if 1lj is an epimorphism then applying the theorem
1
and Proposition 1.30, (A ,iX) < (RoR ,j#) < (RoR,j') < (R,j) so that
(R,j) is reflexive and hence is an equivalence relation. (If T2j is an
1 1 1 1
epimorphism then (Ax,i ) < (R oR,j#*) < (R oR ,j'*) < (R ,j*) and
1
(R ,j*) E (R,j).)
3.10. Corollary. If (R,j) isa quasiequivalence on X then (Rd:) is an
equivalence relation on XR.
Proof. (R,i) is a quasiequivalence on XR (3.6) and (RoR,ijI#) is reflexive
on XR (3.8). Thus (AXR, XR) < (RoR,y#) < (R,w) whence (R,',) is reflexive
and thus is an equivalence relation on XR.
3.11. Proposition. If (R,j) is a quasiequivalence on X then (R,j) and
(RoR,j') are isomorphic relations on X.
Proof. By Corollary 3.10 (R,i) is an equivalence relation on XR whence
(RoR,ir#) and (R,,) are isomorphic relations on XR (1.32). Recall that
there exists a mcrphism E such that the following diagram commutes (3.6).
(RxXR)ri (XRxR) ;> RoR
T'X ( 2Xj 2)
RoR ; XxX
j'
Thus (RoR,(j2xj2)i#) < (RoR,j'). But as mentioned above
(RoR,i#) (R,t) hence there exists an isomorphism A# such that
I^,#jt = So by the definition of the canonical embedding (3.4),
j'gX# = (j2xj2) #lx = (j2xj2)y = j. But this implies that
(R,j) < (RoR,j'). Thus since (R,j) is transitive, (R,j) (RoR,j')
which was to be proved.
3.12. Proposition. Let (R,j) be a relation on X. Then (R,j) < (AX,iX)
if and only if R is symmetric on X and (R,i) < (AXR,iXP).
Proof. If (R,j) < (AX,iX) then there exists a morphism a such that
j = iXa. Thus iTl = rliXa = T2ixa = 72j whence
n <,f2,Tl>j = T2j = IrFj = Tr2<'r2,Ttl>j. Thus by the definition of product
j = j. Consequently the epiextremal mono factorization of
j is (1R,j) and so (R,j) (R1,j*); i.e., (R,j) is symmetric.
Recall that j = (j2xj2)d (3.4). Thus
lj FTl(j2Xj2)' = j2p14 and 72j = 2(j2xj2)) = j2P2P'. But "lj = '2j
hence j2P = j2P2'. Since j2 is a monomorphism it follows that
pl = p21,'. Recall that (AXR,iXR) is the equalizer of pI and p2. Hence
there exists a morphism; B such that iXR5 = 4j'. This implies that
(R,,) < (AXRiXR))
Conversely, if R is symmetric and (R,') < ('XR,iXR) then there
exists a morphism 8 such that d = ixRP; hence
P1 = P liXR = P2iXRB = P2. Since (j2xj2) = j, we have
1 j r "Q(j2xj2) = j 2P1 = J2P2 7T 2'(j 2xj 2) = F2j. Thus r1j = Tr2j
so that there exists a morphism a such that j = iX&. This means that
(R,j) <. (Ax, ix).
3.13. Definition. Let (R,j) be a relation on X. Then R is said to be
a circular relation if and only if RoR < R1
This notion is due to MacLane and Birkhoff [20] exercisee 3,
page 14).
3.14. Proposition. Let (R,j) be a relation on X. Then R is a circular
relation if and only if R1 is a circular relation.
Proof. If R is circular then RoR < R1. Thus
RloR1 = (RoR)1 < (R1)1 R (1.38, 1.;2 and 1.11). Hence R1 is
circular.
Conversely, if R1 is circular then by the above, (R1)'1 =R is
circular.
3.15. Theorem. Let (R,j) be a relation on X. Then R is an equivalence
relation on X if and only if R is reflexive and circular.
Proof. If R is an equivalence relation then R is reflexive. Since R is
transitive and symmetric, RoRR R' hence R is circular.
Conversely, if R is reflexive and circular then R1 is reflexive
(1.17) and R1 is circular (3.14). Hence
R R'1oA.. < 'loR1 < R (1.31 and 1,12) whence R is symmetric. Thus
R R1 (1.11).
Now RoR R1oR1 < R hence R is transitive. Thus R is an equiva
lence relation.
SECTION 4. IMAGES
4.1. Definition. Let (R,j) be a relation from X to Y and let (A,a) and
(B,b) be extremal subobjects of X and Y respectively. Consider
(Rir(AxY),y) and (Rfi(XxB),6). Let (T ,a) and (? .2) be the epiextremil
1 2
mono factorizations of i y and r 6 respectively. Denote the domain of a
2 1
by AR and the domain of B by RB. Thus the following diagrams commute.
R
^ Y 2
axly
Rt(Axy) > Xxy c Y
2 AXY ARIrw,
R
R 1 '(XxB) >.  XxY > X
XWX ^ RB
4.2. Remark. Since (X,1 ) and (Y,I ) are extremal subobjects of X and Y
X Y
respectively, then (R '(XxY),y) (R,j) and (RA (XxY),6) 5 (R, j) whence
(XR,&) is precisely the extremal subobject (XR.,j) used in the canonical
nQboddiin (3.'). Since X = Y in 3.4 then also (RY,3) is precisely
(RX,j ) used inl 3.4.
4.3. Example. In the category Set, for (A,a) < (X,1 ), (B,b) < (Y,1 )
and (R,j) < (XxY,1 ),
XxY
AR j{yY: tL exists acA such that (a,y)ER}
RB {xeX: there exists bcB such that (x,b)eR}.
This is easily seen since R(\(AxY) L {(a,y): acA, (a,y)eR} and
R(I(XxB) {(x,b): beB, (x,b)eR}, and AR is the set of all second terms
of elements of R((AxY) and RB is the set of all first terms of elements
of Rf(XxB).
In the category Top AR and RB have precisely the same underlying
sets as above. They are endowed with the subspace topology determined by
the topology of XxY.
In the category Top2, AR and RB have" precisely the same underlying
sets as in Top for it is easy to verify that AR and RB are closed sub
sets of X and Y respectively. Recall that the image of a morphism in Top
is the closure of the set theoretic image (0.15).
4.4. Theorem. If (R,j) is a relation from X to Y and (A,a) is an extremal
subobject of X then (AR,a) and (R1A,B) are isonorphic extremal subobjects
of Y.
Proof. Consider the following commutative diagrams.
R
A
RAxY)  XxY  Y
AxY _axl,,
R1
6 iT
R 1 (YxA)   YxX 'Y
1 xa
Y
YxA RA
j <7f iT>
j
R  Xx  YxX
R1
It can be shown in a straightforward manner that
(ax] ) = (1 xa)
2 1 Y Y 21
where pl and p2 are the projections of AxY. Hence
(1 xa)
= (axl )X = y = j1 = j* .
Y 12 1 2 2 1 Y2 2 1 2 1 1 1
Thus by the definition of intersection there exists a morphism 5 such
that SS = <2 ,Tr >Y = j*TX (1 xa) X Hence
21 1 Y 2'1 2
f 6 = f Y = y = ai But fi 6 = Bi C. Thus, since (AR,a) is
1 1 21 2 1 1 2
the intersection of all extremal subobjects through which T.2y factors
(0.21), it follows that (AR,o) < (R1A,B).
Similarly, it follows that
< >2'1 *X = > 16 = j (axl,])2,X whence h4rc
<2' 1 2 1 3 (ax] )
exists a moiphisnm such that y'* = 16. Then
2 < ,T7 >16 = f 6 = ST2 = y* = aT ;*. Again, since (R1A, 8) is the
ntrscton f all exremal subobjecs 2
intersection of all extremal suhobjects through which h ,1 6 factors,
1
(RIA,B) < (AR,a). Consequently (RIA,B) E (AR,a).
4.5. Corollary. If (R,j) is a relation from X to Y and (B,b) is an extre
mal subobject of Y then (RB,B) and (BR1,a) are isomorphic as extremal
subobjects of X.
Proof. Recall ((R1)1,j#) E (R,j) (1.11). Letting (Rl,j*) play the role
of (R,j) and (E,b) the role of (A,a) in the theorem, the following is
obtained: (BR1,Q) ((R)1B,B/#) E (RB,B).
4.6. Corollary. If (R,j) is a symmetric relation on X and (A,a) is an
extremal subobject of X then (AR,a) and (RA,B) are isomorphic as extremal
subobjects of X. (In particular, (XR,j ) and (RX,j ) are isomorphic as
extremal subobjects of X as was shown directly in 3.4.)
Proof. Recall that (R1,j*) E (R,j) (1.13). Hence by the theorem
(AR,a) (R1A,B) = (RA, ).
4.7. Proposition. Let (Al,al) and (A2,a2) be extremal subobjects of X and
(R,j) be a relation from X to Y. If (Al,al) < (A2,a2) then
(A1R,al) < (A2R,a2).
Proof. By hypothesis there exists a morphism v so that a2P = a1. Thus,
there exists a morphism ( such that the following diagram commutes.
R('(A1xY) Y xy
A AlXYA axl xy
1 Y 2
R (A,2xy) . .. . ..X XxY
Thus w2Y1 = i2Y25 whence, because (A1R,al) is the intersection of
all extremal subobjects through which T2rY1 factors and r2y2C factors
through (A2R,a2), (AIR,al) < (A2R,a2) which was to be proved.
4.8. Proposition. Let (Bl,bl) and (B2,b2) be extremal subobjects of Y and
(R,j) be a relation from X to Y. If (Bl,bl) < (B2,b2) then
(RBi,B1) < (RB2,B2).
Proof. (RB1 ) 1*)E (B1R(1,12) < (B212) (RB22) (4.5 and 4.7).
4.9. Proposition. Let (R,j) and (S,k) be relations from X to Y and (A,a)
be an extremal subobject of X. If (R,j) < (S,k) then (AR,a) < (AS,a).
Proof. In a manner similar to that in the proof of 4.7 one can establish
the existence of a morphism $ such that the following diagram commutes.
R  S
j t k
RTA(AxY) S A(axy)  Xxy
axi
Axy
Hence the following diagram commutes.
R /(Axy)
S 1 (AxY)  XxY
Thus, since (AR,a) is the intersection of all extremal subobjects
through which yT, factors, and 2 factors through (AS,a), it follows that
(AR,cL) < (AS,5) which was to be proved.
4.10. Proposition. Let (R,j) and (S,k) be relations from X to Y and (B,b)
be an extremal subobject of Y. If (R,j) < (Sk) then
(RB,B) < (SBj).
Proof. (RB,8) (BR1,B*) < (BS1,~*) E (SB,8) (4.5, 1.12, and 4.9).
4.11. Proposition. Let (R,j) be a relation from X to Y and let (Al,al)
and (A2,a2) be extremal subobjects of X. Then
((AlfA^R:A2 ) <_ (AIRAA2R,5).
Proof. Since (AljnA2,a) < (A1,al) and (A1CfA2,a) < (A2,a2) it follows
that ((A,! A2)R,a) < (A1R,a1) and ((A(fA2)R,a) < (A2R,a2) (4.7). Thus
((AiCA2)R,a) < (A1RflA2R,Q) (1.19).
4.12. Proposition. Let (R,j) be a relation from X to Y and let (B.,bl)
and (B2,b2) be extremal subobjects of Y. Then
(P.(B1^ B2),B) < (RB1 nRB2,B).
Proof. (R(B nB2),B) ((B B2 )R1,B*) < (BiRlnB R1,*)
(RB1nRB2,B) (4.5 and 4.11).
4.13. Proposition. Let (RI,jl) and (R2,32) be relations from X to Y and
let (A,a) be an extremal subobject of X. Then
(A(R1AR2),a) < (AR,( AR2, ).
Proof. It is clear that there exist morphisms 1, and C2 such that the
following diagram commutes.
R1 N(AxY) "Y
(r.,fl R2) (AxY) )x > xxY
'R2A (Axy)
69
Thus rr2y = 1r2Yl1 = T2Y2 2. Again since (A(R1(\R2),a) is the inter
section of all extreral subobjects through which ir2y factors it follows
that (A(R(.R2) ,) < (AR1,51) and (A(R1 FR2),a) (AR2,a2). Hence
(A(RI AR2),a) < (AR1 AAR2, ) (1.19).
4.14. Proposition. Let (R1,jl) and (R2,j2) be relations from X to Y and
let (B,b) be an extremal subobject of Y. Then
((R AR )B, ) < (RB 2R B,B).
% 1 2 1 2
Proof. ((RflR2)B,6) E (B(RI.R2))1, *) < (BR3~I BR1 ,1) =
1 2 1 2 i 2
(R1 BR 2B,6) (4.5 and 4.13).
4.15. Proposition. Let (R,j) be a relation from X to Y then (R,j) and
(R1 (RYxY),y) are isomorphic as extremal subobjects of Xxy.
Proof. Consider the following commutative diagrams.
R 1) > Xxy
RY  XxY
T 1 T 1
RY 
ji
RYxY ,

R fA(RY7Y) :>>^ XxY
R
Since (jlxl ) = = j, there exists a morphism E
such that yv = j
R
RYxy
.^^.. yxY YXiy
R n( R ((RYxY) XXY
P, R
Thus (R,j) < (R n(RYxy),y). Clearly the reverse inequality holds
so that (R,j) E (R(T(RYxY),y).
4.16. Proposition. Let (R,j) be a relation from X to Y. Then (R,j) and
(Rr%(XxXR),6) are isomorphic relations from X to Y.
Proof. Analogous to the proof of 4.15.
4.17. Corollary. Let (R,j) be a relation from X to Y. Then (R,j) and
(Rt'(RY^XR),6) are isomorphic relations from X to Y.
Proof. (R.j) : (RO(RYxY),v) E (R(l(XxXR),'6) (4.15 and 4.16). But since
(RY,jl) and (XR,j2) are extremal subobjects of X and Y respectively it
folJows that ((RYxY)A(Xx'R),a) E (RYxXR,). Thus
(R,j) E ((Rr;(RYxY))n(Rn(X\XR)),B) E (RA(RYxXR),).
4.18. Proposition. Let (R,j) be a relation from X to Y and let (A,a) be
an extremal subobject of X. Then (AR,a) ((RYAlA)R,&).
Proof. It follows from Proposition 1.5 that RI(((RYAA)xY) and
RPf((RYxY)A(AxY)) are isomorphic relations from X to Y. By Proposition
4.15. (R,j) and (Ro(RYxY,y) are isomorphic relations from X to Y. Thus
RI~(AxY) and R.i((RYAA)>:Y) are isomorphic relations from X to Y. Conse
quencly by the definition of image (4.1), (AR,a) and ((RYftA)R,&) are
isomorphic as extremal subobjects of Y.
4.19. Corollary. Let (R,j) be a relation from X to Y. Then ((RY)R,a) and
(XR,j,) are isomorphic ns extremal subobjects of Y.
Proof. Let (X,1,) play the role of (A,a) in 4.18.
4.20. Corollary. Let (R,j) be a relation from X to Y and let (B,b) be an
extremal subobject of Y. Then BR1 and (Br.XR)R1 are isomorphic as ex
tremal subobjects of X.
Proof. Immediate.
4.21. Proposition. Let (R,j) be a relation from X to Y. Then (RoR1,j#)
and (RoR71,(RYxX),y) are isomorphic relations on X.
Proof. Consider the following diagram.
RYxX
CTix1
xljxl
X1 01
y
(RxX) ( (XxR1)  ., XxyxX > XxX
02
XxR1 XX(YXX)
T xj*
RoR1
To see the diagram is commutative it need only be observed that
(jxl)(Qrxl) = < lT3>01(jY1). To show this note that
(jxl1)(Qlxl) = (jl'xl) = (Tljxl) and
aT Cl(jxl) = 1 1(jx) !jpl Ir r(jxl),
7r0 1(jxl) = 3C1(jxl) = p2 = T2(rrjxl).
Thus, since (RoR"1 ,j#) is the intersection of all extremal sub
objects through which <'7"l,3>y factors,it follows that
(RoRl,j#) < (RYxX,j xl). Whence (RoR],j#) < (RoR1fI (RYxX),?).
4.22. Theorem. Let (R,j) be a relation from X to Y. Then (RY,jl) and
((RoR)X,B) arc isomorphic as extremal subobjects of X.
Proof. Consider the following products: (XxYxX, 1,2, 3),
(Xx(YxX),p1,'2), ((XxY)xX,Pl *,P*), (RxX,PlP2), and (XxR1, 1, 2).
Referring to the diagram in the proof of 4.21 it is easy to see
that: irlY = 1( = (jxl)Aj = il*(xl)X = r1jp1X1. Thus
<1 'l~' 3 = "> jll = jlT 1 1l
RoR
(RoR )x >>
o;'XxX
1
Since <7 ,I3>' Y = j#i//' and rTlj# 5, the following diagram
cc;Lmu:es.
Cl p
iT#
(R (RoR1)X
XI I
RY .
But since has the diagonal property (0.19) and Tr# is an epi
mcrphism and j is an extremal monomorphism then there exists a morphism
: such that ji = B and pl1?1 = T,. Thus ((RoR1)X,B) < (RY,j).
Next it will be shown that the following diagram is commutative.
. .. 
jxl
RxX  (XxY)xX
<1R'i l
R  X)R (RxX)l(XXR1) .  xxyxX
2 02
XxR1 Xx(YxX)
ixj*
= lP1*(jx1)<1Rlj> = UiP1
2P1 (jx 1)1 lj> = 2j 1~1R j> = 2j.
= 2P ':(jxl) = T2JPl = 2""
= P2*(jx)<1Rlj> = P2 = !j.
= \ 2(!xj*)< ,lj, > = IT J"2 = j =j
= T2p2(]xj*) = 7T2j*2 = I2j*T =
T2< 2',1i j = 1j .
Consequently there exists a morphism * such that the above diagram
commutes and such that yS* = .
Thus <';1,lr,>Y$: = and hence the following diagram is
commutative.
(RxX),(R (KRx)  XxyX
.T I F l1
.______ (' ' X
S(RR 1)X R_ . X
R i;"R ~'"" '
1
2e01(jxl)<1Rnlj>
30 ] (jxl)
"1 2(lxj*)< l 3," >
,2 '02xj*)
i3O2 (lxj)
74
Since (RY,j1) is the intersection of all extremal subobjects
through which irj factors, it follows' that (RY,j1) ((RoR1)X,B). Thus
(RY,j1) ((RoR1)X,B) which was to be proved.
4.23. Corollary. Let (R,j) be a relation from X to Y. Then (XR,j2) and
((RloR)Y,8) are isomorphic as extremal subobjects of Y.
Proof. (XR,j2) (RlX,j3) E (((R1)o(R1)1)Y, ) = ((RloR)Y,P)
(4.4, 1.11 and 4.22).
SECTION 5. UNIONS
5.1. Definition. If {(Ri,ji): icI} is a family of relations from X to Y
then let ((FRi,j) be the intersection of all relations (i.e., extremal
icI
subobjects of XxY) "containing" each (Ri,ji) (where containment is in the
sense of "factoring through" as noted in Remark 0.12). (' Ri,j) shall
ieI
be called the relation theoretic union of the family {(Ri,ji): iIl}.
5.2. Examples. In the category Set the relation theoretic union is the
usual set theoretic union together with the inclusion map.
In the category Top the relation theoretic union is the usual set
theoretic union endowed with the subspace topology determined by the top
ology of XxY together with the inclusion map.
In the category Top2 the relation theoretic union is the closure
of the set theoretic union together with the inclusion map.
In the categories Grp and Ab the relation theoretic union is the
subgroup generated by the set theoretic union of the relations.
5.3. Proposition. Let {(Ri:ji): isl} te a family of relations from X to
Y, let (LRi,k) denote the usual categorical union of subobjects, let
iel
(o.j) be the epiextremal mono factorization of k and let the codomain
of a (domain of j) be denoted R. Then r and ( JRi are isomorphic rela
iel
tions from X to Y.
Proof. Since (~Ri,j) is the intersection of all extremal subobjects
igI
containing each (R.iji) and each (Rj,ji) < ( iRi,k) and
iel
(CjRlk) (Rj) and since j is an extiemal monomorphism then
icI
(UtRi,j) < (R,j).
icd
Since (U R ,k) is the intersection of all subobjects which "con
isI
tain" each (Ri,ji) then (UR,k) < (\jR ,j). Since j is an extremal
isl ieI
monomorphism and (R,j) is the intersection of all extremal subobjects
which "contain" (U R,k) then (R,j) < (Q)R ,j). Thus
ieI iEI
(R,j) E (UR ,j)..
ieI
5.4. Remark. Notice that by the definition of relation theoretic union,
if (R ,j1), (R2,j2), and (S,k) are relations from X to Y and if
(R!,Jl) (S,k) and (R2,j) < (S,k), then (P R2j) < (S,k) (cf. 1.19).
.5.5 Proposition. Let (R ,j ), (R ,j ), (Sk ) and (S ,k ) be relations
= 1 1 2' 2 1 1 2 2
from X to Y. If (R ,ji) < (R2 j2) and (Slkl) < (S2,k2) then
(R 1 S1,) < (R2JS2k).
Proof. (R ,j ) (Rj ) < (R2' S,k) and (S,k ) < (S2,k) < (R2 S2,k)
_of I ] 2,<2 2 1' 1 < 2' 2) 2 2
whence (r ..s ,j) (R2 L S2 k) (5.4).
5.6. .ma.'.rk. The following proposition can be strengthened with the ad
ditional hypothesis that the category has finite coproducts (5.34);
however. J i s included here because it is of interest in its own right.
5.7. Proposition. Let (R,j) be a relation from X to Y and let (S,k) and
(T,m) be relations from Y to Z. Then ((RoS) U (RoT),B) (Ro(S')T), ).
Proof. Consider the following commutative diagrams.
s __ k
^s '^^ 7; '~^"~
S
S T 
S ''"".". Z
rPp
RxZ > (XxY)xZ
S7A jxZz
x 3 z
In ,\  ~
(RxZ)[1Xx(SQ}T) >>
Xxyxz
0
'4 \
Xx(S ) T)  Xx(YxZ)
1XXa
RxZ )'> (XxY)xZ
jxj1
x5
RxZ f XxS
A6
XxS *I ' V XX(YxZ)
lxxk
RxZ >,  (Xxy)xZ
7" jx1Z 31
Y3
R'Z rlXxT
02
XxT > Xx(YxZ)
1Xxm
Yl <1, 7T3>
RxZ rXx (S T) )   XxyxZ  XxZ
I Ro(S T)
Y2
T2 RoS '' 
1401
> IXyxYZ
r'02
XxyxZ
r.
41 
~ ,

.
R.xZfIXxT > XxyxZ XxZ
Y3 < 1'f 3 3>
SRoT '
RoS _
(RoS) ) (RoT)  XxZ
X
RoT B3
Since (S,k) < (S( UT,a) and (T,m) < (S kT,a) it readily fol
lows that ((RxZ)I}(XxS),y2) < ((RxZ)A(Xx(S J T)),y1) and that
((RxZ) n(XxT),Y3) < ((RxZ) n(Xx(S (QT)),yl). Thus there exist imorphis.ns
.! ard 2 such that Yl1, = Y2 and Y12 = Y3. Hence
<~3,'3>' 1 iI Y2 and YiS2 = < ,3>Y3
But (RoS,e2) is the intersection of all extremal subobjects through
which y factors and since Y2 = Y1S1 = 6B'
wv have (RoS,B2) < (Ro(SUT T),B ). And since (RoT, 3) is the intersect
ion of all extremal subobjects through which <1, 3>Y3 factors and since
<'.'3 3 13Y = 122 it follows that
(RoT, 3) < (Ro(S(V T),BI).
Whence ((RoS) k (RoT),P,) < (Ro(SLj T), p) (5.4).
5.8. Proposition. Let (T,m) be a relation from Y to Z and let (R,j) and
(S,k) be relations from X to Y. Then
((RoT)r.a(SoT),R) < ((R.orJS)oT,5).
Prciof. Anaclogour: to the proof of 5.7.
5.9. Lemma. Let {(Ri,Ji): iEl} be a family of relations from X to Y. Then
(*Ri)"l,j*) (Ul (i)1,k).
iel iel
Proof. Consider the following commutative diagram.
is]
j <2.2, > 1
Ri  Y > > YxX
I ( R )1 /
R. k
R 1 (Ri)1
X.* iel
Since (Ril,ji.*) is the intersection of all extremal subobjects
through which Ji factors it follows that
(R 1,j*) < ((iRi)1,j).
i i EI
ieI
Thus
Now (Ri,ji)
"li =i
((Ri 1)k) < ((()Ri)1,j*) (5.4).
iel iel
< ()(Ri1),<2,l>'1k) since
ieI
<2' 1ji*T = i.. Thus
((ljRi),j) < ((1.(Ri1), ,>k)
iel iel
from which it follows that
((JRi),j*) < (r J(Ri' 1),<2,l><2,7 k)
iel icI
whence
(((2Ri )1 *) = (f (R) ).
iEl i il
5.10. Corollary. Let {(Ri,ji). iIl} be a family of symmetric relations
on X. Then ( URi is a symmetric relation on X.
icI
Proof. It is clear that for each icl, (Ri,ji) E (Ri1,ji*) (1.13). Thus
(U Ri,j) = ( (Ri1),k) < (( Ri)1,j*) (5.9).
icI iEI iTl
5.11. Proposition. If (R,j) is a reflexive relation on X and (S,k) is
any relation on X, then (R(t)S,m) is reflexive on X.
Proof. Since (R,j) is reflexive, (Ax,i ) < (R,j). Thus
(AX,iX) < (R,j) < (R:JS,m) hence (R(IS,m) is reflexive on X.
5.12. Definition. Let (R,j) be a relation on X. Consider the relation
(RU RI,j#).
R *
Let (T.i,X) be the epiextremal mono factorization of Tlj/#. The
domain of X (codomain of T#) shall be denoted by XR.
XE R  X
X
According to the notation of Section 4, XR is also denoted by
(RR'XR1)X.
5.13. Examples. In the category Set, XR 1 (R(R1)X = XRU RX.
That is, XR = {xeX: there exists yeX such that (x,y)ER or (y,x)ER}.
In the category Top1, XR is the same set as in Set endowed with
the subspace topology determined by the topology of X.
5.14. Proposition. Let (R,j) be a relation on X and let (RX,jl) and
(R1X,j3) be the images of Rlj and Tlj* respectively. Then
(RX~ R1X,a) < (XR'X) = ((RlUR1)X,X).
Proof. Consider the following commutative diagram.
R RX
jR j T
R (R1  XxX  X
R
Since 1,j = Tlj#XR = XTI#R = JiT1 and (RX,j1) is the intersection
of all extremal subobjects through which Tlj factors then
(PX,jl) (XR,X). Similarly, it can he shown that (R1X,j3) < (XR,X),
whence (RXi'R1X,a) < (XR,X) (5.4).
5.15. Proposition. If (R,j) is a relation on X then (R R ,j/#) is sym
metric on X and (XR,X) (X(R )R1),j2).
Proof. ((RR'])1 ,j*) (Rjql)(Rl)',j) = (Ri1 R,j#) (5.9 and
1.11). Thus (RLiR1,j#) is symmetric so that
(XpX) = ((R)R ')x,x) E (X(R 1),j2) (4.6).
5.16. Proposition. Let (Rj) be a relation on X and let (Ax iR ) be the
diagonal of YjxXR. Then (Ax R (XXX)iX) (AX n (XRxXR),p) where
(A iR(X R
(X X .XX)*
R R R XX
Proof. Consider the following commutative diagram.
XR 7T*
A  XxX  X
R T2 2
I A X xx \X
I t i x xX xX
SR R R
bx' ~ X'xx "'G
iX 2
SinceR =l "," th=t =" Thus,
Observe that E1(XxX)i 2 XX)I.X Xh2 *iX r2(XXX)iXR. Thus,
since iX is the equalizer of 71 and 2, there exists a unique morphism
so that iyX = (XXX)"iX
XR
Thus, since (AX SRXRxXR,p) is the intersection of (Ax,iy) and
(XRnXR,)XX), there exists a morphism B so that pB = ix = (XXX)iX Con
RR R
sequently (A;R(X ^x)  < (A(x (XRXRp) ,p)"
Since irlX = Tr2iX it follows that orp = TliX" = r2iXa i 2p, but
(XxX)A = p so that 71(XXX)X = TT2(XxX)X. IhNence XTI*X = Xv2*X. Recall
that X is an extremal monomorphism, hence a mnonomorphism; so that it
follows that irl*X = R22X. Since (A ) is the equalizer of i* and
fo* there exists a morphism a such that ix o. = A.
Thus (XXX)iX a = (XxX). = p, which means that
(AXP(XRXXp)IpP) < (Ay ,(XxX)j iv). Ihencf
R
(Ax ('(X
5.17. Definition. Let (R,j) b2 a relation on X. Then (R,j) is called
quasireflexive if and only if (A ,(X
X X
R ^R
quasireflexive provided that there exists a morphism X such that the
following diagram commutes.
IXR Tr
A >  X xX X
XR R R 2" R
R XxX ,,X
Ji 2
5.18. Proposition. If (R,j) is a reflexive relation on X, then (R,j) is
quasireflexive on X.
Proof. If (R,j) is reflexive then (RLU)R,j#) is reflexive (5.11);
hence, 7,J# is a retraction (1.24). Thus rTlj# is an epimorphism; so that
if (T~,X) is the epiextremal mono factorization of Tlj#, rlj' = XT# so
that X is an epimorphism as well as an extremal monomorphism. Hence X is
an isomorphism (0.17). Thus (XRx) (X,1X) whence (Ax ,iXR) (A,i ).
Thus, since (Ax,iX) < (R.j), (R,j) is quasireflexive.
5.19. Proposition. Let (R,j) be a relation on X. Then (R,j) < (A ,i ) if
and only if (R,j) < (A, ,(XxX)i ).
R R
Proof. Suppose that (R,j) < (AX,i ). Then there exists a morphism a such
that j = iXa. Thus iJrj = rliXa = v2ixc = r2j; whence
iJ = I2J = Tij = 2j Consequently the unique epiextre
mal mono factorization of j is (1 ,j), and (R,j) (Rj*)
(3.12).
But, since (R,j) (R!,j*), then (R()Ri,ji/) (R,j), and
(X,,X) ((R(~jR)X.x) (RXj ). Thus, since (R,jl) E (X ,x) = (XR,j)
R L
ft follows that (R,j) < (X. x >xx) (3.4).
R I R
However, it has been shown that (AXR,(xxx)iX ) and (A,((XRxXP),p)
are isomorphic relations on X (5.16). So that since (R,j) < (A ,i ) and
(R,j) < ((XRxxR),XXX) it follows that
(R,j) (Ax((XRxXR),p) E (AXR (XXX)iX ) (1.19).
Conversely, if (R,j)< (A ,(XxX)iX ) then since
R R
'IT(XXX)ixR = X'*iXR = X12 *ix = 2(X)iXR it follows that
R R R R
(AXP'(xXX)ix ) < (Ax ,i). Consequently the result (R,j) < (Axi ) fol
lows from the transitivity of < (0.2).
5.20. Theorem. If (R,j) is a relation on X then R, RoA and A oR are
R R
isomorphic relations on X.
Proof. It will be shown that RoA and R are isomorphic relations on X;
X
the proof for A oR and R is analogous and is omitted.
rR
The following products shall be considered: (XRxYE r"I*,2*),
(XxXxX,rP ,,2,3), (XxX,7T,l12), ((XxX)xX,i ,f 2), (Xx(XxX), i1 J ),
(RxX,5p,,2) and (XxAXR 1* 2*).
Consider the following commutative diagram.
jx1X
RxX  (Xx) xX
r. Y I 01
Y 4Z < 1 2,T3>
RxX XxA.,   XxXxX > XxX
RR)
X2 09
i Xx(xxx)ixR
SXXA ^ P'>Xx(X>) 
XR
Next it will be shown that < T,,i'2> = <' ,'i>y.
T < ,T, 2>Y = i1Y = TJ<~I1,7T3>y.
T2<7T2>y = 2Y = 202(1XX(XXX)iXR)2 2(1XX(XXX)iX )X =
T1 (XxX)jxP2*X = X"liXR P2*2 = X2*XRP*2 =
1 XR 2 2 X R 2 2 R
X7T *iXR 2* 2 = 2(XxX)iXR P22 N2 2 (Xx(XxX) iXR 2
3 32(1X(XXX)iXR)A2 = 73Y = T2<7,73>Y
Thus by the definition of product <7 1,)N3>y = <71,T2>y.
Now consider <71,'r2>y. It will be shown that Y = jl1l
T<71T,,2>Y = :FlY = T l01(jx1)A1. Thus Tly lTlm1 (jXl)1 = lJl
T 2<1, 2>Y = I2Y = "201(jxl)1= )2~ (jx1)X1 = 7j2jP11. Hence, by the
definition of product, y = jXl1l.
Since (RoAxR,a) is the intersection of all extremal subobjects
through which <1, 3>y factors and 3>Y = <1l,T2>Y = J1Pil it
follows that (RoAX,a) <_ (R,j).
For the reverse inequality consider (RQJR ,j#). Since
(RLJR,ji'/ ) is syrmmietric, ((R:)]RI)X,ji#) and (X(RC R1),j2#) are
isomorphic extremal subobjects of X (5.15).
x! x
R ^/RI y> xX   X
R 1 J '^< ~ 1 X
x(RIIR 1) '
RR
XR
Thus if (G#,X#) is the epiextremeal mono facrorization of T2j#,.
there exists an isomorphism ( such that g, E/#. So in particular,
S7Tj = Q1 #x = XR #AR"
Let a be the isomorphism for which iXRO = <1XR ,XR> (1.21).
It next will be shown that the following equality holds:
<7Ji2J,2J> = 01 = 02(lx(XX)ix )
so that the following diagram is commutative.
01
jx1X
A^ 
RxX
(XxX)xX
 XxXxXX
i i
1
1 ,
RxX % XxAXR
I X
AX 2
XxA .
XR
51 2(1X(XXX)iXR )<1rj ,a R>
7202 (lX(XX)iXR) <7tlJ,oG~ XR>
0,2(IXx(XXX)iXR )<1j,Gc AR >
SXx(XX)
4. Xx (XxX)
1XX(XXx)iXR
= i1(lXX(XXX)iR ) =
P1* = 7lJ.
= r17 2(1Xx(Xxx)iR ) 
T1(XXX)XR P2 <7t j,oX#R> =
R
XPi XRP2* = XPliR oG/#XR
XR R#R
X011#XR = X1#XR >= T2j.
7 2'2(lxX(XXX)ixR) =
T2(XXX)iXRP2*<1 P Iil R> =
XP2iiX P2,* j,oS#XR> = XP2< 1XR >S R =
X1 XR#X = XW#\R = TT2j
0i1 = 'l l = fl'"
5201 = T2fl = 1r2j.
303j' ,< J> = 2i2<> 2J> = ,2j
sv 
Thus by the definition of product:
= l = 02(1XX(XXX)iXR )
But it is also true that
ol = O1(jx)<1,rTj> (1.31).
Hence by the definition of intersection there exists a unique morphism
E such that yE = . Thus
<1,sf3>yE = <1, 3>< lij, 2ji2j> = = j.
Hence j = atC; and consequently (R,j) < (RoAXR,a).
5.21. Corollary. Let (R,j) be a relation on X. Then R1, AX oR and
RoAXR are isomorphic relations on X.
Proof. Recall that since (RU' R,j#) is symmetric (5.15), (XR1,X*)
and ((RL4R1)X,j1#) are isomorphic as extremal subobjects of X (5.15
and 5.9). Also (X ,x) and ((Rt UR1)X,j,#) are isomorphic as extremal
subobjects of X (5.15) whence,
(X~IX! ,.X*x*) E (XRxXR,XXX) and hence (AXR_1' (*xX*)iX.1) and
(AXR,(XX)i XR). But by the theorem
(R *) = (AXRoR'1,*) E (RoAxR1 ,a*).
Consequently,
(R',j*) (AX oR,a#) (RloAXR,a#).
R R
5.22. Definition. Let R be a relation from X to Y. Then R is said to be
difunctional if and only if Ro(RloR) < R and (RoR1)oR < R.
The term difunctional relation is due to Riguet [22J.
5.23. Proposition. Let R be a relation from X to Y. Then R is difunc
tional if and only if R1 is difunctional.
Proof. If (R,j) is difunctional then since (Ro(PR1oR),k,) < (R,j) we
have ((R!R)oR1, k) E ((RloR)loR'1,k) ((Ro(RoR))1,k *)<(R ,j)
1 1 1
This follows from 1.38, 1.11, and 1.12. Also since ((RoR1)oR,k2) < (R,j)
it follows that
(R7o(RoR1),k2#) (RIo(RoR1)1,k2) E (((RoR )oR)I,k2*) < (R1 j*).
Thus (Rl,j*) is difunctional.
If (Rl,j*) is difunctional then since ((R1)1,j#) (Rj) (1.11)
and since, applying the above to (R1,j*), ((R1)1,j#) is difunctional
it follows that (R,j) is difunctional.
5.24. Proposition. If R is a relation on X then R is a quasiequivalence
(3.2) if and only if R is quasireflexive and difunctional.
Proof. If (R,j) is a quasiequivalence then (R,j) is symmetric hence
(R:^)R,j#) = (R,j). Thus (XR,x) (RX,j1) E (XR,j2) (5.15 and 4.6)
from which it follows that (A R(XxX)i~ ) (XR'(j2xj2)iXR1
1 XR
Since (R,j) is a quasiequivalence then (R,y) is an equivalence
relation on XR (3.10) so ( ,X'iXR) < (R,i). Hence
(AXR,(j2xj2)iR) < (R,j) so that (R,j) is quasireflexive on X. Since
(R,j) is symmetric and transitive then
(Ro(RloR),kl) < (Ro(RoR),k1) < (RoR,j') < (R,j). Similarly,
((RoR1)oR,k2) < ((RoR)oR,k2) < (RoR,j') < (R,j). Hence (R,j) is difunc
ticnal.
Conversely if (R,j) is quasireflexive and difunctional then
(A ,(XxX)i ) < (R,j) so that (A ,(XxX)i ) < (R',j*) (1.16 and 1.12).
XR XR XR XRR
Thus (RcR,j') < (Ro(A oR),k) < (Ro(RloR),k, ) < (R,j) and
(R1,*) < (AXRo(RloAXR),k) < (Ro(RloR),kl) < (R,j) (5.20 and 1.30).
Thus (R,j) is both transitive and symmetric hence a quasiequivalence.
5.25. Proposition. Let R be a relation on X. Then R is an equivalence
relation if and only if R is reflexive and difunctional.
Proof. If (R,j) is an equivalence relation on X then (R,j) is reflexive
and a quasiequivalence on X. Thus (R,j) is difunctional (5.18 and 5.24).
Conversely if (R,j) is reflexive and difunctional then (R,j) is
quasireflexive and difunctional (5.18) hence (R,j) is a quasiequiva
lence (5.24). Since (R,j) is also reflexive it must be an equivalence
relation.
5.26. Theorem. Let (R,j) be a relation from X to Y and (RX,jl) and
(XR,j2) be the usual images (3.4). Then R, RoAXR, and ARXOR are iso
morphic relations from X to Y (cf. 5.20 and 1.31).
Proof. Consider the following commutative diagrams.
J2
XR  Y
T2 T2
R  Xx
Ti Trl
RX >~1X
A RX  XX ___XX
cL iRX t ___________ R
X P12
A >39  XxX X
+ X P2
a 2
xji PJ
RX & P
A . RXxRX ___ RX
RX p2"
Observe that pl(jiXjl)iRX lPliRX = jlP2*i[ = P2(jlxjl)i;
thus there exists a morphism a such that iXa = (j xjl)iRX; i.e.,
(ARX,(' l l)iRX) (AX,ix)
90
Thus: (ARXoR,k2) < (AxoR,j') E (R,j) (1.30 and 1.31).
To see that (R,j) < (ARXoR,k ) consider the following commutative
diagram.
R Xx(XxY) : XxXxy
I 2 2' 3
q X
XxY xy  X
1Xxy
Recall that (ARX,i) E (RX,) (1.21) hence there exists
an isomorphism o such that <1RX,1lRX> = ipRX.
Consider also the products (ARXxY,1,62), (XxR, 1,T2), and
((Xxx)xy, l, ,2)
It will next be shown that the following diagram is commutative.
(J I j )i RX1y
A XY x
R
(XxX)xy
3> XXXXy
^C)2
,T1j,"I2j> = 02
XXi
1,19l((j Ijl)iRjx ly) P 1 ((j 1 xj ) iRXX y) =
Pl QJ 1 RX 1 1 / IL
j I P*< 1RiX N, 1> .= jlT = I j li
q20!o ((jlxjl)iRxXly )<^T1,72'j = P2 l1((jXjl)iRxXIy) =
P2(jlXjl)iRXPl = jP2*iRXoT1 = jlP2*<1RX,'RX> =
Jl1T = T'li
31301((jlxjl)iRXXly) = x2((jlXjl)iRXXly)<0T1,F2j>
ly12 T23.
r1i02(1Xxj) = f1(IxXj)<7rlj,lR> = i = Trlj
fl202(1XXj) = lrlT2(lxxj) = >lj r2< lj,'R> = T!J"
T302(1Xxj) = r2i2(1Xxj) 2= ljJ2<7Tlj,lR> = 2j
1092 = ftl = Tlj.
T2 2< TiJ,j> = '7Tf2 = Trj.
T392 = T2T2<lj j> = r2J"
Consider the following commutative diagram.
(JlXjl)iRXX1
ARXxY > (XxX) xy
'1 13
",
ARXxYA(XxR  XxXxY ~XxY
A 1 xj R k2
ARxOR
By the definition of intersection, there exists a unique morphism
( from R to (A 
R< "
92
Thus 'Y3 = < = <"ljI2j>= j. Hence k2 = j;
that is, (R,j) < (ARXoR,k2). Hence (R,j) (ARXoR,k2).
Similarly it can be shown that (R,j) E (RoAXR,kl).
5.27. Theorem. Let (R,j) be a relation from X to Y. Then
(Ap,,(j1xj1)iyX) < (RoR1,j#) and (AXR,(j2xj2)iXR) < (RloR,j').
Proof. Consider the following products: (XxX,p1,P2)., (RXxPX,*pl ",2*),
(XxY,1T ,2), (RxX,pl,p2), (XxR1,1',2), ((XxY)xX, ~1,2),
(Xx(YxX),'IT*, 2*) and (XxYxX, 1, 2,)3).
Also consider the following commutative diagrams.
R  XxY
T'I jj T
RX  X
j <2T2,7I1>
R   XxY ) YxX
R I
Next, the following diagram will be shown to be commutative.
jxlX
RxX .~ (XxY)xX
< R, IJ> f A X 
SY <
R ~(RxX) (XxR)   XXYXX > XX
X2 2 ^
( \ I Xri* 71.
, i
 J
To see this, it need only be observed that
o1(jxlX) = 2(lXxj*)<Tj,T>.
.i01 1(j1Xx) = Tii (jlX)<1R',7lj> = 1iJP : Tlj.
20! (jxIX)<1R' l j> = T21 I(jXlX)<1R',Ij> = 2jpl = T2j
S30 (jX1X)<1RF"1j> = i2(jxlX) = 2<1R'R1j> = rlj
!G2(1Xxj*)< T> = i*(lXxj) = p1< 1J,T> = Tflj.
02o2(1xXj*)<7!IjT> = Tr712*(xjj*) T 1j*P2 = .lj*T =
1 <2,1T]>J = T2j.
3502(lXxj*) = T 2 2(1xxj*) = 2j = 7lj.
Thus the diagram is commutative and
01(jlX1)<1Rrj> = 02(lxXj*) =
Hence by the definition of intersection there exists a morphism E
such that yE = . Clearly <, 3>YE = thus
= j#r#Z.
It next will be shown that if a is that isomorphism for which
= iRXO then (oTl,(jlxjl)iRX) is the epiextremal mono factor
izauion of <'Trj,'"lj>
R  XxX
1 x 1jR
RX ps A
RX ApRX
o
1i (jl1l)iRX = JiP1RXoi = jlPT 1'*<1RX, RX> = J1"1 = T1iJ
p2 31xj.i)iRXOTI = JlP2*iROTl = j1P2*<1RX,Py > Jl' = li
Thus the diagram above commutes and (orl,(j1xj1)iRX) is the epiex
tremal mono factorization of (0.18). Since (ARX( 1xjl)iRX) is
the intersection of all extremal subobjects through which <'ij,Tlj> fac
tors and since <7lj,Trlj> = j#T#E it follows that
(ARX,(j1xjl)iRX) < (RoR1,j#) which was to be proved. The proof that
(AXR,(j2Xj2)iXR) < (R1oR,j') is similar.
5.28. Theorem. If (R,j) is a relation from X to Y then R is difunctional
if and only if (RoR)oR E R = Ro(RloR).
Proof. If R is difunctional then
(R,j) (RoAXR,kl) < (Ro(R1oR),kl) < (R,j) (5.26, 5.27 and 5.22).
Similarly,
(R,j) = (ARoR,k2) < ((RoR1)oR,k2) < (R,j).
The converse is immediate from the definition (5.22).
5.29. Remark. Let ( be a locally small quasicomplete category having
(finite) coproducts. It is noted in passing that if has arbitrary
products; i.e., is complete, then 1 is also finitelyy) cocompletc 9 9.
Recall that the unique epiextremal mono factorization of a morphism is
obtained by taking the intersection of all extremal subobjects of the
codomain of the morphism through which the morphism factors (0.21). Also
recall that if the intersection of all subobjects of the codomain of the
morphism through which the morphism factors is taken, then the unique
extremal epimono factorization is obtained (0.21). Finally recall that
if {(Ai,ai): isl} is a family of subobjects of a object X, then the
subobject (lkJAi,a) is obtained by taking the intersection of all sub
iE
objects of X which "contain" each (A.,a.).
Full Citation 
Material Information 

Title: 
Relation theory in categories 

Physical Description: 
vi, 122 leaves. : illus. ; 28 cm. 

Language: 
English 

Creator: 
Fay, Temple Harold, 1940 

Publisher: 
University of Florida 

Place of Publication: 
Gainesville, Fla. 

Publication Date: 
1971 

Copyright Date: 
1971 
Subjects 

Subject: 
Categories (Mathematics) ( lcsh ) Mathematics thesis Ph. D Dissertations, Academic  Mathematics  UF 

Genre: 
bibliography ( marcgt ) nonfiction ( marcgt ) 
Notes 

Thesis: 
Thesis  University of Florida. 

Bibliography: 
Bibliography: leaves 120121. 

Additional Physical Form: 
Also available on World Wide Web 

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Manuscript copy. 

General Note: 
Vita. 
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Volume ID: 
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Source Institution: 
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Table of Contents 
Title Page
Page i
Acknowledgement
Page ii
Table of Contents
Page iii
Abstract
Page iv
Page v
Page vi
Introduction
Page 1
Page 2
Page 3
Page 4
Page 5
Preliminaries
Page 6
Page 7
Page 8
Page 9
Page 10
Page 11
Page 12
Page 13
Page 14
Page 15
Page 16
Page 17
Generalities
Page 18
Page 19
Page 20
Page 21
Page 22
Page 23
Page 24
Page 25
Page 26
Page 27
Page 28
Page 29
Page 30
Page 31
Page 32
Page 33
Page 34
Page 35
Page 36
Page 37
Page 38
Page 39
Page 40
Categorical congruences
Page 41
Page 42
Page 43
Page 44
Page 45
Page 46
Page 47
Categorical equivalence relations and quasiequivalence relations
Page 48
Page 49
Page 50
Page 51
Page 52
Page 53
Page 54
Page 55
Page 56
Page 57
Page 58
Page 59
Page 60
Page 61
Page 62
Images
Page 63
Page 64
Page 65
Page 66
Page 67
Page 68
Page 69
Page 70
Page 71
Page 72
Page 73
Page 74
Unions
Page 75
Page 76
Page 77
Page 78
Page 79
Page 80
Page 81
Page 82
Page 83
Page 84
Page 85
Page 86
Page 87
Page 88
Page 89
Page 90
Page 91
Page 92
Page 93
Page 94
Page 95
Page 96
Page 97
Page 98
Page 99
Page 100
Page 101
Page 102
Rectangular relations
Page 103
Page 104
Page 105
Page 106
Page 107
Page 108
Page 109
Page 110
Page 111
Page 112
Page 113
Page 114
Page 115
Page 116
Page 117
Page 118
Page 119
Bibliography
Page 120
Page 121
Biographical sketch
Page 122
Page 123
Page 124

Full Text 
Relation Theory in Categories
By
TEMPLE HAROLD FAY
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF iLOir . IN PARTIAL
FULFILLMENT OS TIH REQUIREMENTS FOR THE DEGREE' OF
DOCTOR OF PHILOSOPHY
UNIVERSITY 0: FL..IDA
To Dr. George E. Strecker, without whose tactful prodding,
infinite patience in proofreadings of handwritten drafts and helpful
suggestions this work would never have been completed.
TABLE OF CONTENTS
Abstract.................................... .................... ... iv
Introduction................................. ...................... 1
Section 0. Preliminaries...................................... 6
Section 1. Generalities ................ ......................... 18
Section 2. Categorical Congruences............................. ]
Section 3. Categorical Equivalence Relations and
QuasiEquivalence Relations........................ 8A
Section 4. Images............................................ 63
Section 5. Unions............................................ 75
Section 6. Rectangular Relations .............................. 103
Bibliography......................................................... 120
Biographical Sketch ...................................... ......... 122
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
RELATION THEORY IN CATEGORIES
By
Temple Harold Fay
March, 1971
Chairman: Dr. George E. Streckcr
Major Department: Mathematics
The purpose of this dissertation has been to systematically
generalize relation theory to a category theoretic context. A quite
general relation theory has emerged which is applicable not only to
concrete categories other than the category of sets and functions, but
also to abstract categories whose objects need have no elements at all.
This categorical approach has provided the opportunity to comprehend
classical relation theory from a new vantage point, thus hopefully
leading to an eventual better understanding of the subject.
A relation from an object X to an object Y is a pair (R,j) where
j is an extreimal monomorphism having domain R and codomain XxY. By
choosing j to be an extremal monomorphism, relations in the category
of sets are the usual subsets of the Cartesian product, relations in
the category of groups are subgroups of the group theoretic product,
and relations in the category of topological spaces are s.bspaccs of
the topological product. This latter fact would not be the case if
relations would Le defined to be merely subobjects of the categorical
product.
Section 0 notes results which are purely categorical in nature
and vhicl.h vill be used extensively throughout the sequel. Particular
emphasis is given to the epiextremal mono factorization property and
necessary and sufficient conditions for the existence of this factori
zation and equivalent forms of the property.
In Section 1, the basic machinery for categorical relation theory
is developed. For example, such notions as inverse relation, reflexive
relation, symmetric relation, and composition of relations are defined
and several important results are obtained.
Section 2 deals with a categorical definition of a congruence
relation. Several algebraic results of Lambek and Cohn are generalized.
Equivalence relations and quasiequivalence relations (symmetric,
transitive relations) are studied in Section 3. A quasiequivalence on
an object X is shown to be an equivalence relation on a subobject of X.
If R is a set theoretic relation from the set X to the set Y and
A is a subset of X then AR = {ycY: there exists acA such that (a,y)eR}.
This definition is generalized in Section 4 and results similar to those
obtained by Riguet are demonstrated.
If {(Ri,ji): iel} is a (finite) family of relations from X to Y
then the relation theoretic union (URi,j) of the family is obtained by
iel
taking the intersection of all relations from X to Y which "contain"
each Ri. If the category being investigated is assumed to have (finite)
coproducts then the union of the family considered as subobjects and the
relation theoretic union of the family considered as extremal sabobjects
turn out to be given by the unique extremal epimono and unique epi
extremal mono factorizations of the canonical morphism from the coproduct
of the family to XxY.
The notion of a (finite) union distributive category is introduced.
Roughly speaking, this property guarantees that unions commutee" with
products and intersections.
Section 5 deals with unions and the importance of the concept of
difunctional relation is brought out.
A well known result in set theoretic relation theory is that a
partition determines an equivalence relation. In order to obtain this
result in its generalized form the existence of an initial object which
behaves similarly to the initial object in the category of sets (namely
the empty set) is postulated and disjointness becomes a useful categor
ical notien. Also the notion of difunctional relations was crucial in
obtaining the above result.
Section 6 deals with rectangular relations and the above result
about partitions is obtained.
INTRODUCTION
The purpose of this work has been an attempt to systematically
generalize relation theory to a category theoretic context. In doing
so, several goals have been realized. Firstly, a quite general rela
tion theory has emerged which is applicable.not only to concrete cate
gories other than the category of sets and functions, but also to
abstract categories whose objects need have no elements at all. Second
ly, taking a categorical approach has provided the opportunity to
comprehend classical relation theory from a new vantage point, thus
hopefully leading to an eventual better understanding cf the subject.
Hany relation theoretic results have been rather straightforward
to prove in an "element free" setting, once the appropriate machinery
has been constructed to handle them. On the other hand it has been
surprising to see that some results which are easy to prove in the set
theoretic context are much more difficult to show categorically.
For example, it is easy to prove that if R is a set theoretic
relation from X to Y such that RY = X then RoR"' = {(x,z): there exists
yeY such that (x,y)eR and (y,z)cR]} is reflexive. This result can be
generalized to categories but is no longer easy to Drove and the result
gains some significance.
Another easy result in set theoretic relatcon theory is that if
AX and Av are the diagonals on X and Y respectively then AxcF. = R = Roby
This result is also generalized to category es bIt "i or.crhic as rela
tions" replaces "equality" and the result is nc longer easy to prove.
Whenever one is generalizing properties care must be taken to be
certain that the generalized definitions are really generalizations of
the notions beig considered and that the proper generalization of the
definition is obtained. This seems to be particularly important in
category theory. Care has been taken when selecting the basic notion
of a relation from an object X to an object Y to be an extremal sub
object of the categorical product XxY; i.e. a pair (R,j) where j is an
extremal monomorphism having domain R and codomain XxY. By doing so
relations in the category of sets are the usual subsets of the carte
sian product, relations in the category of groups are subgroups of the
group theoretic product, and relations in the category of topological
spaces are subspaces of the topological product. This latter fact
would not be the case ii relations would be defined to be merely sub
objects of the categorical product. Much care has also been taken with
the definition of composition of relations (1.26). Using this defini
tion many nice results have been obtained; however, in general, the
composition of relations is not associative (1.36). This, at first
glance, seems to be pathological and casts doubt on the suitability of
the definition of composition of relations. However, the wealth of
other important results obtained belies this doubt (see 1.37). Also,
some further atonement is yielded by trre fact that for rectangular
relations composition is asscciarive (6.15).
Cohn [31 and Lambek i3J define a congruence ii an algebraic
setting to be a sLbalgebra of the cartesian product vhich is "coipat
ible' with the algebraic operations and which is set theoretically an
equivalence relalion. in this work, a generalized noLion of congruence
is given which is equivalent to the above in algebraic categories and
the result that a (categorical) congruence is a (categorical) equi
valence relation is obtained.
It was found that categorical unions were very difficult to work
with. However, by assuming the category being studied had (finite)
coproducts as well as being locally small and quasicomplete the notion
of union became somewhat easier to handle.
For instance, if {(R.,j ): icl} is a (finite) family of relations
from X to Y then the union (CJR.,j) of the family, considered as sub
iT 1
objects of XxY is not necessarily a relation from X to Y,since j is not
necessarily an extremal monomorphism. The relation theoretic union of
the family is obtained by taking the unique epiextremal mono factoriza
tion of j (5. 3) or equivalently by taking the intersection of all rela
tions from X to Y which "contain" each R.. If the category being inves
tigated is assumed to have (finite) coproducts in addition to being
locally small and quasicomplete then the union of the family considered
as subobjects and the relation theoretic union of the family considered
as extremal subobjects turn out to be given by the unique extremal epi
mono and unique epiextremal mono factorizations of the canonical mor
phism from the coproduct of the family to XxY (5.29). It is also snown
that when the category has (finite) coproducts both factorizations
respect unions (5.30 and 5.42).
Unions are still difficult to handle even with the assumption of
(finite) coproduets mentioned above; hence, the notion of a (finite)
union distributive category is introduced (5.31). Roughly speaking,
this property guarantees that unions "commuce" .ith products and inter
scutions and thus unions become "easy" tj handle. Examples of union
distributive categories show that such categories tend to be more of a
topological nature rather than of an algebraic nature.
The set theoretic notion of difunctional relation is due to
Riguet [221 and its importance has been ncted by Lambek 13] and
HacLane 8 A set theoretic relation R is difunctional if and only
1
if RoR oR C R. The categorical definition in view of the fact that
associativity cannot be assumed reads: R is difunctional if and only if
1 1
(RoR )oR < R and Ro(R oR) < R where "<" is the usual order on sub
objects. It is easy to prove, again by choosing elements, that if a
1
set theoretic relation R is difunctional then R = RoR oR. However,
the similar result in the categorical setting is much harder to obtain
1
and is rephrased: if R is difunctional then R (RoR )oR and
_]
R E Ro(R oR) where "" means isomorphic as extremal subobjects (5.28).
A well known result in set theoretic relation theory is that a
partition determines an equivalence relation. In order to obtain this
result in its generalized form additional hypotheses had to be added
to the category being studied. In particular, the existence of an ini
tial object which behaves similarly to the initial object in the cate
gory of sets (namely the empty set) had to be postulated and disjoint
ness became a useful categorical notion. Again, examples of such cate
gories are nonalgebraic. Also the notion of difunctional relations was
crucial in obtaining the above result (6.20).
The excellent reference paper by Riguet 22J has been used as a
guide for the results of set theoretic relation theory. Indeed, most
all of the results contained herein are generalizations of results in
.22 The papers by Lambek 3 14 MacLane 8 and Bednark
aid Wallace provided motivation for many of the generali
,1 1 1,
nations.
The basis for the categorical notions has been taken from the
papers of Herrlich and Strecker 7 ], S8 1 Isbell 11 ], 12 ,
and the forthcoming text by Herrlich and Strecker [9 J (which has
greatly influenced this work). For most of the basic categorical
notions the reader is referred to the texts by Mitchell [211 Freyd
14 and Herrlich and Strecker 9 9.
The work here is begun with a preliminary Section 0 which notes
(often without proof) results which are purely categorical in nature and
which will be used extensively throughout the sequel. Particular empha
sis is given to the epiextremal mono factorization property and neces
sary and sufficient conditions for the existence of this factorization
and equivalent forms of the property. However, it is not intended that
the preliminary section give a complete categorytheoretical background.
It is expected that the reader be familiar with the basic categorical
notions,
SECTION 0. PRELIMINARIES
0.0. Remark. It is assumed that the reader is familiar with the basic
notions of category theory and hence such basic notions as epimorphism,
monomorphism, retraction, section, equalizer, regular monomorphism,
coequalizer, regular epimorphism, subobject, and limits shall not be de
fined. The reader is referred to Mitchell [21 and Herrlich and Streck
er (9] for such notions. All of the following results are proved in
detail in Herrlich and Strecker (9] Since Theorem 0.21 is vital to
this work the proof is sketched here.
0.1. Notation. The category whose class of objects is the class of all
sets and whose morphism class is the class of all functions shall be
denoted by Set.
The category whose class of objects is the class of all groups
and whose morphism class is the class of all group homomorphisms shall
be denoted by Grp.
The category whose class of objects is the class of all tcpological
spaces and whose morphism class is the class of all continuous functions
shall be denoted by Topi.
In a manner similar to that described above, one obtains the fol
lowing categories:
FSet finite sets and functions;
FGp finite groups and grcup homomorphifsms;
Ab Abelian groups and group homomorphisms;
SGp
SG1
Rng
Top2
CT 2
0.2. Proposition.
be m morphisms.
1) If f and
2) If f and
3) If gf is
4) If gf is
5) If gf is
f g
Let : be a category and let XY and YZ
g are monomorphisms then gf is a monomorphism.
g are epimorphisms then gf is an epimorphism.
a monomorphism then f is a monomorphism.
an epimorphism then g is an epimorphism.
an isomorphism then g is a retraction and f is a
section.
0.3. Remark. In general, an equalizer is a limit of a certain diagram.
It is an object together with a morphism whose domain is the object. A
regular monomorphism is a morphism for which there exists a diagram so
that the domain of the morphism together with the morphism is the equal
izer of the diagram.
It is observed in Herrlich and Strecker ,9 J that certain func
tors preserve regular monomorphisms while not preserving equalizers,
hence one reason for the above distinction between equalizers and regu
lar mcnomorphisms.
In this paper, since we shall not deal with functors, no distinc
 semigroups and semigroup homomorphisms;
 semigroups with identity and semigroup homomorphisms which
preserve the identity;
 rings and ring homomorphisms;
 rings with identity and ring homomorphisms which preserve
the identity;
 Hausdorff spaces and continuous functions;
 compact Hausdorff spaces and continuous functions.
tion shall be made between equalizers and regular monomorphisms; i.e.,
between the pair (object and morphism) and the morphism alone. Both will
be called equalizers.
f
0.4. Proposition. Let P be a category and let X Y be a 
morphism. Then the following are equivalent:
1) f is an isomorphism,
2) f is a monomorphism and a retraction,
3) f is an epimorphism and a section,
4) f is a monomorphism and a regular epimorphism,
5) f is an epimorphism and a regular monomorphism.
0.5. Definition. Let {Ai: iIl} be a family of objects then the pro
duct ~tA.,Tr.) of the family is a object T"Ai together with pro
i 1
is 1 iel
section morphisms Ti:TTAi  Ai with the property that if P is
iel
any object for which there exist m morphisms p.: P  'Ai for
each icl, then there exists a unique morphism X: P > Ai such that
iel
ilX = i for each iel.
The dual notion is that of the coproduct (JJ.Aii.).
iel
0.6. Definition. Let {(Ai,ai): iEl} be a family of subobjects of a 
object X. Then the intersection ( ~A,,a) of the family is a object
iel
1 Ai together with a morphism a: Oi1A ^X where for each i there
icl iEl
is a morphism Xi: X f A  Ai such cthat ai.. = a with the property
ic!2
that if P i.' any object for which there exist: morphisms p: P " X
and pi: P  'A1 such that a1ig. p for each isi then there exists a
unique norphism ,: P 1" A. such that aX = p.
It i
It follows that a is a mnonom.orphism.
0.7. Remark. The above two definitions are mentioned because of the
fundamental role they play in the sequel. They are special limits and
are perhaps the most important limits in the categories that will be
considered in tnis work.
The following theorem is a special case of a more general theorem
dealing with the commutation of limits which can be found in Herrlich
and Strecker [9] A variation of the theorem will be proved in
Section 1 (1.5).
0.8. Theorem. Let {(Ai,a.): icI} and {(Bi,b.): icl} be families of sub
objects of 0 objects X and Y respectively. Then if P has finite
products and arbitrary intersections then (( A.i)x( Bi) and ^((A.xB.)
iI 1 iEI il
are canonically isomorphic.
0.9. Notation. Let {Xi: icI} be a family of objects and suppose
f.
{Z   Xi: icl} is a family of 0 morphisms. Then by the defini
tion of product there exists"a unique morphism h from Z to TT Xi such
i.E
that irih fi for each iI. This morphism h shall be denoted by
iET
Let A and B be 0 objects and suppose that a: A  X and
b: B >Y are morphisms. If P1 ard P2 are the projection mor
phisms from AxB to A and B respectively then apl: AXB  X and bP2
AxB ' Y, hence by the definition of product there exists a unique
morphism g from AxB to XxY such that g = a and g = b. Ths mor
1 l and 7..g = b 2P This mor
phism g shall be denoted by axb and shall be called the product of a
and b.
Let f be a ? morphisrm from X to Y. If f is a moncmorphism then
the following notation shall be used:
X  
If f is an epimorphism then the following notation shall be used:
f.
X  Y
If f is an equalizer then the following notation shall be used:
f
X  Y
If f is an isomorphism then the following notation shall be used:
f
X 1v Y
a b c d
0.10. Proposition. Let A X, B  Y, X Z, and Y  W
be ;' morphisms. Then (cxd) (axb) = caxdb.
a b
0.11. Proposition. Let A  X and B >Y be monomorphisms (respec
tively, sections, isomorphisms) then axb is a monomorphism (section,
isomorphism).
0.12. Remark. A partial order may be defined on the subobjects of an
object in in the following way:
If X is a P object and (A,a) and (B,b) are subobjects of X; i.e.,
a and b are monomorphisms with codomain X and domains A and B respec
tively, then (A,a) < (B,b) if and only if there exists a morphism c from
A to B such that be = a.
b
B 1 X
I a
A 7
By an abuse of language, if (A,a) < (d,b) then (B:b) is said to
contain (A,a) and the morphism c is sometimes called the inclusion of
(A,a) into (B,b). It is easy to see that if (Aa) < (B,b) an'
(B,b) < (A,a) then the morphism c is an isomorphisnm. In this case,
(A,a) and (B,b) are said to be isomorphic as subobjects of X. This is
a stronger condition than A and B just being isomorphic objects in the
category The following notation shall be used to denote the case
where (A,a) and (B,u) are isomorphic as subobjects of X:
(A,a) E (B,b).
Sometimes it is written (inaccurately) that A < B or that A and
B are isomorphic as subobjects of X. When this is done, the morphisms
a and b should be clear from the context.
It is immediate that (A,a) E (B,b) if and only if (A,a) < (B,b)
and (B,b) < (A,a). Thus the relation "<" on subobjects is easily seen
to be a partial order up to isomorphism as subobjects.
0.13. Definition. Let f from X to Y be a morphism. f is an extremal
imonomorphism if and only if f is a monomorphism and whenever f = gh and
h is an epimorphism then h is an isomorphism.
If f is an extremal monomorphism the following notation shall be
used:
f
X  >Y
The dual notion is that of an extremal epimorphism and is denoted:
f
X f Y
If f is an extremal monomorphism f: X tY, then (X,f) is
called an extremal subcbject of Y.
0.j4. Remark. The definition of extremal monomornhism is due to Isbell
]i] The concept of extremal monomorphism is important since it
yields what shall be called the "irage" of a morphism (see 0.18),
0.15. Examples. In the categories Set, Grp, Ab and FGp, extremal mono
mcrphis:.s are precisely the onomorphisms (i.e., onetoone morphisms).
In the categories Top and CpT extremal monomorphisms are precisely
the cmbeddings. In the category Top2 they are the closed embeddings.
f
0.16. Proposition. If X. ,Y is a
0.16. Proposition. If X is a morphism such that f = gh
and f is an extremal monomorphism then h is an extremal monomorphism.
f
0.17. Proposition. If X Y is a morphism then the following
are equivalent:
1) f is an isomorphism,
2) f is an epimorphism and an extremal monomorphism,
3) f is a monomorphism and an extremal epirorphism (c.f. 0.3).
0.18. Definition. A category is said to have the unique epiextremal
mono factorization property if for any morphism X Y, there
exist an epimorphism h and an extremal moncmorphism g with f = gh such
that whenever f = g'h' where g' is an extremal monomorphism and h' is an
epimorphism then there exists an isomorphism o such that the following
diagram commutes.
f
'NN
If has the unique epiextremal rnono factorization property and
if f = gh where h is an epimorphism and g is an extremal monomorphism,
then the pair (h,g) shall be used to designate the epiextremal mono
factorization of f. The extremai subobject (Z,g) of Y is called the
inage of X under f. Sometimes (Z,g) is referred to as the image of f.
The notion of the unique extremal epimono factorization property
is defined dually.
If has the unique extremal epimono factorization property and
f = gh where g is a monomorphism and h is an extremal epimorphism then
the pair (h,g) shall be used to designate the extremal cpimono factori
zation of f. The subobject (Z,g) of Y is called the subimage of X under
f. Sometimes (Z,g) is referred to as the subimage of f.
0.19. Definition. A category is said to have the diagonalizing
property if whenever gh = ab such that h is an epimorphism and a is an
extremal monomorphism, then there exists a (necessarily unique) morphism
s such that ( h = b and a g = g.
h
X . Y
b $ 
W z  >a Z
a
0.20. Theorem. Let be a locally small category having equalizers and
intersections. Then the following are equivalent:
1) Y has the unique epiextremal mono factorization property,
2) has the diagonalizing property,
3) the intersection of extremal monoaiorphisms is an extramsa mono
morphism and the composite of extremal monomorphisms is an extremral
monomorphi sm,
4) if has pullbacks and if (P,a,3) is the pullback of f av' g
where f = go and f is an extremal monomorphism then a is an extremal
uionomorphism.
5) if 9 has (finite) products then the (finite) product of
extremal monomorphisms is an extremal monomorphism.
0.21. Theorem. If is locally small and has equalizers and inter
sections then f2 has both the unique epiextremal mono factorization
property and the unique extremal epimono factorization property.
Proof. (sketch). First we will show the existence of the unique extremal
epimono factorization property. If f from X to Y is any morphism
then let ((E.,e) be the intersection of the family {(Ej,e.): jCJ} of
jcj 3 l 
all subobjects of Y through which f factors. Then it follows that e is
a monomorphism and that f factors through e; i.e., there exists a mor
phism h such that f = eh. Now, to see that h is an epimorphism suppose
a and B are morphisms such that ah = Bh. Let (E,k) be the equalizer
of a and Z. It follows from the definition of equalizer that there exists
a morphism g such that kg h.
f
X   Y
g h e
E >'>^ r\ E Z' Z
k jcJ J
Thus it follows that f factors through ek and since ek is a mono
morphism then there exists a morphism ): i% E.  E such that ekX = e.
jJe
From this it follows that k is an iscmcrphism whence a = i and so h is
an epimorphism.
Next it will be shown that h is an extremal epimorphism. Suppose
L: = hlh2 where h1 is a monomorphism. Then eh! is a monomorphism through
which f factors. From this it follows, as above, that h1 is an isomor
phism and hence h is an extremal epimorphism. Suppose f = g'h' where g'
is a monomorphism and h' is an extremal epimorphism. Then since g' is
a monomorphisim through which f factors there exists a morphism i from
( E. to che codomain of h' (domain of g') such that e = g'T. Since e
j J
and g' are monomorphisms, it follows that h' = Th and that T is a mono
morphism. Since h' is an extremal epimorphism it follows that T is an
isomorphism. Thus has the unique extremal epimono factorizaticn
property.
Now suppose that ge = mA where e is an epimorphism and m is an
extremal monomorphism. It will be shown that there exists a morphism
a from the codomain of e to the domain of m such that Ge = h and mo A g.
Let (.( A.,a) be the intersection of the family {(A.,a.): isI}
iT
of all subobjects of the codomain of g (codomain of m) through which
g and m factor. This family is nonempty since both g and m factor
through the identity morphism on the codomain of g. It follows that both
g and m factor through a. Thus there exist morphisms al and a2 such that
the following diagram commutes,
e
X  Y
h C Ai
W Z
It will be shown next that 22 is an epimnorphism. Suppose a* and
are mcoiphisims for which aca2 = hSa2. Let (L*,k"O ) be the equalizer
of 0* and B*. It follows from the definition of equalizer that there
exists a morphism b, such that k*b1 = a2, since o.*a2 = *a2. Since the
diagram conmrutes it follows' that a*al = B*ale. But e is an epimorphism
hence o*al = e1 s,) that by the definition of equalizer there exists a
morphism b2 such that k*b2 = a1. Thus it follows that m = ak*bi and
g = ak*b2 and so both m and g factor through ak* from which it follows
that k* is an isomorphism. Hence a* = B* and a 2 is an epimorphism. But
n is an extremal monomorphism and m = aa2 and a2 is an epimorphism. Thus
a is an isomorphism. Thus defining o = a]al it follows that the fol
2 21
lowing diagram commutes and has the diagonalization property.
e
X ~ Y
w X, Z
111
Hence. has the unique epiextremal rono factorization property
(0.20).
0.22. Theorem. Let ( be any category then the following are equivalent:
1) a is finitelyy) complete,
2) has (finite) products and (finite) intersections,
3) has (finite) products and equalizers,
4) has (finite) products and pullbacks.
0.23. Definition. A category __ is said .o be quasicomplet, if has
finite products and arbitrary intersections.
0.24. Examples. The categories FSet and FGp are quasicomplete cate
gories which are not complete. The categories Set, Top1, Top2, CpT2,
Grp, Ab, Ring, and SGp are quasicomplete.
0.25. Remarks. A quasicomplete category is finitely complete but is uot
necessarily complete as the examples FSet and FGp above show.
Also, a locally small, quasicomplete category has both the unique
extremal epimono factorization property and the unique epiextremal
mono factorization property (0.20 and 0.21).
It can be shown that the unique epiextremal mono factorization
of a morphism can be obtained by taking the intersection of all extremal
monomorphisms through which the morphism factors. It has been shown that
the unique extremal epimcno factorization property is obtained by
taking the intersection of all subobjects through which the morphism
factors (0.20). These characterizations shall be used frequently in the
sequel.
SECTION 1. GENERALITIES
1.0. Standing Hypothesis. Throughout the entire paper it will be assumed
that is a locally small, quasicomplete (finite products and arbitrary
intersections) category.
As noted in the preliminary section s enjoys the unique epi  ex
tremal mono factorization property.
1.1. Examples. Many well known categories are locally small, and quasi
complete. Among such are the categories: Set, Top Top2, Grp, Ab, SGp,
SGp1, Rng, Rny, CpT_ and FGp.
1.2. Definition. Let X and Y be objects. A relation R from X to Y is
an extremal subobject of XxY; i.e., a relation from X tc Y is a pair
(R,j) where R is a object and j is an extremal monomorphism having
dcmain R and codcmain XxY. A relation from X to X is called a relation on
X.
1.3. Definition. Let (R,j) and (S,k) be relations from X to Y. Then (R,j)
and (S,k) are said to be isouorphic relations if and only if they are iso
morphic as extremal subobjects cf XxY.
1.4. Examples. In the categories Set, aid Topi relations are subsets of
the Cartesian product together with the inclusion map.
In the categories Grp, and Ab relations are subgroups of the Car
tesian product together vith the inclusion map.
19
In the categories Top2, and CpT2, relations are closed subspaces
of the Cartesian product together with the inclusion map.
1.5. Proposition. Let X and Y be r objects and let (A,a) and (B,b) be
extremal subobjects of Y. Then Xx(ACIB) and (XxA)n (XxB) are isomorphic
relations from X to Y.
Proof. Consider the following commutative diagrams.
B
A/AB > B
A Y
a
; XxB
1
ixa
Sxb
SX xy
X> XXY
Consider also (Xx(A B), 1 xc = y1). Since extremal subobjects are closed
under intersections and products (0.20) yl and Y2 are extremal monomor
phisms.
Since (lXxa)(1XxXA) = 1Xxc = Y1 and (lxxb) (1XXB) = 1Xxc = Y1 then
by the definition of intersection there exists a unique morphism o from
Xx(AAB) to (XxA)/(XXB) so that y2o = Y1 and the following diagram com
mutes. Thus:
XI
X xA f
(X xA) (xxb)
I .
(Xx(AAB), Y) < ((XxA)n (XXB), y ).
1 Xa
X
] xx
X
Xx(A5 B)    ' (XxA) (XxB)   Xx
B X
XxB
Now let (Trl,T 2), (Tr1,T), (o],P 2) and (pl,p) be the projections of
XxY, Xx(AriB), XxA, and XxB respectively. Observe that:
1Y2 1(Xxa)X = pX l(Xx 2 = 1i 2
2Y2 = 2(] xa)X = aP2X1 = 2(I xbx)X2 = bP2X2
Thus by the definition of intersection there exists a unique morphism z
from (XxA)( (XxB) to AB such that cE = T2y2 and thus by the definition
of product there exists a unique morphism C from (XxA)f/(Xx B) to Xx(AflB)
such that < = ; i.e., Trc = TY2 and 729 = E. Now yS = (1Xxc)E
hence ITYi  1Xi = ~TY2 and 2Yl = cr = c = 2 = 2. Thus "y1E = Y2'
whence:
((XxA) r (XxB), y2) ( (Xx(AriB), y').
1 2
X  (XxA) .jXxB)
A rB XxA Y2 XxB
I/ 3 xa
Xx (, ) ....Yx 
Y1
I. xb
X
1.6. Notation. Let X and Y be objects and let (XxY,T1,T2T) and
(YxX,pl',2) be the indicated products of X and Y. Then there exists a
unique isomorphism from XxY to YxX, denoted by , such that the
following diagram commutes.
XY IY VI
xY   Yx
3
11 1 '
xX
x ) ?x
Note that: = 1x and = 1Xx
1.7. Definition. Let (R,j) be a relation from X to Y and let (T,j*) be
the unique epiextremal mono factorization of j (see 0.18). The
codomain of T (domain of j*) is denoted by R1 and (R1,j*) is called
the inverse relation of (R,j) or more simply, when there is little like
lihood of confusion, the inverse of R.
R 3 ^^ XxY X y ^ 2_._ > yxX
R R "7xj*
1.8. Example. In the categories Set, Top1, Top2, Grp, Ab, and FGp,
<72,~1 >: XxY . YxX
is defined by (x,y) = (y,x); hence, if (R,j) is a relation from
X to Y then R1 g {(y,x): (x,y)eR} with j* the inclusion map of R1 into
YxX.
1.9. Proposition. If (R.j) is a relation from X to Y the R and R" are
.?iso:crphic objectr, of .
Proof. Since <,2,' i> is an isomorphism and j is an extremal monomorphisr:
then <72, l >j is an extremal monomorphism. But <7r2,IT>j = j*T. Thus since
T is an epimorphism then from the definition of extremal monomorphism it
follows that T is an isomorphism.
1.10. Definition. If (R,j) is a relation from X to X then R is said to be
symmetric if and only if (R1,j*) < (R,j).
1.11. Proposition. Let (R,j) be a relation from X to Y. Then the inverse
relation ((R1)1,j#) of (RP,j*) and (R,j) are isomorphic relations.
Proof. Consider the following commutative diagram.
R   X Y
< 7 2 < T2' 1>
R 1 Y x X
`# I V
(R ) Xxy
Since the two inner squares commute the outer rectangle commutes. Both
of T and T# have been shown to be isomorphisms (1.9). And, as also has
been observed: <72,T > = 1Xxy (1.6). Consequently, T#T is an iso
morphism and j = j#(Tr). Thus (R,j) ((R~1)1,j#).
1.12. Proposition. Let (R,j) and (S,k) be relations from X to Y. Then
(S,k) < (Rj) if and only if (S1,k*) i (R',j*).
Proof. Consider the following conmiutative diagram.
j <7;2' 1>
R XxY  YxX
AV
k U R1 k*
S1
If (S,k) < (R,j) then there exists a morphism a: S  R R such that
1
ja = k. Define B = TT1. Then j* = j*Tt1 = <2 >jaT1 k
= k*iT = k*. Thus (Sl,k*) < (R1,j*).
If (S1,k*) < (R,j*) then by the above, ((S1)1,k#) <
((R1)1,j#) thus (S,k) < (R,j) (1.11).
1.13. Corollary. If (R,j) is a symmetric relation on X then
(R,3) < (R,j*) whence (R,j) E (RI,j*).
Proof. Since (R,j) is symmetric (R1,j*) < (R,j). Thus
(R,j) = ((R1)1,j#) < (RIj*) (1.11 and 1.12).
Consequently (R,j) (R1,j*).
1.14. Definition. Recall that since 0 is quasicomplete it has equal
izers, thus for each object X let (AX,iX) denote the equalizer o.f
and Tr2 where Tr and i2 are the projections of XxX. Since iX is an
equalizer it is an extremal monomorphism. Hence (Ax,i ) is always a rela
tion on X (called the diagonal of XxX).
A relation (R,j) on X is said to be reflexive on X provided that
(Axi ) (R,).
1.15. Example. In the categories Grp, Ab, Set, lop,, Top2, and CpT2, it
follows that AX {(x,x): xEX)} XxX with the inclusion map.
1.16. Proposition. For any object X, <12,T'>i' = iX. Thus:
(Ax,ix) (Axl,i*).
Proof. Consider the following commutative diagram.
<1T2' >
XxX >> . XxX
SX x
x .y / i
1 X
TliX = T2iX = TliX = TT2<"'i2,71>iX.
Thus the epiextremal mono factorization of'iX is (1. ,5).
1.17. Corollary. Let (R,j) be a relation on X, then (R,j) is reflexive
on X if and only if (R1,j*) is reflexive on X.
Proof. If (Ax,ix) < (R,j) then (AX,ix) (X,ix*) < (pl,j*) (1.16 and
1.12).
Conversely if (AX,iX) < (R1,j*) then
(Ax,ix) (A ,ix*) < ((1)1 ,j#) E (R,j) (1.16, 1.12. and 1.11).
1.18. Proposition. Let (R,j) and (S,k) be relations from X to Y. Then
the relations (RcS)1 and (R1fS1) are isomorphic relations.
Proof. According to the definitions of intersection and inverse relation
we have the following commutative diagrams.
SR .1
X I " x "*
RA S >^ xY i I s1 ) S  X. YxX
S 2 flk \4 k*
^\~~~ ^ ^ ^ :
9 <'[2 ,'IT1>
R S n s> XxY  YxX
(R A S)11
Observe that =1 j*1iX and = k* T2. Thus by the definition
of intersection: (RAS,i) > (R1A S1, ). However since r"' is an
isomcrphism, (RkS,) E ((RAS)",p*); whence
((RAS)1,I*) < (R1A S1, ).
To obtain the reverse inequality, note that by the definition of
intersection (RIAN SI,) < (R(AS,i) since jT1I3 = and
kT1XL = #. Thus (R1(\S1,) (R/S,).
Whence (R1(1 S~ ,) < (RnS,P) H ((RflS)1,i*). Consequently:
(R1( S1,) = ((Rn S)1, *).
1.19 Remark. It is clear from the definition of intersection (0.6 ) that
if (R,j) < (S,k) and (R,j) < (T,m) then (R,j) < (S(\T,n).
1.20. Proposition. Let (R,j) be a relation on X. Then RAX, R'(IAX, and
RAR1n XA are isomorphic relations on X.
Proof. Consider the following commutative diagram.
.... + R~
X 4
21
/ j
R (R' A  A R  XxX 2> XxX
1 R iX X
b
Note that since ix equalizes I1 and v2', iX = iX (1.16) and
also that < 2' 1 = ; i.e., '< 2,TT 1 > = 1XxX (1.6). Observe
that jTrX4 j* = 1 X3 = i X3. Consequently
JTX4 = i X = 3. Thus by the definition of intersection:
(R7()AXX3) <_ (R(c X,i X 2).
Also observe that j*rX = J = < 2' I>i 2. Whence
j'TXI = i X2 so that by the definition of intersection:
(R1AxiX2) < (R1iAi x 3).
Thus:
(R AxiX 2 ) = (RIfAxix x 3).
Clearly (RCRln Axi X 6) < (R(fAx,i 2). But
(Rt Ax,i X2) < (R'nAxiX 3) and (RftAx X2) < (Rr.A ,iX 2). Thus:
(RA. ) (Aix) (RA,ix 2)/2(RlI AAx,i X3) (Rn/R! fAx,i X6)
Hence,
(R}lAx,iX2) (RfR1('Axi X 6)
<1X, IX
1.21. Lemma. If X is a ( object and X >XxX is the unique
morphism h such that rr h = T2h = 1,, then (X,<1 ,>) and (AXi ) are
isomorphic relations on X.
Proof. Since r = Tr2<,1X > and iX is th3 equalizer of r, and T2,
it follows that (X,<,1 X>) < (AXi ). Since l = l, <1XI > is
a section, hence an extremal monomorphism.
Clearly, z I<1 1 >ri = i1 liX = li and
x 1 X 1 A 1
T2<1X .1 >71iX 1X iX = Tli = T2iX. Hence, by the definition of pro
duct, l i = iX. Thus (A ,i ) < (X,<1,1 >).
X IX xA x A X x
1.22. Example. In the categories Set, Top1, To p,, Grp, Ab, and Rng,
<]X,1 x>: X  XxX can be defined by <1i.1 >(x) = (x,x) eXxX for all
XtX.
1.23. Remark. It is also easy to see that up to isomorphism of extremal
subobjects (X,<]X,1X>) (and thus (Ax,'iX) also) is the equalizer of each
of the following sets of morphisms:
{i] '2 }, {T], <1x,1X> 2}, {<1X,1X>', 1XxX} {"2, 1,,xX and
{T] ,< ]X,1x>2,' 1XxX.
1.24. Proposition. If (R,j) is a reflexive relation on X then T1j and Tn2J
are retractions.
Proof. Since (X,) < (AXx) < (R,j) there exist morphisms a and
B such that ita = and jB = iX. Thus X = "1<1X,1I> = TliXa = i7ljXa.
Thus r1j is a retraction. Similarly T2j is a retraction.
1.25. Remark, Consider the following products: (XxY,p,,p2), (YxZ,1,12),
(XxYxZ,1if J2,3), ((XxY)xZ, ; 2) and (Xx(YxZ),fl*,T 2*). It is easy to
see there exist isomorphisms
01 = and 2 = <1,*' 2' 2*
( _1 02
(XxY)xZ 01 . XxYxZ  XX(YXZ)
such that 0101 = PS1, 21 P271' 301 = T2 and 1 02 7"' 7, 202 P'2
and 1302 = 2 '.
1.26. Definition. Let (R,j) be a relation from X to Y and (S,k) be a re
lation from Y to Z. Consider the following intersection.
lxj
RxZSRxzy  .<( Xxyx
2
Xx S' ^'X>: (YxZ)
Ixk
Let <71i, 3> denote that unique morphism from XxYxZ to XxZ such that
o1 = 1 and 02< 1,3'> = 3 where uJ and o are the projections cf
XxZ to X and Z respectively.
Let (T',j') be the unique epiextremal mono factorization of
<,1'"3>Y, and let the codomain of c' (domain of j') be denoted by RoS.
The relation (RoS,j') is called the composition of R and S.
1.27. Examples. In the categories Set, Grp, Ab, and Topi, the composition
of R and S is isomorphic to the set
{(x,y): there exists a ycY such that (x,y)cR and (y,z) ES}.
This is the usual set theoretic composition of relations (which incident
ally is not the usual notation for the composition of functions when they
are considered as relations).
In the category Top2, the composition of R and S is the closure
of the above set.
1.28. Definition. If (R,j) is a relation on X then K is said to be
transitive if and only if (RoR,j') < (R,j).
A relation on an object X is said to be an equivalence relation
if and only if it is reflexive, symmetric, and transitive.
1.29. Examples. In the categories Set and Top1, transitive relations and
equivalence relations are the usual set theoretic transitive relations
and equivalence relations together with the inclusion maps.
In the category Top2, equivalence relations are closed set theoretic
equivalence relations.
In the categories Grp, and Ab, equivalence relations are subgroups
of the catesian product which are set theoretic equivalence relations.
1.30. Proposition, Let (R, ,j1) and (R2j 2) be relations from X to Y and
let (S ,ki) and (S2,k2) be relations from Y to Z and suppose
(?.,jl) < (K ,2 ) and (S1,k1) < (S2,k2). Then (rl.OSl,j) < (R2oS2,k).
29
Proof. Since (R,jl) < (R2,J2) and (Sl,kl) < (S2,k2) it is immediate that
(RlxZ,jlxl) < (R2xZ,j2xl) and (XxS1,lxkl) < (XxS2,1lxk) whence
((RxZ) f (XxS1),Yl) < ((R2xZ) ( (XxS2),Y2). Consequently there exists a
morphism a such that the following diagram conmutes.
(R2xZ) CI
(RB1Z) n
j2xl
R2xZ >'.  (XxY)xZ
(XxS ) Y2
Xx
(XxS1) *'<
XxS2 1 .2 1xk2 2
XxS1  Xx(YxZ)
ixkI
Thus y 2a = Yl.
Since (RloS1,j) is the intersection of all extremal subobjects
through which Y1 factors (0.21) and since Y1 factors
through (R2oS2,k) it follows that (RloS,j) < (R2oS2,k) which was to be
proved.
1.31. Theorem. Let (R,j) be a relation from X to Y then RoAb, R, and
AxoR are isomorphic relations from X to Y.
Proof. First consider RoAy. From the definition of composition of rela
tions the following commutative diagram is obtained.
jx!
RxY Y  (X>:Y)xY
(RxY) (Xx^y) > .  xYxY
Y'
Xx   * XX (YxY)
Ixi.
Recall that (A ,i ) is the equalizer of the projections pl and p2
from yxy to Y.
It will next be shown that y = y. Let 0i, and 2 be
the projections of XxA to X and A respectively, and let Il"* and IT 2 be
the projections of Xx(YxY) to X and YxY respectively. Then
P1<1' 2 >Y = itY = P 3< 3>Y.
P2<1'pY 2 2 2 202(xi ) 2 1= 2*2 Pliy 2 = 2iY 2 2
P2 2*(]xi )X2 = "32(1xiy )2 = 3= p<2 <1 3>y.
Hence <1, 2I >y y.
Let p1 and P2 be the projections of (Xxy)xY to XxY and Y respec
tively and let p and p2 be the projections of RxY to R and Y respec
tively.
Since ITjp1 = 1P1(jxl)X1 = 0 (jxl)x = iT y = ,y, and
Ti2JP I*I = T2P1J (x)Il = T2rl(jxl)Al = T2Y = T2y, and
< l' I2>y = < 1',T3>y = j T, then the following diagram commutes.
PI*
Rxy   R
(Rxy) (XxAy) < >.T> XXY
^T ROA,
Thus since (RoA ,j') is the intersection of all extremal subobjects
through which y = y factors (0.21) it follows that
(RcA j') < (R,j).
J
To see that (R,j) < (RoAy,ji ) consider the following commutative
diagrams.
I 1
XxY
 :>~ Xx'ixY
<1lW2>
XxY
ly
<1R' 2j
R  RxY
1R 1*
Recall that (Ay,i ) = (Y,<1 ,1 >) (].21), thus there exists a
morphism o:Y >y such that i y = <1 ,1 >.
R 12 XXA
x
`c (yx)xy r
It now will be shown that the following diagram commutes.
jxly
RxY   (XxY)xy
<1R)' 2J> E 01
1 O
R > Xxyxy
<,lj,wT2 ,
XxA >"1 > Xx(Yxy)
Y 1 Xi
yX Y
1 1l(jXl) = Tll(jxl) = IlJP R'2j = 1JlR = r
0 (l(jxl) 2 * = T2j.
3 1 R' 2 22(x1)<12 R 2 2
l0l = lP1 = lj'
i201 = Tr2P = 72j.
<301 j> P=2
1 2 2 1 2j > (2)
_12(1lXiy) = Tf1 ((Xi) = Pl =
T2 2(lxiy)<1l j,o r2j> 1= plI2*(lxiY) = Pliy>2
PliyOTf2j = Pl<1( >IT2j = l = ]2j = T2j.
3 92(1xiY)<7lj,o'r2j> = P2 2*(1xiy)< lj,' m2j> = p2iYP2< 111,o2j> "
P2i 2j = g2T2j 1I 2j = T2j.
Thus by the definition of intersection there exists a unique mor.
phism C from R to (RxY)f1(XxA ) such that
y = e = 01 < > = 1(j ,) = 2(1xiy)<"7Tj:orT2j>. Thus
<1, 3 >Y3 = <13>0 . But since
'<1' l'>01 = 710 = iPl
22<7,T 1 3>l = 30l = 2j it follows from the
definition of product that < , 3>0 j.
Thus j = y = j'T'C whence (R,j) < (RoAy,j).
The proof that (Rj) E (A oR,j"") follows from analogous arguments.
1.32. Proposition. If (R,j) is reflexive and transitive
(RoR,j') E (R,j).
Proof. Since (R,j) is transitive then (RoR,j') < (R,j).
reflexive then (Ax,iX) < (R,j). Thus (R,j) < .(RoA X, )
(1.31 and 1.30). Hence (RoR,j') E (R,j).
on X then
Since (R,j) is
< (RoR,j') (R,j)
1.33. Remark. As has been remarked in (1.27), if (R,j) is a relation from
X to Y and (S,k) is a relation from Y to Z then in the categories Set, Ab,
Grp, and Top (RoS,j') may be taken to be the set
{(x,z): there exists a yeY such that (x,y)cR and (y,z)ES}
together with the inclusion map j^. Thus the categorical definition of
composition (1.26) yields in these special concrete categories the usual
set theoretic composition.
A similar remark can be made about the definition of the inverse
relation. That is, the categorical definition yields the usual set theo
retic definition in the categories Set, Ab, Grp, and Top1 to only men
tion a few. Indeed, the categorical definitions were obtained by analyz
ing the situation in the set theoretic case.
However, in the category Top2 of Hausdorff spaces and continuous
maps the extremal :Tonomorphisms are the closed embeddings which leads to
the following consequences.
1.34. Example. If (R,j) is a relation from X to Y and (S,k) is a relation
from Y to Z for Top2objeccs X,Y, and Z. Let T be the following set.
{(x,z): there exists a yeY such that (x,y)ER and (y,z)ES}
Thaen (RoS,j) (clT,j) where "cl" means closure with respect to the top
ology of XxZ (c.f. 1.27).
Proof. Recall that the extremal monomorphisms are the closed embeddings,
thus RoS is a closed subset of XxZ. Clearly the following diagram
commutes.
Y 
