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







(RxZ) (XxS) -- ---- XxYxZ -- ----> XxZ



cl.T



RoS




It is evident that T RRoS whence clT C RoS. But (RoS,j') is the

intersection of all closed subsets of XxZ through which Y factors.

Thus (RoS,j') C (clT,j). Hence (RoS,j') (clT,j).


1.35. Example. With the hypothesis of Example 1.34, T and clT do not

necessarily coincide.

Proof. Let X = Z be the closed unit interval with the usual subspace

topology induced from the real line. Let Y be the closed unit interval

with the discrete topology. Let R = {(x,y): y = x} considered as a

subspace of XxY. Let S = {(y,z): 0 < y < } considered as a subspace

of YxZ. It is easy to see that both R and S are closed in XxY and YxZ

respectively.

Clearly T = {(x,z): 0 < x < } and c3T = {(x,z): 0 < x < }

whence T clT.









1.36. Example. In the category Top, the composition of relations is
-2
not necessarily associative.

Proof. Let X = Z be the closed unit interval with the usual subspace

topology induced from the real line. Let Y be the closed unit interval

with the discrete topology. Let R = {(,)} considered as a subspace

of XxX. Let Sbe{(x,y): y = x} considered as a subspace of XxY and let

Tbe{(y,z): 0 < y < } where T is considered to be a subspace of YxZ.

Hence, each together with its inclusion map is a relation since each

of R, S, and T is a closed subspace of XxX, XxY, and YxZ respectively.

It follows that RoS = {(,)} and that (RoS)oT = 0. But

SoT = {(y,z): 0 < y < } and from this it follows that

Ro(SoT) = {(,z): zeZ}. Hence Ro(SoT) # (RoS)oT.


1.37. Remark. At first glance, the results of Examples 1.34, 1.35 and

1.36 seem to be pathological, thereby casting doubt on the usefulness

of the categorical definition of composition of relations (1.26). How-

ever, this should cause no more anxiety that1 does the fact that the

set theoretic union of two subgroups of a group is seldom a subgroup.

Furthermore, the results 1.31, 1.38, 1.39, 2.4, 3.1, 3.6, 3.9,

3.10, 3.12, 4.22, 5.20, 5.23, 5.25, 5.26, 5.27, 5.34, 6.13 and 6.27

seem to indicate that this definition (1.26) yields nice theorems which

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 2,' 1 >
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 = P1P> = P2"

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


2 1 p 1> 71P2 = I <7r2,17T>p1 = 72PP.

^72

1 30]p]> = '2P2pl> = 7T2<'r2,7I>PI = n1PI.




I 1 1 2' < -2I' 311 2 1 >(D 1 7 12 1 2 2T12*
" 3<13'T2,1> = 31 2 = 2 22.


i 02<2 1,T1 *> 2= 2 1 T" 2, 1> T<, T l*> f2< 2,']>7n2* = 1712*.



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


3 0222" ,7 i*> = 2T1*T<2',12f>72*,71*> 71= '

3n22 = 212 = 2


Thus 01
    p1 > = 1 and 02 = I2 <2, >>i7r*,71*> .


    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*p2*

    = J-PI*".


    Thus >(jx]) = (Ixj:)


    1<2' 2 I>*>(1xk) = < 2' > ,2*(1Xk) kp2 = k*T

    Il(k*xl) ,1> = k*pl

    = k*-T2.








    2 2<2 T *>(lxk) = 1*(1xk) = PI.

    2(k*xl) = 2 = ~1

    Thus <.T2*1*>(lxk) = (k*xi).

    Hence the diagram is commutative.

    Consequently by the definition of intersection, there exists a

    unique morphism & such that the following diagram commutes.


    Y










    It is easy to see that the following diagram commutes.

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




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





    S-1oR-1



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

    through which y factors, then (RoS, ) < (S oR ,a ).

    -1 -
    Hence there is some morphism p such that a'p = B. Consequently,

    ip=
    S = ts' = t t and the following diagram commutes.










    S- 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 B factors then ((RoS)-1,B*) < (S-loR-1,a').


    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

    fTrlj fr2 = f1r1jT^J Tr2j. Thus flrj*-T f72j*T so that

    since T is an epimorphism it follows that flj* = fj*. Hence, from the

    definition of equalizer, there exists a morphisnm from R-I to R for

    which jn = j*. This implies that (R-",j*) < (R,j) so that (R,j) is sym-

    metric.

    Consider the following products: (XxX,Tl,Tr2), (XXXxX,712,T 23),

    ((XxX)xX,plp2), (Xx(XxX),P1,p2), (RxX,1 *,T2*) and (XxR,I,i2). To see

    that (R,j) is transitive, consider the following commutative diagram.


    jx1X
    RxX >>---- (XxX)xX



    Y
    (RxX) ri (XxR) ------ ------------------- XxxxX




    XxR --------------- Xx(XxX)
    1Xxj


    Let (T#,j#) be the epi-extremal mono factorization of y.

    Recall that the codomain of T# (domain of j#) is RoR.

    Next, it will be shown that fiy = f-2y = fY-.

    y = f 302(]xj)X2 = fT22(1Xxj).2 = fT2ji2,2 = fljt2A2 =

    f-l 2(PX:<)X)2 = f 022(lXxj)'2 = f 2y.

    fly f= 10i(jx-lx)A f1pI1(jxlX)i f1 jTl*'/ =- f2J 1i*l

    fr2p1(jxly)XA = f 12il(jx1X)X1 = f 2y.

    Thus fT,;~i- 'i, 3>"' = fy = fTy fT 2<,3>y; so fTrj/T# fi#2








    Again, since T# is an epimorphism, it follows that firT3j = frr2j#. By

    the definition of equalizer there exists a morphisrm k from RoR to R for

    which jk = j#. This implies that (RoR,j#) < (R,j) so that (R,j) is tran-

    sitive.


    2.5. Theorem. The intersection of any finite family of congruences on

    any -object is a congruence.

    Proof. Let {(Ei,ei): iI)} be a finite family of congruences on X. Then

    there exist morphisms fi with domain X such that (Ei,ei) is the equal-

    izer of fini and fi72 (2.1). Let the codomain of each fi be denoted Yi.

    Consider the morphism from X to TTYi and consider
    iel isI
    the intersection ( ( Ei,e).
    ieI

    It will be shown that (f1 Eie) is the equalizer of 7rl and
    iel iel
    T .2'
    iel


    e 71 f i
    SEi-- XxX ~~_. X ---- "-- Y



    Xi ei fi V Pi

    Ei Yi
    ie



    First observe that: pjlne = fj e = f 72e = p r2e for
    ici isI
    each jIe. Thus 7Te 7 e.
    iel iel

    Now if g is a morphism from W to XxX such that rr2g
    irI ieI
    then fj7Tg = p 1g = PjTr2g = fj2g.Thus by che definition of
    iel iel
    equalizer there exist morphisms ki from W to Ei so that eiki = g for

    each ilI. Thus by the definition of intersection there exists a morphisn








    k from W to (\ Ei such that ek = g. This implies that (f Ei,e) is the
    ieI icI
    equalizer of 72.
    ieI ieI


    2.6. Proposition. If is complete then the intersection of any fam-

    ily of congruences on any -object is a congruence.

    Proof. Repeat the proof of 2.7 assuming I to be infinite.


    2.7. Proposition. Let Q be the family of all congruences on X and let

    (fQ,p) be the intersection of this family. Then ( O and Ax are isomor-

    phic relations on X.

    Proof. If (E,e)eQ then (Ee) is an equivalence relation and hence is

    reflexive (2.4). Thus (xX,ix) < (E,e). Hence (Ax,iX) < (r(Q,p). But

    (Ax,ix) is a congruence; hence (i'P,o) < (AX,iX).


    2.8. Proposition. Let f be a -morphis- from X to Y. Then f is a

    monomorphism if and only if AX and cong(f) are isomorphic relations on X.

    Proof. Since (cong(f),if) is an equivalence relation (2.4) it is reflex-

    ive and hence (Ax,iX) < (cong(f),if). If f is a monomorphism then

    fTlif = fn2if implies that "lif = T2if. Hence there exists a morphism k

    for which iXk = if and consequently (cong(f),if) < (Ax,iX).

    Conversely, suppose that (cong(f),if) (Ax,i') a and a and are

    morphisms having domain Z and codomain X such that fa = f. Consider

    the morphism from Z to XxX. fT = fa = f, = fa2 so that

    there exists a morphism X from Z to AX for which iX = . Thus

    a = 1:1 = nlijXX = 2 = B. Consequently a = 8 so that

    f is a monomorphism.


    2.9. Definition. A -morphism f from X to Y is said to be constant if
    ad only if for all pairs of mrphiss a
    aud only if for all pairs of morphisms Z TLT2ET^. X, fa = f6.








    2.10. Proposition. Let f be a morphism from X to Y. Then f is constant

    if and only if (cong(f),if) F (XxX,1X).

    Proof. If f is constant then fir = f2r so that fIrlx = f2X 1 Thus
    2-X 1 AA 2 XXX
    there exists a unique morphism k from XxX to corig(f) for which ifk = 1XxX

    whence if is a retraction. But since if is an equalizer, it must be an

    isomorphism (0. 4 ) so that (cong(f),if) and (XxX, 1XxX) are isomorphic

    relations on X.

    Conversely, suppose that (XxX,1XxX) (cong(f),if) and that a and

    6 ere morphisms with common domain,and codomain X. Consider from

    Z to XxX where Z is the common domain of a and B. Since

    falXx, = f21X, it follows that fir = fT2 so that

    fa = fT1 = fi2 = fB. Thus f is a constant morphism.


    2.11. Proposition. If f from X to Y, g from Z to Y, and h from X to Z

    are ( -morphisms such that f = gh then (cong(h),ih) < (cong(f),ir).

    Furthermore if g is a monomorphism then (cong(h),ih) (cong(f),if).

    Proof. Since hrlih = h 2ih it follows that ghrlih = ghT2ih so that

    fTlih = fT2ih. Thus there exists a morphism k from cong(h) to cong(f)

    for which ifk = ih. Whence (cong(h),ih) < (cong(f),if).

    If g is a monomorphism then f'lif = fn2if = ghl1if = ghr2if

    implies that hTlif = hi 2if. Thus there exists a morphism k* from cong(f)

    to cong(h) for which ihk* = i,, whence (cong(f),if) < (cong(h),ih). Con-

    sequently (cong(f),if) = (cong(h),ih).


    2.12. Proposition. If ( has coequalizers and f is a I -morphism from

    X to Y and if (f*,Z) is the coequalizer of Tiif and r2if then cong(f)

    and cong(f*) are isomorphic relations on X.

    Proof. Since fnif fiT2if then by the definition of coequalizer there









    exists a morphism k* from Z to Y for which k'f* = f. Since

    f*li = f2if it follows that flif = k*f*rri = k*f*if =

    fr 2i ,. Thus there exists a morphism k from cong(f*) to cong(f) for

    which i k = i Consequently (cong(f*),if,) : (cong(f)i ).
    f f C f
    Now since f* is the coequalizer of r if and i2 i then

    f*7lr i f2r i Hence there exists a morphism k' from cong(f) to

    cong(f*) for which i fk' = i Consequently (cong(f),if) <

    (cong(f*),if,).


    2.13. Proposition. If is complete and 0 is a family of congruences

    on X generated by morphisms f: X ---->Y and if (,r1,p) (A ,iX) then

    the unique morphism 0 from X to TT Y such that rfO = f, is a monomor-

    phism.

    Proof. Observe that for each f, fir i = T~f, r i = 2i = f i 0. Thus
    -- 10 r 2f 10f
    f
    it follows that (cong(),i ) < (cong(f),if) for all X f--- Hence
    (cong(0),i ) < ((n,p) (1.19). Since ((),p) (A,iX) < (cong(0),i)

    (2.4) it follows that (f ,p) (A ,iX) = (cong(O),i). Thus 0 is a

    monomorphism (2.8).


    2.14. Corollary. If is complete and 0 is a family of congruences

    on X generated by morphisms f: X--- Y and for some g: X --- Yg,
    f 0
    g is a monomorphism, then the unique morphism 0 from X to TT Yf such

    that f 0 = f is a monomorphism.

    Proof. Since g is a monomorphism then (cong(g),i ) = (A ,iX) (2.8). Thus
    g X X
    (C0I,p) < (A,i X) by the definition of intersection. But

    (AX,iX) < (f[I,p) (2.4 and 2.6). Consequently (6X.iX) = ((In,p) and the

    result follows from Proposition 2.13.















    SECTION 3. CATEGORICAL EQUIVALENCE
    RELATIONS AND QUASI-EQUIVALENCE RELATIONS


    3.1. Theorem. If {(E.,4 ): icl} is a family of equivalence relations on

    a -object X then their intersection (rf E.,4) is an equivalence rela-
    iel
    tion on X.

    Proof. Since (AX,i ) < (E., .) for each iel it follows that
    X k 1 1
    (A ,i ) < ( Ei.4) (1.19). Hence (\n E.,<) is reflexive.
    iel iel
    Since (fl E.,4) < (E.,V.) for each iel and since each (E.,i.) is
    isl
    symmetric it follows that (( \ Ei)-l,*) < (E -1, i*) < (E.,,.) for each
    I 1 1 1
    iEI
    isl (1.12). Thus ((ft E.)-1,4*) < (r\ Ei,4) and hence (r Ei,r) is sym-
    ilI iel iel
    metric.

    Since ((~l Ei,) < (Ei,4i) for each ilI then
    iEI
    ((C E.)o(R E.),#/) < (E.oE'.,4#) for each ilI (1.30). And since
    idI icI
    (EioEi',i#) < (Ei,4i) for each icI it follows that

    ((f, Ei)o( ( Ei),#) < (fn E. ,) (1.19) whence (f Ei, ) is transitive.
    ieI iel iIe iil
    Thus it is an equivalence relation.


    3.2. Definition. A quasi-equivalence (R,j) ou X is a relation on X which

    is both symmetric and transitive.

    This term is due to Riguet [221 ; however, Lambek [13] calls

    symmetric transitive relations subcongruences. Wiile this term sub-

    congruence is appropriate in the categories Grp and Ab, it does not seen

    co be appropriate in more general categories. MacTane I calls such

    relation .s sy1-Jcric idemporents.








    3.3. Proposition. If (A,a) is an extremal subobject of X then (AxA,axa)

    is a quasi-equivalence on X.

    Proof. Consider the products (AxA,pl,p2) and (XxX,Tr', T2). Since a is an

    extremal monomorphism then axa is an extremal monomorphism (0.20) and

    hence (AxA,axa) is a relation on X.

    Consider the following commutative diagram.


    axa < il ,Il>
    AxA >---- ---- XxX >-- XxX



    T (axa)*



    (AxA)-1


    Since 7l <;T2',~>(axa) = 2(axa) = ap2;

    ir (axa) = ap1 = ap2;

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

    ,2(axa) = ap2 = apl

    then it follows that (axa) = (axa). But is an iso-

    morphism hence an epimorphisn and axa is an extremal monomorphism;thus by

    the uniqueness of the epi-extremal mono factorization of <72',71>(axa)

    (0.13), ((AxA)- ,(axa)*) = (AxA,axa). Thus (AxA,axa) is symmetric.

    To see that (AxA,axa) is transitive, first consider

    (((AxA)xX)tA(Xx(AxA)),y) where y is the unique extremal monomorphism

    induced by the indicated intersection. It will next be shown that

    (AxAxA,axaxa) and (((AxA)xX),1 (Xx(A';A)),y) are isomorphic as extremal sub-

    objects of XxxxX. To show this it will be shown that (AxAxA,axaxa) is

    precisely tha intersection of ((AxA)xX,0 ((axa)xlX)) and

    (X:-(A,-A),02 (l x (ax;a)).







    Consider the products: (XxXxX, 1, 2, 3), (Xx(AxA),plp2),

    ((AxA)xX,I, 2), (Xx(XxX),l '*r *), ((XxX)xX, 11 2*) and

    (AxAxA,p 1,p2'p3

    Observe the following qualities.

    S1] ((axa)x) 1,> = ((axa)xl))<

    ,a3 > =

    1T(axa)1P'P2> ,ap3> = l (axa)2

    ap1 = ap1 = (axaxa).


    T2 1((axa)xl x)<

    ,ap3 >






    3, 1((axa)xl x

    ,ap3>




    102(lx (axa))>




    T202 (2Xx(axa))>






    r302 (x( (axa))>


    -= 2TrI((axa)xl )<

    ,a 3>

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

    ap2 = T2(axaxa).


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

    1XP2<'ap3> = lXap3 = ap3 = '3(axaxa).

    = Ir*(1Xx(axa))
    1Xp = 1 ap1 = aP1 = l(a>aaxa).

    SiTr2*(1Xx(axa))
    7l(axa)p2> = l(axa) =

    ap1 = ap2 = T2(axaxa).


    = T2'r2*(lx(axa))> =

    2(axa) = ao2 = ap3
    77,(axaxa).
    .3


    Thus by the definition of product the following diagram conmlutes.









    (axa)xlX
    (AxA) xX ---------- (XxX) xX

    <<'1 2>,ay axaxa
    AxAxA ,---------------- ----------- XxXxX


    ,P23 Xx (AxA) -------- Xx(XxX)
    lxx(axa)


    Now, if (W,6) is a subobject of XxXxX so that there exist mor-

    phisms y and y such that 0 ((axa)xl )y = 6 = (1 Xx(axa))y then con-
    1 2 1 X i 2 X 2
    side the morphism

    = 2 from W to AxAxA. It will
    1 1 1 2 1 1 2 2 2
    be shown that << ,p >,ap >E = y and > = y
    1 2 3 1 1 2 3 2
    Since p1< ,1'p2> = 1T = pi1Y1 and p2< 1'P2>i = 2 = p2P1Y1 it

    follows that P1<,a3 >E = S P= "

    Now since G((axa)xl1X)Y = 02(1Xx(axa))y2 it follows that

    P2y = f2*((axa)xl )y = 30 1((axa)xlX)y, = ,3 02(l x(axa))y2

    S22*(I (axa))y2 = T2(axa) 2y2 = ap2a2y2.

    ILhence p2 1 p2<<1p,p2>,ap3> = ap3& = aP2P2Y2. Thus

    <

    ,ap >E = y .
    1 2 3 1
    Again since Q ((axa):.l ) = 0 (1 x(axa))y it follows that
    X 2 X 2
    P1 2 1= *(Xx(axa))y2 = 2 0 (1 X(axa))y2 1 11((axa)x1 )yX =

    T Ift 1*((axa)xl )y = Ir (axa)pl = ap

    Hence pl> = ap a = ap 1 Y= 1
    1 3> 1 111 1 2
    Since p 2 P = p2 = P ahd p2P2 = = P2Y 2 2 it

    follows that F p y2. Hence p > =

    = Y
    2 3 2 2 2 3 2 3" 2
    Consequently 1 2 3 2
    From this it follows that (axaxa) = 6. Since (axaxa) is a mono-

    morphism the morphism & is unique. Thus it has been shown that

    (AxAxA,axaxa) is the intersection of (Xx(AxA) ,2 (1 x(axa))) and








    ((AxA)xX,01((axa)x]X)). It next will be shown that the following diagram

    commutes.


    axaxa < Tr1,3
    AxAxA ------ XxXxX --------------- XxX



    -

    "'AxAA A

    (Ti,j#) is the epi-extremal mono factorization of (axaxa).

    Now r1(axa) = ap1 api = 71(axaxa) and

    [2(axa) = ap2< i,P3> = ap3 = 72<1 T, 3>(axaxa). Thus the above

    diagram commutes..

    Since ((AxA)o(AxA),j#) is the intersection of all extremal sub-

    objects through which 3>(axaxa) factors then

    ((AxA)o(AxA),j#) < (AxA,axa)

    whence transitivity is obtained.


    3.4. Canonical Embedding. Let (R,j) be a relation on X. Let (Tl,jIl

    (c2,J2), and (-3,j3) be the epi-extremal mono factorizations of T1j,

    21j, and T2j* respectively. Let RX, XR. and XR-1 denote the domains of

    ji, j2, and j3 (codomains of TIp2 and T3) respectively.







    R -------------- -----> xX ----------- X
    TI
    TTr

    j' "X2 3
    R --------- --------c- XxX -------------- X








    J 12
    R ---------- XXX X



    SXR




    In the categories Set, FGp, Grp, Ab, (RX,j ) may be taken to be

    the set {xcX: there exists ycX such that (x,y)ER} together with the in-

    clusion map. Similarly, in these same categories, (XR,j,) may be taken

    to be the set {ycX: there exists xcX such that (x,y)eR} together with

    the inclusion map and (XR-,j 3) may be taken to be the set

    {xcX: there exists yCX such that (y,x)cR-1} together with the inclusion

    map.

    In the categories Top1, and Top2 the extremal subobjects (RX,j ),

    (XR,j2), and (XR-,j 3) of X have precisely the same underlying sets as

    above endowed with the subspace topology induced by the topology of X.

    See Section 4 (4.1, 4.2, and 4.3) for a more detailed discussion.

    It is easy to see that in the category Set, a symmetric, transi-

    tive relation on a set X is an equivalence relation on a subset of X.

    Recall the discussion in Section 2 (2.0) of the remarks of Lambek who

    obtains the similar result for homomorphic relations on rings with

    identity. This result we wish to generalize. In order to do thils we must

    first be able to pick out the subobject.of X on which the relation is an

    equivalence relation.

    Referring to the above diagrams, since T is an isomorphism (1.9)

    and sialce the eui-extremal mono factorizaticr is unique (0.18) it is

    clear that (RX,j1) = (XR-1,j3). That is, there exists an isomorphism k

    from RX to XR-1 such that j3k = jl (se,- 4.4 and 4.5).








    Consider the product (RXxXR,l,' 2). Also, consider the morphism

    (jlxj2)< r1'2> from R to XxX. Since

    7i(jlxj2)<9lT2> JlP1 = JiTI = 2Tij and

    ^T2( 1xj2) = .222< 1,T2> = j2T2 = 72j it follows from the defini-

    tion of product that (jlxj2) = j. Note, is an extremal

    monomrorphism since j is an extremal monomorphisr (0.16).

    Now suppose that (R,j) is symmetric on X. Then it follows that

    there exists an isomorphism a so that ja = j* (1.13). Thus 72ja = 7T2J*

    and this together with the fact that a is an isomorphism and the unique-

    ness of the epi-extremal mono factorization implies that

    (XR,j2) (XR-l,j3). That is, there exists an isomorphism B so that

    J2 J3. Thus it is routine to see that the following diagram commutes.



    T1





    a XXX X
    R 1-.- XR









    2 12
    SXxX --------------------- X



    R 1 --------- R





    R i








    Consider the following products: (XRxXR,pI,p2) and (XxX,~1I ,2).

    Letting 4 = (kkxlxR) then (2j2x2) = j, since

    11(j2xj2)p = J2P1 = j2BklP = j2BkTl = J1lT = IrTj and

    2(J2xj2) = J2o02 j21XRP2 = 2T2 = 2J. Thus the following

    diagram commutes and the relation (R,i) on XR shall be called the

    canonical embedding of R into XRxXR.




    R *-RXxXR


    Skx1XR

    XRxXR


    I j2xj2

    XxX




    3.5. Lemma. Let (R,j) be a symmetric relation on X. Then (R,i) is a

    symmetric relation on XR.

    Proof. Suppose that (R,j) is symmetric on X and let (R,4*) be the inverse

    of (R.~) on XR. Then Y = i*T*. It is easy to verify that

    p. Thus since (R-l,j*) is the intersection of

    all extremal subobjects through which <72,r1>j factu-rs there exists a

    morphimn X from R-1 to R so that (j2xj2)W*X = j*. But j*'T = <2,7T1.>j

    whence (j2xj22)T* = (j2xj2) = <2,T>j = j*T. So

    j*'T = (1(j2xj2) ) ((2xj2)) *)T*. Since (j2xj2)i* is a mDonomorphism

    it follows that XT = T*. Recall that T and T*- both are isomorphisms (1.9).

    Rence .\ is an isomorphism.








    Recall that by the definition of symmetry (1.10), there exists a

    morphism c from R-1 to R so that ja = j*. Thus

    (j2xj2),"-1 = j 1 = j j= (j2xj2)~. But since (j2xj2) is a monomor-

    phism this implies that *Xca- = Thus since X and a are isomorphisms,

    we have (R,4*) E (R,4). Hence (R,i) is symmetric on XR.


    3.6. Lemma. If (R,j) is a quasi-equivalence on X then (R,i) is a

    quasi-equivalence on XR.

    Proof. In view of Lemma 3.5 it need only be shown that (R,p) is transi-

    tive on XR. To that end first consider the following diagram. It will

    be shown that there exists a morphism X such that the diagram commutes.




    Ox1XR
    RxXR ------------ (XRxXR)xXR


    xP 01 -
    II6 "-
    S(RxXR) (XRxR) XRXPRxXR -------- XRxXR

    2 } x2
    XRxR ,------ >XRx (XRXR) j 2xj2

    X l J2xj2xj2


    RxX :-- X -- (XXX) xx


    SI
    S(RxX) t, (Xx


    XxR -- -------- Xx (XxX)
    1xj


    .--------..- XxX








    Clearly <, 3>(j2xj2xj2) = (j2xj2)<1 ,P3>. Also,

    01 (jxX) (1Rxj2)1 01 = j01 ((j2xj2) jxj2)X =1

    O ((j2xj2)xj2)(x1XRlX) = (j2xJ2xj2)61(x1XR)Xl and

    02(1Xxj)(j 2x1) )2 = 2(j2xj)X2 = 02(j2x(j2x, ) )2 =

    O2(j2x(j2xj2))(IXRpx)X2 (j2xj2xj2)02(lXRX1)X2

    as can be verified in a straightforward manner. Thus the diagram above

    is commutative, and in particular,

    ol(jx1x)(1Rxj2)l = (j2xj2xj2)' = 92(1Xxj)(j2x1R)x2. Hence, by the

    definition of intersection there exists a unique morphism X such that

    yX = (j2xj2xj2)6.

    Let (RoR,i#) denote the composition of (R,p) with (R,u) on XR. Let

    (RoR,j') be the composition of (R,j) with (R,j) on X. Then

    6 !T# where T# is an epimorphism and <1l, 3>y = j'T' where r'

    is an epimorphism. But since yX = (j2xj2xj2)6, it follows that

    <'1f,3>yX = (2xj2)6 so that (j2xj2)0#T# = j'T'A. Hence the fol-

    lowing diagram commutes.




    (RxXR) C (XRxR) -> RoR




    (RxX) n (XxR) XRxXR


    T' j2Xj2

    RoR -----------------'> XxX
    1







    Since T# is an epimorphism and j' is an extremal monomorphism, by

    the diagonalizing property (0.19) there exists a unique morphism & such

    that j'S = (j2xji2)# and CT# = T'X. But this says that

    (RoR,(j2xj2)i#) < (RoR,j'). Since (R,j) is transitive (RoR,j') < (R,j)

    hence (RoR,(j2xj 2)#) < (R,j) =: (R,(j2xj2)i). Hence there exists a mor-

    phism o from RoR to R such that (j2xj2)io = (j2xj2)y1#. Again, j2xj2 is a

    monomorphism so that ipo = p# which says that (RoR,4#I) < (R,y) hence

    (R,y) is transitive.


    3.7. Theorem. If (R,j) is a relation from X to Y and 7rlj is an epi-

    morphism then (RoR-l,j#) is reflexive on X.

    Proof. It will first be shown that the following diagram commutes.




    RxX -------- ---> (XxY) xX
    <1RKlj> 01


    R -- -- > (RxX) r (XxR-l) 1 ;- XXYxX


    < 2

    XxR -------------------- Xx(YxX)




    Consider the following products: (XxY,7Tl,r2), (XxYxX,T~1,7r2, 73),

    (RxX,p1,o2), (XxR-l,pl*,p2*), ((XxY)xX,l*,'2*), (Xx(YxX), 1i 2.),

    (YxX, 1, 12), and (XxX,-T1, 2).

    Now,

    1 (jxlX)= il1*(jxlx) r ljPl
    2O1(jxl) = T27Tl*'(jx = 2" -

    730 (jxl)1R, 1R ,3IJ> = T2*(jxlx)<1R,'1>J = ; 2< R iP> = Tr13








    102(iXxj) = iI(Ixxj*)> = P1*Tj,T> = Tlj.

    72 2(1XxjT)<1]jT> = (T 2(IXXj*) = ,IJ*j * =

    ^Tfj*' = 11j = r2J

    B302(1Xxj*)<3)j.T> = 722(iXxj*)<' .,T> = 22j*T = 2<2,lI>j = 1J"

    Thus by the definition of pr-'uct the diagram commutes. Hence

    there exists a morphism E so that X1E = <1R,'Ij> and X2E = .

    From the above it is easy to see that

    1l< ,1T3>YE = TlY = 1Tlj = T = 3Y 2YE. Recall that (Ax,iX) is

    the equalizer of T1 and T2 hence there exists a morphism such that

    ix = <11l,73>YE.

    Let (RoR ,j#) be the indicated composition of relations and let

    T# denote that epimorphism for which j#T#/ = y. Thus, combining

    the above results, <7Tij,>Tj> = YE -= j#T#E = iX.

    Since (Ax,ix) and (X,) are isomorphic as extremal subobjects

    of XxX (1.21), there exists an isomorphism X such that <1,1x>X = iX.

    Consequently, Xt = iX = Y = .

    Now 7ITX = l XX = X} = Xrl<7T J,7 rj> = 7rlj and by hypothesis

    Trlj is an epimorphism; thus, since is an isomorphism,it follows that

    ( must be an epimorphism.

    Thus has ((,iX) as its epi-extremal mono factcrization.

    But this means that (Ax,ix) is the intersection of all extremal subob-

    j.cts of XxX through which factors (0.21). Recall that

    = j#t#/E, thus (A ,iv) < (RoR- ,ji#) which was to be proved.


    3.8. Corollary. If (R,j) is symmetric on X then (RoR,ipV), the composi-

    tion of (R,i) -,ith (R,.) on XR, is reflexive on XR.

    Proof. Since (R,j) is symmetric on X then (Ri) is symmetric on XR (3.5)

    hence (O,,) 1 (R-1. :*) (1.13). Referring to the diagram in (3.41) fol-








    lowing the c0 'inition of the canonical embedding it is immediate that

    pl is an e; -rphism since plI = Bkt- and each of B, k, and T1 is an
    -1 ,.
    epimorphisr. Thus (RoR ,1) (RoR,/#) is reflexive on XR (3.7).


    3.9. Corollary. If (R,j) is a quasi-equivalence on X then (R,j) is an

    equivalence relation if and only if lrlj is an epimorphism (respectively

    if and only if r2j is an epimorphism).

    Proof. If (R,j) is an equivalence relation then (R,j) is reflexive and a

    quasi-equivalence. Thus by Proposition 1.24, 7rj and T2j are retractions

    hence epimorphisms.

    Conversely, if 1lj is an epimorphism then applying the theorem
    -1
    and Proposition 1.30, (A ,iX) < (RoR ,j#) < (RoR,j') < (R,j) so that

    (R,j) is reflexive and hence is an equivalence relation. (If T2j is an
    -1 -1 -1 -1
    epimorphism then (Ax,i ) < (R oR,j#*) < (R oR ,j'*) < (R ,j*) and
    -1
    (R ,j*) E (R,j).)


    3.10. Corollary. If (R,j) is-a quasi-equivalence on X then (Rd:) is an

    equivalence relation on XR.

    Proof. (R,i) is a quasi-equivalence on XR (3.6) and (RoR,ijI#) is reflexive

    on XR (3.8). Thus (AXR, XR) < (RoR,y#) < (R,w) whence (R,',) is reflexive

    and thus is an equivalence relation on XR.


    3.11. Proposition. If (R,j) is a quasi-equivalence on X then (R,j) and

    (RoR,j') are isomorphic relations on X.

    Proof. By Corollary 3.10 (R,i) is an equivalence relation on XR whence

    (RoR,ir#) and (R,,) are isomorphic relations on XR (1.32). Recall that

    there exists a mcrphism E such that the following diagram commutes (3.6).








    (RxXR)ri (XRxR) --------------;> RoR


    T'X ( 2Xj 2)


    RoR -;- XxX
    j'


    Thus (RoR,(j2xj2)i#) < (RoR,j'). But as mentioned above

    (RoR,i#) (R,t) hence there exists an isomorphism A# such that

    I^,#jt = So by the definition of the canonical embedding (3.4),

    j'gX# = (j2xj2) #lx = (j2xj2)y = j. But this implies that
    (R,j) < (RoR,j'). Thus since (R,j) is transitive, (R,j) (RoR,j')

    which was to be proved.


    3.12. Proposition. Let (R,j) be a relation on X. Then (R,j) < (AX,iX)

    if and only if R is symmetric on X and (R,i) < (AXR,iXP).

    Proof. If (R,j) < (AX,iX) then there exists a morphism a such that

    j = iXa. Thus iTl = rliXa = T2ixa = 72j whence

    n <,f2,Tl>j = T2j = IrFj = Tr2<'r2,Ttl>j. Thus by the definition of product

    j = j. Consequently the epi-extremal mono factorization of

    j is (1R,j) and so (R,j) (R-1,j*); i.e., (R,j) is symmetric.

    Recall that j = (j2xj2)d (3.4). Thus

    lj FTl(j2Xj2)' = j2p14 and 72j = 2(j2xj2)) = j2P2P'. But "lj = '2j
    hence j2P = j2P2'. Since j2 is a monomorphism it follows that

    pl -= p21,'. Recall that (AXR,iXR) is the equalizer of pI and p2. Hence
    there exists a morphism; B such that iXR5 = 4j'. This implies that

    (R,,) < (AXRiXR))

    Conversely, if R is symmetric and (R,') < ('XR,iXR) then there

    exists a morphism 8 such that d = ixRP; hence








    P1 = P liXR = P2iXRB = P2. Since (j2xj2) = j, we have

    1 j r "Q(j2xj2) = j 2P1 = J2P2 7T 2'(j 2xj 2) = F2j. Thus r1j = Tr2j

    so that there exists a morphism a such that j = iX&. This means that

    (R,j) <. (Ax, ix).


    3.13. Definition. Let (R,j) be a relation on X. Then R is said to be

    a circular relation if and only if RoR < R-1

    This notion is due to MacLane and Birkhoff [20] exercisee 3,

    page 14).


    3.14. Proposition. Let (R,j) be a relation on X. Then R is a circular

    relation if and only if R1 is a circular relation.

    Proof. If R is circular then RoR < R-1. Thus

    R-loR-1 = (RoR)-1 < (R-1)-1 R (1.38, 1.;2 and 1.11). Hence R-1 is

    circular.

    Conversely, if R1 is circular then by the above, (R-1)'1 =R is

    circular.


    3.15. Theorem. Let (R,j) be a relation on X. Then R is an equivalence

    relation on X if and only if R is reflexive and circular.

    Proof. If R is an equivalence relation then R is reflexive. Since R is

    transitive and symmetric, RoRR R-' hence R is circular.

    Conversely, if R is reflexive and circular then R-1 is reflexive

    (1.17) and R-1 is circular (3.14). Hence

    R R'1oA.. < '-loR1 < R (1.31 and 1,12) whence R is symmetric. Thus

    R R-1 (1.11).

    Now RoR R-1oR-1 < R hence R is transitive. Thus R is an equiva-

    lence relation.














    SECTION 4. IMAGES


    4.1. Definition. Let (R,j) be a relation from X to Y and let (A,a) and

    (B,b) be extremal subobjects of X and Y respectively. Consider

    (Rir(AxY),y) and (Rfi(XxB),6). Let (T ,a) and (? .2) be the epi-extremil
    1 2
    mono factorizations of i y and r 6 respectively. Denote the domain of a
    2 1
    by AR and the domain of B by RB. Thus the following diagrams commute.



    R


    ^ Y 2


    axly
    Rt(Axy) ----> Xxy c Y

    2 AXY ARIrw,




    R




    R 1 '(XxB) >-.------ -- XxY -----> X




    XWX ^ RB



    4.2. Remark. Since (X,1 ) and (Y,I ) are extremal subobjects of X and Y
    X Y
    respectively, then (R '(XxY),y) (R,j) and (RA (XxY),6) 5 (R, j) whence

    (XR,&) is precisely the extremal subobject (XR.,j) used in the canonical

    nQboddiin (3.-'). Since X = Y in 3.4 then also (RY,3) is precisely








    (RX,j ) used inl 3.4.


    4.3. Example. In the category Set, for (A,a) < (X,1 ), (B,b) < (Y,1 )

    and (R,j) < (XxY,1 ),
    XxY
    AR j{yY: tL exists acA such that (a,y)ER}

    RB {xeX: there exists bcB such that (x,b)eR}.

    This is easily seen since R(\(AxY) L {(a,y): acA, (a,y)eR} and

    R(I(XxB) {(x,b): beB, (x,b)eR}, and AR is the set of all second terms

    of elements of R((AxY) and RB is the set of all first terms of elements

    of Rf(XxB).

    In the category Top AR and RB have precisely the same underlying

    sets as above. They are endowed with the subspace topology determined by

    the topology of XxY.

    In the category Top2, AR and RB have" precisely the same underlying

    sets as in Top for it is easy to verify that AR and RB are closed sub-

    sets of X and Y respectively. Recall that the image of a morphism in Top

    is the closure of the set theoretic image (0.15).


    4.4. Theorem. If (R,j) is a relation from X to Y and (A,a) is an extremal

    subobject of X then (AR,a) and (R-1A,B) are isonorphic extremal subobjects

    of Y.

    Proof. Consider the following commutative diagrams.


    R

    A


    RAxY) -------------- XxY --- -------------Y


    AxY _axl,,










    R1





    6 iT
    R- 1 (YxA) ------- -- YxX -'--Y--

    1 xa

    Y
    YxA RA



    j <7f iT>
    j R ------- Xx ------ YxX




    R-1



    It can be shown in a straightforward manner that

    (ax] ) = (1 xa)


    2 1 Y Y 21
    where pl and p2 are the projections of AxY. Hence

    (1 xa)

    = (axl )X = y = j1 = j* .
    Y 12 1 2 2 1 Y2 2 1 2 1 1 1
    Thus by the definition of intersection there exists a morphism 5 such

    that SS = <2 ,Tr >Y = j*TX (1 xa)

    X Hence
    21 1 Y 2'1 2
    f 6 = f Y = y = ai But fi 6 = Bi C. Thus, since (AR,a) is
    1 1 21 2 1 1 2
    the intersection of all extremal subobjects through which T.2y factors

    (0.21), it follows that (AR,o) < (R-1A,B).

    Similarly, it follows that

    < >2'1 *X = -> 16 = j (axl,])2,-X whence -h4rc
    <2' 1 2 1 3 (ax] )

    exists a moiphisnm such that y'* = -16. Then

    2 < ,T7 >-16 = f 6 = ST2 = y* = aT ;*. Again, since (R1A, 8) is the
    ntrscton f all exremal subobjecs 2
    intersection of all extremal suhobjects through which h ,1 6 factors,
    1







    (R-IA,B) < (AR,a). Consequently (R-IA,B) E (AR,a).


    4.5. Corollary. If (R,j) is a relation from X to Y and (B,b) is an extre-

    mal subobject of Y then (RB,B) and (BR-1,a) are isomorphic as extremal

    subobjects of X.

    Proof. Recall ((R-1)-1,j#) E (R,j) (1.11). Letting (R-l,j*) play the role

    of (R,j) and (E,b) the role of (A,a) in the theorem, the following is

    obtained: (BR-1,Q) ((R-)-1B,B/#) E (RB,B).


    4.6. Corollary. If (R,j) is a symmetric relation on X and (A,a) is an

    extremal subobject of X then (AR,a) and (RA,B) are isomorphic as extremal

    subobjects of X. (In particular, (XR,j ) and (RX,j ) are isomorphic as

    extremal subobjects of X as was shown directly in 3.4.)

    Proof. Recall that (R-1,j*) E (R,j) (1.13). Hence by the theorem

    (AR,a) (R-1A,B) = (RA, ).


    4.7. Proposition. Let (Al,al) and (A2,a2) be extremal subobjects of X and

    (R,j) be a relation from X to Y. If (Al,al) < (A2,a2) then

    (A1R,al) < (A2R,a2).

    Proof. By hypothesis there exists a morphism v so that a2P = a1. Thus,

    there exists a morphism ( such that the following diagram commutes.




    R('(A1xY) Y xy
    A AlXYA axl xy

    1 -Y 2
    R (A,2xy) ------. .. .-- ..-X XxY








    Thus w2Y1 = i2Y25 whence, because (A1R,al) is the intersection of

    all extremal subobjects through which T2rY1 factors and r2y2C factors

    through (A2R,a2), (AIR,al) < (A2R,a2) which was to be proved.


    4.8. Proposition. Let (Bl,bl) and (B2,b2) be extremal subobjects of Y and

    (R,j) be a relation from X to Y. If (Bl,bl) < (B2,b2) then

    (RBi,B1) < (RB2,B2).

    Proof. (RB1 ) 1*)E (B1R(-1,12) < (B212) (RB22) (4.5 and 4.7).


    4.9. Proposition. Let (R,j) and (S,k) be relations from X to Y and (A,a)

    be an extremal subobject of X. If (R,j) < (S,k) then (AR,a) < (AS,a).

    Proof. In a manner similar to that in the proof of 4.7 one can establish

    the existence of a morphism $ such that the following diagram commutes.


    R ---------- S

    j t k


    RTA(AxY) S A(axy) --- Xxy


    axi

    Axy


    Hence the following diagram commutes.


    R /(Axy)




    S 1 (AxY) ---------- XxY



    Thus, since (AR,a) is the intersection of all extremal subobjects

    through which yT, factors, and 2 factors through (AS,a), it follows that

    (AR,cL) < (AS,5) which was to be proved.







    4.10. Proposition. Let (R,j) and (S,k) be relations from X to Y and (B,b)

    be an extremal subobject of Y. If (R,j) < (Sk) then

    (RB,B) < (SBj).
    Proof. (RB,8) (BR-1,B*) < (BS-1,~*) E (SB,8) (4.5, 1.12, and 4.9).


    4.11. Proposition. Let (R,j) be a relation from X to Y and let (Al,al)

    and (A2,a2) be extremal subobjects of X. Then

    ((AlfA^R:A2 ) <_ (AIRAA2R,5).

    Proof. Since (AljnA2,a) < (A1,al) and (A1CfA2,a) < (A2,a2) it follows

    that ((A,! A2)R,a) < (A1R,a1) and ((A(fA2)R,a) < (A2R,a2) (4.7). Thus

    ((AiCA2)R,a) < (A1RflA2R,Q) (1.19).

    4.12. Proposition. Let (R,j) be a relation from X to Y and let (B.,bl)

    and (B2,b2) be extremal subobjects of Y. Then

    (P.(B1^ B2),B) < (RB1 nRB2,B).

    Proof. (R(B nB2),B) ((B B2 )R-1,B*) < (BiR-lnB R-1,*)

    (RB1nRB2,B) (4.5 and 4.11).


    4.13. Proposition. Let (RI,jl) and (R2,32) be relations from X to Y and

    let (A,a) be an extremal subobject of X. Then

    (A(R1AR2),a) < (AR,( AR2, ).

    Proof. It is clear that there exist morphisms 1, and C2 such that the

    following diagram commutes.


    R1 N(AxY) "Y


    (r.,fl R2) (AxY) )x ----> xxY



    'R2-A (Axy)




    69


    Thus rr2y = 1r2Yl1 = T2Y2 2. Again since (A(R1(\R2),a) is the inter-

    section of all extreral subobjects through which ir2y factors it follows

    that (A(R(.R2) ,) < (AR1,51) and (A(R1 FR2),a) (AR2,a2). Hence

    (A(RI AR2),a) < (AR1 AAR2, ) (1.19).


    4.14. Proposition. Let (R1,jl) and (R2,j2) be relations from X to Y and

    let (B,b) be an extremal subobject of Y. Then

    ((R AR )B, ) < (RB 2R B,B).
    % 1 2 1 2
    Proof. ((RflR2)B,6) E (B(RI.R2))-1, *) < (BR3-~I BR-1 ,1) =
    1 2 1 2 i 2
    (R1 BR 2B,6) (4.5 and 4.13).


    4.15. Proposition. Let (R,j) be a relation from X to Y then (R,j) and

    (R1 (RYxY),y) are isomorphic as extremal subobjects of Xxy.

    Proof. Consider the following commutative diagrams.



    R 1) > Xxy
    RY ------ XxY

    T 1 T 1

    RY ---

    ji

    RYxY ,

    - R fA(RY7Y) :>--------------------------->^ XxY




    R



    Since (jlxl ) = = j, there exists a morphism E

    such that yv =- j
    R








    RYxy
    .^^.. yxY YXiy

    R n---( R ((RYxY) XXY


    P, R


    Thus (R,j) < (R n(RYxy),y). Clearly the reverse inequality holds

    so that (R,j) E (R(T(RYxY),y).


    4.16. Proposition. Let (R,j) be a relation from X to Y. Then (R,j) and

    (Rr%(XxXR),6) are isomorphic relations from X to Y.

    Proof. Analogous to the proof of 4.15.


    4.17. Corollary. Let (R,j) be a relation from X to Y. Then (R,j) and

    (Rt'(RY^-XR),6) are isomorphic relations from X to Y.

    Proof. (R.j) : (RO(RYxY),v) E (R(l(XxXR),'6) (4.15 and 4.16). But since

    (RY,jl) and (XR,j2) are extremal subobjects of X and Y respectively it

    folJows that ((RYxY)A(Xx'R),a) E (RYxXR,). Thus

    (R,j) E ((Rr;(RYxY))n(Rn(X\-XR)),B) E (RA(RYxXR),).


    4.18. Proposition. Let (R,j) be a relation from X to Y and let (A,a) be

    an extremal subobject of X. Then (AR,a) ((RYAlA)R,&).

    Proof. It follows from Proposition 1.5 that RI(((RYAA)xY) and

    RPf((RYxY)A(AxY)) are isomorphic relations from X to Y. By Proposition

    4.15. (R,j) and (Ro(RYxY,y) are isomorphic relations from X to Y. Thus

    RI~(AxY) and R.i((RYAA)>:Y) are isomorphic relations from X to Y. Conse-

    quencly by the definition of image (4.1), (AR,a) and ((RYftA)R,&) are

    isomorphic as extremal subobjects of Y.


    4.19. Corollary. Let (R,j) be a relation from X to Y. Then ((RY)R,a) and

    (XR,j,) are isomorphic ns extremal subobjects of Y.








    Proof. Let (X,1,) play the role of (A,a) in 4.18.


    4.20. Corollary. Let (R,j) be a relation from X to Y and let (B,b) be an

    extremal subobject of Y. Then BR-1 and (Br.XR)R-1 are isomorphic as ex-

    tremal subobjects of X.

    Proof. Immediate.


    4.21. Proposition. Let (R,j) be a relation from X to Y. Then (RoR-1,j#)

    and (RoR-71,(RYxX),y) are isomorphic relations on X.

    Proof. Consider the following diagram.


    RYxX



    CTix1

    xljxl
    X1 01
    y
    (RxX) ( (XxR-1) ---- ---.-, XxyxX ---> XxX

    02


    XxR-1 XX(YXX)
    T xj*



    RoR-1



    To see the diagram is commutative it need only be observed that

    (jxl)(Qrxl) = < lT3>01(jY1). To show this note that

    (jxl1)(Qlxl) = (jl'-xl) = (Tljxl) and

    aT Cl(jxl) = 1 1(jx) !jpl Ir r(jxl),

    7r0 1(jxl) = 3C1(jxl) = p2 = T2(rrjxl).

    Thus, since (RoR"1 ,j#) is the intersection of all extremal sub-

    objects through which <'7"l,3>y factors,it follows that








    (RoR-l,j#) < (RYxX,j xl). Whence (RoR-],j#) < (RoR-1fI (RYxX),?).


    4.22. Theorem. Let (R,j) be a relation from X to Y. Then (RY,jl) and

    ((RoR-)X,B) arc isomorphic as extremal subobjects of X.

    Proof. Consider the following products: (XxYxX, 1,2, 3),

    (Xx(YxX),p1,'2), ((XxY)xX,Pl *,P*), (RxX,PlP2), and (XxR-1, 1, 2).

    Referring to the diagram in the proof of 4.21 it is easy to see

    that: irlY = 1( = (jxl)Aj = il*(xl)X = r1jp1X1. Thus

    <1 'l~' 3 = "> jll = jlT 1 1l


    RoR






    (RoR- )x >>


    o;'XxX



    1


    Since <7- ,I3>' Y = j#i//' and rTlj# 5, the following diagram

    cc;Lmu:es.


    -Cl p


    iT#
    (R (RoR-1)X



    XI I


    RY .--


    But since has the diagonal property (0.19) and Tr# is an epi-

    mcrphism and j is an extremal monomorphism then there exists a morphism

    : such that ji = B and pl1?1 = T,. Thus ((RoR-1)X,B) < (RY,j).

    Next it will be shown that the following diagram is commutative.


    . .----.----- ----






    jxl
    RxX ---------- (XxY)xX


    <1R'i l

    R --- --X)R (RxX)l(XXR-1) .- -- xxyxX


    2 02


    XxR-1 Xx(YxX)
    ixj*


    = lP1*(jx1)<1Rlj> = UiP1
    2P1 (jx 1)1 lj> = 2j 1~1R j> = 2j.
    = 2P ':(jxl) = T2JPl = 2""

    = P2*(jx)<1Rlj> = P2 = !j.



    = \ 2(!xj*)< ,lj, > = IT J"2 = --j =j



    = T2p2(]xj*) = 7T2j*2 = I2j*T =

    T2< 2',1i j = 1j .


    Consequently there exists a morphism * such that the above diagram

    commutes and such that yS* = .

    Thus <';1,lr,>Y$: = and hence the following diagram is

    commutative.

    (RxX),(R (KRx)--- -------- XxyX


    .T I F l1

    .______ (' ' X
    S(RR 1)X R_----- --. X
    R ----i;"R -~----'--"" '

    1


    2e01(jxl)<1Rnlj>

    30 ] (jxl)



    "1 2(lxj*)< l 3," >



    ,2 '02xj*)

    i3O2 (lxj-)




    74


    Since (RY,j1) is the intersection of all extremal subobjects

    through which irj factors, it follows' that (RY,j1) ((RoR-1)X,B). Thus

    (RY,j1) ((RoR-1)X,B) which was to be proved.


    4.23. Corollary. Let (R,j) be a relation from X to Y. Then (XR,j2) and

    ((R-loR)Y,8) are isomorphic as extremal subobjects of Y.

    Proof. (XR,j2) (R-lX,j3) E (((R-1)o(R-1)-1)Y, ) = ((R-loR)Y,P)

    (4.4, 1.11 and 4.22).














    SECTION 5. UNIONS


    5.1. Definition. If {(Ri,ji): icI} is a family of relations from X to Y

    then let ((FRi,j) be the intersection of all relations (i.e., extremal
    icI
    subobjects of XxY) "containing" each (Ri,ji) (where containment is in the

    sense of "factoring through" as noted in Remark 0.12). (' Ri,j) shall
    ieI
    be called the relation theoretic union of the family {(Ri,ji): iIl}.


    5.2. Examples. In the category Set the relation theoretic union is the

    usual set theoretic union together with the inclusion map.

    In the category Top the relation theoretic union is the usual set

    theoretic union endowed with the subspace topology determined by the top-

    ology of XxY together with the inclusion map.

    In the category Top2 the relation theoretic union is the closure

    of the set theoretic union together with the inclusion map.

    In the categories Grp and Ab the relation theoretic union is the

    subgroup generated by the set theoretic union of the relations.


    5.3. Proposition. Let {(Ri:ji): isl} te a family of relations from X to

    Y, let (LRi,k) denote the usual categorical union of subobjects, let
    iel
    (o.j) be the epi-extremal mono factorization of k and let the codomain

    of a (domain of j) be denoted R. Then r and ( JRi are isomorphic rela--
    iel
    tions from X to Y.

    Proof. Since (~Ri,j) is the intersection of all extremal subobjects
    igI
    containing each (R.iji) and each (Rj,ji) < ( iRi,k) and
    iel
    (CjRlk) (Rj) and since j is an extiemal monomorphism then
    icI








    (UtRi,j) < (R,j).
    icd

    Since (U R ,k) is the intersection of all subobjects which "con-
    isI
    tain" each (Ri,ji) then (UR,k) < (\jR ,j). Since j is an extremal
    isl ieI
    monomorphism and (R,j) is the intersection of all extremal subobjects

    which "contain" (U R,k) then (R,j) < (Q)R ,j). Thus
    ieI iEI
    (R,j) E (U-R ,j)..
    ieI


    5.4. Remark. Notice that by the definition of relation theoretic union,

    if (R ,j1), (R2,j2), and (S,k) are relations from X to Y and if

    (R!,Jl) (S,k) and (R2,j) < (S,k), then (P R2j) < (S,k) (cf. 1.19).


    .5.5 Proposition. Let (R ,j ), (R ,j ), (Sk ) and (S ,k ) be relations
    = 1 1 2' 2 1 1 2 2
    from X to Y. If (R ,ji) < (R2 j2) and (Slkl) < (S2,k2) then

    (R 1- S1,) < (R2JS2k).

    Proof. (R ,j ) (Rj ) < (R2' S,k) and (S,k ) < (S2,k) < (R2 S2,k)
    _of I ] 2,<2 -2 1' 1 < 2' 2) 2 2

    whence (r ..s ,j) (R2 L S2 k) (5.4).


    5.6. .ma.'.rk. The following proposition can be strengthened with the ad--

    ditional hypothesis that the category has finite coproducts (5.34);

    however. J i s included here because it is of interest in its own right.


    5.7. Proposition. Let (R,j) be a relation from X to Y and let (S,k) and

    (T,m) be relations from Y to Z. Then ((RoS) U (RoT),B) (Ro(S')T), ).

    Proof. Consider the following commutative diagrams.


    s ----__ k

    ^s '^^ 7; '~^"~----------
    S

    S T -----
    S- ''"-"."-. Z
    rPp








    RxZ ---------------> (XxY)xZ
    S7A jxZz
    x 3 z


    In ,\ - ~


    (RxZ)[1Xx(SQ}T) >>


    Xxyxz


    -0


    '4 \
    Xx(S ) T) ----------- Xx(YxZ)

    1XXa


    RxZ )'>- (XxY)xZ


    jxj1


    x5


    RxZ f XxS


    A6

    XxS *I -'- V XX(YxZ)
    lxxk


    RxZ >, ---- -(Xxy)xZ
    7" jx1Z 31-
    Y3


    R'Z rlXxT


    02

    XxT >- Xx(YxZ)
    1Xxm


    Yl <1, 7T3>
    RxZ rXx (S T) )---- --- --- XxyxZ -------- XxZ



    I Ro(S T)



    Y2



    T2 RoS '' -


    1401
    >- IXyxYZ


    r'-02


    XxyxZ


    r.


    41-------------------- -----------------


    ~-------------------- ----------,


    --


    -.







    R.xZfIXxT ->--- XxyxZ XxZ
    Y3 < 1'f 3 3>


    SRoT '





    RoS _



    (RoS) ) (RoT) ------ XxZ
    X
    RoT B3



    Since (S,k) < (S( UT,a) and (T,m) < (S kT,a) it readily fol-

    lows that ((RxZ)I}(XxS),y2) < ((RxZ)A(Xx(S J T)),y1) and that

    ((RxZ) n(XxT),Y3) < ((RxZ) n(Xx(S (QT)),yl). Thus there exist imorphis.ns

    .! ard 2 such that Yl1, = Y2 and Y12 = Y3. Hence

    <~3,'3>' 1 iI Y2 and YiS2 = < ,3>Y3

    But (RoS,e2) is the intersection of all extremal subobjects through

    which y factors and since Y2 = Y1S1 = 6B'

    wv have (RoS,B2) < (Ro(SUT T),B ). And since (RoT, 3) is the intersect-

    ion of all extremal subobjects through which <-1, 3>Y3 factors and since

    <'.'3 3- 13Y = 122 it follows that

    (RoT, 3) < (Ro(S(V T),BI).

    Whence ((RoS) k (RoT),P,) < (Ro(SLj T), p) (5.4).


    5.8. Proposition. Let (T,m) be a relation from Y to Z and let (R,j) and

    (S,k) be relations from X to Y. Then

    ((RoT)r.a(SoT),R) < ((R.orJS)oT,5).

    Prciof. Anaclogour: to the proof of 5.7.








    5.9. Lemma. Let {(Ri,Ji): iEl} be a family of relations from X to Y. Then

    (*Ri)-"l,j*) (Ul (i)-1,k).
    iel iel

    Proof. Consider the following commutative diagram.


    is]


    j <2.2, > 1
    Ri ------------ Y >----- -------> YxX



    I ( R )-1 /

    R. k



    R -1 (Ri)-1
    X.* iel


    Since (Ril,ji.*) is the intersection of all extremal subobjects

    through which Ji factors it follows that

    (R -1,j*) < ((iRi)-1,j).
    i i EI
    ieI


    Thus


    Now (Ri,ji)

    "l-i =i


    (-(Ri -1)k) < ((()Ri)-1,j*) (5.4).
    iel iel

    < ()(Ri-1),<2,l>'-1k) since
    ieI
    <2' 1ji*T = i.. Thus


    ((ljRi),j) < ((1.(Ri-1), ,>-k)
    iel iel
    from which it follows that

    ((JRi)-,j*) < (r J(Ri' 1),<2,l><2,7 k)
    iel icI


    whence


    (((2Ri )-1 *) = (f (R-) ).
    iEl i il








    5.10. Corollary. Let {(Ri,ji). iIl} be a family of symmetric relations

    on X. Then ( URi is a symmetric relation on X.
    icI
    Proof. It is clear that for each icl, (Ri,ji) E (Ri-1,ji*) (1.13). Thus

    (U Ri,j) =- ( (Ri-1),k) < (( Ri)-1,j*) (5.9).
    icI iEI iTl

    5.11. Proposition. If (R,j) is a reflexive relation on X and (S,k) is

    any relation on X, then (R(t)S,m) is reflexive on X.

    Proof. Since (R,j) is reflexive, (Ax,i ) < (R,j). Thus

    (AX,iX) < (R,j) < (R:JS,m) hence (R(IS,m) is reflexive on X.


    5.12. Definition. Let (R,j) be a relation on X. Consider the relation

    (R-U R-I,j#).


    R *








    Let (T.i,X) be the epi-extremal mono factorization of Tlj/#. The

    domain of X (codomain of T#) shall be denoted by XR.








    XE R --------------- X
    X


    According to the notation of Section 4, XR is also denoted by

    (RR'XR-1)X.







    5.13. Examples. In the category Set, XR 1 (R(R-1)X = XRU RX.

    That is, XR = {xeX: there exists yeX such that (x,y)ER or (y,x)ER}.

    In the category Top1, XR is the same set as in Set endowed with

    the subspace topology determined by the topology of X.


    5.14. Proposition. Let (R,j) be a relation on X and let (RX,jl) and

    (R-1X,j3) be the images of Rlj and Tlj* respectively. Then

    (RX~ R-1X,a) < (XR'X) = ((RlU-R-1)X,X).

    Proof. Consider the following commutative diagram.

    R RX

    jR j T
    R (R-1 --------- XxX ------------- X


    R


    Since 1,j = Tlj#XR = XTI#R = JiT1 and (RX,j1) is the intersection

    of all extremal subobjects through which Tlj factors then

    (PX,jl) (XR,X). Similarly, it can he shown that (R-1X,j3) < (XR,X),

    whence (RXi'R-1X,a) < (XR,X) (5.4).


    5.15. Proposition. If (R,j) is a relation on X then (R R- ,j/#) is sym-

    metric on X and (XR,X) (X(R )R-1),j2).

    Proof. ((R-R'-])-1 ,j*) (Rjql)(R-l)-',j) = (R-i1 R,j#) (5.9 and

    1.11). Thus (RLiR-1,j#) is symmetric so that

    (XpX) = ((R)R ')x,x) E (X(R -1),j2) (4.6).


    5.16. Proposition. Let (Rj) be a relation on X and let (Ax iR ) be the

    diagonal of YjxXR. Then (Ax R (XXX)iX)- (AX n (XRxXR),p) where








    (A iR(X R
    (X X .XX)*
    R R R XX


    Proof. Consider the following commutative diagram.


    XR 7T*
    A ----------- XxX ---- X
    R T2 2


    I A X xx \X



    I t i x xX xX

    SR R R
    bx' ~-------- X'xx "'---------G
    iX 2



    SinceR =l "," th=t =" Thus,
    Observe that E1(XxX)i 2 XX)I.X Xh2 *iX r2(XXX)iXR. Thus,

    since iX is the equalizer of 71 and 2, there exists a unique morphism

    so that iyX = (XXX)"iX
    XR
    Thus, since (AX SRXRxXR,p) is the intersection of (Ax,iy) and

    (XRnXR,)XX), there exists a morphism B so that pB = ix = (XXX)iX Con-


    RR R
    sequently (A;R(X ^x) -- < (A(x (XRXRp) ,p)"

    Since irlX = Tr2iX it follows that -orp = TliX" = r2iXa i 2p, but

    (XxX)A = p so that 71(XXX)X = TT2(XxX)X. IhNence XTI*X = Xv2*X. Recall

    that X is an extremal monomorphism, hence a mnonomorphism; so that it

    follows that irl*X = R22X. Since (A ) is the equalizer of i* and

    fo* there exists a morphism a such that ix o. = A.

    Thus (XXX)iX a = (XxX). = p, which means that

    (AXP(XRXXp)IpP) < (Ay ,(XxX)j iv). Ihencf
    R
    (Ax ('(X


    5.17. Definition. Let (R,j) b2 a relation on X. Then (R,j) is called








    quasi-reflexive if and only if (A ,(X X X
    R ^R
    quasi-reflexive provided that there exists a morphism X such that the

    following diagram commutes.


    IXR Tr
    A >- -- X xX X
    XR R R 2" R



    R XxX ,,X
    Ji 2



    5.18. Proposition. If (R,j) is a reflexive relation on X, then (R,j) is

    quasi-reflexive on X.

    Proof. If (R,j) is reflexive then (RLU)-R,j#) is reflexive (5.11);

    hence, 7,J# is a retraction (1.24). Thus rTlj# is an epimorphism; so that

    if (T~,X) is the epi-extremal mono factorization of Tlj#, rlj' = XT# so

    that X is an epimorphism as well as an extremal monomorphism. Hence X is

    an isomorphism (0.17). Thus (XRx) (X,1X) whence (Ax ,iXR) (A,i ).

    Thus, since (Ax,iX) < (R.j), (R,j) is quasi-reflexive.


    5.19. Proposition. Let (R,j) be a relation on X. Then (R,j) < (A ,i ) if

    and only if (R,j) < (A, ,(XxX)i ).
    R R
    Proof. Suppose that (R,j) < (AX,i ). Then there exists a morphism a such

    that j = iXa. Thus iJrj = rliXa = v2ixc = r2j; whence

    iJ = I2J = Tij = 2j Consequently the unique epi-extre-

    mal mono factorization of j is (1 ,j), and (R,j) (R-j*)

    (3.12).

    But, since (R,j) (R-!,j*), then (R()R-i,ji/) (R,j), and

    (X,,X) ((R(~jR-)X.x) (RXj ). Thus, since (R,jl) E (X ,x) = (XR,j)
    R L
    ft follows that (R,j) < (X. x >xx) (3.4).
    R I R








    However, it has been shown that (AXR,(xxx)iX ) and (A,((XRxXP),p)

    are isomorphic relations on X (5.16). So that since (R,j) < (A ,i ) and

    (R,j) < ((XRxxR),XXX) it follows that

    (R,j) (Ax((XRxXR),p) E (AXR (XXX)iX ) (1.19).

    Conversely, if (R,j)< (A ,(XxX)iX ) then since
    R R
    'IT(XXX)ixR = X'*iXR = X12 *ix = 2(X)iXR it follows that
    R R R R
    (AXP'(xXX)ix ) < (Ax ,i). Consequently the result (R,j) < (Axi ) fol-

    lows from the transitivity of < (0.2).


    5.20. Theorem. If (R,j) is a relation on X then R, RoA and A oR are
    R R
    isomorphic relations on X.

    Proof. It will be shown that RoA and R are isomorphic relations on X;
    X
    the proof for A oR and R is analogous and is omitted.
    rR
    The following products shall be considered: (XRxYE r"I*,2*),

    (XxXxX,rP ,,2,3), (XxX,7T,l12), ((XxX)xX,i ,f 2), (Xx(XxX), i1 J ),

    (RxX,5p,,2) and (XxAXR 1* 2*).

    Consider the following commutative diagram.


    jx1X
    RxX ------- (Xx) xX

    r. Y I 01
    Y 4Z < 1 2,T3>
    RxX XxA., ------ --- XxXxX ------> XxX
    RR)
    X2 09
    i Xx(xxx)ixR
    SXXA ^------------ P'>Xx(X>) -
    XR


    Next it will be shown that < T,,i'2> = <' ,'i>y.







    T < ,T, 2>Y = i1Y = TJ<~I1,7T3>y.

    T2<7T2>y = 2Y = 202(1XX(XXX)iXR)2 2(1XX(XXX)iX )X =

    T1 (XxX)jxP2*X = X"liXR P2*-2 = X2*XRP*2 =
    1 XR 2 2 X R 2 2 R
    X7T *iXR 2* 2 = 2(XxX)iXR P2-2 N2 2 (Xx(XxX) iXR 2

    3 32(1X(XXX)iXR)A2 = 73Y = T2<7,73>Y-

    Thus by the definition of product <7 1,)N3>y = <71,T2>y.

    Now consider <71,'r2>y. It will be shown that Y = jl1l-

    T<71T,,2>Y = :FlY = T l01(jx1)A1. Thus Tly lTlm1 (jXl)1 = lJl

    T 2<1, 2>Y = I2Y = "201(jxl)1= )2~ (jx1)X1 = 7j2jP11. Hence, by the

    definition of product, y = jXl1l.

    Since (RoAxR,a) is the intersection of all extremal subobjects

    through which <-1, 3>y factors and 3>Y = <1-l,T2>Y = J1Pil it

    follows that (RoAX,a) <_ (R,j).

    For the reverse inequality consider (RQJR- ,j#). Since

    (RLJR-,ji'/ ) is syrmmietric, ((R:)]R-I)X,ji#) and (X(RC R-1),j2#) are

    isomorphic extremal subobjects of X (5.15).







    x! x
    R ^/R-I y------------->- xX ---- --- X

    R 1 J '-^< ~- 1 X
    x(RIIR 1) '

    RR
    -XR




    Thus if (G#,X#) is the epi-extremeal mono facrorization of T2j#,.

    there exists an isomorphism ( such that g, E/#. So in particular,

    S7Tj = Q1 #x =- XR #AR"








    Let a be the isomorphism for which iXRO = <1XR ,XR> (1.21).

    It next will be shown that the following equality holds:

    <7Ji2J,2J> = 01 = 02(lx(XX)ix )

    so that the following diagram is commutative.


    01


    jx1X
    A------^ -----


    RxX


    (XxX)xX



    --------- XxXxXX


    i i
    1


    1 -,
    RxX % XxAXR

    I X
    AX 2
    XxA .---
    XR


    51 2(1X(XXX)iXR )<1rj ,a R>


    7202 (lX(XX)iXR) <7tlJ,oG~ XR>







    0,2(IXx(XXX)iXR )<1j,Gc AR >


    SXx(XX)
    -4. Xx (XxX)


    1XX(XXx)iXR


    = i1(lXX(XXX)iR ) =

    P1* = 7lJ.

    = r17 2(1Xx(Xxx)iR ) -

    T1(XXX)XR P2 <7t j,oX#R> =
    R
    XPi XRP2* = XPliR oG/#XR

    XR R#R
    X011#XR = X1#XR >= T2j.

    7 2'2(lxX(XXX)ixR) =

    T2(XXX)iXRP2*<1 P Iil R> =

    XP2iiX P2,* j,oS#XR> = XP2< 1XR >S R =

    X1 XR#X = XW#\R = TT2j


    0i1 = 'l l = fl'"

    5201 = T2fl = 1r2j.

    303j' ,< J> = 2i2<> 2J> = ,2j-


    sv ----------







    Thus by the definition of product:

    = l = 02(1XX(XXX)iXR )

    But it is also true that

    ol = O1(jx)<1,rTj> (1.31).

    Hence by the definition of intersection there exists a unique morphism

    E such that yE = . Thus

    <1,sf3>yE = <1, 3>< lij, 2ji2j> = = j.

    Hence j = atC; and consequently (R,j) < (RoAXR,a).


    5.21. Corollary. Let (R,j) be a relation on X. Then R-1, AX oR- and

    R-oAXR are isomorphic relations on X.

    Proof. Recall that since (RU' R-,j#) is symmetric (5.15), (XR-1,X*)

    and ((RL4R-1)X,j1#) are isomorphic as extremal subobjects of X (5.15

    and 5.9). Also (X ,x) and ((Rt UR-1)X,j,#) are isomorphic as extremal

    subobjects of X (5.15) whence,

    (X~IX! ,.X*x*) E (XRxXR,XXX) and hence (AXR_1' (*xX*)iX.1) and

    (AXR,(XX)i XR). But by the theorem

    (R- *) =- (AXR-oR-'1,*) E (R-oAxR-1 ,a*).

    Consequently,

    (R-',j*) (AX oR,a#) (R-loAXR,a#).
    R R

    5.22. Definition. Let R be a relation from X to Y. Then R is said to be

    difunctional if and only if Ro(R-loR) < R and (RoR-1)oR < R.

    The term difunctional relation is due to Riguet [22J.


    5.23. Proposition. Let R be a relation from X to Y. Then R is difunc-

    tional if and only if R-1 is difunctional.

    Proof. If (R,j) is difunctional then since (Ro(PR1-oR),k,) < (R,j) we

    have ((R-!R)oR-1, k) E ((R-loR)-loR-'1,k) ((Ro(R-oR))-1,k *)<(R ,j)
    1 1 1







    This follows from 1.38, 1.11, and 1.12. Also since ((RoR1)oR,k2) < (R,j)

    it follows that

    (R7-o(RoR-1),k2#) (RIo(RoR-1)-1,k2) E (((RoR- )oR)-I,k2*) < (R1 j*).

    Thus (R-l,j*) is difunctional.

    If (Rl,j*) is difunctional then since ((R-1)-1,j#) (Rj) (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-,j#) = (R,j). Thus (XR,x) (RX,j1) E (XR,j2) (5.15 and 4.6)

    from which it follows that (A R(XxX)i~ ) (XR'(j2xj2)iXR1
    1 XR
    Since (R,j) is a quasi-equivalence then (R,y) is an equivalence

    relation on XR (3.10) so ( ,X'iXR) < (R,i). Hence

    (AXR,(j2xj2)iR) < (R,j) so that (R,j) is quasi-reflexive on X. Since

    (R,j) is symmetric and transitive then

    (Ro(R-loR),kl) < (Ro(RoR),k1) < (RoR,j') < (R,j). Similarly,

    ((RoR-1)oR,k2) < ((RoR)oR,k2) < (RoR,j') < (R,j). Hence (R,j) is difunc-

    ticnal.

    Conversely if (R,j) is quasi-reflexive and difunctional then

    (A ,(XxX)i ) < (R,j) so that (A ,(XxX)i ) < (R-',j*) (1.16 and 1.12).
    -XR XR XR XRR
    Thus (RcR,j') < (Ro(A oR),k) < (Ro(RloR),k, ) < (R,j) and

    (R-1,*) < (AXRo(R-loAXR),k) < (Ro(R-loR),kl) < (R,j) (5.20 and 1.30).

    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 if and only if R is reflexive and difunctional.








    Proof. If (R,j) is an equivalence relation on X then (R,j) is reflexive

    and a 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-equiva-

    lence (5.24). Since (R,j) is also reflexive it must be an equivalence

    relation.


    5.26. Theorem. Let (R,j) be a relation from X to Y and (RX,jl) and

    (XR,j2) be the usual images (3.4). Then R, RoAXR, and ARXOR are iso-

    morphic relations from X to Y (cf. 5.20 and 1.31).

    Proof. Consider the following commutative diagrams.


    J2
    XR -------------- Y


    T2 T2

    R ------- Xx

    Ti Trl

    RX >~1------X




    A -----RX ------- XX ___----XX




    cL iRX------ t ___________- R
    X P12
    A >39- --- XxX X
    + X P2
    a 2
    xji PJ
    RX & P
    A -. RXxRX ___ RX
    RX p2"



    Observe that pl(jiXjl)iRX lPliRX = jlP2*i[ = P2(jlxjl)i;

    thus there exists a morphism a such that iXa = (j xjl)iRX; i.e.,

    (ARX,(' l l)iRX) (AX,ix)




    90



    Thus: (ARXoR,k2) < (AxoR,j') E (R,j) (1.30 and 1.31).

    To see that (R,j) < (ARXoR,k ) consider the following commutative

    diagram.



    R -------------Xx(XxY) --:-- XxXxy

    I 2 2' 3

    q X



    XxY xy------- -- X
    1Xxy


    Recall that (ARX,i) E (RX,) (1.21) hence there exists

    an isomorphism o such that <1RX,1lRX> = ipRX.

    Consider also the products (ARXxY,1,62), (XxR, 1,T2), and

    ((Xxx)xy, l, ,2)

    It will next be shown that the following diagram is commutative.


    (J I j )i RX1y


    A XY x



    R





    (XxX)xy


    -3> XXXXy


    ^C)2


    ,T1j,"I2j> = 02


    XXi


    1,19l((j Ijl)iRjx ly) P 1 ((j 1 xj ) iRXX y) =

    Pl QJ 1 RX 1 1 / IL
    j I P*< 1RiX N, 1> .= jlT = I j li








    q20!o ((jlxjl)iRxXly )<^T1,72'j = P2 l1((jXjl)iRxXIy) =

    P2(jlXjl)iRXPl = jP2*iRXoT1 = jlP2*<1RX,'RX> =

    Jl1T = T'li


    31301((jlxjl)iRXXly) = x2((jlXjl)iRXXly)<0T1,F2j>

    ly12 T23.


    r1i02(1Xxj) = f1(IxXj)<7rlj,lR> = i = Trlj

    fl202(1XXj) = lrlT2(lxxj) = >lj r2< lj,'R> = T!J"

    T302(1Xxj) = r2i2(1Xxj) 2= ljJ2<7Tlj,lR> = 2j


    1092 = ftl = Tlj.

    T2 2< TiJ,j> = '7Tf2 = Trj.

    T392 = T2T2<-lj j> = r2J"


    Consider the following commutative diagram.



    (JlXjl)iRXX1
    ARXxY >---------- (XxX) xy


    '1 13
    ",
    ARXxYA(XxR ----- XxXxY ------------~XxY





    A 1 xj R k2



    ARxOR



    By the definition of intersection, there exists a unique morphism

    ( from R to (A -
    R< "--




    92


    Thus 'Y3 = < = <"ljI2j>= j. Hence k2 = j;

    that is, (R,j) < (ARXoR,k2). Hence (R,j) (ARXoR,k2).

    Similarly it can be shown that (R,j) E (RoAXR,kl).


    5.27. Theorem. Let (R,j) be a relation from X to Y. Then

    (Ap,,(j1xj1)iyX) < (RoR-1,j#) and (AXR,(j2xj2)iXR) < (R-loR,j').

    Proof. Consider the following products: (XxX,p1,P2)., (RXxPX,*pl ",2*),

    (XxY,1T ,2), (RxX,pl,p2), (XxR-1,-1',2), ((XxY)xX, ~1,2),

    (Xx(YxX),'IT*, 2*) and (XxYxX, 1, 2,)3).

    Also consider the following commutative diagrams.



    R - XxY


    T'I jj T

    RX ----------- X



    j <2T2,7I1>
    R --- ------- XxY )---- YxX



    R- I



    Next, the following diagram will be shown to be commutative.

    jxlX
    RxX .------------~ (XxY)xX
    < R, IJ-> f A X -

    SY <
    R ----~-(RxX) (XxR-) ----- ------ XXYXX ---------> XX

    X2 2 ^
    ( \ I Xri* 71.


    ,- i


    - -J








    To see this, it need only be observed that

    o1(jxlX) = 2(lXxj*)<-Tj,T>.


    .i01 1(j1Xx) = Tii (jlX)<1R',7lj> = 1iJP : Tlj.

    20! (jxIX)<1R' l j> = T21 I(jXlX)<1R',Ij> = 2jpl = T2j

    S30 (jX1X)<1RF"1j> = i2(jxlX) = 2<1R'R1j> = rlj-


    !G2(1Xxj*)< T> = i*(lXxj) = p1< 1J,T> = Tflj.

    02o2(1xXj*)<7!IjT> = Tr712*(xj-j*) T 1j*P2 = .lj*T =

    1 <2,1T]>J = T2j.

    3502(lXxj*) = T 2 2(1xxj*) = 2j = 7lj.


    Thus the diagram is commutative and

    01(jlX1)<1Rrj> = 02(lxXj*) =

    Hence by the definition of intersection there exists a morphism E

    such that yE = . Clearly <-,- 3>YE = thus

    = j#r#Z.

    It next will be shown that if a is that isomorphism for which

    = iRXO then (oTl,(jlxjl)iRX) is the epi-extremal mono factor-

    izauion of <'Trj,'"lj>



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.).




Title: Relation theory in categories
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 Material Information
Title: Relation theory in categories
Physical Description: vi, 122 leaves. : illus. ; 28 cm.
Language: English
Creator: Fay, Temple Harold, 1940-
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 1971
Copyright Date: 1971
 Subjects
Subject: Categories (Mathematics)   ( lcsh )
Mathematics thesis Ph. D
Dissertations, Academic -- Mathematics -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
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Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 120-121.
Additional Physical Form: Also available on World Wide Web
General Note: Manuscript copy.
General Note: Vita.
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Volume ID: VID00001
Source Institution: University of Florida
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Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
    Abstract
        Page iv
        Page v
        Page vi
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
    Preliminaries
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
    Generalities
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
    Categorical congruences
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
    Categorical equivalence relations and quasi-equivalence relations
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
    Images
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
    Unions
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
        Page 102
    Rectangular relations
        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
        Page 117
        Page 118
        Page 119
    Bibliography
        Page 120
        Page 121
    Biographical sketch
        Page 122
        Page 123
        Page 124
Full Text














Relation Theory in Categories


By

TEMPLE HAROLD FAY











A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF iLOir -. IN PARTIAL
FULFILLMENT OS TIH REQUIREMENTS FOR THE DEGREE' OF
DOCTOR OF PHILOSOPHY















UNIVERSITY 0: FL..IDA















To Dr. George E. Strecker, without whose tactful prodding,

infinite patience in proofreadings of handwritten drafts and helpful

suggestions this work would never have been completed.
















TABLE OF CONTENTS


Abstract.................................... .................... ... iv

Introduction................................. ...................... 1

Section 0. Preliminaries...................................... 6

Section 1. Generalities ................ ......................... 18

Section 2. Categorical Congruences............................. ]

Section 3. Categorical Equivalence Relations and
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 < = ; i.e., Trc = TY2 and 729 = E. Now yS = (1Xxc)E

hence ITYi --- 1Xi = ~TY2 and 2Yl = cr = c = 2 = 2. Thus "y1E = Y2'

whence:

((XxA) r (XxB), y2) ( (Xx(AriB), y').


1 2
X ---------------- (XxA) .jXxB)




A rB XxA Y2 XxB


I/ 3 xa


Xx (, ) ....-Yx------ -
Y1


I. xb
X







1.6. Notation. Let X and Y be -objects and let (XxY,T1,T2T) and

(YxX,pl',2) be the indicated products of X and Y. Then there exists a

unique isomorphism from XxY to YxX, denoted by , such that the

following diagram commutes.



XY IY VI
xY -------- --------- Yx
3
11 1 '
xX
x )------ ---?x


Note that: = 1x and = 1Xx

1.7. Definition. Let (R,j) be a relation from X to Y and let (T,j*) be

the unique epi-extremal mono factorization of j (see 0.18). The

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,y) = (y,x); hence, if (R,j) is a relation from

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: <72,T > = 1Xxy (1.6). Consequently, T#T is an iso-

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




TliX = T2iX = TliX = TT2<"'i2,71>iX.

Thus the epi-extremal mono factorization of'iX is (1. ,5-).


1.17. Corollary. Let (R,j) be a relation on X, then (R,j) is reflexive

on X if and only if (R1-,j*) is reflexive on X.

Proof. If (Ax,ix) < (R,j) then (AX,ix) (X-,ix*) <- (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 =1 j*1iX and = k* T2. Thus by the definition

of intersection: (RAS,i) > (R-1A S-1, ). However since r"' is an

isomcrphism, (RkS,) E ((RAS)",p*); whence

((RAS)-1,I*) < (R-1A S-1, ).

To obtain the reverse inequality, note that by the definition of

intersection (R-IAN S-I,) < (R(AS,i) since jT-1I3 = and

kT-1XL = #. Thus (R-1(\S-1,) (R/S,).

Whence (R-1(1 S~ ,) < (RnS,P) H ((RflS)-1,i*). Consequently:

(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', iX = iX (1.16) and

also that < 2' -1 = ; i.e., '< 2,TT 1 > = 1XxX (1.6). Observe

that jT-rX4 -j* = -1 X3 = i X3. Consequently

JT-X4 = i X = 3. Thus by the definition of intersection:

(R-7()AXX3) <_ (R(c X,i X 2).

Also observe that j*rX = J = < 2' I>i 2. Whence

j'TXI = i X2 so that by the definition of intersection:

(R1AxiX2) < (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 = Tr2<,1X > and iX is th3 equalizer of -r, and T2,

it follows that (X,<,1 X>) < (AXi ). Since l = l, <1XI > is

a section, hence an extremal monomorphism.

Clearly, z I<1 1 >ri = i1 liX = li and
x 1 -X 1 A 1
T2<1X .1 >71iX 1X iX = Tli = T2iX. Hence, by the definition of pro-
duct, l i = iX. Thus (A ,i ) < (X,<1,1 >).
X IX xA x A X x-

1.22. Example. In the categories Set, Top1, To p,, Grp, Ab, and Rng,

<]X,1 x>: X ------- XxX can be defined by <1i.1 >(x) = (x,x) eXxX for all
XtX.








1.23. Remark. It is also easy to see that up to isomorphism of extremal

subobjects (X,<]X,1X>) (and thus (Ax,'iX) also) is the equalizer of each

of the following sets of morphisms:

{i] '2 }, {T], <1x,1X> 2}, {<1X,1X>', 1XxX} {"2, 1,,xX and

{T] ,< ]X,1x>2,' 1XxX.


1.24. Proposition. If (R,j) is a reflexive relation on X then T1j and Tn2J

are retractions.

Proof. Since (X,) < (AXx) < (R,j) there exist morphisms a and

B such that ita = and jB = iX. Thus X = "1<1X,1I> = TliXa = i7ljXa.

Thus r1j is a retraction. Similarly T2j is a retraction.


1.25. Remark, Consider the following products: (XxY,p,,p2), (YxZ,1,12),

(XxYxZ,1if J2,3), ((XxY)xZ, ; 2) and (Xx(YxZ),fl*,T 2*). It is easy to

see there exist isomorphisms

01 = and 2 = <1,*' 2' 2*
( _1 02
(XxY)xZ 01 .-- XxYxZ ------- XX(YXZ)

such that 0101 = PS1, 21 P271' 301 = T2 and 1 02 -7"' 7, 202 P'2

and 1302 = 2 '.


1.26. Definition. Let (R,j) be a relation from X to Y and (S,k) be a re-

lation from Y to Z. Consider the following intersection.


lxj
RxZSRxzy--- ------ .-----<( Xxyx



2
Xx S' -^--------------'X>: (YxZ)
Ixk

Let <71i, 3> denote that unique morphism from XxYxZ to XxZ such that

o1 = 1 and 02< 1,3'> = 3 where uJ and o are the projections cf

XxZ to X and Z respectively.








Let (T',j') be the unique 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 y 2a = Yl.

Since (RloS1,j) is the intersection of all extremal subobjects

through which Y1 factors (0.21) and since Y1 factors

through (R2oS2,k) it follows that (RloS,j) < (R2oS2,k) which was to be

proved.


1.31. Theorem. Let (R,j) be a relation from X to Y then RoAb, R, and

AxoR are isomorphic relations from X to Y.

Proof. First consider RoAy. From the definition of composition of rela-

tions the following commutative diagram is obtained.

jx!
RxY Y------------- --- (X>:Y)xY



(RxY) (Xx^y) >-- ---------.--- --------- xYxY
Y'


Xx -- -------------- ----* XX (YxY)
Ixi.








Recall that (A ,i ) is the equalizer of the projections pl and p2

from yxy to Y.

It will next be shown that y = y. Let 0i, and 2 be

the projections of XxA to X and A respectively, and let Il"* and IT 2 be

the projections of Xx(YxY) to X and YxY respectively. Then

P1<1' 2 >Y = itY = P 3< 3>Y.

P2<1'pY 2 2 2 202(xi ) 2 1= 2*2 Pliy 2 = 2iY 2 2

P2 2*(]xi )X2 = "32(1xiy )2 = 3= p<2 <1 3>y.

Hence <1, 2I >y y.

Let p1 and P2 be the projections of (Xxy)xY to XxY and Y respec-

tively and let p and p2 be the projections of RxY to R and Y respec-

tively.

Since ITjp1 = 1P1(jxl)X1 = 0 (jxl)x = iT y = ,y, and

Ti2JP I*I = T2P1J- (x)Il = T2rl(jxl)Al = T2Y = T2y, and

< l' I2>y = < 1',T3>y = j T-, then the following diagram commutes.


PI*
Rxy ---- ----- R


(Rxy) (XxAy) -< >.T> XXY



^T ROA,


Thus since (RoA ,j') is the intersection of all extremal subobjects

through which y = y factors (0.21) it follows that

(RcA j') < (R,j).
J
To see that (R,j) < (RoAy,ji ) consider the following commutative

diagrams.










I 1
XxY


--- :>-~ Xx'ixY


<1lW2>


XxY


ly


<1R' 2j
R -- RxY


1R 1*


Recall that (Ay,i ) = (Y,<1 ,1 >) (].21), thus there exists a

morphism o:Y ----->y such that i y = <1 ,1 >.


R 12------ XXA






x


---`c (yx)xy r--







It now will be shown that the following diagram commutes.

jxly
RxY ------- -- (XxY)xy
<1R)' 2J> E 01

1 O
R >------- Xxyxy


<,lj,wT2 -,
XxA >"----1- >-- Xx(Yxy)
Y 1 Xi
yX Y


1 1l(jXl) = Tll(jxl) = IlJP R'2j = 1JlR = r


0 (l(jxl) 2 * = T2j.
3 1 R' 2 22(x1)<12 R 2 2

l0l = lP1 = lj'

i201 = Tr2P = 72j.

<301 j> P=2 1 2 2 1 2j > (2)-



_12(1lXiy) = Tf1 ((Xi) = Pl =

T-2 2(lxiy)<1l j,o r2j> 1= plI2*(lxiY) = Pliy>2

PliyOTf2j = Pl<1( >IT2j = l = ]2j = T2j.

3 92(1xiY)<7lj,o'r2j> = P2 2*(1xiy)< lj,' m2j> = p2iYP2< 111,o-2j> "

P2i 2j = g2T2j 1I 2j = T2j.
Thus by the definition of intersection there exists a unique mor.-

phism C from R to (RxY)f1(XxA ) such that

y = e = 01 < > = 1(j ,) = 2(1xiy)<"7Tj:orT2j>. Thus

<1, 3 >Y3 = <13>0 . But since

'<1' l'>01 = 710 = iPl 22<7,T 1 3>l = 30l = 2j it follows from the

definition of product that < , 3>0 j.








Thus j = y = j'T'C whence (R,j) < (RoAy,j).

The proof that (Rj) E (A oR,j"") follows from analogous arguments.


1.32. Proposition. If (R,j) is reflexive and transitive

(RoR,j') E (R,j).

Proof. Since (R,j) is transitive then (RoR,j') < (R,j).

reflexive then (Ax,iX) < (R,j). Thus (R,j) < .(RoA X, )

(1.31 and 1.30). Hence (RoR,j') E (R,j).


on X then



Since (R,j) is

< (RoR,j') (R,j)


1.33. Remark. As has been remarked in (1.27), if (R,j) is a relation from

X to Y and (S,k) is a relation from Y to Z then in the categories Set, Ab,

Grp, and Top (RoS,j') may be taken to be the set

{(x,z): there exists a yeY such that (x,y)cR and (y,z)ES}

together with the inclusion map j^. Thus the categorical definition of

composition (1.26) yields in these special concrete categories the usual

set theoretic composition.

A similar remark can be made about the definition of the inverse

relation. That is, the categorical definition yields the usual set theo-

retic definition in the categories Set, Ab, Grp, and Top1 to only men-

tion a few. Indeed, the categorical definitions were obtained by analyz-

ing the situation in the set theoretic case.

However, in the category Top2 of Hausdorff spaces and continuous

maps the extremal :Tonomorphisms are the closed embeddings which leads to

the following consequences.


1.34. Example. If (R,j) is a relation from X to Y and (S,k) is a relation

from Y to Z for Top2-objeccs X,Y, and Z. Let T be the following set.

{(x,z): there exists a yeY such that (x,y)ER and (y,z)ES}

Thaen (RoS,j) (clT,j) where "cl" means closure with respect to the top-








ology of XxZ (c.f. 1.27).

Proof. Recall that the extremal monomorphisms are the closed embeddings,

thus RoS is a closed subset of XxZ. Clearly the following diagram

commutes.


Y
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