On Coregular Closure Op erators
Gon calo Gutierres
Departamento de Matem atica da Universidade de Coimbra Coimbra Portugal
ggutc mat uc pt
Abstract
Among closure op erators in the sense of Dikranjan and Giuli the regular
ones have a relevant role and have b een widely investigated On the contrary the
coregular closure op erators were intro duced only recently in and they need to
be further investigated In this pap er we study co regular closure op erators in
connection connectednesses and disconnectednesses in the realm of top ological
spaces and mo dules
AMS sub ject classi cation B B E D
Keywords closure op erator co regular closure op erator connectedness
preradical torsion theory
Intro duction
Regular closure op erators were intro duced by Salbany in in the category of
top ological spaces and have been investigated and used by several authors namely
b ecause they play an imp ortant role in the study of epimorphisms They havealsobeen
used in the context of the Diagonal Theorem that is the characterization of delta
sub categories see for instance and
The recent study of nabla sub categories by Clementino and Tholen led these
authors to the de nition of coregular closure op erator whichturnedouttoplay exactly
the role of regular one in this context Besides some interesting examples presented in
not muchisknown ab out these closure op erators even in the category of top ological
spaces
e In this pap er we investigate the b ehaviour nabla sub categories and their resp ectiv
coregular closure op erators in the category T op of top ologoical spaces section and
Mod of mo dules over aringR section
R
In T op we study in particular the least coregular closure op erators and obtain a
prop er class of coregular closure op erators that do not form a chain Prop osition
In Mod we show in Theorem that the regular and coregular closure op erators
R
are exactly the maximal and the minimal closure op erators de ned by radicals and
idemp otent radicals resp ectively
Acknowledgment I thank Professor Maria Manuel Clementino for valuable dis
cussion on the sub ject of this pap er
Preliminaries
We will rst intro duce some notions and techniques that will b e used througout
The author acknowledges partial nancial assistance by the Centro de Matem atica da Universidade
de Coimbra and by Pro jects Praxis PCEX P MAT ACL and XXI MAT
Factorization systems
In the category T op of top ological spaces and continuous maps the class M of em
b eddings has some sp ecial features that can be formulated in a categorical way For
each space X the class M X of embeddings with co domain X can be preordered
by where m M X n N X if there exists t M N such that
n if m n the equivalence relation de ned by m n t m Considering in M X
and n m it is obtains that each equivalence class corresp onds exactly to an inclusion
of a subspace of X In M X one can form arbitrary meets and so also arbitrary joins
and the class M is stable under pullback
Furthermore every morphism f X Y can be factorized as follows
f
X Y
e a
f X
where a is an emb edding and e is a continuous surjection Moreover this factorization
is unique up to isomorphism that is if f a e a e with a a emb eddings and e e
surjections there exists an isomorphism h suchthath e e and a h a The direct
m
f
image of M X under f is obtained by factorizing M X Y as ab ove while the
n
inverse image of N Y under f is exactly the pullback bred pro duct of N Y
along f
This construction can b e generalized straightforward to a general category X given
two classes of morphisms E and M closed under comp osition and containing the iso
morphisms of X one says that E M is a factorization system for morphisms in X if
every X morphism has a unique up to isomorphism E M factorization That is for
each morphism f X Y there exists e X M and m M Y in M such
that f m e
A factorization system is called proper if E is a class of epimorphisms and M is a
class of monomorphisms Consequently E contains the regular epimorphisms and M
the regular monomorphisms cf
The category X is said to b e M complete if X has M pullbacks i e the pullbackof
a morphism in M along any morphism exists and belongs to M and M intersections
i e X has limits of families of morphisms in M with common co domain which be
long to M We remark that the M completeness of X guarantees the existence of a
factorization system E M for morphisms see
Having in mind the behaviour in T op of the factorization describ ed ab ove in an
M complete category X with the factorization system E M a subobject of X X is
a morphism m M X in M Denoting bysubX the class of sub ob jects of X subX
is a p ossibly large complete lattice with its preorder de ned as in the top ological
setting Every morphism f X Y in X induces an image preimage adjunction
f a f sub Y sub X with f n the pullback of n sub Y along f and
f m the M part of the factorization of f m One always has m f f m and
f f n n
For more details on factorization systems see
Closure Op erators
From now on we work in an M complete category X with nite limits and a prop er
factorization system E M
A closure op erator c of the category X with resp ect to the factorization system
E M is given by a family of maps c sub X sub X such that
X X X
m c m for all m sub X
X
if m m then c m c m for all m m sub X
X X
f c m c f m for all m sub X and f X Y in X
X Y
Condition can equivalently b e expressed as c f n f c n for all n in sub Y
X Y
X Y in X and f
A sub ob ject m of X is c closed if c m m and it is c dense if c m
X X X
A closure op erator c is idempotent if c m isc closed for every m M andisweakly
hereditary if j is c dense with m c m j
m m
The preorders of the classes sub X induce in a natural way a partial order in the
conglomerate CL X of all closure op erators in X w r t E M that has meets and
joins formed p ointwise
For additional information on closure op erators see
Regular and coregular closure op erators versus
connectedness
Given a closure op erator c in X an ob ject X of X is called c separated if its diago
nal X X X is c closed and it is called c connected if is
X X X X
c dense This w ay one de nes the sub categories c of c separated ob jects and r c
of c connected ob jects all the sub categories of X we consider are full and closed under
isomorphisms and we denote its conglomerate by SU B X The ob jects that b elong to
c r c are those whose diagonal is an isomorphism which are exactly the preter
minal ob jects
The and r assignments giverise to the functors
op
CL X SU B X
r CL X SU B X
where the partially ordered conglomerates CL X and SU B X are considered as cat
egories
On the other hand each sub category of X de nes two sp ecial closure op erators a
regular and a coregular closure op erator we describ e below
De nitions Let A be a sub category of X The regular and coregular closure
operators induced by A are lo cally de ned by
A
reg m fh j h X A A A and h m g
A A
X
A
coreg m m fh j h A X A A and h mg
A
A
X
for every m sub X and every X X
We remark that every regular closure op erator is idemp otent and every coregular
closure op erator is weakly hereditary
Regular closure op erators were intro duced by Salbany in with a di erent
but equivalent description and were widely used in the literature Coregular clo
were intro duced by Clementino and Tholen in in order to describ e sure op erators
r sub categories
A B A B
Let A B be sub categories of X If A B then coreg coreg and reg reg
hence reg and coreg maybe interpreted as functors
Prop osition
op
The functor reg SU B X CL X is right adjoint to
The functor coreg SU B X CL X is right adjoint to r
Corollary Let A be a subcategory of X and c a closure operator in X Then
A c
a A reg and c reg
A
b A c c reg
r c A
and c coreg a A r coreg
A
b A r c c coreg
From this prop osition one has that there is a bijection between co regular closure
op erators and delta nabla sub categories
Coregular closure op erators in T op
In this section we will present examples of coregular closure op erators and r sub categories
in the category of top ological spaces
It was proved in that r and sub categories in T op extend disconnectednesses
and connectednesses as studied by Arhangel ski and Wiegandt
Prop osition Let A be a subcategory of T op Then
A
r A fX Top j A A g A X g is constant g coreg
A
r reg l A fX Top j A A f X A f is constant g
The sub categories of the typ e l A and r A are called left constant and right
constant resp ectively and in the particular case of top ological spaces they are also
called connectednesses and disconnectednesses
The following examples were studied in
Example Let k b e the Kuratowski closure op erator The sub category r k isthe
class of Hausdor spaces and k is the class of irreducible spaces A top ological space
X is irreducible if for U V X open sets and U V U or V
The class C on of connected spaces is not the nabla sub category of the usual closure
op erator k but as we will see is a nabla sub category
Example Let conn be the connected component closure operator de ned by
S
x where comp x is the connected comp onent of x The conn M comp
X
x M
X X
nabla sub category of conn is the sub category of connected spaces We do not know if
the connected comp onent closure op erator is the coregular closure op erator of C on
Example The path connected component closure operator de ned like the con
nected comp onent closure op erator is the coregular closure op erator de ned by the unit
interval Its nabla sub category is the sub category of path connected spaces
Belowwe outline the b ehaviour of some relevant coregular closure op erators and the
resp ective r sub categories From this study it turns out that r sub categories cover a
muchricher range of sub categories than the connectednesses We fo cus our study in the
least and largest of these closure op erators and sub categories
We will denote by D E and S the discrete space with two p oints and the
indiscrete space with twopoints and the Sierpinski space resp ectively in disc denotes
the in discrete closure op erator
The discrete closure op erator is obviously the coregular closure induced by the sub
category S gl The indiscrete closure op erator is also coregular as we show next
We remark that in T op each nabla sub category is closed under continuous images
Prop osition For a subcategory A of T op closed under images the fol lowing con
ditions are equivalent
A
i r coreg T op
A
ii coreg indisc
iii D A
Proof i ii Obvious
A A
ii iii If coreg indisc then coreg D So there is a continuous map
D
A and h b c for two distinct h A D with A A h a a for all a
points b c of A For g A A de ned by g x b x the function h g is continuous
and h g A D So D is in A b ecause A is closed under images
iii i Let X b e a top ological space For x y X X one de nes h D X
with h and h x y The function h is continuous b ecause its domain
D X
A A
and so X r coreg is a discrete space Hence coreg
X
X
Corollary If A is a nabla subcategory and A T op then A Con
Proof Every nabla sub category containing a disconnected space must contain D since
it is closed under images
Prop osition Let X be a topological space and M X Then
E
coreg M fx X j y M k x k y g
X
E
Proof Let x be an element of coreg M There is f E E X with f x
X
and ff f g M Since E E is indiscrete and f continuous f E E is
indiscrete and so k x k f
Conversely if for x X exists y M such that k x k y then the function
f E E X de ned by f f y and f f x is
continuous
E
Corollary The nabla subcategory inducedby coreg is the subcategory of indiscrete
spaces
E
Corollary If X is a T space and M X then coreg M M
X
Prop osition Let X Top and M X Then
S
coreg M fx X j z w M z k x and x k w g
X
S
Proof Let c coreg and x be an element of c M There is f S S X with
f x f w and f z z w M
From k S S we know that k and by continuity
of f f x k w In the same way k f g implies that
k and nally that z k x
Conversely we have z k x and x k w with z w M and x X and we want
prove that x c M So it is enough to show that the function f S S X with
f f x f w and f z is continuous Let F X b e a closed
if w F x F e z F
S S if w F x F z F
f F
S Snf g if w F e x F z F
F e z F f g if w F x
Since the inverse image of a closed set is closed then f is continuous and the pro of is
complete
S
From the de nition of coreg we may conclude immediately the following results
S
Corollary If X is a T space and M X then coreg M M
X
S
Corollary A space X belongs to r coreg if and only if
x y X z w X z k x k y and fx y g k w
From the results ab ove we have the fol lowing chain of coregular closure operators
S gl E S C on D
disc coreg coreg coreg coreg coreg indisc
Moreover if c is a coregular closure operator di erent from these then
S C on
coreg c coreg
A E
In fact if c coreg with c disc and c coreg then there exists X A such that
X has a non trivial op en set because X can not be a singleton space or an indiscrete
A
space Since nabla sub categories are closed under images s r coreg which implies
S A
that coreg coreg
Note that the trivial closure operator is not a coregular closure operator
E S
Since coreg and coreg are discrete in T spaces and in T spaces resp ectively
we could conjecture that the next coregular closure op erator would be the largest one
r k
discrete in T spaces coreg but that is not true On the contrary there are plenty
of coregular closure op erators
X b e a top ological space X T where X has cardinal For an in nite cardinal let
and T is the co nite top ology
Prop osition If and are two in nite cardinals and then
X X
coreg coreg
X X
which implies that coreg Proof First we will prove that X r coreg
X
Let h X X be acontinuous map Then X coreg h x with h x
x X
a closed set for each x But we know that X is not the union of nite sets b ecause
And so one of the sets h x has to b e equal to X and then h is constant
X X
Hence X r fX g coreg and therefore it cannot belong to r coreg
X X
since only the singleton spaces and the empty space are in coreg r coreg
X
X X
Next we will prove that coreg coreg If x coreg M n M for a subspace
Y
M of Y then there is h X X Y such that h M and h a b x
X
for a b in X Now let us consider a subspace X of X such that a b X Then
X
X X and so x coreg M x hj
X X
Y
The construction of the co nite top ology can b e generalized In fact for two in nite
cardinals we de ne the top ological space X where the cardinal of X is and
A X is closed if its cardinal is less than or A X For the top ology
de ned this way is the co nite one
Prop osition Let and be in nite cardinals If then
X X
coreg coreg
X
X
coreg coreg
Proof If then the identity map f X X is continuous therefore
X X
b ecause the nabla sub categories are closed under images coreg coreg
Next we will show that X r fX g Let g X X be a continuous map
If jg X j then g X has a prop er subset F of cardinal larger or equal to But
jg F j and so jF j jg F j This implies that jg X j and so g X
is a discrete subspace of X A discrete space which is image of X is a singleton
X X X
In conclusion X r coreg and then coreg coreg
The pro of of is similar to the case
S r k
Corollary Between coreg and coreg there is a proper class of
coregular closure operators
A r k
Remark For A fX j is an in nite cardinalg coreg coreg We do not
know if they are equal
Coregular closure op erators in Mod
R
Let Mod be the category of R mo dules with its surjective homomorphisms injective
R
homomorphisms factorization system i e epi mono factorization So in this case
a sub ob ject is up to isomorphism a submo dule
De nition A preradical r in Mod is a subfunctor of the identity functor in Mod
R R
that is r Mod Mod is a map such that r M is a submo dule of M and
R R
f r M r f M for each M N Mod and each homomorphism f M N
R
A preradical r is idempotent if r r M r M for every M Mod and it is a
R
r adical if r M r M O for every M Mod
R
To each preradical r a torsion free subcategory F fM r M Og and a torsion
r
subcategory T fM r M M g are asso ciated
r
Preradicals and closure op erators in Mod are closely connected each closure op
R
erator induces a preradical r by r M c O on the other hand each preradical
M
r
r
de nes in a natural waytwo closure op erators min and max the least and the largest
one such that c O r M for every R mo dule M They are called the minimal and
M
the maximal closure operators resp ectively and de ned by
r
min N N r M
M
r
max N r M N
M
where N is a submo dule of M and M M N is the canonical pro jection
The next results are partially in
Prop osition Let r be a preradical in Mod Then
R
r
r
r min r max T
r
r
r
min max F
r
r r
r r
Proof We already knowthatr min r max b ecause min max Amodule
r r r
N is in r max if and only if max N N The equality max N N
N N
N N
N N N and consequently r N N means that r N
N N N
r
N N b ecause is surjective By the isomorphism N N N N r max
N N
if and only if N T
r
r
At last we show that T r min If N T then r N N A preradical is
r r
nitely pro ductive and so
r
min r N N r N r N N N N N
N N N N
N
r
r
The pro of of F max min is similar to the rst part of the pro of of
r
r
Toshow the remaining inclusion if N is in min then r N N and
N N
consequently r N r N From this fact wehave that r N is a singleton and so
N
r N O
Corollary If c is a closure operator in Mod and r is the preradical induced by c
R
then
r c T e c F
r r
From this result wehave that the torsion sub categories and the nabla sub categories
are exactly the same and at the same time the free torsion and the delta sub categories
coincide
Now we investigate the co regular closure op erators in Mod
R
r F
r
max Prop osition If r is a radical then reg
c r
Proof It is true in general that reg c In particular for c max we know
r F r
r
that max F from Prop osition and so reg max To pro of the other
r
F
r
O r M From the former inequality we inequality is enough to show that reg
M
F
r
r
O reg O In Mod the regular closure op erator may be have r M max
R
M M
computed by
F
r
reg O fker f j f M X X F g
r
M
The quotient mo dule M r M is in F b ecause r is a radical Hence for
r
F
r
O ker r M M M r M the canonical homomorphism we have reg
M
r
T
r
Prop osition If r is an idempotent preradical then cor eg min
r
T
r
Proof That cor eg min may be concluded analogously to the preceding prop osi
r
T
r
O r M cor eg O By de nition of coregular tion We only have to show min
M
closure op erator we have
T
r
coreg O fh X j h X M X T and h Og
r X
M
Let g r M r M M be the homomorphism de ned by g x y x y Since
r M T b ecause r is idemp otent g O and g r M r M r M we
r r M
T
r
conclude that r M coreg O as claimed
M
Since every torsion free torsion sub categoriy of Mod is induced by a radical idem
R
p otent preradical cf and every delta nabla sub category is torsion free torsion
we have
Theorem Let c be a closure operator in Mod
R
r
c is a regular closure operator if and only if c max for a unique radical r
r
c is a coregular closure operator if and only if c min for a unique idempotent
preradical r
Wepoint out that in the radical idemp otent preradical is unique b ecause there
is a one to one corresp ondence between maximal minimal closure op erators and pre
radicals Hence from the results ab ove it follows that there is a one to one corresp on
dence between the conglomerates of regular closure op erators radicals and torsion free
sub categories as well as a one to one corresp ondence between coregular closure op era
tors idemp otent preradicals and torsion sub categories in Mod
R
In it is stated that every sub category A of Mod induces a preradical t de ned
R A
by
t M fkerf j f M A A Ag
A
A
which is exactly the preradical asso ciated to reg
For a sub category A of Mod we de ne a preradical s by
R A
A
s M coreg O
A
M
Prop osition Let A be a subcategory of Mod and r be a preradical of Mod
R R
Then
A F and r t
t F
r
A
A T and r s
s T
r
A
The pro of follows directly from the de nitions
Prop osition For every subcategory A of Mod we have
R
t is a radical
A
s is an idempotent preradical
A
Proof Since every regular closure op erator is maximal and by we know that
a maximal closure op erator is idemp otent if and only if the preradical it induces is a
radical the preradical t is a radical for every sub category A of Mod
A R
For we use a similar result of which says that a minimal closure op erator is
weakly hereditary if and only if it induces an idemp otent preradical
Prop osition Let A be a subcategory of Mod Then
R
F r A fM Mod j A A f A M f A Og
s R
A
T l A fM Mod j A A g M A g M Og
t R
A
Proof Let X be in r A so that for every homomorphism f A X with A A
O we have f A
Let h A X be a homomorphism with A A If we de ne f f A X by
f a h a and f b h b then f A f A O which implies h A A
A
O From this fact we have that coreg O s X O and so X F
A s
X
A
O we Conversely if X F then for all h A X with A A and h
s A
A
have h A O
Let f A X be a homomorphism with A A and consider g A A X
O g A O and consequently f is de ned by g a b f a f b Since g
A
constant
Analogously for the left constant sub categories
Corollary For every subcategory A of Mod
R
A
r A coreg
A
l A r reg
Proof From the preceding prop osition r A F and by Corollary F c
s s
A A
for every closure op erator c such that c O s M for M Mod In particular
M A R
A
coreg F
s
A
The pro of of is similar
If r is an idemp otent radical then the pair T F is a torsion theory in sense of
r r
The torsion and torsion free sub categories of a torsion theory are the left and the right
constant sub categories resp ectively Each pair l A rlA determines an idemp otent
radical r such that T l A and F rlA This idemp otent radical is exactly the one
r r
rlA l A
induced by reg and by coreg
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