Relation theory in categories

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Relation Theory in Categories


By

TEMPLE HAROLD FAY











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















UNIVERSITY 0: FL..IDA















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

infinite patience in proofreadings of handwritten drafts and helpful

suggestions this work would never have been completed.
















TABLE OF CONTENTS


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

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

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

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

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

Section 3. Categorical Equivalence Relations and
Quasi-Equivalence Relations........................ 8A

Section 4. Images............................................ 63

Section 5. Unions............................................ 75

Section 6. Rectangular Relations .............................. 103

Bibliography......................................................... 120

Biographical Sketch ...................................... ......... 122







Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy

RELATION THEORY IN CATEGORIES

By

Temple Harold Fay

March, 1971

Chairman: Dr. George E. Streckcr
Major Department: Mathematics

The purpose of this dissertation has been to systematically

generalize relation theory to a category theoretic context. A quite

general relation theory has emerged which is applicable not only to

concrete categories other than the category of sets and functions, but

also to abstract categories whose objects need have no elements at all.

This categorical approach has provided the opportunity to comprehend

classical relation theory from a new vantage point, thus hopefully

leading to an eventual better understanding of the subject.

A relation from an object X to an object Y is a pair (R,j) where

j is an extreimal monomorphism having domain R and codomain XxY. By

choosing j to be an extremal monomorphism, relations in the category

of sets are the usual subsets of the Cartesian product, relations in

the category of groups are subgroups of the group theoretic product,

and relations in the category of topological spaces are s.bspaccs of

the topological product. This latter fact would not be the case if

relations would Le defined to be merely subobjects of the categorical

product.

Section 0 notes results which are purely categorical in nature

and vhicl.h vill be used extensively throughout the sequel. Particular

emphasis is given to the epi-extremal mono factorization property and







necessary and sufficient conditions for the existence of this factori-

zation and equivalent forms of the property.

In Section 1, the basic machinery for categorical relation theory

is developed. For example, such notions as inverse relation, reflexive

relation, symmetric relation, and composition of relations are defined

and several important results are obtained.

Section 2 deals with a categorical definition of a congruence

relation. Several algebraic results of Lambek and Cohn are generalized.

Equivalence relations and quasi-equivalence relations (symmetric,

transitive relations) are studied in Section 3. A quasi-equivalence on

an object X is shown to be an equivalence relation on a subobject of X.

If R is a set theoretic relation from the set X to the set Y and

A is a subset of X then AR = {ycY: there exists acA such that (a,y)eR}.

This definition is generalized in Section 4 and results similar to those

obtained by Riguet are demonstrated.

If {(Ri,ji): iel} is a (finite) family of relations from X to Y

then the relation theoretic union (URi,j) of the family is obtained by
iel
taking the intersection of all relations from X to Y which "contain"

each Ri. If the category being investigated is assumed to have (finite)

coproducts then the union of the family considered as subobjects and the

relation theoretic union of the family considered as extremal sabobjects

turn out to be given by the unique extremal epi-mono and unique epi-

extremal mono factorizations of the canonical morphism from the coproduct

of the family to XxY.

The notion of a (finite) union distributive category is introduced.

Roughly speaking, this property guarantees that unions commutee" with

products and intersections.







Section 5 deals with unions and the importance of the concept of

difunctional relation is brought out.

A well known result in set theoretic relation theory is that a

partition determines an equivalence relation. In order to obtain this

result in its generalized form the existence of an initial object which

behaves similarly to the initial object in the category of sets (namely

the empty set) is postulated and disjointness becomes a useful categor-

ical notien. Also the notion of difunctional relations was crucial in

obtaining the above result.

Section 6 deals with rectangular relations and the above result

about partitions is obtained.














INTRODUCTION


The purpose of this work has been an attempt to systematically

generalize relation theory to a category theoretic context. In doing

so, several goals have been realized. Firstly, a quite general rela-

tion theory has emerged which is applicable.not only to concrete cate-

gories other than the category of sets and functions, but also to

abstract categories whose objects need have no elements at all. Second-

ly, taking a categorical approach has provided the opportunity to

comprehend classical relation theory from a new vantage point, thus

hopefully leading to an eventual better understanding cf the subject.

Hany relation theoretic results have been rather straightforward

to prove in an "element free" setting, once the appropriate machinery

has been constructed to handle them. On the other hand it has been

surprising to see that some results which are easy to prove in the set

theoretic context are much more difficult to show categorically.

For example, it is easy to prove that if R is a set theoretic

relation from X to Y such that RY = X then RoR"' = {(x,z): there exists

yeY such that (x,y)eR and (y,z)cR-]} is reflexive. This result can be

generalized to categories but is no longer easy to Drove and the result

gains some significance.

Another easy result in set theoretic relat-con theory is that if

AX and Av are the diagonals on X and Y respectively then AxcF. = R = Roby

This result is also generalized to category es bIt "i -or.crhic as rela-









tions" replaces "equality" and the result is nc longer easy to prove.

Whenever one is generalizing properties care must be taken to be

certain that the generalized definitions are really generalizations of

the notions be-ig considered and that the proper generalization of the

definition is obtained. This seems to be particularly important in

category theory. Care has been taken when selecting the basic notion

of a relation from an object X to an object Y to be an extremal sub-

object of the categorical product XxY; i.e. a pair (R,j) where j is an

extremal monomorphism having domain R and codomain XxY. By doing so

relations in the category of sets are the usual subsets of the carte-

sian product, relations in the category of groups are subgroups of the

group theoretic product, and relations in the category of topological

spaces are subspaces of the topological product. This latter fact

would not be the case ii relations would be defined to be merely sub-

objects of the categorical product. Much care has also been taken with

the definition of composition of relations (1.26). Using this defini-

tion many nice results have been obtained; however, in general, the

composition of relations is not associative (1.36). This, at first

glance, seems to be pathological and casts doubt on the suitability of

the definition of composition of relations. However, the wealth of

other important results obtained belies this doubt (see 1.37). Also,

some further atonement is yielded by trre fact that for rectangular

relations composition is asscciarive (6.15).

Cohn [31 and Lambek i3J define a congruence ii an algebraic

setting to be a sLbalgebra of the cartesian product -vhich is "coipat-

ible' with the algebraic operations and which is set theoretically an

equivalence relalion. in this work, a generalized noLion of congruence








is given which is equivalent to the above in algebraic categories and

the result that a (categorical) congruence is a (categorical) equi-

valence relation is obtained.

It was found that categorical unions were very difficult to work

with. However, by assuming the category being studied had (finite)

coproducts as well as being locally small and quasi-complete the notion

of union became somewhat easier to handle.

For instance, if {(R.,j ): icl} is a (finite) family of relations

from X to Y then the union (CJR.,j) of the family, considered as sub-
iT 1
objects of XxY is not necessarily a relation from X to Y,since j is not

necessarily an extremal monomorphism. The relation theoretic union of

the family is obtained by taking the unique epi-extremal mono factoriza-

tion of j (5. 3) or equivalently by taking the intersection of all rela-

tions from X to Y which "contain" each R.. If the category being inves-

tigated is assumed to have (finite) coproducts in addition to being

locally small and quasi-complete then the union of the family considered

as subobjects and the relation theoretic union of the family considered

as extremal subobjects turn out to be given by the unique extremal epi-

mono and unique epi-extremal mono factorizations of the canonical mor-

phism from the coproduct of the family to XxY (5.29). It is also snown

that when the category has (finite) coproducts both factorizations

respect unions (5.30 and 5.42).

Unions are still difficult to handle even with the assumption of

(finite) coproduets mentioned above; hence, the notion of a (finite)

union distributive category is introduced (5.31). Roughly speaking,

this property guarantees that unions "commuce" .ith products and inter-

scutions and thus unions become "easy" tj handle. Examples of union








distributive categories show that such categories tend to be more of a

topological nature rather than of an algebraic nature.

The set theoretic notion of difunctional relation is due to

Riguet [221 and its importance has been ncted by Lambek 13] and

HacLane 8 A set theoretic relation R is difunctional if and only
-1
if RoR oR C R. The categorical definition in view of the fact that

associativity cannot be assumed reads: R is difunctional if and only if
-1 -1
(RoR )oR < R and Ro(R oR) < R where "<" is the usual order on sub-

objects. It is easy to prove, again by choosing elements, that if a
-1
set theoretic relation R is difunctional then R = RoR oR. However,

the similar result in the categorical setting is much harder to obtain
-1
and is rephrased: if R is difunctional then R (RoR )oR and
_]
R E Ro(R oR) where "-" means isomorphic as extremal subobjects (5.28).

A well known result in set theoretic relation theory is that a

partition determines an equivalence relation. In order to obtain this

result in its generalized form additional hypotheses had to be added

to the category being studied. In particular, the existence of an ini-

tial object which behaves similarly to the initial object in the cate-

gory of sets (namely the empty set) had to be postulated and disjoint-

ness became a useful categorical notion. Again, examples of such cate-

gories are non-algebraic. Also the notion of difunctional relations was

crucial in obtaining the above result (-6.20).

The excellent reference paper by Riguet 22J has been used as a

guide for the results of set theoretic relation theory. Indeed, most

all of the results contained herein are generalizations of results in

.22 The papers by Lambek 3 14 MacLane 8 and Bednark

aid Wallace provided motivation for many of the generali-
,1 1- 1,








nations.

The basis for the categorical notions has been taken from the

papers of Herrlich and Strecker 7 ], S8 1 Isbell 11 ], 12 ,

and the forthcoming text by Herrlich and Strecker [9 J (which has

greatly influenced this work). For most of the basic categorical

notions the reader is referred to the texts by Mitchell [211 Freyd

14 and Herrlich and Strecker 9 9.
The work here is begun with a preliminary Section 0 which notes

(often without proof) results which are purely categorical in nature and

which will be used extensively throughout the sequel. Particular empha-

sis is given to the epi-extremal mono factorization property and neces-

sary and sufficient conditions for the existence of this factorization

and equivalent forms of the property. However, it is not intended that

the preliminary section give a complete category-theoretical background.

It is expected that the reader be familiar with the basic categorical

notions,














SECTION 0. PRELIMINARIES


0.0. Remark. It is assumed that the reader is familiar with the basic

notions of category theory and hence such basic notions as epimorphism,

monomorphism, retraction, section, equalizer, regular monomorphism,

coequalizer, regular epimorphism, subobject, and limits shall not be de-

fined. The reader is referred to Mitchell [21 and Herrlich and Streck-

er (9] for such notions. All of the following results are proved in

detail in Herrlich and Strecker (9] Since Theorem 0.21 is vital to

this work the proof is sketched here.


0.1. Notation. The category whose class of objects is the class of all

sets and whose morphism class is the class of all functions shall be

denoted by Set.

The category whose class of objects is the class of all groups

and whose morphism class is the class of all group homomorphisms shall

be denoted by Grp.

The category whose class of objects is the class of all tcpological

spaces and whose morphism class is the class of all continuous functions

shall be denoted by Topi.

In a manner similar to that described above, one obtains the fol-

lowing categories:

FSet finite sets and functions;

FGp finite groups and grcup homomorphifsms;

Ab Abelian groups and group homomorphisms;









SGp

SG1




Rng





Top2

CT 2


0.2. Proposition.

be m -morphisms.

1) If f and

2) If f and

3) If gf is

4) If gf is

5) If gf is


f g
Let : be a category and let X-----Y and Y-----Z



g are monomorphisms then gf is a monomorphism.

g are epimorphisms then gf is an epimorphism.

a monomorphism then f is a monomorphism.

an epimorphism then g is an epimorphism.

an isomorphism then g is a retraction and f is a


section.


0.3. Remark. In general, an equalizer is a limit of a certain diagram.

It is an object together with a morphism whose domain is the object. A

regular monomorphism is a morphism for which there exists a diagram so

that the domain of the morphism together with the morphism is the equal-

izer of the diagram.

It is observed in Herrlich and Strecker ,9 J that certain func-

tors preserve regular monomorphisms while not preserving equalizers,

hence one reason for the above distinction between equalizers and regu--

lar mcnomorphisms.

In this paper, since we shall not deal with functors, no distinc-


- semigroups and semigroup homomorphisms;

- semigroups with identity and semigroup homomorphisms which

preserve the identity;

- rings and ring homomorphisms;

- rings with identity and ring homomorphisms which preserve

the identity;

- Hausdorff spaces and continuous functions;

- compact Hausdorff spaces and continuous functions.








tion shall be made between equalizers and regular monomorphisms; i.e.,

between the pair (object and morphism) and the morphism alone. Both will

be called equalizers.

f
0.4. Proposition. Let P be a category and let X----- Y be a -

morphism. Then the following are equivalent:

1) f is an isomorphism,

2) f is a monomorphism and a retraction,

3) f is an epimorphism and a section,

4) f is a monomorphism and a regular epimorphism,

5) f is an epimorphism and a regular monomorphism.


0.5. Definition. Let {Ai: iIl} be a family of -objects then the pro-

duct ~tA.,Tr.) of the family is a -obj-ect T"Ai together with pro-
i 1
is 1 iel
section morphisms Ti:TTAi ----- Ai with the property that if P is
iel
any -object for which there exist m -morphisms p.: P ----- 'Ai for

each icl, then there exists a unique morphism X: P ---> Ai such that
iel
ilX = i for each iel.

The dual notion is that of the coproduct (JJ.Aii.).
iel

0.6. Definition. Let {(Ai,ai): iEl} be a family of subobjects of a --

object X. Then the intersection ( ~A,,a) of- the family is a --object
iel
1 Ai together with a morphism a: Oi1A ----^X where for each i there
icl iEl
is a morphism Xi: X f A ---- Ai such cthat ai.. = a with the property
ic!2
that if P i.' any object for which there exist: --morphisms p: P ----" X

and pi: P ---- '-A1 such that a1ig. p for each isi then there exists a

unique norphism ,: P --1-" A. such that aX = p.
It i
It follows that a is a mnonom.orphism.









0.7. Remark. The above two definitions are mentioned because of the

fundamental role they play in the sequel. They are special limits and

are perhaps the most important limits in the categories that will be

considered in tnis work.

The following theorem is a special case of a more general theorem

dealing with the commutation of limits which can be found in Herrlich

and Strecker [9] A variation of the theorem will be proved in

Section 1 (1.5).


0.8. Theorem. Let {(Ai,a.): icI} and {(Bi,b.): icl} be families of sub--

objects of 0 -objects X and Y respectively. Then if P has finite

products and arbitrary intersections then (( A.i)x( Bi) and ^(-(A.xB.)
iI 1 iEI il
are canonically isomorphic.


0.9. Notation. Let {Xi: icI} be a family of -objects and suppose
f.
{Z ---- -- Xi: icl} is a family of 0 -morphisms. Then by the defini-

tion of product there exists"a unique morphism h from Z to TT Xi such
i.E
that irih fi for each iI. This morphism h shall be denoted by
iET
Let A and B be 0 -objects and suppose that a: A ---- X and

b: B ----->Y are -morphisms. If P1 ard P2 are the projection mor-

phisms from AxB to A and B respectively then apl: AXB ---- X and bP2

AxB ----' Y, hence by the definition of product there exists a unique

morphism g from AxB to XxY such that g = a and g = b. Ths mor-
1 l and 7..g = b 2P This mor-
phism g shall be denoted by axb and shall be called the product of a

and b.

Let f be a ? -morphisrm from X to Y. If f is a moncmorphism then

the following notation shall be used:

X ---- ---









If f is an epimorphism then the following notation shall be used:
f.
X ---- Y

If f is an equalizer then the following notation shall be used:
f
X ----------- Y

If f is an isomorphism then the following notation shall be used:
f
X 1----------v Y

a b c d
0.10. Proposition. Let A -----X, B ---- Y, X----- Z, and Y --- W

be ;' -morphisms. Then (cxd) (axb) = caxdb.

a b
0.11. Proposition. Let A ---- X and B --->Y be monomorphisms (respec-

tively, sections, isomorphisms) then axb is a monomorphism (section,

isomorphism).


0.12. Remark. A partial order may be defined on the subobjects of an

object in in the following way:

If X is a P -object and (A,a) and (B,b) are subobjects of X; i.e.,

a and b are monomorphisms with codomain X and domains A and B respec-

tively, then (A,a) < (B,b) if and only if there exists a morphism c from

A to B such that be = a.



b
B 1-- --X-


I- a
A 7



By an abuse of language, if (A,a) < (d,b) then (B:b) is said to

contain (A,a) and the morphism c is sometimes called the inclusion of

(A,a) into (B,b). It is easy to see that if (Aa) < (B,b) an'

(B,b) < (A,a) then the morphism c is an isomorphisnm. In this case,









(A,a) and (B,b) are said to be isomorphic as subobjects of X. This is

a stronger condition than A and B just being isomorphic objects in the

category The following notation shall be used to denote the case

where (A,a) and (B,u) are isomorphic as subobjects of X:

(A,a) E (B,b).

Sometimes it is written (inaccurately) that A < B or that A and

B are isomorphic as subobjects of X. When this is done, the morphisms

a and b should be clear from the context.

It is immediate that (A,a) E (B,b) if and only if (A,a) < (B,b)

and (B,b) < (A,a). Thus the relation "<" on subobjects is easily seen

to be a partial order up to isomorphism as subobjects.


0.13. Definition. Let f from X to Y be a -morphism. f is an extremal

imonomorphism if and only if f is a monomorphism and whenever f = gh and

h is an epimorphism then h is an isomorphism.

If f is an extremal monomorphism the following notation shall be

used:
f
X -- >Y

The dual notion is that of an extremal epimorphism and is denoted:
f
X f--- Y

If f is an extremal monomorphism f: X -----tY, then (X,f) is

called an extremal subcbject of Y.


0.j4. Remark. The definition of extremal monomornhism is due to Isbell

]i] The concept of extremal monomorphism is important since it

yields what shall be called the "irage" of a morphism (see 0.18),


0.15. Examples. In the categories Set, Grp, Ab and FGp, extremal mono-

mcrphis:.s are precisely the onomorphisms (i.e., one-to-one morphisms).








In the categories Top and CpT extremal monomorphisms are precisely

the cmbeddings. In the category Top2 they are the closed embeddings.

f
0.16. Proposition. If X.-- ,Y is a
0.16. Proposition. If X is a -morphism such that f = gh

and f is an extremal monomorphism then h is an extremal monomorphism.

f
0.17. Proposition. If X -----Y is a -morphism then the following

are equivalent:

1) f is an isomorphism,

2) f is an epimorphism and an extremal monomorphism,

3) f is a monomorphism and an extremal epirorphism (c.f. 0.3).


0.18. Definition. A category is said to have the unique epi-extremal

mono factorization property if for any --morphism X -----Y, there

exist an epimorphism h and an extremal moncmorphism g with f = gh such

that whenever f = g'h' where g' is an extremal monomorphism and h' is an

epimorphism then there exists an isomorphism o such that the following

diagram commutes.


f








'NN



If has the unique epi--extremal rnono factorization property and

if f = gh where h is an epimorphism and g is an extremal monomorphism,

then the pair (h,g) shall be used to designate the epi-extremal mono

factorization of f. The extremai subobject (Z,g) of Y is called the









inage of X under f. Sometimes (Z,g) is referred to as the image of f.

The notion of the unique extremal epi-mono factorization property

is defined dually.

If has the unique extremal epi-mono factorization property and

f = gh where g is a monomorphism and h is an extremal epimorphism then

the pair (h,g) shall be used to designate the extremal cpi-mono factori-

zation of f. The subobject (Z,g) of Y is called the subimage of X under

f. Sometimes (Z,g) is referred to as the subimage of f.


0.19. Definition. A category is said to have the diagonalizing

property if whenever gh = ab such that h is an epimorphism and a is an

extremal monomorphism, then there exists a (necessarily unique) morphism

s such that ( h = b and a g = g.


h
X --. Y


b $ --


W z ----- ------>a Z
a


0.20. Theorem. Let be a locally small category having equalizers and

intersections. Then the following are equivalent:

1) Y has the unique epi-extremal mono factorization property,

2) has the diagonalizing property,

3) the intersection of extremal monoaiorphisms is an extramsa mono-

morphism and the composite of extremal monomorphisms is an extremral

monomorphi sm,

4) if has pullbacks and if (P,a,3) is the pullback of f av' g

where f = go and f is an extremal monomorphism then a is an extremal









uionomorphism.

5) if 9 has (finite) products then the (finite) product of

extremal monomorphisms is an extremal monomorphism.


0.21. Theorem. If is locally small and has equalizers and inter-

sections then f2 has both the unique epi-extremal mono factorization

property and the unique extremal epi-mono factorization property.

Proof. (sketch). First we will show the existence of the unique extremal

epi-mono factorization property. If f from X to Y is any -morphism

then let ((E.,e) be the intersection of the family {(Ej,e.): jCJ} of
jcj 3 -l -
all subobjects of Y through which f factors. Then it follows that e is

a monomorphism and that f factors through e; i.e., there exists a mor-

phism h such that f = eh. Now, to see that h is an epimorphism suppose

a and B are -morphisms such that ah = Bh. Let (E,k) be the equalizer

of a and Z. It follows from the definition of equalizer that there exists

a morphism g such that kg h.


f
X --- -------- Y



g h e
E >'>^------------ r\ E Z' Z
k jcJ J


Thus it follows that f factors through ek and since ek is a mono-

morphism then there exists a morphism ): i% E. ---- E such that ekX = e.
jJe
From this it follows that k is an iscmcrphism whence a = i and so h is

an epimorphism.

Next it will be shown that h is an extremal epimorphism. Suppose

L: = hlh2 where h1 is a monomorphism. Then eh! is a monomorphism through








which f factors. From this it follows, as above, that h1 is an isomor-

phism and hence h is an extremal epimorphism. Suppose f = g'h' where g'

is a monomorphism and h' is an extremal epimorphism. Then since g' is

a monomorphisim through which f factors there exists a morphism i from

( E. to che codomain of h' (domain of g') such that e = g'T. Since e
j J
and g' are monomorphisms, it follows that h' = Th and that T is a mono-

morphism. Since h' is an extremal epimorphism it follows that T is an

isomorphism. Thus has the unique extremal epi-mono factorizaticn

property.

Now suppose that ge = mA where e is an epimorphism and m is an

extremal monomorphism. It will be shown that there exists a morphism

a from the codomain of e to the domain of m such that Ge = h and mo A g.

Let (.( A.,a) be the intersection of the family {(A.,a.): isI}
iT
of all subobjects of the codomain of g (codomain of m) through which

g and m factor. This family is non-empty since both g and m factor

through the identity morphism on the codomain of g. It follows that both

g and m factor through a. Thus there exist morphisms al and a2 such that

the following diagram commutes,


e
X --------- Y



h C Ai



W Z



It will be shown next that 22 is an epimnorphism. Suppose a* and

are -mcoiphisims for which aca2 = hSa2. Let (L*,k"O ) be the equalizer









of 0* and B*. It follows from the definition of equalizer that there

exists a morphism b, such that k*b1 = a2, since o.*a2 = *a2. Since the

diagram conmrutes it follows' that a*al = B*ale. But e is an epimorphism

hence o*al = e1 s,) that by the definition of equalizer there exists a

morphism b2 such that k*b2 = a1. Thus it follows that m = ak*bi and

g = ak*b2 and so both m and g factor through ak* from which it follows

that k* is an isomorphism. Hence a* = B* and a 2 is an epimorphism. But

n is an extremal monomorphism and m = aa2 and a2 is an epimorphism. Thus

a is an isomorphism. Thus defining o = a]al it follows that the fol-
2 21
lowing diagram commutes and has the diagonalization property.



e
X -~ Y




w X, Z
111


Hence. has the unique epi-extremal rono factorization property

(0.20).


0.22. Theorem. Let ( be any category then the following are equivalent:

1) a is finitelyy) complete,

2) has (finite) products and (finite) intersections,

3) has (finite) products and equalizers,

4) has (finite) products and pullbacks.


0.23. Definition. A category __ is said .o be quasi--complet-, if has

finite products and arbitrary intersections.









0.24. Examples. The categories FSet and FGp are quasi-complete cate-

gories which are not complete. The categories Set, Top1, Top2, CpT2,

Grp, Ab, Ring, and SGp are quasi-complete.


0.25. Remarks. A quasi-complete category is finitely complete but is uot

necessarily complete as the examples FSet and FGp above show.

Also, a locally small, quasi-complete category has both the unique

extremal epi-mono factorization property and the unique epi-extremal

mono factorization property (0.20 and 0.21).

It can be shown that the unique epi-extremal mono factorization

of a morphism can be obtained by taking the intersection of all extremal

monomorphisms through which the morphism factors. It has been shown that

the unique extremal epi-mcno factorization property is obtained by

taking the intersection of all subobjects through which the morphism

factors (0.20). These characterizations shall be used frequently in the

sequel.













SECTION 1. GENERALITIES


1.0. Standing Hypothesis. Throughout the entire paper it will be assumed

that is a locally small, quasi-complete (finite products and arbitrary

intersections) category.


As noted in the preliminary section s enjoys the unique epi -- ex-

tremal mono factorization property.


1.1. Examples. Many well known categories are locally small, and quasi-

complete. Among such are the categories: Set, Top Top2, Grp, Ab, SGp,

SGp1, Rng, Rny, CpT_ and FGp.


1.2. Definition. Let X and Y be -objects. A relation R from X to Y is

an extremal subobject of XxY; i.e., a relation from X tc Y is a pair

(R,j) where R is a -object and j is an extremal monomorphism having

dcmain R and codcmain XxY. A relation from X to X is called a relation on

X.


1.3. Definition. Let (R,j) and (S,k) be relations from X to Y. Then (R,j)

and (S,k) are said to be isouorphic relations if and only if they are iso-

morphic as extremal subobjects cf XxY.


1.4. Examples. In the categories Set, aid Topi relations are subsets of

the Cartesian product together with the inclusion map.

In the categories Grp, and Ab relations are subgroups of the Car-

tesian product together vith the inclusion map.




19


In the categories Top2, and CpT2, relations are closed subspaces

of the Cartesian product together with the inclusion map.


1.5. Proposition. Let X and Y be r -objects and let (A,a) and (B,b) be

extremal subobjects of Y. Then Xx(ACIB) and (XxA)n (XxB) are isomorphic

relations from X to Y.

Proof. Consider the following commutative diagrams.


B
A/AB -------> B


A Y-------
a


-; XxB

1


i-xa


Sxb




SX xy
X> XXY


Consider also (Xx(A B), 1 xc = y1). Since extremal subobjects are closed

under intersections and products (0.20) yl and Y2 are extremal monomor-

phisms.

Since (lXxa)(1XxXA) = 1Xxc = Y1 and (lxxb) (1XXB) = 1Xxc = Y1 then

by the definition of intersection there exists a unique morphism o from

Xx(AAB) to (XxA)/-(XXB) so that y2o = Y1 and the following diagram com-

mutes. Thus:


XI

X xA f


(X xA) (xxb)
I -.


(Xx(AAB), Y) < ((XxA)n (XXB), y ).










1 Xa
X


] xx
X


Xx(A5 B) -- -- -- '- (XxA) (XxB) ------ -- --Xx



B X

XxB




Now let (Trl,T 2), (Tr1,T), (o],P 2) and (pl,p) be the projections of

XxY, Xx(AriB), XxA, and XxB respectively. Observe that:

1Y2 1(Xxa)X = pX l(Xx 2 = 1i 2

2Y2 = 2(] xa)X = aP2X1 = 2(I xbx)X2 = bP2X2
Thus by the definition of intersection there exists a unique morphism z

from (XxA)( (XxB) to AB such that cE = T2y2 and thus by the definition

of product there exists a unique morphism C from (XxA)f/(Xx B) to Xx(AflB)

such that < = ; 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

PAGE 1

Relation Theory in Categories ay TEMPLE HAROLD FAY A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA J.N PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY 07 FLORIDA

PAGE 2

To Dr. George E. Strecker, without whose tactful proddings, infinite patience in proofreadings of handwritten drafts ana helpful suggestions this work would never have been completed .

PAGE 3

TABLE OF CONTENTS Abstract iv Introduction 1 Section 0. Preliminaries 6 Section 1. Generalities IS Section 2. Categorical Congruences I,] Section 3. Categorical Equivalence Relations and Quasi-Equivalence Relations AS Section 4 . Images 63 Section 5 . Unions 75 Section 6. Rectangular Relations 103 Bibliography 120 Biographical Sketch 122 in

PAGE 4

Abstract of Dissertation Presented to the Graduate Counci] of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy RELATION THEORY IN CATEGORIES By Temple Harold Fay March, 1971 Chairman: Dr. George E. Streckcr Major Department: Mathematics The purpose of this dissertation has been to systematically generalize relation theory to a category theoretic context. A quite general relation theory has emerged which is applicable not only to concrete categories other than the category of sets and functions, but also to abstract categories whose objects need have no elements at all. This categorical approach has provided the opportunity to comprehend classical relation theory from a new vantage point, thus hopefully leading to an eventual better understanding of the subject. A relation from an object X to an object Y is a pair (R,j) where j is an extremal monomorphism having domain R and codomain X X Y. By choosing j to be an extremal monomorphism, relations in the category of sets are the usual subsets of the Cartesian product, relations in the category of groups are subgroups of the group theoretic product, and relations in the category of topological spaces are s.ubspaces of the topological product. This latter fact would not be the case if relations would be defined to be merely subobjeccs of the categorical product . Section notes results which are purely categorical in nature and which will be usee 1 , extensively throughout, the sequel. Particular emphasis is give:-; to the epi-extremal mono factorization property and IV

PAGE 5

necessary and sufficient conditions for the existence of this factorization and equivalent forms of the property. In Section 1, the basic machinery for categorical relation theory is developed. For example, such notions as inverse relation, reflexive relation, symmetric relation, and composition or relations are defined and several important results are obtained. Section 2 deals with a categorical definition of a congruence relation. Several algebraic results of Lambek and Cohn are generalized. Equivalence relations and quasi-equivalence relations (symmetric, transitive relations) are studied in Section 3. A quasi-equivalence oi\ an object X is shown to be an equivalence relation on a subobject of X. If R is a set theoretic relation from the set X to the set Y and A is a subset of X then AR = {ycY: there exists aeA such that (a,y)e.R). This definition is generalized in Section 4 and results similar to those r obtained by Riguet are demonstrated. If {(R-jj^): i£l) is a (finite) family of relations from X to Y then the relation theoretic union (^_J^i>J) °^ *-' he f ara ily ^ s obtained by iel taking the intersection of all relations from X to Y which "contain" each Rj. If the category being investigated is assumed to have (finite) coproducts then the union of the family considered as subobjeets and the relation theoretic union of the family considered as extremal subobjeets turn out to be given by the unique extremal epi-mono and unique epiextremal mono factorizations of the canonical morphism from the coproduct of the family to X>
PAGE 6

Section 5 deals with unions and the importance of the concept of difunctional relation is brought out. A well known result in set theoretic relation theory is that a partition determines an equivalence relation. In order to obtain this result in its generalized form the existence of an initial object which behaves similarly to the initial object in the category of sets (namely the empty set) is postulated and disjointness becomes a useful categorical notion. Also the notion of difunctional relations was crucial in obtaining the above result. Section 6 deals with rectangular relations and the above result about partitions is obtained. V3

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INTRODUCTION The purpose of this work has been an attempt to systematically generalize relation theory to a category theoretic context. In doing so, several goals have been realized. Firstly, a quite general relation theory has emerged which is applicable. not only to concrete categories other than the category of sets and functions, but also to abstract categories whose objects need have no elements at all. Secondly, taking a categorical approach has provided the. opportunity to comprehend classical relation theory from a new vantage point, thus hopefully leading to an eventual better understanding of the subject. Many relation theoretic results have been rather straightforward to prove in an "element free" setting, once the appropriate machinery has been constructed to handle them. On the other hand it has been surprising to see that some results which are easy to prove in the set theoretic context are much more difficult to show categorically. For example, it is easy to prove that if R is a set theoretic relation from X to Y such that RY = X then RoR" 1 = {(x,z): there exists ysY such that (x,y)eR and (y,z)eR *} is reflexive. This result can be generalized to categories but is no longer easy to prove and the result gains some significance. Another easy result in set theoretic relation theory is that if t y and A v are the diagonals on X and Y respectively then A cR R = RcA, This result is also generalized to categories but "isomorphic as rela-

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tions" replaces "equality" and the result is nc longer easy to prove. Whenever one is generalizing properties care must be taken to be certain that the generalized definitions are really generalizations of the notions being considered and that the proper generalization of the definition is obtained. This seems to be particularly important in category theory. Care has been taken when selecting the basic notion of a relation from an object X to an object Y to be an extremal subobject of the categorical product X*Y; i.e. a pair (R,j) where j is an extremal monomorphism having domain R and codomain XxY . By doing so relations in the category of sets are the usual subsets of the cartesian product, relations in the category of groups are subgroups of the group theoretic product, and relations in the category of topological spaces are subspaces of the topological product. This latter fact would not be the case if relations would be defined to be merely subobjects of the categorical product. Much care has also been taken with the definition of composition of relations (1.26). Using this definition many nice results have been obtained; however, in general, the composition of relations is not associative (1.35). This, at first glance, seems to be pathological and casts doubt on the suitability of the definition of composition of relations. However, the wealth of other important results obtained belies this doubt (see 1.37). Also, some further atonement is yielded by the fact that for rectangular relations composition is_ associative (6.15). Cohn ^3j and Lambek \ 13 j define a congruence in an algebraic setting to be a sebalgeura of the cartesian product which is "compatible 1 ' with the algebraic operations and which is set theoretically an equivalence relation. In this work, a generalized notion of congruence

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is given which is equivalent to the above in algebraic categories and the result that a (categorical) congruence is a (categorical) equivalence relation is obtained. It was found that categorical unions were very difficult to work with. However, by assuming the category being studied had (finite) coproducts as well as being locally small and quasi-complete the notion of union became somewhat easier to handle. For instance, if {(R.,j ): iel} is a (finite) family of relations i i from X to Y then the union (tjR.J) of the family, considered as subiel 1 objects of X*Y is not necessarily a relation from X to Y, since j is not necessarily an extremal monomorphism. The relation theoretic union of the family is obtained by taking the unique epi-extremal mono factorization of j (5. 3 ) or equivalently by taking the intersection of all relations from X to Y which "contain" each R.. If the category being investigated is assumed to have (finite) coproducts in addition to being locally small and quasi-complete then the union of the family considered as subobjects and the relation theoretic union of the family considered as extremal subobjects turn out to be given by the unique extremal epimono and unique epi-extremal mono factorizations of the canonical morphism from the coproduct of the family to X X Y (5.29) > It is also shown that when the category has (finite) coproducts both factorizations respect unions (5.30 and 5-42). Unions are still difficult to handle even with the assumption of (finite) coproducts mentioned above; hence, the notion of a (finite) union distributive category is introduced (5.31). Roughly speaking, this property guarantees that unions "commute" with products, and intersection? and thus unions become "easy" tj handle. Examples of union

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distributive categories show that such categories tend to be more of a topological nature rather than of an algebraic nature. The set theoretic notion of difunctional relation is due to Riguet [_ 2 ? J and its importance has been nc ted by Lambek \13j and MacLane [18 J . A set theoretic relation R is difunctional if and only -1 if RoR oR C R. The categorical definition in view of the fact that associativity cannot be assumed reads: R is difunctional if and only if -I -1 (RoR )oR _< R and Ro (R oR) _< R where "<" is the usual order on subobjects. It is easy to prove, again by choosing elements, that if a set theoretic relation R is difunctional then R = RoR oR. However, the similar result in the categorical setting is much harder to obtain and is rephrased: if R is difunctional then R ~ (RoR )oR and _] R E Ro(R oR) -..'here " = " means isomorphic as extremal subobjects (5.28). A well known result in set theoretic relation theory is that a partition determines an equivalence relation. In order to obtain this result in its generalized form additional hypotheses had to be added to the category being studied. In particular, the existence of an initial object which behaves similarly to the initial object in the category of sets (namely the empty set) had to be postulated and disjointness became a useful categorical notion. Again, examples of such categories are non-algebraic. Also the notion of difunctional relations was crucial in obtaining the above result (-6.20). The excellent reference paper by Riguet j 22 J has been used as a guide for the results of set theoretic relation theory. Indeed, most all of the results contained herein are generalisations of results in j[22j . The papers by Lambek 113 J , I 14 J ? MacLane I ] 8 \ and Bednarck aiid Wallace s j j , / \ provided motivation for many of the general!-

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zations. The basis for the categorical notions has been taken from the papers of Herrlich and Strecker ]_ 7 J , [s ) , Isbell \_ll"]' L 12 J ' and the forthcoming text by Herrlich and Strecker [_9 J (which has greatly influenced this work) . For most of the basic categorical notions the reader is referred to the texts by Mitchell \1\ J , Freyd \k \ and Herrlich and Strecker \, 9 J • The work here is begun with a preliminary Section which notes (often without proof) results which are purely categorical in nature and which will be used extensively throughout the sequel. Particular emphasis is given to the epi-extremal mono factorization property and necessary and sufficient conditions for the existence of this factorization and equivalent forms of the property. However, it is not intended that the preliminary section give a complete category-theoretical background. It is expected that the reader be familiar with the basic categorical notions ,

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SECTION 0. PRELIMINARIES 0.0. Remark. It is assumed that the reader is familiar with the basic notions of category theory and hence such basic notions as epimorphism, monomorphism, retraction, section, equalizer, regular monomorphism, coequalizer, regular epimorphism, subobject. and limits shall not be defined. The reader is referred to Mitchell £21J and Herrlich and Strecker £9] for such notions. All of the following results are proved in detail in Herrlich and Strecker £$3 . Since Theorem 0.21 is vital to this work the proof is sketched here. 0.1. Nota t ion . The category whose class of objects is the class of all sets and whose morphism class is the class of all functions shall be denoted by Set . The category whose class of objects is the class of all groups and whose morphism class is the class of all group homomorphisms shall be denoted by Grp. The category whose class of objects is the class of all topological spaces and whose morphism class is the class of all continuous functions shall be denoted by Top 1 . In a manner similar to that described above, one obtains the following categories: FSet^ finite sets and functions; FGp finite groups and group homomorphi sras ; Ab Abelian groups and group homomorphxsms ;

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SGp semigroups and semigroup homomorphisms ; SGp semigroups with identity and semigroup homomorphisms which preserve the identity; Rng rings and ring homomorphisms; Rng~ rings with identity and ring homomorphisms which preserve the identity; Top,. Hausdorff spaces and continuous functions; CpT„ compact Hausdorff spaces and continuous functions. f § 0.2. Proposition. Let p be a category and let X* Y and Y— ** Z be &!, -morphisms. 1) If f and g are monomorphisms then gf is a monomorphism. 2) If f and g are epimorphisms then gf is an epimorphism. 3) If gf is a monomorphism then f is a monomorphism. 4) If gf is an epimorphism then g is an epimorphism. 5) If gf is an isomorphism then g is a retraction and f is a section. 0.3. Remark . In general, an equalizer is a limit of a certain diagram. It is an object together with a morphism whose domain is the object. A regular monomorphism is a morphism for which there exists a diagram so that the domain of the morphism together with the morphism is the equal' izer of the diagram. It is observed in Herrlich and Strecker J, 9 j that certain functors preserve regular monomorphisms while not preserving equalizers, hence one reason for the above distinction between equalizers and regular monomorphisms. In this paper, since we shall not deal with functors, no distinc-

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t.ion shall be made between equalizers and regular monomcrphisir.s; i.e., between the pair (object and morphism) and the morphism alone. Both will be called equalizers. , f 0.4. Proposition. Let g be a category and let X —*» 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. Definit ion . Let {A.: iel} be a family of £? -objects then the product (TTA.,i r .) of the family is a fe -object I i A together with proiel iel jection morphisms tt^ :TT*Aj_ — — pAj with the property that if P is iel any j*' -object for which there exist j? -morphisms p.: P — — — **A. for each iel, then there exists & unique morphism X: P — — *>»"f|~A. such that iel ir. A = p. for each iel. y i The dual notion is that of the coproduct (JLLa. ,u.). iel X 0.6. De finitio n. Let {(A., a.): isl} be a family of subobjects of a object X. Then the intersecti on ( O A... ,a) of the family is a ^-object iel « ^ A^ together with a morphism a: C\k. > X where fur each i there id iel is a morphism A.: f'\ A. • H» A, . such that a-X. = a with the property id that if P is any object for which there exist g -morphisms p: P— — *X and p,-: P — -~"V_A ,• such that a,-p,= p for each iel then there exists a unique morphism X: P "O'/jA; such that aA = p. iel It follows that a is a monomorohi sin .

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0.7. Remar k. The above two definitions are mentioned because of the fundamental role they play in the sequel. They are special limits and are perhaps the most important limits in the categories that will be considered in tnis work. The following theorem is a special case cf a more general theorem dealing with the commutation of limits which can be found in Herrlich and Strecker [9J . A variation of the theorem will be proved in Section 1 (1.5) . 0.8. T heore m. Let {(A., a.): iel} and {(B^b.): iel} be families of subobjects of C -objects X and Y respectively. Then if S has finite products and arbitrary intersections then (,C\ A.)x( f\ B. ) and (~\ (A. xB^) iel iel ' iel are canonically isomorphic. 0.9. Notation. Let {X.: iel} be a family of £ -objects and suppose |2 — v X.: iel} is a family of fe -morphisms. Then by the definition of product there exists a unique morphism h from Z to TT Xj such iel that if.h = f . for each iel. This morphism h shall be denoted by < f< > i^I Let A and B be g -objects and suppose that a: A rX and b: B > Y are g -morphisms. If P, and P„ are the projection morphisms from A X B to A and B respectively then aP 1 : A X B — ^ X and bP 2 ^ A*B — — —> Y, hence by the definition of product there exists a unique morphism g from A X B to X X Y such that 7r 1 g = ao and TT g = bp2« This morphism g shall be denoted by axb and shall be called the prod uct of a and b . Let f be a h -morphism from X to Y. If f is a monomorphism then the following notation shall be used: x >„. __: _^ Y

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10 If f is an epimorphism then the following notation shall be used f X ~?oY If f is an equalizer then the following notation shall be used X»>~ -*• i If f is an isomorphism then the following notation shall be used X"*-v.> Y be d — >Y, X — — * Z, and Y— — > W 0.10. P roposition . Let A — S»X, B be £? -morphisms. Then (c*d) (a*b) = ca-db. a b 0.11. Proposition. Let A — — — > X and B > Y be monomorphisms (respectively, sections, isomorphisms) then a*b is a monomorphism (section, isomorphism) . 0.12. Remark. A partial order may be defined on the subobjects of an object in j^ in the following way: If X is a fe -object and (A, a) and (B,b) are subobjects of X; i.e., a and b are monomorphisms with codomain X and domains A and B respectively, then (A,a) <_ (B,b) if and only if there exists a morphism c from A to B such that be = a . B >4 I A .-v X By an abuse of language, if (A, a) < (B,b) then (K.b) is said to contain (A, a) and the morphism c is sometimes called the inclusion of (A, a) into (B,b). It is easy to see that if (A, a) <^ (B,b) an J (B,b) < (A, a) then the morphism c is an isomorphism. In this case,

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11 (A, a) and (B,b) are said to be isomorphic as_ subobjects of X. This is a stronger condition than A and B just being isomorphic objects in the category g . The following notation shall be used to denote the case where (A, a) and (B,u) are isomorphic as subobjects of X: (A, a) I (B,b). Sometimes it is written (inaccurately) that A < B or that A and B are isomorphic as subobjects of X. When this is done, the morphisms a and b should be clear from the context. It is immediate that (A, a) = (B,b) if and only if (A, a) <_ (B,b) and (B,b) <_ (a, a). Thus the relation "_<" on subobjects is easily seen to be a partial order up to isomorphism as subobjects. 0.13. Defini tio n. Let f from X to Y be a fc -morphism. f is an ext remal mo n o mo r p h i s m if and only if f is a monomorphism and whenever f = gh and h is an epimorphism then h is an isomorphism. If f is an extremal monomorphism the following notation shall be used: f X >>— >Y The dual notion is that of an extremal e pimorphism and is denoted: X — «$*> Y If f is an extremal monomorphism f: X — -*-Y, then (X,f) is called an extre mal sub obj ect of Y. 0.14, Remark . The definition of extremal monomorphism is due to Isbell {^llj . The concept of extremal monomorphism is important since it yields what shall be called the "image" of a morphism (see 0.18),. 0.15. Example? . In the categories Set, Grp , Ab and _FGp_, extremal monoiticrpbi.jrr.s are precisely the monomorphisms (i.e., one-to-one morphisms).

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12 In the categories Top and CpT extremal monomorphisms are precisely the embeddings. In the category Top they are the closed embeddings. f . 0.16. Pro posit ion. If X — *-Y is a g -morphism such that f = gh and f is an extremal monomorphism then h is an extremal monomorphism. 0.17. Proposition. If X *-Y is a fe -morphism then the following are equivalent: 1) f is an isomorphism, 2) f is an epimorphism and an extremal monomorphism, 3) f is a monomorphism and an extremal epimorphism (c.f. 0.3). 0.18. Def ini tion. A category j£> is said to have the unique epiex trema l mono facto rizatio n property if for any )° -morphism X— s»Y, there exist an epimorphism h and an extremal inonciuorphism g with f = gh such that whenever f = g'h' where g' is an extremal monomorphism and h ? is an epimorphism then there exists an isomorphism a such that the following diagram commutes. If £ has the unique epi-extremal mono factorization property and if f = gh where h is an epimorphism and g is an extremal monomorphism, then the pair (h,g) shall be used to designate the epi-extremal mono factorization of f. The extremal subobject (Z,e) of Y is called the

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13 image of X under f . Sometimes (Z,g) is referred to as the image of f . The notion of the uniqu e extremal epi-mono factorization proper ty is defined dually. If H has the unique extremal epi-mono factorization property and f = gh where g is a monomorphism and h is an extremal epimorphism then the pair (tug) shall be used to designate the extremal epi-mono factorization of f . The subobject (Z,g) of Y is called the sub image of X u nder f. Sometimes (Z,g) is referred to as the subimage of f . 0.19. Definition . A category g is said to have the diagonalizing pr operty if whenever gh = ab such that h is an epimorphism and a is an extremal monomorphism, then there exists a (necessarily unique) morphism t, such that E, h = b and a £ = g. I **• Y v W >•> > Z 0.20. Theorem. Let C be a locally small category having equalizers and intersections. Then the following are equivalent: 1) ig has the unique epi-extremal mono factorization property, 2) g has the diagonalizing property, 3) the intersection of extremal monouiorphisms is an extremal monomorphism and the composite of extremal monomorphisms is an extremal monomorphism, 4) if g 5 has pullbacks and if (P,q,3) is the puliback of f and g where fg = get and f is an extremal monomorphism then a is an extremal

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14 monomorphism. 5) if ^ has (finite) products then the (finite) product of extremal monomorphisms is an extremal monomorphism. 0.21. Theorem . If Y> is locally small and has equalizers and intersections then t* has both the unique epi-extremal mono factorization property and the unique extremal epi-meno factorization property. Pr oof . (sketch) . First we will show the existence of the unique extremal epi-mono factorization property. If f from X to Y is any fc> -morphism then let (OE.,e) be the intersection of the family {(E.,e.): ieJ} of all subobjects of Y through which f factors. Then it follows that e is a monomorphism and that f factors through e; i.e., there exists a morphism h such that f = eh. Now, to see that h is an epimorphism suppose a and 3 are \% -morphism? such that ah = gh. Let (E,k) be the equalizer of a and (3. It follows from the definition of equalizer that there exists a morphism g such that kg h. »* Y Thus it follows that f factors tHrough ek and sines ek is a monomorphism then there exists a morphism X: f\ E. * E such that ek\ = e From this it follows that k is rn isomorphism whence a = 3 and so h is an epimorphism. Next it will be shown that h is an extremal epimorphism. Suppose h = h 1 h ? where h., is a monomorphism. Then eh., is a monomorphism through

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15 which f factors. From this it follows, as above, that h is an isomorphism and hence h is an extremal epimorphism. Suppose f = g'h' where g' is a monomorphism and h 1 is an extremal epimorphism. Then since g' is a monomorphism vhrough which f factors there exists a morphism x from P\E. to che codomain of h' (domain of g') such that e = g'x. Since e and g' are monomorphisms, it follows that h' = xh and that x is a monomorphism. Since h' is an extremal epimorphism it follows that x is an isomorphism. Thus fe has the unique extremal epi-mono f actorizaticn property. Now suppose that ge = mE where e is an epimorphism and m is an extremal monomorphism. It will be shown that there exists a morphism o from the codomain of e to the domain of m such that oe = h and mo g. Let (f\k.,a) be the intersection of the family {(A., a.): isl} iel of all subobjects of the codomain of g (codomain of m) through which g and m factor. This family is non-empty since both g and m factorthrough the identity morphism on the codomain of g. It follows that both g and m factor through a. Thus there exist morphisms a, and a such that the following diagram commutes. m It will be shown next that £„ is an epimorphism. Suppose ot* and are L> -morphisms for which ot*a„ 3*a.. Let (E*,k*) be the equalizer

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16 of a* and 3*. Jt follows from the definition of equalizer that there exists a morphism b^ such that k*!^ = a 2 since 0*3 = B*a_. Since the diagram commutes it follows' that a*a e = 3*a,e. But e is an epimorphism hence a*aj = £ v a so that by the definition of equalizer there exists a morphism b~ such that k*b~ = a-^. Thus it follows that m = ak*b-, and g = ak*b2 anc ^ so both m ant ^ S factor through ak* from which it follows that k* is an isomorphism. Hence a* = g* and a, } is an epimorphism. But in is an extremal monomorphism and m = aa and a„ is an epimorphism. Thus a is an isomorphism. Thus defining a = a^a it follows that the following diagram commutes and ^ has the diagonal ization property. Hence K has the unique epi-extrema.l mono factorization property (0.20) 0.22. T heorem . Let K be any category then the following are equivalen 1) t^ is (finitely) complete, 2) Q^ has (finite) products and (finite) intersections, 3) fe> has (finite) products and equalizers, A) Jg has (finite) products and pullbacks. 0.23. Definition. A category 7^ is said to be quasi-complete if j^ has finite products and arbitrary intersections. t:

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17 0.24. Exa mples . The categories FSet and FGp are quasi-complete categories which are not complete. The categories Set, Top , Top ^ , CpT ? , Crp, Ah, Ring , and SGp are quasi-complete. 0.25. Re marks . A quasi-complete category is finitely complete but is not necessarily complete as the examples FSe t and FGp above show. Also, a locally small, quasi-complete category has both the unique extremal epi-mono factorization property and the unique epi-extremal mono factorization property (0.20 and 0.21). It can be shown that the unique epi-extremal mono factorization of a morphism can be obtained by taking the intersection of all extremal monomorphisms through which the morphism factors. It has been shown that the unique extremal epi-mcno factorization property is obtained by taking the intersection of all subobjects through which the morphism factors (0.20). These characterizations shall be used frequently in the sequel.

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SECTION 1. GENERALITIES .1.0. Sta nding Hypothesis . Througho ut the entir e paper it will be assumed that /» jls a_ lo cally small , quasic omplete (fi nite prod ucts and arbitrary intersections) c ategory . As noted in the preliminary section fe enjoys the unique epi -extremal mono factorization property. 1.1. Exampl e s. Many well known categories are locally small, and quasicomplete. Among such are the categories: Set , Top. , Top , Grp, Ab, SGp_, SGjp 1 , Rng, Rng 1 , CpT 2 , and FGp . 1.2. Defi nition. Let X and Y be g -objects. A relation R from X to Y is an extremal subobject of X>
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19 In the categories Top ,, , and CpT ? , relations are closed subspaces of the Cartesian product together with the inclusion map. 1.5. Proposition , Let X and Y be K -objects and let (A, a) and (B,b) be extremal subobjects of Y. Then Xx(AftB) and ^X^AjCi (X X B) are isomorphic relations from X to Y. Proof . Consider the following commutative diagrams. (X*A)f\ (XxB) Consider also (Xx(AAB), l y x c = Yi)Since extremal subobjects are closed un-ler intersections and products (0.20; y^ and Yo ar e extremal monomerphisms. Since (l x x a) (! x xA A ) = l x *c = z Yj and (l x xb) (l x x * B ) = l x x c = Yj then by the definition of intersection there exists a unique morphism o from Xx(AAB) to (XxA)/-\(XxB) so that Y 9 °" = Yj and the following diagram cc mutes. Thus: -jm(Xx(AAB), Yi ) 1 ((XxA)n(XxB), Y? ) '2'

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20 X*A Xx(A.^B) XxB Nov let (it , it ), (tt' it'), (o , p ) and (p' pp be the projections of XxY, Xx(AAB), XxA, and XxB respectively. Observe that: V-j " ^(V a)X l = Vl X l = *l (1 X Xh)X 2 'Vl X 2 V2 = 1r 2 (1 X XaU l = Sp 2 X l = ^2 (1 X Xb)X 2 = bp 2 X 2' Thus by the definition of intersection there exists a unique morphism £ from (XxA) A (XxB) to AAB such that r.E = tt„y„ and thus by the definition £ • 2 of product there exists a unique morphism £ from (XxA)A(Xx B) to Xx(AAft) such that £ = ; i.e., tt'^ = 1 y« and ir^ = E. Now y,? = (l„ x c)5 hence tTjY^ ri ^l^ = n l Y 2 and Tr 2 Y l ? = ClT 2 ? = cI = ^2' ThuS Y l^ = ''2' wheuc ((XxA)A (XxB), y) < (Xx(AriB), y) *it 2 (XxA)AXxB)

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21 1.6. Notation. Let X and Y be t -objects and let (X*Y,7T ,it ) and (YxX,p ,p ) be the indicated products of X and Y. Then there exists a unique isomorphism from X*Y to YxX, denoted by , such that the following diagram commutes. Xx Y V> ^ — ~ X »— 4> YxX I VtY Note that: 2 .T^xp^p^ = l y> XxY' 1.7. Definition. Let (R,j) be a relation from X to Y and let (t,j*) be the unique epi-extremal mono factorization of j (see 0.18). The codomain of x (domain of j*) is denoted by R1 and (R~ ,j*) is called the inverse relation of (R,j) or more simply, when there is little likelihood of confusion, the inverse of R. £/YxX 1.8. E xamp le. In the categories S3t , Topi , TopT , Grp , Ab, and FGp ti 1 >: X.xY — — — — — — -> YxX is defined by (x,y) = (y,x); hence, if (R,j) is a relation from X to Y then R _1 {(y,x): (x.y)£R} with j* tiie inclusion map of R -1 into YxX. 1.9. Jrx>p_osit_iop. Ii (R>j) is a relation from X to Y the R and R -i are iso^cr^hic obiects of f* .

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22 Proo f . Since <7i2.ifi> is an isomorphism and j is an extremal monomorphism then j is an extremal monomorphism. But 7r l > J = J* T * Thus since i is an epiruorphism then from the definition of extremal monomorphism it follows that t is an isomorphism. 1.10. Def inition. If (P M j) is a relation from X to X then R is said to be symm etric if and only if (R ,j*) <^ (R,j). 1.11. Proposition. Let (R,j) be a relation from X to Y. Then the inverse relation ((R~ )~ ,j#) of (R ,j*) and (R,j) are isomorphic relation?.. Pro of . Consider the following commutative diagram. R>> R-l >£ J# — *> XxY ->YxX

-*. XxY Since the two inner squares commute the outer rectangle commutes. Both of t and t# have been shown to be isomorphisms (1.9). And, as also has been observed: -po >Pi ><7T 2 j 71 ! > ~ ^-XxY (1*6) • Consequently, t;'/t is an isomorphism and j j#(t#t). Thus (R,j) I ( (R -1 ) _1 , j//) . 1.12. Proposition. Let (R,j) and (S,k) be relations from X to Y, Then (S,k) <_ (R,j) if and only if (S -1 s k*) < (R" 1 ,^*). Proof. Consider the following commutative diagram.

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23 R -*» Y*X If (S,k) < (R,j) then there exists a morphism a: S •* R such that jet = k. Define 3 = tot . Then j*B = j*xaT 1 = jo;t 1 = k*TT -1 = k*. Thus (S -1 ,k*) _< (R -1 ,j*). If (S -1 s k*) <_ (R -1 ,j*) then by the above, ((S -1 ) _1 ,k#) _^ ((R -1 )1 ,^) thus (S,k) < (R,j) (1.11). < TT 2 ,TT 1 >kf 1.13. Corollary. If (R,j) is a symmetric relation on X then (R,j) < (R _1 ,j*) whence (R,j) E (R -1 ,j*). Proof. Since (R,j) is symmetric (R -1 ,j*) ± (R,j). Thus (R,j) = ((R" 1 )1 ^*) 1 (R _1 ,j*) (1.11 and 1.12). Consequently (R,j) = (R -1 ,j*). 1.14. Definition. Recall that since g* is quasi-complete it has equalizers, thus for each g -object X let (A, , i ) denote the equalizer of t-, and 7T where tt-, and tt„ are the projections of X*X. Since i is an equalizer it is an extremal monomorphi sm. Hence (A„.i v ) is always a relaA ' A tion on X (called the d iagonal of X*X) . A relation (R,j) on X is said to be re flexive on X provided chat (A x ,i y ) < (R,j). 1.15. Exa mple . In the categories Grp, Ab, Set, Top , To p ? , and CpT~ , it follows that A v z {(x,x): xeX} C XxX with the inclusion map. A

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24 1.16. Pro position . For any (X -object X, <1, 2> TI l > ix ~ *X' 1' nus; (A x ,i x ) E (A x -l,i x *). Proof. Consider the following commutative diagram. XxX » <7To j TTi > ^>» XXX TT 1 ± x = rr 2 i x = TT X i x = 7t 2 i x . Thus the epi-extremal mono factorization of ' i is (i. ,:'). A Ay A 1.17. Corollary . Let (R,j) be a relation on X, then (R,j) is reflexive on X if and only if (R ,j*) is reflexive on X. Proof. If (A x ,i x ) 1 (R,j) then (A x ,i x ) = (A x _1 ,i x *) < (R _1 , j*) (1.16 and 1.12). Conversely if (A x ,i x ) <_ (R _1 , j*) then ( A X>%) E (V 1 '^")! ((R" 1 )" 1 ,:?/) = (R,j) (1.16, 1.12. and 1.11). 1.18. Propo sition . Let (R,j) and (S,k) be relations from X to Y. Then the relations (RHS)" 1 and (PC l r\S~ l ) are isomorphic relations. Pr oof . According to the definitions of intersection and inverse relation we have the following commutative diagrams. > XxY Nt-. s-r > YxX

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25 RAS >=?» X*Y »^(RAS)1 * 1