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Title:
Categorical embeddings and linearizations
Creator:
McDill, Jean Marie
Publication Date:
1971
Language:
English
Physical Description:
viii, 99 leaves. : illus. ; 28 cm.

## Subjects

Subjects / Keywords:
Automorphisms ( jstor )
Coordinate systems ( jstor )
Embeddings ( jstor )
Hausdorff spaces ( jstor )
Homomorphisms ( jstor )
Isomorphism ( jstor )
Mathematics ( jstor )
Monoids ( jstor )
Morphisms ( jstor )
Topological theorems ( jstor )
Categories (Mathematics) ( lcsh )
Dissertations, Academic -- Mathematics -- UF
Mathematics thesis Ph. D
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis -- University of Florida.
Bibliography:
Bibliography: leaf 98.
Also available on World Wide Web
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Manuscript copy.
General Note:
Vita.

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Full Text

CATEGORICAL EMBEDDINGS
AND
LINEARIZATIONS

By

JEAN MARIE McDILL

A DISSERTATION PRESENTED TO THE GRADUATE C, NCIL OF
fHE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1971

To Bill

and

to Kathleen,

for her birthday

ACKNOWLEDGMENTS

The author would like to thank the Chairman of her Supervisory

Committee, Dr. G. E. Strecker, for his thoughtful encouragement and

many helpful suggestions in preparation of this paper over the

obstacles of distance and time. She would also like to thank

Dr. Z. R. Pop-Stojanovic, Dr. T. O. Moore (who introduced her to

topology), Dr. B. V. Hearsey, Dr. D. A. Drake and Dr. Max H. Kele for

serving on her Supervisory Committee, as well as A. R. Bednarek and

W. E. Clark, who, at various times, have served on the committee.

With gratitude the author wishes to acknowledge the encourage-

ment and unyielding support she has received from her husband. She

would also like to thank the other members of her family who have

aided her work on this dissertation: Mr. and Mrs. P. S. Willmcre,

Mr. and Mrs. J. R. Hutchinson and Kathleen, her daughter, fo ....om

this undertaking has lasted a lifetime.

The author is also very much indebted to Mrs. Karen Walker

for typing this paper.

ACKNOWLEDGMENTS .................................... ......... .

LIST OF CONCRETE CATEGORIES...................................

ABSTRACT ............................................... .......

INTRODUCTION....................................................

CHAPTERS:

1. PRELIMINARIES ........... ....... .......................

1.1 Products and Limits..............................

1.2 Special Categories................................

1.3 Special Morphisms in General Categories.........

1.4 Epireflections................................

1.5 Concrete Embeddings....................... ... ..

2. SOURCES ................................. ............

2.1 Sources in General Categories....................

2.2 Sources in Concrete Categories...................

2.3 r -Sources .......................................

3. EMBEDDINGS INTO PRODUCTS...............................

3.1 Embedding Theorems ..... ...................

3.2 el-T-Embeddable Objects ........................

3.3 7-Coseparating Classes..........................

3.4 )(-Regular and 0(-Compact Objects................

4. LINEARIZATIONS ........................................

4.1 Coordinate Immutors and Permutors ................

4.2 Mt-Linearizations...............................

Page

4.3 Universal M-Linf.rizations..................... 91

BIBLIOGRAPHY............................................ ..... 98

BIOGRAPHICAL SKETCH.................. ............... .............. 99

LIST OF CONCRETE CATEGORIES

Ab The category of Abelian groups and group homomorphisms

AbMon The category of Abelian monoids and monoid homomorphisms

CabMon The category of cancellative Abelian monoids and monoid
homomorphisms

CompT2 The category of compact Hausdorff spaces and continuous
functions

CompRegT1 The category of all completely regular T1 spaces and
continuous functions

Field The category of all fields and field homomorphisms

Grp The category of all groups and group homomorphisms

Haus The category of all Hausdorff spaces and continuous functions

Ind The category of all indiscrete spaces and continuous
functions

Mon The category of all monoids and monoid homomorphisms

POS .The category of all partially ordered sets and order-
preserving functions

RComp The category of all realcompact spaces and continuous
functions

Set The category of all sets and functions

Sgp The category of all semigroups and semigroup homomorphisms

Top The category of all topological spaces and continuous
functions

Top, The category of all T1 spaces and continuous functions

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

CATEGORICAL EMBEDDINGS
AND
LINEARIZATIONS

By

Jean Marie McDill

August, 1971

Chairman: Dr. G. E. Strecker
Major Department: Mathematics

A recent definition for embedding morphisms in concrete

categories is examined. Several types of sources are defined for

which the source morphisms in the aggregate exhibit the character of

single morphisms with 'embedding-like' properties. A theorem giving

.-cessary and sufficient conditions for embedding an object into a

categorical product of objects is proven for a variety of 'embedding-

like' morphisms. The concept of embeddable objects is examined,

and a definition is developed for t-regular and e-compact objects

in concrete categories. In consequence, several characterization

theorems for epireflective subcategories and epireflective hulls,

which previously had been proven only for certain categories of topologi-

cal spaces,are extended to a variety of "reasonable" concrete

categories.

Baayen's generalizations of de Groot's results, on the

existence of universal linearizations for monoids of endomorphisms

on completely regular T1-spaces, are extended. For every isomorphism-

closed, left-cancellative class / of morphisms in a category with

countable products, every endomorphism on an object in the category

is shown to be a restriction, relative to M,of a coordinate-immuting

endomorphism on a power of an object in the category. Under the same

conditions, an automorphism in the category is shown to be the

restriction of a coordinate-permuting automorphism. Furthermore,

simultaneous M-linearizations are shown to exist for certain monoids

of endomorphisms on objects in the category, and universal

t-linearizations are shown to exist for every endomorphism in

certain subcategories of the category.

viii

INTRODUCTION

The "mathematical-man-in-the-street," when asked what he would

consider an embedding morphism to be in any concrete category, would

probably reply, if he replied at all, that it would have to be

injective on the underlying sets and would have to preserve the

structure of the embedded object; i.e., it would have to be an

actual subsystem embedding. It has been somewhat difficult to find a

categorical definition for a morphism that does just that. Several

types of morphisms have been defined which act as subsystem embeddings

in some categories but are too weak or too strong in other categories.

Monomorphisms in algebraic categories usually act as embeddings, but

they are not embeddings in the topological sense in Top; extremal

monomorphisms are precisely the topological embeddings in Top, but

they are too restrictive in Haus where they are homeomorphisms onto

closed subspaces, and consequently both sections and regular mono-

morphisms are too restrictive in Haus. Within the last year, Herrlich

and Strecker [7] developed the definition for concrete embeddings in

a concrete category, which they found to be precisely the topological

embeddings in Top and Haus and to be precisely the monomorphisms in

algebraic categories (1.2.2). In Chapter 1, 1.5, we shall begin to

examine the concept of concrete embeddings; in Chapter 3, we shall begin

to exploit it. Several new results will be proved in 1.5. We will

find that concrete embeddings are always injective on the underlying

sets, and that concrete embeddings tend to have stable hereditary

properties; i.e., that concrete embeddings in many subcategories tend

to be concrete embeddings in the larger category, and conversely.

Frequently, when dealing with concrete embeddings,we restrict

our attention to concrete categories (C ,' ) in which the faithful

functor Ut:C Set preserves products or monomorphisms. This

restriction is fairly weak; in most ordinary concrete categories, the

functor U has a left-adjoint and hence preserves all limits (and

hence all monomorphisms too).

In Chapter 2, we will develop the notion of sources. A

source in a category e is a C-object X together with a family of

C-morphisms having X as their common domain. A categorical product

( n Ei,i) is an example of a source which, in fact, exhibits a
idI
"mono-like" property; i.e., if f and g are morphisms in the category

such that i .f = iT*g for each iEl, then f = g. We call a source,

whose morphisms acting in concert exhibit the property of a single

monomorphism, a mono source. Herrlich and Strecker [7] have developed

definitions for mono sources and extremal mono sources. In Chapter 2,

we shall define several new types of sources (called, collectively,

t-sources) whose morphisms in the aggregate exhibit the properties of

special types of "embedding-like" morphisms (4n-morphisms).

In topology, a long-established method for producing mappings

into topological products has been by use of families of morphisms

into the coordinate spaces. Tychonoff's well-known result, that

for any infinite cardinal number k, a completely regular T1 space X

of weight < k can be topologically embedded into the topological

product [0,1]k of unit intervals, made use of the family C[X,[0,1]]

of all continuous functions from X into the unit interval [0,1].

His result stimulated several mathematicians to investigate this

general procedure for topologically embedding a space into a product

of spaces. In 1956, Mrowka [13] published a theorem giving necessary

and sufficient conditions for a function to be a topological embedding

of a space into a topological power of another space. More recently

[16], he expanded this result to include embeddings of a space into

products of other spaces, as well as characterizing topological embeddings

onto closed subspaces of products of Hausdorff spaces. From our vantage

point in category theory, we can see that these theorems concerned

sources: that certain conditions on the source (X, ) where 3 is a

family of continuous maps from the space X into coordinate spaces of

a topological product are necessary and sufficient to guarantee the

existence of a homeomorphism from X onto a subspace of the product.

In Chapter 3, we will prove a similar embedding theorem for categories

(3.1.1) for a variety of "embedding-type" morphisms.

In 1958, Engelking and Mrowka [3] began to develop the concepts

of E-regular and E-compact spaces in response to two intriguing

questions: (1) for a given space E, when can a space X be topologically

embedded into some topological power of E? (When is X an E-completely

regular space?), and (2) for a given Hausdorff space E, when is a

space X homeomorphic to a closed subspace of some topological power of

E? (When is X an E-compact space?) Later Herrlich [4] expanded

these investigations to include the concepts g-regular (respectively,

e-compact) spaces for collections e of spaces (respectively,

Hausdorff spaces), initiating the use of a category-theoretic approach.

In 3.2, we shall characterize objects which are "embeddable" in

products of other objects. In particular for a concrete category

(C,0L) with a subcategory we will define a C-object X to be

i-regular in C (respectively, (-compact in C ) provided that there

exists a concrete embedding (respectively, an extremal concrete

embedding) from X into the object part of a product of [-objects.

For the first time, the concepts of -regular and &-compact objects

can be applied to categories other than Top or Haus.

The most interesting applications of these concepts will be

in the realm of epireflective subcategories. Recall that a subcategory

( is epireflective in a category C provided that for every C-object

X there exists a pair (r X,), called the 6(-epireflection of X,

where X is an a-object and r :X -* X is an epimorphism with a

maximal extension property for a (In this paper, we will say that

r is O-extendable" provided that for every morphism g:X A for

some Of-object A, there exists a morphismg :X A such that g .r = g.)

Herrlich and Strecker [8] have characterized epireflective subcategories

in the following fashion: in a complete, well-powered and cowell-

powered category C with a full, replete subcategory (%, 8( is epireflec-

tive in C if and only if it is closed under the formation of extremal

subobjects and products in C This theorem gives us a large array of

epireflective subcategories. The actual construction of the epireflection

can be quite difficult. Usually the construction takes one of two

forms: (1) "factoring out" the elements in the object with undesirable

characteristics, e.g., the Abelianization of a group G by factoring

out the commutator subgroup Gc of G (a:G G/Gc), or the Hausdorffication

of a topological space X by finding the appropriate equivalence relation

R so that X/R is a Hausdorff space with the "universal" mapping

property, or (2) constructing a "larger" object with the desired

structure and finding an epimorphism from the object to the "larger"

object, as in the Stone-Cech construction, B:X BX, which gives a

CompT2-epireflection for a Hausdorff space X and which gives a

topological embedding,as well, when X is a completely regular TI space,

or as in the construction of the Grothendieck group K(M) for an Abelian

monoid, k:M N K(M), for which k will be an injection, as well, when

M is a cancellative Abelian monoid. It has always been interesting to

note that in certain cases, the second method described above results

in epireflections that are actual subsystem embeddings. Both methods

of constructing epireflections have been used for both topological and

algebraic categories; yet, until now, there has been no general

categorical way to differentiate between these two methods and to

determine for what objects in the category the epireflection

epimorphism will be an actual subsystem embedding. For the category

Haus, Herrlich and Van der Slot [10] were able to show that a full,

replete subcategory 6( is epireflective if and only if for every

0(-regular object X in Haus there exists an epimorphism r:X A to

some C(-object A such that r is an S(-extendable topological embedding.

In 2.4, we will extend this result to one for "reasonable" concrete

categories, and hence will be able to add a new characterization for

epireflective subcategories in those categories. Furthermore, we will

extend several other theorems (which Herrlich [5] proved for the

category Haus) which will characterize Ot-regular and 8(-compact

objects in "reasonable" concrete categories.

Tychonoff's embedding result, mentioned above, gave rise to

another area of investigation. De Groot [2] proved that for a given

infinite cardinal number k, there exists a monoid F of endomorphisms

on [0,]k which "universally linearizes" any monoid S of at most k

endomorphisms on any completely regular T1 space X of weight < k.

Baayen [1] generalized this result to obtain several theorems in

categorical terms, restricting his considerations to monomorphisms for

general categories and topological embeddings for Top, which he had

to consider separately.

In Chapter 4, we will extend and update Baayen's theorems, but,

in particular, we will change the emphasis of the investigation. We

will not be searching for universal objects in categories as Baayen did;

instead, we will obtain quite general categorical linearization theorems

for endomorphisms (and automorphisms) on objects in a category.

Considering categories in which only certain products are required to

exist, we will find that endomorphisms are actually restrictions of

morphisms of an almost trivial-seeming linear character (i.e.,

morphisms that essentially act only on the coordinates of a power of

an object, serving to "switch or collapse" these coordinates).

We will find that in categories with countable products, every

endomorphism in the category can be linearizedd', i.e., extended to a

coordinate immuting morphism (4.2.10), and that any automorphism can be

considered as the 'restriction' of a coordinate permuting automorphism

on a power. Furthermore, we will find that certain monoids of

endomorphisms, as well as groups of automorphisms, on an object in the

category can be simultaneously linearized (4.2.4). And finally, we

will find that for certain subcategories in a category with infinite

products, universal linearizations may exist for every endomorphism

in the subcategory (4.3.2).

1. PRELIMINARIES

The purpose of this chapter is to list most of the basic

definitions and theorems that will be used in the following chapters.

Proofs will be included only for new results, which will be found in

1.5. Terms from category theory not specifically defined here can be

found in [7], as well as the proofs for those theorems and examples

that are stated here without proof or reference. Algebraic terms can

be found in [12]; topological terms can be found in [11].

1.1 Products and Limits

DEFINITION 1.1.1 A family (Ai)icI indexed by I is a function

A with domain I. For each iel, A(i) is usually written Ai.

Occasionally when I is unimportant or understood, the family (Ai)ic

is written (A ).

DEFINITION 1.1.2 Let 0 be a category. A product in C of

a set-indexed family (Ai)iI of C -objects is a pair (usually denoted

by) ( n Ai, (i ) ) satisfying the following three properties:
i' i iC
iel1
(1) n Ai is a C-object
ieI
(2) for each jel, T.: Ai A is a C-morphism (called the
it1
projection from I A to A ).
il
(3) for each pair (C,(fi)i), where C is a C-object and

for each jel, f.:C A. is a -morphism, there exists a unique

induced C-morphism (usually denoted by) : C -* Ai such that for
each jl, the triangle
each jEl, the triangle

I Ai
f i
C_-----1---- ^

A.
J

commutes.

For notational convenience we may sometimes write (IAi, i) for

the product. Also when Ai = B for each icl, the product (IAiA,i) is

usually written (B Ti) and is called the I'th power of B in C A

category C is said to have products provided that for every set I,

each family of d -objects indexed by I has a product in C The

following categories have products: Set, Grp, Top, Haus, Topl, CompT2,

CompRegT1, Mon, Sg.

THEOREM 1.1.3 (Iteration of Products) Let (Ki) iI be a

pairwise disjoint set-indexed family of sets. Suppose that for each

iEl, (Pi ,(nk)rK ) is the product of a set-indexed family (k)kKi of

S-objects and that (P, ni) is the product of the family (P )iEI.

Then (P,( .i kK) is the product of (X)kV Ki

isomorphism.

PROPOSITION 1.1.4. If (nAi'ni) and (lBi,Pi) are products of the

families (Ai)iCI and (B )ic, respectively, and if for each icI there

is a morphism A Bi, then there exists a unique morphism (usually

denoted by Hfi) which makes the diagram

nfi
"Ai --- B-----i
11A -

i

B B

commute for each jcI.

DEFINITION 1.1.5. The morphism nfi of Proposition 1.1.4 is

called the product of the morphisms (f )iEl.

Products are just a special type of limit which will be defined

next.

DEFINITION 1.1.6. Let I and 0 be categories and let

D:I + C be a functor.

(1) Then a pair (L,(f)i.sb(I)) is called a natural source

for D in C provided that L is a C-object, and for each iOb(I),

fi:L D is a C-morphism, and for all i,jEOb(I) and I-morphisms

m:i j, the triangle

f, D(i)

L D()(m)

Dj(j)

commutes.

(2) A natural source (L,fi)io0b(I)) for D in C is called a

limit for D in 6 provided that if (L ,( i) iOb(I)) is any other natural

source for D in C then there is a unique morphism h:L L such that

for each jEOb(I), the triangle

h D(j)

L fj

commutes.

Some well-known examples of limits are: products, terminal

objects, equalizers, pullbacks, and intersections.

DEFINITION 1.1.7. A category C is said to be complete

provided that for every small category I (i.e., where Ob(I) is a set),

every functor D:I C has a limit.

(The categories Set, Grp, Mon, Top, Haus and CompT2 are

examples of complete categories.)

1.2 Special Categories

DEFINITION 1.2.1. A category C is said to be well-powered

provided that every C-object has a representative class of subobjects

which is a set.

Dual notion: cowell-powered

The categories Set, Grp, Top, Mon, Haus and CompT2 are well-

powered and cowell-powered.

DEFINITION 1.2.2

(1) A concrete category is a pair (C ,U) where C is a

category andU : Set is a faithful functor.

(2) A concrete category (0(,U) is called algebraic provided

that it satisfies the following three conditions:

(a) 3 has coequalizers.

(c) 'U preserves and reflects regular epimorphisms.

Examples of concrete categories include Set, Grp, Top, Haus,

POS, Field and Mon. Top, Haus, POS and Field are not algebraic

categories; however, Set, Grp, Mon and CompT2 are algebraic.

51.3 Special Morphisms in General Categories

DEFINITION 1.3.1. Let C be a category.

(1) A C -morphism m is called a strong orphism provided that

whenever m*k = g*e for some C -morphisms k and g and some C-epimorphism

e there exists a morphism h such that the diagram

e

k g

m
commutes.

(2) A C-morphism m is called a strong monomorphism when it

is both a strong morphism and a monomorphism.

(3) A -morphism f is called an extremal morphism provided

that whenever f = g*e, where g is a C-morphism and e is a C-epimorphism,

e must be a C -isomorphism.

(4) A C -morphism is called an extremal monomorphism provided

that it is both an extremal morphism and a monomorphism.

PROPOSITION 1.3.2. Let C be a category.

(a) The product of strong (mono) morphisms is a strong

(mono) morphism.

(b) If f and g are strong (mono)morphisms and f-g is a

morphism, then f'g is a strong (mono) morphism.

Thus we say that the class of strong morphisms (respectively,

strong monomorphisms) is closed under products in C (1.3.2(a)) and

closed under composition in C (1.3.2(b)). In general, this is not

true for the class of extremal morphisms in C or for the class of

extremal monomorphisms in C.

Extremal monomorphisms are particularly interesting. The

following definitions and propositions will give us sufficient conditions

for which the class of extremal monomorphisms is closed under products

and composition.

DEFINITION 1.3.3. Let C be a category, V be a class of

monomorphisms closed under composition with isomorphisms, and I be

a class of epimorphisms closed under composition with isomorphisms.

(1) Let f be a morphism in C A factorization f = m.e where

e and m are morphisms in C is called an (t,~) factorization of f

provided that eEs and me?7.

(2) An (J ,m) factorization of f, f = mne, is said to be

unique provided that whenever f = m.e is another ( t,Mf) factorization

of f, there is an isomorphism h such that the diagram

jh

commutes.

(3) A category C is called an (,,mn) category provided

that t and M are closed under composition and every morphism in C

has a unique (,71f ) factorization.

(4) A category C is said to have the (t,m) diagonalization

property provided that for every commutative square in C

f g

m

with ec t and me m there exists a morphism k which makes the diagram

e
---g

fk -k
f ^- g

commute.

PROPOSITION 1.3.4. Every (4,7) category has the (.,fl)

diagonalization property.

THEOREM 1.3.5. Every complete, well-powered category C is

an (epi, extremal mono) category. Furthermore the class of extremal

monomorphisms in C is closed under products.

1.4 Epireflections

DEFINITION 1.4.1. Let & be a category and let 0( be a

subcategory of C .

(1) G is a full subcategory of C provided that whenever

f:A B is a t -morphism, and A and B are C(-objects, it follows that

f is an V -morphism.

(2) C is a replete subcategory of r provided that whenever

f:A B is an S -isomorphism and A is an O(-object, it follows that

B is an &(-object.

DEFINITION 1.4.2. Let C be a category and let C be a

subcategory of .

(1) A morphism f:X Y is called (t-extendable provided that

for each Ot-object A and each morphism g:X A there exists a morphism

g :Y A such that the diagram

X /Y

/*
g /g

A
commutes.

(2) Let X be a C -object. The pair (r ,X ) is called an

it-epireflection for X provided that X is an f-object and

r a:X XO is an (-extendable epimorphism.
(3) ., is called an epireflective subcategory of ( provided

that for every C-object X, there exists an G(-epireflection for X.

THEOREM 1.4.3 (Characterization of Epireflective Subcategories)

(Herrlich and Strecker [8]). Let C be a complete, well-powered and

cowell-powered category and let &( be a full, replete subcategory of

C. Then the following statements are equivalent:

(a) S is an epireflective subcategory of C.

(b) ( is closed under the formation of products and extremal

subobjects in C.

PROPOSITION 1.4.4. Let C be a complete, well-powered and

cowell-powered category and let t be any full, replete subcategory ofC.

Then there exists a smallest epireflective subcategory _(t) of C

which contains (.

DEFINITION 1.4.5. Let C be a category and let C be a sub-

category of C. The epireflective hull e(8() of Oin C (if it exists)

is the smallest epireflective subcategory of C, containing OC.

THEOREM 1.4.6 (Characterization of Epireflective Hulls)

(Herrlich [6]). Let C be a complete, well-powered and cowell-

powered category and let &( be any full, replete subcategory of S .

Let X be a -object. Then the following statements are equivalent:

(a) X is a C(ot)-object

(b) X is an extremal subobject of a product of (S-objects

(c) Each 6(-extendable epimorphism is (X}-extendable

(d) Each (O-extendable epimorphism f:X Y is an isomorphism

(e) Each a(-extendable morphism f:X Y is an extremal

monomorphism.

1.5 Concrete Embeddings

In this section, we will define concrete embeddings and will

determine a few of their properties.

DEFINITION 1.5.1. Let (C,u,) be a concrete category.

(1) A morphism f:X Y is called a concrete embedding

provided that it is a monomorphism and whenever there is a morphism

g:Z Y for which there exists a function h: 1(Z) U(X) such that the

diagram

U(z)

U(g)
h

U(x) > M(Y)
14(f)

commutes, there exists a morphism h:Z X such that 9 (h) = h.

(2) A morphism f:X Y is called an extremal (respectively

strong) concrete embedding provided that it is both a concrete embedding

and an extremal (strong) morphism.

PROPOSITION 1.5.2. Let (,'LU) be a concrete category.

(1) The class of all concrete embeddings in C is closed

under composition.

(2) If 9 preserves monomorphisms, the class of all concrete

embeddings in C is closed under products, intersections and pullbacks.

PROPOSITION 1.5.3. Let (C,, ) be a concrete category that is

complete and well-powered for which 1 preserves monomorphisms. Then

every extremal monomorphism is a concrete embedding, hence an extremal

concrete embedding.

COROLLARY 1.5.4. Let (C,U) be a concrete category that is

complete and well-powered for wnich U preserves monomorphisms. Then

(C,U) is an (epi, extremal concrete embedding) category.

In some concrete categories, monomorphisms are not always

injective on the underlying sets. We next show that concrete embeddings

do have the desired property of injectivity.

PROPOSITION 1.5.5. Let (C ,U) be a concrete category. Then

every concrete embedding in C must be injective on the underlying sets.

PROOF: Suppose that f:X Y is a concrete embedding, and hence a

monomorphism (1.5.1), but that 'U(f) is not injective on the underlying

sets. Then there must exist a,bE U(X), a # b such that U(f)(a)=tL(f)(b).

a for x=a, x=b
Define ha: U(X) U(X) such that ha(x) x for all xX, for which
I a x i b

Sb for x=a, x=b
and define hb: U(X) U(X) such that hb(x) = x for all xX
for which a # x t b.

Then both ha and hb make the diagram

U(x)

h I U (f)
ha hb

u(x) (Y)
Uf)

commute. Thus since f is a concrete embedding, there must exist

h :X X such that U(h ) = ha and hb:X X such that U (h) = hb
(1.5.1). But 'L is a faithful functor; hence it reflects commutative

triangles. Thus f.ha = f = f'hb so ha = hb since f is a monomorphism.

But this implies that h hb--a contradiction.

It is known ([7]) that concrete embeddings in Top are

precisely the topological embeddings, and that concrete embedding

in algebraic categories are precisely the monomorphisms. We will show

that these results can be used to characterize concrete embeddings in

PROPOSITION 1.5.6. Let (&,,Ut) be a concrete category, let

8t be any full subcategory of a and let f:X Y be an OS-morphism.

If f is a concrete embedding in C, then it,is a concrete embedding

in SL.

PROOF: Let f:X Y be a concrete embedding in 6 Then it is injective

on the underlying sets (1.5.5), hence it is a monomorphism in S.

Let g:Z Y be an St-morphism such that for some function

h:'U(Z) U(X), U(f).h = 1(g). But g is also a C-morphism and f

is a concrete embedding in C ; hence, there exists a C-morphism

h:Z X such that U1(h) = h. (1.5.1). Since 0(is full, h:Z X is

also an (-morphism.

PROPOSITION 1.5.7. Let C be an algebraic category, let aC be

any subcategory of C and let f:A B be an (-morphism. Then if f

is a concrete embedding in ( it is a concrete embedding in 6 .

Furthermore if a% is a full subcategory of C then f is a concrete

embedding in C if and only if it is a concrete embedding in f(.

PROOF: Suppose f is a concrete embedding in ( Then it is injective

on the underlying sets (1.5.5), and consequently it is a monomorphism

in thus a concrete embedding in (.

The remainder of the proof follows directly from Proposition

1.5.6.

PROPOSITION 1.5.8. Let (6,1) be a concrete category that is

complete and well-powered and for which ti preserves epimorphisms

and monomorphisms. Let a be a full, subcategory of C that is closed

under the formation of extremal subobjects in C and let f:A B

be an a-morphism. Then the following statements are equivalent:

(a) f is a concrete embedding in .

(b) f is a concrete embedding in eL.

PROOF: (a) =*(b): The proof follows directly from Proposition 1.5.6,

since S is a full subcategory of C .

(b) -.(a): Suppose f is a concrete embedding in St. Then,

since C is a complete well-powered category, there exists a unique

(epi, extremal mono) factorization of f, f = m.e (1.3.5). Let L

denote the domain of m.

L

e m

A B
f

By hypothesis, 'U preserves epimorphisms; thus Ul(e):U(A) -+ U(L) is

surjective. But U (f) is injective (1.5.5); so that U(e) must be

injective; hence bijective. Consequently there is a function

h: UZ(L) + t(A) such that tL(e).h = 1(L) and h. U(e) = 1(A). Thus

UL(f).h = U(m).

But (L,m) is an extremal subobject of B which is an 6K-object.

Hence by hypothesis, L is anO5-object, so m is an OC-morphism. Thus

since f is a concrete embedding in CC, there exists an h:L A such

that U 1() = h. Thus h*e:A A and 'U(he) = h. (e) = 1(A) = (IA).

Hence since U is faithful, hbe = 1A. Thus e is a section (and an

epimorphism); hence it is an isomorphism. Thus f is an extremal

monomorphism in C (1.3.1); hence it is a concrete embedding in

(1.5.3).

COROLLARY 1.5.9. In any full, hereditary subcategory & of

Top, the concrete embeddings are precisely the topological embeddings.

PROOF: The epimorphisms in Top are precisely the subjective continuous

functions,

2. SOURCES

In this chapter we will investigate the action of sources;

a source being an object in a category together with a family of

morphisms having the object as a common domain. We will see that in

several special cases the morphisms of a source will act together to

exhibit certain "mono-like" properties--properties which may not belong

to any of the morphisms individually. The definitions and initial results on

mono sources and extremal mono sources were developed by Herrlich and

Strecker [7].

2.1 Sources in General Categories

DEFINITION 2.1.1. Let C be a category.

(1) The pair (A,(fi)i ) is called a source provided that A

is a C-object and (fi)iEl is a family of C-morphisms, each with domain

A. Note that for notational convenience, we will usually write

(A,fi) for a source when the indexing class I is understood or unimpor-

tant.

(2) A source (A,fi) is called a mono source provided that for

any pair of C -morphisms h and k such that fi h fi.k for all iCl, it

follows that h = k; i.e., provided that the family (fi)iC) is

simultaneously left-cancellable.

(3) A source (A,fi) is called an extremal source provided that

for each source (B,gi) and each e-epimorphism e:A B such that for

each i, the diagram

A -~------------ B
A B

ci

commutes, e must be an isomorphism.

(4) A source (A,fi) is called an extremal mono source

provided that it is both an extremal source and a mono source.

(5) A source (A,fi) is called a strong source provided that

for each source (B,g.) and C-morphisms e and k, where e is a

C -epimorphism and gi*e = fi'k for each ilI, there exists a C-morphism

h such that the diagram

e
X --~-------->- B

k gi

A f CI

commutes for each iel.

(6) A source (A,fi) is called a strong mono source provided

that it is a strong source and a mono source.

In the following paragraphs we will examine some fundamental

examples of these sources. First, however, we shall determine their

relative strengths.

PROPOSITION 2.1.2. Let C be a category and let (A,fi) be a

source in C Then each of the following statements implies the

statement below it.

(a) (A,f.) is a strong mono source (resp., strong source).

(b) (A,fi) is an extremal mono source (resp., extremal source).

(c) (A,fi) is a mono source (resp., source).

PROOF: (a) (b): Let (A,fi) be a strong source in Suppose there

exists a family (gi )ic of morphisms in C and an epimorphism e in <

such that fi = gi.e for all icl. Then for all icl, fi.1A = ge.

Therefore, by Definition 2.1.1, there exists a morphism h:Q A

such that the diagram

e
A ------------>- Q

A f 'B- i

commutes for each icl. Hence h-e = 1A, so that e is both a section

and an epimorphism. Thus e is an isomorphism.

(b) ='(c): Clear from Definition 2.1.1.

PROPOSITION 2.1.3. Let I and C be categories and let

D:I -* be a functor. If (L,ti) is a limit of D, then it is a strong

mono source.

PROOF: By the definition of limit (1.1.6), (L,i ) is a natural source

for D; i.e., for each I-morphism m:i j, D(m).ti = j. Let X be a

C-object and let f and g:X L be C-morphisms such that *i*f = ti'g

for each iEOb(I). Then D(m)-(ti.f) (D(m). i).f = zj.f for each

I-morphism m:i j. Hence (X,(i .f)iE0b(I)) is a natural source for D.

But (L,.i) is a limit for D, thus there exists a unique morphism

h:X + L, such that i.*f = i'h for all icOb(I). Therefore f h g,

and consequently (L,1i) is a mono source (2.1.1).

Suppose that (B,gi) is a source and k and e are e-morphisms

where e is an epimorphism such that for each icOb(I) gi.e = i'k.

Then for each I-morphism m:i j, D(m)-gie = D(m)'i'*k = j *k g 'e,

since (L,Z) is a natural source for D. However since e is an

epimorphism, D(m).gi = gj. Thus (B,gi) is a natural source for D.

By Definition 2.1.1, there exists a unique morphism k:B + L such that

ti.k = gi for each icOb(I). Hence, since (L,Pi) is a mono source, the

diagram
e
}B

k / gi

L 0 D(i)

commutes for each icOb(I). Thus (L,ii) is a strong mono source (2.1.1).

It is well known that products, equalizers, terminal objects

and pullbacks are special types of limits; hence they are examples of

strong mono sources, and consequently they are examples of extremal

mono sources. The following corollary provided the motivation for

Proposition 2.1.3.

COROLLARY 2.1.4 (Herrlich and Strecker [7]). Let I and C

be categories and let D:I -* C be a functor. If (L,Li) is a limit of

D, then it is an extremal mono source.

PROOF: Propositions 2.1.3 and 2.1.2.

PROPOSITION 2.1.5. Let C be a category with pushouts. A

source (A,fi) in C is an extremal mono source in C if and only if it

is a strong mono source in C .

PROOF: By Proposition 2.1.3, we need only show that if (A,f ) is an

extremal mono source, then it is a strong mono source. Suppose that

e is an epimorphism and for each icI, the diagram

e
X-------*'

k gi

A rB

A i Ci

commutes. Let

e
X B

k--- P

A q

be the pushout of e and k. Then for each ieI, there exists a morphism

hi:P Ci such that fi = h *q. But since pushouts preserve epimorphisms,

q is an epimorphism; hence q is an isomorphism, since (A,fi) is an

extremal mono source.
-1 -1
Then q- *p:B A and for each icl, fi.q *p.e = hi.q.q -e

= gi.e = fi'k so that since e is an epimorphism and (A,fi) is a mono

source, the diagram

e B
-B

A i Ci

commutes for each icl.

The following proposition was proved for extremal sources by

Herrlich and Strecker [7].

PROPOSITION 2.1.6. Let C be a category with coequalizers.

Then each extremal source (resp., strong source) in C is an extremal

mono source (resp., strong mono source) in C .

PROOF: Let (A,fi) be an extremal source (resp., strong source) in ,

and let p and q be C-morphisms such that fi p = fi*q for each iEl.

Let (k,K) be a coequalizer for p and q. Then k is an epimorphism,

and for every fi,such that f.p = fi.q, there exists a unique morphism

gi:K Ci such that the diagram

K

k I
p I gi
x i
X A C--
q fi

commutes. Since (A,fi) is an extremal source (resp., a strong source,

hence an extremal source by Proposition 2.1.2), k must be an isomorphism

(2.1.1). Hence p = q; so (A,fi) is a mono source.

PROPOSITION 2.1.7. Let C be a finitely cocomplete category and let

(A,fi) be a source in e The following statements are equivalent.

(a) (A,fi) is a strong source

(b) (A,fi) is a strong mono source

(c) (A,f ) is an extremal source

(d) (A,fi) is an extremal mono source.

PROOF: Since e is finitely cocomplete, it has coequalizers and pushouts.Thus

(a) -*(b) and (c) =(d) (2.1.6) and (d) *(b) (2.1.5). Also it is

clear that (b) -(a) and (d) -=(c) (2.1.1) and (b) -(d) (2.1.2).

2.2 Sources in Concrete Categories

DEFINITION 2.2.1. Let (C.,U) be a concrete category.

(1) A source (A,fi) is called a concrete embedding source

provided that it is a mono source and for every source (B,gi) for which

there exists a function h: t'(B) 1t(A) such that the diagram

U(B)

_______(c.)
(U(A)f)

commutes for each iel, there exists a morphism h:B A such that

U(h)= h.

(2) A source (A,f ) is called a strong concrete embedding

source (respectively, an extremal concrete embedding source) provided

that it is both a concrete embedding source and a strong (resp.,

extremal) source.

The properties Of tenerot@g mnbdding soUrces will be examined

in subsequent sections within the generalized framework of rn-sources.

92.3 rn-Sources

Now that we have defined several distinct types of sources, we

will find it convenient to introduce some unifying notation.

DEFINITION 2.3.1. Let e be a category and let f7 be any class

of morphisms in C .

(1) A C -morphism f will be called an -morphism in C

provided that fer?.

(2) 7 will be called isomorphism-closed in e provided that

for any % -morphism m and any C-isomorphisms e and e', such that the

compositions m-e and e'.m are defined in e m-e and e'*m must be

i7-morphisms, and 2f must contain all of the isomorphisms in C.

(3) f will be called left-cancellative in C provided that

whenever p.qs ? for C -morphisms p and q, q must be an M-morphism.

(4) A pair (X,f) will be called an l-subobject of Y provided

that X and Y are e-objects and f:X Y is an rn-morphism.

For certain classes m of morphisms in C we have previously

defined sources whose morphisms in the aggregate exhibit properties

similar to the properties of individual m-morphisms. These sources,

as listed below in Table 2.3.2, will be called "m-sources." We

will refer to Table 2.3.2 frequently throughout the remainder of the

paper. Note that each class of morphisms listed in Table 2.3.2 is

isomorphism-closed in C .

TABLE 2,3.2. Let e be a category, [Let (C,U ) be a concrete

category.]

-m

the class of all
morphisms in C

the class of all
monomorphisms in e

the class of all
extremal morphisms
in e

the class of all
extremal mono-
morphisms in e

the class of all
strong morphisms
in C

the class of all
strong mono-
morphisms in &

r9-morphism m-subobject

morphism weak subobject

monomorphism subobject

extremal
morphism

extremal
monomorphism

strong mor-
phism

strong
monomorphism

extremal weak
subobject

extremal subobject

strong subobject

M-source

source

mono source

extremal
source

extremal
mono source

strong
source

strong mono
source

TABLE 2.3.2 (Continued)

~W1 f-morphism fl-subobject It-source

the class of all concrete concrete embedded concrete
concrete embeddings embedding subobject embedding
in e source

the class of all extremal extremal concrete extremal
extremal concrete concrete embedded subobject concrete
embeddings in C embedding embedding
source

the class of all strong con- strong concrete strong con-
strong concrete create embed- embedded sub- create
embeddings in C, ding object embedding
source

PROPOSITION 2.3.3 (Singleton lt-Sources). Let d be a

category. [Let (. ,U) be a concrete category.] Let W be a class

of morphisms in C listed in Table 2.3.2, and let f:A B be a

morphism in C Then the following statements are equivalent.

(a) (A,f) is an r)-source

(b) f:A B is an fl2-morphism

(c) (A,f) is an V-subobject of B

PROOF: (b) *(c): Apply Definition 2.3.1.

(a) a-(b): Apply Definitions 2.1.1 and 1.3.1 when C is any

category and Definitions 2.2.1 and 1.5.1 when (C,ZL) is a concrete

category. For example: (A,f) is a concrete embedding source provided

that for every source (B,g), for which there exists a function

h: U(B) -+ (A) such that U(f)-h ZL(g), there must exist a morphism

h:A B such that t((h) = h (2.2.1). This condition holds if and

only if f is a concrete embedding (1.5.1).

Thus, from the above result we can see that several of the

propositions on W-sources in 92.1 will automatically yield results on

7I-morphisms; e.g., every strong morphism is an extremal morphism

(2.1.2); in categories with pushouts, every extremal monomorphism is

a strong monomorphism (2.1.5); and in categories with coequalizers,

every strong (resp., extremal) morphism is a strong (resp., extremal)

monomorphism (2.1.6).

PROPOSITION 2.3.4 (Enlargement of f-Sources). Let C be a

category. [Let ( 6,U) be a concrete category.] Let W2 be a class of

morphisms in C listed in Table 2.3.2, (A,fi) be a source and (fk)kcK

be a family of morphisms in C with domain A, having (fi ) as a

subfamily. Then if (A,fi)I is an f-source, so is (A,fk)K.

PROOF: Clearly (A,fk) is a source.

(a) mono: Let p and q be C-morphisms such that fk"p = fk'q

for all kEK. Then f ip = f.*q for all iEt, so that since (A,fi) is a

mono source, p = q. Consequently (A,fk) is a mono source.

(b) extremal: Suppose (gk)kEK is a family of C-morphisms

and e is an epimorphism in C such that fk = gk.e for all keK.

Then fi = gie for all iEl, so that since (A,fi) is an extremal source,

e is an isomorphism (2.1.1). Hence (A,fk) is an extremal source.

(c) strong: Suppose (B,gk) is a source, e is an epimorphism

and r is a morphism such that gk-e fk'r for all keK. Then g *e = fi.r

for all iEl. Thus, since (A,fi) is a strong source, there exists

h:B A such that the diagram

Sht IB
e

h /
r g

A -f Ci
a-J

commutes for all iEI. Hence r = h*e. Now fk'h e = gk'e for all kcK.
But e is an epimorphism, hence fk'h = gk for all keK. Thus (A,fk) is
a strong source.
(d) concrete embedding: Suppose (B,gk) is a source for which
there exists a function h: %(B) U(A) such that U(fk)'h = U(gk)
for all keK. Then 'f(fi)-h U(gi) for all iel. Hence, since (A,fi)
is a concrete embedding source, there exists h:B A such that

U(h) = h (2.2.1), and consequently (A,fk) is a concrete embedding
source.
(e) The remainder of the proof follows directly from parts

(a), (b), (c), and (d).
PROPOSITION 2.3.5 (Left-Cancellation of r-Sources). Let C
be a category. [Let (e ,U) be a concrete category.] And let V be
any class of morphisms in C listed in Table 2.3.2. Let (A,fi) be a
source in C for which there exist families (h i)i and (gi ) of
morphisms in C. such that f = hi'gi for all iI. Then if (A,fi) is
an M-source, so is (A,g.).
PROOF: Clearly (A,gi) is a source in C.
(a) mono: Let p and q be e-morphism such that gi.p = gi'q.
Then fi p = hi.gi.p = hi.gi.q fi.q. Since (A,fi) is a mono source,
p q; thus (A,gi) a mono source (2.1.1).

(b) extremal: Let (ri.)iI be a family of C-morphisms and

let e be an epimorphism such that g = ri'e for all ieI. Then

fi = hi'.r.e for all icl. Since (A,fi) is an extremal source, e is

an isomorphism; thus (A,gi) is an extremal source (2.1.1).

(c) strong: Suppose (B,r.) is a source in C e is an epi-

morphism in C and k is a morphism in C such that r*ie = gi*k for all

iel. Then hi.ri.e = hi gi.k = fi.k. Since (A,fi) is a strong source,

there exists p:B A such that the diagram

e

-

k G.

S{hi

fi
commutes for all icl (2.1.1); hence pe = k and ri.e = gi'(p.e).

However, e is an epimorphism, so ri = gi.p for all i.I, Thus (A,gi)

is a strong source.

(d) concrete embedding: Suppose (B,r.) is a source in C

for which there exists a function h: U(B) -+ (A) such that

y(gi)'h = U(ri) for all icl. Since f, = hi'gi for all iEI and

functors preserve composition, U(fi)-h = U(hi'gi).h = IU(hi). (gi).h

= U(hi)-. (ri) = U(hi.ri) for all icl. And because (A,fi) is a

concrete embedding source, there exists a morphism h:B A such that

U (h) = h (2.2.1) and,consequently,(A,gi) is a concrete embedding

source.

(e) The remainder of the proof follows directly from parts

(a), (b), (c) and (d).

COROLLARY 2.3.6. Let rn be any class of t-morphisms listed

in Table 2.3.2. Then M is left-cancellative in C .

PROOF: Suppose p and q are morphisms in C such that p-q is an

r-morphism. Then (A,p*q) is an rn-source (2.3.3) and consequently

(A,q) is an 12-source (2.3.5). Thus q is an rn-morphism (2.3.3).

PROPOSITION 2.3.7. Let C be a category [( ,U) be a

concrete category] and let #7 be any class of C-morphisms listed in

Table 2.3.2. Then 57 contains all of the sections in ..

PROOF: (a) mono: Clear.

(b) strong: We will show that every section is a strong

monomorphism. Let s be a section. Suppose that k and g are morphisms

and e is an epimorphism such that s-k g.e. Now there exists a

morphism s such that s.s = 1. Thus s.s.g.e = sa.s.sk s-k = g-e, so

that since e is an epimorphism and s is a monomorphism, the diagram

e
----------- >

k g
osg

commutes.

(c) extremal: Every section is a strong monomorphism, hence

an extremal monomorphism (2.1.2). Thus every section is extremal.

36

(d) concrete embedding: We will show that each section is a

concrete embedding. Suppose that s:A B is a section, g:C B is a

morphism and h: U(C) U(A) is a function such that U(s)-h = U(g).

Now there is a morphism s such that s.s = 1A. Thus h = U(s)- U(s)*h

- '(s) 'U(g) = U (s-g). Thus s is a concrete embedding.

The remainder of the proof follows directly from parts (a),

(b), (c) and (d).

3. EMBEDDINGS INTO PRODUCTS

In this chapter we shall investigate the characteristics of

categorical 'embeddings into products,' and prove an embedding theorem,

which interrelates il-sources and the existence of mf-morphisms from

the object part of the source into the product of codomains of the

source morphisms. We shall examine the concept of 'embeddable objects,'

arriving at generalized definitions for O(-regular and O-compact

objects. These definitions will enable us to prove characterization

theorems for epireflective subcategories and epireflective hulls in

certain concrete categories.

53.1 Embedding Theorems

Herrlich and Strecker [7] proved the portions of the following

embedding theorem that deal with mono sources and extremal mono sources.

Their work provided the motivation for the following generalized

result.

THEOREM 3.1.1 (The Embedding Theorem). Let C be a category.

[Let (C ,'u) be a concrete category such that U preserves monomorphisms

and products.] Let M be a class of C-morphisms listed in Table 2.3.2,

and let (X,fi) be a set-indexed source, with each fi having codomain

Ei, such that a product (nEi,ri) of the codomains exists in C.

Then the unique induced morphism :X E i is an M -morphism if and

only if (X,fi) is an fi-source.

PROOF: (a) mono: Suppose that p and q are any C -morphisms such

that fi.p = fi q for all iI. Then i..p = i. cq for each ieI.

But (HEi, i) is a mono source (2.1.3 and 2.1.2), thus 'p = -q.

Consequently, (X,fi) is a mono source if and only if is a mono-

morphism (2.1.1).

(b) extremal: Suppose that is an extremal morphism and

that (B,gi) is a source in d for which there is an epimorphism e in

C such that fi = gi*e for each iel. By the definition of product

(1.1.2), there exists a unique morphism :B HEi such that

T.-gi> = gi for each iEl. Then ri' = fi = gi*e = i" e. Since

(HEi.,i) is a mono source (2.1.3), = e. And because
is an extremal morphism, e must be an isomorphism (1.3.1); consequently,

(X,fi) is an extremal source (2.1.1). Conversely, suppose that (X,fi)

is an extremal source and that g-e = is a factorization of

for which e is an epimorphism. Let Y denote the codomain of e. Then

(Y,Ti.g) is a source and f = i', = (Tnig)'e for each icl. Since

(A,f.) is an extremal source, e must be an isomorphism. Hence

must be an extremal morphism (1.3.1).

(c) strong: Suppose (B,g ) is any source in C and e is any

epimorphism in C and k is any morphism in C such that fi*k = gi'e

for each icl. Then there exists a unique morphism :B HEE such

that i.i = gi for each idl. If (X,f ) is a strong source, there

exists a morphism h:B X such that the inner portion of the diagram
e

commutes for all iEl; i.e., k h'e and fi.h = gi for all iEI. Hence

i.f i>.h = i . for each ieI. Therefore, since (IE i, ) is a

mono source, 'h =. Hence the outer portion of the diagram

above commutes, which implies that is a strong morphism (1.3.1).

Conversely, suppose that is a strong morphism. Then there exists a

morphism h:B X such that the outer portion of the diagram commutes;

i.e., k = h-e and 'h = . Clearly fi.h = wi'.h = ni'
for each icI, so that the inner portion of the diagram commutes for

each iel. Hence (X,fi) is a strong source (2.1.1).

(d) concrete embedding: From part (a), (X,f ) is a mono

source if and only if is a monomorphism. Suppose that (B,gi) is

any source in C for which there exists a function h: U(B) V (X)

such that U(fi).h = IA(gi) for each iEI. Then by the definition of

product (1.1.2), there exists a unique morphism :B HIE such

that Ti'gi> = gi for all ieI. By hypothesis, U preserves products;

hence (U(TE i), U(ti)) is a product in Set, hence a mono source in

Set (2.1.3). Thus U(ni). U()'h = U(i .) h = U(fi).h

= U(gi) = U(i..) = U(ni)-U() for all ieI if and only if

U().h = U(). Consequently (X,fi) is a concrete embedding

source if and only if is a concrete embedding (2.2.1 and 1.5.1).

(e) The remainder of the proof follows directly from parts

(a), (b), (c), and (d).

There are many applications for this theorem since the category

is quite unrestricted. A few of the examples are listed as

corollaries.

COROLLARY 3.1.2. Let A oe any set with at least two elements.

Then for any set S, there exists an injection into some power of A.

PROOF: Let a and b be distinct elements of A, and let I be the set

of all functions from S to A. Then (S,I ) is a mono source in Set. To

see this, suppose that p and q are distinct functions from some set X

into S. Then for some xeX, p(x) i q(x). Define f:S A by

a if s = p(x)
f(s) = b if s = p(x) Then f.p # f.q. Since Set has products,

the power (A3 ,Tf) exists in Set. Let :S A be the unique

induced function. By Theorem 3.1.1, since (S,3 ) is a mono source,

is an injection.

COROLLARY 3.1.3. A continuous map f from a topological space

X into a topological product lEi of spaces (Ei)iCI is a topological

embedding if and only if (X,w.if) is an extremal mono source in Top,

where fi:IEi Ei are the projection maps.

PROOF: f is a topological embedding if and only if f is an extremal

monomorphism in Top ([9]). Since the topological product together

with the projection mappings form a categorical product in Top,

f is an extremal monomorphism in Top if and only if (X,ri'f) is an

extremal mono source in Top (3.1.1).

COROLLARY 3.1.4. A continuous map f from a Hausdorff space

X into a topological product of Hausdorff spaces, nEi, is a homeo-

morphism onto a closed subset of the product if and only if (X,i' f)

is an extremal mono source in Haus (where Ti:IEi I Ei are the projection

maps).

PROOF: Extremal monomorphisms in Haus are exactly the topological

embeddings onto closed subsets ([9]). Since a topological product

together with the projection mappings is the categorical product in

Haus, we have, by Theorem 3.1.1, that f is an extremal monomorphism

if and only if (X,7i.f) is an extremal mono source in Haus.

COROLLARY 3.1.5. Let (C ,U) be a complete, well-powered

concrete category for which U preserves monomorphisms and products.

Let (A,fi) be a set-indexed source in C Then (A,fi) is an extremal

mono source in C if and only if it is an extremal concrete embedding

source in .
f.
PROOF: C is a complete category. Hence (A,A E ) being a set-

indexed source, implies that the product (H Ei i) exists in C .

If (A,fi) is an extremal mono source, the induced morphism ;A IIE

is an extremal monomorphism (3.1.1) and, consequently, is an

extremal concrete embedding (1.5.3). Thus (A,fi) is an extremal

concrete embedding source. Conversely, by definition, every extremal

concrete embedding source is an extremal mono source (2.2.1).

A study of extensions naturally goes hand in hand with the

study of embeddings.

DEFINITION 3.1.6. Let C be a category, T be any isomorphism-

closed class of C -morphisms, e be a subcategory of e and X be a

C-obj ect.

(1) The pair (y,Y) is called an V -extension of X in

provided that Y is an -object, (X,y) is an 7-subobject of Y and

y is an epimorphism.

(2) Let (X,fi) be a source with codomains in C (i.e., for

each icl, the codomain Ei of f is an E -object). (X,fi) is called an

S-nonextendable source with respect to e provided that for any

7 -extension (y,Y) of X in C with the property--for each isl,

there exists a morphism f :Y E for which the diagram

S/ fi

Ei

commutes--y must be an isomorphism (i.e., there must exist no proper

I-extension of X in C with this property).

In particular, if a source (X,f ) contains all the morphisms

from X to S-objects, then (X,fi) is an t-nonextendable source with

respect to C if and only if there exists no proper M-extension

(w,W) for X in C for which w is e-extendable (1.4.2).

Note that every class of 1-morphisms listed in Table 2.3.2 is

isomorphism-closed in C For convenience, when V is the class of all

C -morphisms, we will use the prefix 'weak' to replace '.' in

Definition 3.1.6. Hence we can use the term 'weak-extension' to denote

anR -extension when is the class of all morphisms in C .

The definition for a weak-nonextendable source (X,fi) with

respect to t (3.1.6) corresponds to Mrowka's [16] definition for the

class {fi:ieI} to be an .-nonextendable class for X.'

PROPOSITION 3.1.7. Let C be a category, be a subcategory

of C and (X,fi) be a source with codomains in E Then (X,fi) is

a weak-nonextendable source with respect to & if and only if (X,fi) is

an extremal source.

PROOF: Suppose (X,fi) is a weak-nonextendable source with respect to

8. Suppose that (Y,g ) is a source in e such that for some epi-

morphism e:X Y, the diagram

X '-------Y
Xf e

Ei

commutes for each icl. Then (e,Y) is a weak-extension; hence e must

be an isomorphism (3.1.6). Consequently (X,fi) is an extremal source

(2.1.1).

Conversely, let (X,f ) be an extremal source. Suppose there

exists a weak-extension (y,Y) such that, for each icl, there exists a

morphism gi for which f = g, y. But y is an epimorphism (3.1.6) and

(X,fi) is an extremal source; hence y must be an isomorphism (2.1.1).

Thus (X,fi) is a weak-nonextendable source with respect to .

Corollary 3.1.3 gives us a way to determine whether a continuous

map from a topological space into a topological product of spaces is

a topological embedding. And Corollary 3.1.4 gives us a way to determine

whether a continuous map from a Hausdorff space into a product of

Hausdorff spaces is a homeomorphism onto a closed subspace of the product.

Yet these methods are not as convenient as we might wish. Mrowka's

Embedding Theorem yields a topological characterization of these results.

Parts of this theorem follow directly from Theorems 3.1.1 and 3.1.7.

THEOREM 3.1.8 (Mrowka's Embedding Theorem [16]). Let

S= {fi:iel} be a set of functions with fi:X E where X and E ,

for each ieI, are topological spaces. Let h be the set function from

X into the topological product IEi such that wi.h = fi for each iIl.

We have

(a) h is continuous if and only if each f. is continuous.

(b) h is injective if and only if the set 4 satisfies the

following condition:

Cl. for every p, qeX such that p # q, there is an fie 3 with

fi() f iq).

(c) h is a topological embedding if and only if h is continuous

and injective and the set 3 satisfies the following condition:

C2, for every closed subset A of X and for every peX \A,

there exists a finite system fil...,fi of functions from I such that
1! n

(p) i cl[A] (where cl stands for closure in
E E ).
l
(d) Assume that the spaces Ei are all Hausdorff and assume

that h is a topological embedding. h[X] is closed in E.i if and only

if the set 3 satisfies the following condition:

C3. there is no proper weak extension (y,Y) of X such that

every function fiE 3 admits a continuous extension such that the diagram

Y
X yX

o/

E I
commutes. i

PROOF: Parts (a) and (b) are well-known results. Note that the

topological product with the Tychonoff product topology is a product in

Top. Hence each fi is a Top-morphism, i.e., continuous, if and only

if h is a To-morphism. Also h is injective if and only if it is a

Set-monomorphism, which by Theorem 3.1.1 is true if and only if

condition Cl is fulfilled in Set. (Let p, qeX such that p # q and let

g ,gq:{y) X be defined so that g (y) = p and g (y) = q. Thus (X,fi)

is a mono source in Set if and only if Cl holds.)

(c) The reader is referred to Nrowka's proof [16].

(d) Assume that the spaces E are all Hausdorff and h is a

topological embedding. Then h[X] is a Hausdorff space, so X is also

a Hausdorff space and consequently, h is a concrete embedding. Let

be the full subcategory of Haus having {Ei:icl} as its class of objects.

Then C3 merely states that (X,f ) is a weak-nonextendable source with

respect to t Thus by Proposition 3.1.7, C3 holds if and only if (X,fi)

is an extremal source in Haus, which by Theorem 3.1.1 is true if and

only if h is an extremal morphism in Haus; i.e., if and only if h is an

extremal monomorphism in Haus (2.1.6); i.e., if and only if h[X] is

closed.

COROLLARY 3.1.9. Under the same hypothesis as Theorem 3.1.8,

the following statements hold:

(a) (X,fi) is a mono source in Top if and only if (X,fi) is

a source in Top such that C1 holds.

(b) (X,fi) is a concrete embedding source in Top if and only

if (X,fi) is a mono source in Top such that C2 holds.

PROOF: (a) (X,fi) is a mono source in Top if and only if h is a

monomorphism (3.1.1) which holdsif and only if (X,fi) is a source for

which Cl holds (3.1.8).

(b) (X,fi) is a concrete embedding source in Top if and only

if h is a concrete embedding in Top (3.1.1) [but concrete embeddings

are precisely the topological embeddings ([7])], which holds if and

only if (X,fi) is a source for which condition C2 holds (3.1.8).

3.2 t I)t-Embeddable Objects

Mrowka [15] defined an E-completely regular space to be a

topological space that is homeomorphic to a subspace of some topological

power Em of the space E. For a Hausdorff space E, he defined an E-compact

space to be a space that is homeomorphic to a closed subspace of some

topological power Em of E. For a given full subcategory of the

category Top (respectively, Haus), Herrlich [6] defined an -regular

space (respectively, an E -compact space) to be any object in the

epireflective hull of C in Top (respectively, Haus). First we shall

directly generalize Mrowka's definition in three ways: to arbitrary

categories C (rather than Top or Haus); to arbitrary subcategories

of C (rather than subcategories with a single object E); and to

arbitrary ??-morphisms, for various classes hi of C -morphisms

(rather than topological embeddings) into the object parts of products

of -objects. In later sections, 53.3 and 3.4, on epireflective

subcategories, we will see that our definitions coincide with Herrlich's

definitions in the categories Top and Haus.

DEFINITION 3.2.1. Let C be a category, C be a subcategory of C,

and 7 be any class of C -morphisms that is isomorphism-closed in C .

(1) A C-object X is called e 7ln-embeddable in provided

that there exists a set-indexed family (E )iI of C-objects whose

product (IEi,"i) exists in C and for which there exists an ?l-morphism

f:X HE..

(2) When 9 has a single object E, the term Elf -embeddable in

e will be used interchangeably with jIi -embeddable in C .

(3) Let (C ,U) be a concrete category. Whenever 7 is the

class of all concrete embeddings in C the term t-regular in C will

be used interchangeably with the term t 7-embeddable in C .

Whenever M is the class of all extremal concrete embeddings in C

the term C -compact n will be used interchangeably with the term

E. -embeddable in C .

Note that since the concrete embeddings in Top are precisely

the topological embeddings, our definition for an E-regular space in

Top coincides with Mrowka's definition for an E-completely regular

space. Also our definition for a Hausdorff space to be E-compact in

Haus coincides with Mrowka's definition of E-compactness, since the extremal

concrete embeddings in Haus are homeomorphisms onto closed subspaces of

Hausdorff spaces (1.5.3).

The study of topological spaces has provided the motivation for

these definitions. Clearly, however, by Corollary 3.1.2, for any set

A with at least two points, every set S is A-regular in Set. In order

to illustrate the generality of these concepts, we will construct an

algebraic example. Let AbMon be the category of all Abelian monoids and

monoid homomorphisms. Then Ab, the category of all Abelian groups

and group homomorphisms, is a full subcategory of AbMon. The

following proposition is a well-known result, although it may not be

immediately recognizable in our new terminology.

PROPOSITION 3.2.2. An Abelian monoid is Ab-regular in

AbMon if and only if it is cancellative.

PROOF: Suppose M is Ab-regular in AbMon, then there exists a family

(Ai)ieI of Abelian groups and a concrete embedding morphism f:M HAi.

Suppose a,b,mEM and m+a = m+b. Then f(m+a) = f(m+b) so that since f

is a monoid homomorphism f(m)*f(a) = f(m)*f(b). Thus f(a) = f(b),

since (HAi,*) is a group. Hence a=b, since f is injective on the

underlying sets (1.5.5). Consequently M is cancellative.

Conversely, let M be a cancellative monoid. We will define a

relation R on pairs (x,y) where x,yEM in the following manner:

(x,y)R(x',y.') iff x+y' = y + x'. It is straightforward to show that

R is an equivalence relation. Let R be the set of all equivalence

classes (x,y), and let be the operation of componentwise addition

of pairs, which is well-defined by the definition of R. Since M

is cancellative and Abelian,it is easy to show that (R,*) is an

Abelian group, where (O0,O0) = ((x,x):xeM} is the identity element

and the inverse of any element (x,y) is the element (y,x). Define

h:M -K by h(x) = (OM,x). Since h(x+y) = (OM,x+y) = (OM,x) (OM)

= h(x) h(y), h is a homomorphism. Let zeKer(h). Then h(z)

= (Oz) = (0,OM). Thus (OM,z)R(OM,OM). Hence OM + OM = z+OM; so

z = 0M. Consequently h is injective. Thus h is a monomorphism in Mon,

which is an algebraic category having AbMon as a full subcategory;

hence h is a concrete embedding in Mon, and a concrete embedding in

AbMon (1.5.6). Consequently M is Ab-regular.

THEOREM 3.2.3. Let C be a category with products. [Let

(C,tU) be a concrete category with products such that %U preserves

monomorphisms and products.] Let e be a subcategory of C let 7??

be a class of C -morphisms from Table 2.3.2, and let X be a C-object.

Then X is 7fl-embeddable in C if and only if there exists a

set 4 contained in Ob(t) such that (X, U hom (X,E)) is an
Ec EX

n1-source in .

PROOF: Let X be _I1-embeddable in C Then there exists a set-

indexed family (E )i, of -objects such that there exists an

rn-morphism f:X HE1 (3.2.1). By the Embedding Theorem (3.1.1),

(X,ir'f) is.an lM-source. Since an n -source can be enlarged (2.3.4),

(X, home (X,EI)) is an /1-source.
iEl
Conversely, let be a set contained in Ob(g ) such that

(X, U hom (X,E)) is an W-source. By hypothesis,
EE X -X g>
x
Shas products. Hence the product
hom (X,E) E
( n E ,ni.nE) exists in C (by
EE

the Iteration of Products Theorem (1.1.3)). g
hom, (X,E)
Let :X H E be the unique induced
EE PX
morphism such that r E. * = g for all

ge ( hom (X,E). Thus is an
Ec L X C

P' Ehome(X,E)
e

I E
ShomC(X,E)
Ee
E

-morphism (3.1.1), so that X is P rl-embeddable in r.

COROLLARY 3.2.4. Let ( ,U ) be a concrete category with

products such that U preserves monomorphisms and products. Let

be a full subcategory of C and let X be a C-object.

Then X is C-regular (respectively, -compact) in e if and

only if there exists a set X contained in Ob() such that

(X, U hom (X,E)) is a concrete (respectively, extremal concrete)

embedding source in .

COROLLARY 3.2.5 (Mrowka [16]). A topological space X is

E-regular if and only if the following two conditions are satisfied:

ClI for every p,qeX, p#q, there is a continuous function

f:X E with f(p) # f(q).

C2 for every closed subset A of X and every peX\A, there

is a finite.number n and a continuous function f':X En such that

f' (p) f' [A].

PROOF: Let 3 = hom (X,E). By Corollary 3.1.9, (X, 1) is a concrete

embedding source if and only if conditions Cl' and C2' (which are

precisely Cl and C2 of Theorem 3.1.8 stated for Ei = E for all iel)

hold. By the Embedding Theorem (3.1.1), the induced morphism

:X E is a concrete embedding morphism (hence X is E-regular

(3.2.1)) if and only if (X, 3) is a concrete embedding source.

There are many well-known corollaries to the above theorem.

Topological examples have been collected by Mrowka [13] and [16] and

Herrlich [4]. Many of these corollaries had been established as

separate theorems long before the development of a unifying theory.

Several of these well-known results are listed below as examples and

are stated without proof.

EXAMPLE 3.2.6 (Tychonoff). A topological space is a completely

regular T1-space if and only if it is [0,l]-regular in Top. A

Hausdorff space is compact if and only if it is [0,1]-compact in Haus.

EXAMPLE 3.2.7 (Mrowka [14]). A topological space is a completely

regular T1-space if and only if it is (0,1)-regular in Top. A Hausdorff

space is realcompact if and only if it is (0,1)-compact in Haus.

EXAMPLE 3.2.8 (Alexandroff). Let F be the To-space with two

points {a,b} in which the only proper closed set is {a). A topological

space is a To-space if and only if it is F-regular in Top.

EXAMPLE 3.2.9. Let W be the two-point indiscrete space. A

topological space is indiscrete if and only if it is W-regular in Top.

Recall that a topological space is said to be zero dimensional

provided that it has a base of closed-and-open sets.

EXAMPLE 3.2.10 (Alexandroff). Let D be the discrete space with

two points. A topological space is a zero-dimensional To-space if and

only if it is D-regular in Top. A zero-dimensional Hausdorff space is

compact if and only if it is D-compact in Haus.

EXAMPLE 3.2.11 (Mrowka [13]). Let V be a topological space

with three points {a,b,c} such that (a) is the only proper non-

empty open subset. Every topological space is V-regular in Top.

Let us now consider another result of Mrowka's. Let L
m
denote a space with m-elements and the finite complement topology.

Let Top1 denote the category of T1-spaces and continuous maps. By

Proposition 1.5.8, any Top -morphism f:A B is a concrete embedding

in Topl if and only if it is a concrete embedding in Top.

EXAMPLE 3.2.12 (Mrowka [13]). There exists no Tl-space X such

that every T1-space is X-regular in Top. However t = {L :m is a

cardinal} is a class of T1-spaces such that every T1-space of cardinality

m can be topologically embedded into the topological power (Lm)m.

Note that there exists no T -space X, such that every T1-space

is X-regular in Topl, or X-regular in Top, although by Example 3.2.11,

every topological space is V-regular in Top. Clearly,V is not a

T1-space. Also we have seen that every T1-space is Z-regular in Top~ .

This example illustrates the necessity for our generalized definition.

Trivially, of course, in a category C every C-object is

C i7-embeddable in C if the class M of morphisms in C contains the

identity morphisms.

The remainder of this section will be used to reformulate a

topological result of Mrowka [16] into category-theoretic terms.

Recall that for a category C and C-object E, the contravariant hom

functor of C with respect to E is the functor hom(-,E): C.- Set

defined so that hom ( ,E)(X) = hom (X,E) for every C-object X,

and hom (-,E)([) = _* for each C-morphism .

DEFINITION 3.2.13. Let C be a category and let rf be an

isomorphism-closed class of -morphisms. Let E and X be C-objects.

Then a pair (X ,fX) is called an E17 -transformation of X provided that

the following conditions are satisfied:

Kl: X* is an Ej] -embeddable C -object.

K2: NX:X is an epimorphism in C such that hom(_,E),:

hom (X ,E) + hom (X,E) is bijective.

THEOREM 3.2.14 (The Identification Theorem). Let C be a
category with products [Let (C.,U) be a concrete category with products
where U preserves monomorphisms and products.], let 7 be a class of
C-morphisms from Table 2.3.2 and let C have the (epi,f ) factoriza-
tion property. Let E be a C -object.
Then for any C-object X there exists an El??t-transformation

(X ,X) of X.
PROOF: Let X be a C -object. By hypothesis, the product
hom (X,E)
(Em ( ) exists in C Let the unique induced morphism be
home -X,E)
:X E By hypothesis, = P'*X where P is an
rn-morphism and CX is an epimorphism (1.3.3). Let X denote the
codomain of X Then X is clearly El~ -embeddable in C (3.2.1) and
hom (_,E) (6X) is injective since X is an epimorphism. For every
fshom (X,E), the diagram
'. X

y

X .-. >EhOm,(X,E)

E

commutes; hence Trf'P e hom (X ,E) and f = *f'P'*X = home(_,E)(0X)(Tr fp)
Therefore hom(_,E)(pX) is surjective.
COROLLARY 3.2.15 (Mrowka [16]). For all topological spaces X and E,
there exists an E-regular space X and a continuous surjective map
o:X X such that hoo (_,E) () :homT (X *E) .ho (X,E) is
bijective.

PROOF: Top has the unique (epi, extremal concrete embedding)

factorization property (1.3.5).

53.3 7-Coseparating Classes

In this section we will show that in a category with products,

an object E in the category,with the property that every other object

is Elmono-embeddable,must be a coseparator for the category. Recall

that a -object C is called a coseparator for C provided that for any

two distinct C-morphisms p and q:A B, there exists a C-morphism

x:B C such that x.p i x*q. We shall first prove a proposition

relating coseparators and mono sources, which we will use to develop

the definition of a more general concept, namely an MI-coseparating

class for a category. Then we shall see that in a category C with

products, a class 9 of C-objects is an M-coseparating class for C

if and only if every C-object is t M-embeddable in C.

PROPOSITION 3.3.1 (Herrlich and Strecker [7]). Let C be a

category. A C -object C is a coseparator for C if and only if for

each C-object A, the source (A,hom (A,C)) is a mono source.

PROOF: Let A be any C-object and let p and q be any distinct

C-morphisms having the same domain and having codomain A.

(A,hom (A,C)) is a mono source if and only if there exists fehom (A,C)

such that f*p # f'q, which will happen for all such C-objects A if

and only if C is a coseparator for C .

Consequently,we know,for example,that any set with at least

two points is a coseparator for Set (3.1.2). Herrlich and Strecker [7]

and others have collected many examples of coseparators in categories.

Baayen [1] defined a universal object in a category C to be a

C-object U with the property that for every C-object A, there exists

a monomorphism m:A U Clearly then for every C -object A,

(A,hom (A, U)) must be a mono source (2.3.3); hence U is a coseparator

for C Baayen lists many examples of universal objects in categories.

DEFINITION 3.3.2. Let C be a category, let M be a class of

C-morphisms from Table 2.3.2, and let & be a subcategory of C .

(1) Let E be a subclass of the class of all C-objects in C.

Sis called an 7-coseparating class for 6( provided that for each

OM-object A, there exists a set eA contained in 8 such that

(A, J hom (A,E)) is an -source in C.
EE 'A

(2) A C-object E is called an M-coseparator for Ct provided

that {El is an -coseparating class for Ot.

From Proposition 3.3.1, it is clear that what we now call a

mono-coseparator for C is exactly a coseparator for C by the usual

definition. Note that for every category C the class Ob(C) of

all C -objects forms an M-coseparating class for since any class

5? from Table 2.3.2 contains all the isomorphisms in C .

Propositions 2.1.2 and 3.1.5 state the relative strengths of

M-sources for the various classes rn of C -morphisms from Table 2.3.2;

hence they also imply the relative strengths of n-coseparating classes

for C for different classes R Therefore it is clear that every

strong mono-coseparating class is an extremal mono-coseparating class,

which in turn is a mono-coseparating class (2.1.2). Similarly for

complete well-powered concrete categories (C ,U) for which U

preserves monomorphisms and products, every extremal mono-coseparating

class for C is a concrete embedding-coseparating class for C hence

an extremal concrete embedding-coseparating class for C (3.1.5).

The following theorem is the desired result which relates

1m?? -embeddable objects and m-coseparating classes. It follows

immediately from Theorem 3.2.3 and the above definitions.

THEOREM 3.3.3. Let C be a category with products. [Let

(&,U) be a concrete category with products such that U preserves

monomorphisms and products.] Let m be a class of C-morphisms listed

in Table 2.3.2, and let &L and C be subcategories of .

Then the following statements are equivalent:

(a) Ob( ) is an f/?-coseparating class for &C.

(b) For each CC-object A, there exists a set tA contained in

Ob( ) such'that the unique induced morphism A Ehom (A,E) is an
EC eA
r-morphism in C. .

(c) For each ( -object A, there exists a set CA contained in

Ob(t) for which there is a morphism f such that (A,f) is an 7?T-subobject

of some product of powers of objects in A'

(d) Each 0(-object A is CM 7-embeddable in C

COROLLARY 3.3.4 (Herrlich and Strecker [7]). Let C be a

category with products and let C be a C-object. Then the following

statements are equivalent:

(a) C is a mono-coseparator (respectively, extremal mono-

coseparator) for C .

(b) For each C-object A, the unique induced morphism
home (A,C)
A C is a monomorphism (respectively, an extremal monomorphism).

(c) For each C-object A, there is a morphism f such that (A,f)

is a subobject (respectively, an extremal subobject) of some power of

e.

PROOF: A C-object C is a mono-(respectively, an extremal mono-)

coseparator for if and only if {C} is a mono-(respectively, an extremal

mono-) coseparating class for C ; hence the result follows from

Theorem 3.3.3.

Recall that by Mrowka's result on T1-spaces (3.2.12), Top

has no concrete embedding-coseparator in Topl; however Topl has a

proper concrete embedding-coseparating class in Top1, namely L,

the class of all topological spaces with the finite complement

topology. Note that in Top, the space V with three elements

{a,b,c}, for which {a} is the only proper open set, is a concrete

embedding coseparator for Top (3.2.11) and when Top1 is considered as

a subcategory of Top, V is a concrete embedding-coseparator for Top1.

DEFINITION 3.3.5. Let 6 be a category, let M be a class of

C -morphisms that is isomorphism-closed in C and let C be any

subcategory of C. e( ?l) will be used to denote the full

subcategory of C whose objects are precisely the W7-embeddable

C -objects.

Clearly then Theorem 3.3.3 says that for any category C

(respectively, concrete category (C,U) such that t preserves

monomorphisms and products) which has products, for any class Ml of

C -morphisms listed in Table 2.3.2, and for any subcategory E of e ,

the class Ob(E) is an fi2-coseparating class for C( /(.j).

DEFINITION 3.3.6. Let C be a category and let O( be a full

subcategory of C .

(1) Then \$( will be used to denote the full subcategory of

. whose objects are the object part of products of f-objects that

exist in C .

(2) Let M be any class of \$ -morphisms that is isomorphism-

closed in C ~7twill be used to denote the full subcategory of C

whose objects are the object parts of m-subobjects of aC-objects.

(3) When 6 has a single object A, PA will be used inter-

changeably with O6 and MfA will be used interchangeably with mft.

Then by definition of an ?l -embeddable object in a category

C (3.2.1), the full subcategory C(('i)t) is precisely the full

subcategory i .

The following theorem was proved, for the class of monomorphisms

in a category e by Herrlich and Strecker [7].

THEOREM 3.3.7. Let be any category with products. Let

be any class of C -morphisms that is isomorphism-closed in C and

closed under composition and products, and let Ot and 6 be any full

subcategories of a .

(a) (Pte = 90.

(b) 2447t= mC Z.

(c) 0P72 0 is a full subcategory of 2n.PO1.

(C) Tr70O( is the smallest full subcategory of containing

Ct and closed under the formation of M-subobjects in C and products

in C .

(e) M~1( is a full subcategory of n98 if and only if 5

is a full subcategory of FhS .
PROOF: (a) Clearly products can be iterated; hence 46 = P8 .
(b) By hypothesis, the composition of 2 -morphisms is an
? -morphism; hence fleI = 61 .
(c) Let X be a Pt76( -object. Then X = i Y for a product
(HYi.,i) of some family (Yi)iEI of 7/2ld-objects, and hence for each
isl, there exists an ( -object A. and an ?-morphism mi:Yi Ai.
Let (TAi,Pi) be the product of the family Ai, and let f be the unique
induced morphism that makes the diagram

f
X = IY,---- ------+IHA.
1 1

Yi mi Ai

commute for each isI. Then f is the product of the family (m )iI
of )fn-morphisms (1.1.4). Consequently f is an A -morphism since 71 is
closed under products. Hence X is an ITP t-object.
(d) By part (c), 7?m>\$( is a subcategory of WPf( and
by (a), fVPpo =h?6& Also by part (b), l7RnPL( = flfP So
12fG1( is closed under the formation of (l-subobjects and products.
Clearly any full subcategory containing & and closed under the formation
of f -subobjects and products must contain VIP0( .
(e) Suppose that &( is a full subcategory of MP& Then
S}P. is closed under the formation of products and 7 -subobjects

(by part (d)) and hence tfM0 is a full subcategory of 7/S .

Recall that for a subcategory C1 of e, C(() denotes the

epireflective hull of OL in P (1.4.5).

COROLLARY 3.3.8. Let C be a complete category that is well-

powered and cowell-powered, let ?? be the class of extremal monomorphisms

in C and let f be a full replete subcategory of C Then C (l| extremal

mono) = POt( = -(0(); i.e., the objects in the epireflective hull of

6( are precisely the 6 ,f-embeddable objects in C .

PROOF: By Theorem 1.3.5, M is closed under products and composition.

Thus by Theorqm 3.3.7 rlPCis the smallest full subcategory of .

containing 61 and closed under products and extremal subobjects;

hence llPOtis the smallest epireflective subcategory of C containing

0( (1.4.3). Thus C(! ) = 7at = 9 = e.(clextremal mono), (1.4.4 and

3.3.5).

Although the following results are simply specializations of

Corollary 3.3.8 we will use them frequently throughout the remainder

of this chapter.

COROLLARY 3.3.9. Let 0( be any full replete subcategory of

ToD. Then in Top, Top (0 lextremal mono) = Top (Otextremal concrete

embedding) = Top (aitconcrete embedding) = Top (0); i.e., the 0(-compact

spaces in To are precisely the Q(-regular spaces in Top, and these

are precisely those spaces in the epireflective hull of C6 in Top.

PROOF: Top is complete, well-powered and cowell-powered. Also, the

extremal monomorphisms in Top are precisely the concrete embeddings in

Top ([7]).

COROLLARY 3.3.10. In Haus, let Ot be any full, replete sub-

category of Haus. Then Haus (t lextremal concrete embedding)

= Haus (Mt extremal mono) = Haus (0t), i.e., the t-compact spaces in

Haus are precisely those spaces in the epireflective hull of 0( in

Haus.

Furthermore Haus (0( concrete embedding) = Top (C0), i.e., the

0(-regular spaces in Haus are precisely those spaces in the epireflec-

tive hull of Ot in Top.

PROOF: Haus is complete, well-powered and cowell-powered, and it is

a full, hereditaryy subcategory of Top. Also the extremal concrete

embeddings in Haus are precisely the extremal monomorphisms in Haus

(1.5.3) and tha coftfrte erbeddingg in Hats are doncrete embedding in

Top (1.5.9).

3.4 O(-Regular and Ct-Compact Objects

CONVENTION 3.4.1.

Throughout the remainder of this chapter, all subcategories

will be assumed to be both full and replete.

In the preceding section we found that for every subcategory

ON of Top the OC-regular spaces in Top are precisely the spaces in

the epireflective hull of 61 in Top (3.3.9). Furthermore we discovered

that for each subcategory Ot of Haus, the 0f-compact spaces in Haus

are precisely those spaces in the epireflective hull of 0( in Haus

(3.3.10). Note that when 61 is a subcategory of Haus, the 01-regular

spaces in Haus by our definition are all those Hausdorff spaces which

are C -regular in Top, since concrete embeddings in Haus are concrete

embeddings in Top (1.5.9). Thus our definitions for OC-regular and

(e-compact objects coincide with Herrlich's definitions in Top and Haus

(3.2). Epireflective subcategories have been studied extensively.

Recall that in a complete, well-powered and cowell-powered category

C a subcategory e. of C is epireflective if and only if it is closed

under the formation of products and extremal subobjects (1.4.3).

Consequently, there are many examples of epireflective subcategories:

Haus, Top, CompReT and Ind are epireflective in Top; CompT2,

RComp (i.e., realcompact), CompRegT1 are epireflective in Haus; Ab

is epireflective in Grp, etc.

Recall that an epireflective subcategory fi of a category .

has the property that for every C -object X, there exists an

lt-epireflection (r ,X ) in C., where X is an Ot-object and

r :X X is an Ot-extendable epimorphism (1.4.2).

One of the motivating examples in the study of epireflective

subcategories was the construction of the Stone-Cech compactification

for completely regular T1-spaces. Let X be a Hausdorff space and let

3 = C(X,[0,1]). Then the product ([0,1] ,.i) exists in Haus, and

in CompT2 (by the Tychonoff Theorem). Let the unique induced morphism

be :X [0,1] Let = m.B be the unique (epi, extremal mono)

factorization (1.3.5), and let BX be the codomain of B. Since Haus

has the unique (epi, extremal mono) factorization property and since

every compact space is homeomorphic to a closed subspace of the product

of unit intervals (3.2.6), (B,BX) can be shown to be a CompT2-epireflec-

tion of X. Also if X is completely regular, B is a topological

embedding (3.2.6).

Clearly for CompT2-regular spaces, which are the completely

regular spaces in Haus, there exist topological embeddings into compact

T2-spaces, which are 'universal' in the sense of being CompT2-extendable.

Herrlich and Van der Slot [10] proved a very useful result relating

epireflective subcategories 6( in Haus and the existence of

Ce-epireflections (r) ,X ) for e-regular spaces, for which rt is

a topological embedding. The following theorem is a generalization

of their result to include concrete categories other than Haus. This

generalization and the ones following it came about because of the

development of the definition for concrete embeddings.

THEOREM 3.4.2. Let ( ,U) be a concrete category, which is

complete, well-powered and for which U preserves monomorphisms.

If &, is a subcategory of C then each of the following statements

implies the statement below it.

(a)' % is epireflective in .

(b) Each g(-regular object has a concrete embedding-extension

(w,W) in \$~ for which w is aC-extendable.

(c) CC is closed under the formation of products and extremal

concrete embedded subobjects.

(d) 6t is closed under the formation of products and extremal

subobjects.

Furthermore if C is cowell-powered, then the statements above,

(a) through (d), are equivalent.

PROOF: (c) =(d): The extremal monomorphisms in C are precisely

the extremal concrete embeddings in C (1.5.3).

(a) -(b): Let Ot be epireflective in C Suppose X is an
&-regular object. Then there exists a product (HAi ,i) of -objects

and a concrete embedding f:X HAi Let (r, ,X ) be the O(-epireflec-

tion of X; then X is an A-object and r :X -+ X. is an 0(-extendable

epimorphism. Note that wi.f is a morphism from X to Ai for each ilI.

Thus for each iIe, there exists gi:X A such that the diagram

X 1 X
i

f i
I

NA A.
nAi 7ri A 3 .

commutes. Let :X N H Ai be the unique induced morphism such

that . = gi. Then i'f i gi'r. = i .''r for all icI;

since products are mono sources, f = *r Thus r( is a concrete

embedding,since f is (2.3.5, 2.3.3).

(b) = (c): To show that a is closed under products, let

(Ai)iEI be any set-indexed family of 6(-objects. e is complete, so

the product (nAi.,i) is in C Then RAi is an C -regular object. Thus

by (b),there exists a concrete embedding-extension (w,W),for which W is

an Ot-object and w:HAi W is an Bf-extendable epimorphism. Thus for

every iEI, there exists a morphism i :W Ai such that the diagram

w
EAi

A.
Ai

commutes. Let :W + HAi be the unique induced morphism such that

n < = for each jEI. Then n J1'Ai =n w = n '< .w for each

jel. Consequently, since products are mono sources, 111Ai = w.

Thus w is a section, as well as an epimorphism, hence an isomorphism.

By hypothesis, 1C is replete, and thus EAi is an 6L-object. Also CC

is full; so that i is an Ct-morphism for each isl.

To show that O T is closed under the formation of extremal

concrete embedded subobjects, let (X,m) be an extremal concrete embedded

subobject of an e-object A. Then X is C-compact, hence a -regular.

By (b), there exists (w,W) where W is an CC-object and w:X W is an

c o-extendable epimorphism. Thus there exists m :W o A such that

m *w = m (1.4.2). But m is an extremal concrete embedding, hence w is

(2.3.5, 2.3.3); and consequently, w is an extremal monomorphism, as

well as an epimorphism. Hence w is an isomorphism. Thus, since 6( is

replete, X is an C-object, and because 8t is full, m is an L%-morphism.

Furthermore, if C is cowell-powered, the Characterization Theorem

for Epireflective Subcategories (1.4.3) can be applied so that (d) (a).

Herrlich [5] has collected many examples in Haus which illustrate

the conclusions of this theorem. We will examine an algebraic example.

EXAMPLE 3.4.3. Let AbMon be the category of Abelian monoids

and monoid homomorphisms. Then Ab, the category of Abelian groups and

group homomorphisms, is an epireflective subcategory of AbMon.

Let M be any Abelian monoid. We Aill now construct an Ab-

epireflection fo M, using a method outlined by Lang [12], for the

ccastruction of the Grothendieck group K(M). Let Fab(U M) be the free

Abelian group generated by S(M), the underlying set for M. For

each xeM, let [x] be the generator of Fab (M) corresponding to x.

Clearly we have an injective function f:M Fab(UM) defined by

f(x) = [x] for each xcM. Let B = {[x+y] [x] [y]:x,yeM} and let

be the subgroup of Fab(7M) generated by B. Let

9:Fab(1aM) Fab(UM)/ be the canonical homomorphism. Then

*f:;M K(M) is a monoid homomorphism. Clearly it is an epimorphism.

9-f is Ab-extendable. Thus (.-f, K(M)) is the desired Ab-epireflection

for M. Let us suppose M is Ab-regular (i.e., cancellative (3.2.2)).

Then there exists a set-indexed family (A )iI of Abelian groups, whose

product is (Ai., i), and there is a concrete embedding g:M A i

But HAi is an Abelian group and O'f is Ab-extendable. Thus there

exists a monoid homomorphism g*:K(M) -+ Ai such that g *..f = g.

Consequently, 0-f is a concrete embedding since g is (2.3.5).

Recall that e(CV concrete embedding) is the full subcategory

of t whose objects are the CO-regular objects in 8 .

THEOREM 3.4.4. Let (C,U) be a concrete category that is

complete, well-powered and for which U preserves monomorphisms.

Let Ot and Z be epireflective subcategories of C. such that

&_C C C(eCtlconcrete embedding) and let X be a C-object. If

(r ,XO ) is the Ot-epireflection of X and (r ,X, ) is the 3-epireflection

of X, then there exists a concrete embedding e:X X which is an

(O-extendable epimorphism such that the diagram

X ,XV

/
X e

X

commutes.

PROOF: By hypothesis, X is a 3-object, since it is an Ct-object.

Thus, since ra is O-extendable (1.4.2), there exists a morphism

e:X -+ X such that re = e*r But by hypothesis, XH is 01-regular.

Hence by Theorem 3.4.2, there exists a concrete embedding-extension

(y,Y' ) in CL for which y:Xe Ye is Ot-extendable. But rq is

6t-extendable and y'r :X Y- ; hence there exists a morphism

a:X Y, such that the exterior portion of the diagram

r
x x1 Yo

e

commutes; i.e., yrr = a-r Thus y'rg = a'e-r since r = er .

Now y = -me since r. is a G-epireflection, hence an epimorphism. Thus

e is a concrete embedding because y is a concrete embedding (2.3.5).

Note that e is an epimorphism because r is one (Duals of (2.3.5 and

2.3.3)).

To show that e is t-extendable, suppose that there exists

a morphism f:X -* A for some Ct-object A. Since y is (-extendable,

there exists a morphism g :Yq A such that g *y = g; however,

y = a-e, so that g *a-e = g. Consequently, we can conclude that e

is O-extendable (1.4.2).

Once again, the preceding theorem was proved for the category

Haus by Herrlich [6], who has exhibited several examples. For

instance,the category CompT2 is a full subcategory of RComp, which

is a full subcategory of CompReg 1, whose objects are the CompT2-regular

spaces in Haus. For a given Hausdorff space X, there exists a

CompT2-epireflection (B ,BCX) and a RComp-epireflection (SRBRX).

Then by the conclusion of Theorem 3.4.4, BRX can be densely embedded

in BCX. And if X is completely regular, X can be embedded in

BRX (3.4.2).

Now let us look at our algebraic example. Ab and CAbMon, the

full subcategory of AbMon whose objects are the cancellative Abelian

monoids, are epireflective subcategories of AbMon. In fact the

objects of CAbMon are precisely the Ab-regular objects in AbMon (3.2.2).

And Ab is a full subcategory of CAbMon. Then for any Abelian monoid

M, there exists an Ab-epireflection (A, BAM) and an CAbMon-epireflection

(BC,BCM) of M. From Theorem 3.4.4, there exists anembedding epimorphism

e:3CM BAM. Note that BAM is K(M), the Grothendieck group (3.4.3).

The following theorem was proved by Herrlich [5] for the

category Haus.

THEOREM 3.4.5. Let (C, U) be a concrete category that is

complete, well-powered and cowell-powered, for which U preserves

monomorphisms. If LO is a subcategory of C then the following

statements are equivalent:

(a) X is an & -regular object in .

(b) There exists an Of-compact object Y and a concrete

embedding morphism f:X Y.

(c) There exists a concrete embedding-extension (w,W) for X

where W is qt-compact.

(d) Each G -extendable morphism f:X Y is a concrete embedding.

PROOF: We will show that (b) -(c) (a) -(d) = (b).

(b) (c): Suppose Y is an &-compact object for which there

exists a concrete embedding f:X + Y. By definition of 60-compact,

there exists a product (HA, i,) of 0(-objects and an extremal concrete

embedding m:Y HAi. Now there exists an (epi, extremal concrete

embedding) factorization of f, f = t-w (1.3.5, 1.5.3)

f m
X f Y m EAi

W

Let W denote the codomain of w. Then W is Mt-compact, since the

composition, m.t, of extremal monomorphisms is an extremal monomorphism

in a complete, well-powered category (1.3.5) and mr. is a concrete

embedding,since m and t are (1.5.2). Also w is a concrete embedding,

because f is. Thus since w is an epimorphism and a concrete embedding,

(w,W) is the required extension.

(c) (a): Suppose (w,W) is and-compact concrete embedding-

extension of X. Then since W is C(-compact, there exists a product

(KAi',i) and an extremal concrete embedding m:W HAi. Also the

epimorphism w:X W is a concrete embedding since (w,W) is a concrete

embedding extension. Hence m.w:X -+ Ai is a concrete embedding (1.5.2),

so X is OC-regular.

(a) *(d): Suppose X is Ct-regular and let f:X Y be any

O'-extendable morphism. Then there exists a product (HAi' i) of

Ct-objects and a concrete embedding m:X HAi. Thus for each iel,

there exists a morphism 7im:X Ai; and since f is Mt-extendable,

there exists a morphism hi:Y Ai such that the diagram

IA

X i i
X " Ai

Shi

commutes for each iel (1.4.2). Let the unique induced morphism be

:Y -* Ai, such that ri' hi for each iel. But i.m = hi'f;

hence i'm = r.i'f for each icl. Thus m = .f, since products

are mono sources. Thus f is a concrete embedding since m is.

(d) *(b): Suppose each &1-extendable morphism f:X Y is a

concrete embedding. In Corollary 3.3.8, we have seen that an epireflec-

tive hull C((o) for 1 exists in C. (in fact, it is _(( lextremal

concrete embedding)). Thus there exists (r ,X ), an C(O()-epireflec-

tion (1.4.2). Since re :X X is C((t)-extendable, it is

C(-extendable, and thus it is a concrete embedding (by hypothesis).

Also X is an t(_()-object, hence 0<-compact, so that (b) holds.

The following theorem has been proved by Herrlich [5] and

others for the category Haus. Parts (a), (b), (c), (d) and (e) stem

from the Characterization Theorem for Epireflective Hulls (1.4.6),

given by Herrlich in [6]. Once again our area of discovery lies in

the interplay of it-regular and (t-compact objects in categories

other than Haus.

THEOREM 3.4.6. Let (C ,.L) be a concrete category that is

complete, well-powered and cowell-powered, for which U preserves

monomorphisms. If & is a subcategory of C. then the following

statements are equivalent:

(a) X is in the epireflective hull of O(.

(b) X is CC-compact.

(c) Each 6(-extendable epimorphism is {X}-extendable.

(d) Each 0t-extendable epimorphism f:X Y is an isomorphism.

(e) Each O-extendable morphism f:X Y is an extremal

concrete embedding.

(f) X is (%-regular and for all 6(-regular objects Y, for

which there exists an C(-extendable concrete embedding f:X Y, f

must be an extremal concrete embedding.

(g) X is C(-regular and for each 6(-regular concrete embedding-

extension (w,W) of X such that w is 6(-extendable, w must be an

isomorphism.

PROOF: (a) s (b) *b(c) *C(d) (e): The extremal monomorphisms in

C are precisely the extremal concrete embeddings in C (1.5.3).

Therefore the Characterization Theorem for Epireflective Hulls can be

applied (1.4.6).

(f) (b): Suppose X is Ot-regular. Then there exists a

product (nAi,"i) of (-objects and a concrete embedding f:X HAi.

Let ((tC) be the epireflective hull of Ot (1.4.4) and let (r ,X )

be the C(C()-epireflection of X. But C(OC) is .(Otlextremal concrete

embedding) (3.3.8) and consequently XP is 0(-compact, hence (5-regular.

Note that re is C(O()-extendable, thus &-extendable; and for each icI,

Ti'f:X Ai is a morphism into an a-object. For each icl, there

exists a morphism gi:X Ai such that the diagram

X re-r
x X e
C.

i

HA. A.
f gi

Ai A

commutes (1.4.2). Let the unique product induced morphism be

:Xe -r Ai. Thus vi.f = g.i'r = c.'gi>'r, for each ieI. And

f = .r since products are mono sources. Thus re is a concrete

embedding, since f is. And consequently by (f), r is an extremal

concrete embedding. Hence X is CC-compact (2.3.1).

(b) and (d) -(g): Since X is 8f-compact, it must be O0-regular.

Let (w,W) be an (-regular concrete embedding-extension of X for which

w is Ot-extendable. Then by (d), w is an isomorphism.

(g) -(f): By (g), X is (-regular. Suppose Y is an a-regular

object for which there exists an OC-extendable concrete embedding

f:X Y. Let f = h.w be the (epi, extremal concrete embedding)

factorization of f (1.3.5, 1.5.3). Then w is a concrete embedding,

since f is. Let W denote the codomain of w. Then W is (f-regular

(2.3.1). Let t:X A be a morphism to some e(-object A. Then since f

73

is et-extendable there exists a morphism k:Y A so that the diagram

w
x h

X ~ ~ ~ 1 : -"k--------^
/

A

commutes. Then k.h:W A is a morphism such that (k.h)'w = t. Hence

w is Of-extendable, and (w,W) is an 01-regular concrete embedding

extension. Therefore by (g), w is an isomorphism, and consequently

f is an extremal concrete embedding.

4. LINEARIZATIONS

In this chapter, we will introduce the concept of ?-lineariza-

tions of an endomorphism in a category and will show that for each

endomorphism in a category C with countable products, there exists

an r-linearization of that endomorphism for several classes W of

C-morphisms. We shall show further that by weakening the product

condition on e we can still find -linearizations for certain

endomorphisms in the category. Also, we will generalize de Groot's

result [2] on the existence of 'universal linearizations' for

completely regular spaces X of a weight < k for some infinite cardinal

number k and monoids M of at most k endomorphisms on X. This will

become a category-theoretic result which will extend the generalizations

of Baayen [1] on this same subject.

4.1 Coordinate Immutors and Permutors

In this section, we shall develop some of the mathematical

machinery for later sections. We shall define coordinate immutors and

permutors on powers of objects in a category and shall see that they

are endomorphisms of an interesting linear character, in that they

act only on the coordinates of a power, serving to "switch or collapse"

these coordinates.

DEFINITION 4.1.1. Let C be a category, E be a C-object, S be

any set for which the (Card S)'th power of E exists in and k = Card S.
k k
(1) Then a C -morphism f:Ek Ek is called a coordinate

immutor on Ek provided that there exists a set function a:S S such

that ,'f = 7T a for each #cS; i.e., f must be the unique induced

morphism which makes the diagram

Ek f Ek
--

\_________ B
Ea() E

commute for each cpS.

(2) A coordinate immutor f:Ek Ek is called a coordinate

permutor on Ek provided that a:S S, in the above definition, is

bijective.

PROPOSITION 4.1.2. Let c be a set of coordinate immutors

(respectively, coordinate permutors) on the object part E of some

power in C of some C-object E. If the (k.CardP) 'th power of E

also exists in C then the product of the coordinate immutors

(respectively, coordinate permutors) I p is a coordinate immutor
peP
(respectively, coordinate permutor), up to a natural isomorphism, on
Ek.Card.

PROOF: Without loss of generality we may assume for each p P,

there exists a function a :k k such that 'p = T (M) for each

Xek. By hypothesis (and 1.1.3), the product ( n (Ek) ,p ) is in .
p p
Let 6: E (Ek)p r Ep be the natural isomorphism, and let
pr ? Xk c
pEP

I p be the unique induced morphism that makes the diagram
SP (Ek)

H (Ek) p- AC--
pC -1 / PEC

p

ap(T),p P
p

commute for each p(e [The lower portion of the diagram commutes

for each Xek from the definition of coordinate immutor (respectively,

commuter). 11 p is the product of the family (p) (1.1.4).
pEp
Define a:k'x P kx by a(X,p) = (a (X),p) for each AEk, p6 Then

nXp p6( p).6-1 = (QX) p *( p).6-1= (-X ) pp.-o.6
pee pep

-1
- (A ))p ~.6 = a((A,p)) for each (A,p)ekx(P; consequently
p p p

6* J p.6-1 is a coordinate immutor on Ek*CardP (respectively, a

coordinate permutor on E kCard since a is bijective, if each a

is, for pe ).

PROPOSITION 4.1.3. Let C be a category. Let y be a coordinate

immutor (respectively, permutor) on the object part Ek of some power

of a C-object E, and let E be the object part of the product

( nA i,) of some family (Ai)iE of C-objects. Then there exists an
lel i i i

isomorphism 6:Ek H (Ak) such that 6-Y-6-1 is equal to the product
idE
i i0i of a family (i)ic, where each is a coordinate immutor
iel
(respectively, permutor) on A. for each itl.
k
PROOF: Since p is a coordinate immutor (respectively, permutor) on Ek

where (Ek (P )ck) is a power in then without loss of generality,

we may assume that there exists a set function (respectively, a

bijective set function) a:k k for which P *. = P (i)'

By the Iteration of Products Theorem (1.1.3), there exists a

natural isomorphism 6:Ek 1 (A1). Consider the diagram:
iEI

i (A ) 1 i k -1 k Ek

) 'i T i PiAi P P A )

k-- -- ^ k
1i i { 1/ i E PE Pa (
0A II(x)

o() Pi '/

AA(),i Ai A ,i A(A),i

k k
For each iEl, let pi:A. Ak be the morphism that makes part I of the

diagram commute for each AEk, and let 0Ki be the product of the family

(oi)icI (1.1.4). We need only show that 6.4.6-1= n p.. For each
iEI

Ack and ieI, P ,i'6-1(10),6 ,i'(i ). = n (),i

= P(A),i = Pi' Thus since products are mono sources (2.1.3),

-1
S(l(ni.).6 = ,.

54.2 J-Linearizations

In this section, we will show that endomorphisms in many

categories C can be viewed as restrictions, of some form, of coordinate

immutors on powers of objects in the category, and in similar fashion,

that automorphisms often can be viewed as restrictions of coordinate

permutors on powers of objects. Several theorems will be proved

which establish the existence of various 29-linearizations, for

certain classes ;/ of morphisms in a category C that, depending upon

product properties of C may linearize a given individual endomorphism

in C or simultaneously linearize a monoid of endomorphisms on a

given C-object, or even universally linearize all monoids (of a

certain cardinality) of endomorphisms on an object in a given sub-

category of C For the most part, the results of this section will

generalize results of de Groot [2] and Baayen [1], but with a considerable

change in emphasis.

DEFINITION 4.2.1. Let C be a category, \$4 be a class of

. -morphisms that is isomorphism-closed in C Of and G be subcategories

of C X and L be C -objects and p:X X and ::L L be endomorphisms

in and m:X L be a C -morphism.

(1) The triple (m,L,y) is called an j-lift of (X,) provided

that m:X. L is an 4-morphism such that the diagram

X m >- L

X L
m

commutes.

(2) An mI-lift (m,L,p) of (X,9) is called an fi~-linearization

of (X,c) in n 6 provided that L is the object part of some power of

a product of C -objects and y is a coordinate immutor on L. If

is also a coordinate permutor on L, the triple (m,L,9) is called a

stable 71 -linearization of (X,g) in O6 .

(3) Let S be a monoid of endomorphisms on X. The triple

(m,L,,) is called an S?-lift of (X,S) provided that (m,L,I) is an

it-lift for (X,s) for every seS.

(4) Let S be a monoid of endomorphisms on X. The triple

(m,L,') is called a (stable) J-linearization of (X,S) inm9O provided that

(m,L,v) is a (stable) ? -linearization of (X,s) inIO: for every sES.

Recall that each class i4 of morphisms in a category e ,

listed in Table 2.3.2, is both isomorphism-closed in C and left-cancella-

tive in C (2.3.5).

THEOREM 4.2.2. Let C be a category, k be a cardinal number,

E be a C -object for which the k'th power of E exists in C 7 be

any class of C -morphisms which is both isomorphism-closed in C and

left-cancellative in e and X be any e -object for which there exists
k
an fl-morphism S:X Ek. Let S be any monoid of endomorphisms on X

(respectively, any group of automorphisms on X) for which the (k.Card S)'th

power of E exists in ( .

Then for each scS, there exists a morphism m:X E ardand an

endomorphism ys:Ek.CardSI Ek-CardS such that (m,EkCardSys) is an

;l-linearization (respectively, a stable ,--linearization) of (X,s) in

PE.

Furthermore, if V is contained in the class of C-monomorphisms,

then if s and s' are distinct elements of S, ys # ys,

PROOF: By hypothesis the power ( I E ,s5i,s) exists in t the
Ack
scS

power ( nE ,7X ) exists in C and there exists an 7-morphism
Ark
:X RE X. For each scS and each Ack, there exists a morphism

7T Xr' s:X E ,. Let m be the unique induced morphism which makes

the diagram

m E1,s
x- E>,---- Es
AEk X
sES

X--HE E,
s A .,

commute for each Aek and seS. Thus *,s'm = ix 's for each Ask and

sS.

We wish to show that m is an Mf-morphism. For each seS,

let ps:EE,s HE be the unique induced morphism such that

A 'ps = h,s for Aek. However for each Aek and seS, 71 *m = q.*s
s ,a As A
and 'ps. m = Xsm = x*.s. Since products are mono sources,

Ps*m = S.s for each seS.

By hypothesis, S is a monoid; hence 1XCS. Thus pX.m = .lX =

Moreover is an j -morphism and, by hypothesis, 14 is a left-

cancellative class of C -morphisms; hence m is an M -morphism.

For each yeS, define a :S S by a (s) = s.y for each seS.

Let t be an element of S. Let yt be the unique induced morphism that

makes the diagram

E t E
XEk X,s -- X
seS seS

X,st

EX,s.t EX,s

commute for each seS and kXk. Clearly yt is a coordinate immutor.

[To show that yt is a coordinate permutor when S is a group of

automorphisms on X, we need only show that at:S S, as defined above,

is bijective. Clearly at is subjective: t-lS; thus for any geS,

Sa(gt ) = gt t = g. Suppose at(x) = at(y) then xt = yt; hence x = y,

since S is a group.]

We now will show that (m, f E t) is an Pt-lift for
Xek '
sES

(X,t); i.e., that the diagram

X kEX,s

t t

m ek 1,s
seS

commutes.

For each XEk and seS, (vrs' t)*m = ,st*m
A, X, X,st

= *X'.s-t

= (Xs.m)-t.

And since products are mono sources, yt'm = m-t.

Now suppose that every m -morphism is a monomorphism. Let

s, s'ES for which Ys = s,. Then ysm = y,,'m; hence m-s = mas';

thus s = s'.

This theorem is a generalization of one of Baayen's results

[1] which he stated only in terms of topological embeddings of

completely regular T1-spaces into powers of the real line R.. The

generalization to topological embedding-linearizations in Top is

implicit in his work.

Corollary 4.2.3 (Baayen [1]). Let F be a monoid of continuous

self-maps on a completely regular Tl-space X. Then for each ftF, there

exists a topological embedding-linearization in some pair (R ,6,)

where k = (Card F).weight X.'o z : each 6f is a continuous linear

operator in R k. Furthermore, if f, f' are distinct elements in F,

then 6 6 ,.

PROOF: A well-known result of Tychonoff's states that every completely

regular Tl-space X can be topologicaly embedded into the product space

[0,1] where a = (weight X)*- Of course [0,1]a is a subspace of

R so there exists a topological embedding :X + Rh. The class

of all topological embeddings (i.e., concrete embeddings in Top) is

left-cancellative and contained in the class of monomorphisms in Top.

By Theorem 4.2.2, for each feF, there exists a topological embedding-

linearization (R. CardF, ), such that 6 is a coordinate immutor on
k
S, where k = a-CardF = weight X-*o-CardF, and such that for some

topological embedding m:X + Rk, the diagram

m
x > k
X--------------------

f 6f

V k
X- ------------^-
Xm

commutes.

There are, of course, many additional corollaries to

Theorem 4.2.2. However, we will delay listing them until we have

proved a stronger version of the theorem. The next theorem allows us

to simultaneously linearize a monoid of endomorphisms on an object in

a category, provided that certain products exist in the category.

THEOREM 4.2.4. Let e 'i, E, X, k and C be as in the hypothesis

of Theorem 4.2.2. Let S be any monoid of endomorphisms (respectively,

group of automorphisms) on X for which all subpowers of the

(k.Card S'Card S)'th power of E exist in C Then there exists an

1l4-morphism m:X Ek-CardS-CardS and an endomorphism 1 on Ek'CardSBCardS

such that (m,E kCardS.CardS,) is an R-linearization (respectively,

a stable i7-linearization) of (X,S) in (E.

PROOF: By Theorem 4.2.2 there exists an rn-morphism m:X Ek'CardS

such that (m,E kCardS,y) is an 74-linearization (respectively, a stable

74-linearization) of (X,s) in E. Thus the diagram

X > Ek Card S

X -- Ek.Card S
m

commutes. By hypothesis and the Iteration of Products Theorem (1.1.3),

the product ((E kCardS) ,7s) is in C and there exists a natural

isomorphism 6: I Ek'CardS Ek*CardS.CardS. Let n y denote the
seS scS

product of coordinate immutors (respectively, permutors) (ys)scS'

By Proposition 4.1.2 there exists 4 = 6-Iys*6 which is a coordinate

immutor on Ek-CardSCardS (respectively a coordinate permutor on

EkCardS.CardS.

Let :X -* (E ardS) be the unique induced morphism for
s S

which a * = m for each seS. We shall

<> k (EkCard
X sS
:6 --------- C (
soS

X > H (Ek'Card
seS

show that the diagram

S) 6 k.(Card S)2
S----- E

S_. Ek.(Card S)2
s 6

commutes for each soS. ''s = m's = Ys m = Ys '

= 7 ys' * for each seS; and, since products are mono sources,

.s = rys*. Thus .s = (6-1 *6)., so that 6**s = 0.6-.

Also since m is an 7-morphism and 72 is a class of C-morphisms that

is left-cancellative, must be an f -morphism since s * = m.

Hence (,Ek(CardS)2,0) is an M-linearization of (X,S) in PE

(respectively, a stable r-linearization of (X,S) on PE).

COROLLARY 4.2.5. Let C and i7 be as in the hypothesis of

Theorem 4.2.4, E be a subcategory of C and X be a C-object for

which there exists an M-morphism 4 from X into the product ( IE ,i )
icI
of a set-indexed family of C -objects. Let S be a monoid of

endomorphisms on X (respectively, a group of automorphisms on X) such

that all subpowers of the (CardS-CardS)'th power of ( H E.) exist in C.
icI

Then there exists an TA -morphism m and an endomorphism p on

(iE)(CardS)2 such that (m, (i E) (CardS),4) is an 14-linearization
(respectively, a stable '-linearization) of (X,S) in .

PROOF: Let E = N IEi. Then apply Theorem 4.2.4.

COROLLARY 4.2.6. Let k be any infinite cardinal number, & be

a category with k-fold products, 51 be a class of morphisms in e that

is isomorphism-closed and left-cancellative in C C be a full

subcategory of P and X be a C-object. Then the following statements

are equivalent:

(a) X is e 1[-embeddable in .

(b) For any monoid S of endomorphisms on X for which Card S I k,

there exists an Ri-linearization of (X,S) in PC ,

(c) For any endomorphism t on X, there exists an 7 -linearization

of (X,t) in P .

(d) For any endomorphism t on X of finite order, there exists

an T-linearization of (X,t) in O( .

(e) For any automorphism t on X, there exists a stable

r-linearization of (X,t) in 7 .

(f) For any automorphism t on X of finite order, there exists

an '-linearization of (X,t) in ( .

(g) For any group S of automorphisms on X for which Card S < K,

there exists a stable T7-linearization of (X,S) in j ,

PROOF: (a) -(b) (respectively (a) =(g)): There exists a set-indexed

family (E.)icI of e-objects for which the product (HEi ,i) exists in

C and there is an W-morphism E:X -* EiE (3.2.1). Let S be any monoid

of endomorphisms (respectively group of automorphisms on X) for which

Card S I k. Then (Card S)2 k. Apply Corollary 4.2.5.

(b) =>(c) (respectively (g) (e)): Let ST be the monoid

(respectively, jroup) generated by {t). Then Card ST ~0o < k.

(a) (d) and (e) (f): Clear.

(d) = (a) (respectively (f) -(a)): The identity 1X has order

1. Thus there exists a (stable) A-linearization of (X,MX) in P .

Thus there exists L, a power of C -objects, and an t-morphism

m:X L (4.2.1). Hence X is 1fl-embeddable.

COROLLARY 4.2.7. Let C be a category with products, let

7i be a class of I -morphisms that is isomorphism-closed in C and

left-cancellative in C Let P be any subcategory of C Then X

is j4 -embeddable in C if and only if for every monoid S of

endomorphisms on X (respectively, every group S of automorphisms on

X ), there exists an n-linearization of (X,S) (respectively, a stable

)i-linearization of (X,S)) in P .

PROOF: Apply the previous corollary.

There are three areas of differences between Baayen's

generalizations of de Groot's work and our generalizations. The

first is primarily one of emphasis: Baayen [1] was interested in

the existence of mono-universal objects in a category and universal

mono-lifts for morphisms in the category (i.e., "universal morphisms"),

while our emphasis is on the linear character of the resulting

coordinate immutors. Secondly, we have obtained results on the

existence of P7-linearizations for several classes 11 of morphisms in

a category, while Baayeris results were restricted to monomorphisms in

general categories and topological embeddings in Top (which he had

to consider separately). Thirdly, Baayen's results were restricted to

categories with countable products; our results require only the

existence of certain products in the category. Hence, for example,

we can obtain the following corollary for categories with finite

products.

COROLLARY 4.2.8. Let C be a category with finite products,

Y? be any class of C -morphisms that is isomorphism-closed in C and

left-cancellative in C C be a subcategory of C and X be

a C-object. Then the following statements are equivalent:

(a) X is 7d -embeddable in C .

(b) For any monoid S of endomorphisms on X for which Card S

is finite, there exists an Z-linearization of (X,S) in 8f

(c) For any endomorphism t on X of finite order, there

exists an Y-linearization of (X,t) in P .

(d) For any automorphism g on X of finite order, there

exists a stable 0?-linearization of (X,g) in C .

(e) For any group S of automorphisms on X for which Card S

is finite, there exists a stable t--linearization of (X,S) in P .

PROOF: (a) = (b) (respectively, (a) (e)): There exists a product

( I E, i ) of C -objects and an mr-morphism E:X T E (Card S)2
iEl iel
is finite, since Card S is finite. Apply Corollary 4.2.5 to obtain

the L-linearization of (X,S) in 4C (respectively, to obtain the

stable -linearization of (X,S) in e ).

(b) =(c) (respectively, (e) *(d)): Let ST be the monoid of

endomorphisms on X (respectively, group of automorphisms on X)

generated by {t}. Since t has finite order, Card ST is finite; hence

by (b) there exists an 1l-linearization (respectively, a stable

M-linearization) (m,L,O) of (X,S ) in P Hence (m,L,4) is the

desired linearization for (X,t).

(c) ='(a) (respectively, (d) )-(a)): The identity morphism on

X has order 1. erefore by (c) (respectively, (d)), there exists

an rn-linearization (m,L,4) of (X, X) in PE where L is a power of

a product of I -objects.

A much stronger result can be obtained for a category C with

countable products: every endomorphism 4 on a -object X has a

section-linearization in 4X, and hence an 1h-linearization in OX

for every class 27 of 6 -morphism from Table 2.3.2 (2.3.7).

COROLLARY 4.2.9. In a category C with countable products,

every C -object X and every endomorphism t on X (respectively, every

automorphism t on X) has a section-linearization of (X,t) in X.

PROOF: Let St be the monoid (respectively, group) generated by {t).

Card St 1 The identity morphism 1X:X X is an isomorphism, hence

a section, and the class of all sections in is left-cancellable

and isomorphism-closed in C Thus there exists a section-linearization

of (X,t) in '? X (respectively, a stable section-linearization of (X,t)

in EX) (4.2.6).

It is interesting to note that, in a category C with

countable products and a weakly terminal class F of C-objects

(i.e., for each C -object X, home (X,E) # for some EE ), every pair

(X,4), where X is a \$-object and i is an endomorphism on X, has a

weak-linearization in 6, If a category C has an epireflective

subcategory 1 then Ob(A') is a weakly terminal class for 4 (1.4.2).

Consequently if ( ,U ) is a concrete category that is complete and

well-powered, having a full replete epireflective subcategory C1,

then ('6( = 6 (1.4.3) and for every C-object X and every endomorphism

4 on X, there exists a weak-linearization of (X,p) in ,i; and further-

more, if X is an 6-regular space, there exists a concrete embedding-

linearization of (X,4) in O1 (3.2.1, 4.2.6).

Let us next consider a product BxC of distinct sets B and C

and a Set-monomorphism m:X BxC and an automorphism f:X X. We

know that there exists a coordinate permutor (:(BxC)a (BxC)a,where

a is the order of the automorphism f, such that (m,(BxC)",4) is a

stable mono-linearization of (X,f) in Set (4.2.6). What we will

examine now is the workings of on (ExC)". From the next theorem we

will find that p = BxC, the product of coordinate permutors B on

Be and ,C on C0.

THEOREM 4.2.10. Let C be a category, let M be a class of

C-morphismsthat is isomorphism-closed in C and left-cancellative in

C, let (E )iEI be a set-indexed family of C -objects whose product

(HEi ) exists in p and let X be any ? -object for which there

exists an M -morphism C:X IIEi. If S is any monoid of endomorphisms

on X (respectively, any group of automorphisms on X) such that all

subpowers of the (Card S-Card S)'th power of HEi exist in & then

there exists a triple (m,L,9) which is an 2I-lift for (X,S) such that

L= E C and v is the product of a family (oi)icl of morphisms,
iel
where for each iEI, pi is a coordinate immutor (respectively, permutor)

on E (CardS) 2
on E

PROOF: By Theorem 4.2.4, there exists an E/-linearization (respectively,

stable 2;-linearization) (m,(HEi) (ardS)2 ,) for (nEi,S). The remainder

of the proof follows directly from Proposition 4.1.3.

4.3 Universal ;V-Linearizations

In the previous section, we found that linearizations always

exist for endomorphisms in categories with countable products, and

that for some monoids of endomorphisms on an object in the category,

as well as for some groups of automorphisms on an object in the

category, we could obtain simultaneous linearizations. In this

section, we will restrict our attention to categories with infinite

products in order to obtain some linearizations that are universal

for all endomorphisms (respectively, all automorphisms) on any object

in a given subcategory.

DEFINITION 4.3.1. Let C be a category, M be a class of

e -morphisms that is isomorphism-closed in Q V? be a subcategory

of e and L be a -object where ::L L is an endomorphism in .

(1) A C -object X is called an 1,7-universal object for S

provided that for each Of-object A there exists an l?-morphism

m:A X.

(2) A pair (L,y) is called a universal 3f-lift for End(OM)

[respectively, for Aut(Or)] provided that for each endomorphism

respectively each automorphism] on any Ot-object A, there exists an

r4-morphism m:A L for which (m,L,() is an "-lift for (A,1).

(2) A universal ;;,-lift (L,I) for End(6t) [respectively, for

Aut(*()] is called a universal r-linearization for End(Ot)

[respectively, for Aut(OL)] provided that L is the object part of a

power of a product of C -objects and i is a coordinate immutor. If p

is a coordinate permutor on L, then (L,,) is called a stable universal

l -linearization for End(8() [respectively, for Aut(6&)],

(3) Let k be a cardinal number. A monoid S is called a

k-;,-universal monoid for O provided that card S < k and there exists

a C-object L and a monoid homomorphism a:S hom e(L,L) with the

following property: for each monoid T with card T < k, there exists

a surjective monoid homomorphism y:S T such that for each r -object

A and for each monoid homomorphism \$:T -* hom (A,A) there is some

M-morphism m:A L, such that for each seS, the triple (m,L,u(s))

is an fl-lift of (A,8-y(s)).

A L

S-Y(s) (s)

L
A

(4) Let k be a cardinal number. A group S is called a

k-;i-universal group for & provided that card S < k and there exist

a F-object L and a group homomorphism a:S Aut (L,L) (where

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CATEGORICAL EMBEDDINGS AND LINEARIZATIONS By JEAN MARIE McDILL A DISSERTATION PRESENTED TO THE GRADUATE C^.NCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1971

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To Bill and to Kathleen, ; for her birthday

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ACKNOWLEDGMENTS The author would like to thank the Chairman of her Supervisory Committee, Dr. G. E. Strecker, for his thoughtful encouragement and many helpful suggestions in preparation of this paper over the obstacles of distance and time. She would also like to thank Dr. Z. R. Pop-Stojanovic, Dr. T. 0. Moore (who introduced her to topology), Dr. B. V. Hearsey, Dr. D. A. Drake and Dr. Max H. Kele for serving on her Supervisory Committee, as well as A. R. Bednarek and W. E. Clark, who, at various times, have served on the committee. With gratitude the author wishes to acknowledge the encouragement and unyielding support she has received from her husband. She would also like to thank the other members of her family who have aided her work on this dissertation: Mr. and Mrs. P. S. Wiliircre, Mr. and Mrs. J. R. Hutchinson and Kathleen, her daughter, fo. ., om this undertaking has lasted a lifetime. The author is also very much indebted to Mrs. Karen Walker for typing this paper. iii

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TABLE OF CONTENTS Page ACKNOWLEDGMENTS iii LIST OF CONCRETE CATEGORIES vi ABSTRACT vii INTRODUCTION 1 CHAPTERS : 1 . PRELIMINARIES 8 1.1 Products and Limits 8 1.2 Special Categories 11 1.3 Special Morphisms in General Categories 12 1 . A Epiref lections 15 1 . 5 Concrete Embeddings 17 2. SOURCES 22 2.1 Sources in General Categories 22 2 . 2 Sources in Concrete Categories 28 2.3 W-Sources 29 3. EMBEDDINGS INTO PRODUCTS 37 3 . 1 Embedding Theorems 37 3.2 Â£, [TW-Embeddable Objects. 46 3.3 ?n-Coseparating Classes , 5A 3.4 Ot-Regular and ^-Compact Objects. 61 4 . LINEARIZATIONS 74 4.1 Coordinate Immutors and Permutors 74 4.2 ??1 -Linearizations 78 Iv

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Ab AbMon CabMon Compl y CompRegT Field Grp Haus Ind Mon POS RComp Set Sgp Top LIST OF CONCRETE CATEGORIES The category of Abelian groups and group homomorphisms The category of Abelian monoids and monoid homomorphisms The category of canc^llative Abelian monoids and monoid homomorphisms The category of compact Hausdorff spaces and continuous functions The category of all completely regular T spaces and continuous functions The category of all fields and field homomorphisms The category of all groups and group homomorphisms The category of all Hausdorff spaces and continuous functions The category of all indiscrete spaces and continuous functions The category of all monoids and monoid homomorphisms .The category of all partially ordered sets and orderpreserving functions The category of all realcompact spaces and continuous functions The category of all sets and functions The category of all semigroups and semigroup homomorphisms The category of all topological spaces and continuous functions vi

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Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CATEGORICAL EMBEDDINGS AND LINEARIZATIONS By Je.^n Marie McDill August, 1971 Chairman: Dr. G. E. Strecker Major Department: Mathematics A recent definition for embedding morphisms in concrete categories is examined. Several types of sources are defined for which the source morphisms in the aggregate exhibit the character of single morphisms with 'embedding-like' properties. A theorem giving .. .;cessary and sufficient conditions for embedding an object into a categorical product of objects is proven for a variety of 'embeddinglike' morphisms. The concept of embeddable objects is examined, and a definition is developed for Â£-regular and S-compact objects in concrete categories. In consequence, several characterization theorems for epiref lective subcategories and epiref lective hulls > which previously had been proven only for certain categories of topological spaces, are extended to a variety of "reasonable" concrete categories. Baayen's generalizations of de Groot's results, on the existence of universal linearizations for monoids of endomorphisms vii

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on completely regular T -spaces, are extended. For every isomorphismclosed, lef t-cancellative class ^ of morphisms in a category with countable products, every endomorphism on an object in the category is shovm to be a restriction, relative to W, of a coordinate-immuting endomorphism on a power of an object in the category. Under the same conditions, an automorphism in the category is shown to be the restriction of a coordinate-permuting automorphism. Furthermore, simultaneous /^-linearizations are shown to exist for certain monoids of endomorphisms on objects in the category, and universal /^-linearizations are shown to exist for every endomorphism in certain subcategories of the category. Viii

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INTRODUCTION The "mathematical-man-in-the-street," when asked what he would consider an embedding morphism to be in any concrete category, would probably reply, if he replied at all, that it would have to be infective on the underlying sets and would have to preserve the structure of the embedded object; i.e., it would have to be an actual subsystem embedding. It has been somewhat difficult to find a categorical definition for a morphism that does just that. Several types of morphisms have been defined which act as subsystem embeddings in some categories but are too weak or too strong in other categories. Monomorphisms in algebraic categories usually act as embeddings, but they are not embeddings in the topological sense in Top ; extremal monomorphisms are precisely the topological embeddings in Top , but they are too restrictive in Haus where they are homeomorphisms onto closed subspaces, and consequently both sections and regular monomorphisms are too restrictive in Haus . Within the last year, Herrlich and Strecker [7] developed the definition for concrete embeddings in a concrete category, which they found to be precisely the topological embeddings in Top and Haus and to be precisely the monomorphisms in algebraic categories (1.2.2). In Chapter 1, Â§1.5, we shall begin to examine the concept of concrete embeddings; in Chapter 3, we shall begin to exploit it. Several new results will be proved in Â§1.5. We will find that concrete embeddings are always injective on the underlying sets, and that concrete embeddings tend to have stable hereditary properties; i.e., that concrete embeddings in many subcategories tend to be concrete embeddings in the larger category, and conversely.

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Frequently, when dealing with concrete embed dings, we restrict our attention to concrete categories (Q ,11 ) in which the faithful functor IC : C -Â» Set preserves products or monomorphisms . This restriction is fairly weak; in most ordinary concrete categories, the functor Xi has a left-adjoint and hence preserves all limits (and hence all monomorphisms too) , In Chapter 2, we will develop the notion of sources. A source in a category Â£ is a C-object X together with a family of C-morphisms having X as their common domain. A categorical product ( n E.,7T,) is an example of a source which, in fact, exhibits a iel "mono-like" property; i.e., if f and g are morphisms in the category such that TT.'f = TT -g for each iel, then f = g. We call a source, whose morphisms acting in concert exhibit the property of a single monomorphism, a mono source. Herrlich and Strecker [7] have developed definitions for mono sources and extremal mono sources. In Chapter 2, we shall define several new types of sources (called, collectively, Â•/n-sources) whose morphisms in the aggregate exhibit the properties of special types of "embedding-like" morphisms (Â«)Tl -morphisms) . In topology, a long-established method for producing mappings into topological products has been by use of families of morphisms into the coordinate spaces. Tychonoff's well-known result, that for any infinite cardinal number k, a completely regular T^ space X of weight <_ k can be topologically embedded into the topological product [0,1] of unit intervals, made use of the family C[^Â»[0Â»1]] of all continuous functions from X into the unit interval [0,1],

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His result stimulated several mathematicians to investigate this general procedure for topologically embedding a space into a product of spaces. In 1956, Mrowka [13] published a theorem giving necessary and sufficient conditions for a function to be a topological embedding of a space into a topological power of another space. More recently [16], he expanded this result to include embeddings of a space into products of other spaces, as well as characterizing topological embeddings onto closed subspaces of products of Hausdorff spaces. From our vantage point in category theory, we can see that these theorems concerned sources: that certain conditions on the source (X, "i ) where 3 is a family of continuous maps from the space X into coordinate spaces of a topological product are necessary and sufficient to guarantee the existence of a homeomorphism from X onto a subspace of the product. In Chapter 3, we will prove a similar embedding theorem for categories (3.1.1) for a variety of "embedding-type" morphisms. In 1958, Engelking and Mrowka [3] began to develop the concepts of E-regular and E-compact spaces in response to two intriguing questions: (1) for a given space E, when can a space X be topologically embedded into some topological power of E? (When is X an E-completely regular space?), and (2) for a given Hausdorff space E, when is a space X homeomorphic to a closed subspace of some topological power of E? (When is X an E-compact space?) Later Herrlich [A] expanded these investigations to include the concepts ^-regular (respectively, 6 -compact) spaces for collections Â£ of spaces (respectively, Hausdorff spaces) , initiating the use of a category-theoretic approach.

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In Â§3.2, we shall characterize objects which are "embeddable" in products of other objects. In particular for a concrete category (6>'U.) with a subcategory Â£. , we will define a C-object X to be ^-regular in ^ (respectively, ^-compact in C ) provided that there exists a concrete embedding (respectively, an extremal concrete embedding) from X into the object part of a product of Â£ -objects. For tlie first time, the concepts of Â£ -regular and ^.-compact objects can be applied to categories other than Top or Haus . The most interesting applications of these concepts will be in the realm of epiref lective subcategories. Recall that a subcategory Q{ is epiref lective in a category C provided that for every C-object X there exists a pair (r^ , X^ ) , called the ^-epiref lection of X, where X^ is an (Sf-object and r^ :X -* X.. is an epimorphism with a maximal extension property for QC , (In this paper, we will say that r is " (X~2xtendable" provided that for every morphism g:X -* A for some 0( -object A, there exists a morphism g :X -Â»Â• A such that g .r = g.) Herrlich and Strecker [8] have characterized epiref lective subcategories in the following fashion: in a complete, well-powered and cowellpowered category C Â» with a full, replete subcategory &. , 6( is epireflective in (5 if and only if it is closed under the formation of extremal subobjects and products in (J . This theorem gives us a large array of epireflective subcategories^ The actual construction of the epiref lection can be quite difficult. Usually the construction takes one of two forms: (1) "factoring out" the elements in the object with undesirable characteristics, e.g., the Abelianization of a group G by factoring

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out the commutator subgroup G of G (a:G * G/G ), or the Hausdorf fication of a topological space X by finding the appropriate equivalence relation R so that X/R is a Hausdorff space with the "universal" mapping property, or (2) constructing a "larger" object with the desired structure and finding an epimorphism from the object to the "larger" object, as in the Stone-Cech construction, 3:X -* 3X, which gives a topological embedding, as well, when X is a completely regular T space, or as in the construction of the Grothendieck group K(M) for an Abelian monoid, k:M -Â» K(M), for which k will be an injection, as well, when M is a cancellative Abelian monoid. It has always been interesting to note that in certain cases, the second method described above results in epiref lections that are actual subsystem embeddings. Both methods of constructing epiref lections have been used for both topological and algebraic categories; yet , until now , there has been no general categorical way to differentiate between these two methods and to determine for what objects in the category the epiref lection epimorphism will be an actual subsystem embedding . For the category Haus, Herrlich and Van der Slot [10] were able to show that a full, replete subcategory (91 is epiref lective if and only if for every Â©(-regular object X in Haus there exists an epimorphism r:X A to some Â©(-object A such that r is an (9t -extendable topological embedding. In Â§2.4, we will extend this result to one for "reasonable" concrete categories, and hence will be able to add a new characterization for epiref lective subcategories in those categories. Furthermore, we will extend several other theorems (which Herrlich [5] proved for the

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category Haus ) which will characterize Â©(-regular and 6C-compact objects in "reasonable" concrete categories. Tychonoff's embedding result, mentioned above, gave rise to another area of investigation. De Groot [2] proved that for a given infinite cardinal number k, there exists a monoid F of endomorphisms on [0,1] which "universally linearizes" any monoid S of at most k endomorphisms on any completely regular T space X of weight <_ k. Baayen [1] generalized this result to obtain several theorems in categorical terms, restricting his considerations to monomorphisms for general categories and topological embeddings for Top , which he had to consider separately. In Chapter A, we will extend and update Baayen 's theorems, but, in particular, we will change the emphasis of the investigation. We will not be searching for universal objects in categories as Baayen did; instead, we will obtain quite general categorical linearization theorems for endomorphisms (and automorphisms) on objects in a category. Considering categories in which only certain products are required to exist, we will find that endomorphisms are actually restrictions of morphisms of an almost trivial-seeming linear character (i.e., morphisms that essentially act only on the coordinates of a power of an object, serving to "switch or collapse" these coordinates). We will find that in categories with countable products, every endomorphism in the category can be 'linearized', i.e., extended to a coordinate immuting morphism (4.2.10), and that any automorphism can be considered as the 'restriction' of a coordinate permuting automorphism on a power. Furthermore, we will find that certain monoids of

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endomorphisms, as well as groups of automorphisms, on an object in the category can be simultaneously linearized (A. 2. 4). And finally, we will find that for certain subcategories in a category with infinite products, universal linearizations may exist for every endomorphism in the subcategory (4.3.2).

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1. PRELIMINARIES The purpose of this chapter is to list most of the basic definitions and theorems that will be used in the following chapters. Proofs will be included orly for new results, which will be found in Â§1.5. Terms from category theory not specifically defined here can be found in [7], as well as the proofs for those theorems and examples that are stated here without proof or reference. Algebraic terms can be found in [12]; topological terms can be found in [11]. Â§1.1 Products and Limits DEFINITION 1.1.1 A family (A.) indexed by I is a function Occasionally when I is unimportant or understood, the family (A,), is written (A.) . DEFINITION 1.1.2 Let d be a category. A product in C of a set-indexed family (A.). of (J -objects is a pair (usually denoted by) ( n A., (tt.). ) satisfying the following three properties: iel ^ ^ ^^ (1) n A. is a C-object iel ^ (2) for each jel, tt , : 11 A ^ A is a C-morphism (called the ^ iel J projection from 11 A. to A.) . iel ^ ^ (3) for each pair (C,(f.). ), where C is a (5 -object and for each jel, f.:C -> A. is a G -morphism, there exists a unique induced C -morphism (usually denoted by) 5 C -> n A such that for ^ iel each jel, the triangle

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iel commutes. For notational convenience we may sometimes write (HA , tt ) for the product. Also when A = B for each iel, the product (IIA ,7t ) is usually written (B , tt ) and is called the I ' th power of B in G Â• A category C is said to have products provided that for every set I, each family of C -objects indexed by I has a product in ^ . The following categories have products: Set , Grp , Top , Haus , Top , CompT , CompRegT , Mon , Sgp . THEOREM 1.1.3 (Iteration of Products) Let (K.) be a pairwise disjoint set-indexed family of sets. Suppose that for each iel, (P .("j^)i^ J, ) is the product of a set-indexed family (X,), Â„ of C -objects and that (P, tt ) is the product of the family (P ) . Then (P,(tt, -tt ). j v i' ) is the product of (X,), y^ up to a natural isomorphism. PROPOSITION 1.1. A. If (nA^,TT^) and (nB.,p^) are products of the families (A^)^^j and (B^)^ is a morphism A -*B , then there exists a unique morphism (usually denoted by nf.) which makes the diagram

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commute for each jel. DEFINITION 1.: called the product of the morphisms (fj)j jÂ» Products are just a special type of limit which will be defined next. DEFINITION 1.1.6. Let I and G be categories and let D:I -> G be a functor. (1) Then a pair (L,(f ). ^,1 /-tn) is called a natural source for D in G provided that L is a C.-object, and for each ieOb(I), f .:L -> D is a C-morphism, and for all i,jeOb(I) and I-morphisms m:i -> j, the triangle D(i) D(m) commutes. (2) A natural source (L,f,). Â„, /-r\) for D in G is called a i iEQb(.IJ limit for D in G provided that if (L ,(f.), qt^z-tn) is any other natural source for D in G , then there is a unique morphism h:L ->L such that

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11 for each jeOb(I), the triangle D(j) commutes . Some well-known examples of limits are: products, terminal objects, equalizers, pullbacks, and intersections. DEFINITION 1.1.7. A category C is said to be complete provided that for every small category I (i.e., where Ob(I) is a set), every functor D:I -> <2 has a limit. (The categories Set , Grp , Mon , Top , Haus and CompTÂ„ are examples of' complete categories.) Â§1.2 Special Categories DEFINITION 1.2.1. A category ^ is said to be well-powered provided that every ^ -object has a representative class of subobjects which is a set . Dual notion: cowell-powered The categories Set , Grp , Top , Mon , Haus and CompT. are wellpowered and cowell-powered. DEFINITION 1.2.2 (1) A concrete category is a pair ( (2 , U ) where C is a category and 7>l : C "* Set is a faithful functor.

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12 (2) A concrete category ( Â©( , U ) is called algebraic provided that it satisfies the following three conditions: (a) Â©( has coequalizers. (b) W has a left-adjoint. (c) X\ preserves and reflects regular eplmorphisms. Examples of concrete categories include Set , Grp , Top , Haus , POS, Field and Mon. Top , Haus , PQS and Field are not algebraic categories; however, Set , Grp , Mon and CompT are algebraic. Â§1.3 Special Morphlsms in General Categories DEFINITION 1.3.1. Let C be a category. (1) A fi -fflorphlsm m is called a strong mor|ihi8m provided that whenever m-k = g-e for some C -morphisms k and g and some ^-epimorphis e , there exists a morphism h such that the diagram e commutes. (2) A C -morphism m is called a strong monomorphism when it is both a strong morphism and a monomorphism. (3) A C -morphism f is called an extremal morphism provided that whenever f = g-e, where g is a 6 -morphism and e is a G -epimorphism, e must be a C -isomorphism, (4) A C -morphism is called an extremal monomorphism provided

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13 that it is both an extremal morphism and a monomorphism. PROPOSITION 1.3.2. Let G be a category. (a) The product of strong (mono) morphisms is a strong (mono) morphism. (b) If f and g are strong (mono)morphisms and f-g is a morphism, then f-g is a strong (mono) morphism. Thus we say that the class of strong morphisms (respectively, strong monomorphisms) is closed under products in C (1.3.2(a)) and closed under composition in C (1.3.2(b)). In general, this is not true for the class of extremal morphisms in C or for the class of extremal monomorphisms in C Â• Extremal monomorphisms are particularly interesting. The following definitions and propositions will give us sufficient conditions for which the class of extremal monomorphisms is closed under products and composition. DEFINITION 1.3.3. Let C be a category, 7/1 be a class of monomorphisms closed under composition with isomorphisms, and ^ be a class of epimorphisms closed under composition with isomorphisms. (1) Let f be a morphism in C . A factorization f = m-e where e and m are morphisms in C is called an (it, 771) factorization of f provided that ec i and mcW. (2) An (J., VI) factorization of f, f = m-e, is said to be unique provided that whenever f = in-e is another (i,977) factorization of f, there is an isomorphism h such that the diagram

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14 commutes. (3) A category C is called an (i.W) category provided that jL and m are closed under composition and every morphism in C has a unique (t, ,Tn) factorization. (4) A category C is said to have the (l.,?W) diagonalization property provided that for every commutative square in C with ee
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15 PROPOSITION 1.3.4. Every { L M) category has the (Ji.JS'l) diagonalization property. THEOREM 1.3.5. Every complete, well-powered category C is an (epi, extremal mono) category. Furthermore the class of extremal monomorphisms in C is closed under products. Â§1.4 Epiref lections DEFINITION 1.4.1. Let ^ be a category and let 6( be a subcategory of C . (1) 6( is a full subcategory of fi provided that whenever f :A -Â» B is a (2 -morphism, and A and B are Â©(-objects, it follows that f is an Â©(-morphism. (2) 6t is a replete subcategory of ^ provided that whenever f :A -*B is an U -isomorphism and A is an 0(-object, it follows that B is an Â©(-object. DEFINITION 1.4.2. Let C be a category and let Ot be a subcategory of C . (1) A morphism f :X -^ Y is called g( -extendable provided that for each Cl-object A and each morphism g:X -> A there exists a morphism g :Y -* A such that the diagram commutes.

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16 (2) Let X be a fi -object. The pair (r ,X_ ) is called an Ot 61 Qt -epiref lection for X provided that X is an 5C-object and r :X ^ X is an Â©(-extendable epimorphism. (3) Q{. is called an epiref lective subcategory of \$ provided that for every ^-object X, there exists an <5C-epiref lection for X. THEOREM 1.4.3 (Characterization of Epiref lective Subcategories) (Herrlich and Strecker [8]). Let C be a complete, well-powered and cowell-powered category and let ^ be a full, replete subcategory of C. Then the following statements are equivalent: (a) OL is an epiref lective subcategory of C. . (b) Si is closed under the formation of products and extremal subobjects in C. . PROPOSITION l.A.4. Let C be a complete, well-powered and cowell-powered category and let Qi be any full, replete subcategory ofC. Then there exists a smallest epiref lective subcategory C(Â©<-) of C which contains Si, DEFINITION 1.4.5. Let C be a category and let Â©( be a subcategory of C . The epiref lective hull d(0() of 6t in C (if it exists) is the smallest epiref lective subcategory of C, containing OL. THEOREM 1.4.6 (Characterization of Epiref lective Hulls) (Herrlich [6]). Let C be a complete, well-powered and cowellpowered category and let Â©C be any full, replete subcategory of fi . Let X be a S -object. Then the following statements are equivalent: (a) X is a C(Ot)-object (b) X is an extremal subobject of a product of Â©(-objects (c) Each <9C -extendable epimorphism is {X}-extendable

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17 (d) Each Â©t-extendable epimorphism f :X -> Y is an isomorphism (e) Each Â©(-extendable morphism f :X Y is an extremal monomorphism. Â§1.5 Concrete Embeddings In this section, we will define concrete embeddings and will determine a few of their properties. DEFINITION 1.5.1. Let ( C.. V) be a concrete category. (1) A morphism f :X -> Y is called a concrete embedding provided that it is a monomorphism and whenever there is a morphism g:Z -> Y for which there exists a function h: "UCZ) -> M(X) such that the diagram U(z) M(x) K(Y) 11(f) commutes, there exists a morphism h:Z -Â»Â• X such that *U (h) = h. (2) A morphism f :X -^ Y is called an extremal (respectively strong ) concrete embedding provided that it is both a concrete embedding and an extremal (strong) morphism. PROPOSITION 1.5.2. Let (t yXi) be a concrete category. (1) The class of all concrete embeddings in G is closed under composition.

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18 (2) If "U preserves monomorphisms , the class of all concrete embeddings in C is closed under products, intersections and pullbacks. PROPOSITION 1.5.3. Let (C.ljL) be a concrete category that is complete and well-powered for which '^;J preserves monomorphisms. Then every extremal monomorphism is a concrete embedding, hence an extremal concrete embedding. COROLLARY 1.5.4. Let (C"U) be a concrete category that is complete and well-powered for v. aicii IX preserves monomorphisms. Then (C,^) is an (epi, extremal concrete embedding) category. In some concrete categories, monomorphisms are not always infective on the underlying sets. We next show that concrete embeddings da have the desired piroperty of injeetiVity* PROPOSITION 1.5.5. Let (6,11) be a concrete category. Then every concrete embedding in C must be injective on the underlying sets. PROOF: Suppose that f :X -> Y is a concrete embedding, and hence a monomorphism (1.5.1), but that "U(f) is not injective on the underlying sets. Then there must exist a,be*U.(X), a i?^ b such that U(f ) (a)= *U(f ) (b) , I a for x=a, x=b Define h : U(X) -> U(X) such that h (x) = < ^ ,, ^ ^ , , , ^ i^\ / wÂ«.\ / ^v / 1 Â£^j, ^-^-^ xcX, for which ( a 7^ X ?^ b I b for x=a, x=b and define h : H(X) > U(X) such that h (x) = N ^ for all xeX i for which a 9^ x 9^ b

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U(x) 19 U(Y) Uf) commute. Thus since f is a concrete embedding, there must exist h :X -> X such that U(h ) = h and h, :X -> X such that X^ (h.) = h a a a b b b (1.5.1). But IJL is a faithful functor; hence it reflects commutative triangles. Thus f-h = f = fÂ«h,, so h = h, since f is a monomorphism. a b a b But this implies that h = h Â— a contradiction. It is known ([7]) that concrete embeddings in Top are precisely the topological embeddings, and that concrete embeddings in algebraic categories are precisely the monomorphisms. We will show that these results can be used to characterize concrete embeddings in many additional categories. PROPOSITION 1.5.6. Let (C,U) be a concrete category, let Oi be any full subcategory of 6Â» and let f :X -> Y be an Ol-morphism. If f is a concrete embedding in 6 , then it is a concrete embedding inOC. PROOF: Let f :X -> Y be a concrete embedding in 6 . Then it is infective on the underlying sets (1.5.5), hence it is a monomorphism in & . Let g:Z ->Y be an Â©(-raorphism such that for some function h:1i(z) -* U(X), U(f)-h = 11(g). But g is also a C-morphism and f is a concrete embedding in C ; hence, there exists a C-morphism

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20 h:Z -> X such that U(h) = h. (1.5.1). Since Â©(is full, h:Z ^ X is also an fit-morphism. PROPOSITION 1.5.7. Let C be an algebraic category, let Â©C be any subcategory of C , and let f :A ->Â• B be an Ct-morphism. Then if f is a concrete embedding in 6( , it is a concrete embedding in
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21 By hypothesis, V. preserves epimorphisms; thus tl-(e) : TjL(A) -^ IICL) is surjective. But U (f ) is injective (1.5.5); so that *U.(e) must be injective; hence bijective. Consequently there is a function h: U(L) -^ U(A) such that U(e).h = 1^^^^ and hU(e) = 1^^)Thus U(f)-h = U(in). But (L,m) is an extremal subobject of B which is an Â©1-object. Hence by hypothesis, L is an Â©(-object, so m is an 0C-morphism. Thus since f is a concrete embedding in fit , there exists an h:L -> A such that U (h) = h. Thus h-e:A ^ A and U(h-e) = hU(e) = 1^^^ = U (1^). Hence since ^ is faithful, h-e = 1.. Thus e is a section (and an A epimorphism) ; hence it is an isomorphism. Thus f is an extremal monomorphism in C (1.3.1); hence it is a concrete embedding in (1.5.3). COROLLARY 1.5.9. In any full, hereditary subcategory fit of Top , the concrete embeddings are precisely the topological embeddings. PROOF: The epimorphisms in Top are precisely the surjective continuous functions.

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2. SOURCES In this chapter we will investigate the action of sources; a source being an object in a category together with a family of morphisms having the object as a common domain. We will see that in several special cases the morphisms of a source will act together to exhibit certain "mono-like" properties Â— properties which may not belong to any of the morphisms individually. The definitions and initial results on mono sources and extremal mono sources were developed by Herrlich and Strecker [7]. Â§2.1 Sources in General Categories DEFINITION 2.1.1. Let C be a category. (1) The pair (A,(f.). ) is called a source provided that A is a C-object and (f.). ^ is a family of C-morphisms, each with domain A. Note that for notational convenience, we will usually write (A,f.) for a source when the indexing class I is understood or unimportant. (2) A source (A,f.) is called a mono source provided that for any pair of C -morphisms h and k such that f.'h = f . 'k for all iel, it follows that h = k; i.e., provided that the family (f.) is simultaneously left-cancellable. (3) A source (A,f.) is called an extremal source provided that for each source (B,g.) and each C-epiraorphism e:A -> B such that for each i, the diagram 22

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23 commutes, e must be an isomorphism. (4) A source (A,f ) is called an extremal mono source provided that it is both an extremal source and a mono source. (5) A source (A,f .) is called a strong source provided that for each source (B,g.) and C-morphisms e and k, where e is a Â£ -epimorphism and g.-e = f.'k. for each iel, there exists a C -morphism h such that the diagram commutes for each iel. (6) A source (A,f,) is called a strong mono source provided that it is a strong source and a mono source.

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24 In the following paragraphs we will examine some fundamental examples of these sources. First, however, we shall determine their relative strengths. PROPOSITION 2.1.2. Let C be a category and let (A,f,) be a source in C . Then each of the following statements implies the statement below it. (a) (A,f.) is a strong mono source (resp., strong source). (b) (A,f.) is an extremal mono source (resp., extremal source), (c) (A,f.) is a mono source (resp., source). PROOF: (a) =Â» (b) ; Let (A,f .) be a strong source in C . Suppose there exists a family (g.). of morphisms in C and an epiraorphism e in G, such that f. = g.'e for all ie I . Then for all id, f.'l. = g. 'e. 11 1 A 1 Therefore, by Definition 2.1.1, there exists a morphism h:Q -* A such that the diagram and an epimorphisra. Thus e is an isomorphism. (b) =Â»(c): Clear from Definition 2.1.1.

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25 PROPOSITION 2.1.3. Let I and C be categories and let D:I -> C be a functor. If (L,Si.) is a limit of D, then it is a strong mono source. PROOF: By the definition of limit (1.1.6), (L,l ) is a natural source for D; i.e., for each I-morphism m:i -Â• j, D(m)Â»Â£ = I . . Let X be a Â£ -object and let f and g:X -> L be C-morphisms such that i.-f = i.'g for each ieOb(I), Then D(m)-(!' Â•Â£) = (D(m)'Jl )Â•Â£ = Â£ . Â• f for each I-morphism m:i -+ j. Hence (X,(Â£ "f) f^KCT^^ ^^ ^ natural source for D. But (L,l.) is a limit for D, thus there exists a unique morphism h:X -> L, such that l.-f = Â£ 'h for all ieOb(I). Therefore f h = g, and consequently (L,Jl.) is a mono source (2.1.1). Suppose that (B,g.) is a source and k and e are Â€ -morphisms Then for each I-morphism m:i -Â» j , D(m) -g 'e = D(m) Â•Â£ -k = Â£ 'k = g. "e, since (L,Â£.) is a natural source for D. However since e is an epimorphism, D(m)'g. = g.. Thus (B,g ) is a natural source for D. By Definition 2.1.1, there exists a unique morphism k:B -Â• L such that Â£.'k = g. for each ieOb(I). Hence, since (L,Â£ ) is a mono source, the diagram >. D(i) commutes for each icOb(I). Thus (L,Â£.) is a strong mono source (2.1.1),

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26 It is well known that products, equalizers, terminal objects and pullbacks are special types of limits; hence they are examples of strong mono sources, and consequently they are examples of extremal mono sources. The following corollary provided the motivation for Proposition 2.1.3. COROLLARY 2.1. A (Herrlich and Strecker [7]). Let I and C be categories and let D:I -> t be a functor. If (L,Â£ ) is a limit of D, then it is an extremal mono source. PROOF: Propositions 2.1.3 and 2.1.2. PROPOSITION 2.1.5. Let C be a category with pushouts. A source (A,f .) in C is an extremal mono source in C if and only if it is a strong mono source in C . PROOF: By Proposition 2.1.3, we need only show that if (A,f.) is an extremal mono source, then it is a strong mono source. Suppose that e is an epimorphism and for each iel, the diagram commutes. Let

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27 be the pushout of e and k. Then for each iel, there exists a morphisin q is an epimorphism; hence q is an isomorphism, since (A,f ) is an extremal mono source. Then q Â•p:B -* A and for each id, f.'q -p-e = h.'q*q '6 = g.'e = f.'k so that since e is an epimorphism and (A,f ) is a mono source, the diagram commutes for each iel. The following proposition was proved for extremal sources by Herrlich and Strecker [7]. PROPOSITION 2.1.6. Let C be a category with coequalizers . Then each extremal source (resp., strong source) in C is an extremal mono source (resp., strong mono source) in C . PROOF: Let (A,f.) be an extremal source (resp., strong source) in C, and let p and q be C -morphisms such that f, -p = f.'q for each id. Let (k,K) be a coequalizer for p and q. Then k is an epimorphism, and for every f.,such that f -p = f.-q, there exists a unique morphism g. :K -Â»Â• C, such that the diagram

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28 commutes. Since (A,f.) is an extremal source (resp., a strong source, hence an extremal source by Proposition 2.1.2), k must be an isomorphism (2.1.1). Hence p = q; so (A,f.) is a mono source. PROPOSITION 2.1.7. Let C be a finitely cocomplete category and let (A,f ) be a source in C . The following statements are equivalent. (a) (A,f.) is a strong soutfce (b) (A,f .) is a strong mono source (c) (A,f.) is an extremal source (d)' (A,f,) is an extremal mono source. PROOF: Since C. is finitely cocomplete, it has coequalizers and pushouts .Thus (a) =Â»(b) and (c) =>(d) (2.1.6) and (d) =*Â• (b) (2.1.5). Also it is clear that (b) =*'(a) and (d) =*-(c) (2.1.1) and (b) =*-(d) (2.1.2). Â§2.2 Sources in Concrete Categories DEFINITION 2.2.1. Let (C,U) be a concrete category. (1) A source (A,f.) is called a concrete embedding source provided that it is a mono source and for every source (B,g,) for which there exists a function h: ^l(B) -Â»IL^A) such that the diagram

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U(B) u(A) commutes for each id, there exists a morphism h:B ->A such that U(h) = h. (2) A source (A,f,) is called a strong concrete embedding source (respectively, an extremal concrete embedding source ) provided that it is both a concrete embedding source and a strong (reap., extremal) source. fhe pfopertieÂ§ ef eaficrete etnbeddlng seurceÂ§ will be ejtatniflgd in subsequent sections within the generalized framework of )W -sources. Â§2.3 ^-Sources Now that we have defined several distinct types of sources, we will find it convenient to introduce some unifying notation. DEFINITION 2.3.1. Let C be a category and let 7^1 be any class of morphisms in C Â• (1) A C -iiiorphism f will be called an ^morphism in C provided that ft?ff. (2) 7ft will be called isomorphism-closed in Â£ provided that for any ')7l -morphism m and any fi -isomorphisms e and e', such that the compositions mÂ»e and e'.m are defined in Â£ , m'e and e'Â«m must be ;,/ -morphisms, and ^!f must contain all of the isomorphisms in C. (3) 7^ will be called lef t-cancellative in Â£ provided that

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30 whenever p.qe ^ for C -morphisms p and q , q must be an ^-morphism. (4) A pair (X,f) will be called an Â»/-subobject of Y provided that X and Y are C -objects and f:X -> Y is an W-morphism. For certain classes W of morphisms in C , we have previously defined sources whose morphisms in the aggregate exhibit properties similar to the properties of individual ^-morphisms. These sources, as listed below in Table 2.3.2, will be called " ;!^-sources. " We will refer to Table 2.3.2 frequently throughout the remainder of the paper. Note that each class of morphisms listed in Table 2.3.2 is isomorphism-closed in C . TABLE 2.3.2. Let Â£ be a category. [Let (C,K) be a concrete category, the class of all morphisms in C the class of all monomorphisms in C OTj-morphism subobject morphism weak subobject monomorphism subobject W-source mono source the class of all extremal morphisms in e. extremal morphism extremal weak subobject extremal source the class of all extremal monomorphisms in e. the class of all strong morphisms in C extremal monomorphism strong morphism extremal subobject extremal mono source strong source the class of all strong monomorphisms in CÂ» strong monomorphism strong subobject strong mono source

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31 TABLE 2.3.2 (Continued) ^ rrt -morphlsm ^-subobject yt\the class of all concrete embeddings in e the class of all extremal concrete eiubeddings in C the class of all strong concrete embeddings in C. concrete embedding extremal concrete embedding strong concrete embedding concrete embedded subobject strong concrete embedded subobject concrete embedding extremal concrete extremal embedded subobject concrete embedding source strong concrete embedding source PROPOSITION 2.3.3 (Singleton ?;/-Sources) . Let fi be a category. [Let (C.'ZX) be a concrete category.] Let W be a class of morphisms in C listed in Table 2.3.2, and let f :A ->Â• B be a morphism in^ . Then the following statements are equivalent. (a) (A,f) is an W-source (b) f :A -Â» B is an ^;i -morphism (c) (A,f) is an ;W-subobject of B PROOF: (b)'Â»-fc-(c) : Apply Definition 2.3.1. (a)<Â»=Â» (b) : Apply Definitions 2.1.1 and 1.3.1 when C is any category and Definitions 2.2.1 and 1.5.1 when ( Â€. , Zl) is a concrete category. For example: (A,f) is a concrete embedding source provided that for every source (B,g), for which there exists a function h: U(B) -* 'U(A) such that U(f ) "b = lt(g) , there must exist a morphism h:A -^ B such that U(h) = h (2.2.1). This condition holds if and only if f is a concrete embedding (1.5.1).

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32 Thus, from the above result we can see that several of the propositions on ^//-sources in Â§2.1 will automatically yield results on t/7-morphisms; e.g., every strong morphism is an extremal morphism (2.1.2); in categories with pushouts, every extremal monomorphism is a strong monomorphism (2.1.5); and in categories with coequalizers, every strong (resp., extremal) morphism is a strong (resp., extremal) monomorphism (2.1.6). PROPOSITION 2.3.4 (Enlargement of ^-Sources). Let e be a category. [Let (Â€.,U) be a concrete category.] Let "W be a class of morphisms in Â£ listed in Table 2.3.2, (A,f J be a source and (f, ), i k keK be a family of morphisms in C with domain A, having (f .), j as a subfamily. Then if (A,f.) is an W-source, so is (A,f ) . PROOF: Clearly (A,f ) is a source. (a) mono : Let p and q be C-morphisms such that f, 'p = f, 'q for all keK. Then f^-p = f^-q for all id, so that since (A,f.) is a mono source, p = (b) extremal : Suppose (g, ), v ^^ ^ family of d-morphisms and e is an epimorphism in C such that f = g -e for all keK. Then f^ = g^-e for all iel, so that since (A,f.) is an extremal source, e is an isomorphism (2.1.1). Hence (A,f,) is an extremal source. (c) strong ; Suppose (B,g, ) is a source, e is an epimorphism and r is a morphism such that g -e = f t for all keK. Then g.-e = f.-r for all iel. Thus, since (A,f.) is a strong source, there exists h:B -> A such that the diagram

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33 commutes for all iel. Hence r = h-e. Now f, "h-e = g, *e for all keK. But e is an epimorphism, hence f, 'h = g, for all keK. Thus (A,f, ) is a strong source. (d) concrete embedding ; Suppose (B,g, ) is a source for which there exists a function h: U(B) -Â»K(A) such that tt(f,)'h = "UCg,) for all keK. Then U(f.)'h = U(g,) for all iel. Hence, since (A,f ) is a concrete embedding source, there exists h:B -* A such that U(h) = h (2.2.1), and consequently (A,f,) is a concrete embedding source. (e) The remainder of the proof follows directly from parts (a), (b), (c), and (d) . PROPOSITION 2.3.5 (Left-Cancellation of W -Sources) . Let ^ be a category. [Let (C,U) be a concrete category.] And let W be any class of morphisms in C listed in Table 2.3.2. Let (A,f ) be a source in C for which there exist families (h ) . and (g , ) . ^ of morphisms in C such that f = h.'g for all iel. Then if (A,f,) is an ^-source, so is (A,g.). PROOF: Clearly (A,g.) is a source in C . (a) mono : Let p and q be ^-morphism such that g -p = g. -q. Then f-'P = h -g.-p = h 'g.-q = f^*q. Since (A,f ) is a mono source, p q; thus (A,g.) a mono source (2.1.1).

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34 (b) extremal ; Let (r.).^, be a family of C-morphisms and let e be an epimorphism such that g, = r 'e for all iel. Then ^i '^ ^i*^i'^ ^Â°^ ^^'" ^^'' ^^^'^^ (A,f .) is an extremal source, e is an isomorphism; thus (A,g.) is an extremal source (2.1.1). (c) strong ; Suppose (B,r ) is a source in Â£ , e is an epimorphism in C and k is a morphism in C such that r "e = g. -k for all iel. Then h. -r . -e 1 1 i Â°i i there exists p;B ->A such that the diagram . 'k = f . 'k. Since (A,f.) is a strong source, commutes for all iel (2.1.1); hence p'e = k and r.-e = g.Â«(pÂ«e). However, e is an epimorphism, so r, = g 'p for all iel. Thus (A,g,) is a strong source. (d) concrete embedding ; Suppose (B,r.) is a source in Â£ for which there exists a function h; U(B) -> IKA) such that Â•^(g^)-h = t((r.) for all iel. Since f. = h.-g. for all iel and functors preserve composition, U(f^)*h = ^(.(h.'gj.h = XcCh . ) Â• 'U(g . ) -h = U(h^)U(r^) = U(h^-r^) for all iel. And because (A,f ) is a concrete embedding source, there exists a morphism h;B -> A such that "U (h) = h (2.2.1) and, consequently, (A, g.) is a concrete embedding source.

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35 (e) The remainder of the proof follows directly from parts (a), (b), (c) and (d) . COROLLARY 2.3.6. Let 7}J be any class of C-morphisms listed in Table 2.3.2. Then ?/? is lef t-cancellative in Â£ . PROOF: Suppose p and q are morphisms in Â€ , such that p'q is an J/J-morphism. Then (A,pÂ«q) is an???-source (2.3.3) and consequently (A,q) is an ?.J -source (2.3.5). Thus q is an ^-morphism (2.3.3). PROPOSITION 2.3.7. Let C be a category [(Â€.,U) be a concrete category] and let '}/! be any class of C-morphisms listed in Table 2.3.2. Then "//} contains all of the sections in d . PROOF: (a) mono : Clear. (b) strong : We will show that every section is a strong luonoraorphism. Let s be a section. Suppose that k and g are morphisms and e is an epimorphism such that S'k = g-e. Now there exists a morphism s such that s-s = 1. Thus S'SÂ«g.e Â«= s-s-s-k = sÂ«k = gÂ«e, so that since e is an epimorphism and s is a monomorphism, the diagram commutes. (c) extremal : Every section is a strong monomorphism, hence an extremal monomorphism (2.1.2). Thus every section is extremal.

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36 (d) concrete embedding ; We will show that each section is a concrete embedding. Suppose that s:A -*B is a section, g:C -^ B is a morphism and h: U(C) ^ U(A) is a function such that 'U(s)'h = UKg) . Now there is a morphism s such that sÂ«s = 1 . Thus h = U(s) Â• U(s)'h Â«= 'U,(s)'H(g) = tl(s-g). Thus s is a concrete embedding. The remainder of the proof follows directly from parts (a) , (b), (c) and (d).

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3. EMBEDDINGS INTO PRODUCTS In this chapter we shall investigate the characteristics of categorical 'embeddings into products,' and prove an embedding theorem, which interrelates ?//-sources and the existence of ^-morphisms from the object part of the source into the product of codomains of the source morphisms. We shall examine the concept of 'embeddable objects,' arriving at generalized definitions for (J|f-regular and ^-compact objects. These definitions will enable us to prove characterization theorems for epiref lective subcategories and epiref lective hulls in certain concrete categories. Â§3.1 Embedding Theorems Herrlich and Strecker [7] proved the portions of the following embedding theorem that deal with mono sources and extremal mono sources. Their work provided the motivation for the following generalized result. THEOREM 3.1.1 (The Embedding Theorem). Let C be a category. [Let (C ,*U) be a concrete category such that V. preserves monomorphisms and products.] Let % he a class of C -morphisms listed in Table 2.3.2, and let (X,f.) be a set-indexed source, with each f. having codomain E^, such that a product (nE.,Tr.) of the codomains exists in CThen the unique induced morphism :X -> IIE . is an >7l -morphism if and only if (X,f.) is an W -source. PROOF: (a) mono ; Suppose that p and q are any C -morphisms such that f^-p = f^-q for all iel. Then n^''p = TT,*-q for each iel. 37

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38 But (HE. ,77.) is a mono source (2.1,3 and 2.1.2), thus *p = 'q. Consequently, (X,f ) is a mono source if and only if is a monomorphism (2.1.1) . that (B,g.) is a source in (2 for which there is an epimorphism e in C such that f. = g.'e for each iel. By the definition of product (1.1.2), there exists a unique morphism :B -> TIE. such that (nE.,77,) is a mono source (2.1.3), is a factorization of ^ X 1 for which e is an epimorphism. Let Y denote the codomain of e. Then (Y,T7.'g) is a source and f. = 77,Â« = (77 Â•g)Â«e for each iel. Since (A,f.) is an extremal source, e must be an isomorphism. Hence must be an extremal morphism (1.3.1). (c) strong ; Suppose (B,g.) is any source in fi and e is any epimorphism in C and k is any morphism in C such that f, 'k = g. 'e exists a morphism h:B -> X such that the inner portion of the diagram e

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39 commutes for all iel; i.e., k = h*e and f . 'h = g. for all iel. Hence = TT.' for each iel. Therefore, since (HE,, it. X "i ' i 1 ) is a mono source, 'h =. Hence the outer portion of the diagram above commutes, which implies that is a strong morphism (1.3.1). Conversely, suppose that is a strong morphism. Then there exists a morphism h:B -Â» X such that the outer portion of the diagram commutes; i.e., k = h-e and "h = . Clearly fj'h = iT.'*h = tt Â• = g for each iel, so that the inner portion of the diagram commutes for each iel. Hence (X,f.) is a strong source (2.1.1). (d) concrete embedding ; From part (a), (X,f,) is a mono source if and only if is a monomorphism. Suppose that (B,g,) is any source in C for which there exists a function h: U(B) -^ V (X) such that tX(f.)'h = ^(g.) for each ici. Then by the definition of product (1.1.2), there exists a unique morphism :B -* IIE such hence ( TX (KB . ) , V.(tt . ) ) is a product in Set , hence a mono source in Set (2.1.3). Thus U (tt . ) Â• U() -h = tKn^. ) -h = Â•U(f^).h = U(g^) = *U(TT^-) = U(TT^)-U() for all icI if and only if 'U()'h = 'U(). Consequently (X,f ) is a concrete embedding source if and only if is a concrete embedding (2.2.1 and 1.5.1). (e) The remainder of the proof follows directly from parts (a), (b), (c), and (d) . There are many applications for this theorem since the category C. is quite unrestricted. A few of the examples are listed as corollaries. COROLLARY 3.1.2. Let A . e any set with at least two elements.

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40 Then for any set S, there exists an injection into some power of A. PROOF: Let a and b be distinct elements of A, and let 1 be the set of all functions from S to A. Then (S,^ ) is a mono source in Set . To see this, suppose that p and q are distinct functions from some set X into S. Then for some xgX, p(x) ?^ q(x). Define f:S -> A by Ia if s = p(x) _ _ , .^ / / N . Then f -p 7^ f .q. Since Set has products, the power (A , tt ) exists in Set . Let :S -Â» A be the unique induced function. By Theorem 3.1.1, since (S,3 ) is a mono source, is an injection. COROLLARY 3.1.3. A continuous map f from a topological space X into a topological product IlE. of spaces (E . ) , is a topological embedding if and only if (X,Tr.'f) is an extremal mono source in Top , PROOF: f is a topological embedding if and only if f is an extremal monomorphism in Top ([9]). Since the topological product together with the projection mappings form a categorical product in Top , f is an extremal monomorphism in Top if and only if (X,tt -f) is an extremal mono source in Top (3.1.1). COROLLARY 3.1.4. A continuous map f from a Hausdorff space X into a topological product of Hausdorff spaces, IlE , is a homeomorphism onto a closed subset of the product if and only if (X,7T.'f) is an ( maps) . PROOF: Extremal monomorphisms in Haus are exactly the topological embeddings onto closed subsets ([9]). Since a topological product

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41 together with the projection mappings is the categorical product in Haus , we have, by Theorem 3.1.1, that f is an extremal monomorphism if and only if (X,TT.'f) is an extremal mono source in Haus . COROLLARY 3.1.5. Let (C , U) be a complete, well-powered concrete category for which IL preserves monomorphisms and products. Let (A,f.) be a set-indexed source in C . Then (A,f ) is an extremal mono source in C if and only if it is an extremal concrete embedding source in C . f^ PROOF: C is a complete category. Hence (A, A -> E ) being a setindexed source, implies that the product (IlE ,it ) exists in C . If (A,f ,) is an extremal mono source, the induced morphism :A -> HE, is an extremal monomorphism (3.1.1) and, consequently, is an extremal concrete embedding (1.3.3). Thus (A,f.) is an extremal concrete embedding source. Conversely, by definition, every extremal concrete embedding source is an extremal mono source (2.2.1). A study of extensions naturally goes hand in hand with the study of embeddings. DEFINITION 3.1.6. Let C be a category, T?t be any isomorphismclosed class of C -morphisms, Â£ be a subcategory of C , and X be a C -object. (1) The pair (y.Y) is called an Ififl extension of X in C provided that Y is an Â£ -object, (X,y) is an W -subobject of Y and Y is an epimorphism. (2) Let (X,f.) be a source with codomains in Â€. (i.e., for each id, the codomain E of f is an g, -object). (X,f.) is called an

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42 ^ -nonextendable source with respect to t provided that for any ^ -extension (y.Y) of X in C with the propertyÂ— for each iel, there exists a morphism f^:Y -Â» E^ for which the diagram commutesÂ— Y must be an isomorphism (i.e., there must exist no proper ^7? -extension of X in C with this property). In particular, if a source (X,f ) contains all the morphisms from X to S-objects, then (X,f^) is an ?)I -nonextendable source with respect to Â£. if and only if there exists no proper ^-extension (w,W) for X in C for which w is ^-extendable (1.4.2). Note that every class of 9W-morphisms listed in Table 2.3.2 is isomorphism-closed in C . For convenience, when ^ is the class of all C -morphisms, we will use the prefix 'weak' to replace ';^' in Definition 3.1.6. Hence we can use the term 'weak-extension' to denote an?^ -extension when fA is the class of all morphisms in C . The definition for a weak-nonextendable source (X,f .) with respect to Â£ (3.1.6) corresponds to Mrowka's [16] definition for the class {f^:iel} to be an ' g, -nonextendable class for X.' PROPOSITION 3.1.7. Let C be a category, Â£ be a subcategory of C , and (X,f_^) be a source with codomains in Â£ . Then (X,f ) is a weak-nonextendable source with respect to Â£ if and only if (X,f ) is

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43 an extremal source. PROOF: Suppose (X,f ) is a weak-nonextendable source with respect to S. Suppose that (Y,g ) is a source in C such that for some epimorphism e:X -*Â• Y, the diagram X ' 2 >Y commutes for each id. Then (e,Y) is a weak-extension; hence e must be an isomorphism (3.1.6). Consequently (X,f,) is an extremal source (2.1.1). Conversely, let (X,f ) be an extremal source. Suppose there exists a weak-extension (y,Y) such that, for each iel, there exists a morphism g.. for which f. = g.-y. But y is an epimorphism (3.1.6) and (X,f,) is an extremal source; hence y must be an isomorphism (2.1,1). Thus (X,f.) is a weak-nonextendable source with respect to Â£ . Corollary 3.1.3 gives us a way to determine whether a continuous map from a topological space into a topological product of spaces is a topological embedding. And Corollary 3.1.4 gives us a way to determine whether a continuous map from a Hausdorff space into a product of Hausdorff spaces is a homeomorphism onto a closed subspace of the product, Yet these methods are not as convenient as we might wish. Mrowka's Embedding Theorem yields a topological characterization of these results. Parts of this theorem follow directly from Theorems 3.1.1 and 3.1.7. THEOREM 3.1.8 (Mrowka's Embedding Theorem [16]). Let

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44 3 = {f.:iel} be a set o: functions with f.:X -* E, where X and E,, 1 1 1 i' for each iel, are topological spaces. Let h be the set function from X into the topological product HE. such that ir.-h = f for each iel. We have (a) h is continuous if and only if each f. is continuous. (b) h is injective if and only if the set sf satisfies the following condition: CI. for every p, qeX such that p ?^ q, there is an f .e 3 with f^Cp) ^ f.(q). (c) h is a topological embedding if and only if h is continuous and injective and the set 3 satisfies the following condition: C2. for every closed subset A of X and for every peX \A, there exists a finite system f f, of functions from "J such that 1 n (p) i cl[A] (where cl stands for closure in In In n E. ). Â•' l<_k_
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45 PROOF: Parts (a) and (b) are well-known results. Note that the topological product with the Tychonof f product topology is a product in ToÂ£. Hence each f_^ is a Topmorphism. i.e., continuous, if and only if h is a Top -morphism. Also h is injective if and only if it is a Setmonomorphism. which by Theorem 3.1.1 is true if and only if condition CI is fulfilled in _Set. (Let p, qrX such that p ^ q and let gp.gq:{y) -^ X be defined so that g^Cy) = p and g (y) = q. Thus (X,f ) is a mono source in Set if and only if CI holds.) (c) The reader is referred to Mrowka's proof [16]. (d) Assume that the spaces E are all Hausdorff and h is a topological embedding. Then h[X] is a Hausdorff space, so X is also a Hausdorff space and consequently, h is a concrete embedding. Let Â£ be the full subcategory of Haus having {E^:icl} as its class of objects. Then C3 merely states that (X,f^) is a weak-nonextendable source with respect to'i . Thus by Proposition 3.1.7, C3 holds if and only if (X,f.) is an extremal source in Haus , which by Theorem 3.1.1 is true if and only if h is an extremal morphism in Haus ; i.e., if and only if h is an extremal monomorphism in Haus (2.1.6); i.e., if and only if h[X] is closed. COROLLARY 3.1.9. Under the same hypothesis as Theorem 3.1.8, the following statements hold: (a) (X,f^) is a mono source in ToÂ£ if and only if (X,f ) is a source in ToÂ£ such that CI holds. (b) (X,f^) is a concrete embedding source in ToÂ£ if and only if (X,f^) is a mono source in ToÂ£ such that 02 holds.

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46 PROOF: (a) (X,f ) is a mono source in Top if and only if h is a monomorphism (3.1.1) which holds if and only if (X,f ) is a source for which CI holds (3.1.8). (b) (X,f .) is a concrete embedding source in Top if and only if h is a concrete embedding in Top (3.1.1) [but concrete embeddings are precisely the topological embeddings ([7])], which holds if and only if (X,f.) is a source for which condition C2 holds (3.1.8). Â§3.2 Â£ I ?Â«-Embeddable Objects Mrowka [15] defined an E-completely regular space to be a topological space that is homeomorphic to a subspace of some topological m power E of the space E. For a Hausdorff space E, he defined an E-compact space to be a space that is homeomorphic to a closed subspace of some topological power E of E. For a given full subcategory Â€. of the category Top (respectively, Haus ) , Herrlich [6] defined an Â£ -regular space (respectively, an Â£ -compact space) to be any object in the epiref lective hull of Â£ in Top (respectively, Haus ) . First we shall directly generalize Mrowka' s definition in three ways: to arbitrary categories Â£ (rather than Top or Haus ) ; to arbitrary subcategories Â£ of C (rather than subcategories with a single object E) ; and to arbitrary 5?? -morphisms , for various classes W of C -morphisms (rather than topological embeddings) into the object parts of products of Â£ -objects. In later sections, Â§3.3 and Â§3.4, on epiref lective subcategories, we will see that our definitions coincide with Herrlich 's definitions in the categories Top and Haus . DEFINITION 3.2.1. Let C be a category, Â£ be a subcategory of C,

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and W be any class of C -morphisms that is isomorphism-closed in HE.. 1 (2) When t has a single object E, the term e])}! embeddable in Â£ will be used interchangeably with ^ |M -embeddable in Â£ . (3) Let (C ,U) be a concrete category. Whenever ?M is the class of all concrete embeddings in C , the term ^regular in C will be used interchangeably with the term Â£ |7?J -embeddable in C . Whenever m is the class of all extremal concrete embeddings in C , the term ^ -compact in C will be used interchangeably with the term C l?n-embeddable in C . Note that since the concrete embeddings in Top are precisely the topological embeddings, our definition for an E-regular space in Top coincides with Mrowka's definition for an E-completely regular space. Also our definition for a Hausdorff space to be E-compact in Haus coincides with Mrowka's definition of E-compactness, since the extremal concrete embeddings in Haus are homeomorphisms onto closed subspaces of Hausdorff spaces (1.5.3). The study of topological spaces has provided the motivation for these definitions. Clearly, however, by Corollary 3.1.2, for any set A with at least two points, every set S is A-regular in Set . In order to illustrate the generality of these concepts, we will construct an algebraic example. Let AbMon be the category of all Abelian monoids and

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48 monoid homomorphisms. Then Ab, the category of all Abelian groups and group homomorphisms, is a full subcategory of AbMon . The following proposition is a well-known result, although it may not be immediately recognizable in our new terminology. PROPOSITION 3.2.2. An Abelian monoid is Ab-regular in AbMon if and only if it is cancellative. PROOF: Suppose M is Ab-regular in AbMon , then there exists a family (A^) ^ of Abelian groups and a concrete embedding morphism f:M -> IIA . Suppose a,b,meM and m+a = m+b. Then f(m+a) = f(m+b) so that since f is a monoid homomorphism f(m)*f(a) = f(m)*f(b). Thus f(a) = f(b), since (IIA^,*) is a group. Hence a=b, since f is infective on the underlying sets (1.5.5). Consequently M is cancellative. Conversely, let M be a cancellative monoid. We will define a relation R on pairs (x,y) where x,yeM in the following manner: (x,y)R(x',y') if f x+y ' = y + x' . It is straightforward to show that R is an equivalence relation. Let "r be the set of all equivalence classes (x,y), and let * be the operation of componentwise addition of pairs, which is well-defined by the definition of R. Since M is cancellative and Abelian, it is easy to show that ("r,*) is an Abelian group, where (Oj.j.Oj^) = {(x,x):xeM} is the identity element and the inverse of any element (x,y) is the element (y,x). Define hrM-^Tby h(x) = (0^,x). Since h(x+y) = (Oj_^,x+y) = (ajTx) * (%>y^ = h(x) * h(y), h is a homomorphism. Let zeKerCh). Then h(z) = (S^z) = (^^X Thus (0^,z)R(0^,0^). Hence 0^ + 0^ = z+O^^; so z = . Consequently h is infective. Thus h is a monomorphism in Mon ,

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49 which is an algebraic category having AbMon as a full subcategory; hence h is a concrete embedding in Hon, and a concrete embedding in AbMon (1.5.6), Consequently M is Ab-regular. THEOREM 3.2.3. Let C be a category with products. [Let (CjlA) be a concrete category with products such that \i preserves monomorphisms and products.] Let Â£ be a subcategory of C , let W be a class of C -morphisms from Table 2.3.2, and let X be a C -object. Then X is Â£, |77l -embeddable in Â£ if and only if there exists a set Â£^ contained in Ob ( Â£, ) such that (X, I J horn. (X,E)) is an Â— ^ eVs, ' fA -source in Â£ . PROOF: Let X be Â£ j/Tl-embeddable in C . Then there exists a setindexed family (E . ) . of Â£ -objects such that there exists an jTl-morphism f:X -> IlE. (3.2.1). By the Embedding Theorem (3.1.1), (X.TT.'f) is. an M -source. Since an 5?l -source can be enlarged (2.3.4), (X,(^ hom(X,E.)) is an W -source. iel ^ Conversely, let Â£^ be a set contained in Ob(g ) such that (X, [^ hom(X,E)) is an W-source. By hypothesis. Eeg. Â£ has products. Hence the product hom^(X,E) g ( n E ,77 .77 ) exists in C (by EcÂ£, S the Iteration of Products Theorem (1.1.3)). homg_(X,E) Let :X -Â» n E be the unique induced EcÂ£^ Â£ morphisH'. such that 7t .77 Â• = g for all ge (^ hom (X,E). Thus is an

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50 W-morphism (3.1.1), so that X is Â£ |y??-einbeddable in Â£. COROLLARY 3.2.4. Let (C .W) be a concrete category with products such that IX preserves monomorphisms and products. Let ^ be a full subcategory of C and let X be a C -object. Then X is C -regular (respectively, Â£-compact) in 6 if and only if there exists a set ^ contained in Ob(Â£) such that (X, \j hom (X,E)) is a concrete (respectively, extremal concrete) Ee Â£. *embedding source in C. Â• COROLLARY 3.2.5 (Mrowka [16]). A topological space X is E-regular if and only if the following two conditions are satisfied: Cl^ for every p.qeX, p?^q, there is a continuous function f:X -> E with f (p) ^ f(q). C2': for every closed subset A of X and every pex\A, there is a finite number n and a continuous function f':X -> E such that f'(p) i f'[A]. PROOF: Let 3 = hom^(X,E). By Corollary 3.1.9, (X, ^) is a concrete embedding source if and only if conditions CI' and C2 ' (which are precisely CI and C2 of Theorem 3.1.8 stated for E. = E for all iel) hold. By the Embedding Theorem (3.1.1), the induced morphism :X Â•* E is a concrete embedding morphism (hence X is E-regular (3.2.1)) if and only if (X, 3) is a concrete embedding source. There are many well-known corollaries to the above theorem. Topological examples have been collected by Mrowka [13] and [16] and Kerrlich [4]. Many of these corollaries had been established as separate theorems long before the development of a unifying theory.

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51 Several of these well-known results are listed below as examples and are stated without proof. EXA^IPLE 3.2.6 (Tychonof f ) . A topological space is a completely regular T^-space if and only if it is [0,l]-regular in ToÂ£. A Hausdorff space is compact if and only if it is [0, 1] -compact in Haus . EXAMPLE 3.2.7 (Mrowka [lA]). A topological space is a completely regular T^-space if and only if it is (0,l)-regular in ToÂ£. A Hausdorff space is realcompact if and only if it is (0, l)-compact in Haus . EXAMPLE 3.2.8 (Alexandrof f ) . Let F be the T -space with two o ^ points {a,b} in which the only proper closed set is {a}. A topological space is a T^-space if and only if it is F-regular in Top . EXAMPLE 3.2.9. Let W be the two-point indiscrete space. A topological space is indiscrete if and only if it is W-regular in Top . Recall that a topological space is said to be zero dimensional provided that it has a base of closed-and-open sets. EXAMPLE 3.2.10 (Alexandrof f) . Let D be the discrete space with two points. A topological space is a zero-dimensional T -space if and only if it is D-regular in ToÂ£. A zero-dimensional Hausdorff space is compact if and only if it is D-compact in Haus . EXAMPLE 3.2.11 (Mrowka [13]). Let V be a topological space with three points {a,b,c; such that {a} is the only proper nonempty open subset. Every topological space is V-regular in Top . Let us now consider another result of Mrowka's. Let L m denote a space with m-elements and the finite complement topology. Let ToÂ£^ denote the category of T^-spaces and continuous maps. By Proposition 1.5.9, any Top^-morphism f :A -> B is a concrete embedding in ToD^ if and only if it is a concrete embedding in Top .

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52 EXAMPLE 3.2.12 (Mrowka [13]). There exists no T -space X such that every T -space is X-regular in Tod . However ^ = {L :m is a -IÂ— ^1 m cardinal} is a class of T^-spaces such that every T -space of cardinality m can be topologically embedded into the topological power (L )^ . m Note that there exists no T space X, such that every T -space is X-regular in Top ^, or X-regular in ToÂ£, although by Example 3.2.11, every topological space is V-regular in Top . Clearly, V is not a T^-space. Also we have seen that every T -space is i -regular in Top . This example illustrates the necessity for our generalized definition. Trivially, of course, in a category C , every C -object is C |77?-embeddable in C if the class Vl of morphisms in C contains the identity morphisms. The remainder of this section will be used to reformulate a topological result of Mrowka [16] into categorytheoretic terms. Recall that for a category C and C-object E, the contravariant hom functor of C with respect to E is the functor hom(-,E): C -> Set defined so that hom (_,E)(X) = hom^ (X,E) for every C-object X, C C and hom (_,E)(t{)) = _Â•({) for each C-morphism 4). DEFINITION 3.2.13. Let C be a category and let ?7? be an isomorphism-closed class of Â£ -morphisms. Let E and X be C -objects. Then a pair (X ,
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53 THEOREM 3.2.14 (The Identification Theorem). Let C be a category with products [Let (C,U) be a concrete category with products where V> preserves monomorphisms and products.], let m be a class of C -morphisms from Table 2.3.2 and let C have the (e-pi,7tt ) factorization property. Let E be a C -object. Then for any C -object X there exists an EJTW -transformation (X*,^^) of X. PROOF: Let X be a C -object. By hypothesis, the product homg_(X,E) (E ,TT ) exists in C . Let the unique induced morphism be homg_tX,E) :X ^ E -By hypothesis, = P'^^y where P is an TTJ-morphisn and

) is injective since cf) is an epimorphism. For every fehom (X,E), the diagram ^ commutes; hence tt 'P c hom (X ,E) and f = Tr^'p'(}) = hom(_,E) (<}> ) (ir 'p) . X ^ r X ^ X f Therefore hom(_,E) (cj) ) is surjective. C X COROLLARY 3.2.13 (Mrowka [16]). For all topological spaces X and E, there exists an E-regular space X and a continuous surjective map c;;:X X such that hom^ ( ,E)(
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54 PROOF: Top has the unique (epi, extremal concrete embedding) factorization property (1.3.5). Â§3.3 /Tt -Coseparating Classes In this section we will show that in a category with products, an object E in the category, with the property that every other object is E|mono-embeddable,must be a coseparator for the category. Recall that a C -object C is called a coseparator for C provided that for any two distinct C~morphisms p and q:A -Â»B, there exists a C.-morphism x:B -* C such that xÂ«p ?^ xÂ«q. We shall first prove a proposition relating coseparators and mono sources, which we will use to develop the definition of a more general concept, namely an ^-coseparating class for a category. Then we shall see that in a category C with products, a class Â£. of C-objects is an ?77-coseparating class for C if and only if every C -object is Â£.|??/-embeddable in C. PROPOSITION 3.3.1 (Herrlich and Strecker [7]). Let C be a category. A C -object C is a coseparator for C if and only if for each C-object A, the source (A,hom (A,C)) is a mono source. PROOF: Let A be any C -object and let p and q be any distinct C -morphisms having the same domain and having codomain A. (A.hom -^ (A,C)) is a mono source if and only if there exists fehom-^(A,C) such that f'p =^ fÂ«q, which will happen for all such C-objects A if and only if C is a coseparator for C . Consequently. we know. for example. that any set with at least two points is a coseparator for Set (3.1.2). Herrlich and Strecker [7] and others have collected many examples of coseparators in categories.

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55 Baayen [1] defined a universal object in a category C to be a C -object U with the property that for every C -object A, there exists a monomorphism m:A -* U . Clearly then for every C -object A, (A,hom (A, U)) must be a mono source (2,3.3); hence U is a coseparator for C . Baayen lists many examples of universal objects in categories. DEFINITION 3.3.2. Let C be a category, let W be a class of C-tnorphisms from Table 2.3.2, and let (St be a subcategory of C . (1) Let Â£ be a subclass of the class of all C -objects in C . ^ is called an y}t ~coseparating class for 6i provided that for each Of-object A, there exists a set Â£ contained in Â£ such that (A, [^ horn 3 (A,E)) is an ?7? -source in C. E.Â£^ (2) A Â£ -object E is called an 77^ -coseparator for Ot provided that {E} is an W -coseparating class for Ot. From Proposition 3.3.1, it is clear that what we now call a mono-coseparator for C is exactly a coseparator for C by the usual definition. Note that for every category C , the class Ob(C) of all C -objects forms an ^ -coseparating class f or C , since any class ytl from Table 2.3.2 contains all the isomorphisms in C . Propositions 2.1.2 and 3.1.5 state the relative strengths of //J-sources for the various classes % of C -morphisms from Table 2.3.2; hence they also imply the relative strengths of /)?-coseparating classes for Â£ for different classes J7l . Therefore it is clear that every strong mono-coseparating class is an extremal mono-coseparating class, which in .urn is a mono-coseparating class (2.1.2). Similarly for

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56 complete well-powered concrete categories (C ,U) for which XI preserves monoraorphisms and products, every extremal mono-coseparatlng class for C is a concrete embedding-coseparating class for C , hence an extremal concrete embedding-coseparating class for C (3.1.5). The following theorem is the desired result which relates Â£ IM -embeddable objects and W -coseparating classes. It follows immediately from Theorem 3.2.3 and the above definitions. THEOREM 3.3.3. Let C be a category with products. [Let (C,U) be a concrete category with products such that It. preserves monomorphisms and products.] Let W be a class of C-morphisms listed in Table 2.3.2, and let Â©1 and Â£. be subcategories of C. Then the following statements are equivalent: (a) Ob(Â£) is an 7W-coseparating class f or
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57 (c) For each C -object A, there is a morphism f such that (A,f) is a subobject (respectively, an extremal subobject) of some power of e. PROOF: A C-object C is a mono-(respectively, an extremal mono-) coseparator for C if and only if {C} is a mono-(respectively , an extremal mono-) coseparating class for C ; hence the result follows from Theorem 3.3.3. Recall that by Mrowka's result on T -spaces (3.2.12), Top has no concrete embedding-coseparator in Top ; however Top has a proper concrete embedding-coseparating class in ToÂ£ , namely i/ , the class of all topological spaces with the finite complement topology. Note that in Top , the space V with three elements {a,b,c}, for which {a} is the only proper open set, is a concrete embedding coseparator for Top (3.2.11) and when Top is considered as a subcategory of Top , V is a concrete embedding-coseparator for Top .. . DEFINITION 3.3.5. Let S be a category, let W be a class of C -morphisms that is isomorphism-closed in Â£, and let Â£. be any subcategory of C 6.(g,|7?l) will be used to denote the full subcategory of S whose objects are precisely the ^ ?7J-embeddable C -objects. Clearly then Theorem 3.3.3 says that for any category C (respectively, concrete category (C,2A) such that U preserves monomorphisras and products) which has products, for any class W of C -morphisms listed in Table 2.3.2, and for any subcategory Â£ of (f , the class Ob(Â£,) is an ?/7-coseparating class for (l(Â£.l^).

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58 DEFINITION 3.3.6. Let C be a category and let 0( be a full subcategory of C . (1) Then Â«^^ will be used to denote the full subcategory of C whose objects are the object part of products of ^-objects that exist in Â£ . (2) Let VA be any class of \$ -morphisms that is isomorphismclosed in 6 . /,7^will be used to denote the full subcategory of C whose objects are the object parts of >?t -subobjects of 6C -objects. (3) When Ci has a single object A, ^A will be used interchangeably with (?&i , and TtlA will be used interchangeably with MBl. Then by definition of an C l??l -embeddable object in a category C (3.2.1), the full subcategory C ( S, |W ) is precisely the full subcategory f^fi^. Â• The following theorem was proved, for the class of monomorphisms in a category \$ , by Herrlich and Strecker [7]. THEOREM 3.3.7. Let & be any category with products. Let W be any class of Â£ -morphisms that is isomorphism-closed in C and closed under composition and products, and let Oi and S be any full subcategories of 6 . (a) (?eOl = C^6L. (b) 9/moi.rnoi. (c)
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59 is a full subcategory of tMPlB . PROOF: (a) Clearly products can be iterated; hence A . Let (riA.,p.) be the product of the family A , and let f be the unique induced morphism that makes the diagram X = nY commute for each iel. Then f is the product of the family (m,). of ^ -morphisms (1.1.4). Consequently f is an ^ -morphism since ^i is closed under products. Hence X is an filrOl -object. (d) By part (c), 9%^Q{, is a subcategory of '/^P^Gi, and by (a), Tn^fOi = tn^ei Â• ^^^Â° ^y p^^^ (t), ^nnpet = '^^ot Â• so 7/lfQl is closed under the formation of ^J-subobjects and products. Clearly any full subcategory containing Qt and closed under the formation of y([ -subobjects and products must contain 1/ffi&(_ . (e) Suppose that ^ is a full subcategory of ^(PS Â• Then PiPBi is closed under the formation of products and T'? -subobjects

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60 (by part (d)) and hence plS^Ol is a full subcategory of W^S . Recall that for a subcategory GL of 6, 6(0f) denotes the epireflective hull of Â©I in Â£ (1.4.5). COROLLARY 3.3.8. Let G be a complete category that is wellpowered and cowell-powered, let ?/? be the class of extremal monomorphisms in Q. and let <^ be a full replete subcategory of C. Then C ( ^1 extremal mono) = TfllPOi = C(Cl); i.e., the objects in the epireflective hull of Q{ are precisely the ^1^ -embeddable objects in G, . PROOF: By Theorem 1.3.5, W is closed under products and composition. Thus by Theorem 3.3.7 f/lPS(Â±s the smallest full subcategory of d containing Oi and closed under products and extremal subobjects; hence ?}tPOl is the smallest epireflective subcategory of C containing (S( (1.4.3). Thus C(Ct) = TnfiGi = C(<3/1 extremal mono), (1.4.4 and 3.3.5). Although the following results are simply specializations of Corollary 3.3.8 we will use them frequently throughout the remainder of this chapter. COROLLARY 3.3.9. Let Ot be any full replete subcategory of Top . Then in Top , Top (C(| extremal mono) = Top ((X| extremal concrete embedding) = Top (Ci I concrete embedding) = Top (0(); i.e., the (5t-compact spaces in Top are precisely the 6( -regular spaces in Top , and these are precisely those spaces in the epireflective hull of OL in Top . PROOF: Top is complete, well-powered and cowell-powered. Also, the extremal monomorphisms in Top are precisely the concrete embeddings in Top ([7]).

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61 COROLLARY 3.3.10. In Haus , let Ot be any full, replete subcategory of Haus . Then Haus (0(| extremal concrete embedding) = Kaus (6t I extremal mono) = Haus (Â©1), i.e., the (>t-compact spaces in Haus are precisely those spaces in the epiref lective hull of 0( in Haus . Furthermore Haus (0(| concrete embedding) = Top (Ol) , i.e., the Â©( -regular spaces in Haus are precisely those spaces in the epiref lective hull of QL in Top . PROOF: Haus is complete, well-powered and cowell-powered , and it is a full, t.ereditary subcategory of Top . Also the extremal concrete embeddings in Haus are precisely the extremal monomorphisms in Haus (1.5.3) and the cohCifete eiiibeddingS in HaUS ^te ceilctete embeddihgS in Top (1.5.9). Â§3.4 (X -Regular and 0(-Compact Objects CONVENTION 3.4.1. Throughout the remainder of this chapter, all subcategories will be assumed to be both full and replete. In the preceding section we found that for every subcategory & of Top, the 01 -regular spaces in Top are precisely the spaces in the epiref lective hull of Cl in Top (3.3.9). Furthermore, we discovered that for each subcategory 0^ of Haus , the Ot-compact spaces in Haus are precisely those spaces in the epiref lective hull of 6i in Haus (3.3.10). Note that when (3l is a subcategory of Haus , the Of -regular spaces in Haus by our definition are all those Hausdorf f spaces which are Â£[ -regular in Top , since concrete embeddings in Haus are concrete

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62 embeddings in Top (1.5.9). Thus our definitions for Of -regular and d-compact objects coincide with Herrlich's definitions in Top and Haus (Â§3.2). Epiref lective subcategories have been studied extensively. Recall that in a complete, well-powered and cowell-powered category Â£ , a subcategory 6t of C is epiref lective if and only if it is closed under the formation of products and extremal subobjects (1.4.3). Consequently, there are many examples of epiref lective subcategories: Haus , Top , CompRegT and Ind are epiref lective in Top ; CompT ^, RComp (i.e., realcompact) , CompRegT are epiref lective in Haus ; Ab is epiref lective in Grp , etc. Recall that an epiref lective subcategory 6l of a category 6Â» has the property that for every G -object X, there exists an Ct-epireflection (r ,X^ ) in C, where X ., is an 0( -object and r :X ^ X is an Ot -extendable epimorphism (1.4.2). One of the motivating examples in the study of epiref lective subcategories was the construction of the Stone-Cech compactif ication for completely regular T -spaces. Let X be a Hausdorff space and let ^ = C(X,[0,1]). Then the product ([0,1] ,t;.) exists in Haus , and in CompT ^ (by the Tychonof f Theorem) . Let the unique induced morphism g be :X -> [0,1] . Let = m.3 be the unique (epi, extremal mono) factorization (1.3.5), and let BX be the codomain of 3. Since Haus has the unique (epi, extremal mono) factorization property and since every compact space is homeomorphic to a closed subspace of the product of unit intervals (3.2.6), (3,BX) can be shown to be a _CompT -epiref lection of X. Also if X is completely regular, 3 is a topological embedding (3.2.6).

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63 Clearly for CompT -regular spaces, which are the completely regular spaces in Haus , there exist topological embeddings into compact T. -spaces, which are 'universal' in the sense of being ConvgT. -extendable, Herrlich and Van der Slot [10] proved a very useful result relating epiref lective subcategories G( in Haus and the existence of Â£if-epiref lections (r ,X^ ) for C( -regular spaces, for which r^ is a topological embedding. The following theorem is a generalization of their result to include concrete categories other than Haus . This generalization and the ones following it came about because of the development of the definition for concrete embeddings. THEOREM 3.4.2. Let (C.U) be a concrete category, which is complete, well-powered and for which VL preserves monomorphisms. If (St is a subcategory of G , then each of the following statements implies the statement below it. (a) Si is epiref lective in Â£ . (b) Each Â£v~^egulsr object has a concrete embedding-extension (w,W) in (S( for which w is Â£i -extendable. (c) Ct is closed under the formation of products and extremal concrete embedded subobjects. (d) ol is closed under the formation of products and extremal subobjects. Furthermore if C is cowell-powered , then the statements above, (a) through (d) , are equivalent. PROOF: (c) =Â»(c) : The extreinal monomorphisms in C are precisely the extremal concrete embeddings in C (1.5.3).

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64 (a) =^(b) ; Let &. be epiref lective in C . Suppose X is an ^-regular object. Then there exists a product (HA u.) of C<-objects and a concrete embedding f:X nA^. Let (r^ ,X^ ) be the Â©t-epiref lection of X; then X^ is an A-object and r^ :X -> X^ is an (5(-extendable epimorphisiu. Note that TT^-f is a morphism from X to A. for each iel. Thus for each iel, there exists g :X -^ A such that the diagram f V HA, I ^ commutes. Let -^^ -> HA^ be the unique induced morphism such that TT.- = g.. Then ^^-f = g.-r^^ = TT^--r^ for all iel; since products are mono sources, f = -r^ . Thus r^ is a concrete embedding, since f is (2.3.5, 2.3.3). (b) =Â» (c) : To show that 6( is closed under products, let ^^i^iel ^^ ^^^ set-indexed family of 6C -objects. t is complete, so the product (nA^.u^) is in C. Then HA^ is an <Â« -regular object. Thus by (b), there exists a concrete embedding-extension (w,W),for which W is an ai-object and w:nA^ -> W is an Sf -extendable epimorphism. Thus for every iel, there exists a morphism tt^:W -> A^ such that the diagram

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65 HA.^ W commutes. Let <7T.>:W ->HA, be the unique induced morphism such that TT.* = IT. for each iel. Then it ,Â• 1Â„ . = tt.-w = it,**w for each J 1 J J nA^ J J 1 A jel. Consequently, since products are mono sources, 1 = <7r.>'W. Thus w is a section, as well as an epimorphism, hence an isomorphism. By hypothesis, 6J is replete, and thus IIA is an 6(-object. Also C( is full; so that u . is an C( -morphism for each iel. To show that Ct is closed under the formation of extremal concrete embedded subobjects, let (X,m) be an extremal concrete embedded subobject of an flf-object A. Then X is Oi-compact, hence 6C-regular. By (b), there exists (w,W) where W is an 6( -object and w:X ->W is an fi( -extendable epimorphism. Thus there exists m :W ->A such that m -w = m (1.A.2). But m is an extremal concrete embedding, hence w is (2.3.5, 2.3.3); and consequently, w is an extremal monomorphism, as well as an epimorphism. Hence w is an isomorphism. Thus, since {5t is replete, X is an Ci-object, and because ^ is full, m is an fl(-morphism. Furthermore, if C is cowell-powered, the Characterization Theorem for Epireflective Subcategories (1.4.3) can be applied so that (d) =Â»'(a). Herrlich [5] has collected many examples in Haus which illustrate the conclusions of this theorem. We will examine an algebraic example. EXAMPLE 3.4.3. Let AbMon be the category of Abelian monoids and monoid homomorphisms. Then Ab, the category of Abelian groups and group homomorphisms, is an epireflective subcategory of AbMon .

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66 Let M be any Abelian monoid. We -ill now construct an Abepiref lection fc M, using a method outlined by Lang [12], for the cc. struction of the Grothendieck group K(M). Let F , (IIM) be the free Abelian group genera, ted by 11(M) , the underlying set for M. For each xgM, let [x] be the generator of F (XIM) corresponding to x. Clearly we have an injective function f:M -> F , (UM) defined by ab f(x) = [x] for each xeM. Let B = { [x+y] [x] [y]:x,yeM} and let be the subgroup of F , ('y,M) generated by B. Let (fi:F , (liM) -> F , CUM)/ be the canonical homomorphism. Then <{)'f:M -> K(M) is a monoid homomorphism. Clearly it is an epimorphism. From the 'universal' property for free Abelian groups, it follows that O'f is Ab-extendable. Thus (<^'f, K(M)) is the desired Ab-epiref lection for M. Let us suppose M is Ab-regular (i.e., cancellative (3.2.2)). product is (nA.,-iT.), and there is a concrete embedding g:M -* IIA.. . But IIA. is an Abelian group and <+i*f is Ab-extendable . Thus there exists a monoid homomorphism g :K(M) -> HA. such that g ' X. which is an a >A. 6i -extendable epimorphism such that the diagram

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67 ->.x, commutes. PROOF: By hypothesis, X^ is a 3-object, since it is an Ct-object. Thus, since r is 0-extendable (1.4.2), there exists a morphism e:X -> X such that r^ = e-r . But by hypothesis, X^^ is Cl-regular, Hence by Theorem 3.4.2, there exists a concrete embedding-extension (Y.Ygj ) in Gi for which y:X^ ^ Y^^ is 6Vextendable. But r^ is 6l-extendable and yx :X -*Y^ ; hence there exists a morphism a:X -* Y^ such that the exterior portion of the diagram V'ot commutes; i.e., yx = a-r Thus yr Now Y = ci'e since r^ is a Q-epiref lection, hence an epimorphism. Thus e is a concrete embedding because y is a concrete embedding (2.3.5). 2.3.3)). To show that e is 6!-extendable, suppose that there exists a morphism f:X>j -> A for some 6(,-object A. Since y is C^-extendable, there exists a morphism g :Y^ -* A such that g 'Y = gJ however, Y = aÂ«e, so that g -a-e = g. Consequently, we can conclude that e is Ct-extendable (1.4.2).

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68 Once again, the preceding theorem was proved for the category Haus by Herrlich [6], who has exhibited several examples. For instance, the category GompT is a full subcategory of RComp , which is a full subcategory of CompRegT , whose objects are the CompT -regular spaces in Haus . For a given Hausdorff space X, there exists a CompT ^-epiref lection (6^,3^X) and a RComp -epiref lection (B^.gj^X). Then by the conclusion of Theorem 3. A. 4, g X can be densely embedded in B^X. And if X is completely regular, X can be embedded in B^X (3.4.2). Now let us look at our al:;ebraic example. Ab and CAbMon , the full subcategory of AbMon whose objects are the cancellative Abelian monoids, are epiref lective subcategories of AbMon. In fact the objects of CAbMon are precisely the Ab-regular objects in AbMon (3.2.2). And Ab is a full subcategory of CAbMon . Then for any Abelian monoid M, there exists an Ab-epiref lection (B.,3.M) and an CAbMonepiref lection (3^,B^^M) of M. From Theorem 3.4.4, there exists anembedding epimorphism e:3^M -^ B^^M. Note that B^M is K(M), the Grothendieck group (3.4.3). The following theorem was proved by Herrlich [5] for the category Haus . THEOREM 3.4.5. Let (C,t<) be a concrete category that is complete, well-powered and cowell-powered, for which U preserves monomorphisms. If Ct is a subcategory of C , then the following statements are equivalent: (a) X is an Â£i -regular object in G . (b) There exists an Oi -compact object Y and a concrete embedding morphism f :X -< Y.

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69 (c) There exists a concrete embedding-extension (w,W) for X where W is Ci-cotnpact. (d) Each C? -extendable morphism f:X -> Y is a concrete embedding. PROOF: We will show that (b) =*(c) =* (a) = (d) =*(b). (b) =Â» (c) ; Suppose Y is an (5( -compact object for which there exists a concrete embedding f:X -*Y. By definition of 6( -compact, there exists a product (r[A.,Tr.) of ^-objects and an extremal concrete embedding m:Y -> IIA.. Now there exists an (epi, extremal concrete embedding) factorization of f, f = t-w (1.3.5, 1.5.3) X ^ Y . >. nA^ Let W denote the codomain of w. Then W is ^-compact, since the composition, m> t, of extremal monomorphisms is an extremal monomorphism in a complete, well-powered category (1.3.5) and mÂ«t is a concrete embedding, since m and t are (1.5.2). Also w is a concrete embedding, because f is. Thus since w is an epimorphism and a concrete embedding, (w,W) is the required extension. (c) =Â» (a) : Suppose (w,W) is an Oi -compact concrete embeddingextension of X. Then since W is ^-compact, there exists a product (lA ,77 ) and an extremal concrete embedding m:W -Â» IIA.. Also the epimorphism w:X -^ W is a concrete embedding since (w,W) is a concrete embedding extension. Hence m-wrX -* "lA. is a concrete embedding (1.5.2), so X is w^i -regular.

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70 (a) =^id) ; Suppose X is 5i -regular and let f :X -> Y be any Ci-extendable morphism. Then there exists a product (nA.,TT.) of Ct-objects and a concrete embedding m:X -Â» IIA.. Thus for each iel, there exists a morphism it 'mtX -> A,; and since f is ^-extendable, there exists a morphism h.:Y -* A. such that the diagram commutes for each ie^ (1.4.2). Let the unique induced morphism be :Y -> HA., such that Tr.' = h. for each iel. But ir.'m = h.*f: 11 111 i i ' hence ir.'m = TT.''f for each iel. Thus m = Â«f, since products are mono sources. Thus f is a concrete embedding since m is. (d) =* (b) : Suppose each 6^-extendable morphism f :X -> Y is a concrete embedding. In Corollary 3.3.8, we have seen that an epireflective hull e.(ei) f or 6f exists in C (in fact, it is Â€.(6( j extremal concrete embedding)). Thus there exists (r >X ) , an C.( X^ is C (Â©t) -extendable, it is Cri -extendable, and thus it is a concrete embedding (by hypothesis). Also X^ is an C(<^l)-object , hence 0<-compact, so that (b) holds. The following theorem has been proved by Herrlich [5] and others for the category Haus . Parts (a), (b) , (c) , (d) and (e) stem from the Characterization Theorem for Epiref lective Hulls (1.4.6),

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71 given by Herrlich in [6]. Once again our area of discovery lies in the interplay of 6t-regular and Â©(-compact objects in categories other than Haus . THEOREM 3.4.6. Let (C ,U.) be a concrete category that is complete, well-powered and cowell-powered, for which U. preserves monomorphisms. If 6i is a subcategory of Â£. , then the following statements are equivalent: (a) X is in the epiref lective hull of Oi, (b) X is 6s' -compact. (c) Each 6(-extendable epimorphism is {X}-extendable. (d) Each C(-extendable epimorphism f:X -*Y is an isomorphism. (e) Each 6t-extendable morphism f :X -Â»Â• Y is an extremal concrete embedding. (f) X is C(-regular and for all ^i-regular objects Y, for which there exists an C{ -extendable concrete embedding f :X -*Â• Y, f must be an extremal concrete embedding. (g) X is 6iL -regular and for each ^-regular concrete embeddingextension (w,W) of X such that w is Â©(-extendable, w must be an isomorphism. PROOF: (a) <=Â» (b) *=Â»(c) *^(d) ^^(e) ; The extremal monomorphisms in 5 are precisely the extremal concrete embeddings in C (1.5.3). Therefore the Characterization Theorem for Epiref lective Hulls can be applied (1.4.6). (f ) =Â» (b) : Suppose X is 6( -regular. Then there exists a product (nA.,7T ) of Â©(-objects and a concrete embedding f:X -* IIA..

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72 Let tm) be the epiref lective hull of Â©l (1.4.4) and let (r ,X ) Â€ C be the Â£((Sl)-epiref lection of X. But L(Oi) is C(Otl extremal concrete embedding) (3.3.8) and consequently X^ is 0{ -compact , hence (^-regular. Note that r^ is C(C{) -extendable, thus ^-extendable; and for each iel, IT 'f :X -> A. is a morphism into an flf-object. For each iel, there exists a morphism g.:X -> A. such that the diagram Â„ _ ""C V ^^ V HA,' ^ A. commutes (1.4.2). Let the unique product induced morphism be :X^ -> nA.. Thus Ti.-f = g.Â«r it. Â± f = Â«r since products are mono sources. Thus r. is a concrete concrete embedding. Hence X is Oi-compact (2.3.1). (b) and (d) => (g) ; Since X is Â©/-compact, it must be Cf -regular. Let (w,W) be an ^(-regular concrete embedding-extension of X for which w is Ct -extendable. Then by (d) , w is an isomorphism. (g) =g(f) : By (g) , X is C{-regular. Suppose Y is an 0(-regular object for which there exists an (S\-extendable concrete embedding f :X -> Y. Let f = h-w be the (epi, extremal concrete embedding) factorization of f (1.3.5, 1.5.3). Then w is a concrete embedding, since f is. Let W denote the codomain of w. Then W is 0(-regular (2.3.1). Let t:X -> A be a morphism to some 6(;-object A. Then since f

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73 is 5l-extendable there exists a morphism k:Y -> A so that the diagram W commutes. Then kÂ«h:W -> A is a morphism such that (k.Â«h)'w = t. Hence w is (?(ex tend able, and (w,W) is an (5t-regular concrete embedding extension. Therefore by (g) , w is an isomorphism, and consequently f is an extremal concrete embedding.

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4. LINEARIZATIONS In this chapter, we will introduce the concept of ?^ -linearizations of an endomorphism in a category and will show that for each endomorphism in a category C with countable products, there exists an W -linearization of that endomorphism for several classes ^ of C-morphisms. We shall show further that by weakening the product condition on Â£ , we can still find W1 -linearizations for certain endomorphisms in the category. Also, we will generalize de Groot's result [2] on the existence of 'universal linearizations' for completely regular spaces X of a weight ^ k for some infinite cardinal number k and monoids M of at most k endomorphisms on X. This will become a category-theoretic result which will extend the generalizations of Baayen [1] on this same subject. Â§4.1 Coordinate Immutors and Permutors In this section, we shall develop some of the mathematical machinery for later sections. We shall define coordinate immutors and permutors on powers of objects in a category and shall see that they are endomorphisms of an interesting linear character, in that they act only on the coordinates of a power, serving to "switch or collapse" these coordinates. DEFINITION 4.1.1. Let S be a category, E be a C -object, S be any set for which the (Card S)'th power of E exists in C Â» and k = Card S. k k (1) Then a ^ -morphism f :E -> E is called a coordinate \^ immutor on E provided that there exists a set function ct : S -* S such 74

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75 that TT 'f = IT , > for each 4)eS; i.e., f must be the unique induced morphism which makes the diagram a(4)) ^o.U) commute for each S, in the above definition, is bijective. PROPOSITION 4.1.2. Let (P be a set of coordinate immutors (respectively, coordinate perrautors) on the object part E of some power in C of some C-object E. If the (k'Cardp) ' th power of E also exists in C , then the product of the coordinate immutors (respectively, coordinate permutors) 11 p is a coordinate immutor pc5> (respectively, coordinate permutor), up to a natural isomorphism, on ^k.Cardi5> PROOF: Without loss of generality we may assume for each pe P , there exists a function ct :k -*Â• k such that it -p = tt .,. for each p A '^ a (X) Xek. By hypothesis (and 1.1.3), the product ( H (E ) ,7t ) is in C . Let 6: II (E ) -> n E. be the natural isomorphism, and let pc9 P Xek ^'P pe5>

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76 II p be the unique induced morphism that makes the diagram pe(? n p pe

^ Define a:k 'x e> ^k x (5> by a(X,p) = (a (X),p) for each Aek, peffÂ». Then TT, Â•6-( n p)-6"''" = (^,) Â•" Â•( n p).6~^ = (Tr-)^-p-7 -5" ^'P PE(? ^ P P p. A p p = (it ,,v) Â•7 '6" = 7T ,,^ .. for each (A,p)ekx(p; consequently c^CA; p a((A,p;; -1 k'Card^ 6* n p'6 is a coordinate immutor on E (respectively, a PE(? coordinate permutor on E ' since a is bijective, if each a is, for pei? ) . PROPOSITION 4.1.3. Let ^ be a category. Let tj; be a coordinate immutor (respectively, permutor) on the object part E of some power of a C-object E, and let E be the object part of the product ( n A., IT.) of some family (A.). _ of C-objects. Then there exists an iÂ£l '"'

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isomorphism 6:E 77 k -1 n (A ) such that &'^l)'6 is equal to the product iel IT (j) . of a family (cj).). , where each (J . is a coordinate immutor iel '" " """^ ' (respectively, permutor) on A. for each iÂ£l. PROOF: Since ;> is a coordinate immutor (respectively, permutor) on E where (E Â»(P)^))^,i.) is a power in C , then without loss of generality, we may assume that there exists a set function (respectively, a biiective set function) a:k -> k for which P 'li^ = P ,... A Oi(A) By the Iteration of Products Theorem (1.1.3), there exists a k k natural isomorphism 6:E ^ IT (A.). Consider the diagram: iel ^ ).ii a(A),i .:A. > A, be the morphism that makes part I of the diagram commute for each Xek, and let R . be the product of the family ((?.). -. (1.1.4). We need only show that i'^>6~ = n \$.. For each iel

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78 Xek and iel, P, . Â• d""^(HeJ) . ) Â• 6 = it, '(ni^J.d = tt ,,, '6 A,i ^i' X,Â± ^i' a(X),i = P , . = P .'ij;. Thus since products are mono sources (2.1.3), Â§4.2 Â•??? -Linearizations In this section, we will show that endomorphisms in many categories C can be viewed as restrictions, of some form, of coordinate immutors on powers of objects in the category, and in similar fashion, that automorphisms often can be viewed as restrictions of coordinate permutors on powers of objects. Several theorems will be proved which establish the existence of various /4-linearizations, for certain classes /// of morphisms in a category C that, depending upon product properties of C , may linearize a given individual endomorphism in C , or simultaneously linearize a monoid of endomorphisms on a given C -object, or even universally linearize all monoids (of a certain cardinality) of endomorphisms on an object in a given subcategory of C . For the most part, the results of this section will generalize results of de Groot [2] and Baayen [1], but with a considerable change in emphasis. DEFINITION 4.2.1. Let C be a category, ;;/ be a class of C -morphisms that is isomorphism-closed in C , <% and be subcategories of C , X and L be Â£ -objects and 4>:X -^ X and ij;:L -* L be endomorphisms in Â£. , and m:X -> L be a C -morphism . (1) The triple (m,L,iÂ» is called an y/^lift of (X,) provided that m:X^ L is an J/i-morphism such that the diagram

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79 ->L ** L commutes. (2) An ^-lift (m,L,i>) of (X,v)) is called an )^/ linearization of (.X,<^) Â±n 0C\ provided that L is the object part of some power of a product of Â£{ -objects and i|i is a coordinate immutor on L. If ii is also a coordinate permutor on L, the triple (m,L,ij/) is called a stable PI linearization of (X.O) in (P^ . (3) Let S be a monoid of endomorphisms on X. The triple (m,L,i;) is called an ^lift of (X,S) provided that (m,L,ij;) is an TA-lÂ±ft for (X,s) for every seS. (4) Let S be a monoid of endomorphisms on X. The triple (m,L,v) is called a ( stable ) ^/ -linearization of (X,S) in
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80 lef t-cancellative in Â£ , and X be any 2. -object for which there exists an JP^-morphism 5:X -> E . Let S be any monoid of endomorphisms on X (respectively, any group of automorphisms on X) for which the (k-Card S)'th power of E exists in t . Then for each seS, there exists a morphism m:X ->E ^^ and an k. Cards k'CardS , , , k-CardS , . endomorphism y :E "* E such that (m,E ,Y ) is an KJ -linearization (respectively, a stable ///-linearization) of (X,s) in S>E. Furthermore, if ^ is contained in the class of Â£ -monomorphisms , then if s and s' are distinct elements of S, y ?^ Y i Â• s 's' PROOF: By hypothesis the power (HE ,7t ) exists in C , the Xek ^'^ ^'^ power ( H E ,7T ) exists in C , and there exists an 7W -morphism Aek C:X -> HE . For each scS and each Xek, there exists a morphism t: '^'srX -* E . Let m be the unique induced morphism which makes A '^ Â» V the diagram n E, ',Â£k seS Aik ' X.s -^ h,. ->-^

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81 scS. We wish to show that m is an W-morphism. For each seS, iT^'p^ = 7T for Aek. However for each Aek and seS, it 'A ^s X.s'^ = \'^'^ r "in = IT '^'s. Since products are mono sources, p 'in = E,'S for each seS. By hypothesis, S is a monoid; hence lyCS. Thus p -m =Â£,Â•!= E,. Moreover t, is an ?;i -morphism and, by hypothesis, ^ is a leftcancellative class of C -morphisms; hence m is an /?? -morphism. For each yeS, define a :S -* S by a (s) = s-y for each seS. Let t be an element of S. Let y be the unique induced morphism that makes the diagram X.st Xek seS 'X ,s ^A.s-t: commute for each seS and Xek. Clearly y is a coordinate immutor, [To show that y is a coordinate permutor when S is a group of automorphisms on X, we need only show that a : S -^ S , as defined above, is bijective. Clearly a is surjective: t~ eS; thus for any gcS,

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82 Cj.(gt ) = gt t = g. Suppose a^(x) = a^(y) then xt = yt; hence x = y, since S is a group.] We now will show that (m, II E. , y ) is an ^//-lift for seS \,s' 't' (X,t); i.e., that the diagram Xtk X,s .^II E Xek scS A,s commutes, For each Xek and seS, (it, 'y )'m = it, X,s 't X,st = (IT, 3-in)-t. And since products are mono sources, y *m = m-t. Now suppose that every //I -morphism is a monomorphism. Let s, s'eS for which Y^ = Y iThen y-m = y ,.m; hence m'S = m-s'; thus s = s' . This theorem is a generalization of one of Baayen's results [1] which he stated only in terms of topological embeddings of completely regular T -spaces into powers of the real line R, . The generalization to topological embedding-linearizations in Top is implicit in his work.

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83 Corollary 4.2.3 (Baayen [1]). Let F be a monoid of continuous self-maps on a completely regular T -space X. Then for each feF, there exists a topological embedding-linearization in some pair ( f^ ,6^) where k = (Card F) -weight XÂ•^' ; : each 6^ is a continuous linear operator in R . Furthermore, it f, f ' are distinct elements in F, then &. 7^ 6 ^, . PROOF: A well-known result of Tychonoff 's states that every completely regular T -space X can be topologicaly embedded into the product space [0,1] , where a = (weight X) -^^ . Of course [0,1] is a subspace of R. , so there exists a topological embedding 5:X ^ R. . The class of all topological embeddings (i.e., concrete embeddings in Top ) is lef t-cancellative and contained in the class of monomorphisms in Top . By Theorem 4.2.2, for each feF, there exists a topological embeddinglinearization ( R. ,<5^), such that 6^ is a coordinate immutor on k R , where 'k = a-CardF = weight X-!;! -CardF, and such that for some *'o ^^ topological embedding m:X -Â»R , the diagram commutes. There are, of course, many additional corollaries to Theorem 4.2.2. However, we will delay listing them until we have

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84 proved a stronger version of the theorem. The next theorem allows us to simultaneously linearize a monoid of endomorphisms on an object in a category, provided that certain products exist in the category. THEOREM 4.2.4. Let S , 7/1, E, X, k and 5 be as in the hypothesis of Theorem 4.2.2. Let S be any monoid of endomorphisms (respectively, group of automorphisms) on X for which all subpowers of the (k'Card S-Card S)'th power of E exist in C . Then there exists an ^k. CardsCards _, ^ ^. ^kÂ« CardS* CardS /J?-morphism m:X -> E and an endomorphism (*> on E V ^-u ^ / Â„k' Cards. Cards ^. . n, -, Â• Â• ^ / . -, such that (m,E ,(^) is an >// -linearization (respectively, a stable W -linearization) of (X,S) in ^E. k* Cards PROOF: By Theorem 4.2.2 there exists an ^-morphism m:X -> E k* Cards such that (m,E ,Y ) is an ?/? -linearization (respectively, a stable y^J -linearization) of (X,s) in (?E. Thus the diagram m , X ^>E^' commutes. By hypothesis and the Iteration of Products Theorem (1.1.3), the product (11 (E ) ,tt ) is in C and there exists a natural , . , Â„ ^k' Cards ^k'CardS -CardS ^ Â„ , isomorphism 6: Jl E -> E . Let 11 y denote the seS seS ^ product of coordinate immutors (respectively, permutors) (y ) _c* By Proposition 4.1.2^ there exists <{> = 6*I1y '& which is a coordinate

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85 immutor on E (respectively a coordinate permutor on k 'CardsCards V n (E ' ^^ ) be the unique induced morphism for s.S ^ Let :X which IT Â• = m for each seS. We shall show that the diagr : (Ei^-Card S^ seS ' -> E k.(Card S)' ,k.(Card S)' = Â•TTY Â• for each seS; and, since products are mono sources, s 's 'S = TTY Â•. Thus Â• s = (6 Â•4)'6)', so that 6 Â• Â• s = 4) Â• 6 Â• , Also since m is an 7J?-morphism and 771 is a class of C-morphisms that is lef t-cancellative, must be an 'W -morphism since tt Â• = m. 2 ^ Hence ( ,e''^^^^^'^^^ ,4)) is an ^-linearization of (X,S) in ff>E (respectively, a stable /^-linearization of (X,S) on fiE) . COROLLARY 4.2.5. Let Â£ and //? be as in the hypothesis of Theorem 4.2.4, Â£ be a subcategory of C , and X be a Â£ -object for which there exists an W-niorphism c, from X into the product ( H E.,7t.) iel of a set-indexed family of ^-objects. Let S be a monoid of endomorphisms on X (respectively, a group of automorphisms on X) such that all subpowers of the (CardS* CardS) ' th power of ( 11 E . ) exist in C , iel ^

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86 Then there exists an V4 -morphism m and an endomorphism (^ on 2 2 (.n^E.)^^^''^^^ such that (m.C.H K.)^^^^^^^ ,<^) is an ^-linearization (respectively, a stable ii'? -linearization) of (X,S) in tf*Â£ . PROOF: Let E = .n^E.. Then apply Theorem 4.2.4. COROLL.\RY 4.2.6. Let k be any infinite cardinal number, C be a category with k-fold products, 5?7 be a class of morphisms in C that is isomorphism-closed and lef t-cancellative in 6 , ^ be a full subcategory of Â£ , and X be a C -object. Then the following statements are equivalent: (a) X is Â£.|7;i-embeddable in Â€. (b) For any monoid S of endomorphisms on X for which Card S <_ k, there exists an >/?-linearization of (X,S) in (PÂ£. . (c) For any endomorphism t on X, there exists an 74 -linearization of (X,t) in 5>Â£ . (d) For any endomorphism t on X of finite order, there exists an ^'//-linearization of (X,t) in (?Â£ . (e) For any automorphism t on X, there exists a stable ^-linearization of (X,t) in i?^ . (f) For any automorphism t on X of finite order, there exists an ^/'^-linearization of (X,t) in (?^ . (g) For any group S of automorphisms on X for which Card S <_ k, there exists a stable ^/-linearization of (X,S) in
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87 C and there is an W-morphism E,:X -* HE. (3.2.1), Let S be any monoid of endomorphisms (respectively group of automorphisms on X) for which Card S ^ k. Then (Card S)^ <_ k. Apply Corollary 4.2.5. (b) => (c) (respectively (g) =>(e) ) : Let S be the monoid (respectively, ^^oup) generated by {t}. Then Card S <_'^ i. ^' (a) =^(d) and (e) =Â»(f) ; Clear. (d) =Â» (a) (respectively (f) =*Â» (a) ) : The identity 1^ has order 1. Thus there exists a (stable) ?;? -linearization of (X,l ) in fiÂ£. . Thus there exists L, a power of Â£. -objects, and an ^-morphism m:X -^ L (4.2.1). Hence X is Â£ |?;; -embeddable. COROLLARY 4.2.7. Let 2 be a category with products, let ^ be a class of ^ -morphisms that is isomorphism-closed in C and left-cancellative in G . Let Â£, be any subcategory of Q . Then X is Â£. |7W -erabeddable in C if and only if for every monoid S of endomorphisms on X (respectively, every group S of automorphisms on X ), there exists an ^-linearization of (X,S) (respectively, a stable >J/ -linearization of (X,S)) in iPi, . PROOF: Apply the previous corollary. There are three areas of differences between Baayen's generalizations of de Groot's work and our generalizations. The first is primarily one of emphasis: Baayen [1] was interested in the existence of mono-universal objects in a category and universal mono-lifts for morphisms in the category (i.e., "universal morphisms"), while our emphasis is on the linear character of the resulting coordinate immutors. Secondly, we have obtained results on the existence of /,7-linearizations for several classes T/t of morphisms in

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a category, while Baayeiis results were restricted to monomorphisms in general categories and topological embeddings in Top (which he had to consider separately). Thirdly, Baayen's results were restricted to categories with countable products; our results require only the existence of certain products in the category. Hence, for example, we can obtain the following corollary for categories with finite products. COROLLARY 4.2.8. Let C be a category with finite products, Y/1 be any class of C -morphisms that is isomorphism-closed in C and lef t-cancellative in C , Â£, be a subcategory of C , and X be a C-object. Then the following statements are equivalent: (a) X is t\T/1 -embeddable in C . (b) For any monoid S of endomorphisms on X for which Card S is finite, there exists an /^/-linearization of (X,S) in ^Â£, . (c) For any endomorphism t on X of finite order, there exists an ^-linearization of (X,t) in (PÂ£ . (d) For any automorphism g on X of finite order, there exists a stable ^/-linearization of (X,g) in (P^ . (e) For any group S of automorphisms on X for which Card S is finite, there exists a stable ///-linearization of (X,S) in IPÂ£ . PROOF: (a) =Â»(b) (respectively, (a) ^ (e) ) : There exists a product 2 ( n E, TT.) of Â£ -objects and an /^-morphism E,:X -> li E.. (Card S) id ^ id ^ is finite, since Card S is finite. Apply Corollary 4.2.5 to obtain the ;;;? -linearization of (X,S) in ^Â£. (respectively, to obtain the stable a'? -linearization of (X,S) Â±n Pc. ).

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89 (b) =Â»(c) (respectively, (e) =*-(d) ); Let S be the monoid of endomorphisms on X (respectively, group of automorphisms on X) generated by {t}. Since t has finite order. Card S is finite; hence by (b) there exists an ;,7 -linearization (respectively, a stable ^-linearization) (m, !,<{)) of (X,S ) in ^Â£ . Hence (m,L,(|)) is the desired linearization for (X,t). (c) =*>(a) (respectively, (d) =Â»(a) ): The identity morphism on X has order 1. 1' ,eref ore by (c) (respectively, (d)), there exists an ;77 -linearization (m,L,(}i) of (X,l ) in ^Â£, , where L is a power of a product of E. -objects. A much stronger result can be obtained for a category J? with countable products: every endomorphism ()) on a 6 -object X has a section-linearization in ^ X, and hence an W -linearization in
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90 (i.e., for each ^-object X, hom (X,E) ^ is an endomorphism on X, has a weak-linearization in ^?, . If a category fi has an epiref lective subcategory 61 , then Ob (5/) is a weakly terminal class for \$ (1.4.2). Consequently if ( C yU ) is a concrete category that is complete and well-powered, having a full replete epiref lective subcategory 61, then (PSi = 6( (1.4.3) and for every 5 -object X and every endomorphism (f> on X, there exists a weak-linearization of (X,) in (5i ; and furthermore, if X is an Â£(-regular space, there exists a concrete embeddinglinearization of (X,(f) in (3i (3.2.1, 4.2.6). Let us next consider a product B^C of distinct sets B and C and a Setmonomorphism m:X -> BxC and an automorphism f :X -> X. We know that there exists a coordinate permutor V^:(BxC)'^ ^ (BxC) '^, where a is the order of the automorphism f, such that (m, (BxC)'^,!^^) is a stable mono-linearization of (X,f) in Set (4.2.6). What we will will find that ^ = ij; xi|y , the product of coordinate permutors \li on B and ^^ on C . THEOREM 4.2.10. Let g be a category, let ^ be a class of C-niorphisms that is isomorphism-closed in C and lef t-cancellative in g, let (E^)^^-j. be a set-indexed family of S -objects whose product (nE_j^,TT^) exists in ^ , and let X be any ? -object for which there exists an ^ -morphism Â£;:X ^ HE.. If S is any monoid of endomorphisms on X (respectively, any group of automorphisms on X) such that all subpowers of the (Card S-Card S)'th power of TIE. exist in & , then there exists a triple (m,L,i(^) which is an ;';/-lift for (X,S) such that

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91 2 L = n E; ^^ and >> is the product of a family (()).). .. of morphisms, iel where for each iel, (J) . is a coordinate immutor (respectively, permutor) ^ Â„(CardS)^ on E. 1 PROOF: By Theorem 4. 2. A, there exists an /,7-linearization (respectively, 2 stable /..-linearization) (m, (KE . ) ^'^^^^^'^ ,^) for (nE^,S). The remainder of the proof follows directly from Proposition A. 1.3. Â§4.3 Universal A^ -Linearizations In the previous section, we found that linearizations always exist for endomorphisms in categories with countable products, and that for sor.e monoids of endomorphisms on an object in the category, as well as for some groups of automorphisms on an object in the category, we could obtain simultaneous linearizations. In this section, we will restrict our attention to categories with infinite products in order to obtain some linearizations that are universal for all endomorphisms (respectively, all automorphisms) on any object in a given subcategory. DEFINITION 4.3.1. Let (i be a category, ^ be a class of Q -morphisms that is isomorphism-closed in ^ , 6^ be a subcategory of ^ and L be a \$ -object where v:L -* L is an endomorphism> in Q . (1) A Â£ -object X is called an ^/7 universal object for &i provided that for each Â©^-object A there exists an /i?/-morphism m:A -^ X. (2) A pair (1,1!^) is called a universal ///lift for End (OV.) [respectively, for Aut (C:) ] provided that for each endomorphism [irespectively, each automorphism] p on any Ot-object A, there exists an

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92 W-morphism in:A L for which (in.L,^;) is an ///-lift for (A,*). (2) A universal ;;,'-lift (L,\li) for End(6i) [respectively, for Aut((5()] is called a universal y linearization for End ( Ot) [respectively, fo_r Au_t ( CO ] provided that L is the object part of a power of a product of 6 -objects and i^ is a coordinate immutor. If ip is a coordinate permutor on L, then (.L,\p) is called a stable universal 77? linearization for End ((St) [respectively, for Aut (6<) ] . (3) Let k be a cardinal number. A monoid S is called a k-/// -universal monoid for 0( provided that card S <_ k and there exists a Â£ -object L and a monoid homomorphism a:S -> hom^(L,L) with the following property: for each monoid T with card T <_ k, there exists a surjective monoid homomorphism yjS -> T such that for each Oi -object A and for each monoid homomorphism 3:T -> hom (A,A) there is some >9;-morphism m:A -> L, such that for each seS, the triple (m,L,a(s)) is an ^-li'ft of (A,3-y(s)). Y(s) a(s) (4) Let k be a cardinal number. A group S is called a k-zz-^universal group for C. provided that card S <_ k and there exist a C-object L and a group homomorphism a:S -* Aut (L,L) (where

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93 Aut , (L,L) is the group of automorphisms on L) with the following property: for each group T with card T <_ k, there exists a surjective group homomorphism y = S -Â»T such that for each C{ -object A and for each group homomorphism 3:T -*Â• Aut^(A,A) there is some J^-morphism m:A -> L, such that for each seS the triple (m,L,a(s)) is an ^/-lift of (A,6-y(s)). THEOREM 4,3.2. Let k be an infinite cardinal number, Â£ be a category with k-fold products, and //] be any class of G -morphisms that is isomorphism-closed in v# and lef t-cancellative in C . Then if 01 is any subcategory of C, the following statements are equivalent; (a) There exists an 7?/-universal object for fiC in C . (b) There exists a k-^-universal monoid for fit in C . (c) There exists a k^//-universal group for 5i in C. (d) There exists an universal ^-linearization for End((S() in C . (e) There exists a stable universal J^-linearization for Autcio in e. . (f) There exists an universal ^/7-lift for End((5{) in Â£. Moreover if Ol is closed under the formation of k-fold products in C , then Â£ in statements (a) through (f) may be replaced by 61, PROOF: (a) =Â» (b) : Let S be a free monoid with k generators. Card S = k* ^ = k, since k is infinite. Let K be the ^-universal object tor 01 . By hypothesis, the product (K , (t:,), ,) exists in C Â• For k k each teS, define f :K ->Â• K to be the unique induced morphism that makes the diagram

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94 k k commute for each seS. Define a:S -> hom^ (K ,K ) by a(t) = f for each teS. Note that for given p,qeS and for every scS, it Â• f Â• f s p q = n -f = TT = TT 'f , and since products are mono sources (2.1.3), sp q spq s pq' a(p)a(q) = f 'f = f = a(pq) . Also, clearly a(l) = f-,, where IT 'f= 7T T = IT for every seS; hence a(l) = 1 i.Thus a is a monoid K horaomorphism. Let.T be any monoid such that Card T <_ k. By the universal mappin2 property for free monoids, there exists a surjective monoid homorphism y:S -> T. For some 5t-object A, let 3:T -> hom (A,A) be a monoid homonorphism. Since K is an /)J -universal object for Cl > there exists an ///-morphism m:A -> K. Thus for each seS, there exists a morphism m'S(Y(s)):A -> K. Let q:A -> K be the unique induced morphism such that the diagram B(y(s))

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95 comr/.utes for each scS. Thus tt 'q = in-B(Y(s)) for each seS. T and S are monoids; hence there exist identities e eS and e eT and yCeg) = e^. Thuj 3(Y(eg)) = 6(e^J = 1^, and tt^ -q = m-l^ = m. 7/^ is lef t-cancellative in C ; thus q is an /J-morphism, since m is. Now we need only to show that the diagram q e-Y(t) a(t) commutes for each teS. For every seS, tt Â•a(t)'q = rr 'f -q = t; 'q = m.3(Y(st)) = m-B(Y(s))'3(Y(t)) = Tr^-q3 (Y(t) ) . But products are mono sources (2.1.3), so a(t).q = q-3(Y(t)) for each tcS. Hence (q,K ,a(t)) is an ///-lift of (A,3*Y(t)) for each teS. (b) =^ (a) : Let S be a k-?// -universal monoid for C^ and let a:S -y hom^(L,L) be the monoid homorphism with the property defined in A. 3.1. Let T be a monoid with one element {x}. For each Ct-object A, let 3^:T -^ hom (A,A) be defined by 3^(x) = 1 . Then by A. 3.1, there must exist m:A -^ L. Thus L is an universal object for \$( . (a) =Â»(c) ; This proof is analogous to the proof for (a) =* (b) . Let S be the free group with k generators. Define a:S -Â»Aut . (K ,K^) as in the proof for (a) =* (b) . We need only show that a(t) = f has an inverse for each teT. cx(t~>a(t) = f _Â• f = f -1 -1 ^ ^ ^ ^^ a(ft ) = a(t)a(t ), since S is a group. = a(t ^.t) K

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96 (c) =Â»(a) ; The proof is analogous to the proof for (h) ==*'(a). (a) =Â»Â• (d) : Let S be the free monoid with k generators, let K be the ^?? -universal object for Oi, and construct a:S ^ horn ^ (K ,K ) as in the proof for (a/ =-(b). Clearly a(s) is a coordinate immutor for each seS. Let T be the free monoid with one generator 'x.' As before, there exists a surjective monoid homomorphism y^S ->T. Thus for some s eS, y(s ) = x. We shall now show that (K ,a(s )) is XX X a universal ^//-linearization for End(0^).Let A be any 6j-object and (^ be any endomorphism on A. Since K is an ^-universal object for dl, there is some ?J9-morphism m :A -> L. Define 3.:T -Â» hom (A, A) by 3. (x) = (f. . Then let q :A -> K be the unique induced morphism that makes the diagram Â— >K 8-y(s) ->* K commute for each seS. Then, as before, for every seS the diagram ^ k' S,(y(s)) a(s) -> K^

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97 1q commutes, but S.(y(s )) = B.(x) = <>. Hence (q.,K ,a(s )) is an A X A AX J.; -lift for (A, (J)). (d) =Â»(f) : Clear. (a) "^ (e) : The proof is analogous to the proof for (a) =^ (d) . We need only check that a('c) as defined is a coordinate permutor for each teS. Let p:S -*S be defined by p(s) = st. Now, since S is a group, p is bijective. Thus a(t) is a coordinate permutor (4.1.1) for each teS. Hence (K ,a(s ) ) is a stable universal ^^-linearization for all automorphisms in 64 . (e) =Â»(f) : Clear. (f) -Â»(a) : Clear. Finally we have de Groot's result which initiated the investigations of linearizations and universal objects in categories. COROLLARY 4.3.3 (de Groot [2]). Let k be an infinite cardinal k number, let P be the topological product [0,1] and let F be the free monoid with k generators. Then every completely regular T -space X of weight <_ k and every monoid S of endomorphisms on X admits a universal linearization in the pair (F,P) (i.e., in our terminology, there exists a k-topological embedding -monoid for the subcategory of Top whose objects are the completely regular T -spaces of weight <_ k) . PROOF: By Tychonoff's result, P = [0,1] is a universal object for all completely regular T -spaces of weight <_ k. The class of all topological em.beddings in Top is precisely the class of all concrete embeddings in Top , thus it is let t-cancellative (2.3.5). Consequently Theorem 4.3.2 may be applied.

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BIBLIOGSAPHY [ 1] Baayen, P. C. , Universal Morphisns , Mathematical Centre Tract 9, Amsterdam, 1S64. [ 2] de Groot, J,, Linearization of Mappings, Proc. 1961 Prague Symposium . Prague, 1962. [ 3] Engelking, R. ' and Mrowka, S., On E-compact Spaces, Bull. Acad . Polon. Sci. Ser. Sci. Math. Astr. Phys.^ 6 (1958) , 429-436. [ 4] Herrlich, H. , E-kompakte Raiime, Math. Zeitschr.. 96 (1967), 228-255. [ 5] , Topologische Ref lexionen und Coref lexionen , Lecture Notes in Mathematics, 78, Springer-Verlag, Berlin, 1968. [ 6] , Categorical Topology, Gen. Topology and Its Appl., 1 (1971), 1-15. [ 7] and G. E. Strecker, Category Theory , Allyn and Bacon, to appear. [ 8] and , Coref lective Subcategories, Trans. Amer. Math. Soc , 155 (1971). [ 9] and , Coreflective Subcategories in General Topology, to appear in Fund. Math. [10] and J. van der Slot, Properties l^Jhich Are Closely Related to Compactness, Indag. Math., 29 (1967), 524-529. [11] Kelley, John L., General Topology , Van Nostrand, Princeton, N.J., 1955. [12] Lang, S., Algebra , Addison-Wesley , Reading, Mass., 1965. [13] Mrowka, S., On Universal Spaces, Bull. Acad. Polon. Sci. , CI. Ill, 4 (1956), 479-481. [14] ^ , A Property of the Hewitt Extension vV of Topological Spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. , 6 (1958), 95-96. [15] , On E-Compact Spaces II, Bull. Acad. Polon. Sci., 14 (1966), 596-605. [16] , Further Results on E-compact Spaces I, Acta Math . . 120 (1968), 161-185. 98

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BIOGRAPHICAL SKETCH Jean Marie McDill was born in Trona, California and spent her childhood in various parts of the Southwest. She is married to William R. McDill, an economist, and has one daughter, Kathleen Marie. Mrs. McDill has studied and taught mathematics at the college level, has worked as a computer programmer and has worked in semiconductor research. 99

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Therai 0. Moore Associate Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Docw^..of Philosophy. /n Max H. Kele Assistant Professor of History This dissertation was submitted to the Dean of the College of Arts and Sciences and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August, 1971 Dean, College of Art^s/and Sciences Dean, Graduate School

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertaition for the degree of Doctor of Philosophy. George E/^ Strecker", Chairman Assistant Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. David A. Drake Assistant Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Zoran R. Pop-Stoja^ovic / Associate Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. // // /Bryan V. Hearsey Assistant Professor of Mathematics