Epimorphisms and maximal covers in categories of compact spaces

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1 @ Applied General Topoloy c Universidad Politécnica de Valencia Volume 14, no. 1, 2013 pp Epimorphisms and maximal covers in cateories o compact spaces B. Banaschewski and A. W. Haer Abstract The cateory C is projective complete i each object has a projective cover (which is then a maximal cover). This property inherits rom C to an epirelective ull subcateory R provided the epimorphisms in R are alsoepiinc. Whenthisconditionails, therestill maybesome maximal covers in R. The main point o this paper is illustration o this in compact Hausdor spaces with a class o examples, each providin quite strane epimorphisms and maximal covers. These examples are then dualized to a cateory o alebras providin likewise strane monics and maximal essential extensions MSC: Primary 06F20, 18G05, 54B30; Secondary 18A20, 18B30, 54C10, 54G05 Keywords: Epimorphism, cover, projective, essential extension, compact, stronly riid. 1. Introduction In a cateory, an essential extension o an object A is a monomorphism A m B or which km monic implies k monic. In recent work [3], the authors have considered the inheritance rom a cateory C to a monocorelecive subcateory V o the property that each object has a unique maximal essential extension. The hypothesis each monic in V is also monic in C was crucial. (The property was deployed to similar ends in [9].) This paper is larely directed at exhibitin in a concrete settin some patholoy which can occur in the absence o these hypotheses. But we shall operate in dual, as we now describe briely, and sketch a return to essential extensions in the inal 5.

2 42 B. Banaschewski and A. W. Haer In a cateory, a cover o the object X is an epimorphism Y X or which epi implies that is epi. (This deinition is dual to essential extension ). Any projective cover is also a unique maximal cover (2.3). But there are cateories with no projectives, and still every object has a unique maximal cover ([3], in dual.) In compact Hausdor spaces, Comp, epis are onto and every object has a projective cover (the Gleason cover). For an epirelective subcateory R o Comp, R has a non-void projective i and only i epis in R are onto (3.5) and then the projective covers rom Comp are projective covers in R (3.2). We bein with a necessary discussion o simple cateorical preliminaries, proceed to Comp and two speciic epirelective subcateories, then extract what little can be said or an epirelective R in eneral. Penultimately, we consider a stronly riid E Comp and the epirelective subcateory R(E) which E enerates. There are epis not onto, and any nonconstant E {0, 1} is a maximal cover. Finally, we sketch the dualization o this to a cateory o alebras, in which any proper C(E) R 2 is a maximal essential extension. We thank Horst Herrlich and Miroslav Hušek or alertin us to stronly riid spaces. 2. Preliminaries The context or is a ixed cateory with no hypotheses at all beore 2.4. In the ollowin,,h,k,... are assumed to be morphisms. The terms morphism and map will be interchaneable. Deinition 2.1. (a) A morphism is an epimorphism (epi) i h = k implies h = k. (b) The map is coverin i epi, and epi implies epi. (Such could also be called essential epi (or perhaps co-essential epi).) A cover o object X is a pair (X,) with X coverin. Covers o X, (Y,) and (Y, ) are equivalent i there is an isomorphism h with h =. (c) Object Y iscover-completei, (Z,k)acoveroY impliesk isanisomorphism. A maximal cover o X is a cover (Y,) with Y cover complete. A unique maximal cover o X is a maximal cover which is equivalent to any other maximal cover o X. (d) Object P is projective i whenever X h P and X Y is an epi, Y then there is Y P with = h. A projective cover is a cover (P,p) with P projective. (e) The cateory is called projectively complete i every object has a projective cover, and (weaker) is said to have enouh projectives i or each object X there is X P, epi and P projective. The ollowin two elementary propositions are, except or 2.2 (d) and perhaps 2.3 (b), proved (in dual) in [1], 9.14, 9.19, 9.20.

3 Epimorphisms and maximal covers in cateories o compact spaces 43 Proposition 2.2. Proo. (a) An isomorphism is coverin. (b) The composition o two coverin maps is coverin. (c) I and are coverin, then is coverin. (d) I is coverin and is epi, then is coverin. (d) Given such and, suppose h is epi. Note that is epi (because is). So, (h) is epi, and (h) = ()h shows h is epi, since is coverin. Proposition 2.3. Proo. (a) A projective object is cover-complete. (b) A projective cover is a unique maximal cover. (b) Suppose (P,p) is a projective cover o X. It is a maximal cover by (a). I (Y,) is another cover o X, there is k with k = p (since P is projective and is epi). By 2.2 (c), k is coverin, thus an isomorphism i Y is cover-complete. To proceed urther, we require assumptions. Two Hypotheses 2.4. (to be invoked selectively). Let C be a cateory, and R a subcateory (always assumed ull and isomorphism-closed). The irst condition is on C alone, and is the other ace o 2.2 (d): (F ) I is coverin and is epi, then is coverin The second condition is on R C, and is the (requently invalid) converse to the obvious truth Any C-epi between R-objects is R-epi : (S ) Any R-epi is C-epi. The point o this paper is, in the presence o (F ), what happens when (S ) holds (2.7 and 3), and especially what can happen when it ails ( 5, 6). Proposition 2.5. I C has enouh projectives (in particular i C is projectively complete), then C satisies (F ). Proo. Consider X Y Z with coverin and epi. Since is epi, so is. Suppose Y t T has t epi; we want t epi. Take T e P h epi with P projective. There is Z P with h = te (since is epi).then, ()h = (h) = (te) = (t)e. The last term is epi, and so also the irst term. Thus h is epi (since is coverin), and so also h. Since h = te, t is epi.

4 44 B. Banaschewski and A. W. Haer Proposition 2.6. Suppose (S ). I X, Y R, and X then is R-coverin. Y is C-coverin, Proo. Suppose iven X Y as stated, and Y Z with Z R and R-epi. Then is C-epi (by (S )), so is C-epi (since is C-coverin), thus also R-epi (as desired). We say (R,r) is epirelective in C i R is a subcateory o C, and or each Y C there is ry R (the relection) and epi ry r Y Y (the relection map) or which, whenever X Y with X R, there is with r Y =. (See [11] or a ull account o the theory o epirelective subcateories). Proposition 2.7. Suppose that (R,r) is epirelective in C, and satisies (S ). (a) I P is projective in C, then rp is projective in R. (b) Suppose urther that C satisies (F ). I X R, and (P,p) is a projective cover in C o X, then (rp, p) is a projective cover in R o X. (c) I C is projectively complete, then so is R (with projective covers as in (b)). Proo. (c) (rom (b)). 2.5 says C satisies (F ), so (b) applies. (a) Suppose iven R-epi X Y and any X rp. By (S ), is C- epi, so there is 1 with 1 = r P (since P is C-projective). Next, there is 2 with 2 r P = 1, and we have r P = 1 = ( 2 r P ) = ( 2 )r P. Since r P is C-epi, = 2. (b) By (a), rp is R-projective. We need that the p in pr P = p is R- coverin. Since r P is epi, (F ) says that p is C-coverin, and thus R-coverin by 2.5. Remark 2.8. (a) [3], 1.2 shows (in dual) that i C has unique maximal covers, so does epirelective R, assumin the conditions (S ) and (F ). The proos above o 2.7 (a) and (b) are simpliied versions o those in [3]. For 2.7 (c), the present 2.7 (new here) allows suppression o the hypothesis (F ). (b) I in 2.7, R already contains every C-projective, then 2.7 (a) and (b) simpliy in the obvious way. This is the case or C = Comp, with R havin (S ); see 3.2 below. 3. Compact Hausdor Spaces Comp is the cateory o compact Hausdor spaces with continuous unctions as maps. A map X Y in Comp is called irreducible i (Y) = X, but when F Y (F closed), (F) X. The ollowin is mostly due to Gleason

5 Epimorphisms and maximal covers in cateories o compact spaces 45 [6]. ((a) is a olk item. (e) ollows rom (d) and 2.4; it has a short direct proo, and is noted in [8], 2.5.) Proposition 3.1. In Comp: (a) Epis are onto. (See comment ater 3.3 below.) (b) A map is coverin i it is irreducible. (c) A space is projective i it is extremally disconnected (every open set has open closure). (d) Any object X has a projective cover (PX,p X ); Comp is projectively complete. (e) (F ) holds. The notation (PX,p X ) is reserved or the rest o the paper; this will always denote the projective cover in Comp o X Comp. Also, or brevity, we shall let ED stand or the class o extremally disconnected spaces in Comp. (Considerable literature developed rom Gleason s [6], with various new proos, eneralizations, and variants o the theory. See [2], [8], [14] and their biblioraphies.) Now consider a subcateory A o Comp (which can be identiied with its object class). The amily o all subobjects (resp., products) o spaces in A is denoted SA (resp., P A). (Note that subobjects are closed subspaces.) Kennison [13] has shown that R is epirelective in Comp i R is neither nor { } and R = SPR. For X Comp, let R(X) = SP{X}; this is the smallest epirelective subcateory containin X. Let {0} (resp., {0,1}) denote the space with one (resp. two) points. The smallest epirelective is R({0}) = {,{0}}; here, {0} is epi, so epis are not onto. We comment urther on this shortly. The next larest is R({0,1}): i RisepirelectiveandnotR({0}), thereisx Rwith X 2, thus{0,1} R, so R({0,1}) R. Note that R({0,1}) = Comp, the class o compact zerodimensional spaces [5], and ED Comp [7]. Thus, i R is epirelective and not R({0}), ED R. Corollary 3.2. Suppose R is epirelective and R-epis are onto (i.e., R Comp satisies (S )). Then R is projectively complete. In act, or any X R, the R-(projective cover) is (PX,p X ). Proo. Apply 3.1, 2.4, and the discussion above. Proposition 3.3. Comp -epis are onto. 3.2 applies to Comp. Proo. The ollowin takes place in Comp The only Y has Y = and the map is the identity, which is epi, and technically onto. I X then X is not epi (since there are dierent h {0, 1} X ). k Suppose X, and X Y is epi. Were not onto, there would be p X (Y), and clopen U with p / U (Y). Then h constantly 1 and k the characteristic unction o U has h k but h = k.

6 46 B. Banaschewski and A. W. Haer (To show Comp-epis are onto, arue similarly usin [0,1] instead o {0,1}, and usin complete reularity o X (i.e. the Tietze-Urysohn Theorem).) Remark 3.4. We do not know i there is epirelective R dierent rom Comp and Comp, or which epis are onto. The ollowin(closely related to [3], 4.1) shows that, ailin epis are onto, there are no projectives. But there still may be some maximal covers, o at least two sorts, as the examples in 5 show. Proposition 3.5. Suppose (only) {0} R. The ollowin statements in R are equivalent. (a) Epis are onto. (b) {0} is projective. (c) There is a non-void projective. Proo. (b) (c) obviously, and (c) (b) because {0} is a retract o any X, and a retract o a projective is projective. (a) (b) because {0} is projective in Comp, and i (a) holds, projective in R. (b) (a). I X Y is an epi which is not onto, then there is p X (Y), and or X h {0} deined as h(0) = p, there can be no Y {0} with = h. Finally, we clariy the situation or and or R({0}). Note the ollowin or any R Comp with R. (i) is the initial object o R, i.e., or any X R, there is unique X (namely, the empty map). (ii) is projective in R. (iii) I X is epi in R, then this is a projective cover. Proposition 3.6. Suppose (only) {0} R = SR. The ollowin statements in R are equivalent. (a) R = R({0}) (b) {0} is epi (and thus a projective cover). (c) For any X R, X is epi (and thus a projective cover). Proo. The parenthetical remarks ollow rom the comments above. (a) (c): is epi, and {0} also, since the only map out o {0} is the identity. (c) (b): since {0} R. (b) (a): I R R({0}), then there is X R with X 2, so {0,1} R h (since SR = R). Then there are dierent {0,1} {0} k which compose equally with {0}, so the latter is not epi.

7 Epimorphisms and maximal covers in cateories o compact spaces 47 Corollary 3.7. R({0}) is projectively complete, with epis not onto, and is the only epirelective subcateory with these two properties. Proo. 3.6, (a) (c) yields the irst statement. I epirelective R has epis not onto, then by 3.5, the only projective is. I R is projectively complete, then the projective cover must be X. So these are epi, and 3.6 (c) (a) says R = R({0}). 4. When epis may not be onto Consider R Comp. We localize the condition R-epis are onto. Keep in mind that R miht have no projectives (but any Y R ED is still Compprojective). Deinition 4.1. X has e(r) means X R, and whenever X in R, then is onto. Proposition 4.2. Suppose R = SR. (a) I X R Comp, then X has e(r). (b) I Y R ED, then Y is cover-complete in R. is epi Proo. (a) Identical to the proo o 3.3. (b) I (Z,) is an R-cover o Y, then is onto by (a), so there is with = id Y, since Y is Comp-projective, and R (since Y, Z R). So isan R-section. Also, by 2.2, is anr-coverinmap, thus R-epi. So is an R-isomorphism, sois, and thereore is a Comp-isomorphism, thus a homeomorphism. (The converse to 4.2 (a) ails, with R = Comp. But see 4.5 below.) Proposition 4.3. Suppose X has e(r). (a) I Y R and X Y is irreducible, then (Y,) is an R-cover o X. (b) I also PX R, and supposin R = SR, then (PX,p X ) is the unique maximal R-cover o X. Proo. (a) As in the proo o 2.6, mutatis mutandis (b) By (a), (PX,p X ) is an R-cover, and PX is cover-complete. I (Y,) is anotherr-coverox,then isonto(bye(r)), andthereis Y PX with = p X (since PX is Comp-projective). I Y is cover-complete, is a homeomorphism. Corollary 4.4. Suppose ED R = SR. I X R Comp, then (PX,p X ) is the unique maximal R-cover o X. Proo. 4.2 (a) and 4.3 (b).

8 48 B. Banaschewski and A. W. Haer The ollowin is a qualiied converse to 4.2 (a). Corollary 4.5. Suppose that ED R = SR. For Y R, the ollowin are equivalent. (a) Y is ED. (b) Y is cover-complete and Y Comp. (c) Y is cover-complete and Y has e(r). Proo. (a) (b): 4.2 (b) and ED Comp. (b) (c): 4.2 (a). (c) (a): By 4.3 (b) (usin ED R now), (PY,p Y ) is the unique maximal R-cover o Y, so i Y is cover-complete, p Y is a homeomorphism. Here is one (more) triviality valid in (almost) any R. Proposition 4.6. Suppose {0} R. For any X R, with X 1, there are maps X e {0} (in R). Such an e is R-epi i X = 1. Proo. Given such e, there is (the retraction) X r {0} with re = id {0}, so e is a section. I X = 1, then e is onto, thus R epi. I e is R-epi, it becomes an R-isomorphism, thus a homeomorphism, so X = Epirelectives with Epis not onto, and some maximal covers First, in summary so ar o the situation or R epirelective in Comp: I in R, there are epis not onto, then there are no non-void projectives ( 3.5). That is the case or R = {,{0}}, but here we have the projective (thus unique maximal) covers and {0} ( 3.6). I R contains the two-point space {0,1} then R Comp and at least has unique maximal covers or X Comp, namely the (PX,p X ) ( 4.4). We now display a lare class o such R with some very strane epis, and non-unique maximal covers. This will be the R(E) = SP{E}, or E as ollows. A space E in Comp will be called stronly riid i E 2 and the only continuous E E are id E and constants. Cook [4] has several o these, includin a metric one M 1. Note that i E is stronly riid, then {0,1} E (since E 2), E is connected (since a clopen U, E would yield E {0,1} E ), E c (since there are non-constant E [0,1], usin the Tietze-Urysohn Theorem), and [0,1] E (since [0,1] E would yield non-constant E [0,1] E, and [0,1] is not stronly riid). From Cook s examples, Trnková [15] and Isbell [12] have shown irst, that i n is any cardinal, there is stronly riid E with E n, and second, that i there is no measurable cardinal, there is a proper class E o stronly riid spaces or which, whenever E 1 E 2 in E, the only continuous E 1 E 2 are constants (and thus, or E 1 E 2 in E, neither o R(E 1 ) and R(E 2 ) contains the other).

9 Epimorphisms and maximal covers in cateories o compact spaces 49 Now let E be any stronly riid space. In the ollowin, terms epi, cover,... reer to R(E). O course 4.4 and 4.5 apply here. On the other hand, Proposition5.1. Let F be a closed subspace o E. Label the inclusion E F. (a) i F is epi i F > 1. (b) I F = 2, then (F,i F ) is a cover o E. I F Comp and (F,i F ) is a cover o E, then F = 2. (In the second part o (b), the supposition F Comp cannot be dropped, because E id E is a cover.) Corollary 5.2. Any nonconstant E {0, 1} is a maximal cover o E. Any maximal cover o E is equivalent to one o these. Two o these, with and, are equivalent covers o E i ({0,1}) = ({0,1}). In particular, (PE,p E ) is not a cover o E, and there are at least E c non-equivalent maximal covers o E; Proo. (o 5.1) (a) By 4.6, i F = 1, then i F is not epi. Now suppose F > 1. Suppose, R(E) have common codomain - which miht as well be supposed o the orm E I - and i F = i F, i.e., F = F. Then, or any projection E I π i E, we have π i F = π i F. We want =, which is equivalent to π i = π i i I. Let i I. Then eacho π i and π i is id E or constant. I π i = id E, then π i F is not constant (since F 2), so π i F is not constant, so π i = id E also. I π i is constant, say c, then π i F = c also. So π i F = c, and since F 2, π i = c. (b) Suppose F = 2. By(a), i F isepi. Supposei F isepi. Then isontof (since i not, rane() = 1, since F = 2, but then rane(i F ) = 1 and i F is not epi, by 4.6. So is epi. Suppose F Comp.I there are dierent p 0, p 1, p 2 F, let F {0, 1} be (i) = p i. Then is not epi (by 4.2 (a)), but i F is epi by (a) above. Proo. (o 5.2) I E {0,1} isnonconstant,itisacoverbecausef {(0), (1)} {0,1} is a homeomorphism, and thus a maximal cover because F is cover-complete (bein ED 4.2). h Suppose E Y is a maximal cover. Then h is epi, thus nonconstant ( 6.1). So there are p 0, p 1 Y with h(p 0 ) h(p 1 ). Deine Y {0,1} as (i) = p i. So h is a coverin-map (by the precedin pararaph), thus is a i F

10 50 B. Banaschewski and A. W. Haer coverin map ( 2.2 (d)). Since Y is cover-complete, is a homeomorphism, so (Y,h) and ({0,1},h) are equivalent., Now suppose E {0, 1} are non-constant. There are two homeomorphisms h o {0,1}, the identity and interchane 0 and 1. And, rane() = rane( ) i = h or one o these h. Remark 5.3. Cook s speciic stronly riid M 1 has these urther eatures: M 1 has a countable ininity o disjoint subcontinua; i K is any proper subcontinuum o M 1, the only maps M 1 K are inclusion and constants. (See [4]). Then in the cateory R(M 1 ), in 5.1 and 5.2, E = M 1 may be replaced by any proper subcontinuum K o M 1 (as the proos there show). 6. An application to lattice-ordered roups We now convert the situations o maximal covers in R Comp to situations o maximal essential extensions in subcateories o a cateory o alebras. We use terminoloy cateorically dual to the items in 2.1 (a) - (e), respectively, namely (a) monic, (b) essential extension, (c) essentially complete, maximal essential extension (or, essential completion), (d) injective, injective hull, (e) injectively complete. The cateory o alebras is W, the cateory o archimedean lattice-ordered roups with distinuished stron order unit, and l-roup homomorphisms carryin unit to unit. W has monics one-to-one, and is injectively complete; see [3]. Consequently, the dual o 2.7 applies to W. For X Comp, the continuous real-valued unctions C(X), with unit the constant unction 1, is a W -object, and we have the unctor W C Comp: τ or X Z in Comp, C(X) Cτ C(Z) is Cτ() = τ. This has a let adjoint, the Yosida unctor: For each G W, there is YG Comp and G C(YG) monic in W ; or each G ϕ H in W, there is unique Yϕ YG YH in Comp realizinϕ as ϕ() = Yϕ. Note that YC(X) X, and that ϕ is one-to-one i Yϕ is onto. (See [10]). Basic eatures o (Y, C), and some diaram-chasin, convert the situations in Comp discussed in previous sections to dual situations in W, as ollows. (We omit the calculations). Suppose R is epirelective in Comp, and {0,1} R (so Comp R). For brevity, set R = {G W YG R}. Proposition 6.1. (a) R is monocorelective in W. (b) C(X) ϕ H is monic in R i X YH Yϕ is epi in R. (c) R has an injective other than {0} i monics in R are one-to-one i R-epis are onto. When this occurs, R is injective-complete, with injective hulls G C(YG) C(P(YG)). (d) I X is ED, then C(X) is essentially complete in R.

11 Epimorphisms and maximal covers in cateories o compact spaces 51 (e) I X Comp, then C(D) CpX C(PX) is the unique maximal essential extension o C(X) in R. Now consider, as in 5, stronly riid E Comp and its enerated epirelective R(E). By 5.1 and 6.1 (b), R(E) has monics which are not one-to-one, and thus no {0} injectives. 6.1 (d) and (e) hold in R(E). Notethat{0,1} ComphasC({0,1}) = R 2 W, thesel-homeomorphisms o {0, 1} are the identity and interchane points, and these correspond to the only sel-isomorphisms o R 2, which are the identity, and H(x,y) = (y,x). From 5.2 we obtain Corollary 6.2. In R(E), the maximal essential extensions o C(E) are exactly the W -surjections C(E) ϕ R 2. Two o these, ϕ and ϕ, are equivalent i either ϕ = ϕ, or ϕ = ϕh. Reerences [1] J. Adamek, H. Herrlich and G. Strecker, Abstract and Concrete Cateories, Dover [2] B. Banaschewski, Projective covers in cateories o topoloical spaces and topoloical alebras, pp in General Topoloy and its Relations to Modern Analysis and Alebra, Academia [3] B. Banaschewski and A. Haer, Essential completeness o archimedean l-roups with weak unit, to appear. [4] H. Cook, Continua which admit only the identity mappin onto non-deenerate subcontinua, Fund. Math. 60 (1966) [5] R. Enelkin, General Topoloy, Heldermann [6] A. Gleason, Projective topoloical spaces, Ill. J. Math. 2 (1958), [7] L. Gillman and M. Jerison, Rins o Continuous Functions, Spriner-Verla [8] A. Haer, Minimal covers o topoloical spaces, Ann. NY Acad. Sci. 552 (1989), [9] A. Haer and J. Martinez, Sinular archimedean lattice-ordered roups, Al. Univ. 40 (1998), [10] A. Haer and L. Robertson, Representin and riniyin a Riesz space, Symp. Math. XXI (1977), [11] H. Herrlich and G. Strecker, Cateory Theory, Allyn and Bacon [12] J. Isbell, A closed non-relective subcateory o compact spaces, Manuscript c [13] J. Kennison, Relective unctors in eneral topoloy and elsewhere, Trans. Amer. Math. Soc. 118 (1965), [14] J. Porter and R. G. Woods, Extensions and Absolutes o Hausdor Spaces, Spriner- Verla [15] V. Trnková, Non-constant continuous mappins o metric or compact Hausdor spaces, Comm. Math. Univ. Carol. 13 (1972), (Received December 2011 Accepted November 2012)

12 52 B. Banaschewski and A. W. Haer B. Banaschewski Department o Mathematics and Statistics, McMaster University, Hamilton, ON L8S4K1, Canada. A. W. Haer (ahaer@wesleyan.edu) Department o Mathematics and Computer Science, Wesleyan University, Middletown, CT USA