() @ Applied General Topology c© Universidad Politécnica de Valencia Volume 14, no. 1, 2013 pp. 41-52 Epimorphisms and maximal covers in categories of compact spaces B. Banaschewski and A. W. Hager Abstract The category C is ”projective complete” if each object has a projective cover (which is then a maximal cover). This property inherits from C to an epireflective full subcategory R provided the epimorphisms in R are also epi in C. When this condition fails, there still may be some maximal covers in R. The main point of this paper is illustration of this in compact Hausdorff spaces with a class of examples, each providing quite strange epimorphisms and maximal covers. These examples are then dualized to a category of algebras providing likewise strange monics and maximal essential extensions. 2010 MSC: Primary 06F20, 18G05, 54B30; Secondary 18A20, 18B30, 54C10, 54G05 Keywords: Epimorphism, cover, projective, essential extension, compact, strongly rigid. 1. Introduction In a category, an essential extension of an object A is a monomorphism A B m // for which km monic implies k monic. In recent work [3], the au- thors have considered the inheritance from a category C to a monocoreflecive subcategory V of 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 largely di- rected at exhibiting in a concrete setting some pathology which can occur in the absence of these hypotheses. But we shall operate ”in dual”, as we now describe briefly, and sketch a return to essential extensions in the final §5. 42 B. Banaschewski and A. W. Hager In a category, a cover of the object X is an epimorphism XY g oo for which gf epi implies that f is epi. (This definition is dual to ”essential extension”). Any projective cover is also a unique maximal cover (2.3). But there are cat- egories with no projectives, and still every object has a unique maximal cover ([3], in dual.) In compact Hausdorff spaces, Comp, epis are onto and every object has a projective cover (the Gleason cover). For an epireflective subcategory R of Comp, R has a non-void projective if and only if epis in R are onto (3.5) and then the projective covers from Comp are projective covers in R (3.2). We begin with a necessary discussion of simple categorical preliminaries, proceed to Comp and two specific epireflective subcategories, then extract what little can be said for an epireflective R in general. Penultimately, we consider a strongly rigid E ∈ Comp and the epireflective subcategory R(E) which E generates. There are epis not onto, and any nonconstant {0, 1}E oo is a maximal cover. Finally, we sketch the dualization of this to a category of algebras, in which any proper C(E) R2// is a maximal essential extension. We thank Horst Herrlich and Miroslav Hušek for alerting us to strongly rigid spaces. 2. Preliminaries The context for 2.1 - 2.7 is a fixed category with no hypotheses at all before 2.4. In the following, g, h, k, . . . are assumed to be morphisms. The terms ”morphism” and ”map” will be interchangeable. Definition 2.1. (a) A morphism g is an epimorphism (epi) if hg = kg implies h = k. (b) The map g is ”covering” if epi, and gf epi implies f epi. (Such g could also be called essential epi (or perhaps co-essential epi).) A cover of object X is a pair (X, g) with XY g oo covering. Covers of X, (Y, g) and (Y ′, g′) are equivalent if there is an isomorphism h with g′h = g. (c) Object Y is cover-complete if, (Z, k) a cover of Y implies k is an isomor- phism. A maximal cover of X is a cover (Y, g) with Y cover complete. A unique maximal cover of X is a maximal cover which is equivalent to any other maximal cover of X. (d) Object P is projective if whenever PX h oo and YX g oo is an epi, then there is PY f oo with gf = h. A projective cover is a cover (P, p) with P projective. (e) The category is called projectively complete if every object has a pro- jective cover, and (weaker) is said to have enough projectives if for each object X there is PX f oo , f epi and P projective. The following two elementary propositions are, except for 2.2 (d) and per- haps 2.3 (b), proved (in dual) in [1], 9.14, 9.19, 9.20. Epimorphisms and maximal covers in categories of compact spaces 43 Proposition 2.2. (a) An isomorphism is covering. (b) The composition of two covering maps is covering. (c) If g and gf are covering, then f is covering. (d) If gf is covering and f is epi, then f is covering. Proof. (d) Given such gf and f, suppose fh is epi. Note that g is epi (because gf is). So, g(fh) is epi, and g(fh) = (gf)h shows h is epi, since gf is covering. � Proposition 2.3. (a) A projective object is cover-complete. (b) A projective cover is a unique maximal cover. Proof. (b) Suppose (P, p) is a projective cover of X. It is a maximal cover by (a). If (Y, g) is another cover of X, there is k with gk = p (since P is projective and g is epi). By 2.2 (c), k is covering, thus an isomorphism if Y is cover-complete. � To proceed further, we require assumptions. Two Hypotheses 2.4. (to be invoked selectively). Let C be a category, and R a subcategory (always assumed full and isomorphism-closed). The first condition is on C alone, and is ”the other face” of 2.2 (d): (F ◦) If gf is covering and f is epi, then g is covering The second condition is on R ⊂ C, and is the (frequently invalid) converse to the obvious truth ”Any C-epi between R-objects is R-epi”: (S◦) Any R-epi is C-epi. The point of this paper is, in the presence of (F ◦), what happens when (S◦) holds (2.7 and §3), and especially what can happen when it fails (§5 , §6). Proposition 2.5. If C has enough projectives (in particular if C is projectively complete), then C satisfies (F ◦). Proof. Consider X Y g oo Z f oo with gf covering and f epi. Since gf is epi, so is g. Suppose Y T t oo has gt epi; we want t epi. Take T P e oo epi with P projective. There is Z P h oo with fh = te (since f is epi).Then, (gf)h = g(fh) = g(te) = (gt)e. The last term is epi, and so also the first term. Thus h is epi (since gf is covering), and so also fh. Since fh = te, t is epi. � 44 B. Banaschewski and A. W. Hager Proposition 2.6. Suppose (S◦). If X, Y ∈ R, and X Y g oo is C-covering, then g is R-covering. Proof. Suppose given X Y g oo as stated, and Y Z f oo with Z ∈ R and gf R-epi. Then gf is C-epi (by (S◦)), so f is C-epi (since g is C-covering), thus also R-epi (as desired). � We say (R, r) is epireflective in C if R is a subcategory of C, and for each Y ∈ C there is rY ∈ R (the reflection) and epi rY Y rY oo (the reflection map) for which, whenever X Y f oo with X ∈ R, there is f̄ with f̄rY = f. (See [11] for a full account of the theory of epireflective subcategories). Proposition 2.7. Suppose that (R, r) is epireflective in C, and satisfies (S◦). (a) If P is projective in C, then rP is projective in R. (b) Suppose further that C satisfies (F ◦). If X ∈ R, and (P, p) is a projec- tive cover in C of X, then (rP, p̄) is a projective cover in R of X. (c) If C is projectively complete, then so is R (with projective covers as in (b)). Proof. (c) (from (b)). 2.5 says C satisfies (F ◦), so (b) applies. (a) Suppose given R-epi X Y g oo and any X rP f oo . By (S◦), g is C- epi, so there is f1 with gf1 = frP (since P is C-projective). Next, there is f2 with f2rP = f1, and we have frP = gf1 = g(f2rP ) = (gf2)rP . Since rP is C-epi, f = gf2. (b) By (a), rP is R-projective. We need that the p̄ in p̄rP = p is R- covering. Since rP is epi, (F ◦) says that p̄ is C-covering, and thus R-covering by 2.5. � Remark 2.8. (a) [3], 1.2 shows (in dual) that if C has unique maximal covers, so does epireflective R, assuming the conditions (S◦) and (F ◦). The proofs above of 2.7 (a) and (b) are simplified versions of those in [3]. For 2.7 (c), the present 2.7 (new here) allows suppression of the hypothesis (F ◦). (b) If in 2.7, R already contains every C-projective, then 2.7 (a) and (b) simplify in the obvious way. This is the case for C = Comp, with R having (S◦); see 3.2 below. 3. Compact Hausdorff Spaces Comp is the category of compact Hausdorff spaces with continuous functions as maps. A map X Y f oo in Comp is called irreducible if f(Y ) = X, but when F ( Y (F closed), f(F) 6= X. The following is mostly due to Gleason Epimorphisms and maximal covers in categories of compact spaces 45 [6]. ((a) is a folk item. (e) follows from (d) and 2.4; it has a short direct proof, and is noted in [8], 2.5.) Proposition 3.1. In Comp: (a) Epis are onto. (See comment after 3.3 below.) (b) A map is covering iff it is irreducible. (c) A space is projective iff it is extremally disconnected (every open set has open closure). (d) Any object X has a projective cover (PX, pX); Comp is projectively complete. (e) (F ◦) holds. The notation (PX, pX) is reserved for the rest of the paper; this will always denote the projective cover in Comp of X ∈ Comp. Also, for brevity, we shall let ED stand for the class of extremally disconnected spaces in Comp. (Considerable literature developed from Gleason’s [6], with various new proofs, generalizations, and variants of the theory. See [2], [8], [14] and their bibliographies.) Now consider a subcategory A of Comp (which can be identified with its object class). The family of all subobjects (resp., products) of spaces in A is denoted SA (resp., PA). (Note that subobjects are closed subspaces.) Kenni- son [13] has shown that R is epireflective in Comp iff R is neither ∅ nor {∅} and R = SPR. For ∅ 6= X ∈ Comp, let R(X) = SP{X}; this is the smallest epireflective subcategory containing X. Let {0} (resp., {0, 1}) denote the space with one (resp. two) points. The smallest epireflective is R({0}) = {∅, {0}}; here, {0} ∅oo is epi, so epis are not onto. We comment further on this shortly. The next largest is R({0, 1}): if R is epireflective and not R({0}), there is X ∈ R with |X| ≥ 2, thus {0, 1} ∈ R, so R({0, 1}) ⊂ R. Note that R({0, 1}) = Comp◦, the class of compact zero- dimensional spaces [5], and ED ⊂ Comp◦ [7]. Thus, if R is epireflective and not R({0}), ED⊂ R. Corollary 3.2. Suppose R is epireflective and R-epis are onto (i.e., R ⊂ Comp satisfies (S◦)). Then R is projectively complete. In fact, for any X ∈ R, the R-(projective cover) is (PX, pX). Proof. Apply 3.1, 2.4, and the discussion above. � Proposition 3.3. Comp◦-epis are onto. 3.2 applies to Comp◦. Proof. The following takes place in Comp◦ The only ∅ Yoo has Y = ∅ and the map is the identity, which is epi, and technically onto. If X 6= ∅ then X ∅oo is not epi (since there are different {0, 1} X h oo k oo ). Suppose X 6= ∅, and X Y g oo is epi. Were g not onto, there would be p ∈ X − g(Y ), and clopen U with p /∈ U ⊇ g(Y ). Then h constantly 1 and k the characteristic function of U has h 6= k but hg = kg. � 46 B. Banaschewski and A. W. Hager (To show Comp-epis are onto, argue similarly using [0, 1] instead of {0, 1}, and using complete regularity of X (i.e. the Tietze-Urysohn Theorem).) Remark 3.4. We do not know if there is epireflective R different from Comp◦ and Comp, for which epis are onto. The following (closely related to [3], 4.1) shows that, failing ”epis are onto”, there are no 6= ∅ projectives. But there still may be some maximal covers, of at least two sorts, as the examples in §5 show. Proposition 3.5. Suppose (only) {0} ∈ R. The following statements in R are equivalent. (a) Epis are onto. (b) {0} is projective. (c) There is a non-void projective. Proof. (b) ⇒ (c) obviously, and (c) ⇒ (b) because {0} is a retract of any X 6= ∅, and a retract of a projective is projective. (a) ⇒ (b) because {0} is projective in Comp, and if (a) holds, projective in R. (b) ⇒ (a). If X Y g oo is an epi which is not onto, then there is p ∈ X −g(Y ), and for X {0} h oo defined as h(0) = p, there can be no Y {0} f oo with gf = h. � Finally, we clarify the situation for ∅ and for R({0}). Note the following for any R ⊂ Comp with ∅ ∈ R. (i) ∅ is the initial object of R, i.e., for any X ∈ R, there is unique X ∅oo (namely, the empty map). (ii) ∅ is projective in R. (iii) If X ∅oo is epi in R, then this is a projective cover. Proposition 3.6. Suppose (only) {0} ∈ R = SR. The following statements in R are equivalent. (a) R = R({0}) (b) {0} ∅oo is epi (and thus a projective cover). (c) For any X ∈ R, X ∅oo is epi (and thus a projective cover). Proof. The parenthetical remarks follow from the comments above. (a) ⇒ (c): ∅ ∅oo is epi, and {0} ∅oo also, since the only map out of {0} is the identity. (c) ⇒ (b): since {0} ∈ R. (b) ⇒ (a): If R 6= R({0}), then there is X ∈ R with |X| ≥ 2, so {0, 1} ∈ R (since SR = R). Then there are different {0, 1} {0} h oo k oo which compose equally with {0} ∅oo , so the latter is not epi. � Epimorphisms and maximal covers in categories of compact spaces 47 Corollary 3.7. R({0}) is projectively complete, with epis not onto, and is the only epireflective subcategory with these two properties. Proof. 3.6, (a) ⇒ (c) yields the first statement. If epireflective R has epis not onto, then by 3.5, the only projective is ∅. If R is projectively complete, then the projective cover must be X ∅oo . 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 might have no projectives (but any Y ∈ R∩ED is still Comp- projective). Definition 4.1. ”X has e(R)” means X ∈ R, and whenever X · g oo is epi in R, then g is onto. Proposition 4.2. Suppose R = SR. (a) If X ∈ R∩Comp◦, then X has e(R). (b) If Y ∈ R∩ED, then Y is cover-complete in R. Proof. (a) Identical to the proof of 3.3. (b) If (Z, g) is an R-cover of Y , then g is onto by (a), so there is f with gf = idY , since Y is Comp-projective, and f ∈ R (since Y , Z ∈ R). So f is an R-section. Also, by 2.2, f is an R-covering map, thus R-epi. So f is an R-isomorphism, so is g, and therefore g is a Comp-isomorphism, thus a homeomorphism. � (The converse to 4.2 (a) fails, with R = Comp. But see 4.5 below.) Proposition 4.3. Suppose X has e(R). (a) If Y ∈ R and X Y g oo is irreducible, then (Y, g) is an R-cover of X. (b) If also PX ∈ R, and supposing R = SR, then (PX, pX) is the unique maximal R-cover of X. Proof. (a) As in the proof of 2.6, mutatis mutandis (b) By (a), (PX, pX) is an R-cover, and PX is cover-complete. If (Y, g) is another R-cover of X, then g is onto (by e(R)), and there is Y PX f oo with gf = pX (since PX is Comp-projective). If Y is cover-complete, f is a homeomorphism. � Corollary 4.4. Suppose ED⊂ R = SR. If X ∈ R∩Comp◦, then (PX, pX) is the unique maximal R-cover of X. Proof. 4.2 (a) and 4.3 (b). � 48 B. Banaschewski and A. W. Hager The following is a qualified converse to 4.2 (a). Corollary 4.5. Suppose that ED⊂ R = SR. For Y ∈ R, the following are equivalent. (a) Y is ED. (b) Y is cover-complete and Y ∈Comp◦. (c) Y is cover-complete and Y has e(R). Proof. (a) ⇒ (b): 4.2 (b) and ED⊂Comp◦. (b) ⇒ (c): 4.2 (a). (c) ⇒ (a): By 4.3 (b) (using ED⊂ R now), (PY, pY ) is the unique maximal R-cover of Y , so if Y is cover-complete, pY 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 {0} e oo (in R). Such an e is R-epi iff |X| = 1. Proof. Given such e, there is (the retraction) X r //{0} with re = id{0}, so e is a section. If |X| = 1, then e is onto, thus R − epi. If e is R-epi, it becomes an R-isomorphism, thus a homeomorphism, so |X| = 1. � 5. Epireflectives with Epis not onto, and some maximal covers First, in summary so far of the situation for R epireflective in Comp: If in R, there are epis not onto, then there are no non-void projectives ( 3.5). That is the case for R = {∅, {0}}, but here we have the projective (thus unique maximal) covers ∅ ∅oo and {0} ∅oo ( 3.6). If R contains the two-point space {0, 1} then R ⊇ Comp◦ and at least has unique maximal covers for X ∈ Comp◦, namely the (PX, pX) ( 4.4). We now display a large class of such R with some very strange epis, and non-unique maximal covers. This will be the R(E) = SP{E}, for E as follows. A space E in Comp will be called strongly rigid if |E| ≥ 2 and the only continuous E //E are idE and constants. Cook [4] has several of these, including a metric one M1. Note that if E is strongly rigid, then {0, 1} ⊆ E (since |E| ≥ 2), E is con- nected (since a clopen U 6= ∅, E would yield E // {0, 1} � � // E ), |E| ≥ c (since there are non-constant E // // [0, 1] , using the Tietze-Urysohn Theo- rem), and [0, 1] * E (since [0, 1] ⊆ E would yield non-constant E // [0, 1] � � // E , and [0, 1] is not strongly rigid). From Cook’s examples, Trnková [15] and Isbell [12] have shown first, that if n is any cardinal, there is strongly rigid E with |E| ≥ n, and second, that if there is no measurable cardinal, there is a proper class E of strongly rigid spaces for which, whenever E1 6= E2 in E, the only continuous E1 //E2 are constants (and thus, for E1 6= E2 in E, neither of R(E1) and R(E2) contains the other). Epimorphisms and maximal covers in categories of compact spaces 49 Now let E be any strongly rigid space. In the following, terms epi, cover, . . . refer to R(E). Of course 4.4 and 4.5 apply here. On the other hand, Proposition 5.1. Let F be a closed subspace of E. Label the inclusion E F iF oo . (a) iF is epi iff |F | > 1. (b) If |F | = 2, then (F, iF ) is a cover of E. If F ∈Comp◦ and (F, iF ) is a cover of E, then |F | = 2. (In the second part of (b), the supposition ”F ∈Comp◦” cannot be dropped, because E E id oo is a cover.) Corollary 5.2. Any nonconstant E {0, 1} g oo is a maximal cover of E. Any maximal cover of E is equivalent to one of these. Two of these, with g and g′, are equivalent covers of E iff g({0, 1}) = g′({0, 1}). In particular, (PE, pE) is not a cover of E, and there are at least |E| ≥ c non-equivalent maximal covers of E; Proof. (of 5.1) (a) By 4.6, if |F | = 1, then iF is not epi. Now suppose |F | > 1 . Suppose f, g ∈ R(E) have common codomain - which might as well be supposed of the form EI - and fiF = giF , i.e., f|F = g|F . Then, for any projection EI πi //E , we have πif|F = πig|F . We want f = g, which is equivalent to πif = πig ∀ i ∈ I. Let i ∈ I. Then each of πif and πig is idE or constant. If πif = idE, then πif|F is not constant (since |F | ≥ 2), so πig|F is not constant, so πig = idE also. If πif is constant, say c, then πif|F = c also. So πig|F = c, and since |F | ≥ 2, πig = c. (b) Suppose |F | = 2. By (a), iF is epi. Suppose iF f is epi. Then f is onto F (since if not, |range(f)| = 1, since |F | = 2, but then |range(iF f)| = 1 and iF f is not epi, by 4.6. So f is epi. Suppose F ∈Comp◦.If there are different p0, p1, p2 ∈ F , let F {0, 1} f oo be f(i) = pi. Then f is not epi (by 4.2 (a)), but iF f is epi by (a) above. � Proof. (of 5.2) If E {0, 1} g oo is nonconstant, it is a cover because F ≡ {g(0), g(1)} {0, 1} g oo is a homeomorphism, and thus a maximal cover because F is cover-complete (being ED 4.2). Suppose E Y h oo is a maximal cover. Then h is epi, thus nonconstant ( 6.1). So there are p0, p1 ∈ Y with h(p0) 6= h(p1). Define Y {0, 1} f oo as f(i) = pi. So hf is a covering-map (by the preceding paragraph), thus f is a 50 B. Banaschewski and A. W. Hager covering map ( 2.2 (d)). Since Y is cover-complete, f is a homeomorphism, so (Y, h) and ({0, 1}, hf) are equivalent. Now suppose E {0, 1} g,g ′ oo are non-constant. There are two homeomor- phisms h of {0, 1}, the identity and ”interchange 0 and 1”. And, range(g) = range(g′) iff g′ = gh for one of these h. � Remark 5.3. Cook’s specific strongly rigid M1 has these further features: M1 has a countable infinity of disjoint subcontinua; if K is any proper sub- continuum of M1, the only maps M1 Koo are inclusion and constants. (See [4]). Then in the category R(M1), in 5.1 and 5.2, E = M1 may be replaced by any proper subcontinuum K of M1 (as the proofs there show). 6. An application to lattice-ordered groups We now convert the situations of maximal covers in R ⊂Comp to situations of maximal essential extensions in subcategories of a category of algebras. We use terminology categorically 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 category of algebras is W ∗, the category of archimedean lattice-ordered groups with distinguished strong order unit, and ℓ-group homomorphisms car- rying unit to unit. W ∗ has monics one-to-one, and is injectively complete; see [3]. Consequently, the dual of 2.7 applies to W ∗. For X ∈Comp, the continuous real-valued functions C(X), with unit the constant function 1, is a W ∗-object, and we have the functor W ∗ Comp C oo : for X Z τ oo in Comp, C(X) Cτ //C(Z) is Cτ(f) = f ◦ τ. This has a left adjoint, the Yosida functor: For each G ∈ W ∗, there is Y G ∈ Comp and G //C(Y G) monic in W ∗; for each G ϕ //H in W ∗, there is unique Y G Y H Y ϕ oo in Comp ”realizing ϕ” as ϕ(g) = g◦Y ϕ. Note that Y C(X) ≃ X, and that ϕ is one-to-one iff Y ϕ is onto. (See [10]). Basic features of (Y, C), and some diagram-chasing, convert the situations in Comp discussed in previous sections to ”dual” situations in W ∗, as follows. (We omit the calculations). Suppose R is epireflective in Comp, and {0, 1} ∈ R (so Comp◦ ∈ R). For brevity, set ∗R = {G ∈ W ∗|Y G ∈ R}. Proposition 6.1. (a) ∗R is monocoreflective in W ∗. (b) C(X) ϕ //H is monic in ∗R iff X Y H Y ϕ oo is epi in R. (c) ∗R has an injective other than {0} iff monics in ∗R are one-to-one iff R-epis are onto. When this occurs, ∗R is injective-complete, with injective hulls G //C(Y G) //C(P(Y G)). (d) If X is ED, then C(X) is essentially complete in ∗R. Epimorphisms and maximal covers in categories of compact spaces 51 (e) If X ∈ Comp◦, then C(D) CpX //C(PX) is the unique maximal essen- tial extension of C(X) in ∗R. Now consider, as in §5, strongly rigid E ∈ Comp and its generated epireflec- tive R(E). By 5.1 and 6.1 (b), ∗R(E) has monics which are not one-to-one, and thus no 6= {0} injectives. 6.1 (d) and (e) hold in ∗R(E). Note that {0, 1} ∈ Comp has C({0, 1}) = R2 ∈ W ∗, the self-homeomorphisms of {0, 1} are the identity and ”interchange points”, and these correspond to the only self-isomorphisms of R2, 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 of C(E) are ex- actly the W ∗-surjections C(E) ϕ //R2 . Two of these, ϕ and ϕ′, are equivalent iff either ϕ = ϕ′, or ϕ′ = ϕH. References [1] J. Adamek, H. Herrlich and G. Strecker, Abstract and Concrete Categories, Dover 2009. [2] B. Banaschewski, Projective covers in categories of topological spaces and topological algebras, pp. 63-91 in General Topology and its Relations to Modern Analysis and Algebra, Academia 1971. [3] B. Banaschewski and A. Hager, Essential completeness of archimedean ℓ-groups with weak unit, to appear. [4] H. Cook, Continua which admit only the identity mapping onto non-degenerate sub- continua, Fund. Math. 60 (1966) 214–249. [5] R. 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Univ. Carol. 13 (1972), 283–295. (Received December 2011 – Accepted November 2012) 52 B. Banaschewski and A. W. Hager B. Banaschewski Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S4K1, Canada. A. W. Hager (ahager@wesleyan.edu) Department of Mathematics and Computer Science, Wesleyan University, Mid- dletown, CT 06459 USA Epimorphisms and maximal covers in categories[7pt] of compact spaces. By B. Banaschewski and A. W. Hager