(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

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

re-enforces 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 S-loR-I 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.

ZxR-1

zP

RxZ *y (XXY)xZ '> -- "P ZX (Yxx)/

V01 IV )

(RxZ)CI(xxS) X. : xyxZ >~ -- Zxy>rX

X2 02 02

x XS 1-1') [X(YXyZ) t~--------s (Zxy) cX

k*xl1

"X,

37

01 = < `Ip2',2P2> and 02 = <171 T*, 2 27*,'7 2

11

^1 <3'2 1 = 3l = P2

2 1

^72

1 30]

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 <7T 3 ,1T r> = 12 1 2 2 = 2

3 022

3n2

Thus 01

- p1 > =

0l

p >(jxl) = P2(jx1) = p *.

P(ij*X

p *

= *.

1 2 1 1 2 12

2 2 1 2 2 1 21p

p (lxj*-)

= J-PI*".

Thus

1<2' 2 I>*>(1xk) = < 2' > ,2*(1Xk)

Il(k*xl)

= k*-T2. = ~1

2 2<

2(k*xl)

Thus <

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

(RxZ) r (XxS) >---------- XxxZx

(S-lxX)( (ZxR-1) <--- ~I'3 Z xX

S-1oR-1

Since (RoS,6) is the intersection of all extremal subobjects

through which

-1 -

Hence there is some morphism p such that

S- oR-1

-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

Now applying the above result to (S-1,k*) and (R-1,j*), it fellows

that ((S-loR-1>)-1,a') < ((R-1)-o(S-1)-1,B#) E (RoS,Z) (1.11) whence

(S-loR-1,a') < ((RoS)-1,6*) (1.12), so that

1.39. Corollary. Let (R,j) be a relation from X to Y. Then (RoR-1,j#) is

a symmetric relation on X and (R-IoR,j') is symmetric on Y.

Proof. ((RoR-1)-1,j#*) E ((R-1)-1oR-1,j) (RoR- ,j#) and

((R-loR)-1,j*) = (R-1o(R-l)-I,j) = (R-IoR,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 epi-extremal 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
fTrl*

= ap2;
7r,2 <2'l>(axa) = l (axa) apl; and
,2(axa)

,a3 > =

1T(axa)1P'P2> ,ap3> = l (axa)

2

ap1

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> 'ap3> = lXap3 = ap3 = '3(axaxa).

T2(a-a)< P'P2> = aP2

ap2 = T2(axaxa).

2- *((axa)xl)1',p2>,ap3> =

1XP2<

= Ir*(1Xx(axa))

1Xp

SiTr2*(1Xx(axa))

7l(axa)p2

ap1

= T2'r2*(lx(axa))

2(axa)

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 ,a3 >E =

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<

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

2 1 Y Y 21

where pl and p2 are the projections of AxY. Hence

(1 xa)

R ------------- XxX

1 x 1jR

RX p----------s 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 epi-ex-

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) < (RoR-1,j#) which was to be proved. The proof that

(AXR,(j2Xj2)iXR) < (R-1oR,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(R-loR).

Proof. If R is difunctional then

(R,j) (RoAXR,kl) < (Ro(R-1oR),kl) < (R,j) (5.26, 5.27 and 5.22).

Similarly,

(R,j) = (ARoR,k2) < ((RoR-1)oR,k2) < (R,j).

The converse is immediate from the definition (5.22).

5.29. Remark. Let ( be a locally small quasi-complete 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 epi-extremal 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 epi-mono 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.).

R ------------- XxX

1 x 1jR

RX p----------s 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 epi-ex-

tremal mono factorization of

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) < (RoR-1,j#) which was to be proved. The proof that

(AXR,(j2Xj2)iXR) < (R-1oR,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(R-loR).

Proof. If R is difunctional then

(R,j) (RoAXR,kl) < (Ro(R-1oR),kl) < (R,j) (5.26, 5.27 and 5.22).

Similarly,

(R,j) = (ARoR,k2) < ((RoR-1)oR,k2) < (R,j).

The converse is immediate from the definition (5.22).

5.29. Remark. Let ( be a locally small quasi-complete 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 epi-extremal 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 epi-mono 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.).

*Â£._i ,> y*X Observe that* tt 1 >^ = J*tX, and `y = k^xX^. Thus by the definition of intersection: (R A S,`ifi) <_ (f'AS" 1 ^), However since r* is an isomorphism, (R A S,

`^) E ((R A S)" 1 ,\p*) ; whence ((RASj-^f'O < (R~V\ S" 1 ,^). To obtain the reverse inequality, note that by the definition of intersection (R -1 A S -5 ,`

) <_ (RAS,i|j) since jr -1 *=

4> and kT -1 X,

<^. Thus (R~ ! A S -] ,

`(+>) <_ (R A S , ^) . Whence (R~V\ S -1 ,4>) < (RA S, ^) = ( (R A S)" 1 ,ip*) . Consequently: (R^AS1 ^) E ((R/IS)" 1 ,^). 1.19 Remark. Ic is clear from the definition of intersection (O.G ) that if (R,j) Â£ (S,k) and (R,j) 1 (T,m) then (R,j) < (SAT.n). 1.20. P roposition . Let (R,j) be a relation on X. Then RAA^, R'^Aj,, and R AR~ f\ Â£Â„ are isomorphic relations on X. P roo f. Consider the following commutative diagram. Â— >. R-l S/iR'^lA,,X*X `

PAGE 32

26 Note that since t^ equalizes t^ and 7iÂ„ ,

PAGE 33

27 1.23. Remark . It is also easy to see that up to isomorphism of extremal subobjects (X,<3y,ly>) (and thus (Ay,'iy) also) is the equalizer of each cf the following sets of morphisms: {T7j ,Tr 2 }, {<1 x ,1 x >ttj , <1 x ,1 x >tt 2 }, { <1 x> 1 X >7T l' L XxX*> t<1 X' 1 X >T! 2' X XxX } ' apd {<1 X ,1 X > 1 T 1 ,< J X ' 1 X >TT 2 ' X XxX } ' 1.24. Proposition . If (R,j) is a reflexive relation on X then tt-^j and tt^ j are retractions. Proof. Since (X,

PAGE 34

28 Let (i',j') be the unique epi-extremal mono factorization of

PAGE 35

29 Proof . Since (R-,^) < (R 2 ,J 2 ) and (S^k-^ <_ (S 2 ,k 2 ) it is immediate that (RjXZ.j-jXl) Â£ (R 2 x Z,J 2 x1 ^ and ( XxS iÂ» lxk i) 1 (XxS 2 ,lxk 2 ) whence ((R^OO (XxS 1 ),y 1 ) < ((R 2 xZ)0 (XxS 2 ),y 2 ). Consequently there exists a morphism a such that the following diagram conmutes. j 2 xl > (XxY)xZ (R 9 xZ) H (XxS ) .__/__ (r 1 xz> r, (xxs 1 ) *-^x Xx(YxZ) lxk, Thus

PAGE 36

30 Recall that (A , i ) is the equalizer of the projections p 1 and p from Y>Y to Y. It will next be shewn that y = y. Let p , and p' be the projections of XxA to X and A respectively, and let it * and ir * be the projections of Xx(YxY) to X and Yxy respectively. Then P 1 <7T 1' TT 9 > ^ = *]Y = P 1 <7; 1 > TT 3 > YP 2 < ^1'^2 >Y = V = ^ 2 2 (1>Y. Hence <^,,v Â„>y = YLet. p, and pÂ„ be the projections of (XxY)xy to Xxy and Y respectively and let p * and p * be the projections of RxY to R and Y respectively. Since ^ j 3 P x " x = TjPj^jxl)^ = it.^ Cjxl)X ] = T^y = TT 1 <7T 1 > T1 7 > Y, T, 2^ P 1* X 1 = 1T 2 P 1^ >1 ' )X 1 = 7T 2Â°1 ^ Xl ^ X l = ^2 y = 7T 2 Y ' 3rid y = y = j'x', then the following diagram commutes. and (RxY) H (XxA. ) RxY Pl * -> R fe Y V* XxY fc> RoA, Thus since (RoA , j ") is the intersection of all extremal subobjects through which y =

PAGE 37

31 < JÂ» "2J > (XxY)xY V5~ P2 v YÂ» XxY Â— -;s> XxYxY V -** XxY -& Y

PAGE 38

32 It now will be shown that the following diagram commutes, J*1 Y r*YÂ» Â— >(XxY)xY 1

PAGE 39

33 Thus j =

PAGE 40

34 ology of XxZ (c.f . 1.27) . Proof. Recall that the extremal monomorphisms are the closed embeddi rigs, thus RoS is a closed subset of XxZ. Clearly the following diagram commutes. (RxZ)fUXxS) RoS XxZ It is evident that T CEoS whence clT C RoS. But (RoS.j") is the intersection of all closed subsets of XxZ through which

PAGE 41

35 1.36. Example . In the category Top the composition of relations is 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 = {(h,h)} considered as a subspace of X*X. Let Sbe((x,y): y = x} considered as a subspace of X*Y and let Tbe{(y,z): < 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 X*X, X*Y, and Y*Z respectively. It follows that RoS = {(%,%)} and that (RoS)oT = 0. But SoT = {(y,z): <_ y <_ h) and from this it follows that Rc(SoT) = {(%,z): zeZ}. Hence Ro (SoT) i (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). However, this should cause no more anxiety than 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 re-enforces its appropriateness. 1.38. Theore m. 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 S -1 oR -1 are isomorphic relations from Z to X. Proof . The following products shall be used: (X^ttj .rr.,), (Y

PAGE 42

36 (XxCYxZ),^*,^*), (XxYxZ,7i 1 ,^ 2 ,Tf 3 ), and ( (XxY)xZ ,p 1 ,p 2 ) . The notaLion " * " 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 tij and tt 2 and the projections of ZxY are ft ^ and tt 2 . Consider the following diagram. It will be shown to be commutative. RxZ *y (RxZ)O(XxS) Â»Xx S *>l x xk ZxR> (XxY)xZ V> WZx(YxX) -> XxYxZ >=?0. -> Xx(YxZ) W <'(! i , TTp Â» IT q> <

PAGE 43

37 0, =

` and 9 = . 'j ^i > " ]^2 > "2^2 j_ " 1 > "2 1 ' 2 ^l ] `

` = Pi < P2> `

`p l > = p 2' i 1`0 1 = Vl = V ^2 e i P l > = 7T lP2

P l > = Ti 1 <-iT 2 ,TT 1 >P 1 = T^pj ^y K ~'^ jTTo >^1 >0 1 = TT 2 1 = TT 2 |:> 1* ^3 l

p ] > = Tr 2 p 2

p l > = 7T 2 <1T 2 ' TT 1 > ^1 = ^l^l Tf 3 `0 1 = ij^Oj = TjPj. TT, <71~ , TTÂ« ,71", >0Â„ = Tt^O^ = Tr^TT^*. TT 2 ^ 2 = n 2 Oy = Tr 2 2 = TfiTTn*' 71 3 2 << ''2'^l >Tr 2*' TT l* > = 7T 2* << ^2'^l >1I 2* , '' T l' <> = """l' i 3 `

*0 2 = ^1Â°2 = *!** Thu s 1*

p 1 > =

*0 1 and '^^3 , ^2 > ^3 >0 2 = 2 <* iT i >ir 2*Â» Tr l "> P p >(jxl) = P (jxl) = P *. 12 2 1 ' 1 J 2 2 P (lxj*)

P *

= P= ^2

P 1 > ^ ><1) = `JP l' V = J* Tp l 5 p (lxj*)`

` = j*p *`

` = j*Tp *. Thu S `

`P 1 >(jx]) = (IXJ*)`

` 1 f 1 Â«Tr 2 ,tr 1 >TT 2 *,Trj*>(lxk) = 7r 2 *(lxk) = <;? 2'^l >kp 2 = k * Tp V Tf 1 (k*xl)`

PAGE 44

38 ft 2 <

`-& zxyxx * It is easy to see that the following diagram commutes (RxZ)H (X*S)>*(S } xx) C\ (ZxR l ) ~> XxYxZ <7Tj ir 3> cS` Â— > ZxX S _1 oR l Xxj

` Since (RoS,&) is the intersection of all extremal subobjects through which y factors, then (RoS.P) <_ (S _1 oR _1 ,`

`(0 . Hence there is some morphism u such that `

a'u = B. Consequently,

`f3 u'u = B*x* and the following diagram commutes
PAGE 45
39 RoS Â»S'^oR -1 (RoS) --IT Since ((RoS) -1 , 3*) is the intersection of all extremal subobjects through which `

`B factors then ((RoS) -1 , 3*) <^ (S -1 oR _1 ,a') . Now applying the above result to (S -1 ,k*) and (R -1 ,j*), it follows that ((S1 oR1 )1 ,a'*) <_ ((R -1 ) ^ (S" 1 ) -1 , 3#) = (RoS,E) (1.11) whence (S _1 oR1 1 o') < ((RoS)1 ,3*) (1.12), so that (S-^R-^a') = ((RoS)" 1 , 3"). 1.39. Cor ollar y. Let (R,j) be a relation from X to Y. Then (RoR _1 ,j#) is a symmetric relation on X and (R. _1 oR,j") is symmetric on Y. Proof . ((RoR1 )1 ^/;*) = ((R -1 )1 oR _1 ,j) = (RoR -1 ,j#) and ((R-^R)1 ^'*) = (R~ 1 o(R1 )" 1 ,j) s (R-l R,j') (1.38 and 1.11). 1.40. P roposition . Let (R,j) be a relation from X to Y and lee (S,k) and (T,m) be relations from Y to Z. Then (Ro(SAT),g) < ((RoS)A(RoT),6). P roof . By Proposition 1.5 there exist canonical isomorphisms: ip: (RxZ) A (Xx (S n T) "*/>* (RxZ) A (XxS) A (XxT) Â•jj: Xy(SAT) ^ Â» (XxS)A(XxT). Consider the following commutative diagrams.
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40 SAT >V T Â» > S (Rxz)n(xx(snT)) R*Z ^Â— Â» i*l * (xxy)xz XxS Note that By (lxk) (lxX 2 ) = (lx m )(lxX ) lx a . Let (t,3) be the epi-extremal nono factorization of `

`^ * ^ U S ffflJ*T = f TT^ j *T SO that since x is an epimorphism it follows that fir, j * = f'if 2 j*. Hence, from the definition of equalizer, there exists a morphism n from R to R for which jn = j*. This implies that (R -1 ,j*) <_ (R,j) so that (R,j) is symmetric . Consider the following products: (XxX.ttj ,tt 2 ) , (X^XxX,^ ,t 2 j 7 ^) > ((XxX)xX,p 1 ,p 2 ), (Xx(XxX) ) p 1 ,p 2 ), (RxX,^ 1 *,tt 2 *) and (XxR.ffj ,ff 2 ) . To see that (R,j) is transitive, consider the following commutative diagram. jxl x RxX Â» > (XxX) xX (RxX)A(XxR) Â»Xo XxR Â»1y x X X J -> Xx(XxX) -4sXxXxX Let (r#,j#) be the epi-extremal mono factorization of`

`'T3 > Y Recall that the codomain of i# (domain of j#) is RoR. Next, it will be shewn that fir,y = fT? 2 Y = f^Yfir 3 Y = fTf 3 e 2 (l x xj)X 2 = fir 2 p 2 (l x xj)X 2 = fir 2 jff 2 X 2 = f^jit^ = fn 1 p 2 (l x xj)X 2 = fTr 2 2 (lx x j)X 2 = fi 2 Yfi lY = fTT^^jxl^Xj = f^T 1 P 1 (jxl x )X 1 = fW 1 -jTT 1 *Xj i fTrpjIT^Aj = fir 2 p 1 (jxl x )x 1 = fi 2 1 (jxi x )x 1 = fW 2 Y. Thus ff'' i ''TT i ,Tr 3 >^ = fiT 1 Y = fiT 3 Y = f t, 2`__y; so fir^M = fir 2 j#x#. PAGE 50 44 Again, since t# is an epiniorphism, it follows that fitnj# = fir 2 j#. By the definition of equalizer there exists a morphism k from RoR to R for which jk = j#. This implies that (RoK,j//) <_ (R,j) so that (R,j) is transitive. 2.5. Theorem. The intersection of any finite family of congruences on an Y Â£ -object is a congruence. Proof . Let {(Ei,ei): iel} be a finite family of congruences on X. Then there exist morphisms fi with domain X such that (Li,ei) is the equalizer of fi^i and fiTT 2 (2.1). Let the codomain of each fi be denoted Yi. Consider the morphism__ from X to TT Yi and consider iel iel the intersection ( C\ Ei,e). iel It will be shown that (/I E^e) is the equalizer of "ri and iel iel TT 2 . iel X^X *1 7T 2 -S~ >Y, First observe that: pjTi 1 e = f.r.e = f^Tt 2 e = p.TT2 e for iel J J J iel" each jel. Thus 'u 1 e = TT e. iel iel Now if g is a morphism from W to X*X such that < f'i''T]g = < fi >7T 2S iel iel" then f-Tr.g = p.T^g = p-<Â£-j>Ti2g = f .iTog.Tbus by che definition of 2 l J iel iel" equalizer there exist morphisms kfrom W to E^ so that e^k^ = g for each iel. Thus by the definition of intersection there exists a morphism
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45 k from W to f\ E^ such that ek = g. This implies that ( f\ E-^,e) is the iel iel equalizer of < fÂ± > ^i and TT2. iel iel 2.6. Propositio n. If u is complete then the intersection of any family of congruences on any p -object is a congruence. Proof. Repeat the proof of 2.7 assuming I to be infinite. 2.7. Pro po sition . Let ft be the family of all congruences on X and let (Aft,p) be the intersection of this family. Then C\ ft and A y are isomorphic relations on X. Proof . If (E,e)eft then (E.e) is an equivalence relation and hence is reflexive (2.4). Thus (A x ,i x ) <_ (E,e). Hence (%,i x ) <_ (ftft.p). But (Ay,iy) is a congruence; hence (f\Q,o) <_ (Ay,iy). 2.8. P roposit ion. Let f be a g -morphism from X to Y. Then f is a monomorphism if and only if Ay and cong(f) are isomorphic relations on X. Proof . Since (cong(f),i^) is an equivalence relation (2.4) it is reflexive and hence (Ay,i x ) Â— ( con g Cf ) > if ) < If f is a monomorphism then fTTji^ = fTT 2 ir implies that i'^if ~ 1T 2^f" Hence there exists a morphism k for which iÂ„k = if and consequently (cong(f),if) <^ (A x ,i x ). Conversely, suppose that (cong(f),if) H (Ay,i,,) and a and B are morphisms having domain Z and codomain X such that foe = fBConsider the morphism from Z to X*X. fTi 1 = fa fB = fiT 2 sc that there exists a morphism X from Z to A. for which iyX . Thus a = i: 1 = 7T lyX = Tf 2 "'x^ ~ 1T o

__S> = B. Consequently a = B so that f is a monomorphism. 2.9. D efin ition. A K -morphism f from X to Y is said to be c onstant if and only if for all pairs of morphisms Z Â™~. X, fa = f6. B__

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46 2.10. Propositio n. Let f be a morphism from X to Y. Then f is constant if and only if (cong(f),if) = (X*X, lÂ„ xX ) . Proof. If f is constant then fir. = fir so that fir, 1 = f xr 1 Thus 1 Â£ JX X A ^ A X A there exists a unique morphism k from XxX to eong(f) for which ifk = l^xx whence if is a retraction. But since if is an equalizer, it must be an isomorphism (0.4 ) so that (cong(f),if) and (X X X,1 V ,. V ) are isomorphic 1 A X A relations on X, Conversely, suppose that (X*X,lÂ„ xy ) = (cong(f),if) and that o. and 6 are morphisms with common domain, and codomain X. ConsiderPAGE 52

46 2.10. Propositio n. Let f be a morphism from X to Y. Then f is constant if and only if (cong(f),if) = (X*X, lÂ„ xX ) . Proof. If f is constant then fir. = fir so that fir, 1 = f xr 1 Thus 1 Â£ JX X A ^ A X A there exists a unique morphism k from XxX to eong(f) for which ifk = l^xx whence if is a retraction. But since if is an equalizer, it must be an isomorphism (0.4 ) so that (cong(f),if) and (X X X,1 V ,. V ) are isomorphic 1 A X A relations on X, Conversely, suppose that (X*X,lÂ„ xy ) = (cong(f),if) and that o. and 6 are morphisms with common domain, and codomain X. Consider

__from Z to XxX where Z is the common domain of a and !3. Since f'.i 1 = fir 1 , it follows that fn = fir so that j. A X .A ^ A X A -1 ^fa = fi! i = fT[2 = f3. 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 fc -morphisms such that f = gh then (cong(h) , i,) _< (cong(f ) >if) Furthermore if g is a monomorphism then (cong(h),i n ) ^ (cong(f ) , if ) . Proof . Since hn i^ = hir i, it follows that ghiTji^ = gh^i^ sc that fiiji, = f7r 2 ivThus there exists a morphism k from cong(h) to cong(f) for v.'hich ifk = i^. VThence (cong(h) , i, ) <_ (cong(f ) , if ) . If g is a monomorphism then f^jif = f T, 2^f = 8 n7T lif = S^ l7T 2^f implies that h'tjif = hir 2 if. Thus there exists a morphism k* from cong(f) to cong(h) for which i h k* = i^ , whence (cong(f ) , if ) <_ (cong(h) ,i h ) . Consequently (cong(f),if) = (cong(h) ,i h ) . 2.12. Preposition. If fa has coequalizers and f is a ^ -morphism from X to Y and if (f*,Z) is the coequalizer of Tijif artd "^o^f tnea con g(f) and cong(f*) are isomorphic relations on X. Proof. Since f, : , i f = fll oif then by the definition of coequalizer there__

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47 exists a morphism k* from Z to Y for which k*f* = f . Since f*-n,Â± Â£4c = f*7T i_, it follows that fir,i,. = k*f*ir i = k*f*ir i = If* 2 f* If* 1 f* 2 f* ftr i . Thus there exists a morphism k from cong(f*) to cong(f) for which i_k i.,. Consequently (cong(f *) ,i.. ) < (cong(f),i ). t f * I " Â— r No^v ;ince f* is the coequalizer of tt i, and tt i then f*7r i f*iT i . Hence there exists a morphism k' from cong(f) to If 2 f cong(f*) for which i fA k' = i . Consequently (cong(f ) , i.-) < (cong(f*),i ). f * 2.13. Proposition . If fe is complete and is a family of congruences on X generated by morphisms f: X sÂ»Y and if (r\Q,p) = (A ,i ) then f A A the unique morphism from X to TT Y such that tt = f, is a mcnomorphism. Proof . Observe that for each f, fT..!= ti-Ott i = -n^Q-ny^= f^o^n' Thus it follows that (cong(0),i ) j< (eong(f ) , i c ) for all X Â— '"^ f Hence (cong(C),i ) 5 (f\n,o) (1.19). Since (Ai!,p) = ^ x ^ 1 (cong(e),i Q ) (2. A) it follows that (Hfi,p) = ( A v >i Y ) (cong(0) , i. ) . Thus is a XX ^ monomorphism (2.8). 2.14. Corol lar y. If rf is complete and I! is a family of congruences on X generated by morphisms f : X *Y and for some g: X Â— Â— -*y , g is a monomorphism, then the unique morphism from X to TT Y such that n 9 = f is a monomorphism. Proof . Since g is a monomorphism then (cong(g),i ) = (A . i ) (2.8). Thus g A A (H^,p) < (A i ) by the definition of intersection. But (A v ,i v ) < (fin.p) (2.4 and 2.6). Consequently (A V5 iÂ„) = (A^,p) and the A A Â« A resul t follows from Proposition 2 . 13 .

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SECTION 3. CATEGORICAL EQUIVALENCE RELATIONS AND QUASI-EQUIVALENCE RELATIONS 3.1. Theorem . If {(E.,tf>.): iel} is a family of equivalence relations on a Â£j -object X then their intersection ( M E.,) is an equivalence relaiel tion on X. Proof . Since (Ay,i ) jÂ£ (E.,4>.) for each iel it follows that (A Y ,i Y ) < (AE.,|) (1.19). Kence (Ae. , PAGE 53

47 exists a morphism k* from Z to Y for which k*f* = f . Since f*-n,Â± Â£4c = f*7T i_, it follows that fir,i,. = k*f*ir i = k*f*ir i = If* 2 f* If* 1 f* 2 f* ftr i . Thus there exists a morphism k from cong(f*) to cong(f) for which i_k i.,. Consequently (cong(f *) ,i.. ) < (cong(f),i ). t f * I " Â— r No^v ;ince f* is the coequalizer of tt i, and tt i then f*7r i f*iT i . Hence there exists a morphism k' from cong(f) to If 2 f cong(f*) for which i fA k' = i . Consequently (cong(f ) , i.-) < (cong(f*),i ). f * 2.13. Proposition . If fe is complete and is a family of congruences on X generated by morphisms f: X sÂ»Y and if (r\Q,p) = (A ,i ) then f A A the unique morphism from X to TT Y such that tt = f, is a mcnomorphism. Proof . Observe that for each f, fT..!= ti-Ott i = -n^Q-ny^= f^o^n' Thus it follows that (cong(0),i ) j< (eong(f ) , i c ) for all X Â— '"^ f Hence (cong(C),i ) 5 (f\n,o) (1.19). Since (Ai!,p) = ^ x ^ 1 (cong(e),i Q ) (2. A) it follows that (Hfi,p) = ( A v >i Y ) (cong(0) , i. ) . Thus is a XX ^ monomorphism (2.8). 2.14. Corol lar y. If rf is complete and I! is a family of congruences on X generated by morphisms f : X *Y and for some g: X Â— Â— -*y , g is a monomorphism, then the unique morphism from X to TT Y such that n 9 = f is a monomorphism. Proof . Since g is a monomorphism then (cong(g),i ) = (A . i ) (2.8). Thus g A A (H^,p) < (A i ) by the definition of intersection. But (A v ,i v ) < (fin.p) (2.4 and 2.6). Consequently (A V5 iÂ„) = (A^,p) and the A A Â« A resul t follows from Proposition 2 . 13 .

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SECTION 3. CATEGORICAL EQUIVALENCE RELATIONS AND QUASI-EQUIVALENCE RELATIONS 3.1. Theorem . If {(E.,tf>.): iel} is a family of equivalence relations on a Â£j -object X then their intersection ( M E.,

) <^ (E.,

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49 3.3. Proposition. If (A, a) is an extremal suhobject of X then (AxA,axa) is a quasi-equivalence on X. Proof. Consider the products (AxA,p ,p ) and (XxXjTT, ,tt ) Â• Since a is an extremal menomorphism then axa is an extremal monomorphism (0.20) and hence (Ax A, axa) is a relation on X. Consider the following commutative diagram. a-a ` AxA >V> X> (axa) = fr 2 (axa) =Â• a P ; 1 7 i (axa)`

` = ap 2 ; (ax a )* Tr 2 (axa) = tt (axa) ap^; and Tr 2 (axa)`

` = a Pl then it follows that (axa) (axa)`

. But

`> ap 3 > = Tr 1 (axa)p 1 Â«p 1 ,p 2 > ,ap 3 > '= ^ (axa)`

` = ap 2 < Pl ,p 2 > = apÂ„ = ttÂ„ (axaxa) . Tf 3 e i (( aya ) x l x )<`

` Â» a P 3 > = ^ 2 *((a^a)xl^)Â«p 1 ,p 1 >, a p^ > = l x P2<`

` > a p3 > = 1 x a ^3 = a ^3 = ^3 (axaxa) ~> 1 02( 1 x x ( axa ))< a P 1 >`

`> = Tr 1 *(l x x(axa)) > = ^x ap l = a Pl = iri(axaxa) ^2 e 2^ I X x ^ axa ^ > = Tr i 7r 2*^ 1 x x ^ axa ' ^ `

`> = TT 1 (axa)p2`> = ti j (axa) `,ap >Â£ = y and >Â£ = Y Â• 12 3 1 12 3 2 Since p 1 `

` P 2 >Â£ = P~ 2 C = P 2 Pi Y i it follows that p 1 Â«p 1 ,p 2 >,ap 3 >C = `

`YX = (j 2 x J2) < Pl sP3 >1 ^ so that (j 2 x j 2 )it'#'r# = j't'A. Hence the following diagram commutes. (RxXR) O (XRxR) 1# Â™Â» RoR y s / (RxX) A (XxR) Â£ / 4># XRxXR / RoR V?-Â— i X-i J 2 J 2 -> XxX`

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58 Since x# is an epimorphism and j' is an extremal monomorphism, by the diagonalizing property (0.19) there exists a unique morphism Â£ such that j'C = (j 2 x J2H# and Ct# = x'A. But this says that (RoR, (.i 2 xj 2 )^.:0 _< (RoR.j'). Since (R,j) is transitive (RoR,j T ) <_ (R,j) hence (RoR, (j 2 x j 2H*) Â£ ( R >j) =: ( R > (j 2 X J 2^) Â• Hence there exists a morphism from RoR to R such that (j^^H = Cj 2 X J 2^^ * ^ a ^n ' J2 X J? *" s a monomorphism so that \po = ifi// which says that (RoR, -> (XxY)xX ->(RxX)H (XxR1 ) >* *Â» XxYxX XxR~ J Â»-> Xx(YxX) l Y xj* Consider the following products: (XxYjir, ,ir 2 ) , (XxYxX,^ ,ir 2 , tt 3 ) , (RvX, Pl ,p 2 ), (XxR-l,p 1 *,p 2 *) J ((XxY)xX,ir 1 *,Tr 2 *) 1 (Xx(YxX) ,ff x ,if 2 ) , (YxX,tt 1 ,tt 2 ) , and (XxX,tt 1 ,tt 2 ) . Now, ^l lCj xl x ) <1 RÂ» 7r l3 > = Tr i 7T i A (J xl x) <1 R'"lJ' = TT lJ p l PAGE 64

58 Since x# is an epimorphism and j' is an extremal monomorphism, by the diagonalizing property (0.19) there exists a unique morphism Â£ such that j'C = (j 2 x J2H# and Ct# = x'A. But this says that (RoR, (.i 2 xj 2 )^.:0 _< (RoR.j'). Since (R,j) is transitive (RoR,j T ) <_ (R,j) hence (RoR, (j 2 x j 2H*) Â£ ( R >j) =: ( R > (j 2 X J 2^) Â• Hence there exists a morphism from RoR to R such that (j^^H = Cj 2 X J 2^^ * ^ a ^n ' J2 X J? *" s a monomorphism so that \po = ifi// which says that (RoR,

*= ir 2 7r l*(J xl x) <1 R' ir lJ > = H 23w 3 e 1 (jxl x )* = ir 2 *(jxl x ) = P 2 <1 r Â»tt 1 J> = T^j.

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59 T7 1 2 (l X x j*) = ffl(l x x j")<7T 1 j,T> = Pl* = lTlJ . 7T 2 2 (l x xj*) = TT l ff 2 (1 X X J >V ' ) <7r l J ' T> = ^ 1 J " P * <1T 1J> T> = TT ^ 3 * T = TTi<1T2,Tr 1 >j = Tr 2 3 Â• TT 3 2 (l x XJ*)<'f] ] -T> = TT 2 ff 2 (i x x j ;!: ) < j,T> = TT 2 j*T = U 2

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59 T7 1 2 (l X x j*)

*T < 1 > J = ^ljThus by Lhe definition of p act the diagram commutes. Hence there exists a morphism E so that AjE = < 1^,itiJ > and A 2 E =*. From the above it is easy to see that TTiyE = 1'iYE = ttiJ = ^3Y^ = 7T 2 <7f l Â» Tr 3 > Y^Recall that (A x ,i x ) ^ s the equalizer of ttj and tt 2 hence there exists a morphism <{> such that i x ^ = yZ. 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, = < ^i,'^3 > y^ = j#T#E = ix^' Since (Ay,iy) and (X,) are isomorphic as extremal subobjects of X*X (1.21), there exists an isomorphism A such that A = iy. Consequently, Acf> = iy<{> = < tti,tt3 > yE = . Now tt j A = lyA<Â£ = Ac}) = tti = tt j j and by hypothesis Tijj is an epimorphism; thus, since A is an isomorphisrrij it follows that has (cj>,iy) as its epi-extremal mono factorization. Eut this means that (Ay,iy) is the intersection of all extremal subobjects of X>X through which factors (0.21). Recall that <7V 1 J > fi 1 J > = j#x#E, thus (Ly,Â±v) <_ (RoR _1 ,j#) which was to be proved. 3.3. Coro llary. If (R,j) is symmetric on X ther. (R.OR, ij#) , the composition of (R>i|0 with (Pv. C) on XR, is reflexive on XR. Procf_. Since (R,j) is symmetric on X then (R*i|0 is symmetric on XR (3.5) hence (R,i{0 = (R -1 ,^*) (1.13). Referring to the diagram in (3. A) fol-

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6C lowing the d ! inition of the canonical embedding it is immediate that Pjip is an e t irphism since p^ = 3kx and each of 3, k, and t, is an -1 Â•v epimorphism. Thus (RoR ,i|;) = (RoR, 4)//) is reflexive on XR (3.7). 3.9. Corollary . If (R,j) is a quasi-equivalence on X then (R,j) is an equivalence relation if and only if ir,j is an epimorphism (respectively if ajid only if tt 2 j is an epimorphism). Proof . If (R,j) is an equivalence relation then (R,j) is reflexive and a quasi-equivalence. Thus by Proposition 1.24, ir,j and tt 2 j are retractions hence epimorphisms . Conversely, if ttjj is an epimorphism then applying the theorem -1 and Proposition 1.30, (A ,i ) < (RoR ,j//) < (RoR.j') < (R,j) so that X X (R,j) is reflexive and hence is an equivalence relation. (If tt 2 j is an -1 ' -1 -1 -1 epimorphism then ( A y,i x ) < (R oR,j#*) _< (R oR ,j *) Â£ (R ,j*) and (R~\j*) H (R,j).) 3.10. Corollary. If (R,j) is a quasi-equivalence on X then (R,ij0 is an equivalence relation on XR. Proof . (R,40 is a quasi-equivalence on XR (3.6) and (RoR,^//) is reflexive on XR (3.8). Thus (A YP ,i v D ) < (RoR,^//) < (R.ifi) whence (R,iJ<) is reflexive aK XR Â— Â— and thus is an equivalence relation on XR. 3.11. Prop osition. If (R,j) is a quasi-equivalence on X then (R,j) and (RoR,j') are isomorphic relations on X. P_roo_f. By Corollary 3.10 (R,40 is an equivalence relation on XR whence (RoR,^//) and (R,^) are isomorphic relations on XR (1.32). Recall that there exists a mcrphism E, such that the following diagram commutes (3.6).

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61 t// (RxXR)Ti (XRxR) t'X RoR Â— -l> RoR (J 2 x J 2 )H l V Â— JÂ»XxX , f\S Thus (RoR, (j 2 xj 2 H#) 1 (RoR,j'). But as mentioned above (RoR,tp#) = (R,4<) hence there exists an isomorphism \i\ such that 0\ff = if). So by the definition of the canonical embedding (3.4), j'Â£X# = (J 2 xj 2 )ii;//X// = (j 2 xJ 2 H = 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) <_ (^x^X^ if and only if R is symmetric on X and (R,i|0 <_ (Axr> i-xii' Â• Proof. If (R,j) <_ (AxÂ»ix) then there exists a morphism a such that j = J-x aThus tt,j = Tjix = 1J 2^X a = ^/J wnence 7i , PAGE 66

6C lowing the d ! inition of the canonical embedding it is immediate that Pjip is an e t irphism since p^ = 3kx and each of 3, k, and t, is an -1 Â•v epimorphism. Thus (RoR ,i|;) = (RoR, 4)//) is reflexive on XR (3.7). 3.9. Corollary . If (R,j) is a quasi-equivalence on X then (R,j) is an equivalence relation if and only if ir,j is an epimorphism (respectively if ajid only if tt 2 j is an epimorphism). Proof . If (R,j) is an equivalence relation then (R,j) is reflexive and a quasi-equivalence. Thus by Proposition 1.24, ir,j and tt 2 j are retractions hence epimorphisms . Conversely, if ttjj is an epimorphism then applying the theorem -1 and Proposition 1.30, (A ,i ) < (RoR ,j//) < (RoR.j') < (R,j) so that X X (R,j) is reflexive and hence is an equivalence relation. (If tt 2 j is an -1 ' -1 -1 -1 epimorphism then ( A y,i x ) < (R oR,j#*) _< (R oR ,j *) Â£ (R ,j*) and (R~\j*) H (R,j).) 3.10. Corollary. If (R,j) is a quasi-equivalence on X then (R,ij0 is an equivalence relation on XR. Proof . (R,40 is a quasi-equivalence on XR (3.6) and (RoR,^//) is reflexive on XR (3.8). Thus (A YP ,i v D ) < (RoR,^//) < (R.ifi) whence (R,iJ<) is reflexive aK XR Â— Â— and thus is an equivalence relation on XR. 3.11. Prop osition. If (R,j) is a quasi-equivalence on X then (R,j) and (RoR,j') are isomorphic relations on X. P_roo_f. By Corollary 3.10 (R,40 is an equivalence relation on XR whence (RoR,^//) and (R,^) are isomorphic relations on XR (1.32). Recall that there exists a mcrphism E, such that the following diagram commutes (3.6).

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61 t// (RxXR)Ti (XRxR) t'X RoR Â— -l> RoR (J 2 x J 2 )H l V Â— JÂ»XxX , f\S Thus (RoR, (j 2 xj 2 H#) 1 (RoR,j'). But as mentioned above (RoR,tp#) = (R,4<) hence there exists an isomorphism \i\ such that 0\ff = if). So by the definition of the canonical embedding (3.4), j'Â£X# = (J 2 xj 2 )ii;//X// = (j 2 xJ 2 H = 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) <_ (^x^X^ if and only if R is symmetric on X and (R,i|0 <_ (Axr> i-xii' Â• Proof. If (R,j) <_ (AxÂ»ix) then there exists a morphism a such that j = J-x aThus tt,j = Tjix = 1J 2^X a = ^/J wnence 7i ,

`j = tt 2 j = irjj = Tr 2`j . Thus by the definition of product j = j. Consequently the epi-extremal mono factorisation of <7r 2 ,ir 1 >j is (1r,j) and so (R,j) = (R _1 ,j v ' c ); i.e., (R,j) is symmetric. Recall that j = (j 2 *j 2 )4' (3.4). Thus "lJ = ^l^z^l^^ = 3 2 p lV and ^ = " T 2^2 X J2^^ = J2 p 2^' But "l^ = T ' 2.3 hence j 2 Pii|i = j 2 P 2 '^Â» Since j 2 is a monomorphism it follows that p,i|i = Po'ji. Recall that (Axr^xr) is tne equalizer of p^ and p ? . Hence there exists a morphism g such that ixK.5 = i'This implies that (R,ip) < (Axr.Ixr)Conversely, if R is symmetric and (R,'W Â£_ C^xR'^Xr) tk en there exists a morphism 3 such that il> = ixR^S hence

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62 Pllp = PjixR^ = P2 i XR P = p 2^' Since (J 2 X J 2^ = J> we have ^1J = Ti(J2 x J 2^ = J 2 P l 1 ^ = 3pP2^ = 7T 2^2 X J2^ = lr 2J ' Thus ^ 1 J = ^ so that, there exists a morphisin 2 such that j = iya. This means that (R,j) < (A Xs ix)3.13. Definition . Let (R,j) be a relation on X. Then R is said to be a circular re lation if and only if RoR 5_R . This notion is due to MacLane and Birkhoff ^20 J (exercize 3, page 14) . 3.14. Froposi tion. Let (R,j) be a relation on X. Then R is a circular relation if and only if R is a circular relation. Proof . If R is circular then RoR ^_R -1 . Thus R _1 oR -1 = (RoR) -1 ^(R -1 ) -1 =R (1.38, 1.12 and 1.11). Hence R _1 is circular . Conversely, if R -1 is circular then by the above, (R -1 )" 1 E 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, RoR <_ R = R L hence R is circular. Conversely, if R is reflexive and circular then R 1 is reflexive (1.17) and R _1 is circular (3.14). Hence R~ : = R~ 1 oA y _

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SECTION 4. IMAGES 4.1. De finitio n. Let (R,j) be a relation from X to Y and let (A, a) and (L,b) be extremal subobjects of X and Y respectively. Consider (11 A(AxY) ,y) and (R H (X*B) , 6) . Let (f ,a) and (f .6) be the epi-extremal 1 2 mono factorizations of it y and ir 5 respectively. Denote the domain of a 2 1 Â• by AR and the domain of 6 by RB. Thus the follovring diagrams commute. RA(AxY) Â» A*Y ARV^' -> . Y 4.?. Remark. Since (X.l ) and (Y,l ) are extremal subcbjects of X and Y X Y respectively, then (RA(XxY),y) = (R,j) and (R f\(X*Y) , -5) = (R,j) whence (XR,oO is precisely the extremal subobject (XR ; jÂ„) used in the canonical embedding (3.4). Since X = Y in 3.4 then also (RY,3) is precisely 63

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64 (RX,j ) used in 3.4 4.3. E xamp le. In the category SeX, for (A, a) <_ (X,l ), (B,b) <_ (Y,l ) A Y and (R,j) < (XxY,l ), AR = ; {ycY: tl exists aeA such that (a,y)t:R} RB = {xeX: there exists heB such that (x,b)eR}. This is easily seen since Rf\(AxY) = {(a,y): acA, (a,y)eR} and RA(XxB) = {(x,b): t>Â£B, (x,b)eR), and AR is the set of all second terms of elements of RfV(AxY) and RB is the set of all first terras of elements of RA(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 Top , AR and RB have' precisely the same underlying sets as in Top for it is easy to verify that AR and RB are closed subsets 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. The orem. If (R,j) is a relation from X to Y and (A,a) is an extremal subobject: of X then (AR,a) and (R^Ajg) are isomorphic extremal subobjects of Y. Proof. Consider the following commutative diagrams. Â— > Y -^r AxY ^ r axl v Â£Â» AR *"

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65 R ifMYxA) >Y R>*Â» *^ XxY Â»R 1 <7T 2 ,7T 1 > -*YxX It can be shown in a straightforward manner that(?.X] ) = (1 x a )

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62 Pllp = PjixR^ = P2 i XR P = p 2^' Since (J 2 X J 2^ = J> we have ^1J = Ti(J2 x J 2^ = J 2 P l 1 ^ = 3pP2^ = 7T 2^2 X J2^ = lr 2J ' Thus ^ 1 J = ^ so that, there exists a morphisin 2 such that j = iya. This means that (R,j) < (A Xs ix)3.13. Definition . Let (R,j) be a relation on X. Then R is said to be a circular re lation if and only if RoR 5_R . This notion is due to MacLane and Birkhoff ^20 J (exercize 3, page 14) . 3.14. Froposi tion. Let (R,j) be a relation on X. Then R is a circular relation if and only if R is a circular relation. Proof . If R is circular then RoR ^_R -1 . Thus R _1 oR -1 = (RoR) -1 ^(R -1 ) -1 =R (1.38, 1.12 and 1.11). Hence R _1 is circular . Conversely, if R -1 is circular then by the above, (R -1 )" 1 E 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, RoR <_ R = R L hence R is circular. Conversely, if R is reflexive and circular then R 1 is reflexive (1.17) and R _1 is circular (3.14). Hence R~ : = R~ 1 oA y _

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SECTION 4. IMAGES 4.1. De finitio n. Let (R,j) be a relation from X to Y and let (A, a) and (L,b) be extremal subobjects of X and Y respectively. Consider (11 A(AxY) ,y) and (R H (X*B) , 6) . Let (f ,a) and (f .6) be the epi-extremal 1 2 mono factorizations of it y and ir 5 respectively. Denote the domain of a 2 1 Â• by AR and the domain of 6 by RB. Thus the follovring diagrams commute. RA(AxY) Â» A*Y ARV^' -> . Y 4.?. Remark. Since (X.l ) and (Y,l ) are extremal subcbjects of X and Y X Y respectively, then (RA(XxY),y) = (R,j) and (R f\(X*Y) , -5) = (R,j) whence (XR,oO is precisely the extremal subobject (XR ; jÂ„) used in the canonical embedding (3.4). Since X = Y in 3.4 then also (RY,3) is precisely 63

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64 (RX,j ) used in 3.4 4.3. E xamp le. In the category SeX, for (A, a) <_ (X,l ), (B,b) <_ (Y,l ) A Y and (R,j) < (XxY,l ), AR = ; {ycY: tl exists aeA such that (a,y)t:R} RB = {xeX: there exists heB such that (x,b)eR}. This is easily seen since Rf\(AxY) = {(a,y): acA, (a,y)eR} and RA(XxB) = {(x,b): t>Â£B, (x,b)eR), and AR is the set of all second terms of elements of RfV(AxY) and RB is the set of all first terras of elements of RA(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 Top , AR and RB have' precisely the same underlying sets as in Top for it is easy to verify that AR and RB are closed subsets 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. The orem. If (R,j) is a relation from X to Y and (A,a) is an extremal subobject: of X then (AR,a) and (R^Ajg) are isomorphic extremal subobjects of Y. Proof. Consider the following commutative diagrams. Â— > Y -^r AxY ^ r axl v Â£Â» AR *"

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65 R ifMYxA) >Y R>*Â» *^ XxY Â»R 1 <7T 2 ,7T 1 > -*YxX It can be shown in a straightforward manner that

2 1 Y Y 2. 1 where p and o are the proiections of A*Y. Hence 1 '2 -" (1 x a )

>. = __(a*l )A = y = __

`X . Hence ft dÂ£ = fi y -Â• Tt y = cf . But ff 6Â£ = gi: t" . Thus, since. (AR.et) is i 12 1 7 1 12 the intersection cf all extremal subobjects through which it y factors (0.21), it follows that (AR,a) < (R -1 A,3). Similarly, it follows that <7t .71 > _1 j*A = *" 1 5 = jt -1 X (axl )`

`~*\ whence thare 2' 1 2' 1 J 3 Y 2' 1 4 exists a mciphism E* such that yÂ£* = < it Â»"" >_1 o. Then Tt -1 5 = f 6 =-3t = tt yÂ£* = ax Â£*. Again, since (R'^AjB) is the 1 intersection of all extremal subobjects through which ft 6 factors `

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66 (R 1 A,&) Â± (AR,a). Consequently (R 1 A,&) = (AR,c). 4.5. Corollary . If (R,j) is a relation from X to Y and (B,b) is an extremal subobject of Y then (RB,8) and (BR -1 , a) are isomorphic as extremal subobjects of X. Proof. Recall (CR" 1 )" 1 ,j#) = (R,j) (1.11). Letting (R -1 ,j*) play the role of (R,j) and (B,b) the role of (A, a) in the theorem, the following is obtained: (BR -1 , a) = ( (R~ ] ) _1 B, 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 (R -1 ,j*) E (R,j) (1.13). Hence by the theorem (AR,a) = (R _1 A,B) = (RA,8). 4.7. Pro position . Let (A^a^) and (A 2 ,a 2 ) be extremal subobjects of X and (R,j) be a relation from X to Y. If ^2,3^) <_ (A2,a2) then (AjR^j) <_ (A 2 R,a 2 ). Proof . By hypothesis there exists a morphism u so that a 2 y = a^. Thus, there exists a morphism E, such that the following diagram commutes.

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67 Thus t^Yi = Tr 2^2^ whence, because (AjR,^) is the intersection of all extremal subobjects through which t^Yi factors and tt 2 y 2 Â£, factors through (A 2 R,a 2 ), (AjR.01}) <_ (A 2 R,a 2 ) which was to be proved. 4.8. Proposition . Let (B^jbj) and (B 2 ,b 2 ) be extremal subobjects of Y and (R.j) be a relation from X to Y. If (B^b,) <_ (B 2 ,b 2 ) then (RB^Bj) 1 (RB ? ,6 2 ). Proof . (RB^B^ = (BjR" 1 ^!*) Â± (B 2 R' 1 ,B 2 *) = (RB 2 ,B 2 ) (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 existance of a morphism E, such that the following diagram commutes. RA(AxY) X*Y AxY Kence the following diagram commutes, SH(AxY) 3X*Y Thus, since (AR,a) is the intersection of all extremal subobjects through which * 2 y factors, and ir^y factors through (AS, a), it follows thai (AR,a) <_ (AS, a) which was to be proved.

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68 4.10. Pro position . Let (R,j) and (S,k) be relations from X to Y and (B,b) be an extremal subobject of Y. If (R,j) <_ (S.k) then (RB,3) <_ (SB S 3). Proof. (RB,6) = (BR" 1 , 6*) <_ (BS _1 ,0*) = (SB, 8) (4.5, 1.12, and 4.9). 4.11. P roposition . Let (R,j) be a relation from X to Y and let (Ai,ai) and (A ? ,a 2 ) be extremal subobjects of X. Then ((AjAA 2 )R,a) < (A 1 RAA 2 R,a). Proof . Since (AjAA 2 ,a) < (A^a-,) and (A i r\A 2 ,a) < (A 2 ,a 2 ) it follows that ((A 1 AA 2 )R,a) < (AjR.cxj) and ((A 1 AA 2 )R,a) < (A 2 R,a 2 ) (4.7). Thus ((A 1 AA 2 )R,a) < (A 1 RAA 2 R,a) (1.19). 4.12. Preposition . Let (R,j) be a relation from X to Y and let (B^bj) and (B 2 ,b 2 ) be extremal subobjects of Y. Then (R(B i nB 2 ),6) <_ (RB i nRB 2 ,8). Proof. (R(B i riB 2 ),8) = ((B 1 AB 2 )R" 1 ,B*) < (B 1 R~ 1 AB 2 R~ 1 , 6*) = (RBjARB^f?) (4.5 and 4.11). 4.13. Propo sition . Let (Ri 5 ji) and (R 2 ,j 2 ) be relations from X to Y and let (A, a) be an extremal subobject of X. Then (A(RjAR 2 ).a) Â£ (AR 1 AAR 2 ,a). Proof . It is clear that there exist morphisms ^ and Â£ 2 such that the following diagram commutes. ^ R x A(AxY) (R x nR 2 )r> (axy) Â»Â—

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69 Thus tt 2 y = Tr 2 Y 1 ? 1 = 1T 2 Y 2^2" A Â§ ain since (A(R AR 2 ),a) is the intersection of all extremal subobjects through which irÂ„y factors it follows that (A(R 1 AR 2 ) s a) < (AR^c^) and (AfR^R^a) Â£ (AR 2 ,a 2 ). Hence (A(R 1 AR 2 ),a) < (AR 1 AAR 2 ,5) (1.19). 4.14. Proposi tion. Let (R,,^,) and (R 2 ,j 2 ) be relations from X to Y and let (B,b) be an extremal subobject of Y. Then. ((R 1 AR 2 )B,B) < (RjBARgB.B). Proof. ((R 1 AR 2 )B,6) = (B^/lR,,)'" 1 , g.*) <_ (BR; 1 /^ BR -1 ,g*) = (R BAR B,3) (4.5 and 4.13). 4.15. Prop osit ion. Let (R,j) be a relation from X to Y then (R,j) and (RA(RYxY),Y) are isomorphic as extremal subobjects of X X Y. P roof . Consider the following commutative diagrams. > XxY RYxY RA(RYxY) >$>Â— ?** X X Y Since (j ^xl ) < t, , ., j> =

`= j, there exists a morphism Â£`

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70 RY*Y < ^l^ 2 i > R Thus (R,j) Â£ (RHCRYxY) ,y) Â• Clearly the reverse inequality holds so that (R,j) = (Rn(RYxY),>). 4.16. Prop osition. Let (R,j) be a relation from X to Y. Then (R,j) and (R A (X*XR) , 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 (R A (RY*XR) , B) are isomorphic relations from X to Y. Proof . (R,j) " (RO(RYxY),y) = (R Ci (X*XR) ,'6) (4.15 and 4.16). But since (RY,j|) and (XR,j 2 ) are extremal subobjects of X and Y respectively it follov/s that ((RYxY)A(X>'-Y) are isomorphic relations from X to Y. Consequently by the definition of image (4.1), (AR,a) and ( (RY/1 A)R,c:) 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 as extremal subobjects of Y.

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71 Proof. Let (X,l v ) play the role of (A, a) in 4.18. .A. 4.20. Coro llary. Let (R,j) be a relation from X to Y and let (B,b) be an extremal subobject of Y. Then BR"' 1 and (BOXR)R -1 are isomorphic as extremal subobjects of X. Proof. Immediate. 4.21. Proposition. Let (R,j) be a relation from X to Y. Then (RoR" 1 ^//) and (RoR _1 (RYxX) ,y) are isomorphic relations on X. Proof . Consider the following diagram. RYxX (RxX)H(X^R ! )Â» RoR To see. the diagram is commutative it need only be observed that (j 1 xl)(x i xl) =

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70 RY*Y < ^l^ 2 i > R Thus (R,j) Â£ (RHCRYxY) ,y) Â• Clearly the reverse inequality holds so that (R,j) = (Rn(RYxY),>). 4.16. Prop osition. Let (R,j) be a relation from X to Y. Then (R,j) and (R A (X*XR) , 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 (R A (RY*XR) , B) are isomorphic relations from X to Y. Proof . (R,j) " (RO(RYxY),y) = (R Ci (X*XR) ,'6) (4.15 and 4.16). But since (RY,j|) and (XR,j 2 ) are extremal subobjects of X and Y respectively it follov/s that ((RYxY)A(X>

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71 Proof. Let (X,l v ) play the role of (A, a) in 4.18. .A. 4.20. Coro llary. Let (R,j) be a relation from X to Y and let (B,b) be an extremal subobject of Y. Then BR"' 1 and (BOXR)R -1 are isomorphic as extremal subobjects of X. Proof. Immediate. 4.21. Proposition. Let (R,j) be a relation from X to Y. Then (RoR" 1 ^//) and (RoR _1 (RYxX) ,y) are isomorphic relations on X. Proof . Consider the following diagram. RYxX (RxX)H(X^R ! )Â» RoR To see. the diagram is commutative it need only be observed that (j 1 xl)(x i xl) =

`G 1 (jxl) . To show this note that (ji^UCi^l) = (Ji^xl) = (tt^IxI) an d u 1`G 1 (jxl) = if!Â©! (j x D = TTjjPj fr 1 (TfiJ x D, ^2 < ~l' ?; 3 >e i(j Xl ) = ^3 C j(J xl ) = P2 = ^aC-n"! j x l) Â• Thus, since (RoR" ,j#) is the intersection of all extremal subobjects through which < ~ 1 1 > ^f 3 >_ v factors , it follows that

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72 (Roir^j//) < (RYxX,j xl). Whence (RoR l ,j#) < (RoR^H (RY-X),y). 4.22. Theorem. Let (R,j) be a relation from X to Y. Then (RY,j ) and ((RoR OX, 3) are isomorphic as extremal subobjects of X. Proof. Consider the following products: (X*Y*X, tt ,tt tt ) , (Xx(YxX),P 1 ,p 2 ), ((XxY)xX,p 1 *,p 2 *) s (R^ 3 > Y = i-,^ (j *1) ^ = Tr 1 P 1 *(jxl)X 1 = T^jp.^. Thus TT 1

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72 (Roir^j//) < (RYxX,j xl). Whence (RoR l ,j#) < (RoR^H (RY-X),y). 4.22. Theorem. Let (R,j) be a relation from X to Y. Then (RY,j ) and ((RoR OX, 3) are isomorphic as extremal subobjects of X. Proof. Consider the following products: (X*Y*X, tt ,tt tt ) , (Xx(YxX),P 1 ,p 2 ), ((XxY)xX,p 1 *,p 2 *) s (R

*A = lr^jp^j = Ji^P-^. RoR Â«(RoR _1 )X Hj# '^Â•XxX * X Since <7,' i ,tt >y = j#x// and tt j ty = Bt , the following diagrai commutes . (RxX) H(XxR-l) tt// -J* (RoR -1 )X lf'l A l 1 RY *?Â— * X Jl But since Â£r* has the diagonal propertv (0.19) and it// is an epir-' mocphisra ana j is an extremal monomorphism then there exists a morphism I such that j-,5 = B and TjpjXj = Â£tt#. Thus ((RoR" 1 )X,3) <_ (RY,j ). Next it will be shown that the. following diagram is commutative.*

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73 -*Â• RxX R -3*. (RxX) A(XxR _1 ) Â» jxl lxj* -> (XxY)xX Â•A -** Xx(YxX) w 1 1 (jxl) = Tr 1 P 1 *(jxl) = TT 1 jp 1 = r^j. ^ 2 1 (jxl) = Tr 2 P 1 *(jxl) ir 2 jp 1 = ir 2 j . TT 3 1 (jxl) = P 2 *(jxl) = P 2 <1 R ,TT 1 J> = T^lJ. ir 1 9 2 (lxj*) = P 1 (lxj-0

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73 -*Â• RxX R -3*. (RxX) A(XxR _1 ) Â» jxl lxj* -> (XxY)xX Â•A -** Xx(YxX) w 1 1 (jxl)

*= tt 1* = ir x j Tr 2 2 (lXJ*) = TT J p 2 (lxj*)<7T 1 J J I> = 7r 1 J : ' : ^ 2 = l^j*! = T1 i <1T 2' 7T i > J = T 'ii TT 3 2 (lxj*) = Ti 2 p 2 (lxj*) T> = tT2J"T2 <7r lJ' T> = v 2i'' T = Consequently there exists a morphisra t* such that the above diagram commutes and such that y?* = `Y5* =`l_ Â— ._> xxyxx (RxX)H(XxR 1 )Â»RoR-1 V > (RoR X )X Â— fxx j *1 ->> X .Â«,> py "^

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74 Since (RY,j,) is the intersection of all extremal subobjects through which t^j factors, it follows that (RY.jj) = ((RoR -1 )X, 3) . Thus (RY,jj) = ((RoR -1 )X,8) which was to be proved. 4.23. Corollary. Leu (R,j) be a relation from X to Y. Then (XR,j 2 ) and ((R -1 oR)Y,8) are isomorphic as extremal subobjects of Y. Proof. (XR,j 2 ) = (R1 X,j 3 ) = (((R-^oCR1 )l )Y,l) = ((R^o^Y.P) (4.4, 1.11 and 4.22).

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SECTION 5. UNICES 5.1. Definition. If {(R.,i.): iel} is a family of relations from X to Y then let (^J^R-pj) be the intersection of all relations (i.e., extremal iel subobjects of X>-Y) "containing" each (R-,j.) (where containment is in the sense of "factoring through" as noted in Remark 0.12). (i*/R-:,j) shall iel be called the relation theoretic union of the family {(R. ,j. ): id}. 5.2. E xamp les. 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 subr.pace topology determined by the topology of X X Y together with the inclusion map. In the category Top ? 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. Propos ition . Let {(Ri ; ji): iel} be a family of relations from X to Y, let (LJRi,k) denote the usual categorical union of subobjects, let iel (o.j) be the epi-extremal mono factorization of k and let the codomain of o (domain of j) be denoted R. Then R and KJJ RÂ± are isomorphic relaiel tions from X to Y. Pjroof. Since (i*/Ri,j) is the intersection of all extremal subobjects iel containing each (Ri,ji) and each (RÂ±,ji) < (L/ R iÂ»k) and. . . . iel (vJR i} k) ^_ (Rjj) and since j is an extremal monomorphism then iel

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76 (V*jRi.J) 1 (R,j). iel Since (U R >Â•<Â•) is the intersection of all subobjects which "coniel tain" each (R Â± ,J i ) then (VjR.k) <_ (V*JR ,j). Since j is an extremal iel iel iuonomorphism and (R,j) is the intersection of all extremal subobjects which "contain" (IjK.k) then (R,j) < (\*)K ,j). Thus iel iel (R,j) = (I^JR ,j).. iel 5.4. Remark . Notice that by the definition of relation theoretic union, if (R,Â»j,)) ( R 2 >j ;) )j an d (S,k) are relations from X to Y and ifk) and ( W (S)k) ' then ( R !^ R 2 Â»^ 1 (S ' k) (cf ' 1 ' 19) " 5.5. Proposition . Let (R ,j ), (Rg.jg), (S ,k ) and (S 2> k 2 ) be relations from X to Y. If (R^j^ < (R 2> J 2 ) and (S^k^ <_ (S^k,,) then CRjU/Sj.j) < (R 2 ^S 2 ,k). Proof, (R.,^) < (R 2 ,j 2 ) < (R 2 'o> s 2 > k > and (Sj.kj) 1 ( s 2 ' k 2 ) ( R 2 ^ ,S 2' k) whence (R.^JS^j) < (R^S^k) (5. A) 5.6. Rema rk. The following proposition can be strengthened with the ad-ditioiii'l hypothesis that the category^ has finite coproducts (5.34); however, it is included here because it is of interest in its own right 5.7. P roposition . Let (R 3 j) be a relation from X to Y and let (S,k) and (T,m) be relations from Y to Z. Then ((RoS) [*J (RoT) , g) < (Ro(S \*)T) , g ) , Proof . Consider the following commutative diagrams. S .**' *Y>Z

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77 RxZÂ»(RxZ)nXx(SV*/T) >5> ,\ X*(St*jT) V>l x x a -> (XxY)xZ & Xx(YxZ) RxZ Â»RxZrjXxS Â»XxS Â»J x lj l x xk -*> (XxY)xZ Xx(YxZ) -*> XxYxZ RxZ >V rxzAxxt >y XxT Â»jxl l x x m *Â• (XxY)xZ ^3 * Xx(YxZ) -* XxYxZ RxZ AXx(Sli,'T) >* Yl -v XxYxZ

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74 Since (RY,j,) is the intersection of all extremal subobjects through which t^j factors, it follows that (RY.jj) = ((RoR -1 )X, 3) . Thus (RY,jj) = ((RoR -1 )X,8) which was to be proved. 4.23. Corollary. Leu (R,j) be a relation from X to Y. Then (XR,j 2 ) and ((R -1 oR)Y,8) are isomorphic as extremal subobjects of Y. Proof. (XR,j 2 ) = (R1 X,j 3 ) = (((R-^oCR1 )l )Y,l) = ((R^o^Y.P) (4.4, 1.11 and 4.22).

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SECTION 5. UNICES 5.1. Definition. If {(R.,i.): iel} is a family of relations from X to Y then let (^J^R-pj) be the intersection of all relations (i.e., extremal iel subobjects of X>-Y) "containing" each (R-,j.) (where containment is in the sense of "factoring through" as noted in Remark 0.12). (i*/R-:,j) shall iel be called the relation theoretic union of the family {(R. ,j. ): id}. 5.2. E xamp les. 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 subr.pace topology determined by the topology of X X Y together with the inclusion map. In the category Top ? 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. Propos ition . Let {(Ri ; ji): iel} be a family of relations from X to Y, let (LJRi,k) denote the usual categorical union of subobjects, let iel (o.j) be the epi-extremal mono factorization of k and let the codomain of o (domain of j) be denoted R. Then R and KJJ RÂ± are isomorphic relaiel tions from X to Y. Pjroof. Since (i*/Ri,j) is the intersection of all extremal subobjects iel containing each (Ri,ji) and each (RÂ±,ji) < (L/ R iÂ»k) and. . . . iel (vJR i} k) ^_ (Rjj) and since j is an extremal monomorphism then iel

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76 (V*jRi.J) 1 (R,j). iel Since (U R >Â•<Â•) is the intersection of all subobjects which "coniel tain" each (R Â± ,J i ) then (VjR.k) <_ (V*JR ,j). Since j is an extremal iel iel iuonomorphism and (R,j) is the intersection of all extremal subobjects which "contain" (IjK.k) then (R,j) < (\*)K ,j). Thus iel iel (R,j) = (I^JR ,j).. iel 5.4. Remark . Notice that by the definition of relation theoretic union, if (R,Â»j,)) ( R 2 >j ;) )j an d (S,k) are relations from X to Y and if

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77 RxZÂ»(RxZ)nXx(SV*/T) >5> ,\ X*(St*jT) V>l x x a -> (XxY)xZ & Xx(YxZ) RxZ Â»RxZrjXxS Â»XxS Â»J x lj l x xk -*> (XxY)xZ Xx(YxZ) -*> XxYxZ RxZ >V rxzAxxt >y XxT Â»jxl l x x m *Â• (XxY)xZ ^3 * Xx(YxZ) -* XxYxZ RxZ AXx(Sli,'T) >* Yl -v XxYxZ

`* XxZ "^ Ro(SV*)T) RxZAXxS *-*Â• xxyxz`

`Â— *"XxZ RoS`

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78 RxZflXxT VÂ»*XxYxZ RoT <7T l' Tr 3 > ~>XxZ RoS RoT >XxZ Since (S,k) < (S^T,a) and (T,m) <_ (Sl^T.a) it readily follows that ((RxZ)A(XxS), Y? ) Â± ( (R*Z) C\ (Xx (S \^J T) ) , Yl ) and that ((RxZ) A(XxT),y 3 ) _<_ ((RxZ)0(Xx(S(*;T)) j y 1 ). Thus there exist morphisms c.i and Â£ such that YjCj = Y 2 anc * Y 1^2 = Y 3" ^ encc Y 1 C 1 = <7T iÂ»^3 > Y 2 and Yl^2 = T3 Â• But (RoS,g 2 ) i s the intersection of all extremal subobjects through which PAGE 84

78 RxZflXxT VÂ»*XxYxZ RoT <7T l' Tr 3 > ~>XxZ RoS RoT >XxZ Since (S,k) < (S^T,a) and (T,m) <_ (Sl^T.a) it readily follows that ((RxZ)A(XxS), Y? ) Â± ( (R*Z) C\ (Xx (S \^J T) ) , Yl ) and that ((RxZ) A(XxT),y 3 ) _<_ ((RxZ)0(Xx(S(*;T)) j y 1 ). Thus there exist morphisms c.i and Â£ such that YjCj = Y 2 anc * Y 1^2 = Y 3" ^ encc

`Y2 ^ actors and since ^tF, ,tt 3 >y = <^ -i Â» tt 3 > Yi Â£ Â»i T l5ii we have (RoS,g 2 ) <_ (Ro (S [*J T) , 6, ) Â• And since (IloT.B,) is the intersection of all extremal subobjects through which`y 3 factors and since < ^ 1 ^ 3 > V 3 = '~i ! ^3 > ' r i^2 = ^l" l 1^2 it: follows Lhat (RoT.3 3 ) <_ (RoC-S^T),^). Whence ((RoS) V*>(RoT) ,g) < (Ro(S V*/T) ,3 2 ) (5.4). 5.8. ^^opo_sition. Let (T,m) be a relation from Y to Z and let (R,j) and (S,k) be relations from X to Y. Then ((RoTH*KSoT),3) < ((R^S)oT,3). Proof. Analogous to the proof of 5.7.

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79 5.9. Lemma . Let {(Rj^ji): iel} be a family of relations from X to Y. Then ((^Ri)" 1 ^*) = (^("i) _1 . k )iel iel Proof . Consider the following commutative diagram. iel *<

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79 5.9. Lemma . Let {(Rj^ji): iel} be a family of relations from X to Y. Then ((^Ri)" 1 ^*) = (^("i) _1 . k )iel iel Proof . Consider the following commutative diagram. iel *<

`a.* iel -1 Since (R." 1 ,!.*) is the intersection of all extremal subobiects l ' J i through which`

`j . factors it follows that Thus iel (l^CRi -1 )^) < (((^Rj.)" 1 ^*) (5.4) Now (R i ,j i ) <_ (K*J(R Â± 1 ),``1 k) since iel :ir ,ir,>" i kX.*T. = < tt .-, , tt , > -1 j ,*t . = i.. Thus 21 ii 2' 1 J ii J i iel ' iel from v;hich it follows that iel ' iel whence iel iel`

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80 5.10. Corollary. Let ((R.pj.^): iel} be a family of symmetric relations on X. Then {Â£J R^ is a symmetric relation on X. iel Proof . It is clear that for each iel, (R i ,J i ) = (Ri~ 1 ,j i *) (1.13). Thus (l*jRi,j) = (^(Rr 1 )^) < ((Vi/Ri)1 ^*) (5.9). id iel iel 5.11. Pro positio n. If (R,j) is a reflexive relation on X and (S,k) is any relation on X, then (Ri^jS,m) is reflexive on X. Proof. Since (R,i) is reflexive, (A i ) < (R,j). Thus A A (A x ,i x ) < (R,j) < (R(*JS,m) hence (R^IS,m) is reflexive on X. 5.12. D efinitio n. Let (R,j) be a relation on X. Consider the relation (Rl*>R -1 ,j#). *XxX Let (t#,x) be the epi-extremal mono factorization of 'i,j#. The domain of x (codomain of x#) shall be denoted by X R[*)R l Â»Â» T# X R^" J# R" -> X>

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81 5.13. Examples . In the category Set , X p = (R(*Jr 1 )X = XRU RX. That is X_ = {xeX: there exists yeX such that (x,y)eR or (y,x)?R}. In the category Top -. , X R is the same set as in Set endowed with the subspace topology determined by the topology of X. 5.14. Pro position . Let (R,j) be a relation on X and let (RX,j^) and (R^Xjjo) be the images of tTjJ and tTj j * respectively. Then (RX^R _1 X,a) < (X R ,x) = ((Rl*^R -1 )X,x). Proof. Consider the following commutative diagram. R Â— > RX R Rl*JR-l Â» =*x Since: j = TTjjiUj, = x t #^r = 3i T i and (^.jj) is the intersection of all extremal subobjects through which irjj factors then (RX.j,) < (X R ,x). Similarly, it can be shown that (R _1 X,J 3 ) <_ (X R ,x), whence (RXj^J R _1 X,a) <_ (X R ,x) (5. A). 5.15. Proposition . If (R,j) is a relation on X then (Rl*jR~ ! ,j#) is symmetric on X and (X R ,x) = (X(R \*J R _1 ) , j 2 ) . Proof. ((R^IT 1 )" 1 ,://*) = (^"^(R" 1 )" 1 ^) = (R _1 (^R,J^) (5.9 and 1.11). Thus (Rvi'R _1 ,j//) is symmetric so that (x R ,x) ((RVSJR-^X.x) = (x(R^R _1 ),j 2 ) (4.6). 5.16. Proposition. Let (R,j) be a relation en X and let (A v ,i ) be the a r x R diagonal of X^X^ Then (A x ,(x x x) i x ) = (& x C\ (\*\) ,?) where R X

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82 (AA(X *X ),p) is the intersection of the diagonal (A ,i ) of X*X with A K K XX Proof . Consider the following commutative diagram. TTj* IT o * 7^X R 51X Observe that ^(x x x)i X = Wi* 1 * = X^ 2 *i X = ^ 2 (x x x)i X Â• Thus, K K R R since i x is the equalizer of TTj and tt 2 , there exists a unique morphism Â• so that i x ^ = (x*x)i x R Thus, since (A x H Xr x Xr) p) is the intersection of (A x ,i x ) and (Xn>

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83 quasi reflexive if and only if (A , (x x x)i ) Â± (R.j)That is, (R,j) is X R X R quasi-reflexive provided that there exists a morphism X such that the following diagram commutes. 5.18. Proposition. If (R,j) is a reflexive relation on X, then (R,j) is quasi -reflexive on X. Proof . If (R,j) is reflexive then (R(*jR -1 ,j#) is reflexive (5.11); hence, tt ^ j // is a retraction (1.24). Thus Tr,j// is an epimorphism; so that if (x#,x) is the epi-extremal mono factorization of ttjJ//, ^jj// = X T # so that x is an epimorphism as v;ell as an extremal monomorphism. Hence >; is an isomorphism (0.17). Thus (X R ,x) = (X,lÂ„) whence (A ,i ) = (A ,i ). K R Thus, since (A ,i ) <^ (R.j), (R,j) is quasi-reflexive. 5.19. Pro position . Let (R,j) he a relation on X. Then (R,j) Â£_ (A ,i ) if A A. and only if (R,j) < (A ,(x x x)i v )Â• R R Proof . Suppose that (R,j) < (A ,i ). Then there exists a morphism a such A A that j = ij^ci. Thus tt-^j = ff^i^a = i^^X 01 = ^2^ ' wnence ^Â• i <1T 2Â» Tr l > -3 = ^2^ = ^iJ ~ 7I 2 <1 ' Z'^^J ' Consequently/ the unique epi-extremal mono factorization of PAGE 86

80 5.10. Corollary. Let ((R.pj.^): iel} be a family of symmetric relations on X. Then {Â£J R^ is a symmetric relation on X. iel Proof . It is clear that for each iel, (R i ,J i ) = (Ri~ 1 ,j i *) (1.13). Thus (l*jRi,j) = (^(Rr 1 )^) < ((Vi/Ri)1 ^*) (5.9). id iel iel 5.11. Pro positio n. If (R,j) is a reflexive relation on X and (S,k) is any relation on X, then (Ri^jS,m) is reflexive on X. Proof. Since (R,i) is reflexive, (A i ) < (R,j). Thus A A (A x ,i x ) < (R,j) < (R(*JS,m) hence (R^IS,m) is reflexive on X. 5.12. D efinitio n. Let (R,j) be a relation on X. Consider the relation (Rl*>R -1 ,j#). *XxX Let (t#,x) be the epi-extremal mono factorization of 'i,j#. The domain of x (codomain of x#) shall be denoted by X R[*)R l Â»Â» T# X R^" J# R" -> X>

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81 5.13. Examples . In the category Set , X p = (R(*Jr 1 )X = XRU RX. That is X_ = {xeX: there exists yeX such that (x,y)eR or (y,x)?R}. In the category Top -. , X R is the same set as in Set endowed with the subspace topology determined by the topology of X. 5.14. Pro position . Let (R,j) be a relation on X and let (RX,j^) and (R^Xjjo) be the images of tTjJ and tTj j * respectively. Then (RX^R _1 X,a) < (X R ,x) = ((Rl*^R -1 )X,x). Proof. Consider the following commutative diagram. R Â— > RX R Rl*JR-l Â» =*x Since: j = TTjjiUj, = x t #^r = 3i T i and (^.jj) is the intersection of all extremal subobjects through which irjj factors then (RX.j,) < (X R ,x). Similarly, it can be shown that (R _1 X,J 3 ) <_ (X R ,x), whence (RXj^J R _1 X,a) <_ (X R ,x) (5. A). 5.15. Proposition . If (R,j) is a relation on X then (Rl*jR~ ! ,j#) is symmetric on X and (X R ,x) = (X(R \*J R _1 ) , j 2 ) . Proof. ((R^IT 1 )" 1 ,://*) = (^"^(R" 1 )" 1 ^) = (R _1 (^R,J^) (5.9 and 1.11). Thus (Rvi'R _1 ,j//) is symmetric so that (x R ,x) ((RVSJR-^X.x) = (x(R^R _1 ),j 2 ) (4.6). 5.16. Proposition. Let (R,j) be a relation en X and let (A v ,i ) be the a r x R diagonal of X^X^ Then (A x ,(x x x) i x ) = (& x C\ (\*\) ,?) where R X

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82 (AA(X *X ),p) is the intersection of the diagonal (A ,i ) of X*X with A K K XX Proof . Consider the following commutative diagram. TTj* IT o * 7^X R 51X Observe that ^(x x x)i X = Wi* 1 * = X^ 2 *i X = ^ 2 (x x x)i X Â• Thus, K K R R since i x is the equalizer of TTj and tt 2 , there exists a unique morphism Â• so that i x ^ = (x*x)i x R Thus, since (A x H Xr x Xr) p) is the intersection of (A x ,i x ) and (Xn>

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83 quasi reflexive if and only if (A , (x x x)i ) Â± (R.j)That is, (R,j) is X R X R quasi-reflexive provided that there exists a morphism X such that the following diagram commutes. 5.18. Proposition. If (R,j) is a reflexive relation on X, then (R,j) is quasi -reflexive on X. Proof . If (R,j) is reflexive then (R(*jR -1 ,j#) is reflexive (5.11); hence, tt ^ j // is a retraction (1.24). Thus Tr,j// is an epimorphism; so that if (x#,x) is the epi-extremal mono factorization of ttjJ//, ^jj// = X T # so that x is an epimorphism as v;ell as an extremal monomorphism. Hence >; is an isomorphism (0.17). Thus (X R ,x) = (X,lÂ„) whence (A ,i ) = (A ,i ). K R Thus, since (A ,i ) <^ (R.j), (R,j) is quasi-reflexive. 5.19. Pro position . Let (R,j) he a relation on X. Then (R,j) Â£_ (A ,i ) if A A. and only if (R,j) < (A ,(x x x)i v )Â• R R Proof . Suppose that (R,j) < (A ,i ). Then there exists a morphism a such A A that j = ij^ci. Thus tt-^j = ff^i^a = i^^X 01 = ^2^ ' wnence ^Â• i <1T 2Â» Tr l > -3 = ^2^ = ^iJ ~ 7I 2 <1 ' Z'^^J ' Consequently/ the unique epi-extremal mono factorization of

`j is (1 ,j), and (R,j) = (R _1 ,j*) (3. 12) . But, since (R,j) = (R -1 ,j*), then (R^j R" 1 , j#) = (R,j), and (X ,x) '((R(*>'R _1 )X,x) = (RX,j.). Thus, since (RX,j.) = (X . X ) = (XR,j.) it follows that (R.j) 1 (X, xX ,x*X) (3.4). i X R,`

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84 However, it has been shown that (A Y , (x x x)i Y ) and (i v H (Xt^X,,) ,p) A R A R A K. K are isomorphic relations on X (5.16). So that since (R,j) < (A ,i ) and ""XX (R,j) _< ((X R xX R ),x> (XxXxX,^ 1 ,u 2 ,Tf 3 ) > (XxX j tt 1 ,-t 2 ), ((XxX)xX,fr 1 ,fi z ), (Xx(XxX) , if x ,t 2 ) , (RxX,^,^) and (X x A x , Pl *,p 2 *). R Consider the following commutative diagram. 3*1, RxX >*-> (XxX) xX RxXfiXxA., V>V R ->XxX **Â»Â• RoA... *R Next it will be shown that y = <7r iÂ»' ir 3 > Y'

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85 Ti 1 <71 1 ,TT 2 >Y = ^Y = ^1 < ' ,i l , TT 3 > YTr 2 < : TT 1 /Tf 2 >Y = v 2 y = ^ 2 2( 1 X x (x x x)ix > X 2 = ^1*2 ( 1 X x (x x X>ix > A 2 = R R * (X x x)iv P 9 *A = X^!*iv P ? % = X^o*t. P,*A 2 = 1 A R 2 z R R ^V^ D P 2* X 2 = M* x X)i x P 2 * X 2 "-= u 2^2(3 X x (x x X)i Xp ) A 2 = tt 3 2 (1 x( x x x )i )\ 2 u 3 y = TT 2Y. R Thus by the definition of product

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84 However, it has been shown that (A Y , (x x x)i Y ) and (i v H (Xt^X,,) ,p) A R A R A K. K are isomorphic relations on X (5.16). So that since (R,j) < (A ,i ) and ""XX (R,j) _< ((X R xX R ),x>

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85 Ti 1 <71 1 ,TT 2 >Y = ^Y = ^1 < ' ,i l , TT 3 > YTr 2 < : TT 1 /Tf 2 >Y = v 2 y = ^ 2 2( 1 X x (x x x)ix > X 2 = ^1*2 ( 1 X x (x x X>ix > A 2 = R R * (X x x)iv P 9 *A = X^!*iv P ? % = X^o*t. P,*A 2 = 1 A R 2 z R R ^V^ D P 2* X 2 = M* x X)i x P 2 * X 2 "-= u 2^2(3 X x (x x X)i Xp ) A 2 = tt 3 2 (1 x( x x x )i )\ 2 u 3 y = TT 2

`Y =`YNov; consider

`Y Â• It will be shown that <7Ti,tt 2 >y = jPl^lT ': < ^i>^2 >Y = ^i Y = ^i e i (J xl )^i Â• Thus ,T i <: ^i Â»^2 > 'Y = 1r l^l(j x l)^l = MJPlM 7: 2`Y = if 2 Y = ^2 Q i (j x l )^i= TTofl i (j x 1)a 1 = 7T 2 jp i^j . Hence, by the definition of product,

`y = jpjXj. Since (RoA v ,ct) is the intersection of all extremal subobjects X R through which <^i > ^3 >Y factors and`Y <^\ , ^2 > Y = JPl^l ^t follows that (RoA v ,a) < (R,j). X R For the reverse inequality consider (R^*/R ,j#). Since (Rt*/R -1 ,j# ) is symmetric, ((Rl*/R -1 )X,Ji#) and (X(Rl*J R" 1 ) . J2#) are isomorphic extremal subobjects of X (5.15). V R X X X Â— xCrv^'r" 1 ) * "siJ V R Thus if (Â£#,x#) is the epi-extremal mono factorization of tt _jj#, there exists an isomorphism Â£, such that x ,Â£ > = 5#Â« So in particular, tt 2 jA r tt 2 j = x#5#X R X?C#A R .

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86 Let a be the isomorphism for which i Y a = <1 ,1 > (1.21). 41 x k' X R It next will be shown that the following equality holds: so that the following diagram is commutative. 01 < J, 7T 2J >

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86 Let a be the isomorphism for which i Y a = <1 ,1 > (1.21). 41 x k' X R It next will be shown that the following equality holds: so that the following diagram is commutative. 01 < J, 7T 2J >

`= ff x (l x <(x*x)i x )` R K Pl* = TTij. u 2 2 (l x >: (x x x)i x ) = it^ 2 (l x x( x x x )i x ) a5C#A R > tt : (X x x)i x P 2 * = R XPii v p 2 * `= XPii Y Â°C5//A n = X R K "R Tf 3 2 (l X( X X X )Â± x )`

`= Tl 2 ff 2 (l x x( X X X U )<7T 1 J,05C#X R > = R ** TT 2 ( X x X )i x P 2 *` R XP2 i XR P2*^lJ'Â°^#V xP2 <1 x R > 1 x R > ^ #x : x i Y 5?#x p = xC5#XÂ„ = tt 2 j. *X R ""R ir 1 e 1 = Tr 1 fr 1 = TTjj " 2 S 1 TT 2 ff l < J' 1T 2^ > = 1! 2-i Tf 3 1 = Tr 2 = fr 2 j.

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87 Thus by the definition of pi-oduct: < -"]J, TT 2JÂ» Tr 2J > = 1 < JÂ» T7 2J > = 2( 1 x x(x> = 0i(j x D < l u ,T'2i > (1-3D. Hence by the definition of intersection there exists a unique morphism Z such that yZ = <~n\ j ,^23 > 7T 2J > Â• Thus Yl = < Tfl,iJ3> PAGE 93

87 Thus by the definition of pi-oduct: < -"]J, TT 2JÂ» Tr 2J > = 1 < JÂ» T7 2J > = 2( 1 x x(x>

`= < TT 1 j , TT 2 j > = j Â• Hence j = agS; and consequently (R,j) ^_ (RoA^ ,a). R 5.21. Corol lary . Let (R,j) be a relation on X. Then R _1 , AÂ„ oR -1 and R oA are isomorphic relations on X. X R Proof . Recall that since (R VÂ£J R" 1 ,j#) is symmetric (5.15), (X R -i ,x*) and ((R {*) R -1 )X, j, //) are isomorphic as extremal subobjects of X (5.15 and 5.9). Also (X ,x) and ( (Rli!^ R" 1 )X, j #) are isomorphic as extremal subobjects of X (5.15) whence, (Xgl*XÂ£i,X* x X*) = (\*\,X*x) and hence (A _j , (x* x X*) \l ) and R "R (A Â»(x x x)iv )Â• But ^y tne theorem X R ' X R Consequently, (R l ,3*) = (A v OR" 1 , 3*) 5 (R _1 oA v ,a*). A R _1 A R _1 (R ! ,j*) = (A y oR,a#) E (R ^Ay ,ai7) . A R A R 5.22. Definiti on. Let R be a relation from X to Y. Then R is said to be di functional if and only if Ro(R _1 oR) <_ R and (RoR _1 )oR < R. The term difunctional relation is due to Riguet Â£22 J. 5.23. Proposi t ion . Let R be a relation from X to Y. Then R is difunctional if and only if R is difunctional. Proof . If (R,j) is difunctional then since (Ro (R~ 1 oR) , k, ) <_ (R,j) we have ((R~ 1 oR)oR~ 1 ,k 1 #) = ((R~ 1 oR) -1 oR~ 1 ,1c ) 5 ((Ro(R~ 1 oR))" 1 > k^*)<.(BT 1 ,j)`

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S8 This fellows from 1.38, 1.11, and 1.12. Also since ( (RoR~ * )oR,k 2 ) <_ (R,j) it follows thai (R-MRoR -1 ),^//) = (R'ioCRoR -1 ) -1 ,^) = (((RoR" 1 )oR)~ 1 ) k 2 *) l(R _1 ,j*). Thus (R _1 ,j*) is di functional. If (R _1 ,j*) is difunctional than since ((R _1 ) _1 ,j#) = (R,j) (1.11) and since, applying the above to (R _1 ,j*), ( (R" 1 )~ 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 quasi-equivalence (3.2) if and only if R is quasi-reflexive and difunctional. Proof. If (R,j) is a quasi-equivalence then (R,j) is symmetric hence (R(^R _1 ,j#) E (R,j). Thus (\,x) = (RX.jj) s (XR,j 2 ) (5.15 and 4.6) from which it fellows that (A , (x^x)^ ) = (\rÂ» (J2 x J2) jt y R ) R R Since (R,j) is a quasi-equivalence then (R,^) is an equivalence relation on XR (3.10) so (A ,i ) < (R,i>). Hence XK XK (A ^2 x ^2^vp^ Â— ^>J) sc that (R,j) is quasi-reflexive on X. Since XK AK. (R,j) is symmetric and transitive then (Ro(R~ 1 oR),k 1 ) Â£ (Ro(RoR),k 1 ) < (RoR.j') <_ (R,j). Similarly, ((RoR~ 1 )oR,k 2 ) <_ ((RoR)oR,k 2 ) <_ (RoR,j') Â£ (R,j). Hence (R,j) is difunctional. Conversely if (R,j) is quasi-reflexive and difunctional then (A Y ,(x x x)i ) 1 (R,j) so that (A >(x*x)i Y ) 1 (R _1 ,j*> (1.16 and 1.12). A R A R A R A R Thus (RcR,j') < (Ro(A v oR),k) < (Re (R _1 oR) ,k, ) < (R,j) and X R t. (R'i.j*) < (A o(R -1 oA v ),k) < (Ro(R" 1 oR),k 1 ) < (R,j) (5.20 and 1.30). _ X R X R _ Thus (R,j) is both transitive and symmetric hence a quasi-equivalence. 5.25. Proposition. Let R be a relation on X, Then R is an equivalence relation _f and only if R is reflexive and difunctional.

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89 Froof . If (R,j) is an equivalence relation on X then (R.j) is reflexive and a quasi-equivalence on X. Thus (R,j) is difunctional (5.18 and 5.24) Conversely if (R,j) is reflexive and difunctional then (R,j) is quasi-reflexive and difunctional (5.18) hence (R,j) is a quasi-equivalence (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,^) and (XR,j 2 ) be the usual images (3.4). Then R, RoA^.Â„ , and A R Â„oR are isomorphic relations from X to Y (cf. 5.20 and 1.31). Proof. Consider the following commutative diagrams. A x^~ , I I Â• ^-RX A >*> RX -*XxX t [Ji x Jl * -*RXxRX ~ Pi P 2 Pi* p 2 * Xx Ji ^RX Observe that Pi (j !>

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90 Thus: (A RX oR,k 2 ) <_ (A^RJ') e (R,j) (1.30 and 1.31). To see that (R,j) Â£ (A Rx oR,k ) consider the following commutative diagram. ^Â•XxX^Y *i> XxY ixxy Recall that (A RY ,i KY ) = (RX, <1 RY , 1 PY >) (1.21) hence tl J RX'"RX RX'^RX' nere exiscs an isomorphism a such that ^ryjIr-;^ = ^px Consider also the products (A^xYjp ^ ,o 2 ) , (X*R,ff j ,tt 2 ) , and ((XxX)xY,u 1 ,u 2 ). It will next be shown that the following diagram is commutative. Iwi i RX xl Y (XxX) xY ^l^>^l3,^23> = 02

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S8 This fellows from 1.38, 1.11, and 1.12. Also since ( (RoR~ * )oR,k 2 ) <_ (R,j) it follows thai (R-MRoR -1 ),^//) = (R'ioCRoR -1 ) -1 ,^) = (((RoR" 1 )oR)~ 1 ) k 2 *) l(R _1 ,j*). Thus (R _1 ,j*) is di functional. If (R _1 ,j*) is difunctional than since ((R _1 ) _1 ,j#) = (R,j) (1.11) and since, applying the above to (R _1 ,j*), ( (R" 1 )~ 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 quasi-equivalence (3.2) if and only if R is quasi-reflexive and difunctional. Proof. If (R,j) is a quasi-equivalence then (R,j) is symmetric hence (R(^R _1 ,j#) E (R,j). Thus (\,x) = (RX.jj) s (XR,j 2 ) (5.15 and 4.6) from which it fellows that (A , (x^x)^ ) = (\rÂ» (J2 x J2) jt y R ) R R Since (R,j) is a quasi-equivalence then (R,^) is an equivalence relation on XR (3.10) so (A ,i ) < (R,i>). Hence XK XK (A ^2 x ^2^vp^ Â— ^>J) sc that (R,j) is quasi-reflexive on X. Since XK AK. (R,j) is symmetric and transitive then (Ro(R~ 1 oR),k 1 ) Â£ (Ro(RoR),k 1 ) < (RoR.j') <_ (R,j). Similarly, ((RoR~ 1 )oR,k 2 ) <_ ((RoR)oR,k 2 ) <_ (RoR,j') Â£ (R,j). Hence (R,j) is difunctional. Conversely if (R,j) is quasi-reflexive and difunctional then (A Y ,(x x x)i ) 1 (R,j) so that (A >(x*x)i Y ) 1 (R _1 ,j*> (1.16 and 1.12). A R A R A R A R Thus (RcR,j') < (Ro(A v oR),k) < (Re (R _1 oR) ,k, ) < (R,j) and X R t. (R'i.j*) < (A o(R -1 oA v ),k) < (Ro(R" 1 oR),k 1 ) < (R,j) (5.20 and 1.30). _ X R X R _ Thus (R,j) is both transitive and symmetric hence a quasi-equivalence. 5.25. Proposition. Let R be a relation on X, Then R is an equivalence relation _f and only if R is reflexive and difunctional.

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89 Froof . If (R,j) is an equivalence relation on X then (R.j) is reflexive and a quasi-equivalence on X. Thus (R,j) is difunctional (5.18 and 5.24) Conversely if (R,j) is reflexive and difunctional then (R,j) is quasi-reflexive and difunctional (5.18) hence (R,j) is a quasi-equivalence (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,^) and (XR,j 2 ) be the usual images (3.4). Then R, RoA^.Â„ , and A R Â„oR are isomorphic relations from X to Y (cf. 5.20 and 1.31). Proof. Consider the following commutative diagrams. A x^~ , I I Â• ^-RX A >*> RX -*XxX t [Ji x Jl * -*RXxRX ~ Pi P 2 Pi* p 2 * Xx Ji ^RX Observe that Pi (j !>

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90 Thus: (A RX oR,k 2 ) <_ (A^RJ') e (R,j) (1.30 and 1.31). To see that (R,j) Â£ (A Rx oR,k ) consider the following commutative diagram.

`->XxXxY \ <7r iJ'V X Xx R Â»-* X.x(XxY) l x-i X J v. 1 01 ( ( j i x j i ) i f0( x l y )` P i if i ( ( j i x j 1 ) Irxx l y ) PlOl*Jl)iRXP"l =: JiPi* i RXfV^i^ 2 -i > = JlPl^iRX^l = JiPl* <1 RXÂ» 1 RX >T l Ji T i = M.1-

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91 TT 2 l((Ji x .ii)i RX xl Y ) = P2 Tf l((Jl x 3l) i R X xl Y ) = P2(Jl X Jl) i R,XPl = J]P2* i RX OT l = 3lP2 , ' c<1 RX ! 1 RX >T l J1 T 1 = ^lJT f 30l((Jl x Jl) i RX xl Y) = ^2((Jl x Jl) i RX xl Y ) = l Y P 2 = TT 2 j. ^ 1 ? (l x Xj) PAGE 97

91 TT 2 l((Ji x .ii)i RX xl Y )

`= ff 1 (l x Xj)< 1 r 1 j,l R > = TT 1`

`= TTjj. lf 2 9 ? (l x '` = ir 1 TT 2 (l x XJ)<1T 1 j,l R > = TTlJTf2 <7T l3> 1 R > = *l 3 Â• Tf 3 2 (l x Xj) `= 7T 2 Tf 2 (l x Xj)` = TT 2 j tt 2 <^ \ j , 1 R > = 7T, j . if 1 2 = ff 1 = ir^i. TT 2 e 2 <7T 1 j , j> = TT 1 Tf 2 <7i 1 j,j> = TTjj. Tf 3 2 <7T 1 j,j> = T7 2 TT 2 = 7I 2 j. Consider the following commutative diagram. (Ji x Ji)i-RX xl Y A RX xY ># > t^ x X) *Y Aj^xYHXxR >?V -#Â• XxXxY 1 ^Xx(XxY) * Wfc A RX cP

`X*Y By the definition of intersection, there exists a unique morphism Â£ from R to (A /` Â•

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92 ThusY? = < ifi,Tf 3 >< 7T 1 j,T 1 j,^2J > = <1l iJÂ» 7T 2J > = 3Hence k 2 T = j; that is, (R,j) <_ (A Rx oR,k 2 ). Hence (R,j) e (A RX oR,k 2 ). Similarly it can he shown that (R,j) = (RoA^.kj). 5.27. Theorem. Let (R,j) be a relation from X to Y. Then ^RX'^VV 1 ^) 1 (RoR _1 ,j#) and (A XR , (J 2 x J 2 )1 x r) 1 (R^oR.j'). Proof. Consider the following products: (XxX,pi ,p 2 ) , (BXxRX.pj *,p 2 *) , (XxY.ttj ,ir 2 ), (RxX,p 15 p 2 ), (XxR~ 1 ,p 1 ,p 2 ) J ((XxY)xX,ff 1 ,ff 2 ) , (Xx(YxX) ,ti j * ,tt 2 *) and (XxYxX,tt, ,tt 2 ,tt 3 ) . Also consider the following commutative diagrams. R HT l RX Â»-> XxY Jl -* X R >* Next, the following diagram will be shown to be commutative, jxl v ^r rxx v* <3 R^1J > V (XxY)xX -"''lJ' 1 "V> RoR

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93 To see this, it need only be observed that 0l(jxl x ) = 02(l x xj 5 -)<^r 1 j,T>. Tf 1 G 1 (jXl x ) = Tr l Tf 1 (jxl x ) =TT 1 JP 1 <1 R ,1T 1 J> = TTjj ' t 2 l <: J xl X )<1 R' 1r lJ > = w 2 ff l ^J xl X^ <1 RÂ» ir lJ > = 7T 2JPl <1 R' 7T lJ > = v li ^3 G l ( J xl X )<1 R' TT lJ > = ft 2 ( J xl x :><1 R' 7T lJ > = ^2 <1 R' 7r lJ > = lr lJTT,e 2 (l x x j") TT ? Q 2 (l X >< j") < T 1 J,T> 7T i* A ^ 1 x x J" A '^ <7T iJ' T> = Pl

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92 Thus

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93 To see this, it need only be observed that 0l(jxl x )

`= ^lJTr 1 ir 2 *(l x xj*)< 7 r 1 j,T> = TTj j*P 2``T> = *i$*i = 7T l <7T 2' Tr ] > J = ^?J ' ^3 2^ 1 X X J*^` = ^2 7T 2' : ^ 1 X X J' ! '^

`= ^2 <7I 2' TI 1 > J = '"'l J Â• Thus the diagram is commutative and 1 (jxl x )` 7T 1 J> = 9 2 (1 X XJ*) `=`

`. Hence by the definition of intersection there exists a morphism I such that yZ =``. Clearly`YÂ£ = <'^ 1 .i Â»' ,r iJ > thus

`= j#x#Z. It next will be shown that if a is that isomorphism for which` = ijivO then (Â° T i > (j i x j i)1ry) is the epi-extremal mono factorization of `R Â— ->' X X X 9 (j i x j l)i-RX RX Â»-VsA RX P2 r -Jl x Ji) i RX CT l = JlP2" i R>v OT l = 3iP 2 * <1 RX' 1 RX >T l = J 1 " L 1 = ir lJ`

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94 Thus the diagram above commutes and (ox , , (j , xj , )i ) is the epi-extremal mono factorization of

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94 Thus the diagram above commutes and (ox , , (j , xj , )i ) is the epi-extremal mono factorization of

`(0.18). Since (A_ v , (j , xj , )i ) Â± s the intersection of all extremal subobjects through which``factors and since``= j#x//E it follows that ^ A RX' d lX ^ ^"^RX^ Â— ^ RoR ~ 1 >i J '^ which was to be proved. The proof that (A XR) (J2 x J2)i XR ) 1 (R _1 oR,j*) is similar. 5.28. Theo rem. If (R,j) is a relation from X to Y then R is difunctional if and only if (RoR _1 )oR = R = Rc(R _1 oR). Proof . If R is difunctional then (R,j) = (RoA^kj) < (RoCR^oR),!^) <_ (R,j) (5.26, 5.27 and 5.22). Similarly. (R,j) = (A^oR,^) Â£ ((RoR^oR.k.,) <_ (R,j). The converse is immediate from the definition (5.22). 5.29. Remark . Let c be a locally small quasi-complete category having (finite) coproducts. It is noted in passing that if R 1 has arbitrary products; i.e., is complete, then Â£j i s also (finitely) cocompletc \9 \. Recall that the unique epi-extremal mono factorization of a morphism is obtained by taking the intersection of all extrema l subobj ects of the codcmain 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 epi-mono factorization is obtained (0.21)Finally recall that if {(A., a.): id} is a family of subobjects of a (^-object X, then the subobject (\J\.,s.) is obtained by taking the intersection of all subiel X objects of X which "contain" each (A., a.).`

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95 Now consider the coproduct ( XL A^ ,y^) of the (finite) family iel {(A-,a-): id}. By the definition of coproduct there exists a unique mo rphism y such that pu= afor each Iel. Let A^ be the "inclusion" of (A^,a^) into (i_JA^,a). Again by the definition of coproduct there iel exists a unique morphism A such that Ay= A^ for each iel. Thus the following diagram commutes. -*-X A-: Note that a is a monomorphism. It will be shown that A is an extremal epimorphism. To see this, it will be shown that (U Aj,a) is the iel intersection of all subobjects of X through which y factors. To that end let (Z,g) be any subobject of X through which y factors; i.e., there is a morphism h such that y = gh. Then a^ = yyj_ = ghy^ hence each (A-^,a^) factors through (Z,g). Thus by the definition of union there exists a unique morphism Â£, such that gÂ£ = a. But this is precisely what is required of the intersection of all subobjects of X through which y factors . Now suppose that {(A^,a^): iel} is a (finite) family of relations from X to Y; i.e., each (A 3 -,a^) is an extremal subobject of X*Y. Consider (JLL &Â±,^i) and ( U Aiifl)Again, let y be th

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96 X\i. = X . for each iel, where \. is the inclusion of (A.,a.) into ( (J A .,Â£). Let (r,p) be the epi-extremal mono factorization of a. Recalls I ling Proposition 5.3 it follows that the domain of p (codomain of t) is ^A.. Thus the following diagram commutes, iel ^ XX. A. > XxY iel x u v * ^ A i iel t iel 5.30. Theorem . Let ^ have (finite) coproducts, let {(A^a^: iel} be a (finite) family of subobjects of C, and let f be a fe -morphism from C to D, As above, let (l^A^p) be the extremal mono part of the factoriiel zation of the unique morphism p from JJL.A^ to C with the property that iel Hjij = a^ for each iel. Let (f\E-,e) be the intersection of all extremal jeJ subobjects of D thiough which each fa^ factors. Let (Im(A^) ,p^) and (lw( [Â£) A.. ) .p) denote the extremal mono parts of the epi-extremal mono iel factorizations of fa^ and fp respectively. Finally let ( \*) Im(A^) ,p) iel be the intersection of all extremal subobjects of D through which each pfactors. Then (Im^A^.p) E (flE.,e) = ((*J Im(k L ) ,0) . iel jeJ iel Proof . If (E. ,e.) is an extremal subobject of D through which each fa^ factors, then since (Im(AjO ,0-^) is the intersection of all extremal subobjects through which fa^ factors it follows that (Im(A n .),P,0 < (E.,e-). Thus (Im(A n . ) ,o) < (HE-,e) for each iel. Hence 1 *Â• Â— J j x Â— . T .1 (i v ^Im(A.),p) < (C\E,,e). iel iej J

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97 However, since each fa Â± factors through p and since p is an extremal monomorphism it follows that (f\Ej,e) <_ ( \*J Im(A i ) ,|S ) . jeJ iel Consider the following commutative diagram. ">Â• ^Im(A Â± ) iel iel > D Note that (it, 6) is the epi-extremal mono f actcrization of fu so that (Im((^A i ),p) is the intersection of all extremal subobjects iel through which fjj factors. Let ty be that unique morphism such that ^Uj_ = ^-Â±^Â± fÂ° r each iel. Now pyy i fAjJi = Pi^i = Â£a Â± = fuy Â± . Thus by the definition of coprc-duct it follows that fy = p>. Hence fy factors through (l*J Im(A-|_) ,P) iel whence (Im( \*)A Â± ) , p) Â£ (l*J lm(.AÂ±) ,0) Â• iel iel Now fa-j^ = fuui = fpxpi = pfiy-jj hence fa i factors through p, whence (Hs^e) Â£ (Im( V*/ A Â± ) ,(5) . jeJ " iel Thus: (^ImCAijJ) E (f\E J5 e) E (Im(l&J A Â± ) ,j5) . iel jeJ iel

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93 5.31. Definition . A category is said to be (finitely) union distr ibut ive if the following properties hold: (i) if X and Y are any fa -objects and {(A^a^: iel} is a (finite) family of extremal subobjects of Y, then Xx((*jAjO and ^'J (XxA i ) are iel iel isomorphic relations from X to Y; (ii) if X is any fe -object, {(A-j^a^: iel} is a (finite) family of extremal subobjects of X, and (B,b) is an extremal subobject of X, then BA((^)A i ) and [*} (BftA-^) are isomorphic as extremal subobjects iel iel of X. 5.32. Remark . It can be shown in any quasi-complete category that (5Â» (XxApS v ) are isomorphic relations iel veV (i,v)sI*V from X to Z. In particular if (R,j) is a relation from X to Y and (S,k) is a relation from Y to Z, then Ro(tj/S v ) and ^/(R s v) are isomorphic vev VÂ£ V relations fr:-, X to Z and i\^J?.j)cS and (*/ (R-joS) are isomorphic relaiel iel tions from X to Z.

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99 Proof . From the conditions on fi it is easy to see that: (((^R i )xZ)n(Xx(^S v )),y) = (((V^R Â± )xZ)n(V*)(XxS v )),9) iel veV . iel veV (^(((URi)xZ)n(XxS v )),Y) = (^)((^(Ri x Z))r\(XxS v ),y) veV iel Â„ veV iel (^(^Â»((RiXZ)n(XxS v ))),T) E ( ^*J ((R i xZ)n(XxSv)),y). veV iel (i,v)elxv Hence, from the theorem (with (R-^xZ) C\ (XxS v ) assuming the role of A Â± ) it follows that ((t*^R i )o(l*JS v ),a) = ( \*J (R i oS v ),a). iel veV (i,v)eIxV 5.35. Corollary . If P has finite coproducts and is finitely union distributive and if (R,j) is a quasi-equivalence on X then (Rl*jA x Â»p) is an equivalence relation on X. Proof . Clearly (R\*l hy,p) is both reflexive and symmetric (5.10 and 5.9) Since each of (R,j) and (A x ,i x ) is transitive (2.4 and 2.2) it follows that (R(*M x )o(Rl*M x ),p#) = ((RoR)l*J(RoA x )(*;(A x oR)C*;(A x oA x ),3) Â£ (Rl*jR^R^A x ,p) e (RV*jA x ,p); (5.34 and 1.31). Thus (RV^jA x ,p) is transitive and, consequently, is an equivalence relation. 5.36. Corollary . If JÂ§ has (finite) coproducts and is (finitely) union distributive and if (R,j) is a relation from X to Y and {(A^,a i ): iel} is a (finite) family of extremal subobjects of X, then [Â£) (A^R) and iel (C*jA-)R are isomorphic as extremal subobjects of Y. id Proof . Since (R f\ ( ( [*) A Â± ) xY) , y) = ( V*J (R ACA^Y) ) , y) the result follows iel iel from the theorem. 5.37. Corollary . If Â£ has (finite) coproducts and is (finitely) union distributive and if (R,j) is a relation from X to Y and {(B^,b^): iel} is a (finite) family of extremal subobjects of Y, then V*^(RB.) and iel R(l*^B.^) are isomorphic as extremal subobjects of X. iel

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100 Proof . immediate. 5.38. C orollary . If fÂ£ has (finite) coproducts and is (finitely) union distributive and if {(R.Â£Â»Ji): iel} is a (finite) family of relations from X to Y and if (A, a) Ls an extremal subobject of X then A(\j[jRj) and iel \*J (AR-j^) are isomorphic as extremal subobjects of Y. iel Proof. This result follows immediately from the theorem since ((l^Ri)n(AxY),Y) = (lol(Rin(AxY)), Y ). iel iel on 5.39. Cor ollary. If (J has (finite) coproducts and is (finitely) uni distributive and if ((R^j-^): iel} is a (finite) family of relations from X to Y and if (B,b) is an extremal subobject of X then (tÂ£jRj)3 and iel [Â£j (R-^B) are isomorphic as extremal subobjects of X. iel Proof. Immediate. 5.40. Remark . Without the extra conditions on Â£ ; i.e., only assuming that P is locally small and quasi-complete; it is possible to prove that [*J(A Â± R) < (l^A i )R and that l^J (AR Â± ) ^Ad*^). iel iel iel iel 5.41. R emar k. Recall that if g is a Â£ -morphism from X to Y where jg is locally small and quasi-complete then the intersection of all subobjects of Y through which g factors, (f| Ej,e), yields the extremal jeJ epi-mono factorization of g; i.e., there exists an extremal epimorphism h such that e = eh. Let C\ Zi be denoted Slm(X) and be called the subj jeJ image of g. (Recall that the image comes from the epi-extremal mono factorization of g.)

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101 f Slm(X) >-> Y -& Im(X) Now let {(A^,a-j^: iel} be a (finite) family of subobjects of a Â£ -object C and let f be a fe -morphism from C to D. Then there exists a unique morphism u from the coproduct (_ULA-Â£,Uj_) to C such that iel yuj = a^ for each iel. Let {(E-,e-): jej} be the family of all subobjects of D through which each fai factors. Let (0,6) and (0^,6^) be the extremal epi-mono factorization of u and fa^ respectively. Recall that ( [*} Slm(A^) , I) is the intersection of all subobjects of D through which iel each 5j factors. Let (o*,6*) be the extremal epi-mono factorization of fo. 5.42. Theore m. If fÂ£ has (fiiiite) coproducts then, using the notation above, (^ISImCA^), O E-: and SIm(LJA n -) are isomorphic as subobjects iel jeJ iel of D. Proof. Let if be the unique morphism from JJ.A^ to [^J Slm(A^) such that iel iel ^i ~ >l 'i a Â± ^ or cacn i^I where y^ is the "inclusion" of Slm(Aj) into USIm(A Â± ). iel Thus (as is easily seen) the following diagram commutes. U 5Im(A Â± ) iel "\ >Â» Sim ((J A ; ) ' iel

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102 Since fa i factors through I for each id, it follows that (O^.e) < (U SIm(A i ),5:). jej iel Nov; if (Â£. ,e.) is a subobject of D through which each fa. factors then since (Slm^)^) is the intersectio of all subobjects of D through which fa Â± factors it follows that (SIm(A. ) , 6 . ) <_ (E.,e.) thus (USIm(A.),Z) < (E e .) for each jej. Hence ( \J Sim (A. ) , E) Â£(f\E.,e). iel J J iei X jej 3 Since each fa-j factors through 6* then (PiE:,e) < (SIra( U A^) , 6*) jej J iel Since Z'-p = fy and &*(o*o) = fy is the extremal epi-mono factorization of fu, it follows that (SIm( U A-i_),6*) <_ ( U SIm(A-;) , E) . Thus iel iel OJ SlmCAi) ,E) = (AE js e) E (SIm( (J A Â± ) , 6*) . iel jeJ iel 5.43. Remark. Theorems 5.30 and 5.42 show that the (sub) image of a union is the union of the (sub) images; hence the epi-extremal mono factorization and the extremal epimono factorization properties respect unions in a proper manner.

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104 Since there exists a morphism A for which 4>Â„A = a and a is a monomorphism, A must be a monomorphism. But 4> Y =Â„X<}). = ^yl^Thus it follows that A<}> A = 1. (6.0. i) so that A is a retraction. Hence A is an isomorphism (0.4 ) whence (A, a) = ($,<|>y). 6.3. Proposition . Let (R,j) be a relation from X to Y and let Z be any K -object. Then Ro$ and $ are isomorphic relations from X to Z; and, $oR and 4> are isomorphic relations from Z to Y. Proof. From 6.0.ii, (XxÂ«J>, l x ><<{)y xZ ) and (.^Axx(YxZ)^ are isomorphic as extremal subobjects of X X (Y>,a) = (Â£, v Â„) . Similarly it can be shown that $oR and A X Z $ are isomorphic relations from Z to Y. 6.4. Corollar y. If X is any ^ -object then ($,4>xx:<) is a quasi-equivalence on X.

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105 Proof . By the proposition ($>4>v x y ) = (^^''f'xxx^ hence transitivity is obtained. It is also clear that the following diagram commutes. *X*X X*X Since $ is an extremal monomorphism and 1, is an epimorphism X*>A Â» it follows from the uniqueness of the epi-extremal mono factorization that ($,0 V vv) (^ _1 >4>vÂ„v*) 5 hence symmetry is obtained. A X A AXA 6.5. Definition . Let (R,j) be a relation from X to Y. Then (R,j) is said to be rectangular if and only if there exist extremal subcbjects (A, a) and (B,b) of X and Y respectively such that (R,j) and (A*B,axb) are isomorphic relations from X to Y. 6.6. Remark . Since ($x$ )( f x^) and (y y) are isomorphic as extremal subcbjects of X*Y (6. 0. i and 6 .0. ii) it f ollows that (^^xxY^ ^ s a rec ~ tangular relation. 6.7. P roposition . Let (R,j) be a rectangular relation from X to Y and let (RY,j,) and (XR.j ) be the usual images of tt j and tt j respectively. Then R and RY*XR are isomorphic relations from X to Y. Proof. Since (R,j) is rectangular there exist extremal subcbjects (A, a) and (B,b) of X and Y respectively such that (R,j) = (A*B,axb). hence there exists an isomorphism a such that the following diagram commutes.

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106 -*XxY ->X * AxB -y> A Pi Since Pj is an epimorphism (6.0.iii),' pjO must he an epimorphism, and by the uniqueness of the epi-extremal mono factorization it follows that (A, a) I (RY.jj). Similarly (B,b) = (XR,.j ? ). Thus (AxB,axb) = (R,j) = (RYxXR,j 1 xj 2 ). 6.8. Corollary . Let (R,j) be a relation from X to Y. Then (R,j) is rectangular if and only if (R -1 ,j*) is rectangular. Proof . If (R,j) is rectangular then (R,j) = (RYxXR, j , xj 2 ) (6.7). It is immediate that the following diagram commutes. jl x j2 < ft2Â» TT l > RYxXR Vt -*Â» XxY V5>Â— Â— Â— Â— -**YxX *^ PAGE 101

95 Now consider the coproduct ( XL A^ ,y^) of the (finite) family iel {(A-,a-): id}. By the definition of coproduct there exists a unique mo rphism y such that pu= afor each Iel. Let A^ be the "inclusion" of (A^,a^) into (i_JA^,a). Again by the definition of coproduct there iel exists a unique morphism A such that Ay= A^ for each iel. Thus the following diagram commutes. -*-X A-: Note that a is a monomorphism. It will be shown that A is an extremal epimorphism. To see this, it will be shown that (U Aj,a) is the iel intersection of all subobjects of X through which y factors. To that end let (Z,g) be any subobject of X through which y factors; i.e., there is a morphism h such that y = gh. Then a^ = yyj_ = ghy^ hence each (A-^,a^) factors through (Z,g). Thus by the definition of union there exists a unique morphism Â£, such that gÂ£ = a. But this is precisely what is required of the intersection of all subobjects of X through which y factors . Now suppose that {(A^,a^): iel} is a (finite) family of relations from X to Y; i.e., each (A 3 -,a^) is an extremal subobject of X*Y. Consider (JLL &Â±,^i) and ( U Aiifl)Again, let y be th

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96 X\i. = X . for each iel, where \. is the inclusion of (A.,a.) into ( (J A .,Â£). Let (r,p) be the epi-extremal mono factorization of a. Recalls I ling Proposition 5.3 it follows that the domain of p (codomain of t) is ^A.. Thus the following diagram commutes, iel ^ XX. A. > XxY iel x u v * ^ A i iel t iel 5.30. Theorem . Let ^ have (finite) coproducts, let {(A^a^: iel} be a (finite) family of subobjects of C, and let f be a fe -morphism from C to D, As above, let (l^A^p) be the extremal mono part of the factoriiel zation of the unique morphism p from JJL.A^ to C with the property that iel Hjij = a^ for each iel. Let (f\E-,e) be the intersection of all extremal jeJ subobjects of D thiough which each fa^ factors. Let (Im(A^) ,p^) and (lw( [Â£) A.. ) .p) denote the extremal mono parts of the epi-extremal mono iel factorizations of fa^ and fp respectively. Finally let ( \*) Im(A^) ,p) iel be the intersection of all extremal subobjects of D through which each pfactors. Then (Im^A^.p) E (flE.,e) = ((*J Im(k L ) ,0) . iel jeJ iel Proof . If (E. ,e.) is an extremal subobject of D through which each fa^ factors, then since (Im(AjO ,0-^) is the intersection of all extremal subobjects through which fa^ factors it follows that (Im(A n .),P,0 < (E.,e-). Thus (Im(A n . ) ,o) < (HE-,e) for each iel. Hence 1 *Â• Â— J j x Â— . T .1 (i v ^Im(A.),p) < (C\E,,e). iel iej J

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97 However, since each fa Â± factors through p and since p is an extremal monomorphism it follows that (f\Ej,e) <_ ( \*J Im(A i ) ,|S ) . jeJ iel Consider the following commutative diagram. ">Â• ^Im(A Â± ) iel iel > D Note that (it, 6) is the epi-extremal mono f actcrization of fu so that (Im((^A i ),p) is the intersection of all extremal subobjects iel through which fjj factors. Let ty be that unique morphism such that ^Uj_ = ^-Â±^Â± fÂ° r each iel. Now pyy i fAjJi = Pi^i = Â£a Â± = fuy Â± . Thus by the definition of coprc-duct it follows that fy = p>. Hence fy factors through (l*J Im(A-|_) ,P) iel whence (Im( \*)A Â± ) , p) Â£ (l*J lm(.AÂ±) ,0) Â• iel iel Now fa-j^ = fuui = fpxpi = pfiy-jj hence fa i factors through p, whence (Hs^e) Â£ (Im( V*/ A Â± ) ,(5) . jeJ " iel Thus: (^ImCAijJ) E (f\E J5 e) E (Im(l&J A Â± ) ,j5) . iel jeJ iel

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93 5.31. Definition . A category is said to be (finitely) union distr ibut ive if the following properties hold: (i) if X and Y are any fa -objects and {(A^a^: iel} is a (finite) family of extremal subobjects of Y, then Xx((*jAjO and ^'J (XxA i ) are iel iel isomorphic relations from X to Y; (ii) if X is any fe -object, {(A-j^a^: iel} is a (finite) family of extremal subobjects of X, and (B,b) is an extremal subobject of X, then BA((^)A i ) and [*} (BftA-^) are isomorphic as extremal subobjects iel iel of X. 5.32. Remark . It can be shown in any quasi-complete category that (5Â» (XxAp

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99 Proof . From the conditions on fi it is easy to see that: (((^R i )xZ)n(Xx(^S v )),y) = (((V^R Â± )xZ)n(V*)(XxS v )),9) iel veV . iel veV (^(((URi)xZ)n(XxS v )),Y) = (^)((^(Ri x Z))r\(XxS v ),y) veV iel Â„ veV iel (^(^Â»((RiXZ)n(XxS v ))),T) E ( ^*J ((R i xZ)n(XxSv)),y). veV iel (i,v)elxv Hence, from the theorem (with (R-^xZ) C\ (XxS v ) assuming the role of A Â± ) it follows that ((t*^R i )o(l*JS v ),a) = ( \*J (R i oS v ),a). iel veV (i,v)eIxV 5.35. Corollary . If P has finite coproducts and is finitely union distributive and if (R,j) is a quasi-equivalence on X then (Rl*jA x Â»p) is an equivalence relation on X. Proof . Clearly (R\*l hy,p) is both reflexive and symmetric (5.10 and 5.9) Since each of (R,j) and (A x ,i x ) is transitive (2.4 and 2.2) it follows that (R(*M x )o(Rl*M x ),p#) = ((RoR)l*J(RoA x )(*;(A x oR)C*;(A x oA x ),3) Â£ (Rl*jR^R^A x ,p) e (RV*jA x ,p); (5.34 and 1.31). Thus (RV^jA x ,p) is transitive and, consequently, is an equivalence relation. 5.36. Corollary . If JÂ§ has (finite) coproducts and is (finitely) union distributive and if (R,j) is a relation from X to Y and {(A^,a i ): iel} is a (finite) family of extremal subobjects of X, then [Â£) (A^R) and iel (C*jA-)R are isomorphic as extremal subobjects of Y. id Proof . Since (R f\ ( ( [*) A Â± ) xY) , y) = ( V*J (R ACA^Y) ) , y) the result follows iel iel from the theorem. 5.37. Corollary . If Â£ has (finite) coproducts and is (finitely) union distributive and if (R,j) is a relation from X to Y and {(B^,b^): iel} is a (finite) family of extremal subobjects of Y, then V*^(RB.) and iel R(l*^B.^) are isomorphic as extremal subobjects of X. iel

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100 Proof . immediate. 5.38. C orollary . If fÂ£ has (finite) coproducts and is (finitely) union distributive and if {(R.Â£Â»Ji): iel} is a (finite) family of relations from X to Y and if (A, a) Ls an extremal subobject of X then A(\j[jRj) and iel \*J (AR-j^) are isomorphic as extremal subobjects of Y. iel Proof. This result follows immediately from the theorem since ((l^Ri)n(AxY),Y) = (lol(Rin(AxY)), Y ). iel iel on 5.39. Cor ollary. If (J has (finite) coproducts and is (finitely) uni distributive and if ((R^j-^): iel} is a (finite) family of relations from X to Y and if (B,b) is an extremal subobject of X then (tÂ£jRj)3 and iel [Â£j (R-^B) are isomorphic as extremal subobjects of X. iel Proof. Immediate. 5.40. Remark . Without the extra conditions on Â£ ; i.e., only assuming that P is locally small and quasi-complete; it is possible to prove that [*J(A Â± R) < (l^A i )R and that l^J (AR Â± ) ^Ad*^). iel iel iel iel 5.41. R emar k. Recall that if g is a Â£ -morphism from X to Y where jg is locally small and quasi-complete then the intersection of all subobjects of Y through which g factors, (f| Ej,e), yields the extremal jeJ epi-mono factorization of g; i.e., there exists an extremal epimorphism h such that e = eh. Let C\ Zi be denoted Slm(X) and be called the subj jeJ image of g. (Recall that the image comes from the epi-extremal mono factorization of g.)

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101 f Slm(X) >-> Y -& Im(X) Now let {(A^,a-j^: iel} be a (finite) family of subobjects of a Â£ -object C and let f be a fe -morphism from C to D. Then there exists a unique morphism u from the coproduct (_ULA-Â£,Uj_) to C such that iel yuj = a^ for each iel. Let {(E-,e-): jej} be the family of all subobjects of D through which each fai factors. Let (0,6) and (0^,6^) be the extremal epi-mono factorization of u and fa^ respectively. Recall that ( [*} Slm(A^) , I) is the intersection of all subobjects of D through which iel each 5j factors. Let (o*,6*) be the extremal epi-mono factorization of fo. 5.42. Theore m. If fÂ£ has (fiiiite) coproducts then, using the notation above, (^ISImCA^), O E-: and SIm(LJA n -) are isomorphic as subobjects iel jeJ iel of D. Proof. Let if be the unique morphism from JJ.A^ to [^J Slm(A^) such that iel iel ^i ~ >l 'i a Â± ^ or cacn i^I where y^ is the "inclusion" of Slm(Aj) into USIm(A Â± ). iel Thus (as is easily seen) the following diagram commutes. U 5Im(A Â± ) iel "\ >Â» Sim ((J A ; ) ' iel

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102 Since fa i factors through I for each id, it follows that (O^.e) < (U SIm(A i ),5:). jej iel Nov; if (Â£. ,e.) is a subobject of D through which each fa. factors then since (Slm^)^) is the intersectio of all subobjects of D through which fa Â± factors it follows that (SIm(A. ) , 6 . ) <_ (E.,e.) thus (USIm(A.),Z) < (E e .) for each jej. Hence ( \J Sim (A. ) , E) Â£(f\E.,e). iel J J iei X jej 3 Since each fa-j factors through 6* then (PiE:,e) < (SIra( U A^) , 6*) jej J iel Since Z'-p = fy and &*(o*o) = fy is the extremal epi-mono factorization of fu, it follows that (SIm( U A-i_),6*) <_ ( U SIm(A-;) , E) . Thus iel iel OJ SlmCAi) ,E) = (AE js e) E (SIm( (J A Â± ) , 6*) . iel jeJ iel 5.43. Remark. Theorems 5.30 and 5.42 show that the (sub) image of a union is the union of the (sub) images; hence the epi-extremal mono factorization and the extremal epimono factorization properties respect unions in a proper manner.

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SECTION 6. RECTANGULAR RELATIONS 6.0. S

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104 Since there exists a morphism A for which 4>Â„A = a and a is a monomorphism, A must be a monomorphism. But 4> Y =

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105 Proof . By the proposition ($>4>v x y ) = (^^''f'xxx^ hence transitivity is obtained. It is also clear that the following diagram commutes.

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106 -*XxY ->X * AxB -y> A Pi Since Pj is an epimorphism (6.0.iii),' pjO must he an epimorphism, and by the uniqueness of the epi-extremal mono factorization it follows that (A, a) I (RY.jj). Similarly (B,b) = (XR,.j ? ). Thus (AxB,axb) = (R,j) = (RYxXR,j 1 xj 2 ). 6.8. Corollary . Let (R,j) be a relation from X to Y. Then (R,j) is rectangular if and only if (R -1 ,j*) is rectangular. Proof . If (R,j) is rectangular then (R,j) = (RYxXR, j , xj 2 ) (6.7). It is immediate that the following diagram commutes. jl x j2 < ft2Â» TT l > RYxXR Vt -*Â» XxY V5>Â— Â— Â— Â— -**YxX *^

`J2 x .3i XRxRY Again, by the uniqueness of the epi-extremal mono factorization property it follows that XRxRY and R -1 are isomorphic relations; hence (R ,j*) is rectangular. If (R -1 ,j*) is rectangular then by the above , ( (R _1 )"" 1 , j#) is`

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107 rectangular; but ((R^)" 1 ^/) = (R.J) (1.13). Thus (R,j) is rectangular, 6.9. Proposition . Let (R,j) be a rectangular relation from X to Y and let (C,c) be an extremal subobject of X. Then ' (XR,j 2 ) if (CPlRY.Y) t (*,*x) , ( Y ) if (CrtRY.y) = (*,+ X ). (CR,k) Proof . Since (XR,j 2 ) 1 (Y.ly) it is clear that (XRrtY.y) = (XR,j 2 )Hence it follows that (Rn(CxY),Y ) E ((RYxXR)A(CxY), Yl ) = ( (RYA C) x(XR AY) , T? ) = ((RYrtC)*XR,3) (0. 8 ). Thus there exists an isomorphism a such that the following diagram commutes. (RYAC)xXR Â» -> Y Let (C,I) be the epi-extremal mono factorization of tt 2 p. Since to is an epimorphism and k is an extremal monomorphisia it follows by the uniqueness of the epi-extremal mono factorization that (Z,Z) = (CR,k) Â• But if ((RYnc)xXR,B) t (*,xxY> then the followln S diagram commutes. (RYAC)xXR Â»8 XxY I XR > v J2

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108 Since P 2 is an epimorphism (6.0. iii) it follows that (XR,J 2 ) = (Z,Z) E (CR,k). If ((RYr\C)xXR,g) = (^.^xxy) then there exists an isomorphism X such that the following diagram commutes. $ Â» *> y Thus by the uniqueness of the epi-extremal mono factorization it follows that (Â£,Â„) if (XRrtA.v) = (*,* y ) Proof . The proof is analogous to that of Proposition 6.9. 6.11. P roposition . Let (R,j) be a rectangular relation from X to Y and let (S,k) be a relation from Y to Z. Then RoS <_ RYx(XR)S. Proof . It is easy to see that the following objects are isomorphic as extremal subobjects of X*Y*Z: ( (RYxXR)xZ)A(X*S) , ((RYxXR) *Z) A(RYxS) , and RYx((XRxZ)A S) . Thus there exists an isomorphism o such that the following diagram commutes. RYx((XRxZ)riS) V*~ -V> ((RYxXR)xZ)A(XxS) " Â— *" XxZ (RYxXR)oS

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109 Recall that (R,j) = (RY*XR, j ^j 2 ) (6.7); hence (RYxXR)oS and RoS ire isomorphic relations. Consider the following commutative diagrams. (XRxZ)nS >VP2 -* YxZ -*Z '(XR)S T ' Â©2(Jl xl Y> RYx(YxZ) > XxYxZ l RY x6j

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107 rectangular; but ((R^)" 1 ^/) = (R.J) (1.13). Thus (R,j) is rectangular, 6.9. Proposition . Let (R,j) be a rectangular relation from X to Y and let (C,c) be an extremal subobject of X. Then ' (XR,j 2 ) if (CPlRY.Y) t (*,*x) , ( Y ) if (CrtRY.y) = (*,+ X ). (CR,k) Proof . Since (XR,j 2 ) 1 (Y.ly) it is clear that (XRrtY.y) = (XR,j 2 )Hence it follows that (Rn(CxY),Y ) E ((RYxXR)A(CxY), Yl ) = ( (RYA C) x(XR AY) , T? ) = ((RYrtC)*XR,3) (0. 8 ). Thus there exists an isomorphism a such that the following diagram commutes. (RYAC)xXR Â» -> Y Let (C,I) be the epi-extremal mono factorization of tt 2 p. Since to is an epimorphism and k is an extremal monomorphisia it follows by the uniqueness of the epi-extremal mono factorization that (Z,Z) = (CR,k) Â• But if ((RYnc)xXR,B) t (*,

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108 Since P 2 is an epimorphism (6.0. iii) it follows that (XR,J 2 ) = (Z,Z) E (CR,k). If ((RYr\C)xXR,g) = (^.^xxy) then there exists an isomorphism X such that the following diagram commutes. $ Â» *> y Thus by the uniqueness of the epi-extremal mono factorization it follows that (Â£,

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109 Recall that (R,j) = (RY*XR, j ^j 2 ) (6.7); hence (RYxXR)oS and RoS ire isomorphic relations. Consider the following commutative diagrams. (XRxZ)nS >VP2 -* YxZ -*Z '(XR)S T ' Â©2(Jl xl Y>

`RYx (XR) S >XxZ Let (x,p) be the epi-extremal mono factorization of`

*0 2 (j 1 xi YxZ )(i RY x6 1 ) =**ya. Thus since (P,p) is the intersection of all extremal subobjects through which*7a factors and since o is an isomorphism it follows that ((RoS), a) H ((RYxXR)oS,5) = (P,p) < (RY*(XR)S, j^Oj) . 6.12. Pro position . Let (R,j) be a relation from X to Y and let (S,k) be a rectangular relation from Y to Z. Then RoS <_R(SZ)xYS. Proof. Since (S,k) = (SZxYS .k^k^) (6.7) the result follows from argume nts analogous to those in the proof of Proposition 6.11 6.13. Lemma . Let (R^j^ = (AxB^axb^ be a rectangular relation from X to Y and let (R 2 ,j 2 ) E (R 2 x C,b 2 *c) be a rectangular relation from Y to Z,

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110 Then (R l0 R 2 ,j') E ((Ax3 1 )o(B 2 xC),j#) = |(AxC,ax c ) if (Bif\B 2 ,b) t ($, \( PAGE 116

110 Then (R l0 R 2 ,j') E ((Ax3 1 )o(B 2 xC),j#) = |(AxC,ax c ) if (Bif\B 2 ,b) t ($, \(

*XXZ ) if (Bif\B 2) b) = ($,* Y ) And in either case (RioR2,j') is rectangular. Proo f. It is straightforward (but tedious) to show that: ((AxB 1 )xZ)r>(Xx(B 2 xC)) = (Ax(BixZ))n(Xx(B 2 xC)) = (Ax (BixZ) (\ (Ax (B 2 xZ) ) = Ax((B 1 xZ)f\(B 2 xC)) E Ax((B 1 AB 2 )xC) E Ax(B } A B 2 )xC. If (BjHBpjb) Â£ ( ((AxBi)xZ)Pi (Xx(B2 x C)) * *> Ax(Bir\B2)xC "* => XxYxZ XxJ T# (AxB 1 )c(B 2 xC) It is easy to prove that is an epimorphisw since Ax(BjAB 2 )xC and (AxC)x (B 1 As 2 ) are isomorphic in a canonical way and 6.0.iii holds. Thus since a is an isomorphism it follows by the uniqueness of the epi-extremal mono factorization that Â«AxB 1 )o(B 2 xC),j#) e (AxC,ax c ). Hence (R 1 oR 2 ,j') = (AxC,ax c ) ; so it is rectangular. If (BiHB^b) = (*, VvVv Â„) are isomorphic as extremal subobjects hence by the uniqueness of the epi-extremal mono factorization it follows that ((A*Bi )o(B 2 *C) , j#) = ( Â„) and is rectangular (6.6).

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Hi 6.14. R emar k. As has been noted in Section 1 (1.37) the composition of relations is not necessarily associative. The examples 1.34 1.36 are in the category Top which satisfies the conditions of 6.0. The next theorem shows that the composition of rectangular relations jls associative. Hence, in particular, in Top the composition of rectangular relations is associative. 6.15. Theorem . Let (R,j) be a rectangular relation from X to Y, let (S,k) be a rectangular relation from Y to Z and let (T,m) be a rectangular relation from Z to W. Then Ro (SoT) and (RoS)oT are isomorphic relations from X to W. Proof . Since each of (R,j), (S,k) and (T,m) is rectangular there exist extremal subobjects of X, Y, Z and W such that (R,j) = (Ai *A 2 ,ai*a 2 ) , (S,k) = (B 1 xB 2 ,b 1 xb 2 ) and (T,m) E (Cj xC 2 ,c\ xc 2 ) Â• Then: f(B 1 xC 2 ,b 1 xc 2 ) if (B 2 r\C!,b) f (*,$ ) (SoT,k#) E I V($,^ yxW ) if (B 2 r\C l5 b) E (*,o z ), and '(A 1 xB 2 ,a 1 xb 2 ) if (A 2 flB 1) a) t ($, Y ) ,(*,

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Hi 6.14. R emar k. As has been noted in Section 1 (1.37) the composition of relations is not necessarily associative. The examples 1.34 1.36 are in the category Top which satisfies the conditions of 6.0. The next theorem shows that the composition of rectangular relations jls associative. Hence, in particular, in Top the composition of rectangular relations is associative. 6.15. Theorem . Let (R,j) be a rectangular relation from X to Y, let (S,k) be a rectangular relation from Y to Z and let (T,m) be a rectangular relation from Z to W. Then Ro (SoT) and (RoS)oT are isomorphic relations from X to W. Proof . Since each of (R,j), (S,k) and (T,m) is rectangular there exist extremal subobjects of X, Y, Z and W such that (R,j) = (Ai *A 2 ,ai*a 2 ) , (S,k) = (B 1 xB 2 ,b 1 xb 2 ) and (T,m) E (Cj xC 2 ,c\ xc 2 ) Â• Then: f(B 1 xC 2 ,b 1 xc 2 ) if (B 2 r\C!,b) f (*,$ ) (SoT,k#) E I V($,^ yxW ) if (B 2 r\C l5 b) E (*,o z ), and '(A 1 xB 2 ,a 1 xb 2 ) if (A 2 flB 1) a) t ($,

*XxZ ) if (A 2 AB 1>a ) E (4,* Y ). Thus there are two cases: 1) if (SoT,k#) = (B^xC2 ,bi x c 2 ) then as above it follows from 6.13 that t(AixC 2 , ai xc 2 ) if (A 2 ^B 1 ,a) i ($,<(>Â„) ( *'*XxW> if C A 2ABi,a) = (*,* Y ). If (A 2 AB!,a) t (3>.^) Y ) then (RoS,j#) E (A 1 xB 2 ,a i xb 2 ) hence ((RoS)oT,g) e (A 1 xC 2 ,a 1 xc 2 ) since (B 2 ACi,b) t (.Â§,$%)Â• If (A 2 HB l5 a) E (0,J) V ) then (RoS.i'O e (*~~f Yv . 7 ) hence ((RoS)oT,B) 2 0.
~~

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112 2) If (SoT,k#) E (*,Thus in any case (Ro(SoT),a) = ((RoS)oT.p). 6.16. Proposition . Let {(Ri,ji): iel} be a family of rectangular relations from X to Y. Then (ARj_,j) is a rectangular relation from X to Y. iel Proof. Each (Ri,Ji) is isomorphic to (Aj[xBi ,ai>'ai ) are isomorphic relations on X. a-,xa 2 < ^ 2 >"l > A^A 2 W ' > XxX Â» v*XxX

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112 2) If (SoT,k#) E (*,

^Â»*. ^^ a 2 Xa l ** A 2 xA 2 "* Thus, since (R,j) is symmetric and (AixA 2 ,aixa2) f. (RÂ»j)> it fol-

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113 lows that (A 2 xAi,a 2 xa 1 ) = ( (A 1 xA 2 )" 1 , ( ai x a2 )*) Â± (R -1 ,j*) = (R,j) (1-13 and 1.12). 6.18. D efinition . Let (R,j) be a relation from X to Y and let (Ai,ai) and (A 2 ,a 2 ) be extremal subobjects of X and Y respectively. Then (Ai*A 2 ,a i x a 2 ) is said to be a maximal rectangle in R if and only if (Ai>:A 2 ,aixa 2 ) <_ (R,j) and if (Bjjbi) and (B 2 ,b 2 ) are extremal subobjects of X and Y respectively such that (AixA 2 ,aixa 2 ) <_ (BixB 2 ,b \b 2 ) <_ (R,j), then (B 1 >'B2,b 1 xb 2 ) = (A 1 xA 2 ,a i x a 2 ) . 6.19. Prop osi tion . Let Â£ be finitely union distributive, let (R,j) be a difunctional relation from X to Y (5.22), and let (Ri,ki) and (R 2 ,k 2 ) be maximal rectangles in R such that (Rj,ki) t (R 2 Â»k 2 ) . Then (RiYHRjY.n) = U, $ x ) and (XR 1 nXR 2 ,X) = ( Y ) Â• Hence, in particular, (R 1 nR 2 ,y) e (Â§, cj) XxY ). P roof . If (Rj.kj) H ( $, <Â£ XxY ) then (Ri.kj) Â£ (R 2 ,k 2 ) since the following diagram commutes. XxY Thus, since (Rj,kj) is a maximal rectangle, it follows that (Ri,kj) E (Ps. 2 ,k 2 ) contradicting the hypothesis. Hence (Rj,ki) t ($>

PAGE 120

114 (Rj-^kx*) E (XRxxRiY.ya^l)Suppose that (XR 1 nXR 2 ,X) f ($,'<+>). Then (RloR2 -1 ,j) = (RiYxR 2 Y,y 1 xXi) (6.13) and hence ( (Rj oR 2 _1 )oR 2 ,ct) and (RjYxXR 2 ,uixX 2 ) are isomorphic relations (6.13). Similarly (R 2 oR 1 ~ 1 ,k) = (R 2 YxR 1 Y,A 1 x Ml ) (6.13) and hence ((R 2 oR 1 1 )cR 1 ,3) = (R 2 YxXR 1 ,X 1 xy 2 ) (6.13). Since (Rj.ki) < (R,j) and (R 2 ,k 2 ) Â£ (R,j), (Rx" 1 ,^*) < (R _1 ,j*) and (R 2 1 ,k 2 *) <_ (R _1 ,j*) (1.12) and hence ((R 1 oR 2 _1 )oR 2 ,a) <_ ((RoR -1 )oR,j') and ( (R 2 oRj -1 )oRj , 6) Â£ ( (RoR -1 )oR, j ' ) (1.30). Hence (RlV*/R2VÂ£M(RioR 2 1 )oR 2 )V*/((R 2 oRr 1 )oR 1 ),I 1 ) 1 (RV*J ( (RoR" 1 )oR) ,Z 2 ) (5.5). Since (R,j) is difuncticnal (5.22), (B.[*) ((RoR _1 )oR) ,Z 2 ) <_ (R,j) (5.5). Thus since g, is finitely union distributive, it follows that: ((R 1 Y^R 2 Y)x(XR 1 ^;XR 2 ),C.) E ((RxYxXRx) [*) (R 2 YxXR 2 ) [*J (RiYxXR 2 ) {*J (R 2 YxXRx) ,E) = (Rl\0 , R 2 ^((RloR 2 " 1 )oR 2 )^; ((R 2 oR i 1 )oRi) J I) 1 (R,j). Let K = (R 1 Y(*/R 2 Y)x(XR 1 ^XR 2 ). Since (Ri.ki) Â£ (K,0 and (R 2 ,k 2 ) <_ (K,5) and (K,0 is rectangular, by the definition of maximal rectangle, it follows that (Ri,ki) = (K,Â£) E (R 2 ,k 2 ), contradicting the hypothesis. Thus (XR 1 fiXR 2 ,A) E ($,{>Â„). Now suppose that (RiYf\R 2 Y,y) t (?',*). Then (Rl -1 oR 2 ,j) E (XRixXR 2 ,y 2 xX 2 ) and ((R 1 o(Ri~ 1 oR 2 ) ,o) E (RjYxXR 2 ,ui^X 2 ) (6.13). Similarly (R 2 _1 oR 1 ,k) E (XR 2 xXRi,XiXy 2 ) and (R 2 o(R 2 _1 oRi),B) E (R 2 YxXRi,A : __
PAGE 121
115 and since (Ri,ki) <_ (K,5) and (R2,k2> Â£ (K>5) it follows that (Rl,ki) = (K,5) = (R2,k2) contradicting the hypothesis. Consequently, (R^Hr^u) = (4,* x ). The above implies that (RiHR2 5Y) ($,vv) since ' XX Y ((R 1 YxXR 1 )n(R2YxXR 2 ),Y) -= ( (Ri Y C\ R 2 Y) x (XRjA XR 2 ) ,y*A) = (Sx^^x^) h (*,<(. XxY ) (0.8 and 6.6). 6.20. Proposition . If g has (finite) coproducts and is (finitely) union distributive and if { (A^xB^,a-j_xbi.) : ^e ~^^ ^ s a (finite) family of rectangular relations from X to Y such that (A-jf^A,a) = (O.^y) and (B Â± f\B.,b) = (*,<|> Y ) for i ^ j, i,jel then (R,j) = ( \*) (k Â± *B Â± ) , j ) is a iel difunctional relation from X to Y. Proof . First consider (R -1 ,j ;,; ). Since ((A i xB i )~ 1 , (a i xb i )*) E (B i xA 1 ,b i xa i ) it follows that (R~ ! ,j*) = ((^(AiXBi))1 ^*) = <.\*) (B Â± xA Â± ) ,i) (5.8). Thus iel iel (RoR _1 ,a) 3 ({*) (A i xB i )oV*;(B j xA i ),a). But iel iel (^(A.xB.)oV*;(B i xA i ),a) = ( [*) ( (A^B.) o (B, xA, ) ) ,g) (5.34). From this iel iel (i,j)etxl J J and the fact that (B i AB.,b) = ($,$Â„) for i ^ j, i,jel it follows that (RoR-^.a) = ({*) (A i xA i ),a) (6.13). iel Similarly ( (RoR~ 1 )oR,k 1 ) = ( lÂ£> (A^A^ o l*J (A.xB . ) , y) iel jel J J (^((A i xA i )o(A,xB,)),Y) (5.34). (i.jjetxl 1 J J But since (A^AA,^) E (?,<> x ) for i 4 j, i,jel it follows that (^((A,x/\ ;i )o(A i xB i )),Y) E ((*! (A Â± x?> Â± ),3) (R,j). Thus it has been (i,j)Â£ixi J ^ iel shown that ((RoR _1 )oR,k^ ) = (R,j). Similarly it can be shown that: (Ro(R" 1 eR) ,k 2 ) = (R,j). Hence (R,j) is difunctional. 6.21. Def init ions . Let X be any g -object and let X be a relation on X. Then (R ; j) is a square in X V X if and only if there exists an extremal
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116 subobject (A,a) of X such that R and AxA are isomorphic relations on X. If (S,k) is a relation on X and (R,j) is a square in X*X such that (R>j) Â£ (S,k) then (R,j) is said to be a maximal s quare in (S,k) if and only if for any square (T,m) in XxX for which (B,j) <_ (T,m) <_ (S,k) holds, it follows that (T,m) = (R,j). 6.22. Proposition . Let g be finitely union distributive and let (S,k) be a quasi-equivalence on X. Then (R,j) is a maximal rectangle in (S,k) if and only if (R,j) is a maximal square in (S,k). Proof. Assume (R,j) is a maximal square in (S,k). Suppose (R,j) is not a maximal rectangle in (S,k) then there exist extremal subobjects (B,h) and (C,c) of X such that (R,j) Â£ (B>:C,bx c ) <_ (S,k) and (R,j) f (Bxc,bx c ). Since (S,k) is symmetric and (BxC,b*c) < (S,k), (CxB,cxb) Â£ (S -1 ,k*) = (S,k) (1.12 and 1.13). Since (S,k) is transitive it follows that ( (BxC) o (CxB), a) = (BxB.bxb) <_ (SoS -1 ,a) = (SoS,k#) <_ (S,k) and ((C*B)o(BxC),a) = (CxC,cx c ) < (S -1 oS,a) E (SoS.k//) <_ (S,k) (6.13 and 1.30). Since fc is finitely union distributive it follows that (B^C)x(B^C) = ((BV*JC)xB)\*;((Bi*;C)xC) (BxB) (.*/ (CxB) {*) (BxC) \*J (CxC) . Since each of BxB, CxB, BxC, and C V C is contained in S it follows that (R,j) < (BxC,bx c ) <_ ( (B \*JC) * (B [*) C) . B) < (S.k). Since (R,j) t (BxC,bx c ), (R,j) f ((B'c.JC)x(B^C),6) contradicting the maximality of (R, j ) .. Conversely if (R.j) is a maximal rectangle in (S,k) then (R,j) = (BxC,bxc.) for some paj.t of extremal subobjects (B,b) and (,C,c)
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117 of X. Repeating the above it follows that (R,j) = (BxC,bxc) and (BxC,bx c ) <_ ((B^*JC)x(BlÂ£jC),6) <_ (S,k). Since ((BV*/C)x(B \*) C) , 3) is a rectangular relation then by the raaximality of (R,j) it follows that (R,j) = ((BlÂ£f C)x(Bl*J C),B) and hence is a square. Thus (R,j) is a maximal square in (S,k). 6.23. Example . Consider the following symmetric relation in Top 1 . Let X = Â£.0,1 j with the usual topology, let S = [0,i\ x [%, 3/4*1 U [^,3/4*1 x [p,lj and let k be the inclusion map taking S into XxX. It is clear that (.0,1 J x ^,3/4j together with its inclusion map is a maximal rectangle in (S,k) that is not a maximal square. It is also clear that |*S,3/4J x \^,3/4j together with its inclusion map is a maximal square in (S,k) that is not a maximal rectangle. Note this shows that even in a symmetric relation it may be the case that both maximal squares and maximal rectangles exist and are distinct . Also note that the above example is valid in Top ? and in CpT 2 Â• By neglecting the topology and considering the underlying set, the example is valid in Set. 6.24. Proposition . If g is finitely union distributive, (R,j) is a quasi-equivalence on X, and (Rj.kj) and (R2,k 2 ) are maximal squares in (R,j) such that (R^kj) t (R 2 ,k 2 ) then CR 1 nR 2 ,k) = <-*>$x*x)' Proof. Both of (R ,k ) and (R^k,) are maximal rectangles (6.23). (R,j) is difunctional (5.24) hence the result follows immediately from 6.19. 6.25. Prop o sition . Let fa have (finite) coproducts and be (finitely) union distributive and let {(A-^,a^) : iel} be a (finite) family of
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118 extremal subobjects of X with the property that (Aif\Aj,a) = (__

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## Material Information- Title:
- Relation theory in categories
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- Fay, Temple Harold, 1940-
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- Gainesville, Fla.
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- University of Florida
- Publication Date:
- 1971
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## Subjects- Subjects / Keywords:
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Equivalence relation ( jstor ) Factorization ( jstor ) Isomorphism ( jstor ) Mathematical congruence ( jstor ) Mathematics ( jstor ) Morphisms ( jstor ) Rectangles ( jstor ) Topology ( jstor ) Universal algebra ( jstor ) Categories (Mathematics) ( lcsh ) Dissertations, Academic -- Mathematics -- UF Mathematics thesis Ph. D - Genre:
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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 Quasi-Equivalence 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 epi-extremal 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 quasi-equivalence relations (symmetric, transitive relations) are studied in Section 3. A quasi-equivalence 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 epi-mono 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 relat-con 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 be-ig 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 quasi-complete 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 epi-extremal 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 quasi-complete 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 epi-extremal 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 non-algebraic. 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 epi-extremal 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 category-theoretical 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 X-----Y and Y-----Z 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 -obj-ect 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 1----------v 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., one-to-one 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 epi-extremal 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 epi--extremal 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 epi-extremal 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 epi-mono factorization property is defined dually. If has the unique extremal epi-mono 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 cpi-mono 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 epi-extremal 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 epi-extremal mono factorization property and the unique extremal epi-mono factorization property. Proof. (sketch). First we will show the existence of the unique extremal epi-mono 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 epi-mono 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 non-empty 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 epi-extremal 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 quasi--complet-, if has finite products and arbitrary intersections. 0.24. Examples. The categories FSet and FGp are quasi-complete cate- gories which are not complete. The categories Set, Top1, Top2, CpT2, Grp, Ab, Ring, and SGp are quasi-complete. 0.25. Remarks. A quasi-complete category is finitely complete but is uot necessarily complete as the examples FSet and FGp above show. Also, a locally small, quasi-complete category has both the unique extremal epi-mono factorization property and the unique epi-extremal mono factorization property (0.20 and 0.21). It can be shown that the unique epi-extremal 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 epi-mcno 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, quasi-complete (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 i-xa 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 < = 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 following diagram commutes. XY IY VI xY -------- --------- Yx 3 11 1 ' xX x )------ ---?x Note that: 1.7. Definition. Let (R,j) be a relation from X to Y and let (T,j*) be the unique epi-extremal mono factorization of codomain of T (domain of j*) is denoted by R-1 and (R-1,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 to Y then R-1 g {(y,x): (x,y)eR} with j* the inclusion map of R-1 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 (R-1,j*) < (R,j). 1.11. Proposition. Let (R,j) be a relation from X to Y. Then the inverse relation ((R-1)-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: morphism and j = j#(T-r). 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 (S-1,k*) i (R-',j*). Proof. Consider the following conmiutative diagram. j <7;2' 1> R XxY -- YxX AV- k -U R1 k* S-1 If (S,k) < (R,j) then there exists a morphism a: S -- R R such that -1 ja = k. Define B = TT-1. Then j* = j*-Tt-1 = <2 >jaT1 k= k*i-T = k*. Thus (S-l,k*) < (R-1,j*). If (S-1,k*) < (R-,j*) then by the above, ((S-1)-1,k#) < ((R-1)-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 (R-I,j*). Proof. Since (R,j) is symmetric (R-1,j*) < (R,j). Thus (R,j) = ((R-1)-1,j#) < (R-Ij*) (1.11 and 1.12). Consequently (R,j) (R-1,j*). 1.14. Definition. Recall that since 0 is quasi-complete 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) (Ax-l,i*). Proof. Consider the following commutative diagram. <1T2' > XxX >>----------- ---.--- XxX SX ---x x .y- /-- i 1 X Tl Thus the epi-extremal mono factorization of' 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*) <- (p-l,j*) (1.16 and 1.12). Conversely if (AX,iX) < (R-1,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 (R-1fS-1) 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 s-1 ) S --- X-----. YxX S 2 flk \4 k* ^\~~~ ^ ^ ^ : 9 <'[2 ,'IT1> R S n s------>- XxY -------- YxX (R A S)-11 Observe that of intersection: (RAS, isomcrphism, (RkS, ((RAS)-1,I*) < (R-1A S-1, ). To obtain the reverse inequality, note that by the definition of intersection (R-IAN S-I, kT-1XL = Whence (R-1(1 S~ ,) < (RnS, (R-1( S-1,) = ((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 RAR-1n 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', also that < 2' -1 = that jT-rX4 JT-X4 = (R-7()AXX3) <_ (R(c X,i X 2). Also observe that j*rX = j'TXI = i X2 so that by the definition of intersection: (R1AxiX2) < (R-1iAi x 3). Thus: (R AxiX 2 ) = (R-IfAxix x 3). Clearly (RCR-ln 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(R-lI AAx,i X3) (Rn/R-! fAx,i X6) Hence, (R}lAx,iX2) (RfR-1('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 it follows that (X,<,1 X>) < (AXi ). Since l 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, 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 }, { { 1.24. Proposition. If (R,j) is a reflexive relation on X then T1j and Tn2J are retractions. Proof. Since (X, B such that ita = 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 = ( _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 cfXxZ to X and Z respectively. Let (T',j') be the unique epi-extremal 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 Since (RloS1,j) is the intersection of all extremal subobjects through which 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 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 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 = , Ti2JP I*I = T2P1J- (x)Il = T2rl(jxl)Al = T2Y = T2 < 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 = |

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PAGE 1 Relation Theory in Categories ay TEMPLE HAROLD FAY A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA J.N PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY 07 FLORIDA PAGE 2 To Dr. George E. Strecker, without whose tactful proddings, infinite patience in proofreadings of handwritten drafts ana helpful suggestions this work would never have been completed . PAGE 3 TABLE OF CONTENTS Abstract iv Introduction 1 Section 0. Preliminaries 6 Section 1. Generalities IS Section 2. Categorical Congruences I,] Section 3. Categorical Equivalence Relations and Quasi-Equivalence Relations AS Section 4 . Images 63 Section 5 . Unions 75 Section 6. Rectangular Relations 103 Bibliography 120 Biographical Sketch 122 in PAGE 4 Abstract of Dissertation Presented to the Graduate Counci] 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 extremal monomorphism having domain R and codomain X X Y. 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.ubspaces of the topological product. This latter fact would not be the case if relations would be defined to be merely subobjeccs of the categorical product . Section notes results which are purely categorical in nature and which will be usee 1 , extensively throughout, the sequel. Particular emphasis is give:-; to the epi-extremal mono factorization property and IV PAGE 5 necessary and sufficient conditions for the existence of this factorization 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 or 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 quasi-equivalence relations (symmetric, transitive relations) are studied in Section 3. A quasi-equivalence oi\ 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 aeA such that (a,y)e.R). This definition is generalized in Section 4 and results similar to those r obtained by Riguet are demonstrated. If {(R-jj^): iÂ£l) is a (finite) family of relations from X to Y then the relation theoretic union (^_J^i>J) Â°^ *-' he f ara ily ^ s obtained by iel taking the intersection of all relations from X to Y which "contain" each Rj. If the category being investigated is assumed to have (finite) coproducts then the union of the family considered as subobjeets and the relation theoretic union of the family considered as extremal subobjeets turn out to be given by the unique extremal epi-mono and unique epiextremal mono factorizations of the canonical morphism from the coproduct of the family to X> PAGE 6 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 categorical notion. 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. V3 PAGE 7 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 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. Secondly, 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 of the subject. Many 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" 1 = {(x,z): there exists ysY such that (x,y)eR and (y,z)eR *} is reflexive. This result can be generalized to categories but is no longer easy to prove and the result gains some significance. Another easy result in set theoretic relation theory is that if t y and A v are the diagonals on X and Y respectively then A cR R = RcA, This result is also generalized to categories but "isomorphic as rela- PAGE 8 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 being 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 subobject of the categorical product X*Y; 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 cartesian 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 if relations would be defined to be merely subobjects of the categorical product. Much care has also been taken with the definition of composition of relations (1.26). Using this definition many nice results have been obtained; however, in general, the composition of relations is not associative (1.35). 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 the fact that for rectangular relations composition is_ associative (6.15). Cohn ^3j and Lambek \ 13 j define a congruence in an algebraic setting to be a sebalgeura of the cartesian product which is "compatible 1 ' with the algebraic operations and which is set theoretically an equivalence relation. In this work, a generalized notion of congruence PAGE 9 is given which is equivalent to the above in algebraic categories and the result that a (categorical) congruence is a (categorical) equivalence 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 quasi-complete the notion of union became somewhat easier to handle. For instance, if {(R.,j ): iel} is a (finite) family of relations i i from X to Y then the union (tjR.J) of the family, considered as subiel 1 objects of X*Y 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 epi-extremal mono factorization of j (5. 3 ) or equivalently by taking the intersection of all relations from X to Y which "contain" each R.. If the category being investigated is assumed to have (finite) coproducts in addition to being locally small and quasi-complete 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 epimono and unique epi-extremal mono factorizations of the canonical morphism from the coproduct of the family to X X Y (5.29) > It is also shown 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) coproducts mentioned above; hence, the notion of a (finite) union distributive category is introduced (5.31). Roughly speaking, this property guarantees that unions "commute" with products, and intersection? and thus unions become "easy" tj handle. Examples of union PAGE 10 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 [_ 2 ? J and its importance has been nc ted by Lambek \13j and MacLane [18 J . 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 -I -1 (RoR )oR _< R and Ro (R oR) _< R where "<" is the usual order on subobjects. It is easy to prove, again by choosing elements, that if a set theoretic relation R is difunctional then R = RoR oR. However, the similar result in the categorical setting is much harder to obtain and is rephrased: if R is difunctional then R ~ (RoR )oR and _] R E Ro(R oR) -..'here " = " 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 initial object which behaves similarly to the initial object in the category of sets (namely the empty set) had to be postulated and disjointness became a useful categorical notion. Again, examples of such categories are non-algebraic. Also the notion of difunctional relations was crucial in obtaining the above result (-6.20). The excellent reference paper by Riguet j 22 J has been used as a guide for the results of set theoretic relation theory. Indeed, most all of the results contained herein are generalisations of results in j[22j . The papers by Lambek 113 J , I 14 J ? MacLane I ] 8 \ and Bednarck aiid Wallace s j j , / \ provided motivation for many of the general!- PAGE 11 zations. The basis for the categorical notions has been taken from the papers of Herrlich and Strecker ]_ 7 J , [s ) , Isbell \_ll"]' L 12 J ' 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 \1\ J , Freyd \k \ and Herrlich and Strecker \, 9 J Â• The work here is begun with a preliminary Section which notes (often without proof) results which are purely categorical in nature and which will be used extensively throughout the sequel. Particular emphasis is given to the epi-extremal mono factorization property and necessary 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 category-theoretical background. It is expected that the reader be familiar with the basic categorical notions , PAGE 12 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 defined. The reader is referred to Mitchell Â£21J and Herrlich and Strecker Â£9] for such notions. All of the following results are proved in detail in Herrlich and Strecker Â£$3 . Since Theorem 0.21 is vital to this work the proof is sketched here. 0.1. Nota t ion . 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 topological spaces and whose morphism class is the class of all continuous functions shall be denoted by Top 1 . In a manner similar to that described above, one obtains the following categories: FSet^ finite sets and functions; FGp finite groups and group homomorphi sras ; Ab Abelian groups and group homomorphxsms ; PAGE 13 SGp semigroups and semigroup homomorphisms ; SGp semigroups with identity and semigroup homomorphisms which preserve the identity; Rng rings and ring homomorphisms; Rng~ rings with identity and ring homomorphisms which preserve the identity; Top,. Hausdorff spaces and continuous functions; CpTÂ„ compact Hausdorff spaces and continuous functions. f Â§ 0.2. Proposition. Let p be a category and let X* Y and YÂ— ** Z be &!, -morphisms. 1) If f and g are monomorphisms then gf is a monomorphism. 2) If f and g are epimorphisms then gf is an epimorphism. 3) If gf is a monomorphism then f is a monomorphism. 4) If gf is an epimorphism then g is an epimorphism. 5) If gf is 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 J, 9 j that certain functors preserve regular monomorphisms while not preserving equalizers, hence one reason for the above distinction between equalizers and regular monomorphisms. In this paper, since we shall not deal with functors, no distinc- PAGE 14 t.ion shall be made between equalizers and regular monomcrphisir.s; i.e., between the pair (object and morphism) and the morphism alone. Both will be called equalizers. , f 0.4. Proposition. Let g be a category and let X Â—*Â» Y be a Â£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. Definit ion . Let {A.: iel} be a family of Â£? -objects then the product (TTA.,i r .) of the family is a fe -object I i A together with proiel iel jection morphisms tt^ :TT*Aj_ Â— Â— pAj with the property that if P is iel any j*' -object for which there exist j? -morphisms p.: P Â— Â— Â— **A. for each iel, then there exists & unique morphism X: P Â— Â— *>Â»"f|~A. such that iel ir. A = p. for each iel. y i The dual notion is that of the coproduct (JLLa. ,u.). iel X 0.6. De finitio n. Let {(A., a.): isl} be a family of subobjects of a object X. Then the intersecti on ( O A... ,a) of the family is a ^-object iel Â« ^ A^ together with a morphism a: C\k. > X where fur each i there id iel is a morphism A.: f'\ A. Â• HÂ» A, . such that a-X. = a with the property id that if P is any object for which there exist g -morphisms p: PÂ— Â— *X and p,-: P Â— -~"V_A ,Â• such that a,-p,= p for each iel then there exists a unique morphism X: P "O'/jA; such that aA = p. iel It follows that a is a monomorohi sin . PAGE 15 0.7. Remar k. 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 cf a more general theorem dealing with the commutation of limits which can be found in Herrlich and Strecker [9J . A variation of the theorem will be proved in Section 1 (1.5) . 0.8. T heore m. Let {(A., a.): iel} and {(B^b.): iel} be families of subobjects of C -objects X and Y respectively. Then if S has finite products and arbitrary intersections then (,C\ A.)x( f\ B. ) and (~\ (A. xB^) iel iel ' iel are canonically isomorphic. 0.9. Notation. Let {X.: iel} be a family of Â£ -objects and suppose |2 Â— v X.: iel} is a family of fe -morphisms. Then by the definition of product there exists a unique morphism h from Z to TT Xj such iel that if.h = f . for each iel. This morphism h shall be denoted by < f< > i^I Let A and B be g -objects and suppose that a: A rX and b: B > Y are g -morphisms. If P, and PÂ„ are the projection morphisms from A X B to A and B respectively then aP 1 : A X B Â— ^ X and bP 2 ^ A*B Â— Â— Â—> Y, hence by the definition of product there exists a unique morphism g from A X B to X X Y such that 7r 1 g = ao and TT g = bp2Â« This morphism g shall be denoted by axb and shall be called the prod uct of a and b . Let f be a h -morphism from X to Y. If f is a monomorphism then the following notation shall be used: x >Â„. __: _^ Y PAGE 16 10 If f is an epimorphism then the following notation shall be used f X ~?oY If f is an equalizer then the following notation shall be used XÂ»>~ -*Â• i If f is an isomorphism then the following notation shall be used X"*-v.> Y be d Â— >Y, X Â— Â— * Z, and YÂ— Â— > W 0.10. P roposition . Let A Â— SÂ»X, B be Â£? -morphisms. Then (c*d) (a*b) = ca-db. a b 0.11. Proposition. Let A Â— Â— Â— > X and B > Y be monomorphisms (respectively, sections, isomorphisms) then a*b is a monomorphism (section, isomorphism) . 0.12. Remark. A partial order may be defined on the subobjects of an object in j^ in the following way: If X is a fe -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 respectively, then (A,a) <_ (B,b) if and only if there exists a morphism c from A to B such that be = a . B >4 I A .-v X By an abuse of language, if (A, a) < (B,b) then (K.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 (A, a) <^ (B,b) an J (B,b) < (A, a) then the morphism c is an isomorphism. In this case, PAGE 17 11 (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 g . The following notation shall be used to denote the case where (A, a) and (B,u) are isomorphic as subobjects of X: (A, a) I (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) = (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. Defini tio n. Let f from X to Y be a fc -morphism. f is an ext remal mo n o mo r p h i s m 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 >>Â— >Y The dual notion is that of an extremal e pimorphism and is denoted: X Â— Â«$*> Y If f is an extremal monomorphism f: X Â— -*-Y, then (X,f) is called an extre mal sub obj ect of Y. 0.14, Remark . The definition of extremal monomorphism is due to Isbell {^llj . The concept of extremal monomorphism is important since it yields what shall be called the "image" of a morphism (see 0.18),. 0.15. Example? . In the categories Set, Grp , Ab and _FGp_, extremal monoiticrpbi.jrr.s are precisely the monomorphisms (i.e., one-to-one morphisms). PAGE 18 12 In the categories Top and CpT extremal monomorphisms are precisely the embeddings. In the category Top they are the closed embeddings. f . 0.16. Pro posit ion. If X Â— *-Y is a g -morphism such that f = gh and f is an extremal monomorphism then h is an extremal monomorphism. 0.17. Proposition. If X *-Y is a fe -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 epimorphism (c.f. 0.3). 0.18. Def ini tion. A category jÂ£> is said to have the unique epiex trema l mono facto rizatio n property if for any )Â° -morphism XÂ— sÂ»Y, there exist an epimorphism h and an extremal inonciuorphism 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 a such that the following diagram commutes. If Â£ has the unique epi-extremal mono 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 epi-extremal mono factorization of f. The extremal subobject (Z,e) of Y is called the PAGE 19 13 image of X under f . Sometimes (Z,g) is referred to as the image of f . The notion of the uniqu e extremal epi-mono factorization proper ty is defined dually. If H has the unique extremal epi-mono factorization property and f = gh where g is a monomorphism and h is an extremal epimorphism then the pair (tug) shall be used to designate the extremal epi-mono factorization of f . The subobject (Z,g) of Y is called the sub image of X u nder f. Sometimes (Z,g) is referred to as the subimage of f . 0.19. Definition . A category g is said to have the diagonalizing pr operty if whenever gh = ab such that h is an epimorphism and a is an extremal monomorphism, then there exists a (necessarily unique) morphism t, such that E, h = b and a Â£ = g. I **Â• Y v W >Â•> > Z 0.20. Theorem. Let C be a locally small category having equalizers and intersections. Then the following are equivalent: 1) ig has the unique epi-extremal mono factorization property, 2) g has the diagonalizing property, 3) the intersection of extremal monouiorphisms is an extremal monomorphism and the composite of extremal monomorphisms is an extremal monomorphism, 4) if g 5 has pullbacks and if (P,q,3) is the puliback of f and g where fg = get and f is an extremal monomorphism then a is an extremal PAGE 20 14 monomorphism. 5) if ^ has (finite) products then the (finite) product of extremal monomorphisms is an extremal monomorphism. 0.21. Theorem . If Y> is locally small and has equalizers and intersections then t* has both the unique epi-extremal mono factorization property and the unique extremal epi-meno factorization property. Pr oof . (sketch) . First we will show the existence of the unique extremal epi-mono factorization property. If f from X to Y is any fc> -morphism then let (OE.,e) be the intersection of the family {(E.,e.): ieJ} of 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 morphism h such that f = eh. Now, to see that h is an epimorphism suppose a and 3 are \% -morphism? such that ah = gh. Let (E,k) be the equalizer of a and (3. It follows from the definition of equalizer that there exists a morphism g such that kg h. Â»* Y Thus it follows that f factors tHrough ek and sines ek is a monomorphism then there exists a morphism X: f\ E. * E such that ek\ = e From this it follows that k is rn isomorphism whence a = 3 and so h is an epimorphism. Next it will be shown that h is an extremal epimorphism. Suppose h = h 1 h ? where h., is a monomorphism. Then eh., is a monomorphism through PAGE 21 15 which f factors. From this it follows, as above, that h is an isomorphism and hence h is an extremal epimorphism. Suppose f = g'h' where g' is a monomorphism and h 1 is an extremal epimorphism. Then since g' is a monomorphism vhrough which f factors there exists a morphism x from P\E. to che codomain of h' (domain of g') such that e = g'x. Since e and g' are monomorphisms, it follows that h' = xh and that x is a monomorphism. Since h' is an extremal epimorphism it follows that x is an isomorphism. Thus fe has the unique extremal epi-mono f actorizaticn property. Now suppose that ge = mE where e is an epimorphism and m is an extremal monomorphism. It will be shown that there exists a morphism o from the codomain of e to the domain of m such that oe = h and mo g. Let (f\k.,a) be the intersection of the family {(A., a.): isl} iel of all subobjects of the codomain of g (codomain of m) through which g and m factor. This family is non-empty since both g and m factorthrough the identity morphism on the codomain of g. It follows that both g and m factor through a. Thus there exist morphisms a, and a such that the following diagram commutes. m It will be shown next that Â£Â„ is an epimorphism. Suppose ot* and are L> -morphisms for which ot*aÂ„ 3*a.. Let (E*,k*) be the equalizer PAGE 22 16 of a* and 3*. Jt follows from the definition of equalizer that there exists a morphism b^ such that k*!^ = a 2 since 0*3 = B*a_. Since the diagram commutes it follows' that a*a e = 3*a,e. But e is an epimorphism hence a*aj = Â£ v a so that by the definition of equalizer there exists a morphism b~ such that k*b~ = a-^. Thus it follows that m = ak*b-, and g = ak*b2 anc ^ so both m ant ^ S factor through ak* from which it follows that k* is an isomorphism. Hence a* = g* and a, } is an epimorphism. But in is an extremal monomorphism and m = aa and aÂ„ is an epimorphism. Thus a is an isomorphism. Thus defining a = a^a it follows that the following diagram commutes and ^ has the diagonal ization property. Hence K has the unique epi-extrema.l mono factorization property (0.20) 0.22. T heorem . Let K be any category then the following are equivalen 1) t^ is (finitely) complete, 2) Q^ has (finite) products and (finite) intersections, 3) fe> has (finite) products and equalizers, A) Jg has (finite) products and pullbacks. 0.23. Definition. A category 7^ is said to be quasi-complete if j^ has finite products and arbitrary intersections. t: PAGE 23 17 0.24. Exa mples . The categories FSet and FGp are quasi-complete categories which are not complete. The categories Set, Top , Top ^ , CpT ? , Crp, Ah, Ring , and SGp are quasi-complete. 0.25. Re marks . A quasi-complete category is finitely complete but is not necessarily complete as the examples FSe t and FGp above show. Also, a locally small, quasi-complete category has both the unique extremal epi-mono factorization property and the unique epi-extremal mono factorization property (0.20 and 0.21). It can be shown that the unique epi-extremal 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 epi-mcno 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. PAGE 24 SECTION 1. GENERALITIES .1.0. Sta nding Hypothesis . Througho ut the entir e paper it will be assumed that /Â» jls a_ lo cally small , quasic omplete (fi nite prod ucts and arbitrary intersections) c ategory . As noted in the preliminary section fe enjoys the unique epi -extremal mono factorization property. 1.1. Exampl e s. Many well known categories are locally small, and quasicomplete. Among such are the categories: Set , Top. , Top , Grp, Ab, SGp_, SGjp 1 , Rng, Rng 1 , CpT 2 , and FGp . 1.2. Defi nition. Let X and Y be g -objects. A relation R from X to Y is an extremal subobject of X> PAGE 25 19 In the categories Top ,, , and CpT ? , relations are closed subspaces of the Cartesian product together with the inclusion map. 1.5. Proposition , Let X and Y be K -objects and let (A, a) and (B,b) be extremal subobjects of Y. Then Xx(AftB) and ^X^AjCi (X X B) are isomorphic relations from X to Y. Proof . Consider the following commutative diagrams. (X*A)f\ (XxB) Consider also (Xx(AAB), l y x c = Yi)Since extremal subobjects are closed un-ler intersections and products (0.20; y^ and Yo ar e extremal monomerphisms. Since (l x x a) (! x xA A ) = l x *c = z Yj and (l x xb) (l x x * B ) = l x x c = Yj then by the definition of intersection there exists a unique morphism o from Xx(AAB) to (XxA)/-\(XxB) so that Y 9 Â°" = Yj and the following diagram cc mutes. Thus: -jm(Xx(AAB), Yi ) 1 ((XxA)n(XxB), Y? ) '2' PAGE 26 20 X*A Xx(A.^B) XxB Nov let (it , it ), (tt' it'), (o , p ) and (p' pp be the projections of XxY, Xx(AAB), XxA, and XxB respectively. Observe that: V-j " ^(V a)X l = Vl X l = *l (1 X Xh)X 2 'Vl X 2 V2 = 1r 2 (1 X XaU l = Sp 2 X l = ^2 (1 X Xb)X 2 = bp 2 X 2' Thus by the definition of intersection there exists a unique morphism Â£ from (XxA) A (XxB) to AAB such that r.E = ttÂ„yÂ„ and thus by the definition Â£ Â• 2 of product there exists a unique morphism Â£ from (XxA)A(Xx B) to Xx(AAft) such that Â£ = ; i.e., tt'^ = 1 yÂ« and ir^ = E. Now y,? = (lÂ„ x c)5 hence tTjY^ ri ^l^ = n l Y 2 and Tr 2 Y l ? = ClT 2 ? = cI = ^2' ThuS Y l^ = ''2' wheuc ((XxA)A (XxB), y) < (Xx(AriB), y) *it 2 (XxA)AXxB) PAGE 27 21 1.6. Notation. Let X and Y be t -objects and let (X*Y,7T ,it ) and (YxX,p ,p ) be the indicated products of X and Y. Then there exists a unique isomorphism from X*Y to YxX, denoted by ^ Â— ~ X Â»Â— 4> YxX I VtY Note that: XxY' 1.7. Definition. Let (R,j) be a relation from X to Y and let (t,j*) be the unique epi-extremal mono factorization of j (see 0.18). The codomain of x (domain of j*) is denoted by R1 and (R~ ,j*) is called the inverse relation of (R,j) or more simply, when there is little likelihood of confusion, the inverse of R. Â£/YxX 1.8. E xamp le. In the categories S3t , Topi , TopT , Grp , Ab, and FGp ti 1 >: X.xY Â— Â— Â— Â— Â— Â— -> YxX is defined by (x,y) = (y,x); hence, if (R,j) is a relation from X to Y then R _1 {(y,x): (x.y)Â£R} with j* tiie inclusion map of R -1 into YxX. 1.9. Jrx>p_osit_iop. Ii (R>j) is a relation from X to Y the R and R -i are iso^cr^hic obiects of f* . PAGE 28 22 Proo f . Since <7i2.ifi> is an isomorphism and j is an extremal monomorphism then ->YxX -*. 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: -po >Pi ><7T 2 j 71 ! > ~ ^-XxY (1*6) Â• Consequently, t;'/t is an isomorphism and j j#(t#t). Thus (R,j) I ( (R -1 ) _1 , j//) . 1.12. Proposition. Let (R,j) and (S,k) be relations from X to Y, Then (S,k) <_ (R,j) if and only if (S -1 s k*) < (R" 1 ,^*). Proof. Consider the following commutative diagram. i is (i. ,:'). A Ay A 1.17. Corollary . Let (R,j) be a relation on X, then (R,j) is reflexive on X if and only if (R ,j*) is reflexive on X. Proof. If (A x ,i x ) 1 (R,j) then (A x ,i x ) = (A x _1 ,i x *) < (R _1 , j*) (1.16 and 1.12). Conversely if (A x ,i x ) <_ (R _1 , j*) then ( A X>%) E (V 1 '^")! ((R" 1 )" 1 ,:?/) = (R,j) (1.16, 1.12. and 1.11). 1.18. Propo sition . Let (R,j) and (S,k) be relations from X to Y. Then the relations (RHS)" 1 and (PC l r\S~ l ) are isomorphic relations. Pr oof . According to the definitions of intersection and inverse relation we have the following commutative diagrams. > XxY Nt-. s-r > YxX
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25 RAS >=?Â» X*Y Â»^(RAS)1 * 1 |

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