C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES
G. C. L. B RUMMER
E. G IULI
H. H ERRLICH
Epireflections which are completions
Cahiers de topologie et géométrie différentielle catégoriques, tome 33, no1 (1992), p. 71-93
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© Andrée C. Ehresmann et les auteurs, 1992, tous droits réservés.
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Article numérisé dans le cadre du programme
EPIREFLECTIONS WHICH ARE COMPLETIONS
by G. C. L. BRUMMER, E. GIULI and H. HERRLICH
We dedicate this paper to the memory of Siegfried Grässer
CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE
CATÉGORIQUES
VOL. XXXIII-1 (1992)
Resume. Nous axiomatisons la situation ou tout
ob jet
d’unecategorie
X a uncomplete
et ou toutplongement
dense dansun
objet complet quelconque
est une r6flexion dans la sous-categorie pleine
desobjets complets.
On dit alors que Xadmet une
sous-categorie
S-fermement E-r6flexive.Ici,
S estune classe de
morphismes
de X ayant despropri6t6s analogues
aux
plongements,
et la classe Erepresente
la densite appro-pri6e.
Pour le cas E =EpiX
nous relions cette notion aveccelles de fermeture
S-absolue,
deS-saturation,
et de(E
nS)- injectivit6;
nous en donnonsplusieurs caract6risations,
en par-ticulier la
pr6servation
desS-morphismes;
et nous consideronsbeaucoup d’exemples topologiques
etalg6briques. Quand
Xest une
categorie topologique
on a un contexte naturel pourlequel E
estplus large
que la classe des6pomorphismes.
0. Introduction
Among
the various kinds of extensions that anobject
canhave, compactifications
andcompletions
of spaces exhibit two very different forms of behaviour. ATychonoff
space, say, can have manymutually inequivalent
Hausdorffcompactifications,
among which there is theAMS Subject Classification: Primary 18A40, 18A32; secondary 18B30, 54B30, 54E15, 54E52.
Key words: Firm epireflection, completion, weak injectivity, saturated object, absolutely closed object, complete object, factorization structure, topological cat-
egory.
The first author acknowledges support from the foundation for Research Devel- opment and the hospitality of the University of L’Aquila while working on this
paper. The second author acknowledges a grant form the Italian Ministry of
Cech-Stone compactification
which is the reflection tocompact
spaces.On the other
hand,
a metric space, say,admits,
up toisometry, just
one
completion,
and thatcompletion
si the reflection tocomplete
spaces.
The latter form of behaviour is
paradigmatic
for the kind of com-pletion
ofobjects
that westudy
in this paper. When it occurs in acategory X,
we say that the"complete" objects
form an s-firm E-reflective
subcategory
of X. Here S is a class ofmorphisms
in X ofwhich we like to think as
embeddings,
and Erepresents
the kind ofdensity
that anobject
should have in itscompletion.
Our chosen
setting
then is an(E, M)-category
X with adesignated
class S of
morphisms.
Mildassumptions
on theinterplay
between the(E, M)-factorization
and the class S determine the results. These become trivial when S coincides with M.For the sake of
clarity,
the mainbody
of the paper isdeveloped
forthe case that
(£, M)
=(epi,
extremalmono).
Thecategory
X withthe
given
S admits at most oneS-firmly epireflective subcategory
R. In this case, R consists of those
X-objects
which areinjective
with
respect
toepimorphisms
inS,
and R also coincides with thesubcategory
of S-saturatedobjects;
under an additionalassumption,
R is the
subcategory
ofabsolutely
S -closedobjects.
It is clear thatX admits an S-firm
epireflection
if andonly
if everyX-object
admitsan
epic s-morphism
into anobject
which is(S
nEpiX)-injective.
Weshow
(Theorem 1.6)
that this isequivalent
to Xbeing s-cogenerated by
a class of(S
F1EpiX)-injectives.
The main result(Theorem 2.5)
is that a
given S-epireflector
in S-firm if andonly
if it preserves S-morphisms (equivalently: S-sources)
and reflects into the class of S- saturatedobjects.
Wegive examples
intopology
andalgebra.
Theapplications
totopological categories
alsoprovide
a context for firmE-reflections with E other than
epi.
Injectives
withrespect
toepimorphic embeddings
were studiedby
P.D. Bacsich
[2],
with resultspartly
of the same intent as the presentpaper. R.-E. Hoffmann
[38]
defined andinvestigated
firm reflections intopological categories.
The latter paper was thefirst,
to our knowl-edge,
whichproved
the firmness of the sobrificationepireflector
in theTo-topological
spaces. It may be observed that theprecategorical
pa- per[10] by
G. Birkhoff can still serve as a source of ideas connected with the notion ofcompleteness.
1. Firm
epireflections
Throughout
the paper we consider acategory
X which iscomplete
and
well-powered
and a fixed class S ofmorphisms
of Xsatisfying
thefollowing properties:
(s1)
IsoX C SCMonoX;
(S2)
S is closed undercomposition;
(s3)
If me E S and e EEpiX,
m EExMonoX,
then e E S.MorX, EpiX, MonoX, ExMonoX, RegMonoX,
IsoX denote the class of allmorphisms, epimorphisms, monomorphisms,
extremal mo-nomorphisms, regular monomorphisms
andisomorphisms
ofX,
re-spectively.
If X is concrete then EmbX denotes the class of embed-dings
of X(i.e.
initial maps whoseunderlying
maps aremono);
itsatisfies conditions
(s1)-(s3).
Note that every M which is
part
of an(&, M)-factorization
struc-ture of
X,
withEpiX
C&,
has theproperties (si)-(s3)
and in fact also satisfies the additionalassumptions (s4)
and(s5)
which we im-pose later.
All
subcategories
will be taken as full andisomorphism-closed.
Since X is
complete
andwell-powered,
then(cf. [36, 34A]):
(x1 )
X is an(EpiX,
ExMonoX)-category.
Definitions 1.1. Let X be an
X-object.
(1)
X is said to beS-injective if, for
each e : Y-> Z in S and eachX-morphism f:
Y ->X,
there is anX-morphism
g : Z -> X such that ge =f.
Then g is called(an)
extensionof f (to Z).
(2)
X is said to beweaklyS-injective if
it is(EpiX
flS )-injective.
Inj(S) (WInj(S))
denotes the classof
all(weakly) S-injective objects of X.
(3)
X is said to be S-saturatedif
anX-morphism f :
X ---> Y isan
isomorphisms
wheneverf
EEpiX n
S.Sat(S)
denotes theclas3
of
all S-saturatedobjects of
X.(4)
X is said to beabsolutely
S-closedif
anX-morphism f :
X -Y is a
regular monomorphisyn
wheneverf
E S.AC(S)
denotesthe class
of
allabsolutely
S-closedobjects of
X.We refer to
[36]
and[32]
for othercategorical
terms not definedhere.
Weakly S-injective implies
S-saturated andabsolutely
S-closed im-plies
S-saturated(see Proposition
1.3below).
For the case X a
category
ofalgebras (in particular
ofrings
orsemigroups)
and S the class of allmonomorphisms,
S-saturated andabsolutely
S-closedobjects
were introduced andinvestigated by
Isbell[42] (see
also[39]).
It is shown in[42]
that an S-saturatedalgebra
neednot be
absolutely
S-closed.For the case X an
epireflective subcategory
ofTop
and S =EmbX,
S-saturated and
absolutely
S-closedobjects
were studiedby
Dikran-jan
and Giuli in[16]
and[19].
It was observed in[16]
that absolute S- closedness coincides with X-closedness whenever theX-epimorphisms
coincide with the dense continuous maps. The latter notion was intro- duced
by
Alexandroff andUrysohn [1]
for thecategory
of Hausdorff spaces(H-closedness)
and wasinvestigated by
many authors(see
e.g.[9]
and[51])).
In such a case alsoabsolutely
S-closed = S-saturated.It was also
proved
in[19] that,
for X thecategory
ofUrysohn
spaces(or,
moregenerally
for X thecategory
ofS(n)-spaces),
an S-saturated space need not beabsolutely
S-closed andabsolutely
S-closed need not be X-closed.Weak
S-injectivity
vanishes in some contexts in which theS-inje- ctivity
is consistent(e.g.
in abeliancategories
X with S=MonoX).
In
Unifo,
with S ={Embeddings}, S-injectivity
is a muchstronger property
than weakS-injectivity [41].
The latter notion isequivalent
to
completeness. Analogously
inTop.,
with S ={Embeddings},
S-injective
space means retract of aproduct
ofSierpinski
spaces[49]
while
weakly S-injective
space means sober space[38].
S-injectivity
with no conditions on X and onS,
isinvestigated by
Herrlich
[35] (see
also[34]).
Let X be a
topological
space and let X be theepireflective
hullof X in
Top.
Sobral[50]
showed that theEilenberg-Moore
factor-ization of the functor
Hom(-, X):Top*P
-> Set can be obtained via thecorresponding
factorization of its restriction to thesubcategory AC(EmbX), provided
X satisfies aninjectivity
condition. Thisinjec- tivity
condition is weaker than the(EmbX)-injectivity
notion but it is notcomparable
with our weak(EmbX)-injectivity.
If R is an
epireflective subcategory
ofX,
R : X -> X denotes the reflectionfunctor, and,
for eachX-object X,
rx : X -+R(X)
denotesthe R-reflection
morphism.
For eachX-morphism f :
X ->Y,
Y EObR, f * : R(X)
-> Y denotes theunique X-morphism
such thatf*rx
=f.
Deflnitions 1.2. Let R be a
reflective subcategory of X.
(1)
R is said to beS-epireflective if for
eachX-object X,
rX :X -->
R(X) belongs
toEpiX n S.
(2)
R is said to be anS-firm epireflective subcategory if it is
S-epireflective and, for
eachf :
X ->Y,
with Y ER, f * is an isomorphism
wheneverf
EEpiX n S.
Proposition
1.3. Let R be asubcategory of X
andfor
anX-object
X consider the
following
conditions:(i)
X isS-saturated;
(ii)
Xbelongs
toR;
(iii)
X isweakly S-injective;
(iv)
X isabsolutely
S-closed.Then the
following
hold:(a) Always (iii)=>(i)
and(iv)=>(i);
(b) (i)=>(iv)
whenever ExMonoX =RegMonoX;
(c) (i)=>(ii)
whenever R isS-epireflective;
(d) (ii)=>(iii)
whenever R isS-firrnly epireflective
in X.Proof.
(a) (iii)=>(i):
Since X isweakly S-injective,
then for each(e :
X -Z)
EEpiX
nS,
theidentity
lx has an extension g, so ge =1x, consequently
e E IsoX.(iv)=>(i): Every X-morphism
which is bothepi
andregular
mono is anisomorphism.
(b)
If anX-morphism f :
X -> Ybelongs
to Sthen, by property
(S3),
in the(epi,
extremalmono)-factorization
me =f ,
e E s. IfX E
Sat(S)
then e is anisomorphism,
sof
is an extremal mono,hence, by assumption,
it is aregular
mono,consequently
X EAC(S).
(c)
If R isS-epireflective
then every R-reflection rx : X ->R(X) belongs
toEpiX fl S.
If in addition X ESat (S),
then rx must be anisomorphism, consequently X
E R.(d)
Whenever e : Y --> Zbelongs
toEpiXfls,
since R isS-epireflective
and since S has
property (82),
then rze : Y -> Z ->R(Z) belongs
to
EpiX n S.
Since R issupposed
to beS-firm,
then(rZe)*
in thecommutative
diagram
below is anisomorphism.
Thus,
for everyX-morphism f :
Y -+X,
theX-morphism
g =f*((rZe)*)-1rZ
is the needed extension off.
We refer to
[2, (Theorem 2.2)]
for other conditions under whichSat(S), AC(S)
andWInj(S)
coincide.Corollary
1.4. X admits at most oneS-firm epireflective subcategory
R. In such a case R =
Sat(S)
=WInj(S).
Question
A. Prove ordisprove that, for
each(concrete) category
X(and
S =EmbX), Sat(S)
=AC(S)
holds whenever X admits anS-firm epireflective subcategory.
Question
B. Prove ordisprove that, for
each(concrete) category
X(and S
=EmbX), WInj(S)
cAC(S).
Definition 1.5. Let P be a class
of X-objects.
We say that X is S-cogenerated by P af
everyX-object
is anS-subobject (X
isS-subobject
of Y if
there is s : X -+Y,
with s ES) of
aproduct of objects
in P.The
category
X admits an S-firmepireflection
if andonly
if eachX-object
is the domain of some S nEpiX-morphism
into someobject
of
WInj(S). (This
isimmediately
clear fromProposition
1.3(d).) Wlnj(S)
is closed for thetaking
ofproducts
in X(again clear,
with no
assumptions
on X or S other than S CMorX).
WInj(S)
is closed for thetaking
of extremalsubobjects
in X(this immediately
follows frommerely
the(EpiX, ExMonoX)-diagonaliza-
tion
property).
If X is assumed to be
complete, well-powered
andco-(well-powered),
it follows
by [36, (Theorem 37.1)]
thatWlnj(S)
isepireflective
inX.
Alternatively,
it was shown in[6, (Proposition
1 and pp. 157-158)]
thatWlnj(S)
isepireflective
inX, provided
X is an(Epi, ExMonoSource)-category (cf. [34, (p. 331)].
It is worth
noting
that thefollowing
result does not assume X to beco-(well-powered).
Theorem 1.6. The
category
X admits anS-firm epireflective
sub-category
Rif there
exists a cla33 P CWlnj(S)
whichS-cogenerates
X. In this case, R is the
epireflective
hull in Xof
P.Proof. Suppose
rx : X -->R(X)
is the firmS-epireflection
of X inX.
By definition,
rx ES,
andby Proposition 1.3, R(X)
EWInj(S).
Hence
WInj(S) S-cogenerates
X.Conversely,
suppose there is aclass P C
WInj(S)
whichS-cogenerates
X. Let R be the class of all extremalsubobjects
ofproducts
ofP-objects. By
the aboveremarks,
R C
Wlnj(S).
consider anyX-object
X. Since X isS-cogenerated by P,
there exists anS-morphism s :
X ->IIP,
for some set ofPj
E P.Let X e -+ Mm ->
IIP,
be the(EpiX, ExMonoX)-factorization
of s.Then e E
Epis
n Sby (S3),
and M E Rby
definition of R. Consider anyf :
X --> Y with Y E R. The weakS-injectivity
of Y thengives
a
(unique) f * :
M - Y withf * e
=f .
Thus e : X -> M is anR-reflection of
X;
we choose anequivalent
rX : X -->R(X),
whenceby (s1)
and(S2)
rX EEpiX
fl S. To see that R isS-firm,
considerg:X->ZinEpiXnSwithZER.
We haveg* : R(X) ->
Z withg*rx
= g; the weakS-injectivity
ofR(X)
alsogives h :
Z -R(X)
with
hg
= rX, andclearly h
is inverse tog*. Finally
it is clear that Ris the
epireflective
hull ofP,
also in the sense that R is the smallestepireflective subcategory
of X that contains P.Corollary
1.7.Wlnj(S)
isS-firmly epireflective
in Xif
andonly if X
isS-cogenerated by Wlnj(S).
A result
by
Bacsich[2, (Theorem 3.1)] partly overlaps
with thecontent of Theorem
1.6,
but under differentassumptions.
The corre-sponding
results mentionedby
Kiss et al.[44, (pp. 94-95)]
andgiven by
Tholen[52, (esp.
Lemma7)]
areessentially
different.Example
1.8. In allexamples
below we assume that S is the class ofembeddings
and wedrop
"S" in all terms in which itpreviously appeared.
(1)
Firmness becomes trivial in concretecategories
X in whichepimorphisms
are onto maps(in particular
intopological categories
as well as in abelian
categories).
Indeed in such a case, R = X.(2)
No non-trivial(# Singl
={spaces
with at most onepoint}) epireflective subcategory
ofTop consisting
of Hausdorff spaces admitsa firm
epireflection.
In fact none of thesubcategories
above admitsa class of
weakly injective cogenerators:
if X is asabove,
then thetwo-point
discrete spaceD2(= {O, 1}),
the discrete space of natural numbers N and its Alexandrovcompactification
N*belong
to X.Let
f :
N -D2
be the continuous map definedby f (2n)
= 0 andf (2n +1)==1,nEN.
Thenf
cannot be extended toN*,
whilethe inclusion e : N -> N* is a
dense,
henceX-epi, embedding.
Weconclude that no space with more than one
point
isweakly injective
in
X,
so,by
Theorem1.6,
X does not admit a firmepireflection.
Thesame
proof
also establishes that if Y is anyepireflective subcategory
of Haus
containing N,
then Y has no firmepireflective subcategory.
(3)
inTopo,
thecategory
ofTo topological
space, theSierpinski two-point
space is acogenerator
ofTopo
and it isweakly injective
in
Top..
Theepireflective
hull inTopo
of theSierpinski
space is thecategory
Sob of sober spaces, so, in virtue of Theorem1.6,
Sobis the firm
epireflective subcategory
ofTopo.
This restates a resultwhich Hoffmann
[38]
establishedby
an internalargument.
In virtueof
Proposition 1.3, sober, weakly injective,
saturated andabsolutely
closed coincide in
Topo.
(4) By Unifo
we mean thecategory
ofseparated (i.e. To,
henceHausdorff)
uniform spaces anduniformly
continuous maps. It is well known[53]
that thecomplete
spaces inUmfo
form the firmepireflec-
tive
subcategory
ofUnifo.
(5) Proxo
denotes thecategory
ofseparated proximity
spaces andproximity
maps[41].
Thecomplete (which
are here the same ascompact)
spaces inProxo
form the firmepireflective subcategory
of
Proxo.
This is an instance of Theorem1.6,
therequired weakly injunctive cogenerator being
the compact unit interval.(6)
Thecomplete
metric spaces form a firmepireflective
subcate-gory of the
category
of metric spaces andnon-expansive
maps.(An interesting
extendedsetting
for this classical result isgiven by
Hoff-mann
[38, (p.
321example 3.4)].)
(7) Qun
will denote thecategory
ofquasi-uniform
spaces andquasi-
uniform maps
(see
e.g.[15]
or[24]).
Itssubcategory Qun.
of sepa- ratedobjects
consists of thosequasi-uniform
spaces for which thejoin
of the two induced
topologies
isTo (hence Tychonoff),
orequivalently
the first
topology
isTo,
orequivalently
the secondtopology
isTo (by
the
join
of two structures we shallalways
mean the least fine structure finer thanboth).
Csaszar([14], [15])
showed thatQun.
has a firmepireflective subcategory
whoseobjects
he named"doubly complete".
These are the spaces whose uniform
coreflection,
formedby joining
a
quasi-uniformity
with itsinverse,
is acomplete
uniform space. A convenientconstruction,
andproof
of thefirmness,
isgiven
in[24],
where the
objects
are called"bicomplete" (cf. [13]).
(8) Qprox
denotes thecategory
ofquasi-proximity
spaces, known to beisomorphic
to the fullsubcategory
oftotally
bounded spaces inQun (cf. [47]
or[24]). Qproxo
denotes thecorresponding
sub-category
ofseparated objects. QproXo
has a cogenerator I which isinjective
withrespect
toQproxo-epi embeddings;
I is the closed unit interval with thequasi-proximity
relation 6given by
A6Biff,
for eacha >
0,
there exist x E A and y E B with y > x - a. Thusby
Theo-rem
1.6, Qproxo
has a firmepireflective subcategory.
The spaces inthis
subcategory
are those for which thejoin
of the twounderlying topologies
iscompact; equivalently, they
are thebicomplete (in
thesense of
(7) above) totally
boundedquasi-uniform
spaces.(9) 2Top
denotes thecategory of bitopological
spaces(more briefly,
"bispaces")
in the sense of[43]. Objects
aretriples
of the form X =(|X|, O1X, O2X)
whereIXI
is a set andOlX,
02X aretopologies
on
|X|
amorphism f :
X -> Y is a function withf : (|X|,OiX) ->
(|Y|,OiY)
continuous fori = 1, 2.
The fullsubcategory 2Topo
con-sists of those X with
(|X|, O1X
V02X)
ETopo. 2Topo
has acogenerator Q
which isweakly injective in2Topo:
This was shownin
[28],
where the smallest suchobject
wasnamed
"thequad":
Thus
by
Theorem 1.62Topo
has a firmepireflective subcategory
Rwhose
objects
areprecisely
theabsolutely
closedobjects
and alsoprecisely
theweakly injective objects
of2Topo. (In [28]
the coin-cidence
AC(S)
=WInj(S)
wasproved though
the firmness was notobserved).
It was also shown in[28]
that R is containedin,
but differsfrom,
thesubcategory
of soberbispaces
in2Topo.
The soberbispaces
are
given by
thelargest duality
betweenbispaces
and biframes[4];
they
are thosebispaces
for which thejoin
of the twotopologies
is asober
topological
space. It isnoteworthy
that inQuno
andQProxo
the
"complete" objects
areprecisely
those whosesymmetrization
is"complete"
inUnifo
andProxo respectively,
aphenomenon
whichfails for
2Topo
versusTopo.
(10)
Thecategory RegNear
ofregular
nearness spaces has a firmepireflective subcategory [8],
whoseobjects
are calledcomplete.
Infact the
completion
functor inRegNear
restricts to the one inUnifo.
In the
larger category SepNear
ofseparated
nearness spaces, com-pleteness
misbehaves ininteresting
ways[7], [8].
(11) PTop (PTopo)
will denote thecategory
of(To) pretopological
spaces
(= Cech spaces).
It is shown in[20]
thatPTopo-epis
are onto.So
PTopo trivially
admits a firmepireflective subcategory.
If X is a non trivial
epireflective subcategory
ofPTop consisting
of Hausdorff
pretopological
spaces, then theargument
in(2)
can beused to show that X does not admit a firm
epireflection.
Notice that the class of saturated
(= absolutely
closed =compact)
Hausdorff
pretopological
spaces does not even form(the object
classof)
a reflectivesubcategory
ofPTop (cf. [12]).
The
negative
results above remain valid in othercategories
of filterconvergence spaces which
properly
containTop (e.g.
in thecategory PsTop
ofpseudotopological
spaces and in thecategory
Lim of limitspaces).
It is also easy
to see that 4qmess
becomes trivial in allepireflective subcategories
ofboth-Worn (the category of bornological
spaces[40])
and
Simp (the category
of abstractsimplicial complexes [46]).
(12)
Thecategory
of normed vector spaces over a fixed subfield K ofC,
withnon-decreasing
K-linear maps, has the Banach spaces overK as firm
epireflective subcategory.
This fact isplaced
in thesetting
of
topological categories
in[38].
(13) TopGrpo
denotes thecategory
oftopological
groups withTo-
(hence Hausdorff) topology;
themorphisms
are continuous homomor-phisms. Completion
withrespect
to the two-sideduniformity
wouldprovide
the firmepireflection
if we knew that everyepimorphic
em-bedding
G -> H with Hcomplete
in the two-sideduniformity
wasdense. Since the
epimorphism problem
forTopGrpo
isunsolved,
we
only
know that a fullsubcategory
X ofTopGrpo
admits a firmepireflection
if theepis
of the stated kind in X are dense. This is thecase for the
subcategory TopAbo
of abeliantopological
groups withTo-topology [11].
(14)
Let D be thecategory
of bounded distributivelattices,
withlattice
homomorphisms preserving
0 and 1. EmbD =MonoD. The 2-chain is aninjective cogenerator
of D[5].
Henceby
Theorem 1.6D has a firm
epireflective subcategory,
theepireflective
hull of{2}, consisting
of the Booleanalgebras.
Thus(cf. [2])
the reflection embeds anyobject
of D in its Booleanenvelope.
(15)
Thecategory
of cancellative abelianmonoids,
with homomor-phisms preserving
neutralelement,
has thecategory
of abelian groupsas firm
epireflective subcategory,
the reflectionbeing
the group of dif-ferences
[10], [2].
Thecorresponding
fact is true for the category ofintegral
domains withring homomorphisms preserving 1,
the reflec- tiongiving
the field of fractions(that
thiscategory
does notsatisfy
our initial
assumption
ofcompleteness
is immaterial for the presentpurpose). Similarly,
thecategory
of torsion-free abelian groups admitsa firm
epireflection,
whichgives
the divisible hulls of these groups[2].
This is a
particular
case of thefollowing example.
(16)
Let R-Mod be thecategory
of all left modules over a fixedunitary ring
Rand,
for agiven
radical r inR-Mod,
letFr
be thecorresponding
torsion freeclass, i.e. Fr = {X
ER-Modlr(X)
=0}.
Fr
isepireflective
in R-Mod and everyepireflective subcategory
of R-Mod is of the form
Fr
for aunique
radical r(cf. [21]).
DenotebY.Fjr
the class of all r-torsion free modules which are
U-injective
inR-Mod,
where U is the class of all r-dense
(= Fr-dense) monomorphisms.
Itis shown in
[21]
thatFJr
isfirmly epireflective
inFr
if andonly
if ris a
hereditary
radical.(17)
Nocategory
ofalgebras (as
definedby
Isbell[42]),
inparticular semigroups
orrings,
admits a firmepireflective subcategory.
In factit is shown in
[42] Example 3.2,
that the saturation of analgebra
isnot
unique (and
there is not a universal saturation which maps onto allothers;
so the saturatedalgebras
do not even form anepireflective subcategory).
(18)
Aseparated projection
space[23]
is apair (X, (an :
nE N)) consisting
of a set X and a sequence of maps an : X -+X, subject
tothe
following
conditions:(pro) (sep)
Abbreviated notation for
(X, (an : n E N))
is(X, an).
A
projection morphism f : (X, an) -> (Y, (3n)
is a mapf :
X -> Ysatisfying
the condition(mor)
£RQ8
will denote the(concrete) category
ofseparated projection
spaces and
projection morphisms.
A sequence
(xm)
in(X, an)
is called aCauchy
sequence ifan(xn+1)
= xn, for each n E N. A
Cauchy
sequence(xm)
converges to apoint
x ifan(x)
= xn, for each n E N. Aseparated projection
space is calledcomplete
if eachCauchy
sequence converges.It is shown in
[26]
that thesubcategory
of allcomplete separated projection
spaces isfirmly epireflective
inPRO,.
2. Preservation of S-sources
We now have to extend the notion of
S-morphism
to sources. Weemphasize
that we consider sources indexed overarbitrary
classes.Definitions 2.1.
(1)
A source( f, :
X ->Yi|i
EI)
is called an S-source
if for
some set J C I theX-morPhism fj> :
X -+TI(Y jlj
CJ) belongs
to S.(2)
We say that afunctor
F : X --> X preservesS-morphisms if F(s) : F(X)
-->F(Y) belongs
to S whenever s : X -> Y is in S. Thedefinition for
thepreservation of
S-sources isanalogous.
We shall sometimes need the
following
additional conditions on S:(s4)
Wheneverfi : Xi --> Y
is in S for each i EI,
I any set, theproduct morphism Hfi : TI(Xili
EI)
->TI(Yili
CI)
is inS;
(s5)
ExMonoX C S.Proposition
2.2. Assume condition(si)-(s4).
Let R be an S-epireflective subcategory of
X with R : X -> X thereflector
to R.If R
preservesS-morphism,
then R preserves S-sources.Proof.
Let(fi :
X -+Yi Ii
EI)
be an S-source. Thus there is a set J C I such that themorphism fj> :
X -+TT(Yj| j
EJ)
is in S. LetP =
H(R(Yj) |j
EJ)
withprojection morphisms
pj.By properties (S2)
and(s4),
themorphism h
=rYjfj> :
X-> P is inS,
since eachry j is in S.
Since R is
epireflective P E R,
so that rP is anisomorphism.
We have
R(fj)rx
=rYjfj
=pjh
=pjrp1 R(h)rx,
whenceR(fj) =
pjrp1 R(h),
in other wordsrp’R(h) = (R(f;));
but thismorphism
is in S since both
rp- 1
andR(h)
are in S. Thus(R(ft) : R(X) ->
R(Yi)li
EI)
is anS-source,
asrequired.
Proposition
2.3. Assume conditions(si)-(s3)
and(s5). If
R isan
S-firm epireflective subcategory of X,
theR-reflection functor
Rpreserves
S--morphisms.
e m
Proof.
Letf :
X --> Y be anS-morphism
and let X - M ->R(Y)
be an
(epi,
extremalmono)-factorization
ofry f : X -
Y -->R(Y) (see property (xl)
in§1).
Then e EEpiX
n Sby (s3).
Since M isan extremal
subobject
ofR(Y)
ER,
M E R. Since R is S-firm ande E
EpiX
n shemorphism
e* :R(X) -->
M is anisomorphism,
soit
belongs
toS, by (so ).
We haveme*rx
= me =rY f
=R(f)rx,
whence me*
= R( f ).
Since m E ExMonoX(C
Sby assumption)
ande* E
S, R(f)
= me* E Sby (S2)
and theproof
iscomplete.
Corollary
2.4. Assume conditions(si)-(s5). If R is S-firmdy epire- flective
inX,
then theR-reflection functor
preserves S-sources.Theorem 2.5. Let X be a
complete
andwell-powered category
and Sa class
of X-morphisms satisfying
conditions(si)-(s5).
Let R be anS -epireflective subcategory of X
withR-refiector
R.Then,
thefollowing
conditions areequivalent:
(i)
R isS-firrrcly epireflective
inX;
(ii)
R preservesS-morphisms
and R CSat(S);
(iii)
R preserves S-sources and R CSat(S);
(iv)
R CWInj(S);
(v)
There is a classP C Wlnj(S)
such that R is theepireflective hull of P in X;
(vi)
R preservesS-morphisms
and S nEpiR
= IsoR.When one
of these
conditionsholds,
then R =Sat(S)
=Wlnj(S).
Proof. (i)=>(ii)
follows fromProposition
2.3 and from(ii)=>(i)
ofProposition
1.3.(ii)=>(iii)
follows fromProposition
2.2.(iii)=>(i):
Iff :
X -> Y is inEpiX
n S and Y E R thenf *
E S andf *
EEpiX,
so R CSat(S) gives f *
iso.(i)=>(iv)
isprecisely (ii)=>(iii)
ofProposition
1.3.(iv)=>(i):
In such case X istrivially cogenerated by
a class of weakS-injective objects
so Theorem 1.6applies.
(iv)=>(v)
is trivial and(v)=>(i)
follows from Theorem 1.6.(vi)=(iv):
Let A E R. To show A EWlnj(S),
considerf :
X -> A and e : X -> Y with e EEpiX
n S. since(R(e))rx
= rye,R(e)
isan
X-epimorphism
and hence anR-epimorphism. By (vi), R(e)
ES,
hence
R(e)
EEpiR
nS,
so thatR(e)
E IsoR. Withf
=f*rx
wethen have
f
=f*(R(e))-lrYe,
which proves that A EWinj(S).
(ii)=>(vi):
Observe thatalways EpiR =EpiXflMorR,
which follows at once from the reflection mapsbeing
monic(by (s1)).
Considerf :
X -> Y inEpiR
n S. Thenf
EEpiX
n S withX,
Y E R.Assuming (ii)
we have X ESat(S),
so thatf
is anisomorphism.
Thus
(vi)
holds.The
following example
shows that the condition R CSat(S)
cannotbe deleted form conditions
(ii)
and(iii)
in Theorem 2.5.Example
2.6. Let Alex be thecategory
of Alexandroff spaces[31], [25].
These are the same as the zero-setspaced
of[30].
It is shownin
[25]
that Alex admits no(EmbAlex)-firm epireflection,
but thatthe
realcompact
reflection maps X -> vX in Alex are essential em-beddings.
Thisimplies
that therealcompact epireflector v
preservesembeddings,
so thatby Proposition
2.2 v also preservesembeddings-
sources
(i.e.
initialsources).
Moreover v preservesarbitrary products
in Alex
[30]. (Christopher
Gilmourkindly
reminded us of this exam-ple).
It is worth
noting
that Alex isisomorphic
to the fullsubcategory
SMF of
separable
metric-fine spaces inUnifl [31].
Thecompletion epireflector
inUnifo
then restricts to a firmepireflector
in SMFwith
respect
to uniformembeddings (note
that SMF is stable undercompletion). But, being isomorphic
toAlex,
SMF admits no firmepireflector
withrespect
to thelarger
class ofSMF-embeddings.
Similarly
thecategory Tych,
whichby Example
1.8.(2)
admits nofirm
epireflector
with respect toembeddings,
has manycompletion-
stable full
embeddings
intoUnifo.
Each of the embedded subcate-gories
ofUnifo
then admits a firmepireflector
with respect to uniformembeddings.
Proposition
2.7. Let X admit anS-firm epirefiector
R.If EpiX n S
is closed under the
formation of products,
then R preservesproducts.
Proof. By assumption IIrxj
EEpiX
n S. AlsoR(Xi)
E R. Thenby
the definition of
S-firmness,
the extension ofTTrXj
toR(IIXI)
is anisomorphism.
Definition 2.8.
([3], [52])
AnX-Trtorphism f
E S is called S-essentialif g
E S wheneverg f
E S.If f :
X - Y is S-essential and Y is S-injective,
thenf is
called anS-injective
hullof
X.Remarks 2.9.
(1)
If S satisfies(S5),
then anX-epimorphism
is S-essential if and
only
if it isEpiX
n S-essential.(2)
Let r : 1 -> R be anS-epireflection
in X.If each rx is
S-essential,
then R preservesS-morphisms.
The con-verse
implication
holds under aslight strengthening
of condition(s3),
e.g. that
g f E S implies f
E S which is satisfiedby
e.g. extremalmonomorphisms,
initialmorphisms,
andembeddings.
Proposition
2.10. Let condition(S5)
hold and let X admit an S-firm epireflection
r : 1 -i R.Then, for
each X EX,
rx : X ->R(X)
is an