Applied Categorical Structures 3: 105-118, 1995. 105 @ 1995 Kluwer Academic Publishers. Printed in the Netherlands.

Solid Hulls of Concrete Categories

LURDES SOUSA Instituto PoIitdcnico de Viseu, Escola Superior de Tecnologia, Campus Politecnico, Repeses, 3500 Viseu, Portugal

(Received: 21 December 1993; accepted: 18 October 1994)

Abstract. This paper deals with the problem of the existence of solid hulls for concrete categories. We present sufficient conditions for a concrete category to have a solid hull. For concrete categories over Set with a small finally dense subcategory, we observe that the existence of solid hulls is equivalent to Weak Vopenka's Principle.

Mathematics Subject Classifications (1991). 18B15, 18A40, 18A22, 18A20.

Key words: Concrete category, finally dense extension, MacNeille completion, solid hull, reflective hull, orthogonal closure.

I n t r o d u c t i o n

Solid categories are concrete categories in which every structured sink has a semi- final lift. These categories, introduced, under different names, by V. Trnkov~i [25], M. B. Wischnewsky [27], W. Tholen [21, 22] and R.-E. Hoffmann [10, 1 1], are known to retain properties of the base category, such as completeness, cocom- pleteness and other convenient ones, and yet to be broad enough to encompass all "well-behaved" categories in Topology and Algebra; see [2] for more details. One property is, however, less satisfactory: there seems to be no general procedure for a construction of a solid extension as small as possible, i.e., a solid hull, of an arbi- trary concrete category. This contrasts with the situation of topological categories, i.e., categories in which every structured sink has a final lift: the topological hull, the so-called MacNeille completion, introduced by H. Herrlich [7], was described generally by J. Ad~imek, H. Herrlich and G. E. Strecker [ 1 ] in the sense that, when- ever the described category is legitimate, this is the topological hull, and, whenever it is not legitimate, a topological hull fails to exist.

In the present paper we study conditions under which a given concrete category has a solid hull, and we relate solid hulls to reflective hulls and orthogonal closures. This continues the research initiated by J. Rosick2~ [17, 18, 19] who presented, inter alia, a concrete category over Set which does not have a solid hull, although it has a finally dense solid extension. In [19], Rosick3) shows that, under the set axiom (M) of the non existence of a proper class of measurable cardinals ((M) implies the

Research partially supported by TEMPUS JEP 2692 and by Centro de Matem~itica da Univer- sidade de Coimbra

106 LURDES SOUSA

negation of Weak Vopenka's Principle, see [6]), there is a concrete category over Set with a small finally dense subcategory which does not have a solid hull. Based on results of J. Ad&nek, J, Rosick3~ and V. Trnkov~i ([4], [6], [19]), we show that, furthermore, the existence of solid hulls for concrete categories over Set with a small finally dense subcategory is equivalent to the large cardinal Weak Vopenka's Principle.

Throughout this paper, all subcategories are assumed to be full and isomorphism-closed. By a concrete category over a category X we shall mean a pair CA,U), where U:.A--+X is a faithful functor which is amnestic, i.e., every .A-isomorphism whose U-image is an identity must be an identity. A full concrete embedding E:(.A,U)--+(B,V) is called an extension of (.A,U). We also say that (B,V) is an extension of (A,U).

A convenient reference for background information is [2].

1. Solid Hull

Solid categories are characterized as those concrete categories which have a Mac- Neille completion and are reflective in it (see [12], [22]). In this section, we will show that, if a concrete category has a solid hull, it is its reflective hull in any finally dense solid extension.

If El: (.A,U)'-"~(J~I, V1) and E2:(.A,U)~(B2,V2) are finally dense extensions of (A,U), we say that E1 is smaller than E2 provided that there exists a, neces- sarily unique (see 1.1), full concrete embedding F:(B1,V1)~ (B2,V2) such that FOElffiE2.

The Lemmas 1.1 and 1.3 below, being surely well-known, are included here for the convenience of the reader.

1.1. LEMMA. Given finally dense full concrete embeddings Ei :(,A, U)---~(13i,Vi), i=1,2, there exists at most one full concrete embedding F:(B1,V1)---ffB2,V2) with FoE1 =E2.

Proof. Let F and F ~ be full concrete embeddings such that FOElfFtoE1---E2. For each BEB1, (fi: E 1 A i ~ B ) I is the sink of all morphisms with codomain B and domain in El(A) if and only if (Ffi: E2Ai~FB)x and (F~fi: E2Ai~FIB)I are the sinks of all morphisms with codomain FB and FIB and domain in E2(.A). Since E2 is finally dense, both sinks (Ffi:E2Ai---~FB)I and (F~fi: E2Ai--~FB);r are final. Therefore, from the concreteness of F, F ~ and the fact that V2 is amnestic, we conclude that FB -- FIB. It is now easy to see that F and F ~ coincide on morphisms too. []

From the above lemma, when E1 is smaller than E2 and E2 is smaller than El, there is a concrete isomorphism F such that FoE1-E2, and so we say that the two finally dense extensions of (.A,U) are isomorphic.

SOLID HULLS OF CONCRETE CATEGORIES 107

1.2. DEFINITION. A finally dense solid extension Es:(A,U)-+(As,U s) is called a solid hull of (A,U) if it is smaller than any other finally dense solid extension of (A,u).

Since, by 1.1, a solid hull, when it exists, is unique up to isomorphism, it will often be called the solid hull.

1.3. LEMMA. A solid category is reflective in each of its finally dense extensions. Proof Let (A,U) be solid and E:(A,U)-+(/3,V) a finally dense extension. If

(fi:EAi--+/3)r is the sink of all morphisms with domain in E(A) and codomain B, let p:VB--~UA be the semi-final lift of the U-structured sink (Vfi:UA<-+VB)I. Since (fi)iis final, p :VB-~VEA is a/3-morphism and it is easy to show that it is universal from B to E. []

Let us remark that the problem of the existence of a solid hull or, even, of a solid extension, makes sense only for concrete categories which have a MacNeille com- pletion. Indeed, if E: (A,U)---~(/3,V) is a solid extension, the MacNeille completion of (/3,V) exists and is a topological extension of (A,U) what guarantees that (A,U) has a MacNeille completion ([1]).

Consequently, from now on, we shall always assume that the concrete categories considered have a MacNeille completion.

1.4. PROPOSITION. A concrete category (A,U) has a solid hull if and only if there exists the reflective hull of A in any finally dense solid extension. Furthermore, if the solid hull exists, it is concretely isomorphic to each one of those reflective hulls.

Proof. Let Es:(A,U)-+(As,U s) be the solid hull of (A,U) and E:(A,U)--+(/3,V) a finally dense solid extension. From 1.1 and 1.2, there exists a unique full concrete embedding F:(As,US)-~(/3,V) such that FoE s = E. Since E is finally dense, F is finally dense and F(A s) is reflective in B, because (As,U s) is solid. To show that F(A s) is the reflective hull of E(,A) in/3, let C be a category containing E(A) and reflective in/3. Then (C,V~), where V ~ is the restriction of V to C, is a finally dense solid extension of ,4, because a reflective concrete subcategory of a solid category is solid (see [22]). Therefore, from 1.1 and 1.2, F(A s) is a subcategory of C.

Conversely, let Et:(A,U)-+(At,U t) be the MacNeille completion of (A,U) and let A~be the reflective hull 0fEt(A) in A t. We want to show that the solid extension (A~,U~), where U ~ is the restriction o fU t to ,4 ~, is a solid hull of (A,U). For a finally dense solid extension E:(A,U)-+(/3,V), let Ft :( /3,V)~(/3t ,v t) be the MacNeille completion of (/3,V). Then, FtoE:(A,U)---+(/3t,v t) is a topological extension of (A,U) and so there is a full concrete embedding G: (At ,u t )~( /3 t ,V t) such that GoEt--Ft oE. By hypothesis, FtoE(A) has a reflective hull in/3t. In order to conclude that it is simultaneously concretely isomorphic to A ~ and concretely isomorphic to the reflective hull of E(A) in/3, it is enough to see that, by 1.3, G and Ftare

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reflective embeddings and to use the definition of reflective hull. Therefore, the extension (Ar,U r) of (A,U) is smaller than the extension (/3,V). []

For a given property P on concrete categories, an extension E:(A,U)--+(/3,V) is called a P-extension provided that (/3,V) satisfies the property P. AP-hull of (A,U) is a finally dense P-extension of (,A,U) which is smaller than any other finally dense P-extension. For several properties P and for some classes g of morphisms, the C-reflective hull on every finally dense P-extension of (A,U) is a P-hull of (A,U) ([20]). But there is an important difference between the solid hull and several other P-hulls: Indeed, the existence of the P-hulls considered in [20] is guaranteed by the one of finally dense P-extensions. However, the same does not hold for the solid hull. In fact, J. Rosick3) ([18 1.2) presents an example of a concrete category over Set which has a finally dense solid extension but does not have a solid hull.

2. Or thogonal Closure and Reflective Hull of a Subcategory

In this section, we consider arbitrary categories and develop some tools that will be used in the sequel. In particular, we give an answer to the so-called Ortho- gonal Subcategory Problem ([14], [23], [28]) which enables us to give sufficient conditions for the orthogonal closure of a subcategory to be its reflective hull.

We recall that, for a morphism f : B ~ C and an object A in a category/3, f is said to be orthogonal to A, and it is written f_J_A, if the map/3(f,A):/3(C,A)---,/3(B,A) is bijective. For a class g of morphisms, E±={A E Obj(/3) [ fJ_A, l e g } and, for a class A of objects, A ±-- {fE Mor(/3) I f l A , AEA}. We shall write g±~ and ¢4 ±z every time that the reference to the category/3 is convenient.

2.1. DEFINITION. The orthogonal closure of a subcategory ¢4 of/3 is the subcat- egory of/3 whose c lassof objects is (A±)±.

2.2. REMARK. For each subcategory .At of B, the orthogonal closure of A is closed under limits in B and it is contained in every reflective subcategory of B containing A. (see, e.g., [14])

For a class of morphisms g in a category/3, we are going to consider the following conditions:

(a) Coequalizer condition: If c:C--~D is the coequalizer of a family of morphisms (fi:B~C)iEZ such that, for some egg , fioe--h for all iEI, then cEg.

(b) Fill-in condition: Given morphisms fEg and g with the same domain, there are morphims f and gl such that l e g and g%f--fog.

(c) Pseudoreflectivity condition: For each BEB, the full subcateg~ory of the comma category B.LB whose objects are the g-morphisms has a weakly terminal object. This weakly terminal object is called an g-pseudoreflection of B.

SOLID HULLS OF CONCRETE CATEGORIES 109

2.3. LEMMA. For a subcategory -4 of a category 13, we have that:

(i) .4 ± satisfies the coequalizer condition; (ii) .42 satisfies the fill-in condition whenever 13 has pushouts or .4 is reflective in 13; (iii) i f .4 is reflective in 13 then .42 satisfies the pseudoreflectivity condition and .4 coincides with its orthogonal closure.

Pro@ (i) If eE.4 2, (fi:B---~C)iEX is a family of morphisms such that fioe=h for every iEI, and c : C ~ D is the coequalizer of (fi:B-+C)iEI, let g:C--+A be a morphism with codomain in .4. Then gofioe=gofjoe and, since eE.4 ±, it implies that gofi=gofj for every i,jEI. Thus, there is a unique morphism g~ such that g%c=g.

The proofs of (ii) and (iii) are straightforward. []

2.4. REMARK. The above conditions (a)-(c) are independent, in the sense that none of them is implied by the others. Indeed:

[(a) and (b)] 7~ (c): If 13 is a cocomplete category and -4 is a subcategory of 13 then £---4± satisfies the coequalizer and the fill-in conditions. But, if the ortho- gonal closure of A is not reflective, then -4± does not fulfil the pseudoreflectivity condition, as we can conclude from 2.5 below. An example of a cocomplete category 13 with a subcategory A whose orthogonal closure is not reflective is given by J. Ad~imek and J. Rosick3) [3], where 13 is the category of bitopological spaces and A is the subcategory of all spaces in which both topologies are compact Hausdorff.

[(a) and (c)] ~+ (b): Let/3 be the poset

c 1 ~ °

and A be the subcategory of 13 having only the object a. It is obvious that A ± satisfies the coequalizer and the pseudoreflectivity conditions but it does not fulfil the fill-in condition.

[(b) and (c)] ¢ , (a): Let 13=Set and let £ be the class of injective maps. Clearly £ does not satisfy the coequalizer condition but it satisfies the fill-in condition and, for each set X, the identity map is an g-pseudoreflection of X.

2.5. PROPOSITION. Let 13 be a category with multiple coequaIizers and let g be a class of 13-morphisms. I f g is closed under composition and satisfies the coequalizer, the fill-in and the pseudoreflectivity conditions, then £ ± is g-reflective in 13.

Proof Let BE13 and d:B---~D be an g-pseudoreflection of B. If c:D--.C is the coequalizer of the family (hi)i~r of all morphisms hi:D-+D that satisfy the equality hio d=d, then, from the coequalizer condition and the fact that g is closed under composition, the morphism e=cod belongs to g. Furthermore, e:B-+C is a terminal

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object of the subcategory defined in the pseudoreflectivity condition, as we now show. Let f be a morphism in g, with domain B. Then, there is some morphism t such that tof--e. If a morphism ( also fulfils t%f=e, let g--coeq(t,(). From the coequalizer condition and the fact that g is closed under composition, goeEg. Then there exists a morphism n such that nogoe=d, i.e., nogocod--d. Hence, nogoc belongs to (h i ) iu and so the equality conogoc=c holds which implies that conog=l. Thus, g is an isomorphism and so t=f.

Now, in order to conclude that e:B--+C is a universal morphism from B to g±, it suffices to show that C belongs to g±. Given morphisms f and g with the same domain such that fEg and the codomain of g is C, from the fill-in condition, there are ffEg and g~ such that g%f--fog. Since g is closed under composition, f oe belongs to g. As e:B--+C is terminal, there is a morphism t such that tofoe=e and so tof~--1. Then, for r=tog ~, we have that rof--g. To show the uniqueness, let r%f--rof and q be the coequalizer of (ry). Then qoe belongs to g and there is some morphism p such that poqoe=e. Thus poq--1 and, consequently, r=g. []

2.6. REMARK. The two maps (-)± and (-)± form a Galois correspondence between classes of morphisms and subcategories of a given category.

It is well-known that, if .4 is a reflective subcategory of 13, then .4±={fE M0r(13)[Rf is an iso}, where R is the reflector. Thus, it is easy to prove that, when Iso(13)Cg and g± is g-reflective, g=(g±)± if and only if g is left-cancellable, i.e., if gofEg with gEg then fEg. Using this fact and proposition 2.5, we can conclude the following: For a category 13 with multiple coequalizers, the maps (-)± and (-)± yield a bijection between the collection of all reflective subcategories of 13 and the collection of all classes of morphisms closed under composition which satisfy the coequalizer, the fill-in and the pseudoreflectivity conditions and are left-cancellable.

Therefore, the reflective hull of a subcategory .4 in 13 exists if and only if the conglomerate of all classes of .4±-morphisms closed under composition, which satisfy the coequalizer, the fill-in and the pseudoreflectivity conditions and are left- cancellable, has a greatest element. The following corollary gives a necessary and sufficient condition for .4± to be such greatest element, when 13 is cocomplete.

2.7. COROLLARY. Let -4 be a subcategory of a cocomplete category 13. Then the orthogonal closure of-4 is the reflective hull of -4in 13 if and only if for each BE13, the full subcategory ofB113 whose objects are the -4±-morphisms has a weakly terminal set.

Proof. Let C be the orthogonal closure of ,.4 in/3. If C is reflective then it is clear that -4± satisfies the above condition since, by 2.6, it coincides with C ±. Conversely, for BE13, assume that (fi:B---~13~)z is a weakly terminal set of the full subcategory of B J.13 whose objects are the -4±-morphisms. If f: B---~C is the multiple pushout of (fi:B--~1301, then it is an -4±-pseudoreflection of B. It is clear that -4x

SOLID HULLS OF CONCRETE CATEGORIES 111

is closed under composition and, by 2.3, it fulfils the coequalizer and the fill-in conditions. Therefore, by 2.5 and 2.2, Cis the reflective hull of A in 13. []

2.8. REMARK. The above corollary can be also obtained as a consequence of 1.1(5), 4.3 and 6.1 of Tholen's [23].

2.9. REMARKS.(1) The example 3.11.4 below shows that the reflective hull may be different from the orthogonal closure.

(2) Let 13 be a category such that, for each morphism f, {f}± is a reflective subcategory. This holds, for example, in each locally presentable category and also in the category of topological spaces (see [14] and [26]). Then, the reflective hull of every subcategory A, if it exists, coincides with the orthogonal closure of A. This follows from 2.2 and the fact that the orthogonal closure of A is the intersection of all {f}± with fEA ±.

We finish this section with a proposition which will be very useful in the sequel.

2.10. PROPOSITION. Let A be a subcategory of 13 and 13 be a reflective subcat- egory o f t . Then the orthogonal closure of A in 13 coincides with the orthogonaI closure of A in C.

Pro@ Since (A ±c)j_ c C(A±z)± c and, by 2.2, (A±c)± c is contained in 13, the orthogonal closure of A in C is contained in the orthogonal closure of ¢4 in I3. To prove the other inclusion, let p:C--+D belong to .4 ±c and rc : C --~C / and rD :D---~D ~ be the 13-reflections, being p/:C/---~D ~ the unique morphism such that p%rc--rDop. We are going to show that p~:C~--+D ~ belongs to A ±~. Let AEA and f:C~--~A be given. Then there is a unique morphism f : D ~ A such that fop=fore and, because rD is a reflection, there exists a unique morphism f* such that f*orD= ft. Since rc is A-epimorphic, we get f*op ~-- f and the uniqueness of f* follows from the fact that rDop is A-epimorphic. Now, let B belong to the orthogonal closure of A in 13 and g:C-~B. If g/:C~---~B is the morphism that fulfils g%rc=g and g*:D~---~B is the morphism such that g*op ~= g/, then it is easily seen that the morphism g*orDis the only one that fulfils the equality (g*orD)op=g. []

3. Orthogonal Closure and Solid Hull of a Concrete Category

The following concept will play an important role in the sequel.

3.1. DEFINITION. By the orthogonaI closure of a concrete category (A,U) we shall mean the extension of (A,U) to the orthogonal closure of its image in the MacNeille completion.

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We will see that, under suitable conditions, the orthogonal closure is a solid hull. However, we point out that the solid hull may be different from the orthogonal closure, as it is shown by the example 3.11.4 below.

3.2. PROPOSITION. The orthogonal closure of a concrete category (.A,U) is smaller than any finally dense solid extension of (.A,U).

Proof Let (,A,U) be a concrete category with the MacNeille completion W:(.A,U)~(.At,Ut), let ..4 o be the orthogonal closure of Et(.A) in .A t, U ° the restric- tion of U t to ,,4o and E°:(.A,U)~(.A°,U °) the corestriction of E t to (A°,U°). If E:(.A,U)~(/3,V) is a finally dense solid extension, let .Aa be the orthogonal closure of E(.A) in/3, U 1 be the restriction of V to .A1 and E 1 :(.A,U)--+(.A x ,U 1) be the corre- sponding extension. Let Ft:(/3,V)---~(/3t,V t) be the MacNeille completion of (/3,V); then, there is a full concrete embedding G:(.At,Ut)--+(/3t,V t) such that GoEt--FtoE. Hence, from 2.10 and 1.3, we have that G yields a concrete isomorphism between (.A°,U°) and the orthogonal closure of F%E(.A) in/3t, which, through F t, is con- cretely isomorphic to (.A1 ,U 1); thus the two finally dense extensions E ° and E 1 are isomorphic. Therefore, it is clear that E ° is smaller than E. []

3.3. COROLLARY. I f the orthogonal closure of a concrete category (A, U) is solid then it is the solid hull of (.A,U). []

For the particular case of concrete categories over Set with a fibre-small Mac- Neille completion, the above proposition is stated in [18] as Theorem 1.1 (see also [19], where the translation from model-theoretic terms to categorical ones is mentioned).

3.4. REMARK. 1. The proof of 3.2 shows that we obtain an equivalent definition of the orthogonal closure of a concrete category if, in 3.1, we replace "the MacNeille completion" by "some finally dense solid extension".

2. Consequently, whenever E:(.A,U)-+(/3,V) is a finally dense solid extension of (,A,U), the orthogonal closure E 1 :(.A,U)~(,A 1 ,U 1) as described above is a solid hull if and only if ..41 is reflective in/3, as we can conclude using 1.3 and the fact that a reflective subcategory of a solid category is solid.

3. Let (A,U) be a concrete category over a cocomplete category and E:(A,U)~(/3,V) be a finally dense solid extension of (.A,U). Then, from 2.7 and the fact that a solid category over a cocomplete category is cocomplete ([22]), the orthogonal closure of (.A,U) is its solid hull if and only if, for each BC/3, the full subcategory of B+/3 whose objects are the [E(.A)]±~-morphisms has a weakly terminal set.

3.5. PROPOSITION. Let (A,U) be a concrete category over a complete, wellpow- ered base category. I f (.A,U) has a fibre-small MacNeille completion and .A has a cogenerating set, then (.A,U) has a solid hull.

SOLID HULLS OF CONCRETE CATEGORIES 1 13

Proof. Let Et:(`A,U)--+(At,u t) be a fibre-small MacNeille completion of (`A,U) (see [1]). Then, from the hypothesis over the base category, it follows that .A t is complete and wellpowered. Now, the proof follows from the Special Adjoint Funct0r Theorem: Let .A1 be the closure under limits of Et(A) in .At. Then, it is complete and wellpowered and has a cogenerating set. Consequently, ,,41 is reflective in .At, thus it is a solid hull. []

In [19] it was shown that any small concrete category over Set has a solid hull. From the above proposition we have the following generalization:

3.6. COROLLARY. Every small concrete category over a complete and wellpow- ered category has a solid hull. []

A concrete category over an ($,M)-category is said to be M-topological if every structured source in A.4 has an initial lift. It is well-known that the following implications hold: topological ~ M-topological ~ solid.

The M-topological hull of a concrete category over an (g,M)-category is the smallest finally dense A/l-topological extension. If it exists, it is the g-reflective hull in the MacNeille completion (see, e.g., [20]).

3.7. LEMMA. Let (A,U) be a concrete category over an ($,M)-category P(, with M CMono-Sources( 2( ), and let Em:(`A,U)---+(`Ara,U TM) be the M-topological hull of (`A,U). Then [Era(A)] ±~'~ is contained in the class of initial bimorphisms of A m"

Proof. Let g:B-+C belong to [E'~(A)] ±,t'~ . Let (fi:B--+E'~Ai)x be the source of all `Am-morphisms with domain B and codomain in Era(A) and let (fi:C--+EmAi)I be such that, for each iEI, fiog=fi. Since Em:(A,U)--+(,A'~,U '~) is the M - topological hull of (A,U),, the two sources are initial rnonosources (see [2]).

The morphism g is initial: let h:UmD-+U'~B be an 2(-morphism such that goh is an Am-morphism. Then, for each iEI, fiogoh is an Am-morphism and so fioh is an Am-morphism. Since (f0I is initial, h is an A'~-morphism.

The morphism g is mono: let r,s EMor(,A m) be such that gor=gos. Then the equal- ities fiogor=fiogos, iEI, imply fior=fios, iEI, and, because (fi)I is a monosource, r = s .

The morphism g is epi: if r,s:C--+D satisfy rog=sog, then, for each t:D--+EmA, torog=tosog and, as gE[Em(A)] ±H~, tor=tos. Thus, since the source of all mor- phisms from D to E'~(A) is a monosource, r=s. []

3.8. THEOREM. Let (A,U) be a concrete category over a cocomplete (g,M)- category 2(, with MCMono-Sources(X), and let E'~:(`A,U)-+(A'~,U TM) be the M-topological hull of (A,U).

(1) If`A TM is co-wellpoweredwith respect to initial bimorphisms then (.A,U) has a solid hull.

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(2) I f in X every epimorphism is split and Epi(.Am)=(Um)-l((Epi2()), then Em:(.A,U)--+(.Am,U m) is the solid hull of (.A,U).

Proof (1) It is immediate from 3.4.3 and 3.7. (2) Let g:B--+C be an initial bimorphism in .Am. Since it is initial and Umg is a

split epimorphism in P(, it follows that g is a split epimorphism in .Am. Hence, g is an .Am-isomorphism. Then, from 3.7, [E'~(.A)]~r, consists of isomorphisms only. Therefore, the orthogonal closure of Em(.A) in A m is Am, and, from 3.3 and 3.4.1, Em:(.A,U)---+(.Am,um) is the solid hull of (.A,U). []

3.9. COROLLARY. If(.A, U) is a concrete category over Set with a monotopological hull in which every epimorphism is a surjection then this is also the solid hull of ( .A,u ). []

3.10. COROLLARY. Let ( .A,U) be a concrete category over a cocomplete ( g,Ad )- category 2(, with jtd CMono-Sources(P(). I f in 2( every epimorphism is split and the MacNeille completion of (.A,U) is the M-topological hull of (.A,U), then it is also the solid hull of (.A,U).

Proof It follows from 3.8.2 and the fact that, if Et:(.A,U)--+(.At,u t) is the MacNeille completion, then Epi(.At)--(U t)- 1 ((EpiA')). []

3.11 EXAMPLES. In the following examples, for each category .A equipped with the obvious forgetful functor, we describe the MacNeille completion, the mono- topological hull, the solid hull and the orthogonal closure of .A, which are denoted by .A t, .Am, .As and .A0, respectively. By ..41 we denote the limit closure of .A in .At. We also describe the classes .A~ and .A~m.

1. For the examples described below in (a)-(c), we have that .A~,-- Iso(.At), .A~m= Iso(.A m) and .41 = .A0 = .As _- ,Am = .At.

(a) ([7]) ,.4 is a partially ordered set (P, _<) considered as a concrete category over the one-morphism category. A MacNeille completion of (P, <) when considered as a poset yields a MacNeille completion of the concrete category ,,4.

(b) ([7]) .A is the concrete category over Set consisting of finite topological spaces and continuous maps. Then .At is the category of finitely generated spaces and continuous maps.

(c) ([1]) .A is the concrete category over Set consisting of compact topological spaces and continuous maps. Then .At is the category of compactly generated spaces.

2. ([9]) A quasi-metric space is a pair (X,d) where X is a set and d is a map d:XxX---+[0,oo] such that, for any x,y, zEX, (1) d(x,y)--d(y,x), (2) d(x,x)--0 and (3) d(x,z)<d(x,y)+d(y,z). A map f:(X,d)--+(Y,e) is called non-expansive if e(f(x),f(y))<d(x,y) for any x,yCX. A quasi-metric space (X,d) is called separated if d(x,y)--0 implies x--y for any x,yEX. A separated quasi-metric space (X,d) is called complete if every Cauchy sequence converges. Let .,4 be the concrete category over Set consisting of complete metric spaces and non-expansive maps.

SOLID HULLS OF CONCRETE CATEGORIES 115

Then .A t is the category of quasi-metric spaces and non-expansive maps, .4~ is its full subcategory of separated quasi-metric spaces and .4s is its full subcategory of complete separated quasi-metric spaces.

([13]) Let Fec be the category of vector spaces over K (K=R, C) and linear maps. A quasi-normed space over K is a pair (X,[[.[[) where XEFec and 1[.[I is a map from X to [0,oo] such that ][AxlI--IA [][x[[ and l[x+y[l_<itx [[+[[yl[ for all x,yEX and AEK. A map f:(X,[[.[[)--~ (Y,I[.[I) is non-expansive if [[f(x)il<_iixl] for any xEX. A quasi-normed space is said to be separated if [ix I ]--0 only if x--0 and it is said to be complete if Cauchy sequences converge. Let ..4 be the category Ban of Banach spaces over K and non-expansive maps. Ban is a concrete category over the category Fec. In this case, .At coincides with the category of quasi-normed spaces and non-expansive maps, Am is its full subcategory of the separated quasi-normed spaces and "4s is its full subcategory of the complete separated quasi-normed spaces.

For these two examples, we have that "4±•t consists of all initial and "4- epimorphic .4t-morphisms, i.e., of all initial and dense .4t-morphisms, and .4±~'~ consists of all dense embeddings. Furthermore, .41 = .40 = .4s ¢ A,~ ¢ .At.

3. An example of a concrete category "4 which has a solid hull coinciding with "4o but different from .At, is given in [18,1.4]. "4 is the category whose objects are pairs (X,x) where X is a non-empty set and x = (Xi)iEOrd is a collection of elements of X such that, if xi---xk for some pair (i, k) with i < k, then, for all j_> i, xj =xi; the morphisms f:(X,x)---~(Y,y) are the maps f:X-+Y for which f(xi)=yi,/COrd. With the obvious forgetful functor, ..4 is a concrete category over Set.

The MacNeille completion of .,4 has, as objects, pairs (X,x) with X a set and x = (Xi)iEO~d a collection of subsets of X such that either all Xi are empty or, for all /COrd, Xi¢O and, i fXi n Xk¢O for some pair (i, k) with i < k, then, for all j>_i, Xj=Xi. A morphism f:(X,x)~(Y,y) is a function f : X ~ Y such that f (XdCYi for every i.

The monotopological hull .Am is the full subcategory of .,4 t consisting of all objects of "4 and the objects (X,x)E"4 t such that X~=~,/COrd.

The class A±.~ ~ consists of all At-isomorphisms and all f:(X,x)~(Y,y) such that Xi¢0 and (i) f(Xd---Yi n f(X)(iEOrd), (ii) Y \ Ui Y i c f ( X ) and (iii) if f(x)=f(x/) then x=x / or x,x~E2(i for some i. On the other hand, .4±~'~ is the class of all .4"Msomorphisms.

Since .4 is complete, hence A 1 = A. Thus we have A 1 ¢"40= As= "4~__fi.4t. 4. The following example is a concrete category with the solid hull different

from the orthogonal closure: Let 2( be the poset

a' ob considered as a category, and let .4 be the subcategory {a}. The pair (.4,E), where E is the inclusion functor, is a concrete category over 2(. Up to isomorphism,

116 LURDES SOUSA

(-4,E) has only two finally dense topological extensions: E:( .4 ,E)~(X,lx) and E~:(.4,E)---~(X~,U), where X ~ is the poset

a b

and Uao=Ua=a, Ub--b, Uo=o and E~a--a. It is easily seen that the category X is simultaneously the reflective hull of,4 in X and in X~; then, E:(-4,E)~(X,lx) is a solid hull of (-4,E). But the orthogonal closure of ,.4 in X is .4. Therefore we have .41 ,~ .40 # .4, = At .

The idea that, for concrete categories over Set, the existence of a solid hull depends on a large cardinal principal is due to J. Rosick3~ who showed, in [19], that, under the axiom (M) of non existence of a proper class of measurable cardi- nals, there is a concrete category over Set with a small finally dense subcategory which does not have a solid hull. Now, we are going to see that this result has a refinement: for concrete categories over Set with a small finally dense subcatego- ry, the existence of solid hulls is equivalent to Weak Vopenka's Principle. (This Principle is weaker than the negation of (M), see [6] or [5].)

If (-4,U) is a concrete category over Set with a small finally dense subcategory C, let ~1~ be the category defined by: Objects are pairs (X,a) where X is a set and a is a U-structured sink with domain in C, codomain X and such that, for all morphisms c : C ~ C of C and g :UC~X in a, we have goUc in a. Morphisms f:(X,a)-*(Y,g) are maps f:X--.Y such that, for each gEa, fogEg. The pair (Ac,Uc), where Uc is defined by Uc(X,a)=X and Uc(f)--f, is a concrete category over Set. Let Ec:A---~Acbe the functor such that, for each AE-4, Ec(A)--(UA.a) where a is the sink of all morphisms Ug with (g:C-+A)EMor(-4) and CEC, and, for each fEMor(-4), Ec(f)=Uf.

3.12. REMARK. We point out some important facts: 1. Ec:(A,U)~(-4c,Uc) is a finally dense topological extension of (A,U) (see

[8]). 2. -4c is a locally presentable category (see 1.18(4) in [5] ). Let Graph be the category of graphs, i.e., the category of binary relations and

homomorphisms between them. We are going to consider the following two set- theoretic principles:

Weak Vopenka's Principle: Ord °p cannot be fully embedded into Graph (where Ord °p is the dual of the category of all ordinals).

Vopenka's Principle: Graph does not have a large discrete subcategory. Vopenka's Principle implies Weak Vopenka's Principle and Weak Vopenka's

Principle implies the negation of (M) (see [6]).

SOLID HULLS OF CONCRETE CATEGORIES 1 17

3.13. REMARK. ([6]) Let B be a locally presentable category. Assuming Weak Vopenka's Principle, the limit closure of each subcategory of/3 is reflective.

3.14. THEOREM. The following assertions are equivalent: (a) Every concrete category over Set with a smaIl finally dense subcategory has a solid hull. (b) Weak Vopenka's Principle holds.

Proof Let (,A,U) be a concrete category over Set and C a small finally dense subcategory of ,A. From 3.12.2 and 3.13, we have that, under Weak Vopenka's Principle, the limit closure of Ec(,A) in ,Ac is its reflective hull, and, by 2.2, 3.4.2 and 3.12.1, it yields the solid hull of (A,U).

Conversely, we are going to show that, under the negation of Weak Vopenka's Principle, there is a concrete category over Set with a small finally dense subcate- gory which does not have a solid hull. Our main tool is a construction given in [4] I. 13. Assuming the negation of Weak Vopenka's Principle, we have:

(i) a class of graphs Li=(Yi,/3i), iEOrd, such that

= f ~ if i<k hom(L i, Lk)

{Li---+Lk} if±_> k, and, since it implies the negation of Vopenka's Principle,

(ii) a class of graphs Di=(Xi, c~i),/COrd, such that

hom(Di,Dk) = / 0 i f iCk {1Di} if i--k.

In the category Str(2,2,1) of structures with two binary and one unary relation consider the objects Bi= (Xi+Yi+{ti}, c~iU Yix { ti }, 13iU Xi x { ti }, { ti }), iEOrd. For each ordinal i, put Bi = I_Ik_<i Bk and let M={v~:B0~Bd[ iEOrd}, where vi:Bo--+Bi is the coproduct injection. Following the technique used in [4] 1.13, we can prove that B0 does not have a reflection in the subcategory A d l of Str(2,2,1).

Let $1=( {0,1}, {(0,1)}, 0,(~), $2=( {0,1}, (a, {(0,1)},{~) and $3=( { 0 }, 0, {0, {0} ). The set T={S1, S2, S3} is finally dense in Str(2,2,1). It is clear that S1, $2, $3 EAA± and, thus, T is a finally dense set of 3,l±. Furthermore Str(2,2,1) is solid (even topological) and, then, it is a finally dense solid extension of M ± . By 2.6, the orthogonal closure of A l l in Str(2,2,1) is 3,l± and, since it is not reflective and Str(2,2,1) is locally presentable, it follows, from 2.9.2, that .AA ± does not have a reflective hull in Str(2,2,1). Therefore, using 1.4, we conclude that AA £ does not have a solid hull. []

Acknowledgements

I am indebted to Professors J. Ad~imek and M. Sobral for very valuable comments and suggestions during the preparation of this paper. In particular, I would like to thank Professor J. Adtimek for showing me Proposition 3.5.

118 LURDES SOUSA

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