Relative injectivity as cocompleteness for a class of distributors

RELA TIVE INJECTIVITY AS COCOMPLETENESS F OR A CLASS OF DISTRIBUTORS MARIA MANUEL CLEMENTINO AND DIRK HOFMANN De dic ate d to Walter Tholen on the o c ca sion of his sixtieth bi rthday Abstract. Notions and tec hniques of enriched cat egory theory can b e used to study top ological struc- tures, l ike metric spaces, top ological spaces and approach spaces, in the cont ext of topological theories. Recen tly in [D. Hofmann, Injective space s via adjunction, arXi v:math.CT/0804.0326] the construct ion of a Y oneda embedding allow ed to identify injectivity of spaces as cocompleteness and to sho w monadicit y of the category of injective s paces and left adjoints ov er Set . I n this pap er we generalis e these r esults, studying cocompleteness with respect to a given class of distributors. W e show in particular that the description of several semantic domains presen ted in [ M . Escard´ o and B. Flagg, Seman tic domains, injec- tiv e spaces and monads, Electronic Notes i n Theoretical Computer Science 20 (1999)] can b e translated int o the V -enriched setting. Introduction This w ork contin ue s the research line of previo us pap ers, aiming to use ca tegorica l to ols in the study of top olog ical structures. Indeed, the p ersp ective prop os ed in [2, 6] of lo o king at top ologica l structures as (Eilenberg-Mo o re) lax alg ebras and, simultaneously , a s a monad enrichmen t of V - enriched c ategories , has shown to be very effective in the s tudy o f sp ecial morphisms – lik e effective des c e nt a nd exp onentiable ones – at a first step [3, 4], and recently in the study o f (Lawv ere/Ca uch y- )co mpleteness and injectivit y [5, 11, 10]. The results we pres ent here co mplement this study of injectivity . More pr ecisely , in the spirit of Kelly-Schmitt [1 2] we gener alise the results of [10], showing that injectivity a nd co completeness – when considered re lative to a cla ss o f distributors – s till coincide. Suitable choices o f this class o f dis tr ibutors allow us to r ecov er, in the V -enriched setting, re s ults on injectivity o f Esca rd´ o-Flagg [7]. The sta r ting p oint of o ur study of injectivity is the no tio n of distributor (or bimo dule, or profunctor), which allow ed the s tudy of weigh ted co limit, presheaf catego ry , and the Y o neda embedding. It was then a natur a l step to ‘relativize ’ these ingr edients and to consider c o c ompleteness wi th r esp e ct to a class of distributors Φ. Namely , w e in tro duce the notion of Φ-co complete category , we constr uct the Φ-pres he a f category , a nd we pr ov e that Φ-co completeness is e quiv alent to the existence of a left adjoint of the Y oneda embedding into the Φ-pr e sheaf category . F urthermore, the clas s Φ determines a class of em b eddings so that the injectiv e T -categ ories with resp ect to this clas s ar e precisely the Φ-c o complete ca tegories. This result links our work with [7], where the authors study systematica lly semant ic do mains and injectivit y characterisations with the help of K o ck-Z¨ oberlein monads. 2000 Mathematics Subje c t Classific ation. 18A05, 18D15, 18D20, 18B35, 18C15, 54B30, 54A20. Key wor ds and phr ases. Quantale, V - category , monad, topol ogical theory , distributor, Y oneda lemma, weigh ted colimit. The authors ackno wledge partial financial assistance by Cent ro de Matem´ atica da Universidade de Coimbra/F CT and Unidade de In vest iga¸ c˜ ao e Desenv olvimento Matem´ atica e Aplica¸ c˜ oes da Universidade de Aveiro/F CT. 1 2 MARIA MANUEL CLEMENTINO AND DIRK HOFM ANN 1. The Setting Throughout this paper we consider a (strict) top olo gic al the ory as introduced in [9]. Such a theory T = ( T , V , ξ ) consists of: (1) a commutativ e quantale V = ( V , ⊗ , k ), (2) a Set -monad T = ( T , e, m ), wher e T and m satisfy (BC); tha t is, T sends pullbac ks to weak pullbacks and each naturality squa re of m is a weak pullback, and (3) a T -algebra struc tur e ξ : T V − → V on V such that: (a) ⊗ : V × V − → V and k : 1 − → V , ∗ 7− → k , ar e T -algebra homomorphis ms making ( V , ξ ) a monoid in Set T ; that is , the following diagra ms T 1 !   T k / / T V ξ   1 k / / V T ( V × V ) T ( ⊗ ) / / h ξ · T π 1 ,ξ · T π 2 i   T V ξ   V × V ⊗ / / V are commutativ e ; (b) F or each set X , ξ X : V X − → V T X , ( X ϕ − → V ) 7− → ( T X T ϕ − → T V ξ − → V ), defines a natural transformatio n ( ξ X ) X : P − → P T : Set − → Ord . Here P : Set − → Ord is the V -p ow erset functor defined as follows. W e put P X = V X with the p oint wise order. Each map f : X − → Y defines a mo notone map V f : V Y − → V X , ϕ 7− → ϕ · f . Since V f preserves all infima and all suprema, it has a left adjoint P f . Explicitly , for ϕ ∈ V X we hav e P f ( ϕ )( y ) = W { ϕ ( x ) | x ∈ X , f ( x ) = y } . Examples. Througho ut this pa per w e will keep in mind the following top olo gical theories: (1) The iden tity theo ry I = ( 1 , V , 1 V ), for each quantale V , where 1 = (Id , 1 , 1) denotes the identit y monad. (2) U 2 = ( U , 2 , ξ 2 ), where U = ( U, e, m ) denotes the ultrafilter monad and ξ 2 is essen tially the ident ity map. (3) U P + = ( U , P + , ξ P + ) where P + = ([0 , ∞ ] op , + , 0) and ξ P + : U P + − → P + , x 7− → inf { v ∈ P + | [0 , v ] ∈ x } . (4) The word theory ( L , V , ξ ⊗ ), for ea ch quantale V , where L = ( L, e, m ) is the word monad and ξ ⊗ : L V − → V . ( v 1 , . . . , v n ) 7− → v 1 ⊗ . . . ⊗ v n () 7− → k Every top olo g ical theory T = ( T , V , ξ ) enco mpasses several interesting ingredients. I. The quantaloid V - Rel with sets a s o b j ects and V -r elations (also called V -matric es , see [1]) r : X × Y − → V as mor phisms. W e use the usual notatio n for relatio ns, denoting the V -relation r : X × Y − → V by r : X − → 7 Y . Since every ma p f : X − → Y can b e thought of as a V -relation f : X × Y − → V thro ugh its graph, ther e is a n injective on ob jects and faithful functor Se t − → V - R el , unless V is deg enerate (i.e. k is the bo ttom element) . Mo reov er, V - Re l has an inv o lution ( − ) ◦ : V - R el − → V - Rel , assigning to r : X − → 7 Y the V -r elation r ◦ : Y − → 7 X , with r ◦ ( y , x ) := r ( x, y ). F or e ach V -rela tion r : X − → 7 Y , the maps ( − ) · r : V - Rel ( Y , Z ) − → V - Rel ( X , Z ) and r · ( − ) : V - R el ( Z, X ) − → V - R el ( Z, Y ) RELA TIVE INJE CTIVITY AS COCOMPLETENE SS FOR A CLASS OF DISTRIBUTORS 3 preserve suprema; hence they hav e right adjoints, ( − ) • − r : V - Rel ( X , Z ) − → V - R el ( Y , Z ) and r − • ( − ) : V - Rel ( Z , Y ) − → V - Rel ( Z, X ) . I I. The Set -functor T extends t o a 2-functor T ξ : V - R el − → V - Rel . T o ea ch V -relatio n r : X × Y − → V , T ξ assigns a V -relation T ξ r : T X × T Y − → V , which is the smalles t (order -preser ving) map s : T X × T Y − → V such that ξ · T r ≤ s · h T π 1 , T π 2 i . T ( X × Y ) h T π 1 ,T π 2 i / / ξ X × Y ( r )= ξ · T r $ $ I I I I I I I I I I T X × T Y T ξ r z z V ≤ Hence, for x ∈ T X a nd y ∈ T Y , T ξ r ( x , y ) = _ n ξ · T r ( w )    w ∈ T ( X × Y ) , T π 1 ( w ) = x , T π 2 ( w ) = y o . This 2-functor T ξ preserves the inv olutio n, i.e. T ξ ( r ◦ ) = T ξ ( r ) ◦ (and we write T ξ r ◦ ) for each V -relation r : X − → 7 Y , m b eco mes a na tural tra ns formation m : T ξ T ξ − → T ξ and e an o p-lax natural transformatio n e : Id − → T ξ , i.e. e Y ◦ r ≤ T ξ r ◦ e X for all r : X − → 7 Y in V - Rel . I I I. A V -relation o f the form α : T X − → 7 Y , called a T -r elation a nd denoted by α : X − ⇀ 7 Y , will play an impor tant role here. Given t wo T -relations α : X − ⇀ 7 Y and β : Y − ⇀ 7 Z , their Kleisli c onvolution β ◦ α : X − ⇀ 7 Z is defined a s β ◦ α = β · T ξ α · m ◦ X . This operation is ass o ciative and has the T -rela tion e ◦ X : X − ⇀ 7 X as a lax identit y: a ◦ e ◦ X = a and e ◦ Y ◦ a ≥ a for any a : X − ⇀ 7 Y . IV. T -rela tions satisfying the usual unit and asso ciativity categorica l rules define T -catego ries: a T - c ate gory is a pair ( X , a ) consisting of a s e t X and a T -rela tion a : X − ⇀ 7 X on X suc h that e ◦ X ≤ a and a ◦ a ≤ a. Expresse d element wise, these conditions b eco me k ≤ a ( e X ( x ) , x ) and T ξ a ( X , x ) ⊗ a ( x , x ) ≤ a ( m X ( X ) , x ) for a ll X ∈ T T X , x ∈ T X a nd x ∈ X . A function f : X − → Y b etw een T -catego ries ( X , a ) and ( Y , b ) is a T -fun ctor if f · a ≤ b · T f , which in point wise notatio n r eads as a ( x , x ) ≤ b ( T f ( x ) , f ( x )) for all x ∈ T X , x ∈ X . The c ategory of T -catego r ies and T -functors is denoted by T - Cat . V. In particula r , the quantale V is a T -ca tegory V = ( V , ho m ξ ), where hom ξ : T V × V − → V , ( v , v ) 7− → ho m( ξ ( v ) , v ) . VI. The for getful functor O : T - Ca t − → Set , ( X, a ) 7− → X , is top olo gic al , hence it has a left and a right adjoint. In particular, the free T -catego ry on a one-element set is given by G = (1 , e ◦ 1 ). VI I. A V -r elation ϕ : X − ⇀ 7 Y b etw een T -catego ries X = ( X , a ) and Y = ( Y , b ) is a T -distributor , denoted as ϕ : X − ⇀ ◦ Y , if ϕ ◦ a ≤ ϕ a nd b ◦ ϕ ≤ ϕ . Note that we alwa y s hav e ϕ ◦ a ≥ ϕ and b ◦ ϕ ≥ ϕ , so that the T - dis tributor conditions ab ov e are in fact equalities. T -ca tegories and T -distr ibutors form a 2- category , denoted by T - Mo d , with Kleisli co nv o lutio n as comp o s ition and with the 2-ca tegorical structure inherited from V - Rel . 4 MARIA MANUEL CLEMENTINO AND DIRK HOFM ANN VI I I. Each T -functor f : ( X , a ) − → ( Y , b ) induc es an adjunction f ∗ ⊣ f ∗ in T - Mo d , with f ∗ : X − ⇀ 7 Y and f ∗ : Y − ⇀ 7 X defined as f ∗ = b · T f and f ∗ = f ◦ · b respectively . In fact, these assignment s ar e functorial, i.e. they define tw o functors: ( − ) ∗ : T - Cat co − → T - Mo d and ( − ) ∗ : T - Cat op − → T - Mo d , X 7− → X ∗ = X X 7− → X ∗ = X f 7− → f ∗ = b · T f f 7− → f ∗ = f ◦ · b A T -functor f : X − → Y is called ful ly faithf ul if f ∗ ◦ f ∗ = 1 ∗ X , while it is called dense if f ∗ ◦ f ∗ = 1 ∗ Y . Note that f is fully faithful if a nd only if, for all x ∈ T X and x ∈ X , a ( x , x ) = b ( T f ( x ) , f ( x )). IX. F or a T -distributor α : X − ⇀ ◦ Y , the c omp osition fun ction − ◦ α has a right adjoint ( − ) ◦ − α w her e, for a giv e n T - distributor γ : X − ⇀ ◦ Z , the extensio n γ ◦ − α : Y − ⇀ ◦ Z is constructed in V - Rel as the extension γ ◦ − α = γ • − ( T ξ α · m ◦ X ). T X  γ / / _ m ◦ X   Z. T T X _ T ξ α   T Y K E E The following r ules are easily check ed. Lemma. The fol lowing assertions hold. (1) If α is a right adjoi nt, then α ◦ ( ϕ ◦ − ψ ) = ( α ◦ ϕ ) ◦ − ψ . (2) If γ ⊣ δ , then ( α ◦ − β ) ◦ γ = α ◦ − ( δ ◦ β ) . (3) If γ ⊣ δ , then ( α ◦ γ ) ◦ − β = α ◦ − ( β ◦ δ ) . X. It is also impo rtant the interplay of several functors relating the structures, i.e. Eilenb er g-Mo or e algebr as , T -c ate gories and V -c ate gories . The inclusion functor Set T ֒ → T - Cat , given by regarding the structure map α : T X − → X of an Eilenberg-Mo ore a lgebra ( X , α ) as a T -r elation α : X − ⇀ 7 X , has a left adjoint, cons tructed ` a la ˇ Ce ch-Stone c omp actific ation in [2 ]. Set T   ⊥ / / T - Cat u u W e deno te by | X | the fr e e Eilenberg- Mo ore a lgebra ( T X , m X ) consider ed as a T -catego ry . Making use of the identit y e : Id − → T of the monad, to eac h T -ca tegory X = ( X, a ) w e assign a V -categ o ry str uc tur e on X , a · e X : X − → 7 X . This corre s po ndence defines a functor S : T - Cat − → V - Cat , which has also a left adjoint A : V - Cat − → T - Cat , with A( X , a ) := ( X , e ◦ X · T ξ r ). T - Cat ⊥ S / / V - Cat . A u u F urthermor e, making no w use of the m ultiplica tion m : T 2 − → T of the mo nad, one can define a functor M : T - Cat − → V - Cat which sends a T -catego ry ( X , a ) to the V -catego ry ( T X , T ξ a · m ◦ X ). RELA TIVE INJE CTIVITY AS COCOMPLETENE SS FOR A CLASS OF DISTRIBUTORS 5 W e ca n now define the pro cess of dualizing a T -c ate gory as the compo s ition of the following functors T - Cat M   ( ) op / / T - Cat V - Cat ( ) op / / V - Cat A O O that is, the dual of a T -c ate gory ( X, a ) is defined a s X op = A(M( X ) op ) , which is a structure on T X . If T is the identit y monad, then X op is indeed the dual V -ca tegory of X . XI. The tensor pro duct on V ca n b e tr ansp orted to T - Cat by putting ( X, a ) ⊗ ( Y , b ) = ( X × Y , c ) , with c ( w , ( x, y )) = a ( T π 1 ( w ) , x ) ⊗ b ( T π 2 ( w ) , y ) , where w ∈ T ( X × Y ), x ∈ X , y ∈ Y . The T -categor y E = (1 , k ) is a ⊗ -neutral ob ject, where 1 is a singleton set and k : T 1 × 1 − → V the constant relation with v alue k ∈ V . F or e a ch set X , the functor | X | ⊗ ( − ) : T - Cat − → T - Cat has a right a djoint ( − ) | X | : T - Cat − → T - Cat . Explicitly , the structure J − , − K on V | X | is given by the formula J p , ψ K = ^ q ∈ T ( | X |× V | X | ) q 7− → p hom( ξ · T ev( q ) , ψ ( m X · T π 1 ( q ))) , for each p ∈ T V | X | and ψ ∈ V | X | . Theorem. [5] F or T -c ate gories ( X, a ) and ( Y , b ) , and a T -r elation ψ : X − ⇀ 7 Y , the fol lowing assertions ar e e quivalent. (i) ψ : ( X , a ) − ⇀ ◦ ( Y , b ) is a T -distributor. (ii) Both ψ : | X | ⊗ Y − → V and ψ : X op ⊗ Y − → V ar e T -funct ors. XI I. Hence, e ach T -distributor ϕ : X − ⇀ ◦ Y pr ovides a T -functor p ϕ q : Y − → V | X | which factors throug h the embedding P X ֒ → V | X | , wher e P X = { ψ ∈ V | X | | ψ : X − ⇀ ◦ G } is the T -categ ory o f c ontr avariant pr eshe afs on X : Y p ϕ q / / p ϕ q ! ! C C C C C C C C V | X | P X  ? O O In particula r, for ea ch T -ca tegory X = ( X , a ), the V -re la tion a : T X × X − → V is a T -distributor a : X − ⇀ ◦ X , and therefore we have the Y one da functor y X = p a q : X − → P X . Theorem. [10] L et ψ : X − ⇀ ◦ Z and ϕ : X − ⇀ ◦ Y b e T -distributors. Then, for al l z ∈ T Z and y ∈ Y , J T p ψ q ( z ) , p ϕ q ( y ) K = ( ϕ ◦ − ψ )( z , y ) . 6 MARIA MANUEL CLEMENTINO AND DIRK HOFM ANN Corollary . [1 0] F or e ach ϕ ∈ ˆ X and e ach x ∈ T X , ϕ ( x ) = J T y X ( x ) , ϕ K , that is, ( y X ) ∗ : X − ⇀ ◦ ˆ X is given by the evaluation map ev : T X × ˆ X − → V . As a c onse qu en c e, y X : X − → ˆ X is ful ly faithful. XI I I. T r ansp orting the order- structure on hom-sets fr om T - Mo d to T - Cat via the functor ( − ) ∗ : T - Cat op − → T - Mo d , T - Cat b e c omes a 2-c ate gory . That is, for T -functor s f , g : X − → Y we define f ≤ g whenever f ∗ ≤ g ∗ , which in tur n is equiv alent to g ∗ ≤ f ∗ . W e call f , g : X − → Y e qu ivalent , and write f ∼ = g , if f ≤ g and g ≤ f . Hence, f ∼ = g if and only if f ∗ = g ∗ if a nd only if f ∗ = g ∗ . A T - categor y X is called sep ar ate d (see [11] for details) whenever f ∼ = g implies f = g , for a ll T -functors f , g : Y − → X with co domain X . O ne e a sily v erifies that the T -c ate gory V = ( V , ho m ξ ) is sep ar ate d , and so is each T -categor y of the for m P X for a T -ca tegory X . The full sub catego r y of T - Cat co nsisting of all separa ted T -categor ies is denoted by T - Cat sep . The 2-catego rical structure on T - Cat allows us to c onsider adjoint T -functors: T -functor f : X − → Y is left adjoint if there exists a T - functor g : Y − → X such that 1 X ≤ g · f a nd 1 Y ≥ f · g . Considering the corre sp onding T -distributors , f is left adjoint to g if and only if g ∗ ⊣ f ∗ , that is, if and only if f ∗ = g ∗ . A more co mplete study of this sub ject can be found in [9, 1 0]. 2. The resul ts In the s e quel we consider a class Φ of T -distributors sub ject to the following axioms. (Ax 1).: F or each T -functor f , f ∗ ∈ Φ. (Ax 2).: F or all ϕ ∈ Φ and all T -functor s f : A − → X we hav e f ∗ ◦ ϕ ∈ Φ , ϕ ◦ f ∗ ∈ Φ , f ∗ ∈ Φ ⇒ ϕ ◦ f ∗ ∈ Φ; whenever the compo sitions are defined. (Ax 3).: F or all ϕ : X − ⇀ ◦ Y ∈ T - Mo d , ( ∀ y ∈ Y . y ∗ ◦ ϕ ∈ Φ) ⇒ ϕ ∈ Φ where y ∗ is induced by y : 1 − → Y , ∗ 7− → y . Condition (Ax 2) requires that Φ is closed under certain compositio ns . In fact, in mos t examples Φ will be closed under ar bitrary comp ositio ns . F urthermore, there is a lar gest and a smallest suc h class of T -distributors, namely the class P of a ll T -distributors a nd the class R = { f ∗ | f : X − → Y } of all representable T -distributors. W e call a T -functor f : X − → Y Φ -de nse if f ∗ ∈ Φ. Certainly , if f is a left a djoint T -functor, with f ⊣ g , then f ∗ = g ∗ ∈ Φ, i.e. f is Φ-dense. A T -catego ry X is called Φ -inje ctive if, for all T -functor s f : A − → X and fully fa ithful Φ-dense T -functors i : A − → B , there exists a T -functor g : B − → X such that g · i ∼ = f . F urthermo re, X is called Φ -c o c omplete if each weigh ted dia gram Y h / / ◦ ϕ  X Z with ϕ ∈ Φ ha s a colimit g ∼ = colim( ϕ, h ) : Z − → X . A T -functor f : X − → Y is Φ -c o c ontinuous if f preserves a ll existing Φ-weight ed colimits. Note that in b o th cases it is eno ugh to consider diag rams where h = 1 X . W e denote by T - Co con t Φ the 2-ca tegory of a ll Φ - co complete T -categor ies and Φ- co contin uo us T -functors , and by T - Co cont Φ sep its full sub ca tegory of all Φ-c o complete and s eparated T -ca tegories . If Φ is the cla s s P of all T -distributor s, then T - Co cont Φ is the c a tegory o f co complete T -catego ries and left adjoint T -functors (as s hown in [1 0, Pro p. 2.12 ]). RELA TIVE INJE CTIVITY AS COCOMPLETENE SS FOR A CLASS OF DISTRIBUTORS 7 Lemma. Consider the (u p to isomorphi sm) c ommutative triangle X f   h ∼ = @ @ @ @ @ @ @ Y g / / Z of T -functors. Then t he fol lowing assertions hold. (1) If g and f ar e Φ -dense, then so is h . (2) If h is Φ -dense and g is ful ly faithful, then f is Φ - dense. (3) If h is Φ -dense and f is dense, then g is Φ -dense. Pr o of. The pr o of is stra ightforw ard: (1 ) h ∗ = g ∗ ◦ f ∗ ∈ Φ by (Ax 2 ), since g ∗ , f ∗ ∈ Φ; (2) f ∗ = g ∗ ◦ g ∗ ◦ f ∗ = g ∗ ◦ h ∗ ∈ Φ by (Ax 2), s ince h ∗ ∈ Φ; (3) g ∗ = g ∗ ◦ f ∗ ◦ f ∗ = h ∗ ◦ f ∗ ∈ Φ by (Ax 2), since h ∗ ∈ Φ.  W e put no w Φ X = { ψ ∈ P X | ψ ∈ Φ } considered as a s ub ca tegory of P X . W e have the restr iction y Φ X : X − → Φ X of the Y oneda map, and eac h ψ ∈ Φ X is a Φ-weigh ted co limit o f repre s entables (see [10, Prop osition 2.5]). Lemma. The fol lowing assertions hold. (1) y Φ X : X − → Φ X is Φ - dense. (2) F or e ach T -distributor ϕ : X − ⇀ ◦ Y , ϕ ∈ Φ if and only if p ϕ q : Y − → P X factors thr ough the emb e dding Φ X ֒ → P X . Pr o of. By the Y oneda Lemma (Coro lla ry 1 ), f or any ψ ∈ Φ X we hav e ψ ∗ ◦ ( y Φ X ) ∗ = ψ ∈ Φ, th erefor e ( y Φ X ) ∗ ∈ Φ by (Ax 3) a nd the asser tion (1 ) follows. T o see (2 ), just obser ve that p ϕ q ( y ) = y ∗ ◦ ϕ , and us e again (Ax 3).  Our next result extends Theor e m 2.6 of [10]. W e omit its pro o f because it uses exa c tly the same arguments. Theorem. The fol lowing assertions ar e e quivalent, for a T -c ate gory X . (i) X is Φ -inje ctive. (ii) y Φ X : X − → Φ X has a left inverse Sup Φ X : Φ X − → X . (iii) y Φ X : X − → Φ X has a left adjoint Sup Φ X : Φ X − → X . (iv) X is Φ -c o c omplete. Recall from [10] that, for a giv en T -functor f : X − → Y , w e have an a djoint pair o f T -functors P f ⊣ f − 1 where P f : P X − → P Y and f − 1 : P Y − → P X . ψ 7− → ψ ◦ f ∗ ψ 7− → ψ ◦ f ∗ By (Ax 1) a nd (Ax 2), the T -functor P f : P X − → P Y res tr icts to a T -functor Φ f : Φ X − → Φ Y . On the other hand, f − 1 : P Y − → P X r e stricts to f − 1 : Φ Y − → Φ X pro vided that f is Φ-dense . Prop ositi on. T he fol lowing c onditions ar e e quivalent for a T -funct or f : X − → Y . 8 MARIA MANUEL CLEMENTINO AND DIRK HOFM ANN (i) f is Φ -dense. (ii) Φ f is left adjo int. (iii) Φ f is Φ -dense. Pr o of. (i) ⇒ (ii): If f is Φ-dense, then Φ f ⊣ f − 1 : Φ Y − → Φ X defined a bove. (ii) ⇒ (iii): If Φ f ⊣ g , then (Φ f ) ∗ = g ∗ ∈ Φ, i.e. Φ f is Φ- de ns e. (iii) ⇒ (i): Conside r the diagra m X y Φ X / / f   Φ X Φ f   Y y Φ Y / / Φ Y If Φ f is Φ- de ns e, then y Φ Y · f = Φ f · y Φ X is Φ-dense, and so by 2(2) f is Φ-dense b eca use y Φ Y is fully faithful.  In particular, for e a ch T -categor y X , Φ y Φ X : Φ X − → ΦΦ X has a right adjoint, ( y Φ X ) − 1 . W e show next that ( y Φ X ) − 1 has also a right adjoint, y Φ Φ X : Φ X − → ΦΦ X , so that: Φ y Φ X ⊣ ( y Φ X ) − 1 = Sup Φ Φ X ⊣ y Φ Φ X . Prop ositi on. F or e ach T -c ate gory X , Φ X is Φ - c o c omplete wher e Sup Φ Φ X = ( y Φ X ) − 1 . Pr o of. Since y Φ X is Φ-dense, we may define Sup Φ Φ X := ( y Φ X ) − 1 . W e hav e to sho w that Sup Φ Φ X is a left inv ers e for y Φ Φ X ; that is, ( y Φ X ) − 1 · y Φ Φ X = 1 Φ X : for each ψ ∈ Φ X , (( y Φ X ) − 1 · y Φ Φ X )( ψ ) = ψ ∗ ◦ ( y Φ X ) ∗ = ψ .  In [1 0] we constr ucted P f as the colimit P f ∼ = colim(( y X ) ∗ , y Y · f ), and a straightforward calculation shows that also Φ f ∼ = colim(( y Φ X ) ∗ , y Φ Y · f ), for each T -functor f : X − → Y . T o see this, we consider the commutativ e diagr ams X y Φ X / / f   y X # # Φ X , i X / / Φ f   P X P f   Y y Φ Y / / y Y ; ; Φ Y i Y / / P Y and obtain (Φ f ) ∗ = i ∗ Y ◦ i Y ∗ ◦ (Φ f ) ∗ = i ∗ Y ◦ ( P f ) ∗ ◦ i X ∗ = i ∗ Y ◦ (( y Y ∗ ◦ f ∗ ) ◦ − y X ∗ ) ◦ i X ∗ since P f ∼ = colim(( y X ) ∗ , y Y · f ) = ( i ∗ Y ◦ y Y ∗ ◦ f ∗ ) ◦ − ( i X ∗ ◦ y X ∗ ) by Lemma 1 = ( y Φ Y ∗ ◦ f ∗ ) ◦ − y Φ X ∗ . Prop ositi on. L et f : X − → Y a T -fun ctor wher e X and Y ar e Φ -c o c omplete. (1) The fol lowing assertions ar e e quivalent. (a) f is Φ -c o c ontinuous. RELA TIVE INJE CTIVITY AS COCOMPLETENE SS FOR A CLASS OF DISTRIBUTORS 9 (b) W e have f · Sup Φ X ∼ = Sup Φ Y · Φ f . Φ X Φ f / / Sup Φ X   ∼ = Φ Y Sup Φ Y   X f / / Y (2) If f is Φ - c o c ontinuous, then f is Φ -dense if and only it is a left adjo int. Pr o of. (1) (a) ⇒ (b): Recall that X 1 X / / ◦ ( y Φ X ) ∗  X Φ X (Sup Φ X ) ∗ =1 X ◦ − ( y Φ X ) ∗ = = Hence ( f · Sup Φ X ) ∗ = f ∗ ◦ − ( y Φ X ) ∗ = ((Sup Φ Y ) ∗ ◦ ( y Φ Y ) ∗ ◦ f ∗ ) ◦ − ( y Φ X ) ∗ = (Sup Φ Y ) ∗ ◦ (( y Φ Y ) ∗ ◦ f ∗ ◦ − ( y Φ X ) ∗ ) = (Sup Φ Y ) ∗ ◦ Φ f ∗ . (b) ⇒ (a): Cons ider X ◦ 1 ∗ X / ◦ ϕ  X f / / Y A (Sup Φ X · p ϕ q ) ∗ > > Then ( f · Sup Φ X · p ϕ q ) = Sup Φ Y · Φ f · p ϕ q = Sup Φ Y · p ϕ · f ∗ q ∼ = colim( ϕ, f ) (2) If f is Φ-co contin uous and Φ-dense, from the commut ative dia gram o f (1 )(b) we hav e f ⊣ Sup Φ X · f − 1 · y Φ Y since f · Sup Φ X = Sup Φ Y · Φ f ⊣ f − 1 · y Φ Y and Sup Φ X · y Φ X = 1 X . The c o nv er se is tr ivially true.  Corollary . Φ X is close d in P X under Φ -weighte d c olimits. Pr o of. W e show that the inclusion functor i : Φ X − → P X is Φ-co contin uous, which, b y the prop os ition ab ov e, is eq uiv alent to the commutativit y of the diagram ΦΦ X Φ i / / Sup Φ Φ X   Φ P X Sup Φ P X   Φ X i / / P X . In Prop ositio n 2 we obser ved Sup Φ Φ X = ( y Φ X ) − 1 , and fro m Theorem 2 and [1 0, Theorem 2.8] follows that Sup Φ P X is the res triction of y − 1 X : P P X − → P X to Φ P X . Let Ψ ∈ ΦΦ X . Then i · ( y Φ X ) − 1 (Ψ) = Ψ ◦ ( y Φ X ) ∗ 10 MARIA MANUEL CLEMENTINO AND DIRK HOFM ANN and y − 1 X · Φ i (Ψ) = y − 1 X (Ψ ◦ i ∗ ) = Ψ ◦ i ∗ ◦ ( y X ) ∗ = Ψ ◦ ( y Φ X ) ∗ , and the ass ertion follows.  Theorem 2 says in particula r that, for each T -functor f : A − → X , Φ-injective T -ca tegory X and fully faithful Φ-dense T -functor i : A − → B , we hav e a canonica l extensio n g : B − → X of f along i , namely g ∼ = colim( i ∗ , f ), giving us a n alterna tive description of Φ f . Theorem. Comp osition with y Φ X : X − → Φ X defines an e quivalenc e T - Co co nt Φ (Φ X , Y ) − → T - Cat ( X , Y ) of or der e d sets, for e ach Φ -c o c omplete T -c ate gory Y . The ser ie s of re s ults ab ov e tell us that T - Co cont Φ sep is actually a (non- full) reflective subca tegory of T - Cat , w ith left a djoint Φ : T - Cat − → T - Co cont Φ sep . In fact, Φ is a 2- functor and one verifies as in [10] that the induced monad I Φ = (Φ , y Φ , ( y Φ ) − 1 ) o n T - Cat is of Ko ck-Z¨ oberlein type. Theo r em 2 and Prop ositio n 2 imply that T - Co co nt Φ sep is equiv alent to the ca tegory of Eilenber g-Mo or e algebra s o f I Φ . Finally , w e wish to study monadicity of the canonical forg etful functor G : T - Co cont Φ sep − → Set . Certainly , (a): G has a left adjoint given by the comp osite Set disc − − − − − − → T - Cat Φ − − − − → T - Co cont Φ sep , where disc( X ) = ( X , e ◦ X ), and disc( f ) = f . In order to pr ov e mona dicit y of G w e will imp ose, in addition to (Ax 1)-(Ax 3), (Ax 4).: F or each surjective T -functor f , f ∗ ∈ Φ. Hence, a ny bijective f : X − → Y in T - Co con t Φ sep is Φ-dense and therefor e left a djoint. By [10, Lemma 2.16], f is inv ertible and we have seen that (b): G reflects iso morphisms. In order to c o nclude that G is monadic, it is left to show that (c): T - Co cont Φ sep has and G pr eserves co eq ua liser of G -equiv alence relations (see, for instance, [14, Corollar y 2.7 ]). T o do so, let π 1 , π 2 : R ⇒ X in T - Co cont Φ sep be a n equiv alence relation in Set , where π 1 and π 2 are the pr o jection maps, a nd let q : X − → Q b e its co equaliser in T - Cat . The pro of in [1 0, Section 2.6 ] rests on the observ ation tha t P R P π 1 / / P π 2 / / P X P q / / P Q is a split for k in T - Cat sep . Naturally , we wish to s how that, in our setting, Φ R Φ π 1 / / Φ π 2 / / Φ X Φ q / / Φ Q gives r is e to a split fo rk in T - Cat sep as well. Since π 1 , π 2 and q are sur jective, the T -functors π 1 , π 2 and q are Φ-dense and therefore we hav e T -functors q − 1 : Φ Q − → Φ X and π − 1 1 : Φ X − → Φ R . F urthermore, Φ q · q − 1 = 1 Φ X = Φ π 1 · π − 1 1 . It is left to show that q − 1 · Φ q = Φ π 2 · π − 1 1 , RELA TIVE INJE CTIVITY AS COCOMPLETENE SS FOR A CLASS OF DISTRIBUTORS 11 which can b e s hown with the same calculation a s in [10], based o n the following prop osition. Prop ositi on. C onsider the fol lowing diagr am in T - Cat R π 1 / / π 2 / / X q / / Q with π 1 , π 2 : R ⇒ X in T - Co cont Φ sep , ( π 1 , π 2 ) an e quivalenc e r elation in Set , and q : X − → Q its c o e qualiser in T - Cat . (1) If π 1 , π 2 ar e left adjoints, then q is pr op er. (2) The diagr am Φ R Φ π 1 / / Φ π 2 / / Φ X π − 1 1   Φ q / / Φ Q q − 1   is a split fork in T - Cat . Pr o of. (1) As in [10, Lemma 2.1 9 and Corollar y 2.20 ]. (2) Analog ous to the pro o f pr esented in [1 0, Section 2.6].  Finally , w e conclude that: Theorem. Under (Ax 1)-(Ax 4), t he for getful functor G : T - Co co nt Φ sep − → Set is monadic. Pr o of. In order to show that T - Coco nt Φ sep has and G preserves co equalis er of G -e quiv alence relations, consider again the fir st diagr am of Prop ositio n 2. W e hav e seen that Φ R Φ π 1 / / Φ π 2 / / Φ X π − 1 1   Φ q / / Φ Q q − 1   is a split fork a nd hence a co e qualiser diag ram in T - Cat . Since π 1 and π 2 are Φ-co c o ntin uous, there is a T -functor Sup Φ Q : Φ Q − → Q which, since q : X − → Q is the co e q ualiser of π 1 , π 2 : R ⇒ X in T - Cat , satisfies Sup Φ Q · y Φ Q = 1 Q . The situation is depicted in the following diagra m. R π 1 / / π 2 / / y Φ R   X q / / y Φ X   Q y Φ Q   1 Q y y Φ R Φ π 1 / / Φ π 2 / / Sup Φ R   Φ X Φ q / / Sup Φ X   Φ Q Sup Φ Q   R π 1 / / π 2 / / X q / / Q W e conclude that Q is separated and Φ- co complete, and q : X − → Q is Φ-coc ontin uous. Finally , to see that q : X − → Q is the coeq ualiser of π 1 , π 2 : R ⇒ X in T - Coc ont Φ sep , le t h : X − → Y b e in T - Coco nt Φ sep with h · π 1 = h · π 2 . Then, since Φ q is the co equalis er of Φ π 1 , Φ π 2 : Φ R ⇒ Φ X in T - Co cont Φ sep , there is a Φ-co contin uous T -functor f : Φ Q − → Y such that f · Φ q = h · Sup Φ X . Then f · y Φ Q · q = f · Φ q · y Φ X = h · Sup Φ X · y Φ X = h 12 MARIA MANUEL CLEMENTINO AND DIRK HOFM ANN and Sup Φ Y · Φ f · Φ y Φ Q · Φ q = f · Sup Φ Φ Q · Φ y Φ Q · Φ q = f · Φ q = h · Sup Φ X = f · y Φ Q · q · Sup Φ X = f · y Φ Q · Sup Φ Q · Φ q , hence Sup Y · Φ( f · y Φ Q ) = f · y Φ Q · Sup Φ Q , that is , f · y Φ Q is Φ-co contin uous.  3. The examples 3.1. All distributors. The class Φ = P o f a ll distributors sa tisfies obviously all four axioms. In fact, this is the situation studied in [1 0]. 3.2. Representable distributors. The smalles t p os sible choice is Φ = R being the c la ss o f a ll rep- resentable T -distributors R = { f ∗ | f is a T -functor } . Clearly , R satisfies (Ax 1), (Ax 2) and (A x 3 ) but not (Ax 4). W e hav e R ( X ) = { x ∗ | x ∈ X } , each T -categor y is R -co c o mplete and each T -functor is R -coco ntin uous, a nd therefore T - Co cont R sep = T - Cat sep . This case is certainly not v e r y in ter e sting; how ever, our results tell us that the inclusio n functor T - Cat sep ֒ → T - Cat is mo na dic. In particular, the c ate gory T op 0 of top olo gic al T 0 -sp ac es and c ontinuous maps is a monadic su b c ate gory of T op . 3.3. Almost representable dis tributors. W e can mo dify slightly the example ab ov e and consider Φ = R 0 the class of all almo st representable T -distributors, wher e a T - distributor ϕ : X − ⇀ ◦ Y is called almost r epr esentable whenever, fo r each y ∈ Y , either y ∗ ◦ ϕ = ⊥ or y ∗ ◦ ϕ = x ∗ for some x ∈ X . As ab ov e, R 0 satisfies (Ax 1 ), (Ax 2 ) and (Ax 3) but not (Ax 4). By definition, for a T - categor y X we hav e R 0 ( X ) = { ψ ∈ P X | ψ ∈ R 0 } = { x ∗ | x ∈ X } ∪ {⊥ } , with the structure inherited from P X . F urthermor e, a T -functor f : ( X , a ) − → ( Y , b ) is R 0 -dense whenever, for each y ∈ Y , ∃ x ∈ T X . b ( T f ( x ) , y ) > ⊥ ⇒ ∃ x ∈ X ∀ x ∈ T X . b ( T f ( x ) , y ) = a ( x , x ) . Hence, with Y 0 = { y ∈ Y | ∃ x ∈ T X . b ( T f ( x ) , y ) > ⊥} we can factor ise a n R 0 -dense T -functor f : X − → Y as X f − − → Y 0 ֒ → Y , where Y 0 ֒ → Y is fully faithful and X f − → Y 0 is left a djoint. If w e consider f : X − → Y in T o p , then Y 0 = f ( X ) is the closur e of the image of f , so that each R 0 -dense co nt inuous map factor s as a left adjoint contin uous map follow ed by a clo sed em b edding. Consequently , for a top olo gic al sp ac e X , t he fol lowing assertions ar e e quivalent: (i) X is inje ctive with r esp e ct to R 0 -dense ful ly faithf ul c ontinuous maps. (ii) X is inje ctive with r esp e ct to close d emb e ddings. Note that in this example we are working with the dual or der, compared with [7, Section 11 ]. RELA TIVE INJE CTIVITY AS COCOMPLETENE SS FOR A CLASS OF DISTRIBUTORS 13 3.4. Right adjoint distributors. Now we consider Φ = L the class of all r ight a djoint T -distributor s. This class co n tains all dis tributors of the form f ∗ , for a T -functor f , a nd it is closed under co mpo sition. Since adjo intness of a T -distributor ϕ : X − ⇀ ◦ Y can b e tested p o int wise in Y , the a xioms (Ax 1), (Ax 2) and (Ax 3) are sa tisfied. By definit ion, L ( X ) = { ψ ∈ P X | ψ is right adjoint } , and a T -catego ry is L -co complete if each pair ϕ ⊣ ψ , ϕ : Y − ⇀ ◦ X , ψ : X − ⇀ ◦ Y , o f adjoint T -distr ibutors is of the form f ∗ ⊣ f ∗ , for a T -functor f : Y − → X . F o r V -categor ies, this is precisely the w ell-known notion of Cauch y- completeness as in tro duced by La wvere in [13] as a genera lisation of the clas sical notion for metric spa c e s. How ever, Lawv ere never pr op osed the name “Ca uch y-co mplete”, and, while working on this notion in the context of T -categor ies in [5] and [11], we used instead Lawvere-complete and L- complete, resp ectively . F urthermo re, one ea s ily verifies tha t each T -functor is L -co contin uo us, i.e. (rig ht adjoint)-w eig ht ed colimits ar e absolute, so that T - Co con t L sep = T - Cat cpl is the full sub catego r y of T - Ca t consisting of all separated and Lawvere co mplete T - categor ies. On the other ha nd, for a surjective T -functor f , f ∗ do es not need to b e right adjoint, s o that (Ax 4) is in general no t s a tisfied. This is not a sur prise, since natural instances of this example fail Theorem 2. Indeed, in the categ o ry of ordered sets and monotone maps, an y o rdered set is Lawvere-complete, hence the categ o ry of Lawv ere-complete a nd s eparated or dered sets co incides with the catego ry of anti- symmetric ordered s ets. The canonical forgetful functor from this catego ry to Set is surely not mona dic. Also, the cano nical forgetful functor from the categor y of Lawvere-complete and separa ted to po logical spaces (= so b e r s paces) and con tinuous maps to Set is also no t monadic. 3.5. Inhabited distributors. Another class of distributors cons ider ed in [10] is Φ = I the class of all inhabited T -distributors. Here a T -distributor ϕ : X − ⇀ ◦ Y is called inhabite d if ∀ y ∈ Y . k ≤ _ x ∈ T X ϕ ( x , y ) . (Ax 3) is s atisfied by definition, and in [10] we sho wed a lready the v a lidit y o f (Ax 1) and (Ax 2). F urthermor e, one easily verifies that (Ax 4) is sa tisfied. Hence, as alrea dy obs e rved in [10], a ll results stated in Section 2 a re av aila ble for this class of distributor s. Let us recall that, s pe c ialised to T op , inhabited-dense co nt inuous maps a r e precisely the to p o logically dense contin uo us maps, and the injective spaces with resp ect to top ologica lly dense em be dding s ar e known as Sc ott domains [8 ]. 3.6. “closed” distributors. A further in teresting clas s o f distributors is g iven by Φ = { ϕ : X − ⇀ ◦ Y | ∀ y ∈ Y , x ∈ T X . ϕ ( x , y ) ≤ _ x ∈ X a ( x , x ) ⊗ ϕ ( e X ( x ) , y ) } , that is, ϕ ∈ Φ if and only if ϕ ≤ ϕ · e X · a . Clear ly , (Ax 3) is s atisfied. F or each T - functor g : ( Y , b ) − → ( X , a ) we hav e g ∗ · e X · a = g ◦ · a · e X · a ≥ g ◦ · a = g ∗ , hence g ∗ ∈ Φ. F urthermo re, given T -distributor s ϕ : X − ⇀ ◦ Y and ψ : Y − ⇀ ◦ Z in Φ, then ψ ◦ ϕ = ψ · T ξ ϕ · m ◦ X ≤ ψ · e Y · b · T ξ ϕ · m ◦ X = ψ · e Y · ϕ ≤ ψ · e Y · ϕ · e X · a ≤ ψ · T ξ ϕ · e T X · e X · a ≤ ψ · T ξ ϕ · m ◦ X · e X · a = ( ψ ◦ ϕ ) · e X · a and therefor e als o ψ ◦ ϕ ∈ Φ. W e hav e seen that this class of distributors s a tisfies (Ax 1), (Ax 2) and (Ax 3). On the other hand, (Ax 4) is not satisfied. By definition, a T -functor f : ( X, a ) − → ( Y , b ) is Φ- dense whenever, for all x ∈ T X and y ∈ Y , b ( T f ( x ) , y ) ≤ _ x ∈ X a ( x , x ) ⊗ b ( e Y ( f ( x )) , y ) . 14 MARIA MANUEL CLEMENTINO AND DIRK HOFM ANN Hence, e ach pro p er T -functor (see [3 ]) is Φ-dense. In fac t, Φ-dense T -functors ca n be seen as “pr o p er ov er V - Cat ”, and the condition a b ov e states exactly prop erness o f f if th e underlying V -category S Y of Y = ( Y , b ) is discrete. F urthermore, ea ch surjective Φ-dense T - functor is final wit h resp ect to the forgetful functor S : T - Cat − → V - Cat . T o see this, let f : ( X , a ) − → ( Y , b ) b e a s urjective Φ-dense T -functor, Z = ( Z , c ) a T -categ ory a nd g : S Y − → S Z a V -functor s uch tha t g f is a T -functor. W e hav e to show that g is a T -functor. Let y ∈ T Y a nd y ∈ Y . Since T f is surjective, there is some x ∈ T X with T f ( x ) = y . W e conclude b ( y , y ) = b ( T f ( x ) , y ) ≤ _ x ∈ X a ( x , x ) ⊗ b ( e Y ( f ( x )) , y ) ≤ _ x ∈ X c ( T ( g f )( x ) , g f ( x ) ⊗ c ( e Z ( g f ( x )) , g ( y )) ≤ c ( T g ( y ) , g ( y )) . 3.7. F urther example s. A wide class of examples of injectiv e top o logical spaces is describ ed in [7], where the authors c o nsider injectivit y with resp ect to a class of em b eddings f : X − → Y such that the induced fra me morphis m f ∗ : Ω X − → Ω Y preserves certain s uprema. A similar construction can be done in our se tting; to do so we assume fr om now on T 1 = 1. F or a T -catego ry X , the V -c ate gory of c ovariant pr eshe afs V X is defined as V X = { α : 1 − ⇀ ◦ X | α is a T -distributor } = { α : X − → V | α is a T - functor } , and the V - categor ical structure [ α, β ] ∈ V is given a s the lifting X 1 , ◦ β o ◦ α ⊸ β =:[ α,β ]  1 ◦ α O for all α, β ∈ V X . Since e 1 : 1 − → T 1 is an is o morphism, this lifting of T -distributors does exist a nd can be calculated as the corresp onding lifting of V -distributors X 1 . ◦ β o o ◦ ~ ~ 1 ◦ α O O Each T -dis tr ibutor ϕ : X − ⇀ ◦ Y induces a V -functor ϕ ◦ ( − ) : V X − → V Y , α 7− → ϕ ◦ α, which is r ig ht adjoint if ϕ is a right adjoint T -distributor. Giv en now a c lass Ψ of V -distr ibutors, we may consider the class Φ of a ll those T - distributors ϕ f or whic h ϕ ◦ ( − ) preserves Ψ -weigh ted limits. This class o f T -distributor s is ce r tainly clo sed under c omp osition, and contains all right adjo int T -distributors , hence it includes all r epresentable o nes. Finally , if Ψ-weight ed limits ar e calculated po in twise in V X , then also (Ax 3 ) is fulfilled. As particula r ex a mples w e hav e the class Φ of all T -distr ibutors ϕ : X − ⇀ ◦ Y for which ϕ ◦ ( − ) preserves (1) the top elemen t o f V X , that is, for which ϕ ◦ ⊤ = ⊤ . In po int wise nota tion, this rea ds a s ∀ y ∈ Y . ⊤ = _ x ∈ T X ϕ ( x , y ) ⊗ ⊤ . RELA TIVE INJE CTIVITY AS COCOMPLETENE SS FOR A CLASS OF DISTRIBUTORS 15 If k = ⊤ , then this class of T -distr ibutors c oincides with the class o f inhabited T -distributor s considered in 3.5. (2) cotensor s, that is, for ea ch u ∈ V a nd each α ∈ V X , ϕ ◦ ho m( u, α ) = hom( u, ϕ ◦ α ). (3) finite infima (cf. [7, Section 6 ]). (4) arbitra ry infima (cf. [7, Section 7]). (5) co directed infima (cf. [7, Section 8]). References [1] R. Betti, A. Carb oni, R. Street, and R. W al ters , V ariation thr ough enrichment , J. Pure A ppl. A l gebra, 29 (1983), pp. 109–127. [2] M. M. Clementino and D. Hofmann , T op olo gic al fe atur es of lax algebr as , Appl. Categ . Structures, 11 (2003), pp. 267– 286. [3] , Effe ctive desc ent morphisms in ca te gories of lax algebra s , Appl. Categ. Structures, 12 (2004), pp. 413–425. [4] , Exp onenti ation i n V -c atego ries , T op ology Appl., 153 (2006), pp. 3113–3128. [5] , L awver e c ompleteness in T op olo gy , Accepted for publi cation in A ppl. Categ. Structures, (2008) , arXiv:math.CT/0704.3976. [6] M. M. Clementino and W. Tholen , Metric, top olo g y and multic ate gory—a co mmon appr o ach , J. Pure Appl. Algebra, 179 (2003), pp. 13–47. [7] M. Escard ´ o and B. Flagg , Semantic domains, i nj e cti ve sp ac es and monads. Bro oke s, Stephen (ed.) et al., Mathe- matical foundations of programming semantics. Pro ceedings of the 15th conference, T ulane Univ., New Or leans, LA, April 28 - May 1, 1999. Am sterdam: Elsevier, Electronic Notes in Theoretical Computer Science. 20, electronic paper No.15 (1999)., 1999. [8] G. Gierz, K. H. Hofmann, K. Keimel, J. D. La wson, M. W. Mislove, and D. S. Scott , A c omp endium of c ontinuous lattic es , Springer- V erl ag, Berlin, 1980. [9] D. Hofma n n , T op olo gica l the ories and close d obje cts , A dv. M ath., 215 (2007), pp. 789–824. [10] , Inje ctive sp ac es via adjunction , tech . r ep., Universit y of A veiro, 2008, arXiv: math.CT/0804.0326 . [11] D. Hofman n and W. Tholen , Lawver e co mpletion and sep ar ation via closur e , tec h. r ep., Universit y of A veiro, 2008, arXiv:math.CT/0801.0199. [12] G. M. Kell y and V. Schmitt , Notes on enriche d c ate g ories with co limits of some class , Theory Appl. Categ., 14 (2005), pp. no. 17, 399–423 (electronic). [13] F. W. La wvere , Metric sp ac es, gener alize d lo gic, and close d c ate gories , Rend. Sem. Mat. Fis. Milano, 43 (1973), pp. 135–166 (1974). Al so in: Repr. Theory Appl. Categ. 1:1–37 (electronic), 2002. [14] J. M acDonald an d M. Sob ral , Asp e c ts of monads , i n Categorical foundations, vol. 97 of Encyclopedia M ath. A ppl., Camb ridge Uni v. Press, Cambridge, 2004, pp. 213–268. Dep ar t am ento de Matem ´ atica, Universidade de Coimbra, 3001-45 4 Coimbra, Por tugal E-mail addr ess : mmc@mat.uc.pt Dep ar t am ento de Matem ´ atica, Universidade de A veiro, 38 10-193 A veiro, Por tugal E-mail addr ess : dirk@ua.pt