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 Permanent Link:
 https://ufdc.ufl.edu/UF00097681/00001
Material Information
 Title:
 Categorical embeddings and linearizations
 Creator:
 McDill, Jean Marie
 Publication Date:
 1971
 Copyright 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 )
nonfiction ( marcgt )
Notes
 Thesis:
 Thesis  University of Florida.
 Bibliography:
 Bibliography: leaf 98.
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 Also available on World Wide Web
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 Manuscript copy.
 General Note:
 Vita.
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 University of Florida
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 University of Florida
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 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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13830624 ( OCLC ) ADA1551 ( NOTIS )

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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. PopStojanovic, 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.
TABLE OF CONTENTS
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 elTEmbeddable Objects ........................
3.3 7Coseparating Classes..........................
3.4 )(Regular and 0(Compact Objects................
4. LINEARIZATIONS ........................................
4.1 Coordinate Immutors and Permutors ................
4.2 MtLinearizations...............................
TABLE OF CONTENTS (Continued)
Page
4.3 Universal MLinf.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 'embeddinglike' 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 tregular and ecompact 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 T1spaces, are extended. For every isomorphism
closed, leftcancellative 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 coordinateimmuting
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 coordinatepermuting automorphism. Furthermore,
simultaneous Mlinearizations are shown to exist for certain monoids
of endomorphisms on objects in the category, and universal
tlinearizations are shown to exist for every endomorphism in
certain subcategories of the category.
viii
INTRODUCTION
The "mathematicalmaninthestreet," 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 leftadjoint 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 Cobject X together with a family of
Cmorphisms having X as their common domain. A categorical product
( n Ei,i) is an example of a source which, in fact, exhibits a
idI
"monolike" 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,
tsources) whose morphisms in the aggregate exhibit the properties of
special types of "embeddinglike" morphisms (4nmorphisms).
In topology, a longestablished method for producing mappings
into topological products has been by use of families of morphisms
into the coordinate spaces. Tychonoff's wellknown 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 "embeddingtype" morphisms.
In 1958, Engelking and Mrowka [3] began to develop the concepts
of Eregular and Ecompact 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 Ecompletely
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 Ecompact space?) Later Herrlich [4] expanded
these investigations to include the concepts gregular (respectively,
ecompact) spaces for collections e of spaces (respectively,
Hausdorff spaces), initiating the use of a categorytheoretic 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 Cobject X to be
iregular 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 Cobject
X there exists a pair (r X,), called the 6(epireflection of X,
where X is an aobject and r :X * X is an epimorphism with a
maximal extension property for a (In this paper, we will say that
r is Oextendable" provided that for every morphism g:X A for
some Ofobject 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, wellpowered 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 StoneCech construction, B:X BX, which gives a
CompT2epireflection 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 Otregular 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 trivialseeming 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 setindexed 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 Cobject
ieI
(2) for each jel, T.: Ai A is a Cmorphism (called the
it1
projection from I A to A ).
il
(3) for each pair (C,(fi)i), where C is a Cobject and
for each jel, f.:C A. is a morphism, there exists a unique
induced Cmorphism (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 setindexed family of sets. Suppose that for each
iEl, (Pi ,(nk)rK ) is the product of a setindexed family (k)kKi of
Sobjects 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  Bi
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 Cobject, and for each iOb(I),
fi:L D is a Cmorphism, and for all i,jEOb(I) and Imorphisms
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 wellknown 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 wellpowered
provided that every Cobject has a representative class of subobjects
which is a set.
Dual notion: cowellpowered
The categories Set, Grp, Top, Mon, Haus and CompT2 are well
powered and cowellpowered.
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.
(b) JU has a leftadjoint.
(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 Cepimorphism
e there exists a morphism h such that the diagram
e
k g
m
commutes.
(2) A Cmorphism 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 Cmorphism and e is a Cepimorphism,
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 fg 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, wellpowered 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 (textendable provided that
for each Otobject 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
itepireflection for X provided that X is an fobject and
r a:X XO is an (extendable epimorphism.
(3) ., is called an epireflective subcategory of ( provided
that for every Cobject 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, wellpowered and
cowellpowered 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, wellpowered and
cowellpowered 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, wellpowered 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 (Sobjects
(c) Each 6(extendable epimorphism is (X}extendable
(d) Each (Oextendable 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 wellpowered 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 wellpowered 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 hba 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
many additional categories.
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 OSmorphism.
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 Stmorphism such that for some function
h:'U(Z) U(X), U(f).h = 1(g). But g is also a Cmorphism and f
is a concrete embedding in C ; hence, there exists a Cmorphism
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 wellpowered 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 amorphism. 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 wellpowered 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 6Kobject.
Hence by hypothesis, L is anO5object, so m is an OCmorphism. 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 "monolike" propertiesproperties 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 Cobject and (fi)iEl is a family of Cmorphisms, 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 leftcancellable.
(3) A source (A,fi) is called an extremal source provided that
for each source (B,gi) and each eepimorphism 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 Cmorphisms e and k, where e is a
C epimorphism and gi*e = fi'k for each ilI, there exists a Cmorphism
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 he = 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 Imorphism m:i j, D(m).ti = j. Let X be a
Cobject and let f and g:X L be Cmorphisms such that *i*f = ti'g
for each iEOb(I). Then D(m)(ti.f) (D(m). i).f = zj.f for each
Imorphism 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 emorphisms
where e is an epimorphism such that for each icOb(I) gi.e = i'k.
Then for each Imorphism 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 Cmorphisms 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 rnsources.
92.3 rnSources
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 isomorphismclosed in e provided that
for any % morphism m and any Cisomorphisms e and e', such that the
compositions me and e'.m are defined in e me and e'*m must be
i7morphisms, and 2f must contain all of the isomorphisms in C.
(3) f will be called leftcancellative in C provided that
whenever p.qs ? for C morphisms p and q, q must be an Mmorphism.
(4) A pair (X,f) will be called an lsubobject of Y provided
that X and Y are eobjects and f:X Y is an rnmorphism.
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 mmorphisms. These sources,
as listed below in Table 2.3.2, will be called "msources." 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
isomorphismclosed 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 &
r9morphism msubobject
morphism weak subobject
monomorphism subobject
extremal
morphism
extremal
monomorphism
strong mor
phism
strong
monomorphism
extremal weak
subobject
extremal subobject
strong subobject
Msource
source
mono source
extremal
source
extremal
mono source
strong
source
strong mono
source
TABLE 2.3.2 (Continued)
~W1 fmorphism flsubobject Itsource
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 ltSources). 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 fl2morphism
(c) (A,f) is an Vsubobject 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 Wsources in 92.1 will automatically yield results on
7Imorphisms; 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 fSources). 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 fsource, so is (A,fk)K.
PROOF: Clearly (A,fk) is a source.
(a) mono: Let p and q be Cmorphisms 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 Cmorphisms
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 gke 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
aJ
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 (LeftCancellation of rSources). 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 Msource, so is (A,g.).
PROOF: Clearly (A,gi) is a source in C.
(a) mono: Let p and q be emorphism 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 Cmorphisms 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 tmorphisms listed
in Table 2.3.2. Then M is leftcancellative in C .
PROOF: Suppose p and q are morphisms in C such that pq is an
rmorphism. Then (A,p*q) is an rnsource (2.3.3) and consequently
(A,q) is an 12source (2.3.5). Thus q is an rnmorphism (2.3.3).
PROPOSITION 2.3.7. Let C be a category [( ,U) be a
concrete category] and let #7 be any class of Cmorphisms 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 sk g.e. Now there exists a
morphism s such that s.s = 1. Thus s.s.g.e = sa.s.sk sk = ge, 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 (sg). 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 ilsources and the existence of mfmorphisms 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 Ocompact
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 Cmorphisms listed in Table 2.3.2,
and let (X,fi) be a setindexed 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 fisource.
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 ge = 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 = he 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, wellpowered
concrete category for which U preserves monomorphisms and products.
Let (A,fi) be a setindexed 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
Cobj ect.
(1) The pair (y,Y) is called an V extension of X in
provided that Y is an object, (X,y) is an 7subobject 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
Snonextendable source with respect to e provided that for any
7 extension (y,Y) of X in C with the propertyfor each isl,
there exists a morphism f :Y E for which the diagram
S/ fi
Ei
commutesy must be an isomorphism (i.e., there must exist no proper
Iextension of X in C with this property).
In particular, if a source (X,f ) contains all the morphisms
from X to Sobjects, then (X,fi) is an tnonextendable source with
respect to C if and only if there exists no proper Mextension
(w,W) for X in C for which w is eextendable (1.4.2).
Note that every class of 1morphisms listed in Table 2.3.2 is
isomorphismclosed 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 'weakextension' to denote
anR extension when is the class of all morphisms in C .
The definition for a weaknonextendable 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 weaknonextendable source with respect to & if and only if (X,fi) is
an extremal source.
PROOF: Suppose (X,fi) is a weaknonextendable 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 weakextension; 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 weakextension (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 weaknonextendable 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 wellknown results. Note that the
topological product with the Tychonoff product topology is a product in
Top. Hence each fi is a Topmorphism, i.e., continuous, if and only
if h is a Tomorphism. 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 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 weaknonextendable 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)tEmbeddable Objects
Mrowka [15] defined an Ecompletely 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 Ecompact
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 isomorphismclosed in C .
(1) A Cobject X is called e 7lnembeddable in provided
that there exists a setindexed family (E )iI of Cobjects whose
product (IEi,"i) exists in C and for which there exists an ?lmorphism
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 tregular in C will
be used interchangeably with the term t 7embeddable 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 Eregular space in
Top coincides with Mrowka's definition for an Ecompletely regular
space. Also our definition for a Hausdorff space to be Ecompact in
Haus coincides with Mrowka's definition of Ecompactness, 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 Aregular 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 wellknown result, although it may not be
immediately recognizable in our new terminology.
PROPOSITION 3.2.2. An Abelian monoid is Abregular in
AbMon if and only if it is cancellative.
PROOF: Suppose M is Abregular 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 welldefined 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 Abregular.
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 Cobject.
Then X is 7flembeddable 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
n1source in .
PROOF: Let X be _I1embeddable in C Then there exists a set
indexed family (E )i, of objects such that there exists an
rnmorphism f:X HE1 (3.2.1). By the Embedding Theorem (3.1.1),
(X,ir'f) is.an lMsource. Since an n source can be enlarged (2.3.4),
(X, home (X,EI)) is an /1source.
iEl
Conversely, let be a set contained in Ob(g ) such that
(X, U hom (X,E)) is an Wsource. 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 rlembeddable 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 Cobject.
Then X is Cregular (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
Eregular 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 Eregular
(3.2.1)) if and only if (X, 3) is a concrete embedding source.
There are many wellknown 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 wellknown results are listed below as examples and
are stated without proof.
EXAMPLE 3.2.6 (Tychonoff). A topological space is a completely
regular T1space 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 T1space 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 Tospace with two
points {a,b} in which the only proper closed set is {a). A topological
space is a Tospace if and only if it is Fregular in Top.
EXAMPLE 3.2.9. Let W be the twopoint indiscrete space. A
topological space is indiscrete if and only if it is Wregular in Top.
Recall that a topological space is said to be zero dimensional
provided that it has a base of closedandopen sets.
EXAMPLE 3.2.10 (Alexandroff). Let D be the discrete space with
two points. A topological space is a zerodimensional Tospace if and
only if it is Dregular in Top. A zerodimensional Hausdorff space is
compact if and only if it is Dcompact 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 Vregular in Top.
Let us now consider another result of Mrowka's. Let L
m
denote a space with melements and the finite complement topology.
Let Top1 denote the category of T1spaces 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 Tlspace X such
that every T1space is Xregular in Top. However t = {L :m is a
cardinal} is a class of T1spaces such that every T1space of cardinality
m can be topologically embedded into the topological power (Lm)m.
Note that there exists no T space X, such that every T1space
is Xregular in Topl, or Xregular in Top, although by Example 3.2.11,
every topological space is Vregular in Top. Clearly,V is not a
T1space. Also we have seen that every T1space is Zregular in Top~ .
This example illustrates the necessity for our generalized definition.
Trivially, of course, in a category C every Cobject is
C i7embeddable 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 categorytheoretic terms.
Recall that for a category C and Cobject 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 Cobject X,
and hom (,E)([) = _* for each Cmorphism .
DEFINITION 3.2.13. Let C be a category and let rf be an
isomorphismclosed class of morphisms. Let E and X be Cobjects.
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
Cmorphisms from Table 2.3.2 and let C have the (epi,f ) factoriza
tion property. Let E be a C object.
Then for any Cobject X there exists an El??ttransformation
(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
rnmorphism 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 Eregular 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 7Coseparating 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 Elmonoembeddable,must be a coseparator for the category. Recall
that a object C is called a coseparator for C provided that for any
two distinct Cmorphisms p and q:A B, there exists a Cmorphism
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 MIcoseparating
class for a category. Then we shall see that in a category C with
products, a class 9 of Cobjects is an Mcoseparating class for C
if and only if every Cobject is t Membeddable 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 Cobject A, the source (A,hom (A,C)) is a mono source.
PROOF: Let A be any Cobject and let p and q be any distinct
Cmorphisms 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 Cobjects 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
Cobject U with the property that for every Cobject 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
Cmorphisms from Table 2.3.2, and let & be a subcategory of C .
(1) Let E be a subclass of the class of all Cobjects in C.
Sis called an 7coseparating class for 6( provided that for each
OMobject 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 Cobject E is called an Mcoseparator 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
monocoseparator 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 Mcoseparating 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
Msources for the various classes rn of C morphisms from Table 2.3.2;
hence they also imply the relative strengths of ncoseparating classes
for C for different classes R Therefore it is clear that every
strong monocoseparating class is an extremal monocoseparating class,
which in turn is a monocoseparating class (2.1.2). Similarly for
complete wellpowered concrete categories (C ,U) for which U
preserves monomorphisms and products, every extremal monocoseparating
class for C is a concrete embeddingcoseparating class for C hence
an extremal concrete embeddingcoseparating class for C (3.1.5).
The following theorem is the desired result which relates
1m?? embeddable objects and mcoseparating 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 Cmorphisms 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 CCobject A, there exists a set tA contained in
Ob( ) such'that the unique induced morphism A Ehom (A,E) is an
EC eA
rmorphism 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?Tsubobject
of some product of powers of objects in A'
(d) Each 0(object A is CM 7embeddable in C
COROLLARY 3.3.4 (Herrlich and Strecker [7]). Let C be a
category with products and let C be a Cobject. Then the following
statements are equivalent:
(a) C is a monocoseparator (respectively, extremal mono
coseparator) for C .
(b) For each Cobject A, the unique induced morphism
home (A,C)
A C is a monomorphism (respectively, an extremal monomorphism).
(c) For each Cobject A, there is a morphism f such that (A,f)
is a subobject (respectively, an extremal subobject) of some power of
e.
PROOF: A Cobject 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 T1spaces (3.2.12), Top
has no concrete embeddingcoseparator in Topl; however Topl has a
proper concrete embeddingcoseparating 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 embeddingcoseparator for Top1.
DEFINITION 3.3.5. Let 6 be a category, let M be a class of
C morphisms that is isomorphismclosed 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 W7embeddable
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 fi2coseparating 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 fobjects 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 msubobjects of aCobjects.
(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 isomorphismclosed 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 Msubobjects 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 7/2ldobjects, 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 )fnmorphisms (1.1.4). Consequently f is an A morphism since 71 is
closed under products. Hence X is an ITP tobject.
(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 (lsubobjects 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 cowellpowered, 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 ,fembeddable 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, wellpowered and cowellpowered. 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 tcompact 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, wellpowered and cowellpowered, 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 CtCompact 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 OCregular 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 0fcompact 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 01regular
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 OCregular and
(ecompact objects coincide with Herrlich's definitions in Top and Haus
(3.2). Epireflective subcategories have been studied extensively.
Recall that in a complete, wellpowered and cowellpowered 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
ltepireflection (r ,X ) in C., where X is an Otobject and
r :X X is an Otextendable epimorphism (1.4.2).
One of the motivating examples in the study of epireflective
subcategories was the construction of the StoneCech compactification
for completely regular T1spaces. 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 CompT2epireflec
tion of X. Also if X is completely regular, B is a topological
embedding (3.2.6).
Clearly for CompT2regular spaces, which are the completely
regular spaces in Haus, there exist topological embeddings into compact
T2spaces, which are 'universal' in the sense of being CompT2extendable.
Herrlich and Van der Slot [10] proved a very useful result relating
epireflective subcategories 6( in Haus and the existence of
Ceepireflections (r) ,X ) for eregular 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, wellpowered 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 embeddingextension
(w,W) in $~ for which w is aCextendable.
(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 cowellpowered, 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 Aobject 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 setindexed 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 embeddingextension (w,W),for which W is
an Otobject and w:HAi W is an Bfextendable 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 6Lobject. Also CC
is full; so that i is an Ctmorphism 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 eobject A. Then X is Ccompact, hence a regular.
By (b), there exists (w,W) where W is an CCobject and w:X W is an
c oextendable 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 Cobject, and because 8t is full, m is an L%morphism.
Furthermore, if C is cowellpowered, 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.
From the 'universal' property for free Abelian groups, it follows that
9f is Abextendable. Thus (.f, K(M)) is the desired Abepireflection
for M. Let us suppose M is Abregular (i.e., cancellative (3.2.2)).
Then there exists a setindexed 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 Abextendable. Thus there
exists a monoid homomorphism g*:K(M) + Ai such that g *..f = g.
Consequently, 0f 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 COregular objects in 8 .
THEOREM 3.4.4. Let (C,U) be a concrete category that is
complete, wellpowered 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 Cobject. If
(r ,XO ) is the Otepireflection of X and (r ,X, ) is the 3epireflection
of X, then there exists a concrete embedding e:X X which is an
(Oextendable epimorphism such that the diagram
X ,XV
/
X e
X
commutes.
PROOF: By hypothesis, X is a 3object, since it is an Ctobject.
Thus, since ra is Oextendable (1.4.2), there exists a morphism
e:X + X such that re = e*r But by hypothesis, XH is 01regular.
Hence by Theorem 3.4.2, there exists a concrete embeddingextension
(y,Y' ) in CL for which y:Xe Ye is Otextendable. But rq is
6textendable 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 = ar Thus y'rg = a'er since r = er .
Now y = me since r. is a Gepireflection, 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 textendable, suppose that there exists
a morphism f:X * A for some Ctobject A. Since y is (extendable,
there exists a morphism g :Yq A such that g *y = g; however,
y = ae, so that g *ae = g. Consequently, we can conclude that e
is Oextendable (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 CompT2regular
spaces in Haus. For a given Hausdorff space X, there exists a
CompT2epireflection (B ,BCX) and a RCompepireflection (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 Abregular objects in AbMon (3.2.2).
And Ab is a full subcategory of CAbMon. Then for any Abelian monoid
M, there exists an Abepireflection (A, BAM) and an CAbMonepireflection
(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, wellpowered and cowellpowered, 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 Ofcompact object Y and a concrete
embedding morphism f:X Y.
(c) There exists a concrete embeddingextension (w,W) for X
where W is qtcompact.
(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 60compact,
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 = tw (1.3.5, 1.5.3)
f m
X f Y m EAi
W
Let W denote the codomain of w. Then W is Mtcompact, since the
composition, m.t, of extremal monomorphisms is an extremal monomorphism
in a complete, wellpowered 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 andcompact 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 OCregular.
(a) *(d): Suppose X is Ctregular and let f:X Y be any
O'extendable morphism. Then there exists a product (HAi' i) of
Ctobjects and a concrete embedding m:X HAi. Thus for each iel,
there exists a morphism 7im:X Ai; and since f is Mtextendable,
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 &1extendable 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 itregular and (tcompact objects in categories
other than Haus.
THEOREM 3.4.6. Let (C ,.L) be a concrete category that is
complete, wellpowered and cowellpowered, 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 CCcompact.
(c) Each 6(extendable epimorphism is {X}extendable.
(d) Each 0textendable epimorphism f:X Y is an isomorphism.
(e) Each Oextendable 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 Otregular. 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 (5regular.
Note that re is C(O()extendable, thus &extendable; and for each icI,
Ti'f:X Ai is a morphism into an aobject. For each icl, there
exists a morphism gi:X Ai such that the diagram
X rer
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 CCcompact (2.3.1).
(b) and (d) (g): Since X is 8fcompact, it must be O0regular.
Let (w,W) be an (regular concrete embeddingextension of X for which
w is Otextendable. Then by (d), w is an isomorphism.
(g) (f): By (g), X is (regular. Suppose Y is an aregular
object for which there exists an OCextendable 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 (fregular
(2.3.1). Let t:X A be a morphism to some e(object A. Then since f
73
is etextendable 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 Ofextendable, and (w,W) is an 01regular 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 rlinearization of that endomorphism for several classes W of
Cmorphisms. 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 categorytheoretic 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 Cobject, 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 Cobject 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).61 = (QX) p *( p).61= (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.61 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 Cobject E, and let E be the object part of the product
( nA i,) of some family (Ai)iE of Cobjects. Then there exists an
lel i i i
isomorphism 6:Ek H (Ak) such that 6Y61 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 6.4.61= n p.. For each
iEI
Ack and ieI, P ,i'61(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 JLinearizations
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 29linearizations, 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 Cobject, 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 isomorphismclosed 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 jlift of (X,) provided
that m:X. L is an 4morphism such that the diagram
X m > L
X L
m
commutes.
(2) An mIlift (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
itlift for (X,s) for every seS.
(4) Let S be a monoid of endomorphisms on X. The triple
(m,L,') is called a (stable) Jlinearization 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 isomorphismclosed in C and leftcancella
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 isomorphismclosed in C and
leftcancellative in e and X be any e object for which there exists
k
an flmorphism 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 EkCardS such that (m,EkCardSys) is an
;llinearization (respectively, a stable ,linearization) of (X,s) in
PE.
Furthermore, if V is contained in the class of Cmonomorphisms,
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 7morphism
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
XHE 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 Mfmorphism. 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: tlS; 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 Ptlift 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'.st
= (Xs.m)t.
And since products are mono sources, yt'm = mt.
Now suppose that every m morphism is a monomorphism. Let
s, s'ES for which Ys = s,. Then ysm = y,,'m; hence ms = 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 T1spaces into powers of the real line R.. The
generalization to topological embeddinglinearizations in Top is
implicit in his work.
Corollary 4.2.3 (Baayen [1]). Let F be a monoid of continuous
selfmaps on a completely regular Tlspace X. Then for each ftF, there
exists a topological embeddinglinearization 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 wellknown result of Tychonoff's states that every completely
regular Tlspace 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
leftcancellative 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 = aCardF = weight X*oCardF, 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
1l4morphism m:X EkCardSCardS and an endomorphism 1 on Ek'CardSBCardS
such that (m,E kCardS.CardS,) is an Rlinearization (respectively,
a stable i7linearization) of (X,S) in (E.
PROOF: By Theorem 4.2.2 there exists an rnmorphism m:X Ek'CardS
such that (m,E kCardS,y) is an 74linearization (respectively, a stable
74linearization) 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 = 6Iys*6 which is a coordinate
immutor on EkCardSCardS (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 = (61 *6)., so that 6**s = 0.6.
Also since m is an 7morphism and 72 is a class of Cmorphisms that
is leftcancellative, must be an f morphism since s * = m.
Hence (,Ek(CardS)2,0) is an Mlinearization of (X,S) in PE
(respectively, a stable rlinearization 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 Cobject for
which there exists an Mmorphism 4 from X into the product ( IE ,i )
icI
of a setindexed 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 (CardSCardS)'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 14linearization
(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 kfold products, 51 be a class of morphisms in e that
is isomorphismclosed and leftcancellative in C C be a full
subcategory of P and X be a Cobject. 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 Rilinearization 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 Tlinearization of (X,t) in O( .
(e) For any automorphism t on X, there exists a stable
rlinearization 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 T7linearization of (X,S) in j ,
PROOF: (a) (b) (respectively (a) =(g)): There exists a setindexed
family (E.)icI of eobjects for which the product (HEi ,i) exists in
C and there is an Wmorphism 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) Alinearization of (X,MX) in P .
Thus there exists L, a power of C objects, and an tmorphism
m:X L (4.2.1). Hence X is 1flembeddable.
COROLLARY 4.2.7. Let C be a category with products, let
7i be a class of I morphisms that is isomorphismclosed in C and
leftcancellative 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 nlinearization of (X,S) (respectively, a stable
)ilinearization 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 monouniversal objects in a category and universal
monolifts 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 P7linearizations 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 isomorphismclosed in C and
leftcancellative in C C be a subcategory of C and X be
a Cobject. 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 Zlinearization of (X,S) in 8f
(c) For any endomorphism t on X of finite order, there
exists an Ylinearization 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 tlinearization of (X,S) in P .
PROOF: (a) = (b) (respectively, (a) (e)): There exists a product
( I E, i ) of C objects and an mrmorphism E:X T E (Card S)2
iEl iel
is finite, since Card S is finite. Apply Corollary 4.2.5 to obtain
the Llinearization 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 1llinearization (respectively, a stable
Mlinearization) (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 rnlinearization (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
sectionlinearization in 4X, and hence an 1hlinearization 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 sectionlinearization 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 leftcancellable
and isomorphismclosed in C Thus there exists a sectionlinearization
of (X,t) in '? X (respectively, a stable sectionlinearization 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 Cobjects
(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
weaklinearization 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
wellpowered, having a full replete epireflective subcategory C1,
then ('6( = 6 (1.4.3) and for every Cobject X and every endomorphism
4 on X, there exists a weaklinearization of (X,p) in ,i; and further
more, if X is an 6regular 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 Setmonomorphism 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 monolinearization 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
Cmorphismsthat is isomorphismclosed in C and leftcancellative in
C, let (E )iEI be a setindexed 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 SCard S)'th power of HEi exist in & then
there exists a triple (m,L,9) which is an 2Ilift 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 ;VLinearizations
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 isomorphismclosed 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 1,7universal object for S
provided that for each Ofobject A there exists an l?morphism
m:A X.
(2) A pair (L,y) is called a universal 3flift for End(OM)
[respectively, for Aut(Or)] provided that for each endomorphism
respectively each automorphism] on any Otobject A, there exists an
r4morphism 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 rlinearization 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 Cobject 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
Mmorphism m:A L, such that for each seS, the triple (m,L,u(s))
is an fllift of (A,8y(s)).
A L
SY(s) (s)
L
A
(4) Let k be a cardinal number. A group S is called a
k;iuniversal group for & provided that card S < k and there exist
a Fobject 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
PAGE 2
To Bill and to Kathleen, ; for her birthday
PAGE 3
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. PopStojanovic, 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 WSources 29 3. EMBEDDINGS INTO PRODUCTS 37 3 . 1 Embedding Theorems 37 3.2 Â£, [TWEmbeddable Objects. 46 3.3 ?nCoseparating Classes , 5A 3.4 OtRegular and ^Compact Objects. 61 4 . LINEARIZATIONS 74 4.1 Coordinate Immutors and Permutors 74 4.2 ??1 Linearizations 78 Iv
PAGE 5
TABLE OF CONTENTS (Continued) Page 4.3 Universal WLiiv irizations 91 BIBLIOGRAPHY 98 BIOGRAPHICAL SKETCH 99
PAGE 6
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 'embeddinglike' 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 Scompact 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 tcancellative 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 coordinateimmuting 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 coordinatepermuting 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 "mathematicalmaninthestreet," 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.
PAGE 10
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 leftadjoint 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 Cobject X together with a family of Cmorphisms having X as their common domain. A categorical product ( n E.,7T,) is an example of a source which, in fact, exhibits a iel "monolike" 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•/nsources) whose morphisms in the aggregate exhibit the properties of special types of "embeddinglike" morphisms (Â«)Tl morphisms) . In topology, a longestablished method for producing mappings into topological products has been by use of families of morphisms into the coordinate spaces. Tychonoff's wellknown 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],
PAGE 11
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 "embeddingtype" morphisms. In 1958, Engelking and Mrowka [3] began to develop the concepts of Eregular and Ecompact 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 Ecompletely 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 Ecompact 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 categorytheoretic 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 Cobject 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 Cobject X there exists a pair (r^ , X^ ) , called the ^epiref lection of X, where X^ is an (Sfobject 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, wellpowered 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 StoneCech 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 6Ccompact 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 trivialseeming 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 setindexed 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 Cobject iel ^ (2) for each jel, tt , : 11 A ^ A is a Cmorphism (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 setindexed family of sets. Suppose that for each iel, (P .("j^)i^ J, ) is the product of a setindexed 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 Cmorphism, and for all i,jeOb(I) and Imorphisms 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^ztn) 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 wellknown 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 wellpowered provided that every ^ object has a representative class of subobjects which is a set . Dual notion: cowellpowered The categories Set , Grp , Top , Mon , Haus and CompT. are wellpowered and cowellpowered. 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 leftadjoint. (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 morihi8m provided that whenever mk = ge 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 = ge, 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 fg is a morphism, then fg 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 = me 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 = me, is said to be unique provided that whenever f = ine 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, wellpowered 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 Clobject 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 5Cobject 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 <5Cepiref lection for X. THEOREM 1.4.3 (Characterization of Epiref lective Subcategories) (Herrlich and Strecker [8]). Let C be a complete, wellpowered and cowellpowered 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, wellpowered and cowellpowered 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, wellpowered 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©textendable 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 wellpowered 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 wellpowered 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 fh = 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 Olmorphism. 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 Cmorphism and f is a concrete embedding in C ; hence, there exists a Cmorphism
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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 fitmorphism. 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 Ctmorphism. 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©1object. Hence by hypothesis, L is an Â©(object, so m is an 0Cmorphism. Thus since f is a concrete embedding in fit , there exists an h:L > A such that U (h) = h. Thus he:A ^ A and U(he) = hU(e) = 1^^^ = U (1^). Hence since ^ is faithful, he = 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 "monolike" 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 Cobject and (f.). ^ is a family of Cmorphisms, 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 leftcancellable. (3) A source (A,f.) is called an extremal source provided that for each source (B,g.) and each Cepiraorphism 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 Cmorphisms 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 Imorphism m:i Â• j, D(m)Â»Â£ = I . . Let X be a Â£ object and let f and g:X > L be Cmorphisms 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 Imorphism 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 Imorphism 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 pe = 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 isomorphismclosed 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 tcancellative 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 Wmorphism. 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 isomorphismclosed 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 OTjmorphism subobject morphism weak subobject monomorphism subobject Wsource 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 Wsource (b) f :A Â» B is an ^;i morphism (c) (A,f) is an ;Wsubobject 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/7morphisms; 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 Wsource, so is (A,f ) . PROOF: Clearly (A,f ) is a source. (a) mono : Let p and q be Cmorphisms 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 dmorphisms 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 = he. Now f, "he = 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 (LeftCancellation 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 Cmorphisms 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 Cmorphisms listed in Table 2.3.2. Then ?/? is lef tcancellative in Â£ . PROOF: Suppose p and q are morphisms in Â€ , such that p'q is an J/Jmorphism. 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 Cmorphisms 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 = ge. Now there exists a morphism s such that ss = 1. Thus S'SÂ«g.e Â«= sssk = 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(sg). 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 (Jfregular 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 setindexed 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 = he 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, wellpowered concrete category for which IL preserves monomorphisms and products. Let (A,f.) be a setindexed 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 Sobjects, 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 9Wmorphisms listed in Table 2.3.2 is isomorphismclosed 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 'weakextension' to denote an?^ extension when fA is the class of all morphisms in C . The definition for a weaknonextendable 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 weaknonextendable 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 weaknonextendable 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 weakextension; 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 weakextension (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 weaknonextendable 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 wellknown 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 weaknonextendable 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 Ecompletely 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 Ecompact 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 isomorphismclosed 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?nembeddable in C . Note that since the concrete embeddings in Top are precisely the topological embeddings, our definition for an Eregular space in Top coincides with Mrowka's definition for an Ecompletely regular space. Also our definition for a Hausdorff space to be Ecompact in Haus coincides with Mrowka's definition of Ecompactness, 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 Aregular 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 wellknown result, although it may not be immediately recognizable in our new terminology. PROPOSITION 3.2.2. An Abelian monoid is Abregular in AbMon if and only if it is cancellative. PROOF: Suppose M is Abregular 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 welldefined 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 Abregular. 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/Tlembeddable in C . Then there exists a setindexed family (E . ) . of Â£ objects such that there exists an jTlmorphism 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 Wsource. 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 Wmorphism (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 Eregular 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 Eregular (3.2.1)) if and only if (X, 3) is a concrete embedding source. There are many wellknown 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 wellknown 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 Fregular in Top . EXAMPLE 3.2.9. Let W be the twopoint indiscrete space. A topological space is indiscrete if and only if it is Wregular in Top . Recall that a topological space is said to be zero dimensional provided that it has a base of closedandopen sets. EXAMPLE 3.2.10 (Alexandrof f) . Let D be the discrete space with two points. A topological space is a zerodimensional T space if and only if it is Dregular in ToÂ£. A zerodimensional Hausdorff space is compact if and only if it is Dcompact 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 Vregular in Top . Let us now consider another result of Mrowka's. Let L m denote a space with melements 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 Xregular 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 Xregular in Top ^, or Xregular in ToÂ£, although by Example 3.2.11, every topological space is Vregular 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 Cobject 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 Cobject X, C C and hom (_,E)(t{)) = _Â•({) for each Cmorphism 4). DEFINITION 3.2.13. Let C be a category and let ?7? be an isomorphismclosed 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 (epi,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 TTJmorphisn 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 Eregular 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 Emonoembeddable,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 Cobjects is an ?77coseparating 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 Cobject 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 Cobjects 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 Ctnorphisms 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 Ofobject 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 monocoseparator 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 //Jsources 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 monocoseparating class is an extremal monocoseparating class, which in .urn is a monocoseparating class (2.1.2). Similarly for
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56 complete wellpowered concrete categories (C ,U) for which XI preserves monoraorphisms and products, every extremal monocoseparatlng class for C is a concrete embeddingcoseparating class for C , hence an extremal concrete embeddingcoseparating 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 Cmorphisms 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 7Wcoseparating 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 Cobject 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 embeddingcoseparator in Top ; however Top has a proper concrete embeddingcoseparating 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 embeddingcoseparator for Top .. . DEFINITION 3.3.5. Let S be a category, let W be a class of C morphisms that is isomorphismclosed 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 ^ ?7Jembeddable 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 ?/7coseparating 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 isomorphismclosed 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 ^Jsubobjects 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 cowellpowered, 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 (5tcompact 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, wellpowered and cowellpowered. 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 (>tcompact 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, wellpowered and cowellpowered , 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 Otcompact 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 dcompact 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, wellpowered and cowellpowered 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 Ctepireflection (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 StoneCech 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£ifepiref 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, wellpowered 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 embeddingextension (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 cowellpowered , 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©tepiref lection of X; then X^ is an Aobject 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 ^^ ^^^ setindexed 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 embeddingextension (w,W),for which W is an aiobject 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 flfobject A. Then X is Oicompact, hence 6Cregular. 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 Ciobject, and because ^ is full, m is an fl(morphism. Furthermore, if C is cowellpowered, 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 Abextendable. Thus (<^'f, K(M)) is the desired Abepiref lection for M. Let us suppose M is Abregular (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 Abextendable . 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 3object, since it is an Ctobject. Thus, since r is 0extendable (1.4.2), there exists a morphism e:X > X such that r^ = er . But by hypothesis, X^^ is Clregular, Hence by Theorem 3.4.2, there exists a concrete embeddingextension (Y.Ygj ) in Gi for which y:X^ ^ Y^^ is 6Vextendable. But r^ is 6lextendable 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 = ar Thus yr Now Y = ci'e since r^ is a Qepiref 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 ae = g. Consequently, we can conclude that e is Ctextendable (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 Abregular objects in AbMon (3.2.2). And Ab is a full subcategory of CAbMon . Then for any Abelian monoid M, there exists an Abepiref 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, wellpowered and cowellpowered, 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 embeddingextension (w,W) for X where W is Cicotnpact. (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 = tw (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, wellpowered 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 mwrX * "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 Ciextendable morphism. Then there exists a product (nA.,TT.) of Ctobjects 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 6tregular and Â©(compact objects in categories other than Haus . THEOREM 3.4.6. Let (C ,U.) be a concrete category that is complete, wellpowered and cowellpowered, 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 6textendable morphism f :X Â»Â• Y is an extremal concrete embedding. (f) X is C(regular and for all ^iregular 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 flfobject. 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 Oicompact (2.3.1). (b) and (d) => (g) ; Since X is Â©/compact, it must be Cf regular. Let (w,W) be an ^(regular concrete embeddingextension 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 = hw 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 5lextendable 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 (5tregular 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 Cmorphisms. 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 categorytheoretic 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 Cobject 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)^p7 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 Cobject E, and let E be the object part of the product ( n A., IT.) of some family (A.). _ of Cobjects. 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 /4linearizations, 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 isomorphismclosed 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/imorphism 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 ii 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 TAlÂ±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 tcancellative 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 (kCard 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 , , , kCardS , . 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 Wmorphism. 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) = sy 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.st: 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, 3in)t. And since products are mono sources, y *m = mt. Now suppose that every //I morphism is a monomorphism. Let s, s'eS for which Y^ = Y iThen ym = y ,.m; hence m'S = ms'; 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 embeddinglinearizations 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 selfmaps on a completely regular T space X. Then for each feF, there exists a topological embeddinglinearization 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 wellknown 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 tcancellative 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 = aCardF = 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 SCard 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 Cmorphisms that is lef tcancellative, 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 Wniorphism c, from X into the product ( H E.,7t.) iel of a setindexed 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 kfold products, 5?7 be a class of morphisms in C that is isomorphismclosed and lef tcancellative 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;iembeddable 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 Wmorphism 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 isomorphismclosed in C and leftcancellative 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 monouniversal objects in a category and universal monolifts 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 /,7linearizations 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 isomorphismclosed in C and lef tcancellative in C , Â£, be a subcategory of C , and X be a Cobject. 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 sectionlinearization 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 weaklinearization 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 wellpowered, 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 weaklinearization 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 monolinearization of (X,f) in Set (4.2.6). What we will will find that ^ = ij; xiy , 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 Cniorphisms that is isomorphismclosed in C and lef tcancellative in g, let (E^)^^j. be a setindexed 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 SCard 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 /,7linearization (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 isomorphismclosed 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 Otobject A, there exists an
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92 Wmorphism 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,3y(s)). Y(s) a(s) (4) Let k be a cardinal number. A group S is called a kzz^universal group for C. provided that card S <_ k and there exist a Cobject 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,6y(s)). THEOREM 4,3.2. Let k be an infinite cardinal number, Â£ be a category with kfold products, and //] be any class of G morphisms that is isomorphismclosed in v# and lef tcancellative 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 ^/7lift for End((5{) in Â£. Moreover if Ol is closed under the formation of kfold 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 5tobject 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 = inB(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 = ml^ = m. 7/^ is lef tcancellative in C ; thus q is an /Jmorphism, since m is. Now we need only to show that the diagram q eY(t) a(t) commutes for each teS. For every seS, tt Â•a(t)'q = rr 'f q = t; 'q = m.3(Y(st)) = mB(Y(s))'3(Y(t)) = Tr^q3 (Y(t) ) . But products are mono sources (2.1.3), so a(t).q = q3(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 Ctobject 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 6jobject and (^ be any endomorphism on A. Since K is an ^universal object for dl, there is some ?J9morphism 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 8y(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 ktopological 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 tcancellative (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 Ecompact Spaces, Bull. Acad . Polon. Sci. Ser. Sci. Math. Astr. Phys.^ 6 (1958) , 429436. [ 4] Herrlich, H. , Ekompakte Raiime, Math. Zeitschr.. 96 (1967), 228255. [ 5] , Topologische Ref lexionen und Coref lexionen , Lecture Notes in Mathematics, 78, SpringerVerlag, Berlin, 1968. [ 6] , Categorical Topology, Gen. Topology and Its Appl., 1 (1971), 115. [ 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), 524529. [11] Kelley, John L., General Topology , Van Nostrand, Princeton, N.J., 1955. [12] Lang, S., Algebra , AddisonWesley , Reading, Mass., 1965. [13] Mrowka, S., On Universal Spaces, Bull. Acad. Polon. Sci. , CI. Ill, 4 (1956), 479481. [14] ^ , A Property of the Hewitt Extension vV of Topological Spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. , 6 (1958), 9596. [15] , On ECompact Spaces II, Bull. Acad. Polon. Sci., 14 (1966), 596605. [16] , Further Results on Ecompact Spaces I, Acta Math . . 120 (1968), 161185. 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. PopStoja^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

