What is a Quotient Map with Respect to a Closure Operator?

Applied Categorical Structures9: 139–151, 2001.© 2001Kluwer Academic Publishers. Printed in the Netherlands.

139

What is a Quotient Map with Respect to a ClosureOperator?

Dedicated to our friend Horst Herrlich on the occasion of his sixtieth birthday

MARIA MANUEL CLEMENTINODepartamento de Matemática, Apartado 3008, 3000 Coimbra, Portugal. e-mail: [email protected]

ERALDO GIULIDip. di Matematica Pura ed Applicata, Università degli Studi di L’Aquila, 67100 L’Aquila, Italy.e-mail: [email protected]

WALTER THOLENDepartment of Mathematics and Statistics, York University, Toronto, Canada M3J 1P3. e-mail:[email protected]

(Received: 26 February 1998; accepted: 8 September 1998)

Abstract. It is shown that there is no good answer to the question of the title, even if we restrictour attention toSet-based topological categories. Although very closely related, neither the naturalnotion ofc-finality (designed in total analogy toc-initiality) nor the notion ofc-quotient (modelledafter the behaviour of topological quotient maps) provide universally satisfactory concepts. Moredramatically, in the categoryTop with its natural Kuratowski closure operatork, the class ofk-finalmaps cannot be described as the class ofc-quotient maps for any closure operatorc, and the class ofk-quotients cannot be described as the class ofc-final maps for anyc. We also discuss the behaviourof c-final maps under crossing with an identity map, as in Whitehead’s Theorem. InTop, this givesa new stability theorem for hereditary quotient maps.

Mathematics Subject Classifications (2000):18B30, 54B30, 54C10, 18A20.

Key words: closure operator,c-initial morphism,c-final morphism,c-quotient, Whitehead’s Theo-rem.

1. Introduction

In topology, as in other parts of mathematics, a fundamental device for obtain-ing new structures from old is the formation of quotients. In the categoryT opof topological spaces, quotient maps are described as regular epimorphisms, thatis: as coequalizers of their kernel-pairs. When viewed as a topological categoryover Set (cf. [12, 14]), quotient maps are simply surjections which are final (=co-cartesian) with respect to the forgetful functorT op → Set . Finally, if wethink of T op as a category equipped with a closure operator, as given by the usualKuratowski closure, then we may want to characterize a quotient mapp: X → Y

140 MARIA MANUEL CLEMENTINO ET AL.

as an epimorphism which reflects closedness:B ⊆ Y is closed wheneverp−1(B)

is closed. Is this the ‘right’ general concept?Certainly not. In the topological categorySGph of spatial (= loop-free) graphs

with its natural ‘up-closure’ (which coincides with the Kuratowski closure whenSGph is coreflectively embedded into the category ofCech-pretopological spaces– see [7]), the corresponding notion of quotient no longer characterizes the naturalquotient maps, as given by the regular epimorphisms or the final structures withrespect toSGph→ Set .

On the other hand, there is a natural way of describing quotient maps inSGphvia its natural closure operator↑, namely as↑-final maps. In general,c-final mor-phisms are defined algebraically in total analogy toc-initial morphisms (as in-troduced in [4]). They interact perfectly withc-closed andc-open morphisms viapullback (cf. [11]). However, whilec-initial, c-closed andc-open morphisms de-scribe the ‘right’ maps inT op with c = k the Kuratowski closure,c-finality doesnot characterize quotient maps but hereditary quotient maps. The natural questionthen, whether hereditary quotient maps inT op are describable as thec-final mapsfor any closure operatorc, is answered to the negative (Section 3). Conversely,when definingc-quotients as the ‘surjective’ morphisms reflectingc-closedness forsubobjects, as inT op, then in this way one is not able to characterize the hereditaryquotient maps ofT op (Section 5).

Despite these strikingly negative results, we have a few positive assertions infavour of the notion ofc-finality, in addition to the smooth pullback results givenin [11]. Among these is a general closure-theoretic theorem which, when applied toT op andk, gives the following Whitehead-type result: a mapf : X→ Y for which1Z×f : Z×X→ Z×Y is a hereditary quotient map for every compact HausdorffspaceZ, has this property for every topological spaceZ. We also show that thereis a general concept of localc-compactness which is stable under the formationof c-closed subobjects and ofc-perfectc-final morphisms (Section 7). Finally, ourconsideration of algebraic categories, such as the category ofR-modules with itsnatural closure operators given by preradicals, yields further evidence thatc-finalityis an important and natural categorical notion.

2. Initial and Final Morphisms

2.1. Throughout the paper, we work in a finitely-complete categoryX with aproper and pullback-stable(E,M)-factorization system for morphisms (see, forexample, [1, 3, 7, 10]). For every objectX ∈ X, morphisms inM with codomainX are referred to assubobjectsof X; they are the objects of the full subcategorysubX of X/X, which in fact is a preordered class with finite infima. We use theusual lattice-theoretic notation in subX. Every morphismf : X → Y gives theimage-preimage adjunction

f (−) a f −1(−): subY → subX.

WHAT IS A QUOTIENT MAP WITH RESPECT TO A CLOSURE OPERATOR? 141

Hence, for allm ∈ subX andn ∈ subY , one has the inequalities

(1) m ≤ f −1(f (m)) and (2) f (f −1(n)) ≤ nwith ‘∼=’ holding true in (1) in casef is monic (this is easy to see forf ∈ M; fora proof in casef is any monomorphism, see [7]), and with ‘∼=’ holding true in (2)if and only if f ∈ E .

2.2. (Cf. [5, 7]) A closure operator c of X with respect to(E,M) is givenby a family of functionscX: subX → subX (X ∈ X) such thatm ≤ cX(m),cX(m) ≤ cX(n) if m ≤ n, andf (cX(m)) ≤ cY (f (m)) holds for allf : X → Y inX andm,n ∈ subX; one refers to the last inequality as thec-continuity conditionfor f . A subobjectm ∈ subX is c-closed if m ∼= cX(m), and it is c-denseifcX(m) ∼= 1X. A morphismf : X→ Y is c-denseif f (1X) ∈ subY is c-dense. Theoperatorc is calledidempotentif cX(m) is c-closed for allm ∈ subX, X ∈ X,and it isweakly hereditaryif k with cX(m) · k = m is c-dense for allm ∈ subX,X ∈ X. Finally, c is hereditaryif for all m: M → Y , y: Y → X in M one hascY (m) ∼= y−1(cX(y · m)). At times we omit the subscripts if there is no danger ofconfusion.

2.3. Given a closure operatorc of X w.r.t. (E,M), a morphismf : X → Y iscalledc-initial (cf. [4, 7]) if

cX(m) ∼= f −1(cY (f (m)))

for all m ∈ subX (with ‘≤’ holding true for free, by thec-continuity condition andimage-preimage adjunction); andf is calledc-final if

cY (n) ∼= f (cX(f −1(n)))

for all n ∈ subY (with ‘≥’ being satisfied automatically). We note the followingimportant facts:

(1) if e · m = 1 in X, thenm is a c-initial morphism inM and e is a c-finalmorphism inE (see [7], Ex. 2.L, and [11]);

(2) every morphism inM is c-initial if and only if c is hereditary;(3) in view of (2), we callc cohereditaryif and only if every morphism inE is

c-final;(4) c-final morphisms necessarily belong toE (just considern = 1Y in the defin-

ing property), butc-initial morphisms need not be inM;(5) a c-final monomorphism is automaticallyc-initial and lies inE , but may fail

to be an isomorphism – just consider the discrete closure operator in the cate-gory T op of topological spaces with its (surjective, embedding)-factorizationstructure.


2.4. The following composition-cancellation rules hold true; proofs for the state-ments onc-initiality may be found in [7] (9.2), and the statements onc-finality areproved similarly:

(1) If each of the composable morphismsf andg arec-initial (c-final), theng · fis c-initial (c-final).

(2) If g · f is c-initial, thenf is c-initial, and alsog is c-initial provided thatfbelongs toE .

(3) If g · f is c-final, theng is c-final, and alsof is c-final provided thatg ismonic.

2.5. Consider a pullback diagram

U

p′

f ′V

p

Xf

Y

(1)

in X. The following proposition and its corollary were proved in [11]:

PROPOSITION. (1)If f is c-final andp′ c-initial, thenf ′ is c-final.(2) If f ′ is c-initial and p c-final, thenf is c-initial.

COROLLARY. (1) Let c be hereditary. Then, for everyp: V → Y in M, therestrictionf −1(V )→ V of thec-final morphismf is c-final.

(2) Let c be cohereditary. Then, for everyp: V → Y in E , if the pullbackf ′: U → V of f is c-initial, also f is c-initial.

2.6. As a first example (of a topological category overSet), we consider thecategorySGph of spatial graphs (cf. [7]): objects are sets with a binary reflexiverelation→, and morphisms preserve the relation. With respect to its (surjective,embedding)-factorization structure, consider the up-closure

↑X (M) = {x ∈ X | ∃a ∈ M : a→ x}.↑-initial morphismsf : X → Y are characterized by the condition(x1 → x2 ⇔f (x1) → f (x2)) for all x1, x2 ∈ X (hence these are precisely the initial mapsw.r.t. the forgetful functorSGph→ Set), and the↑-final morphisms are preciselythe natural quotient maps, that is: the surjectionsf : X → Y with (y1 → y2 ⇔∃x1, x2 ∈ X: x1→ x2 andf (xi) = yi) (which are also described as the final mapsw.r.t. SGph→ Set).

We remark that↑-final morphisms do not coincide withc-final morphisms forc the idempotent hull of↑. Indeed,↑-final andc-final morphisms cannot even becompared: consider the set{0,1,2} as a graphY , with the graph structure given


by the natural order, and consider the graphX arising fromY by removing theedge 0→ 2, and the identity maph: X → Y ; h is c-final but it is not↑-final. Onthe other hand, ifZ is the graph{0,1,1′,2} with the nontrivial edges 0→ 1 and1′ → 2, the mapk: Z→ X, with k(0) = 0, k(1) = k(1′) = 1, k(2) = 2, is↑-finalbut it is notc-final.

Finally we note that the operator↑ is hereditary, but neither idempotent norcohereditary. Next we will show that failure of cohereditariness was to be expecteda priori.

2.7. LetX be any topological category overSet with its (surjective, embedding)-factorization structure (see [1, 12, 14]). We consider the discrete operatord (withdX(M) = M), the indiscrete operatorj (with jX(∅) = ∅ and jX(M) = X forM 6= ∅), and the trivial operatort (with tX(M) = X); then:

PROPOSITION.An arbitrary closure operatorc of a topological categoryX overSet is cohereditary if and only ifc ∈ {d, j, t}.

Proof. The operatorsd, j , t are obviously cohereditary. Conversely, considera cohereditary closure operatorc, so that all surjective morphisms arec-final. Weassumec 6= j , c 6= t and showc = d. With D the two-point discrete object inX(obtained by applying the left adjoint ofX→ Set to {0,1}), we first observe:

(1) cD = dD; indeed, assuming the opposite and considering the ‘switch’ mapD→ D, we would either havecD(∅) 6= ∅ and obtaincD = tD, or cD(∅) = ∅,cD({0}) = D and obtaincD = jD, hencecD ≥ jD in any case; this would im-ply cX(M) ⊇ jX(M) for all M ⊆ X ∈ X, as one easily sees when exploitingc-continuity of the mapsfx : D → X with fx(0) ∈ M andfx(1) = x 6∈ M,and thereforec ∈ {j, t}, in contradiction to our assumption.

Next one considers the two-point indiscrete objectJ in X (applying the rightadjoint ofX→ Set to {0,1}) and shows:

(2) cJ = dJ ; this follows immediately from the fact that the identity mapD→ J

must bec-final, by hypothesis.

Now considering mapsX→ J one easily showscX = dX for all X ∈ X. 22.8. In the categoryT op of topological spaces with its (surjective, embedding)-factorization structure andcX(M) = kX(M) = M the usual (Kuratowski-) closureof M ⊆ X ∈ T op, thek-initial mapsf : X → Y are the ones for whichX carriesthe initial topology with respect tof (i.e. A ⊆ X is closed iffA = f −1(B) forsomeB ⊆ Y closed). Thek-final maps are precisely thehereditary quotient maps(cf. [2, 9, 13]), i.e. those surjective mapsf : X → Y for which every subspaceV ⊆ Y carries the final topology w.r.t. the restrictionf −1(V ) → V of f (so thatB ⊆ V is closed ifff −1(B) ⊆ f −1(V ) is closed).


We note that the operatork is idempotent and hereditary but (of course) notcohereditary; more significantly, in general,quotient maps fail to bek-final. Infact, it is well known that for a hereditary quotient mapg: X → Y in T op and aspaceZ, the map

f ′ = 1Z × g: Z ×X −→ Z × Y

need not be a quotient map. For example, consider forg the projectionR → Y ,whereY arises fromR by identifying all the integers, and letZ := R\{12, 1

3, . . .}(cf. [9], 2.4.20). Butf ′ is the restriction off = 1R × g along the embeddingZ × Y → R × Y , and, by Whitehead’s Theorem,f is a quotient map. This alsoshows thatc-initiality of p′ is an essential condition in Proposition 2.5(1). Thatc-finality of p is an essential condition in Proposition 2.5(2) is much easier to see (interms ofX = T op, c = k: restrict a bijective map, which is not a homeomorphism,to a single point).

2.9. In the categoryR-Mod of (left-)R-modules with its (epi,mono)-factorizationstructure, every closure operatorc induces apreradical r (that is: a subfunctor ofthe identity functor), namelyrX = cX(0) ≤ X ∈ R-Mod. Conversely, given anyr ,there is a least closure operator minr and a largest closure operator maxr inducingr , namely

minrX(M) = rX +M and maxrX(M) = π−1(r(X/M)),

with π the projectionX→ X/M for M ≤ X ∈ R-Mod.The minr -initial morphisms are thosef : X→ Y with rX = f −1(rY ), and the

minr -final morphismsf : X→ Y are the epimorphisms withf (rX) = rY . Hence,to say thatr is hereditary(which meansrM = M ∩ rX for M ≤ X ∈ R-Mod) isthe same as to say that every monomorphism is minr -initial; it is also equivalent tothe hereditariness of minr , or of maxr (see [7], 3.4). Similarly,cohereditarinessof r(that is:r(X/M) ∼= (M+ rX)/M forM ≤ X ∈ R-Mod) is equivalently describedby the property that every epimorphism is minr -final, i.e. by the cohereditariness ofthe closure operator minr (cf. 2.3(3)); it is also equivalent to the statement minr =maxr (see [7], 5.12).

The maxr -initial morphismsf : X → Y are characterized by the property thatfor all M ≤ X, the induced morphismX/M → Y/f (M) is minr -initial, and everyepimorphism is maxr -final. We observe that maxr -initial morphisms are minr -initial, but not conversely, in general. Indeed, forR = Z andr the torsion radical,consider the additionf : Z×Z→ Z and letM be the diagonal inZ×Z; thenf isminr -initial althoughZ× Z/M → Z/f (M) is not, since its domain is isomorphicto Z and its codomain is isomorphic toZ2. By contrast, minr -final morphisms are(as epimorphisms) trivially maxr -final, while the converse proposition generallyfails (just consider any projection ofZ onto a finite group forr as above).


We recall that minr is idempotent, and that maxr is idempotent iffr is a rad-ical (i.e. r(X/rX) = 0); by contrast, idempotency ofr (i.e. r(rX) = rX) ischaracterized by weak hereditariness of minr or, equivalently, of maxr (cf. [7],Prop. 3.4).

3. c-Finality does not Describe Topological Quotients

3.1. Since quotient maps in the topological categorySGph are described as the↑-final maps while inT op quotient maps fail to bek-final (cf. 2.6, 2.8), it is natural toask whether there is any closure operatorc of T op such that thec-final morphismsare precisely the quotient maps. Despite the rich supply of closure operators in thecategoryT op, the answer is negative:

THEOREM. The categoryT op has no closure operatorc w.r.t. its (surjective,embedding)-factorization structure, such that thec-final maps are exactly the quo-tient maps.

For the proof we first recall a general fact on closure operators in topologicalcategories overSet (see [7], Theorem 9.1):

3.2. PROPOSITION.If the idempotent hullc of a closure operatorc of a topolog-ical categoryX overSet is hereditary, then alreadyc must be idempotent.

3.3. Proof of the theorem.We assume that there is a closure operatorc suchthat the class ofc-final maps is the class of quotient maps. Thenc 6∈ {d, j, t} (cf.2.7) since otherwise every surjection would bec-final. Now consider the SierpinskispaceS = {0,1} whose only nontrivial closed subset is{1}. Then:

(1) cS 6= jS , since otherwise the identity mapS → J (with J the two-pointindiscrete space) would bec-final;

(2) cS 6= dS , since otherwise the identity mapD → S (with D the two-pointdiscrete space) would bec-final. From (1) and (2) one concludes:

(3) cS = kS (with k denoting the usual closure), sincecS({0}) = S. In fact, if{0} werec-closed, then we would havec ≤ k∗, with k∗ denoting theinverseKuratowski closure, given byk∗X(M) =

⋂ {U |U ⊆ X open,M ⊆ U } = {x ∈X | {x} ∩ M 6= ∅} (cf. [7]); then c would have to coincide with the discreteclosured when restricted to the categoryT op1 of T 1-spaces, so that everysurjection inT op1 would have to be a quotient map. We can now conclude:

(4) cX ≤ kX for everyX ∈ T op. For that it suffices to show that every (k-)closedsetF ⊆ X is c-closed (since thencX(M) ⊆ cX(M) = M = kX(M) forall M ⊆ X). In fact, for a closed setF ⊆ X, we have a continuous mapf : X → S with F = f −1(1), so thatF is c-closed as a pullback of thec-closed set{1}. Next we show:


(5) c = k. In view of (4), for that it suffices to show that everyc-closed subsetG ⊆X ∈ T op is (k-)closed. ForG ⊆ X c-closed andJ the two-point indiscretespace, consider the mapg: X → J with G = g−1(1), and its (quotient,mono)-factorizationg = m · q, wherem: Y → J is an inclusion.c-finality ofq impliescY (1) = 1, and thenY = D or Y = S andcY = kY . In both cases,the setG, as an inverse image of ak-closed subobject, must bek-closed. From(4) we conclude with 3.2:

(6) c = k. But this contradicts our initial assumption that quotient maps arec-final, sincek-final maps are hereditary quotient maps: see 2.8. This concludesthe proof of Theorem 3.1. 2

4. c-Quotient Morphisms

4.1. The characterization ofk-initial morphisms inT op (see 2.8) can be given inour abstract categorical context 2.1, provided that the closure operatorc of X w.r.t.(E,M) is idempotent:

PROPOSITION. Let c be idempotent. Thenf : X → Y is c-initial if and onlyif everyc-closed subobjecta of X is of the forma ∼= f −1(b) for somec-closedsubobjectb of Y .

Proof.The condition is trivially necessary. Conversely, for everym ∈ subX, weknow thatc(m) is of the formf −1(b) with b ∼= c(b) ∈ subY . Hencef (c(m)) ≤b, evenc(f (m)) ≤ c(f (c(m))) ≤ c(b) ∼= b. Consequently,f −1(c(f (m))) ≤f −1(b) ∼= c(m), as desired. 24.2. Of course, idempotency is an essential assumption in 4.1. For instance,the graph morphismh considered in 2.6 is not↑-initial although it satisfies thecondition of Proposition 4.1.

4.3. The attempt to establish an analogous characterization ofc-final morphisms(as is suggested byT op andk) is only partially successful:

PROPOSITION.Letf : X→ Y bec-final. Then a subobjectb of Y is c-closed ifand only iff −1(b) is c-closed inX.

Proof. In fact, if f −1(b) is c-closed, then

cY (b) = f (cX(f −1(b))) ∼= f (f −1(b)) ≤ b.The ‘only if’ part is well-known. 24.4. We call a morphismf : X→ Y in X a c-quotient if it lies in E and reflectsc-closed subobjects, so thatb is c-closed inY wheneverf −1(b) is c-closed inX.Then one may observe immediately:


(1) A c-final map is ac-quotient, with the converse statement failing already forX = T op andc = k (cf. 4.3 and 2.8).

(2) Since the notion ofc-quotient depends only onc-closed subobjects (not on theoperatorc itself), it is no loss of generality to requirec to be idempotent in thiscontext, in contrast with the situation ofc-finality (see 2.6).

(3) If each of the composable morphismsf andg arec-quotients, theng · f isalso ac-quotient.

(4) If g·f is ac-quotient, theng is ac-quotient, and alsof is ac-quotient providedthatg is monic.

(5) The property 2.5(1) on the stability under pullback fails if we replacec-final byc-quotient: considerk in T op. Also 2.5(2) does not hold if one assumesp to bea c-quotient instead of beingc-final: consider again, inSGph, the morphismh: X→ Y defined in 2.6. The pullback ofh along itself is obviously↑-initial,althoughh is ↑-quotient but not↑-initial.

4.5. For a preradicalr of R-Mod as in 2.9 and the closure operators minr andmaxr , one shows quite easily that a minr -quotientf : X → Y is already minr -final; indeed,N ≤ Y is minr -closed iff rY ≤ N , so that it suffices to considerN = f (rX) to see thatf (rX) = rY holds. Since every epimorphism is maxr -final, it is in particular a maxr -quotient. Hence, for the two closure operators inquestion, the notionsc-final andc-quotient are equivalent.

5. c-Quotients do not Describe Hereditary Quotient Maps inT op

5.1. In 3.1 we showed that the class ofk-quotient maps inT op cannot be repre-sented as the class ofc-final maps, for anyc. Now we show that the class ofk-finalmaps cannot be represented as the class ofc-quotient maps, for anyc:

THEOREM. The categoryT op has no closure operatorc such that thec-quotientmaps are precisely the hereditary quotient maps.

Proof. Replacingc-quotient byc-final in the proof of Theorem 3.1, we mayuse exactly the same arguments, since we only deal with quotient maps whosecodomain has two points – and therefore they are also hereditary quotient maps–, to conclude thatc = k. But k-quotient maps do not coincide with hereditaryquotient maps, and the result follows. 2

6. Pullbacks ofc-Final Morphisms

6.1. Proposition 2.5(1) gives a sufficient condition for the pullback of ac-finalmap to bec-final. Unfortunately, the given condition (namely, that in the pullbackdiagram (1) the morphismp′ be c-final) is quite restrictive; in this section we


give sufficient conditions on the objects involved. First we recall that the pullbackdiagram (1) decomposes into two pullback diagrams

U

〈f ′,p′〉

f ′V

〈1V ,p〉

V ×X 1V×fV × Y

Xf

Y

(2)

Since〈1V , p〉 ∈ M, f ′ is c-final if 1V × f is, and ifc is hereditary. Therefore, inwhat follows,we assumec to be hereditaryand consider the problem for whichobjectsZ is 1Z × f c-final wheneverf is c-final.

6.2. Recall (cf. [5, 11]) that a morphismf : X → Y is c-closedif f (cX(m)) ∼=cY (f (m)) for all m ∈ subX. The morphismf is stablyc-closed(stablyc-final)if every pullback off is c-closed (c-final). Note that iff represents ac-closedsubobject,f is actually a stablyc-closed morphism inM; and if f is (stably)c-closed inE , thenf is necessarily (stably)c-final (cf. [11]).

An objectX in X is calledc-Hausdorff (c-compact, resp.) if the morphismδX: X → X × X (!X: X → 1, resp.) is stablyc-closed; in other words ifδX ∈sub(X × X) is c-closed (the projectionX × Y → Y is c-closed for allY ∈ X,resp.). Hence, a stablyc-closed morphismf : X → Y may be described as ac-compact object of the comma-categoryX/Y , andf is c-perfectiff it is c-compactc-Hausdorff inX/Y ; cf. [3].

6.3. For the remainder of this section, we assume that for everyc-HausdorffobjectX there is ac-dense reflexion

βX: X −→ βX

into the full subcategory ofc-Hausdorffc-compact objects ofX. As in [3], we callX c-Tychonoffif X is c-Hausdorff withβX ∈ M. We recall that theHenriksen–Isbell Theoremholds true in this context, which we shall use later on:

PROPOSITION. A morphismf : X → Y with c-Tychonoff objectsX and Y isc-perfect(or stablyc-closed) if and only if

X

f

βXβX

βf

YβY

βY

(3)

is a pullback diagram.


6.4. The following proposition shows in particular thatc-final morphisms withgood pullback behaviour w.r.t.c-compactc-Hausdorff objects behave well w.r.t.c-Tychonoff objects.

PROPOSITION. Let f : X → Y have the property that for everyc-compactc-Hausdorff objectZ, the morphism1Z × f is c-final. Then this property is ac-tually satisfied for every objectZ for which there exists a stablyc-final morphismq: W → Z withW a c-Tychonoff object.

Proof. For every morphismh: U → V we have two consecutive pullbackdiagrams, with the unnamed arrows being projections:

U ×X1U×f

h×1XV ×X

1V×fX

f

U × Y h×1YV × Y Y

(4)

If Z is c-Tychonoff, thenh := βZ: Z → V := βZ and alsoh× 1X lie in M andarec-initial. Since, by hypothesis, 1V × f is c-final, an application of 2.5(1) to theleft pullback of (4) gives that 1Z × f is c-final.

More generally, now letZ be as in the theorem and chooseh := q: W → Z.Thenh×1Y is c-final by hypothesis onq, and 1W×f is c-final as shown previously.An application of the composition-cancellation rules 2.4(1), (3) to the left part of(4) gives that 1Z × f is c-final, as claimed. 26.5. ForX = T op andc = k, we can strengthen the assertion of 6.4 even further:

THEOREM. A map which is a hereditary quotient map when crossed with theidentity map of a compact Hausdorff space is actually a hereditary quotient mapwhen crossed with the identity map of any topological space.

Proof.Following [8] (Lemma 2.4), we can express any spaceX as a hereditaryquotient of a Tychonoff space: for eachx ∈ X, letXx be the space with underlyingsetX and with all points butx isolated; forx one keeps the same neighbourhoodsas inX; the topological sum

∑y∈X Xy is easily seen to be a Tychonoff space, and

the map

p:∑

y∈X Xy −→ X

(x, y) 7−→ x

is obviously a hereditary quotient map. Now the result follows from Proposi-tion 6.4. 26.6. REMARK. We point out that the hypothesis of the theorem is not satisfied foreveryhereditary quotient map: consider again the hereditary quotient mapf : R→Y of 2.8; then, for the unit intervalI , the map 1I×f fails to be a hereditary quotientmap since, withZ := I\{1/n | n ∈ N}, 1Z × f fails to be a quotient map.


7. A Note on Local Compactness

7.1. Finally we indicate that the setting of Section 6 allows, to a certain degree, fora categorical treatment of local compactness. Recall (cf. [7, 11]) that a morphismf : X→ Y is c-openif f −1(cY (n)) ∼= cX(f −1(n)) for all n ∈ subY .

DEFINITION. An objectX is calledlocally c-compactif X is c-Tychonoff withβX: X → βX c-open, andX is a stablec-spaceif X is c-Hausdorff and thereexists a locallyc-compact objectY and a stablyc-final morphismq: Y → X.

We note that forX = T op and c = k the Kuratowski closure operator thenotion of localc-compactness takes on the classical meaning of locally compactHausdorff space, while stablek-spaces form a proper subclass ofk-spaces.

7.2. Localc-compactness has the following closure properties w.r.t. morphisms:

PROPOSITION. Let f : X → Y be a stablyc-closed morphism ofc-TychonoffobjectsX andY .

(1) If Y is locally c-compact andf ∈M, thenX is locally c-compact.(2) If X is locally c-compact andf ∈ E , thenY is locally c-compact.

Proof. Consider the pullback diagram (3). According to Theorem 4.3 of [11],βX is c-open whenβY is c-open andf is c-initial, andβY is c-open whenβX isc-open andβf is c-final. Now, if f ∈M we havef c-initial sincec is hereditary;and if f ∈ E , then composition/cancellation rules show thatβf is c-dense, andsinceβf (as a morphism from ac-compact object to ac-separated object, see [3])is alsoc-closed, it must belong toE and is in factc-final (cf. 6.2). 27.3. From Proposition 6.4 we conclude immediately:

COROLLARY. Let f : X → Y have the property that, for everyc-compactc-Hausdorff objectZ, the morphism1Z × f is c-final. Then this property is satisfiedby any stablec-spaceZ.

We point out that, contrary to first impression, the corollary does not lead toWhitehead’s Theorem in Topology, sincec-final maps cannot describe quotientmaps; see also 6.5(1).

Acknowledgements

This work was supported by a NATO Collaborative Grant (no. 940847) and byVolkswagen-Stiftung (RiP Program Oberwolfach, Germany). The first author wouldlike to thank CMUC (Portugal) for partial financial assistance. The second au-thor acknowledges partial financial assistance by MURST (Italy). The third authorwould like to thank NSERC (Canada) for partial financial assistance. In addition


we thank Alan Dow for drawing our attention to his paper [8] which led us tothe current formulation of Theorem 6.5. We also thank the anonymous referee forher/his careful reading of the paper.

References

1. Adámek, J., Herrlich, H. and Strecker, G. E.:Abstract and Concrete Categories, Wiley, NewYork, 1990.

2. Arhangel’skiı, A.: Bicompact sets and the topology of spaces,Dokl. Akad. Nauk SSSR150(1963), 9–12; English translation:Soviet Math. Dokl.4 (1963), 561–564.

3. Clementino, M. M., Giuli, E. and Tholen, W.: Topology in a category: Compactness,Portugal.Math.53 (1996), 397–433.

4. Dikranjan, D.: Semiregular closure operators and epimorphisms in topological categories,Suppl. Rend. Circ. Mat. Palermo, Serie II29 (1992), 105–160.

5. Dikranjan, D. and Giuli, E.: Closure operators I,Topology Appl.27 (1987), 129–143.6. Dikranjan, D. and Giuli, E.: Compactness, minimality and closedness with respect to a closure

operator, in:Proc. Internat. Conf. on Categorical Topology, Prague 1988, World Scientific,Singapore, 1989, pp. 284–296.

7. Dikranjan, D. and Tholen, W.:Categorical Structure of Closure Operators, with Applicationsto Topology, Algebra and Discrete Mathematics, Kluwer Acad. Publ., Dordrecht, 1995.

8. Dow, A. and Watson, S.: A subcategory ofTop, Trans. Amer. Math. Soc.337(1993), 825–837.9. Engelking, R.:General Topology, revised and completed edn, Heldermann Verlag, Berlin,

1989.10. Freyd, P. J. and Kelly, G. M.: Categories of continuous functors, I,J. Pure Appl. Algebra2

(1972), 169–191.11. Giuli, E. and Tholen, W.: Openness with respect to a closure operator, Preprint, 1997.12. Herrlich, H.: Topological functors,Gen. Topology Appl.4 (1974), 125–142.13. McDougle, P.: A theorem on quasi-compact mappings,Proc. Amer. Math. Soc.9 (1958), 474–

477.14. Wyler, O.: Top categories and categorical topology,Gen. Topology Appl.1 (1971), 17–28.

Documents

What is a Quotient Map with Respect to a Closure Operator?